Mathematical Physics, Analysis and Geometry - Volume 1

Mathematical Physics, Analysis and Geometry1: v, 1998. v

Editorial

This is the first issue ofMathematical Physics, Analysis and Geometryin itsEnglish-language, international form. A journal of the same name in the Russianlanguage, having its roots in the long and splendid tradition of mathematical re-search in the former Soviet Union and, in particular, the Ukraine, was launchedin 1994 by the Kharkov mathematical community and has published papers in theUkrainian, Russian and English languages. The journal, in English only as of 1998,is intended to provide an international forum for important new results not onlyfrom the former Soviet Union but from all over the world.

Mathematical Physics, Analysis and Geometrywill publish research papers andreview articles on new mathematical results with particular reference to complexfunction theory; operators in function space, especially operator algebras; ordinaryand partial differential equations; differential and algebraic geometry; mathemati-cal problems of statistical physics, fluids; etc. The Editors, supported and assistedby an international Editorial Board, will strive to maintain the highest quality.Papers which are too abstract will be discouraged.

It is our purpose to makeMathematical Physics, Analysis and Geometryaleading journal of its kind attracting the best papers in the field.

VLADIMIR A. MARCHENKOANNE BOUTET de MONVEL

HENRY McKEAN

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Mathematical Physics, Analysis and Geometry1: 1–22, 1998.© 1998Kluwer Academic Publishers. Printed in the Netherlands.

1

Homogenization of Harmonic Vector Fields onRiemannian Manifolds with ComplicatedMicrostructure

L. BOUTET DE MONVELUniversité de Paris 6, Institut de Mathématiques de Jussieu (UMR 9994 du CNRS), 4 place Jussieu,F-75252 Paris Cedex 05, France

E. KHRUSLOVB. Verkin Institute for Low Temperatures, Mathematical Division, 47 Lenin Avenue,310164 Kharkov, Ukraine

(Received: 1 October 1997)

Abstract. We study the asymptotic behaviour of harmonic vector fields with given fluxes or periodson special manifolds consisting of one or several copies of the Euclidean space, with a large numberof small holes attached edge to edge by means of thin tubes (wormholes) when the number ofholes tends to infinity. We obtain the homogenized equations describing the leading term of theasymptotics.

Mathematics Subject Classifications (1991):35B27, 35B40.

Key words: electric field, homogenization, wormholes.

According to the well-known ‘Wheeler picture’, electric fields can be representedas harmonic fields on special Riemannian manifoldsMε. Such a manifold canconsist of one or several copies of the Euclidean space with small holes attachededge to edge by means of thin tubes. In geometrodynamics, these tubes are called‘wormholes’ [1]. The flux of a vector field through a wormhole is interpreted as acharge of the electric field. Given fluxes through all wormholes ofMε determinea unique harmonic vector field onMε vanishing at infinity. Such vector fields arealso determined by their periods along cycles passing through wormholes.

In this paper, we consider manifoldsMε depending on a small parameterε > 0such that the number of holes increases and their diameters vanish, asε → 0. Westudy the asymptotic behaviour of harmonic vector fields on these manifolds withgiven fluxes or periods and obtain homogenized equations describing the leadingterm of the asymptotics.

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2 L. BOUTET DE MONVEL AND E. KHRUSLOV

1. Description of the Problem

Let be a fixed bounded domain in the spaceRn (n > 3) andBεi, i = 1 . . . N(ε)be a family of closed pairwise disjoint balls in depending on a small parameterε > 0. We suppose that the total numberN(ε) of balls tends to infinity and theirdiameters tend to zero, whenε → 0, and any open subdomainG ⊂ containssome balls for sufficiently smallε.

Let us consider the infinite domain

ε = Rn \N(ε)⋃i=1

Bεi

and the disjoint union ofm copies (sheets) ofε : ε = ε × 1 . . . m.We construct a new manifoldMε by attaching toε somen-dimension tubes,

gluing their boundaries to those of the holes ofε. More precisely,Mε is definedby the following data:

• for eachε > 0 the numbermN(ε) is even and we are given a partition ofthe set of all pairs(i, k) : i = 1 . . . N(ε), k = 1 . . . m into subsets of twoelements[(i, k), (j, l)]-linked pairs;• for each linked pair[(i, k), (j, l)], letT klεij = Sn−1×[0,1] be a spherical tubeof dimensionn (Sn−1 is unit (n− 1)-sphere). The boundary∂T klεij of this tubeconsists of two components:

∂T klεij = 0kεi ∪ 0lεj ;

• for each(i, k) we are given a diffeomorphism:

hkεi: 0kεi ↔ ∂Bkεi,

where∂Bkεi is a component of the boundary∂kε of kth sheetkε = ε × k.

The manifoldMε is the union of the sheetskε (k = 1 . . . m) and tubesT klεij identi-fying boundary points pairwise according to the diffeomorphismshkεi. We supposethatMε is an orientable manifold. Fragments of such a manifold are shown inFigures 1 and 2.

We equipMε with a differentiable structure inducing its canonical structure ofa differentiable manifold with a boundary on each sheetkε and on each tubeT klεij .We will usually denotex as the points ofMε and, when needed,xα(α = 1 . . . n) asthe coordinates in some local coordinate chart.

Finally, we are given a Riemannian metric onMε with a positively definedsmooth metric tensorgεαβ(x); α, β = 1 . . . n inducing the standard flat metricof Rn outside some sufficiently small neighbourhoods of the tubes onMε. Forsimplicity we will suppose that it induces such a flat metric on each whole sheet

MPAG009.tex; 14/05/1998; 16:03; p.2

HARMONIC VECTOR FIELDS ON RIEMANNIAN MANIFOLDS 3

i jΩ kε

Tkkijε

Figure 1.

i

jΩ lε

Ω kε

Tklijε

Figure 2.

kε. Its dependence on parameterε on the tubes will be quantitatively characterizedbelow (Section 2).

We will study the harmonic vector fields onMε and we may identify them withharmonic differential forms of degree 1 orn − 1. Let us recall some facts andnotations from the theory of differentiable manifolds [2].

Differential forms of degreer (r-forms) are defined in local coordinates asfollows:

ω(x) =∑

16i1<···<ir6nωi1...ir (x)dxi1 ∧ · · · ∧ dxir ,

where functionsωi1...ir (x) are components of a skew-symmetric tensor field of rankr (06 r 6 n) and the sign∧ denotes the exterior product.

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The following three operators act on the set of differential forms. The exte-rior differentiation operator d mapsr-forms into (r + 1)-forms according to theformulae

d(ω ∧ µ) = dω ∧ µ+ (−1)kω ∧ dµ, d2 = 0, df =∑ ∂f

∂xidxi ,

whereω andµ are any smooth forms,k is degree ofω, andf is a function (formof degree 0).

The star operator∗ assigns to anyr-form ω the(n− r)-form ∗ω such that

µ ∧ ∗ω = (µ,ω)√|gε|dxi1 ∧ · · · ∧ dxin

for any r-form µ. Here|gε | is the determinant of the metric tensorgεαβ(x) onMε, and

(µ,ω) =∑

16i1<···<ir6nµi1...ir ω

i1...ir

is the scalar product of forms relative to the metric onMε.The operatorδ mapsr-forms into(r − 1)-forms according to the equality∫

Mε

(δω,µ)dnxε =∫Mε

(ω, dµ)dnxε,

which is valid for anyr-form ω and any smooth(r − 1)-form µ with compactsupport. Here and below we denote dnxε = √|gε |dxi1 · · · dxin the volume elementonMε .

We recall that a formω is called exact, if there is a formφ such thatω = dφ. Itis called closed, if dω = 0, and coclosed, ifδω = 0. A formω is called harmonic,if it is both closed and coclosed.

LetZ be a regular two-dimensional chain onMε, i.e., a formal linear combina-tionZ =∑αkZk, whereαk are real numbers andZk arer-dimensional simplexesin Mε , parametrized by continuously differentiable functionsxαk (u

1 . . . ur) (α =1 . . . n;u = u1 . . . ur ∈ Ur).

The integral of anr-form ω alongZ is defined by the formula:∫Z

ω =∑k

αk∑

16i1<···<ir6n

∫Urωi1...ir (x(u))

∂(xi1k . . . x

irk )

∂(ui1 . . . uir )du1 · · · dur.

A chainZ without boundary (∂Z = 0) is called a cycle. Twor-dimensionalcyclesC1, C2 are homologous if there exists an(r + 1)-dimensional chainZ suchthat the chainC1−C2 is the boundary ofZ (C1−C2 = ∂Z). In particular, a cycleC is homologous to zero if it is a boundary (C = ∂Z).

If ω is a closedr-form andC is anr-dimensional cycle, then the integral∫Cω

is called the period ofω alongZ. According to Stokes’ theorem it depends only

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on the homology classes ofω andZ. In particular, the period∫Cω vanishes ifω is

exact orZ is homologous to zero.It follows from our construction of the manifoldMε , that cycles of dimension

r 6= 1, n−1 are homologous to zero. Besides sincen > 3, one-dimensional cyclesCk situated completely on one sheetkε (k = 1 . . . m) are homologous to zero.

The manifoldMε is not compact. We define its compactification by adjoiningpoints at infinity∞k (k = 1 . . . m) on each of them sheetskε and we will consider1-forms and(n − 1)-forms onMε tending to zero fast enough, i.e., satisfying thefollowing ‘decay condition’ when|x| → ∞:

|ω(x)| = O(|x|1−n), (1.1)

where|ω| = (ω, ω)1/2 is the norm ofω.For such 1-forms the integrals

∫Cklεij

ω are still well defined along non-compact

pathsCklεij that join∞k and∞l through some tubeT klεij :

∞k → ∂Bkεi → T klεij → ∂Blεj →∞l

and are straight lines outside of. Since there are only finitely many holesBkεi andfinitely many sheets, these contours form a basis for one-dimensional homologyclasses. We will still call the integrals

∫Cklεij

ω periods of the formω (althoughCklεijare not cycles) and denote themP klεij . It is clear thatP klεij = −P lkεji if [(i, k), (j, l)]are linked pairs, and we setP klεij = 0 otherwise (consistency conditions).

THEOREM 1. For a given consistent set of periodsP klεij (i, j = 1 . . . N(ε); k, l =1 . . . m) there exists a unique harmonic1-form satisfying the decay condition(1.1).

As a basis for(n− 1)-homology classes we may choose(n− 1)-spheres∂Bkεi.We will denote8k

εi the periods of an(n − 1)-form ω along ∂Bkεi. Accordingto Stokes’ theorem,8k

εi = −8lεj if [(i, k), (j, l)] are linked pairs (consistency

conditions).

THEOREM 2. For a given consistent set of periods8kεi (i = 1 . . . N(ε); k =

1 . . . m) there exists a unique harmonic(n−1)-form satisfying the decay condition(1.1).

Note that ifω is harmonic(n − 1)-form with periods8kεi then 1-formv = ∗ω

is also harmonic and corresponding vector fieldv = vi(x), i = 1 . . . n has fluxes8kεi through(n− 1)-spheres∂Bkεi.Theorems 1 and 2 are variants of Hodge’s theorem for the non-compact man-

ifold Mε . They may be proved by the variational method (see Section 3). Themain goal of this paper is to study the asymptotic behaviour of 1-forms definedby Theorem 1 and the asymptotic behaviour of(n− 1)-forms defined by Theorem2, whenε → 0.

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2. Statements of Main Results

First we describe more notations. Letxεi be the centre of the ballBεi, aεi its radius,rεi the distance fromxεi to the union of the other centres:

rεi = mini 6=j|xεi − xεj |.

We assume that whenε → 0(i)

aεi → 0, rεi → 0,

so that(ii)

aεi 6 Crnn−2εi ,

whereC doesn’t depend onε.LetRεi be the spherical annulus inRn, centered at the pointxεi with inner radius

aεi and outer radiusbεi = rεi/2:

Rεi =x ∈ Rn : aεi < |x − xεi | < rεi

2

.

It is easy to see that these annuli belong to convex non-intersecting polyhedrons5εi (i = 1 . . . N(ε)) such thatRεi ⊂ 5εi ⊂ Rn, ⊂ ⋃i 5εi.

We will assume that(iii)

dεi = diam(5εi) 6 Crεi,

whereC doesn’t depend onε.Let us denoteDkl

εij = T klεij ∪ Rkεi ∪ Rlεj the domain on the manifoldMε, whereRkεi = Rεi × k is thekth copy of the annulusRεi on thekth sheetkε ⊂ Mε , andT klεij is the tube corresponding to the linked pairs[(i, k), (j, l)]. Its boundary∂Dkl

εij

consists of two componentsSkεi ⊂ kε andSlεj ⊂ lε. Let us consider in the domainDklεij the boundary value problem:

1v(x) = 0, x ∈ Dklεij ;

v(x) = 0, x ∈ Skεi;v(x) = 1, x ∈ Slεj ,

(2.1)

wherev(x) is a function,1 = 1Mεis the Laplace operator on the Riemannian

manifoldMε. In local coordinate it has the form

1 = − 1√|gε |n∑

α,β=1

∂

∂xα

(√|gε| gαβε ∂

∂xβ

),

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wheregαβε (x), α, β = 1 . . . n is the inverse tensor to the metric tensor ofMε (it isalso defined on forms:1 = (dδ + δ d)).

There is a unique solution to problem (2.1). We denote itvklεij = vklεij (x) and set

V klεij =

∫Dklεij

dvklεij ∧ ∗dvklεij . (2.2)

These quantities are positive and possess the symmetryV klεij = V lk

εji. They charac-terize the metric on the tubesT klεij . We assume that this metric satisfies an additionalcondition which provides the inequality

(iv)

V klεij > Can−2

εij ,

whereaεij = maxaεi, aεj , C > 0 does not depend onε. Here[(i, k), (j, l)] arelinked pairs.

Note, that solutionvklεij (x) of problem (2.1) minimizes the functional (2.2) overclassH 1(Dkl

εij ) of functions satisfying the boundary conditions onSkεi and Slεj .Taking this into account, it is easy to obtain the opposite inequality

V klεij 6 Can−2

εij , (2.3)

where aεij = minaεi, aεj . Therefore inequality (iv) means that radii of linkedballs have the same order of magnitude whenε → 0.

Now we introduce the distributions onRn × RnVεkl(x, y) =

∑ij

V klεij δ(x − xεi)δ(y − xεj ); k, l = 1 . . . m,

whereδ(x) is the delta-function onRn, V klεij = 0, if pairs [(i, k), (j, l)] are not

linked. These distributions are non-negative and by virtue of (2.3) and (i)–(ii)∫ ∫Vεkl(x, y)dx dy < C||; k, l = 1 . . . m,

where|| is the volume of the domain ⊂ Rn,C does not depend onε. Thus, thesets of distributionsVεkl(x, y), ε > 0 (k, l = 1 . . . m) are compact in the sense ofweak topology of distributions onRn × Rn. We will assume that there exist weaklimits

(v)

w − limε→0

Vεkl(x, y) = Vkl(x, y), k, l = 1 . . . m,

whereVkl(x, y) are non-negative distributions (densities of measures) with sup-ports in × . SymmetryVkl(x, y) = Vlk(y, x) holds true and using (i), (ii), (v)and (2.3), one can show that integrals∫

Vkl(x, y)dy = Ikl(x)are bounded functions ofx: Ikl(x) ∈ L∞().

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We further introduce the distributions onRn:

Pεk(x) =∑ij

V klεijP

klεij δ(x − xεi) (k = 1 . . . m),

and

8εk(x) =∑i

8kεiδ(x − xεi) (k = 1 . . . m),

whereP klεij are periods of harmonic 1-form and8kεi are periods of harmonic(n −

1)-form, which were described in Theorems 1 and 2, respectively.We will also assume that these distributions weakly converge (inD′(Rn))(j)

w − limε→0

Pεk(x) = Pk(x) (k = 1 . . . m),

(jj)

w − limε→0

8εk(x) = 8k(x) (k = 1 . . . m)

and, in addition, that the following inequalities hold(jjj)∑i,j,k,l

an−2εi |P klεij |2 < C1;

(jv)∑i,k

a−n+2εi |8k

εi|2 < C2

with constantsC1, C2 independent ofε. It follows from (jjj), (jv) and (i), (ii) thatlimit distributionsPk(x), 8k(x)(k = 1 . . . m) are functions ofL2().

We denoteL2(Mε)(r) andL2(Rn)(r) the Hilbert spaces ofr-forms(r = 1, n−1)

onMε andRn, respectively, with relation to the metrics scalar products [2].Let us introduce the operatorsQεk (k = 1 . . . m) mappingL2(Mε)

(r) intoL2(Rn)(r), defined by the formula:[

Qεkvε

](x) =

vε(x), x × k ∈ kε,0, x ∈ Rn \ε =⋃i Bεi,

for anyvε ∈ L2(Mε)(r).

The main results of this paper are contained in the following theorems describ-ing the asymptotic behaviour of harmonic 1-forms and(n − 1)-forms onMε, asε → 0.

THEOREM 3. Let vε be the harmonic1-form defined by Theorem1 and let con-ditions (i)–(v) and(j), (jjj) be fulfilled, whenε → 0. Then, for anyk

Qεkvε → vk weakly inL2(Rn)(1),

MPAG009.tex; 14/05/1998; 16:03; p.8


wherevk (k = 1 . . . m) are exact1-forms onRn such thatvk = dUk and the col-lection of functionsUk(x), k = 1 . . . m is the generalized solution of the problem

1Uk −m∑l=1

∫Vkl(x, y)[Uk(x)− Ul(y)]dy = Pk(x), x ∈ Rn;

Uk(x) = O(|x|2−n), x →∞ (k = 1 . . . m).

THEOREM 4. Letvε be the harmonic(n− 1)-form defined by Theorem2 and letconditions(i)–(iv) and(jj), (jv) be fulfilled, whenε → 0. Then, for anyk

Qεkvε → vk weakly inL2(Rn)(n−1),

wherevk (k = 1 . . . m) are (n−1)-forms onRn such thatvk = ∗dUk and functionsUk(x) are the solutions of the problems

1Uk = 8k(x), x ∈ Rn;(2.4)

Uk(x) = O(|x|2−n), x →∞ (k = 1 . . . m).

We will prove these theorems in Sections 3, 4 and 5. In Section 3 we will con-struct suitable representations for harmonic 1-forms and(n−1)-forms onMε . Thenusing these representations, in Sections 4 and 5 we study the asymptotic behaviourof 1-form and(n− 1)-forms, respectively.

3. Representation of Harmonic Forms onMε

Let us make cuts onMε along the middle section on each tubeT klεij , as indicatedin Figure 3. We will denote byT kεi andT lεj the corresponding halves of the tubeT klεij (T

klεij = T kεi ∪ T lεj ).

Thus, we cutMε in m componentsMkε (k = 1 . . . m) whose boundaries con-

sist ofN(ε) spheresSklεij . EachMkε is homeomorphic to the spaceRn with N(ε)

removed ballsBεi. Therefore sincen > 2,Mkε has no 1-homology.

Hence, any closed 1-formvε onMε is exact on eachMkε , i.e.,

vε = duεk, x ∈Mkε (k = 1 . . . m). (3.1)

Hereuεk is a smooth function onMkε such that

uεk = O(|x|2−n), |x| → ∞ (3.2)

andvε satisfying the vanishing condition (1.1).According to the definition of the periodsP klεij of vε and (3.1), (3.2),

uεk(x)|Sklεij − uεl(x)|Slkεji = P klεij . (3.3)

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i

jM lε

M kε

Sklijε

Slkjiε

T ljε

Tkiε

Figure 3.

We also have

1uεk ≡ −(δ d+ dδ)uεk = 0, x ∈Mkε , (3.4)

if vε is a harmonic 1-form.For a given set of numbersP = P klεij (such thatP klεij = −P lkεji), we will denote

H 1(Mε, P ) the set of collectionsuε[m] = uεk, k = 1 . . . m of m functionsuεk(x)defined onMk

ε that have square integrable derivatives and satisfy the conditions(3.2) and (3.3). Sincen > 2 one can show that there exists a unique collectionuε[m] = uεk, k = 1 . . . m ∈ H 1(Mε, P ) which minimizes the functional

J1ε(uε[m]) =∑k

∫Mkε

duεk ∧ ∗duεk (3.5)

overH 1(Mε, P ) and functionsuεk(x) of this collection satisfy Equation (3.4) onMkε (k = 1 . . . m).Thus a 1-formvε onMε is harmonic with the set of periodsP = P klεij if and

only if it is exact on eachMkε and the collection of primitives, as in (3.1), minimizes

the functional (3.5) overH 1(Mε, P ).Now, letwε be a harmonic(n−1)-form onMε with periods8k

εi . Let us considerthe 1-form vε = ∗wε. As is well known, it is also a harmonic form onMε and thefluxes of the corresponding vector field through the spheresSkεi (in the directionof kε) are equal to8k

εi. It is clear thatvε has a represention of form (3.1) withpotentialsuεk(x) = (−1)n−1uεk(x) satisfying conditions (3.2)–(3.4). However, inthis case the constantsP klεij are not known. Thus we have:

wε = ∗duεk onMkε .

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Let us denoteH 1(Mε,⋃P) the set of collections ofm functionsuε[m] = uεk,

k = 1 . . . m ∈ H 1(Mε, P ) with arbitrary setsP = P klεij : P klεij = −P lkεji ofconstant ‘jumps’ in (3.3), i.e.,

H 1(Mε,

⋃P)=⋃P

H 1(Mε, P ).

Sincen > 2 there exists a unique collectionuε[m] = uεk, k = 1 . . . m ∈ H 1(Mε,

P )minimizing the functional

J2(uε[m]) =∑k

∫Mkε

duεk ∧ ∗duεk −∑i,k

8kεiP

klεij (3.6)

overH 1(Mε,⋃P). Here8k

εi (i = 1 . . . N(ε), k = 1 . . . m) are given numberssuch that8k

εi = −8lεj for the linked pairs[(i, k), (j, l)]; P klεij (i, j = 1 . . . N(ε),

k, l = 1 . . . m) are determined by (3.3) for linked pairs[(i, k), (j, l)] andP klεij = 0otherwise.

Using Green’s formula and well-known variational methods one can show thatthe functionsuεk(x) of a minimizing collection satisfy Equation (3.4) onMk

ε andthe corresponding(n− 1)-forms∗duεk(x) have periods8k

εi (i = 1 . . . N(ε)).It follows from the above considerations that an(n− 1)-form vε onMε is har-

monic with periods8kεi if and only ifvε = ∗duεk , where the collection of functions

uε[m] = uεk, k = 1 . . . m minimizes the functional (3.6) overH 1(Mε,⋃P).

Thus we obtain representations for harmonic 1-forms and(n−1)-forms onMε ,which will be used in Sections 4 and 5 to investigate the asymptotic behaviour ofthese forms, asε → 0 (Theorems 3 and 4).

Note that existence and uniqueness of the forms (Theorems 1 and 2) also followfrom these representations if one can prove the solvability of the minimizationproblems (3.5) and (3.6).

4. Proof of Theorem 3

Let P = P klεij be the set of periods of a harmonic 1-formvε, satisfying condition(1.1), and letuε[m] = uεk, k = 1 . . . m be the collection of functions of the classH 1(Mε, P ) minimizing the functional (3.5), i.e., the collection of primitives ofvε .

Let us introduce the collection of functionsu0ε[m] = u0

εk, k = 1 . . . m of theclassH 1(Mε, P ) vanishing outside domainsDkl

εij and set

wε(x) =∑k

(uεk(x)− u0εk(x))χεk(x); (4.1)

v0ε (x) =

∑k

du0εk(x)χεk(x), (4.2)

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whereχεk(x) is the characteristic function ofMkε . It follows from the proper-

ties of the collectionsuε[m], u0ε[m], thatwε(x) is a function with square integrable

derivatives on the manifoldMε, satisfying condition (3.2) and minimizing thefunctional

J1(wε) =∫Mε

dwε ∧ ∗dwε + 2∑k

∫Mkε

du0εk ∧ ∗dwε. (4.3)

Besidesv0ε is a 1-form onMε andvε = v0

ε + dwε . We will choose below thecollectionu0

ε[m] so that, for anyk, Qεkv0ε → 0 weakly inL2(Rn)(1), asε → 0.

Then to prove Theorem 3 we will show that for anyk Qεk dwε → dwk weaklyin L2(Rn)(1), wherewk(x) is a function onRn with square integrable derivativessatisfying the decay conditionwk(x) = O(|x|2−n), asx → ∞, and the collectionof functionsw = wk, k = 1 . . . mminimizes the functional

J (w) =∑k

∫Rn|∇wk|2 dnx +

+∑k,l

∫

∫

Vkl(x, y)[wk(x)−wl(y)]2 dnx dny +

+2∑k

∫

Pkwk dnx. (4.4)

Here we have set

|∇wk|2 =n∑α=1

∣∣∣∣∂wk∂xα

∣∣∣∣2 ,and the functionsPk(x) and the distributionsVkl(x, y) are defined by (j) and (v),integrals are taken with respect to Lebesque measure onRn.

We first describe the abstract scheme of the minimization problems (4.3) and(4.4).

LetHε be a Hilbert space depending on a parameterε > 0, with scalar product(·, ·)ε and norm|| · ||ε, andFε a continuous linear functional onHε uniformlybounded with respect toε.

LetH be another Hilbert space with scalar product(·, ·) and norm|| · ||, F is acontinuous linear functional onH .

Letwε andw be solutions of the minimization problems

infwε∈Hε

[||wε||2ε + Fε(wε)], (4.5)

infw∈H

[||w||2+ F(w)], (4.6)

respectively.

MPAG009.tex; 14/05/1998; 16:03; p.12


The question is to analyse under what conditions and in what sensewε con-verges tow. This is in part answered by the following theorem:

THEOREM 5. Assume that we are given a dense subspaceM ⊂ H and, for eachε > 0, linear operatorsQε : Hε → H , Pε: M → Hε that satisfy conditions(a)–(c):

(a)

||Qεuε || 6 C||uε ||ε;for anyuε ∈ Hε, anyu ∈M and anyvε ∈ Hε such thatQεvε → v weakly in H, asε → 0, we have:

(b1)

QεPεu→ u weakly inH, asε → 0;(b2)

limε→0||Pεu||2ε = ||u||2;

(b3)

limε→0|(Pεu, vε)ε | 6 C ||u|| ||v||;

(c)

limε→0

Fε(vε) = F(v).

Then ifwε is the solution of the minimization problem(4.5),Qεwε converges tow, weakly inH , whenε → 0.

We will apply this theorem, which was proved in [3]. In our situationHε =H 1(Mε) is the Hilbert space of local square integrable functions onMε which havesquare integrable derivatives with scalar product

(u, v)ε =∫Mε

du ∧ ∗dv

and which ‘vanish’ at infinity (at the∞k (k = 1 . . . m)) in the following sense:H 1(Mε) is the completion of the space of smooth functions satisfying the condition(3.1) at∞k with respect to the norm||w||ε = (w,w)1/2ε .

We define the Hilbert spaceH = H 1(Rn)m as the space of locally integrablevector-functionsu = (u1 . . . um) onRn with square integrable derivatives which‘vanish at infinity’. We endow it with the scalar product

(u, v) =∑k

∫Rn

duk ∧ ∗dvk +∑k,l

∫

∫

Vkl(x, y)(uk(x)− ul(y))2 dnx dny.

MPAG009.tex; 14/05/1998; 16:03; p.13


More precisely,H 1(Rn)m is the completion of the space of smoothm-componentvector-functions satisfying condition (3.1) at infinity with respect to the norm||u||= (u, u)1/2.

Note that forn > 3 || · ||ε and|| · || are norms indeed inH 1(Mε) andH 1(Rn)mrespectively. This follows from the well known inequality∫

|x|6R|f |2 dnx 6 R2

2(n− 2)

∫|∇f |2 dnx, (4.7)

which is valid for any functionf ∈ C10.

The linear functionalsFε in H 1(Mε) andF in H 1(Rn)m are defined by theformulae

Fε[wε] = 2∑k

∫Mkε

dwε ∧ ∗du0εk, (4.8)

F [w] = 2∑k

∫

Pkwk dnx. (4.9)

Then the minimization problems (4.3), (4.4) can be reformulated as (4.5), (4.6)respectively. To apply Theorem 5 we must define operatorsQε, Pε and functionsu0εk (k = 1 . . . m) so that conditions (a),(bi), (c) of Theorem 5 are satisfied.

Let us set′ε = ′ \ ⋃i Bεi and′kε = ′ε × k, where′ is a relativelycompact subdomain ofRn such that ⊆ ′.

By virtue of (ii) there exists an extension operatorQ′ε: H 1(′ε)→ H 1(′) suchthat

||Q′εvε ||H1(′) 6 C||vε ||H1(′ε) (4.10)

for anyvε ∈ H 1(′ε) [4]. Here and below we denoteH 1(G) the Sobolev space offunctions on a domainG ⊂ Rn; the constantC does not depend onε.

Such a continuation operator, of course, is not unique. However, we may choosethe unique one that minimizes norms in the spacesH(Bεi). Keeping this in mind,we define the operatorQε : H 1(Mε)→ H 1(Rn)m as follows:[

Qεuε](x) =

(Q′εuε1, . . . ,Q′εuεm) for x ∈ ′;(uε1, . . . , uεm) for x ∈ Rn \′,

whereuεk(x) = uε(x×k) for x ∈ ε, i.e.x×k ∈ kε . It obviously follows from(4.7), (4.10) thatQε is a linear operator satisfying the condition (a) of Theorem 5.

Now we define an operatorPε : M → H 1(Mε) and functionsu0εk(x) rep-

resenting the functionalFε in (4.8). First let us introduce onMε the followingfunctions:

ϕkεi(x) =

1, x ∈ T kεi;ϕ(|x−xεi |rεi

), x ∈ Rkεi;

0, x ∈Mkε \ (T kεi ∪ Rkεi);

MPAG009.tex; 14/05/1998; 16:03; p.14


ϕkεi(x) =

1, x ∈ T kεi;ϕ(|x−xεi |

4aεi

), x ∈ Rkεi;

0 x ∈Mkε \ (T kεi ∪ Rkεi),

whereT kεi is the corresponding half of the tubeT klεij (see Figure 3),ϕ(t) > 0 isa twice continuously differentiable function on real line such thatϕ(t) = 1 fort 6 1/4 andϕ(t) = 0 for t > 1/2, other notations correspond to those consideredin Sections 1 and 2.

We choose the spaceC20(Rn)m of twice continuously differentiable vector-

functionsu = (u1(x) . . . um(x)) with compact support inRn as a dense subsetM in H 1(Rn)m and set:[

Pεu](x) = uk(x)

(1−

∑i

ϕkεi(x)

)+∑i

uk(xεi)ϕkεi(x)+

+∑i

vklεij(ul(xεj )− uk(xεi)

)ϕkεi(x),

u0εk(x) =

∑i

P klεij vklεij (x)ϕ

kεi (x), x ∈Mk

ε , k = 1 . . . m. (4.11)

Here the pairs(i, k) and(j, l) are ‘linked’, vklεij (x) is the solution of the problem(2.1) andP = P klεij is the set of periods of 1-form under consideration along thecontoursCklεij (see Section 1).

Sincevklεij (x) = 1− vlkεji(x), Pεu ∈ H 1(Mε) andu0ε[m] = u0

εik, k = 1 . . . m ∈H 1(Mε, P ).

One can show that conditions(bi) and (c) of Theorem 5 are fulfilled. We onlyhave to check below condition (c).

Let vε(x) be a function ofH 1(Mε) such thatQεvε converges weakly tov inH 1(Rn)m, whenε → 0. Taking into account (4.8), (4.11) and the properties of thefunctionsvklεij (x) andϕkεi(x), and using Green’s formula we obtain

Fε(vε) = 2∑i,k

P klεij

∫Rkεi

1(vklεij ϕkεi) ∧ ∗vε (4.12)

and according to (2.2)∫Rkεi

1(vklεij ϕkεi) ∧ ∗1 =

∫∂Bkεi

∗dvklεij

=∫Slεj

vklεij ∧ ∗dvklεij

=∫Dklεij

dvklεij ∧ ∗dvklεij = V klεij . (4.13)

MPAG009.tex; 14/05/1998; 16:03; p.15


Let us cover (triangulate) by convex polyhedrons5εi containing the setsRkεiand satisfying the condition (iii).

For convex polyhedrons we dispose of the Poincaré inequality:

LEMMA 1. There is a universal constantC = Cn such that, for any boundedconvex domainD ⊂ Rn, we have:

||u||2 6 ||u||2+ Cd2∫D

||du||2

for u ∈ H1(D), whereu denotes the mean of u, andd is the diameter ofD.Proof.This is obviously true ifu = constant , so we may supposeu = 0 (i.e.u

orthogonal to constants – for a givenD the best possible constant is the inverse ofthe first nonzero eigenvalue of the Neuman problem). Ifu = 0 we have∫

D×D|u(x)− u(y)|2 = 2 vol(D)||u||2.

We also have

u(x)− u(y) =∫ 1

0(du(z).(y − x))ds with z = x + s(y − x)

so

2 vol(D)||u||2 6∫D×D×[0,1]

||u′(z)||2||x − y||2 dnx dny ds.

To estimate the last integral, we write everything in polar coordinates centred atz:

x = z+ rω, y = z− ρω (||ω|| = 1), s = r

r + ρwith variablesz ∈ Rn, ω ∈ the unit sphere ofRn, 06 r 6 a, 06 ρ 6 b, whereDis defined byr < a = a(z, ω) in polar coordinates centred atz, andb = a(z,−ω).

The Jacobian determinant of this change of coordinates is given by:

dnx dny ds = (r + ρ)n−2 dnzdn−1ω dr dρ

(with dn−1ω the standard volume element of the sphere), so that (1) can be written

2 vol(D)||u||2 6∫||u′(z)||2||x − y||2(r + ρ)n−2 dnz dn−1ω dr dρ. (4.14)

Now we have||x − y|| = r + ρ 6 d and∫(r + ρ)n−2 dn−1ω dr dρ 6 2C vol(D) (4.15)

for some constantC > 0 (because(r+ρ)n−2 6 2n−3(rn−2+ρn−2) and∫ban−1 dn−1

ω 6∫an dn−1ω + ∫ bn dn−1ω = n vol(D), so one can chooseC = 2n−3 n

n−1). Thelemma follows from (4.14) and (4.15) by dividing by 2 vol(D). 2

MPAG009.tex; 14/05/1998; 16:03; p.16


Let χεi(x) be the characteristic function of the polyhedron5εi and vkεi be themean value of the function(Qεvε)k(x) in 5εi. We set

χεk(x) =∑i

vkεiχεi(x)

and

P εk(x) =∑i

P klεijVklεij |5εi|−1χεi(x).

Then according to (4.12), (4.13) and the properties of the functionsvklεij (x), ϕkεi we

get

Fε(vε) = 2∑k

∫P εk(x)χεk(x)dnx + Eε(vε), (4.16)

where

Eε(vε) = 2∑i,k

P klεij

∫1(vklεij ϕ

kεi)[(Qεvε)k − χεk

]dnx. (4.17)

Since the function(Qεvε)k converges weakly tovk, whenε → 0, it remainsuniformly bounded inH 1(), and therefore converges strongly inL2(). FromLemma 1 and condition (i) we then obtain thatχεk(x) also converges tovk inL2().

Taking into account (j), (jjj) and (2.3), it is easy to show thatPεk(x) convergesweakly toPk(x) in L2(), asε → 0. Hence

limε→0

∑k

∫P εk(x)χεk(x)dnx =

∑k

∫Pk(x)vk(x)dnx. (4.18)

To estimate the termEε(vε) in (4.11) we use the following estimates for thesolutionvklεij of problem (2.1) (see [3, 5]):

∣∣Dαvklεij (x)∣∣ 6 C an−2

εi

|x − xεi |n−2+|α| , x ∈ Rkεi (|α| = 0,1). (4.19)

Then, using (4.18) and the properties of the functionsϕkεi(x), we obtain

|Eε(vε)| 6 C∑

i

a2n−4εi

rnεi

1/2∑k

||(Qεvε)k − χεk||L2().

Thus, in view of (i), (ii) and the convergence of(Qεvε)k andχεk to vk in L2(),we have

limε→0

Eε(vε) = 0. (4.20)

MPAG009.tex; 14/05/1998; 16:03; p.17


It follows from (4.16), (4.18), (4.20) and (4.9) that condition (c) of Theorem 5is fulfilled.

Applying Theorem 5, we conclude thatQεwε converges weakly tow = (w1 . . .

wm) in H 1(Rn)m, wherewε andw are solutions of the minimization problems(4.3) and (4.4) respectively. It means that 1-form d(Qεwε)k converges to 1-formdwk weakly inL2(Rn)(1), whenε → 0. We have

Qεk[dwε] = d(Qεwε)k − d(Qεwε)k ∧ χεwith χε the characteristic function of the union of all ballsBεi. Since∫

χε 6 C∑i

anεi 6 Cmaxia2εi[∑

i

a2(n−2)εi

rnεi

]1/2[∑i

rnεi

]1/2

from conditions (i), (ii) we get

limε→0

∫χε = 0.

Therefore d(Qεwε)∧ χε converges weakly to zero inL2(Rn)(1), soQεk[dwε] con-verges to dwk (k = 1 . . . m). Finally, it follows from (4.2) and (4.11) thatQεk[v0

ε ]converges weakly to zero inL2(Rn)(1) whenε → 0, which can be shown in thesame manner using estimates (4.19) and (jjj).

Thus, the harmonic 1-formvε = v0ε + dwε converges to dwk (k = 1 . . . m) in

the sense defined above, and this proves Theorem 3.

5. Proof of Theorem 4

Let uεk(x), k = 1 . . . m ∈ H 1(Mε,⋃P) be the collection of functions minimiz-

ing the functional (3.6), andu0εk(x), k = 1 . . . m be the collection of functions

defined by equalities (4.11) with a given set of constantsPε = P klεij , satisfyingthe consistency condition (see Section 1). Then the functionwε(x) ∈ H 1(Mε)

defined by formula (4.1) and the collection of constantsPε = P klεij (considered asindependent variables) minimize the functional

J (wε, Pε) =∫Mε

dwε ∧ ∗dwε + 1

2

∑i,k

V klεij (P

klεij )

2+

+2∑i,k

P klεij

∫Rkεi

d(vklεij ϕkεi) ∧ ∗dwε −

∑i,k

8kεiP

klεij , (5.1)

where(i, k) and(j, l) are linked pairs and

V klεij =

∫Rkεi

d(vklεij ϕkεi) ∧ ∗d(vklεij ϕ

kεi)+

∫Rlεj

d(vlkεjiϕlεj ) ∧ ∗d(vlkεjiϕ

lεj ).

MPAG009.tex; 14/05/1998; 16:03; p.18


Taking into account the properties of the functionsvklεij (x) andϕkεi(x) and usingGreen’s formula, we obtain

V klεij =

∫Sklεij

dvklεij ∧ vklεij +∫Slkεji

dvlkεji ∧ vlkεji +

+∫Rkεi

1(vklεij ϕkεi) ∧ ∗(vklεij ϕkεi)+

∫Rlεj

1(vlkεjiϕlεj ) ∧ ∗(vlkεjiϕlεj )

=∫Dklεij

dvklεij ∧ ∗dvklεij +O(a

2(n−2)εi

rn−2εi

).

So, according to (2.2) and (i), (ii), (iv),

V klεij = V kl

εij (1+ o(1)) (ε → 0). (5.2)

We transform the third term in (5.1) using Green’s formula and representJ (wε,

Pε) in the form

J (wε, Pε) = J0(wε, Pε)−∑i,k

8kεiP

klεij (5.3)

with

J0(wε, Pε) =∫Mε

dwε ∧ ∗dwε + 1

2

∑i,k

V klεij (P

klεij )

2+

+2∑i,k

P klεij

∫Rkεi

1(vklεij ϕkεi) ∧ ∗wε. (5.4)

LEMMA 2. There exist positive constantsC0 andε0 such that forε < ε0

J0(wε, Pε) > C0

[ ∫Mε

dwε ∧ ∗dwε +∑i,k

an−2εi (P klεij )

2

].

Proof. Let us suppose that the statement of the lemma is not valid. Then thereexist sequenceswεr , εr → 0, r = 1,2, . . . and Pεr , εr → 0, r = 1,2, . . . suchthat forε = εr → 0

limε→0

J0(wε, Pε) = 0 (5.5)

and ∫Mε



2 = 1. (5.6)

It follows from (5.6) and (4.7) that the sequence of vector-functionsQεwε, ε =εr, r = 1,2, . . . is bounded inH 1(Rn)m, so one can select a subsequenceQεwε,

MPAG009.tex; 14/05/1998; 16:03; p.19


ε = εν → 0, which converges weakly inH 1(Rn)m. According to the embeddingtheorem this subsequence converges inL2, loc. It is this subsequence that we willconsider below.

Let us show thatw ≡ 0. Taking into account (5.5), (5.4), (2.2) and the definitionof the operatorQε (see proof of Theorem 3), we obtain

limε=εν→0

||Qεwε − vε ||2H1(Rn)m = 0, (5.7)

where

vε = Qε

(∑ik

P klεij vklεij ϕ

kεi)

)∈ H 1(Rn)m.

Using estimates (4.20) and the properties ofϕkεi , one can show thatvε convergesweakly to zero inH 1(Rn)m, whenε = εν → 0. Hence, it follows from (5.7) thatw = 0.

Taking into account (5.4), (iv) and the properties of functionsvklεij andϕkεi wewrite

J0(wε, Pε) >∫Mε

dwε ∧ ∗dwε + c0

∑i,k


2+

+∑i,k

P klεij

∫Rkεi

(wε − wkεi)1(vklεij ϕkεi)+

+∑ik

P klεijVklεij w

kεi, (5.8)

wherewkεi is the mean value of the function(Qεwε)k in the polyhedron5εi, c0 > 0.The third term in the right-hand side of this inequality we estimate using (4.20),

Lemma 1 and (ii):∑i,k

P klεij

∫Rkεi

(wε − wkεi)1(vklεij ϕkεi)

6 C1

∑i,k

P klεijan−2εi

rn/2−1εi

√∫5εi

|∇(Qεwε)|2

6 Cmaxir2εi(∑

i,k


2+∫Mε

dwε ∧ ∗dwε

). (5.9)

We estimate the fourth term using Young’ inequality, (2.3) and (ii): we can writefor anyδ > 0∣∣∣∣∑

ik

P klεij Vklεij w

kεi

∣∣∣∣ 6 δ∑i,k


2+ C1

δ

∑i,k

|wkεi|2an−2εi

6 δ∑i,k


2+ Cδ

∑k

∫

|(Qεwεi)k|2. (5.10)

MPAG009.tex; 14/05/1998; 16:03; p.20


Now we chooseδ = C0/3. Then, in view of the convergence ofQεwε to zeroin L2, loc whenε = εν → 0, it follows from (5.8)–(5.10) and (i) that for sufficientlysmallε = εν

J0(wε, Pε) >C0

2

(∑i,k


2+∫Mε

dwε ∧ ∗dwε

).

This inequality contradicts (5.5) and (5.6), so Lemma 2 is proved. 2

SinceJ (wε, Pε) 6 J (0,0) = 0 from (5.3), Lemma 2 and (jv) we have∫Mε



2 6 C1

∑ik

(8kεi)

2

an−2εi

< C, (5.11)

whereC1, C are constants independent ofε.Hence the set of vector-functionsQεwε, ε > 0 is weakly compact inH 1(Rn)m

and we can select a weakly converging subsequenceQενwεν , εν → 0, ν = 1,2:Qενwεν → w weakly inH 1(Rn)m. The compact embedding theorem shows thatthis subsequence in fact converges inL2, loc. Without loss of generality we may alsoconsider condition(v) fulfilled, asε = εν → 0 (see Section 2).

Since derivatives of the functional (5.3) in respect of the variablesP klεij is equalto zero at the point of minimum, we have

V klεijP

klεij = 8k

εi +∫Rkεi

1(vklεij ϕkεi) ∧ ∗wε +

∫Rlεj

1(vlkεjiϕlεj ) ∧ ∗wε, (5.12)

where the consistency conditionsP klεij = −P lkεji,8kεi = −8l

εj for linked pairs[(i, k), (j, l)] are taken into account.

Remembering the definition ofwkεi and using (5.12), (5.2), (4.20) and (4.13) weget

V klεijP

klεij = θklεij

[8kεi + V kl

εij (wkεi − wlεj )

]+ Eklεij , (5.13)

whereθklεij → 1 uniformly with respect toi, j , whenε = εν → 0, andEklεij satisfiesthe estimate

|Eklεij | 6 Can−2εi

rn/2εi

(∫5εi

|(Qεwε)k − wkεi|2 dnx

)1/2

+

+Can−2εj

rn/2εj

(∫5εj

|(Qεwε)l − wlεj |2 dnx

)1/2

. (5.14)

Taking into account (5.13), (5.14), (i)–(iii), (v), (jj) and the fact that the subse-quencesQενwεν converge towk (k = 1 . . . m), we conclude that the distributions

Pεk(x) =∑i,k

V klεijP

klεij δ(x − xεi)

MPAG009.tex; 14/05/1998; 16:03; p.21


converge, whenε = εν → 0, weakly inD′(Rn) to the distributions

Pk(x) = 8k(x)+ Vkl(x, y)(wk(x)−wl(y)). (5.15)

It follows from (5.11) that the collection of constantsP klεij also satisfies thecondition (jjj), so all conditions of Theorem 3 for the 1-formvε with periodsP klεijare fulfilled, whenε = εν → 0. Applying Theorem 3 in view of (5.15), we see thatthe limitw = (w1 . . . wn) corresponding to the subsequenceεν is a solution of theproblem (2.4). Since this problem has a unique solution, it proves Theorem 4.

References

1. Wheeler, J. A.:Geometrodynamics, Academic Press, New York, 1962.2. de Rham, G.:Varietes Differentiables, Actualites Sci. Indust., Hermann, Paris, 1960.3. Boutet de Monvel, L. and Khruslov, E. Ya.: Homogenization on Riemann manifolds, Preprint

BIBOS, Bielefeld, 1993.4. Khruslov, E. Ya.: The asymptotic behaviour of solutions of second order boundary value

problem under fragmentation of the boundary of the domain,Math. USSR-Sb.34(2) (1979).5. Marchenko, V. A. and Khruslov, E. Ya.:Boundary Value Problem in Domains with Fine

Grained Boundary, Naukova Dumka, 1975.

MPAG009.tex; 14/05/1998; 16:03; p.22


23

Hard-core Scattering forN-body Systems?

ANDREI IFTIMOVICIEquipe de Physique Mathématique et Géométrie, Institut de Mathématiques de Jussieu 2, placeJussieu, 75251 Paris Cedex 05, Francee-mail: [email protected]


Abstract. We prove propagation properties (maximal and minimal velocity bounds) for pseudo-resolvents associated toN-body Hamiltonians with short-range potentials that are infinite on a star-shaped domain centred at the origin. Motivated by the fact that the invariance principle holds for usualN-body systems, we define the cluster wave operators in terms of pseudo-resolvents and prove thatthey exist and are asymptotically complete. For any cluster decompositiona, these operators inter-twine the hard-core pseudo-selfadjoint Hamiltonians corresponding to the pair of pseudo-resolventsR, Ra , and equal the Abel operators constructed in terms of Hamiltonians.

Mathematics Subject Classification (1991):81Uxx.

Key words: asymptotic completeness, hard-core interactions,N-body systems, propagation theo-rems, scattering theory.

1. Introduction

One of the most important goals in scattering theory is the study of the asymptoticbehavior (whent → ±∞) of e−itHψ , whereψ is an arbitrary state from theorthogonal complement of the space of eigenvectors of the HamiltonianH . Moreprecisely, we are interested in finding a familyHa of selfadjoint operators, withsimpler (and known) spectral and evolution properties, such that, for any stateψ , afamily of vectorsψ±a should exist, for which the convergences∥∥∥∥e−itHψ −

∑a

e−itHaψ±a

∥∥∥∥t→±∞−→ 0 (1.1)

are satisfied. If this takes place, then we say that the system isasymptoticallycomplete.

The particularity of theN -body Hamiltonians is that they are a sum of a differ-ential operator (with excellent dispersion properties) and a perturbation that doesnot vanish (when|x| → ∞) along certain directions of the configuration spaceX. This makes us think that, if asymptotic completeness holds for such systems,

? Previously published in MAG (Mathematical Physics, Analysis, Geometry),1, no. 2 (1994),265–313.


24 ANDREI IFTIMOVICI

then e−itHaψ±a should be asymptotically localized within some cones centred onthe classical trajectories.

These geometrical ideas allowed V. Enss to show that (1.1) is true for three-body quantum systems with potentials that decay slightly faster than the Coulombinteraction. After that, asymptotic completeness forN -body short-range quantumsystems has been proved in 1987 by I. Sigal and A. Soffer [34], and in the followingyears, many people tried to simplify or to extend their proof for more complicatedmany-body problems. Indeed, there was first an effort of making the theory more‘readable’, done by J. Derezinski in [14]. Then, G. M. Graf, jointly using ideasfrom the earlier works of Enss ([16, 17, 18, 19]) and from [34] but also from themore recent papers of Sigal and Soffer (like [36] and especially [35]), succeeded ingiving in [23] a remarkable time-dependent-like proof of the quoted result, whichdiffered from the previous proofs in several important aspects. We shall emphasizeonly the fact that in [23], some of the main propagation properties have been ob-tained without the use of the Mourre estimate, i.e., independently of an intimateknowledge of the spectral properties of the Hamiltonian. These properties weresufficient for showing the existence of the cluster wave operators but not for theircompleteness. Indeed, for the last result, a propagation property involving jointlya time-dependent localisation in position and a localisation in the total energy wasneeded, and for proving it a good knowledge of the spectrum of theN -body Hamil-tonian was crucial. Actually, this is the only place where Graf invokes the Mourreestimate (in order to obtain (local) positivity for the commutator of the Hamiltonianwith the generator of the dilations group) and by this means he eliminates the decayhypothesis imposed in [34] on the second derivative of the potential. Further, usingrefined results on the Mourre theory due to W. Amrein, Anne Boutet de Monveland V. Georgescu (see [1] and also [11] for optimality) we have shown in [27], onthe lines of [23], that no condition on the derivatives of the potential was needed inorder to prove completeness for the Agmon-type systems. Moreover, since locallythe potentials were allowed to be as singular as the the kinetic energy permits,the question of the validity of a statement on asymptotic completeness for muchsingularly perturbed systems (as the hard-coreN-body quantum systems) arisesnaturally. Indeed, the interest for such problems is rather old, going back, e.g.,to the works of W. Hunziker ([26]) and especially of D. W. Robinson, P. Ferreroand O. de Pazzis (see [32, 21]), where, under rather restrictive assumptions onthe geometry of the potentials (spheric symmetry, the supports of the singularitieswhere cylinders centred on the subspaces of the relative movement of the clusters)and on the forces (repulsivity), the absence of the singular continuous spectrumand the existence and the completeness of the wave operators corresponding totheelasticchannel have been established. But this is, of course, a very simplifiedcase, because even if the problem was posed in anN-body context, the abovehypotheses transformed it in a one-channel scattering problem. Very recently, AnneBoutet de Monvel, V. Georgescu and A. Soffer, using both the locally conjugateoperator method and an algebraic approach (which appears naturally in theN-

MPAG010.tex; 14/05/1998; 16:04; p.2

HARD-CORE SCATTERING FORN -BODY SYSTEMS 25

body context), have succeeded in giving in [12] a complete spectral analysis forthis type of highly singular Hamiltonians. More precisely, it is proven that underquite reasonable smoothness conditions imposed on the border of the supportsof the singularities, the generator of the dilations group is (in some weak sense)conjugated to the hard-coreN-body Hamiltonian, which proves to be sufficient toobtain a limiting absorption principle (even in an optimal form). Then, absence ofthe singular-continuous part of the spectrum and local decay follow in a standardway. Our task is to continue this work by studying the scattering properties of thesesystems.

We shall begin by describing the geometrical particularities of the configurationspaceX (an Euclidian space) related to theN-body problem. Let us denote byLa finite partially ordered index set and demand it to be a lattice. Take then a familyXaa∈L of subspaces ofX such thatXsupa,b = Xa+Xb, and0 andX correspondto minL ≡ amin and maxL ≡ amax respectively. In the usualN -body situation,X is the space of the configurations of the set ofN particles relative to the centerof mass coordinate system,L is the lattice of partitions of the set1, . . . , N, Xa

is the subspace ofX consisting of the configurations which describe the internalmotion of the clusters (fragments) of the partitiona and, finally,Xa (the orthogonalof Xa in X with respect to a well-chosen scalar product) can be identified with thespace of configurations of the relative motions of the clusters. Let us denote foranya ∈ L byπa the orthogonal projection onXa. Further, according to S. Agmon(see [4]) aN-body type Hamiltonian is defined as the sum between the (positive)Laplace–Beltrami operator1 and a familyV (a)a∈L of operators which factorizeasV (a) = V a πa. Here we considered the simplest case whenV a is the operator(in H(Xa)) of multiplication by a function having a good decay at infinity in alldirections ofXa. Although this assumption is currently used in many of the papersdedicated to this subject, in the more recent ones it is shown that the same resultscan be obtained ifV a is a reasonable differential operator.

For the hard-core systems, the physical picture of clusters formed by particlesthat cannot get arbitrarily close to each other is modelled by (positive) singularitiesof the potentials, having as supports cylindersK(a) = Ka ⊕ Xa, whereKa arecompacts ofXa; of course, a short and long range part can be added to thesesingular potentials. Denote byχa : Xa 7→ R the operator of multiplication by thecharacteristic function ofKa and putχ(a) for χa πa. Then, a precise definition ofthe hard-core HamiltonianH is obtained by seeing it as a limit, in strong resolventsense, of the family of self-adjoint operators inH(X):

Hα = 1+∑a∈L

(V (a)+ αχ(a)) = H + α∑

a∈Lχa πa. (1.2)

We have denoted byH a standardN -body Hamiltonian which, under natural as-sumptions on the symmetric operatorsV a, becomes a selfadjoint, bounded frombelow operator with form-domain the Sobolev space of order oneH1(X). Notice

MPAG010.tex; 14/05/1998; 16:04; p.3


that, when tending to infinity, the parameterα > 0 will increase the value of thecylindrically supported perturbationχ(a).

It has been proven ([12], Lemma 3.7 and Proposition 3.8) that foreachz in thecomplement inC of [inf σ(H ),∞), the limit R(z) ≡ limα→∞(z − Hα)−1 existsin the strong sense inB(H−1,H1) and in the norm of each of the Banach spacesB(H s,H t ) for −1 6 s 6 t 6 1 andt − s < 2. Actually,R(z) is aself-adjointpseudo-resolvent family, i.e. it satisfies the first resolvent identity andR(z∗) =R(z)∗. It is known (see [24]) that the closure inH of RanR(z) is apropersubspaceof H (let us denote it byH∞) which does not depend onz, and which coincideswith the closure of the domain of a self-adjoint operatorH for whichR(z)|H∞ =(z−H)−1 andR(z)|HH∞ = 0. We shall callH a pseudo-selfadjointoperator onH , and this will be the Hamiltonian modelling a hard-core Agmon-type problem.We shall refer to its spectral properties as to those of the selfadjoint operatorH

which acts in the proper subspaceH∞.Notice thatH can be explicitly given. Let2 = X \⋃a∈LK(a) andH1

0(2) bethe closure ofC∞0 (2) in H1(X). Then:

D(H) = u ∈ H10(2) | Hu ∈ L2(2) ≡ H(2)

and

(Hu)(x) =(Hu)(x), x ∈ 20, x 6∈ 2 for u ∈ D(H),

soH is the operator1 +∑V (a) in H(2), with Dirichlet boundary conditions.Notice thatH∞ = H(2).

We emphasize that one of the difficulties in studying the spectral and scatteringproperties of such operators is the fact that they are not densely defined. More-over, we cannot say a priori that Hamiltonians constructed as above factorize inthe same tensor product form as those from the familyHα. Actually this holds.Indeed, as in the usualN-body problem, using the limiting process describedbefore, it is possible to construct for eacha ∈ L a pseudo-selfadjoint Hamil-tonianHa, which corresponds to the hard-core problem relative to the sublatticeLa = b ∈ L | b 6 a. On the other hand, it is shown that for anya ∈ L, thefamily of resolventsRa,αα>0 of the (genuine) selfadjoint operators

Ha,α = Haα ⊗a 1+ 1⊗a (πa∇)2

tends in norm inB(H−1(X),H θ (X)), for any θ < 1, to a pseudo-resolventRa(to which corresponds the pseudo-selfadjoint subHamiltonianHa) whenever theconvergenceHa

α → Ha takes place in the norm-resolvent sense inB(H−1(Xa),

H θ (Xa)). This implies

Ha = Ha ⊗a 1+ 1⊗a 1a, (1.3)

where1a denotes the Laplace–Beltrami operator inH(Xa), and where the abovetensor product sum is an operator defined in the proper subspaceD(Ha)⊗H(Xa)

of H(X), with operator domainD(Ha) = D(Ha)⊗H2(Xa).

MPAG010.tex; 14/05/1998; 16:04; p.4


Then, the set ofthresholdsof H is defined as

τ(H) =⋃

a∈L\amaxσp(H

a), (1.4)

whereas the set ofcritical valuesofH , denoted byC(H)will be the union ofτ(H)with σp(H), the point spectrum ofH .

The main ingredient for the study of the spectral and scattering properties of theHamiltonian is the Mourre estimate, which states (see (1.5) below) local positivityfor the commutator ofH with the generator of the dilations groupA, the last onebeing defined as:

A = 1

2(P ·Q+Q · P) = 1

2

dimX∑j=1

(PjQj +QjPj),

whereQj is the operator of multiplication by the coordinatexj (w.r.t. some or-thonormal basis ofX) andPj ≡ −i ∂

∂xj= F ∗QjF , with F the Fourier trans-

form. Actually, this estimate stressesstrict positivity for the lower semicontinuousfunctionρAH : R→ (−∞,+∞] on an open set ofC, where for allλ ∈ R:

ρAH(λ)def= supµ ∈ R | ∃f ∈ C∞0 (R;R), f (λ) 6= 0 s.t.

f (H)[iH,A]f (H) > µf (H)2. (1.5)

In [2, 9, 28], an extensive study of this function is made. It is also importantto point out that under the assumption of strong-C1 regularity of the Hamiltonianw.r.t.A, even ifH is a pseudo-selfadjoint operator, the identity:

[A,R(z)] = R(z)[A,H ]R(z) (1.6)

is valid for anyz ∈ C \ σ(H). We refer to Section 5 from [12] for the precisedefinitions of the above commutators and for the proof of the Mourre estimate inthe context of the hard-coreN-body systems. In fact, the result we needed and thatwe will intensely use is the strict positivity ofρAR(λ) on the setR \ C(R).

Another difficulty arising from the fact that the limit ofHα is only pseudo-selfadjoint inH comes from the way we have to interpret the limit ofeitHα and,correspondingly, the way we have to define the cluster wave operators. It is not atrivial fact (see [13]) that, for anya ∈ L, the family of evolution groups generatedby Ha,α has a limit, but this limit exists only onHa,∞, the closure of RanRa in H .Moreover, taking into account the inclusionHb,∞ ⊆ Ha,∞, true for anya, b ∈ Lwith a 6 b, and the Theorem 3.23(ii) from [13], we see that the domain of the limitlimα→+∞ eitHα is, a priori, included in the range of limα→+∞ eitHa,α even when thisone is applied to the vectors ofH∞.

As for the cluster wave operators, in the usualN -body context (i.e. for any finiteα) they are defined as

±a,α ≡ ±(Hα,Ha,α;Ea,α) = s-limt→±∞ eitHαe−itHa,αEa,α. (1.7)

MPAG010.tex; 14/05/1998; 16:04; p.5


Notice that sinceEa,α = Epp(Haα )⊗a1 commutes withHa,α (which has purely

absolutely continuous spectrum) and also with bounded functions ofHa,α, as aconsequence of the Hilbert space isomorphisms

H(X) ∼= L2((Xa, dξa);H(Xa)) ∼=∫ ⊕Xa

H(Xa)dξa

the limits (1.7) can be seen as wave operators with identificatorEa,α (see [6]).Moreover, for anyz ∈ (−∞, inf σ(H)), the equalityEpp(Ha

α ) = Epp(Raα(z)) is

true, so the identification operator is the same for the wave operators constructedin terms of Hamiltonians and for those constructed in terms of resolvents. For thehard-core case, these identifiers are no more equal, and we shall use the notationEa for Epp(Ha) ⊗a 1, which projectsH into H∞. Then Ea = EHa(R)Ea =ERa(R \ 0)Ea, whereEA(1) denotes the spectral measure of the operatorA

on1 ⊂ R.It seems thus quite natural to redefine the hard-core wave operators in terms

of known objects. This is also suggested by the invariance principle, which istrue in the usualN-body case at least for the admissible function(z − . )−1 (see[6] for definitions), withz chosen as above. Indeed, suppose first thatbothstronglimits ±(Hα,Ha,α;Ea,α) and±(Rα,Ra,α;Ea,α) exist. Then, the correspondingabsolute Abelian limits exist also, and they are equal to the strong ones (see Corol-lary 6.14 in [6]). We can thus use theweakform of the invariance principle (seeTheorem 11.25 in [6]) for this pair of operators, in order to get theirequality, andthus the equality of the strong limits also.

At this level, let us remark that the main purpose of [12] and of our work was totestthe ability of an abstract framework to treat Hamiltonians that have a complexstructure both analytically and algebraically. This partly explains why we will makethe choice to work with wave operators defined in terms of resolvents in place ofHamiltonians. The non-locality of the resolvents will generate a lot of difficulties,which we will overcome using the algebraic framework we wanted to test.

Let us thusdefinethe wave operators corresponding to the hard-core case interms of pseudo-resolvents, and prove existence and asymptotic completeness for±(R,Ra;Ea) ≡ ±a . As we shall see, the way we do it uses the algebraic frame-work, which works identically for the usualN-body resolvents, so, in what follows,we will automatically prove not only the existence of±a , but also the existence of±(Rα,Ra,α;Ea,α), who was previously taken as hypothesis for the weak invari-ance principle in the case of usualN -body systems.

Finally, another reason for the choice we made on±a , is that the intertwiningproperty±a = ER(R\0)±a is valid. Since for alla ∈ L, (z−Ha)Ra = EHa(R),±a are partial isometries with final domainH∞, which intertwine the pairH , Haon the closure of the range ofRa. Also, the connection between±a and the waveoperators defined in terms of Hamiltonians as limits of

Wa(t) = eitHEH (R)eitHaEa,

MPAG010.tex; 14/05/1998; 16:04; p.6


is made by the strong form of the invariance principle. Indeed, letW±a be theabsolute Abelian limits ofWa(t), i.e.

limε→+0

(±ε)∫ ±∞

0e±εt‖Wa(t)ψ −W±a ψ‖2 dt = 0. (1.8)

Then, in Appendix 6.2 we show that the existence of±a implies existence of theAbelian limit W±a and equality. This shows that the strong limits ofWa(t), if theyexist, they are partial isometries with final domainH∞. Actually, this can be shownalso directly using a reasoning similar to that described in the proof of Lemma 4.1.

As we previously said, despite their boundedness, the resolvents are not alwayscomfortable objects to work with, mainly because of their non-local character. Thisfeature becomes critical when one tries to prove, on the lines of [23, 15, 40], thefollowing propagation theorem, which we consider as being the main result of thepaper.

THEOREM 1.1. Let a ∈ L be arbitrarily chosen. Ifθ ∈ C∞0 (R \ C(Ra)) and ifJ ∈ C∞0 (X) has its support localised sufficiently close to the origin and outsideany of the subspaces of the familyXbb∈L\La

, then the estimate:∫ ∞1

dt

t

∥∥∥∥J(Qt)θ(R)e−itRψ

∥∥∥∥2

6 C‖ψ‖2 (1.9)

is true for some positive constantC and for allψ ∈ H(X).

In Section 5 a more precise form of this theorem is stated and proved, by meansof an inductive reasoning on the levels ofL. The first step is to prove it fora =minL, which is the only case when a standard proof can be adapted. Then, theMourre estimate will enter at each level of the induction, both in an explicit wayand implicitly, from the induction hypothesis.

Finally, it is also shown that as a consequence of a particular case of this result,theasymptotic completenessstatement:∑

a∈L\amax±a (

±a )∗ = Ec(H) (1.10)

is valid.Let us pass in review the contents of the following sections. In the following

section we expose (following [7, 9, 10] and [12]) the algebraic framework relatedto anN -body type problem, putting emphasis on the results we need in our paper.We also briefly remind the construction of the partition of unity in the configurationspace due to G. M. Graf and list some of the properties of the vector field attachedto this family of functions. Section 3 is devoted to the proofs of those propagationestimates that can be deduced without the use of the Mourre estimate. In Section 4is shown that the existence of the limits±a is a consequence of these estimates.Finally, Section 5 is mainly devoted to the proof of Theorem 1.1.

MPAG010.tex; 14/05/1998; 16:04; p.7


2. The Algebraic and Geometric Frameworks

As we have already mentioned in the first section, for anya ∈ L the direct sumXa⊕Xa determines a canonical isomorphism of Hilbert spacesH(X) ∼= H(Xa)⊗H(Xa) ∼= L2(Xa;H(Xa)). Let us denote byK(Xa) theC?-algebra of compactoperators onH(Xa) and letT(Xa) be theC?-algebra naturally associated to thetranslation group inB(H(Xa)), i.e. the norm-closure (inB(H(Xa)) ≡ B(Xa))of the ? -subalgebra of operators of the formf (πaP ) = F ∗Xaf (πaQ)FXa , withf ∈ C∞(Xa). Then, the norm-closure inB(X) of the linear space generated by theoperatorsS⊗aT which correspond (through the first of the above isomorphisms)to S ⊗ T ∈ K(Xa)⊗ T(Xa), will be aC?-subalgebra ofB(X), named the algebraof a-semicompact operators and denoted by:

T (a) = K(Xa)⊗a T(Xa). (2.1)

Further, with the aid of the familyT (b)b∈L the vector space sum

Ta =∑b∈La

T (b) (2.2)

(with La = b ∈ L | b 6 a) is constructed for eacha ∈ L. It is shown (see[7, 3]) that the above sum is direct in the topological sense and that the canonicalprojectionsP (b) : T → T (b) (which assign to anyT ∈ T ≡ Tamax a uniqueT (b) ∈ T (b)) are norm-continuous and satisfyP (b)[T ∗] = T (b)∗. We will referto T (b) as theb-connected componentof T . Moreover, for anya ∈ L the projec-

tion Padef=∑b∈La

P (b) is a?-homomorphism betweenT and itsC?-subalgebraTa.For anya ∈ L \ amax a concrete expression forPa was given by W. N. Polyzou(see (6) in [30], and also [31, 7, 3]). Notice that using the Möbius inversion formula(see Theorem 4.18 in [5]) we can retrieve any of the operatorsT (a) ∈ T (a) as aweighted sum of elements of the familyTbb∈La

, the weight being the Möbiusfunctionµ(b, a).

The second important property of the family of algebras of semicompact oper-ators is itsgraduationwith respect to the semi-latticeL, which means that for anya, b ∈ L, the inclusion

T (a)T (b) ⊆ T (supa, b) (2.3)

is valid. We resume all these properties by saying thatT is anL-gradedC?-algebra.Let us go back now to the pseudo-selfadjoint operatorsH defined in Section 1.

According to [9], we shall say thatH is affiliated to theC?-algebraT iff all therealisations of the?-homomorphismφ : C∞(R)→ B(X) belong toT . But sincethe Stone–Weierstrass theorem ensures the existence of a bijection between the?-homomorphismsφ : C∞(R) → B(X) and pseudo-resolvents (such that forany f ∈ C∞(R) we getφ(f )|H∞ = f (H) andφ(f )|HH∞ = 0), and sinceH does not depend on thez from the pseudo-resolvent w.r.t. which it has beendefined, an equivalent definition for the affiliation ofH to T is: R(z) ∈ T for

MPAG010.tex; 14/05/1998; 16:04; p.8


somez ∈ C \ σ(H). Note that ifH is affiliated toT then any of the members ofthe family Haa∈L are affiliated to the elements ofTaa∈L respectively. This isdue to the fact that given twoC?-subalgebrasT and T of B(X), to any pseudo-selfadjoint operatorH affiliated toT it corresponds through the?-homomorphismP : T → T a unique pseudo-selfadjoint operatorP [H ], which is affiliated toTand for whichP [φ] = φ is true on allC∞(R). In [12] several affiliation criteriaare given and it is also proved that the hard-coreN -body Hamiltonian is affiliatedto theN-body algebraT . As a consequence of this, a HVZ theorem is obtained forthis context.

We are particularly interested in establishing the extension of the first resolventidentity to the context ofL-gradedC?-algebras. For the case of the usualN-bodysystems, this is called the Weinberg–Van Winter equation (see [30, 7]). Actually, aswe shall see below, borrowing the algebraical-combinatorial technique of deducingsuch kind of identity, it is possible to give (firstly forN-body HamiltoniansH withbounded perturbations), for anya, b ∈ L, a precise meaning to the differenceRa − Rb and to thea-connected component ofR ∈ T , in terms of a sum ofregularising operators. These results will be important in the effort of getting use-ful commutation relations between hard-core pseudo-resolvents and multiplicationoperators.

Let us begin by introducing some special subsets of the latticeL. We shallcall a totally ordered subsetC a maximal chain inL, iff for any a, b ∈ C, b <a, for which there is noc ∈ C \ a, b such thatb < c < a, there is also nootherd ∈ L \ a, b such thatb < d < a. We define then the rank of the finitelattice L as maxcardC | C ⊆ L,C maximal ≡ |L|. In theN -body case,L isthe lattice of partitions of a set ofN elements, for which|L| = N . SinceL isunion of maximal chainsC, anya ∈ L will belong to at least one of those whichsatisfy maxC = maxL. Then, by convention, therank of a in L is taken to bethe number|a|L ≡ maxcardC | maxC = maxL,minC = a,C maximal. Thisnotion allows us to define for any 16 n 6 |L| ≡ N , the nth level of L asL(n) = b ∈ L | |b|L = n. We thus haveL = ⊔N

n=1 L(n), where⊔

stands for‘disjoint union of sets’. Finally, we introduce for some arbitrarily fixeda ∈ L andanyb ∈ La, two special subsets of the latticeLa:

Lba ≡ (La)

b = c ∈ La | c > b, (2.4)

Lba = c ∈ L \Lb | supb, c = a. (2.5)

It is clear thatLba is a lattice inLa and thatLba is a bilateral ideal inLa. Notice

also that, ifLa denotesLaamax

, then for alla ∈ (L \ Lb) ∪ b we haveLba = ∅and for all a ∈ Lb \ b, we havea ⊆ Lba. Let us put the sign6∼ betweentwo elements of the latticeL whenever they are incomparable. The proof of thefollowing identity will give us more intuition on the sets introduced at (2.5):

La \Lb =⊔

c∈Lba\b

Lbc. (2.6)

MPAG010.tex; 14/05/1998; 16:04; p.9


Indeed, for anyc ∈ Lba \ b, since for alld ∈ Lb

a \ c we have supd, b =d 6= c, we getLb

a ∩ Lbc = c. Secondly, to each of thesec corresponds ind ∈ La | d 6∼ b the remainder of the setLbc, which is disjoint of any of the setsLbc corresponding to anotherc of Lb

a \ b. For, supposing the existence of somed ∈ Lbc ∩ Lbc, b, d will form a pair having (in the casec 6∼ c) no least upperbound inL. But this would mean thatL is not a lattice. Finally, in order to provethe inclusion⊆ in (2.6), note that we have already shown that the r.h.s. of (2.6)includesLb

a \ b, and that for anyd ∈ La, d 6∼ b, we have supb, d ∈ Lba and

d ∈ Lb,supb,d. The inverse inclusion is trivial.Further, corresponding to the setLba we construct the following bilateral ideal

of La:

Tba =∑c∈Lba

T (c), (2.7)

and denote byPba the canonical projection ofT onto it. Then, the resolventidentity

Ra − Rb = Rb(Ha − Hb)Racan be written with the aid of (2.6) as

Ra − Rb =∑c∈La

RbI bcRa,

whereI bc is the sum overLbc of the symmetric operatorsH(d) which all, except1 = Hamin = H (amin) (which is affiliated toTamin) belong toT (d) respectively.Then, iterating the above formula and taking into account that, as a consequence ofthe definition ofLbc, I bc = 0 if c 6 b or c 6∼ b, we getRa − Rb =∑ R(C), with

R(C) = RbI bb1Rb1I b1b2Rb2 · · · I bn−1bnRbn (2.8)

and where the sum is taken over all the (not necessarily maximal) chainsC of La,having the g.l.b.b and as l.u.b. any of the elements ofLa. Moreover, using aninductive argument, it is shown (see Lemma 3.11 in [12]) that for any chainC,R(C) ∈ TminC maxC. This, together with

Pba

( ∑d∈L\La

P (d)+∑d∈Lb

P (d)

)= 0

shows thatPba[R] can be computed as:

Rba = Pba[Ra − Rb] =∑

R(C), (2.9)

where this time the sum is performed over all the chainsC ⊂ L with min C = band maxC = a. Notice that in the caseb = amin we haveTamina = T (a) and

MPAG010.tex; 14/05/1998; 16:04; p.10


thus R(a) = Ramina, which tells us that thea-connected component of the hard-core resolvent is a regularising operator on allH(X), given by the sum (withoutrepetitions)

∑R(C) performed over all the chains of the sublatticeLa.

We are now prepared to state two important consequences of the propertiesdescribed above, which we will use throughout the paper. They were proved in [12]for hard-cores, by extension from the particular usualN -body context, with the aidof the limiting procedure. The first one is the following commutation relation (seeProposition 3.13 in [12])

[g,R(a)] =∑b∈La

R(b)[g,1]Rba + R(a)[g,1]Ra, (2.10)

true for any multiplication operator with functionsg ∈ C2(X) having boundedderivatives of first and second order (we are particularly interested in the case wheng is the identity function). Note that (2.10), (2.9) and (2.8) show that the multiplecommutators[. . . [R,Q],Q], . . . Q], usually denoted byadkQ(R), are inB(H(X))

for any finite orderk.The second result (Theorem 6.5 in [12]) states, for anya ∈ L, decay of the

a-connected component of the hard-core resolvent along all the directions ofXa,with the same rate as that imposed by the hypothesis on the non-hard-core part ofthe potentials. Since in this paper we are only interested in short-rangeN-bodypotentials, the only decay condition we impose on the functionsV a (see (1.2) andthe comments following it) is: for anya ∈ L \ amin, there is someµ > 1 suchthat 〈Qa〉µV a belongs toB(H1(Xa),H−1(Xa)). We used〈x〉 as abbreviation of√

1+ x2. Then the precise result we need states:

〈Qa〉µR(a) ∈ B(H−1(X),H1(X)). (2.11)

In the rest of this section we shall briefly review (following [23] and [15]) thosegeometric particularities of the configuration spaceX which are specific to theN-body problem. Indeed, the need to put in evidence some privileged directionsfor the non-propagation, which are closely related to the particular structure ofthe potentials, suggests the division ofX into disjoint sets, all except one beingneighbourhoods (cone or semi-cylinder-shaped) of these directions. Moreover, asmooth vector field is constructed onX (by convolution or averaging), from the firstdistributional derivative of the convex, locally Lipschitz application% : X→ R

% = 1

2maxa∈L

|πa(.)|2+ νa. (2.12)

In the above definition each of the parametersνa belongs to some interval(ν|a|, ν|a|)with positive bounds conveniently chosen, such that for any 16 n < m 6 N theinequalitiesνm < νm < νn < ν1 be satisfied. In this sense,% can be seen also as afunction of the argumentν ∈∏a∈L(ν|a|, ν|a|) having each of its coordinates doublyindexed, first by the number of the level ofL to which each elementa belongs and

MPAG010.tex; 14/05/1998; 16:04; p.11


second by the number assigned to this element within the level. The total numberof coordinates will thus be cardL.

The identity (2.12) tells us that% is a function almost everywhere differentiableonX, having classical derivative a.e. defined onX which coincides with%′, the dis-tributional derivative of%. Actually, %′ ∈ L∞loc(X). Moreover, it can be shown (seeTheorem 4.1.7 in [25]) that the second distributional derivative of% is a measureonX, taking as valuespositivematrices fromB(X).

In order to see in which way the vector field%′ is a distortion of the identicalmapping onX, we make the connection between (2.12) and the ‘privileged direc-tions’ Xa by the means of a finite, disjoint, a.e.-covering4aa∈L of X, definedby:

4a =⋂b 6=a

x ∈ X | |πa(x)|2 + νa > |πb(x)|2 + νb

=x ∈ X | |πa(x)|2 + νa > max

b 6=a|πb(x)|2 + νb

. (2.13)

This allows us to calculate explicitly% (and its derivatives) with the aid of the par-tition of unity Jaa∈L subordinated to the above open a.e.-covering ofX. Indeed,

%′(x) =∑a∈L

Ja(x)πa(x) (2.14)

holds asX-valued distributions and a.e. as functions. Moreover, since%′′ is apositive measure,

%′′(x) >∑a∈L

Ja(x)πa (2.15)

is satisfied forall x ∈ X.In order to obtain smooth partitions of unity and vector fields, we construct them

from Ja, resp.%, either as in [23, 15], by convolution with aC∞0 -functionϕ (sup-ported in a neighborhood of the origin inX, and satisfying

∫ϕ(x)dx = 1,

∫xϕ(x)

dx = 0), or, as in [39], by an averaging process, performed on the Cartesian productof intervals

∏Nn=1

∏cardL(n)i=1 (νni, νni) previously defined, like in:

g(x) =∫RcardL

ϕ(ν)G(x, ν)dν.

Hereϕ is a positiveC∞0 function supported in the above product set, with‖ϕ‖L1 =1, and in our caseG(. , ν) will be replaced byJa and% respectively (whose de-pendence onν is given in (2.13), resp. (2.12)). We will denote the new, smoothobjects byja andr respectively, and it is not difficult to show that all the propertiespreviously enumerated (especially (2.14) and (2.15)) hold for them also. It is alsoshown that if Id denotes the identity function, then the mappingsr ′ − Id, r ′′ · Id− r ′andr ′′′ are bounded in theL∞ norm.

MPAG010.tex; 14/05/1998; 16:04; p.12


In Appendix 6.1, properties of some subsets ofX which play an important rolein the spectral and scattering analysis, like the open cones (defined for any 06d < 1)

0a(d) =x ∈ X | min

b∈L\La

|πb(x)| = dist

(x,

⋃b∈L\La

Xb

)> d|x|

, (2.16)

or the ‘cells’Xa = Xa \⋃b∈L\La

Xb are given, and the relation between them is

studied. In order to get an intuitive image of the link betweenXa and the charac-

teristic functionJa of the set4a, let us mention only the fact that if the argumentof Ja is multiplied by a positive parameterγ , then the support ofJa(γ .) will tend

to coincide withXa whenγ →∞.

3. Propagation Properties

We begin by the so-called maximal velocity bound theorem (see [23, 34, 35, 36,37]):

PROPOSITION 3.1. Let g be aC∞0 (X \ 0) scalar function. Then there is aconstantλ > 0 (which cannot be made arbitrarily small) and a positive constantC such that:∫ ∞

1

dt

t

∥∥∥∥g(Qλt)

e−itRψ∥∥∥∥2

6 C‖ψ‖2. (3.1)

Proof.Let us choose the propagation observable of the form:

8 = h(Q

λt

)and takeh to be aC∞0 (R) radial function, constant in a neighbourhood of the originand equal to zero at infinity. Then the computation of the Heisenberg derivative of8 with respect to the approximating hard-core resolventRα gives:

DRα8 =1

2λtRα(P · h′ + h′ · P)Rα − 1

t

Q

λt· h′, (3.2)

where the dot means scalar product between two vector operators. We have alsoused the convention according to which, whenever no confusion is possible, wewill omit the arguments of the multiplication operators with (eventually time-dependent) functions. In order to simplify the aspect of some rather complicatedformulae, we will keep this convention in force throughout this paper. Denote nowby h the scalar function Id· h′ and byh(r) ≡ d

dr h(r) 6 0. An obvious calculationshows that for allx ∈ X:

h′(x) = ωh(|x|),h(x) = |x|h(|x|), (3.3)

MPAG010.tex; 14/05/1998; 16:04; p.13


whereω denotes the unit vectorx|x|−1. Finally, takeg(r) = (−h(r))1/2 and noticethatg is a radialC∞0 (X \ 0) function. Then, (3.2) becomes:

DRαh =1

t

∣∣∣∣Qλt∣∣∣∣ g2− 1

tgRα(P · ω+ ω · P)Rα

2λg −

− 1

2λt

([RαP, g

] · ωgRα + h.c.)−

− 1

2λt

(gRαP · ω

[g,Rα

]+ h.c.). (3.4)

The last two terms above are of order O(t−2) uniformly with respect toα becauseof the obvious equality

[Rα, g] = Rα[P 2

2, g

]Rα

and of the fact that for allα > 0,Rα ∈ B(H−1,H1).Denoting〈e−itRαψ, .e−itRαψ〉 by 〈.〉t,α, we estimate the expectation value of

the first two terms from the r.h.s. of (3.4) as follows:

1

t

⟨∣∣∣∣Qλt∣∣∣∣ g2

⟩t,α

> 1

tinf

x∈suppg|x|‖g e−itRαψ‖2,

1

λt〈gRα(P · ω + ω · P)Rαg〉t,α 6 1

t

‖Rα(P · ω + ω · P)Rα‖λ

‖g e−itRαψ‖2.Replacing the above inequalities in (3.4) we get(

infx∈suppg

|x| − 1

λsupα>1‖Rα‖2−1,0

)∫ ∞1

dt

t

∥∥∥∥g (Qλt)

e−itRαψ∥∥∥∥2

6 C‖h‖∞‖ψ‖2.

This shows that ifλ is chosen (with respect to the support ofg) such that

λ > supα>1‖Rα‖2−1,0 sup

x∈suppg|x|−1, (3.5)

applying the usual Fatou lemma (for the integral overt) yields:∫ ∞1

dt

t

∥∥∥∥g(Qλt)


=∫ ∞

1

dt

tlim infα→∞

∥∥∥∥g(Qλt)


6 lim infα→∞

∫ ∞1

dt

t

∥∥∥∥g(Qλt)


6 C‖ψ‖2

so the proposition is proved. 2

Let us make some comments on this first a priori result. From the point of viewof the physical interpretation, it is rather clear why the greatest lower bound of

MPAG010.tex; 14/05/1998; 16:04; p.14


the support ofg is important: the quantum system cannot delocalise arbitrarily fastin time since its energyR is a bounded, decreasing function. There is also thespecial case of the so-called ‘tails’ (scattering states describing the system alreadylocalised at infinity at finite times) which have non null asymptotic probability andthus have to belong to the orthogonal complement of the range of the projectorg. Then, a brutal cutoff introducing a least upper bound for suppg (as in the hy-pothesis of the above proposition) would avoid the problem caused by these ‘tails’but will make us lose the information about those states, describing a system withreally large asymptotic velocities and low energy. This shows that the above resultis far from being optimal. Nevertheless, Sigal and Soffer established in [36] a finerone, which holds for states belonging to a dense set of vectors, in which the upperlimiting cutoff is eliminated and where the norm in the r.h.s. of (3.1) is taken in aweighted Lebesgue space. But this result is no longer an abstract nonsense, since alocalisation in the complement of the set of the critical values of the Hamiltonianis needed and the Mourre estimate has been used in order to prove it.

Let us also note that the lower bound established in (3.5) for the value ofλ

is not optimal. One could have proved the proposition directly, without using theapproximating familyRαα>0 and obtaining the bestλ possible, but this requiresthe result (2.10) concerning commutators with hard-core resolvents (see the wayit is used in the proof of the following proposition). Moreover, we shall see thatnot all the results we need for proving asymptotic completeness can be deduced byworking with a sequence of approximating Hamiltonians, the algebraic propertiesof theN -body (hard-core) resolvents being crucial in the proof of the followingtheorems.

PROPOSITION 3.2. (i)Let f ∈ C∞0 (X) be constant around the origin on aregion with interior diameter not too small(i.e. proportional to the maximal ve-locity bound). Then there existsδ > 0 (depending on the short-range part of thehard-core potential) such that for alla ∈ L and allψ ∈ H(X) one has:∫ ∞

1

dt

t

∥∥∥∥j1/2a,δ

([iR,Qa] − Qa

t

)f

(Q

t

)e−itRψ

∥∥∥∥2

6 C‖ψ‖2, (3.6)

whereja,δ = ja(Qt tδ) denotes the multiplication operator with the smoothed char-acteristic function of the set4a (see(2.13)).

(ii) Moreover, if0a(0) = X \⋃b 66a Xb, then the following estimate is true forall functionsJ in C∞0 (0a(0)) with support not greater than that off :∫ ∞

1

dt

t

∥∥∥∥∣∣∣∣[iR,Qa] − Qa

t

∣∣∣∣J(Qt)


6 C‖ψ‖2. (3.7)

Proof. (i) Let us first introduce a notation concerning only the (vector or scalar)operators of multiplication withCp(X) functionsg: gβ will stand forg(. tβ−1) if

MPAG010.tex; 14/05/1998; 16:04; p.15


β > 0. Notice that this is consistent with the meaning ofja,δ and if (g′)β will bedenoted byg′β , then by iteration we get

g(α)β ≡

(g(α)

)β= t (1−β)|α|(gβ)(α)

for any multiindexα with |α| 6 p 6∞.Further, denote byAδ the operator:

Aδ = 1

2

∑a∈L

([iR,Qa] ·Qaja,δ + h.c.)

= 1

2

t

tδ

([iR,Qa] · r ′δ + r ′δ · [iR,Qa]), (3.8)

wherer ′δ =∑

a∈L πaja,δ is the smooth vector field introduced in Section 2. Usingthis operator we shall construct the propagation observable:

8δ = f(Aδ

t− Q

2

2t2

)f (3.9)

(the central part of8δ will occasionally be calledSδ). Then, we calculate as usualthe Heisenberg derivative of8δ and get:

DR8δ = 2 Re(DRf )Sδf + f (DRSδ)f. (3.10)

The term we need in the conclusion of the proposition (part (i)) will be furnishedby the second term in the r.h.s. of the above equality, while the other terms will beproved as being integrable int on [1,∞). Let us begin with the simplest one:∫ ∞

1dt〈DR8δ〉t 6 2 sup

t>1|〈f Sδf 〉t |, (3.11)

where the notation〈.〉t stands for〈e−itRψ, .e−itRψ〉. SinceSδ is obviously uni-formly bounded in time on the support off , it yields the integrability of DR8δ.

Let us now pass to the second term in the r.h.s. of (3.10) and calculate:

DRSδ =[iR,

Aδ

t

]−[iR,

Q2

2t2

]+ ∂

∂t

(Aδ

t

)+ 1

t

Q2

2t2. (3.12)

We shall estimate each of these terms separately:

1

t[iR,Aδ] = 1

2

∑b∈L

([iR, [iR,Qb]] · Qb

tjb,δ + h.c.

)+

+ 1

2tδ([iR,Q] · [iR, r ′δ] + h.c.

). (3.13)

MPAG010.tex; 14/05/1998; 16:04; p.16


The last line in the above equality can be computed by the repeated use of thecommutation relation (2.10) (notice also that according to it, the first double sumin (3.14) below is precisely[iR,Q]), as follows:

[iR, r ′δ] =∑b∈L[iR(b), r ′δ]

=∑b,c∈L

(R(c)

[iP 2

2, r ′δ

]Rcb + R(b)

[iP 2

2, r ′δ

]Rb

)= 1

ttδ r ′′δ ·

∑b,c∈L

(R(c)PRcb + R(b)PRb)− (3.14)

− i

2t2t2δ

∑b,c,d∈L

R(d)Gδ(RdcPRcb + RdbPRb)−

− i

2t2t2δ

∑b,c∈L

R(c)(r ′′′δ +GδRcP )Rcb −

− i

2t2t2δ∑b∈L

R(b)(r ′′′δ +GδRbP )Rb,

whereGδ has been used as a shorthand forP · r ′′′δ + r ′′′δ · P . Then, because of theboundedness ofr ′′′ (in theL∞-norm) and the fact that alsoP is bounded by theresolvents, the last three terms of the above relation will give rise to an O(t−2(1−δ))contribution; as for the first one, it can be minorated with the aid of

r ′′δ (x) >∑a∈L

ja,δ(x)πa (3.15)

(which is true for allx ∈ X asB(X)-valued measures onX, and for allδ). Finally,we get:

1

t[iR,Aδ] > 1

2

∑a∈L

([iR, [iR,Qa] ] · Qa

tja,δ + h.c.

)+

+ 1

t

∑a∈L[iR,Qa] · ja,δ[iR,Qa] +O(t−2+δ). (3.16)

We pass now to the second term from the r.h.s. of (3.12): using the fact that thefamily ja,δa∈L forms a partition of unity for anyδ > 0 we get:[

iR,−Q2

2t2

]= − 1

2t

∑a∈L

(Qa

t· ja,δ[iR,Qa] + h.c.

)−

− 1

2t1+δ∑a∈L

(Qa

ttδ · ja,δ[iR,Qa] + h.c.

). (3.17)

MPAG010.tex; 14/05/1998; 16:04; p.17


The last term here is O(t−1−δ) because all the components ofxa

ttδ are bounded on

the support ofja,δ ; in other words, if Id denotes the identity function, the last termabove equals

− 1

2t1+δ([iR,Q] · (r ′ − Id)δ + h.c.

)and from the definition of the vector field we know thatr ′ − Id is inL∞(X).

In a similar manner, computing the third term of the r.h.s. of (3.12) one obtains:

∂

∂t

(Aδ

t

)= − 1

2t1+δ([iR,Q] · r ′δ + h.c.)+

+ δ − 1

2t1+δ([iR,Q] · (r ′′ · Id− r ′)δ + h.c.). (3.18)

Sincer ′′ · Id− r ′ is also inL∞(X), the last term above gives also an O(t−1−δ)contribution. Finally, using the same trick, we compute

1

t

Q2

2t2= 1

t

∑a∈L

Qa

t· ja,δQa

t+O

(t−(1+2δ)

)(3.19)

and summing up (3.16) to (3.19) one obtains the estimate:

DRSδ >1

t

∑a∈L

([iR,Qa] − Qa

t

)· ja,δ

([iR,Qa] − Qa

t

)+

+ 1

2

∑a∈L

([iR, [iR,Qa]

] · Qa

tja,δ + h.c.

)+

+O(t−2+δ)+O(t−1−δ)+O(t−1−2δ). (3.20)

It remains to show that the second term from the r.h.s. of this inequality isintegrable w.r.t.t when taken in the mean value〈f . f 〉t . Let us remark first thatthe product of the double commutator[iR, [iR,Qa]] with some multiplicationoperators with functions of argumentx

twill appear quite frequently in this paper.

We will prove below that if these functions are supported outside the subspacesXb for all b ∈ L \ La then the desired decay int of the terms containing suchproducts will be ensured. Roughly speaking, this is due to the good decay alongcertain directions (depending on the givenb ∈ L) of theb-connected componentof our ‘Hamiltonian’ (i.e. the resolvent operator). Let us notice also that sincetheb-connected component of theN-body algebraT is (a semicompact operatoralgebra) of the formT (b) = K(Xb)⊗bT(Xb), its elements will commute withPa ≡ 1⊗aPXa . Hence, taking into account thatT is direct sum overL of T (b),we get:

[iR, [iR,Qa]]ja,δ =∑b∈L

R[iR(b), Pa]Rja,δ

=∑

b∈L\La

R(R(b)Pa − PaR(b))(ija,δR − [iR, ja,δ]). (3.21)

MPAG010.tex; 14/05/1998; 16:04; p.18


Using the commutation formula (2.10) we will be able to compute[iR, ja,δ] as

[iR, ja,δ] =∑c∈L[iR(c), ja,δ]

= tδ

tj ′a,δ

∑c,d∈L

(R(d)PRdc + R(c)PRc)+O(t−2+2δ)

= tδ

tj ′a,δ[iR,Q] +O(t−2+2δ), (3.22)

so (3.14) becomes:

[iR, [iR,Qa]]ja,δ=

∑b∈L\La

RR(b)(ija,δPa + tδ−1j ′a,δ)R + tδ−1j ′a,δPa[iR,Q]

−−

∑b∈L\La

RPaR(b)(ija,δR + tδ−1j ′a,δ[iR,Q]

)+O(t−2+2δ). (3.23)

The r.h.s. of the above equality has the advantage that it contains only products ofR(b) with the functionsja,δ andj ′a,δ. Note also that for allb ∈ L \ La there arestrictly positive constantsCa,b such that

〈xb〉 > |xb| ≡ |xbamin| > |xba | > Ca,b

on the support ofja. This shows that(〈xb〉tδ−1)−µja,δ 6 Ca,b−µja,δ which, joinedto (2.11), gives the desired decay int of the r.h.s. of (3.23) as, e.g., in:

RR(b)ja,δPaR = t−µ(1−δ)R R(b)〈Qb〉︸︷︷︸O(1)

(〈Qb〉ttδ)−µ

ja,δ︸︷︷︸O(1)

PaR, (3.24)

provided that 0< δ < 1 satisfies the supplementary conditionδ < 1− 1/µ.Thus, it only remains to show that|〈(DRf )Sδf 〉t | is integrable int . For this, we

choose an orthonormal basis inX and index byk = 1, . . . ,dimX the componentsof the vector operators asQ,∇f in this basis; actually, we shall denote byf ′k thecomponents of∇f and notice that because of the choice we made onf all of themare negative scalar functions. Denote also byfk theC∞0 (X \ 0)-function

√−f ′kand byχk a smoothed characteristic function of suppf verifying χkfk = fk. Then,commutingfk through[iR,Q] andSδ towards right we get:

(DRf )Sδf = 1

t

∑k

fk χk

(Qk

t− [iR,Qk]

)χkSδf︸︷︷︸

O(1)

fk +O(t−2), (3.25)

where the O(t−2) contribution is brought by the remainders of order two or by the(already) O(t−1) terms containing double commutators of the form[[iR,Q], f ].

MPAG010.tex; 14/05/1998; 16:04; p.19


This shows that there are positive constantsCk andC depending on the support off (and thus on the maximal velocity bound) such that∫ ∞

1dt|〈(DRf )Sδf 〉t | 6

∑k

Ck

∫ ∞1

dt

t

∥∥∥∥f (Qt ) e−itRψ∥∥∥∥2

+ C‖ψ‖2,

which proves the statement (i) of the proposition via the maximal velocity boundtheorem (Proposition 3.1).

We pass now to the proof of (ii): let us fix arbitrarily ana in L andassumefirst(and prove later) that for someJ with support as in the hypothesis (i.e. dependingon the chosena) there is aT > 0 such that for allt > T the equality

J

( ·t

)= J

( ·t

)f 2

( ·t

)∑b∈La

jb

( ·ttδ)

(3.26)

is valid. We stress that this is sufficient for showing that (3.7) is a consequenceof (3.6). Indeed, note first that (changing the lettera by b) an equivalent inequalityfor (3.6) is

dimXb∑k=1

∫ ∞1

dt

t

⟨([iR,Qk] − Qk

t

)f 2jb,δ

([iR,Qk] − Qk

t

)⟩t

6 C‖ψ‖2.

Then, sinceb ∈ La meansXa 6 Xb , the sum of positive terms in the r.h.s. of theinequality (3.27) below will be performed over a bigger index set than that fromthe l.h.s. Hence, using our assumption,∫ ∞

T

dt

t

⟨([iR,Qa] − Qa

t

)· J([iR,Qa] − Qa

t

)⟩t

=dimXa∑k=1

∫ ∞T

dt

t

⟨([iR,Qk] − Qk

t

)Jf 2

∑b∈La

jb,δ

([iR,Qk] − Qk

t

)⟩t

6 ‖J‖∞∑b∈La

dimXb∑k=1

∫ ∞1

dt

t

⟨([iR,Qk] − Qk

t

)f 2jb,δ ×

×([iR,Qk] − Qk

t

)⟩t

. (3.27)

It remains to prove the assumption (3.26). Note that since the familyjb,δb∈Lforms a partition of unity it suffices to show that if suppJ ⊂ suppf then for allt > T , x ∈ suppJ impliesx /∈⋃b∈L\La

suppjb,δ. But this becomes obvious if onethinks ofjb,δ as of a smoothed characteristic function of semi-cylinders centred on

the setsXb which shrink around these sets whent → ∞ (with the ‘velocity’ tδ)

and if one also takes into account that the complementary set for0a(0) is precisely⊔b∈L\La

Xb. 2

MPAG010.tex; 14/05/1998; 16:04; p.20


REMARK. In order to avoid some of the cumbersome calculations we madewhen estimating the terms related to the commutator[iR,Aδ] in the previousproof, we could work with the approximating resolventRα (instead of the hard-core resolventR) and obtain estimates which are uniform with respect to theparameterα. Nevertheless, there will be a difficulty related to the terms containingthe commutator[iRα, Pa], for which the deeper result furnished by Theorem 6.5in [12] has to be used. But this result concerns the hard-core resolvents, so we willhave to pass first to the limitα → ∞ under the integral

∫ T1 dt for someT > 1,

getting (for anya ∈ L)∫ T

1

dt

t

∥∥∥∥j1/2a,δ

([iR,Qa] − Qa

t

)f

(Q

t

)e−itRψ

∥∥∥∥2

6∑b∈L

∫ ∞1

dt

∣∣∣∣⟨R[iR, Pb](itδ−1j ′b,δRP − jb,δ)RQb

tf

⟩t

∣∣∣∣+O(1)‖ψ‖2,

and finally we will letT go to∞ and apply the quoted theorem as in the previousproof (relation (3.24)).

PROPOSITION 3.3.If a is an arbitrary element ofL and ifJ ∈ C∞0 (0a(0)) thenthere is a positive constantC such that∫ ∞

1

dt

t

∥∥∥∥∣∣∣∣[iR,Qa] − Qa

t

∣∣∣∣1/2J(Qt)


6 C‖ψ‖2 (3.28)

for all ψ ∈ H(X).

Before starting the proof let us introduce some notations which we shall usethroughout the rest of the paper:Ta will stand for [iR,Qa] − Qa

tand if A is an

unbounded (eventually time-dependent) operator then its time-dependent regulari-sation(|A|2+ t−4β)1/2 will be denoted by〈A〉β .

Proof. First of all we shall reduce a little the context in which the propositionhas to be proved by assuming thatJ also belongs to a particular class of functionswhich Derezinski denoted byF and defined as being formed byC∞0 (X)-functionsf having the property that for anya ∈ L there is a neighbourhoodVa of thesubspaceXa such thatf = f πa in Va. Obviously, this tells us that in someneighbourhoods of eachXa, the border of the support of each function inF will beperpendicular toXa. It is also easy to check thatF is a?-algebra which separatesthe points ofX and is dense inC∞0 (X) in theL∞-norm. It will thus be sufficient toprove the theorem forJ ∈ C∞0 (0a(0)) ∩ F .

Since0a(0) = ⊔b∈La

Xb, any compact set of0a(0) (and in particular the

support ofJ ) can be partitioned inε-neighbourhoods of a finite number of points

fromXb for someb ∈ La. The arbitrary (but fixed) choice we make forJ will

determine a choice for the diameter of these neighbourhoods and for the num-

bernb of those which are centred on points fromXb. Correspondingly, one shall

MPAG010.tex; 14/05/1998; 16:04; p.21


have a partition of unity on the support ofJ constructed with the aid of a familyjkb | kb = 1, . . . , nb, b ∈ La and satisfying onX:

∑b∈La

J πbnb∑kb=1

j2kb= J. (3.29)

Note that as a consequence of this choice we will also have∇J = (πb∇)J on thesupport of eachjkb .

Secondly, it is clear that the estimate∫ ∞1

dt

t

∥∥∥∥〈Ta〉1/2β J

(Q

t

)e−itRψ

∥∥∥∥2

6 C‖ψ‖2 (3.30)

implies (3.28), so it will be sufficient to prove the above inequality. In order to dothis, let us choose the propagation observable

8 = J 〈Ta〉βJ (3.31)

and compute its Heisenberg derivative:

DR8 = 2 Re(DRJ ) 〈Ta〉βJ + J (DR〈Ta〉β)J. (3.32)

We will show first that the second term in the r.h.s. of the above equality will furnishthe term from the conclusion of the theorem plus some integrable terms. For this,we shall use a particular form of the definition of the fractional power of a positiveoperatorA (see [38])

Aγ = C∫ ∞

0ωγ−1(ω +A)−1Adω, (3.33)

which holds strongly on its domain for some positive constantC and for γ ∈(0,1/2]. Then, an easy computation gives:

DRAγ = C

∫ ∞0ωγ−1(DR(ω+A)−1)A+ (ω+ A)−1DRA

dω

= C

∫ ∞0ωγ−1(ω+ A)−1(DRA)

(1− (ω+ A)−1A

)dω. (3.34)

In our case, we shall replaceA by 〈Ta〉2β and compute its Heisenberg derivative as:

DR〈Ta〉2β = −2

t

(〈Ta〉2β + (2β − 1)t−4β)+

+ ([iR, [iR,Qa]] · Ta + h.c.). (3.35)

Then, takingγ = 1/2 and replacing (3.35) in (3.34) we get:

MPAG010.tex; 14/05/1998; 16:04; p.22


DR〈Ta〉β= −2C

t

(〈Ta〉2β + (2β − 1)t−4β) ∫ ∞0ω−1/2(ω + 〈Ta〉2β)−1×

× 1− 〈Ta〉2β(ω + 〈Ta〉2β)−1dω ++C

∫ ∞0ω1/2(ω + 〈Ta〉2β)−1

Ta · [iR, [iR,Qa]] + h.c.×

× (ω+ 〈Ta〉2β)−1dω. (3.36)

The integral in the first term of the above equality is12〈Ta〉−1

β (see [35, p. 132]), so(3.36) becomes:

DR〈Ta〉β = −1

t〈Ta〉β + (1− 2β)t−4β−1〈Ta〉−1

β +

+2C Re∫ ∞

0dωω1/2(ω+ 〈Ta〉2β)−1

Ta ·[iR, [iR,Qa]

]×× (ω+ 〈Ta〉2β)−1

. (3.37)

Note that the second term in the r.h.s. above is O(t−1−2β) because of the obviousestimate‖〈Ta〉−1

β ‖ 6 t2β , while the first one is exactly the term we need for theestimate (3.30). Thus it will suffice it to show the integrability (w.r.t.t) of theexpectation value〈J . J 〉t of the third term from (3.37). This will be performed asin the proof of the previous theorem, i.e., by taking into account the decay int ofthe double commutator[iR, [iR,Qa]] on the support ofJ . Indeed, the hypothesisJ ∈ C∞0 (0a(0))makes us sure of the existence, for allb ∈ L\La, of some strictlypositive constantsCb for which

〈Qb〉 > |Qb| > Cbton suppJ ( ·

t) for all t > 1. This shows

J

(〈Qb〉t

)−µ6 C−µb J,

thus onceJ being brought nearby the double commutator we can apply the samereasoning as in (3.21) to (3.24) (takeδ = 0 for the present case) in order to obtainan O(t−µ) contribution. Nevertheless, besides the supplementary difficulties raisedby the commutator ofJ with the resolvent(ω+〈Ta〉2β)−1, there is also the problemof the boundedness of the components ofTa (which cannot be pulled out from theintegral overω because they do not commute with the〈Ta〉β from the resolvents).As we shall see, commutingJ with (ω + 〈Ta〉2β)−1 gives rise to O(t−1) factors(which are obviously not enough for the integrability int) containing derivativesof J but also operatorsTa. The strategy will be to continue to commuteJ , J ′through these resolvents until either O(t−2) terms or products of these functionswith [iR, [iR,Qa]] will appear. Then, only the problems of the boundedness of

MPAG010.tex; 14/05/1998; 16:04; p.23


the resultingTa ’s and those of the integrability w.r.t.ω on both[1,∞) and[0,1)will have to be solved. Actually, we shall see that the first two of them are relatedwhile in order to solve the third one we will need to ‘pick’ a little part from thegood decay we have obtained in the variablet and to convert it in decay inω. Letus first compute:[

(ω+ 〈Ta〉2β)−1, J]

= 2

tRe(ω+ 〈Ta〉2β

)−1Ta ·

(J ′ · [iR,Q]PaR + h.c.+ iRJ ′R +O(t−1)

)×× (ω+ 〈Ta〉2β)−1

. (3.38)

This allows us to give a precise formula for the operator which stands in (3.37) inthe r.h.s. of (and in product with) theR(b)’s (whereb ∈ L \ La) yielding from[iR, [iR,Qa]], namely:

R(ω+ 〈Ta〉2β

)−1J

= (JR + J ′[R,Q])(ω + 〈Ta〉2β)−1++ 1

tJ ′R

(ω+ 〈Ta〉2β

)−1(Ta ·O(1)+ h.c.)

(ω+ 〈Ta〉2β

)−1++O(t−2)

1+ (ω + 〈Ta〉2β)−1(Ta ·O(1)+ h.c.)

(ω+ 〈Ta〉2β)−1+

+R(ω+ 〈Ta〉2β)−1(Ta ·O(t−2)+ h.c.

)(ω+ 〈Ta〉2β

)−1−− 1

tRTa ·

(ω + 〈Ta〉2β

)−1O(t−1)(ω+ 〈Ta〉2β)−1O(1)×

× (ω+ 〈Ta〉2β)−1+ h.c.. (3.39)

Note that the last three lines from the r.h.s. above are O(t−2) while the first twoterms have apparently not the desired decay int ; but sinceJ andJ ′ will be nextto someR(b), they also will finally bring an O(t−µ) + O(t−1−µ) contribution.Concerning the boundedness of the components of thoseTa from above whichare taken in scalar product with O(1) terms, it will be no problem to ensure it(uniformly in t andω) with the aid of a square root of the resolvent(ω+ 〈Ta〉2β)−1.Nevertheless, this willnot be the case for theTa which is taken in (3.37) in scalarproduct with[iR, [iR,Qa]], because in order to ensure integrability inω on [1,∞)we need a norm of the resolvent(ω + 〈Ta〉2β)−1 on a power strictly superior to3/2. Thus we prefer to bound it by a〈Ta〉−1

β and to commute the remaining〈Ta〉βtowards left, next to theJ . Finally, using the Schwartz inequality after one hasreplaced (3.39) in (3.37) we obtain the estimate:∫ ∞

1dt∫ ∞

0dωω1/2

∣∣⟨J (ω + 〈Ta〉2β)−1Ta · [iR, [iR,Qa]](ω + 〈Ta〉2β)−1J⟩t

∣∣6 C‖ψ‖

∫ ∞1

dt O(t−µ)∥∥〈Ta〉βJ e−itRψ

∥∥ dimXa∑k=1

∥∥〈Ta〉−1β Tk

∥∥ ∫ ∞1

dωω1/2×

MPAG010.tex; 14/05/1998; 16:04; p.24


×∑

j=0,1,3

∥∥(ω + 〈Ta〉2β)−1∥∥2+j/2+

+ C‖J‖∞‖ψ‖2∫ ∞

1dt O(t−µ)

∫ 1

0dωω1/2×

×∑

j=0,2,3

∥∥(ω + 〈Ta〉2β)−1∥∥3−j/2

, (3.40)

with C, C positive constants depending only onJ , J ′ and whereµ stands formin2, µ. Then, the second term in the r.h.s. of (3.40) will be estimated minoriz-ing ‖(ω + 〈Ta〉2β)−1‖ by t4β , while for the first one we will take advantage of‖(ω+ 〈Ta〉2β)−1‖ 6 ω−1 in order to dominate the r.h.s. of (3.40) by:

‖ψ‖∫ ∞

1dt O(t−µ)‖〈Ta〉βJ e−itRψ‖

∫ ∞1ω−3/2 dω +

+‖ψ‖2∫ ∞

1dt O(t12β−µ)

∫ 1

0ω1/2 dω. (3.41)

Since the choice ofβ > 0 is at our disposal, we shall take it strictly inferior toµ−112 ,

which ensures integrability w.r.t.ω in the second term of the above sum. As for thefirst one, according to the Schwartz inequality, it will be dominated by

‖ψ‖(∫ ∞

1dt O(t1−2µ)

)1/2(∫ ∞1

dt

t

∥∥〈Ta〉βJ e−itRψ∥∥2)1/2

, (3.42)

whose finiteness is a consequence of Proposition 3.2(ii) and of hypothesisµ > 1.It remains to estimate the integrability of the expectation value〈·〉t of the first

term from the r.h.s. of (3.32). For this, we will act exactly as in the proof of Propo-sition 3.2 (relation (3.25)), but use the partition of unity one has introduced at thebeginning of this proof (see (3.29)). We have:

DRJ = 1

t

∑b∈La

nb∑kb=1

([iR,Qb] − Qb

t

)· (πb∇)Jj2

kb+ Remainder, (3.43)

where the ‘Remainder’ will be shown as being an O(t−2) term. Let us for themoment look at the first term above, and estimate:⟨

1

t

∑b∈La

nb∑kb=1

Tb · J ′j2kb〈Ta〉βJ

⟩t

6 1

t

∑b∈La

nb∑kb=1

〈Tb · J ′jkb [jkb , 〈Ta〉β]J 〉t +

+ 1

t

∑b∈La

nb∑kb=1

〈[Tb, jkb ] · J ′〈Ta〉βjkbJ 〉t +

MPAG010.tex; 14/05/1998; 16:04; p.25


+ 1

t

∑b∈La

nb∑kb=1

‖J ′ · Tbjkb e−itRψ‖‖〈Ta〉βJjkb e−itRψ‖. (3.44)

Note that as a consequence of

‖J ′ · Tbjkb e−itRψ‖ 6dimXb∑l=1

‖∂lJ‖∞‖Tl〈Tb〉−1β ‖‖〈Tb〉βjkb e−itRψ‖

and of the Schwartz inequality (applied to the integral∫

dt) we can dominate thelast line from (3.44) by

C∑b∈La

nb∑kb=1

(∫ ∞1

dt

t‖〈Tb〉βjkb e−itRψ‖2

)1/2

×

×(∫ ∞

1

dt

t

∥∥〈Ta〉βJjkb e−itRψ∥∥2)1/2

,

each of the above integrals being6 C‖ψ‖2 by Proposition 3.2. Then, the commu-tator[jkb , 〈Ta〉β ] will be computed with the aid of (3.33) as:

[jkb , 〈Ta〉β ] = 2C Re∫ ∞

0dωω1/2

(ω + 〈Ta〉2β

)−1Ta ·

[[iR,Qa], jkb]×

× (ω + 〈Ta〉2β)−1. (3.45)

The above double commutator gives an O(t−1) contribution, so we will continueto estimate the first two terms from the r.h.s. of (3.44) as before (relations (3.40) to(3.42)), the only difference being thatµ will be replaced by 2. This shows that thel.h.s. of (3.44) is integrable in time; the Remainder from (3.43) can be computedwith the aid of the Fourier spectral formula, as:

1

t2

∫X

dkJ ′′(k)e−ikt Q

∫ 1

0dτ∫ τ

0dσ e−

ikσt Q[ [iR,Q],Q]e+ ikσt Q, (3.46)

where J ′′ is a rapidly decreasing function (as Fourier transform of the smooth,compactly supportedJ ′′).

Finally, the integral overt of the l.h.s. of (3.32) is obviously dominated by

2 supt>1|〈J 〈Ta〉βJ 〉t | 6 2 sup

t>1‖J‖∞‖〈Ta〉βJ e−itRψ‖

= 2 supt>1‖J‖∞

〈JT 2

a J︸︷︷︸O(1)

〉t + t−4β‖J‖2∞‖ψ‖21/2

,

which finishes the proof of the proposition. 2

REMARK. (1) The estimate (3.7) remains valid if one replacesQa by any ofits components relative to some basis fromX. Moreover, this is also true for the

MPAG010.tex; 14/05/1998; 16:04; p.26


sharper estimate (3.28), as a consequence of the implication (see, e.g., Proposition6.2 in [20]):

A2k 6

n∑j=1

A2j ⇒ Ak 6

(n∑j=1

A2j

)1/2

,

valid for the setAj j=1,...,n of (unbounded) positive self-adjoint operators.(2) Since the Propositions 3.2 and 3.3 have been proved for an arbitrary lattice

L, they are also true for any sublattices ofL. More precisely, leta be an arbitrarilyfixed element ofL, and denote byTb,a the operator[iRa,Qb] − Qb

t. Then, due to

(2.2), the estimates (3.7) and (3.28) will also be true if one replacesR by Ra, i.e.for anyb ∈ La and anyJ ∈ C∞0 (0b(0)) there is a positive constantC such that:∫ ∞

1

dt

t

(∥∥∥∥|Tb,a|J(Qt)

e−itRaψ∥∥∥∥2

+∥∥∥∥|Tb,a|1/2J(Qt

)e−itRaψ

∥∥∥∥2)6 C‖ψ‖2. (3.47)

Moreover, taking into account thatb ∈ La implies0b(0) ⊆ 0a(0) and using thereasoning described in the proof of Proposition 3.2 (relation (3.24)) we see that thedifference betweenTb,a andTb is of order O(t−µ+1) on the support ofJ . Then, theobvious inequality(A+B)2 6 2(A2+B2) which holds for any pair of self-adjoint(not necessarily positive) operatorsA, B, shows thatTb,a can be replaced byTb inthe first norm from the above inequality. In what follows we will see that the sameis true for the second norm above, i.e. the estimate:∫ ∞

1

dt

t

∥∥∥∥∣∣∣∣[iR,Qb] − Qb

t

∣∣∣∣1/2J(Qt)

e−itRaψ∥∥∥∥2

6 C‖ψ‖2 (3.48)

holds for allb ∈ La, all J ∈ C∞0 (0b(0)) and allψ ∈ H(X). Indeed, using theformula (3.33), an easy calculation gives:

〈Tb〉β − 〈Tb,a〉β= C

∫ ∞0

dωω1/2(ω + 〈Tb,a〉2β)−1[i(R − Ra),Qb

]2(ω + 〈Tb〉2β

)−1 +

+C Re∫ ∞

0dωω1/2

(ω + 〈Tb,a〉2β

)−1Tb,a ·

[i(R − Ra),Qb

]×× (ω+ 〈Tb〉2β)−1

. (3.49)

This shows that we only have to commute theJ ’s from the left or from the rightthrough the resolvent(ω + 〈Tb〉2β)−1 in order to get either products ofJ , J ′ withR−Ra or O(t−2) terms. The boundedness of the components ofTb,a will be ensuredexactly as in the proof of the previous proposition (see the comments we made afterEquation (3.39)). In this way we show that the above difference is of integrableorder, and thus (3.47) implies (3.48).

MPAG010.tex; 14/05/1998; 16:04; p.27


4. Wave Operators

An important feature of the strategy that Sigal and Soffer imagined in order to proveexistence of the cluster wave operators and asymptotic completeness for usualN-body short-range systems, is that both these problems are treated as if they wereof the same difficulty. Indeed, the existence of the strong limits of exp(itH)Jb ×exp(−itHb), whereJb is a family pseudodifferential operators verifying a par-tition of unity in the phase-space is proven first. In our case, we shall take, as in[23, 15], a time-dependent family of identificators and state:

PROPOSITION 4.1. If a is an arbitrary element ofL and if J is a C∞0 (0a(0))function, then the operator domain of the following limits:

W±(R,Ra; J ) = s-limt→±∞ eitRJ

(Q

t

)e−itRa , (4.1)

W±(Ra,R; J ) = s-limt→±∞ eitRaJ

(Q

t

)e−itR (4.2)

is the wholeH(X). Moreover, the statement is true even whenJ is a boundedcontinuous function with support in0a(0).

Proof. As in the proof of the Proposition 3.3, for showing the first part of theproposition it will be sufficient to takeJ ∈ C∞0 (X) ∩ F . We shall also use thesame partition of unity on the support ofJ constructed with the aid of the familyjkb | kb = 1, . . . , nb, b ∈ La.

In order to prove the existence of the limits (4.1) and (4.2) the Cook criterionwill be used. More precisely, the computation:

d

dt

⟨ψ, eitRaJ e−itRψ

⟩ = ⟨ψ, eitRa DRJ − iJ (R − Ra) e−itRψ⟩

shows that a sufficient condition for the convergence of eitRaJ e−itR in expectationvalue onψ ∈ H(X) is∫ ∞

1

∣∣〈ψ, eitRa DRJ e−itRψ〉∣∣ dt + ‖ψ‖2 ∑b∈L\La

∫ ∞1‖R(b)J‖dt

6 C‖ψ‖2. (4.3)

Then this weaker type of convergence yields (in our special case) strong con-vergence in a standard manner. Reasoning in the same way as in the proof ofProposition 3.2 (see relation (4.3)) we prove that the integrand in the second termfrom the r.h.s. of (3.21) is of order O(t−µ) for all b ∈ L \ La. Further, using theformula (3.43) we show that the first term in (4.3) is dominated (modulo someO(t−2) contributions) by:∫ ∞

1

dt

t

∑b∈La

nb∑kb=1

dimXb∑l=1

∥∥Tl〈Tl〉−1β

∥∥×

MPAG010.tex; 14/05/1998; 16:04; p.28


×∥∥〈Tl〉1/2β jkb e−itRaψ∥∥∥∥〈Tl〉1/2β jkb (∂lJ )e−itRψ

∥∥6 C

∑b∈La

nb∑kb=1

dimXb∑l=1

(∫ ∞1

dt

t

∥∥〈Tl〉1/2β jkb e−itRaψ∥∥2)1/2

×

×(∫ ∞

1

dt

t

∥∥〈Tl〉1/2β jkb(∂lJ )e−itRψ∥∥2)1/2

,

where in the above estimate the Schwartz inequality has been used. Finally let usnote that each of the integrals from the above brackets is6 C‖ψ‖2 as a conse-quence of the remark following Proposition 3.3.

Suppose nowJ ∈ BC(0a(0)). Letχ be the smoothed characteristic function ofa neighbourhood of the origin inX, and denote byχγ the operator of multiplicationwith χ( ·

γ t) , whereγ > 0 is a parameter. Then the productJχγ plays the role of

theJ ’s from the first part of the proposition, so it will be sufficient to prove that forany positiveε we can chooseγ in such a way that

supt>1

∥∥eitRaJ (1− χγ )e−itRψ∥∥ < ε‖ψ‖H1

for allψ belonging to the weighted Lebesgue space of order oneH1(X). Moreover,using the obvious inequality 1− χγ 6 Id, true forγ sufficiently large, we see thata stronger condition is given by

supt>1

1

t‖|Q|e−itRψ‖ < εγ

‖J‖∞ ‖ψ‖H1. (4.4)

But (4.4) is a particular case of a result due to Radin and Simon (see Theorem 2.1in [33]). Indeed, all we have to do is to mimic the proof of the quoted theorem(after one has taken as Hamiltonian the approximating resolventRα) and finallyobtain:

‖|Q|e−itRαψ‖ 6 ‖|Q|ψ‖ +∫ t

0dτ 〈P 2

α 〉1/2t,α ,

wherePαα>0 denotes the uniformly bounded family of operatorsRαPRα. Apply-ing the Fatou lemma to the above inequality proves (4.4), provided thatγ is chosensuperior toε−1‖J‖∞. 2

LEMMA 4.1. If a is an arbitrary element ofL and if Ea denotes the projectionEpp(H

a)⊗a 1, then the limits:

±a = s-limt→±∞ eitR e−itRaEa (4.5)

exist on allH(X). Moreover, ifa 6= b are two elements ofL, then the ranges ofthe corresponding cluster wave operators are orthogonal.

MPAG010.tex; 14/05/1998; 16:04; p.29


Proof. Since the existence of the operatorsW±(R,Ra; J ) has been proved onH(X) for all J ∈ BC(X) with support in0a(0), anε/3 argument shows that inorder to prove the existence of (4.5) the following convergence∥∥∥∥(1− J

(Q

t

))e−itRau

∥∥∥∥ t→±∞−→ 0 (4.6)

has to be true for allu belonging to a dense set ofEaH(X). The choice we willmake forJ in (4.6) depends on vectoru.

For this, let us construct the dense set ofu’s by taking simple tensors of theform wa ⊗a va, wherewa belongs toEpp(Ha)H(Xa) (and thus is an eigenvector

of Ha for the eigenvalueλa) andva = ga(Pa)va with ga ∈ C∞0 (Xa) and

Pa ≡(z− λa − P

2a

2

)−1

Pa

(z− λa − P

2a

2

)−1

. (4.7)

As we explained before we shallchooseJ ∈ BC(0a(0)) ∩ F depending oneachu by requiring:

suppJ |Xa ⊃ suppga,J |Xa = 1 on the support ofga.

(4.8)

Then, the l.h.s. of (4.6) equals:∥∥∥∥(1− J(Q

t

))e−itRawa ⊗a ga(Pa)va

∥∥∥∥=∥∥∥∥(1− J

(Q

t

))e−it (z−λ

a−P 2a /2)

−1wa ⊗a ga(Pa) va

∥∥∥∥ (4.9)

because of relation (1.3), which in turn is a consequence of Theorem 3.10 from[12].

Let us take now one more cutoff functionh ∈ C∞0 (Xa) with h = 1 in a neigh-bourhood of the subspaceXa. The precise choice we will make for the support ofh will depend on the choice we make onJ and hence onga, so for the moment welet the shape of the support ofh entirely at our disposal. In what follows we shalldenote byA . B the inequality (between numbers)A 6 B + o(1), where o(1) isdefined w.r.t.t →±∞. Then the r.h.s. of (4.9) will be majorized by:∥∥∥∥(1− J

(Q

t

))e−it (z−λ

a−P 2a /2)

−1h

(Q

t

)wa ⊗a ga(Pa)va

∥∥∥∥++∥∥∥∥(1− h

(Q

t

))wa ⊗a va

∥∥∥∥.∥∥∥∥(1− J

(Q

t

))h

(Q

t

)ga(Pa)e−it (z−λ

a−P 2a /2)

−1wa ⊗a va

∥∥∥∥6 C

∥∥∥∥(ga(Pa)− ga(Qa

t

))e−it (z−λ

a−P 2a /2)

−1wa ⊗a va

∥∥∥∥+

MPAG010.tex; 14/05/1998; 16:04; p.30


+∥∥∥∥(1− J

(Q

t

))h

(Q

t

)ga

(Qa

t

)e−it (z−λ

a−P 2a /2)

−1wa ⊗a va

∥∥∥∥,whereC is a positive constant depending onJ andh and where in the asymptoticinequality one has supposed thath has the supplementary propertyh = h πa soit commutes with the unitary group exp−it (z− λa − P 2

a /2)−1.

The first term in the above inequality tends towards zero whent → ±∞ as aconsequence of Theorem 7.1.29 of Hörmander (see [25]), and thus one has shownthat: ∥∥∥∥(1− J

(Q

t

))e−itRau

∥∥∥∥.∥∥∥∥(1− J

(Q

t

))h

(Q

t

)ga

(Qa

t

)e−it (z−λ

a−P 2a /2)

−1wa ⊗a va

∥∥∥∥. (4.10)

SinceJ ∈ F one can make our final hypothesis onh, namely: the support ofhis chosen (in function ofJ ) to be included in the neighbourhood ofXa for whichJ = J πa. Then, in the r.h.s. of (4.10) we will have:(

1− J(Q

t

))h

(Q

t

)ga

(Qa

t

)=(

1− J(Qa

t

))ga

(Qa

t

)h

(Q

t

)= 0

because of the second hypothesis in (4.8), and thus the first statement of the lemmais proved.

It remains to show that for arbitrarya 6= b in L and for any vectorsϕ,ψ ∈H(X) we have〈±a ϕ,±b ψ〉 = 0. Actually, it will be sufficient to prove theconvergence⟨

e−itRaEaϕ, e−itRbEbψ⟩ |t |→∞−→ 0

for ϕ andψ belonging to the dense sets previously introduced. This time we don’tneed any functionga so we will take it equal to one. But for these vectors, the l.h.s.of the above relation equals:⟨

Eaϕ, e−it (z−λa−P 2

a /2)−1

eit (z−λb−P 2

b /2)−1

Ebψ⟩,

which tends to zero whent goes to∞ as a consequence of the Riemann–Lebesguelemma. 2

5. Proof of the Minimal Velocity Theorem

Our main result, Theorem 1.1, can be stated in a more precise form as follows:

THEOREM 5.1. Leta ∈ L andε > 0 be arbitrarily chosen. Let(J, θ) and(J , θ )be two couples of functions, withJ, J ∈ C∞0 (0a(0)) andθ, θ ∈ C∞0 (R), such that

MPAG010.tex; 14/05/1998; 16:04; p.31


J , θ equal one on the support ofJ andθ , respectively. Suppose moreover that forarbitrary λ ∈ R, one of the following two conditions:

inf(ρARa(λ)− ε)µ2− x

2a

2| x ∈ suppJ andµ ∈ suppθ

> sup

x∈suppJ

dimXa∑l=1

‖[iR,Ql]‖|xl|2, (5.1)

2 infµ2

supµ2| µ ∈ suppθ

(ρARa(λ)− ε) >

dimX∑l=1

‖Plθ(R)‖ supx∈suppJ

|xl| (5.2)

is satisfied. Then, the estimate:∫ ∞1

dt

t

∥∥∥∥J(Qt)θ(R)e−itRψ

∥∥∥∥2

6 C‖ψ‖2 (5.3)

is true for some positive constantC and for allψ ∈ H(X).

Note that depending on the choice of the pair(J, θ), the above result can be seeneither as a maximal or as a minimal velocity bound theorem. This result gives moreinformation than we need in order to prove asymptotic completeness. Indeed, weshall see later that the following corollary, which corresponds to the particular casea = amax of the above theorem, is sufficient (via a standard induction argument) forthe proof of (1.10). Let us denote byC(R) the set of critical values of the operatorR. Then:

COROLLARY 5.2. Letθ ∈ C∞0 (R\C(R)) and takeJ ∈ C∞0 (X) with the supportsufficiently close to the origin inX. Then,

s-limt→±∞eitRJ

(Q

t

)e−itRθ(R) = 0. (5.4)

Proof.It is sufficient to choose the support ofJ (depending on how close suppθis to the points fromC(R)) such that one among the two conditions (5.1), (5.2)holds. Then (5.3) will be valid, which will imply:

lim inft→∞

∥∥∥∥eitRJ

(Q

t

)e−itRθ(R)ψ

∥∥∥∥ = 0

for all ψ ∈ H(X). But according to Proposition 4.1 the full strong limitsW±(R,R; J ) exist on allH(X), so they equal zero on all setsθ(R)H(X) (whoseunion over allθ as in the hypothesis, is dense inH∞). 2

Proof of Theorem 5.1.Since the theorem should be true for anya ∈ L, duringthe whole proof we will fix an arbitrarily chosena and consider (unless otherwisespecified) that all the operators of multiplication with functions are relative to this

MPAG010.tex; 14/05/1998; 16:04; p.32


a (as it where be indexed bya). Then the proof will be performed by induction overthe levelsLa(n) (n = 1, . . . , rankLa ≡ Na) of the sublatticeLa. Nevertheless,there is a particular case, namelya = amin, for which the proof is simpler andworks exactly in the same manner as for the first step of the inductive processcorresponding to the casesa ∈ L \ amin.

The induction hypothesis will be called(Pn+1) and expressed as follows:

(Pn+1): If for anyb ∈ La(n+ 1) and anyε > 0 there are two couples of functions(J, θ) and(J , θ ) with J, J ∈ C∞0 (0b(0)) andθ, θ ∈ C∞0 (R), such thatJ , θ equalone on the support ofJ andθ respectively, and if for allλ ∈ R, one of the followingtwo conditions:

inf(ρARb(λ)− ε)µ2− x

2a


> sup

x∈suppJ

dimXa∑l=1

‖[iR,Ql]‖|xl|2, (5.5)

2 infµ2

supµ2| µ ∈ suppθ

(ρARb(λ)− ε) >

dimX∑l=1


|xl| (5.6)

is fulfilled, then the estimate(5.3) is true.

The first step of the (weak) inductive process, namely the validity of(PNa), willbe given by the following lemma. As we said before, this lemma also tells us thatTheorem 5.1 is true in the particular casea = amin (see condition (5.7) below).

LEMMA 5.1. Let b = amin in the hypothesis of(Pn+1) or suppose that for thesame couples of functions the following strict inequality

inf(ρARamin

(λ)− ε)µ2− x2


> 0 (5.7)

holds. Then(5.3)holds also.Proof. Let us bring to mind first that given a vector operatorS in H(X), we

shall denote bySa, Sa the operators 1⊗a (πaS) , resp.(πaS) ⊗a 1 in H(X), butwe shall not change the notation where the operators(πaS) acting inH(Xa) and,respectively,(πaS) in H(Xa) will be concerned. As stated before, whenever noconfusion is possible, we shall not mention the usual time-dependent argumentx

t

of the operators of multiplication with functions.Keeping in mind these conventions, let us choose the propagation observable:

8 = θ(R)J[iR,

(Qa)2

2t

]Jθ(R), (5.8)

MPAG010.tex; 14/05/1998; 16:04; p.33


and compute as usual its Heisenberg derivative as:

DR8 = 1

tReθ(R)(DRJ )

[iR, (Qa)2

]Jθ(R)+

+ θ(R)J(

DR

[iR,

(Qa)2

2t

])Jθ(R). (5.9)

Then the obvious estimate∫ ∞1

dt〈DR8〉t 6 2 supt>1|〈8〉t |

6dimXa∑l=1

‖[iR,Ql]‖ supx∈suppJ

|xl|‖J‖2∞‖θ‖2∞‖ψ‖2 (5.10)

shows that we only have to look at the terms from the r.h.s. of (5.9): we shall beginby proving that the first one is integrable w.r.t.t . For this, let us take an orthonormalbasis inX and denote byTk the components of the vector operatorTamin = T in thisbasis. Since the support ofJ is compact, there exists a finite familyji of C∞0 -functions with supports included in suppJ and satisfyingJ = J

∑i j

2i . Then,

proceeding as in the proof of the Proposition 3.3 (see relations (3.43) and (3.44))we get:

(DRJ )

[iR,

(Qa)2

t

]J

= 1

t

dimX∑k=1

∑i

ji〈Tk〉1/2β 〈Tk〉−1β Tk︸︷︷︸

O(1)

Bk〈Tk〉1/2β ji +O(t−2). (5.11)

Sinceji ∈ C∞0 (Xamin) for any i, we can use (3.30) in order to integrate the first

term in the r.h.s. above, provided that

Bknot= 〈Tk〉1/2β (∂kJ )

[iR,

(Qa)2

t

]J 〈Tk〉−1/2

β (5.12)

is shown as being the sum between a uniformly bounded (int ) operator and someintegrable terms. We have thus to commute〈Tk〉1/2β towards left. SinceJ and∂kJbound the above commutator, we will only have to show the integrability of sumsof the form:

1

t

[〈Tk〉1/2β , g][iR,Ql ]g + 1

tg[〈Tk〉1/2β , [iR,Ql]

]g, (5.13)

whereg, g denote the operators of multiplication with the functions∂kJ or (∂kJ )πlandJπl or J , respectively (their argument being as usualx

t). Using the formula

(3.33) withγ = 1/4 and reasoning exactly as in the proof of Proposition 3.3 (see

MPAG010.tex; 14/05/1998; 16:04; p.34


(3.45)) it can be shown that ifβ is appropriately chosen w.r.t.µ, the first term fromabove brings an integrable contribution. Then, in the same manner, the second termof the above sum is computed as:

2C

tRegTk

∫ ∞0

dωω1/4(ω + 〈Tk〉2β)−1[[iR,Qk], [iR,Ql]

]++[[iR,Ql], Qk

t

](ω + 〈Tk〉2β

)−1g.

It is clear that the second term in the curly bracket brings an O(t−2) contribu-tion and, forβ > 0 sufficiently small, it will still be integrable w.r.t.t after theintegration inω is performed. As for the first one, the formula:

1

t

[[iR,Qk], [iR,Ql]]

= 1

2

[[iR, [iR,Ql]], Qk

t

]+ 1

2

[[[iR,Qk], iR], Ql

t

](5.14)

shows that we only have to prove:∫ ∞1

dt∫ ∞

0dωω1/4

∥∥∥∥gTk(ω + 〈Tk〉2β)−1Ql

t[iR, [iR,Qk]] ×

× (ω+ 〈Tk〉2β)−1g〈Tk〉−1/2

β

∥∥∥∥ <∞.The above double commutator can be computed as in the proof of Proposition 3.2,by taking into account that for anyc ∈ L we have[Pk,R(c)] = 0 for all k forwhichXk ⊆ Xc. Note that we refer to⊆ as to a vector space inclusion becauseXkcould not belong to the latticeXbb∈L. We thus have:

[, [iR,Qk], R] =∑c∈LXk 6⊆Xc

R(PkR(c)− R(c)Pk)R.

Note that for allk = 1, . . . ,dimX the inclusionXk ⊂ Xamin shows that the abovesum will be performed over a subset ofL \ amin and since the supports ofg, gare compacts from0amin(0) (for all l = 1, . . . ,dimXa) they are disjoint of anyXcwith c ∈ L \ amin so it will be sufficient to commuteg, g towards right or leftrespectively, in order to obtain either O(t−2) terms, or productsgR(c) ∼ O(t−µ).As in the proof of Proposition 3.3 (see the comments made before (3.40)), therewill be a tribute to pay in order to ensure integrability w.r.t.ω, but for a suitable(small, positive)β there will still remain enough decay int for the convergence ofthe above double integral.

We pass now to the second term in the r.h.s. of (5.9). So far, any of the hypothe-ses (5.5), (5.6) or (5.7) has been used; in what follows, this term will be estimatedin three (somewhat) different ways, each of these variants involving the Mourre

MPAG010.tex; 14/05/1998; 16:04; p.35


estimate and only one of the mentioned hypotheses. Let us start by recalling thatfor anyb ∈ L the differencesR − Rb andθ(R)−θ(Rb) and the double commutator[[iR,Qb], R] are of order O(t−µ) on suppJ ⊂ 0amin(0). Then the obvious equality[

iR,

[iR,

Q2a

2t

]]= Re[iR, [iR,Qa]] · Qa

t+ 1

t[iR,Qa]2 (5.15)

shows that:

θ(R)J

(DR

[iR,

(Qa)2

2t

])Jθ(R)

= −1

tθ(R)J

[iR,

(Qa)2

2t

]Jθ(R)+

+ 1

tJRaminθ(Ramin)[iRamin, A]θ(Ramin)RaminJ −

− 1

tθ(R)J [iR,Qa]2Jθ(R)+O(t−µ)+O(t−2). (5.16)

We start to prove that (5.5) (taken withb = amin) implies (5.3). Since

[iR,Qa]2 = T 2a + 2 ReTa · Qa

t+ Q

2a

t2, (5.17)

the mean value〈.〉t of the last term from the r.h.s. of (5.16) dominates the sum:

−1

t

‖|Ta|Jθ(R)e−itRψ‖2+

dimXa∑l=1

supx∈suppJ

|xl|∥∥〈Tl〉1/2β J θ(R)e−itRψ‖2

+

+O(t−2)‖ψ‖2− 1

tsup

x∈suppJx2a‖J θ(R)e−itRψ‖2 (5.18)

in which the first line is integrable int as a consequence of the propagation Theo-rems 3.2 and 3.3. Further, the first term in the r.h.s. of (5.16) is minorized by:

−1

tsup

x∈suppJ

dimXa∑l=1

‖[iR,Ql]‖|xl |‖Jθ(R)e−itRψ‖2 (5.19)

whereas for the second term from the r.h.s. of (5.16) we apply the Mourre estimate(see def. (1.5) and also (1.6)) in order to dominate it by

2

t

(ρARamin

(λ)− ε) infµ∈suppθ

µ2‖Jθ(R)e−itRψ‖2 +O(t−µ)+O(t−2). (5.20)

Then, summing up (5.18) to (5.20) we obtain a lower bound for the mean value ofl.h.s. of (5.16):⟨

θ(R)J

(DR

[iR,

(Qa)2

2t

])Jθ(R)

⟩t

MPAG010.tex; 14/05/1998; 16:04; p.36


' 2

t

(ρARamin


µ2−

− supx∈suppJ

(x2a

2+

dimXa∑l=1

‖[iR,Ql]‖|xl |2

)‖Jθ(R)e−itRψ‖2,

where' means> modulo addition of some terms of integrable order (wheneverthis type ofequalitywill arise, the sign≈ will be used). But since the support ofJis a dilation of suppJ , the hypothesis (5.5) ensures strict positivity of the quantityfrom the above curly bracket, which proves (5.3).

We pass now to the proof of the implication (5.7)⇒ (5.3) and compute:

θ(R)J

(DR

[iR,

(Qa)2

2t

])Jθ(R)

= −1

tθ(R)J

(ReT a · Q

a

t+ (Q

a)2

t2

)Jθ(R)−

− 1

tθ(R)J

(ReTa · Qa

t+ T 2

a

)Jθ(R)+

+ 1

tJRaminθ(Ramin)[iRamin, A] θ(Ramin)RaminJ −

− 1

tθ(R)J

Q2a

t2Jθ(R)+O(t−µ)+O(t−2). (5.21)

Estimating the terms from the r.h.s. above like in (5.18)–(5.20) we see that the l.h.s.of (5.21) dominates (modulo some integrable terms):

2

(ρARamin


µ2− 1

2sup

x∈suppJx2

‖Jθ(R)e−itRψ‖2

in which the curly bracket contains a strictly positive number as a consequence of(5.7). Finally,

[iR,Qa]2 = T 2a + ReTa · Qa

t+[iR,

Q2a

2t

](5.22)

allows us to calculate:

θ(R)J

(DR

[iR,

(Qa)2

2t

])Jθ(R)

= −1

tθ(R)J

[iR,

Q2

2t

]Jθ(R)+O(t−µ)+O(t−2)−

− 1

tθ(R)J

(ReTa · Qa

t+ T 2

a

)Jθ(R)+

+ 1

tJRaminθ(Ramin)[iRamin, A]θ(Ramin)RaminJ (5.23)

MPAG010.tex; 14/05/1998; 16:04; p.37


which together with the estimate:

1

t

⟨JRθ(R)P · Q

tJRθ(R)

⟩t

/dimX∑l=1


|xl|‖Rθ(R)J e−itRψ‖2

/dimX∑l=1


|xl| supµ∈suppθ

µ2‖Jθ(R)e−itRψ‖2

shows that the l.h.s. of (5.23) is

' 1

t

2(ρARamin


µ2

−dimX∑l=1


|xl| supµ∈suppθ

µ2

‖Jθ(R)e−itRψ‖2.

Using the hypothesis (5.6) (written forb = amin) we see again that the above curlybracket is positive, and this completes the proof of the lemma. 2

We begin now the second step of the inductive reasoning, namely one has toprove the implication(Pn+1)⇒ (Pn) for anyn = 1, . . . , Na−1. Notice that in (5.5)and (5.6) one could have takenρARa(λ) instead ofρARb(λ), but sinceb ∈ La(n+ 1)impliesρARa (λ) 6 ρARb(λ), (Pn+1) would have weakened (anyway, this would notcreate any disadvantage from the point of view of the final result). For proving theabove implication it will suffice to fix arbitrarilyb ∈ La(n) and show that(Pn)holds for thisb. Let thusJ , J andθ , θ be chosen with respect to thisb such that(5.5) or (5.6) be verified. Then Proposition 6.1(iv) shows that for any suchJ we canconstruct (as in the proof of Proposition 3.3) a familyJkc of C∞0 (0b(0))-functionssatisfying:

J = J∑c∈Lb

nc∑kc=1

J 2kc, (5.24)

suppJ ⊃⋃c∈Lb

nc⋃kc=1

suppJkc . (5.25)

Notice that (5.25) is always possible because of the strict inclusion of suppJ insuppJ and that it will be no loss in supposing thatnb = 1 (renoteJkb by Jb). Then,since(Pn+1) is supposed true, (5.24) allows us to reduce the problem a little byestimating:∫ ∞

1

dt

t‖Jθ(R)e−itRψ‖2

MPAG010.tex; 14/05/1998; 16:04; p.38


6 ‖J‖2∞∫ ∞

1

dt

t‖Jbθ(R)e−itRψ‖2 +

+‖J‖2∞∑c<b

nc∑kc=1

∫ ∞1

dt

t‖Jkcθ(R)e−itRψ‖2. (5.26)

Then, as a consequence of (5.25), there is aC∞0 (0b(0)) -function Jkc such thatJkc = 1 on suppJkc and suppJkc ⊂ suppJ . This shows

supx∈suppJ

> supx∈suppJkc

and since as a consequence of the definition of the functionρ:

ρARc(λ) > ρARb(λ) for all c 6 b and allλ ∈ R, (5.27)

the assumptions of the type (5.5) and (5.6) in(Pn) (written forJ ) are stronger thanthe same assumptions in any of the(Pn+1)’s written for Jkc . This shows that thesecond line of (5.27) is6 C‖ψ‖2 so it remains only to estimate the first term inr.h.s. of (5.27). For this, let us consider two smooth functions,Jb andf , having theproperty that for some positive constantC:

Jb 6 CJbf (5.28)

is true on allX. Observe that such functions always exist: it is sufficient to choosethem such that the product be aC∞0 -function and that

suppJb ⊂ suppJb ∩ suppf. (5.29)

We shall, moreover, put some supplementary conditions onJb; namely, we suppose

that there is a set calledcore suppJb (which is centred on the same point ofXb as

suppJb) satisfying:

core suppJb ⊂ suppJb, (5.30)

core suppJb ∩Xb 6= ∅, (5.31)

Jb is constant oncore suppJb, (5.32)

Jb ∈ C∞0 (0b(0)), (5.33)

suppJb ⊂ suppJ . (5.34)

Note that (5.34) is allowed because of the definition ofJ and of (5.25). As forf ,we suppose that:

f ∈ C∞0 (X), (5.35)

suppf ∩ Xb ⊂ core suppJb ∩Xb, (5.36)

f = f πb on suppJ . (5.37)

MPAG010.tex; 14/05/1998; 16:04; p.39


In conclusion, it will be sufficient to prove:∫ ∞1

dt

t

∥∥∥∥f(Qb

t

)Jb

(Q

t

)θ(R)e−itRψ

∥∥∥∥2

6 C‖ψ‖2 (5.38)

which will always be verified provided that∫ ∞1

dt

t

∥∥∥∥f ([iR,Qb])Jb(Q

t

)θ(R)e−itRψ

∥∥∥∥2

6 C‖ψ‖2 (5.39)

and that∫ ∞1

dt

t

⟨Jb

(Q

t

)f 2([iR,Qb])− f 2

(Qb

t

)Jb

(Q

t

)⟩t

6 C‖ψ‖2. (5.40)

In the Appendix 6.4 we prove that estimates of the type (5.40) are consequences ofthe propagation Theorems 3.2 and 3.2, so it remains to prove (5.39). For this, wechoose as propagation observable the bounded operator

8 = θ(R)f([iR,Qb]

)Jb

(Q

t

)×

×[iR,

(Qa)2

2t

]Jb

(Q

t

)f ([iR,Qb])θ(R). (5.41)

As in the proof of Lemma 5.1, an inequality of the same type as (5.10) showsthat it will suffice to estimate each of the terms yielding from the derivation DR8.Starting with the one containing DRf ([iR,Qb]), we shall use the spectral Fourierformula in order to calculate:

[iR, f ([iR,Qb])]=∫X

dsf ′(s)∫ 1

0dτ eis(1−τ)[iR,Qb][iR,[iR,Qb]] eisτ [iR,Qb]. (5.42)

Then, we commuteJb through exp(isτ [iR,Qb]) in order to obtain products ofJb ,J ′b with the above double commutator (which give O(t−µ) contributions). After allthese computations have been performed, we obtain (modulo some O(t−2) terms):

[iR, f ([iR,Qb])]Jb≈∫X

dsf ′′(s)∫ 1

0iτ dτ

∫ 1

0dσ eis(1−τ)[iR,Qb][iR, [iR,Qb]] ×

×(Jb + 1

tJ ′b eisτσ [iR,Qb][[iR,Qb],Q]e−isτσ [iR,Qb]

)eisτ [iR,Qb] (5.43)

which shows∫ ∞1

dt

t

⟨θ(R)(DRf ([iR,Qb]))Jb

[iR,

(Qa)2

2t

]Jbf ([iR,Qb])θ(R)

⟩t

6 C‖ψ‖2.

MPAG010.tex; 14/05/1998; 16:04; p.40


We pass now to the proof of the integrability w.r.t.t of the second term resultedfrom8’s Heisenberg derivation, i.e. the one containing

DRJb = J ′b · Tamin +O(t−2).

Let kc | c ∈ Lb andkc = 1, . . . , nc be a family ofC∞0 (0b(0))-functions having

supports centred on some points ofXc, satisfying:

J ′b =∑c∈Lb

nc∑kc=1

2kcJ ′b (5.44)

and

suppkc ∩Xb = ∅ for all c < b. (5.45)

Note also that according to the assumptions (5.36) and (5.32) one can choosethe sub-familykb | kb = 1, . . . , nb such that:(

nb⋃kb=1

suppkb ∩Xb

)∩ suppf = ∅. (5.46)

Hence, denoting byhkc aC∞0 (0b(0))-function equal to one on the support ofkcand making use of (5.44) we get:⟨

θ(R)f ([iR,Qb])(DRJb)

[iR,

(Qa)2

2t

]Jb f ([iR,Qb])θ(R)〉t

/ 1

t

∑c∈Lb

nc∑kc=1

⟨θ(R)kcf ([iR,Qb])×

× J ′b · Tamin

[iR,

(Qa)2

2t

]hkc︸︷︷︸

O(1)

f ([iR,Qb])kc θ(R)⟩t

/ C

t

∑c∈Lb

nc∑kc=1

‖f ([iR,Qb])kc θ(R)e−itRψ‖2

/ C

t

∑c∈Lb

nc∑kc=1

∥∥∥∥f(Qb

t

)kc

(Q

t

)θ(R)e−itRψ

∥∥∥∥2

, (5.47)

where in the last step Appendix 6.4 has been used. Let us prove that the r.h.s. of(5.47) is6 C‖ψ‖2. For this, observe first that as a consequence of assumption(5.34) we can choose the familykc such that

suppkc ⊂ suppJ for all c ∈ Lb. (5.48)

MPAG010.tex; 14/05/1998; 16:04; p.41


Secondly, (5.46) joined to (5.45) tells us that for allc ∈ Lb

supp(f kc ) ∩Xb = ∅, (5.49)

which together with the inclusion (see (5.48)):

supp(f kc ) ⊂ 0b(0) =⊔c∈Lb

Xc

gives:

supp(f kc ) ⊂⊔

c∈Lb\b

Xc for all c ∈ Lb. (5.50)

On the other hand, sinceb ∈ La(n) is fixed, anyc < b will belong to⋃Nai=n+1

La(i), or, in other words, for anyc < b there is ab ∈ La(n+ 1) such thatc 6 b.This shows that (5.50) can be written as:

supp(f kc ) ⊂⋃

b∈La(n+1)

⊔c∈Lb

Xc ≡

⋃b∈La(n+1)

0b(0). (5.51)

We would like to apply(Pn+1) to any of the functionsf kc . Observe first thatanother way of writing the hypothesis of(Pn+1) (which makes reference to thewhole lattice levelLa(n + 1) and not to the elementsb from it) is to demand tothe support ofJ to be included in

⋃b∈La(n+1) 0b(0) and to replace in (5.5) or in

(5.6)ρARb(λ) by minb∈La(n+1) ρARb(λ). Then, (5.48) tells us that for allc ∈ Lb there

exists aC∞0 -function ˆkc (which equals one on suppkc , and has support included

in suppJ ), which together with (5.26) and (5.51) allows us to conclude that thehypothesis of(Pn+1) written for the product functionf kc is verified, and thus ther.h.s. of (5.47) is integrable.

It remains to estimate the term resulting from DR8 which contains the Heisen-berg derivative of12t [iR, (Qa)2]. It will be computed as follows:

θ(R)f ([iR,Qb])Jb(

DR

[iR,

(Qa)2

2t

])Jbf ([iR,Qb])θ(R)

= θ(R)f ([iR,Qb])Jb(∂

∂t

[iR,

(Qa)2

2t

])Jbf ([iR,Qb])θ(R)−

− θ(R) f ([iR,Qb])Jb[iR,

[iR,

Q2a

2t

]]Jbf ([iR,Qb])θ(R)+

+ f ([iR,Qb])Jbθ(R)R[iR,

A

t

]Rθ(R)Jbf ([iR,Qb])−

−[f ([iR,Qb]), θ(R)]Jb[iR,

[iR,

Q2

2t

]]Jbf ([iR,Qb])θ(R)+

MPAG010.tex; 14/05/1998; 16:04; p.42


+ f ([iR,Qb])θ(R)Jb[iR,

[iR,

Q2

2t

]]Jb[f ([iR,Qb]), θ(R)] +

+ f ([iR,Qb])[θ(R), Jb][iR,

[iR,

Q2

2t

]]Jbθ(R)f ([iR,Qb])−

− f ([iR,Qb])Jbθ(R)[iR,

[iR,

Q2

2t

]][θ(R), Jb]f ([iR,Qb]). (5.52)

Since the commutator[f ([iR,Qb]), θ(R)] is of order O(t−µ) on the support ofJb(which is a compact set of0b(0)), the fourth and the fifth terms from the r.h.s. of(5.52) are integrable. The same is true for the last two terms in the above equality.Indeed, note first that up to addition of some O(t−2) contributions they are of theform:

1

tf ([iR,Qb])J ′b · [θ(R),Q][iR, [iR,Q]] ·

Q

tJb θ(R) f ([iR,Qb]).

Note that there are some components of the above double commutator (namelythose relative to some basis inXb) which arenot small (in the sense that they donot confer integrable decay int) on suppJb. Nevertheless, the above term is of thesame type as the l.h.s. of (5.47), so we can use the same reasoning in order to showits integrability.

It remains to estimate the first three lines from the r.h.s. of (5.52). We shallproceed as in the proof of Lemma 5.1, and minorize the first of these terms by:

−1

t

dimXa∑l=1

‖[iR,Ql]‖ supx∈suppJb

|xl| ‖Jbf ([iR,Qb])θ(R)e−itRψ‖2 +

+O(t−2)‖ψ‖2. (5.53)

Using the Mourre estimate, the mean value〈.〉t of the third one is

' 2

t

(ρARb(λ)− ε

)inf

µ∈suppθµ2‖f ([iR,Qb])Jbθ(R)e−itRψ‖2. (5.54)

Finally, the second term from the r.h.s. of (5.52) is:

≈ −Re⟨θ(R)f ([iR,Qb])Jb[iR, [iR,Qa]] · Qa

tJbf ([iR,Qb])θ(R)

⟩t

−

− 1

t

⟨θ(R)Jbf ([iR,Qb])[iR,Qa]2f ([iR,Qb])Jbθ(R)

⟩t, (5.55)

and since0b(0) ⊆ 0a(0) for anyb ∈ La, the above double commutator will be oforder O(t−µ) on suppJb. The second term of the above sum can be estimated withthe aid of

[iR,Qa]2 = −T 2a + 2 ReTa · [iR,Qa] + Q

2a

t2(5.56)

MPAG010.tex; 14/05/1998; 16:04; p.43


as being (forβ > 0 sufficiently small)

' −1

tsup

x∈suppJb

x2a‖f ([iR,Qb])Jbθ(R)e−itRψ‖2−

− 2

t

dimXa∑l=1

Re⟨θ(R)Jb〈Tl〉1/2β f ([iR,Qb])×

× 〈Tl〉−1β Tl︸︷︷︸

O(1)

[iR,Ql]f ([iR,Qb])︸︷︷︸O(1)

〈Tl〉1/2β Jbθ(R)⟩t+

+ 1

t‖f ‖2∞‖|Ta|Jbθ(R)e−itRψ‖2. (5.57)

Note that in order to obtain the second term above one has shown (in the samemanner as in the Appendix 6.4) that[f ([iR,Qb]), 〈Tl〉1/2β ] brings an integrablecontribution on the support ofJb. Then, the last two lines above are integrable asa consequence of the propagation Theorems 3.2 and 3.3, so summing up (5.53),(5.54) and (5.57) we see that the l.h.s. of (5.52) dominates (in the sense of'):

1

t

2(ρARb(λ)− ε

)inf

µ∈suppθµ2 − sup

x∈suppJb

x2a

−dimXa∑l=1

‖[iR,Ql]‖ supx∈suppJb

|xl|‖f ([iR,Qb])Jbθ(R)e−itRψ‖2.

Since the assumption (5.34) together with the hypothesis of(Pn) ensures strictpositivity for the quantity in the above curly bracket, the estimate (5.39) is provenand hence the first implication of the theorem also.

It remains to prove that (5.3) is a consequence of (5.2). This will be performed inthe same manner as above, the only difference being that we have to invoke relation(5.22) instead of (5.56) when one wants to estimate the second term in (5.55). Thecomments we made after (5.55) show that, up to some integrable terms, the firsttwo lines in the r.h.s. of (5.52) dominate

−1

t

⟨θ(R)f ([iR,Qb])Jb

[iR,

Q2

2t

]Jbf ([iR,Qb])θ(R)

⟩t

which, in turn, can be computed as in the proof of Lemma 5.1 (see the estimatefollowing (5.23)) and thus minorized (modulo O(t−µ)) by:

−dimX∑l=1

‖Plθ(R)‖ supx∈suppJb

|xl | supµ∈suppθ

µ2 ‖f ([iR,Qb])Jbθ(R)e−itRψ‖2.

MPAG010.tex; 14/05/1998; 16:04; p.44


This shows, as before, that the l.h.s. of (5.52) dominates

1

t

2(ρARb(λ)− ε) inf

µ∈suppθµ2− sup

µ∈suppθµ2

dimX∑l=1

‖Plθ(R)‖ supx∈suppJb

|xl|×

×‖f ([iR,Qb])Jbθ(R)e−itRψ‖2,where the curly bracket is strictly positive as a consequence of relation (5.6) corre-sponding to(Pn). 2

The rest of this section will be devoted to the proof of statement (1.10). We willuse a standard induction reasoning (see [34, 2, 23, 27]), performed on the levels ofsome arbitrary latticeL. Let us first denote, for anya, b ∈ L, a 6 b, by±a,b thewave operators±(Ra,Rb;Eb) and notice that they exist as a consequence of theexistence ofW±(Ra,Rb; J ), with suppJ ⊂ 0b(0). Then, since for the rank onelattice statement (1.10), written for±(Ra,Rb;Eb) is trivial, we shall suppose ittrue for any lattice of rankN , and prove it for all latticesL of rankN+1. Actually,it is sufficient to prove it for any stateψ localised with the aid of a smooth cutoffθin a compact ofR \ C(R), because, due to the fact that the set of critical values ofR is countable, a covering argument (see Proposition 4.2.6 in [2]) will allow us toextend the result on allR. Let us begin by computing, using a conveniently chosenpartition of unity inX (e.g., the smooth one constructed in Section 2):

e−itRψ = Jamax e−itRψ +N+1∑n=2

∑a∈L(n)

Ja e−itRψ

hN+1∑n=2

∑a∈L(n)

e−itRa∑b∈La

±a,b (±a,b)∗W±a ψ︸︷︷︸

ψa,b

.

In the above modulo o(1) equalityh,W±a stands for the limits in (4.2), and theCorollary 5.2 and the induction hypothesis were used. We thus have:

ψ hN+1∑n=2

∑a∈L(n)

e−itR e−itRa∑b∈La

±a,bψa,b

h∑

a∈L\amax

∑b∈La

e−itR e−itRbEbψa,b h∑

a∈L\amax

∑b∈La

±b ψa,b

which provesψ ∈⊕b∈L\amax±b H . 2

6. Appendices

6.1. APPENDIX

DEFINITION 6.1. Let us define for alla ∈ L the following sets:

(i) La = b ∈ L | b 6 a,

MPAG010.tex; 14/05/1998; 16:04; p.45


(ii) La = b ∈ L | b > a,(iii)

Xa = Xa \⋃b 66a Xb,

(iv) 0a(0) = X \⋃b 66a Xb.

Notice also the particular casesa = amin, for whichXamin = 0amin(0) = X \⋃

b 66aminXb anda = amax, for which

Xamax = 0 and0amax(0) = X. Denoting by

the signt the disjoint union of sets we have:

PROPOSITION 6.1.For an arbitrary a ∈ L,

(i)Xa = Xa \⋃b>a Xb,

(ii)Xa ∩

Xb = ∅ for all a, b ∈ L with a 6= b,

(iii) Xa =⊔b∈La

Xb,

(iv) 0a(0) =⊔b∈La

Xb.

Remark also that for alla ∈ L, La ∩La = a yieldsXa = 0a(0) ∩Xa.

Proof.Remember that we denoted by6∼ the ‘incomparability’ sign between twoelements of the latticeL. Then, on one handXa \⋃b 6∼a Xb = Xa \

⋃b 6∼aXb∩Xa,

and on the other hand, sinceL is sup-stable there is ac ∈ L with c > a andc > bsuch thatXb ∩ Xa = (Xb + Xa)⊥ = (Xc)⊥ = Xc. This shows the inclusionXa \⋃b 6∼a Xb ⊆ Xa \

⋃c>a Xc and thus (i) is proved. Let nowa ∈ L be arbitrarily

fixed and take firstb ∈ L \ La. Then by Definition 6.1(i),Xb ∩Xa = ∅ and

Xb ⊂ Xb so (ii) is true for theseb. For thoseb belonging toLa \ a notice that

according to (i) for allc > b we haveXc ∩Xb = ∅. But a is one of thesec and

thus (ii) is proved. In order to prove (iii), suppose as usuala is arbitrarily fixedand observe that the rank of an elementb ∈ La, denoted by|b|La , is generallydifferent from |b|L. Denote also for 16 j 6 rankLa the j th level in La asLa(j) ≡ b ∈ La | |b|La = j and the union ofLa(j) with all the levels inLa

which are below it byLaj ≡ b ∈ La | |b|La > j. Denoten ≡ rankLa = |a|La

and take as induction hypothesis the statement

Xa =⊔

b∈Lan−j

Xb ∪

n−j−1⋃l=1

⋃b∈La(l)

Xb. (6.1)

The first step of the induction is given by

Xa =Xa ∪

⋃b∈La\a

Xb (6.2)

MPAG010.tex; 14/05/1998; 16:04; p.46


in which the set inclusion⊇ is ensured by (i). Further, let us suppose that (6.1) istrue for somej < n. Then, using (ii) one gets:

Xa =⊔

b∈Lan−j

Xb ∪

⋃b∈La(n−j−1)

( Xb ∪

⋃c>b

Xc

)∪n−j−2⋃l=1

⋃b∈La(l)

Xb

=⊔

b∈Lan−j−1

Xb ∪

⋃c∈Mj

Xc ∪n−j−2⋃l=1

⋃b∈La(l)

Xb,

where the setMj ≡ c ∈ L | ∃b ∈ La(n − j − 1) such thatc > b obviouslysatisfiesMj ⊆ La \⋃n

l=n−j−1La(l), thus (iii) is proved.

Finally, (iii) and the definition of0a(0) gives:

0a(0) =⊔b∈L

Xb \

⋃b∈L\La

⊔c∈Lb

Xc.

In order to prove (iv) it will be sufficient thus to show the set inclusionLa ⊆L \ c ∈ Lb | b ∈ L \ La, i.e.La ∩ c ∈ Lb | b ∈ L \La = ∅. Suppose theexistence of somec0 ∈ L in this set intersection. Then there is ab0 ∈ L \La suchthatc0 > b0, and on the other handc0 6 a, i.e.b0 ∈ La. Contradiction. 2

6.2. APPENDIX

We have to check (see [6]) that the existence of the Abel operatorsa, which isequivalent to

limT→∞ T

−1∫ T

0‖(EH (R)−a)e−itRaEaψ‖dt = 0,

implies the convergence:

limT→∞

T −1∫ T

0‖Wa(t)ψ −aψ‖dt = 0.

But, using twice the intertwining relation fora, we get for allψ ∈ H(X):

‖Wa(t)ψ −aψ‖ = ‖EH(R)e−itHaEaψ − e−itHEH (R)aψ‖= ‖(EH (R)−a)e−itHaEaψ‖.

Finally, Lemma 1 from [29] can be used in order to get the desired result.

MPAG010.tex; 14/05/1998; 16:04; p.47


6.3. APPENDIX

The following lemma puts in evidence some estimates,uniform in α > 0, con-cerning the approximating family of HamiltoniansHα. Note that for anya ∈ L,(1.2) can be written asHα = Ha,α + I a,α, whereI a,α denotes the sum from the firstequality in (1.2), performed only on the setL \La.

LEMMA 6.1. For any z inf σ(Hα|α=0) and for anya ∈ L, if Ra,α denotes(Ha,α − z)−1, then there is a constantC > 0, independent ofα, such that for anyα > 0:

α‖R1/2α χ(a)R1/2

α ‖ + ‖R1/2α I a,αR1/2

α ‖ 6 C, (6.3)

α

∥∥∥∥Ra,α ∑b∈L\La

χ(b)Rα

∥∥∥∥+ ‖RαI a,αRa,α‖ 6 C. (6.4)

For the proof, note that the first inequality is a consequence of the hypothesismade onz, and of the obvious identity:

1 = R1/2α (Hα − z)R1/2

α

= αR1/2α χ(a)R1/2

α + R1/2α

(Hα|α=0+

∑b 6=a

αχ(b)− z)R1/2α .

Then, using

Rα − Ra,α = −RαI a,αRa,α = −Ra,αI a,αRαwe compute

‖Ra,αI a,αRα‖2 = ‖(Rα − Ra,α)2‖ 6 2‖Ra,α‖‖Rα‖ + ‖Ra,α‖2+ ‖Rα‖2 6 C,where all the norms are inB(H) and where the uniform boundedness (w.r.t.α) ofthe family‖Ra,α‖−1,1, tells us thatC does not depend onα. Finally,

α

∥∥∥∥Ra,α ∑b∈L\La

χ(b)Rα

∥∥∥∥6 ‖Ra,αI a,αRα‖ +

∑b∈L\La

‖Ra,α〈P 〉‖‖V (b)‖1,−1‖〈P 〉Rα‖

finishes the proof of (6.4). 2

Note that uniform estimates of the type (6.4) are useful when one wants toobtain decay of the differenceRα − Ra,α on the support of some time-dependentcutoff J ∈ C∞0 (0a(0)) (as in the proof of Proposition 4.1). As far as the inequality(6.3) is concerned, notice that it is of quadratic type, i.e. ifχ is of the formχ χ∗,

MPAG010.tex; 14/05/1998; 16:04; p.48


then it can be written as√α‖R1/2

α χ‖ 6 C, which is not enough for showing thatthe double commutator

[iRα, [iRα,Qa]] =∑

b∈L\La

αR2α[χ(b), iPa ]R2

α

=∑

b∈L\La

iαR2αχ(b)PaR

2α + h.c.

is bounded uniformly w.r.t.α. This is one of the numerous reasons which makesthe algebraic framework (introduced at Section 2) indispensable.

6.4. APPENDIX

We have to prove that for anya ∈ L and anyb ∈ La, given two functionsJ ∈C∞0 (0a(0)) andg ∈ C∞0 (X) satisfyingg = g πb on a neighbourhood of theintersection of suppJ with Xb, the estimate∫ ∞

1

dt

t

⟨J

(Q

t

)g([iR,Qb])− g

(Qb

t

)J

(Q

t

)⟩t

6 C‖ψ‖2 (6.5)

is true for allψ ∈ H(X).For showing this, we use the Fourier spectral formula in order to compute the

difference:

g([iR,Qb])− g(Qb

t

)=∫X

dsg′(s)∫ 1

0dτ eisτ [iR,Qb]Tb eis(1−τ)

Qbt , (6.6)

whereg′ ∈ S(X). We thus have to look, for anyk = 1, . . . ,dimXb, at:⟨J eisτ [iR,Qb]Tk eis(1−τ)

Qbt J⟩t

=⟨J 〈Tk〉1/2β eisτ [iR,Qb]〈Tk〉−1

β Tk eis(1−τ)Qbt︸︷︷︸

O(1)

〈Tk〉1/2β J

⟩t

+

+⟨J 〈Tk〉1/2β eisτ [iR,Qb] 〈Tk〉−1

β Tk︸︷︷︸O(1)

[〈Tk〉1/2β , eis(1−τ)Qbt

]J

⟩t

+

+ ⟨J [eisτ [iR,Qb], 〈Tk〉1/2β ]〈Tk〉−1/2β Tk eis(1−τ)

Qbt J⟩t. (6.7)

It is clear that the first line above is integrable w.r.t.t as a consequence of Proposi-tion 3.3 (the integrabilities ins andτ are trivial). Further, using formula (3.33) wecompute (as in the proof of Proposition 3.3):[〈Tk〉1/2β , eisτ [iR,Qb]

] = 2sτ ReTk

∫ ∞0

dωω1/4(ω+ 〈Tk〉2β)−1×

×∫ 1

0dσ eisτ (1−σ)[iR,Qb][[iR,Qb], Tk]eisτσ [iR,Qb](ω+ 〈Tk〉2β)−1.

MPAG010.tex; 14/05/1998; 16:04; p.49


Hence, we essentially have to decide if for allk, l = 1, . . . ,dimXb, the commu-tator [[iR,Ql], Tk] confers on the support ofJ enough decay in order to ensureintegrability with respect to botht andω. But this has been already shown inthe proof of Lemma 4.1 with the aid of relation (5.14) (see also the discussionfollowing it), the only difference being that the role ofa was played there byamin,so we could takek, l to run from 1 to dimX. Note also that because of the rapiddecay ofg′ the polynomials ins resulted from the various commutations ofJ withthe unitary groups of the type exp(isτσ [iR,Qb]) do not influence the integrabilityin s. Finally, the second term from the r.h.s. of (6.7) will be treated in the same wayas the previous one. Actually it is even simpler, because we will not be forced touse the good decay along certain directions of the connected components ofR, thebasically O(t−µ) decay being replaced by the better O(t−2). 2

Acknowledgements

I am grateful to Anne Boutet de Monvel, V. Georgescu and A. Soffer for havingcommunicated to me the results they obtained in [12], during their work on thispaper, and for the helpful discussions we held on the hard-core subject. I also thankmy elder colleagues L. Zielinski and M. Mantoiu for having shown me parts of theirworks.

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(1967), 1052–1062.27. Iftimovici, A.: On asymptotic completeness for Agmon type Hamiltonians,C.R. Acad. Sci.

Paris, Série I314(1992), 337–342 (Preprint BiBos-Universität Bielefeld no. 517/1992, 1–18).28. Mantoiu, M.:Contributions à l’analyse spectrale par la méthode des opérateurs conjugués,

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MPAG010.tex; 14/05/1998; 16:04; p.52


75

Onq-Analogues of Bounded Symmetric Domainsand Dolbeault Complexes

S. SINEL’SHCHIKOV? and L. VAKSMAN??

Institute for Low Temperature Physics and Engineering, National Academy of Sciences of Ukraine,Kharkov, Ukraine


Abstract. A very well known result by Harish-Chandra claims that any Hermitian symmetric spaceof non-compact type admits a canonical embedding into a complex vector spaceV . The image of thisembedding is a bounded symmetric domain inV . This work provides a construction ofq-analoguesof a polynomial algebra onV and the differential algebra of exterior forms onV . A way of producinga q-analogue of the bounded function algebra in a bounded symmetric domain is described. All theconstructions are illustrated by detailed calculations in the case of the simplest Hermitian symmetricspaceSU(1, 1)/U(1).

Mathematics Subject Classification (1991):81R50.

Key words: quantum differential calculi, quantum homogeneous spaces.

1. Introduction

Consider an irreducible Hermitian symmetric spaceX of a non-compact type. Letg and g0 denote the complexifications of the Lie algebras of the automorphismgroup ofX and the stabilizer of a pointx ∈ X, respectively. Then the center ofg0 is one-dimensional(Z(g0) = C · H, H ∈ g0), andg = g−1 ⊕ g0 ⊕ g1, whereg±1 = ξ ∈ g | [H, ξ ] = ±2ξ (see, e.g., [8]).

It was shown by Harish-Chandra that a natural embeddingi: X → g−1 existswith iX being a bounded symmetric domain ing−1 [8].

Our purpose is to construct quantum analogues of the (prehomogeneous) vectorspaceg−1, the bounded symmetric domainiX ⊂ g−1 and the differential calculusin g−1.

Normally, we don’t dwell on describing the quantum algebras of functions andquantum exterior algebras in terms of generators and relations, although that couldbe done. (The caseg = slm+n, g0 = s(glm × gln) was partially considered in [21].)

? Partially supported by ISF grant U2B200 and grant DKNT-1.4/12.?? Partially supported by the grant INTAS-94-4720, ISF grant U21200 and grant DKNT-1.4/12.


76 S. SINEL’SHCHIKOV AND L. VAKSMAN

The simplest homogeneous bounded domain is the unit discU = z ∈ C ||z| < 1. It was shown in [12, 14] that the Poisson brackets. , ., that agree withthe action of the Poisson–Lie group SU(1,1) onU , are given by

z, z = i(1− |z|2)(a + b|z|2), a, b ∈ R.Our construction (see Section 9) provides a quantization of this bracket with

b = 0. This ‘simplest’ quantum disc was studied in [18, 23].Most of the constructions of this paper originate from the works of Drinfeld

[6] and Levendorskii and Soibelman [16]. Specifically, we follow [6] in replacingthe construction of algebras by forming the dual coalgebras; also our choice ofa Poisson cobracket, together with the associated quantization procedure, is dueto [16].

The authors are grateful to V. Akulov, V. Lyubashenko, G. Maltsiniotis, and D.Shklyarov for the helpful discussion of the results.

2. Prehomogeneous Vector Spaces of a Commutative Parabolic Type

Everywhere in the sequelC will be the ground field.Let g be a simple complex Lie algebra,h its Cartan subalgebra andαi ∈ h∗, i =

1, . . . , l, a simple root system ofg.Choose an elementα0 ∈ αii=1,...,l and consider the associatedZ-grading

g =⊕j

gj , gj = ξ ∈ g | [H0, ξ ] = 2jξ,

whereH0 ∈ h, α0(H0) = 2, αi(H0) = 0 for αi 6= α0.A subspaceg−1 is called a prehomogeneous vector space of a commutative

parabolic type if the aboveZ-grading breaks off:

g = g−1⊕ g0⊕ g1. (2.1)

The motives that justify this definition and the list of simple rootsα ∈ αii=1,...,l

with (2.1) being valid are given in [2, 19].It is worthwhile noting that all the simple roots of seriesAn Lie algebras possess

the above property, and for the Lie algebra seriesBn,Cn,Dn, together with theexceptional Lie algebrasE6, E7 the set of such roots is non-void.

Setp+ := g0 ⊕ g1, p− := g0 ⊕ g−1. Our purpose is to construct a quantumanalogue of the graded polynomial algebraC[g−1] on the prehomogeneous vectorspaceg−1. For this, it would be useful to have a definition ofC[g−1] in terms of theenveloping algebrasUg ⊃ Up+ ⊃ Ug0 (but not the Lie algebras themselves).

We start with constructing the coalgebraV− dual toC[g−1]. Consider theUg-moduleV− determined by its generatorv ∈ V− and the relations

ξv− = ε(ξ)v−, ξ ∈ Up+, (2.2)

MPAG008.tex; 14/05/1998; 16:05; p.2

ON q-ANALOGUES OF BOUNDED SYMMETRIC DOMAINS 77

whereε: Up+ → C ' End(C) is the trivial representation ofUp+. EquipV−with a structure of a coalgebra [3] by extending the map1−: v− 7→ v− ⊗ v− toa morphism ofUg-modules. The existence and uniqueness of this extension areobvious, and the coassociativity of1− follows from

(1− ⊗ id)1−v− = (v− ⊗ v−)⊗ v−; (id⊗1−)1−v− = v− ⊗ (v− ⊗ v−).It is easy to verify thatV− = ⊕

j (V−)j with (V−)j = v ∈ V− | H0v =2jv, and that the dual algebra

⊕j ((V−)j )

∗ to the coalgebraV− is canonicallyisomorphic toC[g−1].

A replacement of ‘−’ by ‘+’ in the above construction leads to the algebra ofantiholomorphic polynomials ong−1, which will be denoted byC[g−1]. We shallsee in the sequel that these constructions can be transferred to the quantum casewhere they lead to the ‘covariant’ algebrasC[g−1]q,C[g−1]q .

3. Quantum Universal Enveloping Algebras and their ‘Real Forms’

It is well known [20] that a simple complex Lie algebrag admits a description interms of generatorsX±i , Hili=1 and relations

[Hi,Hi] = 0; [Hi,X±j ] = ±aijX±j ;[X+i , X−j ] = δijHi; ad

1−aijX±i

(X±j ) = 0. (3.1)

In the abovei, j ∈ 1, . . . , l, and (aij ) is the Cartan matrix of the simple Liealgebrag, i.e.,aij = αi(Hj).

Let j0 be the number of the simple rootα0. The relations (2.2) can be rewrittenin the form

X−j v− = Hjv− = 0, j = 1,2, . . . , l;X+j v− = 0, j 6= j0.

Consider the real Lie subalgebrag(α0) ⊂ g generated by the elements

X+j −X−j , i(X+j +X−j ), iHj , j 6= j0;X+j0 − X−j0, i(X+j0 +X−j0), iHj0,

wherei = √−1. This subalgebra is interesting because it is the Lie algebra for theautomorphism group of the corresponding bounded symmetric domain ing−1 ⊂ g.We are seeking for the specific ways to distinguishUg(α0) insideUg.

Recall thatUg is a Hopf algebra [3] whose comultiplication1, counit ε andantipodeS are given by

1(Hi) = Hi ⊗ 1+ 1⊗Hi, 1(X±i ) = X±i ⊗ 1+ 1⊗X±i ;ε(Hi) = ε(X±i ) = 0; S(Hi) = −Hi, S(X±i ) = −X±i ,

i = 1,2, . . . , l.

MPAG008.tex; 14/05/1998; 16:05; p.3


It is easy to verify that

Ug(α0) = ξ ∈ Ug | ξ ∗ = S(ξ),with ∗ being the antilinear involution which depends onα and is determined by itsvalues on generatorsX±j ,Hj as follows:?

H ∗j0 = Hj0, (X±j0)∗ = −X∓j0,

H ∗j = Hj, (X±j )∗ = X∓j , j 6= j0.(3.2)

The Hopf algebraUg(α0) does not survive under quantization; in the sequel itwill be replaced by the pair(Ug,∗ ). Now let us consider the quantization of thisHopf ∗-algebra.

We start with Drinfeld–Jimbo formulae [6] which determine a Hopf algebraUhg overC[[h]] complete inh-adic topology (C[[h]] denotes the ring of formalseries). First of all, choose an invariant scalar product ing in such a way thatdi =(αi, αi)/2> 0. Now,X±j ,Hj j=1,...,l work as generators of the topological algebraUhg, and the resulting list of relations is

[Hi,Hj ] = 0, [Hi,X±j ] = ±aijX±j , [X+i , X−j ] = δijsh(djhHj/2)

sh(djh/2),

1−aij∑k=0

(−1)k[

1− aijk

](X±i )

kX±j (X±i )

(1−aij−k) = 0.

Here we use the notation[n

m

]h

=n∏k=1

sh(kh/2)

sh(h/2)

/( m∏k=1

sh(kh/2)

sh(h/2)·n−m∏k=1

sh(kh/2)

sh(h/2)

),

i, j = 1, . . . , l.Comultiplication1, counitε and antipodeS are determined by their values on

the generators

1(Hi) = Hi ⊗ 1+ 1⊗Hi, 1(X±i ) = X±i ⊗ ehHidi/4+ e−hHidi/4⊗X±i ,

ε(Hi) = ε(X±i ) = 0, S(Hi) = −Hi, S(X±i ) = −e∓hdi/2 ·X±i .

An involution in C[[h]] is introduced by settingh∗ = h. We equipUhg withthe structure of∗-algebra overC[[h]] defined by (3.2). The pair(Uhg,∗ ) will bedenoted byUh for the sake of brevity.

A procedure of transition from algebras overC[[h]] to algebras overC is de-scribed in [3]; it allows one to ‘fix the value of the formal parameterh’. Here

? It is implicit that (ξη)∗ = η∗ξ∗, ξ, η ∈ Ug.

MPAG008.tex; 14/05/1998; 16:05; p.4


we only bring to mind the formulae which describe the ‘change of variables’corresponding to the generators of the above algebra

q = e−h/2, K±1i = e∓hdiHi/2,

Ei = X+i e−hdiHi/4, Fi = ehdiHi/4X−i .

In the sequel we fix the value ofq ∈ (0,1). The Hopf algebra overC, given bythe generatorsEi, Fi,K±1

i li=1, and the relations deduced above from the relationsin Uh, will be denoted byUqg, and the Hopf∗-algebra(Uqg,∗ ) byUq.?

The defining relations forUq are similar to (3.1), (3.2). Part of them (the quan-tum analogue of the last among the relations (3.2) can be found in [15]) are listedhere as

KiKj = KjKi, KiK−1i = K−1

i Ki = 1,

KiEj = qdiaij ·EjKi, KiFj = q−diaij FjKi,

EiFj − FjEi = δij Ki −K−1i

qdi − q−di ,

1(Ei) = Ei ⊗ 1+Ki ⊗ Ei, 1(Fi) = Fi ⊗K−1i + 1⊗ Fi,

1(Ki) = Ki ⊗Ki,ε(Ei) = ε(Fi) = ε(Ki − 1) = 0,

S(Ei) = −K−1i Ei, S(Fi) = −FiKi, S(Ki) = K−1

i ,

E∗j =

KjFj j 6= j0,

−KjFj j = j0,F ∗j =

EjK

−1j j 6= j0,

−EjK−1j j = j0,

K∗j = Kj, i, j ∈ 1, . . . , l.We equip the Hopf algebraUqg with a grading

degKj = degEj = degFj = 0, j 6= j0,

degKj0 = 0, degEj0 = 1, degFj0 = −1.

4. Covariant Algebras and Involutions

Remember thatC is endowed with a structure of aUqg-module by means of acounitε: Uqg→ C ' End(C).? See the definition of a Hopf∗-algebra in [3].

MPAG008.tex; 14/05/1998; 16:05; p.5


Let F be a unital algebra overC, which is also aUqg-module. We callF aUqg-module algebraif the multiplication

m: F ⊗ F → F ; m: f1⊗ f2 7→ f1f2, f1, f2 ∈ F ,

and the unit

1: C→ F ; z 7→ z · 1, z ∈ C,are morphisms ofUqg-modules.?

Together with the term ‘Uqg-module algebra’ we shall elaborate the substituteterm‘covariant algebra’for the sake of brevity in the cases when no confusion canoccur.

Covariant modules and covariant bimodules over covariant algebras are definedin a similar way (see [1, 25]).

An involutive (F , ∗) algebra is said to be covariant [22] if it is aUqg-modulealgebra and for allξ ∈ Uqg, f ∈ F one has

(ξf )∗ = (S(ξ))∗f ∗. (4.1)

A linear functionalν: F → C is called an invariant integral if

ν(ξf ) = ε(ξ)ν(f ), ξ ∈ Uqg, f ∈ F .

The ‘compatibility condition’ for involutions (4.1) is extremely important sinceit allows one to use the ‘positive’ invariant integrals for producing∗-representationsof Uqg in the ‘Hilbert function spaces’:

(f1, f2) = ν(f ∗2 f1), f1, f2 ∈ F .

The problem of decomposing such∗-representations is a typical one in har-monic analysis. In this way, for instance, the Plancherel measure for quantumSU(1,1) was found (see [22]).

5. Generalized Verma Modules

Choose a linear functionalλ ∈ h∗ so thatmj = λ(Hj) are non-positive integers forj 6= j0.

Consider the gradedUqg-module determined by the single generatorv+(λ) ∈V+(λ) and the relations

Fiv+(λ) = 0, K±1i v+(λ) = e∓dimih/2v+(λ), i = 1, . . . , l,

E−mj+1j v+(λ) = 0, j 6= j0,

? F ⊗ F becomes aUqg-module by settingξ(f1 ⊗ f2) =∑j ξ′j f1 ⊗ ξ ′′j f2 for ξ ∈ Uqg with

1(ξ) =∑j ξ′j⊗ ξ ′′

j, f1, f2 ∈ F .

MPAG008.tex; 14/05/1998; 16:05; p.6


deg(v+(λ)) = 1

2λ(H0).

Note thatV+(λ) =⊕j V+(λ)j , with V+(λ)j = v ∈ V+(λ) | deg(v) = j, anddimV+(λ)j <∞.

The finite dimensionality of the homogeneous componentV+(λ)j follows fromthe decomposition

V+(λ)j =⊕

µ∈h∗|µ(H0)=2jV+(λ)µ

into a finite sum of the finite dimensional weight subspaces

V+(λ)j = v ∈ V+(λ)|Kjv = e−djµ(Hj )h/2v, j = 1, . . . , l.The graded modulesV−(λ) are defined in a similar way:

Eiv−(λ) = 0, K±1i v−(λ) = e∓dimih/2v−(λ), i = 1, . . . , l,

Fmj+1j v−(λ) = 0, j 6= j0; deg(v−(λ)) = 1

2λ(H0).

Now supposemj0 = λ(Hj0) ∈ Z.Consider the longest elementw0 of the Weil groupW for a Lie algebrag. It

is very well known from [4, 3] that to each reduced decomposition ofw0 onecan associate a Poincaré–Birkhoff–Witt basis inUqg. We demonstrate the reduceddecompositions for which this basis ‘generates’ the bases of weight vectors ingeneralized Verma modules.

Let g′ ∈ g be a Lie subalgebra generated byX±j ,Hj j 6=j0, and letW ′ ∈ W bea subgroup generated by simple reflectionss(αj ), j 6= j0. Obviously,W ′ is a Weilgroup of the Lie algebrag′.

Denote the subset of such elementsu ∈ W byU ⊂ W such that

l(s(αj )u) > l(u) for all j 6= j0.

It is known from [9, p. 19], that, firstly, each elementw ∈ W admits the uniquedecompositionw = w′ · u with w′ ∈ W ′, u ∈ U . Secondly, ifw′ ∈ W ′, u ∈ U ,then one has

l(w′ · u) = l′(w′)+ l(u),with l′(w′) being the length of the elementw′ in W ′, andl(u), l(w′u) the lengthsof u,w′u in W .

That is, inU one can find the unique elementu0 of maximum length such thatw0 = w′0·u0. (w′0 here is the longest element ofW ′.) Now one can derive the desiredreduced decompositions ofw0 by multiplication from the reduced decompositionsof w′0 andu.

MPAG008.tex; 14/05/1998; 16:05; p.7


6. From Coalgebras to Algebras

LetUqgop stand for the Hopf algebra derived fromUqg by replacing its comultipli-cation by the opposite one.

We intend to use the generalized Verma modules for producing coalgebras dualto covariant algebras. To provide a precise correspondence between these two no-tions, we are going to replaceUqg by Uqgop in tensor products of generalizedVerma modules.

Consider theUqg-modulesV±(0). Evidently, the maps

1±: v±(0) 7→ v±(0)⊗ v±(0); ε±: v±(0) 7→ 1

admit the unique extensions to morphisms ofUqg-modules

1±: V±(0)→ V±(0)⊗ V±(0); ε±: V±(0)→ C.

Just as in the caseq = 1, one can verify that the operations1± are coassocia-tive, and thatε± are the counits for coalgebras, respectively, with1±.

Hence, the vector spaces(V±(0))∗def= ⊕

j (V±(0)j )∗ are covariant algebras.? In-

troduce the notation

C[g−1]q = V−(0)∗, C[g−1]q = V+(0)∗.These covariant algebras may be treated asq-analogues of polynomial algebras(holomorphic or antiholomorphic identified by the sign) on the quantum prehomo-geneous spaceg−1.

7. Polynomial Algebra

Consider the algebra Pol(g−1) = C[g−1] ⊗ C[g−1] of all polynomials ong−1.Holomorphic and antiholomorphic polynomials admit the embeddings into thisalgebra as follows:

C[g−1] → C[g−1] ⊗ C[g−1], f 7→ f ⊗ 1,

C[g−1] → C[g−1] ⊗ C[g−1], f 7→ 1⊗ f.Our desire is to obtain that sort of algebra and similar embeddings in the quan-

tum case(q 6= 1). For that, we intend to equip theUqg-module Pol(g1)qdef=

C[g−1]q ⊗ C[g−1]q with a structure of covariant algebra in such a way that themapsf 7→ f ⊗ 1, f 7→ 1⊗ f turn out to be algebra homomorphisms.

Our approach is completely standard [10]. Define the product ofϕ+⊗ϕ−, ψ+⊗ψ− ∈ Pol(g−1)q as

(ϕ+ ⊗ ϕ−)(ψ+ ⊗ ψ−) = m+ ⊗m−(ϕ+ ⊗ R(ϕ− ⊗ ψ+)⊗ ψ−).? The dualUqg-module structure is given byξf (v)

def= f (S(ξ)v), with ξ ∈ Uqg, v ∈ V±(0), f ∈V±(0)∗ [3].

MPAG008.tex; 14/05/1998; 16:05; p.8


Here,m+,m− are the multiplications inC[g−1]q ,C[g−1]q , respectively, andR:C[g−1]q⊗C[g−1]q → C[g−1]q⊗C[g−1]q is the morphism ofUqg-modules definedbelow by Drinfeld’s universalR-matrix [6].

In [6, 7] one can find the description of properties of the universalR-matrixwhich unambiguously determine it as an element of an appropriate completion ofUhg⊗Uhg. In particular,

S ⊗ S(R) = R, R∗⊗∗ = R21. (7.1)

The latter relation involves the elementR21 which is derived fromR by permuta-tion of tensor multiples. The proof of this relation is completely similar to that ofProposition 4.2 in [7].

In [3] there is an explicit ‘multiplicative’ formula for the universalR-matrix.More precisely, any reduced decomposition of the maximum length elementw0

possesses its own multiplicative formula. In the sequel we intend to restrict our-selves to those reduced decompositions which come from Section 5. (Note that the‘multiplicative’ formula was discovered in the papers of Levendorskii and Soibel-man and also by Kirillov and Reshetikhin, see [3]. Its application should take intoaccount the inessential differences in the choice of generators and deformationparameters in this work as compared with [3]. Specifically, one has to substituteX+i , X

−i , Hi, h,Ki, q by−S(Ei),−S(Fi),−S(Hi), h/2,K−1

i , q−1.)It is easy to show that the universalR-matrix determines a linear operator in

C[g−1]q ⊗ C[g−1]q .Now we are in a position to define the operatorR in a standard way:R = σ ·R

with σ : a⊗ b 7→ b⊗ a being a permutation of tensor multiples. Thus,R becomesa morphism ofUqg-modules since [6, 7, 3]

1op(ξ) = R1(ξ)R−1, ξ ∈ Uhg.The associativity of the multiplication in Pol(g−1)q can be derived easily by the

standard argument [10, 11] from the relations

(1⊗ id)(R) = R13R23, (id⊗1)(R) = R13R12.

(HereR12=∑i ai⊗bi⊗1, R23 =∑i 1⊗ai⊗bi, R13=∑i ai⊗1⊗bi wheneverR =∑i ai ⊗ bi , see [6, 7, 3].)

The existence of a unit and covariance of Pol(g−1)q are evident.

8. Involution

Consider the antilinear operators∗: V+(0) → V−(0); ∗: V−(0) → V+(0), whichare determined by their properties as follows. Firstly,v±(0)∗ = v∓(0) and, sec-ondly,

(ξv)∗ = (S−1(ξ))∗v∗ (8.1)

for all v ∈ V±(0), ξ ∈ Uqg.

MPAG008.tex; 14/05/1998; 16:05; p.9


To rephrase the above, we write

(ξv±(0))∗ = (S−1(ξ))∗v±(0)∗.

It follows from the definition ofV±(0) that the involution as above is well de-fined. In particular, (8.1) can be easily deduced; it also follows from the relation(S−1((S−1(ξ))∗))∗ = ξ that the operators constructed above are mutually converse.

The duality argument allows one to form the mutually converse antihomomor-phisms∗: C[g−1]q → C[g−1]q; ∗: C[g−1]q → C[g−1]q :

f ∗(v) def= f (v∗), v ∈ V±(0), f ∈ V±(0)∗. (8.2)

Now we are in a position to define the antilinear operator∗ in Pol(g−1)q by

(f+ ⊗ f−)∗ def= f ∗− ⊗ f ∗+,for f+ ∈ C[g−1]q, f− ∈ C[g−1]q , and also to show that it equips Pol(g−1)q with astructure of covariant involutive algebra.

What remains is to verify that∗ is an antihomomorphism of Pol(g−1)q . The bestway to prove the relation

(f1f2)∗ = f ∗2 f ∗1 ; f1, f2 ∈ Pol(g−1)q

is to apply (7.1) and the duality argument described in details in the concludingsection of the present paper. (Note that it suffices to prove the relation(f1f2)

∗(v) =f ∗2 f

∗1 (v) for the generatorv = v−(0) ⊗ v+(0) of theUqg-moduleV−(0) ⊗ V+(0)

since the mapξ 7→ (S−1(ξ))∗ is an antiautomorphism of the coalgebraUqg.)Verify (4.1). It suffices to consider the casef ∈ V±(0)∗. Application of (8.2)

and the relationS((S(ξ))∗) = ξ ∗, ξ ∈ Uqg, yields

f ((ξv)∗) = f ((S−1(ξ))∗v∗),f (S(ξ)v∗) = f ((ξ ∗v)∗),(ξf )(v∗) = f ∗(ξ ∗v),(ξf )(v∗) = f ∗(S((S(ξ))∗)v),(ξf )∗(v) = ((S(ξ))∗f ∗)(v)

for all v ∈ V±(0), f ∈ V±(0)∗.Thus, in the special casef ∈ V±(0)∗ (4.1) is proved. Hence, it is also valid for

all f ∈ Pol(g)q since the antipode is an antiautomorphism of the coalgebraUqg

and the involution∗ is its automorphism. In fact, iff = f+f−, f± ∈ (V±(0))∗ and1(ξ) =∑j ξ

′j ⊗ ξ ′′j ; ξ ′, ξ ′′j ∈ Uqg then one has

(ξ(f+f−))∗ =∑j

(ξ ′′j f−)∗(ξ ′jf+)

∗,

(S(ξ))∗(f+f−)∗ = (S(ξ))∗(f ∗−f ∗+) =∑j

((S(ξ ′′j ))∗f ∗−)((S(ξ

′j ))∗f ∗+).

MPAG008.tex; 14/05/1998; 16:05; p.10


9. The Simplest Example

Let g = sl2, then one hasg = g−1 ⊕ g0 ⊕ g1, with g0 andg±1 being Cartan andBorel subalgebras ofsl2, respectively. In particular, deg(g−1) = 1.

The algebraUqsl2 is given by its generatorsK±1, E, F and the relations

KK−1 = K−1K = 1, K±1E = q±2EK±1, K±1F = q∓2FK±1,

EF − FE = (K −K−1)/(q − q−1).

Remember that comultiplication1, counitε and antipodeS are defined on theabove generators as

1(E) = E ⊗ 1+K ⊗ E, 1(F) = F ⊗K−1 + 1⊗ F,1(K±1) = K±1⊗K±1;ε(E) = ε(F ) = 0, ε(K±1) = 1;S(E) = −K−1E, S(F) = −FK, S(K±) = K∓.

In the notation

q = e−h/2, K±1 = e∓hH/2, E = X+e−hH/4, F = ehH/4X−

Drinfeld’s formula for the universalR-matrix [6] acquires the form

R = expq2((q−1 − q)E ⊗ F) · exp(H ⊗H · h/4)

with expt (x) =∑∞

n=0 xn(∏nj=1

1−t j1−t )

−1.The involution∗ in Uqsu(1,1) = (Uqsl2,∗ ) is defined on the generatorsE,F,

K±1 by

E∗ = −KF, F ∗ = −EK−1, (K±1)∗ = K±1

(equivalently, on the generatorsX±,H ofUhsl2 it is defined by(X±)∗ = −X∓,H ∗= H ).

Consider theUqsl2-moduleV+(0) determined by its single generatorv+(0) ∈V+(0) and the relationsFv+(0) = 0,K±1v+(0) = v+(0). This module admits thedecomposition

V+(0) =⊕j∈Z+

V+(0)j , V+(0)j = C ·Ej · v+(0).

Hence,Ejv+(0)j∈Z+ is a basis inV+(0).Define a linear functionala− ∈ V+(0)∗ = C[g−1]q by

a−(S(Ej)v+(0)) =

1 j = 1,0 j 6= 1.

MPAG008.tex; 14/05/1998; 16:05; p.11


Prove thata− is a generator ofC[g−1]q , and that for any polynomialP ∈ C[t]K±1: P(a−) 7→ P(q∓2a−), (9.1)

E: P(a−) 7→ (D−P)(a−), (9.2)

F : P(a−) 7→ −q · a2− · (D+P)(a−), (9.3)

where(D±P)(t) = (P (q±2t)− P(t))/(q±2t − t).First note that the relations

Ka−(S(Ej )v+(0)) =q−2 j = 1,0 j 6= 1,

Ea−(S(Ej)v+(0)) =

1 j = 0,0 j 6= 0,

imply that

Ka− = q−2a−, Ea− = 1. (9.4)

Now, apply the covariance ofC[g−1]q to obtain

K±1(P1(a−)P2(a−)) = K±1(P1(a−)) ·K±1(P2(a−)),

E(P1(a−)P2(a−)) = E(P1(a−)) · P2(a−)+K(P1(a−)) · E(P2(a−))

for any polynomialsP1, P2. This already allows one to deduce (9.1), (9.2) from(9.4).

It is worthwhile to note thataj− 6= 0 for all j ∈ Z+ since

Ejaj− =

j∏k=1

((q−2k − 1)/(q−2 − 1)) 6= 0.

This implies that(V+(0)j )∗ = C·aj−. Hence,aj−j∈Z+ is a basis of the vector spaceC[g−1]q . That is,a− is a generator of the algebraC[g−1]q .

Now prove (9.3) in the special caseP(a−) = a−. Specifically, we are going todemonstrate

Fa− = −q · a2−. (9.5)

SinceFa− ∈ (V+(0)2)2 = Ca2− we haveFa− = const· a2−. The fact that theconstant in the latter relation is−q follows easily from

E(Fa−) = K −K−1

q − q−1a− = −(q + q−1)a−,

E(a2−) = (q−2 + 1)a− = q−1(q−1 + q)a−

together witha− 6= 0.

MPAG008.tex; 14/05/1998; 16:05; p.12


The passage from the special caseP(a−) = a− to the general case can beperformed (just as above) by a virtue of covariance. Specifically,

F(P1P2) = F(P1) ·K−1(P2)+ P1F(P2)

for any ‘polynomials’P1(a−), P2(a−).Now turn to the description of the covariant algebraC[g−1]q for the same case

g = sl2. One hasV−(0) =⊕−j∈Z+ V−(0)j , V−(0)−j = C · Fjv−(0). Define alsothe ‘co-ordinate function’a+ by

a+(S(F j )v−(0)) =

1 j = 1,0 j 6= 1.

Now, one can prove, in the same way as above, thata+ is the generator ofC[g−1]q and

K±1: P(a+) 7→ P(q±2a+),F : P(a+) 7→ (D−P)(a+),E: P(a+) 7→ −qa2

+ · (D+P)(a+)for any polynomialP of a single indeterminate.

In particular, one has

K±1a+ = q±2a−, Fa+ = 1, Ea+ = −qa2+. (9.6)

Note that iffi are the generators of a covariant algebraF andaj the gen-erators of a Hopf algebraA, then the action ofA on F can be unambiguouslyretrieved from the action ofaj on fi.

Turn to the description of the covariant algebra Pol(g−1)q in terms of generatorsand relations.

By our construction, the covariant algebrasC[g−1]q andC[g−1]q are embeddedinto Pol(g−1)q .

It follows from the explicit formula for the universalR-matrix and the definitionof the action of exp(H ⊗Hh/4) on the weight vectors that

eH⊗Hh/4a− ⊗ a+ = q− 12 ·2(−2) · a− ⊗ a+,

a−a+ = q2(a+a− + q−1(1− q2)Fa+ · Ea−).

Finally we have

a−a+ = q2a+a− + q(1− q2). (9.7)

Since Pol(g−1)q = C[g−1]q ⊗ C[g−1]q , we deduce that (9.7) gives a completelist of relations between the generatorsa+, a− of Pol(g−1)q , that is the naturalmapC〈a+, a−〉/(a−a+−(q2a+a−+q(1−q2))→ Pol(g−1)q is injective. The action

MPAG008.tex; 14/05/1998; 16:05; p.13


of the generatorsK±1, E, F of Uqg on the generatorsa+, a− of Pol(g−1)q isgiven by (9.4)–(9.6).

What remains is to describe the involution∗.We start with proving that

a∗+ = const· a−, a∗− = const· a+, (9.8)

and then we will find the constants by comparing the explicit expressions forEa∗+ andEa−. The relations (9.8) follow from the decompositionsC[g−1]q =⊕

i(V−(0)i)∗, C[g−1]q =

⊕i(V+(0)i)

∗ and(V∓(0)±1)∗ = C·a±, ∗: (V±(0)i)∗ →

(V∓(0)−i )∗.It was pointed out before thatEa− = 1. Let us computeEa∗+. First use the

relation

(S(F ))∗ = (−FK)∗ = −K∗F ∗ = −K · (−EK−1) = q2E

and the compatibility condition (4.1) for involutions to obtain

q2E · a∗+ = (S(F ))∗a∗+ = (Fa+)∗ = 1∗ = 1.

Thus, we haveq2 ·Ea∗+ = Ea−. Now, (9.8) implies

a∗+ = q−2a−; a∗− = q2a+. (9.9)

The only shortcoming of the definition of the covariant∗-algebra Pol(g−1)qis that it is excessively abstract. In the example forUqsu(1,1) we got anotherdescription of that covariant∗-algebra. Specifically, its generators area+, a−, itscomplete list of relations reduces to (9.7), the action ofUqsu(1,1) is given by(9.4)–(9.7), and the involution is determined by (9.9).

Note that in the work [23] on the function theory in the unit disc the generatorz = q1/2 · a+ was implemented instead ofa+. In this setting, (9.9) implies therelationz∗ = q−3/2a−, and (9.7) can be rewritten as

z∗z− q2zz∗ = 1− q2. (9.10)

(The substitutionq = e−h/2 and theformal passage to a limit ash → 0 yield (cf.(1.1)) limh→0

[z,z∗]ih= i(1− zz∗).)

10. Quantum Disc and Other Bounded Symmetric Domains

Proceed with studying the∗-algebra Pol(g−1)q which was under investigation inthe previous section. Evidently, the formulae

Tϕ(z) = eiϕ, Tϕ(z∗) = e−iϕ, ϕ ∈ R/2πZ,

determine the one-dimensional representation of Pol(g−1)q . We shall also need afaithful infinitely dimensional∗-representationT in the Hilbert spacel2(Z+) givenby

T (z)em = (1− q2(m+1))1/2em+1,

MPAG008.tex; 14/05/1998; 16:05; p.14


T (z∗)em+1 = (1− q2(m+1))1/2em,

T (z∗)e0 = 0,

with emm∈Z+ being the standard basis inl2(Z+). An application of the standardtechniques of operator theory in Hilbert spaces [24] allows one to prove that anyirreducible∗-representation of the above algebra is unitarily equivalent to one ofthe representationsTϕϕ∈R/2πZ, T .

Note that the spectrum ofT (z) is the closureU of the unit discU in C. Just asin [24], we use the notion ‘algebra of continuous functions in the quantum disc’ fora completion of Pol(g)q with respect to the norm‖f ‖ = sup‖ρ(f )‖. Hereρ variesinside the class of all irreducible∗-representations up to unitary equivalence. Onecan easily deduce from the above that‖f ‖ = ‖Tf ‖.

The enveloping von Neumann algebra [5] of the aboveC∗-algebra will be de-noted byL∞(U)q and called the algebra of continuous functions in the quantumdisc. Certainly,L∞ alone is not worthwhile. Only together with a distinguisheddense covariant subalgebraPol(g−1)q (cf. [25]) it is worthwhile.

Note that our quantum disc is only one among those described in [12]. Oth-ers can be derived from this one by a standard argument normally referred to asquantization by Berezin [23].

Also note that the definition ofL∞(U)q, which implements a completion proce-dure and passage to an enveloping von Neumann algebra, does not use the specificfeatures of the special caseg = sl2. That is, to any irreducible prehomogeneousvector space of commutative parabolic type we associate a pair constituted by avon Neumann algebraL∞(U)q and its dense covariant subalgebra Pol(g−1)q .

11. Differential Calculi: the Outline

We follow Maltsiniotis [17] in choosing the basic idea of producing the differentialcalculi. Specifically, we first construct differential calculi of order one, and then weembed them into complete differential calculi by a simple argument described in[17, proof of Theorem 1.2.3].

To outline the construction of order one differential calculi, we restrict ourselvesto the simplest example of a quantum prehomogeneous vector space.

As the first step we consider the type (1,0) forms with holomorphic coefficientsf · dz, f ∈ C[g−1]q , and type (0,1) forms with antiholomorphic coefficientsf ·dz∗, f ∈ C[g−1]q . We prove that

dz · z = q2z · dz; dz∗ · z∗ = q−2z∗ · dz∗. (11.1)

As the second step we assume the consideration of all the forms of types (1,0)and (0,1):f dz, f dz∗, f ∈ Pol(g−1)q . We prove that

dz · z∗ = q−2z∗ · dz; dz∗ · z = q2z · dz∗. (11.2)

MPAG008.tex; 14/05/1998; 16:05; p.15


As the third step we turn to higher forms, which gives the additional relations

dz · dz = 0, dz∗ · dz∗ = 0, dz∗ · dz = −q2dz · dz∗. (11.3)

Of course, the relations (11.1)–(11.3) are well known to the specialists (see, forinstance, [17] and the references therein).

12. Differential Calculi: Step One

We follow the notation of Sections 3, 5, 6.Consider the linear functionalsλ± ∈ h∗ given by

λ±(Hi) = ±aij0,together with the associated generalized Verma modulesV±(λ±). Just as in Section8, define the ‘involutions’∗: V±(λ±) → V∓(λ∓) by (8.1) and∗: v±(λ±) 7→v∓(λ∓).

It follows from the definitions that the maps

v+(λ+) 7→ Ej0v+(0), v+(λ+)∗ 7→ (Ej0v+(0))∗

admit the unique extensions toUqg-module morphisms

δ+: V+(λ+)→ V+(0), δ−: V−(λ−)→ V−(0).

Consider the dual gradedUqg-modules∧1(g−1)q =

⊕j∈Z+

V−(λ−)∗−j ;∧1(g−1)q =

⊕j∈Z+

V+(λ+)∗j .

Our definition of the graded components implies that

δ+V+(λ+)j ⊂ V+(0)j ; δ−V−(λ−)j ⊂ V−(0)j .Now the ‘adjoint’ operators∂ = δ∗−, ∂ = δ∗+ are well defined and becomeUqg-module morphisms

∂: C[g−1]q →∧1(g−1)q; ∂: C[g−1]q →

∧1(g−1)q.

Evidently, the maps

v±(λ±) 7→ v±(0)⊗ v±(λ±); v±(λ±) 7→ v±(λ±)⊗ v±(0)admit the unique extension toUqg-module morphisms

1L±: V±(λ±)→ V±(0)⊗ V±(λ±), 1R

±: V±(λ±)→ V±(λ±)⊗ V±(0).Pass again to the ‘adjoint’ linear operators and observe that they are well defined

and equip∧1(g−1)q with a structure of a covariant bimodule overC[g−1]q , and

MPAG008.tex; 14/05/1998; 16:05; p.16

ON q-ANALOGUES OF BOUNDED SYMMETRIC DOMAINS 91∧1(g−1)q with a structure of a covariant bimodule overC[g−1]q . (The covariance

here means that the actions(1L±)∗, (1R±)∗ of C[g−1]q andC[g−1]q , respectively,areUqg-module morphisms.)

REMARK. With ω ∈∧1(g−1)q orω ∈∧1

(g−1)q one has 1·ω = ω · 1= ω, since

(ε ⊗ id)1L±(v) = v, (id⊗ ε)1R

±(v) = v, v ∈ V±(λ±).

It is easy to show that∂ and∂ are differentiations of the corresponding covariantbimodules

∂(f1f2) = ∂f1 · f2+ f1∂f2; f1, f2 ∈ C[g−1]q,∂(f1f2) = ∂f1 · f2+ f1∂f2; f1, f2 ∈ C[g−1]q .

For example, to prove the latter inequality, it suffices to pass in each its part to theadjoint operators

V+(λ+)→ V+(0)⊗ V+(0)and then to apply both operators to the generatorv+(λ+) of theUqg-moduleV+(λ+).In both cases one obtains

Ej0v+(0)⊗ v+(0)+ v+(0)⊗ Ej0v+(0).In conclusion, let us prove one of the equalities (11.1). Another one can be

derived in a similar way.It follows from z∗dz∗ ∈ (V+(λ+)2)∗, dz∗ · z∗ ∈ (V+(λ+)2)∗,dimV+(λ+)2 = 1

thatz∗ · dz∗ = const· dz∗ · z∗. Thus, it remains to compute the constant.When applying the duality argument, we replacef (v) by 〈f, v〉. Compare

〈z∗dz∗, Ev+(λ+)〉 and〈dz∗ · z∗, Ev+(λ+)〉.Firstly, one has

〈z∗dz∗, Ev+(λ+)〉= 〈z∗ ⊗ dz∗, (1⊗ E + E ⊗K)(v+(0)⊗ v+(λ+))〉= 〈z∗ ⊗ dz∗, (E ⊗K)(v+(0)⊗ v+(λ+))〉= 〈z∗, Ev+(0)〉〈dz∗,Kv+(λ+)〉= q2〈z∗, Ev+(0)〉〈dz∗, v+(λ+)〉 = q2〈z∗, Ev+(0)〉2,

and secondly

〈dz∗ · z∗, Ev+(λ+)〉= 〈dz∗ ⊗ z∗, (1⊗ E + E ⊗K)(v+(λ+)⊗ v+(0))〉= 〈dz∗, Ev+(λ+)〉〈z∗, v+(0)〉 = 〈z∗, v+(0)〉2.

Since〈z∗, v+(0)〉 6= 0, we obtain finally

z∗dz∗ = q2dz∗ · z∗.

MPAG008.tex; 14/05/1998; 16:05; p.17


13. Differential Calculi: Step Two

Consider theUqg-module

(1,0)(g−1)qdef=∧1

(g−1)q ⊗ C[g−1]q.Use the universal R-matrix in the same way as in Section 7 to equip(1,0)(g−1)q

with a structure of a covariant bimodule over Pol(g−1)q .There is a unique extension of the differentiation∂: C[g−1]q → ∧1

(g−1)q toa differentiation∂: Pol(g−1)q → (1,0)(g−1)q such that∂C[g−1]q = 0. Clearly∂(f+ ⊗ f−) = ∂f+ ⊗ f−, f+ ∈ C[g−1]q , f− ∈ C[g−1]q , and∂ is aUqg-modulemorphism.

Turn to the exampleg = Uqsl2. Differentiation of both sides in (9.10) (with theproperties∂: 1 7→ 0, ∂: z 7→ dz being taken into account) yieldsz∗·dz−q2dz·z∗ =0. This is just one of the relations (11.2).

Now consider theUqg-module(0,1)(g−1)qdef= C[g−1]q ⊗ ∧1

(g−1)q togetherwith the morphism ofUqg-modules

∂: Pol(g−1)q → (0,1)(g−1)q; ∂: f+ ⊗ f− 7→ f+ ⊗ ∂f−,wheref+ ∈ C[g−1]q, f− ∈ C[g−1]q . Just as it was done before, one can equip(0,1)(g−1)q with a structure of a covariant bimodule over Pol(g−1)q and provethat ∂ is a differentiation. An application of∂ to both sides of (9.10) gives thesecond one of the relations (11.2).

Finally, set

1(g−1)q = (1,0)(g−1)q ⊕(0,1)(g−1)q, d = ∂ + ∂.

14. Differential Calculi: Step Three

LetA be a unital algebra overC.

DEFINITION. LetM be a bimodule overA andd: A→M a linear operator. Thepair (M, d) is called a differential calculus of order one if

(i) d(a′ · a′′) = da′ · a′′ + a′ · da′′,(ii) A · (dA) ·A = M.

In the case whenA is a covariant algebra,M a covariant bimodule andd: A→M aUqg-module morphism with conditions (i), (ii) being satisfied, the pair(M, d)

is called a covariant differential calculus of order one.The results expounded in the appendix of this work imply that the five covariant

differential calculi of order one:(∧1(g−1)q, ∂

),

(∧1(g−1)q, ∂

),

MPAG008.tex; 14/05/1998; 16:05; p.18


((1,0)(g−1)q, ∂), ((0,1)(g−1)q, ∂), (1(g−1)q, d).

In the sequel we apply to each of those the ‘algorithm of constructing the fulldifferential calculus’ described in [17].

DEFINITION. Let =⊕n∈Z+ n be aZ+-graded algebra andd a linear operatorin of order one. The pair(, d) is called a differential graded algebra if

(i) d2 = 0,(ii) d(a′ · a′′) = da′ · a′′ + (−1)na′ · da′′, a′ ∈ n, a′′ ∈ .

If is a covariant algebra andd a Uqg-module morphism, then under theconditions (i) and (ii) we call the pair(, d) a covariant differential graded algebra.

Let us describe the ‘algorithm’ of construction of the pair(, d) given the pair(M, d). LetM1 = dA ⊂ M.

Equip the tensor algebraT = T (A,M1) with a grading in which dega =0,degm = 1, a ∈ A,m ∈ M1. One hasT0 = T (A) = C⊕A⊕A⊗2⊕ . . . , Tj+1 =T (A)⊗M1⊗ Tj .

There exists a unique operatord: T → T such that

(i) d(t1t2) = dt1 · t2 + (−1)nt1 · dt2, t1 ∈ Tn, t2 ∈ T ,(ii) d|A coincides with the differentiation in the initial calculus of order one,

(iii) d|M1 = 0.

In fact, onT0 we haved1 = 0, d(a1 ⊗ a2 ⊗ · · · ⊗ ak) = ∑j a1 ⊗ · · · ⊗

aj−1 ⊗ daj ⊗ · · · ⊗ ak . From now on we proceed by induction:d(a ⊗ m ⊗ t) =da ⊗m⊗ t − a ⊗m⊗ dt, a ∈ T0,m ∈M1, t ∈ Tj .

(Note thatd is well defined because of the multilinearity of the right-handsides of the above identities in the ‘indeterminates’(a1, . . . , ak) and (a,m, t),respectively.)

Consider the leastd-invariant bilateral idealJ of T which contains all theelements of the form

(i) a1⊗ a2 − a1a2, a1, a2 ∈ A,(ii) 1 ⊗m−m,m⊗ 1−m, m ∈ M1,

(iii) ∑ij a′i ⊗mij ⊗ a′′j | a′i , a′′j ∈ A,mij ∈ M1,

∑ij a′imij a

′′j = 0.

(Note that the left hand side of the latter equality is a sum of elements of theA-bimoduleM.)

From our construction it follows thatJ is a graded ideal:J = ⊕j (J ∩ Tj).

Furthermore,J is aUqg-submodule ofT (due to the covariance of the algebraA,the moduleM and the order one calculus(M, d)).

Hence the quotient algebra = T/J with the differentialdJ : t + J 7→ dt + Jis a covariant graded differential algebra. It is easy to show thatA ' 0,M ' 1,and the initial differentiald: A→M ‘coincides’ with the restriction ofdJ onto0.

MPAG008.tex; 14/05/1998; 16:05; p.19


The five order one differential calculi we have already produced lead to fivecovariant graded differential algebras(∧

q(g−1), ∂),

(∧q(g−1), ∂

),

((∗,0)q , ∂), ((0,∗)q , ∂), (q, d).

In the exampleg = sl2 the relations∂2(z2) = ∂2((z∗)2) = ∂(zdz∗−q−2dz∗z) =

0 imply (11.3).

15. Holomorphic Bundles and Dolbeault Complexes

Just as in Section 5, we choose a functionalµ ∈ h∗ such thatmj = µ(Hj) ∈ Z+for j 6= j0. The linear functional of this formλ− ∈ h∗ was already considered inSection 12.

Consider aUqg-moduleV−(µ) and the associated ‘graded dual’ module0µ.Use the comultiplication1op to equipV−(0)⊗V−(µ) andV−(µ)⊗V−(0) with

a structure of aUqg-module. Also, the morphisms

1L: V−(µ)→ V−(0)⊗ V−(µ); 1L: v−(µ) 7→ v−(0)⊗ v−(µ),

1R: V−(µ)→ V−(µ)⊗ V−(0); 1R: v−(µ) 7→ v−(µ)⊗ v−(0),

(together with the adjoint linear maps1∗L,1∗R) are used to equip0µ with a struc-

ture of a covariant bimodule overC[g−1]q :1∗L: C[g−1]q ⊗ 0µ→ 0µ; 1∗R: 0µ ⊗ C[g−1]q → 0µ.

It follows from the properties of the universalR-matrix overUqg that

σR−1v−(0)⊗ v−(µ) = v−(µ)⊗ v−(0),whereσ : a ⊗ b 7→ b ⊗ a, andR−1 is the universalR-matrix of the Hopf algebraUqg

op. HenceσR−11L = 1R, 1L = Rσ1R,

1∗L = 1∗R · R. (15.1)

HereR: C[g−1]q ⊗ 0µ→ 0µ ⊗ C[g−1]q , R = σR.(15.1) shows how to describe the covariant bimodule0µ in terms of generators

and relations.The standard construction (see Section 7) allows one to equip the tensor product

Mµ = 0µ ⊗ C[g−1]q with a structure of a covariant bimodule over Pol(g−1)q =C[g−1]q ⊗ C[g−1]q .

Consider the simplest caseg = sl2. Denote byγµ the lowest weight vector of theUqg-module0µ such thatγµ(v−(µ)) = 1. Clearlymµ = γµ⊗ 1 is a generator of a

MPAG008.tex; 14/05/1998; 16:05; p.20


covariant bimoduleMµ. It is not hard to deduce the complete list of ‘commutation’relations using the explicit form of the universalR-matrix (see Section 9):

z ·mµ = q−µ(H) ·mµ · z, z∗ ·mµ = qµ(H) ·mµ · z∗. (15.2)

It is easy to prove that

K±1mµ = q±µ(H)mµ, Fmµ = 0,

Emµ = −q1/2 · 1− q2µ(H)

1− q2· zmµ.

(The last equality follows from the covariance of the bimoduleMµ and the relations

Emµ = const· zmµ,FEmµ = −(EF − FE)mµ = −q

µ(H) − q−µ(H)q − q−1

mµ.)

The elements ofMµ could be treated asq-analogues of smooth sections of aholomorphic vector bundle. We are interested in differential forms whose coeffi-cients are such ‘sections’.

Consider the covariant bimodule(0,∗)µ,q = Mµ

⊗Pol(g−1)q

(0,∗)q over Pol(g−1)q .Evidently

(0,∗)µ,q = 0µ⊗C[g−1]q

(0,∗)q = 0µ⊗C

∧(g−1)q. (15.3)

Apply theUqg-module morphismR: C[g−1]q⊗0µ→ 0µ⊗C[g−1]q , R = σRderived from the universalR-matrix to equip(0,∗)µ,q with a structure of a covariantbimodule over(0,∗)q . In the exampleg = sl2 one can readily describe this module:the relation list (15.2) should be completed with one more relation

dz∗ ·mµ = qµ(H) ·mµ · dz∗.It follows from (15.3) and∂C[g−1]q = 0 that the operator

∂µ = id⊗C[g−1]q

∂: 0µ⊗C[g−1]q

(0,∗)q → 0µ⊗C[g−1]q

(0,∗)q .

Certainly,(0,∗)µ,q is a graded bimodule(0,∗)µ,q =⊗

j (0,j)µ,q over(0,∗)q , and∂µ is

its differentiation of order one:

∂µ(am) = (∂a)m+ (−1)dega · a · ∂µm,∂µ(ma) = (∂µm)a + (−1)degm ·m · ∂a

for all homogeneous elementsa ∈ (0,∗)q ,m ∈ (0,∗)µ,q .

Evidently, the differentiation∂µ is determined unambiguously by its values ongenerators.

In the exampleg = sl2 for the generatormµ ∈ Mµ → (0,∗)µ,q as above we have:

∂mµ = 0.

MPAG008.tex; 14/05/1998; 16:05; p.21


Now pass to the homogeneous components

(0,j)µ,q = Mµ

⊗Pol(g−1)q

(0,j)q = 0µ⊗C[g−1]q

(0,j)q

of the graded bimodule(0,∗)µ,q to obtain theDolbeault complex

0→Mµ

∂µ−→(0,1)µ,q

∂µ−→(0,2)µ,q

∂µ−→· · · .Its terms are the covariant bimodules over Pol(g−1)q , and the differentials arethe Uqg-module morphisms which commute with the left and the right actionsof C[g−1]q .

16. Conclusions

Let us now digress from involutions and differentiations and sketch our approachto the construction ofq-analogues of Hermitian symmetric spaces of non-compacttype (one can find more details in Sections 2–10).

Let q = 1. Evidently, for all ξ ∈ g±1 the series exp(ξ)v±(0) converge insome ‘completed’ spacesV ±(0) = ×jV±(0)j . This allows one to elaborate theHarish-Chandra method to produce embeddingsI±: X → V±(0) of an irreducibleHermitian symmetric spaceX. The canonical embeddings can be obtained fromI±by composing them from the right with the projectionsπ±: V±(0)→ V±(0)±1 'g±1.

Our basic observation is that the topologicalUg-modulesV±(0) and hence the

subalgebrasg±1 have the proper quantum analogues?: (g±1)qdef= V±(0)±.

This allows one to imitate the above Harish-Chandra embeddingsi± = π±I±for q 6= 1.

There is a different exposition of our construction forq-analogues of boundedsymmetric domains and prehomogeneous vector spaces. It provides clearer in-terplay between our constructions and the approach of Drinfeld [6] to quantumgroups, and the interpretation of the quantum Weil group described by Leven-dorskii and Soibelman [16].

An alternate approach to introducing Pol(g−1)q is in producing a covariant in-volutive coalgebra and further passage to the dual covariant involutive algebra.This approach requires a more detailed exposition of the ‘duality theory’ forUqg-module algebras andUqgop-module coalgebras. Specifically, we need to equip ouralgebras with the strongest locally compact topologies. The dual coalgebras arethe completions of coalgebras considered in the work above, with respect toW ∗-weak topologies, and their tensor products are replaced by the completed tensorproducts⊗ (see [13]). We describe here the topological covariant∗-coalgebra dualto Pol(g−1)q . Remember (see Section 6) that we replaceUqg by Uqgop in tensorproducts of generalized Verma modules.? Remember [3] that, unlikeUg, the algebrag itself has no ‘good’ quantum analogues.

MPAG008.tex; 14/05/1998; 16:05; p.22


It is easy to show that the vectorv0 = v−(0)⊗ v+(0) is a generator of the topo-logicalUqg-moduleV 0 = V−(0) ⊗V+(0). The structure of a covariant coalgebrain V 0 is imposed by introducing aUqg-module morphism1: V 0 → V 0 ⊗V 0

given by an application of a universalR-matrix:

1v0 = Rv0⊗ v0.

The coassociativity of1 follows from the quasitriangularity of the Hopf algebraUqg. Impose an involution inV 0 by

(ξv0)∗ = (S−1(ξ))∗v0, ξ ∈ Uqg,

which already implies

(ξv)∗ = (S−1(ξ))∗v∗, ξ ∈ Uqg, v ∈ V 0.

(Note that (7.1) provides∗ to be an antilinear coalgebra antihomomorphism ofV 0.)Consider the maps

ε− ⊗ id: V 0→ V+(0); id⊗ ε+: V 0→ V−(0),

with ε± being the counits of the coalgebrasV±(0).It follows from the relations

(ε ⊗ id)(R) = (id⊗ ε)(R) = 1

that these maps are the morphisms of covariant coalgebras (dual to the embeddingsC[g−1]q → Pol(g−1)q, C[g−1]q → Pol(g−1)q).

The relation

R(v0⊗ v0) = v−(0)⊗ Rσ(v−(0)⊗ v+(0))⊗ v+(0)with σ : a ⊗ b 7→ b ⊗ a, demonstrates that the comultiplication1 agrees with themultiplication in Pol(g−1)q introduced in Section 7.

Finally, let us note that the commutation relations between the elements of(C[g−1]q)+1 and(C[g−1]q)−1 are of degree at most two, as follows from the prop-erties of the universalR-matrix.

Appendix

Images of the Differentials∂, ∂

Consider the coalgebrasV−(0) andV−(λ−) and disregard for a moment theirUqg-module structures.

LEMMA 1. The left comoduleV−(λ−) over the coalgebraV−(0) is isomorphic toa direct sum ofm copies ofV−(0), withm = dimV−(λ−)−1.

MPAG008.tex; 14/05/1998; 16:05; p.23


Proof. Remember the decompositionw0 = w′0 · u, with w0 andw′0 being themaximum length elements in the Weil group of Lie algebrasg andg′, respectively(see Section 5). It was our agreement to consider only those reduced decomposi-tions ofw0 which are given by concatenation of reduced decompositions forw′0andu.

Choose such a decomposition. Just as in [4, Proposition 1.7(c)], associate to itthe bases inUqg′, Uqg and the bases of the vector spaces

V−(0) = Uqg · v−(0); V−(λ−)−1 = Uqg′ · v−(λ−);V−(λ−) = Uqg · v−(λ−).

(One can verify that(V−(λ−))−1 is a simpleUqg′-module.) The above bases are ofthe form

ξiv−(0), ηjv−(λ−), ξiηjv−(λ−),with i ∈ Z+, j ∈ 1, . . . ,m, andξi ∈ Uqg, ηj ∈ Uqg′ can be derived from thebases ofUqg andUqg′ described explicitly in [4]. It remains to prove that for eachj the map

πj : ξiv−(0) 7→ ξiηj v−(λ−), i ∈ Z+is a morphism of left comodules. So

1L−πj(ξiv−(0)) = 1(ξi)1

L−(ηjv−(λ−)) = 1(ξi)(v−(0)⊗ ηjv−(λ−))

= id⊗ πj1(ξiv−(0)).

LEMMA 2. Let v ∈ V−(0). Then1−(v) ∈ v−(0) ⊗ V−(0) if and only if v ∈Cv−(0).

Proof.If v = const·v−(0), then1−(v) = v−(0)⊗const·v−(0) ∈ v−(0)⊗V−(0).Conversely, if1−(v) = v−(0)⊗ v1, v1 ∈ V−(0), then

v = (id⊗ ε−)1−(v) = (id⊗ ε−)(v−(0)⊗ v1)

= ε−(v1) · v−(0) ∈ C · v−(0).

LEMMA 3. Let v ∈ V−(λ−). Then1L−(v) ∈ v−(0) ⊗ V−(λ−) if and only ifv ∈V−(λ−)−1.

Proof. Let L = v ∈ V−(λ− | 1L−(v) ∈ v−(0) ⊗ V−(λ−). Evidently,L ⊃V−(λ−)−1, and by Lemmas 1, 2, dimL = dimV−(λ−)−1. It follows that L =V−(λ−)−1. 2

REMARK 4. If 1L−(v) = v−(0)⊗ v1, thenv1 = v sincev1 = (ε ⊗ id)(v−(0) ⊗v1) = (ε ⊗ id)1L−(v).

LEMMA 5. The restriction ofδ− onto(V−(λ−))−1 is an injective linear operator.

MPAG008.tex; 14/05/1998; 16:05; p.24


Proof.This operator is non-zero and is aUqg-module morphism(V−(λ−))−1→V−(0), where(V−(λ−))−1 is a simpleUqg′-module.

PROPOSITION 6.C[g−1]q · ∂C[g−1]q =∧1(g−1)q .

Proof.Assume the contrary. LetV ′ = v ∈ V−(λ−) | 〈f1∂f2, v〉 = 0,∀f1, f2 ∈C[g−1]q. ThenV ′ =⊕i∈Z+ V

′−i 6= 0. It follows from the definitions that1L−(V ′)⊂ V−(0)⊗ V ′.

Let i′ be the least suchi ∈ Z+ thatV ′−i 6= 0. We have

1L−(V

′−i′) ⊂ (Cv−(0)⊕

⊕k>0

V−(0)−k)⊗ V ′

⊂ v−(0)⊗ V−(λ−)+(⊕k>0

V−(0)−k)⊗ V ′.

On the other hand((⊕

k>0V−(0)−k) ⊗ V ′)−i′ = 0, and hence1L−(V ′−i′) ⊂v−(0)⊗ V−(λ−). Now we deduce from Lemma 3 thatV ′−i′ = V−(λ−)−1

⋂V ′.

Let v′ ∈ V−(λ−)−1⋂V ′. It follows from the definition ofV ′ that (id ⊗ δ−)

1L−(v′) = 0. On the other hand, by Lemma 3 and Remark 4 we observe that1L(v

′) = v(0) ⊗ v′. Hence(id ⊗ δ−)(v(0) ⊗ v′) = 0. That is,δ−(v′) = 0 andhence, by Lemma 5,v′ = 0.

Thus we have proved thatV ′−i′ = 0 which makes a contradiction to the contraryof Proposition 6. 2

REMARK 7. One can prove in a similar way that

∂C[g−1]q · C[g−1]q = C[g−1]q · ∂C[g−1]q =∧1(g−1)q,

∂C[g−1]q · C[g−1]q =∧1(g−1)q.

References

1. Abe, E.:Hopf Algebras, Cambridge Univ. Press, Cambridge, 1980.2. Bopp, P. N. and Rubenthaler, H.: Fonction zêta associée à la série principale sphérique de

certain espaces symmétriques,Ann. Sci. École Norm. Sup.(4) 26 (1993), 701–745.3. Chari, V. and Pressley, A.:A Guide to Quantum Groups, Cambridge Univ. Press, Cambridge,

1995.4. de Concini, C. and Kac, V.: Representations of quantum groups at roots of 1, in: A. Connes,

M. Duflo, A. Joseph and R. Rentschler (eds),Operator Algebras, Unitary Representations,Enveloping Algebras and Invariant Theory, 1990, Birkhauser, Boston, pp. 471–506.

5. Dixmier, J.:LesC∗-algèbres et leur représentations, Gauthier-Villars, Paris, 1964.6. Drinfeld, V. G.: Quantum groups, in: A. M. Gleason (ed),Proceedings of the International

Congress of Mathematicians, Berkeley, 1986, American Mathematical Society, Providence,R.I., 1989, pp. 798–820.

7. Drinfeld, V. G.: On almost commutative Hopf algebras,Leningrad Math. J.1 (1990), 321–432.8. Helgason, S.:Differential Geometry and Symmetric Spaces, Acad. Press, NY, London, 1962.

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9. Humphreys, J. E.:Reflection Groups and Coxeter Groups, Cambridge Univ. Press, 1990.10. Joyal, A. and Street, R.: Braided tensor categories,Adv. in Math.102(1993), 20–78.11. Kassel, C.:Quantum Groups, Springer-Verlag, NY, Berlin, Heidelberg, 1995.12. Klimek, S. and Lesniewski, A.: A two-parameter quantum deformation of the unit disc,

J. Funct. Anal.115(1993), 1–23.13. Kelley, J. L. and Namioka, I.:Linear Topological Spaces, Van Nostrand Inc., Princeton, NY,

London, 1963.14. Khoroshkin, S., Radul, A. and Rubtsov, V.: A family of Poisson structures on compact

Hermitian symmetric spaces,Comm. Math. Phys.152(1993), 299–316.15. Lustig, G.: Quantum groups at roots of 1,Geom. Dedicata35 (1990), 89–114.16. Levendorskii, S. Z. and Soibelman, Ya. S.: Some applications of the quantum Weil group,

J. Geom. Phys.7 (1990), 241–254.17. Maltsiniotis, G.: Le langage des espaces et des groupes quantiques,Comm. Math. Phys.151

(1993), 275–302.18. Nagy, G. and Nica, A.: On the ‘quantum disc’ and a ‘non-commutative circle’, in: R. E. Curto,

P. E. T. Jorgensen (eds),Algebraic Methods on Operator Theory, Birkhauser, Boston, 1994, pp.276–290.

19. Rubenthaler, H.: Les paires duales dans les algèbres de Lie réductives,Astérisque219(1994).20. Serre, J. P.:Complex Semisimple Algebras, Springer, Berlin, Heidelberg, New York, 1987.21. Sinel’shchikov, S. and Vaksman, L.: Hidden symmetry of the differential calculus on the

quantum matrix space, to appear inJ. Phys. A.22. Soibelman, Ya. S. and Vaksman, L. L.: On some problems in the theory of quantum groups,

in: A. M. Vershik (ed),Representation Theory and Dynamical Systems, Advances in SovietMathematics 9, American Mathematical Society, Providence, RI (1990), pp. 3–55.

23. Sinel’shchikov, S., Shklyarov, D. and Vaksman, L.: On function theory in the quantum disc:Integral representations. Preprint, 1997, q-alg.

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25. Woronowicz, S. L.: Compact matrix pseudogroups,Comm. Math. Phys.111(1987), 613–665.

MPAG008.tex; 14/05/1998; 16:05; p.26


107

Metastates in the Hopfield Model in the ReplicaSymmetric Regime?

ANTON BOVIERWeierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, D-10117 Berlin,Germany. e-mail: [email protected]

VÉRONIQUE GAYRARDCentre de Physique Théorique – CNRS, Luminy, Case 907, F-13288 Marseille Cedex 9, France.e-mail: [email protected]

(Received: 25 June 1997; accepted: 4 December 1997)

Abstract. We study the finite dimensional marginals of the Gibbs measure in the Hopfield model atlow temperature when the number of patterns,M, is proportional to the volume with a sufficientlysmall proportionality constantα > 0. It is shown that even when a single pattern is selected (bya magnetic field or by conditioning), the marginals do not converge almost surely, but only in law.The corresponding limiting law is constructed explicitly. We fit our result in the recently proposedlanguage of ‘metastates’ which we discuss some length. As a byproduct, in a certain regime of theparametersα andβ (the inverse temperature), we also give a simple proof of Talagrand’s recent resultthat the replica symmetric solution found by Amit, Gutfreund, and Sompolinsky can be rigorouslyjustified.

Mathematics Subject Classifications (1991):82B44, 60K35, 82C32.

Key words: Hopfield model, neural networks, metastates, replica symmetry, Brascamp–Lieb in-equalities.

1. Introduction

Strongly disordered systems such as spin glasses represent some of the most in-teresting and most difficult problems of statistical mechanics. Amongst the mostremarkable achievements of theoretical physics in this field is the exact solution ofsome models of mean field type via the replica trick and Parisi’s replica symmetrybreaking scheme (for an exposition see [15]; the application to the Hopfield model[11] was carried out in [1]). The replica trick is a formal tool that allows to eliminatethe difficulty of studying disordered systems by integrating out the randomnessat the expense of having to perform an analytic continuation of some functioncomputable only on the positive integers to the value zero??. Mathematically, this

? Work partially supported by the Commission of the European communities under contractCHRX-CT93-0411.?? As a matter of fact, such an analytic continuation is not performed. What is done is much

more subtle: The function at integer values is represented as some integral suitable for evaluation


108 ANTON BOVIER AND VERONIQUE GAYRARD

procedure is highly mysterious and has so far resisted all attempts to be put on asolid basis. On the other hand, its apparent success is a clear sign that somethingought to be understood better in this method. An apparently less mysterious ap-proach that yields the same answer is the cavity method [15]. However, here, too,the derivation of the solutions involves a large number of intricate and unprovenassumptions that seem hard or impossible to justify in general.

However, there has been some distinct progress in understanding the approachof the cavity method at least in simple cases where no breaking of the replicasymmetry occurs. The first attempts in this direction were made by Pastur andShcherbina [22] in the Sherrington–Kirkpatrick model and Pastur, Shcherbina andTirozzi [23] in the Hopfield model. Their results were conditional: They assertto show that the replica symmetric solution holds under a certain unverified as-sumption, namely the vanishing of the so-called Edwards–Anderson parameter. Abreakthrough was achieved in a recent paper by Talagrand [24] where he provedthe validity of the replica symmetric solution in an explicit domain of the modelparameters in the Hopfield model. His approach is purely by induction over thevolume (i.e. the cavity method) and uses only some a priori estimates on the supportproperties of the distribution of the so-called overlap parameters as first proven in[6, 7] and in sharper form in [3].

Let us recall the definition of the Hopfield model and some basic notations. LetSN ≡ −1,1N denote the set of functionsσ : 1, . . . , N → −1,1, and setS ≡ −1,1N. We callσ a spin configuration and denote byσi the value ofσ at i.Let (,F ,P) be an abstract probability space and letξ

µ

i , i, µ ∈ N denote a familyof independent identically distributed random variables on this space. For the pur-poses of this paper we will assume thatP[ξµi = ±1] = 1/2. We will write ξµ[ω]for theN-dimensional random vector whoseith component is given byξµi [ω] andcall such a vector a ‘pattern’. On the other hand, we use the notationξi[ω] for theM-dimensional vector with the same components. When we writeξ [ω] withoutindices, we frequently will consider it as anM ×N matrix and we writeξ t [ω] forthe transpose of this matrix. Thus,ξ t [ω]ξ [ω] is theM×M matrix whose elementsare

∑Ni=1 ξ

µ

i [ω]ξνi [ω]. With this in mind we will use throughout the paper a vectornotation with(·, ·) standing for the scalar product in whatever space the argumentmay lie. For example, the expression(y, ξi) stands for

∑Mµ=1 ξ

µ

i yµ, etc.We define random mapsmµN [ω]: SN → [−1,1] through?

mµ

N [ω](σ ) ≡1

N

N∑i=1

ξµ

i [ω]σi. (1.1)

by a saddle point method. Instead of doing this, apparently irrelevant critical points are selectedjudiciously and the ensuing wrong value of the function is then continued to the correct value at zero.? We will make the dependence of random quantities on the random parameterω explicit by an

added[ω] whenever we want to stress it. Otherwise, we will frequently drop the reference toω tosimplify the notation.

MPAG005.tex; 24/08/1998; 10:00; p.2

METASTATES IN THE HOPFIELD MODEL IN THE REPLICA SYMMETRIC REGIME 109

Naturally, these maps ‘compare’ the configurationσ globally to the random con-figurationξµ[ω]. A Hamiltonian is now defined as the simplest negative functionof these variables, namely

HN [ω](σ ) ≡ −N2

M(N)∑µ=1

(mµ

N [ω](σ ))2

= −N2‖mN [ω](σ )‖22 , (1.2)

whereM(N) is some, generally increasing, function that crucially influences theproperties of the model.‖·‖2 denotes the2-norm inRM , and the vectormN [ω](σ )is always understood to beM(N)-dimensional.

Through this Hamiltonian we define in a natural way finite volume Gibbs mea-sures onSN via

µN,β [ω](σ ) ≡ 1

ZN,β [ω] e−βHN [ω](σ ) (1.3)

and the induced distribution of the overlap parameters

QN,β [ω] ≡ µN,β [ω] mN [ω]−1. (1.4)

The normalizing factorZN,β [ω], given by

ZN,β [ω] ≡ 2−N∑σ∈SN

e−βHN [ω](σ ) ≡ Eσ e−βHN [ω](σ ) (1.5)

is called the partition function. We are interested in the largeN behaviour of thesemeasures. In our previous work we have been mostly concerned with the limitinginduced measures. In this paper we return to the limiting behaviour of the Gibbsmeasures themselves, making use, however, of the information obtained on theasymptotic properties of the induced measures.

We pursue two objectives. Firstly, we give an alternative proof (whose outlinewas given in [4]) of Talagrand’s result (with possibly a slightly different rangeof parameters) that, although equally based on the cavity method, makes moreextensive use of the properties of the overlap-distribution that were proven in [3].This allows, in our opinion, some considerable simplifications. Secondly, we willelucidate some conceptual issues concerning the infinite volume Gibbs states in thismodel. Several delicacies in the question of convergence of finite volume Gibbsstates (or local specifications) in highly disordered systems, and in particular spinglasses, were pointed out repeatedly by Newman and Stein over the last years [17,18]. But only during the last year did they propose the formalism of so-called‘metastates’ [19, 20, 16] that seems to provide the appropriate framework to discussthese issues. In particular, we will show that in the Hopfield model this formalismseems unavoidable for spelling out convergence results.

MPAG005.tex; 24/08/1998; 10:00; p.3


Let us formulate our main result in a slightly preliminary form (precise formu-lations require some more discussion and notation and will be given in Section 5).

Denote bym∗(β) the largest solution of the mean field equationm = tanh(βm)and by eµ the µth unit vector of the canonical basis ofRM . For all (µ, s) ∈1, . . . ,M × −1,1 let B(µ,s)ρ ⊂ RM denote the ball of radiusρ centered atsm∗eµ. For any pair of indices(µ, s) and anyρ > 0 we define the conditionalmeasures

µ(µ,s)

N,β,ρ[ω](A) ≡ µN,β [ω](A | B(µ,s)ρ ), A ∈ B(−1,1N). (1.6)

The so-called ‘replica symmetric equations’? of [1] is the following system ofequations in three unknownsm1, r, andq, given by

m1 =∫

dN (g) tanh(β(m1+√αrg)),

q =∫

dN (g) tanh2(β(m1+√αrg)), (1.7)

r = q

(1− β + βq)2 .With this notation we can state

THEOREM 1.1. There exist finite positive constantsc, c′, c0 such that if06 α 6c(m∗(β))4 and0 6 α 6 c′β−1, with limN↑∞M(N)/N = α, the following holds:

Chooseρ such thatc0 6√α

m∗(β) 6 ρ 612m∗(β). Then, for any finiteI ⊂ N, and for

anysI ⊂ −1,1I ,

µ(µ,s)

N,β,ρ(σI = sI )→∏i∈I

eβsi[m1ξ

1i +gi

√αr]

2 cos(β[m1ξ

1i + gi

√αr]) (1.8)

asN ↑ ∞, where thegi, i ∈ I are independent Gaussian random variables withmean zero and variance one that are independent of the random variablesξ1

i , i ∈I . The convergence is understood in law with respect to the distribution of theGaussian variablesgi .

This theorem should be juxtaposed to our second result:

THEOREM 1.2. On the same set of parameters as in Theorem1.1, the followingis true with probability one: For any finiteI ⊂ N and for anyx ∈ RI , there existsubsequencesNk[ω] ↑ ∞ such that for anysI ⊂ −1,1I , if α > 0,

limk↑∞

µ(µ,s)

Nk [ω],β,ρ[ω](σI = sI ) =∏i∈I

esixi

2 cosh(xi)(1.9)

? We cite these equations, (3.3–5) in [1] only for the casek = 1, wherek is the number of theso-called ‘condensed patterns’. One could generalize our results presumably to measures conditionedon balls around ‘mixed states’, i.e. the metastable states with more than one ‘condensed pattern’, butwe have not worked out the details.

MPAG005.tex; 24/08/1998; 10:00; p.4


The above statements may look a little bit surprising and need clarification. Thiswill be the main purpose of Section 2, where we give a rather detailed discussionof the problem of convergence and the notion of metastates with the particularissues in disordered mean field models in view. We will also propose yet a differentnotion of a state (let us call it ‘superstate’), which tries to capture the asymptoticvolume dependence of Gibbs states in the form of a continuous time measure val-ued stochastic process. We also discuss the issue of the ‘boundary conditions’ orrather ‘external fields’, and the construction of conditional Gibbs measures in thiscontext. This will hopefully prepare the ground for the understanding of our resultsin the Hopfield case.

The following two sections collect technical preliminaries. Section 3 recallssome results on the overlap distribution from [3, 4, 5] that will be crucially neededlater. Section 4 states and proves a version of the Brascamp–Lieb inequalities [8]that is suitable for our situation.

Section 5 contains our central results. Here we construct explicitly the finitedimensional marginals of the Gibbs measures in finite volume and study theirbehaviour in the infinite volume limit. The results will be stated in the language ofmetastates. In this section we assume the convergence of certain thermodynamicfunctions which will be proven in Section 6. Modulo this, this section contains theprecise statements and proofs of Theorems 1.1 and 1.2.

In Section 6 we give a proof of the convergence of these quantities and we relatethem to the replica symmetric solution. This section is largely based on the ideasof [23] and [24] and is mainly added for the convenience of the reader.

2. Notions of Convergence of Random Gibbs Measures

In this section we make some remarks on the appropriate picture for the studyof limiting Gibbs measures for disordered systems, with particular regard to thesituation in mean-field like systems. Although some of the observations we willmake here arose naturally from the properties we discovered in the Hopfield model,our understanding has been greatly enhanced by the recent work of Newman andStein [19, 20, 16] and their introduction of the concept of ‘metastates’. We refer thereader to their papers for more detail and further applications. Some examples canalso be found in [14]. Otherwise, we keep this section self-contained and geared forthe situation we will describe in the Hopfield model, although part of the discussionis very general and not restricted to mean field situations. For this reason we talkabout finite volume measures indexed by finite sets3 rather than by the integerN .

METASTATES

The basic objects of study arefinite volume Gibbs measures, µ3,β (which forconvenience we will always consider as measures on the infinite product spaceS∞). We denote by(M1(S∞),G) the measurable space of probability measures on

MPAG005.tex; 24/08/1998; 10:00; p.5


S∞ equipped with the sigma-algebraG generated by the open sets with respect tothe weak topology onM1(S∞)?. We will always regard Gibbs measures as randomvariables on the underlying probability space(,F ,P) with values in the spaceM1(S∞), i.e. as measurable maps→M1(S∞).

We are in principle interested in considering weak limits of these measures as3 ↑ ∞. There are essentially three things that may happen:(1) Almost sure convergence: ForP-almost allω,

µ3[ω] → µ∞[ω], (2.1)

whereµ∞[ω]may or may not depend onω (in general it will).(2) Convergence in law:

µ3D→ µ∞. (2.2)

(3) Almost sure convergence along random subsequences: There exist (at least foralmost allω) subsequences3i[ω] ↑ ∞ such that

µ3i [ω][ω] → µ∞,3i [ω][ω]. (2.3)

In systems with compact single site state space, (3) holds always, and there aremodels with non-compact state space where it holds with the ‘almost sure’ pro-vision. However, this contains little information, if the subsequences along whichconvergence holds are only known implicitly. In particular, it gives no informa-tion on how, for any given large3 the measureµ3 ‘looks like approximately’.In contrast, if (i) holds, we are in a very nice situation, as for any large enough3 and for (almost) any realization of the disorder, the measureµ3[ω] is wellapproximated byµ∞[ω]. Thus, the situation would be essentially like in an orderedsystem (the ‘almost sure’ excepted). It seems to us that the common feeling of mostpeople working in the field of disordered systems was that this could be arrangedby putting suitable boundary conditions or external fields, to ‘extract pure states’.Newman and Stein [17] were, to our knowledge, the first to point to difficulties withthis point of view.In fact, there is no reason why we should ever be, or be able toput us, in a situation where(1) holds, and this possibility should be considered asperfectly exceptional. With (3) uninteresting and (1) unlikely, we are left with (2).By compactness, (2) holds always at least for (non-random!) subsequences3n, andeven convergence without subsequences can be expected rather commonly. On theother hand, (2) gives us very reasonable information on our system, telling us whatis the chance that our measureµ3 for large3 will look like some measureµ∞.This is much more than what (3) tells us, and baring the case where (1) holds, allwe may reasonably expect to know.

We should thus investigate the case (2) more closely. As proposed actually firstby Aizenman and Wehr [2], it is most natural to consider an objectK3 defined as

? Note that a basis of open sets is given by sets of the formsNf1,...,fk,ε(µ) ≡µ′|∀16i6k|µ(fi) − µ′(fi)| < ε, wherefi are continuous functions onS∞; indeed, it is enoughto consider cylinder functions.

MPAG005.tex; 24/08/1998; 10:00; p.6


a measure on the product space⊗M1(S∞) (equipped with the product topologyand the weak topology, respectively), such that its marginal distribution on isP while the conditional measure,κ3(·)[ω], on M1(S∞) given F ? is the Diracmeasure onµ3[ω]; the marginal onM1(S∞) is then of course the law ofµ3.The advantage of this construction over simply regarding the law ofµ3 lies inthe fact that we can in this way extract more information by conditioning, as weshall explain. Note that by compactnessK3 converges at least along (non-random!)subsequences, and we may assume that it actually converges to some measureK.Conditioning this measure onF we obtain a random measureκ on M1(S

∞) (theregular conditional distribution ofK on G given F ). See, e.g., [13]. In a slightlyabusive, but rather obvious notation:K(·|F )[ω] = κ(·)[ω] ⊗ δω(·).

Now the case (1) above corresponds to the situation where the conditionalprobability onG givenF is degenerate, i.e.

κ(·)[ω] = δµ∞[ω](·), a.s. (2.4)

Thus we see that in general evenκ(·)[ω] is a nontrivial measure on the space ofinfinite volume Gibbs measures, this latter object being called the (Aizenman–Wehr) metastate??. What happens is that the asymptotic properties of the Gibbsmeasures as the volume tends to infinity depend in an intrinsic way on the tailsigma field of the disorder variables, and even after all random variables are fixed,some ‘new’ randomness appears that allows only probabilistic statements on theasymptotic Gibbs state.

A TOY EXAMPLE. It may be useful to illustrate the passage from convergence inlaw to the Aizenman–Wehr metastate in a more familiar context, namely the ordi-nary central limit theorem. Let(,F ,P) be a probability space, and letXii∈Nbe a family of i.i.d. centered random variables with variance one; letFn be thesigma algebra generated byX1, . . . , Xn and letF ≡ limn↑∞ Fn. Define the realvalued random variableGn ≡ 1√

n

∑ni=1Xi . We may define the joint lawKn of Gn

and theXi as a probability measure onR⊗. Clearly, this measure converges tosome measureK whose marginal onR will be the standard normal distribution.However, we can say more, namely

TOY-LEMMA 2.1. In the example described above,

κ(·)[ω] = N (0,1), P-a.s. (2.5)

? We write shorthandF for M1(S∞)⊗ F whenever appropriate.

?? It may be interesting to recall the reasons that led Aizenman and Wehr to this construction. Intheir analysis of the effect of quenched disorder on phase transition they required the existence of‘translation-covariant’ states. Such an object could be constructed as weak limits of finite volumestates with, e.g., periodic or translation invariant boundary conditions, provided the correspondingsequences converge almost surely (and not via subsequences with possibly different limits). Theynoted that in a general disordered system this may not be true. The metastate provided a way out ofthis difficulty.

MPAG005.tex; 24/08/1998; 10:00; p.7


Proof.We need to understand what (2.5) means. Letf be a continuous functiononR. We claim that for almost allω,∫

f (x)κ(dx)[ω] =∫

e−x2/2

√2π

f (x)dx. (2.6)

Define the martingalehn ≡∫f (x)K(dx,dω|Fn). We may write

hn = limN↑∞EXn+1 · · ·EXNf

(1√N

N∑i=1

Xi

)

= limN↑∞EXn+1 · · ·EXNf

(1√N − n

N∑i=n+1

Xi

), a.s. (2.7)

=∫

e−x2/2

√2π

f (x)dx,

where we used that for fixedN , 1√N

∑ni=1Xi converges to zero asN ↑ ∞ almost

surely. Thus, for any continuousf , hn is almost surely constant, while limn↑∞ hn =∫f (x)K(dx, dω|F ), by the martingale convergence theorem. This proves the

lemma. 2

The CLT example may inspire the question whether one might not be able toretain more information on the convergence of the random Gibbs state than iskept in the Aizenman–Wehr metastate. The metastate tells us about the probabilitydistribution of the limiting measure, but we have thrown out all information on howfor a givenω, the finite volume measures behave as the volume increases.

Newman and Stein [19, 20] have introduced a possibly more profound conceptof the empirical metastatewhich captures more precisely the asymptotic volumedependence of the Gibbs states in the infinite volume limit. We will briefly discussthis object and elucidate its meaning in the above CLT context. Let3n be anincreasing and absorbing sequence of finite volumes. Define the random empiricalmeasuresκem

N (·)[ω] on (M1(S∞)) by

κemN (·)[ω] ≡

1

N

N∑n=1

δµ3n [ω]. (2.8)

In [20] it was proven that for sufficiently sparse sequences3n and subsequencesNi, it is true that almost surely

limi↑∞ κ

emNi(·)[ω] = κ(·)[ω]. (2.9)

Newman and Stein conjectured that in many situations, the use of sparse sub-sequences would not be necessary to achieve the above convergence. However,

MPAG005.tex; 24/08/1998; 10:00; p.8


Külske [14] has exhibited some simple mean field examples where almost sureconvergence only holds for very sparse (exponentially spaced) subsequences. Healso showed that for more slowly growing sequences convergence in law can beproven in these cases.

TOY EXAMPLE REVISITED. All this is easily understood in our example. WesetGn ≡ 1√

n

∑ni=1Xi. Then the empirical metastate corresponds to

κemN (·)[ω] ≡

1

N

N∑n=1

δGn[ω]. (2.10)

We will prove that the following lemma holds:

TOY-LEMMA 2.2. Let Gn and κemN (·)[ω] be defined above. LetBt , t ∈ [0,1]

denote a standard Brownian motion. Then

(i) The random measuresκemN converge in law to the measureκem= ∫ 1

0 dtδt−1/2Bt ,

(ii) E[κem(·)|F ] = N (0,1). (2.11)

Proof. Our main objective is to prove (i). We will see that quite clearly, thisresult relates to Lemma 2.1 as the CLT to the Invariance Principle, and indeed, itsproof is essentially an immediate consequence of Donsker’s theorem. Donsker’stheorem (see [10] for a formulation in more generality than needed in this chapter)asserts the following: Letηn(t) denote the continuous function on[0,1] that fort = k/n is given by

ηn(k/n) ≡ 1√n

k∑i=1

Xi (2.12)

and that interpolates linearly between these values for all other pointst . Then,ηn(t) converges in distribution to standard Brownian motion in the sense that forany continuous functionalF : C([0,1]) → R it is true thatF(ηn) converges inlaw toF(B). From here the proof of (i) is obvious. We have to prove that for anybounded continuous functionf ,

1

N

N∑n=1

δGn[ω](f ) ≡1

N

N∑n=1

f(ηn(n/N)/

√n/N

)→→

∫ 1

0dtf

(Bt/√t) ≡ ∫ 1

0dtδBt /

√t (f ). (2.13)

To see this, simply define the continuous functionalsF andFN by

F(η) ≡∫ 1

0dtf (η(t)/

√t) (2.14)

MPAG005.tex; 24/08/1998; 10:00; p.9


and

FN(η) ≡ 1

N

N∑n=1

f(η(n/N)/

√n/N

). (2.15)

We have to show that in distributionF(B)− FN(ηN) converges to zero. But

F(B)− FN(ηN) = F(B)− F(ηN)+ F(ηN)− FN(ηN). (2.16)

By the invariance principle,F(B)−F(ηN) converges to zero in distribution whileF(ηN) − FN(ηN) converges to zero sinceFN is the Riemann sum approximationtoF .

To see that (ii) holds, note first that as in the CLT, the Brownian motionBt ismeasurable with respect to the tail sigma-algebra of theXi . Thus

E[κem | F ] = N (0,1). (2.17)2

REMARK. It is easily seen that for sufficiently sparse subsequencesni (e.g.,ni =i!),

1

N

N∑i=1

δGni → N (0,1), a.s. (2.18)

but the weak convergence result contains in a way more information.

SUPERSTATES

In our example we have seen that the empirical metastate converges in distributionto the empirical measure of the stochastic processBt/

√t . It appears natural to think

that the construction of the corresponding continuous time stochastic process itselfis actually the right way to look at the problem also in the context of random Gibbsmeasures, and that the empirical metastate could converge (in law) to the empir-ical measure of this process. To do this we propose the following, yet somewhattentative construction.

We fix again a sequence of finite volumes3n?. We define fort ∈ [0,1]

µt3n[ω] ≡ (t − [tn]/n)µ3[tn]+1[ω] + (1− t + [tn]/n)µ3[tn] [ω] (2.19)

(where as usual[x] denote the smallest integer less than or equal tox). Clearly thisobject is a continuous time stochastic process whose state space isM1(S). We maytry to construct the limiting process

µt [ω] ≡ limn↑∞µ

t3n[ω], (2.20)

? The outcome of our construction will depend on the choice of this sequence. Our philosophyhere would be to choose a natural sequence of volumes for the problem at hand. In mean fieldexamples this would be3n = 1, . . . , n, on a lattice one might choose cubes of sidelengthn.

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where the limit again can in general be expected only in distribution. Obviously,in our CLT example, this is precisely how we construct the Brownian motion inthe invariance principle. We can now, of course, repeat the construction of theAizenman–Wehr metastate on the level of processes. To do this, one must makesome choices for the topological space one wants to work in. A natural possibilityis to consider the spaceC([0,1],M1(S

∞)) of continuous measure valued func-tion equipped with the uniform weak topology?, i.e. we say that a sequence of itselementsλi converges toλ, if and only if, for all continuous functionsf : S∞ → R,

limi→∞

supt∈[0,1]

∣∣λi,t (f )− λt(f )∣∣ = 0. (2.21)

Since the weak topology is metrizable, so is the uniform weak topology andC([0,1],M1(S

∞)) becomes a metric space so we may define the corresponding sigma-algebra generated by the open sets. Taking the tensor product with our old, wecan thus introduce the setM1(C([0,1],M1(S

∞))⊗) of probability measures onthis space tensored with. Then we define the elements

Kn ∈M1(C([0,1],M1(S

∞))⊗)

whose marginals on areP and whose conditional measure onC ([0,1],M1(S∞)),

givenF are the Dirac measure on the measure valued functionµ3[tn] [ω], t ∈ [0,1].Convergence, and even the existence of limit points for this sequence of measuresis now no longer a trivial matter. The problem of the existence of limit pointscan be circumvented by using a weaker notion of convergence, e.g., that of theconvergence of any finite dimensional marginal. Otherwise, some tightness con-dition is needed [10], e.g., we must check that for any continuous functionf ,sup|s−t |6δ |µt3n(f ) − µs3n(f )| converges to zero in probability, uniformly inN ,asδ ↓ 0.??

We can always hope that the limit asn goes to infinity if Kn exists. If thelimit, K, exists, we can again consider its conditional distribution givenF , andthe resulting object is the functional analog of the Aizenman–Wehr metastate. (Wefeel tempted to call this object the ‘superstate’. Note that the marginal distributionof the superstate ‘at timet = 1’ is the Aizenman–Wehr metastate, and the law ofthe empirical distribution of the underlying process is the empirical metastate.) The‘superstate’ contains an enormous amount of information on the asymptotic vol-ume dependence of the random Gibbs measures; on the other hand, its constructionin any explicit form is generally hardly feasible.

Finally, we want to stress that the superstate will normally depend on the choiceof the basic sequences3n used in its construction. This feature is already present? Another possibility would be a measure valued version of the spaceD([0,1],M1(S)) of mea-

sure valued Cadlag functions. The choice depends essentially on the properties we expect from thelimiting process (i.e. continuous sample paths or not).?? There are pathological examples in which we would not expect such a result to be true. An

example is the ‘highly disordered spin glass model’ of Newman and Stein [21]. Of course, tightnessmay also be destroyed by choosing very rapidly growing sequences of volumes3n.

MPAG005.tex; 24/08/1998; 10:00; p.11


in the empirical metastate. In particular, sequences growing extremely fast willgive different results than slowly increasing sequences. On the other hand, the veryprecise choice of the sequences should not be important. A natural choice wouldappear to us sequences of cubes of sidelengthn, or, in mean field models, simplythe sequence of volumes of sizen.

BOUNDARY CONDITIONS, EXTERNAL FIELDS, CONDITIONING

In the discussion of Newman and Stein, metastates are usually constructed withsimple boundary conditions such as periodic or ‘free’ ones. They emphasize thefeature of the ‘selection of the states’ by the disorder in a given volume withoutany bias through boundary conditions or symmetry breaking fields. Our point ofview is somewhat different in this respect in that we think that the idea to applyspecial boundary conditions or, in mean field models, symmetry breaking terms, toimprove convergence properties, is still to some extent useful, the aim ideally beingto achieve the situation (1). Our only restriction in this is really that our procedureshall have somepredictive power, that is, it should give information of the approxi-mate form of a finite volume Gibbs state. This excludes any construction involvingsubsequences via compactness arguments. Thus we are interested to know to whatextent it is possible to reduce the ‘choice’ of available states for the randomness toselect from, to smaller subsets and to classify the minimal possible subsets (whichthen somehow play the rôle ofextremal states). In fact, in the examples consideredin [14] it would be possible to reduce the size of such subsets to one, while inthe example of the present paper, we shall see that this isimpossible. We have todiscuss this point carefully.

While in short range lattice models the DLR construction gives a clear frame-work how the class of infinite volume Gibbs measures is to be defined, in meanfield models this situation is somewhat ambiguous and needs discussion.

If the infinite volume Gibbs measure is unique (for givenω), quasi by definition,(1) must hold. So our problems arise from non-uniqueness. Hence the followingrecipe: modifyµ3 in such a way that uniqueness holds, while otherwise perturbingit in a minimal way. Two procedures suggest themselves:

(i) tilting, and(ii) conditioning.

Tilting consists of the addition of asymmetry breakingterm to the Hamiltonianwhose strength is taken to zero. Mostly, this term is takenlinear so that it has thenatural interpretation of amagnetic field. More precisely, define

µh3,ε[ω](·) ≡

µ3[ω](·e−βε∑i∈3 hiσi )

µ3[ω](e−βε∑i∈3 hiσi )

. (2.22)

Herehi is some sequence of numbers that in general will have to be allowed todepend onω if anything is to be gained. One may also allow them to depend

MPAG005.tex; 24/08/1998; 10:00; p.12


on 3 explicitly, if so desired. From a physical point of view we might wish toadd further conditions, like some locality of theω-dependence; in principle thereshould be a way of writing them down in some explicit way. We should stress thattilting by linear functions is not always satisfactory, as some states that one mightwish to obtain are lost; an example is the generalized Curie–Weiss model withHamiltonianHN(σ ) = −N

4 [mN(σ )]4 at the critical point. There, the free energyhas three degenerate absolute minima at−m∗,0, and+m∗, and while we mightwant to think of three coexisting phases, only the measures centered at±m∗ canbe extracted by the above method. Of course this can be remedied by allowingarbitrary perturbationh(m) with the only condition that‖h‖∞ tends to zero at theend.

By conditioning we always mean conditioning the macroscopic variables to bein some setA. This appears natural since, in lattice models, extremal measurescan always be extracted from arbitrary DLR measures by conditioning on eventsin the tail sigma fields; the macroscopic variables are measurable with respect tothe tail sigma fields. Of course only conditioning on events that do not have a toosmall probability will be reasonable. Without going into too much of a motivatingdiscussion, we will adopt the following conventions. LetA be an event in the sigmaalgebra generated by the macroscopic function. Put

f3,β(A) = − 1

β|3| lnµ3,β[ω](A). (2.23)

We callA admissible for conditioning if and only if

lim|3|↑∞ f3,β[ω](A) = 0. (2.24)

We call A minimal if it cannot be decomposed into two admissible subsets. Inanalogy with (2.22) we then define

µA3,β [ω](·) ≡ µ3,β [ω](·|A). (2.25)

We define the set of all limiting Gibbs measures to be the set of limit points of mea-suresµA

3,β with admissible setsA. ChoosingA minimal, we improve our chancesof obtaining convergent sequences and the resulting limits are serious candidatesfor extremallimiting Gibbs measures, but we stress that this is not guaranteed tosucceed, as will become manifest in our examples. This will not mean that addingsuch conditioning is not going to be useful. It is in fact, as it will reduce the disorderin the metastate and may in general allow to construct variousdifferentmetastatesin the case of phase transitions. The point to be understood here is that within thegeneral framework outlined above, we should consider two different notions ofuniqueness:

(a) Strong uniquenessmeaning that for almost allω there is only one limit pointµ∞[ω], and

MPAG005.tex; 24/08/1998; 10:00; p.13


(b) Weak uniqueness? meaning that there is a unique metastate, in the sense thatfor any choice ofA, the metastate constructed taking the infinite volume limitwith the measuresµA

3,ε is the same.In fact, it may happen that the addition of a symmetry breaking term or condi-

tioning does not lead to strong uniqueness. Rather, what may be true is that sucha field selects a subset of the states, but which of them the state at a given volumeresembles can depend on the volume in a complicated way.

If weak uniqueness does not hold, one has a non-trivial set of metastates.It is quite clear that a sufficiently general tilting approach is equivalent to the

conditioning approach; we prefer for technical reasons to use the conditioning inthe present paper. We also note that by dropping condition (2.24) one can enlargethe class of limiting measures obtainable to includemetastable states, which inmany applications, in particular in the context of dynamics, are also relevant.

3. Properties of the Induced Measures

In this section we collect a number of results on the distribution of the overlapparameters in the Hopfield model that were obtained in some of our previous papers[3, 4, 5]. We cite these results mostly from [5] where they were stated in the mostsuitable form for our present purposes and we refer the reader to that paper for theproofs.

We recall some notation. Letm∗(β) be the largest solution of the mean fieldequationm = tanh(βm). Note thatm∗(β) is strictly positive for allβ > 1,limβ↑∞m∗(β) = 1, limβ↓1 (m

∗(β))23(β−1) = 1 andm∗(β) = 0 if β 6 1. Denoting

by eµ theµth unit vector of the canonical basis ofRM we set, for all(µ, s) ∈1, . . . ,M(N) × −1,1,

m(µ,s) ≡ sm∗(β)eµ, (3.1)

and for anyρ > 0 we define the balls

B(µ,s)ρ ≡ x ∈ RM∣∣‖x −m(µ,s)‖2 6 ρ. (3.2)

For any pair of indices(µ, s) and anyρ > 0 we define the conditional measures

µ(µ,s)

N,β,ρ[ω](A) ≡ µN,β [ω](A | B(µ,s)ρ ), A ∈ B(−1,1N) (3.3)

and the corresponding induced measures

Q(µ,s)

N,β,ρ[ω](A) ≡ QN,β[ω](A | B(µ,s)ρ ), A ∈ B(RM(N)). (3.4)

The point here is that forρ > c√α

m∗(β) , the setsB(µ,s)ρ are admissible in the sense ofthe last section.? Maybe the notion of meta-uniqueness would be more appropriate.

MPAG005.tex; 24/08/1998; 10:00; p.14


It will be extremely useful to introduce the Hubbard–Stratonovich transformedmeasuresQN,β [ω]which are nothing but the convolutions of the induced measureswith a Gaussian measure of mean zero and variance 1/βN , i.e.

QN,β [ω] ≡ QN,β [ω] ?N

(0,

1I

βN

). (3.5)

We recall from [6] thatQN,β[ω] is absolutely continuous w.r.t. Lebesgue measureonRM with density given by

QN,β [ω](dMx)dMx

= e−βN8N,β [ω](x)

ZN,β [ω] , (3.6)

where

8N,β [ω](x) ≡ ‖x‖22

2− 1

βN

N∑i=1

ln cosh(β(ξi, x)). (3.7)

Similarly we define the conditional Hubbard–Stratonovich transformed measures

Q(µ,s)

N,β,ρ[ω](A) ≡ QN,β[ω](A | B(µ,s)ρ ), A ∈ B(RM(N)). (3.8)

We will need to consider the Laplace transforms of these measures which we willdenote by?

L(µ,s)

N,β,ρ[ω](t) ≡∫

e(t,x) dQ(µ,s)

N,β,ρ[ω](x), t ∈ RM(N), (3.9)

and

L(µ,s)N,β,ρ[ω](t) ≡

∫e(t,x) dQ(µ,s)

N,β,ρ[ω](x), t ∈ RM(N). (3.10)

The following is a simple adaptation of Proposition 2.1 of [5] to these notations.

PROPOSITION 3.1. Assume thatβ > 1. There exist finite positive constantsc0, c ≡ c(β), c ≡ c(β) such that, with probability one, for all but a finite numberof indicesN , if ρ satisfies

1

2m∗ > ρ > c

√α

m∗(β)(3.11)

then, for allt with ‖t‖2/√N <∞,

(i)L(µ,s)

β,N,ρ[ω](t)(1− e−cM

)6 e−

12Nβ ‖t‖22L(µ,s)

β,N,ρ[ω](t)6 e−cM +L(µ,s)

β,N,ρ(t)(1+ e−cM

), (3.12)

? This notation is slightly different from the one used in [5].

MPAG005.tex; 24/08/1998; 10:00; p.15


(ii) for anyρ, ρ satisfying(3.11)

L(µ,s)

β,N,ρ[ω](t)(1− e−cM

)6 L(µ,s)

β,N,ρ[ω](t)6 e−cM + L(µ,s)

β,N,ρ[ω](t)(1+ e−cM

), (3.13)

(iii) for anyρ, ρ satisfying(3.11)∣∣∣∣( ∫ dQ(µ,s)N,β,ρ[ω](m)m−

∫dQ(µ,s)

N,β,ρ[ω](z)z, t)∣∣∣∣

6 ‖t‖2 e−cM. (3.14)

A closely related result that we will need is also an adaptation of estimates from[5], i.e. it is obtained combining Lemmata 3.2 and 3.4 of that paper.

LEMMA 3.2. There existsγa > 0, such that for allβ > 1 and√α < γa(m

∗)2,if c0

√α

m∗ < ρ < m∗/√

2 then, with probability one, for all but a finite numberof indicesN , for all µ ∈ 1, . . . ,M(N), s ∈ −1,1, for all b > 0 such thatρ + b < √2m∗,

16Qβ,N(B

(µ,s)

ρ+b )

Qβ,N(B(µ,s)ρ )

6 1+ e−c2βM, (3.15)

where0< c2 <∞ is a numerical constant.

We finally recall our result on local convexity of the function8.

THEOREM 3.3. Assume that1< β <∞. If the parametersα, β, ρ are such thatfor ε > 0,

infτ

(β(1− tanh2(βm∗(1− τ)))(1+ 3

√α)+

+2β tanh2(βm∗(1− τ))0(α, τm∗/ρ)) 6 1− ε. (3.16)

Then with probability one for all but a finite number of indicesN ,8N,β [ω](m∗e1+v) is a twice differentiable and strictly convex function ofv on the setv : ‖v‖2 6ρ, and

λmin(∇28N,β [ω](m∗e1+ v)) > ε (3.17)

on this set.

REMARK. This theorem was first obtained in [3], the above form is cited andproven in [4]. Withρ chosen asρ = c

√α

m∗ , the condition (3.16) means (i) Forβ

close to 1:√α

(m∗)2 small and, (ii) Forβ large:α 6 cβ−1. The condition onα for

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largeβ seems unsatisfactory, but one may easily convince oneself that it cannot besubstantially improved.

4. Brascamp–Lieb Inequalities

A basic tool of our analysis are the so-called Brascamp–Lieb inequalities [8]. Infact, we need such inequalities in a slightly different setting than they are presentedin the literature, namely for measures with bounded support on some domainD ⊂RM . Our derivation follows the one given in [9] (see also [12]), and is in this contextalmost obvious.

Let D ⊂ RM be a bounded connected domain. LetV ∈ C2(D) be a twicecontinuously differentiable function onD, let∇2V denote its Hessian matrix andassume that, for allx ∈ D,∇2V (x) > c > 0 (where we say that a matrixA > c, ifand only if for allv ∈ RM , (v,Av) > c(v, v)). We define the probability measureν on (D,B(D)) by

ν(δx) ≡ e−NV (x) dMx∫D

e−NV (x) dMx. (4.1)

Our central result is:

THEOREM 4.1. Let ν be the probability measure defined above. Assume thatf, g ∈ C1(D), and assume that(w.r.g.)

∫D

dν(x)g(x) = ∫D

dν(x)f (x) = 0.Then∣∣∣∣ ∫

D

dν(x)f (x)g(x)

∣∣∣∣ 6 1

cN

∫D

dν(x)∥∥∇f (x)‖2‖∇g(x)∥∥2+

+ 1

cN

∫∂D|g(x)|‖∇f (x)‖2 e−NV (x) dM−1x∫

De−NV (x) dMx

, (4.2)

wheredM−1x is the Lebesgue measure on∂D.Proof. We consider the Hilbert spaceL2(D,RM, ν) of RM valued functions

onD with scalar product〈F,G〉 ≡ ∫D

dν(x)(F (x),G(x)). Let∇ be the gradientoperator onD defined with a domain of all boundedC1-function that vanish on∂D.Let∇∗ denote its adjoint. Note that∇∗ = −eNV (x)∇ e−NV (x) = −∇ +N(∇V (x)).One easily verifies by partial integration that on this domain the operator∇∇∗ ≡∇ eNV (x)∇ e−NV (x) = ∇∗∇+N∇2V (x) is symmetric and∇∗∇ > 0, so that by ourhypothesis,∇∇∗ > cN > 0. As a consequence,∇∇∗ has a self-adjoint extensionwhose inverse(∇∇∗)−1 exists on allL2(D,RM, ν) and is bounded in norm by(cN)−1.

As a consequence of the above, for anyf ∈ C1(D), we can uniquely solve thedifferential equation

∇∇∗∇u = ∇f (4.3)

MPAG005.tex; 24/08/1998; 10:00; p.17


for ∇u. Now note that (4.3) implies that∇∗∇u = f + k, wherek is a constant?.Hence for real valuedf andg as in the statement of the theorem,∫

D

dν(x)(∇g(x),∇u(x))

=∫D

dν(x)eNV (x) div(e−NV (x)g∇u(x))+ ∫

D

dν(x)g(x)∇∗∇u(x)

= 1

Z

∫D

dMx div(e−NV (x)g∇u(x))+ ∫

D

dν(x)g(x)f (x), (4.4)

whereZ ≡ ∫D

dMx e−NV (x). Therefore, taking into account that

∇u = (∇∇∗)−1∇f,∣∣∣∣ ∫D

dν(x)g(x)f (x)

∣∣∣∣ 6 ∣∣∣∣ ∫D

dν(x)(∇g(x), (∇∇∗)−1∇f (x)) ∣∣∣∣+

+ 1

Z

∣∣∣∣ ∫D

dMx div(e−NV (x)g∇u(x))∣∣∣∣

6 1

cN

∫D

dν(x)‖∇g(x)‖2‖∇f (x)‖2 + (4.5)

+ 1

cNZ

∫∂D

|g(x)| ‖∇f (x)‖2 e−NV (x) dM−1x.

Note that in the second term we used the Gauss–Green formula to convert theintegral over a divergence into a surface integral. This concludes the proof.2

REMARK. As is obvious from the proof above and as was pointed out in [9], onecan replace the bound on the lowest eigenvalue of the Hessian ofV by a boundon the lowest eigenvalue of the operator∇∇∗. So far we have not seen how toget a better bound on this eigenvalue in our situation, but it may well be that thisobservation can be a clue to an improvement of our results.

The typical situation where we want to use Theorem 4.1 is the following: Sup-pose we are given a measure like (4.1) but not onD, but on some bigger domain.We may be able to establish the lower bound on∇2V not everywhere, but only onthe smaller domainD, but such that the measure is essentially concentrated onD

anyhow. It is then likely that we can also estimate away the boundary term in (4.2),either becauseV (x) will be large on∂D, or because∂D will be very small (orboth). We then have essentially the Brascamp–Lieb inequalities at our disposal.

We mention the following corollary which shows that the Brascamp–Lieb in-equalities give rise to concentration inequalities under certain conditions.

COROLLARY 4.2. Let ν be as in Lemma4.3. Assume thatf ∈ C1(D) and thatmoreoverVt(x) ≡ V (x) − tf (x)/N for t ∈ [0,1] is still strictly convex and? Observe that this is only true becauseD is connected. ForD consisting of several connected

components the theorem is obviously false.

MPAG005.tex; 24/08/1998; 10:00; p.18


λmin(∇2Vt) > c′ > 0. Then

0 6 ln∫D

dν(x)ef (x) −∫D

dν(x)f (x) 6 1

2c′Nsupt∈[0,1]

∫D

dνt (x)‖∇f ‖22 +

+ supt∈[0,1]

1

c′N

∫∂D|g(x)|‖∇f (x)‖2 e−NVt (x) dM−1x∫

De−NVt (x) dMx

, (4.6)

whereνt is the corresponding measure withV replaced byVt .Proof.Note that

lnEV ef = EV f +∫ 1

0ds∫ s

0ds′EV[es′f(f − EV es

′f fEV es′f

)2]EV es ′f

= EV f +∫ 1

0ds∫ s

0ds′EVs′ (f − EVs′f )2, (4.7)

where by assumptionVs(x) has the same properties asV itself. Thus using (4.2)gives (4.6). 2

REMARK. We would like to note that a concentration estimate like Corollary4.2 can also be derived under slightly different hypothesis onf using logarithmicSobolev inequalities (see [26]) which hold under the same hypothesis as Theorem4.1, and which in fact can be derived as a special case usingf = h2 andg = ln h2

in Theorem 4.1.

In the situations where we will apply the Brascamp–Lieb inequalities, the cor-rection terms due to the finite domainD will be totally irrelevant. This followsfrom the following simple observation.

LEMMA 4.3. LetBρ denote the ball of radiusρ centered at the origin. Assumethat for all x ∈ D, d > ∇2V (x) > c > 0. If x∗ denotes the unique minimum ofV ,assume that‖x∗‖2 6 ρ/2. Then there exists a constantK < ∞ (depending onlyonc andd) such that ifρ > K

√M/N , then forN large enough∫

∂De−NV (x) dM−1x∫

De−NV (x) dMx

6 e−ρ2N/K. (4.8)

The proof of this lemma is elementary and will be left to the reader.

5. The Convergence of the Gibbs Measures

After these preliminaries we can now come to the central part of the paper, namelythe study of the marginal distributions of the Gibbs measuresµ

(µ,s)

N,β,ρ . Without lossof generality it suffices to consider the case(µ, s) = (1,1), of course. Let us fix

MPAG005.tex; 24/08/1998; 10:00; p.19


I ⊂ N arbitrary but finite. We assume that3 ⊃ I , and for notational simplicity weput |3| = N + |I |. We are interested in the probabilities

µ(1,1)3,β,ρ[ω](σI = sI ) ≡

Eσ3\I e12β|3|‖m3(sI ,σ3\I )‖221Im3(sI ,σ3\I )∈B(1,1)ρ

EσIEσ3\I e12β|3|‖m3(σI ,σ3\I )‖221Im3(sI ,σ3\I )∈B(1,1)ρ

. (5.1)

Note that‖mI(σ )‖2 6√M. Now we can write

m3(σ) = N

|3|m3\I (σ )+|I ||3|mI(σ ). (5.2)

Then

1Im3(sI ,σ3\I )∈B(1,1)ρ 6 1Im3\I (σ )∈B(1,1)ρ+ ,

1Im3(sI ,σ3\I )∈B(1,1)ρ > 1Im3\I (σ )∈B(1,1)ρ− , (5.3)

whereρ± ≡ ρ ±√M |I |N

. Settingβ ′ ≡ N|3|β, this allows us to write

µ(1,1)3,β,ρ[ω](σI = sI )

6

∫B(1,1)ρ+

dQ3\I,β ′(m)eβ′|I |(mI (sI ),m) eβ

|I |22|3| ‖mI (sI )‖22

2|I |EσI∫B(1,1)ρ−

dQ3\I,β ′(m)eβ ′|I |(mI (σI ),m) eβ|I |22|3| ‖mI (σI )‖22

×

×∫B(1,1)ρ−

dQ3\I,β ′(m)∫B(1,1)ρ+

dQ3\I,β ′(m)(5.4)

6 L3/I,β,ρ+[ω](β ′|I |mI(sI ))eβ|I |22|3| ‖mI (sI )‖22

2|I |EσIL3/I,β,ρ−[ω](β ′|I |mI(σI ))eβ|I |22|3| ‖mI (σI )‖22

Q3\I,β ′(B(1,1)ρ+

)Q3\I,β ′

(B(1,1)ρ−

)and

µ(1,1)3,β,ρ[ω] (σI = sI )

>

∫B(1,1)ρ−

dQ3\I,β ′(m)eβ′|I |(mI (sI ),m) eβ

|I |22|3| ‖mI (sI )‖22

2|I |EσI∫B(1,1)ρ+

dQ3\I,β ′(m)eβ ′|I |(mI (σI ),m) eβ|I |22|3| ‖mI (σI )‖22

×

× Q3\I,β ′(B(1,1)ρ−

)Q3\I,β ′

(B(1,1)ρ+

) (5.5)

= L3/I,β,ρ−[ω](β ′|I |mI(sI )) eβ|I |22|3| ‖mI (sI )‖22

2|I |EσIL3/I,β,ρ+[ω](β ′|I |mI(σI ))eβ|I |22|3| ‖mI (σI )‖22

Q3\I,β ′(B(1,1)ρ−

)Q3\I,β ′

(B(1,1)ρ+

) .

MPAG005.tex; 24/08/1998; 10:00; p.20


Now the term |I |2

N‖mI(s)‖22 is, up to a constant that is independent of thesi ,

irrelevantly small. More precisely, we have

LEMMA 5.1. There exist∞ > C, c > 0 such that for allI ,M, and for allx > 0,

P[

supσI∈−1,1I

|I |2N

∣∣∣∣‖mI(s)‖22 − M|I |N

∣∣∣∣ > |I |MN(√ |I |

N+ x

)]6 C exp

(−cM(√1+ x − 1)2). (5.6)

Proof. This lemma is a direct consequence of estimates on the norm of therandom matrices obtained, e.g., in Theorem 4.1 of [4]. 2

Together with Proposition 3.1 and Lemma 3.2, we can now extract the desiredrepresentation for our probabilities.

LEMMA 5.2. For all β > 1 and√α < γa(m

∗)2, if c0

√α

m∗ < ρ < m∗/√

2then, with probability one, for all but a finite number of indicesN , for all µ ∈1, . . . ,M(N), s ∈ −1,1,(i)

µ(1,1)3,β,ρ[ω](σI = sI ) =

L(1,1)3/I,β,ρ[ω](β ′|I |mI(sI ))

2|I |EσIL(1,1)3/I,β,ρ[ω](β ′|I |mI(σI ))

+

+O(N−1/4) (5.7)

and alternatively

(ii)

µ(1,1)3,β,ρ[ω](σI = sI ) =

L(1,1)3/I,β,ρ[ω](β ′|I |mI(sI ))

2|I |EσI L(1,1)3/I,β,ρ[ω](β ′|I |mI(σI ))

+

+O(e−O(M)

). (5.8)

We leave the details of the proof to the reader. We see that the computation of themarginal distribution of the Gibbs measures requires nothing but the computationof the Laplace transforms of the induced measures or its Hubbard–Stratonovichtransform at the random pointst = ∑i∈I siξi. Alternatively, these can be seen asthe Laplace transforms of the distribution of the random variables(ξi,m).

Now it is physically very natural that the law of the random variables(ξi,m)

should determine the Gibbs measures completely. The point is that in a mean fieldmodel, the distribution of the spins in a finite setI is determined entirely in termsof the effective mean fields produced by the rest of the system that act on the spinsσi. These fields are precisely the(ξi,m). In a ‘normal’ mean field situation, themean fields are constant almost surely with respect to the Gibbs measure. In theHopfield model with subextensively many patterns, this will also be true, asm

MPAG005.tex; 24/08/1998; 10:00; p.21


will be concentrated near one of the valuesm∗eµ (see [6]). In that case(ξi,m)will depend only in a local and very explicit form on the disorder, and the Gibbsmeasures will inherit this property. In a more general situation, the local mean fieldsmay have a more complicated distribution, in particular they may not be constantunder the Gibbs measure, and the question is how to determine this. The approachof the cavity method(see, e.g., [15]) as carried out by Talagrand [24] consists ofderiving this distribution by induction over the volume. [23] also followed this ap-proach, using however the assumption of ‘self-averaging’ of the order parameter tocontrol errors. Our approach consists of using the detailed knowledge obtained onthe measuresQ, and in particular the local convexity to determine a priori the formof the distribution; induction will then only be used to determine the remaining fewparameters.

Let us begin with some general preparatory steps which will not yet requirespecial properties of our measures. To simplify the notation, we introduce thefollowing abbreviations:

We write E8N for the expectation with respect to the measuresQ3\I,β,h[ω]conditioned onBρ and we setZ ≡ Z−E8NZ. We will writeEξI for the expectationwith respect to the family of random variablesξµi , i ∈ I , µ = 1, . . . ,M.

The first step in the computation of our Laplace transform consists of centering,i.e. we write

E8Ne∑i∈I βsi(ξi ,Z) = e

∑i∈I βsi(ξi ,E8N Z)E8N e

∑i∈I βsi(ξi ,Z). (5.9)

While the first factor will be entirely responsible for the distribution of the spins,our main efforts have to go into controlling the second. To do this we will useheavily the fact, established first in [3], that onB(1,1)ρ the function8 is convex withprobability close to one. This allows us to exploit the Brascamp–Lieb inequalitiesin the form given in Section 3. The advantage of this procedure is that it allows usto identify immediately the leading terms and to get a priori estimates of the errors.This is to be contrasted to the much more involved procedure of Talagrand [24]who controls the errors by induction.

GENERAL ASSUMPTION

For the remainder of this paper we will always assume that the parametersα andβ of our model are such that the hypotheses of Proposition 3.1 and Theorem 3.3are satisfied. All lemmata, propositions and theorems are valid under this provisiononly.

LEMMA 5.3. Under our general assumption,

(i) EξIE8N e∑i∈I βsi(ξi ,Z) = e

β2

2∑i∈I s2

i E8N ‖Z‖22 × eO(1/(εN)), (5.10)

(ii) There is a finite constantC such that

MPAG005.tex; 24/08/1998; 10:00; p.22


EξI

[ln

(E8N e

∑i∈I βsi(ξi ,Z)

EξIE8N e∑i∈I βsi(ξi ,Z)

)]2

6 C

N. (5.11)

REMARK. The immediate consequence of this lemma is the observation that thefamily of random variables

(ξi, Z)

i∈I is asymptotically close to a family of i.i.d.

centered Gaussian random variables with varianceUN ≡ E8N‖Z‖22. UN will beseen to be one of the essential parameters that we will need to control by induction.Note that for the moment, we cannot say whether the law of the(ξi, Z) convergesin any sense, as it is not a priori clear whetherUN will converge asN ↑ ∞,although this would be a natural guess. Note that as far as the computation ofthe marginal probabilities of the Gibbs measures is concerned, this question is,however, completely irrelevant, in as far as this term is an even function of thesi.

REMARK. It follows from Lemma 5.3 that

lnE8N exp(∑i∈Iβsi (ξi, Z)

)= β2

2|I |E8N‖Z‖22+O

(1

εN

)+ RN, (5.12)

where

EξI R2N 6

C

N. (5.13)

Proof. The proof of this lemma relies heavily on the use of the Brascamp–Lieb inequalities, Theorem 4.1, which are applicable due to our assumptions andTheorem 3.3. It was given in [3] forI being a single site, and we repeat the mainsteps. First note that

EξIE8N e∑i∈I βsi(ξi ,Z) 6 E8N e

β2

2

∑i∈I s2

i ‖Z‖22,

EξIE8N e∑i∈I βsi(ξi ,Z) > E8N e

β2

2

∑i∈I s2

i ‖Z‖22− β4

4

∑i∈I s4

i ‖Z‖44. (5.14)

Note that if the smallest eigenvalue of∇28 > ε, then the Brascamp–Lieb inequal-ities Theorem 4.1 yield

E8N‖Z‖22 6M

εN+O(e−ρ

2N/K) (5.15)

and by iterated application

E8N‖Z‖44 6 4M

ε2N2+O(e−ρ

2N/K). (5.16)

In the bounds (5.14) we now use Corollary 4.2 withf given by β2|I |/2‖Z‖22,respectively byβ2|I |/2‖Z‖22 − β4|I |/4‖Z‖44 to first move the expectation into theexponent, and then (5.15) and (5.16) (applied to the slightly modified measuresE8N−tf/N , which still retain the same convexity properties) to the terms in theexponent. This gives (5.10).

MPAG005.tex; 24/08/1998; 10:00; p.23


By very similar computations one shows first that

E(E8N e

∑i∈I βsi(ξi ,Z) − EξIE8N e

∑i∈I βsi(ξi ,Z)

)6 C

N. (5.17)

Moreover, using again Corollary 4.2, one obtains that (on the subspace whereconvexity holds)

e−β2|I |/2αε 6 E8N e

∑i∈I βsi(ξi ,Z) 6 e+β

2|I |/2αε . (5.18)

These bounds, together with the obvious Lipshitz continuity of the logarithm awayfrom zero yield (5.11). 2

REMARK. The above proof follows ideas of the proof of Lemma 4.1 in [24]. Themain difference is the systematic use of the Brascamp–Lieb inequalities that allowsus to avoid the appearance of uncontrolled error terms.

We now turn to the mean values of the random variables(ξi,E8NZ). These areobviously random variables with mean value zero and variance‖E8NZ‖2. More-over, the variables(ξi,E8NZ) and (ξj ,E8NZ) are uncorrelated fori 6= j . NowE8NZ has one macroscopic component, namely the first one, while all others areexpected to be small. It is thus natural to expect that these variables will actuallyconverge to a sum of a Bernoulli variableξ1

i E8NZ1 plus independent Gaussianswith varianceTN ≡ ∑M

µ=2[E8NZµ]2, but it is far from trivial to prove this. Itrequires in particular at least to show thatTN converges.

We will first prove the following proposition:

PROPOSITION 5.4. In addition to our general assumption, assume thatlim infN↑∞N1/4TN = +∞, a.s. For i ∈ I , setXi(N) ≡ 1√

TN

∑µ=2 ξ

µi E8NZµ.

Then this family converges to a family of i.i.d. standard normal random variables.

REMARK. The assumption on the divergence ofN1/4TN is harmless. We will seelater that it is certainly verified provided lim infN↑∞N1/8ETN = +∞. Recall thatour final goal is to approximate (in law)

∑Mµ=2 ξ

µ

i E8NZµ by√TNgi , wheregi

is Gaussian. So ifTN 6 N−1/4, then∑M

µ=2 ξµi E8NZµ is close to zero (in law)

anyway, as is√TNgi, and no harm is done if we exchange the two. We will see that

this situation only arises in fact ifM/N tends to zero rapidly, in which case all thismachinery is not needed.

Proof. To prove such a result requires essentially to show thatE8NZµ for allµ > 2 tend to zero asN ↑ ∞. We note first that by symmetry, for allµ > 2,EE8NZµ = EE8NZ2. On the other hand,

M∑µ=2

[EE8NZµ]2 6 EM∑µ=2

[E8NZµ]2 6 ρ2 (5.19)

so that|EE8NZµ| 6 ρM−1/2.

MPAG005.tex; 24/08/1998; 10:00; p.24


To derive from this a probabilistic bound onE8NZµ itself we will use concen-tration of measure estimates. To do so we need the following lemma:

LEMMA 5.5. Assume thatf (x) is a random function defined on some open neigh-bourhoodU ⊂ R. Assume thatf verifies for all x ∈ U that for all 0 6 r

6 1,

P[|f (x)− Ef (x)| > r] 6 c exp

(−Nr

2

c

)(5.20)

and that, at least with probability1−p, |f ′(x)| 6 C, |f ′′(x)| 6 C <∞ both holduniformly inU . Then, for any0< ζ 6 1/2, and for any0< δ < Nζ/2,

P[|f ′(x)− Ef ′(x)| > δN−ζ/2] 6 32C2

δ2Nζ exp

(−δ

4N1−2ζ

256c

)+ p. (5.21)

Proof. Let us assume that|U | 6 1. We may first assume that the boundednessconditions for the derivatives off hold uniformly; by standard arguments oneshows that if they only hold with probability 1− p, the effect is nothing morethan the final summandp in (5.21). The first step in the proof consists of show-ing that (5.20) together with the boundedness of the derivative off implies thatf (x)− Ef (x) is uniformly small. To see this introduce a grid of spacingε, i.e. letUε = U ∩ εZ. Clearly

P[supx∈U|f (x)− Ef (x)| > r

]6 P

[supx∈Uε|f (x)− Ef (x)| +

+ supx,y: |x−y|6ε

|f (x)− f (y)| + |Ef (x)− Ef (y)| > r]

6 P[

supx∈Uε|f (x)− Ef (x)| > r − 2Cε

]6 ε−1P

[|f (x)− Ef (x)| > r − 2Cε]. (5.22)

If we chooseε = r4C , this yields

P[supx∈U|f (x)− Ef (x)| > r

]6 4C

rexp

(−Nr

2

4c

). (5.23)

Next we show thatif supx∈U |f (x) − g(x)| 6 r for two functionsf , g withbounded second derivative, then

|f ′(x)− g′(x)| 6 √8Cr. (5.24)

For notice that∣∣∣∣1ε [f (x + ε)− f (x)] − f ′(x)∣∣∣∣ 6 ε

2sup

x6y6x+εf ′′(y) 6 Cε

2(5.25)

MPAG005.tex; 24/08/1998; 10:00; p.25


so that

|f ′(x)− g′(x)| 6 1

ε|f (x + ε)− g(x + ε)− f (x)+ g(x)| + Cε

6 2r

ε+ Cε. (5.26)

Choosing the optimalε = √2r/C gives (5.24). It suffices to combine (5.24) with(5.23) to get

P[|f ′(x)− Ef ′(x)| > √8rC

]6 4C

rexp

(−Nr

2

4c

). (5.27)

Settingr = δ2

CNζ, we arrive at (5.21). 2

We will now use Lemma 5.5 to controlE8NZµ. We define

f (x) = 1

βNln∫B(1,1)ρ

dMz eβNxzµ e−βN8β,N,M (z) (5.28)

and denote byE8N,x the corresponding modified expectation. As has by now beenshown many times [24, 3],f (x) verifies (5.20). Moreover,f ′(x) = E8N,xZµ and

f ′′(x) = βNE8N ,x(Zµ − E8N,xZµ)2. (5.29)

Of course the addition of the linear term to8 does not change its second derivative,so that we can apply the Brascamp–Lieb inequalities also to the measureE8N,x .This shows that

E8N,x(Zµ − E8N,xZµ

)2 6 1

εNβ(5.30)

which means thatf (x) has a second derivative bounded byc = 1/ε.This gives

COROLLARY 5.6. There are finite positive constantsc, C such that, for any0<ζ 6 1/2, for anyµ,

P[|E8NZµ − EE8NZµ| > N−ζ/2] 6 CNζ exp

(−N

1−2ζ

c

). (5.31)

We are now ready to conclude the proof of our proposition. We may choose,e.g.,ζ = 1/4 and denote byN the subset of where, for allµ, |E8NZµ −EE8NZµ| 6 N−1/8. ThenP[cN ] 6 O(e−N1/2

).We will prove the proposition by showing convergence of the characteristic

function to that of product standard normal distributions, i.e. we show that for any

MPAG005.tex; 24/08/1998; 10:00; p.26


t ∈ RI , E∏j∈I eitjXj (N) converges to∏j∈I e−

12 t

2j . We have

E∏j∈I

eitj Xj (N)

= EξIc[1INEξI ei

∑j∈I tjXj (N) + 1IcNEξI ei

∑j∈I tjXj (N)

]= EξIc

[1IN

∏µ>2

∏j∈I

cos

(tj√TNE8NZµ

)]+O

(e−N

1/2). (5.32)

Thus the second term tends to zero rapidly and can be forgotten. On the other hand,onN ,

M∑µ=2

(E8NZµ)4 6 N−1/4M∑µ=2

(E8NZµ)2 6 N−1/4TN. (5.33)

Moreover, for any finitetj , forN large enough,| tj√TNE8NZµ| 6 1. Thus, using that

| ln cosx − x2/2| 6 cx4 for |x| 6 1, and that

EξIc1INEη ei∑j∈I tjXj (N)

6 e−∑j∈I t2j /2 sup

N

[∏j∈I

exp

(ct4j N

−1/4

TN

)]Pξ (N). (5.34)

Clearly, the right hand side converges to e−∑j∈I t2j /2, provided only thatN1/4TN ↑∞. Since this was assumed, the proposition is proven. 2

We now control the convergence of our Laplace transform except for the twoparametersm1(N) ≡ E8NZ1 andTN ≡∑M

µ=2[E8NZµ]2. What we have to show isthat these quantities converge almost surely and that the limits satisfy the equationsof the replica symmetric solution of Amit, Gutfreund and Sompolinsky [1].

While the issue of convergence is crucial, the technical intricacies of its proofare largely disconnected to the question of the convergence of the Gibbs measures.We will therefore assume for the moment that these quantities do converge to somelimits and draw the conclusions for the Gibbs measures from the results of thissection under this assumption (which will later be proven to hold).

Indeed, collecting from Lemma 5.3 (see the remark following that lemma) andProposition 5.4, we can write

µ(1,1)3,β,ρ[ω](σI = sI ) =

eβ′N

∑i∈I si [m1(N)ξ

1i +Xi(N)

√TN ]+RN(sI )

2IEσI eβ′N

∑i∈I σi [m1(N)ξ

1i +Xi(N)

√TN ]+RN(σI )

, (5.35)

where

β ′N → β,

MPAG005.tex; 24/08/1998; 10:00; p.27


RN(sI )→ 0 in probability,

Xi(N)→ gi in law,

TN → αr a.s.,

m1(N)→ m1 a.s.,

for some numbersr,m1 and theregii∈N is a family of i.i.d. standard Gaussianrandom variables.

Putting this together we obtain:

PROPOSITION 5.7.In addition to our general assumptions, assume thatTN →αr, a.s. andm1(N)→ m1, a.s. Then, for any finiteI ⊂ N

µ(1,1)3,β,ρ(σI = sI )→

∏i∈I

eβsi [m1ξ1i +gi

√αr]

2 cosh(βσi[m1ξ1i + gi

√αr]) , (5.36)

where the convergence holds in law with respect to the measureP, andgi∈∈N isa family of i.i.d. standard normal random variables andξ1

i i∈N are independentBernoulli random variables, independent of thegi and having the same distributionas the variablesξ1

i .

To arrive at the convergence in law of the random Gibbs measures, it is enoughto show that (5.36) holds jointly for any finite family of cylinder sets,σi =si,∀i∈Ik , Ik ⊂ N, k = 1, . . . , ` (cf. [13, Theorem 4.2]). But this is easily seen tohold from the same arguments. Therefore, denoting byµ

(1,1)∞,β the random measure

µ(1,1)∞,β [ω](σ ) ≡

∏i∈N

eβσi [m1ξ1i [ω]+

√αrgi [ω]]

2 cosh(β[m1ξ1i [ω] +

√αrgi[ω]]) (5.37)

we have

THEOREM 5.8. Under the assumptions of Proposition5.7, and with the samenotation,

µ(1,1)3,β,ρ → µ

(1,1)∞,β , in law, as3 ↑ ∞. (5.38)

This result can easily be extended to the language of metastates. The followingtheorem gives an explicit representation of the Aizenman–Wehr metastate in oursituation:

THEOREM 5.9. Letκβ(·)[ω] denote the Aizenman–Wehr metastate. Under the hy-pothesis of Proposition5.7,for almost allω, for any continuous functionF : Rk →R, and cylinder functionsfi on −1,1Ii , i = 1, . . . , k, one has∫

M1(S∞)κβ(dµ)[ω]F(µ(f1), . . . , µ(fk))

MPAG005.tex; 24/08/1998; 10:00; p.28


=∫ ∏

i∈IdN (gi)F

(EsI1fi(sI1)

∏i∈I1

eβ[√αrgi+m1ξ

1i [ω]]

2 cosh(√αrgi +m1ξ

1i [ω])

,

. . . ,EsIk fk(sIk )∏i∈Ik

eβ[√αrgi+m1ξ

1i [ω]]

2 cosh(√αrgi +m1ξ

1i [ω])

), (5.39)

whereN denotes the standard normal distribution.

REMARK. Modulo the convergence assumptions, which will be shown to hold inthe next section, Theorem 5.9 is the precise statement of Theorem 1.1. Note thatthe only difference from Theorem 5.8 is that the variablesξ1

i that appear here onthe right hand side are now the same as those on the left hand side.

Proof.This theorem is proven just as Theorem 5.8, except that the ‘almost sureversion’ of the central limit theorem, Proposition 5.4, which in turn is proven justas Lemma 2.1, is used. The details are left to the reader. 2

REMARK. Our conditions on the parametersα andβ place us in the regime where,according to [1] the ‘replica symmetry’ is expected to hold. This is in nice agree-ment with the remark in [20] where replica symmetry is linked to the fact that themetastate is concentrated on product measures.

REMARK. One would be tempted to exploit also the other notions of ‘metastate’explained in Section 2. We see that the key to these constructions would be aninvariance principle associated with the central limit theorem given in Proposition5.4. However, there are a number of difficulties that so far have prevented us fromproving such a result. We would have to study the random process

Xti (N) ≡

M(tN)∑µ=2

ξµ

i E8tNZµ (5.40)

(suitably interpolated fort that are not integer multiples of 1/N). If this processwas to converge to Brownian motion, its increments should converge to indepen-dent Gaussians with suitable variance. But

Xti (N)− Xs

i (N) =M(tN)∑

µ=M(sN)ξµ

i E8tNZµ +

+M(sN)∑µ=2

ξµ

i (E8tNZµ − E8sNZµ). (5.41)

The first term on the right indeed has the desired properties, as is not too hard tocheck, but the second term is hard to control.

To get some idea of the nature of this process, we recall from [3, 4] thatE8NZ isapproximately given byc(β) 1

N

∑j∈3\I ξj (in the sense that the2 distance between

MPAG005.tex; 24/08/1998; 10:00; p.29


the two vectors is of order√α at most). Let us for simplicity consider only the case

I = 0. If we replaceE8NZ by this approximation, we are led to study the process

Y t(N) ≡ 1

t

αtN∑µ=2

ξµ

0

1

N

tN∑i=1

ξµ

i (5.42)

for tN, αtN integer and linearly interpolated otherwise.

PROPOSITION 5.10.The sequence of processesY t(N) defined by(5.42) con-verges weakly to the Gaussian processt−1Bαt2, whereBs is a standard Brownianmotion.

Proof. Notice thatξµ0 ξµi has the same distribution asξµi , and thereforeY t(N)

has the same distribution as

Y t (N) ≡ 1

tN

αtN∑µ=2

tN∑i=1

ξµ

i (5.43)

for which the convergence toBαt2 follows immediately from Donsker’s theorem.2

At present we do not see how to extend this result to the real process of interest,but at least we can expect that some process of this type will emerge.

As a final remark we investigate what would happen if we adopted the ‘stan-dard’ notion of limiting Gibbs measures as weak limit points along possibly ran-dom subsequences. The answer is the following:

PROPOSITION 5.11.Under the assumptions of Proposition5.7, for any finiteI ⊂ N, for anyx ∈ RI , for P-almost allω, there exist sequencesNk[ω] tending toinfinity such that for anysI ∈ −1,1I

limk↑∞

µ(1,1)Nk,β[ω](σI = sI ) =

∏i∈I

eβsi [m1ξ1i [ω]+

√αrxi ]

2 cosh(β[m1ξ1i [ω] +

√αrxi]) . (5.44)

Proof.To simplify the notation we will write the proof only for the casei = 0.The general case differs only in notation. It is clear that we must show that foralmost allω there exist subsequencesNk[ω] such thatX0(Nk)[ω] converges tox,for any chosen valuex. Since by assumptionTN converges almost surely toαr, itis actually enough to show that the variablesYk ≡

√TNkX0(Nk) converge tox. But

this follows from the following lemma: 2

LEMMA 5.12. DefineYk ≡√TNkX0(Nk). For anyx ∈ RI and anyε > 0,

P[Yk ∈ (x0− ε, x0+ ε) i.o.] = 1. (5.45)

MPAG005.tex; 24/08/1998; 10:00; p.30


Proof.Let us denote byFξ the sigma algebra generated by the random variablesξµ

i , µ ∈ N, i > 1. Note that

P[Yk ∈ (x0− ε, x0+ ε) i.o.] = E(P[Yk ∈ (x0− ε, x0+ ε) i.o. | Fξ ]) (5.46)

so that it is enough to prove that for almost allω, P[Yk ∈ (x0 − ε, x0 + ε) i.o. |Fξ ] = 1.

Let us define the random variables

Yk ≡M(Nk)∑

µ=M(Nk−1)+1

ξµ

0 E8Nk Zµ. (5.47)

Note first that

E(Yk − Yk)2 = EM(Nk−1)∑µ=2

(E8Nk Zµ)2 6 M(Nk−1)E(E8NkZ2)

2

6 ρ2Nk−1

Nk. (5.48)

Thus, ifNk is chosen such that∑∞

k=1Nk−1Nk

<∞, by the first Borel–Cantelli lemma,

limk↑∞

(Yk − Yk) = 0 a.s. (5.49)

On the other hand, the random variablesYk are conditionally independent, givenFξ . Therefore, by the second Borel–Cantelli lemma

P[Yk ∈ (x0− ε, x0+ ε) i.o. | Fξ ] = 1 (5.50)

if∞∑k=1

P[Yk ∈ (x0− ε, x0+ ε) | Fξ ] = ∞. (5.51)

But for almost allω, Yk conditioned onFξ converges to a Gaussian of varianceαr(the proof is identical to that of Proposition 5.3), so that for almost allω, ask ↑ ∞

P[Yk ∈ (x0− ε, x0+ ε) | Fξ ] → 1√2παr

∫ x+ε

x−εdy e−

y2

2αr > 0 (5.52)

which implies (5.51) and hence (5.50). Putting this together with (5.49) concludesthe proof of the lemma, and of the proposition. 2

Some remarks concerning the implications of this proposition are in place. First,it shows that if the standard definition of limiting Gibbs measures as weak limitpoints is adapted, then we have discovered that in the Hopfield model all productmeasures on−1,1N are extremal Gibbs states. Such a statement contains some

MPAG005.tex; 24/08/1998; 10:00; p.31


information, but it is clearly not useful as information on the approximate natureof a finite volume state. This confirms our discussion in Section 2 on the necessityto use a metastate formalism.

Second, one may ask whether conditioning or the application of external fieldsof vanishing strength as discussed in Section 2 can improve the convergence be-haviour of our measures. The answer appears obviously to be no. Contrary to asituation where a symmetry is present whose breaking biases the system to chooseone of the possible states, the application of an arbitrarily weak field cannot alteranything.

Third, we note that the total set of limiting Gibbs measures does not depend onthe conditioning on the ballB(1,1)ρ , while the metastate obtained does depend onit. Thus the conditioning allows us to construct two metastates corresponding toeach of the stored patterns. These metastates are in a sense extremal, since they areconcentrated on the set of extremal (i.e. product) measures of our system. Withoutconditioning one can construct other metastates (which however we cannot controlexplicitly in our situation).

6. Induction and the Replica Symmetric Solution

We now conclude our analysis by showing that the quantitiesUN ≡ E8N‖Z‖22,m1(N) ≡ E8NZ1 andTN ≡ ∑M

µ=2[E8NZµ]2 actually do converge almost surelyunder our general assumptions. The proof consists of two steps: First we show thatthese quantities are self-averaging and then the convergence of their mean valuesis proven by induction. We will assume throughout this section that the parametersα andβ are such that local convexity holds. We stress that this section is entirelybased on ideas of Talagrand [24] and Pastur, Shcherbina and Tirozzi [23] and ismainly added for the convenience of the reader.

Thus our first result will be:

PROPOSITION 6.1.LetAN denote any of the three quantitiesUN ,m1(N) or TN .Then there are finite positive constantsc, C such that, for any0< ζ 6 1/2,

P[|AN − EAN | > N−ζ/2] 6 CNζ exp

(−N

1−2ζ

c

). (6.1)

Proof.The proofs of these three statements are all very similar to that of Corol-lary 5.6. Indeed, form1(N), (6.1) is a special case of that corollary. In the two othercases, we just need to define the appropriate analogues of the ‘generating function’f from (5.28). They are

g(x) ≡ 1

βNlnE8NE

′8N

eβNx(Z,Z′) (6.2)

in the case ofTN and

g(x) ≡ 1

βNlnE8NE′8N eβNx‖Z‖

22. (6.3)

MPAG005.tex; 24/08/1998; 10:00; p.32


The proof then proceeds as in that of Corollary 6.6. We refrain from giving thedetails. 2

We now turn to the induction part of the proof and derive a recursion relationfor the three quantities above. In the sequel it will be convenient to introduce a site0 that will replace the setI and to setξ0 = η. Let us define

uN(τ) ≡ lnE8N eβτ(η,Z). (6.4)

We also setvN(τ) ≡ τβ(η,E8NZ) andwN(τ) ≡ uN(τ)− vN(τ). In the sequel wewill need the following auxiliary result:

LEMMA 6.2. Under our general assumptions

(i) 1β√TN

ddτ [vN(τ)− τβη1E8NZ1] converges weakly to a standard Gaussian ran-

dom variable.(ii) | d

dτ wN(τ)− τβ2EE8N‖Z‖22| converges to zero in probability.

Proof. (i) is obvious from Proposition 5.4 and the definition ofvN(τ). To prove(ii), note thatwN(τ) is convex andd2

dτ2wN(τ) 6 βα

ε. Thus,if var(wN(τ)) 6 C√

N,

then var( ddτ wN(τ)) 6

C ′N1/4 by a standard result similar in spirit to Lemma 5.5

(see, e.g., [25, Proposition 5.4]). On the other hand,|EwN(τ)− τ2β2

2 EE8N‖Z‖22| 6K√N

, by Lemma 5.3, which, together with the boundedness of the second deriv-

ative ofwN(τ) implies that| ddτEwN(τ) − τβ2EE8N‖Z‖22| ↓ 0. This means that

var(wN(τ)) 6 C√N

implies the lemma. Since we already know from (5.13) that

ER2N 6 K

N, it is enough to prove var(E8N‖Z‖22) 6 C√

N. This follows just as the

corresponding concentration estimate forUN . 2

We are now ready to start the induction procedure. We will place ourselves ona subspace ⊂ where for all but finitely manyN |UN − EUN | 6 N−1/4,|TN − ETN | 6 N−1/4, etc. This subspace has probability one by our estimates.

Let us note that by (iii) of Proposition 3.1,E8NZµ and∫

dQ(1,1)N,β,ρ(m)mµ differ

only by an exponentially small term. Thus

E8NZµ =1

N

∑i=1

ξµ

i

∫µ(1,1)N,β,ρ(dσ)σi +O

(e−cM

)(6.5)

and, by symmetry,

EE8N+1(Zµ) = Eηµ∫µ(1,1)N+1,β,ρ(dσ)σ0+O

(e−cM

). (6.6)

Using Lemma 5.2 and the definition ofuN , this gives

EE8N+1(Zµ) = EηµeuN(1) − euN (−1)

euN(1) + euN (−1)+O

(e−cM

), (6.7)

MPAG005.tex; 24/08/1998; 10:00; p.33


where to be precise one should note that the left and right hand side are computedat temperaturesβ andβ ′ = N

Nβ, respectively, and that the value ofM is equal to

M(N + 1) on both sides; that is, both sides correspond to slightly different valuesof α andβ, but we will see that this causes no problems.

Using our concentration results and Lemma 5.3 this gives

EE8N+1(Zµ)

= Eηµ tanh(β(η1Em1(N)+

√ETNX0(N))

)+O(N−1/4). (6.8)

Using further Proposition 5.4 we get a first recursion form1(N):

m1(N + 1) =∫

dN (g) tanh(β(Em1(N)+

√ETNg)

)+ o(1). (6.9)

REMARK. The error term in (6.9) can be sharpened to O(N−1/4) by using insteadof Lemma 5.3 a trick, attributed to Trotter, that we learned from Talagrand’s paper[24] (see the proof of Proposition 6.3 in that paper).

We need of course a recursion forTN as well. From here on there is no greatdifference from the procedure in [23], except that theN-dependences have to bekept track of carefully. This was outlined in [4] and we repeat the steps for theconvenience of the reader. To simplify the notation, we ignore all the O(N−1/4)

error terms and put them back in the end only. Also, the remarks concerningβ andα made above apply throughout.

Note thatTN = ‖E8NZ‖22− (E8NZ1)2 and

E‖E8N+1Z‖22 =M∑µ=1

E

(1

N + 1

N∑i=0

ξµ

i µβ,N+1,M(σi)

)2

= M

N + 1E(µ(1,1)β,N+1,M(σ0)

)2++

M∑µ=1

Eξµ0 µ(1,1)β,N+1,M(σ0)

(1

N + 1

N∑i=1

ξµ

i µβ,N+1,M(σi)

). (6.10)

Using Lemma 5.2 as in the step leading to (6.7), we get for the first term in (6.10)

E(µ(1,1)β,N+1,M(σ0)

)2 = E tanh2 (β(η1E8NZ1+√ETN)

) ≡ EQN. (6.11)

For the second term, we use the identity from [23]

M∑µ=1

ξµ

0

(1

N

N∑i=1

ξµ

i µβ,N+1,M(σi)

)=

∑σ0E8N (ξ0, Z)eβσ0(ξ0,Z)∑σ0E8N eβσ0(ξ0,Z)

= β−1

∑τ=±1 uN

′(τ)euN (τ)∑τ=±1 euN (τ)

. (6.12)

MPAG005.tex; 24/08/1998; 10:00; p.34


Together with Lemma 6.2 one concludes that in law up to small errors

M∑µ=1

ξµ

0

(1

N + 1

N∑i=1

ξµ

i µβ,N+1,M(σi)

)= ξ1

0E8NZ1+√ETNX0(N)+

+βE8N ‖Z‖22 tanhβ(ξ1

0E8NZ1+√ETNX0(N)

)(6.13)

and so

E‖E8N+1Z‖22 = αEQN + E[

tanhβ(ξ1

0E8NZ1+√ETNX0(N)

)×× [ξ1

0E8NZ1+√ETNX0(N)

]]++βEE8N‖Z‖22 tanh2β

(ξ1

0E8NZ1+√ETNX0(N)

). (6.14)

Using the self-averaging properties ofE8N‖Z‖22, the last term is of course essen-tially equal to

βEE8N‖Z‖22EQN. (6.15)

The appearance ofE8N‖Z‖22 is disturbing, as it introduces a new quantity into thesystem. Fortunately, it is the last one. The point is that proceeding as above, we canshow that

EE8N+1‖Z‖22 = α + E[

tanhβ(ξ1N+1E8NZ1+

√ETNX0(N)

)×× [ξ1

0E8NZ1+√ETNXN

]]+ βEE8N‖Z‖22EQN (6.16)

so that settingUN ≡ E8N‖Z‖22, we get, subtracting (6.14) from (6.16), the simplerecursion

EUN+1 = α(1− EQN)+ β(1− EQN)EUN. (6.17)

From this we get (since all quantities considered are self-averaging, we drop theEto simplify the notation), settingm1(N) ≡ E8NZ1,

TN+1 = −(m1(N + 1))2+ αQN + βUNQN ++∫

dN (g)[m1(N)+

√TNg

]tanhβ

(m1(N)+

√TNg

)= m1(N + 1)(m1(N)−m1(N + 1))++βUNQN + βTN(1−QN)+ αQN, (6.18)

where we used integration by parts. The complete system of recursion relations canthus be written as

m1(N + 1) =∫

dN (g) tanhβ(m1(N)+

√TNg

)+O(N−1/4)TN+1

MPAG005.tex; 24/08/1998; 10:00; p.35


= m1(N − 1)(m1(N)−m1(N + 1))+ βUNQN ++βTN(1−QN)+ αQN +O(N−1/4),

UN+1 = α(1−QN)+ β(1−QN)UN +O(N−1/4),

QN+1 =∫

dN (g) tanh2 β(m1(N)+

√TNg

)+O(N−1/4). (6.19)

If the solutions to this system of equations converges, then the limitsr = limN↑∞TN/α, q = limN↑∞QN andm1 = limN↑∞m1(N) (u ≡ limN↑∞UN can beeliminated) must satisfy the equations

m1 =∫

dN (g) tanh(β(m1 +

√αrg)

), (6.20)

q =∫

dN (g) tanh2(β(m1 +

√αrg)

), (6.21)

r = q

(1− β + βq)2 (6.22)

which are the equations for the replica symmetric solution of the Hopfield modelfound by Amit et al. [1].

In principle one might think that to prove convergence it is enough to studythe stability of the dynamical system above without the error terms. However, thisis not quite true. Note that the parametersβ andα of the quantities on the twosides of the equation differ slightly (although this is suppressed in the notation). Inparticular, if we iterate too often,α will tend to zero. The way out of this difficultywas proposed by Talagrand [24]. We will briefly explain his idea. In a simplifiednotation, we are in the following situation: We have a sequenceXn(p) of functionsdepending on a parameterp. There is an explicit sequencepn, satisfying|pn+1 −pn| 6 c/n and a functionFp such that

Xn+1(pn+1) = Fpn(Xn(pn))+O(n−1/4). (6.23)

In this setting, we have the following lemma.

LEMMA 6.3. Assume that there exist a domainD containing a single fixed pointX∗(p) ofFp. Assume thatFp(X) is Lipshitz continuous as a function ofX, Lipshitzcontinuous as a function ofp uniformly for X ∈ D and that for allX ∈ D,Fnp (X)→ X∗(p). Assume we know that for alln large enough,Xn(p) ∈ D. Then

limn↑∞

Xn(p) = X∗(p). (6.24)

Proof. Let us choose a integer valued monotone increasing functionk(n) suchthatk(n) ↑ ∞ asn goes to infinity. Assume, e.g.,k(n) 6 ln n. We will show that

limn↑∞Xn+k(n)(p) = X

∗(p). (6.25)

MPAG005.tex; 24/08/1998; 10:00; p.36


To see this, note first that|pn+k(n) − pn| 6 k(n)/n. By (6.23), we have that usingthe Lipshitz properties ofF

Xn+k(n)(p) = Fk(n)p (Xn(pn))+O(n−1/4), (6.26)

where we choosepn such thatpn+k(n) = p. Now sinceXn(pn) ∈ D, |Fk(n)p (Xn(pn)

−X∗(p)| ↓ 0 asn and thusk(n) goes to infinity, so that (6.26) implies (6.25). But(6.25) for any slowly diverging functionk(n) implies the convergence ofXn(p), asclaimed. 2

This lemma can be applied to the recurrence (6.18). The main point to check iswhether the correspondingFβ attracts a domain in which the parametersm1(N), TN ,

UN,QN are a priori located due to the support properties of the measureQ(1,1)N,β,ρ.

This stability analysis was carried out (for an equivalent system) by Talagrand andanswered in the affirmative. We do not want to repeat this tedious, but in principleelementary computation here.

We would like, however, to make some remarks. It is clear that if we considerconditional measures, then we can always force the parametersm1(N),RN,UN,

QN to be in some domain. Thus, in principle, we could first study the fixpointsof (6.18), determine their domains of attraction and then define correspondingconditional Gibbs measures. However, these measures may then be metastable.Also, of course, at least in our derivation, we need to verify the local convexity inthe corresponding domains since this was used in the derivation of the equations(6.18).

Acknowledgements

We gratefully acknowledge helpful discussions about metastates with Ch. Newmanand Ch. Külske.

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145

On Spectral Asymptotics for Domains with FractalBoundaries of Cabbage Type

S. MOLCHANOV and B. VAINBERGDepartment of Mathematics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA

(Received: 4 February 1997; accepted: 13 March 1998)

Abstract. The second term of the asymptotic expansion of the eigenvalue counting functionN(λ)

is found for the Dirichlet Laplacian in a class of domains with fractal boundaries.

Mathematics Subject Classifications (1991):35P, 35J.

Key words: counting function, fractal boundary, spectral asymptotics, eigenvalue, Minkowski di-mension.

1. Introduction

Let N−(λ) be the eigenvalue counting function for the Dirichlet Laplacian in abounded domain ⊂ <d, andN+(λ) be the counting function for the NeumannLaplacian, i.e.

N∓(λ) = #λ∓j : λ∓j 6 λ,whereλ−j are the eigenvalues of the Dirichlet problem

−1ψ = λjψ, x ∈ ; ψ = 0, x ∈ ∂,andλ+j are the eigenvalues of the Neumann problem

−1ψ = λjψ, x ∈ ; ∂ψ

∂n= 0, x ∈ ∂.

The classical Weyl conjecture for domains with smoothboundaries has theform

N∓(λ) = ad ||λd/2∓ bd |∂|λ(d−1)/2+ o(λ(d−1)/2), λ→∞, (1)

where|| is the measure of the domain,|∂| is the (d−1)-dimensional measure ofthe boundary, constantsad andbd depend only on dimensiond of the phase space(in particular(2π)dad is the volume of the unit ball in<d). Weyl justified the firstterm in (1) (Weyl law):

N∓(λ) = ad ||λd/2+ o(λd/2), λ→∞. (2)

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146 S. MOLCHANOV AND B. VAINBERG

Many papers are devoted to the Weyl conjecture for the Laplacian and formore general elliptic operators. Let us mention only the work by Seeley [14] whospecified the estimate of the remainder in (2)

N∓(λ) = ad ||λd/2+O(λ(d−1)/2), λ→∞, (3)

and by Ivrii [7] who showed that the conjecture is valid for smooth domains underadditional ‘billiard condition’.

The main goal of our paper is to find the second term ofN−(λ) for a classof domains withfractal boundaries. Let us mention that Ivrii’s results are notapplicable and the question about billiard condition can not be posed at all if theboundary of the domain is very irregular (fractal) because the billiard trajectoriesare not determined in this case. We consider only the Dirichlet Laplacian becausethe spectrum of the Neumann Laplacian, generally speaking, is not discrete (see[6, 15]), and the counting functionN+(λ) may not exist even when the domainhas only one irregular point on∂. As far as the Dirichlet Laplacian is concernedthe second term of asymptotics ofN−(λ) for domains with fractal boundaries isknown only in the one-dimensional case (for Sturm–Liouville problem) [8] andfor some cases when the variables can be separated or the domain is self-similar[3, 4, 11, 12] (estimates of the second term of asymptotics are known in moregeneral case, see [5, 9, 10]). Our purpose is to find the second term of asymptoticexpansion ofN−(λ) for a broad class of domains with fractal boundaries (we calledthem domains of the cabbage type) where there are no separation of variables orself-similarity.

M. Berry conjectured [1] that in the fractal case the order of the second termof asymptotic expansion ofN−(λ) is one half of the Hausdorff dimension of theboundary. Brassard and Carmona [2] showed that it is wrong. According to thewell-known modified Weyl–Berry conjecture,N−(λ) for domains with fractalboundaries has the form

N−(λ) = ad ||λd/2− c(d,m)|∂|mλm/2+ o(λ(d−1)/2), λ→∞, (4)

wherem is the Minkowski dimension of the boundary,|∂|m is the Minkowskicontent of the boundary and constantc depends only ond andm. In [3, 4, 12] itis shown that for self-similar domains constantc is not universal and may dependon the domain. In our previous work [13] it is shown that in general theorder ofthe second term in (4) is not related to the Minkowski dimension of the boundary.However, we prove here that the modified Weyl–Berry conjecture (4) is valid forthe domains of ‘cabbage type’ which we consider in this paper. These domains havea sequence (or several sequences) of ‘cracks’ which converge to the outer boundaryof the domain. The typical two dimensional cross sections of two different cabbagetype domains are given in Figure 1. The first domain has one sequence of thecracks, the second has two sequences. The rigorous definition of cabbage typedomains is given in Section 3.

MPAG003.tex; 17/08/1998; 14:53; p.2

ON SPECTRAL ASYMPTOTICS FOR DOMAINS WITH FRACTAL BOUNDARIES 147

Figure 1.

The plan of the paper is the following. In the next section we consider the non-fractal domains with smooth and nonsmooth boundaries, and we obtain a Seeleytype estimate (similar to (3)) which is uniform with respect to domains. It is notvery important for us to get the exact rate of decay of the remainder asλ→∞. Sowe get an estimate with an additional logarithmic factor in the remainder, but theconstant in the estimate of the remainder is expressed through simple geometricalcharacteristics of the domain. These estimates could be useful independently ofthe main purpose of the article. We get them for both the Dirichlet and Neumannboundary conditions, and we need both of them to investigateN−(λ) for the Dirich-let Laplacian when the boundary is fractal. In the last section we give the definitionof the cabbage type domains (with fractal boundaries) and find the first two termsof asymptotic expansion ofN−(λ) for this type of domains.

2. Uniform Estimate of the Remainder in Weyl Asymptotics

The goal of this section is to find such an estimate of the remainder in Weyl asymp-totics of the eigenvalue counting function that will be uniform with respect to awide class of smooth domains. The estimate depends on the following geometricalcharacteristic of the domains.

Let (∂)ε beε-neighborhood of the boundary of the domain. In the case of theDirichlet boundary condition we assume that there exist constantsr0 andA suchthat

|(∂)ε| 6 Aε|∂| for 06 ε 6 r0. (5)

Let ν, l be normal and tangent vectors to∂ at pointx andκ = κ(x, l) bethe sectional curvature of∂ at the pointx, i.e. κ is the curvature at the pointxof the planar curve which is the intersection of∂ and the two dimensional planethroughν and l. In the case of the Neumann Laplacian we will assume that thebr -condition (br stands for a ball of radiusr) is satisfied which means that∂ istwice continuously differentiable, all its sectional curvatures do not exceedr, andone can touch any point of∂ by a ball of radiusr from inside of, i.e. for eachpoint x0 ∈ ∂ there is an open ballB of radiusr such thatB ⊂ andx0 ∈ ∂B.Let us mention that the last requirement does not follow from the boundedness of

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the curvatures because it prohibits, for example, the case when has a shape ofa very thin layer in some region of<d. The estimate (5) withr0 = r (and someA) follows from the conditionbr (but not vice versa), so we always will assumethat the estimate (5) is valid and we will user instead ofr0 when we consider theNeumann Laplacian.

THEOREM 1. Let be a bounded domain in<d which satisfies(5). Then

∣∣N−(λ)− αd ||λd2 ∣∣ 6 C(d)λd−12

[A|∂| ln(r0

√λ)+A|∂| + ||

r0

],

λ > 1/r0, (6)

where(2π)dαd is the volume of the unit ball in<d , constantC(d) depends only ond, andA andr0 are defined in(5)

Under additionalbr -condition estimate (6) is valid for the counting function ofthe Neumann Laplacian when

√λ > 20d/r:

∣∣N+(λ)− αd ||λd2 ∣∣ 6 C(d)λd−12

[A|∂| ln(r√λ)+A|∂| + ||

r

],

√λ > 20d/r. (7)

In order to prove the theorem we will need the following three lemmas.

LEMMA 2. Let ⊂ 1 be bounded domains,N+1 (λ) be the counting function forthe Neumann Laplacian in1, andN(λ) be the counting function for the Laplaceoperator in with the Neumann boundary condition on some part(which can beempty) of ∂ ∩ ∂1 and the Dirichlet boundary condition on the remaining partof ∂. ThenN(λ) 6 N+1 (λ), λ > 0.

Proof.Let λ1 6 λ2 6 λ3 6 · · · be the eigenvalues of the Neumann Laplacian in1, andµ1 6 µ2 6 µ3 6 · · · be the eigenvalues of the problem in formulatedin the lemma. Due to the mini-max principle

λj = supHj

inff∈Hj

(∫1|∇f |2 dx∫

1|f |2 dx

)1/2

, µj = supH ′j

inff∈H ′j

(∫|∇f |2 dx∫|f |2 dx

)1/2

, (8)

whereHj are the subspaces of codimensionj of the Sobolev spaceH 1(1),

andH ′j are the subspaces of codimensionj of the spaceH 1,0(). HereH 1,0()

consists of the functionsf ∈ H 1() which vanish on the part of∂ where theDirichlet boundary condition is given. Since any function fromH 1,0() vanisheson ∂ ∩ 1, the function continued by zero onto1\ belongs toH 1(1), andtherefore there is a natural inclusionH ′j ⊂ Hj. From here and (8) it follows thatµj 6 λj for all j. This gives the assertion of the lemma. 2

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In the next lemma it will be convenient for us to write pointsx ∈ <d asx =(x1, x2) wherex1 ∈ <d−1, x2 ∈ <.

LEMMA 3 (Comparison principle for the Neumann Laplacian).Let a boundeddomain ⊂ <d have the form

= (x1, x2) : x1 ∈ ω ⊂ <d−1, 0< x2 < f (x1),whereω is a bounded domain in<d−1, 0< c1 6 f (x1) 6 c2 and |∇f | 6 α.

LetN+ (λ), N+c (λ) be the counting functions for Neumann Laplacian in andthe cylinderDc = (x1, x2) : x1 ∈ ω, 0< x2 < c.

Then

N+c2

(1

2+ α√c1

c2λ

)6 N+ (λ) 6 N+c1

((2+ α)

√c2

c1λ

).

Proof. Let us consider the diffeomorphism of onto the cylinderDc : x1 =x1, x2 = cx2

f (x1)and denote byA the matrix

A =(D(x)

D(x)

)=(

∂x1∂x1

∂x1∂x2

∂x2∂x1

∂x2∂x2

)=(

I 0

− cx2f 2 ∇f c

f

), (9)

whereI is the identity matrix, 0 is zero column. Letu = (u1, u2) ∈ <d be anarbitrary vector withu1 ∈ <d−1, u2 ∈ <. From (9) and inequalities 06 x2 6f (x1), |f | > c1, |∇f | 6 α it follows that

|Au| 6 |u1| +(cα

c1|u1| + c

c1|u2|

)6(

1+ cαc1+ c

c1

)|u|. (10)

Similarly one can rewrite the same diffeomorphism in the formx1 = x1, x2 =f (x1)x2

cand estimate|A−1u| from above. It gives the following inequality

|Au| >(

1+ α + c2

c

)−1

|u|. (11)

One can check directly from (9) that detA−1 = f/c. Thus

c1/c 6 detA−1 6 c2/c. (12)

From the relation∫|∇f |2 dx∫|f |2 dx

=∫Dc|A∇xf |2|detA−1|dx∫Dc|f |2|detA−1|dx

and inequalities (10)–(12) it follows that

(2+ α)−2c1

c2

∫Dc2|∇xf |2 dx∫

Dc2|f |2 dx

6∫|∇f |2 dx∫|f |2 dx

6 (2+ α)2c2

c1

∫Dc1|∇xf |2 dx∫

Dc1|f |2 dx

.

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Together with the mini-max principle (see (8)) this immediately gives the asser-tion of the lemma. 2

The next lemma looks very natural and obvious. However, we couldn’t find anyreference and also didn’t find a short proof. Letν = (ν1, ν2, . . . , νd) be the unitexternal normal to∂. Fix an arbitrary pointx = (x1, . . . , xd) ∈ ∂. Let νk be acoordinate of the vectorν at x such that|νk| > 1/

√d, z = (x1, . . . , xk−1, xk+1, . . . ,

xd) ∈ <d−1 andz = (x1, . . . , xk−1, xk+1, . . . , xd). Clearly,|νk| > 12√d

nearx. Wewould like to show that this inequality is valid in a neighborhood ofx whose sizedepends not on∂, but only on the dimensiond and the constantr which boundsthe sectional curvatures of∂ (see conditionbr ).

LEMMA 4. If sectional curvatures of∂ do not exceedr then there is a part of∂ which containsx and has the form

xk = F(z), |z− z| 6 r

10d

with |∇F | 6 √4d − 1 (and |νk| = 1/√

1+ |∇F |2 > 12√d).

Proof.The proof is based on the classical procedure of the local reconstructionof the surface in terms of the fundamental differential forms (1st and 2nd). Clearly,∂ has the formxk = F(z) in some neighborhood ofx. In local coordinatesz thefirst G1and the secondG2 fundamental forms of the surface∂ are given by theformulas

G1 = gi,j (z)dzi dzj , gi,j = δi,j + ∂F∂zi

∂F

∂zj,

G2 = bi,j (z)dzi dzj , bi,j = (1+ |∇F |2)−1/2 ∂2F

∂zi∂zj.

The Hilbert–Schmidt norm of the matrix [gi,j ] is equal to

‖[gi,j ]‖ =√d − 1+ 2|∇F |2 + |∇F |4.

From the boundedness of the sectional curvatures it follows that−[gi,j ] 6r[bi,j ] 6 [gi,j ] (in the sense of forms), and therefore

r‖[bi,j ]‖ 6 ‖[gi,j ]‖ 6√d − 1+ 2|∇F |2 + |∇F |4.

Let us estimate now the functionL(z) = |∇F |. We have

∇L = L−1

[∂2F

∂zi∂zj

]∇F =

√1+ L2

L[bi,j ]∇F

MPAG003.tex; 17/08/1998; 14:53; p.6


and therefore

|∇L| 6√

1+ L2

L‖[bi,j ]‖L =

√1+ L2‖[bi,j ]‖

6 1

r

√1+ L2

√d − 1+ 2L2+ L4. (13)

The assertion of the lemma follows from this inequality. In order to show that letus mention first of all thatL(z) 6

√d − 1 because|νk| = 1/

√1+ |∇F |2 > 1√

dat

x. If the assertion of the lemma is incorrect then in the ball|z− z| 6 r10d there exist

pointsz whereL = √4d − 1. Let z′ be one of the closest toz points of that type,i.e. ρ := |z′ − z| 6 r

10d , L(z′) = √4d − 1 andL(z) <√

4d − 1 if |z − z| < ρ.Then|∇L| in the ball|z− z| 6 ρ does not exceed the right-hand side in (13) withL = √4d − 1, and therefore

|L(z′)− L(z)| 6 1

r

√4d√d − 1+ 2(4d − 1)+ (4d − 1)2ρ

6 1

5√d

√16d2 + d − 2<

√d.

ThenL(z′) < L(z)+√d 6 √d − 1+√d which contradicts the relationL(z′) =√4d − 1. Lemma 4 is proved. 2

Proof of Theorem 1.Let us recall that the following estimates are valid for thecounting functionsN±(λ) for the Neumann and Dirichlet Laplacian in the unitcube. There is a constantC = C(d) such that

adλd2 + Cλd−1

2 > N+(λ) > N−(λ) > adλd2 − Cλd−1

2 , λ > 1, (14)

where(2π)dαd is the volume of the unit ball in<d . Then the counting functionsN±1(λ) in the cube of size1 satisfy the following estimates

ad(1√λ)d + C(1√λ)d−1 > N+1(λ) > N−1(λ)

> ad(1√λ)d − C(1√λ)d−1, λ > 1. (15)

Let us partition<d into cubes of size10 = r0√d. Let n0 be the number of cubes

(here and below the cubes are assumed to be open) which belong to, m0 be thenumber of cubes which intersect∂. We divide each cube which intersects∂into 2d equal cubes of the size11 = 1

210. Let n1 be the number of the new cubesbelonging to, andm1 be the number of the new cubes which intersect∂. Werepeat the same procedure. We divide each cube of the size11 which intersect∂into 2d equal cubes of the size12 = 1

410 and denote the number of the cubes ofthe size12 which belong to by n2, and the number of the cubes of the same size12 which intersect∂ bym2.We iterate the procedure.

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From (5) withε = √d1k it follows that

mk 6∣∣(∂)√d1k ∣∣(1k)d

6 A√d|∂|(1k)

1−d. (16)

Due to the construction

n0 6||(10)d

, nk 6 mk−12d 6 A2d

√d|∂|(1k−1)

1−d for k > 1. (17)

SinceN−(λ) is not less than the sum of the counting functions for the DirichletLaplacian for any system of nonintersecting cubes which belong to, we have

N−(λ) >K∑k=0

nkN−1k(λ) >

K∑k=0

nk[ad(1k

√λ)d − C(1k

√λ)d−1], λ > 1. (18)

The sum∑K

k=0 nk(1k)d differs from || by not more than|(∂)ε| with ε =√

d1K . Together with (5) and (18) this leads to the following estimate

N−(λ) > ad ||λd2 − adA√d1K |∂|λd2 − Cλd−1

2

K∑k=0

nk(1k)d−1. (19)

Taking into account the inequalities (17) and the fact that1K = 2−K10 =2−Kr0/

√d we get

N−(λ) > ad ||λd2 − adA2−Kr0|∂|λd2 − Cλd−12

[√d||r0+

K∑k=1

2A√d|∂|

]

= ad ||λd2 − adA2−Kr0|∂|λd2 − Cλd−12

[√d||r0+ 2AK

√d|∂|

].

The numberK was an arbitrary integer. We takeK = [log2(r0√λ)] (the maxi-

mum integer which does not exceed log2(r0√λ)). Then we get

N−(λ) > ad ||λd2 − C1(d)λd−1

2

[A|∂| ln(r0

√λ)+A|∂| + ||

r0

],

λ > 1. (20)

In order to estimateN−(λ) from above we use the same partition of<d bycubes of different sizes. This partition allows us to cut into n0 cubes of the size10 = r0√

d, n1 cubes of the size11 = 1

210, n2 cubes of the size12 = 1410, . . . ,

nK cubes of the size1K = 2−K10, andmK domainsj , 1 6 j 6 mK , whichare the intersections of and those cubes of the size1K which have nonemptyintersections with∂. We impose the Neumann boundary conditions on all cutswhich divide on the described set of subdomains. Due to the mini-max principle

MPAG003.tex; 17/08/1998; 14:53; p.8


these cuts with the Neumann boundary conditions on them decrease the eigen-values of the Laplacian in and therefore increase the counting function. Thus,N−(λ) for does not exceed the sum of the counting functions of the Laplacianin all subdomains constructed above with the Dirichlet boundary condition on∂

and the Neumann boundary conditions on all other boundaries of the subdomains.Due to Lemma 2 we can replace here the counting functions of the problems inj by the counting functions of the Neumann Laplacian in the cubes containingj , and also replace the Dirichlet boundary condition by the Neumann boundarycondition on the faces of the cubes which belong to∂. This leads to the followingestimate

N−(λ) 6K∑k=0

nkN+1k(λ)+mKN+1K(λ) (21)

6K∑k=0

nk[ad(1k

√λ)d + C(1k

√λ)d−1] +mK[ad(1K

√λ)d +

+ C(1K

√λ)d−1].

The sum∑K

k=0 nk(1k)d +mK(1K)

d differs from|| by not more than|(∂)ε|with ε = √d1K . Together with (5) and (21) it leads to the following estimate

N−(λ) 6 ad ||λd2 + adA√d1K |∂|λd2 +

+ Cλd−12

[ K∑k=0

nk(1k)d−1+mK(1K)

d−1

]. (22)

The estimate

N−(λ) 6 ad ||λd2 + C2(d)λd−1

2

[A|∂|[ln(r0

√λ)]+ + A|∂| + ||

r0

],

λ > 1, (23)

follows from (22) in the same way as (20) from (19). Together with (20) it provesthe assertion of the theorem aboutN−(λ) (the estimate (6)).

Now we are going to prove (7). First of all let us mention that the estimate (5)with r0 = r follows from thebr condition, so we will replacer0 by r when weconsider the Neumann Laplacian. SinceN+(λ) > N−(λ) the estimate (20) forN+(λ) (from below) is valid, and therefore it is only left to prove the estimate (23)for N+(λ) (from above) when

√λ > 20d/r. The last estimate forN+(λ) can be

proved in the same way as forN−(λ). The main difference is that now we can notuse Lemma 2 in order to estimate the counting functionN−j (λ) of the NeumannLaplacian inj . It will be done using Lemma 3. It leads also to the necessity tochange slightly the partition of which was used earlier.

We start with the same partition of into the set consisting ofn0 cubes of thesize10 = r√

d, n1 cubes of the size11 = 1

210, n2 cubes of the size12 = 1410, . . . ,

MPAG003.tex; 17/08/1998; 14:53; p.9


nK cubes of the size1K = 2−K10, andmK domainsj , 1 6 j 6 mK , whichare the intersections of and those cubesCj of the size1K which have nonemptyintersections with∂. We replace the domainsj in this partition by the set ofdomainsQl, 16 l 6 L, which will be described in the next couple of paragraphs.Eachj will be covered by the union of the domainsQl:

j ⊂⋃Ql, 16 j 6 mK, (24)

and therefore the closures of the cubes and domainsQl cover. Thus, from themini-max principle it immediately follows that

N+(λ) 6K∑k=0

nkN+1k(λ)+

l=L∑l=1

N+Ql(λ), (25)

whereN+1k (λ) andN+Ql (λ) are the counting functions of the Neumann Laplacianin the cube of the size1k and the domainQl, respectively. As earlier, we chooseK = [log2(r

√λ)] (the maximum integer which does not exceed log2(r

√λ)). Then,

in particular,1K 6 2√dλ

.Now we construct the domainsQl. Letν = (ν1, ν2, . . . , νd) be the outward unit

normal to∂. We fix an arbitrary pointx ∈ ∂ ∩ Cj . Let νk be a coordinate ofthe vectorν at x such that|νk| > 1/

√d. Without loss of the generality we can

assume thatνk is positive, i.e.νk > 1/√d. Let x′ = (x1, . . . , xk−1, xk+1, . . . , xd),

x′ correspond to the pointx, andC′j be the projection of the cubeCj on thehyperplanexk = 0. The lengthl of the largest diagonal ofC′j does not exceed√d − 11K < 2√

λ. Thusl 6 r

10d because√λ > 20d/r. Then from Lemma 4 it

follows that there is a part of∂ containingx which is given by the equation

xk = F(x′), x′ ∈ C′j ,andνk > 1

2√d

there. Let

Q1 = x : x′ ∈ C′j , a − 2(d + 1)1K < xk < F(x′), (26)

wherea is thexk coordinate of the lower (if the direction ofxk axis is vertical) faceof the cubeCj . Then we pick another pointx ∈ ∂ ∩ Cj which does not belongtoQ1 and construct the next domainQ2 using this new pointx. We continue thisprocedure until at least in one of the cubesCj there is a pointx ∈ ∂ ∩ Cj whichdoes not belong to the closure of domainsQj already constructed.

Now we are going to derive the following properties of the domainsQj :(1) the following inequality holds:

|F(x′)− a| 6 (2d + 1)1K, x′ ∈ C′j . (27)

In particular, it means that domain (26) is well defined.(2) All domainsQj belong to.

MPAG003.tex; 17/08/1998; 14:53; p.10


(3) Each of the domainsj (see (24)) is covered by the union of not more than2d of the domainsQj and therefore the second sum in (25) contains no more than2dmK terms.

Let us recall that|∇F | 6 √4d − 1 whenx′ ∈ Cj ′ (sinceνk = 1/√

1+ |∇F |2 >1

2√d). From here it follows that|F(x′) − F(x′)| 6 √4d − 1l < 2

√dl < 2d1K .

This proves (27) becauseF(x′) is the coordinatexk of the pointx, and therefore|F(x′)−a| 6 1K . In order to prove the second property let us consider an arbitrarypoint x = (F (x′), x′) on the upper (we consider direction of thexk axis as vertical)boundary ofQ1. Let b be the ball of radiusr which belongs to and touches∂at x. In particular, the vertical secant of this ball with the upper end atx belongs to. Sinceνk > 1

2√d

for the coordinateνk of the normalν to the ball atx, the length

of this secant is not less thanr/√d. Thus

x : x′ ∈ Cj ′, F (x′)− r√d< xk < F(x′)

⊂ ,

and the second property ofQj will be proved once we show that

F(x′)− r√d< a − 2(d + 1)1K.

Due to (27) it is enough to show thatr/√d > 4(d + 1)1K , but this inequality

follows immediately from the facts that1K 6 2/√dλ, and

√λ > 20d/r. Let us

prove the last property ofQj . Let two different domainsQj1 andQj2 correspondto the same cubeCj and have the form (26) with the samek, but different functionsF(x′). Since the lower faces of the domains are the same, and upper parts of theboundaries can’t intersect (∂ is smooth), one of the domains has to contain theother. Thus the upper part of the boundary of the smaller domain belongs to theinterior of the bigger one, and therefore it belongs to and can’t be a part of∂.It contradicts the procedure used for the construction of the domainsQj . Thus thenumber of the domainsQj which are related to the same cubeCj can not be biggerthan the number of the faces of the cube. All the three properties of the domainsQj are now proved.

The first sum in the right-hand side of (25) and the first sum in the first ofinequalities (21) are identical, and since (21) leads to (23), from (25) it follows that

N+(λ) 6 ad ||λd2 + C2(d)λd−1

2

[A|∂| ln(r√λ)+A|∂| + ||

r

]+

+l=L∑l=1

N+Ql(λ), (28)

whereL 6 2dmK andλ > 20d/r. From (26), (27) and the estimate|∇F | 6√4d − 1 it follows that Lemma 3 withα = √4d − 1, c1 = 1K andc2 = 2(d +

MPAG003.tex; 17/08/1998; 14:53; p.11


1)1K can be applied toN+Ql(λ), and therefore

N+Ql(λ) 6 N+1K (c(d)λ) 6 ad(1K

√c(d)λ)d + C(1K

√c(d)λ)d−1

6 C(d) (29)

because1K 6 2/√dλ. The estimate from above forN+(λ) follows from (28),

(29) and (16) withk = K. The proof of the theorem is complete. 2

3. Asymptotics ofN−(λ) for Cabbage Type Domains

This class of domains was singled out in our work [13], and it consists of boundeddomains with smooth outer boundaries and a sequence of ‘cracks’ which convergeto the outer boundary. The exact definition is the following.

DEFINITION 5. We say that is a domain of the cabbage type if

= 0\∞⋃n=1

0n,

where0 is a bounded domain given by the equation0 = x : F(x) > 0 withF ∈ C2 and|∇F | 6= 0 when 06 F(x) 6 1, and the ‘cracks’0n are given by therelations

0n = x : F(x) = n−α, x ∈ 1,whereα > 0 is an arbitrary fixed positive constant, and1 is a domain with asmooth boundary which is transversal to∂0 or0 ⊂ 1.

In fact, all the results below are valid for a wider class of domains. We can con-sider several sequences of the cracks given by the relations0n,j = x : Gj(x)F(x)

= n−α, x ∈ j , 16 j 6 k, whereGj ∈ C2,Gj 6= 0 on∂0, or can admit crackswhich are transversal to∂0 if in some sense their measure is less than the measureof the sequences0n,j . However, for the sake of simplicity we restrict ourselvesto the class of domains given in the definition. Let us stress that the fractality ofthe cabbage type domains has a ‘one dimensional’ structure. This is essential sincethere are examples [11, 13], showing that our main result (Theorem 7 below) failsif the domain contains too many cracks transversal to∂0 such that the fractalityof the domain loses its one dimensional structure.

The following assertion was proved in our paper [13]:

THEOREM 6. If is a cabbage type domain then∂ is Minkowski measurable,its dimension is given by the formula:

m = m(∂) = d − 1+ 1

1+ α

MPAG003.tex; 17/08/1998; 14:53; p.12


and the Minkowski content is

|∂|m = c(α)∫0

|∇F(x)| −11+α dS, 0 = ∂0 ∩1, c(α) = (2/α) α

1+α (1+ α).

The main result of this section is the following.

THEOREM 7. Let be a cabbage type domain. Then

N−(λ) = (2π)−dBd ||λd/2− c(d,m)|∂|mλm/2+ o(λm/2), λ→∞,where

c(d,m) = αB(d−12 ,

12(1+α))Bd−1

2d+α

1+α πd−α

1+α

(− ς

(1

1+ α))

andBp is the volume of the unit ball in<p, B(p, q) is the beta-function,ς(µ) isthe Riemann zeta-function(see formula(57) below).

Proof. The outline of the proof is the following. We get separate estimates forN−(λ) from below and above which together give the asymptotic expansion forN−(λ). In order to get the estimate from below we additionally impose the Dirich-let boundary condition on some hypersurfaces in. The new boundaries allowus to split into λ-dependent domains for which asymptotics of the countingfunctions can be found. To get the estimate from above, we use the Neumannboundary condition on additional boundaries or we delete some part of the interiorboundaries where the Dirichlet boundary condition was imposed. In fact, we splitdomain into a union of smaller domains slightly differently when we prove theestimate from above. We will specify this difference a little later. Now we are goingto give all the details of the estimates forN−(λ) from below, and the first step is todescribe the splitting of.

Let

n∗ = [λ 12(α+1) ln−2 λ], (30)

where[a] is the integer part ofa. Let 0n = x : F(x) = 1/nα. We have0n ⊂ 0n(0n = 0n ∩ 1). We impose the Dirichlet boundary condition on the surfaces0nwith n > n∗. Thus the larger part of is covered by the domainsDn, n 6 n∗, whichare determined by the inequalities:D1 = x : F(x) > 1,Dn = x : 1/(n−1)α >F(x) > 1/nα for n > 1. The numbern∗ is chosen in such a way that the ‘widths’of the domainsDn, n 6 n∗, are much bigger than 1/

√λ. In fact the distance

between0n and0n−1 has order O(n−(α+1)) asn→∞. Thus this distance is greaterthanC ln2(α+1) λ/

√λ if n 6 n∗.

The remaining part′ of is very close to the outer boundary of the domain.In fact, this part belongs to(const. ·L)-neighborhood of∂0 whereL = (n∗)−α =O(λ

−α2(α+1) ln2α λ). Let x = x(t, x0) be solution curves of the system

dx

dt= ∇F(x)|∇F(x)|2 , x(0) = x0 ∈ ∂0. (31)

MPAG003.tex; 17/08/1998; 14:53; p.13


Then the domain′ = 0\⋃n6n∗ Dn can be described by the inequalities 0< t <(n∗)−α, x0 ∈ ∂0 and the surfaces0n, n > n∗, are given by the relationst = n−α,x(t, x0) ∈ 1. We cut∂0 into small domainsus of ‘size’ δ, and then split′ intodomainsUs for which 0 < t < (n∗)−α, x0 ∈ us . We specifyδ and the domainsUs below. We impose the Dirichlet boundary condition on all the new boundaries.Then from the mini-max principle it follows that

N−(λ) >∑n6n∗

N−Dn(λ)+N−′(λ), N−′(λ) >∑s

N−Us (λ). (32)

We apply Theorem 1 in order to findN−Dn(λ). Only the main terms of theasymptotics ofN−Dn(λ) contribute to the asymptotics ofN−(λ). Since the asymp-totic expansions given in Theorem 1 are uniform with respect to domains, we canestimate the sum of the remainders and show that this sum does not exceed theremainder term in the asymptotics ofN−(λ).

Indeed, from Theorem 1 it follows that there exist constantsC andr indepen-dent ofn such that

|N−Dn(λ)− (2π)−dBd |Dn|λn/2| 6 Cλn−12 ln λ, λ > r,

and therefore,∣∣∣∣ ∑n6n∗

N−Dn(λ)− (2π)−dBdλn/2∑n6n∗|Dn|

∣∣∣∣6 Cn∗λn−1

2 ln λ 6 Cλn2− α

2(α+1) ln−1 λ, λ > r. (33)

In order to estimate the counting functions for the Dirichlet Laplacian in do-mainsUs we use the new system of coordinates(t, y) wherey = (y1, . . . , yn−1)

are local coordinates on∂0 and t is the parameter along trajectories of the sys-tem (31). In the new coordinates the Laplacian becomes an elliptic operator withvariable coefficients, but the domainsUs have very simple geometrical shapes.We chooseδ so small that we can fix coefficients of the operator when studyingthe problem in each of these domains. Thus we reduce the problems inUs to theproblems for operators with constant coefficients which can be solved by separationof variables. The first two terms of asymptotics of the counting functions for theseoperators with constant coefficients contribute to the asymptotic expansions forN−(λ). On the other handδ has to be not very small, so that the measure of newboundaries is not very large, and the contribution from the new boundaries does

not affect the main terms of asymptotics ofN−(λ). We chooseδ = O(λ1

4(α+1) ).Now we specify the domainsus (and therefore,Us). We introduce the local coor-

dinatesy = (y1, . . . , yn−1) on∂0 in a special way. We start with a ‘triangulation’of the boundary∂0 of domain0, but we use a cube as a standard polyhedroninstead of a simplex, i.e. we cut∂0 into a finite system of domainsQj ⊂ ∂0,16 j 6 m0, which are diffeomorphic to a cubev of the unit size in<d−1

y :

ϕj : v→ Qj ⊂ ∂0, 16 j 6 m0. (34)

MPAG003.tex; 17/08/1998; 14:53; p.14


We can assume that diffeomorphisms (34) can be extended:

ϕj : v→ Qj ⊂ ∂0, 16 j 6 m0, (35)

wherev andQj are neighborhoods ofv andQj , respectively. Since∂0 ∈ C2 wemay assume that diffeomorphisms (35) belong to the classC2. We divide cubevinto 2d−1 equal cubes of the size 1/2, then we divide each of them into 2d−1 equal

cubes of the size 1/4, and so on, until the sizeδ of the cubes is less thanλ−1

4(α+1) .Thus

1

2λ−1

4(α+1) 6 δ 6 λ−1

4(α+1) . (36)

We denote byus , 16 s 6 M = m0δ1−d , the images of the small cubes of the size

δ under the actions of the diffeomorphisms (34) and byvs the corresponding cubesof the sizeδ. (M = m0δ

1−d since the cubev contains exactlyδ1−d cubes of the sizeδ, and each of them is countedm0 times.) Coordinatesy in v ∈ <d−1

y serve as thelocal coordinates inus ⊂ ∂0. The local coordinates inUs are determined by thefunctionx = x(t, y) which is the solution of the problem

dx

dt= ∇F(x)|∇F(x)|2 , x(0, y) = ϕ(y) ∈ ∂0, y ∈ v ⊂ <d−1

y , (37)

whereϕ is one of the diffeomorphisms (34).Let us denote byVs the image ofUs in the local coordinates(t, y). We have

three types of domainsVs. If Us ∩ 1 = ∅, thenVs is the box which is the directproduct (0, (n∗)−α) × vs of the short interval and one of the small cubesvs ofthe sizeδ. We numerate these boxesVs by subindexess 6 M1. If Us ⊂ 1,thenVs is the similar direct products sliced by the cutst = n−α, n > n∗ (whichcorrespond to0n). We numerate these sliced boxes by subindexess with M1 <

s 6 M2. The remaining domainsVs, M2 < s < M = m0δ1−d , correspond to

Us that have nonempty intersections with∂1, and some of the cutst = n−α areincomplete there. We extend the cuts in these type of domains (and we extend0nin the correspondingUs) to all values ofy ∈ vs . We impose the Dirichlet boundarycondition on all additional cuts. The inequalities (32) are still valid. Thus all thedomainsVs, s > M1, have a similar structure which differs of the structure of thedomainsVs with s 6 M1. To keep this difference in mind we rewrite (32) in theform:

N−(λ) >∑n6n∗

N−Dn(λ)+∑s6M1

N−Us (λ)+

+∑

M1<s6MN−Us (λ), M = m0δ

1−d. (38)

Now we show that the sum of(d − 1)-dimensional measures of the domainsuswithM2 < s 6 M does not exceedCδ:∑

M2<s6M|us | 6 Cδ. (39)

MPAG003.tex; 17/08/1998; 14:53; p.15


In particular, from here it follows that∣∣∣∣( ⋃s>M2

us

)\0∣∣∣∣ 6 Cδ, (40)

where0 = ∂0∩1. To show (39) let us consider0′ = ∂0∩∂1 (the edge of0)and let (0′)ε be the set of pointsx on∂0 such that the distanceρ(x, 0′) < ε. Since∂0 and∂1 are transversal there exists a constantA such that the trajectories ofthe problem (37) emitted from∂0\(0′)ε with ε = A(n∗)−α do not intersect∂1

when 06 t 6 (n∗)−α. Thus domainsus with M2 < s 6 M have nonemptyintersections with (0′)ε, ε = A(n∗)−α. If we also take into account that domainsus are small (they are the images of cubesvs under the action of diffeomorphisms(34)) we will get thatus withM2 < s 6 M belong to (0′)ε with ε = A(n∗)−α+Cδ.Sinceδ > (n∗)−α for λ large enough (see (36) and (30)) we obtain (39).

Now we study the counting functionN−Us (λ) for the Laplacian in small domainsUs were local coordinates can be used. The first step is to write the Laplacian in(t, y) coordinates. Let

gi,j = gi,j (t, y) = 〈xyi , xyj 〉(=∑k

(xk)yi (xk)yj

)(41)

and let[gi,j ] = [gi,j (t, y)] = [gi,j (t, y)]−1 be the inverse matrix. Let

J = J (t, y) = 1

|∇F |√

det[gi,j ]. (42)

Then

1 = P(t, y, ∂t , ∂y) = 1

J

[∂

∂tJ |∇F |2 ∂

∂t+ ∂

∂yiJgi,j

∂

∂yj

]. (43)

The important feature of this formula is the absence of the mixed derivatives∂2

∂t∂yiin the right-hand side. This formula can be found in many books, but for the

sake of completeness we shall prove it here. Letz = (t, y) andA be the JacobymatrixA = [xz] =

[xtxy

]. Then dx = A∗ dz and

|dx|2 = 〈A∗ dz,A∗ dz〉 = 〈AA∗ dz,dz〉.Similarly∇x = A−1∇z and

1 = (∇x)2 = 〈A−1∇z, A−1∇z〉 = 〈∇z, (AA∗)−1∇z〉 +Q, (44)

whereQ is an operator containing only the first order derivatives. Since1 is asymmetric operator, it is a symmetric operator in the new coordinates with the dotproduct〈u, v〉 = ∫ uv|detA|dz. Together with (44), this leads immediately to

1 = 1

detA〈∇z, (detA)(AA∗)−1∇z〉. (45)

MPAG003.tex; 17/08/1998; 14:53; p.16


Sincext andxy are orthogonal we have

AA∗ =[ |xt |2 0

0 gi,j

]=[ |∇F |−2 0

0 gi,j

]. (46)

From here it follows that detA = J . Together with (45) and (46) it proves (43).Thus

N−Us (λ) = N−Vs (λ), (47)

whereN−Vs (λ) is the counting function for the Dirichlet problem inVs for theoperator (43).

We would like to compare the eigenvalues of the Dirichlet problem inVs foroperatorsP(t, y, ∂t , ∂y) andP(0, y0, ∂t , ∂y), wherey0 is the center of the cubevswhich is the base forVs. We will use the following simple consequence of themini-max principle. Let

Pi = 1

bi(z)〈∇z, Bi(z)∇z〉, z = (t, y) ∈ V, i = 1,2, (48)

be two elliptic operators in a bounded domainV such that 0< b1(z) 6 b2(z),the matricesB1(z), B2(z) are symmetric and positive,B1(z) > B2(z). Let λi,1 <λi,2 6 λi,3 6 · · · be eigenvalues of the Dirichlet problem for operatorsPi in V .Thenλ1,j > λ2,j , j = 1,2,3, . . . . This assertion follows immediately from thefact that

λi,j = infHj

supf∈Hj

(∫V〈Bi(z)f, f 〉dz∫V|f |2bi(z)dz

)1/2

,

whereHj are j -dimensional subspaces of the spaceH 0,1. Heref ∈ H 0,1 if fbelongs to the Sobolev spaceH 1(V ) andf = 0 on ∂V . Thus ifN−i (λ) are thecounting functions for the Dirichlet problem for operatorsPi in V , then

N−1 (λ) 6 N−2 (λ). (49)

Let us write operatorP (see (43)) in the form (48):

P = 1

J (t, y)〈∇z, B(t, y)∇z〉, B =

[ |∇F |2 00 gi,j

], z = (t, y) ∈ Vs,

where 0< t < (n∗)−α and|y − y0| 6 δ for z ∈ Vs. From (30), (36) it follows thatδ > (n∗)−α, and therefore there is a constantc such that

J (t, y) > J (0, y0)(1− cδ), B(t, y) 6 B(0, y0)(1− cδ)−1 as(t, y) ∈ Vs. (50)

This leads to (49) for the operatorsP1 = (1− cδ)−2P(0, y0, ∂t , ∂y), P2 = P(t, y,∂t , ∂y). Thus

N−Vs (λ) > N−s ((1− cδ)2λ), λ > 1, (51)

MPAG003.tex; 17/08/1998; 14:53; p.17


whereN−s (λ) is the counting function for the Dirichlet problem for the operatorP(0, y0, ∂t , ∂y) in Vs. Together with (47) and (36) this gives

N−Vs (λ) > N−s

((1− cλ −1

4(α+1))2λ), λ > 1. (52)

Now we are going to study the counting functionN−s (λ) for the operator

P(0, y0, ∂t , ∂y) = (|∇F |2)(0, y0)∂2

(∂t)2+ gi,j (0, y0)

∂2

∂yi∂yj(53)

in Vs. First we consider the more complicated case whens > M1. The variablest andy in the Dirichlet problem for operator (53) in the domainVs can be sep-arated, i.e., the eigenvalues have the formτk + νl whereτk are the eigenvaluesof the corresponding one-dimensional (Sturm–Liouville) problem for the operatorσ 2 d2

dt2 , σ = |∇F |(0, y0), andνl are the eigenvalues of the Dirichlet problem for theoperator ∂

∂yigi,j (0, y0)

∂∂yj

in the cubevs . ThusN−s (λ) is the convolution

N−s (λ) =∫ λ

0N1(λ− τ)dN2(τ)

(=∫ λ

0N2(λ− τ)dN1(τ)

)(54)

of the counting functions (we denote them byN1(λ) andN2(λ), respectively) forthe corresponding one-dimensional problem and the problem in the cubevs .

Let us specify the one-dimensional problem. DomainVs is sliced into the thin-ner domains by the cutst = n−α, n > n∗. ThusN1(λ) is the eigenvalue countingfunction for the problem

σ 2d2u

dt2= λu, 0< t < (n∗)−α, t 6= n−α;

u(0) = u(n−α) = 0, n > n∗. (55)

Problem (55) is the Sturm–Liouville problem on the set of the intervals(n−α,(n+ 1)−α), n > n∗. Two terms of the asymptotic expansion ofN1(λ) are found byLapidus [9] in the case whenn∗ does not depend onλ. The expansion is expressedthrough the Minkowski measure of the sequence of the pointsn−α, n > n∗, and ithas the following form

N1(λ) = (πσ )−1L√λ+ π−µζ(µ)(α/σ )µλµ/2+ o(λµ/2), λ→∞. (56)

HereL = (n∗)−α is the length of the set of the intervals in (55),µ = 1/(α + 1)is the Minkowski measure of the end points of the intervals,ζ(·) is the Riemannzeta-function, and coefficient forλµ/2 can be expressed through the Minkowskicontent of the sequencen−α. Since 0< µ < 1 the value of the functionζ(τ) =∑∞

j=1 j−τ , Reτ > 1, at the pointτ = µ can be written in the form

ζ(µ) = 1

µ− 1+∫ ∞

1([x]−µ − x−µ)dx, (57)

MPAG003.tex; 17/08/1998; 14:53; p.18


where[x] is the integer part ofx. One may verify that the Lapidus result and itsproof remain valid ifn∗ = n∗(λ) tends to infinity sufficiently slow. In particular,formula (56) is valid whenn∗ is given by (30).

The standard Weyl formula is valid for the eigenvalue counting functionN2(λ)

for the operatorQ = gi,j (0, y0)∂2

∂yi∂yjin vs :

N2(λ) = (2π)1−dBd−1|vs |√det[gi,j (0, y0)]

λd−1

2 +O(λd−2

2)

asλ→∞, (58)

whereBd−1 is the volume of the unit ball in<d−1, |vs | is the volume of the cubevs ⊂ <d−1 (of the sizeδ). It is not important for us to have a sharp estimate ofthe remainder in (58), but we need an estimate which is uniform with respect tooperatorsQ. We may get it with the help of Theorem 1 from which it follows that∣∣∣∣N2(λ)− (2π)

1−dBd−1|vs |√det[gi,j (0, y0)]

λd−1

2

∣∣∣∣ 6 Cλd−22

(|∂vs | ln(δ

√λ)+ |vs |

δ

),

λ > λ0, (59)

whereC andλ0 do not depend ons. However we can not get (59) by direct ref-erence to Theorem 1 since the estimate (59) concerns to operatorQ but not theLaplacian.

Let us show that (59) holds. SinceQ has constant coefficients there is a lineartransformationT of <d−1 onto itself which reducesQ to the Laplace operator.Let v′s be the image ofvs , ρ be the distance between any two points andρ ′ bethe distance between their images. The transformationT changes the volumesby (det[gi,j (0, y0)])−1/2 times (in particular,|v′s | = |vs |/

√det[gi,j (0, y0)]), and

it changes the measure of the boundary|∂vs| and the distances in the followingway:√

λmax

det[gi,j (0, y0)] |∂vs | > |∂v′s | >

√λmin

det[gi,j (0, y0)] |∂vs |,ρ√λmin

> ρ ′ > ρ√λmax

,

whereλmax, λmin are the maximal and the minimal eigenvalues of[gi,j (0, y0)]. Inparticular, from here it follows that(∂v′s)ε (ε-neighborhood of∂v′s) is contained inthe image of (∂vs)ε/√λmin

. Estimate (5) withr0 = 1 is valid for the cubev of the unitsize, and therefore it is valid forvs with r0 = δ and with the same constantA = A0.

Then from the properties of the transformationT , it follows that forε 6√λminδ

|(∂v′s)ε| 6∣∣T [(∂vs)ε/√λmin

]∣∣ 6 ∣∣(∂vs)ε/√λmin

∣∣√det[gi,j (0, y0)]

6 Aε|(∂vs)|√λmin det[gi,j ]

6 Aε

λmin|∂v′s |,

MPAG003.tex; 17/08/1998; 14:53; p.19


i.e. (5) is valid forv′s with r0 = √λminδ andA = A0/λmin. It allows us to applyTheorem 1 to the Dirichlet Laplacian inv′s . Taking into account the existence ofconstantsa, b independent ofy0 such thatb > λmax > λmin > a > 0 we get (59)whereλ0 can be found from the inequality

√λ > 1/(

√aδ) (see (6)) and (36).

Due to (36) we can replace ln(δ√λ) by lnλ in the right-hand side of (59). Using

also relations,|vs | = δd−1, |∂vs | = 2d−1δd−2 we get

N2(λ) = (2π)1−dBd−1|vs |√det[gi,j (0, y0)]

λd−1

2 + n(λ); |n(λ)| 6 Cδd−2λd−2

2 ln√λ,

λ > λ0, (60)

whereC andλ0 do not depend ons, andBd−1 is the volume of the unit ball in<d−1.

Now we substitute (56) and (60) into (54). SinceN1(λ) is monotonic from (56)and (60) it follows that∫ λ

0N1(λ− τ)dn(τ) 6 N1(λ)

∫ λ

0dn(τ) 6 N1(λ)n(λ) (61)

6(c1L√λ+ c2λ

µ2)Cδd−2λ

d−22 ln λ 6 Cλd−1

2 ln λ.

For the remainder term o(λµ/2) in (56) we have∣∣∣∣ ∫ λ

0o((λ− τ)µ2 )dτ

d−12

∣∣∣∣ 6 d − 1

2λd−3

2

∫ λ

0

∣∣o((λ− τ)µ2 )∣∣ dτ (62)

= d − 2

2λd−3

2

∫ λ

0

∣∣o(τ µ2 )∣∣ dτ= o

(λd−1+µ

2)

asλ→∞.There are no difficulties in evaluating the convolution of the main terms of (56),(60):∫ λ

0(λ− τ) k2 dτ

d−12 = d − 1

2B

(d − 1

2,k + 2

2

)λd−1+k

2

= k

2B

(d + 1

2,k

2

)λd−1+k

2 . (63)

Herek = 1 or k = µ, andB is the beta-function, i.e.B(p, q) = ∫ 10 x

p−1(1−x)q−1 dx. Thus, from (60), (56) and (54)–(63) it follows that

N−s (λ) = As,1(πσ )−1Lλ

d2 +As,µπ−µζ(µ)(α/σ )µλd−1+µ

2 ++ δn−2O

(λd−1

2 ln λ)+ |vs |o(λd−1+µ

2), λ→∞, (64)

wheres > M1 and

µ = 1

α + 1, σ = |∇F |(0, y0), As,k = kB(d+1

2 ,k2)Bd−1|vs |

2(2π)d−1√

det[gi,j (0, y0)].

MPAG003.tex; 17/08/1998; 14:53; p.20


It is important that here and in all formulae below the estimates of the remain-ders O(·) and o(·) are uniform with respect tos.

We are going to express the coefficients in (64) through the volume|Us| of thedomainUs ⊂ ′ and (d − 1)-dimensional measure of its baseus ⊂ ∂0. Recallthat the Jacobian detA, whereA = [ dx

dtdy ], is equal to the function (42) (see (46)),

and one can replace det[gi,j ] by (det[gi,j ])−1 in (42). Taking also into account thatdiameter of the domainUs and diameter of its imageVs in (t, y) coordinates do notexceedCδ (see the arguments used for (50)) we get

detA = 1

σ√

det[gi,j (0, y0)]+O(δ) for (t, y) ∈ Vs,

and therefore

|Us| = |Vs|σ√

det[gi,j (0, y0)]+O(δ|Vs|). (65)

SinceL = (n∗)−α (see (56)) is the height of the domainVs andvs is its base we canreplace|Vs | in (65) byL|vs| and then specify the remainder with the help of (36)and relations|vs | = δd−1 and (30). Thus O(δ|Vs|) = O(δdL) = δd−1o(λ

−α2(α+1) ), and

from (65) it follows that

L|vs|σ√

det[gi,j (0, y0)]= |Us| + δn−1o

(λd−1+µ

2), λ→∞. (66)

Now if we also take into account thatB(d+12 ,

12)Bd−1 = Bd we can rewrite the first

term in the right-hand side of (64) in the form

(2π)−dBd |Us|λd2 + δd−1o(λd−1+µ

2), λ→∞.

In order to simplify the coefficient of the second term in the right-hand side of(64) we have to note first of all that from (37) it follows that dS = (detA)|∇F |dy,where dS is (d − 1)-dimensional measure of an element of the surface∂0. Thus∫

us

1

|∇F(0, y)|µ dS =∫vs

1

|∇F(0, y)|µ√det[gi,j (0, y)] dy.

From here similarly to (66) it follows that

|vs|σµ√

det[gi,j (0, y0)]=∫us

1

|∇F(0, y)|µ dS + δd−1o(1), λ→∞.

We use this relation to simplify the second term in the right-hand side of (64).Since the last two terms in (64) can be written in the formδd−1o(λ

d−1+µ2 ) formula

(64) implies

N−s (λ) = (2π)−dBd |Us|λd2 − as(d, µ)λd−1+µ2 +

+ δd−1o(λd−1+µ

2), λ→∞, (67)

MPAG003.tex; 17/08/1998; 14:53; p.21


where

as(d, µ) = −µB(d+1

2 ,µ

2 )Bd−1

2d(π)d−1+µ ζ(µ)αµ∫us

1

|∇F(0, y)|µ dS > 0

(the constant is positive becauseζ(µ) < 0).Now let us take the sum of equalities (67) with respect to alls ∈ (M1,M].

Since the decay of the remainders o(·) in (67) is uniform with respect tos andM = m0δ

1−d (see (38)), the sum of the remainders has the order o(λd−1+µ

2 ). From(40) it follows that∑

s<M1

∫us

1

|∇F(0, y)|µ dS =∫0

1

|∇F(0, y)|µ dS +O(δ).

Thus∑s>M1

N−s (λ) = (2π)−dBd( ∑s>M1

|Us|)λd2 − a(d,µ)λd−1+µ

2 +

+ o(λd−1+µ

2), λ→∞, (68)

where

a(d,µ) = −µB(d+1

2 ,µ

2 )Bd−1

2d(π)d−1+µ ζ(µ)αµ∫0

1

|∇F |µ dS > 0. (69)

The asymptotic expansion forN−s (λ) with s 6 M1 can be obtained similarly to(67). The only difference is that the one dimensional Sturm–Liouville problemnow is much simpler. It is the problem on the interval(0, L), but not on the systemof intervals as in (55). The eigenvalue counting functionN1(λ) for this problemcan be found immediately, and it has the form (56), but without the middle termin the right-hand side (in fact, the remainder also can be specified). It leads to theanalog of (67), but without the middle term in the right-hand side. Correspondingly(68) holds if the limits of the summations are changed tos 6 M1 and the middleterm in the right-hand side is omitted. Together with (68) it gives the followingresult:∑

s6MN−s (λ) = (2π)−dBd |′|λd2 − a(d,µ)λd−1+µ

2 +

+ o(λd−1+µ

2), λ→∞. (70)

Let us note that|′| has the order O((n∗)−α) = O(λ−α

2(α+1) ln λ2α) which is the‘thickness’ of the domain′. Thus

∑s6M N

−s ((1− λ

−14(α+1) )2λ) also has the form

(70). Together with (52) this implies∑s6M

N−Vs (λ) > (2π)−dBd |′|λd2 − a(d,µ)λd−1+µ2 +

+ o(λd−1+µ

2), λ→∞. (71)

MPAG003.tex; 17/08/1998; 14:53; p.22


Together with (32) and (33) this gives the estimate forN−(λ) from below:

N−(λ) > (2π)−dBd ||λd2 − a(d,µ)λd−1+µ2 + o

(λd−1+µ

2), λ→∞. (72)

Hereµ = 11+α (see (56)), anda(d,µ) is given by (69) and (57). It is obvious that

Theorem 7 will be proved if we get the same estimate forN−(λ) from above.As we mentioned in the beginning of the proof of Theorem 7 the estimate of

N−(λ) from above can be proved absolutely similarly to (72). Now we describe thechanges which we need to make to get the estimate from above. First of all we im-pose the Neumann but not the Dirichlet boundary condition on0n = x : F(x) =n−α, n 6 n∗. Then from the mini-max principle we have the following inequalityinstead of (32):

N−(λ) 6∑n6n∗

N+Dn(λ)+N′(λ),

whereN′(λ) is the counting function of the Laplacian in′ with the Dirichletboundary condition on0n, n < n∗, and the Neumann boundary condition on0n∗ .Then similarly to (33) (using the second assertion of the Theorem 1 instead of thefirst one), we have∣∣∣∣ ∑

n6n∗N+Dn(λ)− (2π)−dBdλn/2

∑n6n∗|Dn|

∣∣∣∣ 6 Cλn2− α2(α+1) ln−1 λ, λ > r.

In order to get an estimate forN′(λ) from above we could try to split′ intothe set of domainsUs which was used earlier and impose the Neumann boundarycondition instead of the Dirichlet condition on all additional boundaries. However,this approach will not work because the boundaries of the basesus of the domainsUs are not smooth, and we will not be able to use Theorem 1 to get an analog of(64) in the case of the Neumann boundary conditions. Thus we use a covering of′ by domainsUs with basesus instead of the splitting of′.

It is not difficult to construct a family of neighborhoodsvh, 0 < h < 1, ofthe cubev ∈ <d−1 of the unit size such that the following conditions hold: (1)∂vh ∈ C2, (2) sectional curvatures of∂vh do not exceed 1/h, (3) each point of∂vh can be touched by a ball of radiush from inside, (4)(d − 2)-dimensionalmeasure of∂vh does not exceed a constant independent ofh, i.e. |∂vh| < C, (5)(d − 1)-dimensional measure ofvh\v does not exceedCh where the constantCdoes not depend onh, (6) Inequality (5) is valid with someA and r0 = 1. Weput h = 1/(δ

√λ) and consider a homothety with the coefficientδ. It gives us the

neighborhoodv = v(δ, λ) of the cubev(δ) ∈ <d−1 of the sizeδ such that (1)∂v ∈ C2, (2) sectional curvatures of∂v do not exceed

√λ, (3) each point of∂v

can be touched by a ball of radius 1/√λ from inside ofv, (4) (d − 2)-dimensional

measure of∂v does not exceedCδd−2, i.e. |∂v| < Cδd−2, (5) (d − 1)-dimensionalmeasure ofv\v(δ) does not exceedCδd−2/

√λ, i.e. |v\v(δ)| < Cδd−2/

√λ, (6)

Inequality (5) is valid with someA andr0 = δ. We construct such a neighborhood

MPAG003.tex; 17/08/1998; 14:53; p.23


for each cubevs , 1 6 s 6 M = m0δ1−d , of the sizeδ, introduced earlier, and

denote these neighborhoods byvs . Let us be the image ofvs under the action ofthe corresponding diffeomorphism (35), and letUs be the subdomain of′ givenin local coordinates (see (37)) by the relations 0< t < (n∗)−α, ϕ(y) ∈ us . It isobvious that′ ⊂ ⋃s6M Us. We impose the Neumann boundary condition on thelateral sides (ϕ(y) ∈ ∂us) of Us. Then

N′(λ) 6∑s6M

NUs (λ),

whereNUs (λ) is the eigenvalue counting function for the Laplacian inUs withthe Neumann boundary conditions on the top part (which belongs to0n∗) of theboundary and on the lateral part of the boundary, and with the Dirichlet boundaryconditions on the base and the cuts0n, n > n∗.

We denote byVs the image ofUs in the (t, y) coordinates. As earlier we havethree types of domainsVs. The domains of the first type are cylinders(0, (n∗)−α)×us (it is the case whenVs ∩1 = ∅), the domains of the second type are similarcylinders sliced by the cutst = n−α, n > n∗ (it is the case whenVs ⊂ 1) andthe domains of the third type are the cylinders in which some of the cuts are notcomplete, i.e. the cuts exist not for all values ofy ∈ us (if Vs ∩ ∂1 6= ∅). Wedelete all the cuts in the third type of domainsVs, so they will have the same formas the domains of the first type (earlier we continued these cuts to get the estimateof N−(λ) from below). Then we get the analog of (38):

N−(λ) 6∑n6n∗

N+Dn(λ)+∑s6m1

NUs (λ)+∑

m1<s6MNUs (λ),

where the middle term in the right-hand side corresponds to domainsUs of thefirst and third type and the last term corresponds to domainsUs of the second type(which are sliced by the cuts).

After the change of the variablesx → (t, y) we get the analog of (47):

NUs (λ) = NVs (λ),whereNVs (λ) is the counting function for the operator (43) inVs with the Dirichletboundary condition on that part of the boundary where we had this conditionin x coordinates, and with the Neumann type boundary condition of the form〈∇z, (AA∗)−1ν〉 = 0 on the other part of the boundary (where we had the usualNeumann boundary condition inx coordinates). Hereν is the normal to the bound-ary of Vs.

Inequality (49) for the counting function holds for operators (48) when theDirichlet boundary condition is imposed on some part of the boundary and theNeumann type boundary condition〈∇z, B−1

i ν〉 = 0 holds on the remaining part ofthe boundary. Thus similarly to (52) we have:

NVs (λ) 6 Ns((

1+ cλ −14(α+1)

)2λ), λ > 1,

MPAG003.tex; 17/08/1998; 14:53; p.24


whereNs(λ) is the counting function for the operator (53) inVs with the Neu-mann type boundary condition of the form〈∇z, [(AA∗)−1(0, y0)]ν〉 = 0 on thelateral side and on the top ofVs and with the Dirichlet boundary condition on theremaining part of∂Vs.

As earlier the variablest andy can be separated when we studyNs(λ), and itleads to an analog of (54), (56) and (59):

Ns(λ) =∫ λ

0N1(λ− τ)dN2(τ).

HereN1(λ) has the form (56) ifs > m1 or the same form without the middle termin the right-hand side ifs 6 m1, and estimate (59) with|vs | and |∂vs | instead of|vs| and |∂vs | is valid for N2(λ). To get the expansion (56) forN1(λ), we haveto note only that the change of the boundary condition (from the Dirichlet to theNeumann) in (55) at one pointt = (n∗)−α does not effect the main terms of the ex-pansion (56). To get the estimate forN2(λ) we use the same linear transformationT : <d−1 → <d−1 which transfers the second term in the right-hand side of (53)into the Laplacian. This transformation also reduces the Neumann type boundarycondition mentioned above to the usual Neumann boundary condition. After thisreduction the estimate forN2(λ) follows from the second assertion of Theorem 1(and from the properties ofvs described earlier).

We can replace|vs | by |vs | and |∂vs| by |∂vs | in the right-hand side of theestimate forN2(λ) because|vs | ∼ |vs| and |∂vs | ∼ |∂vs | asλ → ∞. Then wecan replace|vs | by |vs | in the left-hand side of the estimate forN2(λ) because|vs|− |vs | = Cδd−2/

√λ and therefore the error due to the replacement does not

exceed the right-hand side of (59). Thus (56) and (59) are valid in the case underconsideration. After this all the arguments used to get the estimate from belowwork without any changes and lead to the same estimate forN−(λ) from above.

The proof is now complete. 2

References

1. Berry, M. V.: Some geometric aspects of wave motion: Wave front dislocations, diffractioncatastrophes, diffractals, inGeometry of the Laplace Operator, Proc. Symp. Pure Math., Vol.36, Amer. Math. Soc., Providence, 1980, pp. 13–38.

2. Brossard, J. and Carmona, R.: Can one hear the dimension of a fractal?,Comm. Math. Phys.104(1986), 103–122.

3. Fleckinger, J. and Vasil’ev, D.: Tambour fractal: example d’une formule asymptotique à deuxtermes pour la ‘foction de comptage’,C.R. Acad. Sci. Paris, Série I311(1990), 867–872.

4. Fleckinger, J. and Vasiliev, D.: An example of a two-term asymptotics for the ‘countingfunction’ of a fractal drum,Trans. Amer. Math. Soc.337(1) (1993), 99–117.

5. Hua, C. and Sleeman, B. D.: Fractal drums and then-dimensional modified Weyl–Berryconjecture,Comm. Math. Phys.168(1995), 581–607.

6. Jaksic, V., Molchanov, S. and Simon, B.: Eigenvalue asymptotics of the Neumann Laplacian ofregions and manifolds with cusps,J. Funct. Anal.106(1) (1992), 59–79.

MPAG003.tex; 17/08/1998; 14:53; p.25


7. Ivrii, V. Ja.:Precise Spectral Asymptotics for Elliptic Operators, Lecture Notes in Mathematics1100, 1984.

8. Lapidus, M. L.: Fractal drum, inverse spectral problems for elliptic operators and a partialresolution of the Weyl–Berry conjecture,Trans. Amer. Math. Soc.325(1991), 465–529.

9. Lapidus, M. L.: Spectral and fractal geometry: From Weyl–Berry conjecture for the vibrationsof fractal drums to the Riemann zeta-function, inDifferential Equations and MathematicalPhysics, Proc. UAB Intern. Conf. (Birmingham, 1990), Academic Press, New York, 1992, pp.151–182.

10. Lapidus, M. L. and Fleckinger-Pellé, J.: Tambour fractal: vers une résolution de la conjecturede Weyl–Berry pour les valeurs propresdu Laplacien,C.R. Acad. Sci. Paris Sér. I Math.306(1988), 171–175.

11. Lapidus, M. L. and Pomerance, C.: Countrexamples to the modified Weyl–Berry conjecture onfractal drums,Math. Proc. Cambridge Philos. Soc.119(1996), 167–178.

12. Levitin, M. and Vassiliev, D.: Spectral asymptotics, renewal theorem, and the Berry conjecturefor a class of fractals,Proc. London Math. Soc. (3)72 (1996), 188–214.

13. Molchanov, S. and Vainberg, B.: On spectral asymptotics for domains with fractal boundaries,Comm. Math. Phys.183(1997), 85–117.

14. Seeley, R. T.: A sharp asymptotic remainder estimate for the eigenvalues of the Laplacian in adomain of<3, Adv. in Math.29 (1978), 244–269.

15. Simon, B.: The Neumann Laplacian of a jelly roll,Proc. Amer. Math. Soc.114(3) (1992),783–785.

MPAG003.tex; 17/08/1998; 14:53; p.26


171

Minimal TerminalQ-Factorial Models of DrinfeldCoarse Moduli Schemes

IGOR YU. POTEMINE?Laboratoire de Mathématiques Emile Picard, Université Paul Sabatier, 118 route de Narbonne,31062 Toulouse, France, e-mail: [email protected]

(Received: 1 October 1997; accepted: 22 January 1998)

Abstract. In this article we formally prove that the coarse moduli schemeMr(1) of rational Drinfeldmodules of rankr is an affineQ-factorial toricA-variety of relative dimensionr−1. The correspond-ing coordinate ring and the simplicial cone are constructed viaJ -invariants. Thej -invariants playthe role of extremal rays of this cone and define a finite flat coveringMr(1) → Ar−1

Aétale over

Gr−1m,A. The ramification of this covering over the divisor complementary toGr−1

m,A is described. It isalso shown that the singular locus ofMr(1) coincides with the locus of elliptic points Ell(Mr(1)) forr greater than 2. We construct a minimal terminalQ-factorial equivariant compactification ofMr(1)and we prove the uniqueness of such a compactification. Finally, the coarse moduli surfaceM3(1) isdescribed in detail.

Mathematics Subject Classifications (1991):Primary: 11G09; secondary: 14D22, 14E30, 14M25.

Key words: Drinfeld module, coarse moduli scheme, minimal model, toric variety.

1. Introduction

LetA = Fq[T ] be the ring of polynomials over the finite fieldFq andK = Fq(T )its quotient field. LetL be a field equipped with a non-trivial morphismαL: A→ L

andE a Drinfeld module of rank 3 overL:

T 7→ TE = αL(T )+ a1τ + a2τ2+ a3τ

3, a3 ∈ L∗. (1.1)

In the previous article [Po], we have defined thej -invariant as the pair:

j (E) = (j1(E), j2(E)) =(aq2+q+11

a3,aq2+q+12

aq+13

). (1.2)

This invariant gives a finite flat covering

M3(1)→ SpecA[j1, j2] (1.3)

? Partially supported by KIAS research grant M-97009.

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172 IGOR YU. POTEMINE

of degreeq2+ q+1 étale outside of the divisorj1j2 = 0. In general, we define thebasicJ δ1δ212 -invariant by the formula:

Jδ1δ212 =

aδ11 a

δ22

aδ33

, (1.4)

where integersδ1, δ2, δ3 verify the equation

(A1) δ1+ (q + 1)δ2 = (q2+ q + 1)δ3

and satisfy the conditions

(A2) 06 δ1, δ2 6 q2+ q + 1, g.c.d.(δ1, δ2, δ3) = 1.

We are able to prove that

M3(1) = SpecA[Jδ1δ212

], (1.5)

whereJδ1δ212

is the system of all basicJ -invariants. The covering (1.3) is the pro-

jection to the coordinatesj1, j2. We can describe equations definingM3(1), its min-imal equivariant smooth compactificationM3

min(1), the zeta function ofM3min(1),

etc.The main goal of this article is to obtain an analogous description of the coarse

moduli schemeMr(1) of rational Drinfeld modules for anyr. A sketch of the con-struction of the Satake compactification of Drinfeld modular varieties was given byE.-U. Gekeler [Ge2]. The compactification of modular varieties of rational Drinfeldmodules into complete normal varieties is constructed by M. Kapranov [Ka]. Wecan formally prove that the coarse moduli scheme of rational Drinfeld modules ofrank6 r is the weighted projective space:

Mr(1) = PA(q − 1, q2 − 1, . . . , qr − 1) (1.6)

(cf. [Ka], 1.6). In general, R. Pink has constructed a smooth compactification ofDrinfeld modular varieties (cf. [Pi]). Our aim is to prove the existence of a uniqueminimal proper modelMr

min(1) withQ-factorial terminal singularities. The explicitconstruction is given in Section 6. The proof of the minimality and the uniquenessof this model uses a theorem of V. Danilov [Da2] (for threefolds) and of M. Reid[Re] (in general) on the decomposition of toric morphisms. This result also im-plies the existence and the uniqueness of the smoothG-desingularization and ofthe smoothG-compactification of the coarse moduli threefoldM4(1) ([BGS]) al-though we do not construct it explicitly in this article. We consider neither rigidanalytic uniformization nor algebraic Drinfeld modular forms here.

2. J -Invariants of Drinfeld Modules

Let K = Fq(T ) be a rational function field of characteristicp andA = Fq[T ].Furthermore, letS be anA-scheme,L a lineA-bundle overS andGa(L) the addi-tive group scheme ofL overS. Let αL: A→ 0(S,L) be a structural morphism.

MPAG012.tex; 4/09/1998; 11:15; p.2

MINIMAL TERMINAL Q-FACTORIAL MODELS 173

A DrinfeldA-module of rankr > 1 overS is a morphismE: A→ EndS(Ga(L)),T 7→ TE, such that

TE = αL(a)+∑

16k6rak,Eτ

k, (2.1)

whereak,E ∈ 0(S,L1−qk ), ar,E ∈ 0(S, (L1−qr )∗) andτ is the relative Frobeniusmorphism of degreeq. We write ak instead ofak,E if there is no confusion andwe denote1(E) = ar,E. A morphismof Drinfeld modules overS is a morphismof the additive group schemes commuting with theA-action. Any isomorphism ofDrinfeld modules overS is given by someu ∈ 0(S,L∗) ([Dr], §5). On the otherhand,D = u−1Eu is a Drinfeld module for anyu ∈ 0(S,L∗). We have

ak,D = uqk−1ak,E. (2.2)

For integersk1, . . . , kl one denotes

d(k1, . . . , kl)def= g.c.d.(k1, . . . , kl). (2.3)

We also suppose thatr is greater than 2.

DEFINITION 2.1. For any multi-index(k1, . . . , kl) where 16 k1 < · · · < kl 6r − 1 theJ δ1...δlkl ...kl

(E)-invariant is the global section

Jδ1...δlk1...kl

(E) = aδ1k1· · · aδlkl

1(E)δr, (2.4)

where the integersδ1, . . . , δl verify the equation

(B1) δ1(qk1 − 1)+ · · · + δl(qkl − 1) = δr(qr − 1).

If δ1, . . . , δl satisfy also the conditions

(B2) 06 δi 6 (qr − 1)/(qd(i,r) − 1), 16 i 6 l; d(δ1, . . . , δl, δr) = 1

we say thatJ δ1...δlk1...kl(E) is thebasicJ -invariant. Denote

Jδ1...δlk1...kl

the system of all basicJ -invariants. By definition, the global sections

jk(E) = J δkk (E) =a(qr−1)/(qd(k,r)−1)k

1(E)(qk−1)/(qd(k,r)−1)

, 16 k 6 r − 1, (2.5)

are called thejk-invariantsof E. Moreover, the(r − 1)-tuple

j (E) = (j1(E), . . . , jr−1(E)) (2.6)

is called thej -invariant of E.

MPAG012.tex; 4/09/1998; 11:15; p.3


THEOREM 2.2. (i)If two Drinfeld modulesD, E of rankr over anA-schemeSare isomorphic then allJ δ1...δlk1...kl

-invariants coincide.(ii) LetD, E be Drinfeld modules of rankr over a separably closedA-fieldL

having the samej -invariant j (D) = j (E). Then there exists a Drinfeld moduleD′isomorphic toD such that

ak,E = ξk · ak,D′, ξ (qr−1)/(qd(k,r)−1)k = 1, 16 k 6 r − 1. (2.7)

(iii) LetD,E be Drinfeld modules of rankr over a separably closedA-fieldL.If their basicJ -invariants coincide:

Jδ1...δlk1...kl

(D) = J δ1...δlk1...kl(E) (2.8)

for all integersδ1, . . . , δl satisfying(B1) and(B2) then these modules are isomor-phic.

Proof.(i) This is the immediate consequence of the definition of theJ -invariants.(ii) One can always findD′ isomorphic toD such that1(D′) = 1(E). In fact,

any root of the separable equation

Xqr−1 = 1(D)−11(E)

gives such an isomorphism. Since

jk(D′) = a

(qr−1)/(qd(k,r)−1)k,D′

1(E)(qk−1)/(qd(k,r)−1)

= a(qr−1)/(qd(k,r)−1)k,E

1(E)(qk−1)/(qd(k,r)−1)

= jk(E),

for any 16 k 6 r−1, thejk-invariant definesak up to an elementξk ∈ F∗qr /F∗qd(k,r) .In particular, thej -invariant defines a finite number of the isomorphism classes ofDrinfeld modules.

(iii) If (2.8) is true for all integersδ1, . . . , δr satisfying (B1) and (B2) then itis also true for all integers satisfying only (B1). Indeed, anyJ -invariant satisfying(B1) moduloj -invariants gives an element lying in the ‘fundamental parallelotope’corresponding to the system

Jδ1...δlk1...kl

.

According to the previous section of this proof we may suppose that1(D) =1(E) and

ak,E = ξk(D,E) · ak,D, ξ (qr−1)/(qd(k,r)−1)k = 1, 16 k 6 r − 1. (2.9)

If there is no confusion we writeξk instead ofξk(D,E). Let k1, . . . , kl be the setof all indices such that

jk(D) = jk(E) 6= 0 for k ∈ k1, . . . , kl. (2.10)

If l = 0 there is only one isomorphism class corresponding to the Drinfeld module

TD = αL(T )+ τ r .

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If l = 1 there exists an elementu ∈ F∗qr such thatuqk1−1 = ξk1. The Drinfeld

moduleD′ = u−1Du coincides therefore withE since

ak1,D′ = uqk1−1ak1,D = ξk1ak1,D = ak1,E

andak = 0 for anyk 6= k1.Thus, we can suppose thatl > 2. For all integersδ1, . . . , δl verifying (B1) we

obtain

Jδ1...δlk1...kl

(D) = J δ1...δlk1...kl(E) = ξ δ1k1

· · · ξ δlkl · J δ1...δlk1...kl(D)

and, consequently,

ξδ1k1· · · ξ δlkl = 1. (2.11)

For anym such that 16 m 6 l there exists anl-tuple (δ′1, . . . , δ′l ) verifying (B1)

such that

δ′m = (qd(k1,...,km,...,kl ,r) − 1)/(qd(k1,...,kl ,r) − 1).

The signkm means that this term is omitted. It follows from the fact that the greatestcommon divisor of integers may be represented as some integral combination ofthese integers (the Bezout identity). Moreover, we can find a ‘neighbour’l-tuple(δ′′1, . . . , δ

′′l ) verifying (B1) such that

δ′′m = 2δ′m and δ′′i = δ′i for i 6= 1,m

for the same reason. Since

ξδ′1k1· · · ξ δ′lkl = 1 and ξ

δ′′1k1· · · ξ δ′′lkl = 1

we obtain

ξδ′′1−δ′1k1

ξδ′mkm= 1

for all 16 m 6 l. Acting by an isomorphismu ∈ F∗qr which keeps1(D) invariant,we obtainξk1(D1, E) = 1 whereD1 = u−1Du. Hence

ξkm(D1, E)δ′m = 1 (2.12)

for all 16 m 6 l. We shall now reason by induction onm. Form = 2 we have

ξk2(D1, E)(qd(k1,r)−1)/(qd(k1,k2,r)−1) = 1.

This equation is weaker than (2.12) and can be obtained supposingδi = 0 for i > 2.Acting by an isomorphismu1 ∈ F∗qd(k1,r) which keeps1(D1) and thek1-coefficient

of D1 invariant we obtainξk1(D2, E) = ξk2(D2, E) = 1 for D2 = u−11 D1u1.

Suppose thatm > 2 and

ξk1(Dm−1) = · · · = ξkm−1(Dm−1) = 1

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for some Drinfeld moduleDm−1∼= D overL. We have

ξkm(Dm−1, E)(qd(k1,...,km−1,r)−1)/(qd(k1,...,km,r)−1) = 1.

Acting by an isomorphismum ∈ F∗qd(k1,...,km−1,r)

which keepsak1, . . . , akm−1 and

1(Dm−1) invariant we obtain thatξkm(Dm,E) = 1 forDm = u−1m Dm−1um. Finally,

we can find a Drinfeld moduleDl∼= D such that

ξk1(Dl, E) = · · · = ξkl (Dl, E) = 1.

In view of (2.9) and (2.10) we obtain thatDl∼= D coincides withE and this

finishes the proof. 2

By their definition theJ -invariants are algebraic weakly modular functions([Go], Def. 1.14).

3. Coarse Moduli Schemes and Canonical Compactification

LetL be a separably closedA-field andM anA-scheme. We denoteML the schemeoverL obtained fromM by base change. Consider two contravariant functors fromthe category ofA-schemes to the category of sets:

D r : A-schemeS 7→ isomorphism classes

of Drinfeld modules of rankr overS

and

hM : A-schemeS 7→ Hom(S,M).

A schemeM = Mr(1) is called thecoarse moduli schemes of Drinfeld modules ofrank r if there exists a morphism of functorsf : D r → hM such that

(1) D r (L) ' hM(L) for any separably closedA-fieldL;(2) For anyA-schemeN and for any morphism of functorsg: D r → hN thereexists a unique morphismχ : hM → hN such that the following diagram

D r //f

!!g CCCCCCCC hM

χ

hN

(3.1)

commutes.

It follows from the Theorem 2.2(ii) thatMrL(1) is the factor of the varietyV r

given by the equations

X(qr−1)/(qd(k,r)−1)k = jk, 16 k 6 r − 1, (3.2)

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by the action of the finite groupF∗qr /F∗q such thatξ(Xk) = ξqk−1Xk for any ξ ∈

F∗qr /F∗q. The varietyV r is affine and toric, consequently,MrL(1) is also an affine

toric variety overL. ThereforeMrL(1) is the spectrum of anL-algebra generated

by invariant monomials. Thus we have

MrL(1) = SpecL

[Jδ1...δlk1...kl

]. (3.3)

Using the descent of the ground field we obtain thatMr(1) is the spectrum of anA-algebra generated by the same system of invariants. A more formal proof is givenbelow.

THEOREM 3.1. The affine toricA-variety of relative dimensionr − 1

Mr(1) = SpecA[Jδ1...δlk1...kl

](3.4)

is the coarse moduli scheme of DrinfeldA-modules of rankr.Proof. In virtue of Theorem 2.2(iii) the isomorphism classes of Drinfeld mod-

ules of rankr overL correspond bijectively to the geometricL-points ofMr(1).Thus, the condition (1) above is verified.

One can define a natural transformationf : D r → hM in the following way.Let L be a line bundle over anA-schemeS andE a Drinfeld module of rankrover (S,L). Moreover, letS = ⋃

Si be a covering trivializingL, Si = SpecBi,then

Jδ1...δlk1...kl

(E)∣∣Si= γ δ1...δlk1...kl

(E) ∈ Bi.Define the morphism ofA-algebras

A[Jδ1...δlk1...kl

]→ Bi

by the specialization

Jδ1...δlk1...kl= γ δ1...δlk1...kl

(E).

This defines a morphismS → Mr(1), that is, a geometricS-point ofMr(1). Ifλ: S′ → S is a morphism ofA-schemes andE′ = λ∗(E) then the diagram

E′

// E

S′ //λ

##fE′ FFFFFFFFF S

fE

Mr(1)

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commutes. It means that the geometricS′-point ofMr(1) defined byfE′ coincideswith the geometricS′-point defined by the compositionfE λ. Consequently, thediagram

D r (S)

// hM(S)

D r (S′) // hM(S

′)

also commutes. We have proved thatf is the natural transformation. The univer-sality of scheme (3.4) follows from geometric invariant theory. 2

Let n be some admissible ideal ofA, that is, divisible by at least two primedivisors. LetMr(n) be the fine moduli scheme of Drinfeld modules withn-levelstructure (in the sense of Drinfeld) ([Dr], §5). It is known thatMr(n) is a non-singular affineA-variety of relative dimensionr − 1 ([Dr], Cor. of Prop. 5.4). Wehave also the forgetful morphismMr(n)→Mr(1). In virtue of ([KM], (7.1), (8.1))Mr(n) is the PGL(r,A/n)-torsor (in the f.p.p.f. topology).

Any geometricA-pointP ofMr(1) defines a unique isomorphism class of Drin-feld modules over the separable closureKs . Let EP be some DrinfeldA-modulerepresenting this class. Denote alsod(EP ) the greatest common divisor ofr andall natural integersk < r such thatjk(EP ) 6= 0. We have

Aut(EP ) = F∗qd(EP ) . (3.5)

A geometricA-pointP of Mr(1) is calledelliptic if Aut(EP ) strictly containsF∗q .Let Sing(Mr(1)) and Ell(Mr(1)) be the loci of singular points and elliptic pointsresp. SinceMr(1) is the quotient of the non-singular varietyV r by the finite cyclicgroupF∗qr /F∗q all the singularities are cyclic quotient singularities.

THEOREM 3.2. For r > 2 we have

Sing(Mr(1)) = Ell(Mr(1)). (3.6)

Proof. Let Q be a geometricA-point of Mr(n) over P . The inertia group isisomorphic to:

I (Q/P ) = Aut(EP )/F∗q. (3.7)

If Aut(EP ) = F∗q then I (Q/P ) is trivial andP is non-singular ([Oo], Th. 2.7).On the other hand, ifP is non-singular there are two possibilities by the theoremon ‘purity of branch locus’ ([Oo], Th. 2.7; [AK], Ch. 6, Th. 6.8). The morphismMr(n)→ Mr(1) is non-ramified atP andI (Q/P ) = 1 orP is ramified in codi-mension 1. The second case is impossible because Ell(Mr(1)) is of codimensionstrictly greater than 1 forr > 2. Indeed, if Aut(EP ) 6= F∗q thenjk(EP )-invariantswith k prime tor are equal to zero by (3.5). 2

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Consider the following contravariant functor from the category ofA-schemesto the category of sets:

D r : A-schemeS 7→ isomorphism classes

of Drinfeld modules of rank6 r overS

Thecoarse moduli schemeMr(1) of rational Drinfeld modules of rank6 r isdefined in the same manner as in the beginning of this section.

PROPOSITION 3.3 (cf. [Ka], 1.6).The weighted projective space

Mr(1) = PA(q − 1, q2 − 1, . . . , qr − 1)

is the coarse moduli scheme of rational Drinfeld modules of rank6 r.Proof.The reasoning analogous to the proof of the Theorem 3.1 shows that the

affine subvariety ofPA(q − 1, q2 − 1, . . . , qr − 1) corresponding to the non-zerokth coordinate is the coarse moduli scheme of Drinfeld modules of rank6 r withnon-zerojk-invariant. The gluing finishes the proof. 2

COROLLARY 3.4. We have the following description of the cuspidal divisor:

Cusp(Mr(1)

) def=Mr(1)\Mr(1) =⋃

16k6r−1

Mk(1).

COROLLARY 3.5. We have

Sing(Mr(1)

) = Ell(Mr(1))\Ell(M2(1)

) = ⋃36k6r−1

Ell(Mk(1)).

4. Ramification of thej -Covering

By Theorem 2.2(ii) thej -invariant defines the finite flat covering of the affine spaceof dimensionr − 1 byMr(1).

PROPOSITION 4.1.The finite flat covering

j : Mr(1)→ Ar−1A , (4.1)

is étale over

Gr−1m,A = SpecA[j1, . . . , jr−1, j

−11 , . . . , j−1

r−1] (4.2)

and tame. The degree of this covering is equal to:

N =r−1∏i=2

qr − 1

qd(i,r) − 1. (4.3)

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Proof. It follows from the Theorem 2.2(ii). 2

Let (i1, . . . , is) be a multi-index such that 16 i1 < · · · < is 6 r − 1. WedenoteAA(i1, . . . , is) ⊂ Ar−1

A the affine subvariety generated by the coordinatesji1, . . . , jis andGm,A(i1, . . . , is) the corresponding subtorus. We also denote byMr(1)[i1, . . . , is] the subvariety ofMr(1) corresponding to Drinfeld modules suchthat their coefficients different fromi1, . . . , is are zero.

PROPOSITION 4.2.Mr(1) is regular in relative codimensionr−φ(r)−2 whereφ(r) is the length of a maximal chain(i1, . . . , is) such thatd(i1, . . . , is, r) > 1.Furthermore,

Sing(Mr(1)) =⋃

d(i1,...,is ,r) > 1

Mr(1)[i1, . . . , is]. (4.4)

Proof.The result follows immediately from (3.5) and (3.6). 2

COROLLARY 4.3. If r is a prime integer thenMr(1) is regular outside of theorigin.

PROPOSITION 4.4.The finite flat covering

j (i1, . . . , is): Mr(1)[i1, . . . , is] → AA(i1, . . . , is) (4.5)

is étale overGm,A(i1, . . . , is) and tame. The degree of this covering is equal to:

N(i1, . . . , is) = (qr − 1)s−1(qd(i1,...,is ,r) − 1)

(qd(i1,r) − 1) · · · (qd(is ,r) − 1). (4.6)

In particular,N(i1) = 1 for any16 i1 6 r − 1.Proof. The first part is analogous to the first part of the Proposition 4.1 if we

consider Drinfeld modules such that their coefficients different fromi1, . . . , is arezero. It suffices therefore to prove (4.6). Notice thatN(i1, . . . , is) is equal to thenumber of non-isomorphic Drinfeld modules with the samej -invariant such thattheir non-zero components are exactlyi1, . . . , is . According to Theorem 2.2(ii) onecan suppose that thek-coefficients of such Drinfeld modules belong toF∗qr /F∗qd(k,r) .We reason by induction ons. If s = 1 via an isomorphismu ∈ F∗qr one cansuppose that thei1-coefficient is equal to 1. Thus,N(i1) = 1. If s = 2 we put thei1-coefficient equal to 1. Thei2-coefficient may be written as

tk(qd(i2,r)−1), 16 k 6 (qr − 1)/(qd(i2,r) − 1),

wheret is some generator ofF∗qr . On factorizing by the action ofF∗qd(i1,r)

we obtain

N(i1, i2) = g.c.d.

(qr − 1)

(qd(i1,r) − 1),

(qr − 1)

(qd(i2,r) − 1)

= (qr − 1)(qd(i1,i2,r) − 1)

(qd(i1,r) − 1)(qd(i2,r) − 1).

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In general, we have

N(i1, . . . , is) = N(i1, . . . , is−1) ·[g.c.d.

(qr − 1)

(qd(i1,...,is−1,r) − 1),

(qr − 1)

(qd(is ,r) − 1)

]= N(i1, . . . , is−1) · (qr − 1)(qd(i1,...,is ,r) − 1)

(qd(i1,...,is−1,r) − 1)(qd(is ,r) − 1).

The formula (4.6) is an immediate consequence. 2

COROLLARY 4.5. The covering(4.1) is tamely ramified overGm,A(i1, . . . , is) oframification index

e(i1, ..., is) = N

N(i1, . . . , is)(4.7)

for any multi-index(i1, . . . , is). Therefore this covering is totally ramified overAA(i1) for any16 i1 6 r − 1.

COROLLARY 4.6. If r > 3 is prime then

N(i1, . . . , is) =(qr − 1

q − 1

)s−1

; e(i1, . . . , is) =(qr − 1

q − 1

)r−s(4.8)

for any multi-index(i1, . . . , is).

EXAMPLE 4.7. If r = 4 we have

N = (q4 − 1)2

(q2− 1)(q − 1); N(1,2) = N(2,3) = (q4 − 1)

(q2 − 1);

N(1,3) = (q4 − 1)

(q − 1)

and Sing(M4(1)) = M4(1)[2].

5. Rational Polyhedral Cone and Its Dual

For anyr > 3 we fix some latticeNr of rank(r−1) and letNr∗ = HomZ(Nr,Z) beits dual. We write simplyN andN∗ if there is no confusion. There exists a naturalcorrespondence between(r − 1)-dimensional rational strictly convex polyhedralcones inN∗R and(r − 1)-dimensional affine toric varieties ([Da1], [Fu], [Od]).

THEOREM 5.1. The rational simplicial cone generated by the following vectors

e∗1 = (1,0, . . . ,0),

e∗k =(

(qk − 1)

(qd(k,r) − 1),0, . . . ,0︸︷︷︸

k−2

,(qr − 1)

(qd(k,r) − 1),0, . . . ,0

)(5.1)

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for 2 6 k 6 r − 1, is the dual rational polyhedral coneσ ofMr(1). The rationalpolyhedral coneσ ofMr(1) is generated by:

e1 =(qr − 1

q − 1,−q − 1, . . . ,−q

k − 1

q − 1, . . . ,−q

r−1− 1

q − 1

),

ek = (0, . . . ,0︸︷︷︸k−1

,1,0, . . . ,0︸︷︷︸r−k−1

), 26 k 6 r − 1. (5.2)

Proof.We know that

Mr(1) = SpecA[Jδ1...δlkl ...kl

].

For any 16 k 6 r − 1, letJk be an element verifying (3.2), i.e. such that

J(qr−1)/(qd(k,r)−1)k = jk.

Using the transformations

U1 = j1, Uk = Jk

J(qk−1)/(q−1)1

(5.3)

for 26 k 6 r − 1 ([Fu], Sect. 2.2), we obtain that

jk = J(qk−1)(qr−1)

(q−1)(qd(k,r)−1)

1 U

qr−1qd(k,r)−1k = U(qk−1)/(qd(k,r)−1)

1 U(qr−1)/(qd(k,r)−1)k .

There is a bijective correspondence between the integral points of this cone and theJ -invariants. Indeed, any monomial ofU1, . . . , Ur−1 belonging to the coneσ givesaJ -invariant by the formula (5.3).

On the other hand, the invariantJ δ1...δlk1...klverifying (B1) determines the monomial

Uδr1 U

δ1k1. . . U

δlkl.

The coneσ is obtained by taking suitable orthogonal vectors to the facets ofσ . 2

COROLLARY 5.2. The rational simplicial fan generated by the ray(−1,0, . . . ,0)and by the rays(5.2) is the rational polyhedral fan ofMr(1).

Proof. In virtue of Proposition 3.3 we have

Mr(1) = PA(q − 1, q2 − 1, . . . , qr − 1)

= PA(

1,q2 − 1

q − 1, . . . ,

qr − 1

q − 1

). (5.4)

ThereforeMr(1) is the equivariant compactification of the affine spaceAr−1A ([Do],

1.2.4). Moreover, the weighted projective space (5.4) is the gluing ofr affine toric

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Figure 1. Dual rational coneσ of M3(1) (on the left) and ofM4(1).

varieties with simplicial cones. Thus, we have to add only one ray to the coneσ inorder to form a fan ofMr(1). It is easy to see that adding the ray(−1,0, . . . ,0)we obtain the result. 2

Remark.This result may be also deduced applying ([Od], Th. 2.22) to the polytopecorresponding toσ (cf. [Do], 1.2.5).

6. Minimal Terminal Q-Factorial Compactification

We shall now construct the minimal simplicial terminal subdivision of the coneσ

of Mr(1). We suppose here thatr > 4 andq is big enough. The unique equivari-ant minimal smooth compactification of the coarse moduli surfaceM3(1) will beconstructed in the next section.

We denote Sk1σ the set of the extremal rays of a simplicial coneσ and lσ alinear form onNQ such thatlσ (Sk1σ) = 1. The convex polytopeσ ∩ l−1

σ [0,1] iscalled theshedof σ and the convex polytopeσ ∩ l−1

σ (1) in codimension 1 is calledtheroof of the shedof σ (cf. [Re], [BGS]). Theshed(resp. theroof of the shed) of afan6 is the union of the sheds (resp. of the roofs of the sheds) of its cones. A coneis terminal if its shed does not contain integral points distinct from its vertices.Finally, a fan isterminal if it is the union of terminal cones.

THEOREM 6.1. The consecutive star subdivisions centered in the rays

(qr−m−2 + qr−m−4 + qr−m−5 + · · · + q + 1,0, . . . ,0︸︷︷︸m

,−1,−q,

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Figure 2. Rational polyhedral cone ofM3(1) (on the top) and ofM4(1).

− q2 − 1, . . . ,−qr−m−3 − qr−m−5 − qr−m−6 − · · · − q − 1) (6.1)

for 0 6 m 6 r − 4 (in ascending order) and in the ray(1,0, . . . ,0) define theunique minimal terminalQ-factorial equivariant modelMr

min(1) ofMr(1).Proof. TheQ-factoriality follows from the fact that star subdivisions are sim-

plicial (cf. [Br], Sect. 4.2). We shall check the terminality of singularities. Let6min denote the fan ofMr

min(1). The extremal rays of the cones of6min are calledterminal rays ofσ . A point of the shed ofσ generating a terminal ray will be calleda terminal point.

The coordinates of terminal rays in the interior of the shed ofσ may be foundby consecutive projections to the coordinatese1, er−1 and −ek+1, ek for 2 6k 6 r − 2. The projection one1, er−1 defines the two-dimensional cone⟨

(0,1),(qr − 1

q − 1,

1− qr−1

q − 1

)⟩(6.2)

which is the rational cone of the surfaceMr(1)[1, r−1]. The points(lq+1,−l) for06 l < (qr−1−1)/(q−1) are the only terminal points in the shed of this cone apartfrom the extremal rays (see Figure 3). They define the minimal desingularizationof the surfaceMr(1)[1, r − 1].

A projection of the second type defines the two-dimensional fan⟨(−1,0), (0,1),

(qk+1− 1

q − 1,

1− qkq − 1

)⟩

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Figure 3. Minimal desingularization of the surfaceMr(1)[1, r − 1].

Figure 4. Minimal smooth compactification of the surfaceMk+1[1, k].

which is the rational fan of the surfaceMk+1(1)[1, k]. The points(lq + 1,−l) for0 6 l < (qk − 1)/(q − 1) and the point(q,−1) are the only terminal points inthe shed of this fan (apart from the extremal rays). These points give the minimalsmooth compactification of the surfaceMk+1(1)[1, k] (see Figure 4).

We obtain, consequently, that a point(x1, . . . , xr−1) distinct from the originand lying strictly inside of the shed ofσ is terminal only if one of the followingconditions is satisfied:

06 −x2 6 q, xk+1 = xkq − 1 and x1 = 1− xr−1q, (6.3)

for 26 k 6 r − 2, or

x2 = · · · = xm+1 = 0 (if m > 1), xk+1 = xkq − 1 and x1 = 1− xr−1q, (6.4)

for 06 m 6 r − 2 andm+ 26 k 6 r − 2, or finally

x2 = · · · = xm+1 = 0 (if m > 1), xm+2 = −1, xm+3 = −qxk+1 = xkq − 1 and x1 = 1− xr−1q (6.5)

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for 06 m 6 r − 2 andm+ 46 k 6 r −2. The relations (6.3) and (6.4) define thepoints

(lqr−2 + qr−3+ · · · + q + 1,−l,−lq − 1, . . . ,−lqr−3− qr−4 − · · · − q − 1)

for 06 l 6 q + 1 and the points

(qr−m−2 + qr−m−3 + · · · + q + 1,0, . . . ,0︸︷︷︸m

,−1,−q − 1,

− q2 − q − 1, . . . ,−qr−m−3 − qr−m−4 − · · · − q − 1)

respectively. These points except for(1,0, . . . ,0) lie above the hyperplane inNRpassing throughe1, . . . , er−1 (see (5.2)) which is easy to prove by straightforwardcomputation. Therefore these points do not belong to the shed ofσ . The point(q,0, . . . ,0,−1) corresponding tom = r − 3 in (6.5) can no more belong to thisshed. Indeed, its projection one1, er−1 which is (q,−1) does not belong to theshed ofMr(1)[1, r − 1] (see Figure 3).

Thus, the point(x1, . . . , xr−1) lying strictly inside of the shed ofσ is terminalif and only if it is (1,0, . . . ,0) or

x2 = · · · = xm+1 = 0 (if m > 1), xm+2 = −1, xm+3 = −q,xk+1 = xkq − 1 and x1 = 1− xr−1q (6.6)

for 0 6 m 6 r − 4 andm + 4 6 k 6 r − 2. The varietyMr(1) obtained bythe consecutive star subdivisions centered in these rays (in ascending order withrespect tom) has the shed with concave roof along the internal walls (see [Re] forterminology). It follows from the Reid theorem ([Re], Th. 0.2) that it is a minimalmodel. Any other minimal model with terminalQ-factorial singularities has thesame shed. The roof of this shed is strictly concave along the internal walls and,consequently, constructed minimal model is unique. 2

THEOREM 6.2. The consecutive star subdivisions of the rational polyhedral fanσ ofMr(1) by the following rays

(qr−m−2 + qr−m−4 + qr−m−5 + · · · + q + 1,0, . . . ,0︸︷︷︸m

,−1,−q,

− q2 − 1, . . . ,−qr−m−3 − qr−m−5 − qr−m−6 − · · · − q − 1) (6.7)

for 0 6 m 6 r − 2 (in ascending order) define the unique minimal terminalQ-factorial equivariant compactificationMr

min(1) ofMr(1).Proof. It suffices to prove that the points (6.7) are the only terminal points

strictly inside of the shed ofσ apart from the origin. As in the proof of the pre-vious theorem take the consecutive projections to the coordinatese1, er−1 and−ek+1, ek for 2 6 k 6 r − 2. These projections define the two-dimensionalfans ⟨

(−1,0), (0,1),

(qk+1− 1

q − 1,

1− qkq − 1

)⟩

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for 26 k 6 r − 1.We obtain, consequently, that a point distinct from the origin and lying strictly

inside of the shed ofσ is terminal if and only if the following condition is satisfied:

x2 = · · · = xm+1 = 0 (if m > 1), xm+2 = −1, xm+3 = −q,xk+1 = xkq − 1 and x1 = 1− xr−1q (6.8)

for 06 m 6 r−2 andm+46 k 6 r −2. These points without(q,0, . . . ,0,−1)(corresponding tom = r − 3) define the unique minimal simplicial terminalsubdivision of the rational cone ofMr(1) by the previous theorem. The point(q,0, . . . ,0,−1) lie in the subcone〈(−1,0, . . . ,0), e1, . . . , er−2〉 of the fanσ(cf. (5.2)). The star subdivision centered in the corresponding ray together withconsecutive star subdivisions of Theorem 6.1 define a unique minimal terminalQ-factorial equivariant compactification ofMr(1). 2

Remark.We supposed thatq is sufficiently big. In fact, ifq 6 r−3 then the ray of(6.1) and of (6.7) corresponding tom = 0 does not necessarily belong to the shedof the coneσ of Mr(1).

7. Drinfeld Coarse Moduli Surface

7.1. EQUATIONS DEFININGM3(1)

First of all, we shall construct a regular subdivision of the dual coneσ (see Fig-ure 5).

Let χu0, . . . , χuq+1 be the characters of the torusTN corresponding to the raysu0, . . . , uq+1 of the regular subdivision ofσ . Using the property

χuχu′ = χu+u′

Figure 5. Regular subdivision of the dual cone ofM3(1).

MPAG012.tex; 4/09/1998; 11:15; p.17


valid for any elementsu, u′ ∈ N we deduce in our case that

χuq+1χuq−1 = χ(q+2)uq , χui+1χui−1 = χ2ui , 16 i 6 q − 1. (7.1)

DenoteXi = χui . We obtain thatM3(1) is defined as a scheme over SpecA bythe followingq equations:

Xq+2q = Xq+1Xq−1,

X2i = Xi+1Xi−1, 16 i 6 q − 1,

(7.2)

whereXq+1 = j2 and

Xi = U1Ui2 = j1

(J2

Jq+11

)i= J q2+q+1−i(q+1)

1 J i2 (7.3)

for 06 i 6 q in notations (5.3).

7.2. MINIMAL SMOOTH COMPACTIFICATION OF M3(1)

In order to find a resolution of singularities of an affine toric variety it sufficesto find a regular subdivision of the corresponding rational cone. In our case theminimal regular subdivision is given by Figure 3 sinceM3(1) = M3(1)[1,2].

The minimal resolution of singularities is given by a chain of blowing-upsM3

min(1)→M3(1) atTN -invariant centers. The exceptional divisor

E = C1+ C2+ · · · + Cq+1

is described by Figure 6. HereC1, . . . , Cq+1 are rational curves with the followingindices of self-intersection:

(C1)2 = −q − 1, (Ci)

2 = −2, for i > 1.

The minimal smooth compactification is represented by Figure 4 fork = 2. Therational polyhedral fan ofM3

min(1) is the subdivision of the subfans corresponding

to the Hirzebruch surfacesHq andHq+1. Consequently,M3min(1) may be obtained

by a succession of blowing-ups atTN -stable points of any of these surfaces.We see further that the rational fan ofM3

min(1) containsd2 = q + 5 two-dimensional regular subcones andd1 = q + 5 one-dimensional subcones. In par-ticular, the Euler characteristic is equal to:

χ(M3

min(1))= d2 = q + 5

([Fu], Ch. 4.3).

MPAG012.tex; 4/09/1998; 11:15; p.18


Figure 6. Exceptional divisor and its weighted graph.

Figure 7. Weighted circulated graph ofM3min(1).

LetD1, . . . ,Dq+5 be the irreducible invariant divisors onM3min(1). They corre-

spond to the rays (one-dimensional subcones) in Figure 4 fork = 2. It is knownthat

K = −q+5∑i=1

Di

is the canonical divisor. Its self-intersection index is given by the formula:

(K)2 = 12− d2 = 7− q.It is also possible to calculate the (l-adic) Betti numbers and the Poincaré poly-

nomial using ([Fu], §4.5). We putd0 = 1 (the number of zero-dimensional sub-cones). Thenβ3 = β1 = 0, β0 = β4 = 1 and

β2 = d1 − 2d0 = q + 3.

Furthermore, the Poincaré polynomial is equal to:

PM(t) = β4t4 + β2t

2+ β0 = t4+ (q + 3)t2 + 1.

MPAG012.tex; 4/09/1998; 11:15; p.19


7.3. ZETA-FUNCTION OF M3min(1)

First of all,

Card(M3

min(1)(Fqm))= β4q

2m + β2qm + β0 = q2m + (q + 3)qm + 1.

We recall now that ifM is anA-variety of relative dimensionr − 1 and if thenumber of its geometric points overFqmn is equal to:

νn =r−1∑i=0

µi(qmn)i

then

ζ(M/Fqm, s) = exp

(∑n>1

νn(qm)−sn

n

)= exp

(∑n>1

r−1∑i=0

µi(qm)in

(qm)−sn

n

)

= exp( r−1∑i=0

∑n>1

µi(qm)(i−s)n

n

)=

r−1∏i=0

exp(∑n>1

µi(qm)(i−s)n

n

)

=r−1∏i=0

exp(− µi ln(1− qm(i−s))) = r−1∏

i=0

(1− qm(i−s))−µi

(cf. [MP], Ch. 4, §1). In our case we have:

µ0 = β0 = µ2 = β4 = 1, µ1 = β2 = q + 3.

Consequently,

ζ(M3

min(1)/Fqm, s)= (1− q−ms)−1(

1− qm(1−s))−q−3(1− qm(2−s))−1

.

In addition,

ζ(M, s) =∏m>1

∏p∈SpecmAdegp=m

ζ(M ⊗A Fqm, s) =∏m>1

∏p∈SpecmAdegp=m

r−1∏i=0

(1− qm(i−s))−µi

=r−1∏i=0

∏m>1

∏p∈SpecmAdegp=m

(1− qm(i−s))−µi = r−1∏

i=0

ζA(s − i)µi ,

whereζA(s) = (1− q1−s)−1 is the Dedekind zeta-function ofA = Fq[T ]. In ourcase we have:

ζ(M3

min(1), s)= ζA(s)ζA(s − 1)q+3ζA(s − 2)

= ((1− q1−s)(1− q2−s)q+3(

1− q3−s))−1.

MPAG012.tex; 4/09/1998; 11:15; p.20


Acknowledgements

I am very grateful to Michel Brion for valuable remarks, to Catherine Bouvier forhelpful discussions concerning toric varieties and to Alexei Panchishkin for con-stant encouragement and advice. I am thankful also to Karsten Bücker for readingthis text and correcting some spelling mistakes.

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Br. Brion, M.: Variétés sphériques et théorie de Mori,Duke Math. J.72(2) (1993), 369–404.Da1. Danilov, V. I.: Geometry of toric varieties,Uspekhi Mat. Nauk33(2) (1978), 83–134 (in

Russian); Engl. transl.:Russian Math. Surveys33(2) (1978), 97–154.Da2. Danilov, V. I.: Birational geometry of toric 3-folds,Izv. Akad. Nauk SSSR, Ser. Mat.46(5)

(1982), 971–982 (in Russian); Engl. transl.:Math. USSR-Izv.21 (1983), 269–280.Do. Dolgachev, I.: Weighted projective varieties, Proc. of a Polish-North American Seminar,

Vancouver, 1981,Lect. Notes Math.956(1982), 34–71.Dr. Drinfeld, V. G.: Elliptic modules,Mat. Sbornik94 (1974), 594–627 (in Russian); Engl.

transl.:Math. USSR S.23 (1974), 561–592.Fu. Fulton, W.:Introduction to Toric Varieties, The William H. Roever Lectures in Geometry,

Princeton University Press, 1993.Ge1. Gekeler, E.-U.: Moduli for Drinfeld modules, inThe Arithmetic of Function Fields, Walter

de Gruyter, Berlin, New York, 1992, pp. 153–170.Ge2. Gekeler, E.-U.: Satake compactification of Drinfeld modular schemes, inProc. Conf. on

p-adic Analysis held in Hengelhoef (Houthalen), Belgium, 1986, pp. 71–81.Go. Goss, D.:π-adic Eisenstein series for function fields,Comp. Math.1 (1980), 3–38.

KM. Katz, N. M. and Mazur, B.:Arithmetic Moduli of Elliptic Curves, Ann. of Math. Stud. 108,Princeton University Press, 1985.

Ka. Kapranov, M. M.: On cuspidal divisors on the modular varieties of elliptic modules,Izv.Akad. Nauk URSS, Ser. Mat.51(3) (1987), 568–583 (in Russian); Engl. transl.:Math. USSR- Izv.30(3) (1988), 533–547.

MP. Manin, Yu. I. and Panchishkin, A. A.: Number theory I, in A. N. Parshin and I. R. Shafarevich(eds.),Encyclopaedia of Math. Sciences49, Springer-Verlag, 1995.

Mu. Mumford, D.:Geometric Invariant Theory, Springer-Verlag, 1965.Od. Oda, T.:Convex Bodies and Algebraic Geometry, Ergebnisse der Math. 15, Springer-Verlag,

1988.Oo. Oort, F.: Coarse and fine moduli spaces of algebraic curves and polarized Abelian varieties,

in Sympos. Math.XXIV, Academic Press, London, New York, 1981.Pi. Pink, R.: On compactification of Drinfeld moduli schemes, in:Moduli Spaces, Galois

Representations andL-functions, Surikaisekikenky¯usho Kokyuroku 884 (1994), 178–183(Japanese).

Po. Potemine, I. Yu.:J -invariant et schémas grossiers des modules de Drinfeld, Séminaire deThéorie des Nombres, Caen, Fascicule de l’année 1994–95, 15 pp.

Re. Reid, M.: Decomposition of toric morphisms, Arithmetic and geometry, Papers dedicatedto I. R. Shafarevich on the occasion of his 60th birthday, Birkhäuser,Progress in Math.36(1983), 395–418.

MPAG012.tex; 4/09/1998; 11:15; p.21


193

Stability Criteria for the Weylm-Function

W. O. AMREINDepartment of Theoretical Physics, University of Geneva, CH-1211 Geneva 4, Switzerland

D. B. PEARSON?Department of Mathematics, University of Hull, Cottingham Road, Hull, U.K.

(Received: 29 October 1997; in final form: 15 July 1998)

Abstract. This paper presents a new approach to spectral theory for the Schrödinger Operator onthe half-line. Solutions of nonlinear Riccati-type equations related to the Schrödinger equation at realspectral parameterλ are characterised by means of their clustering properties asλ is varied. A familyof solutions exhibiting a so-calledδ-clustering property is shown to imply precise estimates for thecomplex boundary value of the Weylm-function and the spectral measure, and leads to an analysisof the absolutely continuous component of the spectral measure in terms of stability criteria for thecorresponding Riccati equations.

Mathematics Subject Classifications (1991):34B25, 47E05.

Key words: m-function, Schrödinger operator, spectrum.

1. Introduction

This paper presents a new approach to the spectral theory of the Schrödinger oper-ator on the half-line, based on an analysis of the Weyl–Titchmarshm-function andits boundary values.

It is well known (see, for example, [1–5]) that them-function, defined in termsof two solutionsu(·, z), v(·, z) of the Schrödinger equation at complex spectral pa-rameterz as the coefficientm(z) for whichu(·, z)+m(z)v(·, z) is square integrableover the half-line, is a Herglotz function (analytic in the upper half-plane withpositive imaginary part), the boundary behaviour of which determines the spectralproperties of the differential operator−d2/dx2 + V (x) in L2(0,∞). Here we areassuming a real locally integrable potentialV , and limit-point case at infinity; werefer the reader to [2, 6, 7] for a treatment of the Weyl limit-point/limit circle theoryand note that the limit-point case, which holds for any potential bounded at infinityand more generally for a wide range of unbounded potentials as well, is the caseof physical interest in most applications to quantum mechanics and elsewhere, andallows one to dispense with a boundary condition at infinity.

? Partially supported by the Swiss National Science Foundation and by EPSRC.


194 W. O. AMREIN AND D. B. PEARSON

We do, however, need to impose a boundary condition atx = 0, and a one-parameter family of boundary conditions, parametrised by a real parameterα inthe rangeπ/2 < α 6 π/2, leads to a one-parameter familymα of m-functions,and correspondingly a one parameter familyTα = −d2/dx2+V (x) of Schrödingeroperators inL2(0,∞), each with its associated spectral properties. The caseα = 0,withm(z) ≡ m0(z), corresponds to the Schrödinger operator with Dirichlet bound-ary condition atx = 0; it should, however, be noted that it is usually necessary,in developing spectral theory for such operators, to deal with a family of operatorsTα rather than just a single operator.

Since the spectrum of each of these Schrödinger operatorsTα is a subset ofR, and the spectral measureµα is a measure on Borel subsets ofR, one mayexpect that in principle it is better to deal with the Schrödinger equation at realspectral parameterλ, rather than at complex spectral parameterz. This will havethe additional advantage that we can then call upon the variety of methods (orthog-onality properties, oscillation and separation theorems) which apply to solutions ofreal Sturm–Liouville equations. Various theoretical ideas and methods have beenintroduced, particularly in recent years, which allow one to pass from a treatmentof them-function as a function of a complex variablez in the upper half-plane,to an analysis of the boundary valuesm+(λ) ≡ limε→0+m(λ + iε), defined foralmost allλ ∈ R. This leads to a link between spectral behaviour and the a-symptotic properties in the limitx → ∞ of solutionsf (x, λ) of the Schrödingerequation at real spectral parameterλ. As examples of such developments, we maycite the application of the notion of subordinacy, introduced first of all in [8–10], and recently developed still further in [11–14], as a powerful tool of spectralanalysis, the treatment of absolutely continuous spectrum in [15, 16], using anasymptotic condition for the squared wave-function, recent results in [17–20] onthe absolutely continuous spectrum, and new techniques for problems of singularspectra in [21, 22].

A novel feature of the approach presented here, applied in particular to a studyof the absolutely continuous component of the spectral measureµα of Tα, is thatit is based on an analysis ofcomplexsolutions of the Schrödinger equation atrealspectral parameterλ.

At first sight, this approach seems a little unusual, not to say perverse, sincefor λ real, the solution space for the Schrödinger equation−d2f /dx2 + Vf = λfis spanned at eachλ by just two solutionsu(·, λ), v(·, λ), which may be taken tobe real, and any complex solutionf will just be a complex linear combinationof these two real solutions. Nevertheless, as is already suggested for example in[23, 24] and [25], complete spectral information cannot be extracted from a studyof the asymptotics of two solutionsu andv in isolation, but depends rather on aknowledge of theirrelative asymptotics, for example of their relative amplitudesand phases. This information appears to be encapsulated in a particularly crucialway, for spectral analysis, in the largex asymptotics of complex solutions at realλ.It should also be noted that to consider complex solutions is equivalent to consider-

MPAG013.tex; 5/11/1998; 8:38; p.2

STABILITY CRITERIA FOR THE WEYLm-FUNCTION 195

ing simultaneously apair of solutions, which is very much in line with recent ideas,expressed for example in [26], which stress the analogy between the asymptoticanalysis of the Schrödinger equation and the large time behaviour of dynamicalsystems. The current paper, in drawing on notions such as stability, recurrence, andclustering, is a continuation of this line of development.

Rather than dealing with complex solutionsf (x, λ) = Au(x, λ) + Bv(x, λ)per se(A,B ∈ C, and dependent onλ), we consider instead, for such solutions,the ratioh(x, λ) = f ′(x, λ)/f (x, λ), wheref ′ denotes differentiation with respectto x. This functionh(x, λ) is thus a particular rational combination ofu, v, u′ andv′ which contains more spectral information than, for example, the real functionsu′/u and v′/v considered separately. It is well known, and follows easily as aconsequence of the Schrödinger equation satisfied byf (x, λ), thath(x, λ) satisfiesthe nonlinear Riccati differential equation

d

dxh(x, λ) = V (x)− λ− (h(x, λ))2. (1)

This equation is appropriate to the study of them-function and spectral propertiesfor T = −d2/dx2+ V (x) subject to Dirichlet boundary condition atx = 0; this isthe special caseα = 0, and for general boundary condition we will have a relatedRiccati equation, of which the solutionshα are related toh by explicit rationaltransformations.

The principal aim of this paper will be to show how the largex asymptotics offamilies of solutions of the above Riccati equation, asλ is varied, imply explicitbounds ofm+(λ), the boundary value of them-function (or ofmα if Neumann orother boundary conditions are imposed); these estimates ofm+ can be used to gen-erate information about the spectral properties of the corresponding Schrödingeroperator, in particular as relates to the absolutely continuous part of the spectrum.

As an example which will provide a flavour of the kind of result we shall obtain,may be cited the following, which applies to all real-valued, locally integrablepotentials in the limit-point case at infinity:

Let h(·, λ) be any (complex-valued) solution of the above Riccati equation,measurable as a function ofλ and satisfying, for allx sufficiently large and forall λ belonging to some finite intervalI, the bound

|h(x, λ)− i√λ| < δ√λ,whereδ is a constant in the range 0< δ < 1/2. Then, for almost allλ ∈ I, wehave the estimates:

|h(0, λ)−m+(λ)| 6 δ

1− 2δImm+(λ),

| Imh(0, λ)− Imm+(λ)| 6 δ

1− δ Imm+(λ).

MPAG013.tex; 5/11/1998; 8:38; p.3


These estimates allow us to deduce from the initial valuesh(0, λ) of our given fam-ily of solutionsh(·, λ) precise upper and lower bounds for the value of Imm+(λ).Sinceπ−1 Imm+(λ) is the density function for the spectral measureµa.c. of the ab-solutely continuous component for the Dirichlet Schrödinger operator inL2(0,∞),we can deduce corresponding estimates for the spectral measure of the intervalIitself. We can also estimate, to orderδ, the value of the complex limitm+(λ) itself,and similar results apply to them-function for all other boundary conditions atx = 0.

A particular consequence of this result applies if the value ofδ may be madearbitrarily small. For this to be so, we require limx→∞ h(x, λ) = i

√λ, and such

a solution must then satisfy the initial conditionh(0, λ) = m+(λ) exactly. (It alsofollows, for generall > 0, thath(l, λ) is then the boundary value atλ for them-function of the Schrödinger operator acting inL2(l,∞), with Dirichlet boundarycondition atx = l.)

For a wide class of short – and long – range potentials (for example in the casesV ∈ L1(0,∞), orV of bounded variation withV → 0 at infinity), it is indeed thecase that a solutionh(·, λ) of the Riccati equation exists with limx→∞ h(x, λ) =i√λ, for anyλ > 0. Such solutions can then be used to determine the boundary

values of them-function and related spectral behaviour. It should, however, benoted that our results are extremely general, and apply to a far wider class of poten-tials, and under much weaker assumptions which do not require the convergence ofh(x, λ) asx → ∞. The particular notion which we are able to isolate, and whichseems to govern all spectral behaviour on the absolutely continuous part of thespectrum, is that of (recurrent) clustering. Roughly, a familyh(·, λ) of solutionsof the Riccati equation is said to beδ-clustering, forλ in some setS, providedrecurrently at a sequence of pointsx = x1, x2, x3, . . . , with xj → ∞, solutionsasλ varies overS are within distance of orderδ of each other. The family is saidto be clustering ifδ can be made arbitrarily small. Precise definitions of these twoconcepts are given in Section 5.

The main results of the paper, stated below as Theorem 1, provide estimatesto orderδ of the boundary value of them-function, spectral density function, andspectral measure of a set, based on the hypothesis that a given family of solutionsof the Riccati equation isδ-clustering, and imply that the only solutions havingthe clustering property must be subject to initial conditionsh(0, λ) = m+(λ).All of these results are extended to arbitrary values of the boundary conditionparameterα.

As a consequence, one can use the behaviour of a family of solutions of theRiccati equation, forλ in some setS and at an increasing sequence of pointsxj ,to derive precise bounds on them+-function, forλ ∈ S. It appears to us that the the-oretical framework which leads to the derivation of such bounds provides a viablebasis for a possible numerical approach to spectral analysis, in which results of thekind described here are coupled with ideas of interval analysis. Quite apart fromsuch developments, we believe that the characterisation of the boundary values of

MPAG013.tex; 5/11/1998; 8:38; p.4


them-function in terms of cluster properties of families of solutions of Riccati-typeequations is of theoretical interest.The paper is organised as follows:

Section 2 begins with an introduction to the general properties of Herglotzfunctions. Any Herglotz function has a unique representation ([27]) in terms ofa corresponding measureν on the Borel subsets ofR. Just as is the case forSchrödinger operators, the spectral analysis of Herglotz functions is best carriedout with afamily Fy(y ∈ R) of Herglotz functions, rather than a single function.

Given any Herglotz functionF , one can define for almost allλ ∈ R a Cauchymeasureω(λ, ·). Given any Borel subsetS of R, πω(λ, S) is almost everywhereequal to the angle subtended by the setS at the boundary valueF+(λ) of F at λ.Thus theω-measure carries important information relating to the boundary behav-iour of the Herglotz function and has the added advantages, as compared to themeasureν, of both being a finite measure and of behaving well under variouslimiting operations. For the general background to theω-measure and familiesof Herglotz functions, see [24]. In order to make full use of theω-measure inspectral analysis, it is necessary to transfer between estimates of angles subtendedby setsS at pointsw in the upper half-plane and estimates of the location of thepointsw themselves. Section 2 concludes with a general lemma which providesthe necessary theoretical basis for doing this.

In Section 3, we define the family ofm-functionsmα, as well as a related familyof m-functions for the differential operator−d2/dx2 + V (x) acting inL2 of afinite interval [0,N] (see also [2, 25]). In this latter case, it is necessary to allowfor complex boundary conditions atx = N , which may even beλ-dependent,and under these general conditions we prove in Lemma 2 a number of formulaerelating averages over the parameterα of the spectral measures with integrals overλ of the correspondingω-measures. These formulae allow us, by taking the limitN →∞, to relate theω-measures for differential operators acting inL2(0,∞) toω-measures for operators inL2(0,N) for N finite. The main idea of the proof ofLemma 2 is to make a change of variables between the parametery for a generalHerglotz family and the parameterα for a family ofm-functions, and use a generalspectral averaging formula for Herglotz functions to be found in [24].

Section 4 of the paper is concerned mainly with the Riccati equation and itssolutions. Here we deal principally with the Riccati equation most appropriate tothe caseα = 0 for which the differential operator is subject to Dirichlet boundarycondition. In Lemma 3 we derive some estimates which are used in the sequeland which relate to the question of stability with respect to changes in initialcondition.

In Section 5, we give precise definitions of the notions ofδ-clustering and ofclustering for families of solutionsh(·, λ) of the Riccati equation, and illustratethese definitions with reference to some of the standard classes of potentials(L1,bounded variation, and periodic). This is followed by the main theorem of thepaper, which shows how the hypothesis ofδ-clustering leads to estimates of the

MPAG013.tex; 5/11/1998; 8:38; p.5


boundary values of them-function, the spectral density, the spectral measure andthe ω-measure for Schrödinger operators. A corollary to Theorem 1 provides acharacterisation of clustering families of solutions in terms ofm+(λ), and theseresults are then extended to general (real) boundary conditions atx = 0.

2. Herglotz Functions

Given a Herglotz functionF (analytic in the upper half-plane with positive imagi-nary part), we have the representation ([27])

F(z) = a + bz+∫ ∞−∞

(1

t − z −t

t2+ 1

)dν(t) (Im z > 0). (2)

Herea = ReF(i), b = lims→+∞ s−1 ImF(is), andν = dν(t) is the uniquelydetermined spectral measure corresponding toF .

In terms of a real parametery, we can define a one-parameter family of HerglotzfunctionsFy(·) by

Fy(z) = F(z)

1− yF(z) (Im z > 0). (3)

We denote respectively byay, by andνy the constantsa, b and the measureν forthe functionFy .

For anyw in the upper half plane, we can associate, as in [24], a Cauchymeasure| · |w by

|A|w = 1

π

∫A

Imw

|t −w|2 dt,

for any Borel subsetA of R. Thenπ |A|w is the angle subtended at the pointw bythe subsetA of R.

For almost allλ ∈ R, we can define, for the Herglotz functionF , a Cauchymeasureω(λ, ·) atλ by

ω(λ, S) = limε→0+|S|F(λ+iε). (4)

F has a boundary valueF+(λ) = limε→0+ F(λ + iε) for almost allλ ∈ R.The decomposition of the measureν into its singular and absolutely continuouscomponents is determined by the boundary behaviour ofF(λ+ iε); thus (see, forexample, [28])

νs = ν λ ∈ R : lim

ε→0+ImF(λ+ iε) = ∞

;

νa.c. = ν λ ∈ R : lim

ε→0+ImF(λ+ iε) exists finitely

.

MPAG013.tex; 5/11/1998; 8:38; p.6


For a given measurable subsetS of R, we haveω(λ, S) = |S|F+(λ) for all λ atwhichF+(λ) exists with ImF+(λ) > 0, and

ω(λ, S) =

1 (F+(λ) ∈ S),0 (F+(λ) /∈ S)

for almost allλ at whichF+(λ) exists with ImF+(λ) = 0.The following integral identity relates theω-measureω(λ, ·) for a given Her-

glotz functionF to the corresponding one parameter familyνy of measures:∫S

y−2νy−1(A)dy =∫A

ω(t, S)dt, (5)

whereA, S are arbitrary measurable subsets ofR. (For a proof see [24].)The ω-measure for a given Herglotz functionF may be used to investigate

the boundary valuesF+(λ) of F . The following application of this idea will bedeveloped in this paper and used to study the boundary behaviour of the Weyl–Titchmarshm-function for a differential operator:

SupposeF+(λ0) exists at someλ0 ∈ R. Then Theorem 3 of [23] implies theresult

limδ→0+

1

2δ

∫ λ0+δ

λ0−δω(t, S)dt = ω(λ0, S), (6)

for any measurable subsetS of R. (In [24], this is stated under the hypothesis thatF+(λ0) is real, but the proof easily extends to the general case.) We shall applythis result, using a limiting argument together with detailed bounds for solutionsof the appropriate differential equations, to estimate theω-measure on the righthand side of (6) for general measurable setsS, often taken for convenience to beintervals. Sinceπω(λ0, S) is just the angle subtended by the setS at the pointF+(λ0), this will lead to an estimate for the value ofF+(λ0), the boundary value ofthe Herglotz function/m-function. The viability of such an approach to boundaryvalues of Herglotz functions depends on the following fundamental question:

What are the implications for the value ofω(λ0, S) of a given estimate of theboundary valueF+(λ0), and conversely what consequences for the value ofF+(λ0)

follow from detailed bounds onω(λ0, S) asS is varied? An answer to this questionrelies on a study of the relationship between the location of points in the upperhalf-plane and estimates for the corresponding angles subtended by subsets ofR,and is provided by the following lemma. Observe in this connection and in thelater analysis presented in this paper that an appropriate measure of the separationbetween two pointsw1, w2 in the upper half-plane is provided by|w1−w2|/ Imw2

rather than by|w1−w2|. The proof is given in the Appendix.

LEMMA 1. Letw1, w2 be two complex numbers and denote byθ1(S), θ2(S) theangles subtended by a given measurable subsetS ofR atw1 andw2, respectively.Then, for anyδ with 0< δ < 1,

MPAG013.tex; 5/11/1998; 8:38; p.7


(i) |w1 − w2| 6 δ Imw2 ⇒ |θ1(S) − θ2(S)| 6 δ∗θ2(S), for all S ⊆ R, whereδ∗ = δ/(1− δ).

(ii) |θ1(S)− θ2(S)| 6 δθ2(S), for all S ⊆ R, ⇒ |w1−w2| 6 δ Imw2.

[This implication requires onlyδ > 0 rather than0< δ < 1].

3. m-Functions and their Properties

We consider the differential operatorTα = −d2/dx2 + V (x), acting inL2(0,∞),subject to the boundary condition

(cosα)ϕ + (sinα)dϕ

dx= 0 atx = 0. (7)

HereV is assumed real and locally integrable, with no further conditions imposedon the behaviour ofV at large distances, apart from the requirement that we are inthe limit-point case at infinity.

Associated with the differential expression−d2/dx2 + V (x) is the differentialequation

− d2

dx2f (x, z)+ V (x)f (x, z) = zf (x, z), (8)

wherez is a complex spectral parameter; we take for convenience Imz > 0. In thecase of real spectral parameterλ we write the differential equation

− d2

dx2f (x, λ)+ V (x)f (x, λ) = λf (x, λ). (8′)

Solutionsuα(·, z), vα(·, z) of (8) and correspondinglyuα(·, λ), vα(·, λ) of (8′), aredefined subject to the initial conditions

uα(0, z) = cosα, vα(0, z) = − sinα,u′α(0, z) = sinα, v′α(0, z) = cosα.

The so-called Weyl–Titchmarshm-functionmα(z) for the differential operatorTαis then uniquely defined, for Imz > 0, by the condition that

uα(·, z)+mα(z)vα(·, z) ∈ L2(0,∞). (9)

In the caseα = 0, we shall write simplyu andv for u0, v0 andm(z) for m0(z); wethen havem(z) = f ′(0, z)/f (0, z), wheref (·, z) is any nontrivial solution of (8)for whichf (·, z) ∈ L2(0,∞).

Them-function mα is a Herglotz function (i.e.mα is analytic with positiveimaginary part in the upper half-plane) having the dependence onα given by

mα(z) = (cosα)m(z)− (sinα)

(cosα)+ (sinα)m(z).

MPAG013.tex; 5/11/1998; 8:38; p.8


We shall denote byµα the spectral measure defined in terms of the Herglotz repre-sentation formα (cf. (2)).

We shall also need to consider the Herglotz function for the differential operator−d2/dx2 + V (x) defined inL2(0,N), with boundary conditions at each end ofthe finite interval[0,N]. Suchm-functions have often been considered (see forexample [2, 25]), but here we deviate slightly from usual practice in imposinga complexboundary condition at the right-hand endpointx = N . Thus, for anyHerglotz functionη(·), define them-functionmNα,η by the condition that

fα(x, z) ≡ uα(x, z)+mNα,η(z)vα(x, z) (10)

satisfy atx = N the condition

f ′α(N, z) = η(z)fα(N, z) for Im z > 0.

Considering first the caseα = 0, and using the initial conditions foru andv, wesee thatmN0,η(z) = f ′(0, z)/f (0, z), wheref (·, z) is any (nontrivial) solution ofthe differential equation (8), subject to the prescribed condition atx = N .

Since ddx Im(f ′f ) = −(Im z)|f |2 by the standard Lagrange identity ([2]), we

have, on integrating with respect tox from 0 toN and using the condition thatIm(f ′(N,Z)f (N, z)) = (Im η(z))|f (N, z)|2 > 0, the result that Im(f ′(0, z)f (0,z)) > 0. Hencef cannot be zero atx = 0, and also

ImmN0,η(z) =Im(f ′(0, z)f (0, z))|f (0, z)|2 > 0.

It follows thatmN0,η is a Herglotz function. Forα 6= 0, the solutionfα used to deter-minemNα,η(z)must be a constant multiple of the solutionf used forα = 0; hence

mN0,η(z) =f ′(0, z)f (0, z)

= f ′α(0, z)fα(0, z)

,

which on substituting forfα, f ′α and using the initial conditions foruα, vα, implies

mN0,η(z) =(sinα)+ (cosα)mNα,η(z)

(cosα)− (sinα)mNα,η(z). (11)

HencemNα,η(z) has the sameα dependence,

mNα,η(z) =(cosα)mN0,η(z)− (sinα)

(cosα)+ (sinα)mN0,η(z), (12)

as in the case ofmα(z). It follows easily thatmNα,η is a Herglotz function for generalα. On combining Equation (11), withα replaced byβ, and Equation (12), we havethe equation

mNα,η(z) =(cos(α − β))mNβ,η(z)− sin(α − β)(cos(α − β))+ (sin(α − β))mNβ,η(z)

, (13)

MPAG013.tex; 5/11/1998; 8:38; p.9


which relates these functions for different values ofα andβ. Using the Wronskianidentityuαv′α − u′αvα = 1 atx = N , we may verify the explicit expression for them-function

mNα,η(z) =uα(N, z)η(z)− u′α(N, z)−vα(N, z)η(z)+ v′α(N, z)

. (14)

This may be verified by checking the boundary condition atx = N for the functionfα defined by (9), making use of the Wronskian identity. Here we shall mainly beconcerned with the special case in whichη is a constant function having posi-tive imaginary part. More generally, we assume that the boundary valueη+(λ) ≡limε→0+ η(λ + iε) satisfies Imη+(λ) > 0 for almost allλ ∈ R. In that case, foralmost allλ ∈ R the functionmNα,η(z) also has boundary valuemN+α,η (λ) havingstrictly positive imaginary part; we have, in fact

ImmN+α,η (λ) =Im η+(λ)

|vα(N, λ)η+(λ)− v′α(N, λ)|2.

An alternative characterisation ofmN+α,η (λ) is by the condition thatfα(x, λ) ≡uα(x, λ) + mN+α,η (λ)vα(x, λ) satisfy atx = N theλ-dependent complex boundaryconditionf ′α(N, λ) = η+(λ)fα(N, λ).

The following lemma extends to the case of complex boundary condition resultsalready known ([25]) for real boundary condition, and will be the basis for theestimates which we shall carry out in Section 5.

LEMMA 2. LetµNα,η denote the spectral measure for the Herglotz functionmNα,η,and letωNα,η(λ, ·) denote theω-measure for this Herglotz function, defined as inEquation(4). (Thus, forS ⊆ R, πωNα,η(λ, S) is the angle subtended by the setS atthe pointmN+α,η (λ).) Letµα andωα(λ, ·) denote respectively the spectral measureandω-measure formα, wheremα is them-function for the differential operatorTα = −d2/dx2 + V (x) in L2(0,∞) with boundary condition(7) at x = 0. Thenfor any Lebesgue measurable subsetsA, S ofR,

(i)∫S

(1+ y2)−1µN− cot−1 y,η(A)dy =

∫A

ωN0,η(t, S)dt,

(ii) limN→∞

∫S

(1+ y2)−1µN− cot−1 y,η(A)dy = lim

N→∞

∫A

ωN0,η(t, S)dt

=∫A

ω0(t, S)dt.

Proof. We start from the identity (5), which holds for arbitrary Herglotz func-tionsF , and take the special caseF(z) = mN0,η(z). Equation (3) then implies

Fy−1(z) = yF(z)

y − F(z) =ymN0,η(z)

y −mN0,η(z),

MPAG013.tex; 5/11/1998; 8:38; p.10


which on substitutingy = − cotα becomes

Fy−1(z) = (cosα)mN0,η(z)

(cosα)+ (sinα)mN0,η(z).

However, from the dependence (12) ofmNα,η(z) onα, we may verify thatFy−1(z) =(cos2 α)mNα,η(z)+sinα cosα. Since, withy = − cotα, the Herglotz functionsFy−1

and(cos2 α)mNα,η differ by a constant, they must have the same spectral measures,so thatνy−1 = (cos2 α)µNα,η.

With y = − cotα, we havey−2 cos2 α = sin2 α = (1+ y2)−1, so that∫S

y−2νy−1(A)dy =∫S

(1+ y2)−1µN− cot−1 y,η(A)dy,

implying (i) of the lemma.The proof of (ii) follows closely the arguments of [25, p. 4074]. First fixz in the

upper half plane. For thisz, the value ofη(z) determines the boundary condition atx = N for the functionfα in (10). Standard limit point/limit circle theory, withηreplaced by a real valued function, shows that in that case the set of pointsmNα,η(z)

lie on a circleCα,N(z) in the upper half plane, the circle depending on the values ofα andN , as well as onz. In our case, withη a Herglotz function and hence Imη >0, a minor modification of this theory implies that these points lie in the open discenclosed byCα,N(z). ForN1 > N2, theN = N1 disc is contained in theN = N2

disc. AsN →∞ the disc shrinks to a single point, which is the pointmα(z).One may verify that convergence ofmNα,η(z) tomα(z) is uniform inz over com-

pact subsets of the upper half-plane. This implies convergence of the correspondingspectral measures for finite intervalsA, thus

limN→∞µ

Nα,η(A) = µα(A) (15)

provided neither endpoint ofA is a discrete point of the measureµα. For a givenfinite intervalA, an endpoint can be a discrete point ofµα for at most one valueof α. We also have positive upper and lower bounds for|mN0,η(i)| which, on usingtheα dependence ofmNα,η given by (12), may be made uniform inα for |mNα,η(i)|.Using the Herglotz representation as in (2) formNα,η yields a uniform estimate

ImmN− cot−1 y,η(i) =

∫ ∞−∞

dµN− cot−1 y,η(t)

(t2+ 1)6 const., (16)

provided them-function has no linear term inz in its representation. The coefficientof the linear term formNα,η(z) is given bybNα,η = lims→+∞ s−1 ImmNα,η(is). Hencefrom Equation (13) we see that ifbNβ,η 6= 0 for someβ thenbNα,η = 0 for allα 6= β. Itfollows that the estimate (16) is uniform iny over all except for at most one possiblevalue ofy, for each value ofN . For a given finite intervalA, this leads to a bound

µN− cot−1 y,η(A) 6 const., (17)

MPAG013.tex; 5/11/1998; 8:38; p.11


holding except for at most one value ofy for eachN .Using now (15) and (17), we may apply the Lebesgue dominated-convergence

theorem to obtain, for any finite intervalA and measurable subsetS of R,

limN→∞

∫S

(1+ y2)−1µN− cot−1 y,η(A)dy =

∫S

(1+ y2)−1µ− cot−1 y(A)dy. (18)

The equation extends readily to general measurable subsetsA ofR, using countableadditivity.

To complete the proof of (ii) of the lemma, it remains only to use Equation (12)of [25], which states that∫

S

(1+ y2)−1µ− cot−1 y(A)dy =∫A

ω0(t, S)dt. 2

REMARK 1. In relation to the proof of (ii) of Lemma 2, we point out that in factthe estimate (16) holds uniformly forall values ofy. To see this, one has to showthat the Herglotz coefficientbNα,η mentioned in the proof is zero for allα. That thisis so is a consequence of (12) and of the following asymptotic formula:

s−12mN0,η(is) = (−1+ i)/√2+O

(s−

12)

ass → +∞ through real values; the proof of this formula uses the analogue ofEquation (171) of [1] for the solutionf (x, is) of the Schrödinger equation (8)with complex boundary conditionf ′(N, is)/f (N, is) = η(is). (See also [29] forthe special case in whichη is a real constant.)

The following Corollary extends the results of Lemma 2 to the functionmNβ,ηfor general values ofβ.

COROLLARY 1. For any Lebesgue measurable subsetsA, S ofR, we have

(i)∫S

(1+ y2)−1µN(− cot−1 y+β),η(A)dy =

∫A

ωNβ,η(t, S)dt,

(ii) limN→∞

∫S

(1+ y2)−1µN(−cot−1y+β),η(A)dy = lim

N→∞

∫A

ωNβ,η(t, S)dt

=∫A

ωβ(t, S)dt.

Proof.Again we start from the identity (5), taking in this caseF(z) = mNβ,η(z).We have, then,

Fy−1(z) = ymNβ,η(z)

y −mNβ,η(z),

which on settingy = − cotα becomes(cosα)mNβ,η(z)/((cosα)+ (sinα)mNβ,η(z)).Substituting

mNβ,η(z) =(cosβ)mN0,η(z)− (sinβ)

(cosβ)+ (sinβ)mN0,η(z)

MPAG013.tex; 5/11/1998; 8:38; p.12


from (12), we find, withy = − cotα,

Fy−1(z) = (cosα cosβ)mN0,η(z)− (cosα sinβ)

(cos(α + β))+ (sin(α + β))mN0,η(z).

Noting that

mNα+β,η(z) =(cos(α + β))mN0,η(z)− sin(α + β)(cos(α + β))+ (sin(α + β))mN0,η(z)

and using standard trigonometric identities, it is straightforward to verify that

Fy−1(z) = (cos2 α)mNα+β,η(z)+ sinα cosα.

As in the proof of the lemma, this leads to an analogous relation betweenmeasures, giving hereνy−1 = (cos2 α)µNα+β,η.

The proofs of (i) and (ii) of the corollary now follow similar lines to those ofthe corresponding results of Lemma 2, of which they are a generalisation.2

REMARK 2. We shall actually need the results of Lemma 2 and Corollary 1 inslightly greater generality, such that the complex boundary condition defined byη

is allowed to depend on the value ofN . It may be verified that the proofs of (i) and(ii) may in each case be carried through in this more general case, with the role ofthe Herglotz functionη taken by a familyη

N of Herglotz functions.

4. The Riccati Equation

Given any solutionf (·, z) of the Schrödinger equation (8) at complex spectralparameterz, such thatf (x, z) 6= 0 for x > 0, we can define a corresponding func-tion h(x, z) = f ′(x, z)/f (x, z) such thath(·, z) satisfies the well-known Riccatidifferential equation

dh(x, z)

dx= V (x)− z− (h(x, z))2. (19)

We assume here Imz > 0. Given the value of the solutionh at x = N , for someN > 0, with Imh(N, z) > 0, the solution is well defined and has positive imagi-nary part for allx in the interval[0,N]. We can construct the solution explicitly interms of any solutionf (·, z) of the Schrödinger equation subject to the conditionf ′(N, z)/f (N, z) = h(N, z); we have, in that case, for 06 x 6 N ,

Imh(x, z) = 1

|f (x, z)|2|f (N, z)|2 Im h(N, z)+ Im z

∫ N

x

|f (t, z)|2 dt,

from which the positivity of the imaginary part is easily seen.

MPAG013.tex; 5/11/1998; 8:38; p.13


On the other hand, forx > N , positivity of the imaginary part of the solutionh(x, z) will not necessarily be preserved. In fact it is well known (and indeed maybe proved from the above identity for Imh(x, z)), that the only solution of theRiccati equation such that Imh(x, z) > 0 for all x > 0 is the solution subject tothe initial conditionh(0, z) = m(z),wherem is the Weyl–Titchmarshm-function.For all other initial conditions, the solutionh(x, z)will either diverge at some finitepositive value ofx, or will have negative imaginary part for allx sufficiently large.

In the case of the Riccati equation at real spectral parameterλ,

dh(x, λ)

dx= V (x)− λ− (h(x, λ))2, (19′)

the situation is completely different. Hereeverysolution subject to an initial con-dition satisfying Imh(0, λ) > 0 will be well defined and have positive imaginarypart for allx > 0. An explicit expression for the solution of the Riccati equation,subject to such an initial condition, ish(x, λ) = f ′(x, λ)/f (x, λ), wheref (x, λ) isgiven in terms of the standard solutionsu, v(≡ u0, v0) of the Schrödinger equationdefined in Section 3, by

f (x, λ) = u(x, λ)+ h(0, λ)v(x, λ).From the Wronskian identity, we then have

Imh(x, λ) = Imh(0, λ)

|f (x, λ)|2 ,

exhibiting clearly the positivity of the imaginary part of the solution.In this paper, we are particularly interested in the case in which the differential

operatorTα = −d2/dx2 + V (x) has absolutely continuous spectrum, though notnecessarilypurely absolutely continuous spectrum. A support of the absolutelycontinuous part ofµα can be defined as the set of allλ ∈ R at which the boundaryvaluemα+(λ) ≡ limε→0+mα(λ + iε) exists with strictly positive imaginary part.Equation (13) then implies that this set is in fact independent ofα. Forλ belong-ing to this set, a particularly significant solution of the Riccati equation is thatsolution which we shall denote byh+(x, λ), which satisfies the initial conditionh+(0, λ) = m+(λ) ≡ m0+(λ). Thus

h+(x, λ) = u′(x, λ)+m+(λ)v′(x, λ)u(x, λ)+m+(λ)v(x, λ) =

f ′+(x, λ)f+(x, λ)

,

wheref+(x, λ) is the boundary value atλ of theL2 solutionf (x, z) = u(x, z) +m(z)v(x, z) of the Schrödinger equation at complex spectral parameterz.

Just asm+(λ) = f ′+(0, λ)/f+(0, λ), so

m+(N, λ) = h+(N, λ) = f ′+(N, λ)f+(N, λ)

MPAG013.tex; 5/11/1998; 8:38; p.14


is the boundary value atλ of them-functionm(N, z) for the differential opera-tor −d2/dx2 + V (x), acting inL2(N,∞) with Dirichlet boundary condition atx = N . Hence the single solutionh+(x, λ), with the appropriate initial condi-tion h+(0, λ) = m+(λ), determines the boundary value of them-function for theDirichlet Hamiltonian inall intervals[N,∞) for N > 0.

The main purpose of this paper will be to identify criteria which will char-acterise the particular solutionh+(x, λ) of the Riccati equation at real spectralparameterλ which determines this family ofm-functions (and hence also theirrelated spectral measures, densities, and so on). Such criteria are to be found inan analysis of the clustering properties of solutions of the Riccati equation as thespectral parameterλ is varied.

As a preliminary to this analysis, to be carried out in the next section, thefollowing result deals with a different but related question, that of stability withrespect to changes in initial condition. As in Section 2, we estimate the separationof two complex numbersw1, w2 through a comparison of|w1−w2| with Imw2 (orImw1) rather than a bound on the magnitude of|w1−w2|.LEMMA 3. Leth1(·, λ), h2(·, λ) be two solutions of the Riccati equation(19′), atreal spectral parameterλ, over an interval[0,N]. Then, for anyδ in the interval0< δ < 1,

|h1(0, λ)− h2(0, λ)| 6 δ Im h2(0, λ)

⇒ |h1(N, λ)− h2(N, λ)| 6 δ

(1− δ) Imh2(N, λ).

Moreover,

|h1(N, λ)− h2(N, λ)| 6 δ Imh2(N, λ)

⇒ |h1(0, λ)− h2(0, λ)| 6 δ

(1− δ) Imh2(0, λ).

Proof.We have already written down the solution of the Riccati equation subjectto a given initial condition, from which we have, for any solutionh(·, λ) over theinterval,

h(N, λ) = u′(N, λ)+ h(0, λ)v′(N, λ)u(N, λ)+ h(0, λ)v(N, λ) .

For fixedN andλ, the transformation which takesh(0, λ) into h(N, λ) is a so-called Möbius or fractional linear transformation of the upper half plane, of theformw→ (aw + b)/(cw + d), wherea, b, c, d are real and the Wronskian iden-tity implies ad − bc = 1. For the properties of Möbius transformations, see forexample [30]. For such a transformation, suppose thatw1 → ξ1 andw2 → ξ2,where|w1−w2| 6 δ Imw2, with 0< δ < 1. Then

|ξ1− ξ2| =∣∣∣∣aw1+ bcw1+ d −

aw2+ bcw2+ d

∣∣∣∣ = |w1−w2||cw1+ d||cw2 + d| ,

MPAG013.tex; 5/11/1998; 8:38; p.15


and Imξ2 = Imw2/|cw2+ d|2. Hence

|ξ1− ξ2|Im ξ2

=∣∣∣∣cw2+ dcw1+ d

∣∣∣∣ |w1− w2|Imw2

6 δ∣∣∣∣cw2+ dcw1+ d

∣∣∣∣.However,∣∣∣∣cw1+ d

cw2+ d∣∣∣∣ = ∣∣∣∣1+ c(w1−w2)

cw2+ d∣∣∣∣ > 1− |w1−w2|

|w2+ d/c|> 1− δ Imw2

|w2+ d/c| > 1− δ,

providedc 6= 0. Hence∣∣∣∣cw2+ dcw1+ d

∣∣∣∣ 6 1

1− δ ,

the inequality holding trivially in the casec = 0, and we have

|ξ1− ξ2|Im ξ2

6 1

(1− δ)|w1−w2|

Imw26 δ

(1− δ).The first implication of the lemma now follows by taking the appropriate Möbiustransformation for the Riccati equation across the interval[0,N], withw1 = h1(0,λ), w2 = h2(0, λ) and ξ1 = h1(N, λ), ξ2 = h2(N, λ). To prove the secondimplication, observe that the inverse transformation toξ = (aw + b)/(cw + d)is a transformation of the same form, given byw = (dξ − b)/(a − cξ), and repeatthe previous argument. 2

REMARK 3. The multiplicative constantsδ/(1− δ) in the inequalities of thelemma are optimal, and rely on the bound|(w2+ x)/(w1+ x)| 6 1/(1− δ) forall x ∈ R, whenever|w1−w2| 6 δ Imw2.

REMARK 4. One could use|w1−w2|/√(Imw1)(Imw2) rather than, say,|w1−w2|/Imw2, as a measure of separation for complex numbersw1, w2 through-out this paper. This has the advantage of being symmetric betweenw1 andw2, andalso it follows from the conservation of cross ratios ([30]) that this quantity isconserved by Möbius transformations. Nevertheless, in our view these advantagesare outweighed by the relative simplicity in the estimates of angle subtended andω-measures which we shall derive in the following section, through the use of the|w1−w2|/Imw2 estimate. In the current context such alternative estimates differin any case only to orderδ2.

5. δ-Clustering Solutions of the Riccati Equation

The following definitions express more precisely the notion that a solutionh(·, λ)(more precisely a family of solutions) of the Riccati equation may to orderδ be

MPAG013.tex; 5/11/1998; 8:38; p.16


asymptotically independent ofλ, for λ belonging to suitable setsE , and for asequence of large values ofx. For such a solution, we use the term “recurrentlyδ-clustering”, abbreviated for convenience to “δ-clustering”.

We shall also make use of the terms “point of density” and “approximate con-tinuity”. (See [31, 32].) A real numberλ0 is said to be a point of density of ameasurable setE ⊆ R provided limK→0+ |E ∩ [λ0−K,λ0+K]|/2K = 1, where| · | stands for Lebesgue measure. A measurable functionF fromR toR is said tobe approximately continuous at a pointλ0 of its domain if, for anyδ > 0, λ0 is apoint of density of the set ofλ for which |F(λ)− F(λ0)| < δ.

Thus, “point of density” expresses the idea that ‘most’ points nearλ0 belong toa given setE , and “approximate continuity” expresses the idea thatF(λ) is closeto F(λ0) for ‘most’ pointsλ nearλ0. Given a measurable setE , almost allλ ∈ Ewill be points of density ofE , and given a measurable functionF , almost allλ ∈domain(F ) will be points of approximate continuity.

We are now ready to define the notion ofδ-clustering.

DEFINITION 1. LetE ⊆ R be measurable and letδ be a positive constant. Wesay that a family of solutionsh(·, λ) of the Riccati equation

d

dxh(x, λ) = V (x)− λ− (h(x, λ))2; x ∈ [0,∞), λ ∈ E

is δ-clustering onE if there exists a functionH : [0,∞) → C, with ImH > 0,such that

lim infx→∞ sup

λ∈E|h(x, λ)−H(x)|

ImH(x)< δ. (20)

The familyh(·, λ) is said to beclusteringat someλ0 ∈ R, if there is a measurablesubsetE ofR, with all λ ∈ E points of density ofE andh(0, λ) approximately con-tinuous at allλ ∈ E as a function ofλ, and such that for anyδ > 0 an open intervalI containingλ0 can be found, with the solutionh(·, λ) δ-clustering onE ∩ I.

The solution is said to be clustering onE if it is clustering at allλ ∈ E .

REMARK 5. Given a setE and correspondingly a solutionh(·, λ) of the Ric-cati equation for eachλ ∈ E , we may try to minimise the value ofδ in (20),by choosing the value ofH(x) at eachx > 0 to minimise the supremum of|h(x, λ)−H(x)|/ ImH(x) asλ is varied overE . This minimisation, though possi-ble in principle, is not usually practical, and in practice, in verifying theδ-clusteringproperty, it is simpler (though not optimal) to take for exampleH(x) = h(x, λ0)

for some fixedλ0 ∈ E .

REMARK 6. The property of a solution to beδ-clustering on a setE will hold ifand only if a functionH and an increasing sequence`1, `2, `3, . . . can be found,with `j →∞, such that, for allj = 1,2,3, . . .

|h(`j , λ)−H(`j )| < δ ImH(`j)

MPAG013.tex; 5/11/1998; 8:38; p.17


for all λ ∈ E .On the other hand, if theδ-clustering propertyfails for E , then for anyδ0 in the

interval 0< δ0 < δ we have supλ∈E |h(x, λ)−H(x)|/ImH(x) > δ0 for all suf-ficiently largex and for all choices of the functionH . In particular, we then have,for any fixedλ0 ∈ E , and for large enoughx, |h(x, λ)−h(x, λ0)| > δ0 Imh(x, λ0)

for someλ ∈ E .

REMARK 7. The clustering property at a pointλ0 implies that a family of solu-tions clusters arbitrarily closely (i.e. to orderδ with δ arbitrarily small) forλ suffi-ciently close toλ0. Since the inequality|h(`, λ)−H(`)| < δ ImH(`), for all λ ∈ E ,implies also|h(`, λ) − h(`, λ0)| < (2δ/(1− δ)) Im h(`, λ0), for all λ, λ0 ∈ E , inconsidering the clustering property at a pointλ0 one may take without loss of gen-eralityH(x) = h(x, λ0), and this is often a convenient choice of theH function.

Before proceeding to the main results of this paper, it may be helpful to con-sider briefly the application of the term “δ-clustering” to some simple classes ofpotentials. The simplest case is that in whichV ≡ 0. The Riccati equation thentakes the form dh/dx = −λ− h2, which has the solutionh = i√λ for anyλ > 0.Note thati

√λ is the boundary valuem+(λ) of them-function, and that this solution

is a constant function for eachλ > 0 because−d2/dx2 has the samem-function(with Dirichlet boundary condition) as a differential operator inL2([`,∞)) for any` > 0.

For all initial conditions other thanh(0, λ) = i√λ , such that Imh(0, λ) > 0,the orbit of the solutionh(x, λ) asx is varied, for fixedλ > 0, is a circle|h|2+λ =const. Imh, with the pointi

√λ in its interior.

In the case of zero potential, it is relatively straightforward to determine whethera given solution of the Riccati equation isδ-clustering or not, since exact solutionsof the equation may be written down, for arbitrary initial conditions.

As an example, consider the solutionh(·, λ) of the Riccati equation (withV =0), subject to the initial conditionh(0, λ) = i√λ0, whereλ0 is an arbitrary positivenumber.

For 0< δ < 1, if E is any closed subset of the intervalE ′ = λ : |λ−λ0| < δλ0,we may verify for allx > 0 the estimate, forλ ∈E ′,

|h(x, λ)− h(x, λ0)|Imh(x, λ0)

= |λ− λ0|λ0

(1+ cot2 x

√λ)−1/2

< δ,

and it follows thath(·, λ) is δ-clustering onE , with H(x) = h(x, λ0). This maynot, however, be the optimal choice ofH to achieveδ-clustering, if we can varyE . Taking againh(0, λ) = i

√λ0, one may verify that anyλ in the interval

λ0(1− δ)/(1+ δ) < λ < λ0(1+ δ)/(1− δ), which is larger than the intervalE ′, will belong to an open intervalE on whichh(·, λ) is δ-clustering, by suitablechoice ofH .

The above example also provides an illustration of the clustering property atλ0;the solutions subject to initial conditionh(0, λ) = i√λ0 are clustering atλ0 (and

MPAG013.tex; 5/11/1998; 8:38; p.18


at no other point), since by shrinking the intervalE containingλ0 we can ensuretheδ-clustering property onE with an arbritrarily small value ofδ.

In Theorem 1 below, we shall use theδ-clustering property to obtain an estimateof the closeness ofh(0, λ), the initial value of the solution, tom+(λ), the boundaryvalue of them-function, for values ofλ in a given setE . Roughly, this estimate willstate that if the familyh(·, λ) is δ-clustering onE thenh(0, λ) is within distanceof orderδ of m+(λ), for almost allλ ∈ E . Although such estimates are of limitedinterest wherem+(λ) is known exactly, we believe that the general result, whichmakes no special assumptions on the form of the potential, is of both theoreticaland practical interest in the study of them-function and its boundary values.

The class of potentialsV ∈ L1(0,∞) may be treated as a perturbation of thecase of zero potential. By regarding the Riccati equation as a pair of coupled differ-ential equations for the real and imaginary parts ofh, and evaluating the derivative,one may verify the identity, for any solutionh(·, λ),

d

dx

|h(x, λ)|2 + λImh(x, λ)

= 2V (x)Reh(x, λ)

Imh(x, λ)

=[

2V (x)Reh(x, λ)

|h(x, λ)|2+ λ] |h(x, λ)|2+ λ

Imh(x, λ)

. (21)

ForV ∈ L1(0,∞), it follows that(|h|2+ λ)/Imh converges to a limit asx →∞,for eachλ > 0. This limit may be zero, in which caseh(x, λ) → i

√λ, and the

solution converges to the value ofm+(λ) for the unperturbed problemV ≡ 0; orthe limit may be nonzero, in which case the orbit of the solution asymptoticallyapproaches a circle in the upper half-plane. It will be a consequence of Theo-rem 1 below (which, however, is stated in much greater generality) that for theunique initial condition which yields the asymptoticsh(x, λ) → i

√λ we have

h(0, λ) = m+(λ), where nowm+(λ) is the boundary value of them-function forthe perturbed Dirichlet operator−d2/dx2 + V (x) in L2(0,∞); correspondingly,h(`, λ)will then be the boundary value of them-function for the Dirichlet operator−d2/dx2 + V (x) in L2(`,∞).

The case of a potentialV of bounded variation (assuming for convenience thatV → 0 asx → ∞) may be treated in a similar way. Using the identity (21)and noting that 2 Reh/Imh = d/dx(1/Imh), one may integrate the identity byparts and using the estimate that Imh is bounded below by a positive constant, forfixed λ > 0, deduce that again in this case(|h|2+ λ)/Imh converges to a limitasx → ∞. Hence for potentials of bounded variation we may identify, as in theL1 case, a particular solution of the Riccati equation which leads to the asymptoticpropertyh(x, λ) → i

√λ, and satisfying the initial conditionh(0, λ) = m+(λ);

the analysis may be extended without difficulty to potentials which are a sum of anL1 component and a component of bounded variation, covering in this way a wideclass of decaying potentials, both of short and long-range. In all such cases, theseparticular solutions of the Riccati equation satisfying specific asymptotics are infact precisely the clustering solutions we have already defined above.

MPAG013.tex; 5/11/1998; 8:38; p.19


An interesting class of nondecaying potentials in this context is provided by theclass of periodic potentials. For a potential of periodT , a family of solutionsh(·, λ)of the Riccati equation will be clustering on a setE if h(·, λ) is itself periodic, inthe sense thath(x+T, λ) = h(x, λ) for λ ∈ E ; for such solutions we will then haveh(0, λ) = m+(λ). Theδ-clustering solutions may then roughly be characterised asthose which stay within distance O(δ) of a periodic solution, at a sequence`j ofpoints with`j →∞.

As a typical application to a class of potentials tending to−∞ at large distances,consider the one-parameter family of potentialsV = Vβ(x) = −βx2 (β > 0). Inthat case, forλ > 0, the transformation

s =∫ x

0(λ+ βt2)1/2 dt, g = (λ+ βx2)1/4f

reduces the Schrödinger equation to the standard form

d2g

ds2+ 1+ Rβg = 0,

withRβ ≡ Rβ(s, λ) satisfying∫∞

0 |Rβ(s, λ)|ds <∞. (See [33] for further details.)If λ is allowed to be negative, sayλ ∈ [−λ−,0] for some fixedλ− > 0, a

similar transformation can be carried out for allx > x0 sufficiently large, by takingthe transformed variables to bes = ∫ x

x0(λ+ βt2)1/2 dt , with x0 >

√−λ−/β.The ratiok = (dg/ds)/g then satisfies a Riccati equation dk/ds = −1 −

Rβ − k2, where, for a given solutionf (x, λ) of the Schrödinger equation,h(x, λ)(= (df /dx)/f ) is given in terms ofk(s, λ) by

h(x, λ) = (λ+ βx2)1/2k(s, λ)− βx2(λ+ βx2)−1.

As x ands tend to infinity, this enables us to pass easily from bounds onk, of theform

| k(s, λ)−K(x) |ImK(x)

< δ′,

to corresponding bounds onh, of the form

| h(x, λ)−H(x) |ImH(x)

< δ,

providedH andK are related by

H = (λ0+ βx2)1/2K − βx2(λ0+ βx2)−1, for some fixedλ0.

By a natural extension of Definition 1, the notion ofδ-clustering may be applied tofamilies of solutionsk(·, λ) of the transformed Riccati equation, andh(·, λ) will beδ-clustering whenever the familyk(·, λ) is δ′-clustering for someδ′ < δ. Moreover,

MPAG013.tex; 5/11/1998; 8:38; p.20


with Rβ(·, λ) ∈ L1(0,∞), we may adapt the same techniques as forL1 potentialsto determineδ′-clustering solutions fork(·, λ).

We omit, here, the detailed consequences of this approach, but note that thereis a solutionh(·, λ) of the Riccati equation (19′) which is clustering onR, with theasymptotic behaviourh(x, λ) ∼ iβ1/2x asx → ∞, and that the solutionhN(·, λ)of (19′), subject to the condition

hN(N, λ) = i(λ+ βN2)1/2− βN2(λ+ βN2)−1

is δ-clustering withδ = tanhIN(β, λ), where

IN(β, λ) =∫ ∞N

β

4|(3βx2 − 2λ)(λ+ βx2)−5/2|dx (N2 > −λ/β).

The following theorem derives some consequences of the clustering property, andapplies to all locally integrable potentials under the sole hypothesis that we are inthe limit-point case at infinity.

THEOREM 1. LetV be any real-valued potential, integrable on compact subin-tervals of[0,∞), and in the limit-point case at infinity. LetE be a measurablesubset ofR, with eachλ ∈ E a point of density ofE . For λ ∈ E , let h(·, λ) satisfythe Riccati equation(19′) subject to initial conditions withImh(0, λ) > 0 andh(0, λ) approximately continuous at allλ ∈ E . Suppose that the solutionh(·, λ) isδ-clustering onE , for someδ in the interval0< δ < 1/2.

Let m(z) denote them-function, with boundary valuem+(λ), ω(λ, ·) the ω-measure,µ the spectral measure with absolutely continuous componentµa.c. forthe differential operator−d2/dx2 + V (x) in L2(0,∞), with Dirichlet boundarycondition atx = 0.

Then the functionh(0, λ) provides the following orderδ estimates ofm+(λ),ω(λ, S) andµa.c.(E) valid for almost allλ ∈ E , and holding in particular at allλ ∈ E at whichm+(λ) exists:

(i) |h(0, λ)−m+(λ)| 6 (δ/(1− 2δ)) Imm+(λ);(ii) |π−1ψ(λ, S)− ω(λ, S)| 6 (δ/(1− 2δ))ω(λ, S), whereψ(λ, S) is the angle

subtended byS at the pointh(0, λ);

(iii)

∣∣∣∣ ∫E

π−1 Im h(0, λ)dλ− µa.c.(E)

∣∣∣∣ 6 δ

1− 2δµa.c.(E).

Moreover, ifE is an interval, the multiplicative constant on the right handside of this last estimate may be improved toδ/(1− δ), under the weakerhypothesis0 < δ < 1, and we have the following additional estimate of1π

Imm+(λ), the spectral density function:(iv) | Imh(0, λ)− Imm+(λ)| 6 (δ/(1− δ)) Imm+(λ).

Proof. Supposeh(·, λ) is δ-clustering onE , with 0 < δ < 1/2. FollowingRemark 6 after Definition 1, let`j be an increasing sequence, with`j → ∞,

MPAG013.tex; 5/11/1998; 8:38; p.21


such that|h(`j , λ) − H(`j)| < δ ImH(`j ), for λ ∈ E . Let ` denote`j for somefixed j , and define the constant functionη = H(`). Now define the correspondingHerglotz functionmN0,η(z) as in Equation (10).

Then the solution of the Riccati equation, which atx = 0 has the valuemN+0,η (λ),will have the valueH(`) atx = `. We also know that the solution, which atx = 0has the valueh(0, λ), will have the valueh(`, λ) at x = `. Moreover, atx = ` wehave the estimate|h(`, λ) − H(`)| < δ ImH(`), which by Lemma 3 implies theestimate atx = 0∣∣h(0, λ)−mN+0,η (λ)

∣∣ 6 δ

1− δ ImmN+0,η (λ), (22)

for all λ ∈ E . We can now apply Lemma 1, which translates an estimate of theseparation between two points in the upper half-plane into an estimate of the dif-ference between the angles subtended by a given measurable subsetS of R at thesetwo points. Noting that in this case

δ∗ = δ

1− δ/(

1− δ

1− δ)= δ

1− 2δ,

from the definitions ofψ(λ, S) andωN0,η(λ, S), we then have

|π−1ψ(λ, S)− ωN0,η(λ, S)| 6δ

1− 2δωN0,η(λ, S),

for all λ ∈ E and all measurableS ⊆ R.Integrating over the setE now gives∣∣∣∣ ∫

E

π−1ψ(t, S)dt −∫

E

ωN0,η(t, S)dt

∣∣∣∣ 6 δ

1− 2δ

∫E

ωN0,η(t, S)dt,

which from Lemma 2, on lettingN tend to∞ and taking note of Remark 2 at theend of Section 3, yields∣∣∣∣ ∫

E

π−1ψ(t, S)dt −∫

E

ω(t, S)dt

∣∣∣∣ 6 δ

1− 2δ

∫E

ω(t, S)dt. (23)

This inequality holds also withE replaced byEK ≡ E ∩ [λ − K,λ + K], foranyλ ∈ E andK > 0. Takeλ ∈ E to be a point at whichm+(λ) exists; this willbe so for almost allλ ∈ E . By hypothesis,λ will be a point of density ofE anda point of approximate continuity ofh(0, λ). Hence alsoψ(λ, S) will be approxi-mately continuous at this point. So we have limK→0+ 1

2K

∫EKψ(t, S)dt = ψ(λ, S),

and limK→0+ 12K

∫EKω(t, S)dt = ω(λ, S). We donot, here, need to assume that

EK covers the whole of the interval[λ − K,λ + K]; since bothπ−1ψ(t, S) andω(t, S) are bounded by 1, andλ is a point of density ofE , the contributions to

12K

∫ λ+Kλ−K ψ(t, S)dt and 1

2K

∫ λ+Kλ−K ω(t, S)dt from integrating over points not inE

would in any case vanish in the limitsK → 0+. We may therefore conclude, on

MPAG013.tex; 5/11/1998; 8:38; p.22


settingE = EK in (23), dividing by 2K, and proceeding to the limit, that, at almostall λ ∈ E ,

|π−1ψ(λ, S)− ω(λ, S)| 6 δ

1− 2δω(λ, S). (24)

This proves (ii) of the theorem. The proof of (i) now follows immediately from (ii)of Lemma 1, which allows us to proceed from an estimate of angles subtended toan estimate of distances of points in the upper half-plane.

Note that, under the hypotheses of the theorem,m+(λ) cannot be real for anyλ ∈ E . If m+(λ) were real we should haveω(λ, S) = 0 for any closed intervalSwith m+(λ) /∈ S. For such intervalsS, (24) then impliesψ(λ, S) = 0, which isnot possible with Imh(0, λ) > 0. A similar argument shows that, forλ ∈ E , wecannot have limε→0+ Imm(λ+ iε) = ∞, since this would implyω(λ, S) = 0 forany finite intervalS. Since the singular partµs of the measureµ is supported onthe set ofλ at which limε→0+ Imm(λ + iε) = ∞, it follows thatµs(E) = 0, andhence thatµ(E) = µa.c.(E).

Part (iii) of the theorem now follows from (i) and the fact thatπ−1 Imm+(λ) isthe density function forµa.c. [28].

The proof of the stronger version of (iii), under the hypothesis thatE is aninterval, follows from (15) on integrating the inequality (22) directly and notingthat π−1 ImmN+0,η (λ) is the density function for the measureµN0,η. Inequality (iv)follows as before from a limiting argument. 2

The following corollary is a straightforward consequence of the main theorem:

COROLLARY 2. Under the same hypotheses on the potentialV as for Theo-rem1, supposeh(·, λ) is a family of solutions of the Riccati equation(19′) whichis clustering atλ0. Thenh(·, λ0) satisfies the initial conditionh(0, λ0) = m+(λ0),providedm+(λ0) exists.

Proof.From Definition 1, we may assume the existence of a measurable setE ,with λ0 ∈ E , and a familyIδ of open intervals containingλ0, such thath(·, λ) isδ-clustering onE ∩ Iδ.

Then the estimate (i) of Theorem 1 holds atλ = λ0 for all δ in the interval0< δ < 1/2. It follows immediately thath(0, λ0) = m+(λ0). 2

Theorem 1 and its corollary provide a general criterion for distinguishing thesolutionh+(·, λ) of the Riccati equation which agrees atx = 0 with the boundaryvaluem+(λ) of them-functionm(z) and atx = `with the boundary valuem+(`, λ)of them-function for the differential operator acting inL2(`,∞). Thus,h+(·, λ)is clustering for eachλ and any family of solutions which is clustering at a pointλ allows us to determine the boundary value at this point of them-function for thedifferential operator inL2(`,∞) for ` > 0.

REMARK 8. One may easily convert (i) of the theorem into the specific bound|m+(λ) − h(0, λ)| 6 (δ/(1− 3δ)) Imh(0, λ) for m+(λ), providedδ < 1/3. This

MPAG013.tex; 5/11/1998; 8:38; p.23


bound may be slightly improved by definingh(λ) = Reh(0, λ) + (i/(1− δ2))

Imh(0, λ).Then we have|m+(λ)− h(λ)| 6 (δ/(1− 2δ)) Im h(λ). Bounds similar to those

above may also be obtained as a consequence of (ii), (iii) and (iv) of the theorem.

REMARK 9. Estimates (i)–(iv) of the theorem, based on the hypothesis ofδ-clustering on a setE , are close to optimal, in the sense that any improvement in thevalues of the multiplicative constantsδ/(1− 2δ) or δ/(1− δ) can at best providean orderδ2 correction. To illustrate this point, consider the simple caseV (x) ≡ 0,taking as before the solution subject to initial conditionh(0, λ) = i

√λ0. Taking

λ0 = 1 andδ = 1/10, we can find an interval, containing the pointλ = 1, onwhich h(·, λ) is δ-clustering. Part (iv) of the theorem then provides the followingestimate form+(1) : 9/10 6 Imm+(1) 6 9/8. These bounds on the spectraldensity function atλ = 1 provide an accuracy of up to 10%. On the other hand,still with δ = 1/10 but taking a different family of solutions which leads toδ-clustering in a neighbourhood ofλ = 1, namelyh(0, λ) = i

√λ0 andλ0 slightly

larger than 9/11, we deduce the upper bound Imm+(1) 6 1.018. A third family ofsolutions withδ = 1/10 andλ0 slightly smaller than 11/9 gives rise to the lowerbound Imm+(1) > 0.995. It is interesting to note, here, that two estimates to orderδ, taken together, give rise to the single estimate 0.995 6 Imm+(1) 6 1.018,which determines the spectral density atλ = 1 to orderδ2. Note also that wereit possible to use theδ-clustering hypothesis to derive the bound (iv) withδ (orevenδ/(1− 1/2δ)) in place ofδ/(1− δ) the last stated inequalities for Imm+(1)would be replaced in the case of the lower bound by an inequality that is in factcontradicted by the exact result Imm+(1) = 1. Henceδ/(1− δ), or something likeit, is really needed.

REMARK 10. For the class of potentialsVβ(x) = −βx2 (β > 0) the resultsof Theorem 1 lead to explicit bounds form+(λ), using theδ-clustering familiesmentioned earlier. As an example, one finds the interesting estimate∣∣∣∣m+(λ)√

λ− i cosh

(cβ1/2λ−1)∣∣∣∣ 6 tanh

(cβ1/2λ−1),

valid for all λ > 0, where the constantc can be precisely determined. In theasymptotic limitλ→∞, one may use the method of Harris ([34]) to write downa series for the solution of the Riccati equation, of which the first few terms give

m+(λ) = i(

1− β

4λ2+ o

(1

λ2

)).

See [33] for another approach to the asymptotic expansion in inverse powers ofλ.Theδ-clustering familieshN(·, λ) referred to earlier, taking as an exampleN =

10, β = 4, and making a crude estimate of the resulting integral, already lead touniform estimates ofm+(λ) to within a tolerance 2×10−3 over the range| λ |< 65.These estimates can be further improved, either by controlling more precisely the

MPAG013.tex; 5/11/1998; 8:38; p.24


solution of the Riccati equation, or, in the case of largeλ, making use ofλ→ ∞asymptotics. See [35] for the numerical computation ofm(z) for Im z > 0, usingrepeated diagonalisation to control the largex asymptotics.

REMARK 11. Although we have dealt mainly with the caseα = 0, we can useCorollary 1 to cast the theory into a form which applies to general values ofα. Letus define solutionsuNα (·, λ), vNα (·, λ) of (8′), subject to the conditions atx = N

uNα (N, λ) = cosα, vNα (N, λ) = − sinα,uN′

α (N, λ) = sinα, vN′

α (N, λ) = cosα.

Given a complex solutionf (·, λ) of Equation(8′) with Im(f ′/f ) > 0, wedefine a corresponding functionhα(·, λ) by the equation

f (x, λ) = CNα uNα (x, λ)+ hα(N, λ)vNα (x, λ),for all x > 0,N > 0.

In the caseα = 0 we haveh(x, λ) ≡ h0(x, λ) = f ′(x, λ)/f (x, λ) andh(·, λ)then satisfies the Riccati equation(19′). For generalα, hα is related toh, as inEquations (11), (12), by

h(x, λ) = (sinα)+ (cosα)hα(x, λ)

(cosα)− (sinα)hα(x, λ),

hα(x, λ) = (cosα)h(x, λ)− (sinα)

(cosα)+ (sinα)h(x, λ).

We can substitute the latter expression into the Riccati equation forh(·, λ) to obtainthe Riccati equation forhα(·, λ):

d

dxhα(x, λ) = [V (x)− λ][(cosα)− (sinα)hα(x, λ)]2 −

− [(sinα)+ (cosα)hα(x, λ)]2. (25)

If, now, we define a familyhα(·, λ) of solutions of (25) to beδ-clustering onE if there exists a functionHα such that lim infx→∞ supλ∈E |hα(x, λ)−Hα(x)|/ImHα(x) < δ, then the proof of Theorem 1 will proceed as before, with the obvi-ous changes ofh replaced byhα,m bymα,µ byµα, and so on. By suitable choiceof theHα function, we may then obtain estimates analogous to those of (i)–(iv) ofTheorem 1, which will relate to properties of the absolutely continuous spectrum ofTα. Since the transformation betweenhα andhβ is a Möbius transformation, we canalso use Lemma 3 to show that ifhα(·, λ) is δ-clustering thenhβ(·, λ) is δ/(1− δ)-clustering. In particular, this shows that the property of a family of solutions beingclustering is in fact independent ofα.

MPAG013.tex; 5/11/1998; 8:38; p.25


Appendix: Proof of Lemma 1

(i) The density function for the measureS 7→ θ(S) corresponding to a pointwin the upper half-plane is given (see p. 56 of [30]) by

q(t) = Imw

|t −w|2 = Im1

t −w.

Settingqj (t) = Im 1/(t − wj) (j = 1,2), we shall derive the bound|q1(t) −q2(t)| 6 δ∗q2(t). The result will then follow from the inequality|θ1(S)− θ2(S)| =| ∫S(q1(t)− q2(t))dt| 6 ∫

S|q1(t)− q2(t)|dt .

Assuming, then,|w1−w2| 6 δ Imw2, we have

|q1− q2| =∣∣∣∣ Im(

1

t −w1− 1

t − w2

)∣∣∣∣ = ∣∣∣∣ Im(w1−w2

(t −w1)(t −w2)

)∣∣∣∣6 |w1−w2||t −w2||t −w2| .

With q2 = Imw2/|t −w2|2, this gives∣∣∣∣q1 − q2

q2

∣∣∣∣ 6 |w1− w2|Imw2

∣∣∣∣ t −w2

t −w1

∣∣∣∣ 6 δ∣∣∣∣ t − w2

t − w1

∣∣∣∣.Now ∣∣∣∣ t −w1

t −w2

∣∣∣∣ = ∣∣∣∣1− w1−w2

t −w2

∣∣∣∣ > 1− |w1− w2||t −w2| > 1− |w1−w2|

Imw2> 1− δ.

Hence∣∣∣∣ t −w2

t −w1

∣∣∣∣ 6 1

1− δ ,

and we have∣∣∣∣q1 − q2

q2

∣∣∣∣ 6 δ

1− δas required.

(ii) The density functionq for the measureS 7→ θ(S) may be expressed as

q(t) = limε→0+

1

2ε

∫ t+ε

t−εq(x)dx = lim

ε→0+1

2εθ(t − ε, t + ε).

Hence from the assumed inequality(1− δ)θ2(S) 6 θ1(S) 6 (1+ δ)θ2(S), validin particular if S is an interval, we can derive, by a limiting argument for smallintervals, the corresponding inequality for density functions

(1− δ)q2 6 q1 6 (1+ δ)q2,

MPAG013.tex; 5/11/1998; 8:38; p.26


implying that∣∣∣∣q1 − q2

q2

∣∣∣∣ 6 δ.To make full use of this inequality for the density functions, we first derive anidentity for (q1− q2)/q2. To do so, we make the substitutiont = Rew1+α Imw1,whereα is an arbitrary real parameter. Then|t − w1|2 = (1+ α2)(Imw1)

2, andsettingw1−w2 = ρeiφ (ρ, φ, real withρ > 0), we get

|t −w2|2 = |t −w1+ ρeiφ|2 = |(α − i) Imw1+ ρ cosφ + iρ sinφ|2= (1+ α2)(Imw1)

2+ ρ2+ ρ Imw12α cosφ − 2 sinφ.With qj (t) = Imwj/|t −wj |2, we have

q1(t)− q2(t)

q2(t)= (Imw1)|t −w2|2− (Imw2)|t −w1|2

(Imw2)|t −w1|2 ,

and substituting the above expressions for|t − w1|2 and |t − w2|2 in terms ofαresults inq1− q2

q2

= (1+ α2)(Imw1)2+ ρ2 + ρ Imw12α cosφ − 2 sinφ − (Imw1)(Imw2)(1+ α2)

(Imw1)(Imw2)(1+ α2).

The first and last terms in the numerator on the right hand side together give(1+ α2)(Imw1) Im(w1−w2) = (1+ α2)ρ(Imw1) sinφ, so that we now have

q1 − q2

q2

= ρ2

(Imw1)(Imw2)(1+ α2)+ ρ

(Imw2)

2α

(1+ α2)cosφ − (1− α

2)

(1+ α2)sinφ

.

Now setα = tan γ2 (−π < γ < π), and use the trigonometric identities cosγ =(1− α2)/(1+ α2), sinγ = 2α/(1+ α2), to give

q1− q2

q2= ρ2 cos2 γ2(Imw1)(Imw2)

+ ρ

(Imw2)sin(γ − φ). (26)

Sinceq1(t)−q2(t)

q2(t)converges to a limit as|t| → ∞ (i.e. asγ → ±π), Equation (26)

also makes sense forγ = ±π .By hypothesis, we now have|(q1 − q2)/q2| 6 δ for all γ ∈ [−π, π ]. Setting

γ = π2 + φ (mod 2π), one may deduce the inequalityρ/Imw2 6 δ. (Equality

can occur only in the case sinφ = 1, for whichw1 is vertically abovew2 in thecomplex plane.)

Thus|w1−w2| 6 δ Imw2, completing the proof of (ii) of the lemma. One mayverify that this part of the lemma holds without the assumptionδ < 1.

MPAG013.tex; 5/11/1998; 8:38; p.27


References

1. Titchmarsh, E. C.:Eigenfunction Expansions, Part I, 2nd edn., Oxford University Press, 1962.2. Coddington, E. A. and Levinson, N.:Theory of Ordinary Differential Equations, McGraw-Hill,

New York, 1955.3. Chaudhury, J. and Everitt, W. N.: On the spectrum of ordinary second order differential

operators,Proc. Roy. Soc. Edinburgh Sect. A68 (1968), 95–115.4. Bennewitz, C. and Everitt, W. N.: Some remarks on the Titchmarsh–Weylm-coefficient, in:A

Tribute to Ake Pleijel, Proc. Pleijel Conference, University of Uppsala, 1980, pp. 49–108.5. Eastham, M. S. P. and Kalf, H.:Schrödinger-type Operators with Continuous Spectra, Pitman,

Boston, 1982.6. Weyl, H.: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen

Entwicklungen Willkürlicher Funktionen,Math. Ann.68 (1910), 220–269.7. Atkinson, F. V.: On the location of the Weyl circles,Proc. Roy. Soc. Edinburgh Sect. A88

(1981), 345–356.8. Gilbert, D. J.: PhD Thesis, University of Hull, 1984.9. Gilbert, D. J. and Pearson, D. B.: On subordinacy and analysis of the spectrum of

one-dimensional Schrödinger operators,J. Math. Anal. Appl.128(1987), 30–56.10. Gilbert, D. J.: On subordinacy and analysis of the spectrum of Schrödinger operators with two

singular endpoints,Proc. Roy. Soc. Edinburgh Sect. A112(1989), 213–229.11. Stolz, G.: Bounded solutions and absolute continuity of Sturm–Liouville operators,J. Math.

Anal. Appl.169(1992), 210–228.12. Kiselev, A.: Absolutely continuous spectrum of one-dimensional Schrödinger operators and

Jacobi matrices with slowly decreasing potentials,Comm. Math. Phys.179(1996), 377–400.13. Jitomirskaya, S. and Last, Y.: Dimensional Hausdorff properties of singular continuous spectra,

Phys. Rev. Lett.76(11) (1996), 1765–1769.14. Jitomirskaya, S. and Last, Y.: Power law subordinacy and singular spectra, in preparation.15. Al-Naggar, I. and Pearson, D. B.: A new asymptotic condition for absolutely continuous

spectrum of the Sturm–Liouville operator on the half line,Helv. Phys. Acta67 (1994),144–166.

16. Al-Naggar, I. and Pearson, D. B.: Quadratic forms and solutions of the Schrödinger equation,J. Phys. A29 (1996), 6581–6594.

17. Last, Y. and Simon, B.: Eigenfunctions, transfer matrices and absolutely continuous spectrumof one-dimensional Schrödinger operators, Caltech Preprint, 1996.

18. Remling, C.: The absolutely continuous spectrum of one-dimensional Schrödinger operatorswith decaying potentials, Caltech Preprint, 1997.

19. Simon, B.: Bounded eigenfunctions and absolutely continuous spectra for one-dimensionalSchrödinger operators,Proc. Amer. Math. Soc.124(1996), 3361–3369.

20. Christ, M. and Kiselev, A.: Absolutely continuous spectrum and Schrödinger operators withdecaying potentials; some optimal results, MSRI Preprint.

21. Kiselev, A., Last, Y. and Simon, B.: Modified Prüfer and EFGP transforms and the spectralanalysis of one-dimensional Schrödinger operators, Caltech Preprint, 1997.

22. Simon, B. and Zhu, Y.: The Lyapunov exponents for Schrödinger operators with slowlyoscillating potentials,J. Funct. Anal.140(1996), 541–556.

23. Simon, B. and Wolff, T.: Singular continuous spectrum under rank one perturbations andlocalisation for random Hamiltonians,Comm. Pure Appl. Math.39 (1986), 75–90.

24. Pearson, D. B.: Value distribution and spectral theory,Proc. London Math. Soc. (3)68 (1994),127–144.

25. Pearson, D. B.: Value distribution and spectral analysis of differential operators,J. Phys. A26(1993), 4067–4080.

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26. Pearson, D. B.: Sturm–Liouville theory, asymptotics, and the Schrödinger equation, in: D. Hin-ton and P. W. Schaefer (eds),Spectral Theory and Computational Methods of Sturm–LiouvilleProblems, M. Dekker Inc., New York, 1997, pp. 301–312.

27. Akhiezer, N. I. and Glazman, I. M.:Theory of Linear Operators in Hilbert Space, Pitman,London, 1981.

28. Pearson, D. B.:Quantum Scattering and Spectral Theory, Academic Press, London, 1988.29. Everitt, W. N.: On a property of them-coefficient of a second-order linear differential equation,

J. London Math. Soc. (2)4 (1972), 443–457.30. Conway, J. B.:Functions of One Complex Variable, 2nd edn., Springer-Verlag, New York,

1978.31. Saks, S.:Theory of the Integral, 2nd edn., Hafner Publishing Company, New York, 1937.32. Munroe, M. E.:Measure and Integration, 2nd edn., Addison-Wesley, Reading, MA, 1971.33. Eastham, M. S. P.: The asymptotic form of the spectral function in Sturm–Liouville problems

with a large potential like−xc(c 6 2), Proc. Roy. Soc. Edinburgh Sect. A128(1998), 37–45.34. Harris, B. J.: The form of the spectral functions associated with Sturm–Liouville problems with

continuous spectrum, Northern Illinois University Preprint, 1996.35. Brown, B. M., Eastham, M. S. P., Evans, W. D. and Kirby, V. G.: Repeated diagonalisation and

the numerical computation of the Titchmarsh–Weylm(λ) function,Proc. Roy. Soc. London Ser.A 445(1994), 113–126.

MPAG013.tex; 5/11/1998; 8:38; p.29


273

On a Counterexample ConcerningUnique Continuation for EllipticEquations in Divergence Form

NICULAE MANDACHEInstitute of Mathematics, Romanian Academy, P.O. box 1-764, 70700 Bucharest, Romania.e-mail: [email protected]

(Received: 1 October 1997; accepted: 4 November 1997)

Abstract. We construct a second order elliptic equation in divergence form inR3, with a nonzerosolution which vanishes in a half-space. The coefficients areα-Hölder continuous of any orderα < 1.This improves a previous counterexample of Miller (1972, 1974). Moreover, we obtain coefficientswhich belong to a finer class of smoothness, expressed in terms of the modulus of continuity.

Mathematics Subject Classifications (1991):35A05, 35J15 (35B60, 35K10).

Key words: elliptic equations, partial differential equations, unique continuation.

Introduction

The aim of this paper is to improve a counterexample by Keith Miller [3, 4]. Partof the results presented here belong to the author’s Ph.D. thesis [2, Section 3.4].

The first who constructed an elliptic second order equation for which the Cauchyproblem does not have the uniqueness property is Pliš [5]. The first and zero or-der coefficients of his equation are smooth, but the leading coefficients are onlyα-Hölder continuous of any orderα < 1. This result is optimal, since for Lipschitz-continuous coefficients we always have uniqueness in the Cauchy problem (and foreven stronger results, see [1]).

Miller was concerned with the nonuniqueness in the Cauchy problem for theelliptic equation in divergence form

n∑i,j=1

∂iaij ∂ju = 0, (1)

and the backward nonuniqueness for the corresponding parabolic equation

∂tu =n∑

i,j=1

∂iaij ∂ju. (2)


274 NICULAE MANDACHE

Here the matrix of coefficients(aij ) is supposed real, symmetric, continuous anduniformly positive, i.e.,

n∑i,j=1

aij xixj > C|x|2, C > 0, for anyx ∈ Rn.

The interesting aspects of Equations (1) and (2) lie in the fact that the first corre-sponds to symmetric operators inL2(Rn) and the second is the evolution equationfor such operators. They also have a physical meaning: (2) is the heat equation in amedium with specific heat 1 and with the thermic conductivity given by the matrix(aij ). See [4] for further comments.

Our example is better than the one in [4] in the following ways:

(1) It allows optimal regularity by a precise choice of the parameters used in theconstruction. We obtain Hölder continuous coefficients of any orderα < 1,whereas in [4], Miller obtained only the orderα = 1/6. We also obtain a finerresult:Suppose thatω: [0,∞) → [0,∞) is concave, continuous, nondecreasing,ω(0) = 0,ω(1) > 0 andω satisfies:∫ 1

0

dt

ω(t)<∞.

Then we can choose the coefficients of our equation such that their moduli ofcontinuity are majorated byω.

(2) It is interesting to rephrase the problem into a system of inequations for se-quences of numbers. The inherent limits of the construction below suggestthat the unique continuation property for Equation (1) might hold under theassumption thataij ∈ W1,1.

(3) There is a simplification in the technical part which allows us to give explicit(though complicated) expressions of the coefficients.

THEOREM 1. There exist a smooth functionu, smooth functionsb11, b12, b22, andcontinuous functionsd1, d2 defined onR3 3 (t, x, y), with the following properties

(i) u is the solution of the equation:∂2t u+ ∂x((b11+ d1)∂xu)+ ∂y(b12∂xu)+ ∂x(b12∂yu)++ ∂y((b22+ d2)∂yu) = 0. (3)

(ii) There is aT > 0 such thatsuppu = (−∞, T ] ×R2.(iii) u, bij anddi are periodic inx and iny with period2π .(iv) For anyt ∈ R, u(t, ·, ·) satisfies the Neumann boundary condition on(0,2π)2

with respect to Equation(3) (seen as an equation in the variablesx andy).(v) d1 andd2 do not depend onx andy and are Hölder continuous of orderα for

all α < 1.

(vi)1

2<

(d1 + b11

b12

b12

d2+ b22

)< 2 onR3.

MPAG011.tex; 5/11/1998; 8:53; p.2

ON A COUNTEREXAMPLE CONCERNING UNIQUE CONTINUATION 275

Furthermore, there are also functions as above, satisfying conditions(i)–(vi) exceptthat (3) is replaced with the parabolic equation:

∂tu = ∂x((b11+ d1)∂xu)+ ∂y(b12∂xu)+ ∂x(b12∂yu)+ ∂y((b22+ d2)∂yu). (4)

REMARK. Equation (3) can be seen, given the periodicity condition (iii), as anabstract equation for anL2(R2/2πZ2)-valued function:

u′′ = A(t)u.HereA(t) is an elliptic operator on the torus, which is positive inL2(R2/2πZ2).Thus our theorem asserts the existence of anA(t) such that the Cauchy problem forthe above equation does not have the uniqueness property. The interesting aspectof point (iv) of the theorem is that the aboveA can be replaced with an ellipticselfadjoint operator onL2((0,2π)2), with Neumann boundary condition.

Idea of the proof.We start from operator1 = ∂2t + 1xy , and from its solu-

tions e−λt cosλx and e−λt cosλy. It is convenient to view the operator as beingconstructed (appearing in (3)) as a perturbation of1. The above solutions of1decay witht , the biggerλ is the faster is the decay. We will ‘glue’ an infinitenumber of them, corresponding to the frequenciesλ = λk, such thatλk → ∞ ask→∞. In this way, ast ↑ T the solution will be, for shorter and shorter intervalsof time, proportional with e−λkt cosλkx, then with e−λk+1t cosλk+1y and so on. Inthese intervals the equation is∂2

t u+1xyu = 0. In the gaps between them, we willmodify the coefficients such as to fit a prescribed solution, which passes smoothlyfrom e−λkt cosλkx to e−λk+1t cosλk+1y. Choosing suitableλk and suitable lengthsof the intervals and of the gaps, we obtain a smooth solution which vanishes infinite time. In fact the solution is also decaying in the gaps and we can chooseintervals of length zero.

The first part of the proof consists of constructing generic functionsv, Bij ,Di: [0,5a]×R2→ R, i, j = 1,2, which describe the solution and the coefficientsin a gap. They depend on the following parameters:

a > 0 gives the length (in time) of the domain of definition,λ > 1/a is the old frequency,λ′ > λ is the new frequency andρ ∈ (0, λ/λ′) is a technical parameter.

These functions satisfy the equality

∂2t v + ∂x((B11+D1)∂xv)+ ∂x(B12∂yv)++ ∂y(B12∂xv)+ ∂y((B22+D2)∂yv) = 0 (5)

on [0,5a] ×R2, and do the required job of gluing, i.e., there is anε > 0 such that:

Bij = δij ,Di = 0 for t ∈ [0, ε) ∪ (5a − ε,5a],v(t, x, y) = e−tλ cosλx for t ∈ [0, ε),v(t, x, y) is proportional to e−tλ

′cosλ′y for t ∈ (5a − ε,5a].

MPAG011.tex; 5/11/1998; 8:53; p.3


In the second stage of the proof we will construct the functionsu, bij , di : R3→R, which satisfy the conclusions of the theorem. This is done by putting togetheran infinite number of instances of this generic construction, with appropriate valuesfor the parameters.

Construction of the genericv,Bij ,Di. Letχ : R→ [0,1] be a smooth functionwith χ(t) = 1 in a neighbourhood of[1,∞) andχ(t) = 0 in a neighbourhood of(−∞,0]. Each of the intervals[(i − 1)a, ia], with i = 1, . . . ,5 (henceforth calledsteps) will have a precise job. We will describe the functionsv, Bij andDi in eachof them.

The first step serves to a smooth decay ofB22+D2 from 1 toρ2:

v = e−λt cosλx, B11= B22 = 1, B12 = D1 = 0,

D2 = χ(t

a

)(ρ2− 1). (6)

Sincev does not depend ony, the last term in the l.h.s. of (5) vanishes and therefore(5) is satisfied for arbitraryD2.

The second step is the ‘seed’ step and serves to introduce a tiny component ofthe solution oscillating iny.

v = e−λt cosλx + c χ(t − aa

)e−ρλ

′t cosλ′y. (7)

The constant factor

cdef= e

5a2 (ρλ

′−λ) (8)

serves to make the two components of the solution (one oscillating inx and one iny) of equal amplitude att = 5a

2 , the middle of the third step. We put

B22= 1, D1 = 0, D2 = ρ2− 1

and we construct belowB11def= 1+ B andB12. Equation (5) reads:

λ2e−λt cosλy + c(

1

a2χ ′′(t − aa

)− 2

aχ ′(t − aa

)ρλ′+

+ χ(t − aa

)ρ2λ′2

)e−ρλ

′t cosλ′y+

+ ∂x((1+ B)∂xe−λt cosλx

)+ ∂x(B12∂yc χ

(t − aa

)e−ρλ

′t cosλ′y)+

+ ∂y(B12∂xe−λt cosλx)+ ∂y

(ρ2∂yc χ

(t − aa

)e−ρλ

′t cosλ′y)= 0.

After reductions:

c

(1

a2χ ′′(t − aa

)− 2

aρλ′χ ′

(t − aa

))e−ρλ

′t cosλ′y + ∂x(−Be−λtλ sinλx)+

+ ∂x(− B12c χ

(t − aa

)e−ρλ

′t λ′ sinλ′y)+ ∂y(−B12e

−λtλ sinλx) = 0.

MPAG011.tex; 5/11/1998; 8:53; p.4


Simplifying this equation by e−λt and using the notation

χ(t) = c e(λ−ρλ′)t(

1

a2χ ′′(t − aa

)− 2ρλ′

aχ ′(t − aa

)), (9)

we obtain the equivalent relation

χ(t) cosλ′y = λ∂x(B sinλx)+ λ′cχ(t − aa

)e(λ−ρλ

′)t sinλ′y ∂xB12++ λ sinλx∂yB12.

Then if we choose firstB12 from the above relation,Bλ sinλx has to be the primi-tive of some function (depending ony andt as parameters). But this is only possibleif that function has zero integral fromkπ/λ to (k + 1)π/λ, in order to allow theprimitive to have zeros atx = kπ/λ. To this end we take

B12(t, x, y) = χ (t)2 sinλx sinλ′yλλ′

. (10)

Then the above relation becomes:

λ∂x(B sinλx) = χ (t) cosλ′y − λ sinλx χ(t)2 sinλx λ′ cosλ′y

λλ′−

− λ′c χ(t − aa

)e(λ−ρλ

′)t sinλ′y χ(t)2λ cosλx sinλ′y

λλ′,

and this yields further simplifying byλ:

∂x(B sinλx) = χ(t) cosλ′y1− 2 sin2 λx

λ−

− c χ(t − aa

)e(λ−ρλ

′)t χ(t)2 cosλx sin2 λ′y

λ,

and since∫(1− 2 sin2 λx)dx = sinλx cosλx/λ + C, we obtain by integration

from 0 tox with respect tox and then simplification by sinλx:

B(t, x, y) = χ (t)(

cosλ′y cosλx

λ2− ce(λ−λ′ρ)tχ

(t − aa

)2 sin2 λ′yλ2

). (11)

The third step has the coefficients

B11= 1, B12= D1 = 0, B22= 1, D2 = ρ2− 1

and the solution is

v = e−λt cosλx + ce−ρλ′t cosλ′y.

This step serves to propagate the two components with different speeds. Althoughthe second term (depending ony) has a space frequencyλ′ > λ, its decay rate issmaller than that of the term depending onx, sinceρλ′ < λ.

MPAG011.tex; 5/11/1998; 8:53; p.5


The fourth step is symmetric to the second one and the construction is similar.Its purpose is to remove the component ofv oscillating inx, which has becomesmall with respect to the other component.

v = χ(

4a − ta

)e−λt cosλx + ce−ρλ′t cosλ′y.

Here the roles ofx andy have changed. We have

B11= 1, D1 = 0, D2 = ρ2− 1

andB12, B22def= 1+ ˜B are computed below.

Equation (5) gives

∂2t

(χ

(4a − ta

)e−λt cosλx + ce−ρλ′t cosλ′y

)+

+ ∂x(

1 · ∂xχ(

4a − ta

)e−λt cosλx

)+

+ ∂y(B12∂xv)+ ∂x(B12∂yv)+ ∂y((ρ2+ ˜B )∂yce−ρλ′t cosλ′y

) = 0

(we substituted the actual value ofv only in the terms which are subject to reduc-tions) and after reduction we obtain(

1

a2χ ′′(

4a − ta

)+ 2λ

aχ ′(

4a − ta

))e−λt cosλx+

+ ∂y(B12χ

(4a − ta

)e−λt∂x cosλx

)+

+ ∂x(B12ce−ρλ′t ∂y cosλ′y)+ ∂y

( ˜B ce−ρλ

′t ∂y cosλ′y) = 0.

Simplifying by ce−ρλ′t and using the notation

˜χ (t) = e(ρλ′−λ)t

c

(1

a2χ ′′(

4a − ta

)+ 2λ

aχ ′(

4a − ta

)), (12)

the relation becomes

˜χ (t) cosλx = ∂y

(B12χ

(4a − ta

)e(ρλ

′−λ)t

cλ sinλx

)+

+ λ′ sinλ′y∂xB12+ ∂y( ˜B λ′ sinλ′y

).

We choose

B12= ˜χ (t)sinλx

λ

2 sinλ′yλ′

, (13)

MPAG011.tex; 5/11/1998; 8:53; p.6


and taking the second term in the r.h.s. to the left in the relation above we obtainthe equivalent relation

˜χ cosλx(1− 2 sin2 λ′y) = ∂y

(˜B λ′ sinλ′y +

+ ˜χ (t)sinλx

λ

2 sinλ′yλ′

×

× χ(

4a − ta

)e(ρλ

′−λ)t

cλ sinλx

).

Since∂ysinλ′y cosλ′y

λ′ = 1 − 2 sin2 λ′y, the following relation ensures that (5) isfulfilled for t ∈ [3a,4a] (after simplification by sinλ′y):

˜χ cosλxcosλ′yλ′= ˜B λ′ + ˜χ (t)2 sin2 λx

λ′χ

(4a − ta

)e(ρλ

′−λ)t

c,

that is,

˜B =

˜χ (t)λ′ 2

(cosλx cosλ′y − 2χ

(4a − ta

)e(ρλ

′−λ)t

csin2 λx

). (14)

The aim of the fifth step is to increase the coefficientB22+D2 from the valueρ2

to 1, in order to get back to the valuesB11= B22= 1 andB12= D1 = D2 = 0 (thisensures the continuity of coefficients in the final construction). As in the previoussteps, it is simpler first to choosev and then construct the coefficients accordingly.

Let us defineχ1(t) =∫ t

0 χ(s)ds. Then we haveχ1(t) = t + χ1(1) − 1 ina neighbourhood of[1,∞) andχ1(t) = 0 in a neighbourhood of(−∞,0]. Thesolution is

v = c cosλ′y exp

(− λ′ρt − λ′(1− ρ)aχ1

(t − 4a

a

)).

The coefficients are

B11= B22 = 1, B12 = D1 = 0 and

D2 = −∂2t v

∂2yv− 1= ∂2

t v

λ′2v− 1 =

(ρ + (1− ρ)χ

(t − 4a

a

))2

−

− 1− ρaλ′

χ ′(t − 4a

a

)− 1. (15)

We will now eliminate one of our parameters. The constantρ is very sensitivein our construction; in fact 1−ρ2 is the order of magnitude of the coefficientD2. Insteps 2 and 4 there is an exponential factor inχ(t) and in ˜χ (t), which will manage

MPAG011.tex; 5/11/1998; 8:53; p.7


to make the coefficientsBij (more precisely,Bij − δij ) small at little expense.Therefore, since we have the restrictionρ < λ

λ′ , which gives 1−ρ2 > 1− (λ/λ′)2,we cannot do better (modulo a multiplicative constant) than chooseρ = (λ/λ′)2.We then have 1−ρ2 ≈ 2(1−(λ/λ′)2) for λ/λ′ close to 1. In order to keep formulasto a reasonable complexity we will continue to use the constantρ, substitutingλ2/λ′2 for it when needed.

We can express the solution in a single formula:

va,λ,λ′(t, x, y) = χ

(4a − ta

)e−λt cosλx+

+ c χ(t − aa

)e−

λ2

λ′ t−(

1−(λλ′)2)

λ′aχ1

(t−4aa

)cosλ′y.

(16)

Let us notice here that

va,λ,λ′(t, x, y) =

e−λt cosλx in the neighbourhood of 0,α(a, λ, λ′)e−λ′(t−5a) cosλ′y in the neighbourhood of 5a.

(17)

The constantα(a, λ, λ′) is given by (8) and (16) witht = 5a:

α(a, λ, λ′) = e−5a(λ+λ2/λ′)/2−(1−λ2/λ′2)λ′aχ1(1) 6 e−5aλ/2. (18)

Estimates for the derivatives.We now compute the size of the derivatives ofv

andBij constructed above. ForDi, only the first order derivative is needed in theproof of Theorem 1, and we give a bound for it.

Let k, l andm be three natural numbers,k+ l+m > 0. Then during the secondstep,B11= 1+ B, whereB is given by (11) and we have

∂kt ∂lx∂my B11 = ∂kt ∂

lx∂my B

= ∂kt χ(t)∂lx∂my

cosλ′y cosλx

λ2−

− ∂kt χ (t)cet (λ−λ′ρ)χ

(t − aa

)∂lx∂

my

2 sin2 λ′yλ2

. (19)

Thekth derivative ofχ is (see (9))

∂kt χ(t) = ck∑j=0

(k

j

)∂jt e(λ−ρλ

′)t ∂k−jt

(1

a2χ ′′(t − aa

)− 2ρλ′

aχ ′(t − aa

)),

and its absolute value is bounded by

ce(λ−ρλ′)t

k∑j=0

(k

j

)(λ− ρλ′)j

(1

ak−j+2

∣∣∣∣χ(k−j+2)

(t − aa

)∣∣∣∣++ 2ρλ′

ak−j+1

∣∣∣∣χ(k−j+1)

(t − aa

)∣∣∣∣).

MPAG011.tex; 5/11/1998; 8:53; p.8


Let us setCχ,k = supi6k,t∈R |χ(i)(t)|. Using λ > ρλ′ and recalling thatc =e−5a(λ−ρλ′)/2, we infer that

cet (λ−ρλ′) 6 e−a(λ−ρλ

′)/2 for any t ∈ [a,2a].Now we useλ > 1/a and obtain:

∣∣∂kt χ ∣∣ 6 ce(λ−ρλ′)t k∑j=0

(k

j

)(λ− ρλ′)j

(1

a

)k−j( 1

a2Cχ,k+2+ 2ρλ′

aCχ,k+1

)6 ce(λ−ρλ′)t

(λ+ 1

a

)k3λ2Cχ,k+2 6 e−a(λ−ρλ

′)/2 · 3 · 2kCχ,k+2λk+2.

(20)

The same kind of computation will give∣∣∣∣∂kt χ (t)ce(λ−λ′ρ)tχ( t − aa)∣∣∣∣

=∣∣∣∣∣c2

∑i+j+h=k

(k

i j h

)∂it e

2(λ−ρλ′)t ∂jt(

1

a2χ ′′(t − aa

)−

− 2ρλ′

aχ ′(t − aa

))∂ht χ

(t − aa

)∣∣∣∣∣6 c2e2(λ−ρλ′)t ∑

i+j+h=k

(k

i j h

)(2λ− 2ρλ′)i(1/a)j×

×(

1

a2Cχ,k+2+ 2ρλ′

aCχ,k+1

)(1/a)hCχ,k

6 c2e2(λ−ρλ′)t(

2λ+ 2

a

)k· 3 · λ2Cχ,k+2Cχ,k

6 e−a(λ−ρλ′)C2

χ,k+2 · 3 · 4kλk+2. (21)

Now we can estimate the derivatives ofB11 (see (19)) using (20) and (21):∣∣∂kt ∂lx∂my B11(t, x, y)∣∣

6 e−a(λ−ρλ′)/2 · 3 · 2kCχ,k+2λ

k+2λlλ′m

λ2+ e−a(λ−ρλ

′)C2χ,k+2 · 3 · 4kλk+22m+1λ′m

λ2

6 e−a(λ−ρλ′)/2C′χ,k,mλ

k+lλ′m. (22)

Here the constantC′χ,k,m depends only onχ, k andm. For coefficientB12 thecomputation is simpler and we obtain in view of (10) and using estimate (20) andλ′ > λ:∣∣∂kt ∂lx∂my B12(t, x, y)

∣∣ 6 e−a(λ−ρλ′)/2 · 3 · 2kCχ,k+2λ

k+22λlλ′m

λλ′6 e−a(λ−ρλ

′)/2C′′χ,k,mλk+lλ′m. (23)

MPAG011.tex; 5/11/1998; 8:53; p.9


For the fourth step the estimate is similar. We use

et (ρλ′−λ)

c6 e−a(λ−ρλ

′)/2 for any t ∈ [3a,4a],

and obtain in view of (12) that∣∣∂kt ˜χ (t)∣∣ 6 e−a(λ−ρλ′)/2 · 3 · 2kCχ,k+2λ

k+2.

The computation is made in the same way and we obtain thatB22 satisfies (22)(replaceB11 by B22 and [a,2a] by [3a,4a]) andB12 satisfies (23) for anyt in[3a,4a] (and anyx, y ∈ R). SinceBij = δij during the first, third and fifth steps,we conclude from relations (22) and (23), that we have for anyt ∈ [0,5a]:∣∣∂kt ∂lx∂my Bij (t, x, y)∣∣ 6 e−a(λ−ρλ

′)/2C′′′χ,k,l,mλk+lλ′m. (24)

Now we turn to the derivatives ofv. We have from (16):

∂kt ∂lx∂my v = ∂kt χ

(4a − ta

)e−tλ∂lx∂

my cosλx+

+ c∂kt χ(t − aa

)e−ρλ

′t−(1−ρ)λ′aχ1(t/a−4)∂lx∂my cosλ′y.

(25)

We first take care of thet derivatives. Usingt > 0:

∣∣∂kt χ(4− t/a)e−tλ∣∣ 6 e−tλ

k∑j=0

(k

j

)(1/a)jCχ,jλ

k−j

6 e−tλ(2λ)kCχ,k 6 (2λ)kCχ,k. (26)

By induction we prove the existence of a constantCχ,k, depending only onχ andk, such that∣∣∂kt e−ρλ

′t−(1−ρ)λ′aχ1(t/a−4)∣∣ 6 Cχ,kλ′k. (27)

This is true fork = 0 since the exponent is negative. We prove that if (27) holdsfor k = 0,1, . . . ,m, then it also holds, for a certainCχ,m+1, for k = m+1. Indeed,∣∣∂m+1

t e−ρλ′t−(1−ρ)λ′aχ1(t/a−4)

∣∣= ∣∣∂mt (−ρλ′ − (1− ρ)λ′χ(t/a − 4))e−ρλ

′ t−(1−ρ)λ′aχ1(t/a−4)∣∣

6 λ′m∑j=0

(m

j

)∣∣(∂jt (−ρ − (1− ρ)χ(t/a − 4)))∂m−jt e−ρλ

′t−(1−ρ)λ′aχ1(t/a−4)∣∣

6 λ′m∑j=0

(m

j

)∣∣(1/a)jCχ,j Cχ,m−j λ′m−j ∣∣ 6 Cχ,m+1λ′m+1.

MPAG011.tex; 5/11/1998; 8:53; p.10


We usedλ′ > 1/a. Applying (27) we obtain:∣∣∂kt χ(t/a − 1)e−ρλ′t−(1−ρ)λ′aχ1(t/a−4)

∣∣6

k∑j=0

(k

j

)∣∣(1/a)jCχ,j Cχ,k−jλ′k−j ∣∣ 6 ˜Cχ,kλ′k. (28)

Using (25), (26) and (28), we conclude that∣∣∂kt ∂lx∂my v∣∣ 6 Cχ,kλ′k+mλl. (29)

It remains to estimate the derivative ofDi. The functionD1 is identically 0, andD2 is constant during the second, the third and the fourth steps (i.e., on[a,4a]).We have, in view of (6) and (15):

|∂tD2| 6 Cχ,1(1− ρ2)/a 6 2Cχ,1(1− ρ)/a for any t ∈ [0, a],|∂tD2| =

∣∣∣∣2ρ(1− ρ)aχ ′(t − 4a

a

)+ 2(1− ρ)2

aχ ′(t − 4a

a

)χ

(t − 4a

a

)−

−(1− ρ)a2λ′

χ ′′(t − 4a

a

)∣∣∣∣6 (1− ρ)(2Cχ,1/a + 2Cχ,1Cχ,0/a + Cχ,2/a)6 5Cχ,2(1− ρ)/a for any t ∈ [4a,5a],

and we can conclude that

|∂tDi| 6 5Cχ,2(1− (λ/λ′)2)/a for any t ∈ [0,5a]. (30)

Boundary conditions.Functionu satisfies the Neumann boundary condition forEquation (1) in the open set ⊂ Rn if and only if

n∑i,j=1

niaij ∂ju(x) = 0 for anyx ∈ ∂,

where(n1, . . . nn) is the normal vector to∂.We want our functionv to satisfy this condition for Equation (5), seen in the

variablesx andy, in the open set(0,2π)× (0,2π). In this case, the above relationreads:

(B11+D1)∂xv + B12∂yv = 0 on0,2π × [0,2π ], (31)

B12∂yv + (B22+D2)∂xv = 0 on[0,2π ] × 0,2π. (32)

We have

∂xv = χ(4− t/a)e−tλ(−λ sinλx),

∂yv = cχ(t/a − 1)e−tρλ′−(1−ρ)λ′aχ(t/a−4)(−λ′ sinλ′y).

MPAG011.tex; 5/11/1998; 8:53; p.11


SinceB12 is a multiple of sinλx sinλ′y (see (10) and (13)), the conditions

λ ∈ N, λ′ ∈ N (33)

are sufficient for ensuring the boundary conditions (31) and (32). These relationswill imply that u, bij anddi constructed below fulfill condition (iv) of Theorem 1.They satisfy also the periodicity condition (iii).

Proof of Theorem 1.Let akk>1 and λkk>1 be two sequences of positivenumbers. We will suppose

∞∑j=1

aj <∞ and 1/ak < λk < λk+1. (34)

We denoteTk = ∑k−1j=1 aj for k > 1 andT = ∑∞

j=1 aj . The sequenceρkk>1 isdefined byρk = λ2

k/λ2k+1. We postpone the choice of these sequences as much as

we can, in order to first derive all the conditions they have to fulfill. We shall usethe indicesa, λ, λ′ for the functionsBij andDi, with i, j = 1,2 (similarly to (16))since we will use them for different values of these parameters. Letk0 > 0 beanevennatural number, to be chosen later. We are ready for the definition of thefunctionsu, bij anddi.

u(t, x, y) =

e−(t−Tk0)λk0 cosλk0x for all t ∈ (−∞, Tk0],ckvak,λk,λk+1(t − Tk, x, y)ckvak,λk,λk+1(t − Tk, y, x)

for k even

for k odd

∀t ∈ [Tk, Tk+1],∀k > k0,

0 for all t ∈ [T,∞).(35)

Hereck are constants which ensure the continuity (and therefore, the smoothness)of u. They are defined by the relations

ck0 = 1,

ck+1

ck= α(ak, λk, λk+1),

whereα(a, λ, λ′) is defined by relation (18). We have therefore (see (18)):

ck 6 exp

(− 5

2

k−1∑j=k0

ajλj

). (36)

The coefficients are

bij (t, x, y) =

δij for any t 6∈ [Tk0, T ),

Bij ak,λk,λk+1(t − Tk, x, y) for t ∈ [Tk, Tk+1] with k even,

Bji ak,λk,λk+1(t − Tk, y, x) for t ∈ [Tk, Tk+1] with k odd,

MPAG011.tex; 5/11/1998; 8:53; p.12


for all i, j = 1,2 with i 6 j , wherei = 3− i andj = 3− j . This inversionis necessary since the derivatives with respect tox and y – and therefore thecoefficients – swap their roles in the odd intervals. The singular coefficients aredefined in a similar manner:

di(t, x, y) =

0 for anyt 6∈ [Tk0, T ),

Di ak,λk,λk+1(t − Tk, x, y) for t ∈ [Tk, Tk+1] with k even,

Di ak,λk,λk+1(t − Tk, y, x) for t ∈ [Tk, Tk+1] with k odd.

The aboveu, bij anddi fulfill Equation (3): indeed, they are obtained by simplechanges of variables (the translationt → Tk + t and the symmetry which reversesthe roles ofx andy) from the functions satisfying (5).

Notice thatBij a,λ,λ′ = δij for t in a neighbourhood of 0 or in a neighbourhoodof 5a, and thereforebij are smooth inR\T × R2. In order to obtainbij beingsmooth att = T too, it is enough that all their derivatives are continuous and havethe limit 0 ast ↑ T . In view of (24), we have for anyi, j = 1,2:

supt∈[Tk,Tk+1]x,y∈R

∣∣∂pt ∂lx∂my bij (t, x, y)∣∣ 6 C′′′χ,p,l,me−a(λk−λ2k/λk+1)/2λ

p+lk λmk+1

and due to the monotony ofλk the following condition ensures thatbij are smoothonR3:

limk→∞

e−ak(λk−λ2k/λk+1)/2λmk+1 = 0 for any m ∈ N. (37)

Note that if we supposedi continuous, then limk→∞(1−ρ2k ) = 0, sincedi takes

the value(1− ρ2k ) on a subset of[Tk, Tk+1], for i = 2 for evenk and i = 1 for

odd k, anddi = 0 for t > T for i = 1,2. This implies thatρk → 1, and sinceρk = λ2

k/λ2k+1, we have

limk→∞

λk/λk+1 = 1. (38)

For the smoothness ofu we use relation (29), and obtain∣∣∂pt ∂lx∂my u(t, x, y)∣∣ 6 ckCχ,pλp+lk λmk+1 ∀k > k0,∀t ∈ [Tk, Tk+1],∀x, y ∈ R,and in view of (36) a sufficient condition for the smoothness ofu is

limk→∞ exp

(− 5

2

k−1∑j=k0

ajλj

)λmk+1 = 0 for any m ∈ N. (39)

Due to relation (38), we can replace in the limit aboveλmk+1 by λmk or, equiva-lently, take the sum under exponential fromk0 to k. We have

−5

2

k∑j=k0

ajλj 6 −akλk/26 −ak(λk − λ2k/λk+1)/2

MPAG011.tex; 5/11/1998; 8:53; p.13


and therefore (39) is a consequence of (37). Since we will put conditions onλkandak that ensure the continuity ofd1 andd2 (hence (38) holds), we will omitcondition (39).

Continuity ofdi . We will prove that fori ∈ 1,2 we have:

|di(t1)− di(t2)| 6 10Cχ,2 supk>k0

((1− λ2

k/λ2k+1

)min(5, |t1 − t2|/ak)

),

∀t1, t2 ∈ R. (40)

In order to do so, we show that for anyt1 andt2 there is ak > k0 such that:

|di(t1)− di(t2)| 6 10Cχ,2(1− λ2

k/λ2k+1

)min(5, |t1 − t2|/ak). (41)

Since⋃∞k=k0[Tk, Tk+1] = [Tk0, T ), there are three cases to treat:

(a) There is ak > k0 such thatt1, t2 ∈ [Tk, Tk+1].(b) One of theti belongs toR\[Tk0, T ).(c) t1 ∈ [Tk1, Tk1+1] andt2 ∈ [Tk2, Tk2+1], with k1 6= k2.

Case(a). Using the theorem of Cauchy, and the upper bound of the derivativeof di given by (30), we obtain:

|di(t1)− di(t2)| 6 |t1− t2|5Cχ,2(1− λ2

k/λ2k+1

)/ak.

Using further that

t1, t2 ∈ [Tk, Tk+1] ⇒ |t1− t2| 6 Tk+1− Tk = 5ak,

we obtain

|di(t1)− di(t2)| 6 5Cχ,2(1− λ2

k/λ2k+1

)min(5, |t1 − t2|/ak).

Case(b). Suppose thatt1 6∈ [Tk0, T ). Thendi(t1) = 0. If t2 is also outsidethis interval, thendi(t2) = di(t1) = 0 and there is nothing to prove. So, we maysuppose thatt2 ∈ [Tk, Tk+1], with k > k0. Then one ofTk andTk+1 (let us denote itby t ′1) must lie betweent1 andt2 (or equalt2). Then|t1 − t2| > |t ′1 − t2| and sincedi(Tk) = di(Tk+1) = 0, we havedi(t ′1) = 0 = di(t1). Applying the case (a) tot ′1andt2 we obtain:

|di(t1)− di(t2)| = |di(t ′1)− di(t2)| 6 5Cχ,2(1− λ2

k/λ2k+1

)min(5, |t ′1− t2|/ak)

6 5Cχ,2(1− λ2

k/λ2k+1

)min(5, |t1 − t2|/ak).

Case(c). The method is similar to the one used in case (b). Supposet1 ∈[Tk1, Tk1+1] and t2 ∈ [Tk2, Tk2+1] with k1 6= k2. By symmetry we may supposethat t1 < t2, hencek1 < k2. Let t ′1 = Tk1+1 andt ′2 = Tk2. Then we havedi(t ′j ) = 0for j = 1,2 and

|di(t1)− di(t2)| 6 |di(t1)| + |di(t2)| = |di(t1)− di(t ′1)| + |di(t2)− di(t ′2)|6 5Cχ,2

(1− λ2

k1/λ2

k1+1

)min(5, |t1− t ′1|/ak1)+

+ 5Cχ,2(1− λ2

k2/λ2

k2+1

)min(5, |t2 − t ′2|/ak2)

6 10Cχ,2 maxj=1,2

((1− λ2

kj/λ2

kj+1

)min(5, |tj − t ′j |/akj )

).

MPAG011.tex; 5/11/1998; 8:53; p.14


The proof of (41) is complete.We turn back to Theorem 1, condition (v). In order to obtain Hölder continuous

coefficients of any order 0< α < 1 our sequencesak andλk must satisfy (inview of (41)):

∀α ∈ (0,1) ∃C > 0 s.t.(1− λ2

k/λ2k+1

)min(5, |t|/ak) 6 Ctα,∀t > 0, ∀k > k0.

Since the r.h.s. is concave and increasing, while the l.h.s. is linear on[0,5ak] andconstant on[5ak,∞] and is continuous, it is enough to check the inequality att = 5ak. In this way we obtain the condition:

∀α < 1 ∃C > 0 s.t.(1− λ2

k/λ2k+1

)6 Caαk , ∀k > k0.

Summarising, we need two sequencesakk>1 andλkk>1 which must satisfy:

(α)∑∞

1 ak <∞ (the construction is to be achieved in finite time).(β) 1/ak < λk < λk+1 (technical condition).(γ ) λk ∈ N (in order to ensure the 2π -periodicity and the boundary conditions).(δ) limk→∞ e−ak(λk−λ2

k/λk+1)/2λmk+1 = 0 for anym ∈ N (to ensure the smoothnessof bij and implicitly that ofu).

(ε) ∀α < 1 ∃C > 0 s.t.(1 − λ2

k/λ2k+1

)6 Caαk for any k > k0 (the Hölder

continuity ofd1, d2, of any orderα < 1).

The following sequences satisfy all these conditions:

λk = (k + 1)3,

ak = (k ln2(k + 1))−1.

Condition (α) is easy to prove, and also (β), and (γ ). We have for (δ):

e−ak(λk−λ2k/λk+1)/2λmk+1 = e

− (k+1)3−(k+1)6/(k+2)3

k ln2(k+1) (k + 2)3m

= e−(k+1)3 3k2+9k+7

(k+2)3k ln2(k+1) (k + 2)3m.

The exponent is asymptotically

−(k + 1)33k2+ 9k + 7

(k + 2)3k ln2(k + 1)= −(1+O(1/k))3k ln−2(k + 1)

and therefore the whole expression above has limit zero ask →∞. For condition(ε) we have:(

1− λ2k/λ

2k+1

) = (1− (k + 1)6/(k + 2)6) = 6k5 + 45k4+ · · · + 63

(k + 2)66 Ck−1,

C > 0,

and since limk→∞ k−1+α ln−2α(k + 1) = 0 for anyα < 1, we have

∀α < 1 ∃Cα > 0 such that(1− λ2

k/λ2k+1

)6 Cαk−α ln−2α(k + 1).

MPAG011.tex; 5/11/1998; 8:53; p.15


It remains to choosek0. We must ensure the uniform ellipticity of Equation (3), asrequired in point (vi) of the theorem. This is possible since the coefficients that wehave constructed are uniformly continuous:di andbij − δij have compact supportin the t variable and are periodic inx andy. Now, passing from ak0 to a biggerk0

has the only effect that these functions become zero fort ∈ [Tk0, Tk0] and remain

as they were fort ∈ [Tk0, T ]. Since they tend uniformly to zero ast ↑ T , we can

choosek0 such that|di | 6 1/18 and|bij − δij | 6 1/18 and then∣∣∣∣∣∣∣∣ (b11− 1+ d1

b12

b12

b22− 1+ d2

) ∣∣∣∣∣∣∣∣ 6 6× 1/18= 1/3

and we obtain

1− 1/36(b11+ d1

b12

b12

b22+ d2

)6 1+ 1/3.

The proof is complete.

The construction for the parabolic problem (4) is similar to the one for theelliptic equation and will be not done here.

REMARK. From the conditionλk →∞ we infer that

limk→∞ λ

−4k = λ−4

k0

∞∏k=k0

λ4k

λ4k+1

= λ−4k0

∞∏k=k0

ρ2k = 0,

and sinceρ2k ∈ (0,1) for any k, we can pass to the infinite sum associated to the

infinite product, and obtain from the relation above:

∞∑k=k0

(1− ρ2k ) = ∞.

Since in each of the intervals[Tk, Tk+1] one of the functionsd1, d2 takes the value−(1−ρ2

k ) and gets back to the value 0 at the end of the interval, the relation aboveimplies that eitherd1 or d2 have unbounded variation. Thus, we cannot obtainW1,1

coefficients in the construction above.Professor N. Lerner raised the problem of the refinement of the result above,

considering the continuity moduli of the coefficients. He asked in particular whetherthe results below hold. The following corollary is actually a corollary of theproofof Theorem 1.

COROLLARY 1. Letω: [0,∞) → [0,∞) be a continuous, nondecreasing andconcave function such thatω(0) = 0 andω(1) > 0. Suppose that∫ 1

0

dt

ω(t)<∞. (42)

MPAG011.tex; 5/11/1998; 8:53; p.16


Then there existu, bij and di, wherei, j = 1,2, satisfying all the conditions ofTheorem1, except(v), which is replaced by:

|di(t1)− di(t2)| 6 ω(|t1− t2|), ∀t1, t2 ∈ R, i = 1,2. (43)

REMARK. If f : Rn→ R then the modulus of continuity off is by definition thefunction

ωf : [0,∞)→ [0,∞), ωf (t) = sup|x1−x2|6t

|f (x1)− f (x2)|.

It is easy to prove thatωf is nondecreasing and satisfies the relation

ωf (αt1+ (1− α)t2) > 1/2(αωf (t1)+ (1− α)ωf (t2)

),

∀t1, t2 > 0, ∀α ∈ [0,1]. (44)

This shows that there is a concave functionωf , more precisely,

ωf (t) = sup06t1<t<t2

(t2− t)ω(t1)+ (t − t1)ω(t2)t2− t1 ,

such that12ωf 6 ωf 6 ωf . It follows that the restriction to concave functionsω inthe above corollary does not affect the generality.

Proof of Corollary 1.We may suppose that

ω(t) 6√t . (45)

Indeed, replacingω by the function

ω(t) = min(ω(t),√t)

the hypotheses of the corollary remain true:ω is continuous, nondecreasing, con-cave and∫ 1

0

dt

ω(t)=∫ 1

0max

(1

ω(t), t−1/2

)dt 6

∫ 1

0

dt

ω(t)+∫ 1

0t−1/2 dt <∞.

We will make another choice of the constantk0, of the sequencesakk>1 andλkk>1 in the proof of Theorem 1, such that the conditions (α)–(δ) and the relation(43) are satisfied. We choose

λk = k4.

Let

δkdef= 1− λ2

k/λ2k+1 =

(k + 1)8− k8

(k + 1)8.

MPAG011.tex; 5/11/1998; 8:53; p.17


We have to make some preparations in view of the construction of the sequenceak. Let a = supx ∈ [0,1] | ω(x) < ω(1). Sinceω(0) = 0 andω(1) > 0,we havea ∈ (0,1]. Then by continuity we haveω(a) = ω(1) and the functionω: [0, a] → [0, ω(1)] is bijective. Indeed, suppose 06 x < y 6 a. Sinceω isnondecreasing,ω(x) < ω(a) by the definition ofa. Using thatω is concave,

ω(y) > (a − y)ω(x)+ (y − x)ω(a)a − x >

(a − y)ω(x) + (y − x)ω(x)a − x = ω(x),

which proves thatω is strictly increasing on[0, a]. We put

ak = 1/5ω−1(50Cχ,2δk) for anyk > k0.

This requires that the argument ofω−1 lies in [0, ω(1)]. To this end, we impose

50Cχ,2δk0 6 ω(1).

This relation is satisfied fork0 big enough sinceδk → 0. Since 0< ak1 6 ak0 foranyk1 > k0, we obtain from the concavity ofω:

50Cχ,2δk1 = ω(5ak1) >(5ak0 − 5ak1)ω(0)+ 5ak1ω(5ak0)

5ak0

= ak1

ak0

50Cχ,2δk0

and we inferak1

δk1

6 ak0

δk0

for all k1 > k0. (46)

Now we will check in order the conditions (α), (β), (γ ) and (δ) stated at the endof the proof of Theorem 1.

We first prove that∑ai <∞. Using the monotony ofω and then relation (42):

k1∑k=k0

1

δk(ak − ak+1) = 50Cχ,2

k1∑k=k0

1

50Cχ,2δk(ak − ak+1)

= 10Cχ,2

k1∑k=k0

1

ω(5ak)(5ak − 5ak+1)

6 10Cχ,2

k1∑k=k0

∫ 5ak

5ak+1

dt

ω(t)

6 10Cχ,2

∫ 5ak0

0

dt

ω(t)= M <∞ for anyk1 > k0.

We will associate differently the terms in the first sum above, in order to obtaininformation about the series

∑ak. We have

k1∑k=k0

1

δk(ak − ak+1) =

k1−1∑k=k0

(1

δk+1− 1

δk

)ak+1+ ak0

δk0

− ak1+1

δk1

MPAG011.tex; 5/11/1998; 8:53; p.18


and we obtain using (46):

k1−1∑k=k0

(1

δk+1− 1

δk

)ak+1 6 M − ak0

δk0

+ ak1

δk1

6 M for anyk1 > k0.

Sinceδk is decreasing,( 1δk+1− 1

δk)ak+1 > 0 for anyk > k0. We obtain that the

series∞∑k=k0

(1

δk+1− 1

δk

)ak+1

is convergent. It remains now to use the fact that

limk→∞

(1

δk+1− 1

δk

)= 1/8 (47)

and the positivity ofak to conclude that

∞∑k=k0

ak <∞.

In order to show that relation (47) holds, we compute

1

δk= (k + 1)8

8k7 + 28k6+O(k5)= 1

8

k8+ 8k7+O(k6)

k7+ 7/2k5+O(k5)= 1

8(k + 9/2+O(1/k)).

The proof of condition (α) is complete.Due to relation (45), we haveω−1(t) > t2 for t ∈ [0, ω(1)] and in particular

5ak = ω−1(50Cχ,2δk) > (50Cχ,2δk)2 for anyk > k0. Sinceδk = 8/k + O(1/k2),we obtain the existence of aC > 0 such that

ak > Ck−2. (48)

Choosingk0 big enough, we obtain 1/ak < k4 = λk for anyk > k0 and condition(β) is fulfilled.

Condition (γ ) is obviously satisfied:λk = k4 ∈ N.We have from (48):

e−ak(λk−λ2k/λk+1)/2λmk+1 6 e−Ck

−2k4(1−k4/(k+1)4)/2(k + 1)4m

6 e−Ck2(4k3/(k+1)4)/2(k + 1)4m.

The limit of the above expression is 0 ask→∞ since the exponent is−2Ck(1+O(1/k)), hence condition (δ) is satisfied.

It remains to prove inequality (43). In order to do so it is enough to prove that

10Cχ,2 supk>k0

((1− λ2

k/λ2k+1)min(5, t/ak)

)6 ω(t) for any t ∈ [0,∞),

MPAG011.tex; 5/11/1998; 8:53; p.19


since (43) is then a consequence of (40). We will prove the inequality for eachk > k0:

ω(t) > 10Cχ,2δk min(5, t/ak).

We use the concavity ofω and the fact that it is nondecreasing. This implies that itis enough to prove the above inequality at the pointt = 5ak where the r.h.s. passesfrom a linear function to a constant one. Indeed, suppose the inequality proved att = 5ak . Then the result is, on the one hand, because of the monotony ofω, thatthe inequality holds in the interval[5ak,∞). On the other hand, it is obviously truefor t = 0 and from the concavity ofω it is true in the interval[0,5ak].

We have to check that

ω(5ak) > 10Cχ,2δk · 5,in fact by the definition ofak we have equality. The proof is complete. 2

Acknowledgements

I thank Professor Anne Boutet de Monvel for drawing my attention to the work ofMiller. I am also indebted to Professor Vladimir Georgescu for valuable remarkson the paper.

References

1. Hörmander, L.: Uniqueness theorems for second order elliptic differential equations,Comm.Partial Differential Equations8(1) (1983), 21–64.

2. Mandache, N.: Estimations dans les espaces de Hilbert et applications au prolongement unique,Thèse, Université Paris 7, 1994.

3. Miller, K.: Non-unique continuation for certain ode’s in Hilbert space and for uniformly par-abolic and elliptic equations in self-adjoint divergence form, in:Symposium on Non-Well-PosedProblems and Logarithmic Convexity(Heriot-Watt Univ., Edinburgh, 1972), Lecture Notes inMath. 316, Springer, 1973, pp. 85–101.

4. Miller, K.: Non-unique continuation for uniformly parabolic and elliptic equations in self-adjointdivergence form with Hölder-continuous coefficients,Arch. Rational Mech. Anal.54 (1974),105–117.

5. Pliš, A.: On non-uniqueness in Cauchy problem for an elliptic second order differential operator,Bull. Acad. Polon. Sci.11 (1963), 95–100.

MPAG011.tex; 5/11/1998; 8:53; p.20

Mathematical Physics, Analysis and Geometry1: 293, 1999. 293

Editorial

It is with great sadness that we learned of the unexpected and untimely death onNovember 27, 1998, of Moshé Flato.

Moshé was extremely supportive of the launching ofMathematical Physics, Analy-sis and Geometryand he provided much useful advice as to the development ofour journal. We are extremely pleased that he agreed to be on our Editorial Board– despite his longstanding commitment as founding Editor of the journalLetters inMathematical Physics.

His creative energy and loyalty will be sorely missed.

VLADIMIR MARCHENKOANNE BOUTET de MONVEL

HENRY McKEAN

MPAGED2.tex; 6/04/1999; 9:34; p.1VTEXJu PIPS No: 230386 (mpagkap:mathfam) v.1.15


295

Arnold’s Diffusion in Isochronous Systems?

G. GALLAVOTTIUniversità di Roma 1, Fisica, Italy

(Received: 16 January 1998; in final form: 30 October 1998)

Abstract. I discuss an illustration of a mechanism for Arnold’s diffusion following a non-variationalapproach, and an improvement of the related estimates for the diffusion time.

Mathematics Subject Classifications (1991):34C15, 34C29, 34C37, 58F30, 70H05.

Key words: Arnold’s diffusion, homoclinic splitting, KAM.

1. Introduction

Arnold’s diffusion was established in simple paradigmatic examples by Arnold [A].Since that paper several methods aiming at extending its validity to more generalsystems have been developed: this was done either by following methods some-times called “geometric methods” close to the original approach, [CG], [C], [M],or by “variational methods”, [Be], [Br]. In the approach [CG] one finds estimates,for the time necessary for a diffusion of O(1) in the space of the action variables,which are terribly big as functions of the sizeε of the perturbation when it ap-proaches 0 (their order is exp O(ε−1)); the variational method instead givesbetter estimates, “fast”, ([Be], their orders is exp O(ε−1/2)), and even very good,“polynomial”, ones ([Br], their order is O(ε−2)).

Recently remarkable progress has been made in the geometric approach via thepapers [M] and the impressive “summa” [C], who have been able to recover notonly the best variational results but to extend them to the cases discussed in [CG],greatly improving the bounds obtained there, and to many substantially new casesof applicative interest. The work [C] gives an extensive bibliography to whichI refer. However, the subject is still presented at a very technical level, and therelation of the new methods with those in [CG] is not transparent.

Here I first illustrate (Section 5) the method of [CG] by developing it withthe aim of showing existence of diffusion. This may lead to a clarification of amethod not appropriately quoted in the literature and which maintains its interestbecause of its relative simplicity, in spite of the better estimates coming from thequoted alternative methods. If explicit estimates are avoided one gains enormouslyin simplicity: this kind of approach was probably the one meant in [A] where the

? The first version of this paper is archived in: [email protected]#9709011.

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296 G. GALLAVOTTI

problem was first posed and solved without bothering to give the (fairly obvious,see Section 5) details. What follows in Section 5 also applies to the Arnold’s case,but I prefer to illustrate it in a case that is even simpler.

Furthermore I show (Section 6) that if a new idea is added to the method of[CG], then one can get a “fast” (still exponential) estimate for the drift time at leastin the “isochronous” cases considered here, see (1.1) below. This bound is derivedin detail and is conceptually independent of the other works.

Consider HamiltoniansH with three degrees of freedom described by coordi-natesI ∈ R,A0 = (A01, A02) ∈ R2 and anglesϕ ∈ T 1, α = (α1, α2) ∈ T 2:

H = ω ·A0+I 2

2+ g2(cosϕ − 1)+ εf (ϕ, α) def= H0+ εf (ϕ, α), (1.1)

whereω = (ω1, ω2) ∈ R2 is a vector with Diophantine constantsC, τ , i.e., suchthat for all integer components vectorsν = (ν1, ν2) it is |ω · ν|−1 6 C|ν|τ ifν 6= 0; theperturbationf is supposed to be a (fixed) trigonometric polynomial ofdegreeN : f (ϕ, α) =∑06|ν|<N,|n|<N fn,ν cos(nϕ+ν ·α). The subject being fairlywell understood we do not need to be really very careful about units so that somecoefficients in (1.1) have been set equal to 1.

One can also use Jacobi’s hyperbolic coordinates, which we denotep0, q0, todescribe the pendulum1

2I2 + g2(cosϕ − 1) near the unstable pointI = ϕ =

0, see for instance Appendix A1 of [G4]. In the new coordinates the pendulumHamiltonian becomesJ (p0q0) with

J ′(x) def= dJ (x)

dx= g +

∞∑n=1

= gnxn def= g(x)

and the total Hamiltonian becomes:

H = ω ·A0+ J (p0q0)+ εf0(α, p0, q0), (1.2)

wheref0(α, p0, q0) = f (ϕ, α).The functionf0, still a trigonometric polynomial inα, has the property:f0(α, p,

q) = f0(−α, q, p) = f0(−α,−p,−q) and we shall callparity the 4-elementsgroup of transformations generated byP1: (α, p, q) ↔ (−α, q, p) andP2: (α,p, q)↔ (−α,−p,−q). We callP0,P1, P2, P the group elements; and we say that

f0 has “even parity”. IfF(α, p, q) = −F(Pj (α, q, p)) def= PjF(α, q, p), j = 1,2,we say thatF has odd parity: for instance∂αf0 has “odd parity”. Thep derivativeF or theq derivativeG of an even function have the propertyF = P1G = −P2F ,G = P1F = −P2G. The Jacobi’s map in general transforms functions ofϕ, α withgiven parity inα, ϕ in the ordinary sense into functions with the same parity in the(α, p, q) variables.

MPAG024.tex; 6/04/1999; 8:19; p.2

ARNOLD’S DIFFUSION IN ISOCHRONOUS SYSTEMS 297

2. Invariant Tori and Nearby Flow

We look for a change of coordinates(A0, α, p0, q0) ↔ (A,ψ, p, q) which inte-grates locally (1.2) near the unstable equilibrium of the pendulum. More preciselyso that in the new coordinates the motion is:

A = const, ψ → ψ + ωt, p→ pe−(1+γ )gt, q → qe(1+γ )gt , (2.1)

whereg = g(pq), γ = γ (pq) andx → γ (x) is a suitable function analytic inxnearx = 0 whileg(x) = J ′(x) (see (1.2)). We shall attempt to write the change ofcoordinates:

A0 = A+H(ψ,p, q), p0 = p + L(ψ,p, q),α = ψ, q0 = q + L(ψ, p, q) def= q + L(−ψ, q, p), (2.2)

whereH(ψ,p, q) has even parity, and zeroψ average atp = q = 0. Setting

∂/def= q∂q − p∂p and imposing that (2.2) and (2.1) verify the equations of motion

one gets:

(g(x)∂/+ ω · ∂ψ)H = −ε∂αf0(ψ, p0, q0)− g(x)γ (x)∂/H,(g(x)+ g(x)∂/+ ω · ∂ψ)L = −ε∂q0f0(ψ, p0, q0)− (g(x0)− (2.3)

− g(x))p0+ γ (x)g(x)p − g(x)γ (x)∂/L,wherex = pq, x0 = p0q0, p0 = p + L, q0 = q + L. In fact, the second equationis independent of the first. Both can be shown to admit, forε small enough as weshall always suppose below, a solution analytic inε and divisible byε. Note thatthe unknowns areH , L, γ . This is a very simple theorem. A “classical” (i.e., byquadratic iterations) proof can be derived from Section 5 in [CG] where the harderanisochronous case is detailed; or see [G4] (p. 2–3 and p. 9) for a self-containedelementary analysis (see also [DGJS] where the same theorem is also explicitlyproved in p. 52–62). A better proof is essentially in the basic paper [Ge]: basic inthe sense that it discusses the general theory of the manifolds from what I think is a“basic” point of view: i.e., it follows the Eliasson’s method using bounds on powerseries expansions, [E] (as developed in [G2], [G3], [GG]).

In what follows wefix ε so small that the above functionsH , L, L are welldefined by their power series inε. The variablesA will be calledaverage actionsbecause they are the average values with respect to time of the actions of themotions that take place on the corresponding invariant tori.

3. Stable and Unstable Manifolds

From (2.2) we can read the following facts (of course ifε is small enough):

(1) Phase space contains a family of invariant toriT (A) parameterized byA ∈ R2

and obtained by settingp = q = 0. The average position of the tori is precisely

MPAG024.tex; 6/04/1999; 8:19; p.3

298 G. GALLAVOTTI

A, becauseH can (and will) be chosen to have zero average ifp = q = 0:average with respect toψ or to time.?

(2) GivenA and settingq = 0,p 6= 0 one obtains a surface whose points are para-meterized byψ , p and which is a local piece of the stable manifoldWs(A) ofT (A). The quantity−(1+ γ (x))g(x) is the “Lyapunov exponent” ofWs(A).Likewise, settingp = 0, q 6= 0 one defines a local piece of the unstablemanifoldWu(A) of A and(1+ γ (x))g(x) is the corresponding exponent.

(3) GivenA and settingq 6= 0, p 6= 0 one parameterizes the rest of phase spacenearT (A). In this part of phase space the motion is in some sense very regular.

(4) However, the motions, just described locally, are globally more interestingand chaotic. In fact genericallyWu(A), Ws(A′) do intersect transversally ifA, A′ are close enough (depending onε) andT (A), T (A′) have the sameenergy. In such casesWu(A)∩Ws(A′) consists of trajectories, orheteroclinicintersections, running asymptotically aroundT (A) as t → −∞ and aroundT (A′) ast →+∞.The symmetry of the problem implies (as is well known, see for instance Sec-tion 9 of [CG]), that ifA = A′ andϕ = π then the point ofWs(A)with ϕ = π ,α = 0 is homoclinic, i.e., it is on a trajectory onWu(A) ∩ Ws(A) (providedε is small so that the results of Section 2 apply). Generically the intersectionWu(A) ∩ Ws(A′) exists for allA, A′ close enough and istransversalin thesense that if we fixϕ = π (or ϕ to any other value6= 0,2π ) then any pair oftangents toWu(A) andWs(A′) at common points form an angle> µ > 0; theboundµ depends onε, of course, and generically can be taken proportionalto ε.

(5) Analytically we can writeA′ + As(α, ϕ) andA + Au(α, ϕ) the parametricequation of the manifoldsWu(A′) andWs(A), so representable at least forϕaway from 0 or 2π , see (2.2) (e.g., see [CG] or [G3]). The functionsAu, As

do not depend onA because of isochrony:?? to see this note that the evolutionequations forI , ϕ, α do not involve theA’s; explicit expressions forAs , Au

can be found in [G3] or [GGM].The equation for theα value of an intersection point inWu(A)∩Ws(A′) with

ϕ = π (say) is justQ(α)def= As(α, π) − Au(α, π) = A′ − A, where usually

Q(α) is called thesplitting vectoratα (andϕ = π ). The angles between thetangent planes toWu(A) andWs(A) at the homoclinic intersection atϕ =π , α = 0 are related to the eigenvalues of theintersection matrixDij

def=? This is a simple special case of a property which becomes rather nontrivial in more interesting

“anisochronous” Thirring models. Such models (see [T] and [G2, G3]) differ from (1.1) because ofa possible addition toH0 of an extra term 1

2KA2 with K constant; then the average actionA of

the motion on a torus is directly related to the gradient of theunperturbedHamiltonian viaω =∂AH0(A), i.e., the frequencies are not “twisted” by the perturbation (a fact apparently “discovered”in [G3]).?? Note that the parametric equation for theI variable needs not to be specified as it follows from

the ones for theA’s via the energy conservation.

MPAG024.tex; 6/04/1999; 8:19; p.4


∂iQj(α)|α=0 which isalso the Jacobianof the implicit equationQ(α) = A′ −A nearα = 0.It follows from the classical Melnikov theory of splitting (see for instance[G3]) that the eigenvalues ofD generically have values of order O(ε) sothat the angles between tangents toWu(A) andWs(A) at α = 0, ϕ = π

will have size O(ε) (and detD = O(ε2)). The genericity condition is a verysimple algebraic condition on the coefficients of the polynomialf and is eas-ily verified in many examples: the very simplest being perhapsf (α, ϕ) =cos(α1+ ϕ)+ cos(α2+ ϕ).The non-vanishing of the intersection matrix determinant, and its interpre-tation as Jacobian of the implicit equation for the heteroclinic intersections,implies that the latter exist always, as soon as the average actions of the toriare close enough (and the tori have the same energy, of course).

(6) One can define also the splitting in theϕ variables: callXu−(t), Xs−(t) thevalues at timet of the ϕ coordinate of the point on the unstable or stablemanifold which at timet = 0 has coordinates (Au(α, π), α, I u(α, π), π ) or(As(α, π), α, I s(α, π), π ) and one sets1(t) = Xs−(t) − Xu−(t), which alsodepends onα.

(7) Finally a definition: LetA0, A1, . . . , AN be a sequence such that|Aj − Aj+1|is so small thatWu(Aj ) ∩Ws(Aj+1) have a transversal heteroclinic intersec-tion, in the above sense, with intersection angles> µ atϕ = π . We call such achain aheteroclinic chainor ladder. As remarked in (5) one finds genericallyand in most simple examplesµ = O(ε), henceN = O(ε−1).

We shall prove the following theorem (“Arnold’s diffusion” or “drift”):

THEOREM 1. LetA0, A1, . . . , AN be a heteroclinic chain: for anyδ > 0 thereare trajectories starting withinδ of T (A0) and arriving after a finite timeTdrift

within δ of T (AN ).

I shall give a complete proof of it (Section 5), again, along the lines of [CG]for the sake of illustrating thesimplicity of the method (due to Arnold). The pur-pose being of showing the conceptual difference with respect to the variationalapproaches, which accounts for the impressive difference in the time scale ofTdrift

compared with [Be, Br] or with the estimate in Theorem 2 below (see (6.9)). InSection 6 I give a more refined, yet very simple and detailed, proof getting explicitand much better bounds (“fast”), although still far from the best in the literature.

4. Geometric Concepts

Let 2κ > 0 be smaller than the radius of the disk in the(p, q)-plane where thefunctions in (2.2) are defined. We callκ a “target parameter”.

To visualize the geometry of the problem involving 2-dimensional tori and their3-dimensional stable and unstable manifolds, in the 5-dimensional energy surface,

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300 G. GALLAVOTTI

we shall need the following geometric objects:

(a) a pointXi, heteroclinic betweenT (Ai) andT (Ai+1), which has local coordi-nates, see (2.2),Xi = (Ai, ψi

,0, κ).(b) the equations, at fixedq = κ, of the connected part ofWs(Ai+1) containing

Xi, in the local coordinates nearT (Ai); they will be written as:

Yi(ψ) = (Asi+1(ψ),ψ, psi+1(ψ), κ) (4.1)

with |ψ − ψi| < ζ for someζ > 0 (i-independent): it isAsi+1(ψi

) =Ai, p

si+1(ψi

) = 0 because we requireYi(ψi) = Xi. There are constantsF ′, F

such that|Asi+1(ψ)−Asi+1(ψi)| and max|ψ−ψ

i| = fixed|psi+1(ψ)| are bounded,

for ζ small enough, below byF ′|ψ−ψi| and above byF |ψ−ψ

i|; the constants

F ′, F have size bounded below by O(µ) (by the transversality assumption inthe definition of heteroclinic chain, (7) of Section 3).Note thatWs(Ai+1) also contains a part with local equations(Ai+1, ψ, p,0)which should notto be confused with the previous one described by the func-tion Yi(ψ). This is more easily understood by looking at the meaning of theabove objects in the original(A, α, I, ϕ) coordinates: in a way the first partof Ws(Ai+1) is close toϕ = 0 and the second toϕ = 2π . They can be closebecause of the periodicity, but they are conceptually quite different.

(c) a pointPi = Yi(ψi) with |ψ

i− ψ

i| = ri , whereψ

i, ri will be determined

recursively, and a neighborhoodBi:

Bi = |A−Asi+1(ψ)| < κ2r ′i , |ψ − ψ i| < r ′i , |p − psi+1(ψ)| < κr ′i ,

q = κ, (4.2)

wherer ′i < ri are scales< 1 and to be determined recursively. Ifg, 2g arelower/upper bounds to(1+ γ (x))g(x) for |x| < 4κ2, the pointPi evolves ina timeTi ' g−1 logκ−1 into a pointX′i nearT (A′i+1) which has local coor-dinatesX′i = (Ai+1, ψ

′i, κ,0). Note that any point(A,ψ, p, q) will evolve,

provided it does not exit the neighborhood where the local coordinates aredefined, into a point of the form(A,ψ + ωTin, q, p) after a timeTin =−g(x)−1(1+ γ (x))−1 logqp−1 if x = pq, because of the special hyperbolicform of the time evolution, see (2.1): we shall call this time theinterchangetimeof the “last two coordinates” and we shall repeatedly use it.The choice ofB0 is rather arbitrary and we taker0 = ζ (ζ is defined after(4.1)) andr ′0 = 1

2r0, choosingψ0

arbitrarily (at distancer0 fromψ0).

(d) The pointsξ of the setBi are mapped by the time evolution to points that, atthe beginning at least, come close toT (Ai+1) and in a timeτ(ξ) acquire localcoordinates nearT (Ai+1) with p = κ exactly: the timeτ(ξ) is of the order ofg−1 logκ−1.

MPAG024.tex; 6/04/1999; 8:19; p.6


If St is the time evolution flow for the system (1.1) we writeSξ = Sτ(ξ)ξ (notethatS depends also oni). ThenS maps the setBi into a setSBi containing:

B ′i =|A− Ai+1| <

1

Eκ2r ′i , |ψ − ψ ′i| <

1

Er ′i , p = κ, |q| <

κ

Er ′i

(4.3)

because all the points inBi with A = Asi+1(ψ), p = psi+1(ψ), q = κ evolve(each taking its own time) to points withA = Ai+1, p = κ, q = 0 andψclose toψ ′

i, by the definitions. HereE is a bound on the Jacobian matrix ofS.

The latter, being essentially a flow over a time O(g−1 logκ−1), has derivativesboundedi-independently: since we suppose thatε is “small enough” we couldtakeE = 1+ bε for someb > 0 if, as often the case,|Ai −Ai+1| < O(ε).

5. The [CG]-method of Proof of the Theorem

Consider the pointsYi+1(ψ) ∈ Ws(Ai+2) with coordinates (Asi+2(ψ), ψ , psi+2(ψ),κ). They will evolve backwards in time so thatA stays constant,ψ evolvesquasi-periodically hence “rigidly”, andpsi+2(ψ) evolves toκ while theq-coordinateevolves fromκ to q = psi+2(ψ) (becausepq stays constant, see (c) above). Thetime for this evolution isTψ ' g−1 logκ|psi+2(ψ)|−1−−−−→

ψ→ψi+1

+∞.

Therefore there is a sequenceψn−−−−→n→+∞ ψ

i+1such that|psi+2(ψ

n)| > 0, psi+2

(ψn) → 0, Asi+2(ψn) → Ai+1 andψn − ωTψn −→

n→∞ψ′i, as a consequence of the

Diophantine properties ofω. So that there isψi+1

def= ψn with n suitable and a point

Pi+1 = (Asi+2(ψi+1), ψ

i+1, psi+2(ψi+1

), κ) ∈ Ws(Ai+2) (actually infinitely many)which evolves, backwards in time, fromPi+1 to a point ofB ′i.

Hence we can defineri+1 = |ψi+1− ψ

i+1| andr ′i+1 small enough so that the

backward motion of the points inBi+1 enters in due time intoB ′i. It follows that thesetBi evolves in time so that all the points ofBi+1 are on trajectories of points ofBi, for all i = 1, . . . ,N . Hence all points ofBN will be reached by points startingin B0.

This completes the proof. All constants can be estimated explicitly, even thoughthis is somewhat long and cumbersome, see [CG]. The result is an extremely largedrift timeTdrift (namely the value atN of a composition ofN exponentials! at leastthis is the estimate I get after correcting an error in Section 8 of [CG]: the error isminor but leads to substantially worse bounds).

Nevertheless the estimate that comes out of the above scheme seems essentiallyoptimal. And then the problem is: “how is it possible that by other methods (e.g.,variational methods of [Be], [Br]) one can getfar betterestimates?

The above argument is quite close to the proof of the “obstruction property” in[C], p. 34: hence the latter work shows that the above analysis misses some key

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302 G. GALLAVOTTI

idea that is exploited in the papers [M], [C]; perhaps the possibility of setting up asymbolic dynamics around the tori and exploiting it in the bounds.

The difference with respect to the variational methods may be due to the fact thatthey are “less constructive”: less so than the above. The “fast drifting” trajectoryexists but there seems to be no algorithm to determine it, not even the sequence ofits “close encounters” with the invariant tori that generates drift: which is in factpreassignedin the above method. This certainly might account for a difference inthe estimates. In fact the above construction is far too rigid: we pretend not onlythat drift takes place but also that it takes place via a path that visits closely aprescribed sequenceof tori in an essentiallypredeterminedway. In Section 6 aless constructive method is proposed and used to obtain bounds: which, however,turn out to be still far from polynomial. Of course a better understanding of whythe results are so different with the different methods is highly desirable: but myefforts to understand satisfactorily this point only led to the improvement in Sec-tion 6 below, which has nevertheless some interest as it introduces the notion ofelastic heteroclinic chain which I think might be useful also for the analysis of theanisochronous cases.

6. Fast Diffusion: Elastic Heteroclinic Chains

The following adds a new idea to the method exposed in Section 5, allowing usto improve the super-exponential estimate of [CG] mentioned there. Belowε willbe fixed small enough, andg, 2g will be lower and upper bounds, respectively, tog(x)(1+ γ (x)), see (2.1).

Let y be the curvilinear abscissa of a curvey → A(y), y ∈ [0, ymax], inthe “average action space” such that the toriT (A(y)) have fixed energy. Thenevaluating the energy at the homoclinic pointα = 0, ϕ = π and using that theI coordinate of the points onWu(A), Ws(A) do not depend onA, because ofisochrony, (2.2) one sees thatω · A(y) is constant so that the liney → A(y) isparallel toω⊥ = (ω2,−ω1).

HenceA(y) = A0+w⊥y with w⊥ def= ω⊥|ω| andA′(y) def= ∂A(y)

dy ≡ w⊥.Define y → A(y), y ∈ [0, ymax], to be anelastic heteroclinic chainwith

flexibility parametersβ, ϑ > 0 and splittingµ if:(i) for all |y − y′| < ϑµ there is a heteroclinic intersection between the stable

and unstable manifolds ofT (A(y)) andT (A(y′)) with splitting angles> µ atϕ = π .

(ii) The intersection matrixDdef= µDo atϕ = π , α = 0 verifies:

(w⊥ ·D−1o w

⊥) def= β 6= 0, w⊥ = ω⊥

|ω| . (6.1)

REMARKS. (a) The above definition is a special case of a natural more generaldefinition relevant for higher dimensions and for anisochronous systems. For in-stance in the case of anisochronous systems, in which a termA2/2K, withK > 0

MPAG024.tex; 6/04/1999; 8:19; p.8


constant, is added to (1.1) one has to require thaty → A(y) is a simple rectifiablecurve and that, uniformly iny ∈ [0, ymax], (6.1) holds withD replaced by theintersection matrixDy, andω replaced byω(A(y)) = ω +A(y)K−1.

But in the anisochronous cases the condition that for ally there is the torusT (A(y)), called “no gap property”, is strongly restrictive and quite artificial (al-though it is verified in the example in [A], see also [P]).

Below we consider, without further mention, only the isochronous models in(1.1) and in this caseD, Do are y-independent because of isochrony, see(5) inSection3.

Condition (6.1) is a transversality property: in the case of (1.1) it holds generi-cally (in the perturbationf and forε small) and in this case it is a consequence of(i). Thus examples exist and are elementary, and genericallyµ = O(ε). A simpleconcrete example is provided by the already mentioned perturbationf (α, ϕ) =cos(α1+ ϕ)+ cos(α2+ ϕ).

Below we shall also suppose, without further mention, thatµ = O(ε), i.e., thatwe consider a generic case.The greater generality of the above definition is meantto clarify a notion that might seem special for the isochronous case, and for futurereference.

(b) Thus every sequencey0, y1, . . . , yN with |yi − yi+1| < ϑµ is a heteroclinicchain in the sense of Section 3, and the theorem proved in Section 5 applies to it. Aelastic heteroclinic chain with parameterϑ is also elastic with parameterϑ ′ < ϑ .Hence it will not be restrictive to suppose thatϑ is as small as needed.

(c) If ϑ is small enough so that the first order Taylor’s expansions of the splittingvectorQ(α), see Section 3, (5), are “good” approximations we deduce (by applyingthe implicit functions theorem) that a heteroclinic intersection atϕ = π betweenWs(A(y)) andWu(A(y + δ)) takes place at:

αy(δ) = D−1o w

⊥ϑ ′ +O(ϑ ′2) for δ = µϑ ′, |ϑ ′| 6 ϑ,(6.2)1

2β|ϑ | < |(αy(δ′)− αy(δ′′)) ·w⊥| < 2β|ϑ |, ∀δ′ = µϑ ′, δ′′ = µϑ ′′

for ϑ small enough, for|ϑ ′|, |ϑ ′′| < ϑ , and having setϑ = ϑ ′ − ϑ ′′.(d) A geometrical consequence of (6.1), (6.2) is that wheny varies byδ (so that

A(y) varies inR2 orthogonally to ω by O(δ)), then the heteroclinic intersectionαy(δ) betweenWs(A(y)) andWu(A(y + δ)) is displaced away from 0with acomponentin the direction orthogonal toω of size O(δµ−1), providedδµ−1 = ϑ ′is small enough.

(e) The valueϕ = π is not special in many respects and the same remains trueif one looks at the displacement of the heteroclinic intersection at any other sectionlocated away from the tori by a fixed distanceκ > 0, if ε is small enough. In factconsider the intersection matrixD(t) evaluated along the heteroclinic trajectory

MPAG024.tex; 6/04/1999; 8:19; p.9

304 G. GALLAVOTTI

at a timet after the passage throughϕ = π . From the equations of motion itsevolution is:

D(t) = D − ε∫ t

0∂αϕf (ϕ(τ), ωτ)∂α1(τ)dτ, (6.3)

where1(t) denotes the splitting in theϕ-coordinates (and|1(t)| < O(ε)), definedin remark (6) of Section 3, andϕ(t) is the heteroclinic evolution ofϕ: henceD(t) =D +O(ε2) (whileD = O(ε)) for t bounded, by Melnikov’s theory (see also (5.5)in [GGM]).

In particular if we look at theψ-coordinateψy(δ) of the heteroclinic intersec-

tion point atq = κ, on the same heteroclinic trajectory, and compare it with theposition of the homoclinic pointψ

y(0) of T (A(y)) atq = κ then we can say that,

for some constants 2b1, 2b0 (the factor 2 is for later convenience) it is:

|w⊥ · (ψy(δ′)− ψ

y(δ′′))| ∈ [2b1ϑ,2b0ϑ] (6.4)

with ϑ = (δ′ − δ′′)µ−1; the constantsb0, b1 depend on the constantκ prefixed atthe beginning of Section 6, and onβ.

THEOREM 2. Suppose thaty → A(y) is elastic in the above sense, then fixeda,b there exist heteroclinic chainsA0 = A(y0),A1 = A(y1), . . . , AN = A(yN ) withy0 = 0, yN = ymax along which the drift time isg−1eO(µ−1).

The estimates proceed by performing the construction of Section 5withoutfix-ing a priori the heteroclinic chain: we construct it inductively, by trying to optimize(as well as we can) various choices.The proof below is divided, into several trivialstatements, into a few propositions and lemmata each of which is marked by a•.

Using the notations of Section 4, assume thatyj have been constructed forj 6i + 1 together withψ

j, rj , r ′j , Bj , ψ

′j, B ′j for j 6 i, verifying ri < ϑ . We must

defineyi+2, ψi+1

, ri+1, r ′i+1. The setB0 is fixed as in the paragraph following (4.2)

above, andy1−y0 = µϑ , r0, r ′0 = 12r0 are arbitrarily chosen (positive) and we also

requirer0 < ϑ andϑ small.

• 1. LetE be as in Section 5 and letE′ be so large that ifT 0 = g−1 logE′E−1 thepointsωt , t ∈ [0, T 0], fill the torus within1

2b1ϑ (see (6.4) for the definition ofb1).This means thatE′ is very large.?

• 2. LetXi+1(y) be heteroclinic betweenT (Ai+1) andT (A(y)) for y ∈ [yi+1 +12µϑ, yi+1+µϑ]. We choose to look foryi+2 among suchy’s to be sure that every

? One can takeE′ = E exp O(Cϑ−τ ) estimating by O(Cδ−τ ) the time needed to a quasi periodicrotation of the torus with vectorω, Diophantine with constantsC, τ , to fill with lines parallel toωand withinδ the whole torusT 2. I discuss this estimate in Appendix A1 as an aside, since here onlyfiniteness ofE′ is required (a trivial fact): this gives me the chance of discussing a simple conjecture.

MPAG024.tex; 6/04/1999; 8:19; p.10


time i increases by one unit thenyi increases by12µϑ at least, so that afterN steps,with N = O(µ−1) we shall have reached the “upper extremeymax of the chain”.

Let the local coordinates ofXi+1(y) be(Ai+1, ψi+1(y),0, κ) (see Section 4 for

the notations). Let, see (4.1):

Yi+1,y(ψ) = (Asi+2,y(ψ),ψ, psi+2,y(ψ), κ) (6.5)

be the equation ofWs(A(y)) in the local coordinates around the torusT (Ai+1)

nearXi+1(y). We may suppose that:

|Asi+2,y(ψ)−Ai+1|, |psi+2,y(ψ)| < b3µ|ψ − ψi+1(y)| (6.6)

for someb3 of O(1) and we may supposeb3 > 1, for simplicity. This simplyexpresses the analyticity inψ andε of the stable manifold (note thatAsi+1,y(ψ) −Ai+1 andpsi+1,y(ψ) vanish atψ

i+1(y), i.e., at the heteroclinic point).

• 3. Supposer small: a first approximation toψi+1

will be obtained byfixing y at

the left extremey of its interval of variation(which is[yi+1+ 12µϑ, yi+1+µϑ]) and

by choosing a pointψi+1,y,r

at distancer from ψi+1(y) along a line3 on which

we can be sure thatpsi+2,y(ψ) does not vanish. For instance we can take the straightline 360 at 60 from the gradientai+2(y) of psi+2,y (ψ) at ψ = ψ

i+1(y). In this

way (cos 60 = 12):

λdef= |psi+2,y(ψi+1,y,r

)| ' 1

2max

|ψ−ψi+1(y)|=r|psi+2,y(ψ)| (6.7)

and λ ∈ [b2µr, b3µr], for someb3, b2 = O(1) > 0, by the assumption on

the splitting angles (which implies that the modulus of the gradientai+2(y)def=

∂ψpsi+2,y(ψi+1

(y)) is in [b2µ, b3µ] for some constantsb2, b3 > 0 of O(1)). Theconstantsb2, b3 depend on the “target” parameterκ, fixed once and for all, seebeginning of Section 4, andb3 can be taken to be the same constant in (6.6). Letd = b2/2b3.

• 4. To improve the approximation forψi+1

note that asr varies in the range

dr ′i

4b3E′ < r <

r ′i4b3E

the pointψi+1,y,r

varies andλ varies by a factor not smaller

than 2E′/E by our definition ofd. Hence the timeT (r) necessary in order thatthe backward evolution of the pointYi+1,y(ψi+1,y,r

) = (Asi+2,y(ψi+1,y,r), ψ

i+1,y,r,

psi+2,y(ψi+1,y,r), κ) interchangesthe last two coordinates, will vary by an amount

> T 0 = g−1 logE′/E = O(Cϑ−τ ), see footnote? and comment (c) in Section 4.

• 5. This implies, by continuity and by the size ofT0, that there will be a valuer(y)such that the “backward motion”ψ

i+1(y)→ ψ

i+1(y)−ωt of durationT (r(y)) has

ψ-coordinate close toψ ′i

within 12b1ϑ and on the line orthogonal toω through

ψ ′i.

MPAG024.tex; 6/04/1999; 8:19; p.11

306 G. GALLAVOTTI

We can alsoprefix on which sideof it will be. Remark that asy increases pastythe pointψ

i+1(y) moves with a displacement having a nonzero component in the

direction parallel to (by the second of (6.1) and the first of (6.2)). Therefore weshall choose the side so that the component of the displacement along` is towardsψ ′i: this is convenient for reasons that will become clear below.

• 6. I now imagine varyingy in its interval of variation[yi+1 + 12µϑ, yi+1 + µϑ]

and selectr(y), henceψi+1,y,r(y)

, so that the time of interchange of the last twocoordinates of the pointYi+1,y(ψi+1,y,r(y)

) does not change. The latter time is,

by (2.1), g(pκ)−1(1 + γ (pκ))−1 logκ|p|−1 if p = psi+2,y(ψi+1,y,r(y)), because

the motion is “exponential” and preserves the product of the last two coordinates(see (2.1)).

Hence fixing the interchange time means determiningψ = ψi+1,y,r(y)

so thatp is constant. Although it might be clear that this can be done I describe in somedetail the way that I follows in Appendix A2 which also gives the quantitativeinformation that:

d

8b3E′ r′i < r(y) <

1

2b3E′ r′i (6.8)

and, therefore, the pointYi+1,y(ψ) remains in the neighborhood where the localcoordinates are defined.

• 7. For eachy the pointψi+1(y) − ωT (r(y)) will either “fall short” or “long”

of the line ` orthogonal toω throughψ ′i: but “only by a length bounded by6

2|D−1o w

⊥|ϑ”. In fact by construction the vectorψi+1(y)−ωT (r(y)) is exactly on

the line` and the maximum variation that|ψi+1(y) − ψ

i+1(y)| can undergo asy

varies by at most12µϑ is bounded by the first of (6.2).

• 8. Henceby a suitable rotation of the direction of the line360 along which wechoose the pointψ

i+1,y,r(y)we can change the size ofpsi+2,y by a factor 1+O(ϑ)

and arrange that at the timeTin when the last two coordinates are interchangedψi+1(y)− ωTin is exactlyon the line` orthogonal toω throughψ ′

i.

In fact our choice of the line360 on whichψi+1,y,r(y)

is selected, neither or-thogonal nor parallel to the gradient ofpi+2,y(ψ), shows that we can change in thisway |psi+2,y| by up toa factor about 2, i.e., by a factor 1+ O(ϑ) if ϑ is small,anddown toa factor 0.?

So that, by continuity, we can find a line3′ slightly off 360 by an angle ofO(ϑ) and a pointψ on it such that in its interchange timeTin the (other) pointψi+1(y) ends on the target line.

? Note thatpsi+2,y(ψ) vanishes in correspondence of the heteroclinic point, i.e., atψ

i+1(y), aswell ason a curve through the heteroclinic point valueψ

i+1(y) by the transversality of the splittingand by the implicit functions theorem.

MPAG024.tex; 6/04/1999; 8:19; p.12


We still call ψi+1,y,r(y)

the new (and final) choice ofψ on3′. Note, however,that the pointψ

i+1,y,r(y)will not end on the line but it will miss it by at most a

distancer(y) < r ′i /(2b3E), becauseψi+1,y,r(y)

is r(y) apart fromψi+1(y), which

by the construction ends onat the interchange timeTin: the point reached onwill be away by at mostb1ϑ fromψ ′

i.

The angle of the needed rotation will be of O(ϑ) off the line360 at 60 de-grees to the gradient∂ψpi+2,y(ψ

i+1(y)), because the velocity of the quasi periodic

motion has size of order O(1) (i.e., it is |ω|) so that a time variation of up to O(ϑ)suffices for a displacement ofr(y) < ϑ (recall thatr ′i < ri < ϑ , as stipulated atthe beginning).

• 9. As we varyy we find, by continuity, a pointy∗ such thatψi+1(y∗)−ωTin = ψ ′i

becauseψi+1(y) has a component alongw⊥ which varies byb1ϑ at least, see (6.4),

and in the right directiontowardsψ ′iby the above proposition• 5.

• 10. Settingr∗ = r(y∗) and ψi+1= ψ

i+1,r∗,y∗ we see that the evolution of

Yi+1,y∗(ψi+1) leads to a point which hasψ-coordinate close within

r ′i2b3E

to thecoordinateψ ′

iof the pointX′i = (Ai+1, ψ

′i, κ,0) (around which the already induc-

tively known setB ′i is constructed, see (4.3)), becauseψi+1

is within r ′i/2b3E ofψi+1(y) by construction.Note that this is just a continuity statement: hence it is

nonconstructive, as much as the other continuity arguments used above.

We setri+1 = r∗ > dr ′i

8b3E′ , r ′i+1 = γ r2

i+1 with γ small enough see that thepoints ofBi+1 defined by such parameters via (4.2) evolve backward in time to fallinsideB ′i at their last interchange time.

However the interchange timeTin varies whenψ varies in the disk of radiusr ′i+1 centered at a point at distanceri+1 from the heteroclinic point. And it is propor-tional to the logarithm of the inverse of|pi+2(ψ)|; the latter is a function essentially

linear inψ .? Hence it varies bounded by a factor proportional to logri+1+r ′i+1ri+1−r ′i+1

.

The latter variation has size O(r ′i+1ri+1) which must be< O(r ′i ) becausewe want

that the backward evolution of the points inBi+1 is, at their interchange time, insideB ′i and the velocityω of the quasi periodic motion is of O(1). Hence if the timevaries by O(r ′i ) the resulting displacement of the final value ofψ will be of O(r ′i ):recalling thatri+1 ∈ [ d

8b3E′ r′i ,

12b3E

r ′i ] we get a quadratic recursion for the definitionof theri : unavoidable in the above scheme.

• 11. We find that, from the above proposition, thatr ′i , ri = O((0r0)2i

) for some0(one can take0 = dγ/(8b3E

′)); so that the timeTi = O(g−1 log r−1i ) necessary to

hop one step along the chain is O(g−12i ) and the timeTdrift for drifting along the

? Because the pointψi+1 is still on a line3′ very close, by the last remark in the preceding

proposition, to360 , at 60 degrees to the gradient∂ψpi+2,y(ψi+1(y∗)).

MPAG024.tex; 6/04/1999; 8:19; p.13

308 G. GALLAVOTTI

chain is bounded above by O(g−12N ):

Tdrift 6 g−12O(µ−1). (6.9)

Recalling thatϑ is fixed, ifµ = O(ε) (generic) this is const econstε−1.

REMARK. Therefore the exponential bound (6.9) is due to the rapid convergenceto 0 of ri, i.e., with a logarithm exponentially diverging, which arises from thequadratic recursionr ′i+1 = O(r ′2i ).

7. Concluding Remarks. Very Fast Diffusion?

For a review on diffusion see [L]: in this paper the possibility of estimates of sizeof an inverse power ofε is proposed and discussed.

(1) The above non-variational proof gives results not directly comparable tothe best known, [Be], [Br], based on a variational method and giving (in [Br]) apolynomial drift time of O(µ−2).

The papers [Be], [Br], deal with Arnold’s example, [A], i.e., with a differentcase. However, they make use in an essential way of the very similar structureof the model, i.e., of the fact that it admits a “gap-less” foliation into stable andunstable manifolds of invariant tori, see also [P]. It is hard to see how to improvethe bounds of Section 6 in Arnold’s example, if it is studied along the same lines.

The recent works [M], [C], also lead very close by to the estimates in [Be],[Br] and, if I understand them correctly, they should also apply to the cases treatedhere and give polynomial estimates: hence the difference between the sizes of thebounds obtained by our approach and the ones obtained via variational methodsor via geometric methods alternative to the ones exposed here remains (for me) apuzzle that I hope to understand in the future.

(2) It is worth stressing again that the methods of Section 5 apply every timethere are “no gaps” around resonant tori and the homoclinic angles admit a nonzerolower bound: therefore they apply to the case in [A] with, in the notations of [A],µ = εc andc large enough.

In the isochronous models they apply, immediately, to a variety of cases: a non-trivial one is the Hamiltonian (1.1) withω = (ηa, η−1/2), a > 0, ε = µηc withc large enough and,possibly, even a further“monochromatic, strong and rapid”perturbationβf0(ϕ, λ) like β cos(λ+ ϕ) with β = O(1).

Consider only values ofη such that|ω · ν| > Cηd |ν|−τ for all 0 6= ν ∈ Z2, andfor someC, d > 0, see Section 2 in [GGM]. Thenby using the results of[GGM](Section 8) we see that ifη is fixed small enough the homoclinic splitting is analyticin β for |β| < O(η−1/2), while it does not vanishfor β small (i.e.,β = O(ηc)),generically inf (see [GGM], Section 6).

Hence it is not 0 for allβ < 2 (say)except, possibly, for finitely many valuesof β. This means that in suchstrongly perturbed systems(β = O(1)) one stillhas elastic heteroclinic chains of arbitrary length, see Section 8 of [GGM], and

MPAG024.tex; 6/04/1999; 8:19; p.14


therefore there is diffusion provable by the methods of Sections 5 and 6, exceptpossibly in correspondence of finitely many values ofβ.

Furthermore theA-independent (because of isochrony, see [GGM]) homoclinicangles can become large whenβ,µ approach their convergence radii and this givesus the possibility of “very fast” drift on time scales of∼ O(1). In fact I think that thehomoclinic splitting might be a monotonic function ofε, β for interesting classesof perturbations.

(3) An advantage of the technique of Section 5 is its flexibility which makes itimmediately applicable, essentially without change, to anisochronous systems, see[CG] as corrected in [CV].

(4) Constructivity, even partial (see comments in Section 5), seems the key tounderstanding the huge difference between the results of Section 5 and the vari-ational results, or those of Section 6 above: diffusion time bounds in an inversepower ofε (in [Br] and Section 6) versus a super-exponential in the more con-structive proposal in Section 5. A hint in this direction is provided by the bound inSection 6: by adding a new idea to the method of Section 5, i.e., of [CG], one canget a drift time estimate of 2−O(ε−1) instead of the super-exponential of [CG], andSection 5.But the theory becomes now less constructive: not even the sequenceof close encounters with invariant tori is determined constructively as continuityarguments are used.

(5) The method of Section 5 and of Section 6 seems related to the “windowing”analysis in the early work [Ea] and in [M], [C] as pointed out to me by P. Lochakand J. Cresson.

(6) Finally only drift in phase space is discussed here: but it is clear that hete-roclinic chains do not need to “advance” at each step (e.g., aA-coordinate needsnot to increase systematically): we can use heteroclinic chains that advance andback up at our prefixed wish (e.g., randomly). In this sense, there is no differencebetween drift and diffusion.

Acknowledgements

I am indebted to P. Lochak stimulating comments and, in particular, to G. Gentileand V. Mastropietro for many discussion and help in revising the manuscript. Thiswork is part of the research program of the European Network on: “Stability andUniversality in Classical Mechanics”, #ERBCHRXCT940460.

Appendix A1. Filling Times of Quasi Periodic Motions: A Conjecture

Let (ω1, . . . , ωd) = ω ∈ Rd be such that|ω · ν|−1 6 C|ν|τ . Let χ(x), χ⊥(x) beC∞-functions even and strictly positive for|x| < 1

2π , vanishing elsewhere and with

integral 1. Letψ ,ψ0∈ T d andx(ψ)

def= ε−(d−1)χ(ω·(ψ−ψ0)/|ω|)·χ(ε−1|P⊥(ψ−

ψ0)|), P⊥ = orthogonal projection on the plane orthogonal toω.

MPAG024.tex; 6/04/1999; 8:19; p.15

310 G. GALLAVOTTI

The functionx can be naturally regarded as defined and periodic onT d : if χ(σ )is the Fourier transform ofχ as a function onR then the Fourier transform ofxis χ (ν||)χ(ε|ν⊥|), ν integer components vector,ν|| = ω · ν/|ω|, ν⊥ = P⊥ν. The

averageX = T −1 ∫ T0 x(ωt)dt is:

X = 1+∑ν 6=0

x(ν)e−iψ0·ν 1

T

eiω·νT

iω · ν

> 1− 2C

T

∑ν 6=0

∣∣χ (ν||)χ(ε|ν⊥|)∣∣|ν|τ . (A1.1)

Since the last sum is bounded above bybε−(τ+d−1) the averageX is positive, e.g.,> 1

2, for all ψ0

if T > 4bCε−(τ+d−1). This means that forT > 4bCε−(τ+d−1) +π/|ω|, hence forT > BCε−(τ+d−1) with B a suitable constant depending only ond, the torus will have been filled by the trajectory of any point within a distanceε.This proof is taken from (5), p. 111, of [G1], see [BGW] for an alternative proofand a much stronger result (i.e., withτ replacingτ + d − 1).

Of course the above estimateT > O(ε−τ−(d−1)) really deals with a quantitydifferent from the minimum time of visit. It is an estimate of the minimum timebeyond which all cylinders with height 1 (say) and basis of radiusε have not onlybeen visited but they have been visited with a frequency that is, for all of them,larger than1

2 of the asymptotic value (equal toεd−1): we can call the latter time thefirst large frequency of visit time. The difference between the two concepts explainsthe difference between the two estimates which are equally good, i.e., alternative,for the purposes of our analysis (and both too detailed since we only need that theminimum time of visit is finite). And Iconjecturethat both are optimal: the first isoptimal as an estimate of the first visit time and the second as an estimate of thefirst large frequency of visit time.

Appendix A2. Fixing the Time of Interchange

The differential condition onψ is ddy (p

si+2,y(ψ))−psi+2,y(ψi+1

(y)) = 0 (having in-sertedpsi+2,y(ψi+1

(y)) = 0 for convenience) or, if prime denotesy-differentiation:

0 = ∂ψpsi+2,y(ψ) · (ψ ′ − ψi+1

(y)′)+ (∂ψpsi+2,y(ψ)− ∂ψpsi+2,y(ψi+1(y))) ·

(A2.1)· ψi+1(y)′ + (∂ypsi+2,y)(ψ)− (∂ypsi+2,y)(ψi+1

(y)),

which means thatr(y)def= |ψ ′−ψ

i+1(y)| verifies|r(y)′| < C1r(y) for someC1 > 0

independent ofµ because:

(1) all derivatives ofpsi+2,y with respect to the argumentsψ , y are of orderµ.

MPAG024.tex; 6/04/1999; 8:19; p.16


(2) The vectorψ − ψi+1(y) has the formr(y)w60(y) wherew60(y) is the unit

vector parallel to the axis forming 60 degrees with∂ψpsi+2,y(ψi+1(y)), so that

(ψ − ψi+1(y))′ = r ′(y)w60(y)+ r(y)w′60(y).

(3) ψi+1(y)′ has size O(1); the second derivatives ofpsi+2,y(ψ) with respect to

ψ have size O(µ) and the derivative ofw60(y) that can be computed bydifferentiating its expression (namely∂ψpsi+2(ψi+1

(y))/|∂ψpsi+1(ψi+1(y))|) is

of O(1).

This fixes they-derivatives ofr(y)def= |ψ − ψ

i+1(y)| to have size O(r(y)) so

that the variation ofr(y), asy varies in its interval of size12µϑ and starting at

y = yi+1 + 12µϑ , is bounded by(eC1

12µϑ − 1) 6 C1µϑr(y) if ϑC1 <

12. Hence

d8b3E

′ r′i < r(y) < 1

2b3E′ r′i and the pointYi+1,y(ψ) remains in the neighborhood

where the local coordinates are defined.

References

[A] Arnold, V.: Instability of dynamical systems with several degrees of freedom,Sov.Mathematical Dokl.5 (1966), 581–585.

[Be] Bessi, U.: An approach to Arnold’s diffusion through the Calculus of Variations,Nonlinear Analysis, 1995.

[Br] Bernard, P.: Perturbation d’un hamiltonien partiellement hyperbolique,C.R. Academiedes Sciences de Paris323(1) (1996), 189–194.

[BGW] Bourgain, J., Golse, F. and Wennberg, S.: The ergodisation time for linear flows ontori: Application for kinetic theory, Preprint, 1995, to appear inCommunications inMathematical Physics.

[C] Cresson, J.: Symbolic dynamics for homoclinic partially hyperbolic tori and “Arnolddiffusion”, Preprint of Institut de mathematiques de Jussieux, 1997. And, mainly: Pro-priétés d’instabilité des systèmes Hamiltoniens proches de systèmes intégrables, Doctoraldissertation, L’Observatoire de Paris, Paris, 1997.

[CG] Chierchia, L. and Gallavotti, G.: Drift and diffusion in phase space,Annales de l’InstitutPoincarè B60 (1994), 1–144.

[CV] Chierchia, L. and Valdinoci, E.: A note on the construction of Hamiltonian trajectoriesalong heteroclinic chains, to appear inForum Mathematicum.

[DGJS] Delshams, S., Gelfreich, V. G., Jorba, A. and Seara, T. M.: Exponentially small splittingof separatrices under fast quasiperiodic forcing,Communications in Mathematical Physic189(1997), 35–72.

[Ea] Easton, R. W.: Orbit structure near trajectories biasymptotic to invariant tori, in R. De-vaney, Z. Nitecki (eds.),Classical Mechanics and Dynamical Systems, Dekker, 1981, pp.55–67.

[E] Eliasson, L. H.: Absolutely convergent series expansions for quasi-periodic motions,Mathematical Physics Electronic Journal2 (1996).

[G1] Gallavotti, G.:The Elements of Mechanics, Springer, 1983.[G2] Gallavotti, G.: Twistless KAM tori,Communications in Mathematical Physics164

(1994), 145–156.[G3] Gallavotti, G.: Twistless KAM tori, quasi flat homoclinic intersections, and other cancel-

lations in the perturbation series of certain completely integrable Hamiltonian systems.A review,Reviews on Mathematical Physics6 (1994), 343–411.

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312 G. GALLAVOTTI

[G4] Gallavotti, G.: Hamilton–Jacobi’s equation and Arnold’s diffusion near invariant tori in apriori unstable isochronous systems,Rendiconti del seminario matematico di Torino, inprint; also in [email protected]#9710019.

[Ge] Gentile, G.: A proof of existence of whiskered tori with quasi flat homoclinic intersec-tions in a class of almost integrable systems,Forum Mathematicum7 (1995), 709–753.See also: Whiskered tori with prefixed frequencies and Lyapunov spectrum,Dynamicsand Stability of Systems10 (1995), 269–308.

[GG] Gallavotti, G. and Gentile, G.: Majorant series convergence for twistless KAM tori,Ergodic Theory and Dynamical Systems15 (1995), 857–869.

[GGM] Gallavotti, G., Gentile, G. and Mastropietro, V.: Pendulum: Separatrix splitting, Preprint,chao-dyn #9709004: this paper will appear with a different, more informative, title“Separatrix splitting for systems with three time scales”. And G. Gallavotti, G. Gentileand V. Mastropietro: Melnikov’s approximation dominance. Some examples, chao-dyn#9804043, in print inReviews in Mathematical Physics.

[L] Lochak, P.: Arnold’s diffusion: A compendium of remarks and questions,Proceedings of3DHAM, s’Agaro, 1995, in print.

[Ea] Easton, R. W.: Orbit structure near trajectories biasymptotic to invariant tori, in R. De-vaney, Z. Nitecki (eds.),Classical Mechanics and Dynamical Systems, Dekker, 1981, pp.55–67.

[M] Marco, J. P.: Transitions le long des chaines de tores invariants pour les systèmeshamiltoniens analytiques,Annales de l’Institut Poincaré64 (1995), 205–252.

[P] Perfetti, P.: Fixed point theorems in the Arnol’d model about instability of the action-variables in phase space, [email protected], #97-478, 1997, in print inDiscreteand Continuous Dynamical Systems.

[T] Thirring, W.: Course in Mathematical Physics, vol. 1, p. 133, Springer, Wien, 1983.

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313

The Inverse Spectral Method for CollidingGravitational Waves

A. S. FOKASDepartment of Mathematics, Imperial College, SW7 2BZ U.K.

L.-Y. SUNGDepartment of Mathematics, University of South Carolina, Columbia, SC 29208, U.S.A.

D. TSOUBELISDepartment of Mathematics, University of Patras, 261 10 Patras, Greece

(Received: 2 February 1998; in final form: 18 February 1998)

Abstract. The problem of colliding gravitational waves gives rise to a Goursat problem in thetriangular region 16 x < y 6 1 for a certain 2×2 matrix valued nonlinear equation. This equation,which is a particular exact reduction of the vacuum Einstein equations, is integrable, i.e. it possessesa Lax pair formulation. Using thesimultaneousspectral analysis of this Lax pair we study the aboveGoursat problem as well as its linearized version. It is shown that the linear problem reduces to ascalar Riemann–Hilbert problem, which can be solved in closed form, while the nonlinear problemreduces to a 2× 2 matrix Riemann–Hilbert problem, which under certain conditions is solvable.

Mathematics Subject Classifications (1991):83C35, 35Q20, 58F07, 65.

Key words: colliding gravitational waves, Ernst equation, boundary-value problem, inverse spectralmethod, Riemann–Hilbert problem, Goursat problem, Einstein equations.

1. Introduction

One of the most extensively studied problems in general relatively is the collisionof two plane gravitational waves in a flat background. Assuming that the two ap-proaching waves are known, it can be shown ([1] and appendix) that the problemof describing the interaction following the collision of the two waves is closelyrelated to the following boundary value problem: Letg(x, y) be a real, symmetric,2× 2 matrix-valued function ofx andy for (x, y) ∈ D, whereD is the triangularregionD = (x, y) ∈ R2,−1 6 x < y 6 1 depicted in Figure 1. Let subscriptsdenote partial derivatives. The functiong(x, y) solves the PDE

2(y − x)gxy + gx − gy + (x − y)(gxg−1gy + gyg−1gx) = 0, (1.1)

with the boundary conditions

g(−1, y) = g1(y), −1< y 6 1;g(x,1) = g2(x), −16 x < 1, (1.2)

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314 A. S. FOKAS ET AL.

Figure 1. The regionD = (x, y) ∈ R2 : −1 6 x < y 6 1 corresponding to the Goursatproblem defined by (1.1) and (1.2).

where the functionsg1(y) andg2(x) are uniquely specified by the approachingwaves.

Equation (1.1), which is a particular exact reduction of the vacuum Einsteinequations, is equivalent to the celebrated Ernst equation [2]. Belinsky and Zakharov[3] have shown that Equation (1.1) is integrable, in the sense that it admits the Laxpair formulation [4]

∂9

∂x+ 2κ

κ + x − y∂9

∂κ= (x − y)gxg−1

κ + x − y 9, (1.3a)

∂9

∂y+ 2κ

κ + y − x∂9

∂κ= (y − x)gyg−1

κ + y − x 9, (1.3b)

where9(x, y, κ) is a 2× 2 matrix-valued function of the arguments indicated andk ∈ C. An alternative Lax pair of Equation (1.1) is [5, 6]

∂9

∂x= 1

2

(1− (y − λ)

1/2

(x − λ)1/2)gxg−19, (1.4a)

∂9

∂y= 1

2

(1− (x − λ)

1/2

(y − λ)1/2)gyg−19, (1.4b)

whereλ ∈ C. For integrable equations it is usually possible to: (i) Construct alarge class of particular explicit solutions, using a variety of the so-called directmethods, such as Bäcklund transformations [7], the dressing method [8], the directlinearizing method [9], etc. (ii) Investigate certain initial-value problems using theso-called inverse spectral method [10 – 12]. Solving an initial-value problem ismore difficult than deriving particular solutions. The problem of colliding grav-itational waves is a boundary-value problem and such problems are even moredifficult than initial-value problems. Indeed, regarding the interaction of planegravitational waves, although many classes of particular exact solutions have been

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THE INVERSE SPECTRAL METHOD FOR COLLIDING GRAVITATIONAL WAVES 315

found (see [1, 13 – 17]), the initial-value problem has only been addressed byHauser and Ernst [18, 19]. These authors did not investigate Equation (1.1) di-rectly. Instead, they have shown that, in the particular case of gravitational waves,Equation (1.1) can be related to the equation

(y − x)Gxy + [Gx,Gy] = 0, (1.5)

where[ , ] denotes the matrix commutator. Equation (1.5) has been studied in[18, 19] using indirectly the fact that Equation (1.5) possesses the Lax pair

∂9

∂x= Gx9

x − λ,∂9

∂y= Gy9

y − λ, (1.6)

where9(x, y, λ) is a 2× 2 matrix-valued function of the arguments indicated.In this paper we use the inverse spectral method to solve the boundary value

problem defined by Equations (1.1) and (1.2) as well as the following linear bound-ary value problem: Let the matrix-valued functionγ (x, y), (x, y) ∈ D, satisfy thelinear PDE

2(y − x)γxy + γx − γy = 0, (1.7)

whereγ (−1, y) and γ (x,1) are given functions ofy and x respectively. Thisboundary value problem can be considered as the smallg limit of the boundaryvalue problem defined by Equations (1.1) and (1.2). Indeed, substitutingg = I+εγin Equation (1.1) (whereI is the identity matrix) and keeping only O(ε) terms,Equation (1.1) becomes Equation (1.7).

We now state the main result of this paper:

THEOREM 1.1. Assume that the derivative ofg1(y) and ofg2(x) areC2 in [−1,1],sufficiently small, andg1(1) = g2(−1) = I , whereI is the2× 2 identity matrix.Then the Goursat problem defined by(1.1)and(1.2)has a uniqueC2 classical solu-tion inD. This solution can be obtained by solving the following Riemann–Hilbertproblem for the2× 2 matrix-valued functions9 and8:

9+(x, y, λ) =

8−(x, y, λ), x 6 λ 6 y,9−(x, y, λ), −∞ < λ 6 −1, 16 λ <∞,9−(x, y, λ)Gl(λ), −16 λ 6 x,9−(x, y, λ)Gr (λ), y 6 λ 6 1,

(1.8a)

8+(x, y, λ) =

9−(x, y, λ), x 6 λ 6 y,8−(x, y, λ), −∞ < λ 6 −1, 16 λ <∞,8−(x, y, λ)Gl(λ)

−1, −16 λ 6 x,8−(x, y, λ)Gr (λ)

−1, y 6 λ 6 1,

(1.8b)

limλ→∞9 = I, lim

λ→∞8 = g, (1.8c)

where

9±(x, y, λ) = 9(x, y, λ ± i0), 8±(x, y, λ) = 8(x, y, λ ± i0), λ ∈ R,

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and the2× 2 matrix-valued functionsGl(λ),Gr(λ) are defined in terms ofg1(y)

andg2(x) as follows:

Gl(λ) = (L−(λ, λ))−1L+(λ, λ), Gr(λ) = (R−(λ, λ))−1R+(λ, λ), (1.9)

where

L±(x, λ) = I + 1

2

∫ x

−1

(1∓ i (1− λ)

1/2

(λ− ξ)1/2)×

× dg2(ξ)

dξg−1

2 (ξ)L±(ξ, λ)dξ, −16 x 6 λ, (1.10)

R±(y, λ) = I − 1

2

∫ 1

y

(1± i (λ+ 1)1/2

(η − λ)1/2)×

× dg1(η)

dηg−1

1 (η)R±(η, λ)dη, λ 6 y 6 1. (1.11)

We conclude this introduction with some remarks.(1) It can be shown that ifg ∈ R, and ifg(x, y) satisfies the equationgCgC =

ρ(x − y)2, whereρ is a real constant andC is a real, nonsingular, constant matrix,then Equation (1.1) is simply related to (1.5). This relationship is valid for theparticular case of gravity. Thus the initial-value problem for the colliding gravita-tional waves can also be investigated by applying the inverse spectral method toEquations (1.6). The inverse spectral method for the Lax pair (1.6) involves thetechnical difficulty of analyzing eigenfunctions with Cauchy type singularities; arigorous investigation of this problem remains open.

(2) The boundary value problem of Equation (1.7) mentioned above was firstsolved by Szekeres [20] using the classical Riemann function technique. The sameproblem was later solved by Hauser and Ernst using separation of variables and theAbel transform [21, 22]. The linear problem has also been discussed in [23]. It wasemphasized in [24] that before solving a given nonlinear integrable equation, itis quite useful to use the inverse spectral method to solve the linearized versionof this nonlinear equation. This is carried out in Section 3, using the fact thatEquation (1.7) possesses the Lax pair

∂9

∂x+ 9

2(x − λ) =γx

2(x − λ),∂9

∂y+ 9

2(y − λ) =γy

2(y − λ). (1.12)

(3) When analyzing a Lax pair, it is customary to study the two equations form-ing this pairindependently. Indeed, one usually studies one of the two equationsto formulate an inverse problem in terms of appropriate spectral data, and thenone uses the second equation to determine the “evolution” of the spectral data.Actually, this philosophy is precisely the one used for solving linear equations.However, it turns out that for solving the boundary value problem (1.1) and (1.2),it is more convenient to study both equations forming the Lax pairsimultaneously.This important insight was gained from the inverse spectral analysis of the linear

MPAG025.tex; 19/04/1999; 16:12; p.4


Equation (1.7). The simultaneous spectral analysis of the Lax pair has led to aunified transform method for solving initial-boundary-value problems for linearand for integrable nonlinear PDE’s [25].

(4) The Riemann–Hilbert (RH) problem (1.8) has the technical difficulty that8(x, y,∞) = g(x, y) is unknown. This difficulty can be bypassed by formulatingan equivalentRH problem for some other sectionally analytic functionalµ (seeEquation (3.16) for the relationship between8,9 andµ). The functionµ satisfiesµ(x, y,∞) = I . Furthermore, it is shown in [27] that the RH problem satisfiedby µ is solvablewithouta small norm assumption ofg1(y) and ofg2(x) providedthat they satisfy a certain symmetry condition. This is a consequence of the factthat there exists atopological vanishing lemmafor this RH problem (see [27] fordetails). We emphasize that the solvability of the RH problem (1.8) is based on theproof presented in [27] of the solvability of the equivalent RH problem forµ.

2. The Lax Pair Representation

PROPOSITION 2.1.Letg(x, y) be a matrix-valued function belonging toC2(R2).

(i) The nonlinear Equation(1.1)is the compatibility condition of Equations(1.3),where9(x, y, κ) is a 2× 2 matrix-valued function belonging toC2(R× C)andκ ∈ C.

(ii) Equation(1.1) is also the compatibility condition of Equations(1.4).(iii) Under the transformation

Gx = (x − y)gxg−1, Gy = (y − x)gyg−1, (2.1)

Equation(1.1)becomes

2(y − x)Gxy +Gy −Gx + [Gx,Gy] = 0. (2.2)

Proof. (i) and (iii). Let9 satisfy

9x + 2κ

κ + x − y9κ =A9

κ + x − y , (2.3a)

9y + 2κ

κ + y − x9κ =B9

κ + y − x , (2.3b)

whereA(x, y) andB(x, y) ∈ C1(R2). It can be verified that the compatibility ofEquations (2.3) yields

Ay = Bx, (2.4a)

(x − y)(Bx +Ay)+A− B + [B,A] = 0. (2.4b)

Indeed, if

D1 := ∂x + 2κ

κ + x − y ∂κ, D2 := ∂y + 2κ

κ + y − x ∂κ,

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it is straightforward to show that

D1(D29) = D2(D19). (2.5)

Then the compatibility of Equations (2.3) yields

D2

(A9

κ + x − y)= D1

(B9

κ + y − x),

which implies Equations (2.4).Integrating Equation (2.4a) it follows thatA = Gx andB = Gy. Then Equa-

tion (2.4b) becomes Equation (2.2). Using Equations (2.1) in Equation (2.2), Equa-tion (1.1) follows. We note that the compatibility condition of Equations (2.1) isEquation (1.1) itself, thus the transformation (2.1) is well-defined.

(ii) It can be verified directly that the compatibility of Equations (1.4) is Equa-tion (1.1). 2

REMARK 2.1. Equation (2.2) is the compatibility condition of

9x = 1

2

(1−

(y − λx − λ

)1/2)Gx9, (2.6a)

and

9y = 1

2

(1−

(x − λy − λ

)1/2)Gy9. (2.6b)

REMARK 2.2. It is straightforward to obtain the Lax pair (1.4) from the Lax pair(1.3): Indeed, one can introduce characteristic coordinates in (1.3) if and only if

∂κ

∂x= 2κ

κ + x − y ,∂κ

∂y= 2κ

κ + y − x . (2.7)

These equations are compatible sinceκxy = 2κ/(κ2 − (x − y)2) = κyx. Theirsolution is

κ2+ 2κ(2λ− (x + y))+ (x − y)2 = 0,

whereλ is a constant. Thus

κ = x + y − λ+ 2(x − λ)1/2(y − λ)1/2. (2.8)

Using this equation it follows that

κ + x − y2

= (x − λ)1/2((x − λ)1/2+ (y − λ)1/2),

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and

κ + y − x2

= (y − λ)1/2((x − λ)1/2+ (y − λ)1/2).Substituting these expressions into the right-hand side of Equations (1.3), and using

x − y = (x − λ)− (y − λ)= (

(x − λ)1/2− (y − λ)1/2)((x − λ)1/2+ (y − λ)1/2),Equations (1.3) become Equations (1.4).

PROPOSITION 2.2.Letg(x, y) satisfy Equation(1.1). Assume that

g ∈ R, gCgC = ρ(x − y)2, (2.9)

whereρ is a real constant, andC is a real nonsingular constant matrix. DefineG(x, y) by

G = iαgC + βfC, α2 = − 1

16ρ, β = − 1

4ρν, ν = constant, (2.10)

wheref (x, y) is defined by

fx = νgCgx

x − y , fy = νgCgy

y − x . (2.11)

ThenG(x, y) solves Equation(1.5).Proof.Equation (2.9) implies

gxCgC + gCgxC = 2ρ(x − y), (2.12)

gyCgC + gCgyC = 2ρ(y − x). (2.13)

Usingg−1 = CgC/ρ(x − y)2, Equation (1.1) becomes

(x − y)(

2gxy − gxCgCgy

ρ(x − y)2 −gyCgCgx

ρ(x − y)2)+ gy − gx = 0.

Multiplying this equation bygC and using Equations (2.12) to replacegCgxC andgCgyC, it follows that

(x − y)(2gCgxy + gyCgx + gxCgy)+ gCgx − gCgy = 0. (2.14)

This equation can be written as(gCgx

x − y)y

=(gCgy

y − x)x

,

which shows thatf is well-defined by Equation (2.11).

MPAG025.tex; 19/04/1999; 16:12; p.7


Figure 2. The cut complexλ-plane used in Theorem 3.1.

SubstitutingG = iαgC+βfC into (1.5) one finds two equations. One of themis

(y − x)gxy + βgxCfy − βgyCfx + βfxCgy − βfyCgx = 0.

Replacingfx andfy (see Equations (2.11)),gCgxC andgCgyC (see Equations(2.12)), andCgC by g−1ρ(x − y)2, this equation becomes (1.1) if and only if4ρβν = −1. The other equation is

β(y − x)fxy − α2gxCgy + α2gyCgx + β2fxCfy − β2fyCfx = 0.

Replacingf, gCgxC andgCgyC, this equation becomes (2.13) if and only if 4α2 =νβ. 2

3. The Spectral Theory of a Boundary Value Problem of the Ernst Equation

We first discuss the linear equation (1.7).

THEOREM 3.1. Let the matrix-valued functionγ (x, y), where−16 x < y 6 1,satisfy Equation(1.7). Letγ (−1, y) andγ (x,1) be given differentiable functionsof y andx respectively. The solution of this boundary value problem is given by

γ (x, y) = γ (−1,1)+ 1

π

∫ x

−1

(1− λ)1/2(x − λ)1/2(y − λ)1/2 γ1(λ)dλ−

− 1

π

∫ 1

y

(λ+ 1)1/2

(λ− x)1/2(λ− y)1/2 γ2(λ)dλ, (3.1)

where

γ1(λ) =∫ λ

−1

ddx γ (x,1)

(λ− x)1/2 dx, γ2(λ) =∫ 1

λ

ddy γ (−1, y)

(y − λ)1/2 dy. (3.2)

MPAG025.tex; 19/04/1999; 16:12; p.8


Proof. Let the function(λ − α)1/2, α ∈ R, be defined with respect to a branchcut from−∞ to α (see Figure 2).

The common solution of Equations (1.12) satisfying9(−1,1, λ) = 0, pos-sesses two different representations,

9 =

1

(λ− x)1/2∫ x

−1

γx ′ dx′

2(λ− x′)1/2−

− (λ+ 1)1/2

(λ− x)1/2(λ− y)1/2∫ 1

y

γy ′(−1, y′)2(λ− y′)1/2 dy′, (3.3a)

− 1

(λ− y)1/2∫ 1

y

γy ′ dy′

2(λ− y′)1/2+

+ (λ− 1)1/2

(λ− x)1/2(λ− y)1/2∫ x

−1

γx ′(x′,1)

2(λ− x′)1/2 dx′. (3.3b)

Using

−16 x′ 6 x < y 6 y′ 6 1,

it follows that:

(i) If λ > 1, the square roots appearing in (3.3) have no jumps, hence9 has nojumps.

(ii) If λ < −1, all the square roots have jumps, which however cancel, and hence9 has no jumps.

(iii) If −1 6 λ 6 x, then(λ − y)1/2, (λ − y′)1/2, (λ − 1)1/2, (λ − x)1/2 havejumps,(λ+ 1)1/2 has no jump, and(λ− x′)1/2 has no jump ifλ > x′ but hasa jump ifλ < x′. Thus, if the superscripts+ and− denote the limit of9 asλapproaches the real axis from above and below respectively, Equation (3.3b)implies

9+ −9− = (1− λ)1/2i(x − λ)1/2(y − λ)1/2

∫ λ

−1

γx ′(x′,1)

(λ− x′)1/2 dx′,

−16 λ 6 x. (3.4)

(iv) Similarly if y 6 λ 6 1, Equation (3.3a) yields

9+ −9− = − (λ+ 1)1/2

i(λ− x)1/2(λ− y)1/2∫ 1

λ

γy ′(−1, y′)(y′ − λ)1/2 dy′,

y 6 λ 6 1. (3.5)

Thus9 is a sectionally holomorphic function ofλ, with jumps only in[−1, x] and[y,1], given by Equations (3.4) and (3.5) respectively. Also Equations (3.3) implythat9 = O( 1

λ) asλ → ∞, λI 6= 0. This information defines a Riemann–Hilbert

problem [26] for9. Its unique solution is given by

9 = − 1

2π

∫ x

−1

(1− λ′)1/2(x − λ′)1/2(y − λ′)1/2 γ1(λ

′)dλ′

λ′ − λ +

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+ 1

2π

∫ 1

y

(λ′ + 1)1/2

(λ′ − x)1/2(λ′ − y)1/2 γ2(λ′)

dλ′

λ′ − λ, λI 6= 0. (3.6)

Equation (3.3) implies that

9 = 1

2λ

[γ (x, y) − γ (−1,1)

]+O

(1

λ2

), λI 6= 0, λ→∞. (3.7)

Using Equation (3.6) to compute the O( 1λ) term of9 and comparing with Equa-

tion (3.7), Equation (3.1) follows.The rigorous justification of the above formalism involves the following steps:

(i) Given γ (−1, y) andγ (x,1) in C1, Equations (3.2) defineγ1(λ) andγ2(λ) inC1.

(ii) Given γ1(λ) and γ2(λ) in C1, defineγ (x, y) by Equation (3.1). Use a di-rect computation to show thatγ (x, y) satisfies Equation (1.7) and the givenboundary conditions. 2

Derivation of Theorem 1.1. We first assumethat g(x, y) exists and show thatg(x, y) can be obtained through the solution of the RH problem (1.8). We then dis-cuss the rigorous justification of this constructionwithout the a priori assumptionof existence.

Let9(x, y, λ) be the unique matrix-valued function defined by

9x = 1

2

(1− (λ− y)

1/2

(λ− x)1/2)gxg−19, (3.8a)

9y = 1

2

(1− (λ− x)

1/2

(λ− y)1/2)gyg−19, (3.8b)

9(−1,1, λ) = I. (3.8c)

9 possesses the two different integral representations

9 = I +

∫ x

−1

(1− (λ− y)1/2

(λ− x′)1/2)a(x′, y)9(x′, y, λ)dx′−

−∫ 1

y

(1− (λ+ 1)1/2

(λ− y′)1/2)b(−1, y′)9(−1, y′, λ)dy′,

−∫ 1

y

(1− (λ− x)1/2

(λ− y′)1/2)b(x, y′)9(x, y′, λ)dy′+

+∫ x

−1

(1− (λ− 1)1/2

(λ− x′)1/2)a(x′,1)9(x′,1, λ)dx′,

(3.9)

wherea(x, y) andb(x, y) are defined by

a(x, y) = 1

2gxg−1, b(x, y) = 1

2gyg−1. (3.10)

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Note that9(x,1, λ) and9(−1, y, λ) satisfy

9(x,1, λ) = I + 1

2

∫ x

−1

(1− (λ− 1)1/2

(λ− x′)1/2)×

×(gxg−1)(x′,1)9(x′,1, λ)dx′ (3.11)

and

9(−1, y, λ) = I − 1

2

∫ 1

y

(1− (λ+ 1)1/2

(λ− y′)1/2)×

×(gyg−1)(−1, y′)9(−1, y′, λ)dy′. (3.12)

Let8(x, y, λ) be the unique matrix-valued function defined by

8x = 1

2

(1+ (λ− y)

1/2

(λ− x)1/2)gxg−18, (3.13a)

8y = 1

2

(1+ (λ− x)

1/2

(λ− y)1/2)gyg−18, (3.13b)

8(−1,1, λ) = I. (3.13c)

8(x, y, λ), 8(x,1, λ) and8(−1, y, λ) satisfy equations analogous to Equations(3.9), (3.11) and (3.12).

We now compute the jumps of9.

(i) λ < −1 orλ > 1.The integral representations (3.9) imply that9+ = 9−. Indeed, ifλ > 1none of the square roots appearing in (3.9) has a jump; ifλ < −1 all of thesquare roots have a jump, which however cancel.

(ii) −16 λ 6 x.Both (λ− y)1/2 and(λ− x)1/2 have a jump, hence(λ− y)1/2/(λ− x)1/2 hasno jump and both9+ and9− satisfy Equations (3.8). Thus

9+(x, y, λ) = 9−(x, y, λ)Gl(λ).

In order to compute the matrixGl(λ) we evaluate this equation atx = λ andy = 1,

Gl(λ) = (9−(λ,1, λ))−19+(λ,1, λ).

LetL±(s, λ) = limz→λ±i09(s,1, z) for −16 s 6 λ. Equation (3.11) yields

L±(s, λ) = I + 1

2

∫ s

−1

(1∓ i (1− λ)

1/2

(λ− s′)1/2)×

×(gxg−1)(s′,1)L±(s′, λ)ds′ (3.14)for −1 6 s 6 λ and9±(λ,1, λ) = L±(λ, λ). We have thus established

(1.10) and the first half of (1.9).

MPAG025.tex; 19/04/1999; 16:12; p.11


(iii) y 6 λ 6 1.Both (λ − y)1/2 and(λ − x)1/2 have no jumps, hence(λ − y)1/2/(λ − x)1/2has no jump and both9+ and9− satisfy Equations (3.8). Thus

9+(x, y, λ) = 9−(x, y, λ)Gr(λ).

In order to compute the matrixGr(λ) we evaluate this equation aty = λ andx = −1,

Gr(λ) = (9−(−1, λ, λ))−19+(−1, λ, λ).

LetR±(t, λ) = limz→λ±i09(−1, t, z) for λ 6 t 6 1. Equation (3.12) yields

R±(t, λ) = I − 1

2

∫ 1

t

(1± i (λ+ 1)1/2

(t ′ − λ)1/2)×

×(gyg−1)(−1, t ′)R±(t ′, λ)dt ′ (3.15)for λ 6 t 6 1 and9±(−1, λ, λ) = R±(λ, λ). We have thus established

(1.11) and the second half of (1.9).(iv) x 6 λ 6 y.

The ratio(λ− y)1/2/(λ− x)1/2 has a jump, thus9+ and8− satisfy the samesystem of integrable equations. Since9+(−1,1, λ) = I = 8−(−1,1, λ),we have9+ = 8−.

The jumps of8 can be computed in a similar way. Also note that for−16 x 6λ or λ 6 y 6 1.

9±(x,1, λ) = 8∓(x,1, λ), 9±(−1, y, λ) = 8∓(−1, y, λ).

Equations (3.9) and the analogous equation for8 imply Equation (1.8c).We now discuss the rigorous justification of the above construction:

(i) Equations (1.10) and (1.11) are Volterra integral equations. Thus ifg1 andg2 ∈ C2, the jump matricesGl andGr are well-defined.

(ii) It can be shown that the RH problem (1.8) has a unique global solution. Thisfollows from the fact that this RH problem is simply related to a RH problemsatisfied by the functionµ(x, y,w) defined by

µ(x, y,w) =8(x, y, f (x, y,w)),

∣∣w − 12

∣∣ 6 12,

9(x, y, f (x, y,w)),∣∣w − 1

2

∣∣ > 12,

(3.16)

whereλ = f (x, y,w) is the rational function defined by(1− 1

w

)2

= λ− yλ− x . (3.17)

The functionµ satisfiesµ(x, y,∞) = I , furthermore it turns out that the RHproblem forµ satisfies a vanishing lemma [27], i.e., the homogeneous RHproblem has only the zero solution, provided thatg andg2 satisfy a certainsymmetry condition.

MPAG025.tex; 19/04/1999; 16:12; p.12


(iii) Using direct differentiation it can be shown that if8 and9 solve the RHproblem (1.8), then

9x = 1

2

[1− (λ− y)

1/2

(λ− x)1/2]α(x, y)9,

9y = 1

2

[1− (λ− x)

1/2

(λ− y)1/2]β(x, y)9,

(3.18)

8x = 1

2

[1+ (λ− y)

1/2

(λ− x)1/2]α(x, y)8,

8y = 1

2

[1+ (λ− x)

1/2

(λ− y)1/2]β(x, y)8,

(3.19)

whereα andβ are someλ-independent functions. Let91 andg be defined by

9(x, y, λ) = I + 91(x, y)

λ+ o

(1

λ

)asλ→∞,

8(x, y, λ) = g(x, y) + o(1) asλ→∞.(3.20)

Then

(91)x = (y − x)4

α(x, y), (92)x = (x − y)4

β(x, y), (3.21)

α(x, y) = gxg−1, β(x, y) = gyg−1. (3.22)

The compatibility condition of Equation (3.21) implies thatg solves the Ernstequation.

(iv) The proof thatg(x, y) satisfiesg(−1, y) = g1(y) andg(x,1) = g2(x) isgiven in [27].

(v) Equation (1.1) is invariant under

g→ gA, g→ g, g→ gT , g→ g−1,

whereA is a nonsingular matrix. Thus without loss of generality we can as-sume thatg(−1,1) = I . Furthermore, ifg1(y) andg2(x) are real, symmetric,positive definite matrices, then the solution also has the same properties.2

Appendix. The Collision of Two Plane Gravitational Waves

The spacetime manifold representing the collision of plane gravitational waves invacuum is characterized by the presence of two spacelike, commuting and hy-persurface orthogonal Killing vector fields. This allows one to write the metricas

ds2 = gab dxa dxb − 2f dudv, a, b = 1,2, (A.1)

MPAG025.tex; 19/04/1999; 16:12; p.13


where the 2× 2 symmetric matrix functiong := (gab) and the scalar functionfdepend only on the null coordinatesu, v, and satisfy the constraints

f (u, v) > 0, det(g(u, v)) > 0. (A.2)

Hence, one can introduce a pair of scalar functionsα and0 such that

detg = α2, α(u, v) > 0 and f = α−1/2e20. (A.3)

Thus, the matrixg can be written as

g = αS, detS = 1, (A.4)

and Equation (A.1) takes the form

ds2 = αSab dxa dxb − 2α−1/2e20 dudv. (A.5)

In this form the four degrees of freedom characterizing the geometry of thespace-time manifold of plane gravitational waves are expressed by the two scalarfunctionsα and0 and the unimodular, symmetric 2×2 matrixS. The two degreesof freedom incorporated in the latter can be expressed by a pair of real valuedfunctionsF andω. Thus,S can be written as

S = F−1

[EE ω

ω 1

], E := F + iω, (A.6)

whereE denotes the complex conjugate ofE.The functionsα,0 andE are determined by solving the Einstein field equations

in the vacuum, namely the systemRij (u, v) = 0, i, j = 1,2,3,4, whereRij isthe Ricci tensor corresponding to Equation (A.5). The components of this systemwhich do not vanish identically yield

αu,v = 0, (A.7)

F(2αEuv + αuEv + αvEu) = 2αEuEv, (A.8)

0u = 1

2

αuu

αu+ α

αu

∣∣∣∣Eu2F

∣∣∣∣2, (A.9a)

0v = 1

2

αvv

αv+ α

αv

∣∣∣∣Ev2F

∣∣∣∣2, (A.9b)

0uv = −Re

(EuEv

4F 2

). (A.10)

The fundamental components of the above system of field equations are Equa-tions (A.7) and (A.8). This follows from the fact that Equations (A.7) and (A.8)

MPAG025.tex; 19/04/1999; 16:12; p.14


Figure 3. The domainW = I ∪ II ∪ III ∪ IV of the (u, v)-plane corresponding to a space-timemanifold which represents the collision of a pair of plane gravitational waves. Region Irepresents the initially flat (gravity free) domain into which the waves propagate. The twoincoming pulses of gravitational radiation are represented by region II and III, respectively.Their interaction is represented by region IV, which corresponds to regionD of Figure 1.

are the integrability conditions of Equations (A.9) and (A.10). Hence, givenα andE,0 can be found by quadrature.

The matrix equation

(αgug−1)v + α(gvg−1)u = 0, (A.11)

called theErnst equation, is equivalent to the system of Equations (A.7) and (A.8).In particular, taking the trace of Equation (A.11) one finds Equation (A.7).

Let us now consider the following adjacent regions of the(u, v)-plane (seeFigure 3), where(u0, v0) is a pair of positive numbers,

I = (u, v) ∈ R2 : u 6 0, v 6 0

,

II = (u, v) ∈ R2 : u 6 0,06 v < v0

,

III = (u, v) ∈ R2 : 06 u < u0, v 6 0

,

IV = (u, v) ∈ R2 : 06 u < u0,06 v < v0, α(u, v) > 0

.

It will be assumed that the metric coefficients are continuous in the domainW :=I ∪ II ∪ III ∪ IV, with α(u, v) > 0 for all (u, v) ∈ W , andα(u, v) = 0 for(u, v) ∈ ∂W . Moreover, the same symbols I–IV will be used in the following forthe corresponding regions of space-time. For example, II denotes the set II× R2,whereR2 represents the extent of the ignorable coordinatesx1 andx2.

Region I represents a domain free of gravity into which a pair of gravitationalwaves impinge from the left and from the right. The latter are represented byregions II and III, respectively. Thus, in region I the line element is given by

ds2I = −2 dudv + (dx1)2+ (dx2)2. (A.12)

MPAG025.tex; 19/04/1999; 16:12; p.15


In region II the metric coefficients depend only onv. They are specified by a givenu-independent solution of the field equations (A.7)–(A.10). Similarly, the metriccoefficients in region III depend only onu, and follow from a givenv-independentsolution of the same equations. By continuity, the given solutions in regions II andIII determine the initial values of the metric coefficients in region IV, i.e., their val-ues along the null hypersurfacesu = 0, 06 v < v0 and 06 u < u0, v = 0. Thus,taking into account the earlier remarks regarding the function0, one can formulatethe problem associated with the process of colliding plane gravitational waves asfollows. Find(α(u, v),E(u, v)) which: (i) satisfy Equations (A.7) and (A.8) in theinterior of region IV, and (ii) take preassigned values along the boundary∂IV of theabove region, where∂IV = (u, v) ∈ R2: u = 0, 06 v < v0∪(u, v) ∈ R2: 06u < u0, v = 0. It is assumed that the boundary data setsα(0, v), α(u,0) andE(0, v),E(u,0) consist of functions which belong to the differentiability classesC2 andC1, respectively.

Following [21], let us introduce the functionsr, s defined by

r(u) := 1− 2α(u,0), 06 u < u0, (A.13a)

s(v) := 2α(0, v)− 1, 06 v < v0. (A.13b)

Then it is easily verified that the unique solution of Equation (A.7) in region IVwhich satisfies the given initial conditions is given by

α(u, v) = 1

2

(s(v)− r(u)). (A.14)

It turns out that the field equations themselves determine a set of junction con-ditions along the null hypersurfacesu = 0 andv = 0. Following [21] theseconditions can be written in the following form

(i)dr

du(u) > 0, for 0< u < u0, (A.15a)

ds

dv(v) < 0, for 0< v < v0, (A.15b)

(ii)dr

du(0) = ds

dv(0) = 0. (A.16)

(iii) The following limits exist

limu→0+

[d2r

du2(u)− 4L(u,0)

2drdu(u)

], whereL := α

∣∣∣∣Eu2F

∣∣∣∣2, (A.17a)

limv→0+

[d2s

dv2(v)− 4K(0, v)

2dsdv (v)

], whereK := α

∣∣∣∣Ev2F

∣∣∣∣2. (A.17b)

MPAG025.tex; 19/04/1999; 16:12; p.16


Conditions (ii) and (iii), calledcolliding wave conditionsby Hauser and Ernst,must be satisfied in order for a solution of the associated boundary-value problemto admit the interpretation of a colliding plane gravitational wave model. Condition(i), on the other hand, allows one to introduce a new pair of null coordinatesx, y

by setting

x = r(u), y = s(v). (A.18)

These equations define a one-to-one, bicontinuous mapping of region IV of the(u, v)-plane onto the triangular regionD = (x, y) ∈ R2: −1 6 x < y 6 1 ofthe(x, y)-plane.

In the new coordinate systemα = 12(y − x), and Equation (A.11) becomes(

1

2(y − x)gxg−1

)y

+(

1

2(y − x)gyg−1

)x

= 0. (A.19)

Thus, the boundary-value problem reduces to solving Equation (A.19), which isequivalent to Equation (1.1), in the interior ofD for specified boundary dataE(−1, y) andE(x,1).

Global aspects of this problem and the singularity structure of the correspondingspace-time manifolds are discussed in [28, 29].

Acknowledgements

The authors wish to thank J. B. Griffiths for valuable discussions, D.T. grate-fully acknowledges the hospitality of the Department of Mathematical Science,Loughborough University of Technology. This research was supported by GrantNo MAJF2 from EPSRC.

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1. Griffiths, J. B.:Colliding Plane Waves in General Relativity, Oxford University Press, 1991.2. Ernst, F. J.:Phys. Rev.168(1968), 1415.3. Belinsky, V. A. and Zakharov, V. E.: Integration of the Einstein equations by means of the

inverse scattering problem technique and construction of exact soliton solutions,Sov. Phys.JETP48 (1978), 985–994.

4. Lax, P. D.: Integrals of nonlinear equations of evolution and solitary waves,Comm. Pure Appl.Math.21 (1968), 467–490.

5. Neugebauer, G.:Proc. Workshop on Gravitation, Magneto-Convection and Accretion(ed. B.Schmidt, H. U. Schmidt and H. C. Thomas), MPA/P2, Max-Planck-Institut für Physik undAstrophysik, Garching, Germany38 (1989).

6. Manojlovic, N. and Spence, B.: Integrals of motion in the two-Killing-vector reduction ofgeneral relativity,Nuclear PhysicsB423(1994), 243–259.

7. Rogers, C. and Shadwick, W. F.:Bäcklund Transformations and Their Applications, AcademicPress, 1982.

8. Zakharov, V. E. and Shabat, F. B.: A plan for integrating the nonlinear equations of mathemat-ical physics by the method of the inverse scattering problem. I,Funct. Anal. Appl.8 (1974),

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226–235; Integration of the nonlinear equations of mathematical physics by the method of theinverse scattering problem. II,J. Funct. Anal. Appl.13 (1979), 166–173.

9. Fokas, A. S. and Ablowitz, M. J.: Linearization of the Korteweg de Vries and Painlevé IIequations,Phys. Rev. Lett.47 (1981), 1096–1100.

10. Ablowitz, M. J. and Segur, H.:Solitons and the Inverse Scattering Transform, SIAM, 1981.11. Newell, A. C.,Solitons in Mathematics and Physics, SIAM, 1985.12. Fokas, A. S. and Zakharov, V. E. (eds):Important Developments in Soliton Theory, Springer-

Verlag, 1993.13. Nutku, Y. and Halil, M.:Phys. Rev. Lett.39 (1977), 1379.14. Chandrasekhar, S. and Xanthopoulos, B. C.: The effect of sources on horizons that may develop

when plane gravitational waves collide,Proc. Roy. Soc. A414(1987), 1–30.15. Ferrari, V., Ibanez, I. and Bruni, M.: Colliding gravitational waves with non-collinear po-

larization: a class of soliton solutions,Phys. Lett.A122 (1987), 459–462; Colliding planegravitational waves: a class of nondiagonal soliton solutions,Phys. Rev. D.36 (1987),1053–1064.

16. Ernst, F. J., Garcia-Diaz, A. and Hauser, I.: Colliding gravitational plane waves with non-collinear polarization. III,J. Math. Phys.29 (1988), 681–689.

17. Tsoubelis, D. and Wang, A. Z.: Asymmetric collision of gravitational plane waves: a new classof exact solutions,Gen. Rel. Grav.21 (1989), 807–819.

18. Hauser, I. and Ernst. F. J.: Initial value problem for colliding gravitational plane waves. III,J.Math. Phys.31 (1990), 871–881.

19. Hauser, I. and Ernst. F. J.: Initial value problem for colliding gravitational plane waves. IV,J.Math. Phys.32 (1991), 198–209.

20. Szekeres, P.: Colliding plane gravitational waves,J. Math. Phys.13 (1972), 286–294.21. Hauser, I. and Ernst. F. J.: Initial value problem for colliding gravitational plane waves. I,J.

Math. Phys.30 (1989), 872–887.22. Hauser, I. and Ernst. F. J.: Initial value problem for colliding gravitational plane waves. II,J.

Math. Phys.30 (1989), 2322–2336.23. Yurtsever, U., Structure of the singularities produced by colliding plane waves,Phys. Rev. D38

(1988), 1706–1730.24. Fokas, A. S. and Gel’fand. I. M.: Integrability of linear and nonlinear evolution equations and

the associated nonlinear Fourier transforms,Lett. Math. Phys.32 (1994), 189–210.25. Fokas, A. S.: A unified transform method for solving linear and certain nonlinear PDEs,Proc.

R. Soc. Lond. A453(1997), 1411–1443.26. Ablowitz, M. J. and Fokas, A. S.:Complex Variables with Applications, Cambridge University

Press, 1997.27. Fokas, A. S. and Sung, L.-Y.: Preprint, 1999.28. Penrose, R.: A remarkable property of plane waves in general relativity,Rev. Modern Phys.37

(1965), 215–220. The geometry of impulsive gravitational waves, inGeneral Relativity: Papersin honor of J. L. Synge(ed. L. O’Raifeartaigh), Oxford University Press (1972), 101.

29. Yurtsever, U.: Colliding almost-plane gravitational waves: colliding plane waves and generalproperties of almost-plane-wave spacetimes,Phys. Rev. D37 (1988), 2803–2817; Singularitiesin the collisions of almost-plane gravitational waves,Phys. Rev. D38 (1988), 1731–1740;Singularities and horizons in the collisions of gravitational waves,Phys. Rev. D40 (1989),329–359.

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331

Product Cocycles and the Approximate Transitivity

VALENTIN YA. GOLODETS and ALEXANDER M. SOKHETInstitute for Low Temperature Physics and Engineering, Academy of Science, 46 Lenin Avenue,310164 Kharkov, Ukraine

(Received: 25 June 1997; accepted: 25 March 1998)

Abstract. Some criteria of the approximate transitivity in the terms of Mackey actions and productcocycles are proved. The Mackey action constructed by an amenable type II or III transformationgroupG and a 1-cocycleρ × α, whereρ is the Radon–Nikodym cocycle whileα is an arbitrary1-cocycle with values in a locally compact separable groupA, is approximately transitive (AT) ifand only if the pair(G, (ρ, α)) is weakly equivalent to a product odometer supplied with a productcocycle. Besides, in the case when the given AT action from the very beginning was a range of a typeII action and a nontransient cocycle, then this cocycle turns out to be cohomologous to aθ-productcocycle. An example is constructed that shows that it is necessary to consider the double Mackeyactions since they can not be reduced to the single ones.

Mathematics Subject Classifications (1991):Primary 46L55; Secondary 28D15, 28D99.

Key words: ergodic theory, approximate transitivity, product cocycle, Mackey action.

Introduction

The class of approximately transitive (AT) actions was introduced by A. Connesand E. J. Woods [3] in connection with the characterization problem for the factorswhich are infinite tensor products of type I factors. These actions have turned out tobe very interesting from the ergodic theory point of view. Papers [14, 9, 10, 4, 5, 12]and some others were devoted to studying these actions.

The result proved by A. Connes and E. J. Woods in [3] states that a typeIII 0 hyperfinite factor is ITPFI if and only if its flow of weights is AT. As thesefactors appear as Krieger factors constructed by a product odometer, their resultbeing translated to the measure-theoretic language meant that an amenable er-godic transformation group is orbit equivalent to a product odometer if and onlyif its associated flow is AT. A ‘pure ergodic’ proof of this theorem was found byT. Hamachi [12]. Therefore, all AT flows obtained their exact characterization.

The natural direction to generalize this result was to obtain a characterizationof AT actions of arbitrary groups, not onlyR. To do that, instead of the associatedflow (also called Poincaré flow) one needs to consider the associated action (alsocalled the Mackey action) constructed by a given action and its 1-cocycle.

Hence, two natural directions of generalization can arise.


332 VALENTIN YA. GOLODETS AND ALEXANDER M. SOKHET

First, one can consider a type II transformation groupG acting on a Lebesguespace and supply it with a 1-cocycleα ∈ Z1(,G;A) with values in a groupA.One can try to prove that the pair(G, α) is weakly equivalent to a pair consistingof a measure-preserving product odometer and a product cocycle if and only if theassociated action is AT. This situation is referred to below asthe typeII caseforbrevity.

Second, one can consider a type III transformation groupG acting on a Lebesguespace and supply it with a 1-cocycleα with values in a l.c.s. groupA. Constructa Mackey action byG and by the double cocycle(α, ρ), whereρ is the Radon–Nikodym cocycle, and prove that the pair(G, α) (or – which is the same – the pair(G, (α, ρ))) is weakly equivalent to a pair consisting of a product odometer and aproduct cocycle if and only if this double Mackey action is AT. This situation isreferred to below asthe typeIII casefor brevity.

We have to comment here that it becomes natural to consider the double Mackeyactions due to paper [1]. It was shown there that, for the type II case, two pairsare stably weakly equivalent if and only if their Mackey actions are metricallyisomorphic, and for the type III case, that two pairs are weakly equivalent if andonly if their doubleMackey actions are metrically isomorphic. The result provedthere for the case of an Abelian groupA was then generalized in [7] for the case ofany l.c.s. groupA.

In this paper, both the type II case and the type III case are studied. The mainresult, Theorem 4.1 for the type III case and Theorem 5.1 for the type II case,states that a pair(G, α) is weakly equivalent to a product odometer with a productcocycle if and only if the (double – for the case ofG of type III) Mackey action isAT.

Note that an important corollary of our result is that any AT action can be con-structed as an associated action to an action of any prescribed type and its productcocycle.

Section 2 contains two technical criteria of the decomposability of the givenpair (G, α) to an infinite product. The first of these is valid both for type II andIII transformation groupsG, while the second one is a corollary of the first oneapplicable for the type III case. They are quite similar to Propositions 6 and 7 of[12], but here we deal with a more complicated situation than Hamachi did: westudy not only an action but an action and a cocycle together.

Sections 3 and 4 are devoted to the type III case. In Section 3, the countabletransformation groupG is introduced, and some auxiliary technical lemmas areproved. Note that these Lemmas 3.2–3.8 correspond to Lemmas 11–16 of [12] withthe necessary complications. Then, in Section 4, we prove our main Theorem 4.1.The main idea dates back to Hamachi’s proof: it turns out to be possible to makean arbitrary cocycle the same as was made for the Radon–Nikodym cocycle.

Section 5 is devoted to the type II case. The new proof of our main Theorem5.1 is presented here for the first time. Of course, historically the type II case wasstudied earlier than the type III case: this was done in [4] for the case of a discrete

MPAG007.tex; 6/04/1999; 8:14; p.2

PRODUCT COCYCLES AND THE APPROXIMATE TRANSITIVITY 333

groupA, and in [5] and [9] for the general case. The short proof presented belowreplaces all the intricate technical considerations of these papers.

(Here a funny situation arises. Though the type III case seems to be ‘the generalcase’, and the type II case seems a particular case when our measure is invariant,the ‘general’ proof, however, is not valid for the ‘particular’ case of type II. Forexample, the main approximation Lemma 3.2 is invalid for the type II case. Thishappens because it is impossible to use partial transformations in the type II case.It is well known that any two measurable sets are equivalent for any given ergodicaction of type III, but in the case of type II transformations this statement is false,and hence we must present a separate proof for the type II case.)

Section 5 also contains a statement that is valid for the type II case only. Sup-pose that the given AT action was represented as a cocycle range from the verybeginning. The main theorem implies that this cocycle is weakly equivalent to aproduct cocycle, but for the type II case we may sharpen this result and prove thatit is not only weakly equivalent, but even cohomologous to aθ-product cocycle(Theorem 5.7).

Finally, in Section 6, we compare the double Mackey action constructed by(α, ρ) with two single ones constructed byα andρ, respectively. It is easy to seethat when the double Mackey action is AT, the two single ones are also AT. But theconverse statement is false, and we construct an appropriate example.

All our considerations have been taken into account for a more general casethan those of the cocycles with values in Abelian groups, while we always keep inmind the Abelian case as the most simple and natural. The requirements for groupA where the cocycles take their values are formulated in Section 1.4.

The results of this paper are included in the Ph.D. thesis of the second author[21]. There one can find a more detailed comparison of the Abelian case and thenon-Abelian one.

1. Notation and Definitions

1.1. APPROXIMATE TRANSITIVITY

The following notion was introduced by A. Connes and E. J. Woods in [3]:

DEFINITION 1.1. An action of a groupG on a Lebesgue space(,B, µ) iscalledapproximately transitive(AT) if for any ε > 0 and an arbitrary finite familyf1, f2, . . . , fN ∈ L1+(,µ) there exist a single functionf ∈ L1+(,µ), andgj ∈G, and coefficientsλij > 0 (herei = 1, . . . , n; j = 1, . . . , Ni) satisfying theinequality∥∥∥∥∥fi −

Ni∑j=1

λij · f (gjω) · dµ gjdµ

∥∥∥∥∥1

< ε.

There are a lot of reformulations of this definition, and the reader can find themin [3]. As the Radon–Nikodym derivative provides a one-to-one correspondence

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between functions∈ L1+(,µ) and finite measures absolutely continuous withrespect toµ, one can easily transform this definition into the language of approx-imation of a given finite family of measures by a single measure; in fact, this wasthe initial Connes–Woods’ formulation, but this will not be used below.

Connes and Woods proved in [3], in particular, that the AT property impliesergodicity, and a measure preserving AT transformation is of zero entropy. Themain Connes–Woods’ result states that a type III hyperfinite factor is an infinitetensor product of type I factors if and only if its flow of weights is AT, and in thispaper we intend to present a generalization of this result.

In the proof of Theorem 4.1 below we will use the following special refor-mulation of the definition of the AT property. LetA be a l.c.s. group, and considernonsingular joint action(Wa, Fr) of the product groupA×R, wherea ∈ A, r ∈ R.

PROPOSITION 1.1.A nonsingularA×R-action(Wa, Fr) is AT if for anyε > 0and for any given finite familyf1, f2, . . . , fn ∈ L1+(,µ) there exist a functionf ∈ L1+(,µ) and a finite collection ofr(i, j) ∈ R and a(i, j) ∈ A (here1 6j 6 Li) such that∥∥∥∥∥fi(x)−

Li∑j=1

exp(−r(i, j)) · f (Fr(i,j)Wa(i,j)ω) · dµ Fr(i,j)Wa(i,j)

dµ

∥∥∥∥∥1

< ε

for anyi, 16 i 6 n.Proof.This condition differs from the general one only by the following: we see

exp(r(i, j)) instead of arbitraryλi,j . So theif part is obvious.To make theonly if part obvious as well, note that the coefficientsλi,j can be

selected to be rationally commensurable with the relative exp(r(i, j)) without lossof the precision:λ(i, j) = exp(r(i, j)) ·q(i, j), q(i, j) ∈ Q. Then, dividef by theleast common multiple of the denominators of theseq(i, j). Thus we obtain someinteger multiples of exp(r(i, j)) instead ofλi,j . Finally, replace integer multiplesby a sum of equal addends. 2

1.2. COCYCLES, WEAK EQUIVALENCE AND MACKEY ACTIONS

LetG be a transformation group acting on a Lebesgue space(,B, µ), andA be alocally compact separable group. A measurable mappingα: ×G→ A is calleda cocycle(or, more exactly, a 1-cocycle) if it satisfies the following equation:

α(ω, g1g2) = α(ω, g2) · α(g2ω, g1).

Hereg1, g2 ∈ G, ω ∈ . The set of all these cocycles is usually denoted byZ1(,G;A).

Two cocycles, namelyα andβ, are said to becohomologouswhen there existsa Borel functionf : → A such that

β(ω, g) = f −1(ω) · α(ω, g) · f (gω).

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The functionf providing this equality is calledan intertwining function. Obviously

it is not unique. Sometimes we writeβf∼ α, and it means thatf intertwinesα

with β.The most important and natural example of a cocycle is the so-calledRadon–

Nikodym cocycle of the measureµ,

ρ(ω, g) = logdµ g

dµ(ω).

It takes values in the additive groupR.Below we studydouble cocycles.This means that we consider the given action

as being supplied with two cocycles simultaneously:ρ with values inR, andα withvalues inA. In other words, we deal with a single cocycleρ × α with its values inR×A.

When a free countable transformation groupG and a cocycleα are given, it ispossible to extendα onto the whole[G]. Namely, letω ∈ , p ∈ [G] andg ∈ Gbe such thatgω = pω; then we defineα(ω, p) to be equal toα(ω, g).

For a measurable setE ⊂ , 0 < µ(E) < ∞, and any given cocycleα ∈Z1(, [G];A) we define the restrictionαE ∈ Z1(E, [G]E;A) which acts onlyupon the induced transformations.

Now let us define an important notion providing a comparison of pairs consist-ing of an action and a cocycle.

DEFINITION 1.2. Let two pairs(Gi, αi), i = 1,2, be given, whereGi are trans-formation groups acting on Lebesgue spaces(i,Bi , µi), andαi be 1-cocyclesi×Gi → A. We call these pairsweakly equivalentif there exists a mapθ : 1→2, which ensures the orbital equivalence of the automorphism groupsG1 andG2

and is such that the cocycleα2(θω1, θg1θ−1) ∈ Z1(1,G1;A) is cohomologous

to α1(ω1, g1).

(Recall that orbital equivalence means thatθ[G1]θ−1 = [G2] andµ2 θ ∼ µ1.As θg1θ

−1 ∈ [G2], it is possible to computeα2 for it.)Of course, it is necessary but not sufficient for the orbit equivalence that the

countable groupsG1 andG2 are of the same type. However, a simple analogueof this notion provides a possibility of comparing type II1 and type II∞ actions.I.e., let (G, α) be a given pair acting on(,B, µ). Consider the space × Zwith product measure (Z is endowed with the counting measure, of course). Thefull transformation group generated byg′(ω, z) = (gω, z), τ(ω, z) = (ω, z + 1)and supplied with the cocycleα′((ω, z), (g′, τ )) = α(ω, g) is called the trivialexpansionof the given pair. Two pairs are said to bestably weakly equivalentiftheir trivial expansions are weakly equivalent.

In the case when[G1] = [G2] act on the same space and the pairs(G1, α1) and(G2, α2) are weakly equivalent, we shall also say that our cocyclesα1 andα2 areweakly equivalent.

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Note that it is easy to see that the two statements ‘the pairs(Gi, αi) are weaklyequivalent’ and ‘the pairs(Gi, (αi, ρi)) are weakly equivalent’ mean the same.

See [1] for more discussion of the notion of the weak equivalence. A powerfulcriterion of the weak equivalence proved there (for the Abelian case; see also [7]for the general one) will be used several times below. This criterion says that theassociated actions must be isomorphic; let us now define this term.

DEFINITION 1.3. LetG be a transformation group acting on a Lebesgue space(,B, µ), andα a 1-cocycle with values in a groupA endowed with its left Haarmeasure. Consider the product measure space× A. The action

g(ω) = (gω, a · α(ω, g))is calledthe skew productof G andα. Consider also the following action ofA onthe product space (hereb ∈ A):

Vb(ω, a) = (ω, a · b).These two actions commute.This allows us to consider the quotient spaceX of

× A by the partition into the ergodic components of the skew product actiong.We can correctly define the quotient action ofVb onX, and this quotient action isusually calledthe action associated with(G, α) or the Mackey action.

In the case whenA = R andα = ρ, this quotient action is also calledtheassociated flowor the Poincaré flow.

1.3. TOWERS AND AMENABILITY

A single tower ofG is a finite collectionζ = er,s ∈ [G]∗ : r, s ∈ 3 of partialtransformations, together with a finite collectioner : r ∈ 3 of measurable subsetssuch that the setser are disjoint,er,s mapses ontoer ander,s · es,t = er,t . The setser are calledlevelsor floors.

The set⋃r∈3 er is calledthe support ofζ and will be denoted by supp(ζ ). For

anyω ∈ es , the seter,sω : r ∈ 3 is calledthe orbit ofω and will be denoted byOrbζ (ω). WhenE ⊂ es , Orbζ (E)means

⋃ω∈E Orbζ (ω).

A finite union of single towers with disjoint supports is calleda multiple tower.Let

∑ni=1 ζi be a multiple tower withζi = eir,s : r, s ∈ 3i andζ = er,s :

r, s ∈ 3 be a single tower. If

3 = (i, r, ξ) : 16 i 6 n, r ∈ 3i, ξ ∈ 4i,n⋃i=1

supp(ζi) = supp(ζ ), and

eirξ ⊂ er , eirξ,isξ = er,s on eisξ ,

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thenζ is calleda refinement of∑n

i=1 ζi. Wheni = 1, fix an arbitrary floorer of ζ1.Thenζ is called to bea productof ζ1 andζ |er ; we denote it byζ = ζ1⊗ ζ |er .

Let α be a given cocycle andζ be a given tower. Ifα(ω, er,s) is a constant onesfor eachr, s ∈ 3, we say thatζ has constantα-passage values.

In the case of the Radon–Nikodym cocycleρ of any given measureQ equiv-alent tom, suppose thatζ has constantρ-passage values. Then the vectorq =(

dQ·er,sdQ (ω) : r ∈ 3) is called the distribution ofQ relative to ζ . As one can

restoreQ by its distribution vectorq and its restrictionν on es , we sometimeswriteQ = νq.

In the general case of an arbitrary cocycleα with constant passage values, thevectorα(ω, er,s) : r ∈ 3 can be also calledthe distribution ofα relative toζ .The easiest way to construct another cocycle, namelyβ, cohomologous toα andalso with constant passage values, is the following definition of the intertwiningfunctionf :f is equal toeA everywhere outside of supp(ζ );f |es for some fixeds ∈ 3 is equal to any given measurable function, and

f (er,sω) = α−1(ω, er,s) · f (ω) · α(ω, er,s)for anyω ∈ es, r ∈ 3.

Then the cocycle

β(ω, g) = f −1(ω) · α(ω, g) · f (gω)has the same constant passage values asα with respect toζ . (In the Abelian casewrite f (er,sω) = f (ω).)

Now we recall a very important general notion.

DEFINITION 1.4. The transformation groupG is said to beapproximately finite

if there exists a nonsingular transformationT such that[T ] def= [T i : i ∈ Z] =[G].

Obviously a product-odometer is approximately finite sincethe adding machinetransformationalso generates it.

It is well known (see [17] and also [12]) thatG is approximately finite if andonly if for anyε > 0 and any finite collection of partial transformationsg1, g2, . . . ,

gn ∈ [G]m∗ there exists a single towerζ satisfying the following two properties:

(a) Domgi, Im gi ∈m,ε B(ζ ).

HereB(ζ ) is the sub-sigma-algebra generated by all levels ofζ , and∈m,ε meansthat the set on the left-hand side can be approximated by a set from the right-handside up toε in the sense of the measurem of their symmetric difference.

(b) m(ω ∈ Domgi : giω ∈ Orbζ (ω)) > (1− ε)m(Domgi), where 16 i 6 n.

Following T. Hamachi [12], instead of two words ‘approximately finite’ we shalluse below one single word ‘amenable’. (The reader can find the definition of

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amenability in [22], for example, but it will not be used below. A. Connes, J. Feld-man and B. Weiss proved in [2] that the definition of amenability is equivalent tothe requirement to the approximate finiteness for free l.c.s. group actions.)

1.4. PRODUCT ODOMETER, PRODUCT COCYCLES, AND THE REQUIREMENTS

FOR GROUPA

Let n, n ∈ N, be a finite set0,1, . . . , Rn − 1 ⊂ N. Let mn be a probabilitymeasure onn such thatmn(k) > 0, 0 6 k 6 Rn − 1. Take the infinite productspacepr = ∏∞i=1n with the product measurempr = ∏∞i=1mn. The permutationλn acts onn by λn(k) = k + 1 mod(Rn). Theseλn generate a free countabletransformation groupGpr on the spacepr, and this groupGpr is calledthe productodometer.

To define a product cocycle, let us start from the Abelian case.

DEFINITION 1.5. Letαpr be a 1-cocyclepr × Gpr → A with values in anAbelian groupA such thatαpr(ω, λ

ln) (0 6 l < Rn) depends only on thenth

coordinate of the pointω = (ω1, ω2, . . . , ωn, . . .) ∈ pr. A cocycleαpr havingsuch a form is usually calleda product cocycle.

And now the general (non-Abelian) case.Suppose thatA is an arbitrary l.c.s. group, not necessarily Abelian. In this case,

a natural analogue of a product cocycle can be defined in the following way. Letpr,Gpr, λn, etc., be as above.

DEFINITION 1.6. A cocycleαpr ∈ Z1(pr,Gpr;A) is calleda product cocycleif αpr(ω, λ

ln) (0 6 l < Rn) depends only on the 1st, 2nd,. . . , nth coordinates of

the pointω = (ω1, ω2, . . . , ωn, . . .) ∈ pr.

This definition can be easily reformulated in a following form:α: ×G→ A

is a product cocycle if it possesses the following six properties:

(1) for eachj ∈ N, there exists a partitionEjk , 06 k < pj , that is,Ejk1∩Ejk2

= ∅for k1 6= k2, and

⋃pj−1k=0 E

j

k = ;(2) and there exists a type I transformationTj that permutes the setsEjk :

T lj · Ejk = Ejk+l (modpj ), T

pjj = id, so that

(3) T lj · Eik = Eik for all l, if only i 6= j , and(4) the group generated byTj ∞j=1 coincides with[G];(5) dµTj

dµ is equal to a constant on eachEjk ;(6) andα(ω, Tj) is a map from to A measurable with respect to theσ -algebra

generated byEjk : k = 0, . . . , pj − 1; j = 1, . . . , l.One easily sees that conditions (1)–(5) define the same as the product odometer,

while condition (6) requires that the cocycles have constant passage values oncertain towers.

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A l.c.s. groupA where our cocycles take their values will be assumed every-where below to satisfy the following requirements:

(1) A is an amenable l.c.s. group;(2) A contains a countable amenable dense subgroupB (this property holds, for

example, for any solvable Lie group), and(3) the given cocycleα ∈ Z1(,G;A) is such that log1(α(ω, g)) is a coboundary,

where1 stands for the modular function ofA (this property holds triviallywhenA is unimodular).

DEFINITION 1.7. When properties (1)–(3) are satisfied, we shall say that thegroupA is admissible.

Here is a simple and natural generalization of the notion of a product cocyclethat is convenient for the cocycles classification problem.

DEFINITION 1.8. Let(,B,m) be a Lebesgue space, andG an ergodic freecountable transformation group acting of this space. Letα: × G → A be a 1-cocycle. If there exists a measure-preserving orbit equivalence mappingθ : (,m)→ (pr,mpr),m θ = mpr, such thatθ[G]θ−1 = [Gpr] and

α(θ−1ω, g) = αpr(ω, θgθ−1), whereg ∈ [G],

then the cocycleα will be called belowa θ-product cocycle. In fact, it is equivalentto a product cocycleαpr with respect to certain equivalence relations.

2. Two Auxiliary Criterions of the Product Property

2.1. FIRST DECOMPOSITION CRITERION

PROPOSITION 2.1.Let a countable amenable groupG act by ergodic transfor-mations on a Lebesgue space(,B,m). Suppose this action to be supplied witha cocycleα taking values in an amenable groupA. The pair (G, α) is weaklyequivalent to a pair consisting of a product odometer and a product cocycle if andonly if for anyε, θ, σ > 0 and any partial transformationsg1, g2, . . . , gn ∈ [G]m∗there exist a finite measureP ∼ m, a cocycleβ cohomologous toα, a functionfintertwining β with α and a simple towerζ with constant(β, P )-passage valuessuch that:

Domgi, Im gi ∈m,ε B(ζ );m(ω ∈ Domgi ∩ supp(ζ ) : giω ∈ Orbζ (ω)

)> (1− ε) ·m(Domgi);

‖P −m‖supp(ζ )∩Edef=∫

supp(ζ )∩E

∣∣∣∣1− dP

dm

∣∣∣∣dm < ε

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(hereE =⋃i(Domgi ∪ Im gi)); and

m(ω ∈ supp(ζ ) ∩ E : dist(eA, f (ω)) > σ

)< θ ·m(supp(ζ ) ∩ E).

REMARK. We may slightly modify the statement of this theorem by replacing thelast inequality with an estimate like the following one:

m(ω ∈ supp(ζ ) ∩ E: f (ω) 6= eA

)< θ ·m(supp(ζ ) ∩ E).

Proof. With the aid of a standard routine calculation, it is easy to see that thiscondition is necessary. We shall only prove the nontrivial part of this statement,i.e., suppose that the condition written above is valid and prove that(G, α) reallyis weakly equivalent to a product odometer supplied with a product cocycle.

We may suppose thatm() < ∞. SinceG is amenable, there exists a non-singular transformationT such that[T ] = [G]. Take three sequences of positivenumbers, denoted by(εk), (σk), and(θk), such that

∑∞n=1 εn < m(),

∑∞n=1 σn

and∑∞

n=1 θn converge. Form a sequence of sets(An) ⊂ B which is dense inBand such that each set appears in it infinitely often.

We shall prove by an induction argument the existence of a sequence(Ek) ofmeasurable sets, whereE1 = , Ek+1 ⊂ Ek , and a sequence(Qk) of measures onEk, whereQk ∼ m, and a sequence(ζk) of simple towers, and a sequence(γk) ofcocycles cohomologous toα, satisfying the following conditions:

(a) ζk has constant(Qk, γk)-passage values;(b) supp(ζk) = Ek;(c) the setEk is ζk−1-invariant, and the towerζk is a refinement of the towerζk−1

in the sense thatζk = ζk−1|Ek ⊗ ηk;(d) dQk−1er,s

dQk−1(ω) = dQkert,st

dQk(ω), whereω ∈ es ∩ Ek , er,s ∈ ζk−1, ert,st ∈ ζk, and

γk−1(ω, er,s) = γk(ω, ert,st);(e) m(Ek−1 \ Ek) < εk;(f) exp(−εk) < dQk

dQk−1(ω) < exp(εk), whereω ∈ Ek, and there exists a function

fk intertwiningγk−1 with γk such that

m(ω ∈ Ek : dist(eA, fk(ω)) > σk

)< θk ·m(Ek);

(g) Ai ∩ Ek ∈m,εk B(ζk), 16 i 6 k;(h) m(ω ∈ Ek : TEkω ∈ Orbζk (ω)) > (1− εk) ·m(Ek).HereTE means the induced transformation.

We set:A1 = , ζ1 is trivial, Q1 = m, γ1 = α. Now suppose that the setsE1 ⊃ E2 ⊃ · · · ⊃ En, the measuresQ1,Q2, . . . ,Qn, the towersζ1, ζ2, . . . , ζn andthe cocyclesγ1, γ2, . . . , γn have already been constructed, while according to theconstruction

ζi = η1⊗ η2⊗ · · · ⊗ ηi =er1r2···ri ,s1s2···si : r1r2 · · · ri, s1s2 · · · si ∈

i∏j=1

3j

.

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Let us fix r = r1r2 · · · rn ∈ ∏nj=13j . Let ω ∈ En, s = s1s2 · · · sn and t =

t1t2 · · · tn ∈∏nj=13j are such thatω ∈ et andTEnω ∈ es . Then it is possible to find

j = j (ω) ∈ Z such thatTEnω = es,r · T jer · er,tω. Now we apply the assumption ofthis theorem. For anyθ ′, σ ′, ε′ andε′′ > 0, and anyN ∈ Nwe apply it to the partialtransformationsT jer , −N 6 j 6 N , and id|er,s (es∩Ak). This allows us to obtain ameasureQ ∼ m, a cocycleγ ∼ α, and a simple towerηn+1, supp(ηn+1) ⊂ er ,with constant(Q, γ )-passage values, such that

er,s(es ∩ Ak) ∈m,ε′ B(ηn+1), 16 k 6 n+ 1, s ∈n∏j=1

3j ;

m(ω : T jerω ∈ Orbηn+1(ω)

)> (1− ε′′) ·m(er), −N 6 j 6 N;

m(er \ supp(ηn+1)

)< ε′;

exp(−ε′) < dQ

dQn

(ω) < exp(ε′), ω ∈ supp(ηn+1).

Besides, there exists a functionϕn intertwining γn with γ and satisfying thecondition

m(ω ∈ supp(ηn+1) : dist(eA, ϕk(ω) > σ

′) < θ ′ ·m(supp(ηn+1)).

Now we put the setEn+1 = Orbζn(supp(ηn+1)), and construct the product towerζn+1 = ζn|En+1 ⊗ ηn+1. The finite measureQn+1 onEn+1 will be defined asQq,whereq = (dQnes,r

dQn: s ∈ ∏n

j=13j). The cocycleγn+1 will be defined as follows:set the functionfn+1 which intertwines it withγn to be equal toeA outside ofsupp(ζn), and for anyω ∈ er, s ∈ 3n set

fn+1(es,rω) = f −1n (ω, es,r) · ϕn(ω) · fn(ω, es,r).

Now we have to verify whetherEn+1, Qn+1, ζn+1, γn+1 satisfy the conditions(a)–(h). This can be done rather straightforwardly, whenε′ 6 εn+1, σ

′ 6 σn+1, ε′′

is sufficiently small andN is sufficiently large. For example, let us check the secondpart of condition (f). We see that the values taken byfn+1 on eaches reproduce thevalues taken byϕn on the fixed leveler . Sinceζn+1 has constantQn+1-passagevalues, we obtain:

Qn+1(ω ∈ En+1 : dist(eA, fn+1(ω)) > σ

′)= card(ζn) ·Qn+1

(ω ∈ supp(ηn+1) : dist(eA, ϕn(ω)) > σ

′).Hence,

m(ω ∈ En+1 : dist(eA, fn+1(ω)) > σ

′)< card(ζn) ·Const(Qn+1) · θ ′ ·m(supp(ηn+1)).

As the choice ofθ ′ is to hand, the checking condition is valid.

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We see thatE = ⋂∞k=1Ek has a positive measure. It easy to show that, accordingto the Poincaré return lemma, for almost everyω ∈ E and for all but a finite numberof k ∈ N we haveTEω = TEkω. Applying Borel–Cantelly’s lemma to the setswhere the condition (h) is false, we obtain that for almost everyω ∈ E and for allbut a finite number ofk ∈ N,

TEω = TEkω ∈ Orbζk (ω).

Following the first part of condition (f) we may define a functionF : E → R+by

F(ω) =∞∏k=2

dQk

dQk−1(ω),

and a new measureµ ∼ m onE by

dµ(ω) = F(ω)dm(ω)∫EF(ω)dm

.

Similarly, by the second part of condition (f) we may define a function8: E →A by

8(ω) =∞∏k=1

fk(ω).

This product converges in measurem (while the product definingF convergesalmost everywhere). There exists a subsequence of partial products convergingalmost everywhere and giving a pointwise definition of8(ω). This function allowsus to construct a new cocycleβ which is cohomologous toα and is intertwined by8 with it.

And now it is easy to check that the transformationTE acting on the space(E,µ) and supplied with the cocycleβ satisfies conditions (1)–(6) of the reformu-lation of the Definition 1.6; hence,β is a product cocycle. 2REMARK. The pair(G, ρ) (whereρ is the Radon–Nikodym cocycle of the mea-surem), of course, by the same proof, has turned out to be weakly equivalent tothe pair consisting of the constructed product odometer and the (product) Radon–Nikodym cocycle of the product measure

∏∞n=1 νn.

2.2. SECOND DECOMPOSITION CRITERION

PROPOSITION 2.2. Let G be a typeIII countable ergodic amenable group ofnonsingular transformations on(,B,m), supplied with a cocycleα with valuesin an amenable groupA. The given pair(G, α) is weakly equivalent to the pairconsisting of a product odometer and a product cocycle, if the following conditionis valid:

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for any finite measureP equivalent tom, for any cocycleβ equivalent toα, andfor any multiple tower

∑ni=1 ζi with constant(P, β)-passage values, for anyε > 0

andσ > 0 there exist a finite measureQ ∼ m, a cocycleγ ∼ α, a simple towerξ with constant(Q, γ )- passage values being a refinement of the given multiple

tower, and a functionf , γf∼ β, so that

‖P −Q‖⋃ni=1 supp(ζi)

def=∫⋃ni=1 supp(ζi)

∣∣∣∣1− dP

dQ

∣∣∣∣ dQ < ε,

m

(ω ∈

n⋃i=1

supp(ζi) : dist(eA, f (ω)) > σ

)< ε ·m

(n⋃i=1

supp(ζi)

).

REMARK. The condition formulated here is not only sufficient but also necessary.This will be shown later (see Corollary 4.3).

Proof. One must check the conditions of the previous criterion. Letε > 0, g1,

. . . , gn ∈ [G]m∗ . SinceG is amenable, there exists a single towerζ = er,s : r, s ∈3 such that(16 i 6 n)

Domgi, Im gi ∈m,ε B(ζ ),

m(ω ∈ Domgi ∩ supp(ζ ) : giω ∈ Orbζ (ω)

)> (1− ε) ·m(Domgi).

Take an arbitrary leveler ∈ ζ and divide it into a finite number of disjoint setsAj , 0 6 j 6 N , such that for almost everyω ∈ Aj ⊂ er , 1 6 j 6 N , and anys ∈ 3,

cs,j exp(−ε) < dm · es,rdm

(ω) < cs,j exp(ε);dist(as,j , α(ω, es,r)) < ε,

m(Orbζ (A0)) < ε.

Herecs,j ∈ R, as,j ∈ A, cr,j = 1, ar,j = eA. Define a new measureP byP(es,rE) = cs,jm(E), whenE ⊂ Aj , andP(es,rE) = m(E), whenE ⊂ A0. Wesee thatP is equivalent tom.

Define a new cocycleβ by βh∼ α, whereh|er ≡ eA, and forω ∈ Aj we put:

h(es,rω) = α(ω, es,r)−1 · as,j , when 16 j 6 N,h(es,rω) = eA, whenω ∈ A0.

Restrictζ to Orbζ (Aj) and denote the restrictions byζj , 1 6 j 6 N . Wesee that the multiple tower

∑Nj=1 ζj has constant(P, β)-passage values. Note that

‖P − m‖ is small due to the definition ofP , and the set of those points wheredist(h(ω), eA) > ε is contained in Orbζ (A0) and has a small measure.

Now we apply the given condition to our criterion and see that there exist ameasureQ equivalent tom andP , a cocycleγ cohomologous toα andβ, and a

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single towerξ refining∑N

i=1 ζi with constant(Q, γ )-passage values satisfying theestimates written above. This implies that the condition of the previous criterion istrue. 2REMARK. See remark after the proof of Criterion 2.1.

3. Type III Case – Auxiliary Results

3.1. THE DOUBLE MACKEY ACTION

Let our amenable countable groupG act freely on a Lebesgue space(,B,m),and assume this action to be supplied with the Radon–Nikodym cocycleρ and withone more cocycleα with its values in an admissible groupA. The pair(ρ, α) willbe considered as adouble cocycle. Define the product space × A × R with thefollowing measure:

dν(ω, a, u) = dm(ω) · da · exp(u)du;hereω ∈ , a ∈ A, u ∈ R.

The natural projection maps from(,B,m) onto,A andR will be denotedby π, πA andπR, respectively.

The triple(ω, a, u) will sometimes be denoted byz.Introduce theskew actionof the groupG on this space:

g(ω, a, u) =(gω, a · α(ω, g), u+ log

dm gdm

(ω)

).

Also, consider the following actions ofA andR on the product space:

Tt(ω, a, u) = (ω, a, u+ t), t ∈ R,Vb(ω, a, u) = (ω, ba, u), b ∈ A.

We see thatg andVb preserve the measureν, whileTt does not. These three actionsare permutable.

Let G = g : g ∈ G. Consider the quotient spaceX of × A × R by theσ -algebra of allg-invariant sets. Letπ be the natural projection from×A×R ontoX. Take an arbitraryσ -finite measureµ onX which is equivalent withν0 π−1,whereν0 is any finite measure equivalent toν. Then, we can write the followingdecomposition:∫

×A×Rk(ω, a, u)dν(ω, a, u)

=∫X

dµ(x) ·∫π(ω,a,u)=x

k(ω, a, u)dν(ω, a, u | x)

MPAG007.tex; 6/04/1999; 8:14; p.14


for anyk ∈ L1(×A×R; ν), where dν(ω, a, u | x) denote the uniquely-definedσ -finite nonatomicG-invariant measures, satisfying

ν((ω, a, u) ∈ ×A×R : π(ω, a, u) 6= x | x) = 0

for almost everyx ∈ X.

DEFINITION 3.1. Consider the quotient actions

Ft(π(ω, a, u)) = π(Tt(ω, a, u)) and

Wb(π(ω, a, u)) = π(Vb(ω, a, u))

onX of R andA, respectively. The joint action(Ft ,Wb) will be calledthe doubleMackey action.

DEFINITION 3.2. Let1 be a countable dense subgroup ofR, andB a countabledense subgroup ofA. Define the following countable nonsingular transformationgroupG on (× A× R,B×B(A)×B(R), ν):

G = g · Tδ · Vb : g ∈ G, δ ∈ 1, b ∈ B.It is easy to see that ifG is amenable, countable and of type III, thenG is orbit

equivalent withG. This is not sufficient for our purposes, and we shall now provethe following:

PROPOSITION 3.1.The pair(G, (ρ, α)) is weakly equivalent to the pair(G, (ρ1,

α1)), whereρ1 andα1 are the following cocycles:

ρ1(ω, a, u; g, Vb, Tt ) = −t;α1(ω, a, u; g, Vb, Tt ) = b−1.

Recall that(ρ, α) ∈ Z1(,G;R×A). Due to the latter definitionρ1 ∈ Z1(×A× R, G× B ×1;R) andα1 ∈ Z1(×A×R, G× B ×1;A).

Proof. Note that in the Abelian case the cocycleρ1 ∈ Z1( × A × R, G ×B × 1;R) defined to be equal to(−t) is exactly the Radon–Nikodym cocycle ofthe joint action(g, Tt , Vb). This allowed us to apply the weak equivalence theoremproved in [1] to the Abelian group case, and, hence, to prove our statement, weonly have to check that the Mackey action constructed by the pair(G, (ρ, α)) isisomorphic to the Mackey action constructed by the pair(G, (ρ1, α1)).

The cited theorem of [1] was generalized by Golodets and Sinelshchikov [7]to state that ifG1,G2 are free countable amenable transformation groups suppliedwith cocyclesα1, α2 with values in a l.c.s. groupA, andρ1, ρ2 are the Radon–Nikodym cocycles ofG1,G2, then the double Mackey actions associated with(Gi, (αi, ρi)) are isomorphic if and only if the pairs(Gi, αi) are weakly equivalent.In the case of a unimodular groupA we can use this result directly in the proof of

MPAG007.tex; 6/04/1999; 8:14; p.15


Proposition 3.1 when comparing(G, α) and(G, α1), but in the general caseρ1 isnot equal to the Radon–Nikodym cocycle of the actionG, and the result of [7]is inapplicable. However, when log1(α(ω, g)) is a coboundary, one can easilychange the measure on so that this case is reduced to the unimodular one. Thisexplains the admissibility conditions for groupA that ensure the existence of acountable amenableB together with the possibility to apply the result of [7].

To prove our statement, we only have to check whether the Mackey action con-structed by the pair(G, (ρ, α)) is isomorphic to the Mackey action constructed bythe pair(G, (ρ1, α1)). To construct the latter Mackey action, we write the followingfive actions being permutable one with one (belowg, Vb, Tt stand for the skewproduct constructed byG and(ρ1, α1), whileω ∈ , g ∈ G, a, a1, a2 ∈ A, b ∈B ⊂ A, u, u1, u2 ∈ R, t ∈ 1 ⊂ R).

(a) g(ω, a, u, a1, u1) = (gω, aα(ω, g), u+ ρ(ω, g), a1, u1);

(b) Vb(ω, a, u, a1, u1) = (ω, ba, u, a1 · b−1, u1);

(c) Tt (ω, a, u, a1, u1) = (ω, a, u+ t, a1, u1 − t);(d) Va2(ω, a, u, a1, u1) = (ω, a, u, a2 · a1, u1);

(e) Tu2(ω, a, u, a1, u1) = (ω, a, u, a1, u1+ u2).

According to the definition, to construct the Mackey action in this case, wehave to find the quotient space of × A × R × A × R by theσ -algebra of all(g, Vb, Tt )-invariant sets and then consider the quotient action of (d) and (e) in thisspace.

Note that the conditiona · a1 = Const, for any given constant value belongingtoA, picks out an invariant subset for actionVb, andVb acts ergodically inside eachof these subsets because of the density ofB in A.

Similarly, the conditionu+u1 = Const, for any given constant value belongingto R, picks out an invariant subset for the actionTt , andTt acts ergodically insideeach of these subsets because of the density of1 in R.

This allows us to define the quotient space whose elements have the form(ω, a,

u), wherea = a · a1 = Const∈ A andu = u+ u1 = Const∈ R, together with thequotient actions

(a′) g(ω, a, u) = (gω, a · α(ω, g), u+ ρ(ω, g));(d′) Va2(ω, a, u) = (ω, a2 · a, u);(e′) Tu2(ω, a, u) = (ω, a, u+ u2).

We see that this space can indeed be identified with the quotient space of ×A×R×A×R by the ergodic components of the actions (b) and (c), and the actions(a′), (d′) and(e′) are exactly the quotient actions of (a), (d), (e), respectively. Butnow we only have to note that the definition of the quotient action of(d′) and(e′)by the ergodic components of(a′) coincides with the construction ofFt andWb

verbatim. 2

MPAG007.tex; 6/04/1999; 8:14; p.16


REMARK. The same argument implies, in particular, that the transformation groupsG andG are orbit equivalent and the pairs(G, α) and(G, α1) are weakly equiva-lent.

REMARK. An observation similar to this proposition was first made in [11].

3.2. THE MAIN APPROXIMATION LEMMA

DEFINITION 3.3. Leth ∈ [G]ν∗, andE ⊂ Domh be a measurable set.fh(x) andfE(x) will be nonnegative integrable functions∈ L1(X,µ) such that

fh(x) = ν(Im h | x);fE(x) = ν(E | x).

ObviouslyfE = fid|E , fh = fIm h, ‖fE‖1 = ν(E).

LEMMA 3.2. Let ε > 0, E be a measurable subset of × A × R, andf ∈L1(X,µ)+ such that‖f − fE‖1 < ε. Then there exists a measurable setE1 ⊂×A×R and a maph ∈ [G]ν∗ fromE ontoE1 such thatfE1 = f ,

‖ν(h·)− ν(· ∩ E)‖ 6 ‖f − fE‖ + ε < 2ε, and

ν(z ∈ E : α1(z, h) 6= eA

)< 2ε.

Note that the cocycleα1 is defined in Proposition 3.1.

REMARK. Our proof almost reproduces the proof of Lemma 13 in [12], (whichwas presented there for a simpler case), but the basic idea of this proof dates backto Lemmas 5.9 and 6.4 in [3].

Proof.Decompose the spaceX into three disjoint subspacesX−,X0,X+ in thefollowing way:

X− = x ∈ X: f (x) < fE(x),X0 = x ∈ X: f (x) = fE(x),X+ = x ∈ X: f (x) > fE(x).

Whenµ(X−) = µ(X+) = 0 we may seth = id, E1 = E.Sinceν(· | x) is an infiniteσ -finite measure, it is possible to find a measurable

subsetE0 ⊂ × A× R such that

ν(E0 | x) = f (x) for almost everyx,

E0 ∩ π−1(X−) ⊂ E ∩ π−1(X−),E0 ∩ π−1(X0) = E ∩ π−1(X0),

E0 ∩ π−1(X+) ⊃ E ∩ π−1(X+).

MPAG007.tex; 6/04/1999; 8:14; p.17


Case(i). Letµ(X−) > 0. In this case we can find some measurable setsE′0 ⊂ EandF ⊂ × A× R such that

E′0 ∩ π−1(X+ ∪X0) = E0 ∩ π−1(X+ ∪X0),

ν(E′0 | x) < f (x)(= ν(E0 | x)),F ∩ E = ∅,ν(F | x) = ν(E0 | x)− ν(E′0 | x),‖f (x)− ν(E′0 | x)‖ < ε/2.

SinceG is of type III, we can obtain a partial transformationh ∈ [G]ν∗ suchthat Domh = E \ E′0, Imh = F . Then, extendh to the whole setE by settingh|E′0 = id. Now we see thathmapsE ontoE′0∪F . DenoteE′0∪F byE1. Obviously,fh = fE1 = f andν(z ∈ E : h·z 6= z) 6 ν(E\E′0) 6 ν(E\E0)+ν(E0\E′0) < 3

2ε.Then

‖ν(h·)− ν(id|E·)‖ = ‖ν(h|(E\E′0)∩π−1(X−)·)− ν((E \ E′0) ∩ π−1(X−) ∩ ·)‖6 ν(h((E \ E′0) ∩ π−1(X−)))+ ν((E \ E′0) ∩ π−1(X−))

6∫X−ν(F | x)dµ(x) + 3

2ε 6 ε

2+ 3

2ε = 2ε.

Case(ii). Let µ(X+) > 0. Similarly to the case considered above we canconstruct some measurable setsE′0 ⊂ E, andF ⊂ ×A×R such that

E′0 ∩ π−1(X− ∪X0) = E0 ∩ π−1(X− ∪X0),

ν(E′0 | x) < fE(x) for almost everyx ∈ X+,

F ∩ E0 = ∅,ν(F | x) = ν(E | x)− ν(E′0 | x),‖ν(F | x)‖1 < ε/2.

SinceG is of type III, we can obtain a partial transformationh ∈ [G]ν∗ such thatDomh = E \E′0, Imh = (E0 \E)∪F . Note that the last union is disjoint. Defineh|E′0 = id. Then we obtain a maph which transformsE ontoE1 = E′0∪F ∪ (E0 \E); hereE1 is represented as a disjoint union. Thus, we see that

fh = fE′0 + fF + fE0 − fE = fE0 = f,the set of those points whereh differs from id is contained inE \ E0 and hence itsmeasure is less thanε/2, and

‖ν(h·)− ν(id|E·)‖ = ‖ν(h|(E\E′0)∩π−1(X+)·)− ν((E \ E′0) ∩ π−1(X+) ∩ ·)‖6 ν(h((E \ E′0) ∩ π−1(X+)))+ ν((E \ E′0) ∩ π−1(X+))

6 ν(E0 \ E)+ ν(F )+ ε2< ε + ε

2+ ε

2= 2ε.

MPAG007.tex; 6/04/1999; 8:14; p.18


3.3. PROPERTIES OF THE TRANSFORMATION GROUPg.

DEFINITION 3.4. Two setsE andE′ ⊂ B are called to beG-Hopf equivalentifthere exists a partial transformationh ∈ [G]∗ such that Domh = E, Imh = E′.LEMMA 3.3. The correspondenceE 7→ fE betweenG-Hopf equivalence classesin B and functions fromL1+(X,µ) is bijective and additive and, moreover,‖fE −fE′‖ 6 ν(E 4E′).

Proof. It is clear that forE,E′ ∈ B such thatν(E) < ∞ andν(E′) < ∞,E andE′ areG-Hopf equivalent if and only ifν(E | x) = ν(E′ | x) for almosteveryx. Since eachν(· | x) is an infinite andσ -finite measure, the mapE 7→fE ∈ L1+(X,µ) is onto. The additivity, i.e. thatfE∪F = fE + fF whenE andF are disjoint, is trivial. The estimate‖fE − fE′‖ 6 ν(E 4 E′) follows from thedefinition ofν(· | x). 2LEMMA 3.4. For anyδ ∈ 1, b ∈ B, h ∈ [G]ν∗, f ∈ L+1 (X,µ) we have:

fT −1δ ·V−1

b ·h(x) = exp(−δ) · fh(FδWbx) · dµ FδWb

dµ(x).

Proof.Let g(x) be some function∈ L∞(X,µ). Then∫X

g(x) · fT −1δ ·V−1

b ·h(x)dµ(x)

(according to the definition of our measures)

=∫X

g(x)dµ(x) ·∫π(ω,a,u)=x

χT −1δ V−1

b (Im h)(ω, a, u)dν(ω, a, u | x)(due to the construction ofν)

=∫

dm(ω) ·∫A×R

g(π(ω, a, u)) · χIm h(ω, ab, u+ δ) · exp(u)duda

(changing variables in×A× R)

=∫

dm(ω)∫A×R

g(π(ω, ab−1, u− δ))χImh(ω, a, u)exp(u)duda exp(−δ)(according to the definition of our measures)

= exp(−δ)∫X

g(F−δW−1b x)dµ(x)

∫π(ω,a,u)=x

χIm h(ω, a, u)dν(ω, a, u | x)

=∫X

exp(−δ) · g(x) · dµ FδWb

dµ· fh(FδWb · x) · dµ(x).

LEMMA 3.5. Let h ∈ [G]ν∗ and fi ∈ L1+(X,µ) (1 6 i 6 N) be such thatfh =∑N

i=1 fi. Then there exist partial transformationshi ∈ [G]ν∗ such that

Domh =N⋃i=1

Domhi (disjoint union),

fi = fhi ,

MPAG007.tex; 6/04/1999; 8:14; p.19


ν(h·) =N∑i=1

ν(h·), and

z ∈ Domh: α1(z, h) 6= eA =N⋃i=1

z ∈ Domhi: α1(z, hi) 6= eA.

Proof.Decompose the set Imh into a finite number of disjoint measurable setsEi (16 i 6 N) such that

ν(Ei | x) = fi(x) for almost everyx ∈ X.

Define partial transformationshi ∈ [G]ν∗ by

hi(ω, a, u) = h(ω, a, u), where(ω, a, u) ∈ h−1(Ei).

Then it is easy to check that thesehi satisfy the desired conditions. 2Let B denote the transformation groupVb : b ∈ B.

LEMMA 3.6. [G× B] = h ∈ [G]∗ : ν(h·) = ν(·).Proof. We only have to prove that ifh ∈ [G]∗ is ν-preserving thenh ∈ [G ×

B]. Since for almost every(ω, a, u) ∈ Domh we see thath(ω, a, u) = g ·VbTδ(ω, a, u) for someg ∈ G, b ∈ B ⊂ A, and δ ∈ 1 ⊂ R, and sincedνg·Vb·Tδ

dν = exp(δ), the conditionν(h·) = ν(·) impliesδ = 0. 2LEMMA 3.7. [G] coincides with the set of allh ∈ [G]∗ possessing the followingproperties:

ν(h·) = ν(·) and

for almost every pointz ∈ × A × R there existsg ∈ G such thatπ(h · z) =gπ(z) andπA(h · z) = α(π(z), g) · πA(z).

Proof. It is evident thath = g possesses these properties. Conversely, the con-dition ν(h·) = ν(·), according to the previous lemma, implies thath is (locally)equal tog · Vb. The second condition is valid only whenb = eA. 2LEMMA 3.8. Leth, h′ ∈ [G]ν∗. The following statements are equivalent:

(a) there existsv ∈ [G]ν∗ such thatDomv = Domh′, Im v = Domh, ν(hv·) = ν(h′·),

and for any(ω, a, u) ∈ Domh′ there exists someg ∈ G such thatgπ(ω, a, u) = gω = π(hvh′−1(ω, a, u)) and

πA(hvh′−1(ω, a, u)) = α(ω, g);

(b) the setsImh and Imh′ are G-Hopf equivalent;

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(c) fh = fh′ .Proof. The equivalence between (a) and (b) follows from Lemma 3.7; we only

need to note thatv = h−1gh′, g = hvh′−1. The equivalence between (b) and (c)was proved in Lemma 3.3. 2

4. Type III Case – Proof of the Main Theorem

4.1. FORMULATION AND COMMENTARY

THEOREM 4.1. LetG be a typeIII amenable ergodic countable group of nonsin-gular transformations on(,B, µ), and letα be a1-cocycle for this action withvalues in an admissible groupA, and ρ the Radon–Nikodym cocycle. The pair(G, (ρ, α)) is weakly equivalent to a pair consisting of a product odometer and aproduct cocycle if and only if the double Mackey action(Ft ,Wb) is AT.

Proof of sufficiency.The if part of this theorem is rather well known. It followsfrom the fact that transitive actions are approximately transitive (for Abelian groupactions it was shown in [8] and independently in [15]; for the general case, it wasnoted in [14]; see the complete proof in [10]?) and from the fact that a quotientaction of an AT action is also AT ([3, Remark 2.4]). Indeed, the definitions of aproduct cocycle and a product odometer imply that the joint action(g, Tt , Vb) istransitive and hence AT by an straightforward computation; but the double Mackeyaction is only its quotient action. 2

A nontrivial part of the theorem is the fact that approximate transitivity of thedouble Mackey action implies that the given pair is weakly equivalent to a productodometer supplied with a product cocycle. It will be proved below.

In view of Proposition 3.1, instead of the given pair(G, (ρ, α))we may considerthe pair(G, (ρ1, α1)) and try to prove that this pair is weakly equivalent to a pairconsisting of a product odometer and a product cocycle.

To do so, we are going to apply the criterion proved in Proposition 2.2. Thus,letZ =∑n

i=1 ζi be an arbitrary multiple tower forG with constant(P, β)-passage

values, whereP ∼ ν, β h∼ α1. Our purpose is to constructξ, Q, γ as in Proposi-tion 2.2. (See also the remarks after its proof and after the proof of Proposition 2.1.)

4.2. REDUCTION TO A PARTICULAR CASE

Take an arbitrary floorer(i) from eachζi . Consider the setE = ⋃ni=1 er(i) and call

it the base ofZ.

? Even a stronger fact is valid, namely, the transitive actions not only are AT but also have thefunny rank one. See [20].

MPAG007.tex; 6/04/1999; 8:14; p.21


LEMMA 4.2. In the further proof of Theorem4.1we may assume thatP = ν onE and thatβE = (α1)E. This means that when the statement of our theorem turnsout to be true for this particular case, it will imply its validity for the general case.

Proof will be presented in two steps. First, we construct another multiple towerZ′ instead ofZ, together with another measureP ′ and another cocycleβ ′ that willbe used instead ofP andβ, respectively, and that really possess the propertiesP ′ = ν on E andβ ′

E= (α1)E. Second, suppose that the theorem is true for this

particular case. This means that we can find someξ, Q′ andγ ′ as in Proposition2.2 suitable for the triple(Z′, P ′, β ′). The same single towerξ together withQ andγ that we present here will be suitable for the initial case of the triple(Z, P, β).

Step 1.For anyδ > 0 we may decompose the setser(i) into a finite number ofdisjoint setsEij (06 j 6 J ) such that

P(Ei0) < δ,

ν(Ei0) < δ,

cij exp(−δ) < dP

dν(z) < cij exp(δ), 16 j 6 J,

dist(h(z), aij ) < δ/2.

Herecij ∈ R+, aij ∈ A, z ∈ Eij . Taking ζi-invariant sets Orbζi (Eij ) andrestricting the towersζi to these sets we obtain the towersζij , 1 6 j 6 J . Let qibe the distribution vectors ofP relative toζi . Construct the measureP ′ as

∑ni=1 νqi .

Recall that (hereg is any element of[G])β(z, g) = h−1(z) · α1(z, g) · h(gz).

Define a new cocycleβ ′ by setting

β ′(z, g) = h′−1(z) · α1(z, g) · h′(gz),where the functionh′ will be constructed now. Note thatβ andβ ′ are intertwinedby h−1 · h′.

Let h′(z) = eA for z ∈ E. Then the propertyβ ′E= (α1)E is already true. To

obtainβ ′ having constant passage values on eachζi we require that the values ofh−1 · h′ be the same on all the levels ofζi . This leads us to the following:

h′(er,r(i)z) = h(er,r(i)z) · β−1(z, er,r(i)) · h(z)−1 · β(z, er,r(i))for eachz ∈ supp(ζi), 16 i 6 n.

To complete the definition ofh′, put it to be equal toeA everywhere outside of⋃ni=1 supp(ζi).Now we may and do replaceZ = ∑n

i=1 ζi by Z′ = ∑ni=1

∑Jj=1 ζij , P andβ

byP ′ andβ ′ respectively. Indeed,Z′ has constant(P ′, β ′)-passage values, and therequired properties are valid.

MPAG007.tex; 6/04/1999; 8:14; p.22


Step 2.When proving Theorem 4.1 for the described case of

Z′ =n∑i=1

J∑j=1

ζij ,

P ′ andβ ′, we will show the existence of a single towerξ , of a finite measureQ′equivalent toν and of a cocycleγ ′ cohomologous toα1, that satisfy the followingproperties:

(i) ξ refinesZ′ =∑ni=1

∑Jj=1 ζij ,

(ii) ξ has constant(Q′, γ ′)-passage values,(iii) ‖Q′ − P ′‖⋃n

i=1 supp(ζi) < δ′, and

(iv) there existsf ′ intertwiningγ ′ with β ′ such that

ν

(ω ∈

n⋃i=1

supp(ζi) : dist(eA, f′(ω)) > δ/2

)< δ′′ · ν

( n⋃i=1

supp(ζi)).

Now we can construct a (uniquely defined) measureQ together with a (uniquelydefined) cocycleγ in such a way that

(a) Q(E) = cij ·Q′(E), whenE ⊂ Eij , 16 j 6 J ;(b) Q(E) = Q′(E), whenE ⊂ Ei0;(c) the distributions ofQ andQ′ relative to

∑ni=1 ζi coincide;

(d) the functionk(z) which intertwinesγ with γ ′ is equal toaij on eachEij , 16j 6 J , and toeA onEi0;

(e) the distributions ofγ andγ ′ relative to∑n

i=1 ζi coincide.

Then we see thatξ has constant(Q, γ )-passage values, and

‖Q− P‖⋃ni=1 supp(ζi) 6 ‖Q− P‖⋃n

i=1⋃Jj=1 Orbζi (Eij )

+ ‖Q− P‖⋃ni=1 Orbζi (Ei0)

6n∑i=1

J∑j=1

cij exp(δ) · δ′ + 2δ

can be done as small as we need because the choice ofδ, δ′ is to hand.Moreover, note thatf (z) = h−1(z) · h′(z) · f ′(z) · k(z) intertwinesβ with γ ,

and

ν

(z ∈

n⋃i=1

supp(ζi) : dist(f (z), eA) > δ

)

6 δ′′ · ν( n⋃i=1

supp(ζi))+ ν

( n⋃i=1

Orbζi (Ai0))

also can be made as small as we need. 2Note that this lemma implies, in particular, that for any partial transformation

v ∈ [G] such that Domv ⊂ er(1), Im v ⊂ er(i) and forz ∈ er(1) we may regardβ(z, v) as being equal toα1(z, v).

MPAG007.tex; 6/04/1999; 8:14; p.23


4.3. PROOF OF THE MAIN THEOREM

We know that the double Mackey action(Ft ,Wb) (t ∈ R, b ∈ A) is AT. Since foranyf ∈ L1(X,µ) the mappingR×A→ L1(X,µ), (t, b) 7→ f (FtWbx) · dµFtWbdµis continuous and since1 andB are dense subsgroups inR andA, respectively,then the restriction of the double Mackey action onto1×B is also AT. This meansthat for the above chosener(i), 1 6 i 6 n, and anyθ > 0, we can findf ∈L1+(X,µ) and a finite number ofδ(i, l) ∈ 1 ⊂ R, b(i, l) ∈ B ⊂ A, 1 6 l 6Li,16 i 6 n, such that∥∥∥∥∥fer(i) (x)−

Li∑l=1

exp(−δ(i, l)) · f (Fδ(i,l)Wb(i,l)x)dµ Fδ(i,l)Wb(i,l)

dµ(x)

∥∥∥∥∥ < θfor any 16 i 6 n. Here, as above,fer(i) = ν(er(i) | x).

Applying Lemma 4.2n times to each seter(i) and to the function

Ri(x) =Li∑l=1

exp(−δ(i, l)) · f (Fδ(i,l)Wb(i,l)x)dµ Fδ(i,l)Wb(i,l)

dµ(x),

we obtain each time a partial transformationhi ∈ [G]ν∗ such that

Domhi = er(i), Imhi will be denoted byZi,

fhi (x) = Ri(x),‖ν(hi·)− ν(er(i) ∩ ·)‖ < 2θ,

ν(ω ∈ er(i) : α1(ω, hi) 6= eA

)< 2θ, 16 i 6 n.

Applying Lemma 3.5, for eachi, to hi andRi(x), we obtain that there existpartial transformationshli ∈ [G]ν∗ together with the corresponding setsY li =Domhli, Z

li = Imhli so that

er(i) =Li⋃l=1

Y li ,

fhli= fZli = exp(−δ(i, l)) · f (Fδ(i,l)Wb(i,l)x)

dµ Fδ(i,l)Wb(i,l)

dµ(x),

ν(hi·) =Li∑l=1

ν(hli·), and

Li∑l=1

ν(z ∈ Y li : α1(z, h

li) 6= eA

)< 2θ.

Now we can use Lemma 3.4. We see that

fhli(x) = exp(−δ(i, l)) · f (Fδ(i,l)Wb(i,l)x)

dµ Fδ(i,l)Wb(i,l)

dµ(x)

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= exp(−δ(i, l)) · f (Fδ(1,1)Wb(1,1)(Fδ(i,l)−δ(1,1)Wb(1,1)−1b(i,l)x)) ·

· dµ Fδ(1,1)Wb(1,1)

dµ(Fδ(i,l)−δ(1,1)Wb(1,1)−1b(i,l)x) ·

· dµ Fδ(i,l)−δ(1,1) ·Wb(1,1)−1b(i,l)

dµ(x)

= fh11(Fδ(i,l)−δ(1,1)Wb(1,1)−1b(i,l)x) ·

dµ Fδ(i,l)−δ(1,1)Wb(1,1)−1b(i,l)

dµ(x) ·

· exp(−δ(i, l)+ δ(1,1))= fT−1

δ(i,l)−δ(1,1)·V−1b(1,1)−1b(i,l)

·h11(x).

This allows us to apply Lemma 3.8 to the partial transformationshli andT −1δ(i,l)−δ(1,1) · V −1

b(1,1)−1b(i,l)· h1

1 ∈ [G]ν∗ to obtain partial transformationsvli ∈ [G]ν∗such that

Domvli = Domh11 = Y 1

1 ,

Im vli = Domhli = Y li ,ν(hliv

li (·)) = ν(T −1

δ(i,l)−δ(1,1) · V −1b(1,1)−1b(i,l)

· h11(·))

(by the definitions ofT, V andν)

= exp(−δ(i, l)+ δ(1,1)) · ν(h11·),

andhli · vli · (h11)−1 · Vb(1,1)−1b(i,l) · Tδ(i,l)−δ(1,1) ∈ [G].

4.4. CONSTRUCTION OF THE TOWERξ

Let z ∈ es, which is a floor ofζj , and leter(j),sz ∈ Ymj . Define

eirl,jsm(z) = er,r(i) · vli · (vmj )−1 · er(j),s(z).It is easy to see that

eirl,jsm : 16 i, j 6 n, 16 l 6 Li, 16 j 6 Lj, r ∈ 3i, s ∈ 3j form a tower which we denote byξ . The levels ofξ are the setser,r(i)Y li . Obviouslythe simple towerξ makes a refinement of the given multiple tower

∑ni=1 ζi .

4.5. CONSTRUCTION OF THE MEASUREQ

LetE ⊂ Y li . Let us define, for anys ∈ 3i,

Q(es,r(i)E) = P(es)

P (er(i))· ν(hliE).

This means that the values of the Radon–Nikodym cocycle forQ reproduce thecorresponding values of the Radon–Nikodym cocycle forP on eacher,s belonging

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to the initial multiple tower. ObviouslyQ is equivalent toν andP . Since we seethat for anyz ∈ Y li ,

dQ vlidQ

(z) = dν hlivlidν h1

1

(z) = exp(−δ(i, l)+ δ(1,1)),

thenξ has constantQ-passage values.

4.6. CONSTRUCTION OF THE COCYCLEγ

We will search for the appropriateγ in the following form:

γ (ω, a, u;h) = f (ω, a, u) · β(ω, a, u;h) · f −1(h(ω, a, u)).

This ensures thatγ is cohomologous toβ andα.The intertwining functionf can be found in the following form:

f (er,r(i)z) = β−1(z, er,r(i)) · f (z) · β(z, er,r(i)),wherez ∈ er(i) and r ∈ 3i . This guarantees that the initial multiple tower hasconstantγ -passage values.

Let f |Y 11≡ eA, and for anyz ∈ Y 1

1

f (vliz) = α1(z, hlivli)−1 · α1(z, h

11) · β(z, vli ).

In other words, forz ∈ Y 11 we have just defined that

γ (z, vli) = α1(z, h11)−1 · α1(z, h

livli ).

Let us compute this value and prove that it is a constant. Lemma 3.8 allows usto write (locally)

hli · vli · (h11)−1 · Vb(1,1)−1b(i,l) · Tδ(i,l)−δ(1,1) = gli ,

and hence, for anyz2 ∈ V −1b(1,1)−1b(i,l)

· T −1δ(i,l)−δ(1,1) · Y li ,

α1(z2, hli · vli · (h1

1)−1 · Vb(1,1)−1b(i,l) · Tδ(i,l)−δ(1,1)) = eA.

By the definition of a cocycle, we obtain:

α1(z2, Tδ(i,l)−δ(1,1)) · α1(z1, hli · vli · (h1

1)−1 · Vb(1,1)−1b(i,l)) = eA,

wherez1 = Tδ(i,l)−δ(1,1) · z2 ∈ V −1b(1,1)−1b(i,l)

· Y li .As the first multiplier here is also equal toeA, we write:


1)−1 · Vb(1,1)−1b(i,l)) = eA.

Using the definition of a cocycle again, we obtain:

α1(z1, Vb(1,1)−1b(i,l)) · α1(z0, hli · vli · (h1

1)−1) = eA,

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wherez0 = Vb(1,1)−1b(i,l) · z1 ∈ Y li .As the first multiplier here is known due to the construction ofα1, we write:


1)−1) = b(1,1)−1b(i, l) = Const.

But the latter expression can be transformed in the following way:

α1(z0, (h11)−1) · α1((h

11)−1z0, h

li · vli) = b(1,1)−1b(i, l),

and the first multiplier here is evidently equal to

(α1((h11)−1z0, h

11))−1.

As a result of this, forz = (h11)−1z0 ∈ Y 1

1 , we can write that

α1(z, h11)−1 · α1(z, h

livli ) = b(1,1)−1 · b(i, l).

Thus, we have proved thatξ has constantγ -passage values.

4.7. ESTIMATES FOR THE MEASUREQ

We have to estimate‖Q− P‖⋃ni=1 supp(ζi).

As P = ν on eacher(i) due to Lemma 4.2, it suffices to estimate

‖Q− νq‖⋃ni=1 supp(ζi)

(hereq is the distribution vector ofP relative to∑n

i=1 ζi). SinceP andQ havethe same distributions, the required estimate can be obtained as‖Q(er(i) ∩ ·) −ν(er(i) ∩ ·)‖ multiplied by the number of levels of the given multiple tower.

But we see that

‖Q(er(i) ∩ ·)− ν(er(i) ∩ ·)‖

=∥∥∥∥∥

Li∑l=1

ν(hli·)− ν(er(i) ∩ ·)∥∥∥∥∥ = ‖ν(hi·)− ν(er(i) ∩ ·)‖ < 2θ.

Since the choice ofθ is to hand,‖Q − P‖⋃ni=1 supp(ζi ) can be done less than any

givenε > 0.

4.8. ESTIMATES FOR THE COCYCLEγ . A COROLLARY

We have already constructed the functionf which intertwinesβ with γ , and wemust estimate the set of points wheref differs fromeA. Asβ andγ have the samedistributions relative to

∑ni=1 ζi, to do so we have only to estimateν(z ∈ Y 1

1 :f (vliz 6= eA). Then,

f (vliz) = α1(vliz, h

li)−1 · α1(z, v

li )−1 · α1(z, h

11) · β(z, vli ).

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The first and the third multipliers here differ fromeA only inside the setsz ∈Y li : α1(z, h

li) 6= eA whose summary measure is small. Outside these sets the

product of the second and the fourth multipliers is equal toeA due to Lemma 4.2.This completes the proof of Theorem 4.1. 2

COROLLARY 4.3. The sufficient condition for the product property proved inProposition2.2 is also a necessary condition.

Indeed, the double Mackey action for a product odometer and a product cocycleis AT. We have already checked in the proof of Theorem 4.1 that approximatetransitivity implies the required condition of Proposition 2.2. 25. Type II Case

5.1. PROOF OF THE MAIN THEOREM

THEOREM 5.1. LetG be a countable amenable transformation group of typeIIacting on the Lebesgue space(,B,m), andα: × G → A be a1-cocycle ofthis action with values in an admissible groupA. The pair(G, α) is stably weaklyequivalent to the pair consisting of a product odometer and a product cocycle ifand only if the associated action is AT.

Proof.Consider the product space×A. Denote the product measure dm×daby dν. LetG be a transformation group acting on this space, defined as follows:

G = g · Vb : g ∈ G, b ∈ B,whereB is a countable dense subgroup ofA, and the actionsg andVb, as usually,are defined by

g(ω, a) = (g · ω, a · α(ω, g)),Vb(ω, a) = (ω, b · a),

and the corresponding Mackey action will be denoted byWb, as above.Assume this action to be supplied with a cocycleα1: (× A)×G→ A:

α1(ω, a; g · Vb) = b−1.

It is rather easy to see that the pair(G, α) is stably weakly equivalent to thepair (G, α1). The proof can be done in the same manner as in Proposition 3.1, andthe main idea is to use the fact that the Mackey actions associated with these pairsare isomorphic. So we have to prove that the pair(G, α1) is weakly equivalent to aproduct odometer and a product cocycle.

We are going to apply Proposition 2.1. To do that, suppose that a finite collectionof partial transformationsh1, h2, . . . , hn ∈ [G]ν∗ is given. (Of course, they areν-preserving.) Denote Domhi by Ei and Imhi by Fi; Ei andFi ⊂ × A. Our

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purpose is to construct someG-stackζ and some cocycleβ ∼ α1 so thatζ wouldhave constantβ-passage values, and the functionf intertwiningβ with α1 wouldbe close toeA, andEi andFi ∈ν,ε B(ζ ).

To do that, notice from the very beginning that we may suppose that there existai ∈ A such that

ν(ω ∈ Ei : dist(ai, α1(ω, hi)) > σ

)< ε′ν(Ei)

for any pre-givenσ, ε′. (If this is wrong, splitEi into smaller sets.)Recall that(X,µ) is the quotient space of × A by the ergodic components

of g. Let fi(x) = fEi (x) = ν(x | Ei): X → R+. Since the Mackey actionWb on X is approximately transitive, for any givenε > 0 it is possible to findf ∈ L1(X,µ), b(i, l) ∈ B andλi,l ∈ Z+ (herei = 1, . . . , n; l = 1, . . . , Li) suchthat ∥∥∥∥∥fi(x)−

Li∑l=1

λi,l · f (Wb(i,l) · x) · dµ Wb(i,l)

dµ(x)

∥∥∥∥∥1

< ε.

Hence, there exist setsRi ∈ ×A such thatν(Ri 4 Ei) < ε and

fRi (x) =Li∑l=1

λi,l · f (Wb(i,l) · x) · dµ Wb(i,l)

dµ(x).

Note that it is possible to extend the given partial transformationshi so that theywould be defined forEi ∪ Ri and would simultaneously belong to[G]ν∗.

Decompose these setsRi into a disjoint union of setsRi,l so that⋃Lil=1Ri,l = Ri

and

fRi,l (x) = λi,l · f (Wb(i,l) · x) · dµ Wb(i,l)

dµ(x)

and hence

Li∑l=1

fRi,l (x) = fRi (x).

Since we might assume from the very beginning thatλi,l are positive integers,it is also possible to decomposeRi,l into a disjoint union of setsRi,l,j , wherej =1, . . . , λi,l, so that

⋃Lil=1Ri,l,j = Ri,l and

fRi,l,j (x) = f (Wb(i,l) · x) · dµ Wb(i,l)

dµ(x).

LEMMA 5.2. fV−1b E(x) = fE(Wbx) · dµWb

dµ (x) for any measurable subsetE ⊂×G.

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Proof.This can be shown very easily. A more difficult but similar equality wasproved in Lemma 3.4 above.

Applying this lemma, we immediately obtain:

fRi,l,j (x) = f (Wb(i,l) · x) · dµ Wb(i,l)

dµ(x)

= f (Wb(1,1) ·Wb(1,1)−1b(i,l)x) · dµ Wb(1,1)

dµ(Wb(1,1)−1b(i,l)x) ·

· dµ b(1,1)−1b(i, l)

dµ(x)

= f1,1,1(Wb(1,1)−1b(i,l)x) ·dµ Wb(1,1)−1b(i,l)

dµ(x)

= fV−1b(1,1)−1b(i,l)

R1,1,1(x).

Hence, the setsRi,l,j andV −1b(1,1)−1b(i,l)

R1,1,1 not only have equal measures, but alsothe same conditional measures for a.e.x. Therefore, there exist partial transforma-tions vi,l,j such that Domvi,l,j = V −1

b(1,1)−1b(i,l)R1,1,1, Im vi,l,j = Ri,l,j , and these

vi,l,j belong not only to[G]ν∗, but even to[G]ν∗. Note thatα1(·, vi,l,j ) = eA.DenotehiRi,l,j by Si,l,j . The desired stackζ is now ready: it consists of the sets

Ri,l,j andSi,l,j , and the setsRi,l,j are intertwined by the transformationsvi,l,j V −1b(1,1)−1b(i,l)

: R1,1,1→ Ri,l,j . It is evident that this stack provides a good approxi-mation of the initial partial transformationshi in the sense of Proposition 2.1.

Note that forz ∈ R1,1,1 we have

α1(z, vi,l,j V −1

b(1,1)−1b(i,l)R1,1,1

) = b(1,1)−1b(i, l) = Const.

Now we are ready to define the cocycleβ. Let the intertwining functionfbe equal toeA on eachRi. Further, we must have forz ∈ Ri,l,j , that f (hiz) =α−1

1 (z, hi)·β(z, hi), whereβ(z, hi) are not defined yet but must be constants. Sincefor z ∈ Ei the valuesα1(z, hi) are close toai , let for z ∈ Ri the valuesβ(z, hi)be equal to the theseai . This defines functionf completely and correctly togetherwith β, and the stackζ has constantβ-passage values according to the definition.Finally,

ν(z ∈ supp(ζ ) : dist(f (z), eA) > σ

)6 ν

(n⋃i=1

hiRi \ hiEi)+

n∑i=1

ν(z ∈ Ei : dist(f (z), eA) > σ

)6 n · ε + n · ε′ · ν(Ei) 6 n · ε + ε′.

Asn is given, and the choice ofε, ε′ is to hand, the conditions of Proposition 2.1are satisfied. 2

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5.2. A REPRESENTATION OF APPROXIMATELY TRANSITIVE GROUP ACTIONS

AS A PRODUCT COCYCLE RANGE

Now we suppose an AT action to be given. The following is obvious:

COROLLARY 5.3. Each admissible group AT action can be represented as aproduct cocycle range.

Proof. Really, according to [6], each l.c.s. group action can be represented ascocycle range with the base action of any prescribed type. Now this statementfollows directly from Theorem 5.1 (type II case) and Theorem 4.1 (type III case).2

Here we are going to strengthen the result of Corollary 5.3 and to prove thatif the initial AT action of an admissible group was from the very beginning rep-resented as a Mackey action associated with a type II action and a cocycle, it ispossible to choose aθ-product cocycle generating this action to be cohomologousto the initial one. Now we only know that these cocycles are weakly equivalent.

So, we deal with the following situation: a type IIG-action on(,m) suppliedwith cocycleα generates an AT Mackey action. The pair(G, α) is stably weaklyequivalent to the pair consisting of a product odometer that we denote byTpr and aproduct cocycle that we denote byαpr, and the space whereTpr acts will be denotedby (pr,mpr). Note that our product odometer is of type II1. Hence, two cases canarise: eitherG is also of type II1 and they are weakly equivalent, orG is of typeII∞ and is weakly equivalent to the trivial expansion of our product odometer.

In the case of type II1 actions, there exists a transformationθ : → pr thattransforms[G] to [Gpr] andα to a cocycle cohomologous to a product cocycleαpr.Note thatmpr θ turns out to be an invariant probability measure on and hencempr θ = m. Hence,α is cohomologous to aθ-product cocycle – see Definition1.8.

Now let us consider the type II∞ case in more detail.Introduce the trivial expansion of our product odometer and consider the prod-

uct spacepr × Z with product measurem′ = mpr × mZ. The following actionswill be considered:

g′(ω, z) = (gω, z),τ(ω, z) = (ω, z+ 1),

(hereω ∈ pr, z ∈ Z, g ∈ Gpr) together with the following cocycle:

α′(ω, z; g · τn) = α(ω, g).The action associated with the pair(G′pr, α

′pr), whereG′pr = g′τn : g ∈ Gpr, n ∈

Z, is the same as for the pair(Gpr, αpr) because of [1, Proposition 2.3].The weak equivalence relation is provided byθ : → pr × Z such that

θ[G]θ−1 = [G′pr], mpr θ ∼ m, andα′pr(θω, θgθ−1) is cohomologous toα(ω, g),

whereω ∈ , g ∈ G. Note thatm andmpr θ both areG-invariant infinitemeasures and hence differ by a constant multiplier:mpr θ = λ ·m.

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Now let us take the following setP ⊂ : P = θ−1(pr× 0). Thenm(P ) =1/λ. For eachi ∈ Z, let Pi = θ−1(pr × i). ObviouslyPi are disjoint and⋃i∈Z Pi = . So the space is represented in the formP×Z by settingP×i =

Pi. FormP being the restriction ofm on P , we can see thatm = 1/λmP × mZ;this follows from the fact thatθ preserves the measure. Letτ1 = θ−1 τ θ ; τ1 bean automorphism of the space = P × Z, it preserves the measurem and has thepropertyτ1(Pi) = Pi+1.

LEMMA 5.4. There exists a cocycleβ ∈ Z1(P × Z,G;A) cohomologous toαand taking its unit value atτ1.

Proof.We shall construct a functionf (p, z) such that the cocycle

β((p, z), g) = (f (p, z))−1 · α((p, z), g) · f (g · (p, z))possesses the required property. Hereg ∈ [G]. We putf (p,0) = eA, f (p, z) =α((p,0), τ z1)

−1. Then we immediately obtain

β((p,0), τ z1) = (f (p,0))−1 · α((p,0), τ z1) · f (p, z) = eAand hence

β((p, z), τ1) = β((p,0), τ z1)−1 · β((p,0), τ z+11 ) = eA.

We see also from the proof thatαP = βP . We are now ready to formulate thefollowing result.

THEOREM 5.5. An Abelian group AT action being a Mackey action associatedwith a typeII action and a cocycle is also a range of aθ-product cocycle cohomol-ogous to the initial one.

Proof. Let us consider again the restriction ofθ to P that mapsP to pr andmP tompr. As β andα′pr do not depend onτ1 andτ , respectively,θ transformsβPto a cocycle cohomologous toαpr. Hence,βP = αP is aθ-product cocycle. So, theexistence ofβ andP proves our theorem. 2

6. The Double Mackey Action and Two Single Ones

Let us now compare the double Mackey action considered in the above with twosingle ones. Namely, we intend to considerα andρ separately; this allows us tointroduce the following actions:

g(ω, a) = (gω, a · α(ω, g)),Vb(ω, a) = (ω, a · b).

They act on the product space× A, and, as usual, we define the quotient actionof Vb by the ergodic components ofg by Wb.

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Besides, introduce two actions

g(ω, u) =(gω, a + log

dm gdm

),

Tt (ω, u) = (ω, u+ t)acting on the product space × R; and similarly we define the quotient action ofTt by the ergodic components ofg by Ft .

The relation between the double Mackey action(Ft ,Wb) considered in theabove (see Definition 3.1) and the single Mackey actionsFt andWb introducedhere is not clear because of the fact that theg-, g- andg-orbits are very different.However, the following is true:

PROPOSITION 6.1.When the double Mackey action(Ft ,Wb) is AT, the two sin-gle Mackey actionsFt andWb are also AT.

Proof. Indeed, when(Ft ,Wb) is AT, thenα×ρ is a product cocycle. Accordingto the definition of product cocycles we see thatα, i.e. its component with valuesin A, as well asρ, i.e. its component with values inR, are both product cocycles.This implies the approximate transitivity ofWb andFt , respectively. 2

Is the reverse statement true? The following example provides the negativeanswer to this question for the general case.

EXAMPLE 6.2. (We reproduce this construction from [5] with a little correction.)Let = 0,1Z andm be a product measure,m = ∏∞

i=1mi, mi(0) = mi(1) =1/2. Let θ be the Bernoulli transformation. Consider the spaceX = × withthe product measureµ = m × m and two measure-preserving automorphisms onthis space:θ1 = θ × θ, θ2 = id× θ . Let (Y, ν) be a Lebesgue space with aσ -finitemeasureν. Let S be anyν-preserving ergodic transformation onY , andu1, u2 betwo automorphisms permutable one with one and possessing the property:

ν u1 = exp(τ1) · ν,ν u2 = exp(τ2) · ν,

whereτ1 andτ2 are two rationally incommensurable numbers, and belong to thenormalizer of[S].

Introduce a Lebesgue space(X × Y,µ × ν) and construct the three followingactions:

Q1(x, y) = (θ1x, u1y),

Q2(x, y) = (θ2x, u2y),

S0(x, y) = (x, Sy).

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These actions are permutable. They generate the full group which we denote byG.Their joint action is of type III1 because of the rational incommensurability, and isorbit equivalent toZ-action.

Define the cocycleα ∈ Z1(X × Y,G;Z) in the following way:

α(x, y;Q1) = 0, α(x, y;Q2) = 1, α(x, y;S0) = 0.

The Mackey action constructed byG andα is trivial and hence AT. Indeed, it iseasy to see that the skew action acts ergodically on the product spaceX × Y × Z.(In this caseα is said to bea cocycle with dense range.)

On the other hand, the Mackey action constructed byG and the Radon–Nikodymcocycleρ of the measureµ × ν, i.e. the associated flow, is also trivial and henceAT. This easily follows from the fact thatG is an action of type III1.

Now consider the double Mackey action constructed byG and the double cocy-cleα× ρ. To do so, we write the following five actions being permutable one withone:

Q1(x, y; z, t) = (θ1x, u1y; z, t − τ1),

Q2(x, y; z, t) = (θ2x, u2y; z + 1, t − τ2),

S0(x, y; z, t) = (x, Sy; z, t),z1(x, y; z, t) = (x, y; z − z1, t),

t1(x, y; z, t) = (x, y; z, t − t1).Herez, z1 ∈ Z, t, t1 ∈ R. We have to construct the quotient space ofX × Y ×Z × R by the orbits of(Q1, Q2, S0) and obtain the quotient action of(z1, t1) onthis quotient space.

Fix y0 ∈ Y . It is easy to check that the set

X × y0 × 0 × [0; τ1]can be identified with this quotient space. A straightforward computation showsthat the Mackey action can be realized in this space in the following way:

z1(x, t) = (θz12 · θ [(t−z1τ2)/τ1]1 · x, (t − z1τ2)/τ1 · τ1),

t1(x, t) = (θ [(t−t1)/τ1]1 · x, (t − t1)/τ1 · τ1).

Here the brackets mean the integral part and the braces mean the fractional part ofa real number.

Finally, consider theσ -algebra containing all sets of the form× A× [0; τ1],whereA is a measurable subset of. The quotient action of(z1, t1) on the quo-tient space by thisσ -algebra is essentially the action ofZ on generated by theBernoulli (1/2,1/2)-action. It has positive entropy, and it now follows from [3,Theorem 3.5 and Remark 2.4], that our double Mackey action is not AT.

This completes the consideration of this example. In other words, though the co-cyclesα andρ are separately isomorphic to product cocycles, these isomorphismshave turned out to be incompatible.

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References

1. Bezuglyi, S. I. and Golodets V. Ya.: Weak equivalence and the structures of cocycles of anergodic automorphism,Publ. Res. Inst. Math. Sci.27(4) (1991), 577–625.

2. Connes, A., Feldman, J. and Weiss, B.: An amenable equivalence relation is generated by asingle transformation,Ergodic Theory Dynamical Systems1 (1981), 431–450.

3. Connes, A. and Woods, E. J.: Approximately transitive flows and ITPFI factors,Ergodic TheoryDynamical Systems5(2) (1985), 203–236.

4. Golodets, V. Ya. and Nessonov, N. I.: Approximately transitive actions and product cocycles ofan ergodic automorphism, Preprint of Institute for Low Temperature Physics and EngineeringNo. 4, Kharkov, 1987.

5. Golodets, V. Ya. and Nessonov, N. I.: Approximately transitive actions of Abelian groups andproduct cocycles, Preprint of Institute for Low Temperature Physics and Engineering, No. 20,Kharkov, 1991.

6. Golodets, V. Ya. and Sinelshchikov, S. D.: Amenable ergodic group actions and ranges ofcocycles,Soviet Math. Dokl.41 (1990), 523–526.

7. Golodets, V. Ya. and Sinelshchikov, S. D.: Classification and structure of cocycles of amenableergodic equivalence relations,JFA121(2) (1994), 455–485.

8. Golodets, V. Ya. and Sokhet, A. M.: Ergodic actions of an Abelian group with discrete spec-trum, and approximate transitivity,J. Soviet Math. 52(6) (1990), 3530–3533; translated fromTeor. Funktsi˘ı, Funktsional. Anal. i Prilozhen.51 (1989), 117–122.

9. Golodets, V. Ya. and Sokhet, A. M.: A representation of approximately transitive group actionsas a product cocycle range, Preprint of Institute for Low Temperature Physics and EngineeringNo. 2, Kharkov, 1991.

10. Golodets, V. Ya. and Sokhet, A. M.: Cocycles of type III transformation group and AT prop-erty for the double Mackey action, Preprint of the Erwin Shrödinger International Institute forMathematical Physics, ESI 97, 1994.

11. Hamachi, T.: The normalizer group of an ergodic automophism of type III and the commutantof an ergodic flow,J. Funct. Anal.40(3) (1981), 387–403.

12. Hamachi, T.: A measure theoretical proof of the Connes–Woods theorem on AT flows,PacificJ. Math.154(1) (1992), 67–85.

13. Hamachi, T. and Osikawa, M.: Ergodic groups of automorphisms and Krieger’s theorems,Sem.Math. Sci.3 (1981), 113.

14. Hawkins, J. M.: Properties of ergodic flows associated to product odometer,Pacific J. Math.141(2) (1990), 287–294.

15. Hawkins, J. M. and Robinson, E. A.: Approximately transitive (2) flows and transformationshave simple spectrum,Lecture Notes Math.1342 (1988), 261–280.

16. Hewitt, E. and Ross, K.:Abstract Harmonic Analysis,Vol. 1, Springer, Berlin, 1963; Vol. 2,Springer, Berlin, 1970.

17. Krieger, W.: On nonsingular transformations of a measure space I, II,Z. Wahrsch. verw. Gebiete11 (1969), 83–97, 98–119.

18. Mackey, G. W.: Ergodic transformation groups with a pure point spectrum,Ill. J. Math. 8(2)(1964), 593–600.

19. Rohlin, V. A.: On the fundamental ideas of measure theory,Mat. Sb.25(67)(1949), 107–150(in Russian).

20. Skandalis, G. and Sokhet, A. M.: Transitive actions have funny rank one, to appear.21. Sokhet, A. M.: Les actions approximativement transitives dans la théorie ergodique, Thèse de

doctorat de l’Université Paris VII, soutenue le 26 juin 1997.22. Zimmer, R. J.: Amenable ergodic group actions and an application to Poisson boundaries of

random walks,Ann. Sci. École Norm. Sup.11 (1978), 407–428.

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367

The Quantum Commutator Algebra of a PerfectFluid

M. D. ROBERTSFlat 5, 17 Wetherby Gardens, London SW5 OJP, U. K.? e-mail: [email protected]

(Received: 2 December 1997; in final form: 15 December 1998)

Abstract. A perfect fluid is quantized by the canonical method. The constraints are found and thisallows the Dirac brackets to be calculated. Replacing the Dirac brackets with quantum commutatorsformally quantizes the system. There is a momentum operator in the denominator of some coordinatequantum commutators. It is shown that it is possible to multiply throughout by this momentum op-erator. Factor ordering differences can result in a viscosity term. The resulting quantum commutatoralgebra is unusual.

Mathematics Subject Classifications (1991):81S05, 81R10, 82B26, 83CC22.

Key words: quantum commutator algebra, perfect fluid.

1. Introduction

It has been known for some time [1] that a perfect fluid has a Lagrangian for-mulation. The Lagrangian is taken to be the pressure and variations are achievedthrough an infinitesimal form of the first law of thermodynamics. A perfect fluid’sstress is described using the vector field comoving with the fluid. This vector fielddefines an absolute time for the system. Furthermore, this absolute time can thenbe used to define canonical momenta and canonical Hamiltonians. This is donehere for the first time. There are equivalences between scalar fields and fluids,[2]; more generally, the comoving vector field can be decomposed into scalarfields resulting in a description of a perfect fluid employing only scalar fields.Previously, this decomposition has been investigated by choosing anad hocglobaltime rather than absolute time and defining canonical momenta and other quan-tities with respect to the global time. Typically, the resulting theory is applied tocosmology [3, 4].

Once the constrained Hamiltonian has been calculated by the standard canonicalmethod [7, 8], the Dirac brackets can be replaced by quantum commutators. Theoriginal motive for investigating this was to find a fluid generalization of Higg’smodel [5, 9]. A quantum treatment is required to estimate the VEVs (quantum

? From 1 Jan. 98: Department of Mathematics and Applied Mathematics, University of CapeTown, Rondebosh 7701, South Africa. e-mail: [email protected]

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368 M. D. ROBERTS

vacuum expectation values) of the scalar fields which are related to the inducednonzero mass. The quantum commutator algebra is unusual, perhaps reflecting thestructure of the scalar field decomposition of the comoving vector field [6, 10]. It ishoped that eventually the present theory will be applied to low temperature superfluids. To do this, it probably will be necessary to include a chemical potential termin the first law of thermodynamics (2.1).

2. Lagrangian and Hamiltonian Formulation of a Perfect Fluid’s Dynamics

A perfect fluid has a Lagrangian formulation in which the Lagrangian is the pres-surep. Variation is achieved by using the first law of thermodynamics

dp = n dh− nT ds, (2.1)

wheren is the particle number,T is the temperature,s is the entropy, andh theenthalpy. The pressure and the density are equated to the enthalpy and the particlenumber by

p + µ = nh. (2.2)

In four dimensions, a vector can be decomposed into four scalars, however thefive-scalar decomposition

hVa = Wa = φa +6(i)θ(i)S(i)a, VaVa = −1, (2.3)

(i) = 1,2 is often used, because fori = 1, s and θ = ∫T dτ have interpre-

tation as the entropy and the thermasy, respectively. From now on, the index(i)

is suppressed as it is straightforward to reinstate. There are other conventions forthis scalar field decomposition, for example witha− instead ofa+ before thesummed fields. “q” is used to notate an arbitrary scalar field, i.e.,q = φ, θ or s.The coordinate space action is taken to be

I =∫ √−gp dx4. (2.4)

Replacing the first law with dp = −nVa dWa − nT ds, variation with respect tothe metric andφ, θ , ands gives

Tab = (p + µ)VaVb + pgab,(2.5)

(nV a)a = n +n2 = 0,s= 0,

θ = T,

respectively.2 = V a·a is the expansion of the vector field.The canonical momenta are given by5i = δI /δqi and are

5φ = −n, 5θ = 0, 5s = −nθ. (2.6)

MPAG021.tex; 15/04/1999; 14:15; p.2

THE QUANTUM COMMUTATOR ALGEBRA OF A PERFECT FLUID 369

The Hamiltonian density is usually defined in terms of components of the canonicalstress asθ t·t . In the present case, the canonical stress is not defined so that the metricstressT a·b is used instead; also 4-vectors are used rather than components, resultingin

Hd = V a· V

b· Tab = µ. (2.7)

The standard Poisson bracket is

A,B = δA

δqi

δB

δ5i

− δA

δ5i

δB

δqi, (2.8)

wherei, which labels each field, is summed; the integral sign and measure havebeen suppressed and the variations are performed independently. When absolutetime is used, Hamiltons equations have an additional term in the expansion [12],explicitly

q = δHc/δ5,

5 + θ 5= −δHc/δq, (2.9)

whereHc is the canonical HamiltonianHc =∫Hd√−g dx4. From (2.6), the

momenta are constrained

ϕ1 = 5s· − θ5φ

· , ϕ2 = 5θ· . (2.10)

The initial Hamiltonian isHI = 5i

q i · −L, replacing the dependent5’s gives

H0 = 5ϕ(φ + θ s)−L, (2.11)

adding the constraints gives the Hamiltonian density

Hλ = H0+ λα· ϕα, λ1· =

s, λ2

· =θ, (2.12)

Hd = 5ϕ· (φ + θ s)+ λ1

· (5s· − θ5ϕ

· )+ λ2·5

ϕ· −L,

where theλ’s are the Lagrange multipliers. Substituting the values of the momen-tum the Hamiltonian density is still weakly the fluid density. Using (2.9), the timeevolution of any variableX is given by

dX

dτ= ∂X

∂τ+ X,Hλ −25i δX

δ5i, (2.13)

replacing the Hamiltonian densityH byHλ and then holding the multipliers con-stant so that

X,Hλ = X,H0 + λα· X,ϕα (2.14)

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370 M. D. ROBERTS

gives the time evolution

dX

dτ= ∂X

∂τ−25i

·δX

δ5i+ X,H0 + λα· X,ϕα (2.15)

= ∂X

∂τ+ ( ϕ + θ(s −λ1

· ))δX

δϕ+ λ1

·δX

δs+ λ2δX

δs+

+ δX

δ5φ·((V a· 5

φ· )a −25φ

· )+δX

δ5θ·((− s +λ1

· )5φ −25φ)+

+ δX

δ5s·((V a· θ5

φ· )a −25s

· )

≈ ∂X

∂τ+ ϕ δX

δϕ+ θ δX

δθ− n δX

δ5φ·− n δX

δ5φ·− (θn)· δX

δ5s·.

Letting X equal the constraints gives dϕα/dτ = 0, this shows that there are nofurther constraints so that the Dirac brackets can now be constructed.

A quantityR(q,5) is first class [8] if

R, ϕα ≈ 0, α = 1,2, (2.16)

otherwise it is second class. TheCαβ matrix, cf. [11, p. 10], is

Cαβ = ϕα, ϕβ = −iσ 2· 5

ϕ· , C−1

αβ = +iσ 2· /5

ϕ· , (2.17)

σ 2· =

(0 −i+i 0

),

where is a Pauli matrix. The Dirac bracket is defined by

A,B∗ = A,B − A,ϕαC−1αβ ϕβ,B. (2.18)

In the present case, this gives the Dirac bracket

A,B∗ = A,B − 1

5φ

δB

δϕ

(δA

δs− θδA

δϕ+5ϕ δA

δ5θ

)+ (2.19)

+ 1

5φ

δA

δϕ

(δB

δs− θδB

δϕ+5a δB

δ5θ

).

Now

Hλ = H0− H0, ϕαC−1αβ , λβ = −H0, ϕαCλ−1

β , (2.20)

from whichHλ given by (2.11) can be recovered with the correctλ’s.

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3. Quantization

To quantize a classical dynamical system the Dirac bracket is replaced by thecommutator

A,B∗ → ih[AB − BA], (3.1)

where h is Planck’s constant divided by 2π and the hat “∧” signifies that thevariable is now an operator. There are various correspondence criteria which canbe investigated, for example: ash → 0, there should be (a) the same time evo-lution, (b) the same stress, and (c) the first law should be recovered. Anothercorrespondence criteria can be called the particle number criteria: the particle num-ber n should bear a relation to the quantum particle number constructed fromcreation and destruction operators. An intermediate aim, between formal quanti-zation achieved by replacing field and momenta Dirac brackets with commutators,and establishing contact with applications, is to produce a quantum perfect fluid.This could be obtained from brackets involving the numbered field, the angularmomentum and so on, or from brackets involving a mixture of these and geometricobjects. However, no progress has been made so far in finding a quantum perfectfluid, so that attention is restricted to implications of replacing brackets consistingsolely of individual components of fields and momenta with commutators. Effect-ing the replacement of the 15 Dirac brackets between the fields and momenta thereare four nonvanishing commutators

5φ· ϕ − ϕ5φ

· = −ih, 5θ· θ − θ 5θ

· = 0, 5s· s − s5s

· = −ih, (3.2)

ϕθ − θ ϕ = − ihθ5φ

, θ s − sθ = − ih5φ·, ϕs − sϕ = 0.

The last two commutators have the operator5φ· in the denominator. This mightnot be well-defined. To avoid5φ in the denominator we multiply by the operator5ϕ, using the first commutation of (3.2) it turns out that multiplying on the left ormultiplying on the right are equivalent so that

[ϕθ − θ ϕ]5q· = 5q

· [ϕθ − θ ϕ] = −ihθ , (3.3)

[θ s − sθ]5ϕ· = 5ϕ

· [θ s − sθ] = −ih.These results are in accord with the equations deduced if the Dirac bracketsqi· ,qj· 5k· ∗ are replaced by commutators. Left and right multiplication by5k· are alsoequivalent if anti-commutation rather than commutation is considered.

The quantum Hamiltonian is

Hq = l15ϕ·ϕ + l2 ϕ5ϕ

· − l35ϕ· θs − l45ϕ

·sθ − (3.4)

− l55ϕ·s − l6θ s5ϕ

· − l7s5ϕ· θ − l8

sθ5ϕ

· − p,

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372 M. D. ROBERTS

where thel’s are constant and obeyl1+ l2 = 1, l3+ l4+ l5+ l6+ l7+ l8 = 1, usingthe commutation relations the quantum Hamiltonian (3.3) is

Hq = 5ϕ· (φ − θ s)− ih2l − p, (3.5)

wherel = l2+ l4+ l7+ l8 is called the ordering constant: it is of undefined size butis can be taken to be of order unity. Because the Dirac bracket ofp with anythingvanishes the commutators withp also vanish andp can be taken to bep⊥, where⊥ is the identity element.

To investigate the algebraic implications of (3.2) and (3.3), label the six opera-tors byv’s,

φ s θ 5ϕ· 5s· 5θ·v1 v2 v3 v4 v5 v6

(3.6)

v6 commutes with everything and can be disregarded. Of the remaining com-mutators, only four are nonzero. In unitsh = 1 (3.2). (3.3) and (3.7) give thealgebra

v4(v3v2− v2v3) = −i, (3.7)

v4(v1v3− v3v1) = −iv3,

v4v1− v1v4 = −i,v5v2− v2v5 = −i.

This algebra does not seem to be realizable in terms of matrices and differentialoperators, the closest algebras are found in [11]. If a commutator is constructedwith a time derivative of the field or momenta, the same algebra results but multi-plied by a term in the expansion. Similarly ifm time derivatives occur, the algebrais multiplied by the expansion to the power ofm.

Acknowledgements

I would like to thank Prof. T. W. B. Kibble for discussion of some of the pointsthat occur. This work was supported in part by the South African Foundation forResearch and Development (FRD).

References

1. Hargreaves, R.:Philos. Mag.16 (1908), 436.2. Tabensky, R. and Taub, A. H.:Comm. Math. Phys.290(1973), 61.3. Tipler, F.:Phys. Rep. C137(1986), 2314. Lapchiniskii, V. G. and Rubakov, V. A.:Theoret. Math. Phys.33 (1977), 1076.5. Roberts, M. D.:Hadronic J.20 (1997), 73–84.6. Schutz, B.:Phys. Rev. D4 (1971), 3559.7. Dirac, P. A. M.: Lectures on quantum mechanics, Belfor Graduate School of Science, Yeshiva

University, New York, 1963.

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8. Hanson, A. J., Regge, T. and Teitelboim, C.:Constrained Hamiltonian Systems, AccademiaNazionale die Lincei Rome, 1976.

9. Roberts, M. D.: A generalized Higg’s model, Preprint.10. Eckart, C.:Phys. Fluids3 (1960), 421, Appendix.11. Ohaki, Y. and Kamefuchi, S.:Quantum Field Theory and Parastatistics, Springer-Verlag,

Heidelberg, 1982.12. Roberts, M. D.: An expansion term in Hamilton’s equations,Europhys. Lett.45 (1999), 26–31.

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Mathematical Physics, Analysis and Geometry1: 375–376, 1999.

Contents of Volume 1

Volume 1 No. 1 1998

Editorial v

L. BOUTET DE MONVEL and E. KHRUSLOV / Homogenizationof Harmonic Vector Fields on Riemannian Manifolds withComplicated Microstructure 1–22

ANDREI IFTIMOVICI / Hard-core Scattering forN-body Systems 23–74

S. SINEL’SHCHIKOV and L. VAKSMAN / Onq-Analogues of BoundedSymmetric Domains and Dolbeault Complexes 75–100

Instructions for Authors 101–106

Volume 1 No. 2 1998

ANTON BOVIER and VÉRONIQUE GAYRARD / Metastates in theHopfield Model in the Replica Symmetric Regime 107–144

S. MOLCHANOV and B. VAINBERG / On Spectral Asymptotics forDomains with Fractal Boundaries of Cabbage Type 145–170

IGOR YU. POTEMINE / Minimal TerminalQ-Factorial Models ofDrinfeld Coarse Moduli Schemes 171–191

Volume 1 No. 3 1998

W. O. AMREIN and D. B. PEARSON / Stability Criteria for the Weylm-Function 193–221

M. A. FEDOROV and A. F. GRISHIN / Some Questions of theNevanlinna Theory for the Complex Half-Plane 223–271

NICULAE MANDACHE / On a Counterexample Concerning UniqueContinuation for Elliptic Equations in Divergence Form 273–292

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CONTENTS OF VOLUME 1

Volume 1 No. 4 1998/1999

Editorial 293

G. GALLAVOTTI / Arnold’s Diffusion in Isochronous Systems 295–312

A. S. FOKAS, L.-Y. SUNG and D. TSOUBELIS / The Inverse SpectralMethod for Colliding Gravitational Waves 313–330

VALENTIN YA. GOLODETS and ALEXANDER M. SOKHET /Product Cocycles and the Approximate Transitivity 331–365

M. D. ROBERTS / The Quantum Commutator Algebra of a PerfectFluid 367–373

Volume Contents 375–376

Instructions for Authors 377–382

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Mathematical Physics, Analysis and Geometry1: 377–382, 1999. 377

Mathematical Physics, Analysis and Geometry

INSTRUCTIONS FOR AUTHORS

EDITORS-IN-CHIEF

VLADIMIR A. MARCHENKO, B.I. Verkin Institute for Low Temperature Physicsand Engineering, Academy of Sciences of Ukraine, Kharkov, Ukraine

ANNE BOUTET DE MONVEL, Université de Paris 7 – Denis Diderot, Institutde Mathématiques, Paris, France

HENRY McKEAN / New York University, Courant Institute of MathematicalSciences, NY, U.S.A.

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The journal will publish papers presenting new mathematical results in mathemat-ical physics, analysis, and geometry, with particular reference to:

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References to articles in periodicals should include the author’s name; year ofpublication; title of article; full or abbreviated title of periodical; volume number(issue number where appropriate); first and last page numbers. For example:

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4. Schwartz, J. H.: Evidence for nonperturbative string symmetries,Lett. Math.Phys.34 (1995), 309–317.

References to technical reports or doctoral dissertations should include the author’sname; year of publication; title of report or dissertation; institution; location ofinstitution. For example:

5. Ramaroson, R. A., 1989: Modeling in one and three dimensions, PhD thesis,Paris VI.

PROOFSProofs will be sent to the corresponding author. One corrected proof, together withthe original, edited manuscript, should be returned to the Publisher within threedays of receipt by mail (airmail overseas).

OFFPRINTS50 offprints of each article will be provided free of charge. Additional offprints canbe ordered by means of an offprint order form supplied with the proofs.

PAGE CHARGES AND COLOUR FIGURESNo page charges are levied on authors or their institutions. Colour figures are pub-lished at the author’s expense only.

COPYRIGHTAuthors will be asked, upon acceptance of an article, to transfer copyright of thearticle to the Publisher. This will ensure the widest possible dissemination of infor-mation under copyright laws.

PERMISSIONSIt is the responsibility of the author to obtain written permission for a quotationfrom unpublished material, or for all quotations in excess of 250 words in oneextract or 500 words in total from any work still in copyright, and for the reprintingof figures, tables or poems from unpublished or copyrighted material.

ADDITIONAL INFORMATIONAdditional information can be obtained fromMathematical Physics, Analysis andGeometry, Science and Technology Division, Kluwer Academic Publishers, P.O.Box 17, 3300 AA Dordrecht, The Netherlands; fax 078-6932388; e-mail [email protected]

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Mathematical Physics, Analysis and Geometry - Volume 1

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