r AD-A093 056 WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER F/S 12/1THEORETICAL AND PRACTICAL ASPECTS OF SINGULARITY AND EIGENMODE -ETC(U)MAR 80 A 6 RAMM F49620 79-C-0128

NCLASSIFIED RC-TSR-2051 AFOSR-TR-80-1171 NL

-EEEEE~hNE

4 AFOSR.TR. 8 0- 117 1

MRC Technical Summary Report #2()51

!i THEORETICAL AND PRACTICAL ASPECTS OF

SINGULARITY AND EIGEN140DE EXPANSION~METHODS

~A. G. Ram

Mathematics Research CenterUniversity of Wisconsin-Madison

610 Walnut StreetMadison, Wisconsin 53706

March 1980

1eived February 25, 1980)

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* ~'aWIERETICAL AND.ACTICAL 4SPECTS OF \Interim-" INGULARITY AND IGENMODE EXPANSION MIrrHODS, 76PERFORMING01G. REPORT NUMBER

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9. PERFORMING ORGANIZATION NAME AND ADDRESS EMRT0OE7 TAS

University of Michigan AE OKU4~U8R

Department of MathematicsAnn Arbor, MI 48109 612

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17. DISTRIBUTION STATEMENT (of the abstract entered in Block 20, if differ'nl froom Report)

16. SUJPPLEMENTARY NOTES

19. KEY WORDS (Continue on reverse side if necessary and identity by block number)

Singularity iAhdeigenmcde expansion, nonself-adjoint operators, Riesz basis,integral equation, complex poles of Green functions, inverse problems,transient fiel, scattering theory

20. ABSTRACT (Continue on trerse side If necessary and identify by block number)

Foundations of singularity and eigenmode expansion methods and somepractical questions related the theory are discussed.

DD I'jAN 73 1473 EDITION OF I NOV 65 IS OBSOLETE

SE C 1 ASSIFIIC TI T11 GM eD d

UNIVERSITY OF WISCONSIN-MADISON

MATHEMATICS RESEARCH CENTER

THEORETICAL AND PRACTICAL ASPECTS OF SINGULARITYAND EIGENMODE EXPANSION METHODS

A. G. Ramm

Technical Summary Report #2O151

March 1980

ABSTRACT

Foundations of singularity and eigenmode expansion method3 and

some practical questions related to the theory are discussed.

AMS (MOS) Subject Classifications: 47A40, 35P25, 35P05, 35P10, 30E25,

78A45, 81F20, 81F99

Key Words: Singularity and Eigenmode Expansion, Nonself-adjoint Operators,Riesz Basis, Integral Equation, Complex Poles of Green Functions,Inverse Problems, Transient Field, Scattering Theory

Work Unit Number 1 (Applied Analysis)

"' ie,,e n:d is

,-i L, .. lA,," , £.)u- (7b).A. D. ELOS;Technical InfoOwa=tion Officer

Address: Department of Mathematics, University of Michigan, Ann Arbor, MI48109. -

Sponsored bv the United States Army under Contract No. AG29-75-C-0024 a dthe Air Force Office of Scientific Research under F49620-79c00-1

-I F~: TFCANC E AND EXPIANATInN

Inr ,m-t .nunr of literaturf, li,. bprY wriffen on

In i. t-: .; ;. ,trj x: drISjelI moth-ds unA id 11r1l The main rirti-dl

it K-- t-~ -v whiul, ..j] lows us t-( identify ai fl,-in4 (or

-,ti 110 : rrem tho nhl,-; r\vcd t rats 'nt fi -1 1 t t -ered :-: t ht-

t r-)1 : r' m is4 di-su scd in Ili- :i~n r our~ A

x rI I t h( r. hII Iw~~' Si .tn li. F,~ III,. m M t

* ~~~I M, t- r~ I0 to.1.l:ttV r niitn rl 2 l -i i1l

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Accession ForNTIS GRA&IE71C TAB

I th:?n-ourlced

C: rJ~ ion/A -t)libility Codes

jAvail anid/orDist Special

THEORETICAL AND PRACTICAL ASPECTS OF SINGULARITY

AND EIGENMODE EXPANSION METHODS

A. G.Ram

V7:1sL 1 ltcrat!re has been wriL'ton on iclrt 'c

0-:,nson methods during the last aucaJ; rg ,osi:

PhYsiciStsq stimulated interest in the-- subject-se [1], (2),

[3] and references given in this review). atcaicx4ar :iS

of the? problems was irnitiated in [4] , [5] and uas puslmdc: con-

siderably further by M. Agranovich (see [13) . Ne xe rthele -10s s

many quest'Cions in the theory are open and of considerabLie intor -<;t

to engineers and mc-ithematicians. The purpose of this paper can

be summarized as follows: WNe are going to explain in a siimOpe~

way the principal features of the singularity7 and ei~jnno c-

expansion methods and to formulate exliity ) hat hsLC

iused by engineers with proof, 2) v'hat is irimrait " or pract" C!c,

3) %-h1atL has hceirigorok-sl.y ecs'aD1J-S~i-cJ th,-2

Ur SO1Vvo. mat;bema Li ccm problcem ,s in this fi E1'] .

Address: Department of Mathematics, University of Michigan,* Ann Arbor, MI 48109.

Sponsored by the United States Army under Contract No. DAAG29-75-C-0024 and the Air Force office of Scientific Research underF 4962 0-7 9CC)128.

The main results obtained in scalar wave scattering theory

were generalized to electromagnetic wave scattering without

much difficulty. That is why and also for simplicity wo restrict

ourselves by the presentation of the theory for scalar wave

scattering.

§2. What are singularity and eiqenmode expansion methods

(SEM and EEM)?

1. What is EEM?

Consider the problem

(V 2 + k2 )u = 0 in P IR3 D, k 2 > 0 , (i)

ulr = , (2)

u = u0 + v =exp{ik (i ,x)} + v (3)

and v satisfies the radiation condition

- k v = o(IxI a I - (4)

Here r is th, siooth surface of a finite obstacle D.

If we look for a solution of (1) - (4) in the form

x zJr t)

4 u 0 : ir 9 f(t)dt, r - ., t (5)

-2-

i

then

Sexp (ikrst

A(k)f H 4 ex pirst f(t) dt = - u 0 (s), s e r. (6)

The EEM method can now be explained as follows. Suppose that

the set of eigenvectors of operator A(k)

A(k)¢j = )j (k)Cj , j = 1,2 ...... (7)

forms a Riesz basis of L2 (r) H. This means that any g e H

can be expanded in the series

CO

9 g.~ (8)j=l j (

and

c gl2 _< Z j 2 ]2C< 2 g , c1 > 0 , (9)

where iIgII is the norm in L 2(F) . The inequality (9) substi-

tutes the Parseval equality for orthonormal bases. A complete

system in H does not necessarily form a basis of H (example:

H = L 2[0,1], the system j(x) = x j , j = 0,1,2,...: Not

every L2[0,1] can be expanded in the series g(x) = gx3

If the assunc:tion made is true, then equation (6) can be

solved by the l'ic'-rd formula

Co U

f(s) = - Is ( (10)j=l ,.(k) ]

-ii

- - ~ ~ 77 4w6L.-------

t/

where the coefficients are uniquely defined by the equality

0= X ~iI

j=l ujlJ

This method of solution of the scattering problem (1-

is called EEM. It was used without mathematical analysis

by cl,,i Lers 1I1, 13). The k;uestions, which immediately

aiasc, can be [ormuljild a.3 follows: 1) Is it true that the

nonselfadjoint operator A(k) has eigenvectors (e.g. Volterra

operator has no eigenvectors) ; 2) Is it true that the set of

eigenvectors of A(k) forms a Riesz basis of H; 3) suppose

that the set of eigenvectors (a eigensystem) of A(k) does

not form a Riesz basis of H. Is it true that the root system

of A(k) forms a Riesz basis of H ?

Let us explain the root system. Let A be a linear operator

on H 4= , ¢ 0. Consider equation API -i = X . If

this i'- as solvable, Ol is called a root vector of A

corresponding to eigenvalue X and eigenvector . If ! exists

consider equations A() k -X = k-l' k > 1. It is known [7]

that onlya finite number r of root vectors . 'I associated

with (j, exist. The chain( q,,q ~l, ... is ca led a Jordan Chain

with the length r + 1. The union of all root vectors of a

linear operator A corresponding to all eigenvalues of A is

called the root system of A. It is well known from linear algebra

that the eiqensystem of a nonselfadjoint operator may not

form a basis. For example if the operator A is an operator

-4-

in JR2 with the matrix A = , then A has only one0 1

eigenvector so that the eigensystem of A does not form a

basis of IR 2 . It is also known that the root system of any

linear operator (matrix) in fn forms a basis of JRn Of

course in IRn any basis is Riesz basis. In a Hilbert space

(infinite dimensional space) this is not true. For practice

it is important to have affirmative ans'-ocrs for questions 1)

and 3). Indeed, if the eigensystem of A(k) does not form a

Riesz basis but its root system forms a Riesz basis of H, then

it is still possible to solve equation (6) using the root

system of A.

2. What is SEM?

In order to explain what SEM is, consider the problem

- Au = 0 in Q ; ulF= 0; ult=0 = 0, utlt=0 = f(x), (12)

where f e C0 (a smooth function which vanishes for large IxL.0 2

If G(x,y,-p ) is the Green function of the problem

(-A + p2 )G = 6(x - y) in P, GIr = 0, Rep > 0 (13)

then the solution of (12) can be written as

- J exp(pt)u(x,p) dp, (1i ,2,i c - ic-

-5-

where

u(x,p) = f G(x'y'-p 2 )f(y)dy (15)

Suppose that u(x,p) is meromorphic in p on the whole

compl:x plane p (this is actually true [8"-[12]) and suppose

the following estimate is valid

IU(x,p) < c , a > 1/2, c = const.> 0, Rep > -A,(16)

jImpj > NA

where A > 0 is arbitrary.

This estirite follows from the Lax-Phillips result [12]

and the argu:cnts given in [9], [10], [13]. Under the assump-

tions made, the contour of integration in (14) can be moved to

the left and one gets

N b.-1u = ) c.t 3 exp(pjt) +o(exp(-Re pNt)) (17)

j=l

N

where pj are the poles of G(x, y, -p ), b. are the mul-

tiplicity of the pole p. Expansion (17) is called the SEP

expansion. Actually such expansions were known for a long time

for various concrete ptoble-s of mathematical physics (especially

in cases when the solution can be represented explicitly in

the form of series). The main difficult , , is to prove estimate

(16) which illows us 1Lo move the contour on intc:;r,: 7 (c4),

Tf only th,_ morri'ori-,, ic n-tuTo of n (,).

is established , then n c.... . . o. of the tyi. (17) can

be proved in general bcuse thCe -s a possibi1ity tlat pole-s

-C + i.. with very small > 0 and very large cann n n1 n

exist. The general Mit;:ag-Iffer representation crn not he

applied for derivation of SEM expansion (17), because this

representation uses special sys tern of expanding corntonrs, 1..14-le

the derivation of (17) requires the possibility to move our

particular contour (c -i *, c +i-.,) to the left. From a prac-

tical point of view, the SEN. expansion is usedl at present I

according to the following scheme:

Suppose that only a few terms in (17) are essentJal, e.g.

1-3. This will be true if IRe p.I >> Pep for j > 3. Then

in experiments the transient field u (x, t) is ;ea ured ann

each p., j = 1,2,3 is determined. It is assu.,u tht th,2

location of these complex poles of the Green fu:,cLio:! :*:,

can give information enough to identify the ostacle (the

scatterer D). This assumption has not been backed theoretically.

Nevertheless, if there is a finite sot of scatters (say flyng

targe ts) it is possiblc to hal i ve that a one to one cnrre-

spondence can be est;.b!.)hed empi-ically be tvnc the scatterers

and the corres .c 4r-,: C :.,] ); I o .

An Jntere. Li,. vi' pro 1 , can be formulated in con-

nection with ,

Inverse problei: Civ ni ' , of o:..........:- 7' .

P e. 0, i 'c:

Green fu . 2 . .: -u :. L

1 S

W¢hat restrictions must be imposed on the set {p }, Re ]. < 0

in order that this set will be the set of cor.:l:ex poles of the

Green functinn of a scatterer?

If the scatterer is a star-like body and the boundary con-

dition is the Dirichlet condition then the set {p_} must

satisfy Lhe condition IRe P.j > a In jlir pj + b, a > 0 [12].

It seems that no other information on the problem is available.

From a practical point of view this problem may not be so

important as it seems. First, only a few complex poles are

available. It seems hopeless to make any general conclusions

about the scaterer from this information without severe restric-

tions on the set of scatterers. (For example, if it is apriori

known that the scatterer is a ball, it is possible to determine

its radius from the above information.) That is why this author

thinks that from a practical point of view in order to use the

SEM for identification of scatterers it is more useful to work

out tables of responses of the typical scatterers, then to try

to develop a theory of the posed (which is very interesting

from a theoretical point of view) inverse problem.

3. T f ic* n.. .-':-* , . ;s artse naturally in comnection

with the E, .

1) rs integral opLrators in

.. . ..o a Riesz ba;is of .?

3) Do the complex poles of the Green function depend

continuously on the obstacle? In more detail: suppose that

xj = xj(tl,t2 ), 1 < tilt 2 < 1 are parametric equations of

p, yj = x + EZ (tlt2 0 < tilt 2 < 1, 0 < C < !

are parametric equations of the surface of perturbed scatterer.

Let us assume that xj(t), zj(t) e C2 ( , t = tilt 2, A =

{t . < tilt 2 < 1}. Let us fix an arbitrary number R > 0 and

let pj, 1 < j < r(R) be the complex poles of the unperturbed

Green function which lie in the circle Pj.1 < R. Let pj(c)

be the complex poles of the perturbed Green function. Our

question can now be formulated as follows: is it true that

p(C) - pj, as £ 0, uniformly in 1 < j < n(R) provided

that Lhe numeration of pU() is properly done?

4) How can one calculate the complex poles?

5) What are sufficient conditions for the validity of SEM

expansion (17)?

6) Is it possible to calculate complex poles via calcula-

tion of zeros of some functions?

§3. What has been riqorously established in EEM and SEM methods?

In this section we give answers to questions l)-6) of

Section 2.3. No proofs will be given but the results obtained

will be formulated and references will be given. Proofs are

-9-

C)-i L t Ed Co 112- rc S I ) tli 11 e i f Ai Cn -I I c0 i.I! 'z:

2) the-2y 1011(jiq, 171C 3) thecan ,1 oundC in h itdppr

1. In orde~r to or lt ?the an svir t~o jueF~t.on 1) of, E SctiCn

2. 3 we miust, expin ,Tl.ant a Riesz bas~ s %.i Lb brackets.- is. Le t

{f.}be anl orthononn;, 1hA. IDSs of H, {h bec a co!.31 r:t-c. and

iinmlS"'Stce in 11. (A co1-,i )c1:u Systcrt {I is called, mnj~r!I

if th(2 system {b. -\h kis not coupet.n for a ny k, k =I, 2, 3..

In oth(er V.oCrdS -iL we roil:ove any elemocnt h kof our svs Ler-, we

obtain an ineompot(- system). Let " -11 < m 2 < be an arral-

trary increasing sccjucnce of integ!(er s; F. is the lincar s'prce

with the basis {f lit f 1" 1 f m, -1 ! is the linearj-1 -1

space with the basi~s [h m . .1 ,.h 1 } Suppose-, that the,-rc

exists a linear bour!dClci operator B with bounded B3 n~

on all. 11, such thaF-t B11 F . j 1,2....Theal C- svs,.efl

{h. is called a Riesz b-asis of H with bracket s. rp.'is Clefi-

nition is enUnivaIent1 (see 1141I t~o the follo-,oanq, " Lt P. be

prjectorcs in HI on tc, H.. Suppose Lbe.t for ay f C 11,

then the sys ter I. is Called a lRicsz hbaSis l vJ .',

It is- prov':dtha thz ront syFstem o-f op o .(k)(see

f orrmila (6) )o io:- / n bo' V: wi. I brackj-c t-S ].Thsa

is true for the operator arising in the exterior Neumann

boundary value problem (1], [6). The same is also true for the

electrodynamics scattering problem [1].

2. If the integral operator of the diffraction problem is

normal then its eigensystem coincides with its root system.

This is the case in the problem (l)-(4) for a spherical surface

r and for a linear antenna. First this was observed in [5].

An operator is called normal if A*A AA*, where A* is the

adjoint operator. The condition AA* A*A is the condition on

r provided that the kernel of the integral operator A is given.

(see [5] for details).

3. The answer to question 3 from section 2.3 is affirmative

(see [61 for details).

4. A general method (with the proof of its convergence) for

calculating the complex poles of the Green's functions in dif-

fraction and potential scattering theory was given in [4], [15]

(also see [61). The method can be explained for the problem

with the impedance boundary condition

(V 2 + k2)u = 0 in , 'u - h 0 (18)

aN I

where h = const., Re h > 0, N is the outer normal to r

we look for a solution of the problem without sources in the

form

-41-

U I exp (ik i -ti) i(t) dt (19)JF 4 f X L t

From (19) and (18) it follows that

a = Oc , (20)

where

exp(ikr C) exp(ikrQS t (s)ds - h stNt 2nr strr - (

2Let {0j} be a Riesz basis of H = L(F),

Na j=c. c . (22)

Substituting (22) in (20) and multiplying by . in 11 we

obtain:

Nlj=l b j (k)cj = 0, 1 _< i < N ; bij (k) = 6ij -(Qqj, I). (23)

This system has nontrivial solutions if and only if

det bij (k) = 0. (24)

The left-hand side of (24) is an entire function of k. Let

k (N) be its roots. Then there exist the limits lim k(N) = kM8 N-*- M1 )1I

and k are the plo 2: o the Green function corresponding torn

the problem (18). ...... , all the complex poles can be obtain)DJ

by this method. Proofs are given in [41, [151, [61. From a

practir] : . .f m-; tice are two nontrivial points in prer-

~ 2 .-

forming this method. 1) calculations, of b.. (k) by fo -i..ul.

(23) and 2) numerical solution of equation (24). For hot-h

steps thcre are methods available in the liter-ature on

numerical analysis.

5. Sufficient conditions foi- the validity of SEP" ex:ansion

(17) were given in Scction 2.

6. The set of the complex poles of the Green function of

the problem (1) - (4) coincides with the set of the complex

zeros of the eigenvalues p (k) of the operator A(k):

A k) n = 1,2,... . (25)

Inc.]:!ed, let G = R(x,y) + . i.e. z is a pole of the Green(k-z)r

function G, GI 0,

IFexp (ikr xy)G = g (x,t,l)p (t,y,k)dt, g = 4 (26)

4rxy

3N

Multiplying (26) by (k - z) r and taking k z, we get

g(x,tz) DR(t y) dt = 0, x e r (27)

The kernel R(t,y) is degenerate. Thus a function c 0 exists

such that

r g(x,t;z) 0(t)dt 0(28)

-13-

This m.~st!at K z i S a 0'r of t0 Ctic' f tl

* U). o~vrsc ,if f0 is I SClILlLiOfl OF (28),

U f ~x~~z )<(tcli(29)

-is a Solution~ of, the p ro 1,1 C,

+ z )u =0 in uH 0(0

-7ith the outgoing. osympto&.1c at infi ni Cy. fTn1 u 0in ?

i 1 f z. is not a pole of G. Since z 2is comlplex ,U (X) 0

in D (as a solution of the homoqeTneous interior problem)

By the boundary value junmp relation 0. This contradiction

proves th,-,t z i.s a pol1.e of G. A viari ationiAl i-othod,: for cal-

culatLion of eigen-valuos of nonsc-l facjoinit comnpact operators

is given in [6].

4. Open p)roblems.

1 ) Tile inver-;e prOb)em fori~ii L-Ated in Sea--tioii 2 is of

interest. -It is vary iinLc rcmi nu to hove parL jal answc'ii.~,.

vw1hoL in forr.,at-.ior about Lhe l oc of a_ scat ifre r cain I)-

obtaineod from thc oree of the corlplex ol

2) There is a con1_jeC.ture7 [31 tiat. Lie cnlxpolces of

the Green function of t'e (,r-cblew ( 1) - (4) for a ckro ye sl':o )L1~

colupac L bwwndarv arc Sicr:-1ce. It %-oulId ho i L rps Li nc o

it or to give a cconLtc cxim~.l

-14-

) It C):iU .2 t7.KSi~i ~, to r) mr m nr 1&- r c ii ly ;m.. lj,

d c ion 3. 6 .in sorPK. pr~cLIcClI probn;

4) 1 :1 11 .;OIIQ proxcrtie. of t hc purel'y real poles, P, a < 0 /

* I ~ .. 0 ve re us tali i ~hed- .it would IJo in tozestino~ to tell1

W~i~nnrii ndib-uL Lhc Ic)o. L ry of an obst'ad 0 can bhe-

ohL iJ£~~II oca L.I )n of t I e purcely re(al pol-()Ies. n t6 - :)iysi c-F

literature LhU: h'~l> =1;u~1 ip is usual] V used. On Lb; is

plane the purl y rcal c-ic lox poles;' Ore 0111:01" l mw ry,

Re k. 0, In 1-. 0 .

References

(1] V. Voitovich, B. KaCcnelenbaum, A. Sivov, Generalized

method of eigenoscillations in diffraction theory, with

I Appendix written by M. Agranovich, Nauka, Moscow, 1978

(in Russian).(2] C.L. Dolph, R. Scott, Recent developments in the use of

complex singularities in electromagnetic theory and

elastic wave propagation, in "Electromagnetic scattering",

Ac. Press, N.Y., 1978, 503-570.

[31 C.E. Baum, Emerging technology for transient and broad

band analysis and synthesis of antennas and scaterers,

IEEE, 66, (1976), 1598-1614.

[4] A.G. Ramm, On exterior diffraction problems, R. Engr. and

Electr. Phys., 7, (1972), 1031-1034.

[5 A.G. Ramm, Eigenfunction expansion corresponding the dis-

crete spectrum. Radiotech. i Electron., 18, (1973),

496-501.

[6] A.G. Ramm, Nonselfadjoint operators in diffraction and

scattering, Math. methods in applied sciences. (1980)

[71 Dunford, T. Schwartz, Linear operators, vol. 1, Wiley,

N.Y., 1958.

(81 A.G. Ramm, On analytic continuation of the solutions to

the Schrodinger equations in spectral parameter and the

behavior of the solution to nonstationary problem as

t - 0. Uspechi Mat. Nauk, 19, (1964), 192-194.

(91 A.G. Ranmi, Analytical continuation of the resolvent kernel

of the Schrodinger Operator in spectral parameter and limit

amplitude principle in infinite domains, Doklady Ac. Sci.

Azerb. SSR, 21, (1965), 3-7.

[10] A.G. Raom, Some the.oroms on c Co ,_

the Schrodinger operator resolvent her e in.spxicr.3

parameter. Izv. Ac. Nauk . Arl. SSR, ..' 1'- ;.. :- .

(1968) , 780-783.

[11] P.D. Lax, R.S. Phillips, Scattering Theory, Sri~'..r,

N.Y., 1967.

[12] P.D. Lax, R.S. Phillips, "A logarithmic bound on, the

location of the poles of the scattering mtri.," Arch.

Rat. Mech. Anal. 40, (1971), 268-280.

[13] A.G. Ramm, Exponential decay of solution of hynabolic

equation. Diff. Eq. 6, (1970), 2099-2100.

[14] I. Gohberg, M. Krein, Introduction to the theory of linea

nonselfadjoint operators, AMS translation NJ8, Providence,

1969.

[15] A.G. Ramm, Calculation of the quasistationary states in

quantum mechanics. Doklady, 204, (1972), 1071-1074.

[16] P.D. Lax, R. Phil]ips, Decaying modes for the wave cmua-

tion in the exterior of an obstacle, Comm. Pure Appl.

Math., 22, (1969) , 737-787,

-17-