NEW ZEALAND JOURNAL OF MATHEMATICS Volume 25 (1996), 199-216

SPLITTING OF ACOUSTIC RESONANCES IN DOMAINS CONNECTED BY A THIN CHANNEL

B. P a v l o v

(Received October 1994)

Abstract. For two-dimensional domains connected by a thin channel the perturba­tion of eigenvalues and resonances are considered. The techiques used are based on integral equations, containing contracting operators.

The calculation of shifts and splittings of resonances and eigenvalues for weakly- connected domains is rather a classical problem of mathematical physics. We men­tion here only the latest publications, concerning this question for a homogeneous Dirichlet boundary condition, [1 - 6 ]. All of them give realistic estimates for cor­responding shifts and splittings, but the proper calculation of them or even lower estimates for shifts and splittings meets serious obstacles.

It appears, that the straightforward approach to the calculation of shifts and splittings of resonances and eigenvalues, suggested in a more simple problem for two domains connected by a small small opening (see [7]) can be used to calculate all spectral characteristics of the acoustic problem described above, if acomplished with asymptotic estimates for Green functions, based on the iterated Hilbert iden­tity. The corresponding integral equations can be written directly in terms of Green functions of unperturbed problems and the channel. The integral equations, which appear in this way, contain an integral operator, which is generally represented in every bounded domain of the complex plane of the spectral parameter A as a sum of contracting operator and a finite dimensional rational function of A. The corre­sponding equations can be solved easily by combination of iterations and matrix algebra methods. In what follows, we suppose that the boundaries of the nonper­turbed domain U O3 and of the channel are smooth, and have inner angles at the points of contact.

1. Gotlib’s Type Integral Equations for a Thin ChannelLet = Qjn , 0 3 = Clout be inner and outer domains, joined by a thin

“e-channel” O2 = which is formed by two parallel curves lo, Zi, dist(/o,/i) = e, e <C 1. We assume, that the channel ^2 meets <9£7i and Oils in an orthogonal way at the small flat pieces 7 1 , 73 of dtli and <903 . The Laplace operator we consider here is defined as Friedrichs’s extension of —A in = £ 2(^1 U ^ U O 3). Itmeans,that the homogeneous Dirichlet boundary condition at the common part of boundaries <9f2i fl 00,2 = 7 i, fl <9f}3 = 73 is replaced by the “non-jump” conditions for Ui = w|n :

d^i,3 dn "Y1 Q

du2dn

= 0 ;

Uldf2\(7iU72)

1991 AMS Mathematics Subject Classification: Primary 35P15.

2 0 0 B. PAVLOV

The domain of the corresponding Laplace operator is contained in the Sobolev class W2 ,0(fi) of all functions which posess square integrable first order derivatives and vanish on dti; thus Meixner conditions at the inner angles between <9Qi, and 80,2 are fulfiled automatically. From the very beginning we suppose, that all spectral properties of the Laplacians A* in L2(^i) with homogeneous boundary conditions on d f1i are known. Note, that the spectral properties of Laplacian A 2 in the thin channel can be calculated by a perturbation procedure, if e is small enough. In the two dimensional case, which we discuss here, they can be deduced from the integral equations of Lippmann-Schwinger type, containing the derivative of the holomorphic function, which maps the curved channel fi2 onto the corresponding direct channel.

We consider two cases:1. O3 is a bounded domain.2 . O3 is an unbounded domain with a bounded border dCl3 (O3 contains a

neighborhood of 00).Both fti, fI2 are supposed to be bounded. In both cases the integral equation

for the resolvent kernel can be deduced for any value of the spectral parameter. But in the second case we can deduce for real values of the spectral parameter A the inhomogeneous equation for scattering waves as well.

Let us denote by K f, i f f two discs, centered at Xq, zj^in the middle of 71 , 73 , diamK f = |7*| = e, i = 1,3, d K f = En U E12, E is = 8 K\ fl Cla, s = 1,2; d i f f = E32 U E33, E3s = dKs f l Q s , s = 2,3.

In what follows we omit the parameter e in the notations. We construct the integral equation for the “compensating” term gn of the Green function in Q

G(x, y) = G0{x, y) + gn(x, y),

which satisfies the in homogeneous boundary conditions' - A xg = k2g,

<

, 9n(x ,y )|xean = ~Go(x ,y ) |xGafi-Here Go is the singular solution of the Helmholtz equation in R2 ,

Go{\x ~y\) = l-H l(k \ x -y\ ).

Actually we derive the integral equation for the pair { 51, 52}, 9i = 9 \71» g3 = g |73. The contracting property of the corresponding integral operator we get as a corollary of results of Sections 2,3,4. Now we deduce this equations just formally.

If 9i ) 92 are known, then the values of g on can be calculated as the convolutionof

9 i|71 + (-G °)|aniVri with the Poisson kernel of the Helmholtz equation in fl\ :

. . dG\ . .Vi(x,s) = - — (x ,s ).

o n s

ACOUSTIC RESONANCES IN DOMAINS CONNECTED BY A THIN CHANNEL 2 0 1

Here G\ is the Green-function of the Laplacian in fii with the homogeneous Dirich- let boundary condition at the boundary dCli, y E Qi;

9(x,y) |a:GEi I Vi{x,s)gi(s)ds+ I V i(x ,s)[-G 0{s,y)]dsJ71 Jd f2i\7i

Similarily we get for domains Q3,

^(x’ )̂lxeE33 = / 'P3(x,s)g3{s)ds,

U s 12 = / 'P2(x,s)gi{s)ds+ V2(x,s)g3(s)dsL J 'V j J~y3

xEEn

9(x ’ y)\xez32 = / 'P2{x,s)gi(s)ds+ / V2(x,s)g3{s)d,LJ71 7 73

(1)

(2)

(3)

Here V2, V3 are Poisson kernels in ^2, ^3. If y G Q3, then an inhomogeneous term, similar to the second addenda in the right part of g(x,y)\xe^1, appears in the representation formula (1).

Considering the reduced values of g onto 7 1 ,7 3 we can calculate gi, g3 in form of convolutions with the Poisson kernels VJ* 3 of the Helmholtz equation in the discs Kn

9ifay)\x€yi = { f V i {x , s )g ( s , y )d s + f V 2 {x,s)g(s,x)ds Vt/Sj i JT.2l x€7i

and K 3

53(^,y)|x€73 = { [ V i{x ,s)g(s,y)d s+ f V?(x,s)g{s,x)ds\V. J E 32 " E 33 )

(5)

xG73

Combination of the equations (1 — 5) gives a system of integral equations for <71, <73. This system can be written in the form

+ F , (6)

where T is an inhomogeneous term, defined by the position of y, and K is an integral operator, which proves to be contracting one in the Banach space of all continuous two-component vector functions, defined onto 7 1 , 7 3 :

C7 = C (7i) + C (7 3),

for the values of the spectral parameter k not equal to the eigenvalues of inner or outer problem and the width e of the channel fi2 small enough.

2 0 2 B. PAVLOV

The case y E Cl3 can be treated in a similar way. Generally the integral equations for the perturbed scattered waves in the unbounded domain can be derived the same way with the only difference that the nonperturbed Green function Gq(x , y) should be replaced by the nonperturbed scattered waves in ^3 or just by the exponentials e-ik(x,v) (jf contains some neighborhood of 00).

Formally in the last case the corresponding equation can be deduced from (6) by using the asymptotics of the Green function at infinity in the direction u:

Go{x, y ) ~ e~tk{x'u)Go(0, y), y -> v 00,

Actually it contains the same contracting integral operator /C. In the case when the value of the spectral parameter is singular for one of the domains (fii), the integral equations need some regularization, which includes explicit substraction of singular terms (see below).

2. Estimates of Green Functions and Poisson Kernels of the Helmholtz Equations in fi1? fi2

Lemma 1. The Green function G i(x,y,X) of the homogeneous Dirichlet problem for the Helmholtz equation in fii, —Aw = Au, admits the representation in the following form for A approaching the isolated eigenvalue Ai of the corresponding operator

Gx{x, y) = G - i(ar, y) + — =1 + g(x , y), (7)

where {</?is} is the complete orthogonal system of eigenvectors of the problem

—A ip = Ai <p

in the corresponding eigenspace = Ker{—A — Ai I), and g(x,y,X) is a smooth function, which admits uniform estimates, when x —> y, A —* Ai.

Proof. The proof is based on the iterated Hilbert identity which has the following form:

AT

Gx = G-x + £ ( A + + (A + 1)w+1G "+1Ga , (8)s = 1

Gs_i = G - 1 x G - 1 x . . . G - 1 (s-times). Using the spectral representation for G\, we can write the last term in the right part of (9) as a sum of a singular and regular terms

fvN+ ls~< _ \ __________I_________ ^2________ • /'Q'i(a* + V N+1& ' - A) (A1 + 1 ) N + 1 & 1 - A) ’ 1 jAs

ACOUSTIC RESONANCES IN DOMAINS CONNECTED BY A THIN CHANNEL 2 0 3

here V s are the orthogonal projectors onto the eigenspace jVs, corresponding to the eigenvalues Xs of the Laplacian in Oi. Using the fact, that even the first iteration of the resolvent kernel of the Laplacian in the two-dimensional case gives a continuous function, and the following iterations give smooth functions, we see, that the first addenda in the right part of (9) is an integral operator with the smooth kernel in Qi x Ol5 provided the eigenvalue Ai is isolated. Thus we have the representation:

n , \ r ( \ i ' A + 1 ^ +1 T,t=iVu{x)Vit(y)G\{x,y) = G - i { x , y ) + ' 1 — ^ -------------------Ai + 1J Ai — A

V""' X /r=l lPrp{x)iPrp{y) ( A + 1A p -A VAp + 1

+ f > + l)*GL+1(x,y). (10)

The third and the fourth members in the right side of the last equation are contin­uous functions in f2i x fi2, hence

G\(x,y) = G -i(x ,y ) + ^ ++ \N +1 ^ N i

E t = i <pu(x)<pit(v) Ai - A

N

+ + l ) sGrI'\1(a:, y) + 0 (1),1

where 0(1 ) is bounded uniformly in x ^ , when A is in some neighbourhood of the isolated eigenvalue Ai, which does not contain other eigenvalues. I

Rem ark. The last statement is true actually for any regular differential operator, for which the series of some power N, N > 0 of eigenvalues is convergent

w " < ° ° ’Ap̂ O

and the eigenfunctions are smooth functions. The precise formulations and proofs of relevant results will be published elswhere.

Corollary. A similar representation holds for the Poisson kernel of the Helmholtz equation:

dG\(x, y)r\ r

dnn

N

d G ~ 1 (x,y) + £ > + i r ^ i X G U (x,y)xGdfli ®nx i

N +1 Ni, / A + i Y 1 ^ d < p lr(x) ^( aT + i J a

+ 0 (1). (11)

204 B. PAVLOV

In the case, when O3 is a bounded domain, the same sort of asymptotic esti­mates can be written in O3. The case of an unbounded domain O3 containing a neighborhood of infinity is described by the following:

Lemma 2. Let us consider the Helmholtz equation in O3, with the homogeneous Dirichlet conditions at the the boundary. Let us denote by the scattered waves of the exterior problem for initiated by the plane wave e%k'v'x' , incoming from infinity along the direction v.

Then the behaviour of the Green function of the Helmholtz equation in CI3 , for A approaching Aq ± iO, Aq > 0 is defined by the following formula

Here gd is the contribution, which comes from the discrete spectrum of the problem,

estimate in some neighborhood of any “generic” point of the absolutely continu­ous spectrum of the unperturbed Laplacian, where the scattered waves are smooth functions of the spectral parameter.

The proof will be published elswhere. Actually it follows the same idea of using iterated Hilbert identity, as the previous one (Lemma 1, concerning the nonsingular case) with the only difference, that now the standard formula for the boundary values of the Cauchy-type integral must be used.The statement of the Lemma 2 holds for Schrodinger operators in the exterior domain with the real integrable potentials potentials, and can be generalized for quasiconical domains as well.

Corollary. In the generic case when the scattered waves of the nonperturbed spec­tral problem are smooth near the real positive point A of the spectral parameter, then the Poisson kernel admits the following representation

3. “ A coustic C onductivity” o f the Thin ChannelNote first, that scattering problems for curved channels were studied intensely

last years, see [8 ] and references there. We discuss here the case of the thin channel only. This case is selected by the condition, that the spectral parameter A is chosen below the spectral treshold of the channel.

N + 1 rjjv{x,z) ipv(y,z)z - A0

(12)

and is the integral in a sense of “principal value” over [0 ,oo) with singularity at Aq, Aq > 0. The members, which are staying in the bracket admit the uniform

d G - 1dnx (x'y)±\LX dnx

^ ipv(yAo)dv + 0 ( 1).

ACOUSTIC RESONANCES IN DOMAINS CONNECTED BY A THIN CHANNEL 205

Acoustic conductivity of a thin channel is defined by the behaviour of the solution of the Dirichlet problem in it for the Laplacian with a nonvanishing boundary data only on one of the end sections 71 , or 72 , for instance:

The simple method of estimating “conductivity” of a thin channel is based on maximum principle for elliptic equations, see [9], where A.M. Iljin applies it to the transformed elliptic equation. In our case this approach looks as follows.

Supposing, that the equation of the border of the channel l\ is given in the natural form

where s is the length of arc l\ between xq and x, and n(s) is the normal to h at the point x, which forms a positive frame with the tangent vector, t =

we can pass to the coordinates (s, h), connected with this frame and corresponding to the metric form

^1 — —̂ ds2 + dh2.

The Laplacian in the channel Q2 = {0 < s < L, 0 < h < s} looks now like

Considering an enlarged channel f22 = { 0 < s < L , —<5 < /i < £ + <5}, we define the function

—Aui = Xui,

“ i|7I = ¥ ’i “ i|ana\7.= 0 -

■00 = sin 7r <

which is obviously positive in and vanishing on the part of the boundary fl2l not containing 71. For small e, 6 we have

Replacing u — u\ by v'tpo, we get for v the following equation

1 * ,Av — Xv — —A'tbov — ^0

v Un2\7i = 0 , v|7i = ^ 0 V i .

206 B. PAVLOV

Using the fact, that ^Aipo is a large negative number for thin channel, we see, that the the maximum principle for the solutions of the last equation is fulfilled for any fixed real A if e, 8 are small enough, 0 < e + 28 < 1, since

—— "I- A — A + A;2Wo

7r(e + 2 8)2 < - 1.

Thus we get

|u| < max l^l^o = max sin 1 7rIn i+ i

In 1- !1 + S

7i

Inx sin7r

i+ i

In i+*

sinh k — s + h f* * j

sinh k^L + HJq ^ |

(13)< max |<£>i| Const e hf° dp \ 0 < s < L.

Here Const is defined by the first factor of ipo inside ^ 2-This exponential estimate for the solution of the Dirichlet problem u = u\ serves

to prove the exponential decay of splittings and shifts of resonances for narrow channels. In particular it follows from the estimate above, that the Poisson kernel Px,y decreases exponentially if x € 71 and y is running along the channel to 72 (or vice versa):

\V\(x,y)\ < Const e - fc£dist(x’y),where k£ > — A > 0 and the distance between x, y is measured along the channel. This exponential estimate for the Poisson kernel shows, that the matrix elements of the integral operator K, in the equation (6 ), which correspond to the transmittance of exitations from 71 to 72 and vice versa, are exponentially small in a sense of the C (7 ) norm. Together with the compressing property, which will be veryfied in the next section, it means, that the perturbations of all eigenvalues of the auxiliary problems in fii U ^2 and in Q2 U (see below) produced by the opening the opposite end of the channel O2 are exponentially small.

But the evaluation of these perturbations needs more information. Let us denote by fe the function, which maps conformally the direct channel f 2̂ = W '• 0 < flew < L, 0 < Imuj < e} onto ^ 2- Supposing this function is known, we can replace the Dirichlet problem for u = u\ in O2 by the Dirichlet problem for the conformally transformed problem in the direct channel: u — u o f ein the direct channel fij]

-Au — \u = — - ■AJa — \u — 0 .1/ i M I 2

Then the Green function for the perturbed problem can be found as a solution of the corresponding Lippman-Schwinger equation

G = G0 - \ [ G0\f',\2 Gd2 u.

Investigating this equation by standard methods, we get the following statement.

ACOUSTIC RESONANCES IN DOMAINS CONNECTED BY A THIN CHANNEL 2 0 7

Lemma 3. For any A and e small enough the Green function G of the transformed problem and the corresponding Poisson kernel V satisfies the following asymptotical conditions:

G ~ G o + o f l A I - 4 - A \ min pl )

P ~ P o + o ( \ \ \ - 4 - i ) .\ mm p1J

In particular the leading exponential term of the Poisson kernel V(0, h'\ s, h), s ~ L — e, e/ p <C 1, is given by the expression

1 . 7 rh . nti sinh \ f W ~ XiL - s)- sm — sin-------------- ----- . -------.e e £ sinh^ /fj - A L

4. Auxiliary Problem s and Estimates o f Integral Operators

The integral operator, which appeared in equation (6) is constructed of Poisson kernels in matrix form with elements, which are bounded operators in the space C ( t i ) + C ( t s ):

f P ° (P 1 + P 2) P ? P 2 1

\ V l V 2 V l(V 3 + V 2) } ' ( '

The diagonal terms have a special sense, being connected with two auxiliary prob­lems, which should be discussed before solving the central problems about splitting resonances and eigenvalues. The auxiliary problems are the following:

(i) Problem of attaching of the thin channel to the inner domain Qi. It is the spectral Dirichlet problem in Qi U with the homogeneous Dirichlet boundary condition on d(£2i U f22) = (dCli U dQ 2) \ 7i-

(ii) Problem of attaching the thin channel f]2 to the outer domain CI3 . It is the spectral Dirichlet problem in fi2 U O3 with the homogeneous Dirichlet boundary condition on d(fJ2 U 1)3) = (df22 \ 73 .

The terms staying in (14) outside of the diagonal describe the interaction between these problems. We suppose, that the eigenfunctions of the outer problem are smothly dependent of the spectral parameter, so we discuss only generic case now. At the moment we do not have a proper description of conditions, which guarantee the generic case for a given value A, corresponding to the outer (or auxiliary outer II) problem.

The analysis of auxiliary spectral problems is described in Appendix I and in Appendix II. Here base on this material to describe solutions of two main problems.

208 B. PAVLOV

I. Let us suppose, that the auxiliary spectral problem is solved in fix U O2, and we have a group of eigenvalues {As} of it, which are separated form the rest of the spectrum {Ap}, dist|{As}, {Ap}| > 8e > 0. We suppose, that the separation of eigenvalues of the auxiliary inner problem takes place, if for decreasing e the variable 8e has a lower bound

We call this group {As} asymptotically separated. Practically it means, that the group {As} is formed by a family of “genetically” close eigenvalues of the unperturbed inner problem, which is separated from the rest of the spectrum by the distance ~ 60. We suppose also, that no resonance states of the outer problem are situated in the neighbourhood of the group. Based on the accomplished spectral analysis of the inner perturbed spectral problem, we can study the splitting of resonances, caused by the removing of the wall 73 . The Laplace operator in Cti U O2 U O3 with no walls 71, 73 we call just the “perturbed operator” .

Theorem 1. If there exists a group of eigenvalues of the auxiliary inner problem, which is asymptotically separated from the rest of the spectrum of the auxiliary inner problem and from the resonances of the outer problem, then removing the wall 73 causes the appearance a series of resonances (or, possible, eigenvalues) of the perturbed operator which can be calculated from the integral equation (6). The total number of them is equal to the total multiplicity of the eigenvalues in the separated group.

P roof. Suppose that the spectral analysis of the auxiliary inner problem is accom­plished, and we have a resolvent kernel G™ and Poisson kernel V™ for it, provided A is not an eigenvalue of the auxiliary inner problem. Then we can represent the resolvent kernel G™ in a form similar to one given in Lemma 1,

Using the representation of the Poisson kernel of the outer problem, described in Lemma 2, we get the following integral equation for the boundary value of the scattered wave of the perturbed operator on the wall 73:

6e > 60, e —> 0 .

N E ^ l Vatiy)A — As

, E & i vtP(x) vtpjy) { A + 1 \A — Ap \\p + l )

N

N

5 = 1

y€c (̂niUf22)

(15)

ACOUSTIC RESONANCES IN DOMAINS CONNECTED BY A THIN CHANNEL 209

The operator VQ{V Zg \ ^ + V m g|Sm} is represented in form of a sum of thecontracting operator

if0p = p ° { p ^ | s j3 + ^ 9 ^ } + 0(1)9

and the finite dimensional one with the singular coefficient

A + lM

S=1 Ao + 1

N y-viVs difist d ipatdnx dn

A -A ,— ECOS0 ,

where z = ecos0 is a point on the circle £§, 6 = (n X0, x — x q ), and the inho- mogeneous term is combined of the boundary values of the unperturbed scattered wave.

The proof of the fact, that the operator K q is a contracting one is based on the maximum principle for the Helmholtz equation. Really, for given function g, defined on 73 and A = - 1 we have due to the maximum principle:

^ - i 5|E33 < o sup |y|73

73IE23 2

hence once more using maximum principle we have

IV -ig + V -ig W ^ ^ ^ supM

It means, that the operator K ° is contracting

73

\K°g\ = O 00 <0 + V™xg } 1 , , < 0 SUP 1̂1

£ 3 3 £ 2 3 ) L73

Now the equation (15) can be rewritten in the form

9 = (1 - K 0) - 1^ 9 + (1 - i f 0) " 1 V

(16)

(17)

including a finite dimensional operator K \, dim K\ = Y^L i Na, which depends on A as a rational function. The equation (15) has an unique solution, if

det[l - (1 - K °)~ lK x} ^ 0. (18)

The perturbed scattered waves exist for real values of the spectral parameter, A which satisfy the last condition (18).

2 1 0 B. PAVLOV

Vice versa, it is easy to check, that the complex roots of the determinant (18) coincide with resonances, and real roots can be both resonances or eigenvalues. Prom the Rouchet theorem follows, that the total multiplicity of them is equal to £ s=1 Ns. The corresponding resonance states <p\ can be produced from eigenstates of the “nonlinear” (in A) eigenvalue problem

g = ( l - K ° r 1K x g,

det [1 - (1 - •K'V'A'a] = 0

by Poisson integrals,V s g, x e 03,V mg, i 6 ( f l iU n 2),

which give growing functions, called resonance states, for complex roots of the determinant or bounded and/or decreasing ones for real roots. These functions are solutions of the homogeneous problem, when the spectral parameter coincides with the roots.

The existence and smoothness of the Poisson kernel of the nonperturbed outer problem V 3 is guaranteed, if no resonances of the problem are situated in a neigh­bourhood of the group.

II. The shifts and splittings of eigenvalues produced by attaching the channel can be calculated in a similar way. Let us suppose, that the outer domain is bounded, and there exists a separated group of eigenvalues of the perturbed inner problem and a group of eigenvalues of the outer problem, which both are situated in a domain G on the complex plane of the spectral parameter A and asymptotically separated from the rest of the spectrum of these problems.

Theorem 2. If the group of eigenvalues of the auxiliary inner problem and the group of eigenvalues of the outer problem exist in G and both of them are asymp­totically separated from the rest of the spectrum of both problems, then for the cor­responding joint perturbed problem with the narrow channel in the domain G, there exist a group of eigenvalues, whose common multiplicity is equal to the multiplicity of joint initial group.

Proof. The integral equation for the boundary values of the resolvent can be represented in the form used in the course of the proof of the preceeding theorem, with the only difference, that now we consider the homogeneous equations for the boundary values g% of the solution on the wall 73

ACOUSTIC RESONANCES IN DOMAINS CONNECTED BY A THIN CHANNEL 2 1 1

Combining the contracting part of the corresponding integral operator in form

Kg = [V-x + O(e)] [{P™ + Oin(e)}a|S33 + { V h + 0 3(£)}9|.IS33

and the finite dimensional part in the form

N V ' diPat d Vatdi d n.

A - A tcos Og +

£33

(P - 1 + 0 ( e ) ) £ ( A + 1 A3 + 1

V' 9<Prt ̂<p*t d n x d ny

A - A t= K xg,

£33

we reduce the problem to the situation, observed in Theorem 1. The further proof follows the same way.

The described method of analysis of the perturbation of resonances and eigenval­ues can be applied to the Schrodinger equation in domains, joined by thin channels. Similar problems arise in nanoelectronics, when considering the ballistic movement of electrons in “horizontal” nanoelectronic channels, see [8 ]. One can show, that the shifts and splittings of resonances and eigenvalues are defined in the main order, by the auxiliary spectral problem, and the removing of the “last wall” produces only exponentially small perturbations. In particular, the shifts of resonances, generated by the eigenvalues of the inner problem from the real axis will be expo­nentially small in comparing to the shifts of them along the real axis. The analysis of examples will be published later.

Appendix I. Auxiliary Spectral Problem for the Laplacian in U O2 with Dirichlet Boundary Condition at (df2i Udf^ATi*

In this case the system of integral equations containing the contracting operator, is simplified, since g3 is absent, so we get only one equation to solve for g\. We write it down for the resolvent kernel’s boundary values g:

. 9x = 'P0{V 1g). + Gi}\Si2 + P 0 V 2 9x = Kg + P 0 G i k 12. (19)

If this equation is solved, then the perturbed resolvent can be reconstructed by the solution of the auxiliary Dirichlet problems in f^i, ^2

„ , «>. j C unperturbed(A) ~|- 'Pl(^)gXt % £ ^1>perturbed( ) = j ^ ^ X G Q 2 ,

provided, that A is not eigenvalue of the perturbed problem.

Theorem 1.1. Equation (19) is an equation with a compact operator represented in form of a sum of contracting operator and a finitedimensional operator, de­pending on the spectral parameter as a rational function, if A is situated in a neighbourhood of multiple eigenvalue X\ of the unperturbed inner problem.

2 1 2 B. PAVLOV

P roof. Using formula (11) for the Poisson kernel in the disk K f and a similar formula for fii, we get the representation for the operator K

K = { P _ a + 0 ° (e )} | P i , + ( ^ - )a + i \ w+1

Ai - A+ 0 \ e ) \ ,

where {</?si } are the eigenvectors of the inner operators and O0(e), 0 1(e) are inte­gral operators with small norm, ||0°(e)||c < const £, ||01(e)||c < const e. Using the boundednes of the Poisson operators V°_i, V 2_i in C{6) ^ C'(Ei) we can rewrite the operator K in form

K = V°_1 V 2_ 1

+ V \A + 1 Ai + 1

N +1

+ O0(e)Ai - A

,8 = 1

9 <fsi(y)dn,.

71

+ 0 ° (e )P i , |Sl2 + {P _ ! + O V tfO H e).

Replacing ^si(^)|x€S by cos#, 9 = (x — xo,nxli), we get the following formof the equation for g:

{ l - p ° _ 1-p l1 + 0 (e )}g =

V \A + 1 Ai + 1

N +1

+ O0(e) cos 9 dipsi(x)Ai - A dna

d<Pai(y)71 dny

9 + V °G |7i

The operator on the left part is obviously invertible, hence the whole equation can be reduced to the finite dimensional one

9 = { I - V 0_ 1V 2_1 + O (e)}-1 A + 1 N +1

+ O0(£) cos 9 •

d<Psl(x)Ai - A dnn

9<Psi(y)71 d n V

Ai + 1

9 + { I - ■P°_1P 2_ 1 + 0 ( £ ) } -1 V° G|Ss.7i

Solution of this equation demands calculation of the corresponding eigenvalues as roots of characteristic determinant

det -1 A + 1 Ai + 1

N +1

+ 0 °(e) cos 9 ■

d<ps i(x)Ai - A dn3

9<Psi(y)71 dny 7i

The last can be easily calculated by reducing it to the corresponding matrix prob­lem. If the A under consideration is not an eigenvalue of the described finite di­mensional problem, we have the existence and uniqueness of the solution g.

Conversely, the eigenvalues of the finite-dimensional matrix problem prove to be eigenvalues of the perturbed operator and the corresponding eigenfunctions can be reconstructed from the eigenvectors of the matrix problem.

ACOUSTIC RESONANCES IN DOMAINS CONNECTED BY A THIN CHANNEL 213

Appendix II. Auxiliary Spectral Problem in 0,2 U 3̂Let us consider the spectral problem with the homogeneous Dirichlet boundary

conditions on 8 (0,2 U Q3) = {dfl^ U dCl2)\73- In this case the system of integral equations (7) is simplified due to the absence of <71, g\ = 0, so we get only one equation for the scattered wave

The scattered wave in U ^3 is constructed as a two-component function:

Theorem 1.2. The integral equation (20) possesses a unique solution in the space of continuous functions C(73), provided X is a generic point of continuous spec­trum of the outer problem, i.e. the scattered waves depend smoothly on the spectral parameter in a neighbourhood of A.

Proof. If the spectral parameter A is chosen in such a way that maximum principle is true, e.g. A = — 1, then the operator

is contracting. This fact can be veryfied the same way as the similar statement in the previous section. On the other hand, using the representation (11) for

the Poisson kernel V 3 (A) the following formula, y G dfl3, a: G O2 ^ 3 (A)(x, y) = v t (A + i0)(x, y), A e [0, oo):

g = V0 { V i g + ^ } \ Z3a + V °V 2 g. (20)

Here Vv is the scattered wave in Q3 , and is the boundary value of the Poisson kernel

^ l = tpu + V i g,

I = 9-

Poisson kernels and the boundary values theorem for Cauchy integrals we get for

N + 1 ^ u(x,z) ^ ( y ,z )z — A

and the similar formula for inside the disk

214 B. PAVLOV

All the eigenvalues Ap of Laplace operator in the disk K 3 = i f f with zero bound­ary conditions are much greater, than A, if e is small enough. Hence using the smoothness of iterated resolvent kernels, we can conclude, that in both cases the main part of (A), is given by ^ (_ i) respectively , the rest beingrepresented by operators with smooth kernels on a small domain, since the singu­larities do not destroy the convergence of the integrals:

and hence a small norm ~ s. The same arguments could be applied to ^ 2(A)- Taking into account, that all Poisson operators are bounded in the space of con­tinuous functions, we get

Here 0(e) is an operator with the small norm in C , 3, \0( e ) \ c < const e. Since the sum of a contracting operator and a small operator is a contracting operator as well, we see, that for small e the equation (20) has a unique solution g, which posesses the same generic properties, as the scattered wave i\)v.

Having the representation for g via the contracting operator K :

g = [ I - P ° { V t * \ ^ + ' P 2 * k 33}] 1 P 0 V>,|2, = ( / - A ' r 1P ° ^ | S3, (21)

we can reconstruct the inner and outer components of the perturbed scattered waves, just by solving the corresponding Dirichlet problems in ^ 3- Note, that

the main singular solution.One can calculate the scattering amplitude from In fact, using the asymp­

totics for the resolvent kernel for x —> ujoo

Then the formula for the amplitude follows immediately from the representation for the perturbed scattered wave:

g — {V 0_ 1 + Oo(e)} (_!)<? + 0 3(s)g + Vvj

+ { 'P -i + ^o(e)} { ^ ( - 1 ) 5 + 0 2(e)g|

= P°-i )9 + 0(e)g-

the Poisson kernels for A ^ 0 are decreasing at infinity generally not slower than

G3,x(X’ V) -^ M z ,A )G + A(0 ,y)

and the symmetry of Green functions we have

ACOUSTIC RESONANCES IN DOMAINS CONNECTED BY A THIN CHANNEL 215

Note, that the perturbed amplitude is symmetrical in respect to the unit vectors uj, v. Really, we have the following asymptotics

dipu

gv{x) = [I - K\ 1 V°y/ £ 2 - \ y - X 0\2dny

Here [I — K] 1 = I + K + K 2 + . . . is the integral operator (21), and we can assume, that ~ q-^u- in the generic case. Hence for small e we get

fp ert{v ,u ) = /u n p e rt(^ ,^ ) + on x d, nx

8f = f [I - K] 1 V °\ /e 2 - |y - x0\2 dx3 = j V °y /s 2 - \y - x0\2dx3 + . . . .

Note, that the first member in the right side is the solution of the Dirichlet problem with the given boundary values on the circle E3S

y/e2 - Iy - xQ\2 = ecos[nXo,Q), 9 < fo, e > f

Being reduced onto the diameter 73 of the disk i f f it gives a standard function

V °yje 2 - \ y - x 0\2 = -00 2 . 1

i + l + E m2 ^ (m2 - l )2

which arises as a necessary detail of the suggested construction.

Acknowledgem ent. This work was partly supported by the International Solvay Institute, Brussels, in frames of the contract ESPRIT P 9282 ACTOS, and by the staff research grant from the University of Auckland.

This paper was started during one month stay of the author at the Department of Mathematics of the University Paris 7 with Professor A. Boutet de Monvel, whom the author is deeply indebted for the profound and encouraging discussion of the problem. The first version of the text was presented to the participants of the semester of the Harmonic Analysis and Differential equations at the International Centre for Mathematical Sciences in Edinburgh, July 1994.

The author is grateful to Professors R. Gadylshin, P. Hislop and A. Martinez for materials supplied.

216 B. PAVLOV

References

1. P.D. Hislop and A. Martinez, Scattering Resonances of a Helmholtz Resonator, Indiana Math. J. 40 (1991), 767-788.

2. P.D. Hislop, Singular Perturbations of Dirichlet and Neumann Domains Res­onances for Obstacle Scattering, S.M.F. Asterisque 210**, 1992.

3. R.M. Brown, P.D. Hislop, A. Martinez, Lower bounds on the interaction be­tween cavities connected by a thin tube, Duke Math. J. N 1 (1994), 163-176.

4. R.M. Brown, P.D. Hislop, Eigenvalues and Resonances for Domains with Tubes: Neumann boundary conditions, Ed. Ames, Harrel, Herod, Math, in Sciences and Engineering 192 (1993), in Differential Equations with Applica­tions to Mathematical Physics, Acad. Press,.

5. S. Jimbo and Y. Morita, Remarks on the behaviour of certain eigenvalues on a singularily perturbed domain with several thin channels, Comm. Partial Diff. Eq. 17 (1992), 523-552.

6 . R. Gadylshin, On acoustic Helmholtz resonator and on it’s electromagnetic analog, J. Math. Phys. 35 No. 7 (1994), 3464-3481.

7. V. Gotlib, Scattering by a resonator with the narrow opening as a problem of perturbation theory, DAN USSR 287 N6 (1986), 1109-1113. (in Russian).

8 . P. Duclos, P. Exner, Curvature induced bound states in quantum wavwguides in two and three dimensions, Rev. Math. Phys. 7 NO. 1 (1995), 73-102.

9. A.M. Iljin, Soglasovanie asimptoticheskih razlozhenij kraevyh zadach, M. Nauka (1989), p. 334. (in Russian).

B. PavlovThe University of AucklandPrivate Bag 92019AucklandN E W ZEALANDpavlov@math.auckland.ac.nz