Asymptotic behalrior of the scattering phase for exterior domains

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Asymptotic behalrior of the scattering phase forexterior domainsArne Jensen a , Arne Jensen b & Tosio Kato ca Matematisk Institut , Aarhus Universitet , DK 6000 Aarhus C, Denmarkb Department of Mathematics , University of California Berkeley , 94720,Californiac Department of Mathematics , University of California Berkeley , 94720 ,CaliforniaPublished online: 14 May 2007.

To cite this article: Arne Jensen , Arne Jensen & Tosio Kato (1978) Asymptotic behalrior of the scatteringphase for exterior domains, Communications in Partial Differential Equations, 3:12, 1165-1195, DOI:10.1080/03605307808820089

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COMM. IN PARTIAL DIFFERENTIAL EQUATIONS, 3(12), 1165-1195 (1978)

ASYMPTOTIC BEHAlrIOR OF THE SCATTERING PHASE

FOR EXTERIOR DOMAINS

Arne Jensen

Matematisk I d s t i t u t , Aarhus Univers i te t DK 6000 Aarhus C , Denmark

and Department of Mathematics, Universi ty of Ca l i fo rn ia

Serkeley, Ca l i fo rn ia 94720

Tosio Kato

Department of Mathematics, Universi ty of Ca l i fo rn ia Berkeley, Ca l i fo rn ia 94720

1 , In t roduct ion .

Let " C Rm be an e x t e r i o r domain, by which we mean a connected

m open s e t with C = R \ R nonempty and compact, except when m = 1 . I f m = l , Re i s t h e complement of a compact i n t e r v a l 1 , which

may reduce t o a po in t , C w i l l be c a l l e d t h e obs tac le .

Let He be t h e s e l f a d j o i n t r e a l i z a t i o n of -A (negat ive Laplac-

2 i a n ) i n H = L ( R e ) with t h e Di r i ch le t boundary condit ion. Thus

--e

He > 0 with D(~:/2) = H ~ ( R ~ ) ( t h e Sobolev space ) . Let Ho be t h e

2 m canonical s e l f a d j o i n t r e a l i z a t i o n of -A i n g = L ( R ) , so t h a t

1 m Ho 0 with I)(Hi/') = H ( R ) .

Copyright O 1978 by Marcel Dekker, Inc. All Rights Reserved. Neither this work nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.

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1166 JENSEN AND KATO

I n what follows we s h a l l cons t ruc t t h e S-matrix

f o r t h e p a i r I H ~ , He) , and t h e t t o t a l ) s ca t t e r ing - e ( h ) given

by

Since it t u r n s out t h a t det S (h ) e x i s t s and i s continuous (even ana-

l y t i c ) i n h > 0 with ldet SCA) I = 1 , (1 .2) w i l l def ine 0(A) as

a real-valued continuous function up t o an add i t ive constant VT wi th

V an in t ege r ( c f . [ l ] ) . The p rec i se value of v i s not important

s ince we a r e mainly i n t e r e s t e d i n t h e asymptotic values of @ ( A ) , which a r e l a r g e . Actually a complete determination of V w i l l be

achieved by iden t i fy ing @ ( A ) with - ~ < ( h ) , where < i s t he s p e c t r a l

s h i f t f'unction t o be introduced i n sec t ion 3. - We s h a l l prove t h e following r e s u l t s .

THEOREM I. Suppose t h a t C i s s t r i c t l y s t a r l i k e i n t h e sense t h a t - a Z i s represented by an equation 1x1 = f (w) > 0 , w E Sm-l , with f

continuous. Then

O,3) - i m / 2 e ( ~ ) = nc,lZl r o ( l ) as A + .I ,

-1 where c = ~ 4 ~ 1 ~ / ~ r h / 2 + 11 and I Xi i s t h e Lebesgue measure of Z . PI

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ASYMPTOTIC BEHAVIOR OF THE SCATTERING PHASE 1167

If i n addi t ion f i s smooth of c l a s s c2 , then o ( 1 ) i n ( 1 , 3 ) can

be strengthened t o log A)-') ,

For obs tac les C which a r e not s t a r l i k e we have not been able t o

prove an asymptotic formula l i k e C1.31, but we do have an analogous for -

mula i n t h e averaged sense.

THEOREU 11. Suppose t h a t a C i s smooth of c l a s s c2 . Then

where ~ ( 1 ) i s t he number of t h e eigenvalues not exceeding A of t h e

D i r i c h l e t problem f o r t he i n t e r i o r of 2 .

FEMAFXS. 1. Since it i s known t h a t ~ ( h ) = c m l ~ l h m/2

+ O((10g A)A - ) , L l . 4). shows t h a t -B(X)/* has t h e same asymp-

t o t i c behavior i n an averaged sense.

2. A r e s u l t of t h e form (1 .3) with a smaller remainder term has

been proved by Majda and Ralston [ 2 ] i n t h e case when m i s odd and C

i s s t r i c t l y convex and smooth. Stronger r e s u l t s with higher order ex-

pansions were announced by Buslaev [ 3 ] , but i t seems t h a t a complete

proof has not been published. Our proof given below depends on t h e

combination of various ideas , inc luding Krein 's s p e c t r a l s h i f t funct ion

[4, 5 , 6, 71 es t imates f o r t h e t r a c e norm of t h e per turbat ion f o r

t h e heat kernel extending the r e s u l t s of De i f t and Simon [ 8 ] , t h e invar-

iance p r i n c i p l e and monotonicity f o r t h e s c a t t e r i n g matrix 11 , 9 , 101,

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and a Tauberian theorem due t o Freud 1111. The proof i s q u i t e simple,

but it seems d i f f i c u l t t o go beyond Q ( (Jog h )-l) i n the e r r o r es t imate

i n (2 .3 ) by our method,

3 , Our s p e c t r a l parameter h represents t h e "energy" and equals

t h e square of t h e wave number used i n [ 2 ] . The S-matrix (1.1) i s defin-

ed by abs t r ac t s t a t iona ry s c a t t e r i n g theory without d i r e c t reference t o

t h e wave o r t h e ~chr 'ddinger equation o r anything depending on t ime,

Since t h e wave equation i s of t h e second order i n time and i s "doubled"

when transformed i n t o a f i r s t - o r d e r system, t h e phase 8(X) may appear

doubled in t h e theory of t h e wave equation [ c f . 21.

2 Basic es t imates & const ruct ion of t h e S-matrix, .,.-. ..,.

I n what follows nothing i s assumed on t h e smoothness of acie = a z

unless e x p l i c i t l y s t a t e d otherwise.

To avoid t h e inconvenience t h a t t h e s e l f a d j o i n t opera tors Ho and

He a c t i n d i f f e r e n t Hi lber t spaces, we introduce t h e following bounded

opera tors on & :

where $ c o r r e s p ~ n d s t o the d i r e c t sum decomposition

2.1. Let A = e- I X I (operator of mul t ip l i ca t ion ) --.c.

w e have f o r s u f f i c i e n t l y small t > 0 , Then

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ASYMPTOTIC BEHAVIOR OF THE SCATTERING PHASE

( 2 . 3 ) Got - Gt = ADtA w i t h Dt E Eil(N) ,

where B ( g ) i s t h e t r a c e c l a s s , I n p a r t i c u l a r Got - Gt E ~ ~ ( & l . 1

L E ~ 2.2. ~ f ac i s of c l a s s c2 , then

The proof of t h e s e lemmas i s p u r e l y t e c h n i c a l and w i l l be given

i n Appendix a t t h e end of t h e paper . ( s e e ( ~ 6 . 3 ) , (A5,5) . )

S i n c e Got - Gt ~ ~ ( f l ) by Lemma 2.1, t h e ( g e n e r a l i z e d ) wave o p e r a t o r s W: = Wi(Gt, G ) e x i s t and a r e complete ( s e e , e , g . , [ g ] ) . O t

S i n c e Got i s s p e c t r a l l y a b s o l u t e l y con t inuous , Wf: a r e i s o m e t r i c

on w i t h ranges i d e n t i c a l w i t h t h e subspace of a b s o l u t e cont inu-

i t y H f o r Gt . S i n c e Gt = 0 on $+ = g 3 & , we have I i -&c a c

C . i io t ing t h a t t n e map e d t I-+ ), i s monotone d e c r e a s i n g , he con-

c lude from t h e i n v a r i a n c e p r i n c i p l e t h a t t h e complete ( two-space) wave

o p e r a t o r s W+ = W+ (lie, h9; 3) e x i s t and e q u a l W; , where J i s t h e - -

p r o j e c t i o n of & onto $ , The s c a t t e r i n g o p e r a t o r S = W:W- ex-

i s t s a s a u n i t a r y o p e r a t o r on i . The S-matrix S(X) = S ( h ; Ife, H ~ )

i s t h e n g iven a s a d i r e c t i n t e g r a l decomposition of S adap ted t o

t h e one f o r Ho . F u r t h e r necessary p r o p e r t i e s of S ( X ) w i l l be deduced f r ~ m t h e

s tudy of t h e s p e c t r a l s h i f t f u n c t i o n ( s e c t i o n 3 ) .

RE.% 2 , 3 . A g e n e r a l t h e o r y of two-space s c a t t e r i n g was g iven i n

1101. The e x i s t e n c e and completeness of W+ were proved i n [ a ] under

t h e assumption t h a t aZ bas measure z e r o . A c t u a l l y t h i s c o n d i t i o n i s

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not necessary s ince Lema 2 , l i s t r u e f o r any 51, , I n i81 t h e ope ra to r -tH1

Glt = e (see (-5.21) i s used in s t ead of Gt .

3. The s p e c t r a l s h i f t T ~ c t i ~ n ,

For a summary of Kre in l s theory of t h e s p e c t r a l s h i f t funct ion used

below, we r e f e r t o Appendix, s ec t ion A l .

Since Gt - Got E B ~ ( K ) by Lemma 2 .1 , t h e s p e c t r a l s h i f t funct ion

1 i s defined, St i s red-Valued and belongs t o L (.-w, m) , with

( 3 . 2 ) St&] = 0 f o r v '$ 10, 11 ,

s ince t h e s p e c t r a of Got and Gt a r e subse ts of [ 0 , 1 1 .

LEMMA 3.1. There i s a unique real-valued func t ion 5 wi th t h e - fol lowing p r o p e r t i e s . (We c a l l 5 t h e s p e c t r a l s h i f t funct ion f o r C ) .

w i t h t r a c e e q u a l t o

Q

.ne 5 unique-

l y . Indeed, t h e d i f f e r ence 9 o f m y two E I S with t h e s e p rope r t i e s

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m

w i l l s a t i s f y ( ( ' (h)i l(h)dh = 0 f r a $ - ) , Hence i s - IX)

a c ~ n s t a n t , which must be zero because ~ ( 1 ) = 0 f o r 1 < 0 by ( a ) .

To prove t h e exis tence of 5 , it s u f f i c e s t o f i x a t > 0 and

s e t < ( A ) = -St(e -It) so t h a t LC) i s s a t i s f i e d . Then (a) follows

from ( 3 . 2 ) . To v e r i f y (.b), s e t

w Then $ l E ~ o ( - w , m) with supp $l C ( 0 , W) . Application o f ( ~ 1 . 5 ) i n

Appendix then g ives

a s requi red . Because of t h e uniqueness proved above, 5 thus defined

does not depend on t ,

Proof. I n view of Lemma 3 . l ( c ) , it s u f f i c e s t o show t h a t i s - r ea l - ana ly t i c i n p f o r 0 < p C 1 . t t ( p ) i s given by ( s e e ( ~ 1 , 2 ) )

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Due t o the f a c t o r i z a t i o n (-2.31, we may wr i t e

But

where E ~ ( A ) i s t h e s p e c t r a l family f o r Ho . Since M ~ ( X ) has ana-

l y t i c continuation i n a neighborhood of t h e p o s i t i v e r e a l a x i s , it f o l -

lows t h a t A ( G ~ ~ - ,)-'A can be continued a n a l y t i c a l l y i n z from the

upper half-plane across t h e r e a l i n t e r v a l ( 0 , 1) . We denote t h e re-

+ suiting ana ly t i c funct ion by QOt(.z) .

Since Dt E B ~ ( K ) by Lemma 2.1, D ~ Q ; ~ ( Z ) i s a 3 (&)-valued ana. 1

l y t i c funct ion and [ I - D ~ Q ; ~ ( Z ) ] - ~ i s meromorphic. I n view of ( 3 . 4 ) ,

t h e lemma w i l l be proved i f we show t h a t t h i s meromorphic funct ion has

no poles on t h e i n t e r v a l 0 < P < 1 .

by t h e second resolvent equation, it su f f i ces t o show t h a t

-tH e But t h i s i s obvious because Gt F e @ 0 has no p o s i t i v e eigenvalues;

indeed, He has no e i ~ e n v a l u e s due t o Re l l i ch ' s theorem.

Inc iden ta l ly , we have shown t h a t

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(3,i') E t ( P ) = n - l a r g d e t [ l - D ~ Q : ~ @ ) I , 0 < P < 1 ,

where t h e determinant i s nowhere ze ro ,

LEPIMA 3.3 ( s i m i l a r i t y l aw) . Let t a denote t h e s p e c t r a l s h i f t - f u n c t i o n f o r ax ( t h e d i l a t i o n o f C by a f a c t o r a > 0 1 . Then

<"(A) = t ( a 2 ~ ) , x > o .

P r o o f . Let Ua be t h e u n i t a r y d i l a t i o n o w : -

2 2 a I t i s easy t o s e e t h a t u ~ H ~ u ; ~ = a HO and uaHeuil = a He , where

HE i s t h e e x t e r i o r o p e r a t o r f o r t h e o b s t a c l e d ; n o t e t h a t HZ a c t s

i n t h e subspace L * ( & ~ ) = UaI& C & . Thus t r a n s f o r m a t i o n by ua sends Got = e - t H ~ i n t o G w i t h

0 s 2

s = 8 t and Gt = e-"k 8 0 i n t o G: ( w i t h obvious n o t a t i o n ) , s o t h a t

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Proof. I f m = 1 , C i s an i n t e r v a l w i th l eng th a 2 0 , - A d i r e c t computation shows t h a t

from which t h e a s s e r t i o n fol lows. Assume now t h a t m P 2 , I n view

of Lemma 3 , 3 , it s u f f i c e s t o show t h a t l i m < ' (A) = 0 f o r a f i x e d a+ o

A > 0 . Lemma A7.1 ( ~ p p e n d i x ) shows t h a t l l~tal l -+ 0 a s a 0

f o r a f i x e d t > 0 , where D: i s t h e Dt f o r t h e obs t ac l e ax . Since <'(A) = -~:(e-'~) , t h e des i r ed r e s u l t fol lows from ( 3 . 7 ) .

2 L E : ~ 3,5; 1f ac i s smooth (o f c l a s s C , then -

= trtGOt - Gt ) by (.Ale 3 1, Thus (3,9) fol lows from Lemma 2.2.

4. Proof of Theorem 1.

According t o K r e i n t s theory b e e ( ~ 1 . 6 ) ) we have

(4.1) det S (V; G ~ , G ~ ~ ) = exp[-2.rriSth)l f o r a . e . p E [0 , 1 1 ,

where 10, 11 i s t h e spectrum of Got , which i s abso lu t e ly continu-

-1 - A t ous. Since d e t S ( . P ; G t , G o t ) = d e t S ( ? \ ; H e p H 0 ) f o r P < e by

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t h e invar iance p r inc ip l e f o r t h e s c a t t e r i n g operator ( see s e c t i ~ n 21, we

obta in by (1 .2) and Lemma 3 . l ( c )

Since <(A) i s continuous (even a n a l y t i c ) i n A E ( 0 , a ) by Lemma 3.2,

we may s e t

Actually t h i s i s a de f in i t i on of 0 a s f a r a s t h e determination of an

i n d e f i n i t e constant inherent i n 0 i s concerned ( see sec t ion 1). (4.3)

impl ies , f o r example, t h a t 0 ( A ) + 0 as A C 0 i f m > 2 ( s e e Lemma

3 . 4 ) .

Suppose now t h a t i s s t r i c t l y s t a r l i k e wi th a C smooth. By a

monotonicity theorem b e e 111; c f . a l so Helton and Ralston 112] ) , -6(X)

i s monotone increasing i n A SO t h a t t h e same i s t r u e of < ( A ) , Hence

5' (J ) 2 0 and

by Lemmas 3.4 and 3.5. It follows from a Tauberian theorem by Freud

1111 t h a t

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with c a s given i n Theorem I, I n view of (4 .3 ) , t h i s proves the m

second p a r t of Theorem I.

The f i r s t p a r t i s then proved by approximating C by obs tac les

C1, C2 with smooth boundaries such t h a t C l C C Z 2 and lZ1l , I C 2 1

a re a r b i t r a r i l y c lose t o 121 ; t h i s i s poss ib le because 1 i s s t r i c t -

l y s t a r l i k e . Since -el < -8 < -e2 by t h e monotonicity theorem men-

t i oned above (wi th obvious n o t a t i o n ) , t h e des i red r e s u l t follows e a s i l y

from t h e second p a r t of t h e theorem j u s t proved.

4.1. Since Krein ' s genera l r e s u l t ( ~ 1 . 6 ) on the r e l a t ion -

sh ip between < ( A ) and det s C A ) was s t a t e d i n 161 without proof, it

would he w ~ r t h while t o note t h a t i n the present case, ( 4 , l ) follows

d i r e c t l y from ( 3 , ~ ) and a fornula f o r t h e S-matrix given i n [l].

4 , 2 , Because of t h e monotonicity theorem, < ( A ) must be

m~nwtone increas ing i n C whether Qr not it i s s t a r l i k e . This f a c t i s

not v i s i b l e i n ( 3 . 7 ) , i n which Dt = A ( G ~ ~ - G ~ ) A i s not necessa r i ly

monotone i n C , It i s t r u e t h a t t h e kernel f o r Gt i s pointwise mono..

tone decreasing i n C , but t h i s does not imply t h a t Gt i s monotone

i n C i n t h e sense of t h e operator order ( see [8]). The proof of mono-

t o n i c i t y f o r @(A) given i n [ l ] i s based on another expression i n which

the semigroups Got , Gt a r e replaced by t h e resolvents , which have

t h e monotonicity proper ty . On t h e o ther hand, t h e analog of Dt f o r

t h e resolvents i s not i n B ~ C ~ ) Cexcept f o r m = 1 ) and i s u se l e s s i n

Kre in f s theory

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5 , Proof of Theorem I.

I n what follows we assume t h a t C has c2 boundary, but i t need

not be s t a r l i k e ; i n p a r t i c u l a r C may consis t of a f i n i t e number of

separa ted p a r t s . We denote t h e i n t e r i o r of C by R i . We introduce t h e se l f ad jo in t operator

where He i s a s before and Hi i s t h e s e l f a d j o i n t r e a l i z a t i o n of -A 2 2

i n gi = L ( ~ 1 L (Qi) with t h e Di r i ch le t boundary condit ion. We s e t

L E W 5.1. With A = e-'XI as i n Lemma 2.1, we have -

The proof i s given i n Appendix b e e (A5.2) 1.. It should he noted

t h a t Glt i s c lo se r t o Got than Gt i s , so t h a t l l ~ ~ ~ l l ~ i s smaller

t h a t l l ~ ~ l l ~ (which i s ~ ( t - ~ ' ~ ) see (A5.3)) .

We now repeat t h e argument i n sec t ion 3 with Gt replaced by

%t ' On s e t t i n g <l t (y j = < ( y ; Glt, G o t ) and then Sl(h)

= -<,,(e -") , w e see t h a t Lemma 3.1 i s t r u e f o r c l (h) wi th

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@(He) @ 0 i n (b 1 replaced by $ ( H ~ ) . Lemma 3.2 i s no longer t r u e ,

however, s ince t h e meromorphic funct ion [ l - D ~ ~ Q : ~ ( Z ) ] - ~ has poles

on t h e r e a l a x i s ,

The r e l a t i o n s h i p between 5 and t1 can be deduced from Lemma

3 . l ( b ) and i t s analog f o r El j u s t mentioned. We have

Since Hi has a pure d i s c r e t e spectrum bounded from below and s ince m

$ E c0(-m, m ) i s arbitrary, it follows t h a t < ( A ) - Cl(h) = N ( X ) i s

t h e number o f eigenvalues f o r Hi not exceeding ? . Thus we have

by (4.31

To complete t h e proof of Theorem 11, it s u f f i c e s t o show t h a t

To t h i s end we use t h e es t imate

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But t h e l e f t member of ( 5 . 7 ) i s l a r g e r than

Thus t h e des i r ed r e s u l t ( 5 . 5 ) follows on s e t t i n g t = h-I . 5 It i s of sdme i n t e r e s t t o note t h a t when rn = 1 , -

I C I = a 0 , S1(A) i s a sawtooth-like bounded funct ion wi th 1 / 2

2 2 2 < S1(?,) 3/2 f o r A > 0 , with jumps by -1 a t ?, = A n = n T / a

n = 1, 2, . , . , ( see ( 3 . 8 ) and (-5.4)) . More i n t e r e s t i n g perhaps i s t o

change t h e boundary condit ion f o r t h e e x t e r i o r pa r t He (but not t h e in-

t e r i o r pa r t Hi ) from t h e Di r i ch le t t o Neumann condi t ion . The r e s u l t -

M ing s h i f t funct ion C1(A) d i f f e r s from S1(A) by t h e negat ive of t h e

Heaviside function ( c f . [ b ] ) , so t h a t 2 ( A ) 1 2 . The Laplace

transform ~:(h)e- '~dh has asymptotic expansion o as t 1 0 . I 0 -1 t

This means t h a t t h e t h e t a funct ion C e f o r t h e i n t e r i o r D i r i c h l e t

problem has p rec i se ly t h e same asymptotic s e r i e s as t h e " e x t e r i o r t h e t a

function" SN(h) e - h t d ~ f o r t h e Neumann boundary condi t ion . r 0

APPENDIX

A l . The s p e c t r a l s h i f t funct ion ,

Fle c o l l e c t here bas i c r e s u l t s on t h e s p e c t r a l s h i f t funct ion due t o

Krein 14, 51 and Birman-Krein [ 6 ] , i n s o f a r a s t h e s e a r e necessary f o r

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our purposes. (We note t h a t t h e s e papers conta in more gene ra l r e s u l t s . )

Let A1 , A2 be bounded s e l f a d j o i n t ope ra to r s i n H i lbe r t space

wi th

( A l . 1 ) A2 - Al = Y E %(&) ( t h e t r a c e c l a s s ) .

Then t h e s p e c t r a l s h i f t funct ion

-1 (m.2) f ( h ) = 5 th ; A2, All = T l i m a rg d e t [ l + V ( A - h - is)- ']

ES 0 1

e x i s t s f o r a . e . ), E (.-a, W ) and has t h e fol lowing p r o p e r t i e s .

outs ide t h e smal les t i n t e r v a l conta in ing t h e s p e c t r a of A1 , A2 -

For any complex-valued funct ion 4 E C;(-W, w ) , @(A2) - $(A1) i s i n B1@) ,

with t h e t r a c e equal t o r 4 L ( h ) f ( h ) ~ . rFo

(~1.6) e x p ( - m i f ( A ) ) = det S ( h ) f o r a . e . A E ul,aC ,

where s (A) = S(X ; A*, ,A1) i s t h e S-matrix f o r t h e p a i r A1 , A2 and

'1 ,ac i s t h e spectrum ~f t h e abso lu t e ly continuous p a r t of A1 . Here

it i s implied t h a t S (A) - 1 i s i n t h e t r a c e c l a s s f o r a . e , E O l m a c ,

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ASYMPTOTIC BEHAVIOR OF THE SCATTERING PHASE 118 1

RFMARK Al .1 , 1, I f dim 5 < , < ( A ) i s e q u a l t o N1(X) - N2(A) - where N (1) i s t h e number ~f e igenva lues A f o r A

J '

2 , (A1,2) i s not e x p l i c i t l y g i v e n i n 14 , 5, 61 bu t i t s ana log f o r

a p a i r of u n i t a r y o p e r a t o r s i s g i v e n , I n any c a s e it f o l l o w s e a s i l y

from o t h e r formulas proved t h e r e . Note t h a t a r g d e t l l + Y ( A ~ - L)-'] i s w e l l d e f i n e d by t h e condi t ion t h a t it should t e n d t o z e r o as

Im 5 -+ + m .

3 , I n ( ~ 1 , 6 ) it should be no ted t h a t S ( h ) makes s e n s e o n l y f o r

%,ac whi le <(A) i s d e f i n e d f o r a . e . E (-W, .

A2. A lenma f o r t h e h e a t equa t ion .

We need t h e fo l lowing lemma i n s e c t i o n A 3 .

LEMMA A2.1, Le t R b e a domain ( e x t e r i o r o r i n t e r i o r ) i n R~ -L-

w i t h compact smooth boundary r , Let u be a s o l u t i o n of Lu

= u - Au = 0 on ( 0 , m ) x fL , bounded and cont inuous on [ o , x 0 t -m/2

w i t h u ( 0 , X) = 0 . I f u t t , x) <t e ~ ~ ( - f 3 ~ / 4 t ) on 1 0 , W) x r -m/2

w i t h some cons tan t 6 > 0 , t h e n u ( t , x ) 4 c ' t exp(.(l + < ) / c t )

on 10, a ) x , where Px = d i s t t x , r ) and c , c1 a r e p o s i t i v e con-

s t a n t s depending only on fi and @ .

P r o o f , S ince i s smooth and compact, t h e r e i s a f i n i t e s e t

( $ 1 C f\ 5 such t h a t each z E r i s w i t h i n d i s t a n c e !$ o f some of

t h e yj . S e t Dow

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Then Lv = 0 on (-0, m , x with v ( 0 , x ) = 0 , Moreover, u < v on 2

10, m) x r becmse z E r implies t h a t 1 z - yj 1 < b2 t o r spme j . It follows from a maximum pr inc ip l e ( see P r o t t e r ana Weinberger 113, p .

1831) t h a t u v on 10, x 6 . This g ives t h e des i red r e s u l t

2 s ince t h e r e i s c > 0 such t h a t c l x - yJ l 2 2 4(l + px) f o r a l l x E b

and a l l j .

A3, Estimates f o r t h e heat kernels .

I n t h i s s ec t ion we assume t h a t fie C F? i s an e x t e r i o r domain w i t h

boundary I. (compact and) smooth of c l a s s c2 . Let 4 = F?l\ 4 so

t h a t r i s t h e common boundary of Re and ni . ni need not be con-

nected but must cons is t of f i n i t e l y many components.

Let go , ge , and gi be t h e heat kernels f o r t h e domains R~ , fie , and f$ respect ive ly , f o r t h e Di r i ch le t boundary condi t ion . I n

p a r t i c u l a r

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t h a t

For s m d i t > 0 , k ( t ; x , y ) i s l a r g e i f x , y a r e c l o s e t o r bu t

smal l o therwise . It i s t h e purpose of t h i s s e c t i o n t o e s t i m a t e k

more p r e c i s e l y .

Let

S i n c e r i s c2 , t h e r e i s 6 > O such t h a t x * px i s c1 on

If 6 i s chosen s u f f i c i e n t l y s m a l l , we can d e f i n e t h e image x* o f

x E r6 such t h a t

The msp x I+ X* i s c1 and i s o n t o r w i t h x** = x . I n what 6

f o l l o w s we fix a 6 with t h e s e p r o p e r t i e s . Also we w r i t e

Then x E R: impl ies x* E f2; and v i c e versa.

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JENSEN AND KATO

LEWlAA3.1. There a r e c , c' > 0 such t h a t f o r t > 0 ; -

2 2 ( i i1.8) k ( t ; x , Y ) < ~ ~ t - ~ / ~ l e ~ ~ ( . - l x - y*[ / c t ) + exp(- (1 + p x ) / c t ) ]

f o r x E Q e , y E R C , o r x E n i , y t f i ; ;

CA3.9) k ( t ; x , y ) <c ' t -m/2exp(- (2 + px 2 + ~ , 2 ) / 2 C t )

f o r x, y E R" o r x , y E Qr .

This lemma i s not sharp regarding t h e cons tants c , c ' , but it

s u f f i c e s f o r our purpose. The fo l lowing proof i s admittedly not very

e legant but i s elementary.

F i r s t we no te t h a t ( ~ 3 . 9 ) fo l lows from ( ~ 3 . 7 ) because k ( t ; x , Y )

i s symmetric i n x, y , To prove (J.3.7) we note t h a t i f y E 0; ,

k ( t ; X , y ) = g o ( t ; x, y j G ( 4 ~ t ) - ~ ' ~ e x ~ ( - 6 ~ / 4 t ) , x E I. . Since uCti XI = k(.t; x , y ) i s a s o l u t i o n of Lu = u -A u = 0 on t

(0, m, ne wi th uCQ, x). -- 0 , b 3 . 7 ) fol lows from LeramaA2.1. The

same i s t r u e w i th subsc r ip t e rep laced by i . The proof of W.8) i s more complicated and r equ i r e s some prepara-

t i o n s . S ince t h e image y* of y r6 i s not a well-adapted t o o l f o r

t h e heat equat ion , we f i n d it necessary t o supplement it wi th a d i s t r i b -

u t ed imaue - {y*u)} . This i s a f i n i t e s e t of po in t s near yX , de-

f i n e d by

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where E i s an a r b i t r a r y (but f i xed ) number such t h a t 0 5 E < 2

{q(l) , . . . , q(m-l)) i s an orthonormal system i n the t a n g e n t i a l hyper-

plane t o r a t 7 , and where '(.-j) = - q ( j )

j = 1, ..., m - 1 . The s e t { Y * ( ~ ) I depends on t h e choice of t h e orthonormal system

{ q ( J ) } , but t h i s does not matter . It may be chosen a r b i t r a r i l y and

a t random f o r each y E r 6 '

LEW A3.2. There i s 6 > 0 (depending on r , 6 , and E ) - 1 ,'

such t h a t f o r any y E and z E r with l z - < 61 , we have

The proof i s simple and may be omitted; it i s e s s e n t i a l here t h a t

r i s c2 . LEMMA A3.3. There i s 6 > 0 (.depending on r , 6 , and E )

such t h a t f o r any y r6 . z E r , and t > O ,

P r ~ o r . I f I z - C S1 , t he r e s u l t follows from Lema A 3 . 2 ,

If I Z - , then l z - yl 2 6 f o r some constant 8 > 0 so

t h a t t h e r e s u l t i s t r u e .

We now complete t h e proof of (A3.8). Let y E Q: and consider

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JENSEN AND KATO

v i s a s o l u t i o n o f Lv = 0 on ( 0 , m, x Re w i t h v ( 0 , x ) = 0 . Fur-

-m/2 2 t h e r more, Lemma A3.3 shows t h a t v ( t , e ) ( 4 W ) exp( -6 / h t ) f o r

z E r , I t fo l lows from L m a A2.1 t h a t v ( t , x )

c't 'm'2exp(-(l + ( ) / c t ) f o r sone c , c 1 > 0 . On t h e o t h e r hand it

i s e a s y t o s e e t h a t lx - y* I / lx - y*( 1 i s bounded f o r x Qe and

y E fi: , Thus ( ~ 3 . 8 ) fo l lows a f t e r an ad jus tment of t h e c o n s t a n t c > 0

~ 4 . Some I i i lber t-Schmidt norms (smooth boundary) .

We now e s t i m a t e t h e Hilbert-Schmidt norms 11 11 f o r some opera-

t o r s . As i n s e c t i o n s 2 and 5 , we d e f i n e t h e i n t e g r a l o p e r a t o r s A H O - t H - t H - t H -t1-; . G o t

= e , Gt = e @ O , and Glt = e l = e e @ e f o r

t > 0 . Gat has k e r n e l go , Gt has k e r n e l ge @ 0 , Glt has

k e r n e l ge @ gi , and Kit = G o t - Glt had k e r n e l k g iven by

b 3 . 2 ) ,

We i n t r o d u c e .an o p e r a t o r of m u l t i p l i c a t i o n ;

Our b a s i c e s t i m a t e s a r e g iven by

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To prove ( ~ 4 . 2 ) ~ we compute

We d i v i d e t h e i n t e g r a t i o n d ~ n a i n i n t o f o u r p a r t s Re X Re , Ri X Ri ,

ile x Ri and Ri x Re . ( a ) F i r s t we c o n s i d e r t h e i n t e g r a i on Re X " . Looking a t

t h e e s t i m a t e s g iven by Lemma A3.1, we s e e t h a t only t h e c o n t r i b u -

t i o n from x , y E RL i s in ipor tan t . Indeed , i f x , y a r e bo th i n

a:, ( ~ 3 . 9 ) shows t h a t t h e c o n t r i b u t i o n t o (~4.4) i s of t h e o r d e r

If x c Q; a d , Re1 , we have (~3.8) w i t h I X - y * I > & > 6 s o t h a t t h e i n t e g r a t i o n i n x g i v e s a q u a n t i t y of o r d e r (~4.5).

S i n c e Q L i s bounded, i n t e g r a t i o n i n y does no t change t h i s

o r d e r . The sene i s t r u e by symmetry f o r t h e c o n t r i b u t i o n from

x E a : , y € R e U *

I n t h e remaining p a r t x , y E ) we a g a i n u s e (~3,8) b u t

n o t e t h a t t h e l a s t t e r m on t h e r i g h t of (~3.8) a g a i n c o n t r i b u t e s a

q u a n t i t y of o r d e r ( ~ 4 ~ 5 ) . Thus it remains t o e s t i m a t e

If we i n t e g r a t e i n

from t h o s e x w i t h

x E f o r a f i x e d y E fi' t h e c o n t r i b u t i o n e '

Ix - I 8 26 i s a g a i n o f t h e o r d e r (.~4.5) and

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t h i s does no t change a f t e r i n t eg ra t ion i n y E R: because 0 ' i s

bounded. Thus it s u f f i c e s t o consider those x with Ix - y1 <26 ,

For t h i s small domain f o r x , we may introduce a coordinate

t ransformat ion so as t o make t h e r e l a t e d pa r t of r f l a t . I n t h e

new coordinate system we may s t i l l use t h e same expression a s i n

( ~ 4 . 6 ) , s ince / x - y* I f o r t h e new and o ld system do not d i f f e r i n

t h e order of magnitude and t h e same i s t r u e of PY ' In the new

system, t h e in t eg ra t ion va r i ab le s a re separated. In t eg ra t ion i n x

i n t h e t a n g e n t i a l d i r ec t ion cont r ibutes a f a c t o r t(m-1)'2 , The

remaining i n t e g r a l i s rnajorized by

wi th i n t e g r a t i o n in dpy already included. The remaining in tegra-

t i o n i n y i n t h e t a n g e n t i a l d i r e c t i o n , which i s roughtly equiva-

l e n t t o mul t ip ly ing by t h e surface a r e a of r , does not change

t h e o rde r . ( T O be more p r e c i s e , one shouldin t roduce an appropr ia te

p a r t i t i o n of un i ty on r6 . ) Altogether we obta in

~ ( t ' ~ t ( ~ - ~ ) / ~ t ] = ~ ( . t - ( " - ~ ) / ~ ) - a s t h e cont r ibut ion t o ( ~ 4 . 4 ) from

9 " Q e . (4) Contribution from ni x ni can be handled exac t ly i n t h e

same way as above.

(.c) I n t h e pa r t s a x fLi and ni x fie , we have

Hence t h e computation i s almost t h e same as i n ( a ) ; note t h a t x , y

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a re now s i t u a t e d on t h e oppos i te s i d e of r . We may omit t h e de-

t a i l .

2 S w i n g up, a l l con t r ibu t ions t o I ~ M ~ K ~ ~ M ~ ~ ~ a r e of t h e order

o(t-(m-l) /2 ) . This proves ( ~ 4 . 2 ) .

To prove (p4 .3 ) , we have t o compute

The expression i n [ 1 on t h e r i g h t

Hence in t eg ra t ion i n y i n (~4.7) produces a dominant f a c t o r cons t .

1/ 2 tm/2 , Subsequent i n t e g r a t i o n i n x gives a f a c t o r c o n s t , t . -m m/Zt1/2) = OCt'("-1)/2) , This Thus ( ~ 4 . 7 ) i s of t h e o rde r 0 ( t t

proves (Ah. 3 ) .

A5. Some t r a c e n o m s and t r a c e s (smooth boundary).

We a r e now i n a pos i t i on t o e s t ima te t h e t r a c e norms r equ i r ed

i n t h e t e x t , We have as t + 0

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( ~ 5 . 3 ) IIealx ( G ~ ~ - G~ )ealxl = oit-m12) 9 (a 0)

( ~ 5 ~ 5 ) t r ( ~ ~ ~ - G t ) has t h e same asymptotic form a s ( ~ 5 . 4 ) .

The proof of ( ~ 5 , l ) depends on a t r i c k due t o Deift-Simon [8].

Usin@; t h e senigroup proper ty of Got and G w e ob t a in , wi th t It

= 2s ,

Since M 2 t ' = M4s = ~i~~ , it fa l lows from (114.2-3) t h a t

note t h a t IIMG:II < 1 and t h a t IIK'G 14 11 < I I M ~ ~ G ~ ~ M ; ~ ~ ~ ~ because s 1 s 4s 2

0 .< g1 < go pointwise. This proves ( ~ 5 . 1 ) .

( ~ 5 . 2 ) fol lows from ( ~ 5 . 1 ) s i nce eaJXIM-I 2 t

expIalx 1 - px/(2t)1'2] i s a uniformly borinded opera tor f o r small

Before proceeding f u r t h e r , i t i s convenient t o no te t h a t i f

Pi denotes t h e p ro j ec t ion of K onto H+ ,

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Also we i n t r g d u c e t h e decompositions

(A5.8

where

(A5.7

t h e remainder term Yt i s t h e same i n t h e two e q u a l i t i e s of - t H e - t H - t H

) , s i n c e Glt = e $ e and G = e e ~ O . t

s i n c e ealXlpi i s a bounded o p e r a t o r , ( ~ 5 . 6 ) and (115, €3) inply

Also (A5,2) w i t h a = 0 impl ies t h a t I I x 11 = I I P ~ C G ~ ~ - Glt)pilll It 1

G I I G ~ ~ - G ~ ~ I I ~ = O W m 1 ' 2 ) . S i n c e ea lx lp i i s bounded,

~ e ~ l ~ l ~ e a l x i n l i s of t h e same o r d e r . I n view of ( ~ 5 . 2 ) and It

(.A5.7 ) , we have

( ~ 5 ~ 3 ) i s a d i r e c t consequence o f ( ~ 5 . 9 ) and ( ~ 5 . 1 0 ) . S i m i l a r l y

( ~ 5 . 4 ) fo l lows from ( ~ 5 . 6 ) and ( ~ 5 . 1 0 ) w i t h a = 0 , s i n c e Ini 1 = IcI 9

F i n a l l y ( ~ 5 ~ 5 1 fo l lows from t r ( ~ ~ ~ - G t ) = tr X t + t r Yt , where t r Xt i s given by ( ~ 5 . 6 ) and ltr yt 1 < Ilyt ~ o ( t - ( ~ - ' ) / ~ )

by ( ~ 5 . 1 0 ) .

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JENSEN AND KATO

A6. Nonsmootk domains ,

I n t h i s s e c t i o n Re i s an a r b i t r a r y e x t e r i o r domain, s o t h a t

Z = F . ~ \ R e i s compact bu t need n o t b e e q u a l t o t h e c l o s u r e of i t s

i n t e r i o r . Thus t h e r e i s no p o i n t i n i n t r o d u c i n g -the o p e r a t o r s Zi -tH

and Glt ) but He and Gt = e e @ Q make s e n s e ,

I n t h i s c a s e we have, as t J- 0 ,

To prove (A6.1) we may aga in assume t h a t 2 1 i s smooth, by r e v

p l a c i n g i f necessary by a l a r g e r smooth o b s t a c l e , f o r which

Got - Gt has a (po in twise) l a r g e r k e r n e l . Then we e s t i m a t e t h e I i i l -

bert-Schmidt norm of t h e k e r n e l involved i n ( ~ 6 . 1 ) . I n t h i s computa-

t i o n , we may r e p l a c e t h e f a c t o r e 2 a / x l w i t h i t s majoran t

exp(2px/t1/2) f o r x % ( f o r s m a l l t ) and s i m i l a r l y f o r

e2aly 1 , The c o n t r i b u t i o n of t h e resu l t ing ; majorant i n which e i t h e r

x o r y i s i n Re i s e x a c t l y t h e same a s i n ( ~ 4 . 2 ) and i s of t h e

o r d e r ~ ( t - ( ~ - ~ ) ' ~ ) , The c o n t r i b u t i o n from x , y E pi , where

t h e k e r n e l f o r Gt i s ze ro , i s e q u a l t o

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T h i s proves ( ~ 6 . 1 ) .

The computation f o r t h e proof of ( ~ 6 . 2 ) i s s i m i l a r t o ( ~ 4 , 7 ) .

A f t e r t h e e s t i m a t e e x p [ a ( l x l - 2 I y 1 ) ] exp[-alxl + 2 a l x - y l ] , in-

t e g r a t i o n i n y g i v e s ~ ( t ~ ' ~ ) , and a subsequent i n t e g r a l of

e x p ( - a / x l ) g i v e s 0 ( 1 ) . Hence we o b t a i n ( ~ 6 . 2 ) .

To deduce ( ~ 6 . 3 ) from ( ~ 6 . 1 - 2 ) , we aga in use t h e t r i c k of

Deift-Simon. Here it should be no ted t h a t Gt i s n o t a C -semi- 0

group on b u t iiS2 = iiZs i s t r u e f o r any s > 0 .

miWK A6.1. It i s l i k e l y t h a t (~5.4-5) a r e t r u e i n t h e non- - smooth c a s e w i t h t h e remainder t e n s ~ ( t - ( ~ - ~ ) / ~ ) r e p l a c e d by

~ ( t - ~ ~ ~ ) , but we have no proof . The d i f f i c u l t y i s t h a t t h e r e i s

no s imple m a j o r a t i o n f o r t h e t r a c e norm.

A7. Small o b s t a c l e s .

LEMMA AT. 1. Let in 2 and l e t {Zn) be a sequence of ob- - s t a c l e s such t h a t Z n -t (01 ( i n t h e s e n s e t h a t f o r any b a l l B

about 0 , z n B f o r s u f f i c i e n t l y l a r g e n ) . Then

a x 11, I 1 ( G ~ ~ - G:)ealx1~l l‘f 0 , n - t w , f o r each t > 0 and a 2 0 , where G: i s t h e o p e r a t o r G~ f o r z = C" .

P r o o f Let gn be t h e k e r n e l f o r G: . Using t h e maximu - p r i n c i p l e , it i s easy t o s e e t h a t gn go and l i m gn = go e x i s t s

n-

e . e , p o i n t v i s e . go i s t h e k e r n e l f o r t h e o p e r a t o r G: a s s o c i a t e d

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0 wi th C = (01 . I f m 2 2 , however, Gt = Got because (01

has capacity 0 (see Kato i 1 4 1 , Rauch-Taylor [15] ) . Thus we have gn -+ go pointwise and dominatedly. It follows

t h a t Ilealx1 (G O t - On)ealxl t \I2 + 0 as n + . Anorher app l i ca t ion

or t he Deift-Simon t r i c k then leads t o t h e des i r ed r e s u l t .

WJ4Q.K A7.2, 1. The lemma i s f a l s e f o r m = 1 , - 2. More gene ra l ly , one can prove t h a t I I ~ ~ ' ~ I ( G - G;)ealXIII t

.-+ 0 i f zn + C with zn 3 C .

This work was p a r t i a l l y supported by NSF Grant M c S ~ ~ - 0 4 6 5 5 .

The authors a re indebted t o Preben Alsholm f o r he lp fu l comments and

discussions.

[ l ] T. Kato, Monotonicity theorems i n s c a t t e r i n g theory , Hadronic S.

1 (1978), 134-154. - [2 ] A. Maj da and 9. Ralston, An analogue of Weyll s theorem f o r un-

bounded domains, I , Duke Math. J. 5 (1978), 183-196; 11, t o

appear.

[31 V,S. Buslaev, Scat tered plane waves, s p e c t r a l asymptotics and

t r a c e formulae i n e x t e r i o r problems, Dokl. Akad. Nauk SSSR

(.l971), 999-1002; Soviet Math, Dokl. 12 C19711, 591-595,

[ b ] M.G. Krein, On t h e t r a c e formula i n t h e theory of pe r tu rba t ion ,

Mat. Sb. 33(751 (19531, 597-626.

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M.G. Krein, On pe r tu rba t ion determinants and a t r a c e formula

f o r un i t a ry and s e l f a d j o i n t ope ra to r s , Dokl. Akad. Nauk SSSR

144 (1962) , 260-271. - M.Sh. airman and M.G. Krein, On t h e theory of wave ope ra to r s

and s c a t t e r i n g opera tors , Dokl. Akad. Nauk SSSR 144 (1962) ,

475-478.

L.S. Koplienko, Local condi t ions f o r t h e exis tence of t h e func-

t i o n of s p e c t r a l s h i f t , Zap. ~ a u z n . Sem. LOMI (1977), 102-

118.

P. Dei f t and B. Simon, On t h e decoupling of f i n i t e s ingular -

i t i e s from t h e ques t ion of asymptotic completeness i n two body

quantum systems, J. Functional Anal. 3 (1976), 218-238.

T, Kato, Pe r tu rba t ion theory f o r l i n e a r ope ra to r s , Second Edi-

t i o n , Spr inger 1976.

T. Kato, Sca t t e r ing theory wi th two Hi lbe r t spaces, J. Func-

t i o n a l A n d . 1 (1967) , 342-369.

G. Freud, Res tg l ied e ines Tauberschen Sa t zes , I , Acta Math.

Acad. S c i . Hungar, 2 (1951), 299-308 . J . W . Helton and Z.V. Rals ton , The f i r s t v a r i a t i o n of t h e s ca t -

t e r i n g matr ix , J. Dif f . E q . 21 (1976), 378-394.

M,H, P r o t t e r and K.F , Weinberger, Meximum p r i n c i p l e s i n d i f -

f e r e n t i a l equations, Prentice-Hall 1967.

T. Kato, On some Schr$dinger opera tors wi th a s ingu la r complex

p o t e n t i a l , Ann. Scuola Normale Sup. Ser . IT, 2 (1978)~ 105-

1 1 4 .

J , Rauch and M. Taylor, P o t e n t i a l and s c a t t e r i n g theory on

wi ld ly per turbed domains, J . Functional Anal. 18 (1975),

27-59.

Received June, 1978

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Asymptotic behalrior of the scattering phase for exterior domains

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Transcript of Asymptotic behalrior of the scattering phase for exterior domains