Transmutation Theory and Applications (Notas De Matematica, 105)

s AND APPLICATIONS NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (105) Editor: Leopoldo Nachbin Centro Brasi...

Author: R. Carroll

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s AND APPLICATIONS

NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (105)

Editor: Leopoldo Nachbin Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro and University of Rochester

NORTH-HOLLAND-AMSTERDAM

NEW YORK

OXFORD

117

TRANSMUTATION THEORY AND APPLICATIONS Robert CARROLL University of Illinois Urbana, Illinois

1985 NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD

@

Elsevier Science Publishers B.V., 1985

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0 444 87805 X Publishers:

ELSEVIER SCIENCE PUBLISHERS B.V. P.O. BOX 1991 1000 BZ AMSTERDAM THE NETHERLANDS

Sole distributors for the U.S.A. and Canada:

ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 52 VAN DER BILT AVE N U E NEW YORK, N.Y. 10017 U.S.A.

Librarj o f Congress Cataloging in Publiralion D a l a

Carroll, Robert Wayne, 1930!Transmutation theory and applications. (Eurth-Holland mathematics studies ; 117) (Notas de m a t e d t i c a ; 105) Bibliography: p. Includes index. 1. Transmutation operators. 2. C i f f e r e n t i a l operators. I. T i t l e . 11. Series. 111. Series: Notas de m a t d t i c a (Rio de Janeiro, Brazil) ; 105. QAl.NB6 no. 105 C€@.329.43 510 s t515.7'2421 65-12928 ISBN bh44-87805-X (U.S. )

PRINTED IN THE NETHERLANDS

PREFACE

We use t h e word t r a n s m u t a t i o n o p e r a t o r o r t r a n s m u t a t i o n t o r e f e r t o operat o r s B which i n t e r t w i n e two second o r d e r l i n e a r d i f f e r e n t i a l o p e r a t o r s P and Q ( u s u a l l y on [O,CD))

i n t h e sense t h a t QB = BP, a c t i n g on s u i t a b l e o b j e c t s .

One can a l s o deal w i t h d i f f e r e n t i a l o p e r a t o r s o f d i f f e r e n t o r d e r s and we r e f e r t o [C29] and r e f e r e n c e s t h e r e f o r t h i s aspect o f t h e t h e o r y .

Such

o p e r a t o r s a r e o f t e n c a l l e d t r a n s f o r m a t i o n o p e r a t o r s by t h e Russian school ( L e v i t a n , Naimark, MarEenko, e t . a1 . ) , b u t t r a n s f o r m a t i o n seems t o o broad a term, and, s i n c e some o f t h e machinery seems "magical" a t times, we have f o l l o w e d L i o n s and D e l s a r t e i n u s i n g t h e word t r a n s m u t a t i o n .

L e t us empha-

s i z e t h a t t h e i n t e r t w i n i n g above i s n o t o p e r a t o r s i m i l a r i t y i n Lp t y p e spaces ( t h e s p e c t r a can be d i f f e r e n t ) ; B i s u s u a l l y an i n t e g r a l o p e r a t o r w i t h a d i s t r i b u t i o n k e r n e l and, when t r i a n g u l a r , i t w i l l be i n v e r t i b l e (as a Volterra operator).

Such t r a n s m u t a t i o n s a r i s e and can be c h a r a c t e r i z e d

i n v a r i o u s ways v i a Cauchy problems, Goursat problems, G e l f a n d - L e v i t a n (G-

L) equations, m i n i m i z a t i o n procedures, s p e c t r a l e i g e n f u n c t i o n p a i r i n g s , Thus, l e t P and 0 be o f t h e form P Qu = ( A o u ' ) ' / A Q - q ( x ) u f o r example. Then i f e.g. b, = -A2v!, v r ( 0 ) = 1, Dxqh(0) P =, 0, and b! = pi,Q where p y has s i m i l a r p r o p e r t i e s r e l a t i v e t o Q,

e i g e n f u n c t i o n c o n n e c t i o n formulas, e t c .

one w i l l have a t r a n s m u t a t i o n B: P

+

Q and t h e formal c o n n e c t i o n i s expres-

sed v i a q 4x ( y ) = ( e ( y , x ) , p ~ ( x ) ) w i t h 6(y,x) = (!Ji(x),;:(y))v P P A p ( x ) q p x ( x ) and v denotes a s p e c t r a l p a i r i n g f o r (P,pA). t h a t i n c e r t a i n circumstances (e.g.

Ap = A

q and, f o r s i m p l i c i t y , p = 0 so P = D',

Q

where 3 P( x ) =

As a model we n o t e

= 1 with suitable potentials

P

Cosxx = p x ( x ) , and dv = (2/T)dx)

one has @(y,x) = 6 ( x - y ) t K(y,x) w i t h K(y,x) = 0 f o r x > y ( c a u s a l i t y ) and t h e connection l e a d s t o a Goursat problem f o r K o f t h e form Q(Dy)K(y,x) = P(Dx)K(y,x)

w i t h q ( x ) = 2DxK(x,x) and Kx(y,O) = 0.

a l s o has k e r B - l = y(x,y)

= 6(x-y)

Q

+ L(x,y) and t h e s p e c t r a l p a i r i n g f o r

( Q , p x ) i s g i v e n v i a a Parseval f o r m u l a (Qf,@,g)w

QfQg), where Qf = 9:

fr

Q

=

f ( x ) n x ( x ) d x ( w i t h say f E L

= Cf here i f we t a k e A

Q

=

I n such s i t u a t i o n s one

2

fr f ( x ) g ( x ) d x

=

( R4 ,

h a v i n g compact s u p p o r t

-

1 ) , and R Q i s a d i s t r i b u t i o n ( t h e g e n e r a l i z e d

vi

ROBERT CARROLL

s p e c t r a l function of MarEenko), RQ = (2/7)[1 + FCL(y,o)] ( F c

Q

Fourier co-

s i n e transform). Thus the transmutation theory, B: D2 + Q here, expressed through connections ( k e r n e l s ) K and L "sees" both t h e potential q (via K(x,x)) and t h e spectrum ( v i a L(y,O)); when q i s real R (I d w = a spectral Q

measure. Moreover t h e development uses very l i t t l e information i n a canonical way reminiscent of category theory. The crucial ingredients a r e hyperbolic d i f f e r e n t i a l equations, Riemann functions, generalized t r a n s l a t i o n , eigenfunctions p X Q ( x ) which a r e e n t i r e functions of X of exponential type x , Paley-Wiener ideas and contour i n t e g r a t i o n , G-L equations, e t c . (a G-L equation a r i s e s from p X 4 ( y ) = (S(y,S),q!(S)) by taking w s c a l a r products w i t h ((XI - thus r ( y , x ) = ( ~ ~ ( x ) , ~ ~ =( (Y ~) ~) (~ Y , PS ) , ~P ~ ~) u( )S ) ~ ~ ( X In f a c t one has a l s o z ( y , x ) (R(Y,s),A(s,x))). as the kernel of a transmutation " a d j o i n t " t o B-'.

Ap(x) =

= aP(x)aQ1(y)y(x,y)

Now t h e above leads t o many developments of e s s e n t i a l l y mathematical i n t e r e s t r e l a t e d t o special functions and d i f f e r e n t i a l operators b u t t h e r e i s another s i d e of the story. One knows t h a t many l i n e a r physical processes based on Newton's second law f o r example a r e governed ( a t l e a s t t o f i r s t approximation) by second order 1 i n e a r d i f f e r e n t i a l operators and equations. T h u s i t should come as no s u r p r i s e t h a t t h e mathematical machinery useful (and e s s e n t i a l ) i n studying such processes has s i m i l a r p a t t e r n s and s t r u c t u r e f o r various problems a r i s i n g i n d i f f e r e n t d i s c i p l i n e s . Moreover transmutation can be p a r t i a l l y regarded i n the context of studying an operat o r Q i n terms of a "known" operator P and i t i s possible t o t r a n s p o r t various types of P machinery t o Q via B (e.g. t h e Fourier cosine transform and i t s inverse correspond t o t h e Q transform and i t s inverse f ( x ) = ( RQ, Qf(X) C QP ~ ( X ) ) via ~ B[Cosxx](y) = p QX ( y ) ) . In this s p i r i t v a r i o u s formulas a n d procedures i n quantum s c a t t e r i n g theory and geophysical a c o u s t i c wave theory f o r example have s i m i l a r s t r u c t u r e (based on transmutation connections) and the transmutation machine;-y i s e s p e c i a l l y useful in studying inverse problems. A t another level ( i n t e g r a l equations of G-L and Wiener-Hopf (W-H) type) one encounters c e r t a i n aspects of l i n e a r estimation and f i l t e r i n g theory w i t h t h e underlying s p e c t r a l i z a t i o n based on Fourier theory f o r s t a tionary processes. There a r e many f a s c i n a t i n g and useful connections here with techniques from s c a t t e r i n g theory f o r example in analyzing t h e various f i l t e r i n g and smoothing kernels. Thus i n a c e r t a i n meaningful way t h e transmutation theme, i n the context of second order linear d i f f e r e n t i a l operators, can be thought of q u i t e generally as a d e f i n i t i v e way of studying

PREFACE

" a l l " such operators i n a unified and canonical manner.

vii The machinery aoes

t o a reasonable enough depth in t h a t i t sees t h e c o e f f i c i e n t s and spectrum and one expects t o study f u r t h e r the " s e n s i t i v i t y " o f t h e machinery in variour senses. Thus although we f e e l t h a t t h e theory has reached a stage where

a s o r t of d e f i n i t i v e presentation i s possible ( a n d hopefully embodied here i n p a r t ) we a l s o suggest t h a t t h e theory and methods can be developed f u r t h e r in various ways. Let us i n d i c a t e now t h e r e l a t i o n of t h i s book t o t h e a u t h o r ' s previous two books on transmutation ( 1 ) Transmutation and operator d i f f e r e n t i a l equat i o n s , North-Hol land, 1979 and ( 2 ) Transmutation, s c a t t e r i n g theory, and s p e c i a l f u n c t i o n s , North-Holland, 1982. There is very l i t t l e i n t e r s e c t i o n w i t h ( 1 ) since i n ( 1 ) we were primarily i n t e r e s t e d in s o l u t i o n s o f ordinary d i f f e r e n t i a l equations w i t h operator c o e f f i c i e n t s and the transmutation 2 methods were b a s i c a l l y only of the most elementary s o r t , connecting D with 2 2 2 2 2 D, D with - 0 , and D with Qm = D + [(2m+l)/x]D o r w i t h D + (2m+l)CothxD ( t h e l a t t e r t h r e e connections occur of course a l s o i n t h e present volume). Also contained i n ( 1 ) was a d e t a i l e d account of the Hutson-Pym development of generalized t r a n s l a t i o n i n a t e n s o r product context (not repeated here). However a t l e a s t one half of ( 1 ) was mainly concerned w i t h operator d i f f e r e n t i a l equations as such and questions of existence of s o l u t i o n s , uniqueness, e t c . On t h e other hand, although we have w r i t t e n rouqhly 25 papers s i n c e w r i t i n g ( 2 ) , and of course o t h e r work h a s appeared in the intervening t h r e e y e a r s , we will n a t u r a l l y include here some basic material from ( 2 ) i n a reorganized form. The present book i s designed more as a " t e x t " on t r a n s mutation ( a s well as a research monograph) and t h e f i r s t chapter in f a c t s t a r t s with an introduction t o d i s t r i b u t i o n s and Fourier a n a l y s i s . Then t h e r e a r e 5 s e c t i o n s on basic s p e c t r a l a n a l y s i s (from several points of view), and on transmutation f o r operators D2 - q , where t h e presentation i s s s s e n t i a l l y s e l f contained a n d t h e d e t a i l s a r e s p e l l e d out completely. Then in s e c t i o n s 9-12 we deal with s i n g u l a r operators a n d t h e general Parseval formula v i a transmutation methods following ( 2 ) ; the procedure i s designed t o d i s p l a y the e s s e n t i a l canonical s t r u c t u r e without becoming immersed in excessive d e t a i l (which can be found i n ( 2 ) ) . Chapter 2 begins with a treatment of general transmutation theory via s p e c t r a l pairings and develops t h e i n t e r p l a y between various c h a r a c t e r i z a t i o n s of transmutation in terms of connection formulas, Goursat problems, Cauchy problems, s p e c t r a l p a i r i n g s , and minimization. General G-L a n d Marzenko ( M ) equations a r e developed

viii

ROBERT CARROLL

from various points of view with use of generalized translation as an essential ingredient in the theory. The canonical M equation connecting full line Fourier type operators P with general Q uses a new form of operational calculus in its development and reveals the intrinsic structural form of such equations as factorizations related t o the general G-L factorization. Sections 8-9 involve new results on Bergman-Gilbert (8-G) operators and related operators arising in transmutation theory via "complex angular momentum". A general Kontorovi&Lebedev (K-L) theory is developed and applied in the study of generatinq functions as transmutation kernels. Section 10 is also new and uses transmutation techniques intrinsically in the development of orthogonal functions relative to general measures. Section 1 1 is mostly new material on the construction o f transmutations with emphasis on the relations between kernels and coefficients. Chapter 3 consists of applications in several areas to show most clearly the intrinsic and canonical nature of transmutation methods in studying physical problems governed by second order linear differential equations. The first section is an introduction to stochastic ideas (with definitions and basic background information from probability theory). Then in §§3-5 we review some main lines of work on linear stochastic estimation and filtering in the sense of extracting and studying the structure of the basic integral equations and relating this t o underlying differential problems. Theorems are proved in detail and various techniques of use in electrical engineering for computation are indicated (although we do not say anything specific about numerical procedures). In §§6-7further connections of this work to transmutation theory and scattering theory are developed and we show how the minimizing procedure characterizing transmutation kernels in 57.7 i s equivalent to linear least squares estimation when there is an underlying stochastic process. In 558-9 we show how transmutation methods play an intrinsic role in the study of one dimensional geophysical inverse problems (such techniques can also be used for certain three dimensional problems as reported on in (2)). Section 8 is largely taken from (2) (with considerable refinement) and 59 contains new material involving transmission readout. Many new mathematical features (e.g. splitting of spectral measures) arise and have structural similarity to topics in estimation theory (e.g. Wiener filtering). In 570 we briefly survey some information on random evolutions related t o transmutation and as a separate topic make some remarks on the Darboux transformation. 511 is about canonical equations in the context of

PREFACE

ix

t r a n s m u t a t i o n w i t h o p e r a t o r c o e f f i c i e n t s and some a p p l i c a t i o n s t o t r a n s m i s sion l i n e s a r e indicated.

The a u t h o r would again l i k e t o thank Leopoldo Nachbin f o r h i s s u p p o r t and encouragement over the p a s t y e a r s . My w i f e Joan has been p a t i e n t l y support i v e a g a i n through months of s c r i b b l i n g and t y p i n g and I am g r a t e f u l . I would a l s o l i k e t o acknowledae some c o n v e r s a t i o n w i t h a number of people on v a r i o u s a s p e c t s o f this and r e l a t e d t h e o r y ; i n p a r t i c u l a r l e t me mention r e c e n t (5 1982) d i s c u s s i o n s w i t h A. B r u c k s t e i n , S. Dolzycki, M. Faierman, T. K a i l a t h , I . Knowles, B. Levy, J . McLaughlin, E. Robinson, E. Rosinger, P. S a b a t i e r , F. S a n t o s a , W. Symes, and B. Whiting.

This Page Intentionally Left Blank

CABLE Of CONCENCS

PREFACE

1, I n t m d u c t i a n

1

2, Distribufinn thenry

3

3, Fnurier a n a l y s i s

8

4, Basic transmutatinns

14

5- Parseval fnrmulas v i a transmutation ana the generalizea

s p e c t r a l function 6, S p e c t r a l thenry i n the energy v a r i a b l e 7. S p e c t r a l fhenry i n the momentum v a r i a b l e 8, C l a s s i c a l s p e c t r a l theory and r e l a t i o n s t n f u l l l i n e s c a t t e r ing 9, I n t r o a u c t i a n t o s i n g u l a r uperators and s p e c i a l functinns 10, Paley-Wiener thenrems, s p h e r i c a l t r a n s f n m , and Parseval formulas f u r s i n g u l a r aperatnrs 11- E x p l i c i t cnnstructions nf generalizea t r a n s l a t i o n s and transmutatinns f n r s i n g u l a r nperatars 12, canonical fnrmulatiun nf Parseval fnrmulas ana transfarms

I, m t m a u c t i n n

21

29 37

47

58 69 80

90

103

2, S p e c t r a l pairings f o r generalized t r a n s l a t i n n and trans-

mutatinn kernels

105

3, &he general extenaed &elfand-Levitan equation

117

4 , Quantum s c a t t e r i n g t h e m y

126

5. &he marcenkn equation v i a transmutatinn

136

6, &he marcenkn equatinn f o r Fnurier type nperatnrs

147

7. minimization a s a hireckive i n characterizing transmu-

t a t i n n kernels 8, cnnstructinn nf transmutatinns far 5 type nperatnrs 9, &he Bergman-Gilbert (B-c) a p e r a t n r ana generating f unct inns 10. Orthngnnal pnlynnmials ana transmutation xi

159

171 187 208

xii

ROBERT CARROLL

11, Relatims behueen kernels ana p o t e n t i a l s

1- Intrnhuctinn

220

229

2- Prnbahility t h e n q ana rananm prncesses

230

3, Ginear s t u s h a s t i c estimation

237

4, F i l t e r i n g ana i n t e g r a l eqnatinns

244

5- Znnouatinns ana s c a t t e r i n g

252

6, &ransmutatinn ana l i n e a r s t n c h a s t i c estimatinn

259

7. Randm € i e l a s ana s i n g u l a r nperators

270

8, Geophysical inverse pruhfems ( r e f l e c t i o n data)

275

9,

Geophysical inverse prnblems (transmissinn aata)

10, Same miscellaneous tupics 11, Equatinns with operator c o e f f i c i e n t s

288 300 308

REFERENCES

323

INDEX

347

CHAPEER 1

BACKGR0LIND IIIAEERIAL: AND BASIC IDEM

1, I N E R 6 D U ~ Z B N . This chapter i s designed t o serve as a source of basic information f o r t h e r e s t of t h e book. I t contains s e c t i o n s on d i s t r i b u t i o n s , Fourier transforms, eigenfunction theory, e t c . and i s l a r g e l y s e l f contained (some basic information on p r o b a b i l i t y theory and s t o c h a s t i c processes appears in Chapter 3). Naturally some f a m i l i a r i t y w i t h basic functional ana l y s i s , t h e Lebesgue i n t e g r a l , complex a n a l y s i s , e t c . will be helpful b u t

i t i s l e s s necessary than one m i g h t imagine. W e have found f o r example t h a t b r i g h t engineering o r physics students without a g r e a t deal of mathematical s o p h i s t i c a t i o n a r e o f t e n the best audience f o r " i n t e r d i s c i p l i n a r y s t u d i e s " of t h i s type; t h e i r physical i n t u i t i o n and general good sense allow them t o see through a l o t of "axiomatic t r a s h " and come t o terms w i t h t h e real i s sues immediately. On t h e o t h e r hand mathematics students often have t o overcome t h e p a r a l y s i s induced by too many E ' S , 6 ' s , a-rings, e t c . before their i n t u i t i o n can f l o u r i s h . In any event, c e r t a i n ideas from point set topology and basic functional a n a l y s i s (e.g. open s e t ) will n o t be defined b u t otherwise we will t r y t o be as complete as possible. Let us give a preview of t h e f i r s t chapter a s follows. §§2-3 involve i n troductory material on d i s t r i b u t i o n s and Fourier transforms. The material on boundary values of a n a l y t i c functions i s only included because of i t s int r i n s i c i n t e r e s t and o t h e r possible a p p l i c a t i o n s ( c f . here t h e material on e l l i p t i c transmutation in [C35,40]). Next §§4-5: Theorem 1.4.3 i s a basic theorem showing a technique f o r constructing transmutations via Cauchy problems. The i n t e r a c t i o n of t h i s theorem w i t h the construction o f kernels v i a Riemann functions and Goursat problems t o a r r i v e a t Theorems 1.4.8 and 1.4. 9 i s p a r t i c u l a r l y i n s t r u c t i v e . The use of these kernels i n t h e subsequent machinery t o prove t h e Parseval formula in Theorem 1.5.8 shows repeatedly 2 how information based on D and t h e Fourier transform can be transmitted 2 ( o r perhaps transmuted!) t o t h e development of theory f o r Q = D - q. The

1

2

ROBERT CARROLL

f o r m u l a RQ = ( 2 / ~ ) [ 1 + cLh(y,o)]

i n Theorem 1.5.5 shows how t r a n s m u t a t i o n

"sees" t h e spectrum w h i l e q ( y ) = 2D K (y,y) from Theorem 1.4.9 e x h i b i t s how Y h t r a n s m u t a t i o n "sees" t h e c o e f f i c i e n t q. §§6-8:

S e c t i o n 6 shows how t o determine t h e s p e c t r a l measure and i n v e r s i o n

formula f o r " s p h e r i c a l f u n c t i o n s " based on Q(D)u = ( A u ' ) ' / A where A has p r o p e r t i e s o f i n t e r e s t i n a p p l i c a t i o n s (see Chapter 3, § 8 ) . The expression 9 ( y ) + c-aQ (y) o f ( 6 . 2 6 ) leads t o dv = dX/2alcQ(X)I 2 as i n (6. qQ(y) = c A Q h Q -A 37) and t h e i n v e r s i o n (6.35)-(6.36). C l a s s i c a l c o n t o u r i n t e g r a l techniques u s i n g a Green's f u n c t i o n a r e e x p l o i t e d .

S e c t i o n 7 uses e s s e n t i a l l y t h e

-

same k i n d o f c o n t o u r i n t e g r a l technique w i t h c e r t a i n e i g e n f u n c t i o n s based

-

2

2 2

on o p e r a t o r s Qu = x u" + 2xu' + x [k

-

q(x)]u where t h e s p e c t r a l parameter

now corresponds t o complex a n g u l a r momentum ( i n s t e a d o f energy).

Section 8

reviews t h e c l a s s i c a l f o r m u l a t i o n o f e i g e n f u n c t i o n expansions f o l l o w i n g e.g. Titchmarsh and develops some f a c t s about " F o u r i e r t y p e " o p e r a t o r s D2 on

(-m,m).

-Sinhx/x,

-

p(x)

Such o p e r a t o r s posses e i g e n f u n c t i o n s @; 5 e x p ( i A x ) , x! P (pX n, CosXx, and Z A n, e x p ( - i x x ) and t h e o p e r a t i o n a l c a l c u l u s based Q

P

on these f u n c t i o n s i s r e l a t e d t o f u l l l i n e s c a t t e r i n g t h e o r y s i m u l t a n e o u s l y w i t h t h e c l a s s i c a l e i g e n f u n c t i o n expansion t h e o r y . 509-12 a r e on s i n g u l a r o p e r a t o r s

= (A u ' ) ' / A

Q

Q

+

p

2

Q

u - { ( x ) u modeled on

t h e r a d i a l Laplace-Beltrami o p e r a t o r i n a rank one noncompact Riemannian symmetric space.

The s p h e r i c a l f u n c t i o n s i n v o l v e e.g.

Bessel f u n c t i o n s ,

a s s o c i a t e d Legendre f u n c t i o n s , Jacobi f u n c t i o n s , e t c . and t h e i n t e g r a l t r a n s forms i n c l u d e t h e Hankel and g e n e r a l i z e d Mehler t h e o r y . o n i c a l technique o f 854-5 t o such s i n g u l a r o p e r a t o r s .

We extend t h e canNumerous examples a r e

g i v e n and t y p i c a l k e r n e l s f o r t r a n s m u t a t i o n and g e n e r a l i z e d t r a n s l a t i o n a r e displayed.

P r o p e r t i e s o f t h e s p h e r i c a l f u n c t i o n s and J o s t s o l u t i o n s a r e

proved as needed f o r l a r g e classes o f t y p i c a l s i t u a t i o n s and general cons t r u c t i o n s a r e i n d i c a t e d w i t h some s k e t c h o f t h e p r o o f a t l e a s t .

The main

theme i s t h a t t h e r e i s a canonical i n t r i n s i c procedure e x p l i c i t l y based on t r a n s m u t a t i o n f o r d e t e r m i n i n g Parseval formulas and e i g e n f u n c t i o n expansion h

theorems f o r general o p e r a t o r s Q i n terms o f s u i t a b l e p r o t y p i c a l model opera t o r s Qo whose t h e o r y i s known.

The t r a n s m u t a t i o n machine t r a n s p o r t s t h e

necessary p r o p e r t i e s and s t r u c t u r e around and produces e x p l i c i t c o n s t r u c t i o n s from which t h e g e n e r a l i z e d s p e c t r a l f u n c t i o n RQ a r i s e s i n terms o f a transmutation kernel.

DISTRIBUTION THEORY

3

DZ5ERZBUE'I0)N CHE0Rg. A g r e a t deal of the progress in studying p a r t i a l d i f f e r e n t i a l equations over t h e l a s t 35 years o r so has been due t o t h e de-

2.

velopment and systematic use of t h e theory of d i s t r i b u t i o n s (and i t s extensions t o boundary values of a n a l y t i c functions, hyperfunctions, e t c . ) . There a r e many treatments of t h i s theory a v a i l a b l e ( c f . [Bgl; Bzl; C19,29; Gfl; H11; Hml; Jb3; Nal; Rb1,2; Th2,3; Yal; S j l ] ) . One can deal with c l a s s i c a l d i s t r i b u t i o n theory as a ( b e a u t i f u l and s i g n i f i c a n t ) t o p i c i n l o c a l l y convex topological vector spaces b u t this i s f o r t u n a t e l y unnecessary i n p r a c t i c e f o r a n a l y s i s and applied mathematics. In f a c t i t i s s u r p r i s i n g l y easy t o approach t h e s u b j e c t honestly and almost immediately begin t o use

T h i s i s the approach we will adopt here and f o r our purposes we can e s s e n t i a l l y confine our a t t e n t i o n t o 1 R . Thus

d i s t r i b u t i o n s and t h e r e l a t e d Fourier theory.

Let S? be Rn o r an open s e t i n Rn. Define C; as t h e vector DEFlNIElBN 2.1. space of Cm functions i n R with compact support (support q = supp q is t h e s m a l l e s t closed s e t o u t s i d e of which q :0 ) . A d i s t r i b u t i o n T in R i s a l i n e a r map T: C: C such t h a t f o r any compact s e t K c R t h e r e e x i s t cons t a n t s C and k (depending on K ) with (*) I T ( 9 ) ) I = I ( T , q ) l 5 C 1 suplD'q1, -f

Ci;

c K), Daq = D;',..D>p, Dk = a / a x k , a = ( a l , . . . , a ), 1 ~ =1 1 a k ) . I f k i s t h e same f o r a l l K one says T i s of n order 5 k. One w r i t e s D ' ( S ? ) f o r t h e vector space of such d i s t r i b u t i o n s T. la1 5 k ( q E C;(K)

=

19 E

SUPP q

This can be s t a t e d i n terms of sequential c o n t i n u i t y as follows.

Given a

compact s e t K c W l e t DK be t h e vector space of C: functions in P w i t h supp o r t i n K. One places a topology on DK by specifying t h a t a sequence q € j DK converges t o 0 provided suplD'q. I -+ 0 uniformly on K f o r each f i x e d a. J Clearly i f T s a t i s f i e s t h e condition of Definition 2.1 then ( T , p ) -f 0 when j -+ 0 i n DK ( i . e . T: D K + C i s continuous). On the other hand i f (*) i n

'j

Definition 2.1 does not hold f o r some K = ^K, while ( T,q ) -+ 0 whenever q j j 0 i n D K a r b i t r a r y , then, f o r any j , taking C = k = j i n Definition 2.1, we have I( T , q j ) I > j 1 suplDaqj I ((a15 j ) f o r some p E Di. One can assume j ( T , q j ) = 1 (by l i n e a r i t y ) and then I0"q.I 5 l / j f o r la1 5 j ( i . e . q + 0 i n J j ~i when we l e t j r u n ) although ( T , q J. ) + 0. This c o n t r a d i c t s and hence one can s t a t e -+

EHE0REm 2.2, A l i n e a r map T: C;(R)

+

C i s a d i s t r i b u t i o n ( T E D ' ( S ? ) ) i f and

only i f T i s a continuous l i n e a r map DK REIRARK 2.3,

+

C f o r every K c R compact.

By Theorem 2 . 2 in order t o t e s t whether o r not a s p e c i f i c o b j e c t

4

(e.g.

ROBERT CARROLL

a d e l t a o b j e c t d e f i n e d b y ( 8 , ~ =) ~ ( 0 )i s a d i s t r i b u t i o n one needs

o n l y check i t s a c t i o n on convergent sequences o f t e s t f u n c t i o n s q I n p r a c t i c e t h i s i s a l l we need.

j

E

UK).

However l e t us mention t h a t t h e r e i s a

m

t o p o l o g y on U = C o y c a l l e d a s t r i c t i n d u c t i v e l i m i t topology, which i s chara c t e r i z e d by t h e p r o p e r t y t h a t a l i n e a r map T: U

+

F,

F a l o c a l l y convex

t o p o l o g i c a l v e c t o r space, i s continuous i f and o n l y i f T: U t i n u o u s f o r each Kn i n any " d e t e r m i n i n g " sequence Kn

a

which exhaust 52 ( i . e .

= UK,).

C

Kn+l

Kn

-+

F i s con-

o f compact s e t s

We remark t h a t a l o c a l l y convex t o p o l o g i c a l

v e c t o r space F i s a t o p o l o g i c a l v e c t o r space whose t o p o l o g y i s determined by a ( n o t n e c e s s a r i l y c o u n t a b l e ) f a m i l y o f seminorms p,.

T h i s means t h a t a

fundamental system o f neighborhoods o f 0 i n F i s determined by f i n i t e i n t e r s e c t i o n s of s e t s V,$E)

= I x E F, p ( x ) < a

-

€1.

R e c a l l t h a t a seminorm p

on F i s a r e a l v a l u e d f u n c t i o n on F such t h a t p(x+y) 5 p ( x ) + p ( y ) and p(ax) = IciIp(x);

i f p ( x ) = 0 i m p l i e s x = 0 t h e n p i s c a l l e d a norm.

This allows

us t o s p e c i f y d i s t r i b u t i o n s T E U'(C)as continuous l i n e a r maps T: U when

U

-f

C

has t h e s t r i c t i n d u c t i v e l i m i t t o p o l o g y (and accounts f o r t h e

= C:

"duality" notation U

EHWLE 2-4. L e t clearly (8,q

.) =

- U').

= R ' and l e t q + 0 i n U be a g e n e r i c sequence. Then j K ~ ~ ( +0 0 ) so t h e 6 o b j e c t i s a d i s t r i b u t i o n . F o r any Llo1c

J function f define (f,q) =

rZ f ( x ) q ( x ) d x

so f determines a d i s t r i b u t i o n .

f o r IP E C i .

Evidently

(f,q.)

+

J I n p a r t i c u l a r one d e f i n e s t h e Heavyside

0

f u n c t i o n Y by Y(x) = 0 f o r x < 0 and Y(x) = 1 f o r x > 0. Now t h e main reason f o r c o n s t r u c t i n g a t h e o r y o f d i s t r i b u t i o n s was t o be a b l e t o d i f f e r e n t i a t e enough o b j e c t s so t h a t a t h e o r y o f l i n e a r p a r t i a l d i f f e r e n t i a l equations was p o s s i b l e .

Thus U

i s constructed v i a a topology

based on d i f f e r e n t i a t i o n and by d u a l i t y we w i l l be a b l e t o d i f f e r e n t i a t e ob-

U'. More p r e c i s e l y l e t T E 0' and c o n s i d e r t h e map M: q + Dkq -(TyDkq): U U + C. C l e a r l y M i s l i n e a r and Dk: UK + U K i s continuous;

jects i n

-f

-f

hence ( g i v e n t h a t t h e t o p o l o g y o f UK i s i n f a c t t h e t o p o l o g y induced b y U ) by Remark 2.3 Dk: finition,

U

-f

U i s continuous.

+

C i s continuous by de-

M i s continuous and hence determines an element i n

-(T,Dkq)).

UI(a)(M(lp)

=

This leads t o

DEFZNZCZ0N 2-5- Given T (2.1)

Since T: U

( D k T,q) =

EHilPCE 2.6.

-

E

U' one d e f i n e s DkT by t h e f o r m u l a

(q E

U)

(T,Dkq)

Given T = f E C'(n) we see t h a t (2.1) reduces t o t h e standard

5

DISTRIBUTION THEORY formula of i n t e g r a t i o n by p a r t s . DY = 6 s i n c e

(2.2)

(DY,?)

= -(Yp') =

-

Applied t o T = Y of Example 2.4 one has

q'(X)dX = r ~ ( 0 =) ( 6 , ~ )

jOrn

DEFZNZ&ZBN 2-7, Let E denote Cm(R) with t h e topology of uniform convergence on compact s e t s of functions a n d a l l d e r i v a t i v e s . T h i s will be a metrizable space ( t h e topology i s defined by a countable number of seminorms) and convergence can always be r e f e r r e d t o sequences. If Kn C Kn+l w i t h a = UKn i s a "determining" sequence of compact s e t s then a sequence 9 k + 0 i n E means t h a t f o r any p and n , s u p lDaqkl -+ 0 (x E Kn) f o r l a \ 5 p. The dual space E ' (= the space of continuous l i n e a r maps E -+ C ) i s in f a c t the space of d i s t r i b u t i o n s T with compact support (we omit t h e proof of t h i s b u t i t i s r o u t i n e - see t h e references c i t e d e a r l i e r ) . Here one says t h a t T = 0 in The complement i n 52 of an open s e t A c a i f ( T , 9 ) = 0 f o r a l l q E C:(A). t h e union of a l l such A where T = 0 is c a l l e d s u p p T. DEFZNZCZBN 2-8, For B

t h a t s u p Ix Da9(x)J

-

=

R we l e t

Ra).

CY + m

R l ~ \ - ( x Y ~ )

.

L

(l+lc))-Ndc

t o obtain f ( x ) = 0 ((x,n)

Hence supp f C B(0,R).

=

a l x l while

Rlql =

An e n t i r e f u n c t i o n F ( r ; ) i s t h e F o u r i e r t r a n s f o r m o f T E E ' i f N and o n l y i f f o r some c o n s t a n t s R, N, and C one has (*) IF(r)l 5 C ( l + ( < I )

&MBREm 3.7expR1 Imr;I .

Ptrool;: F o r n e c e s s i t y l e t K = supp T and t a k e some f i x e d $ E C E equal t o 1 on K w i t h supp $ = K ' 3 K. Then one sees e a s i l y t h a t f o r any cp E E , (T,cp) = ($T,q)

= (

Since T E

T,cp$).

D'

by D e f i n i t i o n 2.1 t h e r e e x i s t c o n s t a n t s c

1 sup

and k (depending on K ' ) such t h a t I T ( x ) ( 5 c

DKl. C'

1

But sup

take K'

x =kE IOCYcpI (1.1

C

B(0,R)

O K , f o r any

5 k, x

E

f o r some R.

cp E

E so

( ( T,cp)l

(1.1

ID%(

5c

1

5 k) for x

SUP

E

5

ID"(cpPJ,)I

K ' ) where c ' depends on c and $.

We can a l s o

Now t a k e cp = e x p ( i ( x , < ) )

=

s o (T,cp)

FT and A

one o b t a i n s (+).

F o r s u f f i c i e n c y we know F

f o r some T

L e t Gk be an approximate i d e n t i t y as i n (2.11) and one

E

S'.

has FTk = F(T ($k( 0).

By P a r s e v a l ' s

formula ? ( z ) = ( F ( t ) , ( l / Z d ( t - z ) ) )

= ( F f , l / 2 ~ i ( t - z ) ) = (f( 0 ) o r = ( f ( E ) , - Y ( - E ) e x p ( i E z ) )

(y < 0) .

S i m i l a r l y one can e a s i l y shownthat i f G

E

L

T h i s i s Theorem 3.10). 2

and g ( t ) = F-l(G,t)

then

Such r e p r e s e n t a t i o n s as i n Theorem 3.10 or (3.21) can be extended i n an obv i o u s manner t o f u n c t i o n s ( c o n t i n u o u s ) f and G which a r e bounded by Itla f o r 4

some

~1

as t

-+ m

n

and F(z) = f ( f , z )

i s c a l l e d a g e n e r a l i z e d F o u r i e r transform,

"-1

F (G,z) i s t h e g e n e r a l i z e d i n v e r s e transform. L e t us c a l l a continuous f u n c t i o n f w i t h If(t)l = O ( l t 1 " ) a tempered f u n c t i o n . Then i t i s

while $(z) =

i m n e d i a t e t h a t t h e f o l l o w i n g theorem h o l d s .

FOURIER ANALYSIS

13

KHE0REm 3-11. If f is a tempered function then f o r

I

q E S

m

(1/2n)

- j ( f , x - i ~ ) ] e - ~ ~=~ ed-x€ l t l f ( t ) ;

[;(f,x+is)

n ,

E+O lim

- ;(f,x-ie)]

F-'[;(f,x+is) 1i m

E+o

=

f(t);

[ [f(f,x+ie) - i(f,x-i~)]q(x)dx

=

(F(f),q)

a r e a n a l y t i c f o r y # 0. Also 4 c l e a r l y from t h e d e f i n i t i o n s , F ( f , x + i e ) = F [ f ( t ) Y ( t ) e x p ( - ~ t ) , x ] and F ( f , x - i s ) = - F [ f ( t ) Y ( - t ) e x p ( e t ) ,XI. This leads t o t h e f i r s t equation d i r e c t l y . Next by t h e Parseval formula and the f i r s t equation t h e r i g h t s i d e of t h e P J L V V ~ : One notes t h a t j ( f , z ) and ? ' ( f , z ) 4

second equation tends t o lim

(

e x p ( - E ( t l ) f ( t ) , f q )= ( f ( t ) , F q )

= (Ff,q).

Now f o r T E S ' one has DmT E S ' and a simple d u a l i t y argument y i e l d s F(DmT) = (-ih)mFT while DmFT = F[(ix)'"T] ( c f . Example 3 . 4 ) . Further one can prove Let T E S ' . Then t h e r e e x i s t s a tempered function f and a n i n t e g e r n such t h a t Dnf = T . Writing ;(T,z) = (-iz)";(f,z) i t follows t h a t F(T,z) i s an a n a l y t i c representation of FT i n t h e sense t h a t f o r q E S,

KHE0RZm 3.12,

I\

( F ( T ) , q ) = lim

(

[F(T,x+iE)

A

- F(T,x-ie)],q)

(E +

0).

Ptlvo6: From Definition 2.8 ( c f . a l s o Remark 2.3) t h e topology of S i s determined by seminorms p ( q ) = sup ((l+x2)mDkqIso t h a t a fundamental sysm,k tern of neighborhoods of 0 i n S c o n s i s t s e.g. of s e t s V (6) = { q E S ; m,k 2mL Ilqll = s u p I ( l + x ) D qpI 5 6 ; 0 5 L 5 k; x E R Consequently, given T E m yk S ' and E > 0 t h e r e e x i s t s m , k such t h a t (( T,q ) I 5 E when q E V m Y k ( B ) . For any J, E S w r i t e = 6$/11J,llm,k; then q E V ( 6 ) and (( T y q ) I = ( ( TyJ,6/lIJ,ll ) I m,k < E which means (*) (( T,IL)I < ( ~ / 6 ) I I $ l l = c s u p I ( l + x2 )mD P$ 1 f o r 0 5 p 5 m,k k. A simple a p p l i c a t i o n of Hahn-Banach ideas and some r o u t i n e c a l c u l a t i o n y i e l d s now T = DnF where ( l + x 2 ) - p F i s continuous and bounded ( e x e r c i s e - c f . [Gfl; Jb3; S j l l ) . In this connection l e t us note t h a t s u p (l+x2)ml$ql 5 2 m+j L+j ( D q I and hence i n ( * ) t h e r e e x i s t s c ' s u c h t h a t ( ( T , J , ) I 5 k . sup(l+x ) J c ' sup I (1+x2)m+kDk$l5 c ' /tID[(l+x2)m+kDk$Jdx< ~'IlD[(l+x~)~~D~J,]ll~l. Thus T is defined and continuous on a subspace A = C D [ ( ~ + X ' ) ~ + ~ DIL~ E$ ; Sl of L 1 and by Hahn-Banach t h e r e e x i s t s g E Lm such t h a t ( T , x ) = I gxdx. From 2 m+k Dg] a s a d i s ( T , J , ) = I g D [ ( l + ~ ~ ) ~ + ~ D ~we J , ]have d x T = (-l)k+lDk[(l+x ) 2 t r i b u t i o n . Finally one can represent (-l)k+1(l+x2)m+kDga s D f ( i n various ways) w i t h f continuous and tempered. Hence (with n = k+2) we have

.

(3.22)

;(T,x+iE) - i ( T , x - i e )

=

(-i)"[(x+iE)'f(f,x+iE)

- (x-iE) n FA ( f , x - i ~ ) ]

14

ROBERT CARROLL

m

= (-ix)n

(3.22)

[ f(t)e-€ltleitxdt

+ O ( E ~ )+

LW

+ n ( -i)"x"-'

( iE )

Imf (t)e-'

1 I [ U( t ) -Y ( - t ) l e i Xtdt

im

.

I f we l e t (3.22) a c t on a t e s t f u n c t i o n 'P E S and

w i l l follow.

E

t e n d t o z e r o t h e theorem

EMCAFRPCE 3-13, One can check e a s i l y t h a t t h e f o l l o w i n g formulas h o l d . T = 6 ; T ( z ) = -1/2niz;

(3.24)

T = Y(t);

FT = 1; i ( ~ , z ) =

(

6+,rp) = l i m

= -(l/hi)log(-z)

BMZt

-

=

{

-l/iz

(y > 0 )

0

(Y < 0 )

We w i l l be concerned f i r s t w i t h d i f f e r e n t i a l op-

q ( x ) on [O,m)

( c f . g e n e r a l l y [Ael;

;(T,z)

-m

tmmmuc~cI0NSi.

e r a t o r s Q f D ) = DL

(y # 0 ) ; FT = 2n6+

(1/2ai) j ( t ) d t / ( t + i E ) ;

E O '

4,

(Y < 0)

-1/2

I\

T(z

' 0)

1 / 2 (Y

A

(3.23)

and t h e i r e i g e n f u n c t i o n expansion t h e o r y Dcl; CL1; Chl; Cml; Cgl-4;

C29,30,37,39,40;

2; Lll-3,6-10; Lvl-3; Mcl-4; Nb1,2; Stb2; Te2; T j l ] ) . 2 e r a t o r i s Q = D and t h e a s s o c i a t e d F o u r i e r t h e o r y .

Gel-4;

Gf

The p r o t o t y p i c a l opThus from t h e F o u r i e r

i n v e r s i o n formulas i n §3 one has immediately.

CHZ0RZm 4.1. (4.1)

2 F o r f E S ( o r L ) one can w r i t e (du = (Z/n)dA) FCf =

c

f(x)Coshxdx;

f(x) =

I

(FCf)(x)CoshxdA

S i m i l a r formulas h o l d o f course w i t h COSAX r e p l a c e d by SinAx and we r e f e r t o dv = (2/n)dA as a s p e c t r a l measure.

L e t us c o n s i d e r t h e b a s i c e i g e n f u n c t i o n

equation

S t r i c t l y speaking t h e Q and h s h o u l d be c a r r i e d a l o n g as i n d i c e s i n c a l c u l a t i o n s b u t we w i l l o f t e n w r i t e rpQ ( x ) = rp(A,x,h) o r even rpQ ( x ) = v(A,x) i f A,h AYh no c o n f u s i o n can a r i s e . When h = 0 we speak o f " s p h e r i c a l f u n c t i o n s " r p Y y 0 = ( f o r reasons t o appear l a t e r ) and when h -+ w we d i v i d e by h approprpA(x) Q r i a t e l y and d e a l w i t h so c a l l e d ( i n p h y s i c s ) " r e g u l a r " s o l u t i o n s rp! = OT = 1. The " p o t e n t i a l " q ( x ) i s assumed t o s a t i s f y i n g O ,Q( O ) = 0 w i t h Dxe,(0) Q vanish s u i t a b l y a t

m

( f o r now assume e.g.

t h e r e w i l l be s o l u t i o n s ( J o s t s o l u t i o n s )

( * ) Jm x l q ( x ) l d x < -) so t h a t 0 2 o f Qip = - A rp s a t i s f y i n g

15

BASIC TRANSMUTATIONS

(4.3)

akX(x) Q exp(+ixx) a s x Q

m

-f

We will discuss various ways of developing t h e expansion theorems analogous t o Theorem 4.1 f o r t h e operator Q a n d eigenfunctions P ~ , ~O u. r basic technique here i s transmutational i n nature and will serve a s a model f o r o t h e r developments l a t e r . T h u s 2 DEFINZCIBN 4-2, Let P and Q be two operators of t h e form D2 - q ( x ) = Q ( D - p ( x ) = P ) where p and q will be assumed continuous f o r the moment. An opQ , i f BP = QB, a c t i n g on e r a t o r B i s s a i d t o transmute P i n t o Q , B: P s u i t a b l e functions. There a r e normally many such B which usually w i l l be i n t e g r a l operators w i t h d i s t r i b u t i o n kernels. -f

One c o n s t r u c t s such transmutations i n several ways and we give two points of view here ( c f . a l s o Chapter 2 f o r f u r t h e r information). T h u s f i r s t we give a general procedure, somewhat formally, i n order t o i n d i c a t e d i r e c t i o n s ( c f . [C38-40; Ho2-4; L13; Lpl -3; Mc3,4] ) . CHE@)REm 4.3- Let A and C be l i n e a r operators commuting with P ( a c t i n g on

s u i t a b l e o b j e c t s ) and assume the Cauchy problem (y 2 0, (4.4)

P(Dx)lp = Q ( D )v; q ( x , O ) Y

=

-m

A l i t t l e c a l c u l a t i o n y i e l d s then

c f . LMc41)

i

& ( x+ t 1

(4.20)

K(x,t)

(1/2)

=

j0

t+x-u

q(u)du + ( 1 / 2 ) J:q(u)

K(u,t ) & d u t-x+u

The r e g i o n o f i n t e g r a t i o n f o r t h e second i n t e g r a l i s shown i n (4.21 )

I n t h e r e g i o n s 1 and 2 , 151 > l u l so K(u,() = 0 and a change o f v a r i a b l e s u+c = 2a, u-5 = 2 ~ ,x + t = 2w, x - t = 2v y i e l d s ( H ( ~ , B ) = K ( c c + ~ , a - ~ )e t c . )

1

w

(4.22)

H(w,v)

(1/2)

=

q(y)dy +

da Ivq(a+B)H(a,B)dB 0

0

0

One solves t h i s by successive approximations i n w r i t i n g

1

W

W

(4.23)

HO(w,v) = ( 1 / 2 )

q(y)dy; Hn(W,V) =

jo da l:q(a+B)Hn-1(a,B)dB

0

and t h e n H =

lm Hn(w,v) 0

w i l l be u n i f o r m l y convergent f o r say 0 5 w,v 5 a

and hence r e p r e s e n t s a c o n t i n u o u s f u n c t i o n s a t i s f y i n g (4.22). w r i t e u o ( x ) = Ix I q ( t ) \ d t , u,(x) 0

=

Indeed i f we

IX u o ( t ) d t , and D ( u ) = max 1 1 ' 0

f o r 0 5 g 5 u t h e n one can show e a s i l y t h a t (4.24)

I

IHn(w,v)

5 (1/2)n(w)[ul (w+v) -

ul (w)

-

u1 ( v ) l n / n !

(q(a+B)ldB = uo(v+a) C l e a r l y \HO(w,v)I 5 (1/2)n(w) and, s i n c e IV Da[ul(v+a)

-

(4.25)

\H(w,v)

- u,(v)],

u1(a)

I

q(t)dtl

0

0

-

(4.24) i s s t r a i g h t f o r w a r d ( e x e r c i s e ) .

5 (1/2)n(w)exp[ol (w+v) -

u1 (w)

-

ul

(v)l

uo(a) =

Hence

20

ROBERT CARROLL

(some v a r i a t i o n s on t h i s e s t i m a t e a r e a l s o i n d i c a t e d i n [ M c ~ ] ) .

F i n a l l y we

n o t e t h a t from (4.22) w i t h continuous q and H i t f o l l o w s t h a t Hw and Hv a r e continuous as w e l l as Hwv = q(w+v)H(w,v). = 0.

and H(0,v)

IIHE0RElll 4.8,

F u r t h e r H(w,O) = (1/2) I W q ( y ) d y 0

Consequently

The c o n t i n u o u s k e r n e l K o f (4.15) and Theorem 4.7 can be a l s o

c o n s t r u c t e d by successive approximations from (4.20) o r e q u i v a l e n t l y (4.22) and e s t i m a t e d v i a (4.25) x-t).

( w i t h H(w,v) = K(w+v,w-v) = K ( x , t ) ,

2w = x+t, 2v =

K has continuous f i r s t p a r t i a l d e r i v a t i v e s and s a t i s f i e s Hwv =

q(w+v)H g e n e r i c a l l y w i t h

rx

(4.26)

K(x,x) = (1/2)

J

q(S)dS; K(x,-X)

= 0

0

I f q has n continuous d e r i v a t i v e s t h e n K has n + l continuous p a r t i a l d e r i v a -

t i v e s and i n p a r t i c u l a r t h e e q u a t i o n Hwv = q(w+v)H can t h e n be w r i t t e n ( f o r n 21)

2 2 DtK = [Dx

(4.27)

-

q(x)]K

Thus these e q u a t i o n s r e p r e s e n t necessary and s u f f i c i e n t c o n d i t i o n s f o r a t r a n s m u t a t i o n k e r n e l K as i n (4.15) i . e . I + Kh o r I + K, i n an obvious n o t a t i o n , have i n v e r s e s d e f i n e d by Neumann s e r i e s . We

One knows t h a t V o l t e r r a t y p e o p e r a t o r s o f t h e form (4.17), write I

+

Lh = ( I + Kh)-'

e t c . so t h a t e.g.

Now u s i n g (4.16) and (4.26) we see t h a t K h ( x y x ) =,h + K(x,x)

I x q(S)dS w h i l e DxKh(y,O)

= 0.

= h

+ (1/2)

On t h e o t h e r hand Kh w i l l s a t i s f y an equa-

0

t i o n o f t h e form (4.27) which we d e r i v e as f o l l o w s (assume q

E

C

l

so t h a t

t h e second p a r t i a l d e r i v a t i v e s a r e a l l d e f i n e d ) . Take (4.17) and w r i t e 2 2 2 down D 9 - 9p + A cp = 0 = -qCosAy - q ( y ) I KhCos + A' J KhCos + I D K Cos Y 2 Y h Then observe t h a t A I KhCos = + (Kh(y,y)CosAy)' + DyKh(y,y)CosAy. 2 Thus - I KhD 2 Cos = ASinAyKh(y,y) - J DxKhCos + DxKh(y,y)CosAy - DxKh(y,O). (4.29)

2 [Dy

-

2 q ( Y ) l K h ( y ~ x ) = DxKh(Y,X);

2DyKh(Y,Y)

which i s c o n s i s t e n t w i t h o u r o b s e r v a t i o n s above. c a r r i e d o u t i n [Mc4] r e l a t i v e t o ,L,

= q(Y);

DxKh(Y,O)

= 0

Similar calculations are

and one shows (as f o r K above) t h a t

s o l u t i o n s Kh and Lh t o t h e corresponding problems (4.29) and (4.30)

(below)

BASIC TRANSMUTATIONS

can be constructed by successive approximations.

21

Hence

Transmutation kernels Kh(y,x) f o r (4.17), s a t i s f y i n g (4.29) (with Kh(y,y) = h t ( 1 / 2 ) I Y q ( c ) d E ; ) and L h ( x y y ) f o r (4.28) s a t i s f y i n g

EHE0REm 4.9,

0

2 2 D x L h ( X , Y ) = EDY - q ( Y ) l L h ( X , Y ) ; Lh(xYx)

(4.30)

=

-h

- (1/2)

q(t)dt; JOX

D L ( ~ ~ -0 h L) h ( x , O ) = 0

Y h

can be constructed by successive approximations ( q

E

1 C ).

One sees now t h a t i f Kh i s constructed via (4.29) and f thenfor Bh = I + Kh

E

C

2

with f ' ( 0 ) = 0

Q ( D ) B f = BhD:f Y h 2 Similarly i f f E C s a t i s f i e s f ' ( 0 ) = h f ( 0 ) and L h i s constructed v i a (4.30)

(4.31)

then f o r Bh (4.32)

=

I + Lh

D:Bhf

= BhQ(D )f

Y

W e note t h a t f E C:(O,-) s a t i s f i e s both requirements so t h a t generally one can find a l a r g e c l a s s of functions on which various transmutations B: P + Q can a c t and intertwine P and Q. REmARK 4-10, There a r e analogous r e s u l t s t o Theorem 4.9 f o r Km and Lm ( c f .

( 4 . 1 7 ) ) b u t we will omit t h e d e t a i l s here.

Similarly in what follows we

o m i t the s e p a r a t e c a l c u l a t i o n s needed f o r t h e Bm - Bm s i t u a t i o n . P A ~ E V A I :F ~ R ~ L IUZA L Mewczrnucmm AND EHE GENERACZZED S ~ P E ~ R A IFLINC: We will f i r s t develop t h e s p e c t r a l theory and eigenfunction expan2 sions following [Mc4] ( c f . a l s o [C38-40; Mc31). T h u s write P(D) = D and 2 1 Q ( D ) = D - q (where q can be complex valued and q E C o r Co will be assumed whenever convenient - a c t u a l l y t h e theory can be developed f o r say P JOmx l q ( x ) l d x < e t c . b u t we abstain for now). One w r i t e s q p , ( x ) = CosXx and 5.

&LON,

-

s e t s f o r s u i t a b l e f ( c f . Theorem 4 . 1 ) (5.1)

rm

P f ( h ) = Cf(X) =

f(x)CosAxdx

JO

We take P

= P-'

(5.2)

PF(x) = IT IT)

so t h a t m

F(A)CosXxdA 0

(and w r i t e a l s o d v = (2/n)dh).

For Q we w r i t e f o r s u i t a b l e f

22

ROBERT CARROLL

We w i l l show how an i n v e r s i o n f o r (5.3) can be o b t a i n e d i n t h e form

a!h

=

Q-,1

where RQ i s a c e r t a i n d i s t r i b u t i o n c a l l e d t h e g e n e r a l i z e d s p e c t r a l f u n c t i o n by Marc'enko.

Note t h a t s i n c e q i s p o s s i b l y complex valued t h e o p e r a t o r Q

i s n o t n e c e s s a r i l y s e l f a d j o i n t and R4 may n o t be a measure. 2 2 2 L e t K ( r e s p . K ( u ) ) be t h e space o f L f u n c t i o n s on C O Y - ) 2 w i t h compact s u p p o r t (resp. w i t h s u p p o r t i n [O,a]). The space CK (u) o f 2 Cosine t r a n s f o r m s o f K ( u ) c o n s i s t s o f even e n t i r e f u n c t i o n s g(X) w i t h g E 2 L f o r X r e a l and Is(?,)] 5 cexpulImXl ( b y a v e r s i o n o f t h e Paley-Wiener 1 theorem 3.8). L e t Z(u) be t h e even e n t i r e f u n c t i o n s g w i t h g E L f o r 2 1 2 r e a l and I g ( X ) l 5 cexpulImXI. Put L (resp. L ) t y p e t o p o l o g i e s on C K ( a ) 2 (resp. Z ( u ) ) . L e t Z = U Z ( U ) and CK2 = UCK ( u ) w i t h s t a n d a r d i n d u c t i v e l i m i t

DEFZNZCZBN 5.1.

Thus a sequence gn .+ g i n o r countable u n i o n t o p o l o g i e s ( c f . [C19; G f l ] ) . 2 Z (resp. CK ) i f t h e e x p o n e n t i a l t y p e o f a l l gn i s bounded by some u and gn 2 g i n L1 ( r e s p . L ); such s e q u e n t i a l convergence i s a l l we need c o n s i d e r 2 2 (as for U - c f . Remark 2.3). E v i d e n t l y Z C CK and i f g1,g2 E CK t h e n g =

-f

g 1g 2 E Z ( i n f a c t t h e v e c t o r space o f such p r o d u c t s forms a dense s e t i n Z ) .

DEF'INICZBN 5.2,

The dual Z ' o f Z i s a space o f g e n e r a l i z e d f u n c t i o n s ( i n

which t h e so c a l l e d g e n e r a l i z e d s p e c t r a l f u n c t i o n R w i l l be found) w i t h act i o n on Z denoted by ( R , v ) o r

(

R,V)~.

g i v e n by a f u n c t i o n i n t h e form

(

R,P)

R E Z ' i s called regular i f i t i s =

Im R(X)'+'(X)dX f o r R

E Lm.

The co-

0

s i n e t r a n s f o r m i s d e f i n e d i n Z ' by d u a l i t y , i . e . ( q , c ( T ) ) = (T,Cv) where

op

= jm v(X)CosXxdX. 0

REmARK 5.3.

I n v o k i n g t h e Banach-Steinhaus theorem ( c f . [C19])

t h a t i f a sequence Rn E Z ' converges weakly ( i . e . then R

E

Z ' and Rn

-f

R weakly.

(

Rnv)

-+ (

one can say

R,v) f o r p E Z )

F o r o u r purposes such s e q u e n t i a l convergence

w i l l s u f f i c e and t h e r e i s no need t o go i n t o more d e t a i l i n d e s c r i b i n g t h e topologies o f Z o r Z ' . Now suppose we have c o n s t r u c t e d t r a n s m u t a t i o n s Bh and Bh = Bh' as i n Theorem 4.9 and w r i t e &(yyX) = 6(y-x) + Kh(y,x) w i t h yn(x,y) Define then f o r s u i t a b l e g

= 6(x-y) + Lh(x,y).

23

PARSEVAL FORMULAS

(5.5)

Btg(X) =

(

Bh(YYx)Yg(Y)) = g ( X )

+

J

m

Kh(Y,x)g(Y)dY

X

S i m i l a r l y Big(y) = ( y h ( x ' Y ) Y g ( x ) ) = g ( y ) + 1 ; Lh(xyy)g(x)dx. EHEBREm 5-4- For f , g E K2 ( u ) , Bif and B i g belong t o K2 ( u ) and (P (5.6)

PBif = Qhf; Qh"g

Q

C)

= Pg

*

*

P t r o o ~ : Using (4.17) one has PBhf = (CosAx,B,,f(x))

=

(CosXx,( B h ( y Y x ) , f ( y)) )

@h(yYx)yCoshx)yf(y))= ( q Q ( y ) , f ( y ) ) = Qhf(A). Similarly one has A ,h Q = ( ( y h ( x Y Y ) Y 9 h y h ( Y ) ) Y g ( x ) ) = ( cosXxYg(x)) QhBtg = ( q XQ, h ( Y ) , ( Y h ( x Y Y ) , g ( x ) ) ) = Pg. = ((

This kind of r e s u l t (due here t o Martenko) was generalized in LC39,40] and provides a useful ingredient in proving Parseval formulas of the type

1

m

(5.7)

Q

f ( x ) g ( x ) d x = ( R YQhfQhg)

*

0

where f , g E K2 and RQ E Z ' (note QhfQ,,g E Z by Theorem 5.4 since e.g. Bhf * 2 2 and Bhg E K ( u ) ( f o r some 0 ) so Qhf = PBif E C K ( u ) and by Definition 5.1 t h e product is i n Z ) . Another ingredient involves t h e idea of a generalized t r a n s l a t i o n S: which can be defined v i a Theorem 4.3. Thus i f P = Q i n Theo-

rem 4.3 we write q ( x , y ) = S{f(x). I f we now want S; t o have s u i t a b l e prope r t i e s however we must examine the extension problem more c a r e f u l l y . T h u s

we will be concerned with F ( x )

= Af(x) = f ( x ) and G(x) = Cf(x) = h f ( x ) in (4.4) f o r P = Q. Now t h e construction leading t o Theorem 4.5 and Theorem 4.6 can a l s o be used p r a c t i c a l l y verbatum f o r t h e equation Q(DX)q = Q ( D )q Y ( c f . [ M c ~ ] ) and one obtains

x-y One notes here t h a t i f we take x ~y J

where w i s continuous f o r q E Co say. (i 2; i n ( 4 . 8 ) ) so x-y 2 0 then t h e extension of q t o ( - m , m ) does not a r i s e Generally one thinks of data f in ( 5 . 8 ) s a t i s f y i n g f ' ( 0 ) = h f ( 0 ) following t h e construction f o r Theorem 4.9 and i n order t o have q ( 0 , y ) = f ( y ) an extension of such data t o ( - m , m ) i s suggested i n t h e f o r m (w(O,y, l / n we have en(x,y) = 0 f o r ( x - y l > l / n . le,(x,y)\

Given f,g

2

E K

(0)

consider

I

Iom Imen(x,y)f(x)g(y)dxdyl 5 c(a) I I If(x)g(y) 0 0 5 x,y 5 a1 whose two dimensional

dxdy o v e r a r e g i o n On = { \ x - y ( 5 l / n ,

I

26

ROBERT CARROLL

measure tends t o 0 as n

-+

On t h e o t h e r hand i t i s obvious t h a t (1/2)

m.

From t h i s we can conclude t h a t t h e l e f t s i d e o f (5.17)

2 Iomf(x)g(x)dx).

f o l l o w s , which we w r i t e s y m b o l i c a l l y as (U:(x,y),f(x)g(y)) -+ ( a ( x - y ) , f ( x ) 2 K ( a ) . Next we observe t h a t (5.18) i s i n f a c t r i g o r o u s ( r e -

g(y)) f o r f , g E

c a l l Lh i s continuous e t c . ) .

F u r t h e r as u

-f

U

m,y

(y)Lh(y,O)

-+

Lh(y,O)

uni-

formly on compact s e t s f o r example and t h e n i t s Cosine t r a n s f o r m as an e l To s p e l l t h i s o u t l e t us w r i t e i n D e f i -

ement o f Z ' tends t o CILh(y,O)]. n i t i o n 5.2,( q,CT) = (C q,T)

Z ( T ) f o r some 'I, and supp

f o r T E Z ' and q E Z. C9

c

continuous f u n c t i o n s converging u n i f o r m l y t o (

Tn,Q)

-+

(

T , Q ) so (q,CTn)

(weakly) as u

+ m

q E

Z, q E

T on compact s e t s t h e n c l e a r l y

converges t o what must be (q,CT)

T h i s proves t h a t Ra

t i o n s 5.2 and 5.3).

Note t h a t f o r

I f CTn E Z ' and Tn i s a sequence o f

[o,T].

+

(cf. Defini-

RQ = ( 2 / ~ ) [ 1 +CLh(y,O)]

in Z'

and s i n c e QhfQhg E Z i n (5.17) we can pass t o l i m i t s i n

t h e r i g h t hand s i d e .

Consequently one o b t a i n s (5.7).

We can e x p l i c i t l y i n d i c a t e t h e a c t i o n o f R4 on a t e s t f u n c t i o n q = C q E Z by ( n o t e supp q i s compact) (5.19)

( R Q , Q ) = (2/.rr)(Q,CS q

(O)

'

t

CLh) = ( q y 6

t

L ) = h

Lh(YSo)q ( Y l d Y

Now i n t h e Parseval f o r m u l a (5.7) l e t us s e t g ( x ) = l / S on [x,x+S] 0 elsewhere so t h a t as 6

mally.

+

0, Imf ( x ) g ( x ) d x = ( l / S ) JXtSf(x)dx 0

T h i s p o i n t v a l u e may n o t make sense f o r f E

f so t h a t Qf,

*

xrK however sof (t ax k) e f oe.g.r -+

*

Then by Theorem 5.4 we have PBhf E Z and B h f i s con* 1 t i n u o u s (PBgf = C B h f E L 1. Hence f ( x ) = B t f K h ( y , x ) f ( y ) d y i s conE

Z.

and g =

t i n u o u s and p o i n t values make sense.

On t h e o t h e r hand qhg = ( l / S )

c+*

Q Y h ( y ) d y q Q ( x ) p o i n t w i s e and Q g f -+ 9 XQ , h ( ~ ) Q h f ( A ) i n Z when Qhf E Z qX Ash2 * hp% ( n o t e here g E K ( a ) i m p l i e s Bhg E K (u) and f o r g as above Bhg i s i n f a c t -f

'I B;gCosAxdx i s i n f a c t 0 bounded f o r X r e a l (and a n a l y t i c i n A ) - t h i s i m p l i e s ( qQA y h ( x )I 5 M f o r r e a l so q Q ( x ) can a c t as a m u l t i p l i e r i n Z - t h e l i m i t i n g process can a l s o piecewise continuous and bounded so !2+g = PBtg =

XYh be used t o e s t a b l i s h a n a l y t i c i t y ) .

We a r r i v e t h e n a t

CHE0REN 5-6- I f C?+,fE Z one has an expansion (5.20)

Q f ( x ) = (RQ,C!+,f(h)qX,h(X))

27

PARSEVAL FORMULAS

RElllARK 5-7- As i n d i c a t e d above a t t h e c o n c l u s i o n o f t h e p r o o f o f Theorem 5.6

one can deduce p r o p e r t i e s o f t h e

(x) from t h e i r representation v i a h,h D i r e c t estimates are also possible v i a

lpQ

t r a n s m u t a t i o n k e r n e l s and cosines.

i n t e g r a l equations o f t h e form (4.18) and t h i s w i l l be examined l a t e r . remark a l s o t h a t (5.20) i s v e r y general, a l t h o u g h f i s r e s t r i c t e d .

We

The po-

t e n t i a l q may be complex valued and R 4 i s n o t n e c e s s a r i l y a measure ( c f . The d i s t r i b u t i o n a c t i o n o f RQ i n v o l v e s X r e a l

[C40; Mc4] f o r examples).

g e n e r a l l y and one p i c k s up complex eigenvalues f o r example i n t h e sense o f c o n t o u r i n t e g r a t i o n ( c f . [C40]). w r i t t e n f o r Q h f = F,

qhF

(5.21)

=

qh

=

The i n v e r s i o n expressed by (5.20) can be

~il

( RQ,F(i)lpQ ( x ) ) A,h

I f now q and h a r e r e a l one has a s e l f a d j o i n t s i t u a t i o n w i t h $

‘ PQ ~ , ~ ( X ) Hence . i f F ( X ) = Qhf i t f o l l o w s t h a t F(A)) =

0 for f

jm f(x)T(x)dx 0

K2 .

E

F(x) =

(x) = A,h Q h F a n d hence (RQ,F(x)

Such RQ w i l l be c a l l e d p o s i t i v e and

one proves i n [Mca]

CHEBRElll 5-8, I f q and h a r e r e a l t h e r e e x i s t s a nondecreasing f u n c t i o n -a < 1-1 < m , such t h a t f o r f , g E K2 m

(5.22)

f(x)g(x)dx = 0

I n f a c t f o r f,g E L

w i t h phf(J!J)

=

Jm 0

2

(

I

p(u),

m

Q

R QhfQhg) =

phf(J!J)Qhg(JF)dP(!J)

-m

one has ( w i t h i n t e g r a l s convergent i n s u i t a b l e senses)

f(xk(JP,,(x)dx

(f,g)

and

= 1 : Qhf(d!J)QhS(J!J)dP(!J).

The p r o o f o f Theorem 5.8 i s based on a c l a s s i c a l r e p r e s e n t a t i o n theorem o f F. Riesz f o r p o s i t i v e f u n c t i o n a l s ( c f . LGf3; Mc4; Rml; R o l l ) . Z’ i s positive i f

(

R,f(x))

>

One says R E

0 f o r a l l f ( - ) E Z satisfying f(dv) L 0 f o r

( R,F(X)F(x)) 5 0 f o r F E 2 Q . CK as above so R i s p o s i t i v e and induces a p o s i t i v e homogeneous, a d d i t i v e f u n c t i o n a l RQ [ g ( p ) ] = RQ [ f ( d u ) ] = ( R Q , f ( A ) ) on t h e s e t A o f g ( u ) = f ( h ) -m

< 11
0 occur

Q

Q on t h e imaginary a x i s . A t such a point one would have cp,(y) = c ( X ) @Q~ ( Y ) 2 Q which by (6.11) belongs t o L Such eigenfunctions would correspond t o what a r e c a l l e d bound s t a t e s b u t we can show t h a t t h e r e a r e n ' t any. Indeed 2 (with obvious n o t a t i o n ) , given q = c ( A ) @ E L w i t h ( 6 . 1 1 ) , multiply (6.1) Q by AG and i n t e g r a t e t o obtain ( A = i h 2 ) +

(6.29)

h2

l~\cp12Ady= -

c

lo m

( b ' ) ' l p d y = -Ap'GIm

Since DxvA(0) Q = 0 and ADxq:(y)$:(y)

+

0 as y

-f

m

+

(A
0 and does n o t v a n i s h t h e r e . A l s o c ( - 1 ) = 0 f o r r e a l x # 0. The func-

Q

Q Q t i o n s c ( 1 ) and c (-A) can be expressed v i a Dx@-x(0) and DxaX(0) as above.

Q

Q

It i s p o s s i b l e t o g i v e some f u r t h e r formulas f o r c

Q Thus from (6.5) w i t h q,(y)

v a r i o u s ways. (6.30)

q'

- ixq

=

-

ixy

+

%

'Q

which a r e o f use i n

q(X,y), @.,(y)

rY e - ix (y-rl)

%

a+, e t c .

q(rl )q ' (A ,rl )drl

JO

As Y m 3 A(Y)W(V,@+) Theorem 6.5; hence

A)d_exp(ixy)[iAq

-t

-+

- q'],

(-A)by

Note also f r o m ( 6 . 5 ) "

so t h a t t h e i n t e g r a l i n (6.31) makes sense.

x

9

cQ(-X) = (

(6.31)

Let

which equals 2 i h c

-+

0 so t h a t J/(O,y) = Jy q(q)$(O,q)dn w i t h J / ( O , O ) = J/,

=

0 since 0

5

0

I t f o l l o w s t h a t J/(O,y) = &,exp(JY

$(x,O). let

x

+

0 one o b t a i n s c (-A)

Q

(6.6) and (6.27);

q(rl)drl) f 0 and i n (6.31) if we 0 Another form f o r c (-A) f o l l o w s f r o m

i2/2.

Q

indeed

c (-A) =

(6.33)

-+

Q

1/2)Ai3[1

- (l/ix)

~osxrlq(n)@~(n)d~J 0

CHE0REiR 6.6, One can r e p r e s e n t c (-A) by (6.31) o r (6.33) ( I m x 2 0 ) and Q (1/2)i: as x + 0. Consequently ( c f . Theorem 6.5) from (6.31) c Q ( - x ) -+

c ( - 1 ) # 0 f o r Imx 2 0.

Q

R?EiMRK 6-7- F o r v a r i o u s purposes one would l i k e an e s t i m a t e I c (-A)[

Q

f o r Imx (6.33).

n ,

q(q)(l/Z)[exp(2ix,-,)

cQ(-x)

-+

-+

m

t h e n from t h e c o n s t r u c t i o n i n Theorem

and ( r e c a l l q = - A ' / A )

A?ihexp(ihq)

Hence f o r I m x > 0 and Imh (6.34)

E

0 and some ( h e u r i s t i c ) i n f o r m a t i o n i n t h i s d i r e c t i o n f o l l o w s from Thus i f I m x > 0 and Imh

6.2 e t c . @: "J,"

2

+ l]dq +

%

( l / i x ) f m CosAqq(n)@idn 0

( 1 / 2 ) A 2 J m q(n)dn = -(1/2)A~'logAm. 0

m

(l/Z)A?[l

and t h i s vanishes o n l y when log(:

+

(1/2)A~31~gAm] = -A4.m

Except f o r such i s o l a t e d cases

%

SPECTRAL THEORY I N ENERGY

5 c f o r Imh

t h e n one would expect I l / c (-X)l

Q

35

0.

Now one can develop an expansion t h e o r y f o r Q j u s t as i n §5 ( c f . Theorem We want t o i n d i c a t e here another method

5.6) and t h i s w i l l be done l a t e r .

o f d e t e r m i n i n g t h e s p e c t r a l measure by c o n s t r u c t i n g a Green's f u n c t i o n and u s i n g c o n t o u r i n t e g r a t i o n ( c f . [C40,67;

Thus we w i l l es-

Dcl; Nel; S e l l ) .

For s u i t a b l e f

t a b l i s h the following inversion. m

j

(6.35)

Qf(h) =

(6.37)

d v ( x ) = ^v(h)dh = d h / 2 a l c (A)[

f(x)A(x)q:(x)dx

= F(X)

2

Q

9-l).

The technique which we d e s c r i b e now can ob i o u s l y be ap2 Consider t h e p l i e d t o Q(D) = D - q b u t we o m i t t h e d e t a i l s ( c f . [ D c l l ) . (thus

=

so c a l l e d r e s o l v e n t k e r n e l o r Green's f u n c t i o n (q((A,x) a QA ( x ) , x < = m i n ( x , x ' ) , (6.38)

RCX2,x,^X)

Q

Q XI,

(pA

@(h,X)

%

and x, = max(x,x')) =

(p(X,x,)a(x,x,)/A(x)W(~,a)

L e t JI E C 2 , 4x+ -( r e c a l l f r o m Theorem 6.5 t h a t A(x)W((p,@) = 2 i h c (-A)). 2Q 2 :+O, and ;= ;-0 so f o r I = jm J,(x)[Q(DX) + X ]R(h ,x,$)A(x)dx one has 0

I

(6.39)

=

+:

JI(x)[Q(Dx) + x 2 ] R ( h 2 , x , ~ ) A ( x ) d x = $(X)A(X)R,(~x+ x-

-A

(x)~(x)~l;

$1

h

-

+

J; i+ R(h2,x,?)[Q(Dx)

-

+ X27JI A(x)dx

-

so t h e l a s t two terms v a n i s h w h i l e t h e f i r s t

Now R i s continuous and J, E C' term gives (6.40)

I

s i n c e AR

X

=

= J,(i)A($)[Rx(h2,;+,?)

W(q,@)/A(x)W((p,@)

-

Rx(h2,C-,"x)1

=

JI($)

( w i t h W e v a l u a t e d a t ?).

Consequently one

can make an i d e n t i f i c a t i o n (6.41)

A ( x ) [ Q ( D ~ )+

S i m i l a r l y A(x)[Q(Dx)

x 2 [ R ( A2 ,x,?)

+ x2 ]R(x2,;,x)

=

6(x-j;)

= 6(x-*x).

L e t now 5 be a smooth f u n c -

t i o n v a n i s h i n g n e a r 0 and m (e.g. 5 E CE(0,w)) and t h e n f o r e = Q(D)c, 2 2 2 (A(y)5(y),[Q(Dy) + A l R ( X ,x,Y)) = S(x1 = (A(y)R(X ,x,Y),[Q(DY) + A215(Y))

(
0 w i t h a zero o n l y a t A = 0 f o r Imh 2 0 w h i l e

i s a n a l y t i c f o r Imh > 0. Also by 2 Theorem 6.2 and (6.41) i n t h e numerator R(h ,x,y) w i l l have e x p o n e n t i a l = @

bounds exp(y-x)Imh f o r x > y and exp(x-y)Imh f o r y > x s i d e r R as a f u n c t i o n o f E = h

2 (E

T,

(ImA

Con-

> 0).

Except f o r a c u t on

energy).

i n t h e E p l a n e R w i l l be a n a l y t i c i n E ( c f . [Dcl;

[O,m)

Nel] f o r d i s c u s s i o n

-

the

upper h a l f p l a n e i n h i s mapped o n t o t h e E plane). Now t a k e a l a r g e c i r c u l a r c o n t o u r o f r a d i u s Y i n t h e E p l a n e and i n t e g r a t e (6.42) around t h i s c o n t o u r t o o b t a i n

(note g e n e r a l l y IR/EI

1 ',

0(1/E3/2;at

least

-

c f . Remark 6.7).

On t h e o t h e r

hand i f one takes a c o n t o u r as i n d i c a t e d i n (6.44) i n t h e E p l a n e ( a v o i d i n g the c u t )

1

t h e n upon i n t e g r a t i n g (6.42) around t h i s c o n t o u r t h e r e r e s u l t s (6.45)

I

dE

EI=r

Put t h i s i n (6.43),

rm

c(y)A(y)R(x2,x,y)dy

1;

'0

-

2 E A R ( 1 +ie,x,y)dy

dE

2

with y

2 n i ~ ( x )=

loy

0

dE j o m c ( ~ ) A ( ~ ) R (-iE,x,y)dy h -+

= 0

t o obtain

(4,

jo dE l O c ( ~ ) A ( y ) [ R (2h-ie,x,y) m

(6.46)

+

m

-

Now pass t o t h e h plane, o b s e r v i n g t h e p o s i t i o n s o f A

2

R(h + i ~ , x , y ) l d y

2

+iE

and l e t t i n g

E

-f

0,

we o b t a i n

lo m

(6.47)

Y o r I 1 = @ ( h , x ) C ~ ( - x , y ) / ( - ~ c q ( h ) )

-

-

*(A,x)/

@(h,Y)/AcQ(-A)l

for

SPECTRAL THEORY IN MOMENTUM y > x.

37

The equation (6.47) follows then upon using (6.26

.

Since 5 i s an

a r b i t r a r y t e s t function we have proved CHE0RElll 6-8, The spectral measure f o r t h e eigenfunctions

1

'(x) A

i s given by

( 6 . 3 7 ) and t h e inversion (6.35)-(6.36) holds f o r s u i t a b l e f . REmARK 6-9, The transmutation theory f o r (more general) operators Q of the

form ( 6 . 1 ) will be developed l a t e r and we will a r r i v e a t a Parseval formula e t c . a s i n 55. Thus from ( 6 . 3 5 ) - ( 6 . 3 6 ) one has f o r s u i t a b l e f , g

=lo m

(6.48)

(

w i t h f(x) 7,

=

R Q ,Qfqg)

(k5f)(x)(A'g)(x)dx

Thus R Q ( R Q ,Qf(A)qA(x)). Q

?1

dv i n (6.37).

SPECCRAI; CHEBRU I N CHE lIl0mENClllll UARZABCE,

Let us give a v a r i a t i o n on

S6 t o cover operators of t h e form

-

Qu

(7.1)

2

=

2 x u " + (n-1)xu'

t x2[k2

-

G(x)]u

= X

2

u

E i s fixed and A 2 = L(L+l) ( L corresponds t o complex angular momentum) i s e s s e n t i a l l y t h e s p e c t r a l v a r i a b l e (see below). The case o f gene r a l n w i l l be t r e a t e d l a t e r and we deal here only w i t h n = 3. Note t h a t t h e n-dimensional Laplace operator has t h e form where k

(7.2)

%

( rn-1 ur)r/rn-l + A i u ; r 2A u

An u =

=

r 2 urr + ( n - l ) r u r

t Dnu S

2 s

where 0;" = r An depends only on "angle" v a r i a b l e s . Thus operators such as ( 7 . 1 ) will a r i s e i n s p h e r i c a l l y symnetric problems of physics and s e t t i n g n = 3 w i t h 'P = xu (7.1) becomes (7.3)

x

2

q"

+ x2[k2 -

{(x)]lp

2

= A q

( n o t e ( x L u ' ) ' = ~ ( x u ) " ) . This equation a r i s e s e.g. i n studying s c a t t e r i n g problems a t f i x e d energy k2 = E in quantum mechanics and one can e x t r a c t a wealth of information from the physics l i t e r a t u r e ( s e e e.g. [Bdl; Bel; Bbdl, 2 ; C43-46,54; Cnl-7; Cel; Crl; Dc1,2; Gjl-3; L17; L r l ; Jb2,4; J d l ; Ne1,6; Rfl; Sa10-121). Later in connection w i t h special functions e t c . we will have occasion t o study s i n g u l a r operators of t h e form (7.4)

Gu

=

(xn-'u')'/xn-'

- G(x)u

= -k

2

u 2 A

with s p e c t r a l v a r i a b l e k and f o r q^(x) = A2/x2 - 0 with P continuous up t o Rev = 0. The J o s t s o l u t i o n f ( v , k , x ) is a n a l y t i c i n (u,k) f o r v E C and Imk < 0 (and continuous up t o Imk = 0). Correspondingly f ( v , k ) is a n a l y t i c i n ( v , k ) f o r Rev > 0, Imk < 0 , and i s continuous up t o the boundaries Rev = 0 and Imk = 0. v

ZIEIRARK 7-4- Another assumption of

often made ( c f . [Eel; C e l l ) which allows

one t o enlarge the a n a l y t i c i t y region involves t h e assumption t h a t G ( x ) = (m > 0 ) so t h a t q ( x ) can be a n a l y t i c a l l y continued i n t o

( o(p)exp(-px)du/x

t h e half plane Rex > 0. Also on any ray argx = 8, 1 0 1 5 ( ~ / Z ) I T - E , one requires Jm I x q ( x ) ) d x < M < m. T h i s c l a s s includes t h e Yukawa p o t e n t i a l s of 0

i n t e r e s t i n physics.

Other work involving p o t e n t i a l s

having a n a l y t i c o r

meromorphic continuations i n t o regions i n C can be found in e.g. [Bdl; F n l ; Ne6; Sa11,131; the results a r e important i n terms o f locating Regge poles o r zeros of f ( v , - k ) . There i s of course an enormous wealth of information i n the physics 1i t e r a t u r e concerning the r e l a t i o n between hypotheses on and t h e r e s u l t i n g p r o p e r t i e s of the regular s o l u t i o n , J o s t s o l u t i o n s , and t h e J o s t function. We have c i t e d a few sources but make no attempt t o survey t h e l i t e r a t u r e . We go now t o t h e e x p l i c i t development i n [Bbdl] f o r t h e so-called inversion

i n the v-plane ( k w i l l be f i x e d ) . Much of t h e c a l c u l a t i o n t h e r e i s repeated here and f u r t h e r d e t a i l s a r e supplied i n order t h a t we can expand upon t h i s and modify ( n o n t r i v i a l l y ) c e r t a i n techniques. With t h e proper interp e r t a t i o n this will t h e n lead t o transmutation kernels e t c . as desired. The c l a s s i c a l Green's function is

where r< = m i n ( r , r ' ) and r, = m a x ( r , r ' ) . Then a s o l u t i o n JI of (7.3) can be = 0 via expressed i n terms of a s o l u t i o n JIo of ( 7 . 3 ) w i t h JI(v,k,r) = Jl0(w,k,r) +

(7.13)

c

G(v,k,r,r'):(r'

)Jlo(v,k,r')dr'

Consider now formally f o r s u i t a b l e h

(7.14) where

vdv ~ ~ m h ( r ' ) [ - G ( u , k , r , r ' ) / r r ' ] d r '

I(k,r) =

r

r i s a semicircle

Q

of " i n f i n i t e " radius i n t h e half plane Rev > 0

w i t h v e r t i c a l s i d e the axis Rev = 0. This can be evaluated i n two ways as follows. First t h e only poles of the integrand occur a t t h e zeros of the J o s t function a t v = v j ( t y p i c a l l y simple) so t h a t

SPECTRAL THEORY I N MOMENTUM

Note here t h a t t h e f r e e J o s t f u n c t i o n f o ( v , - k )

41

f o r example has no such

zeros so t h i s c o n t r a s t s s i g n i f i c a n t l y from t h e s i t u a t i o n i n [ L r l ; g a r d i n g t h e d i s c r e t e spectrum.

One can show t h a t ( c f . [Bel;

v and s i n c e ~ ( ,k,r) j

j'

(7.17)

I(k,r)

(1/2ik)f(v

=

1

= ri

-k,r)

L l l ] re-

Bbdl])

we have

lorn*

f(v.,-k,r)f(v

-k,r') dr'

j'

J

M2 ( v j , k )

Next we e v a l u a t e I a l o n g t h e c o n t o u r

r.

Since f ( v , - k , r )

i s even i n v and

(7.11) h o l d s one has f i r s t , f o r m a l l y ,

F o r t h e l a r g e s e m i c i r c l e SF. one r e c a l l s t h a t ( c f . [Bel; v(v,k,r)

%

exp( -+in),

Ipo(v,k,r)

rv+', f ( u , - k )

%

%

fo( v,-k )

and ( u s i n g (7.11 ) ), f ( v , - k, r )

%

%

+

m,

( kr/v)+exp ( % i r v ) [ (2v/ek)"

exp(-+ia)/rv

-

c f . "e61).

Hence ( Y d e n o t i n g t h e Heavyside f u n c t i o n )

(ek/2v)"r"exp(+in)]

B b d l l ) as 1v1

2(~kf'(2v/ek)~exp(+iav)

(such e s t i m a t e s a r e standard i n p h y s i c s

-

We n o t e here t h a t i n t e g r a l s o f t h e form 9 ( e k / 2 ~ ) ~ " ( r s ) " d v v a n i s h s i n c e IvI

-f

(observe ( e k / 2 ~ ) ~ " ( r s ) "= ( a / 2 ~ ) ~ f' o r a = ek(rs)'

l i n g ' s formula aZ/zZ

4271 azexp(-z)z-'/r(z)

%

etc.).

(7.19) one r e c a l l s t h a t l i m SinRy/y = 1~6(y)as R d e l t a f u n c t i o n and we r e f e r t o [C40,67]

-+

-

also by S t i r -

I n order t o evaluate m

( t h i s i s a two-sided

f o r a d i s c u s s i o n o f one and two

-

c f . a l s o pp.150 and 292). Set y = l o g ( s / r ) f o r exiR iR ample So when s 5 r, I-iR ( s / r ) " d v = J-iRexp(vy)dv = 2iSinyR/y + 271iS(y) iR e r e - m < y 5. 0) w h i l e f o r s ? r t h e o t h e r t e r m i n (7.19) y i e l d s I-iR (r/s)" (h iR dv = J-iRexp(-yv)dv = 2iSinyR/y 271i6(y) a g a i n and h e r e r 5 s < p u t s US i n t h e range 0 5 y m. Hence (7.19) becomes

sided d e l t a functions

-+

(7.20)

.I.

jOr+ [ ] @ G ( l o g ( s / r ) ) d s rij

m

=

T i [

=

$th(ret)6(t)dt

-m

which equals r i h ( r ) .

Set now ( v

j

d e n o t i n g t h e zeros Z of f ( v , - k )

(Rev > 0)

ROBERT CARROLL

42

(7.21)

dp(v) = (2i/~)vLdv/f(v,-k)f(-vy-k) dp(v) =

1 ~ ( v - Jv . )

( v E [O,i-);

( v E Z)

Then from ( 7 . 1 7 ) , ( 7 . 1 8 ) , and ( 7 . 2 0 ) one o b t a i n s ( c f . [Bbdl]) &HE0REm 7-5, The " c o m p l e t e n e s s " r e l a t i o n f o r the J o s t s o l u t i o n s can be writ-

ten s y m b o l i c a l l y a s ( f ( v , - k , r ) (7.22)

s(r-s)

=

I

= rg(v,-k,r)

etc.)

g(v,-k,r)g(v,-k,s)dp(v)

In o t h e r words, f o r s u i t a b l e f , from G f ( v ) = fn(v) = J; obtains f ( x ) = ( ? ( v ) , g ( v , - k , x ) )p.

Phuod:

f ( x ) g ( v , - k , x ) d x one

T h i s c o n c l u s i o n a r i s e s from the e q u a t i o n h ( r ) = Jm h ( s ) E d s where 0

lo

i m

(7.23)

E =

1 g ( v j , - k y r ) g ( v j y - k y s ) / M 2 +-2 i

2

g(v,-k,r)g(v,-k,s)v f(v,-k)f(-v,-k)

dv

The f u n c t i o n h can be q u i t e g e n e r a l here - h E Lz i s s u g g e s t e d i n [Bbdl] and c e r t a i n l y h E is a d m i s s a b l e . 1 Let now f ( v , - k , r ) refer t o e q u a t i o n ( 7 . 3 ) w i t h p o t e n t i a l G, and c o n s i d e r

Ci

f o r s u i t a b l e H(u,r) ( t h e model here i s H ( u , r ) = f ( u , - k , r ) / r (7.24)

JH(vyk,r) =

I

i n [Bbdl])

m l 1 H(uyr)dpl( v ) k g (u,-k,s)g (v,-k,s)ds

1 1 1 ( r g ( v , - k , r ) = f ( v , - k , r ) e t c . ) . One knows t h a t ( $ ( v ) 2. f ( v , - k , r ) ) 2 2 DrW($(v),$[v)) = ( v 2 - u ) $ ( u ) $ ( v ) / r and W 0 as r m y so 1 1 (7.25) J H ( v , k , r ) = - H ( u , r ) W(f ( v , - k , r ) , f ( u y - k y r ) ) d p l ( p ) -+

I

u

-+

- v

1

Now d e n o t e by u . the z e r o s o f f ( u , - k ) f o r Reu > 0 and p u t ( 7 . 2 1 ) i n ( 7 . 2 5 ) J t o o b t a i n ( u s i n g (7.11) a g a i n ) i m 1 W(f ( v , - k y r ) ~ ( u y k , r ) L uH(u,r)du (7.26) JH = ( i / r ) 2 1 - i m ( ~ 2- v If ( u , - k ) W(f'(v,-k,r)f 1(Ujy-k,r))

I

-1

2 2 2 ( u j - v )Ml(pjyk)

H(vj,r)

(symmetry p r o p e r t i e s o f t h e J o s t s o l u t i o n s a r e worked i n h e r e and v i s t a k e n s l i g h t l y away from t h e imaginary p a x i s i n t o the r i g h t h a l f p l a n e

43

SPECTRAL THEORY IN MOMENTUM Rev > 0 ) . Add now t o (7.26) an i n t e g r a l over a l a r g e semicircular a r c of the same integrand so t h a t a term /-r a r i s e s which can be evaluated as before (-r denotes r traversed in the opposite d i r e c t i o n )

( t h e H ( v , r ) t e r n comes from t h e pole a t u (7.28)

= v).

Consequently

JH = H ( v , r ) + ( i / n ) f I H d u

where IH i s t h e integrand i n (7.27).

To evaluate k b now one uses asymptot i c estimates f o r l a r g e 1u1 ( c f . (7.19) e t c . ) and here s p e c i f i c assumptions about H(u,r) a r e needed. Typically one can s t a t e

Take H(v,r)

EHEOREm 7-6,

(7.29)

(i/r) t I H d u

(7.30)

JH

(7.31) Pmad:

=

= f ( u , - k , r ) / r in J =

H and IH above. Then

-(1/2)f 1(v,-k,r)/r

f ( v , - k , r ) / r - (1/2)f1 ( v , - k y r ) / r =

N

B(r,s)

=

The i n t e g r a l

1

1 1 g ( u , - k , r ) g ( v , - k , s ) d p (u)

lrm

B(r,s)f 1(v,-k,s)ds/s

9 1 H d p i s s i m i l a r t o (7.19) i n some respects.

For

1 ~ we 1

have W(f 1 ( v , - k , r ) , p ’ ( p , k , r ) f 1 (v,-k,r)(v+%)r’-$ - Drf 1 ( v , - k , 1 r)r”+$ so t h a t estimating f ( p , - k ) and f ( u , - k , r ) as before one has ( R + m) large

2 2 Note here t h a t (u+$)/(p -v ) = A/(u-v) + B / ( p + v ) with A+B = 1 and log[(iR+v)/(-iR+v)] -t log(-1) = in a s R 2 2 + m. On t h e other hand l / ( u - V ) = (1/2v)[l/(u-v) - l / ( u + v ) ] so t h e i n terms cancel. The r e s t follows immediately.

so ( i / n ) ’I* IHdu ’L - ( 1 / 2 ) f 1 ( v , - k , r ) / r .

REmARK (7.33)

7-7- The p r o p e r t i e s of

H

%

f(p,-k,r)/r

%

H used here were p r e c i s e l y r-1(kr/p)4e5iiau[

1;

44

ROBERT CARROLL

-

[ ] = (2u/ek)ve-’iTr-u

(7.33)

and o n l y t h e f i r s t t e r m i n [ 1 L e t us w r i t e ( f l % f ( v , - l , r ) (7.34)

I made

1-1

kim u

(ek/Zp) e

r

a c o n t r i b u t i o n t o t h e i n t e g r a l i n (7.32)

etc.)

IHn.(ek/2v)’e‘LrinyH/2~)(v%/(v2-v2)[f1

Thus i f e.g.

H(ekr/Zv)’exp(-%iav)

-f

0 as l u l

-f

(v+3)rU-4 m

-

Drflru%]e’iT

t h e r e w i l l be no c o n t r i b u -

t i o n from t h e IH i n t e g r a n d and JH = H. Now go back t o I i n (7.14) and observe t h a t t h e c a l c u l a t i o n i s b a s i c a l l y t h e same i f one t a k e s

and d p ( v ) i n (7.21) w i l l be t h e same.

The d i f f e r e n c e which a r i s e s due t o

t h e i n t e g r a l frm i n s t e a d o f f m i s however o f c o n s i d e r a b l e i n t e r e s t and i t 0

appears o n l y i n t h e t e r m (7.19)% corresponding t o (7.19).

Thus t h e t e r m

(7.19)% w i l l be

.f.=

(7.36)

f(r/s)vY(s-r)dv

%

C a l c u l a t i o n s such as (7.20) a p p l y again o f course b u t now we o b t a i n a two s i d e d 6 e x p r e s s i o n a c t i n g on one s i d e o n l y which must be i d e n t i f i e d w i t h $6+ where 6, denotes a one s i d e d 6 f u n c t i o n ( c f . [C40,67]

E m m A 7.8. so t h a t ( s (7.37)

REmARK (7.38)

F o r s u i t a b l e f u n c t i o n s h (as i n Theorem 7.5),

y(k,r)

Hence

= (1/2)h(r)

r) (1/2)6+(s-r)

7.9.

and 53.8).

=

g(u,-k,r)g

1

(v,-k,s)dp(u)

L e t us d e f i n e now ( c f . (7.31)) B(r,s) =

g(v,-k,r)g

1

(u,-k,s)dp(u)

and Lemna 7.8 demonstrates t h a t p ( r , s ) = 0 f o r s > r which we r e f e r t o as a t r i a n g u l a r i t y p r o p e r t y ( c f . [C40]).

The a c t i o n o f p ( r , s )

f o r s < r w i l l be

i n d i c a t e d below. I f one combines Lemma 7.8 w i t h Theorem 7.6 we o b t a i n a r e s u l t o f [ B b d l l

which can be s t a t e d as f o l l o w s ( t a k e h = f 1( v , - k , r ) / r

i n Lemma 7.8)

SPECTRAL THEORY IN MOMENTUM

EHE0REFII 7-10,

(7.39)

45

Let b ( r , s ) = ;(r,s) - B(r,s) ( r 5 s); then

g(v,-k,r)

1

= g (v,-k,r)

-t

b(r,s)gl(v,-k,s)ds

REmARK 7-11- The kernel b ( r , s ) can be w r i t t e n as b ( r , s ) = I g ( v , - k , r ) 1 1 1 g (v,-k,s)[dp ( v ) - d p ( v ) ] and t h e use of both s p e c t r a l measures d p and d p in i t s d e f i n i t i o n renders i t i n e f f e c t i v e f o r c e r t a i n purposes. In [Bbdl]

b ( r , s ) i s t r e a t e d a s a function with a j u m p a t r = s and various formulas a r e derived ( c f . (7.47)). This appears however t o be i l l advised and we w i l l discuss t h e matter below i n Remark 7.13. 1 Referring t o Remark 7.11 and Theorem 7.10 t h e presence of dp and dp i n -1 b ( r , s ) mixes t h e operators Q and Q i n an unwieldy manner and we want t o remove t h i s ( a l s o i t i s i n c o r r e c t t o t r e a t b as a f u n c t i o n ) . The key t o and 6 on (0,~)r a t h e r achieving t h i s i s t o consider t h e kernel a c t i o n of T h u s B ( r , s ) = 0 f o r s > r and a c t i n g on s E [ r , m ) , than only on [ r , m ) . B ( r , s ) = ( 1 / 2 ) 6 + ( s - r ) by (7.37). To see t h e r e s t of t h e B a c t i o n we s e t again g ( v , - k , r ) = f ( v , - k , r ) / r (so g s a t i s f i e s (7.1) w i t h n = 3, L(L+l) = 2 2 h2 = v -4,e t c . i n t h e form Qu = h u ) and interchange the r o l e s of ql and 1 1 1 -1 q in JH ( s o p ct p , f c+ f , e t c . where p l y f r e f e r t o Q w i t h p o t e n t i a l -1 q ) in order t o consider ( c f . ( 7 . 2 4 ) - ( 7 . 2 8 ) ) Iu

Iv

N

(7.41)

IH = [W(f(v,-k,r),~(~,k,r))/(p

2

-v

2

)f(~,-k)l~H(~,r)

1 Upon taking then H = g ( p , - k , r ) one obtains ( i / n ) 9 1 H d p = - 4 g ( v , - k , r ) h.

(7.42)

“ 1 J H = g ( v , - k , r ) - ( l / 2 ) g ( v y - k y r )=

e

and

g(v,-k,s)B(s,r)ds

Now combine (7.42) w i t h (7.37) and w r i t e formally m

(7.43)

(g(v,-k,s),B(s,r))

=

1 g(v,-k,s)B(s,r)ds = g (v,-k,r) -

( i . e . B ( s , r ) = (1/2)6+(r-s) f o r r 2 s ) .

Consequently 1 and i i ( r , s ) = EHE0REm 7-12. Define B ( r , s ) = ( g ( v , - k , r ) , g ( v , - k , s ) ) P ( g ( v , - k , r ) , g 1 ( ~ , - k , s ) ) ~ l .Then B(r,s) = 0 f o r s > r and F(r,s) = 0 f o r

46

s

ROBERT CARROLL

r ; in f a c t f o r s > r B(r,s) (1/2)6+(r-s). Define
r by (7.37) and thus b ( r , s ) = ; ( r , s ) - B ( r , s ) = g ( r , s ) f o r s > r. I f one wanted t o regard b ( r , s ) in ( 7 . 3 9 ) a s a function 6 ( r , s ) f o r s > r w i t h t h e 6 function action removed ( a s in LBbdl]) i t would have t o be ident i f i e d w i t h g ( r , s ) t h e r e and g - g1 = i ( r , s ) g1 ( s ) d s ( g ( r ) = g ( v , - k , r ) ,

fr

On the o t h e r hand t h e r e a r e two o t h e r "equally l e g i t i m a t e " candiL ( r , s ) g ( s ) d s . I f t h e argument of [Bbdl] were c o r r e c t 1 1 we should simply be a b l e t o e x t r a c t g - ( 1 / 2 ) g = 1 ; bv(r,s)g ( s ) d s from J H in (7.30). Perhaps more compelling i s t o t h i n k of F ( r , s ) = bY(r,s) i ( r , s ) g1 ( s ) d s . Thus ( l / Z ) & + ( s - r ) in J H o r i n (7.46) and t o w r i t e g = ,--1 t h e r e a r e c o n t r a d i c t i o n s and consequently we will t r e a t B[g ] = ( ; ( r , s ) , g 1 ( s ) ) = g ( r ) a s t h e basic transmutation here and dismiss (7.39) a s heurist i c (and c o r r e c t ) b u t misleading s i n c e b ( r , s ) i s not a function. etc.).

d a t e s t o represent

I,"

8.

CLABSZCAL SPEtXRAL CHEsRM AND RELACZ0w C 0

FULL LINE BCACCERZNG,

The

development of s p e c t r a l ideas i n §§6-7 i s e s p e c i a l l y important f o r applicat i o n s in physics and special functions. The connection of J o s t functions w i t h s p e c t r a l measures and t h e i m p l i c i t f a c t o r i z a t i o n of t h e s p e c t r a l meas u r e in terms o f e.g. c ( A ) and ? ( A ) = c (-A) will have f a r reaching s i g Q Q Q n i f i c a n c e in succeeding chapters. I t will a l s o be useful t o connect this material w i t h t h e c l a s s i c a l theory expounded i n [TeZ] ( c f . a l s o [L19]). T h u s we w i l l r e c a l l and sketch here t h e formulation of [TeZ] f o r [0,m) and (leaving some of the d e t a i l s a s e x c e r c i s e s ) . tinuous and r e a l ) (-my-)

(8.1) Let

q

(8.2)

QU = U "

= q ( x , x ) and

-

q(X)U

e

=

One considers ( q con-

= -XU

e ( x , X ) s a t i s f y (8.1) w i t h

q ( 0 ) = S i n a ; q ' ( 0 ) = -Cosa; e ( 0 ) = COSU; e ' ( 0 ) = Sina

Thus W ( q , e ) = q e ' - q ' e = 1 . The general s o l u t i o n of (8.1) has then t h e form u = e + Lq and i f one imposes a boundary condition a t some x = b of the form u(b)Cosg + u ' ( b ) S i , n ~= 0 then s e t t i n g ctnB = z

48

ROBERT CARROLL

For b f i x e d , as z v a r i e s

l d e s c r i b e s a c i r c l e Cb w i t h Cb

C

f o r b' < b

Cbl

+ m Cb tends t o a l i m i t c i r c l e o r a l i m i t p o i n t . I f m = m(A) i s 2 t h e l i m i t p o i n t o r any p o i n t on t h e l i m i t c i r c l e , Jm Ie+mpI dx 5 -1m m/ImA

and as b

0

Consequently f o r Imx # 0 (8.1) has a s o l u t i o n $(x,x) 2 2 = e(x,A) + m(x)v(x,A) i n L (0,m) (and i n f a c t J t I$(x,X)l dx = - Im m ( A ) / 2 Imh). I n f a c t i n t h e l i m i t c i r c l e case a l l s o l u t i o n s o f (8.1) a r e i n L

(sgn I m m = -sgn Imx).

.

One w r i t e s now f o r f E L 2 (no c o n f u s i o n w i t h @(A,x) = a QA ( x ) as i n 56 should a r i s e here) (8.4)

@(x,X) = $(x,x)

i"

v(y,h)f(y)dy

+

v(X,A)

0

$G ( x ,Y¶ A 1f ( Y )dY

1

$(y,h)f(y)dy

=

so O(0,x)Cos~r t @'(O,x)Sina = 0 and f o r f E C say, Qip t A@ = f ( G i s c a l l e d a Green's f u n c t i o n

-

The f u n c t i o n m(A) i s e a s i l y seen t o be

as b e f o r e ) .

a n a l y t i c i n e i t h e r h a l f p l a n e Imx > 0 o r ImA < 0. F u r t h e r i f f, q f , and f " 2 0 (Imx f 0) w i t h f ( 0 g o s a + f ' ( 0 ) S i n a = 0 w h i l e , as x -+ m, W($,f)

E L

-+

then an i n t e g r a t i o n by p a r t s i n (8.4) y i e l d s

@(x,A)

(8.5)

*

where @

= (l/x)[f(x)

-

@*(x,x)I

has t h e same form as @ i n (8.4), w i t h f r e p l a c e d by Qf= f "

-

qf.

Note here t h a t (8.5) has t h e f o r m f(x) =

(8.6)

lom

G(xyy,A) [Af(y)

A more o r l e s s r o u t i n e e s t i m a t e on P(x,x)

=

O ( 1/

@,

Qf(y)ldy u s i n g Q€J + A@ = f, shows now t h a t

f o r Imx # 0 ( e x e r c i s e - c f . [Te2, p. 341). A p p l y i n g 2 w i t h f and QfE L , one o b t a i n s from (8.51, ( * ) @(x,X) = f ( x ) / A

O(lA\'/lImA\)

t h i s t o @*, L

+

I x I 3/4 I I m x I 1.

low l e t I' be a s e m i c i r c l e o f r a d i u s R and c e n t e r i s w i t h base segment ( - R + i s , R + i G ) . here on

r, x

=

By (*) above Ir @(x,A)dx + n i f ( x ) as R

i s + Rexp(ie), 0 5 e

-

6 > 0 one can e s t i m a t e e.c(. t &fn/2de/Re

6/R

= O(R-3/4)

+

< n, and

iF [F?*/(s+RSine)]de

O(R-3/410gR)

i s a n a l y t i c i n t h e upper h a l f R+i 1i m I m C IT) (8.7) f ( x ) = R+= -R+i Define next formally Since

@

Ir dh/A

-+

T i

B the l i n e -+

Note

i n terms o f say I"RF?'de/s

(and s i m i l a r l y f o r a/2 5 p l a n e one o b t a i n s 6 @(x,X)dxl 6

-.

while f o r fixed 0

e

5 T).

49

FULL LINE SCATTERING

1

x

lim 6+0

(8.8)

[ - I m m(u+i6)]du = k ( x )

0

I t can be shown t h a t k ( x ) makes sense and i s a non-decreasing f u n c t i o n o f A

-

( c f . [TeZ]

t h i s i s somewhat more t h a n an e x e r c i s e b u t we w i l l o m i t t h e Given ( 8 . 8 ) one can use ( 8 . 4 ) and w r i t e o u t ( 8 . 7 ) as

d e t a i l s here).

R+i 6 I m [+ IR+A!x,i)dA] R t i6

(8.9)

1 1 loX c (Y, A

f

(Y d y l

t

1

C-

( 1/a)

IRt

jOm

+ m(x)c(x,x)ld

I

P(x,h)dk(x)

c(Y*x)f(Y)dY 0

e and c a r e r e a l f o r x r e a l ) .

(8.10)

[e(x,h)

LRLi 6 c (x, A 1d l [e (Y A )+m( A )c (Y )If (Y )dy

id m

-m

(recall

[ -(1/1~)1

R+i 6

m

(]/IT)

+

Im

= Im

F u r t h e r one o b t a i n s immediately

m

l f ( x ) I 2 d x = ( l / n ) lm1F(h)12dk(h);

F(X)

=

jomP(Y,x)f(Y)dY

= c(A,f)

CHEtIREIII 8-1. Given t h e nondecreasing f u n c t i o n k(A) d e f i n e d by (8.8), has f o r f E L2, f ( x ) = We go n e x t t o (8.11)

(8.12)

and l e t q and

(-m,m)

~ ( 0 =) 0; ~ ' ( 0 )= -1;

(so w(p,e) m2(A),

(l/1~)/1 q(x,x)c(h,f)dk(x),

= 1).

e

one

and (8.10) holds.

be t h e s o l u t i o n s o f Qu = -xu s a t i s f y i n g

e ( 0 ) = 1; e ' ( 0 ) = 0

By t h e h a l f l i n e t h e o r y t h e r e w i l l be f u n c t i o n s m,(x)

and

a n a l y t i c f o r Imx > 0 say, such t h a t $l(xyx)

= e(x,A)

+ m,(A)c(x,x)

9 2 ( x y x ) = e(x,x)

E

+ m2(A)c(x,h)

L2 ( - m , O ) ; E

L 2 (0,m)

One has I m ml > 0 and Im m2 < 0 f o r Imx > 0 w h i l e W(IL1y$2) = ml(h) - m 2 ( X ) . and i n f a c t l$l(x,A)l 2 dx = Im(ml)/ImX w i t h Im I$,(x,X)[ 2 dx = - I m ( m 2 ) /

lL

Imx. (8.13)

D e f i n e now 9(x,A)

G

=

=

if

$2(x,A)+,(y,A)/(ml

0

G(x,y,x)f(y)dy

-

m2)

G = ILl(xy~)$2(yy~)/(ml

-

f o r s u i t a b l e f where

(Y 1x1;

m2)

(Y > x )

We n o t e i n p a r t i c u l a r t h a t i f q i s even t h e n c i s even and follows that

m,(x)

=

-m,(x).

D e f i n e now

e i s odd.

It

50

ROBERT CARROLL

(8.14)

Thus 5 and 5 a r e nondecreasing and

i s o f bounded v a r i a t i o n .

Given 5, TI, 6 d e f i n e d as i n (8.14) one has f o r f

tHEe)REIII 8.2-

e(x,A)e(A,f)dc(A)

+

( l / n ) l z e(x,A)p(x,f)dn(A)

IT)/: p(x,A)p(h,f)dc(x)

e(x,f)dT-(A)

+

REMARK 8.3,

The case o f even q, where m,(A)

f ( x ) = (1/n)

F u r t h e r 5 ' = -Im(l/2ml)

:1

e(x,h)e(A,f)dch)

and 5'

+

E

LL, f ( x ) =

(1/~)_/:V ( X , A )

and (8.16) h o l d s .

importance l a t e r and we have t h e n ml/(ml-m2) (8.17)

One argues

2 I f ( x ) l dx =

(8.16)

(l/n)lz

T-

w i l l be o f p a r t i c u l a r

= -m,(A),

= 1 / 2 so ~ ( 1=) 0. +

IT)

Im(ml/2).

Hence

c

p(x,A)v(A,f)dc(A)

EXNWCE 8-4- It i s worth w h i l e showing how t h e s e formulas r e l a t e t o t h e standard F o u r i e r t h e o r y on [0,-) say when q = 0. Thus one has p = Sina C o s ( x J ~ ) - (x-')CosaSin(xJA) function $ =

e +

and

e

= CosaCos(xJA)

+ (x-')SinaSin(xJx).

mp must be a m u l t i p l e o f e x p ( i x J A ) i f ImA > 0 and one

It f o l l o w s t h a t -1m m ( x ) = f i n d s m(A) = [Sina-iJACosa]/[Cosa+ iJASina]. 2 2 Jh/[Cos a + ASin a3 f o r A > 0 and - I m m ( A ) = 0 f o r A < 0. One has t h e n

The


dx f ( x ) = ( l / n ) jm"p 2( x Y )p ( Y f, 2 oCos ~1 + xSin ~1

(8.18)

Consider 2h4

(A

51

~1

=

n / 2 so p = CosxJx and e = x-'SinxJA

x

0) with k(A) = 0 f o r

The s t a n d a r d F o u r i e r t h e o r y f o l l o w s e a s i l y .

RENARK 8.5. Take q even again and r e f e r t o Remark 8.3. some n o t a t i o n used i n [C47,48,80] (8.20)

P

x x(0)

=

P

0; Dxx x ( 0 )

2

-

F o r comparison w i t h

we w i l l w r i t e here i n s t e a d o f (8.11)

P P -1; ~ ~ ( =0 1;) 0 X'p A ( 0 )

=

Thus p p and xhp s a t i s f y P u = ( 0

e

=

=

0

2 2 p ) u = - A u ( a r e p l a c e s A ) and q

T,

x xP w i t h

F o r convenience here we w i l l assume p ( x ) i s even, r e a l , p o s i t i v e , 1 and continuous w i t h p(x)exp(2Hx) E L (0,m) f o r some H > 0. Operators P = 2 0 - p w i t h such p w i l l be c a l l e d F o u r i e r t y p e o p e r a t o r s ( c f . [C47,48,80; 'L

p!?

Hol; S t b l ] ) .

Much o f t h e development i n t h e f o l l o w i n g remarks f o r F o u r i e r

t y p e o p e r a t o r s i s v a l i d f o r weaker growth hypotheses on p (e.g. Im Ip(x)l 2 0 ( l + x )dx < m w i l l do). The e x p o n e n t i a l growth c o n d i t i o n above was used i n [Hol] t o g i v e a n a l y t i c i t y i n a s t r i p \ I m h ( < 6 , which we do n o t need ( c f . a l s o [Nbl]);

however i t i s convenient here t o use t h i s h y p o t h e s i s i n o r d e r

t o connect t h e m a t e r i a l t o [Hol, S t b l ] . -1m(1/2m1)

and t' = Im(ml/2)

As i n Remark 8.3 one has now 5' =

w h i l e (8.17) w i l l be w r i t t e n now

I n t h e p r e s e n t s i t u a t i o n t h e spectrum o f P w i l l be a b s o l u t e l y continuous, P P p x ( x ) i s even i n x, x X ( x ) i s odd i n x, and b o t h end p o i n t s on (-m,m) a r e P P l i m i t p o i n t ( c f . [Hol; S t b l ; TeZ]). The f u n c t i o n s p A and x h can be cons t r u c t e d f r o m h a l f l i n e c o n s i d e r a t i o n s (one r e f e r s here t o CTe2, Chapter51

ROBERT CARROLL

52

a s well as [Hol; S t b l ] f o r some of t h e c a l c u l a t i o n s which follow). P us s e t x ( x ) = x A ( x ) w i t h lp(x) = l pPh ( x ) and w r i t e

x(x)

(8.22) As

x

-f

m

+ (l/x)

= -[Sinxx/x]

c

Sinh(x-y)p(y)x (y)dy

r e a l ) , x ( x ) = u(A)Coshx + v(A)SinAx + o ( 1 ) where

(A

Thus l e t

p(A) =

- 1/A)

( u and v a r e (A r e a l ) , q ( x ) = u ( A )

IO SinAyp(y)x(y)dy m and v(X) = - ( l / A ) + ( l / A ) l m Coshyp(y)x(y)dy 0

a c t u a l l y functions of A' = z ) . Similarly as x m Coshx + v,(A)SinAx + o ( 1 ) where p 1 ( A ) = 1 - ( 1 / ~ ) / " Sinhyp(y)lp(y)dy a d -f

0

Coshyp(y)lp(y)dy. Since W(x,lp) = xlp' - v ' x = 1 , FI(A)V~(X) On t h e o t h e r hand f o r Imh > 0 one obtains as x -+ m, x(x) - u l ( h ) v ( h ) = 1/A. = exp(-ixx)[M(A) + o ( l ) ] and ~ ( x =) exp(-ixx)[Ml(x) + o ( l ) ] ( n o t e t h a t

v l ( A ) = (l/A)$

exp(iAx)

0 in this s i t u a t i o n ) where

(8.23)

M(A) = (1/2iA) - (1/2iX) M1(A)

must have ml

=

w i t h Im[m;']

=

eihYp(y)x(y)dy;

- (1/2iA) ~omeiAypb)lp(y)dy

+ %iv and M1 ( A ) + %pl + 4 i v l . If we r e q u i r e lp + n ~ ( > 0 i n t h e s p e c t r a l theory ( i . e . m % m2 = -m, above) we 2 2 2 -m = M1/M. I t follows t h a t Im[m,] = l / A ( v +v ) = 1/4A1Ml 2 -1/4x1M11 . Consequently ( w r i t i n g e.g. d g ( z ) = g ' ( z ) d z =

0, M ( A ) As ImA 2 E L (0,~) f o r ImA -+

= (1/2)

:j

-f

%p

2x5 ' ( z ) dh ) (8.24)

2

dg = dh/41M11 ; d r = dA/41MI

2

We note a l s o t h a t i n f a c t from (8.22), f o r 0 5 ImA < H ( A # O), x ( x ) = exp(-iix)M(A) + exp(iAx)M(-A) + E w i t h M g i v e n i n (8.23) and s i m i l a r l y (8.25)

q ( x ) = CosAx + ( l / x )

r

Sinh(x-y)p(y)lp(y)dy =

0

e-iAxM1(h) + eiAxMl(A) + I where I % O(exp(-xlPH-ImXI)). Formulas such as ( 8 . 2 2 ) and (8.25) can be solved i t e r a t i v e l y here f o r Imh > -2H say. From [Hol; S t b l ] one knows t h a t M and M1 a r e a n a l y t i c f o r ImA > -H (except f o r a simple pole a t A = 0 ) and n e i t h e r functions vanishes f o r ImA > - 6 . Further, uniformly in Imh > -H+E,

IM(A) - (1/2iA)l REMARK 8.6,

(8.26)

= O(/h(-2)

and I M 1 ( A ) - (1/2)1 = O(IAl-') as

Now define two functions as follows ( c f . [Holl)

u1 = ZiA[M(A)lp

- M1(A)x]; u 2

=

ZiA[M(A)lp + M1(1)x]

1x1

+ m.

FULL LINE SCATTERING E v i d e n t l y u1 and u2 a r e d e f i n e d f o r a r e a l s o a n a l y t i c f o r Imh > -H. r e a l , as x

-f

since, f o r

x

-

MIM-

=

m

( $ ) u1

r e a l , by (8.23)

=

(8.27)

u2(x) p(x)

ul(-x) 'L

m

-

w i t h ul(-x)

= u2(x)

and t h e y

Using (8.22) and (8.25) we see t h a t f o r X

-

2iAexp(iXx)[M(A)M1(-x)

k [ ( u + i v ) ( q - i v 1)

from u,(x)

where

?,

< x
-6.

appesrs t o be a standard one i n t h e f u l l l i n e s c a t t e r i n g problem as i n [Cel; Ddl; F a l ; K f l ] and we w i l l d i s c u s s r e f l e c t i o n c o e f f i c i e n t s e t c . below. e v e r i n view o f t h e way u, and u2 a r e formed i n (8.26),

How-

which d i f f e r s f r o m

t h e s t a n d a r d c o n s t r u c t i o n , some e x p l i c i t connections w i t h standard n o t a t i o n e t c . must be developed. P

REI1IARK 8-7- We r e c a l l t h e d e f i n i t i o n o f @.,(x) i n [C40] and e a r l i e r i n t h i s P P P P c h a p t e r t o see immediately t h a t u1 = Q A ( x ) . Now vA = cPaA + c;@-~ f o r A = Ikpx(x) ((

r e a l and .t,(x) P

= @hp/cp) w i t h

T h i s l e d us ( i n c o r r e c t l y ) t o w r i t e { ( x ) [C40, p. 3261.

=

d P ( x ) v PA ( x ) -

P 2 i h x A ( x ) i n e.g.

-A)

However (as w i l l be shown below) M 1 ( X ) = cp

and (8.26)

says t h a t P g ( x ) = 2iA[(M/Ml)vA(x)

(8.29)

-

x,(x)] P

A w i t h 2Re[PiAM/M1]

= 2iA[MM;

-

2 = l/IMII 2 = l / l c p

MIM-]/IMII

(cf.

(t) -

in

P + Im[2ihM/Ml]vA(x) P But I m g ( x ) = -2AxX(x) = -2Ax;(x) PA I n general A ( x ) , which w i l l correspond t o a r e [2XA(X)M(h)/M1(-X)lvA(x).

Remark 8.6).

f l e c t i o n c o e f f i c i e n t , i s n o t z e r o ( c f . Remark 8.11), [Hol]

s h o u l d s t i p u l a t e Ims > 0 i n (2.17)

and Theorem 2 . 4 i n

( c f . 22.6 f o r f u r t h e r c l a r i f i c a -

t i o n about t h i s p o i n t ) ; f o r ImA = 0 one has (8.27) above. passing t h a t -2AAM/M; = i A / 2 p l c p l 2 .

REmARK 8-8-

We n o t e a l s o i n

L e t us examine b r i e f l y t h e f u l l l i n e s c a t t e r i n g problem i n

o r d e r t o g a i n some p e r s p e c t i v e here.

The c o n n e c t i o n o f c l a s s i c a l s p e c t r a l

q u a n t i t i e s w i t h t h e parameters o f s c a t t e r i n g t h e o r y on t a i n i n t e r e s t i n i t s e l f and i t c o u l d

(-my=)

i s o f a cer-

be i n d i c a t e d f u r t h e r i n a system

54

ROBERT CARROLL

context l a t e r .

One d e f i n e s ( c f . [Cel; F a l l )

1

m

-

(8.30)

f+(A,x)

= e ixx

(8.31)

f-(X,x)

= e -ihx

Thus f+ % e x p ( i x x ) as x as above t h e n f + ( x , - t )

+

+ m

[SinA(x-t)/~]p(t)f+(A,t)dt

x

I,

X

[Sinx(x-t)/A]p(t)f_(A,t)dt

and f -

= f (A,t)).

'L

e x p ( - i h x ) as x

-f

-m

( a l s o i f p i s even

By i t e r a t i o n one can c o n s t r u c t f+ and

f- v i a t h e i n t e g r a l equations (8.30)-(8.31) and g e n e r a l l y f o r reasonable

Imx

p o t e n t i a l s i n p h y s i c s f+ w i l l be a n a l y t i c f o r also).

(8.32)

f-(x,x)

> 0 (here f o r

Imx

>

-H

= cij(x))

One w r i t e s (cij

= Cllf+(A,x)

f+h,x)

+ cl2f+(-x,x);

= c22f-(Lx)

+

c21

+ 2 i h w i t h c12 = cZ1 = W(f+(x , x ) , f - ( x , x ) ) / 2 i h ; Also cl1(x) = c1 = w(f-(x ,x) ,f+(-x ,x) ) / 2 i h ; c Z 2 = W(f-(-x ,x , f + ( x , x ) ) / Z i x . 2 2 2 E x p l i c i t formulas can be ob- c Z 2 ( - x ) and Ic121 = 1 + lcllI = 1 + IcZ2( t a i n e d by w r i t i n g f o r example as x + m and W(f,(x,x),f+(-x,x))

(8.33)

f-(A,x)

=

=

e- i x x

+

(,ixx P i x ) jme-i

j w e i A t p(t)f-(x,t)dt

(,-i~x/~~~)

-

tp( t ) f -( A , t ) d t

-03

+

o(1); f-(A,x)

%

+

clle ixx

C12e-ixx

-m

( t h e l a t t e r from (8.32)). f-(x,t)dt

and c12 = 1

-

It f o l l o w s t h a t cll

(l/Zix)/:

problem i n v o l v e s f i n d i n g s o l u t i o n s

(8.34)

x1 x2

Q ,

'L

{ {

=

(1/2ix)lI exp(-ixt)p(t)

exp(ixt)p(t)f-(h,t)dt.

x1

and

x2

The s c a t t e r i n g

such t h a t

exp(ixx) + s12exp(-ixx)

X

s1 exp( ixx)

x+m

s22exP(-iAx)

X

e x p ( - i x x ) + sZlexp(ixx)

X'm

-f

+

-m

-m

Here s12 and sZ1 a r e r e f l e c t i o n c o e f f i c i e n t s (resp. s l l and sZ2 a r e t r a n s -

I f we w r i t e now x1 = f - ( - x , x ) + s 1 2 f - ( h , x ) = f (x,x) = sll[cZ2f-(x,x) + c,,f-(-x,x)] t h e n one f i n d s t h a t s l l = l / c Z 1 sll+

mission c o e f f i c i e n t s ) .

and s12 = c ~ ~ / c ~ ,S.i m i l a r l y sZ2 = l / c 1 2 and sZ1 = cll/c12 and ( s l , ( ' + (s2,I2 = 1 f o r example ( a l s o s,*

REmARK 8.9.

=

so s l l = sZ2

sZ1 h e r e ) .

By way o f connecting t h e f u l l and h a l f l i n e s i t u a t i o n s ( f o r p


55 ihx

-

even) let us write now as x - m (cf. (8.30) and (8.33)), f+(A,x) = e jm -m [ Si n x (x-t )/Alp ( t ) f+ ( A, t ) dt = exp ( i AX) - [exp (i xx)/2i A] exp ( - i At ) p (t) f+(h,t)dt + [exp(-ihx)/2ih]lI exp(iht)p(t)f+(A,t)dt. From (8.32) f + cZ2 and cZ1 = exp(-ihx) + cZlexp(ixx) so cZ2 = (l/Zih)/: exp(iht)p(t)f+(h,t)dt 1 - (l/Zih){I exp(-ixt)p(t)f,(A,t)dt. Note that c12 = c21 implies that I - mm exp ( i At)p (t) f- (A, t )dt = iz exp (-i At )p (t)f+( A, t )dt = iz exp ( i At) p (-t) f+(h,-t)dt, which is clear for p even s o that f+(A,-t) = f-(x,t). Now if (8.30) represents u1 = P and u2 = f- (ul f+) then from (8.26) and (8.23) -+

/I

Q ,

I

a

(8.35)

ul(0) = 2ihM(h)

1 -

=

u 2 (0) = 1 +

[Sinht/h]p(t)u,(t)dt

=

0

jmeixtp(t)x(t)dt 0

Note also -LL Sinxtp(t)u2(t)dt u,(-t) = u,(t) (p being even). (8.36)

Dxul(0)

=

=

,)i

=

-I: SinAtp(t)ul (t)dt automatically when Similarly

-2ihM1~'(0)= 2ixM1

-

Im

= ih

-

:1

Cosxtp(t)ul(t)dt

eihtp(t)p(t)dt

0

We observe a l s o that (8.26) is compatible in structure with (8.30)-(8.31). Thus recall (8.22) and (8.25) for x and p and write e.g. (using (8.35)(8.36)) (8.37)

lox

f,

=

eixx

,

Sinhlx-t) pf+dt

jox- g 1 SinA(x-t) p(t)f+dt

=

,i~x

+

A

2ihM1) - Cosxx(2ihM - l)]

(ix- [-r

=

X

2ihMCosxx + 2ihM1 - +

j

[Sinh(x-t)/x]p(t)f,(X,t)dt

0

Now substitute f+ = u1 =_2ix[b - Mlx] into (8.37) and everything fits together. Further since Lm ppexp(iAt)dt = 2Jm WCoshtdt and pxexp(iht)dt 0 = 2iI" pxSinhtdt we have, cZ2 = (1/2ih)Jm -m exp(iht)pu,dt = [b - Mlxlp(t) 0 exp(iht)dt = 2M/" ppcoshtdt - 2iMlim BSinhtdt. But 1 - 2ixM = Imexp(ixt) 0 0 0 pxdt and i x - 2iAM1 = j m ppexp(iht)dt so ImSinhtpxdt = Im[l - 2ihMI with 0 0 Im Cosxtbdt = Re[ih - 2ixM1] and it follows that cZ2 = -2ih[2MM1A] = -A/p 0 so that the full line picture is consistent with the view o f A as a reflection coefficient (cf. (8.27)). In order now to explicitly compare with (8.34) write for p = 1/4ixMM1 exp( ihx) X'W (8.38) U, = p u l = %[(p/M1) - (x/M)I { exp(ihx) - Aexp(-ixx) x --

r:

_/I

Q

-f

56

ROBERT CARROLL

( $ ) i n Remark 8.6 and (8.27) p l u s u,(-x) = u 2 ( x ) ) . Thus i s e x h i b i t e d as a t r a n s m i s s i o n (resp. r e f l e c t i o n ) c o e f f i c i e n t (cf.

A = -s12 i n (8.34) and

p =

s22y A = -s

0 f o r 1x1 > R say, so t h a t i n f a c t

f o r x > R,

7

(8.39)

(v/M1) = e-ihx

= exp(-iAx)M1

T h i s r e q u i r e s t h a t M;/M1 M-M1]/2MM1

f

PI,

(resp. - A ) sll,

(p =

A l s o n o t e t h a t i f we assume p =

21 ). becomes = i n (8.38),

exp(ihx)M;.

+ eixx(M-/M1 1 )

p

t h e n u s i n g (8.25)

Consequently we must have f o r x > R = pul

+ pu2

= (p-A)eihx

+

e-ixx

I f we w r i t e t h i s o u t w i t h A = -[MM;

= p-A.

and p = 1/4ihMM1 one a r r i v e s a t z-z = 1 / 2 i h where z = MM;

+ and

t h i s i s known t o be t r u e . I n c o n n e c t i o n w i t h t h e s p e c t r a l measures o b t a i n e d i n (8.24)

REmARK 8.10-

t h e expansion theorem (8.21) f o r even f ( w i t h p even) so t h a t P = 0 ( s i n c e x,(x) i s odd) and

t a k e e.g. x!(f) (8.40)

f(x) =

lo -

P P q,(f)q,(x)dC;

P qx(f) =

2 2 where d g ( h ) = dv(h) = dX/2.rr[M1(

.

m

f(y)v!(y)dy 0

Then f o r M1 = c p ( - h ) we have t h e c o r -

r e c t h a l f l i n e measure dv = dA/2rlcp12 ( c f (6.37)).

We can d e r i v e t h e r e -

s u l t M1(A) = cp(-A) d i r e c t l y as f o l l o w s . Thus r e c a l l f o r W(f,g) = f g ' P - P P P P f ' g ( q = cPbx + c ~ @ - ~ W(*A,@-x) ), = - 2 i x w i t h W(IP,@~) = 2ihcp(-A) and P W ( V J , @ - ~ ) = -2iXcp ( c f . (6.27)). Now use (8.25) w i t h t h e asymptotic r e l a P P t i o n s ah plr e x p ( i h x ) and DXaA i h e x p ( i h x ) t o o b t a i n as x m -f

(8.41)

2ihcp(-x) =

qD

*p

X h

-

qlah P = ih

-

lom weiAYdy

Hence, d i r e c t l y , f r o m (8.23) we have c p ( - h ) = M 1 ( X ) . know t h e s p e c t r a l measure f o r

x

I n a s i m i l a r s p i r i t we

transforms v i a c l a s s i c a l s c a t t e r i n g theory

on t h e h a l f l i n e i n terms o f J o s t f u n c t i o n s F ( A ) e t c .

Thus f o r o u r x ( c f .

[C40, p. 222 o r C e l l (8.42)

x

= [F(A)@-

-

F(-h)@+]/2ix

) = - 2 i x , F ( A ) = W(x,@+)). By formulas i n [ C e l l , w h i c h a r e o b t a i n ed i n t h e same way as (8.41) one has F ( h ) = 1 - Im p ( t ) e x p ( i A t ) x ( t ) d t so (W(@+,@

t h a t F ( x ) = 2ihM(h) by (8.23).

0

The s p e c t r a l measure f o r t h e e i q e n f u n c t i o n


57

t h e o r y i n [ C e l l f o r example i s g i v e n as f o l l o w s

1

m

w

(8.43)

f(X) =

f(y)x[(y)dy;

f(x) =

0

lom

7(X)x~(x)[2h2dh/nlF121

Now go t o (8.21) and proceed as i n (8.40) t o o b t a i n v i a (8.24) ( u s i n g odd functions f )

2

d - 6 (except f o r a s i m p l e

Then M, M1,

x

Set p = 1/4ixMM1 = +[(M-/M ) - (M-/M)] and A 1 1 + (M-/M)] ( t h u s p+A = -M-/M and p - A =

= -+[(M;/M1)

= 0 i n M and M1)

and n e i t h e r M n o r M1 vanishes f o r ImX > - 6 .

The

f u n c t i o n s u1 and u2 a r e a n a l y t i c f o r ImA > -H and f o r X r e a l as x m , u1 = P % e x p ( i X x ) w i t h pu2 1~ e x p ( - i A x ) - A(X)exp(iAx). The f u n c t i o n q~~ = a XP / c -P -f

has t h e f o r m (8.29) and c p ( - x ) = Ml(h) w i t h F ( X ) = 2iXM(x) where F = P The c o n n e c t i o n w i t h t h e f u l l l i n e s c a t t e r i n g problem as i n ReW(x,$). mark 8.8 g i v e s f+ = ul,

f - = u2, and e x h i b i t s -A (resp. p ) as a r e f l e c t i o n

(resp. t r a n s m i s s i o n ) c o e f f i c i e n t f o r u+ = pul and u- = pu2 ; thus p = sll

sZ2 and A

=

-s12 = -sZ1.

We n o t e a l s o t h a t A/A-

= -p/p-

and AA- = 1.

The f o l l o w i n g i n v e r s i o n f o r m u l a ( c f . [Hol; S t b l ] ) w i l l be v e r y h e l p f u l i n d i s c u s s i n g t h e Marzenko (M) e q u a t i o n i n Chapter 2 . v a t i o n i n [C40,34]

We gave a f o r m a l d e r i -

w h i c h was b a d l y phrased and t h i s was r e v i s e d and c o r -

r e c t e d i n [C47;48].

iTHE0REII 8-13. F o r s u i t a b l e f one has m

(8.45)

f(x)ul(x,x)dx;

F(A) = -m

Prrooa:

Consider f o r m a l l y

f ( x ) = (1/2n)

p(A)F(X)u2(A,x)dA

=

58

ROBERT CARROLL

]

(i/2n)

[ ( M / M 1 ) ~ ~ - ( M 1 / M ) x x l ~+ d ~( 1 / 2 n )

-m

1

[X(X)V(Y)

-

- mm q(x)x(y)hdA = 0 = Now I

[I q ( y ) x ( x ) A d A s i n c e

r e c a l l t h a t ( c f . (8.24),

Remark 8.10,

I

q

and x a r e even i n A and we

and [C40; C e l l )

I

m

(8.47)

~ ( x - Y )= ( 1 / 4 ~ )

m

q(x)q(y)dA/M,M;

= (1/4a)

Then observe t h a t

il

(M/Ml)qqAdh

= 1 / 2 i x so t h a t (i/Zn)/;

dh = (i/4n)L:

-

qqh[MM;

x(x)x(Y)dAlMM-

-m

m

M-M,

~(x)x(~)l(ihdh)

-m

=

-/I(M-/M;)qqxdA

qqx(M/Ml)dA

and r e c a l l t h a t MM;

-

= ( i / 4 n ) j z qqph[(M/M1)

-

(M-/M;)]

MIM-]dA/MM;

= ( 1 / 8 n ) l I qqdh/M M- = ( 1 / 2 ) 6 ( x - y ) . lml (M1/M)xxAdx = - ( i / 4 n ) L m xxh[(M1/M) -

S i m i l a r l y one o b t a i n s -(i/Zn)/:

(M;/M-)]di = -(i/4n)j_: xxAIM-M1 - MM;]dA/MM= (1/8n)/: xxdA/MM- = ( 1 / 2 ) ~ ( x - Y ) . The i n v e r s i o n (8.45) f o l l o w s immediately v i a an i d e n t i f i c a t i o n

.

~ ( x - Y )= ( 1 / 2 ~ ) j : pU2(X)U1(Y)dA. 9,

INCR0DlltXIBN CO 5ZNGlltAR 0PERAE0G AND 5PECIAL fllNmI0W-

The s t u d y o f

s i n g u l a r d i f f e r e n t i a l o p e r a t o r s i s connected w i t h v a r i o u s s p e c i a l f u n c t i o n s and t h i s p r o v i d e d t h e i n i t i a l impetus f o r o u r work on t r a n s m u t a t i o n t h e o r y i n 1978-79.

I t was p o s s i b l e t o develop a general t h e o r y i n v o l v i n g connec-

t i o n s between Operators, e i g e n f u n c t i o n s , i n t e g r a l transforms, e t c . and p r o v i d e a u n i f i e d f o r m u l a t i o n f o r many r e s u l t s (new and o l d ) .

Much o f t h i s

w i l l be sketched i n t h e p r e s e n t book and some d e t a i l s w i l l be repeated

-

( c f . [C27-49,54,63-65,741

f o r s i n g u l a r problems cf.

Fhl; D i l - 9 ; Dul; F i l , 2 ; F j l ; 5; Ocl; Oel; Pb2-4;

4

p

Q

= $lim[A'/A

Q

Q

9;

] as x

Gql-3; Gt1,2;

S r l - 3 ; Sx1,2;

Bbal,2;

He1,2;

Sy1,2;

B b f l ; Cpl-4;

La1,2;

L12; Lpl-3,

S t e l ; Tc1,2;

Tql; Wbl,

The model o p e r a t o r s w i l l have t h e form

Ybl-51).

Qou = (A u ' ) ' / A

where

F11,2;

Pd2; Rk1,2;

2; Wcl; Wel; Wil-14;

(9.1)

a l s o [C1-5,9-11,19-

B j l ; Bkl; B11; B r l ; Bul; By1,7;

22,25,26,56,60-62;Spl;

Z A A. i o u = Qou + pqu; Qu = Q u +

m.

The

pQ

-

A

q(x)u

f a c t o r i s p u t i n as i n d i c a t e d i n

order t h a t various spectral regions fit together. and A

Q

are possible.

(m > +) A

Q

t

-

where C as x

3

Q

> 0 and

w i t h A'/A

singularities i n 1; Sz1,2;

For example i n [Tj1,2]

Vd1,2]).

Q

a r e Cm and even.

Q

Various hypotheses on 2m+l Cq(x) Q G e n e r a l l y we a l s o t h i n k of

one takes A ( x ) = x

3. 2p 2 0 as i n [Cgl-4]

6 are permitted The o p e r a t o r s

( c f . a l s o [Bxl-5;

â" = Go

where i n a d d i t i o n s u i t a b l e Cf';

Ge2,3;

Ff1,5,6;

Kp

a r e modeled on t h e r a d i a l Laplace-

59

SINGULAR OPERATORS

B e l t r a m i o p e r a t o r i n a noncompact r a n k one Riemannian symmetric space ( c f . [Fcl-3;

Ff2-4; C f l ; Hbl; Hkl; Hcl-7;

T11; Wgl; Mkl,2])

Kp2-13; L b l ; Gbl; Snl; T a l ; Tj1,2;

and n a t u r a l l y t h i s embodies a l s o t h e t y p i c a l s i n g u l a r op-

e r a t o r s a r i s i n g i n many problems i n d p p l i e d mathematics i n v o l v i n g s p h e r i c a l

*'+'

o r c y l i n d r i c a l symmetry. T y p i c a l examples a r e A = x h2m+l x -x 2 o t l (ex+e-x Q2B+1 x (p, = mtk), and A, = ( e -e ) ) (p,

(pQ =

=

o),

AQ

a+B+l).

=

For A

s i m p l i c i t y i n t h i s s e c t i o n we w i l l u s u a l l y exclude s t r o n g s i n g u l a r i t i e s q = B/X

2 near x

= 0 i n o r d e r t o deal w i t h t r a n s f o r m s based on " s p h e r i c a l func-

t i o n s " p! s a t i s f y i n g

&

(3.2)

= = 1; DXpA(0) Q

= - i 2 p ; p!(O)

0

( c f . however Example 9.5 and S e c t i o n 10 f o r s t r o n g s i n g u l a r i t i e s ) . We remark here t h a t r r h i i e hypothesss on general A w i l l be ex-

FEmAZUC 9.1.

4

p l i c i t l y p r o v i d e d l a t e r we w i l l d e l i b e r a t e l y n o t be t o o s p e c i f i c about The reason f o r t h i s i s t w o f o l d .

t.

F i r s t we observe t h a t t h e r e a r e numerous

t r e a t m e n t s o f s i n g u l a r problems i n t h e l i t e r a t u r e i n v o l v i n g v a r i o u s t y p e s o f hypotheses on Co,

6 and c o r r e s p o n d i n g l y d i f f e r e n t

L2, weak, e t c . types o f s o l u t i o n s - see e.g.

Sz1,2;

Tj1,Z;

Vd1,2])

[Bx1,2;

Ge2,3;

Cgl-4; S o l ;

These a r e a l l o f i n t e r e s t i n t h e i r own r i g h t and

sometimes n o t comparable another.

t y p e s o f r e s u l t s (e.g.

-

i.e.

one t y p e o f r e s u l t i s n o t " b e t t e r " t h a n

Thus l i s t i n g a l l t h e types o f r e s u l t s i s e x c e s s i v e and u n r e a l -

i s t i c w h i l e attempting t o e x t r a c t a "best" r e s u l t o f a given type requires i s o l a t i n g the type o f r e s u l t .

T h i s l e a d s t o t h e second p o i n t which we want

The t r a n s m u t a t i o n "machine" by means o f which we can r e l a t e

t o emphasize.

d i f f e r e n t i a l o p e r a t o r s and c o r r e s p o n d i n g s p e c i a l f u n c t i o n ,

"runs" by means

o f v a r i o u s p r o p e r t i e s o f e i g e n f u n c t i o n s and t r a n s m u t a t i o n k e r n e l s .

Any which produces such p r o p e r t i e s a r e thus considered s a t i s -

hypotheses on

f a c t o r y o r d e s i r a b l e here.

There a r e v a r i o u s hypotheses which work ( f o r

which one r e f e r s t o [Bx1,2;

Ge2,3;

Cgl-4; S o l ; Sz1,Z;

Tj1,2;

Vd1,2]

b u t we

p r e f e r n o t t o c o n t i n u o u s l y c i t e such hypotheses i n o r d e r t o be a b l e t o emphasize t h e p r o p e r t i e s o f e i g e n f u n c t i o n s and t r a n s m u t a t i o n k e r n e l s which f u e l t h e t r a n s m u t a t i o n machine. dition that

jm(l+x)l:(x)ldx 0

-(1/2) b u t one can a l s o t r e a t m E C with Rem > - ( 1 / 2 ) ) (9.3)

;iou

m

= Q O ~= ( x 2 m + 7 u ~ ) ~ / x 2 m=+ u1" + [ ( ~ m + l ) / x l u '

m

4o 2 Then Qmu - -A u has spherical function s o l u t i o n s

(9.4)

P?(X) = 2 ~ ( 1 n + l ) ( x x ) - ~ J ~ ( ~ x )

We w r i t e g e n e r i c a l l y O,(x) Q = A ( x ) p9h ( x ) and expect generally t o find " J o s t " Q 2 Q solutions akX(x)f o r ije = - A e s a t i s f y i n g Q

(9*51

%

&+(x)e-+ i h x

(x

Q

-+

-1 A

( c f . [Cgl-4; C f l ; Tj1,2; Kpl; Ffl]). (9.6)

In the present s i t u a t i o n f o r Qm

q QA ( x ) = i m+l5x -rn+ (vhx/2f5 HA(Ax)

where H A denotes the Hankel function of f i r s t k i n d . (*) P ~ ( x )= cQ(X)@x(x) Q + c Q ( - A ) @ - ~Q( x ) and (9.7)

Then a s before one has

w(@;(x)y@!h(x))x 2m+l = - 2 i h

From (*) and (9.7) follows (9.8)

cQ(-A) = 2~(m+l)A-m-4im~'/J2n

(note -2ixc

Q as x

pA]

(9.9)

+

= x 2m+l c ( - x ) w ( @ Q ~ ,Q@ - ~=) x2m+1~(@f,p;) = Q0).(-A) Q From this one obtains (cm = 1/2"r(m+l))

Ro(x) =

tQ(x)

lim x 2m+ 1 [-ox@:

= cih2m+1 = 1 / 2 n l c Q ( h ) 2 (

Ro a r i s e s i n the inversion theory as i n (6.35)-(6.37),

Remark 8.10, Remark 10.8, Theorem 10.12 - c f . a l s o 511. I t can o f course a l s o be produced v i a the c l a s s i c a l theory of Hankel transforms. (9.10)

q4 h ( x ) = Oh(x)/c Q (-A) =

4 aO-(A/X)~+'

HA(xx)

2"+(m+l) As w i l l be seen l a t e r functions o f t h e type q xQ ( x ) i n (9.10) play an importa n t r o l e i n the general theory. We note here t h a t J ( z ) / z p i s e n t i r e in z P and WAQ can be regarded as a n a l y t i c in the A plane c u t e.g. along t h e nega1 f a c t o r ( r e c a l l here Hm(z) = t i v e imaginary axis t o accomodate a = [~-,,(z) - e-imTJm(z)l/iSinmn and consequently one can write q lQh ( x ) (1

SINGULAR OPERATORS

61

~ , [ A X - ~ ~ J - , ( A X ) / ( A X ) - ~ - e - imnA2m+lJ , ( x x ) / ( ~ x ) ~ ]

Q

remark however t h a t n e a r A = O,aA(x)

-

-2mA+m

c f . [C40,

(+

kmx

%

p. 1271).

We

f o r m > +) which

m

t h e r e f o r e d i f f e r s i n b e h a v i o r from some o t h e r t y p i c a l examples ( c f . Example 2 L e t us r e c o r d here a l s o f o r f u t u r e r e f e r e n c e t h a t i A ( z ) = H,(z) for 1 1 z r e a l whi 1e Hm(zexp ( v i n ) ) = [ S i n (1 -v)mn/Sinmn]H,( z ) - exp (-ima) [ Sinvmn/ 2 S inmn] Hm ( z )

9.4).

.

EXAmPfE 9-3, F o l l o w i n g [Kpl; C f l ; F f l ] we c o n s i d e r A 4 = AaYB= (ex-e-x ) 2a+1(ex+e-x)2B+1 w i t h p = p Q = a + ~ + 1 . It i s e q u i v a l e n t here t o work w i t h A p y q = (ex-e-x)p(e2x-e-2x)q

with

p =

(p+2q)/2 and we w i l l use

whichever n o t a t i o n makes t h e formulas appear t h e most simple. +1, p = ~ ( w B ) , B = ( q - l ) / Z , [ c t n h x t tanhx]/2. L e t us w r i t e t h e n

a l s o coth2x = c t n h Z x =

We a r e m a i n l y i n t e r e s t e d i n r e a l p,q 2 = Qou + p u w i t h

&I

+ [A' /A

= u"

c4B

]u';

A'/A = ( 2 a + l ) c t n h x + ( 2 ~ + l ) t a n h x

a7B

&

One d e f i n e s t h e s p h e r i c a l f u n c t i o n s as b e f o r e ( i . e . and ~ ' ( 0 =) 0 ) and t h e r e s u l t i n g t h e f i r s t kind.

(a+l =

q y y BE q p y q a r e

Thus (sh = Sh = sinh, ch = Ch = cosh) 2

q A ( x ) = ~ " ' ~ ( x=) F[%(p+iA),%(p-iA),a+l,-sh (p+q+1)/2 and a # -1,

r(a+l)-'q:(x)

2

= -1 q , q ( 0 ) = 1,

c a l l e d Jacobi f u n c t i o n s o f

4

(9.12)

0 as i n [ F f l ] .

4

Qou = Q:,Bu

(9.11)

and a = (p+q-1)/2;

Note q = 28

-2,

...

It f o l l o w s e a s i l y t h a t

i s required).

i s e n t i r e i n a, B, and A.

XI

For J o s t s o l u t i o n s one has t h e

Jacobi f u n c t i o n s o f t h e second k i n d (9.13)

Q

a,(x)

= (ex-e

(where A # -i,- 2 i ,

... )

-x

)

iA-p

-2

F[~(~-a+l-i~),+(~+a+l-ih),l-iA,-sh

Q( x )

and

@

w r i t e i n o u r s t a n d a r d manner

q!

%

e x p ( i h - p ) x as x

+ c-@'

= cQ@!

Q

where ( c f . L e m a 9.9)

-A

-f

m.

[Kpl] c Q i s used somewhat d i f f e r e n t l y

-

A l s o one can

f o r 1 # 0,Li , f 2 i ,

The l a t t e r form i s used i n [ F f l ] and t h e former i n [Kpl] one uses

Q

x]

. ..>

(note t h a t i n

= 2JncQ/r(a+l) f o r c

4;

n o t e a l s o here t h a t i f one w r i t e s a = % ( p + i A ) , b = % ( p i x ) , and c = %(p+q+ Q 2 1) t h e n i n (9.13) 'px(x) = F(a,b,c,-sh x ) ) . The f o l l o w i n g i m p o r t a n t propert i e s a l s o f o l l o w immediately f r o m t h e above formulas

62

ROBERT CARROLL

and c-+,-+(A) = 1 / 2 . Let us use S t i r l i n g ' s formula log r ( z + y ) = (z+y-$) logz - z + +1og2n + 0 ( 1 z 1 - ~ ) , uniformly i n largzl 5 n-6, t o estimate : Q ( ~ ) = 1/2nlc,(A)12

= 1/2nc (A)c (-A)

9

s

f o r real

and a simple c a l c u l a t i o n shows t h a t + m as 1 x 1 -+ m.

G,

w"Q

%

x

with

1x1 large.

Thus

T h u s f o r Re(Za+l) > 0,

klx12at1.

EMlnPCE 9-4, A special case of p a r t i c u l a r iimportance involves A. = Am,-+ A = ( e - e ) -- ZZm+' shZm+'x. Then Ak/Am = (2m+l)cothx and we a r e

-'"+'

m

=

dealing with the r a d i a l Laplace-Beltrami operator in spaces l i k e SL(Z,R)/ S O ( 2 ) ( c f . [C25,26,60-63; Spl; Vfl]). In t h i s s i t u a t i o n

v Qx

(9.17) where P!;-+

= Zmr(rn+l)sh-"x P;T-+(chx)

denotes t h e associated Legendre function of t h e f i r s t kind (cf

[Rel]) and, w i t h

p =

m++,

i s t h e associated Legendre function of the second k i n d .

where Q-+i (9.19)

c (1) = 22mr(m+l)r(ix)/Jnr(p+ix)

(9.20)

\IrQ(x) x = [ - i ~ 2 - ~ ~ s h r-( p~ -xi x ) / r ( m + l ) ] T + - i A ( c h x )

Also

Q

hl

Note where Q:(z) = e x p ( i ~ n ) Q ~ ( z ) / r ( u + ~is+ le)n t i r e i n 1-1 and v ( c f . [Rel]). here t h a t r ( p - i x ) becomes i n f i n i t e f o r p - i x = -n o r x = - i ( p + n ) . We note a1 so (9.21) A 2

Ic,(x)

= Ac,(h/n)sh(nX)r(p+ix)r(p-iX) 2

where c, = aZ-4?-2(m+l)

(recall r ( i x ) r ( - i x ) = n/hsh(ax)).

MAmPCE 9.5, The following example w i t h s i n g u l a r 2e t i v e ( c f . a l s o [ C f l l ) . Take AQ = sh 2 ch x and

from [Cg3] is i n s t r u c 2 2 2 2 = ( B /sh x ) - ( c /ch x ) .

4

SINGULAR OPERATORS

2 One s e t s v = J l ( i - Z e + [ ( 2 e - l ) +4c2I4) and

63

4(1-2a+[(Za-l) 2+4~']') 2 2 L here 2a % 2m+l s o 2a-1 = 2m and T = -m t (m +B ) '). Then (note spherical function b u t we use t h e same notation here) (9.22)

q!

=

chvxshvx

4

F['(p+~+v-iX),%(p+T+u+ih),atT+4,-Sh

-p-~+ih

a A ( x ) = shTxch (note here

p =

(9.23)

qQ =

A

X F[%(p+T+v-ih)

(note

T =

2

i s not a

x];

, % ( a - e + ~ - v +-iA), l 1-ix,ch

-2

X]

ate). Another, more revealing,form f o r q! i s

shTxch-p-Ttih X F[%(

p+T+v-i A )

,%( a-e+T-V+l -i A )

2

,~l+~q t h3 ,X ]

Q 2 The eigenvalues A a r e characterized by @.,(x) being L near x = 0 which rej quires t h a t %(n-e+=-v+l-iA) = -n, n E Z. Hence f o r Imx > 0 (9.24)

cQ(-A) -

x

r ( - i )r( a+T+%) r(%(a-e+T-v+l-iA)r(%(a+e+T+v-ix)

-

f o r i x $ Z. Note t h a t % ( a - e + ~ - v + l ) -4ix = -y/2 - i x / 2 = - n in (9.22) corresponds t o i x = 2n - y while in c (-1) t h e gamma function w i t h argument Q -4y - g i x becomes i n f i n i t e f o r -b- $ i x = -n which is the same s i t u a t i o n . Thus t h e eigenvalues A,, have t h e form i A n = 2n -

y

or An = (y-2n)i.

In [C40] we displayed a g r e a t deal of d e t a i l e d information from [Bxl,2; Ge 2,3; F f l ; Kpl; Sz1,2; T j 1 , 2 ; Cgl-4; Vd1,2] concerning t h e construction of Riemann functions, transmutation kernels, generalized t r a n s l a t i o n s , e t c . f o r s i n g u l a r operators of t h e type indicated (containing t h e generic singul a r i t y ( 2 m + l ) / x i n t h e u ' term). We w i l l n o t repeat a l l of t h e technical d e t a i l here and will organize t h e material i n a somewhat d i f f e r e n t manner. A c e r t a i n amount of t h e d e t a i l was needed in order t o give an extension o f

t h e Marzenko technique of 55 t o cover s i n g u l a r operators. Other d e t a i l s were developed in order t o e s t a b l i s h p r o p e r t i e s of eigenfunctions, transmut a t i o n kernels, e t c . i n order t o deal w i t h e.g. i n t e g r a l transforms and connection formulas between special functions. In t h e remainder of t h i s s e c t i o n we w i l l sketch some r e s u l t s from [ F f l ; Kpl] which e s t a b l i s h c e r t a i n p r o p e r t i e s o f eigenfunctions based on Example 9.3 and provide a model s i t u a t i o n f o r constructing general transmutation kernels l a t e r v i a s p e c t r a l i n t e g r a l s ( c f . Chapter 2). The technique f o r obtaining general Parseval formul a s f o r s i n g u l a r operators, of the type i n §5, will then be developed i n §§lo-12. (9.25)

Let us f i r s t note t h a t f o r Qo a s i n ( 9 . 1 )

64

ROBERT CARROLL

( s o (Qo)*Q:

= -A2Sf

for

ff = A@:)

while a useful transformation w i l l a r i s e

from t h e formulas

( n o t e a l s o t h a t Qou = (A u ' ) ' / A

Q

Q

2 i s i n formal s e l f a d j o i n t form on L (A d x )

Q

I n particub u t we p r e f e r t o work w i t h Qo and (ao)* f o r v a r i o u s reasons). 2 2 l a r f o r Q: o f (9.3) one has = D - (m - k ) / x 2 . Thus g e n e r a l l y a t r a n s -

5:

f o r m a t i o n o f t h i s t y p e i n t r o d u c e s s p e c i f i c " s t r o n g " s i n g u l a r i t i e s 8/x2 i n Moreover i f one begins w i t h a Q' f o r which s p h e r i 2 2 c a l f u n c t i o n s o l u t i o n s o f Qou = - A u e x i s t ( c f . ( 9 . 2 ) ) t h e n f o r 6Ov = - A v t h e p o t e n t i a l t e r m q"(x).

one has p a r t i c u l a r corresponding s o l u t i o n s v = L'u

Q

w i t h A%

1 as x

-f

Q "0

Thus c e r t a i n p a r t i c u l a r s t r o n g s i n g u l a r i t i e s and o p e r a t o r s Q

+

0.

w i t h "non-

s p h e r i c a l " b a s i c s o l u t i o n s w i l l always a r i s e and must be accomodated. The q" c o n t a i n s general terms B / X 2 i s r e a l l y n o t t o o much d i f -

general case when

f e r e n t b u t we p r e f e r t o d e f e r i t f o r t h e moment ( c f . [ C f l ;

REIIIARK 9.6,

Cgl-411 and

To c l a r i f y t h i s a l i t t l e here we n o t e t h a t t h e r e a r e v a r i o u s

ways o f h a n d l i n g t h e p r o t o t y p i c a l s i n g u l a r i t y (2m+l)/x i n t h e u ' term. one works w i t h A ( x ) = C

example i n [ T j l ] ( c f . a l s o [C40]) C

Q

10).

E Cm i s even and p o s i t i v e w h i l e

4 E. Cm i s even

Q

and r e a l

For

' 4 (, X ) X ~ ~ + where (Q = ^s" -

6).

Q ( x ) a r e t h e n compared t o those o f AQ: i n (9.4). The s p h e r i c a l f u n c t i o n s p A A

A.

I n [ S o l ] a general t h e o r y i s g i v e n f o r Qu = Q u,

-

t u w i t h q u i t e general

q^

( a d m i t t i n g s t r o n g s i n g u l a r i t i e s ) w h i l e i n [Cg3] one works w i t h e s s e n t i a l l y t h e same s i t u a t i o n b u t expressed d i f f e r e n t l y v i a hypotheses on A [C40]);

Q

(cf. also

i n b o t h of these t r e a t m e n t s t h e b a s i c e i g e n f u n c t i o n s may n o t be

s p h e r i c a l f u n c t i o n s however.

We w i l l comment on t h i s l a t e r ( c f . 910).

us n o t e here t h a t i f one begins w i t h terms Qu = u " + (Zm+l)u'/x s u i t a b l e a ) t h e n f o r A = xZm+'exp( l a ) t h i s i s ( A u ' ) ' / A = Qu.

Let

+ au' ( f o r V i a (9.26)

2

- %A"/A. q v where q = -A-'(A')" = +(A'/A) 2 2 But A ' / A = (2m+l)/x + and A"/A = (2m+1)(2m)/x + Za(Zm+l)/x + ( a ' f a ) 2 2 2 2 so q = -(m - k ) / x 2 - a(m+*)/x - CY / 4 - a'/2 = - ( m - k ) / x + q. Hence Qu = 2 2 xZm+' one has -A-'(&)" - A u i s e q u i v a l e n t t o bv = - A v and s i n c e f o r A 2 2 -m-4 Q = m+' Q Q = -(m +)/x i t follows that w = x v satisfies x Q [: +q]w = 2 2 + ';i)[xm+l"w] = bv = - A v so [Qo + {]w = -1 w. Hence a n o n s i n g u l a r au' m added t o t h e g e n e r i c s i n g u l a r i t y ( 2 m + l ) u ' / x can be passed t o a p o t e n t i a l w i t h a s i n g u l a r p a r t l / x which does n o t c o n t r i b u t e any new q u a l i t a term

we have A5Qu = ~ [ A % J ] = Gv = v "

(x

+

SINGULAR OPERATORS

65

t i v e features t o the solution. One can use t h e e x p l i c i t Now go t o Example 9.3 and we f o l l o w [Kpl; F f l ] . formulas f o r p QA and Q o f course t o determine p r o p e r t i e s b u t one can a l s o proceed v i a general a n a l y t i c a l techniques which g e n e r a l i z e t o o t h e r s i t u a Thus one proves

t i o n s (complex a , can ~ a l s o be a d m i t t e d b u t we o m i t t h i s ) . ( c f . [C40; F f l ; Kpl]) EHE@RE:1 9-7- F o r x i s e n t i r e i n A. = C-1-1")

For x E

as x

n E Z+ t h e r e e x i s t s Kn such t h a t f o r X = c+iu,

+

QQA ( x ) = e ( i h - p ) ~

where I D F ( A , x ) l

Qu = Qou +

p

2

u, Qo = Q:,B)

Q Q ( g i v e n by ( 9 . 1 3 ) ) i s a n a l y t i c f o r A

(O,m),

and a Q X ( x ) = [l + o ( l ) ] e x p ( i A - p ) x

(9.27)

-

( g i v e n by (9.12)

q!

E [O,m),

e-2'

-f

ri

m.

L

For c > 0,

-1Sle,

and x

> 0, and

E

E

E 0

[c,m)

Q(X,x)l

5 Kn.

I n o r d e r t o e s t a b l i s h t h a t p QA ( x ) i s an e n t i r e f u n c t i o n o f e x p o n e n t i a l t y p e we p r e f e r t o make c o n t a c t here w i t h t h e f o r m u l a t i o n of [ T j l ] and w i l l prove &HEOREM 9.8.

L e t A ( x ) = X ~ ~ + ' C ~ ( Xm ) >, -4, w i t h C

s t r i c t l y positive.

Q

Q

( n )

A

E

Cm, even, and

L e t q E Cm be even ( r e a l ) and s e t Qu = (A u ' ) ' / A Q

-

qu

corresponds t o Theorem 9.7). L e t pQQA be t h e (so q = - p 2 w i t h A = A Q Q, asB2 Q unique s o l u t i o n o f Qu = -1 u w i t h p QX ( 0 ) = 1 and DxqA(0) = 0. Then p QA ( x ) i s e n t i r e i n X and s a t i s f i e s f o r x L 0,

5 K ( x ) e x p ( ( I m h ( x ) ( K E Co[O,-)) 2 Phaad: Set VJx) = A??xkX(x) and we o b t a i n (*) V; - [ % ( A ' / A + 4(Al/A ) ' 2 Q 2 Q Q Q Q + q]VA+ A V, = 0 ( n o t e ( A ' / A ) I = A"/A - ( A ' / A ) ) . Since A = xZm+lcQ we Q Q Q Q Q Q 2 can s e t X ( x ) = (Zm+l)C'/ZxC + $ ( C ' / C ) I + + ( C ' / C ) + q and w r i t e (*) as Q Q Q Q Q Q 2 2 V i - [(m -$)/x2 + X ( x ) - A ]Vx = 0 (9.28) Iq!(x)l

Q

Note t h a t s i n c e C %

-4 d e f i n e F

d t (Abel t r a n s f o r m ) where A = Aa,B. Recl > Re8 >

-4one

[ f ] ( s ) = :J f ( t ) A ( s , t ) a,B i s a n a l y t i c i n (a&) and i f

F,,,[f](s)

has by (9.39) (C(t,s)

=

Ch2t

-

Ch2s, d c = d(Chw))

ROBERT CARROLL

70

On t h e other hand combining Lemma 9.11, ( l O . l ) , and t h e d e f i n i t i o n of F a,% above we obtain

Actually (10.3) t u r n s out t o be a very special case of a genREmARK 10.2. e r a l formula in transmutation theory which we develop l a t e r ( c f . [C40,64, 651). I t a l s o has a version in t h e theory of Lie groups and symmetric spaces where exp(-ps)F [ f ] ( s ) can be i n t e r p e r t e d a s a Radon transform of a a,B radial function f ( c f . [Hc2,5]) and we l e t i t s u f f i c e f o r now i n t h i s d i r ection t o w r i t e i n standard Lie theory notation ( c f . [Hcl-91 f o r example) (10.4)

Ff(a)

=

e'(logs) I N f ( a n ) d n ; F*(?I)

=

1,

F ( a ) e-iA(loga)da ;

Then ? = (Ff)* corresponds t o (10.3) a n d our transmutation version of (10.3) l a t e r will have t h e form PF [ f ] = Q f . In [Lbl] one speaks of f a c t o r i n g t h e 4 spherical transform S as S = MH where H i s c a l l e d a Harish transform and M is a Mellin transform.

To analyse F Koornwinder works with Weyl f r a c t i o n a l i n t e g r a l transformaa,B t i o n s ( f o r which we give transmutation versions l a t e r - c f . a l s o [C40; Mkl; T j l ] and see [KplZ] f o r f u r t h e r group t h e o r e t i c meaning). Thus DEFZNIBZ0N 10.3- For a

E

One shows e a s i l y t h a t W l J o

R, g

WY

C:([a,

E

=

?J+Y)

) ) , and Reu

W lJ [g](y) E

>

0 define

C:[a,m),

W o = i d e n t i t y , 113-1

[g] = - g ’ , and WIJ[g](y) i s e n t i r e in l~ w i t h (p,y) WIJ[g](y) continuous. Rev > 0, Thus U p : C o [ a , m ) -+ Co[a,m) i s 1-1 onto. Define next f o r f E ,C: -f

u > 0, s 2 0 , m

(10.6)

!.Uz[f](s) = r(V)-’

f ( t ) [ C h u t - Chus]’-’d(Chat)

W“[f](s) can be extended t o be e n t i r e in and Nu: C: + C; i s 1-1 o’nto w i t h lJ kJ inverse W y Applying t h e s e constructions t o (10.2) one sees t h a t f o r f E lJ C,: F,,,[f](s) has an a n a l y t i c continuation t o an e n t i r e function i n ( a , ~ ) given by

.

71

SPHERICAL TRANSFORMS

For a , @ E C, Fa,B:

C:

-f

C E i s 1-1 o n t o and t h e i n v e r s e i s

Combining (10.3) w i t h t h e above b i j e c t i o n s and t h e Paley-Wiener theorem f o r t h e Cosine t r a n s f o r m we have a Paley-Wiener theorem f o r t h e Jacobi t r a n s form (10.1)

EHE6REN 10-4. For a,B E C t h e map f

+

ia,B

i s 1-1 f r o m C:

o n t o H. b m

Now f o r t h e i n v e r s i o n f o r m u l a we have i n (10.1) ?-$,-$(A)

= (2/i~)'/

0

COS t

d t so t h a t

S e t t i n g Cosht = [ e x p ( i x t ) + e x p ( - i A t ) ] / 2

and changing t h e i n t e g r a t i o n p a t h

i n (10.9) one o b t a i n s irl+m,,

(10.10)

f-, (h)eixtdx in-m 2,-%

f ( t ) = (l/ZIT)+

.

where

rl

i s a r b i t r a r y (note

t h e change o f c o n t o u r s

LI

I

ii s

even, 1- ;exp(-iht)dh = i : t e x p ( i x t ) d h , and 0 in+m t o fifl-- i s j u s t i f i e d by Cauchy ' s theorem).

The i d e a now i s t o g e n e r a l i z e t h i s f o r m u l a i n u s i n g f o r g

where e.g.

50, Q

a n a l y t i c f o r Imh >

> -Re(a+B+l), Q

and

r\

> -Re(a-B+l)

SO

H, t > 0, and

that c

( c f . here Lemma 9.10 and n o t e t h a t c

have zeros where a - B + l + i h = -2n o r a + B + l + i h = -2n).

E

a,B

(-A)-'

is

~ i n, (9.14) ~ will

Now f o r g E H ( g even,

e n t i r e , r a p i d l y d e c r e a s i n g o f e x p o n e n t i a l t y p e ) t h e r e i s an A such t h a t On t h e o t h e r hand by I g ( X ) l f K n ( l + l h l ) - n e x p ( A I I m h I ) f o r any n = 0,1,

....

Theorem 9.12 f o r c > 0 t h e r e e x i s t s K such t h a t when t 5 c and Imx 2 0, \@:'@(t)\ 5 Kexp[-t(Imh+Rep)]

w h i l e t o e s t i m a t e ca,B(-h)-l

v a r i a t i o n o f Lemma 9.9 f o r complex

one can use a

which i s e s t a b l i s h e d from (9.14) u s i n g Thus r e c a l l i n g t h a t

C Y , ~

u s i n g S t i r l i n g ' s f o r m u l a i n t h e same way ( c f . [ K p l ] ) . p+q = 2a+l

I;EmmA 1 0 - 5 - For each a , @ E C and y > 0 t h e r e e x i s t s K such t h a t i f E C and A i s a t a d i s t a n c e > y f r o m t h e p o l e s o f ?a,B(-A)I then l ~ C Y , B ( - ~ ) l - '
-Re(a-B+l) +

(a+B+l) + y and

1 g (A

(10.12 )

y

( - 1)1 5

B(t)/zaJ

-4 and IReB( < R e ( a t 1 ) . However f i s a n a l y t i c i n (a,a,A) and 0,B a l y t i c continuation ( f o r Rea > - n - 1 ) described a f t e r (10.1) can be sed in the form

a , E~ C

f o r Rea t h e anexpres-

0 as an a n a l y t i c function of a , @ . Consequently t h e r e l a t i o n f = (Fa,@),',, follows by a n a l y t i c continuation by extension from t h e region of Lemma 10.6 (note i s a n a l y t i c in ( a , ~ )( A + - i Y - 2 i , . . . ) by (9.13).

(n

=

Suppose t h a t

REIRARK 10.8,

-$and Re161

Rea+l).

(-A)-'

Then from (10.13) and (10.1) one has ( f

E

, C:

g

E

H)

This follows from estimates Ig(A)I 5 Kn(l+lXI)-neexp(AIImhI), Ic"a,@( - h ) I - ' A < K(l+lhl)Rea+4, e t c . as above. S e t t i n g g = h with h = we will r e f e r t o ~ say a (10.19) a s a Parseval formula even when a , E~ C . For real a , with

4

IBI

a+l (10.19) i s a standard Parseval formula w i t h Ic" ( h ) I 2 = _ Q The formula becomes e.g. 1- f F A d t = lm?;,; 4-2 o 1 2 4 0 \FQ(h)\ dA f o r f l y f 2 E :C a n d t h e transform f + ? can then be extended as 2 2 2 L (dw ) where dw ( A ) = d h / I ? Q ( l ) I (cf. an isometric isomorphism L (A d t ) 4 4 4 a l s o [Ffl] - t h e formulas have t o be adjusted when we use Q i n (10.1) and dwq = dX/2alcQ(i)I 2 ).

>

-4 and

0 ( p = p ). More p r e c i s e l y ( c f . [Cg3] f o r d e t a i l s ) Q Q Q A A DEFINI&IBN 10-9. Assume A > 0 and :(x) qo (9, 5 0 g e n e r a l l y ) . L e t b be Q-

x

-f

m

and A ' / A

an odd a n a l y t i c f u n c t i o n , f an even e n a l y t i c f u n c t i o n , g1 and g2 bounded

f u n c t i o n s on any i n t e r v a l [xO,m), bounded f u n c t i o n on [xo,m)

along w i t h t h e i r d e r i v a t i v e s , and h a

( x o > 0) .

One s t i p u l a t e s e i t h e r o f t h e f o l l o w -

i n g s i t u a t i o n s , denoted by H1 and H2 r e s p e c t i v e l y . i n v o l v e s (rn

5 -4,

B

(10.20)

Al/A

= 2mt1 -+ 2b(x);

Near x = (10.21)

Q Q

m,

= (B~/x') + f(x)

!(x)

x

H1 r e q u i r e s (a > 0, 6 > 0, B~ 1 A;7/AQ = 2al/x

+ e-"gl(x);

A ; ~ / A=~ 2p

2 0,

y >

$ ( x ) = a:/x2

+ e-"g2(x)

0) + emYxh(x)

4 as w i t h H1 p l u s

w h i l e H2 r e q u i r e s t h e same h y p o t h e s i s f o r (10.22)

Near x = 0, H1 :H2

L 0)

(p >

0, 6 > 0 )

&

2

REMARK 10.10, I n [Cg3] t a k e s2 = -A w i t h s % - i A . Then t h e e q u a t i o n = 2 2 2 2 - A u becomes n e a r x = 0 (*) u " + ( ( 2 m + l ) / x ) u ' + 2bu' + p u - ( a / x )u - f u 2 = - A u so t h a t x = 0 i s a r e g u l a r s i n g u l a r p o i n t . The Fuchs-Frobenius t e c h n i q u e leads t o c o n s i d e r a t i o n o f an i n d i c i a 1 e q u a t i o n

+

'T

2mT

(m +a 2) 4.

-

= 0 with

2

r o o t s T + = -m ? (m2 + B')'. Then t h e r e L e t T = T+ and s e t a = a r e t w o - l i n e a r l y independent s o l u t i o n s q Q and II,Q o f (*) such t h a t (m,B 0) (10.23)

q Q (x,A)

= xTUl(x,A);

ILQ (x,A)

=

(xT'/2a)Vl(x,A)

w i t h DxqQ = T X ~ - ' U ~ ( X , A ) and D x i Q = (T-/2a)xT--1V2(x,A) a n a l y t i c i n ( x , ~ ) and t e n d t o 1 as x -+ 0. and t h u s i f m < 0,

T+ =

-2m and

T-

= 0.

+

where Ui and Vi a r e

Note here if a = 0, I n o r d e r t o have

qQ

T+

=

- m + Iml

be t h e s o l u -

+

0. It t i o n equal t o 1 a t x = 0 i n t h i s s i t u a t i o n one assumes m 2 0 o r a i s a l s o necessary t o change t h e f o r m o f Dxq Q above i n case B = 0 and T+ = 0.

The case

a

= 0 and rn = 0 i s t r e a t e d i n [Cg3]

t i o n o f i t here.

Note t h a t f o r

T

b u t we o m i t any d e s c r i p -

# 0, qQ(x,A) d i f f e r s f r o m a s p h e r i c a l

75

SPHERICAL TRANSFORMS f u n c t i o n 9 hQ ( x ) s i n c e

xT near x = 0; i f B = 0 we can s i m p l y deal w i t h

'L

ip

t h e s p h e r i c a l f u n c t i o n ~ Qp ~ ( as x ) before. S i m i l a r l y one has two l i n e a r l y i n A dependent ( J o s t ) s o l u t i o n s @Q (x,*h) o f Qu = - h 2 u such t h a t ( n o t e @Q (x,X) Q

@J

Q

-1) = Ag5(x)exp(-ixx)W1 (x,X)

(x w i t h Wi(x A ) @

+

1 as x

ip

Q (x,h)

P

2u

(x)W(ip 4 (x,h),@ Q (x,A))

=

= A04(x)exp(ihx)W2(x.h)

[ [ ( B2~ / x2 ) + e - Y x h ( x ) ] ] u = -12 u;

-

For H2: [ ] =

cQ(A)@Q (x,A)

=

@Q (x,X)

The p o i n t x = m i s an i r r e q u l a r s i n g u l a r p o i n t A 2 t h e e q u a t i o n Qu = -1 u t a k e s t h e form

+ e-"gl(x);

[ ] = (2nl/x)

A

=.

u" + [ I u ' +

(10.24)

One has

+

and near x =

i n genera

and

-c

+ c

4 (x,-X)

(-A)@

Q (-x)2ix,

+

2p

For H1:

e-"g2(x)

with A

M(@y,@Q)

4 the

i n analogy t o

= 2 i x and

s i t u a t i o n f o r spher-

(x)ip 4 (x,A)

i cQ a l f u n c t i o n s . One a l s o w rQi t e s RQ (x,X) = A and dw ( A ) = dh/ r) Q Z I T / C ~ ( X ) I 2 w i t h q f ( h ) = Jm f ( x ) nQ (x,A)dx. I n general, besides a continuous 02 2 spectrum on 10,~)( i . e . X LO), t h e o p e r a t o r -$ ( i n L (A d x ) ) w i l l have a 2 Q f i n i t e number o f eigenvalues p = -yj = - s 2 ( s > 0 ) i n t h e i n t e r v a l [(op2,o)

( ~ y =

-ujip:

= y?ipQ J J =

-

359:;

xj

=

i:j;

dq

=

po(x,iyj))

L Z W 10.11- F o r x 2 0 t h e r e e x i s t c o n s t a n t s K and N such t h a t f o r IwQ(x,h)l 5 Kexp( I q l - p ) x where h = s + i v .

1x1 2 N

I f H2 holds w i t h B~ = 0 t h e n f o r

xo > 0, x 2 x and Imh > - 6,, t h e f u n c t i o n A ( x ) @9 (x,x) i s holomorphic i n Q X and as 1x1 + O' =, @Q ( x , x ) = 8 ( x ) e x p ( i A x ) [ l + O ( l / h x ) ] and Dx@9 (x,X) = -i(%/%)aQ(x,x)

Q

+ ih~Si(x)exp(ihx)[l+O(l/Ax)].

I f H2 holds w i t h B~

+ 0 or

Imx > 0 and continuous 2 0 which, as I h l + m w i t h I m x 2 nl > 0 and x 2 xo > 0, has t h e form v ( x , x ) = cl(~)(-ih)~Siexp(ixx)[l+O(l/hx)] where c1 (1) = l / l o g ( - i h ) i f 2 2 al = 0 (a, -~ = + B: - m - k ) ; cl(h) = ( - i x l a i f a, 0. A - + ( X ) V ( X , X ) = Q c1 ( A ) ( - i X ) % (x,h). F u r t h e r f o r Imx 2 0 t h e r e e x i s t c o n s t a n t s and such t h a t f o r 1x1 2: one has ( C ~ ( - A ) ) - ~5 z l X l y under hypotheses H1 o r H2 (y can be made p r e c i s e b u t t h i s i s n o t needed). F i n a l l y f o r x > 0, @Q (x,x)/ H1 h o l d s one has a f u n c t i o n v, holomorphic i n h f o r f o r Imh

;,

+

c (-1) i s holomorphic f o r Imh > 0, h # A . (= i y . ) ; t h e zeros o f c,(-A) in Q J J t h e upper h a l f p l a n e ( i f any) correspond t o t h e A j' T h i s lemma p l u s Remark 10.10 i n d i c a t e some o f t h e b a s i c i n f o r m a t i o n going i n t o t h e f o l l o w i n g theorem.

We t a k e DT = I f ; x - T f ( x ) E D,;

I,

= even Cm

L e t H denote even r a p i d l y decreasing e n t i r e f u n c t i o n s g o f 2 m e x p o n e n t i a l t y p e as b e f o r e ( i . e . t h e r e e x i s t s R such t h a t ( l + l h ( ) exp functions).

(-RIImX()Ig(x)(
- % a n d t h e k e r n e l y(x,y,n) 0 2m+l Q = ,Q, A,,, = x

By known formulas ( c f . [Fbl;

and rl > x+y w h i l e f o r I x - y l

2 2 2 where z = ( x +y -n )/2xy. a l s o [Cpl;

i n (11.3) becomes f o r

Bbel]) one has y(x,y,n) < t-

= 0 f o r 0 < r7 < I x - y l

< x+y

2 2 Hence s e t t i n g n = ( x +y -2xyz)'

we o b t a i n ( c f .

L121)

i?HEB)%?Zm 11-2- The g e n e r a l i z e d t r a n s l a t i o n S:

associated w i t h

Qi Qm has t h e =

form g i v e n i n (11.6) below

We r e c a l l a l s o t h e model s i t u a t i o n f o r 0 = CosiyCoshzdh ( f o l l o w i n g (11.3)).

+ 6(x-y+n) + S(x-y-rl)] (1/2)[f(x+y)

+ f(x-y)]

D2 w i t h y(x,y,n)

Thus y(x,y,n)

=

= (2/n)Jm CosXx 0

(1/4)[6(x+y+n)

+ 6(x+y-r7)

which upon a c t i o n on even f u n c t i o n s f g i v e s S;f(x)

=

( t h e w e l l known d'Alembert s o l u t i o n of t h e wave equa-

tion).

REmARK 11-3- The general s i t u a t i o n here a measure dw (A) =

Q

I:Q (A)dh -

-

f o r a w s p e c t r a l p a i r i n g g i v e n by

i n v o l v e s f o r m u l a s o f t h e t y p e ( c f . (11.3))

82

ROBERT CARROLL

(11.8)

= A i l (~)Y(x,Y,o) =

y(x,Y,rl) = Q 9 gQ ( x ) =

Now ;:(A)

m a l l y sinceZ$:(A)

1 ,ro Q (x)fiT(x)dx

‘

= rop;(x) O =

= 6 (A-5)

( A ) must h o l d f o r -

= 6(A-5)/$

0

w

1m -r o0 c ( A ) .Qp A ( x ) ~(A)dA.

I n f a c t more g e n e r a l l y

t h e e q u a t i o n Q q F = F f o r 9 determined 0 by R QQ as i n Theorem 12.12 o r (10.38) (and s u i t a b l e F ) g i v e s F(A) = (Cf(y),( R Q F(~),ro:(y))) = ( F ( V ) R 0 ,,(~~(Y), SO t h a t R?CA(y),roP(y)) Q Q roP(y))) Q

= ~(A-P)

i n terms o f a c t i o n o f

F. Hence

f o r m a l l y ( c f . a l s o Theorem 12.5 f o r a general p r e s e n t a t i o n )

” and t h i s equals r oQL ; ( x bQp L ; ( y ) . One d e f i n e s now a g e n e r a l i z e d c o n v o l u t i o n v i a

1m

(11.10)

(f

*

g)(X) =

U

U

and thus, f o r s u i t a b l e f,g, l i k e (11.9), SY ro Q ( x ) = ro Q ( x Im~ ( x , y , r l ) A ( O ) V‘Q (rl)dn a5 r e

4

0

‘

g(y)s~f(x)Aq(x)dY =

-

U

?.:

(f * g ) = We remark t h a t p r o d u c t formulas ) Q~( y ) , when w r i t t e n o u t as P Q ( x b Q ( y ) = ‘ 5 5 o f i n t e r e s t i n s t u d y i n g s p e c i a l f u n c t i o n s and

m o t i v a t e d some o f t h e work on g e n e r a l i z e d t r a n s l a t i o n s ( c f . [Ak1,5; D j l ; Ff5; Gdl; Kp2-8,11,12;

Sy1,2;

Cal;

Cgl-41).

Now we s h a l l c o n s i d e r t h e o p e r a t o r A

(11.11)

Qu = u ” + ( ( 2 m + l ) / x ) u ’

-

q q ( x ) u = Qmu

and g i v e a b r i e f d e s c r i p t i o n o f c o n s t r u c t i o n s v i a Riemann f u n c t i o n s e t c . which produce g e n e r a l i z e d t r a n s l a t i o n s and t r a n s m u t a t i o n s as i n 554-5.

The

techniques f o l l o w [ B x ~ ; S o l ] and were g i v e n w i t h t h e e s s e n t i a l d e t a i l s i n [C40].

One should a l s o r e f e r here t o [Ge1,2;

Fi1,2;

Cpl-3; Sz1,2;

f o r r e l a t e d work, some o f which i s reproduced i n [C40].

Vd1,2]

I n view of Remark

A

9.6 t h e o p e r a t o r 0 o f (11.11) w i l l be a p p r o p r i a t e f o r o u r g e n e r i c s i n g u l a r -

i t y (2m+l)/x i n t h e u ’ t e r m a r i s i n g from (A u ’ ) ’ / A

Q

Q’

The c o n s t r u c t i o n s here

w i l l p e r m i t c e r t a i n s i n g u l a r i t i e s i n q as i n d i c a t e d below b u t we exclude 2 2 s i n g u l a r i t i e s o f t h e t y p e B / x f o r now. L e t us r e c a l l some f a c t s about Riemann f u n c t i o n s t o expand upon t h e c o n s t r u c t i o n s of 94 where o p e r a t o r s

D2

-

q were t r e a t e d ( c f . Theorem 4.5 f o r example).

f o l l o w i n g [Cp2-4]

are collected i n

The general f a c t s here,

83

EXPLICIT CONSTRUCTIONS

L e t t h e e q u a t i o n be g i v e n i n t h e form

REmARfi 11.4.

LU = u

(11.12)

- u + 2gux + 2 f u YY Y

xx

L*V = v

- v

xx

YY

-

2gvx

cu =

F

The a d j o i n t o p e r a t o r i s

w i t h c h a r a c t e r i s t i c s x+y = c o n s t a n t . (11.13)

f

-

2fv

+ ( c - 2gx - 2fy)V

Y

- uv + Zguv w i t h K = -vu + uv + Zfuv one has t h e X x* Y Y L e t C be a n o n c h a r a c t e r i s t i c curve, standard formula vLu - uL v = Hx + K

and s e t t i n g H = vu A

( i , ; )a

P =

Y'

p o i n t n o t on C, and c o n s i d e r t h e r e g i o n

c h a r a c t e r i s t i c s from

*P A

$0

bounded by C and t h e

Thus l e t t h e c h a r a c t e r i s t i c y-x =

c u t t i n a C. 4

c u t C i n Q and y+x = y+x c u t C i n R so t h a t t h e boundary A

*

A

r

;-;

o f R consists o f

I f L v = 0 one has t h e n

t h e segment PQ, t h e a r c QR, and t h e segment RP.

vFdxdy = lr (-Kdx + Hdy) by t h e divergence theorem and hence standard

f,

A

h

c a l c u l a t i o n s ( u s i n 9 dy = dx on PO and dx = -dy on R P ) y i e l d

u(;,$)

(11.14)

=k[(uv),

provided t h a t (note A

on PQ and vx

);,;

-

v

Y*

=

$1

+ (uv),]

+

(-Kdx

f

Hdy) -

QR v ( x ,y ,?

R

6Q

,? A

F ( x ,Y) dxdy A

y - x = y - x and R F

%

+;

A

A

A

= yfx) A

( g + f ) v on R^P w i t h v(x,y,x,y)

=

1.

(6)

vx

vy = ( g - f ) v

f

The f u n c t i o n v(x,y,

satisfying L v = 0 with the characteristic conditions ( 6 ) i s called

t h e Riemann f u n c t i o n v = R and we n o t e t h e r e i s agreement w i t h t h e R of 54. Indeed i n 54 w i t h o p e r a t o r s Dz - q we have g = f = 0, c = q ( y ) F = 0 w i t h c o n d i t i o n s R = 1 on t h e c h a r a c t e r i s t i c l i n e s . s t i p u l a t e s here t h a t vx

n

+

v

=

0 on PQ and vx

-

-

q ( x ) , and

The c o n d i t i o n ( 6 ) A

v

= 0 on RP; t h e s e a r e ac-

Y Y A 4 t u a l l y d i r e c t i o n a l d e r i v a t i v e s and s p e c i f y t h a t v = c o n s t a n t ( = 1 = v(x,y, A

A

x , y ) ) on t h e c h a r a c t e r i s t i c s . "0 2 Now c o n s i d e r f i r s t o p e r a t o r s o f t h e f o r m (11.11) and w r i t e Om = D + ((2mf

We a r e p r i m a r i l y concerned here w i t h t h e equations f o r g e n e r a l i z e d l)/x)D. t r a n s l a t i o n s S i associated w i t h and thus c o n s i d e r

ti

(11.15)

^Qi(Dx)u = {i(Dy)u;

9 --/ \Qm 0

-

q; u(x,O) = f ( x ) ; uY (x,O)

=

0

The case q = 0 a l r e a d y t a k e s account o f t h e s i n g u l a r i t y i n u ' and once t h e Riemann f u n c t i o n R (C,n,x,y) A

function R (11.16)

9

0

-

f o r Q:(Dx)

Rq(S,n,x,y)

=

6:(Dy)

-

f o r ?:(Dx)

C:(Dy)

i s known t h e n t h e Riemann

has i n f a c t t h e form

Ro(C,n,x,y)

-

$f

Ro(S,n,s,t)Q(s,t)Rq(s,t,x,Y)dsdt

84

ROBERT CARROLL

= ( S n / ~ y ) ~ + ' f o r Ix-El = l y - n l , Q ( s , t ) = q ( s ) - q ( t ) , and t h e i n 9 < s+t < x+y w i t h x - y 5 s - t 5 S-n i n t h e ( s , t ) p l a n e t e g r a l i s o v e r 5 : E+n -

where R

which i s shown i n (11.17)

( c f . here [Bx2, Cp2-4, L12, S o l ] and t h e p r o o f o f Theorem 11.5 below f o r t h e method o f p r o o f ) .

Moreover u s i n g R

one can g i v e a " u n i f i e d " formula 9 f o r g e n e r a l i z e d t r a n s l a t i o n s a r i s i n g f r o m such s i n g u l a r problems. To see t h i s suppose u s a t i s f i e s (11.15),

where S c i s t h e

so t h a t u(x,y) = S:f(x)

generalized t r a n s l a t i o n associated w i t h

ti; t h e n

( f o r s u i t a b l e f), v =

u -

f satisfies

= 0. Now use Riemann's method f r o m Remark 11.4 t o Y s o l v e (11.18) where t h e i n i t i a l c u r v e i s t h e l i n e y = 0. L e t ,r2= D = D

w i t h v(x,O)

= v (x,O)

XY

be t h e t r i a n g l e w i t h v e r t i c e s (x-y,O), u s i n g (11.14) w i t h u = v and v = R (11.19)

V(X,Y)

(since v = v

-

(11.20)

Then

Rq(S,n,x,Y)ii(S,n)dEdn

Rqf(2m+l)/n and

W(X,Y,S)

again.

one o b t a i n s

Now i n f a c t ?(E,n)

= vx = 0 on t h e l i n e y = 0 ) .

0 9 ((2m+l ) / s ) $ - ((2m+l

and (x+y,O)

-%jD

= - L f so v(x,y)

?!i(Dn)]f(c) = (D R )f

Y

=

q

(x,y),

)/n)Dn). =

lim

S&

=

=

-[$i(DS);

4JaD (-cdg + z d n ) by Remark 11.4 where now K = R f ' - f D R + (2m+l)Rqf/5 ( n o t e L i n v o l v e s 9 Using

E 9

s+n = x i y

2mtl [yRq(S,n,xYy)

and s e t t i n g

- D n Rq ( E y n y X , Y ) I

(which w i l l be seen t o make sense) one o b t a i n s an e q u a t i o n (m > -%)

Here one needs m > -S i n o r d e r t o have R (xty,O,x,y) 2 9 E C implicit

=

0.

Hence f o r m a l l y ,

with f

&HE@REEI 11.5.

Generalized t r a n s l a t i o n s S:

as above f o r s i n g u l a r o p e r a t o r s

85


o f t h e form (11.11) can be expressed i n t h e f o r m (11.21) f o r w as i n (11.20). N

Phoud:

L e t us check t h e passage from 4faD (Hdn - r d t ) t o (11.20)-(11.21).

On t t n = x+y one has dg = Tdn and dR

= DgRqdg + D R dn = (D R

D R )dg. 114 5.9 0 9 From ( 6 ) we have on ^PQ Q g-n = c o n s t a n t , D R + D R = +[(Zm+l)/g + (2m+l)/ 5 9 n 9 s+n = c o n s t a n t , D 5R 9 - Dn R 9 = & ( 2 m + l ) [ ( l / E ) - (l/n)]Rq. n]R w h i l e on RE q A Thus on PO, dR = ( m + % ) [ ( l / c ) + ( l / n ) ] R dg w h i l e on R;, dR = ( m + $ ) [ ( l / t ) q 9 9 (Kdn(l/n)]Rqdg. Consequently, w r i t i n g o u t Hdn - i?dc one has f i r s t &IaD

9

Q

N

Kdc) = 4faD [-f(t)DQRq + ( 2 m + l ) f ( ~ ) R ~ / n l d !+ [ f ’ ( c ) R q - f ( 5 ) D 5 R q + (2m+l) f ( 6 ) R /c]dn so on RP where dn = -dg t h e i n t e g r a n d i s 2 1 = [ f ( D R - D R )

5 9

9

n q

-

w h i l e dR = ( D R - DnRq)dg = (m+&)(l/g-l/n) 9 4 5 9 Rqdc. Hence 21 = -D ( f R ). The i n t e g r a l o v e r (x-y,x+y) on t h e a x i s r e 5 9 duces i m n e d i a t e l y t o (11.21) and f o r ?‘Q where do = dg we have 21 = [ - f ( D R

f ’ R q + f R (2m+l)(l/r1-1/5)]dg

+ DnRq) + f ’ R q + fRq(2m+l)(l/n+l/g)dg w h i l e dR = (D5Rq + D R )dc 9 n 9 (l/c+l/n)Rqdg. Hence on ?Q, 21 = D ( f R ) , and (11.19) becomes 5 9 (11.22)

V(X,Y)

=

x =

U(X,Y)

rty x-Y

+ 4fR

- 4fR

( i n obvious n o t a t i o n ) . t h e consequence.

IX-’

+

EHEOREN 11.6. ( 1 1 .23)

9,

-

But R (x+y,O,x,y) 9

+ 4fR

9

(x-y)

= 0 ( c f . below) and (11.21)

g i v e n i n [Bx2]

is

( c f . a l s o [CpZ-4;

Thus

Ro (5, n ,x ,y ) = ( ~ n / x y ) ~ + ’ (1 -z )-m-4F (5n/xy)m+’t?i-mF

- (s-n) 2 ][(x+y)

The f u r t h e r a n a l y s i s o f (11.21) lation.

X

D5(fRq)dg - 4jx+yDg(fRq)dr

f + 4fRq(x+y)

The Riemann f u n c t i o n Ro(~,n,x,y)

where z = [(x-y)’

(m+k)

X

u

=

L e t us r e c o r d here t h e form o f RO(g,n,x,y) De4; L11; Sol; F i l , Z ] ) .

$1

X -Y

w(x,y,c)f(c)dc

5 9

=

(+m,$-m, 2

-

for 6i(Dx)

-

Ôo(D ) i s m y

(m++ ,m+LL ,1 ,(z/z- 1 ) )

=

1 ,1- < )

(5+n)2]/16xycn

(and 5 = ( l - z ) - ’ ) .

r e q u i r e s many e s t i m a t e s and e x t e n s i v e c a l c u -

We r e f e r t o fBx2; S o l ] f o r d e t a i l s , many o f which a r e reproduced

i n [C40]. We w i l l be c o n t e n t here t o i n d i c a t e t h e main r e s u l t s . One obt a i n s e s t i m a t e s on Ro(g,n,x,y) and s o l v e s (11.16) by successive approximat i o n s i n a s t a n d a r d manner ( o b t a i n i n g e s t i m a t e s on Ra i n t h e process).

Then from (11.23) one can show t h a t as rl

-f

0 (*) C ( 2 m + l ) / ~ l R o ( ~ , n , x , ~ )-

Set

86

-

ROBERT CARROLL

DnRO(t,~,x,y)

+

2w0(x,y,s).

Thus w o ( x , ~ , s ) must be t h e k e r n e l Y(x,Y,E;)

determined i n Remark 11.1 and Theorem 11.2; we check t h i s as f o l l o w s . F i r s t 2 2 2 2 2 4 4 4 2 2 2 2 2 2 2 n o t e t h a t 4x y (1-2 ) = 2x y - x -y - 5 +2x 5 +2y 5 where z = ( x +y - 5 )/2xy. Hence i n (11.24) we have wo(x,y,s) = [ 2 1 - 2 v ( m + l ) / ~ ~ r ( m t l ? ) ] ~ ( x y ) - ~ ~ ( l - z ~ ) 2 2 m-+ 2 m-J-, ( 1 - ~ ~ ) ~ - % (y4 )x - (c/xy)r(mtl)(l-z ) /JTr(m+$) = y(x,y,s). Next we n o t e t h a t (11.16) can be w r i t t e n as

Using t h e d e f i n i t i o n s (11.20) and (*) and p r o p e r t i e s o f Ro i t f o l l o w s from (11.25) e a s i l y t h a t f o r m > -% (11.26)

= wo(x,~,s)

W{X,Y,C)

-

no

wo(s,t,s)Q(s,t)Rq(s,t,x,y)dsdt

%.

N

where

1-

r e f e r s t o t h e f i g u r e (11.17) w i t h

EHE6REM 11.7- For m

>

q =

0.

-4 t h e t r a n s m u t a t i o n k e r n e l w(x,y,s)

i s determineddby (11.16). q Simultaneously one o b t a i n s e s t i m a t e s f o r w(x,y,c)

has a representa-

t i o n (11.26) where R

-

wo(x,y,s)

and u = S:f(x)

d e f i n e d by (11.21) (we r e f e r t o [BxZ; C40; S o l ] f o r t h e d e t a i l s ) . t i o n q ( x ) can have s i n g u l a r i t i e s q we o m i t m = i n §§4-5.

-5 s i n c e

'L

O ( X - ~ - ~ )( E < 1

-

The f u n c -

c f . Remark 11.14) and

i t i s n o n s i n g u l a r and has t h u s a l r e a d y been covered

S: determined by (11.21) s t i l l r e p r e s e n t s a g e n e r a l i z e d t r a n s l a 2 t i o n when f C b u t u may become i n f i n i t e as y + 0. Y REMARK 11-8, The a n a l y s i s o f [ S o l ] extends t h e t e c h n i q u e o f t h i s s e c t i o n ,

+

w i t h some improvements and s i m p l i f i c a t i o n s , t o equations ( c f . (11.15)) (11.27)

$:l(Dx)u

=

tq2(D ) u ; P Y

where m and p a r e s u i t a b l e complex numbers. p r o v i d e d and o f course s i n c e p = compl i c a t e d .

-

(DX2 -DY2 + 2m+l + f y y ) u

[q;

-

qg]u = 0

Considerably more d e t a i l i s

m t h e p r o o f s and r e s u l t s a r e somewhat more

Formulas such as ( 1 1.20)- ( 1 1.21 ) a r e c o n s t r u c t e d and t h e r e -

l a t e d Cauchy problems f o r u i n v o l v e (11.27) w i t h i n i t i a l c o n d i t i o n s u(x,O) = f ( x ) and uy(x,y)

= o(y-')

as y

+

0 where

y =

1 + Rep

-

]Rep\ (so y = 1

f o r real p 2 0).

T h i s l a s t c o n d i t i o n on u can be improved when f i s s u i t Y a b l y d i f f e r e n t i a b l e t o u (x,O) = 0 ( c f . a l s o Theorems 11.10 and 11.13) Y

I n t h e l a s t s e c t i o n of [ S o l ] some o f t h e r e s u l t s a r e p a r t i c u l a r i z e d t o t h e

87

EXPLICIT CONSTRUCTIONS A

/

c o n t e x t o f t r a n s m u t a t i o n s Q2

\

+

Q, w i t h m = p and we w i l l i n d i c a t e some o f

t h e s e r e s u l t s here.

R e l a t e d r e s u l t s a r e c o n t a i n e d i n e.g. [Ge1,2; Sz1,Z; 2 Vd21. R e c a l l f i r s t ( c f . (9.26)) t h a t Om = = D + ((2m+l)/x)D i s r e l a t e d to = D2 - (m 2 -$)/x2 by a t r a n s f o r m a t i o n Gm(D)[xm"f] = xm+'Qm(D)f. We

Qi

&

2 c o n s i d e r t i 1 = D2 - (m -+)/x2 - q ( x ) and 6 i 2 = D2 corresponding o p e r a t o r s and 20, as i n (11.11).

ail

4

-

2 (m -+)/x* - q,(x) w i t h We n o t e t h a t i t i s na-

6;

on a subspace Dm o f Em = I f ; xm+'f E L 2 ) w i t h xrn+'$;f = tural t o define 2 ?)i[xmq'f] E L (i.e. E Em f o r f E Dm C Em corresponds t o mapping 2 s u i t a b l e f u n c t i o n s g = xm+'f E L i n t o L2). Then i f B i s a t r a n s m u t a t i o n

:6

6f:

B: a'2 + t i 1 ( i . e . v mm+' -m-+ B = x By (y

B$:2f

=

t:lBf

(y

-f

x) f o r f

Dm say)

E

it follows that

x ) transmutes Gq2 i n t o 6'1. Indeed xm+'4qlBf = 6:l[xm+' = ijqliYm!kf; xm+
-m-?i m+fl

6ii

and s e t

DEFINZCZBN 11-9, On an i n t e r v a l [o,a] a] (AC Dx[x-m-4g]

?);is

absolutely continuous); o(x-')

=

an o p e r a t o r

Pi -+ m

as x L

2

+

let E

E:i

ig E

=

2 L (0,a);

c 1 (0,al;

g'

E

A C ~ ~ ~ ( O ,

g = xm++[1 + o ( 1 ) l ;

6:i

0 where y = l+Rem-IRem\}.

i s t h o u g h t o f as

.

The k i n d o f theorem r e s u l t i n g f r o m t h e method o f a n a l y s i s i n [ S o l ] i s L e t n-% < Rem < n+4, m # 0 o r m = n+%, P = max(Z,n),

CHZ0REM ll,lO,

I n a d d i t i o n i f n = 1 assume

> 0.

P

Suppose qi E C (0,al and qi

(j)

> 3/2-Rem and i f n = 0 assume a > 4-Rem.

( x ) = O(x there e x i s t s a transmutation operator o n l y ) , ~ q 1 =i i4':2 and

g:

~ ( x + )

Em2

I,"

on

i22, i;

and a

a-j-1)

i: 26:

as x +

i s continuous L2

6'1

+

+

0 (0 5 j 5 L ) .

Then

( i n a sense used here

? (on

[o,al),

5-1 e x i s t s ,

E q l i s expressed v i a a c o n t i n u o u s k e r n e l z(x,y) as b ( x ) = m f o r o 5 y 5 x. i(x,y)q(y)dy. F u r t h e r I ~ ( x , Y ) ( 5 Mx'(y/x) +

REmARK 11-11. We want t o emphasize again t h a t g e n e r a l l y we do n o t want t o 2 work w i t h t r a n s m u t a t i o n o p e r a t o r s i n L t y p e spaces; t h e i r n a t u r a l h a b i t a t seems r a t h e r t o be i n spaces o f Cp f u n c t i o n s where i n v e r t i b i l i t y o f B does 2 n o t g e t t i e d i n w i t h t h e L t y p e s p e c t r a l t h e o r y ( a c t u a l l y and L:oc c o n t e x t 2 o r L (0,a) as i n Theorem 11.10, i s a c c e p t a b l e f o r some aspects o f t h e t h e o r y 2 t h e o r y ) . Note t h a t i n t h e c o n t e x t o f L ( 0 , m ) o p e r a t o r s , i f we have BP = QB w i t h B - l p r e s e n t t h e n Q = BPB-' o f s p e c t r a would be i m p l i c i t .

would be " s i m i l a r " t o P and some i d e n t i t y However we have seen i n 54-5 t h a t transmuta-

t a t i o n s e x i s t between o p e r a t o r s w i t h v a s t l y d i f f e r e n t s p e c t r a . One notes 2 2 a l s o t h a t i n an L o r Lloc c o n t e x t t h e q u e s t i o n o f domains and ranges would

88

ROBERT CARROLL

have t o be examined v e r y s e r i o u s l y a t e v e r y stage o f t h e a n a l y s i s and t h i s s i m p l y g e t s i n t h e way.

It i s much more n a t u r a l t o work i n a C p c o n t e x t

and subsequent passage t o l i m i t s i n s u i t a b l e weighted L be e n v i s i o n e d l a t e r i f needed o r d e s i r a b l e .

2

spaces can always

We w i l l however d i s c u s s l a t e r

and d i s p l a y v a r i o u s connections between t r a n s m u t a t i o n s and r e l a t e d t r a n s forms i n t h e c o n t e x t o f weighted spaces and maps s i m i l a r t o t h e s i t u a t i o n o f D e f i n i t i o n 11.9.

1

K y ( x y ~ ) ~ ( ~ ) l = [q2(x)-q1 ( x ) + 2 D x K ( x y x ) l ~ ( x )+ [K(x,O)v' (0)-Ky(x.O)v(O)l y= 0 Consequently one seeks K(x,y) (11.30)

?);l(Dx)K

= 6;2(Dy)K;

-

q,(x) Further for

B' =

satisfying ( f o r suitable

q,(x)

K(x,O)q'(O) =

Set B f ( x ) =

know g[ym+'h]

= xm+%h

(

E

K (x,O)v(O) Y

i:2) = 0;

2DxK(xYx)

X ~ ' ' B ~ - ~ - ' we have x"+'{;lBf

kGi2[ym+'f].

-

v

B(x,y),f(y))

= 6:li[ymt'f]

and xm+'Btq2f m

and gg(x) = ( g ( x , y ) , g ( y ) ) .

=

We

so a p p a r e n t l y f o r g = ymt4h

and f = h above one has -m-$ v = x ( B(x,y),h(y)) = x-"-'( E(x,y)ym+',h(y) ). B(X,Y) Note a l s o i n t h i s c o n n e c t i o n t h a t one expects ~ ( x , y ) (y x) t o saty""". 2m+lAq -2m-1 i s f y (cf.554-5) ( D x ) ~ ( x , y ) = ^ O q 2 ( D y ) * ~ ( x , ~ )= Y Q 2(Dy)[y B(x,Y)] s i n c e Q*(A v ) = A Qv. Thus y-2m-'~~l(Dx)[x-m-'~(x,y)y m&! ] = AQ i 2 ( D ) [ x - ~ - ' Q shows t h a t t h e f u n c t i o n ?(x,y) = ~ - ~ - ~ g ( x , y-m&) y s a t i s ~ ( ~ , y ) y - ~ - 'which ] Consequently B(X,Y)

-f

6l;

fies

5il ( D x ) 2 = ?$.2;

The p r o o f i s t h e n reduced t o s t u d y i n g t h e a p p r o p r i a t e


89

M

There a r e many technical d e t a i l s ( c f .

Goursat problem f o r K = ( X ~ ) - ~ - ' K . [C40; Sol]).

The following theorem i s i n s t r u c t i v e i n several ways; in p a r t i c u l a r i t i n -m -% CJ] = o(x-') as x 0 can a r i s e . d i c a t e s how t h e condition Dx[x 1 EHE0RER 11-13- Let m E C , m f 0, q measurable on ( 0 , a l w i t h t Y q ( t ) E L (0,a) = 0 ; q = xm+'[l + O(X'-~)] where y = 1 + Rem - IReml. Then t h e problem 2 as x 0 = D2 - (m -%)/x2 - q ( x ) ) has a unique s o l u t i o n . This s a t i s f i e s -f

-f

6:~

(Gq

-4 o r even m

lem f o r

= -A2q

G,$

=

>

-f

o r eventually f o r

= -A2$

(q =

xm+%).

Prrooh: For q = 0 a fundamental s e t o f s o l u t i o n s i s x"+' by v a r i a t i o n of parameters (11.31)

q ( x ) = axm+'

2

6

0. Note t h a t f o r q = - A and s u i t a b l e - 1 ) we can deal here with an eigenfunction prob-

o(xmY)as x

a l s o Dx[x-m'5q] m (e.g. m >

+ Bx4-m

+

(1/2m)

i,x

and

xm+4t4-m-p+G;J-*

i'-m so t h a t mlq(t)v(t)dt

S e t t i n g q = xm+'$ one has $ ( x ) = a + ~ x + -(1/2m)JX ~ ~ t [ l - ( t / ~ ) ~ ~ ] q $and dt 0 = 0. Then J, $ = 1 + o(xl-') i s required as x + 0 so we want ~1 = 1 and s a t i s f i e s a Volterra i n t e g r a l equation with kernel ( 1 / 2 1 n ) [ l - ( t / x ) ~ ~ ] t q ( t ) 1 E L . There i s a unique s o l u t i o n $, continuous on [O,a], and i t is seen e a s i l y t h a t $ = 1 + o(x'-') as x 0 (note e.g. 1-y x ItYq(t)ldt 5 x lo I t Y q ( t ) l d t ) . Finally -f

(11.32)

$ ' ( x ) = x-2m-1r t2"+'q(t)J,(t)dt

IJx t q ( t ) d t l < JX tl-' 0

= o(x-')

0

as x

-f

0

' 0

Note here t h a t i f Rem > 0 then y = 1 whereas f Rem < 0 then y = 1 - 2IReml = 1 + 2Rem. T h u s in p a r t i c u l a r , taking m rea f o r s i m p l i c i t y i n i l l u s t r a t i o n , i f m > 0, y = 1 and ( t / x ) 2 m 5 1 w i t h t q t ) E L1 i n (11.32). I f m < 0, y = 1 + 2m i n (11.32) w i t h t Y q ( t ) E L'.

2 A s ' 2 Consider $ in t h e case q = q' - A so Qm$ - - A $ and f o r m > -%, 0 < y 5 1 , so t Y q ( t ) E L1 i s equivalent t o t Y { ( t ) E L1. Then $ = 1 + O ( X ' - ~ ) tends t o 1 as x 0 b u t J,' = O(X-') m i g h t become i n f i n i t e a s x 0. 1 However note from (11.32) t h a t i f 2m+l L O and q E L then $' = o ( 1 ) as x 2 1 0 and J/ will be a spherical function when q = q' - A w i t h q ' E L . We note t h a t f o r y = 1 f o r example a s i n g u l a r i t y $ = O ( l / t ' + € ) i s permitted f o r E < 1 . In p a r t i c u l a r i f one had an a n a l y t i c s i t u a t i o n w i t h 6 ( t ) = g/t + $ ( t ) the corresponding i n d i c i a 1 equation f o r A Q, q remains s(s-1) + (2m+l)s = 0

REEWRK 11-14.

-f

-f

-f

w i t h s = 0, o r s = -2m, and f o r s = 0 a s o l u t i o n w i t h J , ( O ) = 1 a r i s e s ; howe v e r $ ' ( O ) = ;/(2m+l) and t o produce a spherical function we would need

ROBERT CARROLL

90

= 0 (note t h a t i f d " ( 0 ) = 0 the d i f f e r e n t i a l equation i s n o t s a t i s f i e d a t

x = 0 unless - as would occur here - t h e s i n g u l a r terms can be c a n c e l l e d 1 o u t ) . Thus E L seems i n t i m a t e l y r e l a t e d t o t h e e x i s t e n c e o f s p h e r i c a l function type solutions.

12,

CAN0NZCAC F0RiWCACZ0N OF PAlGEVAC F'P)RI!IUCAS AND CRAWF@RW, We w i l l

c o n t i n u e here w i t h t h e development o f S e c t i o n 10 b u t f i r s t l e t us g i v e a summary k i n d o f p i c t u r e o f t h e v a r i o u s maps a s s o c i a t e d w i t h two o p e r a t o r s

?

P and Q l i n k e d by a t r a n s m u t a t i o n B: P -+ Q. Thus t a k e two o p e r a t o r s and A Q as i n (9.1) w i t h g e n e r a l i z e d s p e c t r a l f u n c t i o n s R P and R9 as i n (10.38) A

and l e t B: P (12.1)

A

+

P = Q be t h e t r a n s m u t a t i o n c h a r a c t e r i z e d by D X

Pf(A) =

1;

PF(x) =

(

f ( x ) n XP ( x ) d x ; q f ( X ) =

Q vX.

Then

f(x)q(x)dx; 0

qF(x)

= (

Pf(X) =

P P P R , F ( X ) P ~ ( X ) )=~ ( F(h),vX(x)),; R Q , F ( ~ ) QY J ~ ( X =) )( ~F(X),P,(X))~; Q

1"-

Qf(A) =

f ( x ) v XQ ( x ) d x

n

B F ( x ) = ( RP , F ( X ) ~ : ( X ) ) ~ = ~ F ( X )= (

RQ,F(X ) f ( x ) ) X

PF(X) =

F(X ),P). (XI),

(

j

m

f(x)P:(x)dx;

P

(

F(X ) , a : ( ~ ) ) ~ ;

=(

F(A ) , ~ : ( x ) ) ~ ; P

= ( RQ,F(X)qX

XI)^ ;

$F(X) = ( F ( X ) , VQ ~ ( X ) ) =~ ( R P , F ( X ) P Q~(X))~

Then, working on s u i t a b l e f and F, one has by c o n s t r u c t i o n

B

= p-',

4

= Q-',

etc.

P = P-',

P = Q-',

We w i l l p r o v i d e c o n s i d e r a b l e d i s c u s s i o n l a t e r t o

show t h a t t h e f o l l o w i n g ( f o r m a l l y e v i d e n t ) s p e c t r a l p a i r i n g s make sense and a r e c o r r e c t under n a t u r a l hypotheses ( c f . i n p a r t i c u l a r Theorem, 2.2.2, C o r o l l a r y 2.2.3, (12.2)

(12.3)

etc).

Thus ( B = B - l )

ker B = ~ ( y , x ) =

(nX P (x),v,

k e r B = y(x,y)

(vXP (x),aXQ (y)),

B =

pP; B

=

Q (y)),

=

( R p ,QXp (x)v, Q ( Y ) ) ~;

= ( R Q ,vXP ( X ) ? ~ ( Y ) ) ~

= IPQ

Now i n general we do n o t want t o s p e c i f y p r e c i s e domains f o r o u r transmutat i o n s s i n c e i n p a r t i c u l a r t h e y a c t on v a r i o u s t y p e s o f o b j e c t s a t v a r i o u s

CANONICAL FORMULATION

times.

91

S i m i l a r l y o u r t r a n s f o r m s P, P, P, e t c . can be d e f i n e d on v a r i o u s

t y p e s o f o b j e c t s and we do n o t want t o impose l i m i t a t i o n s on t h e i r a c t i o n We would have t o keep i n s e r -

b y a r t i f i c i a l l y s p e c i f y i n g some f i x e d domain.

t i n g n o t a t i o n a t a r a t e f a r exceeding t h e r a t e o f theorem p r o d u c t i o n .

On

t h e o t h e r hand o f course p r e c i s e domains can be s p e c i f i e d when i t seems des i r e a b l e and we r e c a l l e.g.

D e f i n i t i o n 11.9 i n t h i s d i r e c t i o n .

So, i n t h i s

s p i r i t , l e t us d e f i n e some n a t u r a l spaces whose c o n s t r u c t i o n i s m o t i v a t e d

ô

by t h e o p e r a t o r

REWRK 12-1- S e t =

[i; hm+'f"(h)

(12.4)

where

E

=

$

=

Qi and D e f i n i t i o n 11.9. Thus as o u r model c o n s i d e r Qi and s e t Em I f ; xm+'f(x) L2(0,-)}with Fm QEm =

L2(0,m)}.

Qf(h)hm+'

=

I n t h i s connection note t h a t

c m- l Hm [xm+'f(x)];

IQF(x)x"+'

H, denotes t h e Hankel t r a n s f o r m .

forms ( c f . [Dsl;

=

E

=

c mHm[km+'F(h)]

Standard theorems on Hankel t r a n s -

L19] f o r example) g i v e Hm: L2

m e t r i c ) f o r s u i t a b l e m (and hence xm+'f(x)

+

+

L2 as an isomorphism ( i s o -

hm+'Qf(x)

modulo a f a c t o r o f

I n s t e a d o f always w o r k i n g w i t h Em as a H i l b e r t space ( w i t h s c a l a r proc,). d u c t (f,g), = Imx 2 " + ' f ( x ) 6 ( x ) d x ) we w i l l f r e q u e n t l y use EA = Em = i f ; -m-+ 20 A f ( x ) E L 1 i n a n a t u r a l d u a l i t y . S i m i l a r l y Em has a n a t u r a l H i l b e r t x A . 4

structure with (f,g) E

{?; A

= Iml Z m + ' ? ( h ) i ( h ) d l as w e l l as a n a t u r a l dual space

z m 0 L 1 ; however

M

A

$, here w i t h ( f , g ) *

A

=

and $ E Em = Em f o r reasons i n d i c a t e d below. We r e c a l l m++ m++ 2 f ] and = {x f, f E Em} = L . Note a l s o t h a t Qm[x

(?,?j),,, for

f E E

t h a t xmyGif

=

g e n e r a l l y i f p:

A

we w i l l use EIF, = Em =

M

A

Em

v'6

A

i s a spherical function f o r

0 then

4"

AQq:

=

T7

i s a corres-

ponding fundamental o b j e c t f o r r e l a t i v e t o an L2 expansion t h e o r y ( i . e . "vQ 2vQ 0 vQ QpA = - A p h ) . However l e t us emphasize t h a t t h e p h and ppha r e themselves g e n e r a l i z e d e i g e n f u n c t i o n s and one c e r t a i n l y does n o t expect ( n o r have) pV x4 L2 f o r example. I n any event one has Qf = f m f ( x ) p h4( x ) d x = jm(k'f)$Q(x) 2 0 0 '2Q dx w i t h E L t h e n a t u r a l desideratum; we w r i t e i g ( x ) = Im g ( x ) ;h(x)dx

E

Lf'Q

w i t h Qf(ph) = z [ b f ] ( h ) .

Here

$!

xm+?i2"r(m+l)(hx)-mJm(~x)

Jm(hx) = ~ ~ ~ h - ~ - ' ( h x ) ~ J , , , ( h x % ) ^ , ~ 5 ( h x ~ J m ( x xwhich ) suggests t h a t t h e na-

t u r a l g e n e r a l i z e d e i g e n f u n c t i o n s a r i s i n q i n an L2 t r a n s f o r m t h e o r y when R Q

Q

=

:Qdh w i l l be $!(x)$;(l) ^ w z ( h ) i f ( h ) so t h a t Qf =

= $:(x).

{[k;f]

Then we w r i t e i f ( X ) = ir f(x)?:(x)dx

=

A

-4"

uQ

@J[L;f]o r

92

ROBERT CARROLL

A 2 The 9 transform theory i n L f o r example should then correspond t o the HanA kel transform theory f o r Q = and one i s led t o t h e general question of equiconverqence theorems f o r eigenfunction expansions ( c f . [ F f l ; Kpl] f o r " 2 t h e L isometry b f L'? f o r c e r t a i n n, T h u s f o r example i f one

6;

Q

-f

6

Q

6').

4

knows the Hankel theory and can transmute into ( s u i t a b l y ) then the transform theory should be "isomorphic" t o t h e Hankel theory. Conversely given an equiconvergence s i t u a t i o n one expects t o be a b l e t o construct a s u i t a b l e transmutation ( c f . f o r example [Bhl-3; Rsl]).

Now more generally we consider t h e following basic spaces (note t h a t t h e A operator Q f o r example i s t o be thought of as defined on a s u i t a b l e domain in EQ - c f . Definition 11.9).

DEFZNICI0N 12.2. Given % A as i n (9.1) s e t EC = { f ; supp f i s compact a n d Q Q kf'Q E L21 with EQ = { f ; L t f E L']. E Q i s not a good domain space in genera1 b u t since e.g. B does n o t map E F + EC one must use t h e l a r g e r format t o Q f i t things together. We r e a l l y do not want t o work in E unless we have Q e.g. a theory isomorphic t o the Hankel theory as i n Remark 1 2 . 1 ; i n p a r t i c u l a r we do not know a p r i o r i even t h a t q i s defined on a l l E and even i f i t Q w e r e , 9 , expressed via R Q , generally would not be defined on BE as such.

Q

One can work w i t h t h e obvious Hilbert s t r u c t u r e i n EC and expect t o t r a n s 4 port t h i s t o = QE;. We will eventually be dealing however w i t h countable Q AC unions of H i l b e r t spaces, EC = U E C ( o ) f o r example, and thus E i s not t o be Q Q Q thought of a s a p r e h i l b e r t space. Thus ( f , g ) Q = Im AQ(x)f(x)g(x)dx and t h e

tc

0

G0dA natural t r a n s p o r t i s ( f , g ) - ( f , g ) f o r i= Qf and $ = Qg. When R Q Q-, Q ,,PA A 1 t h i s corresponds by (12.5) t o f = 9f :$f = Q[A'f] and (?, l / n ,

Then w r i t e Un(x,y) = SY6'(x) X Q

Cm w i t h compact s u p p o r t ( e x t e n d i t as even t o PR6: to

in

Fn =

(12.15)

(m)

a f t e r (10.31).

where $ ( x ) (-m,m)).

We w r i t e a l s o R;

=

I n p a r t i c u l a r t h e n Theorem 12.5 i s a p p l i c a b l e

6n E EC so t h a t f o r a r b i t r a r y G E EC one has

Q

Q

Q

(

SY6n(x),AQ~) x Q = (6:

*

-%) c h a r a c t e r i z e d by P [Coshx] = q QA ( y ) .

Q,

rl

DL

4:

-+

We w i l l p u t a s u b s c r i p t A

A

Q on o p e r a t o r s and k e r n e l s r e f e r r i n g t o a t r a n s m u t a t i o n P -+ Q o r P Q when 2 2 P = D R e c a l l h e r e t h a t f o r Ap = 1,D has t h e form Pu = @ p u ’ ) ’ / A p and P P q P , ( x ) = Coshx, a k x ( x ) = exp(?iAx), cp(A) = 1/2, dvp(A) = Cp(X)dX = (2/?r)dh, -f

.

etc.

T h i s t r a n s m u t a t i o n B was s t u d i e d e.g.

Kdl-3;

Kel-9;

Consider f i r s t t h e s p e c t r a l k e r n e l s as i n Theorems 2.2 and 2.4.

Thus from (2.2) and Theorem 2.2 k e r B

Q

(cf.

i n [C30,33,40;

Q

Lpl-31.

[C2,3,30,36,40,63]

should have t h e form

f o r such c a l c u l a t i o n s ) .

w i l l be g i v e n by (2.5) as yQ (x,y)

(2.9)

!

W

m ‘

=

The k e r n e l y (x,y)

Q

(Cosh~,E!(y))~ = AQ(y)

=

k e r Bo

= U

( hy)m+l Jm( hy)Cosxxdx = ~ ~ ~ + % ~ [ h ~ + ~ C o s A x ]

0

where c,

=

1 / 2 v ( m + l ) and Hm denotes a ( g e n e r a l i z e d ) Hankel t r a n s f o r m ( c f .

110

ROBERT CARROLL

[Shl;

There a r e s e v e r a l ways t o d e s c r i b e (2.9) i n e x p l i c i t

Tkl; Zbl]).

terms and we w i l l do t h i s below.

CHEORET!! 2-6. The k e r n e l s B

4 and y 4 o f B9 and B4 a r e

g i v e n by (2.8)-(2.9).

The f o r m u l a (2.8) i s q u i t e u s e f u l e s s e n t i a l l y as i t i s . t o i t however goes as f o l l o w s .

(2.10)

(

p4(y,x),f(x))

=

One useful a d j u n c t 2 L e t km = 2r(m+l)/Jd'(m+%), x2 = 5, y =

kg-2m jy(y2

-

x2)m-?if(x)dx =

0

-

(1/2)km

c)m-4f(Jc)dc/Jc

0

We r e c a l l now t h e d e f i n i t i o n o f t h e pseudofunctions Y w i t h Y-n

E fm

Thus f i r s t w r i t e f o r p E

Sjl]).

p(k)(0)xk/k!]dx (

xy,p

DmT = Y-m

*

=

(

0

p ( k,

-

I!-'

Since fEmxap(x)dx ( 0 )/ k ! (a+k+l ) ] one

Di i s t h e n d e f i n e d by Y p = [ l / r ( B ) ] P f xB-' f o r a # !(n) f o r n 2 0 ( n = i n t e g e r ) . One has Y * Yq = Yp+q and E

*

T; ImT = Ym

B0(Y,x),f(x)

q

Yp

we go back t o (2.10) as

* I:f(JE )/Jc1 3 ( 0 )

"n+l )/JvI~-"IY,+%

) =

(where y2 *

Now b e f o r e

T i s a l s o a f r e q u e n t l y used n o t a t i o n .

CbROtGARy 2-7. The k e r n e l B form (2.12)

Jm x a [ p ( x )

as

g o i n g f u r t h e r w i t h o p e r a t i o n s on t h e (2.12)

In-' [E 0

-

p(k)(0)xk/k!]dx )

The d i s t r i b u t i o n Y -n o r 0 and Y-n

D, ( x y , ~ )=

when -n-1 < Rea < -n ( s o Re(n+a) > - 1 ) .

-

xa[p(x)

can w r i t e

= (l/r(B))Pf xB-l

= 6 ( n ) and some r e l e v a n t f a c t s which w i l l be needed l a t e r as w e l l

(cf. [Gfl; =

R

Q

of B

and x2

g i v e n by (2.8) can a l s o be w r i t t e n i n t h e

c).

REmARK 2-8, The f o l l o w i n g i n f o r m a t i o n w i l l be needed l a t e r and i t seems app r o p r i a t e t o r e c o r d i t now. i s given i n [ G f l ; =

Bbel].

The F o u r i e r t r a n s f o r m o f t h e pseudofunction Y

ble r e c a l l f i r s t t h a t x:

0 f o r x 2 0; thus ( x ~ , p ( x ) ) = (x'?,p(-x)

i 0 ) " by (x+iO)" = x y

+

exp(iua)x!;

).

One d e f i n e s d i s t r i b u t i o n s ( x &

( x - i 0 ) " = x+" + e x p ( - i a T ) x r .

f o l l o w i n g formulas h o l d (where F f = ( f ( x ) , e x p ( i s x ) ) , ~ ( x y )= i e x p ( + i a n ) r ( a + l ) (u+io)-a-l; F ( l x ( " ) = -2Sin+an I'(a+l)lsl-'-'

F(X:)

8

= \ x I a f o r x < 0 and x:

s =

u+iT,

Then t h e

.):

a # -1,-2,.

= - i e x p ( - + i a a ) r ( a + l )(u-io)-a-l

;

F(lxIaSgnx) = 2 i C o s h Note here i n p a r t i c u l a r t h a t t h e s e a r e c o n s i s t e n t ( f o r s = u) i n w r i t i n g 1x1' = x; + x,: 1 0 1 -a-1 -- u+-a-1 + u--a-1

r(a+l)ls/-"-lsgns

( a # -2,-4,...).

(a f -1,-3,...);

.

SPECTRAL P A I R I N G S

Also since Y formally

B

= [l/r(B)]x:-'

jmxaCosXxdx =

0

we have FYatl %[Jmx"exp(ixx)dx 0

111

iexp(%av)(otiO)-"-'

=

+ 1:

and w r i t i n g

+

( - g ) a e x p ( i h g ) d g = %[FxY

F x r ] = % F I X \ " ( x - 7 1 ) one o b t a i n s Jm xaCosXxdx = FC[xa] = -r(a+l)Sin%m 0 L e t us n o t e t h a t f o r i n t e g r a l ~ 1 , ~1 = n 2 0, one has F(xY) = in+' IA\-"-'. n! (o+iO)-"-'; F(X;) = -1.nt+ln! ( o - i o ) - n - l , Now g o i n g back t o (2.9) one e x p r e s s i o n f o r y (x,y) due t o [ L p l - 3 1 was l i s t e d

Q

b u t we o m i t i t here ( c f . a l s o [Kdl-3;

i n [C29,40]

more u s e f u l e x p r e s s i o n f o r y r y o f Euler-Poisson-Darboux

Q [ p QX ( y ) ] ( x ) = (yQ(x,y),q1(y) g i v e n by (1.9.4)

i n u s i n g a f o r m u l a o f W e i n s t e i n from t h e theo-

RQ

Thus CosXx =

(EPD) e q u a t i o n s (see [C63]).

)

lle can o b t a i n a

4

and one knows t h a t f o r - % < m < n - 4 and

Iph

1 0 2m+l 2 2 n-rn-3/2 CosXx = ~ i x ( ( s ; O ~j o) v~i ( Y ) Y (X -Y dY

(2.13) where

Q

Kel-91).

~1

=

r(%)/Zn-lr(m+l)r(n-m-~). Consequently ( t a k i n g n

> m+3/2

i f de-

s i r e d and s e t t i n g x2 = 5 , y2 = n )

'/2 = Y -m-$ (2.13) says t h a t (BQg)(J y (such t r i a n g u l a r i t y p r o p e r t i e s a r e d e r i v e d i n 54 i n c o n s i d e r a b l e g e n e r a l i t y ) . Thus pp,(y) Q = Coshy + JY ?(y,S)CoshgdE and one o b t a i n s a G-L e q u a t i o n immed0

i a t e l y i n t h e form

Q

N

(3.19)

B(Y,x) = ( V ~ ( Y ) , C O S X X =) ~ A(y,x)

where A(y,x) = (Coshy,Coshx) on

+

loy

t(y,S)A(S,x)dS

6 ( x - y ) -+ sZ(y,x)

=

(assume dw = (2/v)dh

-+

du

Now by a n a l y s i s such as t h a t i n Theorem 1.4.9 one knows t h a t

[O,m)).

^K(y,y) = Wy q ( x ) d x ( t h i s i s dependent o n l y on t h e c o n n e c t i o n and on t h e 0

d i f f e r e n t i a l e q u a t i o n (0'

-

q)q!

Now f o r m a l l y i f one deals w i t h

= -h2q?).

N

N

f u n c t i o n s f ( x ) = ( f ( A ) , C o s h x ) v (dv = (2/v)dh) where f = F C f t h e n Q B f = " Q hl 2 Q 2 2 4(f(A),qX(y)), = ( f ( A ) , - A pp,(y)), w h i l e BD f = B ( 3 h ) , - A C O S A X ) ~ = (?(A), 2 4 - 2 - A ~ ~ ( y ) ) Thus ~ . f o r such f, g i v e n t h a t ( f , - A Coshx)" makes sense e t c . , B i s a t r a n s m u t a t i o n . We can a r r i v e a t t h i s c o n c l u s i o n a l s o i n another way v i a (3.19) and a Goursat problem.

Thus i n (3.19) one can deduce t h a t T(y,

x ) = 0 f o r x < y by s e v e r a l arguments (e.g. z ( y , x )

= y(x,y)

o r a contour

i n t e g r a l - P a l e y - W i e n e r argument based on t h e form o f F ( y , x ) as a s p e c t r a l pairing

-

Hence f o r x < y (3.19) becomes

see S4).

h A 2 2 + R + ? ' ( y , y ) n ( y , x ) + K(y,y)-Cy(y, E v i d e n t l y DxS = D 0 and one o b t a i n s K E A YY YY A x ) t iy(y,y)QA+ / Y K (y,S)fL(E,x)dS = 0 w i t h 0 = K + P + Iy t(y,c) 0 YY A xx XY, 0 R (S,x)dS = Kxx + ARxx + K(y,y)fiY(Y,x) - K(y,O)Pj,(O,x) - K,(Y,Y)E(Y,X)~+ ASS Kc(y,O)fL(O,x) + KSc(y,S)fL(t,x)dS. Now f? (0,x) = 0 and we w i l l s e t K (y, Y n E 0 ) = 0 (see below). Then s u b t r a c t t h e above two equations and s e t = K =

/d'

A

A

A

A

K - Kxx ( r e c a l l here ?'(y,y) = %q(y) and n o t e t h a t K'(y,y) = KY (y,y) + KE(y,y)). There r e s u l t s

AYY

(3.21)

0 =

=;

+

q(y)a(y,x)

-+

joy=c(y,S)n(c,x)dc

A

It f o l l o w s t h a t =K(y,x)/q(y)

s a t i s f i e s (3.20) which we can assume t o have a A

unique s o l u t i o n (see Chapter 3).

I\

Hence =K/q = K o r

GELFAND-LEVITAN EQUATION

Given a c o n n e c t i o n p QX ( y ) = CosXy

CHEOREDI 3.10,

-I-

123

f Y ?(y,x)Cosxxdx @

as i n d i -

A

c a t e d i t f o l l o w s t h a t B w i t h k e r n e l ~ ( y , x ) = 6 ( x - y ) + K(y,x)

i s a transmutaA

t i o n B: D2

Q ( a c t i n g on f u n c t i o n s w i t h f ' ( 0 ) = 0 ) .

-f

A

The k e r n e l K s a t i s f i e s

2^

t h e Goursat problem Q(D )K = DxK w i t h q ( y ) = 2D K(y,y) and Kx(y,O) = 0. Y Y A A Ptiuud: I n o r d e r t o deduce t h a t Kx(y,O) = 0 s i m p l y l o o k a t K(y,x) = (Z/IT) inim[ q X Q ( y ) - CosxylCoshxdA and d i f f e r e n t i a t e i n x. To see t h a t one has @ A t e r t w i n i n g l o o k a t Q B f and B f " t o o b t a i n r e s p e c t i v e l y O B f = f " -qf t K ' f f A

.

h

h

A

A

K f ' + K f - q i K f + f Kyyf and B f " = f " + K f ' Y Q B f - B f " = 0.

A

4

- K5f

f

f K f.

Consequently

55

We n o t e a l s o t h a t if i n f a c t we express ~ ( y , x ) = ( q Qp ( y ) y C o s x x ) v t h e n t ( y , x ) a u t o m a t i c a l l y s a t i s f i e s t h e d i f f e r e n t i a l e q u a t i o n i n Theorem 3.10 f o r y # x n

(and Kx(y,O)

=

I n any e v e n t we see t h a t connections o f t h e t y p e w i t h

0).

which we a r e d e a l i n g a u t o m a t i c a l l y a r e t r a n s m u t a t i o n formulas (see here a l so 52.12 f o r f u r t h e r i n f o r m a t i o n ) .

Note a l s o t h e assumption dw = ( 2 / 1 ~ ) d x

do i s n a t u r a l f o r i l l u s t r a t i v e purposes

t

-

see e.g.

t h e m a t e r i a l on geo-

p h y s i c a l i n v e r s e problems i n Chapter 3.

REmARK 3-11. One can develop a G-L t y p e e q u a t i o n f o r any s i t u a t i o n where, Q P P i n s t e a d o f q: = b P one begins w i t h ILx(y) = ( B $ ( y , t ) , p x ( t ) ) = B q where Q PA ' Q A $: = * ( A ) q x and v x ( x ) = ( y $ ( ~ , t ) , $ ~ ( t ) )f o r Y~ = k e r R$, B$ = BJ, (here

-?

B+(y,t) = ( $ Q x ( y ) , q xP( t ) ) v f o r example). F o l l o w i n g t h e procedure o f (3.8)(3.9) one o b t a i n s ( m u l t i p l y by R%-'qp(x) and R 0* -1 q Q ( y ) r e s p e c t i v e l y )

,*

+) - 2 P P (Y) = ( B$(y,t),A$(t,x)) where A$(t,x) = ( R Px(xk,(t)). SetN t i n g A$(t,x) = Ap(x)A ( t , x ) and ;$(y,x) = A p ( ~ ) A ~ l ( y ) y , ( ~ y y )we o b t a i n B$ = # -1 #$ B A w i t h R = (B ) = B$. $ $ $ J /

Y (x,Y)/A $

Q

L e t us g i v e now a model v e r s i o n o f t h e extended G-L e q u a t i o n which e x h i b i t s various t y p i c a l features f o r the singular situation. A

Q

= A

m

recall for B x2

Q ,

For t h e model problem

= xZm+' we a l r e a d y have numerous formulas (see 551.9 and 2.2).

5, and

:z

.

Qi

Thus

A

D2 Q

gl(JS) = [Jn/r(m+l

-z

n,

= Q one has e.g.

il

Theorem 2.9 where f o r -% < rn < n-%,

[ B Q f ] ( J q ) = [r(m+l)/h~][Y,+%

)lDn"un-m-+

*

nmg(v'n)].

w i t h 5-'[B0

Now r e c a l l t h a t t h e extended G-L

e q u a t i o n i s g i v e n by say (3.11) i n t h e form

(r

* f(Js)/Js]

(

BQ(y,t),A(t,x))

4'

= &(y,x)

(Y,x) = Ap(x)Aql ( Y ) Y ~ ( x , Y ) = ( Y ) Y ~ ( x , Y ) ) and A(t,x) = ( RQ,q;(t)Jf(x)) = ( QP v A ( t ) , qP( x ) ) W = (pPX(t),r$(x)W(X))v P P = (2/71)Jr CosxtCosxxW(x)dx where W(x) =

6

(W

4

/v

Q

(3.22)

P

) = %ITR ( A ) 0

A(t,x)

=

= + I T C ~ A ~ ~ +w ' ith

m

c, = 1 / 2 9 ( m + l ) .

% c i j;~~~+'[Cosx(x-t)

Thus

+ Cosx(x+t)]dx

124

ROBERT CARROLL

Now such i n t e g r a l s were discussed i n 52 f o r example and we r e c a l l (*)

ImA'CosxydA

Set B,

= 2 J ~ / r ( m + l ) r ( - + - m ) so t h a t we o b t a i n then

tEmmA 3-12, The G-L k e r n e l f o r B -2m-2

0 and i s continuous and bounded f o r Imk 0.

One has e s t i m a t e s

F u r t h e r w i t h q r e a l one has ip(x,-k) @(x,-I),

=

ip(x,k),

ip(x,-k)

= Z(x,k),

g(x,k) =

and F ( - C ) = F ( k ) .

Now t h e s p e c t r a l t h e o r y f o r an o p e r a t o r 0 = D2 - q, q r e a l , i s c l a s s i c a l ( c f . Chapter 1, §§5-8).

We assume F ( 0 ) = 0 f o r convenience and one o b t a i n s 2 a s e l f a d j o i n t o p e r a t o r i n L ( 0 , m ) r e l a t i v e t o boundary c o n d i t i o n s ip(0,k) = 2 0 w i t h ip'(0,k) = 1. There i s a continuous spectrum i n t h e energy o r E = k plane f o r E

0 and p o s s i b l y a f i n i t e number o f d i s c r e t e eigenvalues a t

2 p o i n t s E = - y j ( k k = i y j and F ( k . ) = 0 - t h e s e correspond t o what a r e c a l j J l e d bound s t a t e s i n p h y s i c s ) . One has t h e f o l l o w i n g t y p e o f theorem express i n g a symbolic completeness r e l a t i o n

&HE@REEI 4.2, (4.6)

Setting

j

=

J ( 2 / ~ ) rip(x,k)ip(y,k) 0

where c

ip.(x)

2

ip(x,k.) J k2dk

one has f o r m a l l y

I F(k) I*

+

c

Cjipj(X)ipj(Y)

= S(X-Y)

= '/lom I i p j ( x ) l dx. F o r s u i t a b l e f t h i s l e a d s t o an expansion

128

ROBERT CARROLL

and t j ( k ) = Jr f ( y ) v . ( y ) d y . J Now s e t F ( k ) = I F ( k ) l e x p ( - i S ( k ) ) which d e f i n e s a so c a l l e d phase s h i f t s ( k )

where ?(k) = Jm f ( y ) v ( y , k ) d y 0

(one can t a k e 6 ( - k ) = - & ( k ) f o r k > 0). IF(k)lSin(kx + s(k))/lkl

+ o(1).

Then f o r r e a l k, as x + m,q(x,k)

?J

The theme o f i n v e r s e s c a t t e r i n g t h e o r y i n

quantum mechanics i s t h a t i f one knows t h e phase s h i f t (measurable from s c a t t e r i n g experiments), t h e bound s t a t e energies E . ( i . e . t h e k j ) , and t h e J normalization constants c t h e n one can r e c o v e r t h e p o t e n t i a l q. I n f a c t jy

t h e passage f r o m 6 ( k ) and t h e b i n d i n g energies t o F ( k ) , which t h u s c o n t a i n s a l l t h i s i n f o r m a t i o n , can be achieved v i a a formula ( c f . [ C e l l ) F ( k ) =

n[l-(Ej/E)]exp[-(2/n)~om { 6 ( ~ ) ~ d ~ / ( ~ ~ - k I~n ) pl a] r. t i c u l a r i f t h e r e a r e no bound s t a t e s t h e n one can pass d i r e c t l y f r o m s ( k ) t o F ( k ) and hence t o t h e 2 2 s p e c t r a l measure d p ( k ) = 2k d k / a ( F ( k ) l . The a c t u a l machinery f o r r e c o v e r i n g t h e p o t e n t i a l i n v o l v e s two main procedures based on e i t h e r t h e G-L o r M equation.

L e t us s k e t c h some o f t h e

background and develop t h e m a t t e r here f o l l o w i n g s t i l l [Cel; F a l l .

We use

t h e c l a s s i c a l Paley-Wiener t y p e theorems f r o m Chapter 1, S3 as needed.

A

standard procedure now i s t o l o o k a t (4.4) f o r example and deduce t h a t t h e e n t i r e f u n c t i o n o f e x p o n e n t i a l t y p e x, q ( x , k ) = q ( x , k ) - [Sinkx/k], belongs 1 f o r k r e a l , and hence Theorem 3.9 o f Chapter 1 i m p l i e s t h e e x i s t e n c e

to L

o f a function $(x,t) (4.8)

q(x,k)

Here $ ( x , t )

=

such t h a t

:1

$(x,t)Cosktdt

i s continuous i n x and t w i t h $(x,+x)

Ik o f t y p e (4.4) jm *(x,k)Cosktdk

v(x,k)

= 0 and t h e e s t i m a t e s on

a l l o w one t o d i f f e r e n t i a t e t h e f o r m u l a $ ( x , t ) = ( l / a ) under t h e i n t e g r a l s i g n . Then one can produce a formula

0

(4.9)

= 2

rX$(x,t)eiktdt

= [Sinkx/k]

+

I"

K(x,t)[Sinkt/k]dt

'0

from (4.8) where K ( x , t ) = -2Dt$(x,t) t i c u l a r K(x,O) = 0 ) .

has reasonably n i c e p r o p e r t i e s ( i n par-

From o u r p o i n t o f view t h e formula (4.91,

called the

Povzner-Levitan r e p r e s e n t a t i o n f o r q , i s a t r a n s m u t a t i o n formula. It ex2 presses t h e a c t i o n o f a t r a n s m u t a t i o n o p e r a t o r B: 0 = P + D2 - q = Q c h a r a c t e r i z e d by i t s a c t i o n on e i g e n f u n c t i o n s ( i . e . k e r n e l r e p r e s e n t a t i o n k e r B = B(x,t)

=

B[Sinkt/kf

6(x-t) + K(x,t).

=

v)

through a

A p r i o r i such a

t r a n s m u t a t i o n o p e r a t o r B would be an i n t e g r a l o p e r a t o r w i t h a d i s t r i b u t i o n k e r n e l @ ( x , t ) a c t i n g on LO,..); e n f u n c t i o n s 7 and [Sinkx/k]

t h e a n a l y s i s based on p r o p e r t i e s o f t h e e i g a l l o w s one t o deduce t r i a n g u l a r i t y ( i . e .

~(x,y)

QUANTUM SCATTERING THEORY

=

129

0 f o r t > x ) t o g e t h e r w i t h o t h e r n i c e p r o p e r t i e s o f B.

This i s a t y p i c a l

s i t u a t i o n a l t h o u g h i n general f o r s i n g u l a r problems a decomposition B(x,t) = s(x-t)

+ K(x,t) i s n o t n a t u r a l (as i n d i c a t e d e a r l i e r ) .

Now l e t us i n d i c a t e a d e r i v a t i o n o f t h e G-L e q u a t i o n f o l l o w i n g [ F a l l which Thus f i r s t we i n -

s p e l l s o u t t h e d i s c r e t e spectrum i n t h e d e r i v a t i o n o f 53. v e r t (4.9) i n t h e s p i r i t o f V o l t e r r a o p e r a t o r s t o o b t a i n [ S i n k y / k l = Ip(y,k) +

(4.10)

f

L(y,t)cp(t,k)dt

0

where L i s j u s t a r e s o l v a n t k e r n e l o b t a i n e d i n a standard manner ( c f . f o r Now i n (4.6) we w r i t e cp (x,k)

example [ T i l l ) .

W (k) = 1/IF(k)I2, andWp(k)

Q

f o r p(x,k),

pP(x,k) f o r

1 where t h e completeness r e l a t i o n 2 ( k ) k We mulf o r qp(x,k) i s t h e n (+) 6(x-y) = ( 2 / n ) f m c p p ( ~ , k ~ p ( y , k ) ~ ~ p dk. 0 2 2 t i p l y t h e e q u a t i o n s (4.9) and (4.10) by vp(y,k)W ( k ) k and v (x,k)W ( k ) k

Sinkx/k,

=

Q

r e s p e c t i v e l y and i n t e g r a t e i n k.

4

4

rl

A f t e r some c a l c u l a t i o n s u s i n g (4.6) and

( 4 ) one o b t a i n s t h e G-L e q u a t i o n ( x > y )

(4.11)

0 = c ( x , y ) + K(x,y) +

K(x,t)n(t,y)dt

where t h e k e r n e l R i s g i v e n by

CHE0REm 4.3,

The G-L e q u a t i o n f o r P = D

and =;

D2

-

q i s g i v e n by (4.11)

K i s t h e t r a n s m u t a t i o n k e r n e l from (4.9) and

f o r x > y where (4.12).

2

,O.

i s defined by

It w i l l have a unique s o l u t i o n K and t h e p o t e n t i a l q can be r e -

covered f r o m t h e r e l a t i o n q ( x ) = 2DXK(x,x).

Pmal;:

We a c t u a l l y know t h a t K e x i s t s from (4.9) and t o show uniqueness we

suppose two s o l u t i o n s o f (4.11) e x i s t so t h a t f o r t h e i r d i f f e r e n c e K(x,y)

+

0 f o r x > y. M u l t i p l y by K(x,y) and x x i n t e g r a t e t o o b t a i n (*) E = lo K2(x,y)dy x + la lo o(t,y)K(x,t)K(x,y)dtdy = 0.

one has K(x,y)

Jx K ( x , t ) R ( t , y ) d t 0

Now w r i t e (4.12) as n(x,y)

=

=

~1cpp(x,k)cpp(y,k)dp(E)

-

s ( x - y ) = A(x,y)

2 &(x-y) where dp(E) = (2/n)W ( k ) k dk f o r E 2 0 and dp(E) Q 2

-

1

= c.s(E-E.) f o r J J E < 0 (where E = - y . w i t h k = i y . , and we s e t ( s ( E - E . ) q ( k ) ) = cp(k.) w i t h j~ j J J J some abuse o f n o t a t i o n ) . The p o i n t here i s t h a t dP i s a p o s i t i v e measure

and E i n (*) can be w r i t t e n as (4.13)

=

1; JY

1

m

A( t ,y 1K( x ,t 1K( x ,y ) d t d y =

m

dP ( E ) [ r K ( x ,Y 0

)q p

2 d1~

130

ROBERT CARROLL

By Paley-Wiener ideas, f i r s t t h e i n t e a r a l I = Ix K(x,y)p 0

t i r e f u n c t i o n o f k (and o f E s i n c e i t i s a f u n c t i o n o f =

(y,k)dy i s an en-

5 k -

r e c a l l qPp(y,k)

F u r t h e r , s i n c e I i s d e f i n e d f o r a l l E and i s r e a l f o r E r e a l

Sinky/k).

w i t h dp(E) a p o s i t i v e measure i t f o l l o w s t h a t I = 0 and thus f x K(x,y) 0

Consequently f o r each x, K(x,y) = 0 f o r a l l y E [0,

Sinkydy = 0 f o r a l l k. x],

which e s t a b l i s h e s uniqueness.

To prove t h e statement t h a t q ( x ) = 2Dx

K(x,x) p u t (4.9) i n t o (4.2) and t a k e F o u r i e r S i n e t r a n s f o r m s ( t h i s a l s o connects (4.9) t o t h e Schrodinqer e q u a t i o n

-

c f . a l s o Chapter 1, 94 and

Chapter 2 , 58, and Chapter 3, 58 f o r d i f f e r e n t t y p e s o f p r o o f s - we i n Thus f i r s t

c l u d e t h i s p r o o f here t o i l l u s t r a t e a v a l u a b l e t e c h n i q u e ) .

1

m

(4.14)

K(x,t)

NOW ( c f . [C40])

= (2/n)

for

a,@ >

Sinkt

Sink(x- 0 ( t 2 x ) .

Assume

f o r s i m p l i c i t y t h a t t h e r e a r e no bound s t a t e s and s e t S(k) = F ( - k ) / F ( k ) so

S(k) = exp(-Zis(k)).

t h a t t h e phase s h i f t i s 6 ( k ) = ( i / Z ) l o g S ( k ) ( i . e .

The

s c a t t e r i n g f u n c t i o n S ( k ) i s t h u s determined e x p e r i m e n t a l l y from t h e phase s h i f t 6(k).

Now w r i t e t h e completeness r e l a t i o n (4.6) i n t h e form m

(4.17)

~ ( x - Y )= ( 1 / 2 ~ ) *(X,k)[@(y,-k)

- S(k)@(y,k)]dk

m !

Then, as f o r (4.10) one has f r o m (4.16) (4.18)

e

ikx - @(x,k) +

m

i(x,t)*(t,k)dt

t o g e t f o r x < y, :-f @ ( x , k ) [ e x p ( - i k y )

Now combine (4.17)-(4.18) e x p ( i k y ) ] d k = 0.

Then p u t t h i s w i t h (4.16) t o o b t a i n f o r x < y

(4.19)

=

V(x,y)

=

S(k)

I

V(x,t)Vo(y+t)dt

c o u l d be d e f i n e d f o r m a l l y by e i t h e r V o ( t )

where V o ( t )

or Vo(t)

Vo(x+y) +

-

( 1 / 2 1 ~ ) i I[S(k) - l ] e x p ( i k t ) d k .

= (1/271)jm S(k)eiktdk -m

The second form i s used i n phy-

s i c s and d i f f e r s from t h e f i r s t by a t e r m ( 1 / 2 1 ~ ) j Ie x p ( i k t ) d k = B ( t ) so t h a t t h e g r a t u i t o u s l y added terms s ( x + y ) o r 6 ( y + t ) c o n t r i b u t e n o t h i n g i n

- 1

i s t h a t i t behaves b e t t e r as k + m 1 I n sumand V o ( t ) w i l l t h e n be i d e n t i f i e d w i t h an L f u n c t i o n ( c f . [ F a l l ) . (4.19).

The reason f o r u s i n g S ( k )

mary (and o m i t t i n g some f u r t h e r d e t a i l s ) The

&HEOREB 4.5,

M equation f o r P

= D

2

and Q = D2

-

q i s g i v e n by (4.19)

w i t h V t h e k e r n e l from (4.16) and V o ( t ) = ( 1 / 2 ~ ) i I [ S ( k )

-

l]exp(ikt)dk.

There w i l l be a unique s o l u t i o n and t h e p o t e n t i a l q can be recovered from t h e r e l a t i o n q ( x ) = -2DxV(x,x)

(we assume here no bound s t a t e s ) .

R€FRARK 4-6, L e t us denote t h e t r a n s m u t a t i o n B o f (4.9) by U so t h a t U f ( x ) = f ( x ) + :1

Then w r i t e t h e map determined by (4.16) as V so

K(x,t)f(t)dt.

that Vf(x) = f ( x ) +

I,” V ( x , t ) f ( t ) d t

i n g on s u i t a b l e o b j e c t s as a t r a n s m u t a t i o n ) . = 1/lF(k)I2.

-

( V w i l l a c t u a l l y be a t r a n s m u t a t i o n a c t -

which however d i f f e r from those on which U a c t s

Assume t h e r e a r e no bound s t a t e s and s e t a g a i n Wo(k)

2

R e c a l l dF(k) = dp(E) = (2/a)Wo(k)k dk and d e f i n e ( A = 6 tC2)

132 Set

ROBERT CARROLL N

i? = UWQ

$Sinkx/k]

and t h i s w i l l t u r n o u t t o be a t r a n s m u t a t i o n U: P =

?(x,k)

= W (k)q(x,k)

Q

(in fact

r~

from 5 3 ) .

+

Q satisfying

Further ;will

have an o p p o s i t e s o r t o f t r i a n g u l a r i t y p r o p e r t y from U i n t h a t m

N

Uf(x) = f ( x ) +

(4.21)

K(x,t)f(t)dt X

analogous t o V.

-

N

N

Moreover U l i n k s U and V v i a a r e l a t i o n U = LZ where:

*

is -1

an o p e r a t o r t o be discussed l a t e r and i t w i l l t u r n o u t a l s o t h a t U = ( U ) N

( c f . Theorem 3.1). i z e d as B and

I n p a r t i c u l a r t h e o p e r a t o r s U and U ( s u i t a b l y general-

r)w i l l

be o f g r e a t use i n e s t a b l i s h i n g c o n n e c t i o n formulas

between s p e c i a l f u n c t i o n s o f R i e m a n n - L i o u v i l l e ( R - L )

and Weyl t y p e ( c f .

r~Ak3; C40; F f l ; Kpl; T j l ] ) . E i t h e r t h e G-L o r M e q u a t i o n can be used t o determine t h e p o t e n t i a 7 i n t h e i n v e r s e s c a t t e r i n g problem b u t t h e y r e f l e c t somewhat d i f f e r e n t aspects o f t h e p h y s i c a l problem ( t h e ample).

M e q u a t i o n i n v o l v e s hypotheses on q a t

m

f o r ex-

The experimental i n f o r m a t i o n g o i n g i n t o t h e d e t e r m i n a t i o n o f e i t h e r

e q u a t i o n i s b a s i c a l l y t h e same however; e.g. t h e phase s h i f t s ( k ) determines S(k) i n t h e

i n t h e absence o f bound s t a t e s

M

method o r t h e s p e c t r a l mea-

sure dp(k) i n t h e G-L method ( t h e J o s t f u n c t i o n F ( k ) i s t h e common i n g r e d i ent).

Now one expects t h e methods t o be e q u i v a l e n t i n some sense and t h e r e

a r e v a r i o u s ways o f c o n n e c t i n g t h e two approaches.

I n particu!ar

one can

accomplish t h i s by l i n k i n g t h e two o p e r a t o r s U and V and t h i s was done i n a r e v e l a i n g way i n [ F a l l .

We w i l l s k e t c h Fadeev's t e c h n i q u e f o r t h e quantum

s i t u a t i o n and t h e n show how i t can be c o n s i d e r a b l y g e n e r a l i z e d and an i n t r i n s i c meaning can be e s t a b l i s h e d f o r such formulas.

The l i n k i n g t r a n s f o r -

0

N

mation U w i l l generalized t o

B ( c f . Theorem 3.1) which serves a l s o as a

Weyl t y p e i n t e g r a l i n p r o v i d i n g c o n n e c t i o n formulas f o r s p e c i a l f u n c t i o n s . We n o t e t h a t o f course an a d j o i n t t o

B s h o u l d have c e r t a i n i n t e r e s t i n g pro# .

perties.

However t h e m o t i v a t i o n f o r i n t r o d u c i n g U, and hence o u r eventual

w

B, a r i s e s from [ F a l l , and was q u i t e d i f f e r e n t t h a n mere a d j o i n t n e s s ; t h e o p e r a t o r has t r a n s m u t a t i o n a l s i g n i f i c a n c e and i s i m p o r t a n t i n c o n n e c t i n g d i f f e r e n t i a l o p e r a t o r s v i a s c a t t e r i n g i n p u t (cf. We work w i t h Q = D2

-

[C47,48]).

q as above a t f i r s t , assuming f o r convenience t h a t

t h e r e a r e no bound s t a t e s and t h a t F ( 0 ) # 0. so t h a t from Theorem 4.2 one can w r i t e

W r i t e $+(x,k)

= q(x,k)/F(k)


*

*

and T+T+ = T+T+ = I.

Here we keep g r e a l b u t use complex L

t h e corresponding c o n j u g a t i o n i n s e r t e d i n T .:

2

IT)/:

and G E Lu = I G ;

lo

IG(k)I2k2dk < m l .

m

Tog(k) =

(4.23)

133

g(x)

Sinkx 7 dx;

2

spaces w i t h

Thus i n (4.22) t a k e g g L

2

S i m i l a r l y one w r i t e s m

T:G(x)

=

(2/71)

G(k)

Sinkx k2dk 7

0

2 one has ToPg = - k 2 Tog ( w h i l e T+Qg = - k 2 T+g) and TOT: .= * T T = I. L e t now x be any e i g e n f u n c t i o n o f 4 r e l a t i v e t o t h e i n i t i a l con0 0 so t h a t f o r P = D

d i t i o n x ( 0 , k ) = 0 and w r i t e ( c f . Remark 3.11)

lo m

(4.24)

Txg(k) =

g(x)x(x,k)dx

so t h a t i n f a c t x = X(k)J/+ by uniqueness (x(0,k)

Set X(k) = x'(O,k)/$;(O,k) #(k)$+(O,k)). - -1 o r T T* =

Then c l e a r l y T

=

f

1x1';

x x

*

*-

Tx = T X ( k ) ; I = T+TZ = X - l T T x x * x *--I -1 and I = T+T+ = TxX X Tx = T2x l X [ - 2 T In particular

.

x -

f o r x = rp w i t h v ' ( 0 , k ) k real.

*

= X(k)T+;

=

1 one has X ( k ) = F ( k ) and %(k) = F ( k ) = F ( - k ) f o r 2 = l / ( X l and we o b t a i n T T*W(k) = W(k)

W r i t e W(k) = [ F ( k ) F ( - k ) ] - l

T T* = I;T*W(k)Tq = I. Next one assigns an o p e r a t o r EX i n rp2p L t o an opera-

%

PIP

t o r Ek i n L

by t h e r u l e E~ = T~E~T:; E'

*

= ToEkTo.

For example t h e opera-

t o r W(k) above i n L2 i s a s s o c i a t e d w i t h

L e t us w r i t e h e r e ( r e c a l l W(k) = l / I F ( k ) l 2 )

and r e c a l l t h a t i n t h e absence o f bound s t a t e s ( c f . (4.12)) A(x,y) = iz ~ p ( x , k ) r p p ( y , k ) d p ( E ) = (2/71)1; rpp(xyk)rpp(y,k)WQ(k)k 2 dk where WQ = 1 / I F I 2 and pp(x,k)

= Sinkx/k.

Thus W(x,y)

=

A(x,y)

which i s t h e known i n g r e d i e n t

i n t h e G-L e q u a t i o n (4.11) (n(x,y) = A(x,y) - 6 ( x - y ) ) . We c o n s i d e r now t h e 2 t r a n s m u t a t i o n o p e r a t o r U o f (4.9) and w r i t e t h i s as U = Ux = T i T o i n L and * 2 Uk = T T ( = TOUT:) i n Lu. To c o n f i r m t h i s we n o t e f i r s t t h a t g e n e r a l l y O l p

(4.27)

T*Tof(y)

= ( 2 / n ) jom$+(y,k)?(k)k2

&F(x)

/omf(x)[Sinkx/k]dxdk

=

[ ( 2 / ~ ) jm[Sinkx/k]9(y,k)k 2 d k l dx 0

( r e c a l l IP =

7

f o r k real).

t i o n (*) To[Sinix/c]

^k]

=

=

F u r t h e r i n a formal and e a s i l y checked c a l c u l a -

10" [Sincx/z][Sinkx/k]dx

JF lp(y,k)k26(k-i)dk/kc

=

~(y,;).

= (n/2ki)6(k-o

same a c t i o n S i n k x / k

+

i s a t r a n s m u t a t i o n D2

9(y,k)

Q c h a r a c t e r i z e d by t h e

+

as U; consequently U = T T

Ip 0 '

EHEB)REEI 4-7- The t r a n s m u t a t i o n U o f (4.9), c h a r a c t e r i z e d by U[Sinkx/k] = * * 2 2 can be w r i t t e n as U = TqTo i n L o r Uk = T T i n L It s a t i s f i e s

v(y,k)

.

O P

U*UWx = I and UkW(k)Ui = I where Ker Wx = W(x,S)

i s g i v e n by (4.26).

The

Iu

e q u a t i o n U*UWx w r i t t e n as UWx = (U*)-'

Pmud:

*

*

*

*

*

T?F(k)T and U*UI/Ix =~,T,T,ToToWTo = To(TqT,W) us, 0 x *o To = I. S i m i l a r l y UkWUk = ToTqWT,To = To(T,*WTq)To = I. F i n a l l y i f we w r i t e W(x,y)

Note t h a t U = T*T

= U i s t h e G-L equation.

=

+ 6 ( x - y ) and U f ( y )

= n(x,y)

=

f ( y ) +

(since U = B i s

g i v e n by ( 4 . 9 ) ) t h e n

(n(y,E),f(S))

+

f(y)

+ (

K(Y,x),(

fi(x,c),f(E)

(K(y,x),f(x))

)) +

On t h e o t h e r hand ( r e c a l l K(y,x) = 0 f o r x > y ) (U*)-'

= ( I

+ K*)-l

=

I+

N

N

K i n t h e sense of Neumann s e r i e s and K(x,y) w i 1 have t h e same t r i a n g u l a r i t y

*

N

as K (x,y) = K(y,x) ( y + x ) . Thus K(y,x) = 0 f o r x > y and K(x,y) = 0 f o r y < x. We w r i t e t h e n = (U*)-' and have c f ( y ) = f ( y ) + C r ( y , n ) f ( n ) ). Equating t h i s w i t h (4.28) one o b t a i n s ( r ) [ y , ~ ) + K ( y , S ) , f ( S ) )

+

((

K(y,x),

N

S(x,S) ) , f ( S ) ) = ( K(y,n),f(n) 9. Consequently f o r 5 < y we have t h e standard G-L e q u a t i o n (4.11), namely, N y , 5 ) + K(y,5) + Jdy K(y,x)dx,S)dx = 0 ( n o t e t(Y,n)f(n)dn),

(K(YYn),f(n)

) =

REINARK 4.8,

It i s i m p o r t a n t t o n o t e t h a t t h e G-L e q u a t i o n UWx = U has i n

N

Iv

Ad

f a c t t h e form sdy,s) + K(Y,s) + J{ K(y,x)sl(x,S)dx

0 f o r 5 < y.

= K(y,c) where K(y,S)

=

T h i s v e r s i o n , which we sometimes c a l l an extended G-L equa-

t i o n , i s more u s e f u l i n t h e general t h e o r y i n v o l v i n g s p e c i a l f u n c t i o n s .

It

w i l l be s t u d i e d l a t e r more e x t e n s i v e l y f r o m v a r i o u s p o i n t s o f view ( c f .

Theorem 3.4). v

The o p e r a t o r U i s o f c o n s i d e r a b l e i n t e r e s t i n i t s e l f as i n d i c a t e d above. note f i r s t t h a t

= (U*)-'

have f o r m a l l y r[[Sin^kx/^k] = T A

*

*

We

= T T T W(k)To = T*W(k)To so from t h e above we

* q o o A b.l(k)[ 6N (k-t)/2kk] 9

q

=

1 ; q(y,k)I/l(k)[k26(k-i)dk/

knk] = W(?)q(y,t) = q(y,k). F u r t h e r U i s a t r a n s m u t a t i o n s i n c e as b e f o r e * 2 * 2 We n o t e t h a t i n general i f i s g i v e n as Q ( T F o ) = TQ(-k WTo) = ToWToD

.

X

C

2kAk] = ;+(y,E')X(k) = Tz/F(k)

A

A

w i l l be a t r a n s m u t a t i o n w i t h T:To[Sinkx/k]

above t h e n T*T (i.e.

A

= q ( y , ~ ) ~ ( ~ ) / ? ( ~Observe ). that

X ( k ) = l / F ( k ) = 1/F(-k) and

%J/?=

?

%

T4c

= T+g(k)[r6 ( k - t ) /

T*W(k) v = T:qk)W(k)

q/FF = W q ) .


Czmm

4 - 9 - Any T as above g i v e s r i s e t o a t r a n s m u t a t i o n X

U [Sinkx/k]

c h a r a c t e r i z e d by t h e p r o p e r t y N

X

*

= p

U = TqWTo (corresponding t o X ( k ) = l / F ( - k ) ) /v

=

135

q(y,k)

N

f m E(k)exp(iky)dk/k =

-m

Q, U

x

= T*T

x

0'

In particular

i s c h a r a c t e r i z e d by c [ S i n k x / k ]

W

U now as was done i n (4.9) f o r K.

-iI E ( k ) [ e x p ( - i k y ) / k ] d k

y(x,k)

=

[T(x,k)

-

[Sinkx/k] +

I.

Sinkx

Thus ( n o t e

f o r E even)

m

(2/n)

+

= W(kb(y,k).

L e t us express t h e k e r n e l K o f

(4.29)

'0

(y,k)i( k)/F(k).

N

t(x,y)[Sinky/k]dy; k2dk =

K(x,y) =

ikx]

= ( -m

Now f o r x+y > 0 an i n t e g r a l o f t h e f o r m

-

( i / r ) f [my ( x , k )

F]kebikYdk

,-ikydk

/I exp - i k ( x + y ) ) d k

can be thought

o f i n terms o f a l a r g e s e m i c i r c u l a r c o n t o u r i n t h e l o w e r h a l f p l a n e where Imk 5 0 and can be equated t o zero. @(x,k)exp(-ikx)/F(k) f o r Imk dk =

F u r t h e r one knows ( c f . L e m a 4.1) t h a t

(resp. @ ( x , - k ) e x p ( i k x ) / F ( - k ) ) Hence one can s e t

0 ( r e s p . Imk < 0 ).

/I [ @ ( x , - k ) e x p ( i k x ) / F ( - k ) ] e x p ( - i k ( x + y ) ) d k

c o n t o u r i n t e g r a t i o n w i t h Imk 5 0.

i s a n a l y t i c and bounded

,I [@(x,-k)exp(-iky)/F(-k)]

= 0 b y a s i m i l a r recourse t o

D e t a i l s f o r such arguments w i l l be q i v e n

l a t e r and we emphasize t h a t we a r e working i n a d i s t r i b u t i o n c o n t e x t .

Thus

t h e p r o p e r t r e a t m e n t o f such i n t e g r a l s r e q u i r e s t e s t f u n c t i o n s (and ParseVal formulas). K(x,y)

I

m

N

(4.30)

@*

T h e r e f o r e (4.29) becomes = (1/21~)

-m

[

-

eikx]e-ikydk

Again c o n t o u r i n t e g r a t i o n , now i n t h e h a l f p l a n e Imk

2 0, l e a d s t o an ab-

s t r a c t proof o f the t r i a n g u l a r i t y r(x,y) = 0 f o r x > y (thus

-/f e x p ( i k ( x - y ) )

dk = 0 f o r x > y and i n t h e same s p i r i t

LI [ @ ( x , k ) e x p ( - i k y ) / F ( k ) ] d k

f m [@(x,k)exp(-ikx)/F(k)]exp(ik(x-y))dk

= 0 f o r x > y).

-03

=

Such a b s t r a c t

p r o o f s o f t r i a n g u l a r i t y w i l l be e s p e c i a l l y u s e f u l l a t e r i n a general cont e x t o f s p e c i a l f u n c t i o n s where t r i a n g u l a r i t y r e s u l t s had o n l y p r e v i o u s l y been d e r i v e d by e x p l o i t i n g f o r example s p e c i a l p r o p e r t i e s and f o r m u l a s f o r hypergeometric f u n c t i o n s .

4-10, The k e r n e l ? o f

Summarizing we have can be w r i t t e n as (4.29) o r as (4.30) and

f r o m t h e l a t t e r form, u s i n g a n a l y t i c i t y p r o p e r t i e s o f

@

and F one can de-

e

duce immediately t h a t K(x,y) = 0 f o r x > y. N

I n o r d e r t o r e l a t e U and V v i a U Fadeev i n [ F a l l p u t s t o g e t h e r a f a s c i n a t i n g

p a t t e r n o f F o u r i e r a n a l y s i s and o p e r a t o r t h e o r y t o produce t h e SO c a l l e d

136

ROBERT CARROLL

Marc'enko (M) equation.

Thus f o r y > x

I.

m

(4.31)

V(X,Y)

= V0(x+y) +

V(x,t)Vo(y+t)dt;

-A

ik t d k where S(k) = F ( - k ) / F ( k ) appears i n (4.16).

i s t h e one dimensional s c a t t e r i n g m a t r i x and V(x,t)

T h i s i s t h e same r e s u l t as (4.19)

proof i s very d i f f e r e n t .

(Theorem 4.5) b u t t h e

Furthermore t h e i m p o r t a n t f o r m u l a

i s a l s o proved i n [ F a l l by these t r a n s m u t a t i o n methods.

We have extended

these procedures i n two stages ( c f . [C31,32,40,47-49,80])

t o a canonical

general v e r s i o n which i s presented i n Sections 55-6 and t h u s we w i l l o m i t t h e d e t a i l s h e r e f r o m [ F a l l l e a d i n g t o (4.31)-(4.32) i n which we p r e s e n t t h e

( c f . [C40]).

The f o r m

M e q u a t i o n l a t e r (Theorem 6.23 f o r example) i s a l s o

i n t r i n s i c i n t h e sense t h a t i t a r i s e s as a m i n i m i z i n g c r i t e r i o n ( c f . 57).

5.

&HE M A R E N K @ EQI.lA&Z@N UZA &RAW~TIUCA&Z~N. We go now t o t h e M equation,

a f o r m o f which was i n d i c a t e d i n 54 f o r t h e quantum s c a t t e r i n g s i t u a t i o n . A f i r s t g e n e r a l i z a t i o n o f t h e Fadeev procedure was developed by t h e a u t h o r i n [C31,32,40]

and a subsequent f u r t h e r e x t e n s i o n was g i v e n i n [C47-49,801.

The l a t t e r p r e s e n t a t i o n , a l t h o u g h more general, d i s p l a y s t h e m a t e r i a l much more i n t r i n s i c a l l y and c a n o n i c a l l y and i n f a c t i t i s t h i s v e r s i o n which a l s o a r i s e s as a m i n i m i z i n g c r i t e r i o n ( c f . 57).

and t h e n w i l l g i v e t h e general method i n

b r i e f l y t h e method o f [C31,32,40] detail.

T h e r e f o r e we f i r s t s k e t c h

B e f o r e d o i n g t h i s however i t w i l l be u s e f u l t o r e c a l l some t y p i c a l

p r o p e r t i e s of s p h e r i c a l f u n c t i o n s , e s t a b l i s h some r e s u l t s o f t r i a n g u l a r i t y f o r kernels, develop some techniques f o r m a n i p u l a t i n g s p e c t r a l i n t e g r a l s , and prove c e r t a i n c o n n e c t i o n formulas.

I n p a r t i c u l a r the operator

studied

e a r l i e r p r o v i d e s a f a s c i n a t i n g complement t o B i n terms o f mapping propert i e s f o r special functions.

We w i l l see t h a t ~ ( y , x ) w i l l g e n e r a l l y be

t r i a n g u l a r i n t h e sense t h a t a ( y , x ) t h i s z(y,x) = AP(x)Ai1(y)y(x,y) type f r a c t i o n a l integrals.

= 0 f o r x > y and as a complement t o

= 0 f o r y > x.

T h i s l e a d s t o R-L and Weyl

We r e c a l l now some formulas f o r k e r n e l s i n t h e

general form ( f r o m 52) (5.1)

B(Y,x)

= ( " xP( x ) y v : ( ~ ) ) v ;

;(Y,X)

= (QP x(x)yv:(~))u;

MARCENKO EQUATION

(Standard p r o p e r t i e s )

REmARK 5.1,

137

L e t us r e c a l l t h a t H i s t h e space o f even

e n t i r e r a p i d l y d e c r e a s i n g f u n c t i o n s o f e x p o n e n t i a l t y p e w h i l e 3T c o n s i s t s o f even e n t i r e f u n c t i o n s o f e x p o n e n t i a l t y p e and o f slow growth ( c f . Chapter 1, The general r e s u l t o f Paley-Wiener t y p e which we developed i n

§§9-10).

Chapter 1 i s t h a t

q

i s an isomorphism 27

e r t i e s o f and e s t i m a t e s f o r pi,

@pi-X,

+

H and E’

A n a l y t i c i t y prop-

3T.

-f

and c (+A) were a l s o d e s c r i b e d t h e r e

4-

and i n a l l cases p 4x ( x ) w i l l be an e n t i r e f u n c t i o n o f e x p o n e n t i a l t y p e w i t h an e s t i m a t e [ p4A ( x ) I 5 K(x)exp( ( I m h l x ( x 5 0, K E Co[O,m) Kexp(-Repx) e.g.

$

i n the basic s i t u a t i o n

= Qo n~

-

K has a bound

w i t h AB:

+

%

P ~ Q

Q,).

How-

e v e r t h e development o f [ T j l ] does n o t e x p l o i t t h e @A o r c (A) and t h e o n l y

Q

4

i n f o r m a t i o n recorded so f a r i n t h i s d i r e c t i o n a p p l i e s t o t h e b a s i c case

Q

Qo +

=

gR

Qa,

o f [Ffl;

p2

f

$

for c

r e g i o n n (e.g.

Kpl] o r t o t h e [Cg3] hypotheses.

> 0 and x

in

complex a , i ~n A

aB’

e x c l u d ng c e r t a i n p o l e s (a r e g i o n

i n AclB where X = 5 + i n ) and Q,.,(x) 4

A l s o I+,(x)l 9

exp(-Zx)@(A,x)].

=

5 c e.g. we expect a 4A ( x ) t o be a n a l y t i c i n a

= C/{-iN})

i s used f o r r e a l ,B

4

Thus f o r t h e case

For r e a l

rl

-\E,(E

= exp(ix-p)x[1

+

I m h L 0 even f o r n 2 - / E , l c , I A c Q ( - h ) l 5 K ( l + l A l ) 1-?JP+9 1

5 Kexp -x(ImA + Rep)] f o r U,B

and

and IcQ(-A)1-’ -i[;,m)

< K(l+IAI)%(p+q) ( w i t h Xc ( - A ) a n a l y t i c i n n having zeros i n Q r e c a l l 2a+l = p+q and 2 ~ + 1= 9 ) . F o r complex n,B a s i m i l a r t y p e

-

o f estimate f o r c-’(-x)

holds i f one s t a y s away f r o m poles.

4

For more gen-

era1 a 9A ( x ) we can r e f e r t o [Cg3] however ( c f . Lemma 1.10.11);

f o r now we

exclude s i n g u l a r i t i e s i n t h e p o t e n t i a l which do n o t l e a d t o s p h e r i c a l funct i o n s o l u t i o n s . Thus p 4( x , ~ ) i n (1.10.23) w i l l be a s p h e r i c a l f u n c t i o n ( w i t h a bound Ip Q ( x , h ) I 5 kexp( I n l - p ) x f o r x 2 0 and I h l 2 N - T = 0, B

Q( x , ~ ) p l a y t h e r o l e p r o p e r t i e s f o r cb4( x , ~ ) and

The f u n c t i o n s a t e s and

@

o f @Jx) 4

i n t h i s case and one has e s t i m -

c ( A ) d e l i m i t e d i n L e m a 1.10.11.

4

g i v e n hypotheses H2 w i t h B~ = 0, f o r x 2 xo > 0 and I m x > -s,/2 A

4 Q (x)aP,(x)

a n a l y t i c i n h and as ( X I +m,aA(x) 4

;aq’hx)exp(iAx).

t h e s i s H2 holds w i t h B~ # 0 o r H1 h o l d s (and n1 + B~ # f o r ImA

0; as

1x1

-f

m

w i t h Imh

+ o(l)].

A:(x)exp(ihx)[l

Q x4 4A )

4 has @.,(x) =

5, > 0 and x ? x 0 > 0 one has a 4 A(x)

Q 4

%

rlAlyf o r

We r e c a l l a l s o t h a t c ( - x ) 2 i x = -A,(x)

s o hc (-A) i s a n a l y t i c where

Q

I f hypo-

F i n a l l y f o r ImA 5 0 one has I c ( A 1 I - l 5

[ A \ 2 N under h y p o t h e s i s H1 o r H2. W(p ,Q,

m+k) one

Thus

one has

where v i s holomorphic i n A f o r ImA > 0 and continuous

(-iA)YA-”I(x)v(x,A)

4

4 0).

Q (x,A)

@

i s analytic.

I n particular

i s a n a l y t i c f o r I m A > 0 except f o r a f i n i t e number o f poles (x,A)/c (-A) 4 where c (-A) = 0 (A = h = i y j ) . 4 j W i t h t h i s k i n d o f background i n f o r m a t i o n f r e s h i r , mind now l e t us go t o some

@

138

ROBERT CARROLL Q and w r i t i n g r ( x , y ) = Consider f o r example q PA = EipA

t r i a n g u l a r i t y theorems. y(x,y)/ag(y)

we express t h i s as

P (x) = qA

(5.2)

(

Y ( X ~ Y ) ~Q~ , ( Y ) )= Q.Y(x,.)

= w(x,-)

P P We know q A ( x ) i s e n t i r e i n A f o r x > 0 w i t h Iq ( x ) I < K(x)exp( l ~ l x )and A K( - ) continuous w i t h say I K ( x ) ( 5 r a s s u m e d here ( t h i s h o l d s under t h e hypotheses o f [Cg3] f o r example). IK(x) I (Kx

A c u t a l l y f o r any f i n i t e x we can say P Thus q A ( x ) i s o f ex-

so no a d d i t i o n a l hypotheses a r e necessary.

p o n e n t i a l t y p e x i n A ( o f s l o w growth) and consequently v i a q i t comes from a distribution r(x,-)

E ' w i t h supp r ( x , - ) c [O,x].

E

T h i s i s b a s i c Paley-

Wiener i n f o r m a t i o n f o l l o w i n g Chapter 1, 549-10 and [ F f l ; y(x,

0 )

Kpl; T j l ] .

Since

may be i n f a c t a f u n c t i o n o r a d i s t r i b u t i o n we w i l l have t o have a

convention here and t h u s we w i l l r e f e r t o y ( x , - ) as a d i s t r i b u t i o n i n E ' . I f i n f a c t y(x,y)

i s a f u n c t i o n t h e n t o say ~ ( x , . ) E E ' w i l l mean r ( x , * )

as a f u n c t i o n i s a d i s t r i b u t i o n (under t h e map r ( x , - ) a f u n c t i o n f determines a d i s t r i b u t i o n by t h e r u l e f o r A = Ap o r A

9

as i s a p p r o p r i a t e

CHE0REm 5-2- y(x, -)

E

El

-

Q

pp,(Y) =

(

w i t h supp y ( x , - ) c [O,x]

P B(Y,x),vA(x))

-f

r(x,=)A (-)

10"

Q

(i.e.

-

i.e.

f(x)p(x)A(x)dx Thus

c f . Theorem 1.10.13).

Now c o n s i d e r ~ ( y , x ) i n t h e same s p i r i t . (5.3)

q

-f

y(x,y)

= 0 f o r y > x).

One has

= PB(Y,*)

= P[A(Y,.)l

where A(y,x) = ~ ( y , x ) / A ~ ( x ) . E x a c t l y t h e same r e a s o n i n g as f o r Theorem 5.2 again i s a p p l i c a b l e ( w i t h f ' i d e n t i f i c a t i o n o f f u n c t i o n s i n v o l v i n g A p ) ; thus

CHEaREM 5-3- ~ ( y , . )

E

E ' w i t h supp B(Y,-) c [O,y]

(~(y,x) = 0 f o r x > y).

Now combine these r e s u l t s w i t h t h e formulas (5.1) ( t h u s r e c a l l i n p a r t i c u l a r i;iy,x)

= Ap(x)A;

(Y)Y(x,Y) and ~ ( X , Y ) = A';

COR0tLARij 5-4, Apl(x)A A,'(y)Y(.,y)

E

R€ARK 5.5, When R l c P l 2 and (5.4)

9

(-)r(.,x)

E ' w i t h ';(x,y) P Q

E

El

(x)AQ(Y)6(Y,x))

w i t h T(y,x)

= 0 f o r y > x and A,(-)

= 0 f o r x > y.

dvp = Gp(A)dA and RQ

Q

duQ = cQ(?,)dh

;Q=

1/2nlcQ12 we can w r i t e P B(Y,x) = 7 Ap(x) @$yA(Y)dA; 9

jrn

t o Obtain

Apb) B(Y,X)

N

-m

=

with

I

m

GP

=

1/2~

@.,(Y) Q p ~ ) q ~ ( x ) d A

-m

These formulas i l l u s t r a t e n i c e l y t h e r o l e r e v e r s a l between x and y i n B and A,

6 and w i l l be examined l a t e r i n more d e t a i l .

MARCENKO EQUATION

139

The n e x t k i n d o f formula we want t o examine i n v o l v e s a g e n e r a l i z a t i o n o f t h e r e l a t i o n G[exp(ikx)] = @(y,k)/F(k) o f (4.32). (4.32) w i l l f o l l o w as a s p e c i a l case. on [C40,64,65] take P =

We g i v e s e v e r a l v e r s i o n s and

The f i r s t two techniques a r e based

and t h e n a new p r o o f based on [C47,80]

2 D (and B: P

n

Q i s t h e n denoted by B ).

+

4

i s given l a t e r .

First

Then u s i n g a t e c h n i q u e

modeled on c o n t o u r i n t e g r a t i o n as i n 64 we w i l l p r o v e t h a t

,-.,

(5.5)

BQ[exp(i”x)/%l

=

0 @A(Y)/cQ(-’)

Then u s i n g a d i f f e r e n t t e c h n i q u e o f p r o o f we w i l l demonstrate a more generP P a1 f o r m u l a (u,(x) = *)I,h(x)/cp(-~)) “ P “X(-)l(Y)

(5.6)

= +Y)

F i n a l l y a new p r o o f o f ( 5 . 6 ) i s g i v e n i n 86. A

REIllARK 5-6, L e t us p o i n t o u t t h a t ( 5 . 6 ) was e s t a b l i s h e d i n [Kpl]

f o r P and

A

Q o f t h e f o r m PaB

%

w i t h no p o t e n t i a l , u s i n g known formulas f o r hyper-

AaB,

Indeed r e f e r r i n g t o Chapter 1, 559-10

geometric f u n c t i o n s (as i n [Ak3]). we r e c a l l ( c f . (1.9.37)-(1.9.38))

I

Y

(5.7)

qy+’”+’(y)

R e c a l l here t h a t

=

Ta8 =

2 J n c a 8 / r ( a + l ) so from (1.9.38)

A

Here we t h i n k o f P

80(y,x)v:B(x)dx;

A

A

and Q

and one has ~ ~ ( y , x= ) A

(x) aB . , (y)yo(x,y)i Then we want t o i d e n t i f y 6, w i t h B and To w i t h y where a+u, B+!J P To do t h i s s i m p l y compare t h e 6 = k e r B, €3: + Q w i t h &PA = q Q , e t c . P Q f i r s t e q u a t i o n i n (5.7) ( i . e . q + , ~ + v ( Y ) = P I B o ( Y A l ( h ) ) w i t h &Pi = By uniquew r i t t e n i n t h e f o r m ,p~+’lYB+’(y) = ( ~ ( y , x ) , q : ~ ( x ) ) = P[B(Y, * ) ] ( A ) . %

aB

%

Aa+u,8+!J

A-1

ness i n t h e P - P o r ?? - P t r a n s f o r m t h e o r y one has 6 = B, Taking

~1

=

6

=

-+

A

-

and hence =

i n (5.7) ( i . e . P % D2 ’ L A -$,-$’ c-+,-+ s i n c e ??’ = pQ w i t h k e r n e l yo = 7.

Q

yo = ?.

(1/2)) we o b t a i n

2 L e t us go now t o an a b s t r a c t p r o o f o f (5.5) when R 4 % dwQ = d h / 2 n l c o ( X ) I . A 2 P Here P = D w i t h ,ph(x) = C O S X X , A, = 1, e t c . so u s i n g yQ(x,y) = AQ(y)BQ(y,x) and (5.4) we can w r i t e

140

ROBERT CARROLL

A). We w i l l show i n Lemma 5.9 below

even i n

(5.10)

I

E =

['P:(y)/cQ(-x)]eihxdi

0

=

-02

so t h a t (5.9) becomes m

(5*11)

YQ(xYY) = [AQ(Y)/4711

[ ' 4 ~ ( Y ) / c Q ( ~ ) l e iXxdh -m

Lemna 5.9 ( o r Theorem 5.2) y

shows t h a t y (x,y)

2 0 i n o u r arguments) and changing

4

-x

h to

=

0 f o r y > x (note x

0 and

which i s c l e a r l y

i n (5.11),

p e r m i t t e d , we o b t a i n by F o u r i e r i n v e r s i o n (9: = @:/cQ(-h))

I

m

Yq(x,y)eiAxdx = 2 Bg[e * i x xI ( Y ) Y * One knows f u r t h e r ( c f . Theorem 1.12.3) t h a t 8 = qP and 'ij = IQP ( c f . a l s o A Q ( Y ) *Q~ ( Y ) = 2

(5.12)

Theorem 3.1). (E'(y,x),f(x))

4

Q

I n f a c t l e t us n o t e t h a t i n general ( c f . ( 3 . 4 ) ) Ef(y) = = ((RA(x),qA(y))u,f(x)) P Q = ~ ~Q P X ( yP ) , ~ ~ x ( ~ ) , fS(i m ~ i)l a~r ~ W .

c a l c u l a t i o n s h o l d f o r 7 3 ( c f . (3.2))

and we mention i n p a s s i n g (as an ad-

j u n c t t o Theorem 3.1)

LEiiUW 5.7.

= A

=WE' and % = PQ.

R P and RQ one has

F o r general

has B*[Apf]

cf and B*[A Qf] Q

F u r t h e r one

= A$f.

*

Q L e t us w r i t e o u t t h e a c t i o n as B f ( y ) = ( y ( x , y ) , f ( x ) ) = (n,(y), P A/ 4 P (qh(x),f(x)))w; Bg(Y) = ( g ( y , x ) , q ( x l ) = ( I p x ( y ) y ( ~ h A ( x ) , g ( x ) ) ) oHence . one has B*[Apf] = A S i m i l a r l y B*[AQf](x) = ( ~ ( y , x ) , A f ( y ) ) = (Qh(x), P Prraal;:

rf.

.

Q

( q Qh (Y) YAQ(Y)f(y) )),,,

Ap(x)gf(x).

= 'p(X

In t h e p r e s e n t s i t u a t i o n

)( q Ph ( x )

¶(

Q

A ' (!Q)

Ap = 1 and

P

,f(y)

=

(5.12) one can w r i t e A (y)['PX(y)/cQ(-A)] Q (5.5).

4

) )u

= Ap(x)(y(x,y) ,f(y)

*

P so we have BQ

=

N

= AQB.

= 2A (y)F[exp(iAx)](y)

4

)

Hence i n and t h i s i s

Thus, modulo Lemma 5.9 t o f o l l o w , we have proved

EHE0REm 5-8. The e q u a t i o n (5.5) i s v a l i d when RQ

%

d h / 2 n ] c Q ( h ) 1 * under t h e

hypotheses o f Lemma 5.9 below. LEllUllA 5-9- Assume s t a n d a r d hypotheses f o r y L C > 0 and Imh 0 o f t h e form and I@A(y) Q I 5 cexp(-yImh) w i t h ' ~Q ~ ( y ) / c ~ ( - ahn)a l y t i c IcO(-h) 1-l5 k ( l + l h l f o r Imh > 0.

Then X = 0 i n (5.10)

f o r x 5 0 and y > 0, and one can show

141

MARCENKO EQUATION

d i r e c t l y t h a t y ( x , y ) = 0 f o r y > x when yQ i s g i v e n by (5.11).

Q

Pmad:

We t a k e

x

= s+in,

0, and y > c > 0.

r~

The i n t e g r a n d I ( h , y ) = a QA(y)/

i n arguments below i n s t e a d o f x 5 0, y > 0. c (-A)

q xr) ( y ) i n (5.10) i s bounded by a polynomial i n 111 f o r A r e a l so we

=

Q

are i n t h e context o f Fourier transforms i n S ' . for

Imx

Then one c a l a l l o w x > -4c

F u r t h e r I(X,y)

i s analytic

5 p ( l x l ) e x p ( - n y ) (p a p o l y n o m i a l ) . To see i n t u i I 5 exp(-qx) and approximate

> 0 with lI(A,y)l

t i v e l y t h a t Z = 0 use t h e f a c t t h a t l e x p ( i x x )

a l a r g e s e m i c i r c u l a r c o n t o u r i n t h e upper h a l f p l a n e by a sequence o f cont o u r s w i t h base l i n e s

Q =

€ 1 ~ 1 so

More p r e c i s e l y s e t Z ( x , y ) = F I ( A , y ) a n y t h i n g about Z(x,y)

for x

=

Lr

I(A,y)$(A)dA

( t r e a t y as a parameter).

makes sense f o r r e a l A s i n c e

D w i t h supp v

For 9

E

1x1

i,

=

€

$

C (-%c,R)

E

S and I ( . , y )

S,

From t h e Parse-

= Fp € S, CZ(x,y),

The i n t e g r a l on t h e r i g h t

has o n l y polynomial qrowth.

we have f o r II L 0 on a S e m i c i r c u l a r c o n t o u r

L$exp(+crI)(l+lA\)-N f o r N arbitrary.

1$(A)1

$

0.

Hence I I ( A , y ) $ ( X ) l

5

p ( I A l ) e x p [ - n ( y - 4 c ) l ( l + l h l )-N where y L c and t h e corresponding c o n t o u r i n t e g r a l vanishes. consequently ( Z ( x , y ) , q ( x ) ) = 0 and E ( * , y ) = 0 i n D l ( - + c ,

-) which means i n p a r t i c u l a r Z ( x , y ) = 0 f o r x 5 0 and y 5 c ( c b e i n g a r b i trary).

T h e r e f o r e Z ( x , y ) = 0 f o r x L 0 and y > 0.

F i n a l l y t o show t h a t

t h e f o r m u l a (5.11) i m p l i e s t r i a n g u l a r i t y use a c o n t o u r i n t e g r a l argument i n t h e l o w e r h a l f p l a n e where l a Q- A ( y ) l 5 cexp(ny) (rl 5 0) and l e x p ( i x x ) l 2 ( A ) w i l l be bounded by $(lXl) The i n t e g r a n d J(A,y) = @-,(y)/c Q exp(-nx). exp[n(y-x)] =

with

$

4

.

a polynomial and an argument as above w i l l y i e l d y ( x , y )

0 f o r y > x ( w h i c h o f course we a l r e a d y know from Theorem 5.2.

Q

We now develop an a b s t r a c t procedure f o r p r o v i n g (5.6) ( t h e above t e c h n i q u e 2 f o r (5.5) does n o t e x t e n d d i r e c t l y ) . We assume a g a i n Rp % dA/2alcp(X)I and RQ 1 ' , d x / 2 a l c Q ( x ) (2 . Since = ! x = W ( X ) q Xr) now ( c f . Theorem 3.7 -

&!

W(x)dvp = dwQ w i t h W(X) = I c p ( x ) / c

we have

4 2 P ~ ( x , y k ~ ( y ) d =y I c Q ( A ) / c p ( A ) l P P , ( ~ )

(5.13) (note

2

Q (A)])

&!

ru

= Mp!

andBq!

=

&!

where

$(A)=

W-'(A)).

Hence f o r X r e a l

142

ROBERT CARROLL

For t h e d i s c u s s i o n t o f o l l o w we t r e a t y ( x , y )

as a f u n c t i o n n o t a t i o n a l l y ( 6

f u n c t i o n components can a l s o be so w r i t t e n i n o u r standard manner); i n t h e event t h a t ?(x,y) i s a d i s t r i b u t i o n o f h i g h o r d e r we know t h a t ?(x,y) w i l l be a c o r r e s p o n d i n g l y smooth f u n c t i o n and one c o u l d work w i t h %! = WP, Q i n P stead o f = %!. Under s t a n d a r d hypotheses as i n Lemma 5.9, *,(x) and

&:

$(x,A)

a r e a n a l y t i c f o r I m h > 0 and p o l y n o m i a l l y bounded t h e r e ( u n i f o r m l y P i s bounded by p ( l X l ) e x p ( - x I m A ) b u t i n c > 0 ) . Note t h a t *,(XI

for x

$(x,A) we o n l y have t h e polynomial bound on I c ( - A ) I - '

Q

a t our disposal a f t e r

i n t e g r a t i o n . We assume t h e i n t e g r a l (5.15) converges s u i t a b l y (hypotheses P t o f o l l o w ) and w r i t e now J/(x,X) = J/+ and q X ( x ) = w i t h J/- = $(x,-A) and

*+

*-.

*+

Then (5.14) can be w r i t t e n as @+ = J/+ = - ( J / - - *-) = -0= *!,(x) f o r 1 r e a l , and t h i s i s r e m i n i s c e n t o f t h e Riemann problem f o r s e c t i o n a l l y Thus we have

holomorphic f u n c t i o n s ( c f . [Gal; Mpl]). > 0 and

and

a n a l y t i c f o r Imh

= -0- f o r

@+

means

@-

@+

@+

a n a l y t i c f o r ImA

0 ( w i t h polynomial bounds i n b o t h h a l f planes)

0).

Consequently by a v e r s i o n o f L i o u v i l l e ' s theorem

i n A o f f i x e d degree f o r a l l x L c.

polynomial p(x,X) @

+

= 0

-

=

-@

+

so Re@+ = Re p(x,A) = 0 b u t we d o n ' t need t o use t h i s .

us s t a t e now ( n o t e A-'/yy) = exp(-pqy) i n t h e s i t u a t i o n o f [ F f l ;

Q

e.g.

@+

i n (5.8) ?(x,y)exp(-pY)

Q,

is a

(Further f o r

real

.

Kpl] and

exp[-(a+B+l)y]).

IIHEBREI 5.10, Assume hypotheses as i n Lemma 5.9 w i t h t h e bound on @,(y) a,P o r @A) Q expressed f o r y 2 c > 0 and I m A > 0 as l@,(y)I 5 '&-'(y)exp Q,

c

(-yImA) and suppose

Pmab:

Take A =

irl

5

(?(x,y)lb:(y)dy

I t remains t o prove t h a t

plying 8).

Let

@+

c^.

(@,

Then (5.6) i s v a l i d . = 0 which i s (5.6)

= p(x,A)

(upon ap-

f o r example and w r i t e

-4

. The exp(-nx) 5 cap ( x ) e x p ( - r l x ) / l c p ( - A ) l 5 Fexp(-nx)/lc,(-h) terms i n these e s t i m a t e s w i l l dominate t h e polynomial bounds on l c p ( - 1 ) ~ - ' c and on I c Q ( - X ) I - ' so b o t h and w i l l be bounded by ?exp(-Enx) f o r x

w h i l e I*!(x)

I

N

*+

> 0.

Hence Ip(x,A)l

*+

(?exp(-Erlx)

-

f o r A = in.

I f we w r i t e p(x,A)

cn(x)Xn ( w i t h cn r e a l by an e a r l i e r remark) t h e n l c ( x , n ) 1 =

n n 1 5 ?exp(-Enx)

+

0 as n

-+

f o r each x.

c n ( x ) must be i d e n t i c a l l y z e r o f o r each n.

=

'1,N

ll," i c n ( x ) i n

It f o l l o w s t h a t t h e c o e f f i c i e n t s

=

MARCENKO EQUATION

143

Connection formulas between s p e c i a l f u n c t i o n s r e a t e d t o these r e s u l t s app e a r from time t o time i n t h e t e x t . We turn now t o t h e M e q u a t i o n and s k e t c h f i r s t the approach of [C31,32,40]. T h u s c f . (4.16) and ( 4 . 1 8 ) ) one s e e k s an analogue f o r V-l in t h e form

Q

We c o n t i n u e t o write i n t e g r a l s f o r F o u r i e r t r a n s f o r m s even when d e a l i n g w i t h BC! has kernel d i s t r i b u t i o n s p a i r i n g s . Next BQ = N-1 given by ( E P, ( x ) , l 0p ~ ( y ) ) ~ P Q ( w i t h Ap(x) = 1 , q x ( x ) = Cosxx, e t c . ) and ( y , x ) = 0 f o r y > x. Hence lY

4

( c f . ( 5 . 5 ) ) . Now assume ( c f . Remark 5.1) @,(y) Q i s a n a l y t i c i n A f o r say -4 Imh > 0 and I@!(y)I 5 CA (y)exp(-yImX) f o r Imx 2 0 and y 2 c > 0. Then i n ,4 -b (5.18) the i n t e g r a l f o r V ( y , x ) has i n t e g r a n d bounded by cAQ’(y)exp[-n(y-x)] N

N

Q

( n = Imx) and r e f e r r i n g t o a contour i n t e g r a l i n the h a l f p l a n e Imx 2 0 we A Q o b t a i n V ( y , x ) = 0 f o r y > x. Hence i n (5.18) we have @,(y) = Ir ?n (y,x) Q Y Q e x p ( i h x ) d x . Now using t h i s with ( 5 . 1 9 ) one o b t a i n s

I f we can w r i t e now

1

m

(5.21)

(1/2)/c

Q (-A)=

Fq = Q

qQ(c)eihEdg

-m

t h e n from (5.20) we w i l l have ‘u

(5-22)

eg(Y3x) =

A

* v ~ ( Y , * ) l ( x )=

I n t h i s connection l e t us r e c o r d

c

A

qQ(x-E)Vg(y, 0 on a semic i r c l e 1x1 = $(A)/ 5 cexp(-brt), and Ic-'(-h)$(h)l 5 p ( l h l ) e x p ( - s n ) . Q Consequently f o r such ~p t h e x i n t e g r a l in (5.24) vanishes so ( \ k ( x ) , q ( x ) )

Rh,

Q

= 0 and hence t h e d i s t r i b u t i o n rI, (x) has support in [O,-).

Q

Using now Lema 5.11 we can w r i t e (5.22) i n t h e form

( t h e i n t e g r a l i s formal of course) and t h i s y i e l d s again >

x.

We summarize i n

LEl!UW 5-12, The kernels

G Q and FQ a r e r e l a t e d

4 (y,x)

= 0 for y

by (5.25).

Now define an operator

I

a,

(5.26)

EQf(5) =

qQ(x-c)f(x)dx

5

Then, W r i t i n g out the

action from (5.25) we have

Q

I, m

(5.27)

(rQ(y,x),f(x))

C AVQ ( Y I S )

=

[Q

X

A

f ( x ) L*Q(x- 0 we have (+)

i:

=

= (]/ZIT) @A(y)p(A)

Q

0 by a n a l y t i c i t y i n t h e upper h a l f p l a n e and a c o n t o u r i n t e g r a l

argument (as i n ( = ) above f o r

z)

so t h a t g(y,x)

T h i s f o r m u l a (6.12) r e p r e s e n t s g(y,x)

can be i d e n t i f i e d w i t h

= 0 f o r y - x > 0 and thus i s t h e n a t u r -

a l form t o use on t h e f u l l x a x i s i n o u r t h e o r y . Moreover t h e c a l c u l a t i o n g(y,x),@,(x) P ) = (1/2n)jm m Qh(y)p[: Q BAfx)@p(x)dxdh P P = G4W ( y ) f o l l o w s immedia-

(

t e l y from (6.8) i n t h e f u l l l i n e sense (whereas b e f o r e one used a K-L h a l f l i n e formula p l u s a s t i p u l a t i o n F(y,x) = o f o r x 5 0).

&%0RZm

!:

@'

A

6.7, -f

Thus

For f u l l l i n e a c t i o n i n x w i t h F o u r i e r t y p e

aQ and A

i(y,x)

=

ker

(x

-+

*P

we w r i t e

i:

+

y ) i s g i v e n by (6.12).

RZmARK 6-8, I t w i l l be necessary below i n c o n s t r u c t i n g

b-'

(y x, y 0, t o l o o k a t formulas o f t h e f o r m (6.11) and t h e corresponding K-L h a l f l i n e i n v e r s i o n . Thus c o n s i d e r ( ~ ( Y , X ) , @ P ~ ( X )=) @.,(y) Q when i s P F o r ( ~ ( y , x ) , @ ( x ) ) we must beware o f w r i t i n g t h i s as g i v e n by (6.11).

-m

< x
y. =

There i s some i l l u s t r a t i v e m a t e r i a l i n [C40] c o n c e r n i n g t h e

REmARK 6.11.

s p l i t t i n g o f g X i n t o r e a l and i m a g i n a r y p a r t s ( c f . (1.8.29) form).

Thus

( generally

i s a n a l y t i c i n t h e upper h a l f p l a n e I m x > 0 and

i s s u i t a b l y bounded t h e r e (e.g.

The d i s t r i b u t i o n a l

as i n Remark 5.1).

H i l b e r t t r a n s f o r m t h e o r y o f e.g. one o b t a i n s e.g.

f o r the correct

[Od1,2]

t h e n a p p l i e s ( c f . a l s o [C40])

and

I T $ ~ ( X ) PV , ( X )= -H[2px P ( x ) + Z U ( A M / M ; ) P~ ~ ( X ) ] ( X ) w i t h H2 = -I. !J

Now we r e f e r t o Theorem 1.8.13 and r e c a l l

1

m

(6.14)

F ( A ) = @ ( f )=

P f ( x ) @ h ( x ) d x ; f ( x ) = (1/21~)

P

and we c o n t i n u e t o w r i t e u2 = gp.

also d e f i n e ( r e c a l l cp m P

=

M,),*(f)

(1/2r)Lm *(f)[z,(x)/4ixMld~

m

F(X)p(h)z!(x)dh

m

m

where u1 =

1

=

( 1 / 2 ~ ) L : *(f)[z;(x)/M;]p

=

I n t h i s c o n n e c t i o n we c o u l d

lI f ( x ) q APA( x ) d x

= @(f)/M1 w i t h f ( x ) =

IM1 I2dX b u t g e n e r a l l y

we p r e f e r t o work w i t h (6.14).

DEfZNZCI0)N 6-12, As a k i n d o f g e n e r a l i z e d t r a n s l a t i o n s e t now

I

m

(6.15)

C i f ( x ) = (1/21~)

*(f)@!(y)z!(x)p(A)dh

m

154

ROBERT CARROLL

and f o r a g e n e r a l i z e d c o n v o l u t i o n we s e t

*

I t i s immediate t h a t ( n o t e c c f ( x ) f E X f ( y ) however) @(f 9) = @ ( f ) @ ( g ) Y s i n c e i n d e a l i n g w i t h (6.14) as an i n v e r s i o n one has f o r m a l l y (6.8), i . e . P P (1/2a)l: @,,(x)xx( x ) d x = 6 ( X - P ) / P ( A ) .

Some c a l c u l a t i o n s based on (6.15)-(6.16)

and t h e i n t e r a c t i o n w i t h a~ and rk

t r a n s f o r m s w i l l o c c u r below a l t h o u g h we w i l l n o t always s p e l l o u t t h e notation via

*

Thus we w i l l o m i t becoming e n t a n g l e d i n a n o t a t i o n a l

and E;.

maze here b u t remark t h a t t h e n o t a t i o n i n d i c a t e d perhaps should be event u a l l y w r i t t e n o u t and used s y s t e m a t i c a l l y .

REmARK 6.13,

L e t us r e c a l l h e r e t h a t g e n e r a l i z e d t r a n s l a t i o n can be u s e f u l l y

expressed v i a k e r n e l a c t i o n . ( x ) = (T"(y,x,s),f(s))

I n keeping w i t h t h a t s p i r i t we can w r i t e

el[f

where f o r m a l l y m

(6.171

r(y,x,s)

= (1/2~)

@~(s)@~(~)~~(x)~(l)dh

m

M

We w i l l now develop a g e n e r a l

e q u a t i o n f o l l o w i n g t h e g u i d e l i n e s o f 55 b u t A

i n a more i n t r i n s i c manner.

6

L e t P be a F o u r i e r t y p e o p e r a t o r w i t h Q o f

standard t y p e and from Theorem 6.1 one has

I, m

(6.18)

(Z(Y,X),((X)) =

k,x) y.

FOURIER TYPE OPERATORS

LEMMA 6-18. The k e r n e l o f W* i s $(y,c) ( 5 =

+

157

y ) g i v e n by (6.28) and i ( y , c )

0 f o r 5 > y.

(6.29) where

;(X,Y)

=

v

B' t o

&HEOREM 6-19. An i n v e r s e

B can be produced v i a t h e k e r n e l

(1 /2.ir)

= A p Q mi n p a r t i c u l a r Q A'

=

@!. 4

m

-P

Pxood: We r e f e r t o Remark 6.8 and observe t h a t (;(x,y),* (y)) = j qA(x) = izx(x)'?(1,u)dA = $'(XI where k* A P ( x ) = GTA ( x ) / Q Q ( 1 / 2 r ) f t .S21(y)*u(y)dydA u There i s no a p r i o r i f u l l l i n e c (-1) and ?((A,p) = ( 1 / 2 r ) f r C?,,(y)\Iu(y)dy. Q 4

t h e o r y f o r Q so t h e formal K-L f o r m u l a ? ( h , u ) = s ( 1 - p ) i s needed and by Re-P mark 6.8 t h i s i s c o r r e c t f o r a c t i o n on q A ( x ) . Thus, u s i n g t h e f u l l l i n e form o f

F((x,u)

g i v e n by (6.12) and t h e

i ( y , t ) d y = ( 1/ 2 r

T(x,y)

O!)m

)iz p (LJ

/I p(u)ZL(t)@'(x)du

a c t i o n j u s t described, : f

( X ) ( 1/ 2 a $(Y)'$(Y )dydxdu = (1 / 2 r t )c$," S i m i l a r l y iz g(y,x)T(x,t)dx = ( 1 / 2 v ) j m dx

= 6(x-t).

Ym

4

7

A ( x ) d A ( l /2a)iI @L(x)[a:( )/cQ(-LJ ) I d V = ( 1 / 2 V ) j I @A (Y) [ Q ~( t ) / - m @T(Y ) P ( x ) z P Q Q c$dA = (1/21r)L: QA(y)ClA(t)d1 = s(y-t). We n o t e a l s o by c o n t o u r i n t e g r a t i o n t h a t ;(x.y)

= 0 f o r x > y as d e s i r e d .

.

Now t h e k e r n e l o f t h e M e q u a t i o n w i l l i n v o l v e t h e f o l l o w i n g terms which we d i s p l a y here f o r reference (s (6.30)

S(t,x)

=

(1/2~)

Q

= c /c-)

Q Q

P P SQ(A)@l(t)@A(X)dA;

P P A(A)@l(t)@A(x)dA

=

J(t,x)

+

= 6(t-x)

6 ( x - t ) + M(t,x)

m

( n o t e by Remark 6.15 t h a t J ( t , x )

=

m P P ( l / 2 r ) f -m @ A ( t ) @ - A ( x ) d x ) . Next one can N

*

e x p l i c i t l y w r i t e o u t t h e upper l o w e r f a c t o r i z a t i o n k-': o f rrwJc. v-14 EHEOREI 6.20- The o p e r a t o r B B has k e r n e l S + J f r o m (6.30).

Pxood: We w r i t e r ( t , x )

=

($(t,c),;(c,x))

ft/f 9,P(t)[s2!(c)/cq]dx(l/Zr)~f

o u t t h e r(A,u) = (1/2r)(s2!(c),pQ(c)) lJ

Remark 6.8).

fi

action t o

-m

2 9

Q

r(t,X)

2

= (c0I [ ~ ( A - L J )

m

as b e f o r e ( c f .

~ Q( E , ) QV(E,)de = ( 1 / 2 1 ~ )

+ 6(A+u)]

u

(another approach

= (1/271) j m ~ ~ ( t ) ( l / c g ) [ c Q @ ! ( x )+ c$rA(x)]dA m

= (1/2r)

Now one must spread

< A,u
w ( ~ ( y , x ) = 0 f o r x < y and One w r i t e s E(T,K) (7.9) where ;(T)

has a 6 ( x - y ) t e r m a l o n g t h e d i a g o n a l ) .

e x a c t l y as i n (7.3) and by t h e same arguments a r r i v e s a t makes sense as b e f o r e .

The unique m i n i m i z i n g k e r n e l KO

s a t i s f i e s t h e G-L e q u a t i o n o f Theorem 7.3 ( t h e e q u a t i o n has t h e same appearance) and thus Theorem 7.1 h o l d s f o r t h i s s i t u a t i o n . We go n e x t t o t h e o p e r a t o r s Qu = ( A u ' ) ' / A where 0
y a " r e v e r s e " G-L e q u a t i o n (7.25)

+ ?o(x,y)A-l(y)

A'(x)$x,y)

+

1'

!?o(x,t)6(t,y)dt

=

0

0

One s h o u l d check t h i s w i t h (7.21) w r i t t e n o u t as ( r e c a l l F ( t , x ) = A - ' ( t ) y(x,y)

= A-'(t)G(x-t)

+ A - l ( t ) L ( x , t ) ) ; t h u s ( z ( t , x ) , 6 ( t - y ) + A(t)s((t,y)) + L ( X , ~ ) A -(y) ~ + ( L ( x , t ) , E ( t , y ) ) = B(Y,X) = o

=

A + ( Y ) ~ ( ~ - Y )+ ~'(x)E(x,y) f o r x > y.

Consequently we have f o r m a l l y proved ( n o t e t h a t uniqueness f o r

t h e r e v e r s e G-L e q u a t i o n h o l d s s i n c e L o r e q u i v a l e n t l y y i s k e r B-').

EHEoREm 7-9, The s o l u t i o n

z0 o f t h e m i n i m i z a t i o n problem f o r " i s (uniqueZ

l y ) determined as t h e s o l u t i o n L of t h e r e v e r s e hr

G-L e q u a t i o n (7.25).

CJ

We c o n s i d e r n e x t t h e t r a n s m u t a t i o n B where Ba = Ws ( W ( 1 ) = ;/$ = I c / c I P Q 2 = ( 1 / 4 ) / l c Q l 1, Z(y,x) = A - ' ( ~ ) ~ ( X - Y ) + 't?(y,x), e t c . F o r some s u i t a b l e

2

class o f anticausal kernels we t r y t o m i n i m i z e T m m 2 (7.26) E(T,g) = [A-+a-Ws + E(y,x)a(x,x)dx] dudy '0 '0 'Y 2 = ( ~ / I T ) ~ cdx~ \a r i s e s i n t h e t h e o r y o f t h e M e q u a t i o n where du = (;'/c)dx

[

[

N

( c f . 555-6).

= BA i n t h e form B = 'iiA"i n s t e a d o f 2 Thus ( f o r A,, = 1 when P = D ) one has W = A w i t h A(x,y) =

(a(x,x),a(x,y)

)w

when u s i n g t h e general G-L eq:ation

N

while

= A-l

has k e r n e l A(x,y)

= ( a(h,x),a(x,y))

v

t h e r f r o m A(x,y)

= 6(x-y)

t h e G-L e q u a t i o n fi(y,x) from (7.21) and (7.25))

+ n ( x , y ) one o b t a i n s A(x,y) cv

= (T(y,c),A(t,x))

= 6(x-y)

Nu

.

Fur-

+ n ( x , y ) and

l e a d s t o ( n o t e how t h i s d i f f e r s

167

M T 2 Now w r i t i n g forma l y E = fo Jr (A-'a-Ws) dudy = -* r,a12dudy = 1 ; lm Im~ ( y , x ) ~ ( y , s ) ~ c)dxdcdy (x = T r t ( l + f i ) K we have

which i s z e r o f o r x > y.

6 1;

N

(

Y

Y

S)dxdEdY

-

N

M

x)A(y,x)dxdy where

=

6(y,x)).

= E

*,,

+ T r izK

[n"(l+z) t A-15(y)2z] ( s i n c e ( Ws(A,y),a(A,x))u = ( s(h,y),a(h,x))v = The c a l c u l a t i o n s a r e formal b u t under s u i t a b l e hypotheses everyA v a r i a t i o n a l argument g i v e s then

t h i n g makes sense ( c f . Remark 7.11).

&HEe)REm 7-10. The s o l u t i o n

go o f

t h e m i n i m i z a t i o n problem f o r Z(T,K)

i s de-

t e r m i n e d as t h e s o l u t i o n o f t h e G-L e q u a t i o n (7.27) f o r x > y.

REmARK 7-11- The q u a n t i t y

rv

M

Z which a r i s e s i n m i n i m i z i n g Z i s n o t perhaps ob-

v i o u s l y meaningful and we w i l l make a few comments here t o show t h a t i t makes sense a t l e a s t f o r a l a r g e c l a s s o f problems. ple that A t o (**) W " as x + m ) .

Thus assume f o r exam2 C 2 w i t h A(0) = 1 and r e c a l l t h a t u = Au4w i n Ou = - A u leads - {W = -A 2 w w i t h = A-'(i')" (recall also that A' + 0 rapidly We s e t aQ = A%: w i t h *A '4 t h e s o l u t i o n o f (**) s a t i s f y i n g ah YQ %

E

A

e x p ( i A x ) and DXaA -4 % i A e x p ( i A x ) as x

+

m.

Note t h e n a!

%

Az5exp(ihx) and

A t 5 i x e x p ( i A x ) s i n c e A ' + 0 ) . Moreover i f A ' ( 0 ) = 0 f o r example we Dx@Y w i t h $?(O) = 1 and Dx$:(0) = 0 ( n o t e CxpA(0) 4 = A-'(O) can s e t v: = A-%! DxgY(0) - A'(0)A-3/2(O)$:(O)). We can t h e n i d e n t i f y cQ and c* where ( 1 ) = cq(A)aA Q + cQ(-A)@yA and ( 2 ) $Q = c' :A(;) + EQ(-h)6sA ( s i n c4 e A - + ( x ) ( l ) " A2 Now f o r t h e o p e r a t o r Q = D - q one has formulas f o r E as i n t h e = (2)). Q m case o f F o u r i e r t y p e o p e r a t o r s (namely, c* ( - A ) = ( 1 / 2 ) - ( 1 / 2 i x ) f o { ( y )

v!

3

4

c f . Chapter 1, 58 and Chapter 2, 96) showing t h a t c" ( - A ) 2 4 + 1/2 s t r o n g l y ( i n L ) as h m ( A real). T h e r e f o r e c ( - A ) + 1/2 ( a l o n g w i t h c p ( - h ) ) and W ( A ) = ;/$ = Icp/cQ12 + 1 w i t h du = ($ 42 /d)dA + dA/2nlcpl 2

-

$:(y)exp(iAy)dy

-f

=

(2/n)dA.

M

Hence f o r A l a r g e i n E we a r e t a l k i n g about 1

I (s-A-'a)

2

dhdy

and t h i s i s known t o make sense by c o n s i d e r i n q t h e t r a n s m u t a t i o n B w i t h k e r n e l 5 = A-%

REmARK 7.12.

+ K as b e f o r e .

We show now t h a t t h e y i n t e g r a l i n (7.26) f o r example may be

removed and we c o n s i d e r t h e problem o f m i n i m i z i n g (7.29)

' r( y, % ) =

fm

0

-

Ws + ( z , a l I 2 ( y ) d u

ROBERT CARROLL

168

Given a n operator A w i t h a (function) kernel A(y,x) l e t us w r i t e A(y,y)

=

Then t h e c a l c u l a t i o n s leading t o (7.28) y i e l d ( f o r y f i x e d )

Aly.

(7.30)

J .

N

Y*

T(y,K) = [Z(l+n)K

The v a r i a t i o n a l argument with (7.31)

Iy %

v

*,

+ [c(l+O)K =

Iy

+ 2A-'(y)%l

zo + EJ gives

2[(~o(l+~)+ A-''(Y)E)J*]~

Y

then

= 0

Let us note here t h a t f o r anticausal J w i t h J f ( y ) = JmJ ( y , x ) f ( x ) d x one has y J*g(x) = g(y)J(y,x)dy so e.g. (KoJ s)(Y)*= ly K0(y,x)$ g ( n ) J ( n , x ) d q d x = 10" s ( n ) ~ ~ x ( , ~ ~ ~ o ( ~ , x ) J ( n y x ) d Hence xdn. I = Jm ( y , x ) J ( y , x ) d x and m ,Y Y 0 -* N * s i m i l a r l y (JK,*)(Y) = g(n)hax(,,y)J(y.x)Ko(n,x)dxdn so JKoly = KoJ ly. In the same way JZ/ = OJ I f o r example s i n c e is formally s e l f a d j o i n t Y Y A(y,x)J(y,x)dx = and t h e conclusion (7.31) f o r admissable J in the form Im Y 0 implies A(y,x) = 0 f o r x > y (which i s t h e G-L equation of Theorem 7.10).

Jt

N

*

03,-

Z0J

A2

&I*

Thus t h e t r a c e s t e p i n our minimization theorems i s not necessary and i t was included b a s i c a l l y i n order t o compare with the formulation of [Dafl,2].

W e can now go t o t h e general M equation of 56 and show t h a t i t a l s o can be characterized as a minimizing c r i t e r i o n ( c f . [C52,53]). Thus l e t P be a Fourier type operator as i n Chapter 1 , 58 w i t h B: P + Q: q Px ( P = D2 - q here). For Q we take Qu = (Au')'/A w i t h A as above f o r a typical model A and i n order t o f a c i l i t a t e t h e inclusion o f operators Qu = Qu - $(x)u w i t h say ( l + x ) l t ( x ) l d x < m (and the a n a l y s i s o f kernels) we will assume A 6 2 ( f o r s i m p l i c i t y we will a l s o assume here t h a t QA is absolutely continuous C i n t h e sense t h a t dw = dw = Gdh on [ O , m ) ) . Hence s e t t i n g u = A-W ' an equaQ t i o n 4 u = -1'" becomes -+

fy

(7.32)

V

QW =

wll

- q'w

=

-x 2w ; q'

=

A-+(A')~' +

&!

q*

Q = 0 t h e n $! = A%! satisFurther i f = -h2q! w i t h q Qx ( 0 ) = 1 and Dxpx(0) f i e s g:(O) = 1 and D$!(O) = h = (l/Z)A'(O). On t h e o t h e r hand f o r J o s t = A%! % A-'(x)exp(ixx) and Dx(P! % s o l u t i o n s V Q of (7.32) one has e . g . A-'(x)ixexp(ixx) ( s i n c e A' + 0 as x -+ m ) . Between Fourier type operators 2 Pu = (D - q)u and operators one has a v a i l a b l e the Marrenko transmutation of Chapter 1 , 514-5 f o r example and standard s p e c t r a l i n t e g r a l s f o r kernels will be appropriate ( c f . a l s o §§2-3 of this chapter). In p a r t i c u l a r one v P VQ VQ P will have transmutations g: P Q: p x q A w i t h q (y) = q P , ( y ) + .f[ k(y.x) Consequently f o r p!(x)dx so t h a t q!(y) = A-'(y)v!(y) + {I K(y,x)px(x)dx. Ph

4"

-f

-f

MINIMIZATION

B: P

+

B

Q: L PP + ~ LQ P k~e r,

6 - l has k e r n e l v ( x , y )

=

=

169

+ K(y,x).

B(y,x) = A-'(y)6(x-y)

+ L(x,y)

A5(y)6(x-y)

e r a t o r ) and z ( y , x ) = A - l ( y ) y ( x , y )

=

Therefore R =

(by i n v e r s i o n o f a V o l t e r r a op-

+ r ( y , x ) ( c f . 552-3).

A-'(y)s(x-y)

Now l e t us use t h e m i n i m i z a t i o n (7.26) as a p o i n t o f d e p a r t u r e and reorgani z e i t i n terms o f t h e MarEenko t h e o r y i n 56.

We r e c a l l t h a t

(;i'= JC*).

The k e r n e l K(x,s)

(7.33)

JC(x,s) = (

(X(x,s)

= 0 for x >

E= .(

"yr

&where PJ,w

v

and t h e general M e q u a t i o n i s 6% = BAJC = B(JCAJC) = BK

JCf(x) = (JC(x,s),f(s))

i s g i v e n by

s and g f ( x )

= C%(x,s),f(s))

w i t h z(x,s)

= K(s,x).

?&!

Note

a l s o t h a t by s p e c t r a l forms f o r t h e k e r n e l s B and y we o b t a i n a g a i n = Q 2 Wp, where Id = G/v^ = c / c I . The i d e a now i s t o w r i t e = b C i n (7.26) P Q and rephrase t h e m i n i m i z a t i o n a c c o r d i n g l y so as t o t r e a t t h e Marc'enko k e r nel

as unknown.

From t h e s p e c t r a l f o r m u l a f o r g(y,x) one has a rough b u t

u s e f u l decomposition ( y > 0 ) (7.34)

;(Y,X)

=

(1/271) ~ m @ ~ ( ~ ) P[ w , ( x ) / c p ( - h ) 3 d =h -m

m

+ c(y,x)

A-'(y)eixy[CosXx/(?i)]dX

(1/2n)

=

+ t(y,x)

A-'(y)6(x-y)

-m

P P s i n c e e.g. wP,(x) = CosXx + (l/A)J: Sinh(x-y)q(y)qh ( y ) d y g i v e s an e s t i m a t e f o r / LPP ~ ( X- ) CosXx(, cp(-X) = k - ( l / Z i X ) f : q ( y ) w Ph ( y ) e x p ( i k y ) d y , @,(y) (I =

+ @(y,X)] w i t h cb(y,A) bounded f o r I m X 0 ( t h i s f o l l o w s A-'(y)exp(iXy)[l from t h e c o n s t r u c t i o n s i n Chapter 1, 585-6 f o r example), and L z @X(y)CosXxd Q =

im% m @QX ( y ) e x p ( - i h x ) d X .

there.

Note a l s o i ( y , x )

=

0 f o r x < y since g(y,x) = 0

Corresponding t o t h e expressions above f o r z a n d

t h a t 3c has t h e form JC(x,s)

= 6(x-s) + h(x,s)

B'

one can deduce

where h i s a n t i c a u s a l a l o n g

= b C i n k e r n e l form l o o k s l i k e A-'(y)6(y-s) + ~ ( ' ( Y , =s ) + z ( y , x ) , 6 ( x - s ) + h ( x , s ) ) x ) . . From t h i s one o b t a i n s a l s o 3% and one knows x ( x , y ) = 6(x-y) + fL(x,y))

w i t h Jc ( n o t e t h a t CA-'(y)s(y-x)

N

(recall K =

= 6(t-X)

+

K(t,x)

and we n o t e t h a t K ( t , x ) = J J i i ( s , t ; ) X ( t , s ) J c ( x , ~ ) d ~ d si s a symmetric k e r n e l Consequently t h e general M e q u a t i o n i n k e r n e l form may be w r i t t e n as k e r BK =

[A-%(y)s(y-s)

+ c(y,s)]o[6(s-x)

t K(s,x)]

= 0

for x > y or

170

ROBERT CARROLL

(7.36)

+ z(y,x)

A-'(y)K(y,x)

C'

+

K(y,s)K(s,x)ds

= 0

We w i l l a l s o have use f o r t h e e x p r e s s i o n N

(7.37)

k=

ker B = ker + N(y,x);

A-'(y)6(y-x)

+ l?(y,s)l[S(s-x)

[A-'(y)6(y-s)

[N = ?](y,x)

=

[z

+ h(s,x)]

:j

+

+ A-'(y)h](y,x)

=

?(y,s)h(s,x)ds

N

We go now t o t h e m i n i m i z a t i o n problem f o r E i n (7.26) and ( f o r F i n some c l a s s o f admissable a n t i c a u s a l k e r n e l s ) we c o n s i d e r a g a i n (7.28). here t h a t B(y,s) = 0 f o r 5 > y w i t h a t e r m A-'(y)b(y-s)

Recall

along t h e d i a g o n a l .

We r e w r i t e t h e l a s t e q u a t i o n i n (7.28) now as

'i: = 2

(7.38)

loT[

+ 2

A-'(y)

( t h e t r a c e depending on T).

#

Y

Here one notes t h a t

*,,,

ru*

+ T r %(l+CL)K

z(y,s)z(y,c)dcdy

%* i s

causal w i t h % * f ( x ) =

L e t us has k e r n e l Jm la, ii(y,s)x(s,x)"ic~,x)dxds. . - Y n t h i n k now o f our t r i a l o p e r a t o r s X as a r i s i n g f r o m a c o n s t r u c t i o n as i n and KAK

g(y,x)f(y)dy

(7.37),

i . e . :(y,x)

+ A-"(y)h(y,x)

= c(y,x)

+ i(y,-)oh(. ,x) w i t h t h e $(y,x)

as t h e fundamental o b j e c t s i n t h e m i n i m i z a t i o n .

+ g(y,x)

o p e r a t o r w i t h k e r n e l A-$(y)S(y-x) 6(y-x) + g(y,x).

Tr

(7.39)

We can w r i t e e.g.

and t h e n

5=

'i f o r

the

k X has k e r n e l A-'(y)

By an easy computation we n o t e now t h a t

v

"V*

NU

_*

T r BW =

=

JI

+

A-'(y);(y,y)dy

0

2 j)-&(y)

rz(y,t)z(y,t)dtdy + Y O so t h a t i n (7.38) one o b t a i n s N

(7.40)

M

B = E

~(y,t)z(y,T)~(t,~)dTdtdY Y

urv*

Y

..led

+ T r KK - T r

v*

+ Tr i ( X W ) B

Observe n e x t t h a t ( u s i n g t h e same symbol f o r o p e r a t o r s and k e r n e l s when no " ,"* + %]O[A-'s +%*I = + c o n f u s i o n can a r i s e ) ?3mB = %* has k e r n e l [A% u u* A-%C* i A - % + KeK so (7.40) becomes ( n o t e K = = %)

s+

+

N

(7.41)

v

M

v*

E = Z + Tr[BKB

- Am',

- A-'Z - A-'g*

- ZA-']

V

F i n a l l y t o p u t e v e r y t h i n g i n terms o f K we r e f e r t o (7.37) and w r i t e (7.42)

Tr[A-%?

+

%I =Tr[A-%? ']

+

iA-']

+5

N

Q TYPE OPERATORS

2

171

c.

I t i s important t o note here t h a t and h are both anticausal and hence ( c f . ( 7 . 3 7 ) ) ker $41= J X i ( y , c ) h ( c , x ) d c . This Y means t h a t :oh has t r a c e zero (along w i t h i t s a d j o i n t ) . Hence where

does not depend on

EHEBREIII 7-13. Under t h e hypotheses indicated the minimizing procedure f o r N

E reduces t o minimizing ( r e c a l l K

= 6

+ K w i t h K symmetric)

i.

over a s u i t a b l e c l a s s o f anticausal kernels on t h e kernel g.

Here

and

do not depend

Now s e t = io+ EC in a standard manner, where go designates a minimizing o b j e c t (e.g. ko = ?i where i i s t h e M a r h k o k e r n e l ) . Then d i f f e r e n t i a t i n g in E a n d s e t t i n g E = 0 one obtains (7.44)

2Tr [2,(6

+ K ) + A-%]C*

=

0

This i s t o hold f o r a s u i t a b l y l a r g e c l a s s of anticausal kernels I: so we conclude t h a t ( c f . ( 7 . 3 6 ) )

i0f o r E

N

CHEBREIII 7-14, The minimizing kernel

i s characterized as t h e (uni-

que) s o l u t i o n of t h e M equation A-'(y)K(y,x) + g o ( y , x ) + Jm Y" = 0 f o r x > y and thus coincides w i t h t h e Marzenko kernel K.

k

(y,s)K(s,x)ds 0

For f u r t h e r r e s u l t s on minimization s e e S52.10 and 3.6. 8- C ~ W C R U ~ I B OF N E R A W ~ U E A E ~F0R BW

5 E ~ P E0PERACBRB-

I t will be i n -

s t r u c t i v e t o consider f i r s t some constructions via Goursat problems of ext e r i o r and i n t e r i o r transmutations a r i s i n g in acoustic s c a t t e r i n g problems The operators which a r i s e a r e s i m i l a r t o those of following [Cnl-4,7]. Chapter 1 , 57 ( f o r which some corresponding transmutations were developed i n s p e c t r a l form) and t h e r e w i l l be connections t o t h e Bergman-Gilbert (BG) operator of 59 which i s useful in studying special functions as well. One considers (8.1)

Anu + k 2 [l

- q(r)]u

= 0;

1r i ~m m,4(n-1

- iku]

=

Here An = A i n Rn and t h e condition a t m i s t h e Somnerfeld r a d i a t i o n condit i o n which s p e c i f i e s the wave as outgoing. For convenience take f i r s t q = 0 f o r r > a. In t h e notation of Chapter 1 , 57 we w i l l be dealing with op-

172

ROBERT CARROLL

2 2 2 2 operators Qu = r u" + (n-1)ru' + r k [l - q ( r ) l u ( i . e . y ( r ) = k q ( r ) ) . One t r i e s now t o find a solution of (8.1) in the form (the represents spherical variables) m

u(r,-) = Be[h](r,-

(8.2)

h(r,-) +

=

~"~K(r,s)h(s,-)ds

2 where h s a t i s f i e s (An + k ) h = 0 (here Be refers t o "exterior"). The kernel K = Ke can be constructed by successive approximations as the solution of a Goursat type problem

-QrK(r,s)

(8.3)

-1 n- 2 Q,K(r,s); 2r K(r,r)

=

m

k2sq(s)ds

=

,.,

-1 2 2 2 2 where Q,u = s u" + (n-1)su' + s k u. Let us write = P and = k q for simplicity and we will show f i r s t how (8.3) arises i f (8.2) represents a transmutation (acting on functions h ( r ) ) . Thus consider Be: F + and for functions h = h ( r ) one wants t B h = R,(Mn(c,n)l

/I:

< a(l+i+a2/R2)$

I Mn( 5 , n ) I

(8.18)

IMn-l ( s , t ) l d t d s ;

hence by i n d u c t i o n

5 [a2;/

1+i+(a2/R2) ( a % ) (1 -n)In

(n ! )'][a(

Consequently t h e i n f i n i t e s e r i e s f o r M converges u n i f o r m l y and one has Under t h e c o n d i t i o n s i n d i c a t e d t h e k e r n e l K ( r , s ) f o r Be w i t h

CHE0RER 8.5,

n = 3 can be c o n s t r u c t e d v i a t h e Goursat problem (8.15)

REmARK 8.6.

( o r (8.3) f o r n = 3)

A s i m i l a r c o n s t r u c t i o n can be used i n a s i t u a t i o n where e.g.

k2/o" s ( q ( s ) l d s
0 and t h e n t h e Mn s e r i e s above begins w i t h a d o m i n a t i o n ] M o t

r close t h e contour t o the

202

ROBERT CARROLL

right.

From

o r f-ro

- fv-

= f(v

-1;

r

f(v,-k)ro(-v,k,p)

=

-2vf(v,-k,p)

A t v = vn = (n+1)/2 we w r i t e l i m ( v - v n ) f ( - v , - k )

= -2vf.

as b e f o r e and l i m ( v - v n b ( - v , k , p ) f(v,-k)

-

we have f(-v,-k)ro(v,k,p)

( 0 )

=

=

?_"

$:.

Since f(v,-k,p), ro(v,k,p), and An nn n n we have f-ro, = f+p- where ron = ro(wn,k, ) and f, Qn = ronf-/f: and c a l c u l a t i n g g ( r , p ) by r e s i d u e s we o b t a i n

are a n a l y t i c a t v

-k). Ri2.n

Then

f-ron/pf:

$!

-L(r,p).

=

REmARK 9-24, L e t us make a few remarks h e r e about t h e use o f t h e formula T(v,p) = -(iv/v)$

g(v,-k,r)*(v,k,r)dr

a l s o [CSO] and Remark 8.18). ing that

(T =

tions

"Q ) dT( P 1 )

v-4, u

Recall t h a t v / r = i m

=

Li

= v-%).

?(u)=

Such a f o r m u l a i s used f o r m a l l y i n e s t a b l i s h -

d e f i n e d by (9.30) maps

y(r,p)

culations there). ( ;(r,p

= ~ ( v - u ) i n t h e K-L i n v e r s i o n ( c f .

$!

$:

+

and

r' (see here (9.27) and t h e c a l -

* = $:/f(v,-k)

so t h e r e f o r e

r'[ ( - ~ P / v 9 ( v , -k,p ,*(v, k, P 1 ) [ f ( u ,- k ) / f ( v ,- k ) I( v / p )dv Now i n t h e K-L t h e o r y T ( v , u ) = 6 ( v - v ) a c t i n g on funcwhich a r e even i n v .

(f(s),g(u,-k,s))

Note t h a t T ( v , v ) i s

even i n v i t s e l f b u t now symmetrical i n v , ~ . The a c t i o n we want i n d i f f e r e n t (namely ? ( v , p ) rv-%) N

and i t i s s i m i l a r t o t h e a c t i o n ( r e c a l l N

W

@

is

= 6 ( v - v ) a c t i n g on

= [f(v,-k)/f(v,-k)l(v/v)T(v,v)

= Ip/f(v,-k)

and

* = @/r)

N

W1 = * w h e r e B: Q, -+ Q, Bg' = ($/$l)g,

B(r,s) = ker B ( s

( c f . Remark 8.18). P - ( i / n ) l i I vg 1 (u,-k,s)*(v,k,r)dv and f o r m a l l y

= ( g(v,;k,r),g'(y,-k,s))

(

+

r ) , and p ( r , s )

Thus one can w r i t e a ( r , s ) = B(rYs),ql (v,k,s)) = i i z * ( v , k ,

Thus we want Tl(v,v)(w/v) = ~ ( v - P ) a c t i n g on *(v,k,r). r)'Tl(vyp)(v/p)dv. 2 Now modeled on Remark 8.18 we can w r i t e dp = $dv w i t h ^p = ( 2 i / T ) v / f ( v , - k ) f(-v,-k)

and u s i n g

( 0 )

one has Gg(v,-k,x)

L e t us r e p e a t t h e argument o f Remark 8.18.

Thus s e t t i n g

'u

=

*(r,v)

s)

-

k,r)

one has f r o m Bgl = (g/;l)g,

\~r (-v,k,s))

= G(r,v)

,1 - *(r,-v)]

-

f o r Rev = 0.

-

P(v,k,r)

-

o r 0, = *(v,k,r)

*(-v,k,x)]. (v,k,s))

B(r,s.),ql

(

*(-v,k,r)

Y

*(r,-v)

-

= (-iv/v)[*(v,k,x)

= ( B(r,s),*,(v,k,

-*(r,v)

= 0- = [*(-v,

Here 0, i s a n a l y t i c f o r Rev > 0 and 0- i s an-

a l y t i c f o r Rev < 0 so t h e y r e p r e s e n t p a r t s o f an a n a l y t i c f u n c t i o n 0. By such t h a t t h e a s y m p t o t i c beassumption here we a r e d e a l i n g w i t h o p e r a t o r s h a v i o r o f "wave" f u n c t i o n s i s t h e same as t h a t f o r Now q0(v,k,r) for a l l

V,

= (n/2kr)4i-v+4Jv(kr)

co ( c f .

Example 9.16).

and, a l t h o u g h t h i s i s i n f a c t a n a l y t i c

i t i s t h e e s t i m a t e f o r Rev > 0 which determines e s t i m a t e s f o r e .

One knows e.g.

Jv(z)

0 so l i v * o ( v , k , r ) /

2,

(z/Z)'/r(v+i)

2,

(2~v)-~(z/2)~exp[v-vlog~] f o r Rev 5

5 c f o r Rev > 0 (and say r

has s i m i l a r l y q0(-",k,r)

= (n/2kr)Jliv+35J-v(kr)

> 0 fixed).

so l;*'-il

For Rev < 0 one

2 c.

Hence iv*+ N

and i-%(resp. i%+ and i-%-) a r e bounded and we w r i t e iv@+ = @+ (resp. h

i-'0-

= 0-1.

We see t h a t t h e corresponding

6

w i l l be a bounded a n a l y t i c

GENERATING FUNCTIONS

203

To see t h a t

f u n c t i o n and hence a c o n s t a n t by L i o u v i l l e ' s theorem. look a t v

+ m

and hence T1(v,p)(w/u) *(v,k,r).

=

Iiv*oI

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

a c t i n g on *(v,k,r)

= a(w-1~)

= 0

It follows that 0 = 0

0.

+

with

(

o(r,s),*l(v,k,s))

T h i s v e r i f i e s Theorem 8.17 as i n Remark 8.18.

The same pro-

cedure can now be used t o check t h e a c t i o n on o t h e r f u n c t i o n s ( c f . [ C S O ] ) . Thus one w r i t e s

Jr

f(v,-k)f(-v,-k)

on [O,im).

It follows t h a t

(

-

-k)r'-'

Jo" ;(r,p)g(u,-k,p)dp

=

and r e c a l l t h a t *(u,k,p)

i n g G ( r , u ) = (T(r,p),*(u,k,p))

=

-

we o b t a i n 0, = ? ( r , p )

g+ i s

- G(r,-p)

0.

Hence t h e y a r e p a r t s o f an a n a l y t i c f u n c t i o n

f o r Reu = 0 where

i';(u,r)

Q

Li,

i m

&

= 0- = p ( r ,

G- f o r

Rep
0 and

,lJ-' = T

Hence d e f i n -

< c f o r Reu > 0 and r > 0 f i x e d as b e f o r e and,

Thus \ i ' G ( u , r ) l

bounded as b e f o r e .

Set now ? ( r , u )

p(p,k,p)/pf(u,-k).

-u)

from Example 9.16,

= -(1/2u)[f(-u,

- f(u,-k)p(-u,k,~)l/p). N

before.

2

(2iv /n)/

=

w h i l e on t h e o t h e r hand from (a) t and one o b t a i n s (10.5). +

Then a Vol-

has t h e form ( * ) i n Theorem 10.3. t t e r r a i n v e r s i o n a l l o w s one t o w r i t e Cosxt = f ( x , t ) + so L ( t , s

REMARK 10.4.

Suppose f ( A , t )

ds.

If

t h e f ( A , t ) a r e o r t h o g o n a l one o b t a i n s then (10.10)

z(T,t)

Consequently, f o r

CosAt,f(X,T))w

= ( T

> t, B(T,t)

=

&(t-T)

+ L(t97)

0 which i s (10.6).

=

Thus g i v e n o r t h o g o n a l -

i t y o f t h e f expressed i n t h e form ( * ) one o b t a i n s t h e G-L e q u a t i o n v i a

Theorem 10.1, and conversely, g i v e n t h e G-L e q u a t i o n , one c o n s t r u c t s o r t h o g Hence f o r f i n t h e f o r m (*) t h e G-L e q u a t i o n i s

onal f i n t h e f o r m (*).

equivalent t o orthogonality. REMARK 10-5. The G-L e q u a t i o n i s o f course analogous t o a c o e f f i c i e n t d e t e r m i n a t i o n procedure i n t h e d i s c r e t e case ( c f . [ A l l ;

Cd41).

Thus t o d e t e r -

such t h a t p o l y n o m i a l s p n ( x ) = 1 : a m n r are orthogonal r e l a t i v e t o mn k dw one uses ( 4 ) t o g e t L[X pn(X)] = Kmfirnn = 1, amnuk+m. T h i s i s a k i n d o f

mine a

d i s c r e t e G-L equation. r e p r e s e n t e d i n t h e f o r m (*) i n Theorem 10.3 w i t h K s a t i s f y -

Now g i v e n f ( x , t )

i n g t h e G-L e q u a t i o n o f Theorem 10.1 l e t us assume n ( t , s ) i s t w i c e c o n t i n u 2 2 o u s l y d i f f e r e n t i a b l e ( w i t h DSa = D p ) . Then K w i l l be d i f f e r e n t i a b l e i n Theorem 10.1 and we n o t e a l s o t h a t K(t,T) = ( Z / T ) / ; so t h a t K T ( t , O )

=

[f(A,t)-Cosht]CoSA~d~

A s t r a i g h t f o r w a r d c a l c u l a t i o n f o l l o w i n g [Lxl;

0.

C511

( c f . a l s o [Mc41) y i e l d s t h e n

UtEORElIl 10.6,

i n t h e form ( * ) w i t h K t h e unique s o l u t i o n o f t h e

Given f ( h , t )

G-L e q u a t i o n i n Theorem 10.1 and r2 t w i c e d i f f e r e n t i a b l e i t f o l l o w s t h a t K 2 2 s a t i s f i e s a Goursat t y p e problem Q(Dt)K(t,T) = D T K ( t , T ) where Q(Dt) = Dt q ( t ) , q ( t ) = 2DtK(t,t),

and K T ( t , O )

C O S ~ T ) = Cosxt + 1 ; K(t,T)Cosxrdr

=

0.

The c o n n e c t i o n f ( A , t )

= Ca(t,T),

2 then determines a t r a n s m u t a t i o n B: D

+

a c t i n g on f u n c t i o n s 9 w i t h g ' ( 0 ) = 0.

Pma6: (10.11)

S e t now DK

mK =

Ktt

-

KTT so a f t e r a c a l c u l a t i o n from t h e G-L e q u a t i o n

+ q(t)a(t,T) +

where q ( t ) = 2DtK(t,t).

.K(t,S)a(S,r)dS

jot

=

0

By uniqueness o f s o l u t i o n s f o r t h e G-L e q u a t i o n

Q

212

ROBERT CARROLL

2 one o b t a i n s t h e d i f f e r e n t i a l e q u a t i o n QtK = D K.

I n o r d e r t o show t h e i n -

: t e r t w i n i n g p r o p e r t y one s i m p l y l o o k s a t B g ( t ) = g ( t ) t J

computes QBg u s i n g t h e procedure which l e d t o (10.11). e s t a b l i s h e s t h e theorem (see a l s o [C51]).

K

and

Comparison w i t h Bg"

represented by (*) w i t h L2 t w i c e d i f f e r e n t i a b l e 2 as i n Theorem 10.6 i t f o l l o w s t h a t Q ( D t ) f ( x , t ) = - h f ( A , t ) and f ( A , O )

E0R@CCAR!.J10.7. and

K(t,T)g(T)dr

Given f ( A , t )

= 1 w i t h f'(h,O)

Phaa6:

= -n(O,O) =

-1; do(h)

=

-gr(0)

=

h.

The c a l c u l a t i o n i s r o u t i n e w h i l e f o r f ' ( h , t )

(10.12)

fl(h,t)

Hence f ' ( A , O )

= K(0,O)

-xSinht

-k

K(t,t)CosAt +

one has f r o m ( * )

lot

Kt(t,T)COShTdT

and t h e r e s t f o l l o w s f r o m t h e G-L equation.

.

REMARK 10.8,

I t i s c l e a r l y o f i n t e r e s t t o s t a r t i n general w i t h some " a r b i P t r a r y " f u n c t i o n s p A ( x ) i n s t e a d o f Cos x and form " p o l y n o m i a l s "

U

P One can t h e n begin w i t h minimal knowledge and s t r u c t u r e r e g a r d i n g b o t h p x

and f ( h , t )

and g r a d u a l l y i n s e r t v a r i o u s i n g r e d i e n t s such as measures dv P (resp. dw) under which t h e p A ( r e s p . f ( h , t ) ) a r e t o be o r t h o g o n a l , d i f f e r -

e n t i a l equations, e t c . i n o r d e r t o show p r e c i s e l y what depends on what. T h i s theme w i l l be p a r t i a l l y developed l a t e r ( c f . a l s o [Cd4]). The r o l e o f m i n i m i z i n g procedures i n c h a r a c t e r i z i n g t r a n s m u t a t i o n k e r n e l s

In this ( c f . a l s o [Cd4; D a f l ] ) . was developed i n S7 and [C51-53,74,78,80] 2 s p i r i t , f o r a t r a n s m u t a t i o n P = D + Q = D2 - q f o r example w i t h v!,h(t) = t h e G-L k e r n e l c o n n e c t i n g p Q and Cosht v i a (*) i n Theorem 10.3 can x,h be c h a r a c t e r i z e d as t h e m i n i m i z i n g k e r n e l f o r ( T = f h e r e ) f(h,t)

(10.14)

[T(A,t)

where K(t,T) for

T

- COSht

-

2 K ( t , ~ ) c O S h r d ~ ]dwdt

runs o v e r a s u i t a b l e c l a s s o f causal k e r n e l s ( i . e .

K(t,-r)

= 0

> t); we r e f e r t o Chapter 3 f o r f u r t h e r d e t a i l s i n t h i s p r e s e n t s i t u a -

4

The " E u l e r " e q u a t i o n f o r t h i s as a v a r i a t i o n a l problem i s T i n f a c t t h e G-L e q u a t i o n (we r e c a l l a l s o t h a t t h e i n t e g r a l lo d t i s n o t need-

tion with

ed i n (10.14)

and when t h e r e i s an u n d e r l y i n g s t o c h a s t i c process t h e m i n i m i -

zw i s e q u i v a l e n t t o t h e l e a s t squares e s t i m a t i o n t e c h n i q u e t o det e r m i n e a f i l t e r i n g k e r n e l - c f . Chapter 3 ) . I n t h i s s p i r i t l e t us t r y t o n o t n e c e s s a r i l y o f t h e form m i n i m i z e z f o r some general f u n c t i o n T ( x , t ) , w zation o f

ORTHOGONAL POLYNOMIALS

(10.2),

21 3

and a p r i o r i h a v i n g no p a r t i c u l a r r e l a t i o n t o dw except we r e q u i r e ( ~ ( x , t ) , C o S x S ) ~= F ( t , s ) = 6 ( t - S ) + Z ( t , s )

(10.15)

where g ( t , s ) = 0 f o r s < t.

I f V(X,t) = f ( A , t ) we r e f e r t o (10.6) and w r i t e

N

U(t,s) = ~ ( L - s ) + K ( t , s ) .

I f we w r i t e o u t now (10.14) one o b t a i n s (7.5) 2 T = JO :1 [ . ; r ( h , t ) - Cosxt] dwdt (which we as-

i n p l a c e o f %) where

(with

sume makes sense) and A(t,s) an obvious n o t a t i o n E w =

s ( ! - ~ ) + .Q(t,s). Using (10.15) we o b t a i n i n

=

Gw +

2Tr Kn + T r KK

*

+ TrKOX

*

(cf. (7.9)).

The

c r i t e r i o n f o r KO t o be a m i n i m i z i n g k e r n e l i s t h e n t h e G-L e q u a t i o n o f Theorem 10.1 and one has

CHE0REfl 10.9,

Given a general . ( X , t )

gw above

f o r which

makes sense (dw be-

i n g a measure as b e f o r e ) , and f o r which (10.15) h o l d s w i t h ;;7 a n t i c a u s a l , t h e best approximation t o V(x,t)

(= c(t,s))

Ko(t,s)

by f u n c t i o n s o f t h e form (10.2) r e q u i r e s

K i s t h e G-L k e r n e l ( i . e .

= K ( t , s ) where

o f Theorem 10.3 t h e b e s t a p p r o x i m a t i o n t o V ( h r t ) i s f ( x , t ) , :”I I [T ” ( x , t: ) - f ( x , t ) l 2 dwdt.

REmARK 10-10. The m i n i m i z i n q procedure o f [C52,53,74,75,78]

=

i.e.

(

z

~ =)

and 57 charac-

0 can be g i v e n a n o t h e r i n t e r p e r t a 2 We suppose f o r convenience Q = D2 - q and P = D w i t h

t e r i z i n g KO = K when t i o n as f o l l o w s . B: P + Q: cosxx

+ KO + KoR

r e l a t i v e t o dw a r e r e p r e s e n t e d by (*)

Thus s i n c e t h e orthogonal f ( x , t )

0).

.Q

+

II =

f and

qy ( ~ Y ( o )

=

=

1; ~

-

Q~

~ = ~0 ) . ( W0r i t e) ~ ( y , x ) = 6 ( x - y ) +

K(y,x) w i t h K causal, Bg(y) = ( B ( y , x ) , g ( x ) ) ,

B-l = B w i t h ker B = v(x,y) =

6(x-y) + L(x,y),

=

and g(y,x)

( q x4( y ) , C o s x x ) w ( 6 = R ). N

= ker B = y(x,y)

measure do can be a s s o c i a t e d t o t h e t r a n s f o r m t h e o r y f o r p! = :”I g ( x ) p x4( x ) d x

A

= g(x)

(g s u i t a b l e ) w i t h g(y) = 6

A

one has a Parseval f o r m u l a ( f , g ) w = ( f ( x ) , g ( x ) ) . w i t h integrand i n v o l v i n g

(T

Q

f

‘L

,, 9:) p ( X , t

*

The

i n t h e form Qg

( $ ( i ) , q4 x ( y ) ) w=l!4$(y)

and

Now i n an e x p r e s s i o n

= p Qx ( t )

-

COSht

-

(I{(t,T),

C O S ~ T ) we use t h e Parseval f o r m u l a f o r m a l l y t o w r i t e ($ = q q )

(10.16) N

8(X,t)

Ew

-

=

joT

(K(t,T),?(X,T))

(note q ( x , t )

Ip(x,t)l

2

dxdt; p ( x , t

= -L(t,X)

-

K(t,x)

= 0 automatically f o r x > t)

=(

-

&t),q$x)

)w = 6 ( x - t )

(K(~,T),L(T,X))

But (1+K)-’

= l + L which means

rt

(10.17)

0 = L ( t , x ) + K(t,x)

Setting K =

Ko+EJ

+

1,-

-

K(t,T)L(T,x)dT

f o r J causal we g e t ( c f . h e r e a l s o Remark 3.8)

~

~

~

21 4

ROBERT CARROLL

CHEBREIII 10.11- M i n i m i z a t i o n o f Zw v i a (10.16) c r i t e r i o n KO

f

f = q Qh ) l e a d s t o t h e

L + K L = 0 f o r x < t which c h a r a c t e r i z e s t h e G-L k e r n e l K. 0

For completeness l e t us c o n s i d e r ( f o r general

zv(t)

(10.18)

(II=

=

rm [ n ( A , t ) -

COSht

IT)

2 K ( t , ~ ) c O S h - r d ~ dv ]

-

where dv = (2/11)dh ( t h i s c o u l d c l e a r l y be g e n e r a l i z e d - c f . Remark 3.8). L e t (10.19)

(

II(X,t),Coshs)v

= s(t-s)

+ a(t,s)

where a p r i o r i a need n o t be t r i a n g u l a r .

Set now

2

( t ) = :/

2 [~~(h,t)-Cosxt]

dw which we assume t o make sense and w r i t e o u t Z v t o o b t a i n (10.20)

.,(t) c

=

gv(t) +

(note the s i m i l a r i t y t o c o e f f i c i e n t estimation i n Fourier series).

EHEBREN 10.12,

Given a general r ( X , t )

w i t h (10.19) (a n o t n e c e s s a r i l y t r i a n -

A

g u l a r ) and f o r which Z w ( t ) makes sense, t h e c o e f f i c i e n t s K o ( t , s ) = a ( t , s ) f o r s 5 t p r o v i d e t h e b e s t a p p r o x i m a t i o n o f t h e form (10.2) t o IT ( i n terms t 2 o f m i n i m i z i n g P ( t ) ) and one has a "Bessel" i n e q u a l i t y Jo a ( t , s ) d s 5 v 2 :J [ r ( A , t ) - Cosht] dv. P I n t h e s p i r i t o f Remark 10.8 t a k e f u n c t i o n s q h and qhQ w i t h Q P P qP,(y) = ( R ( y Y x ) , v ~ ( x ) )= &PA and v X ( x ) = (Y(X,Y),V:(Y)) (where R(Y,X) = s ( x - y ) + c ( y , x ) and y(x,y) = s ( x - y ) + L ( x , y ) - no t r i a n g u l a r i t y i s assumed).

REIIIARK 10.13,

Then v o r t h o g o n a l i t y o f t h e q PX i m p l i e s ~ ( y , x ) = ( q XQ ( y ) , q AP ( x ) ) and w o r t h o g P ,vv o n a l i t y o f t h e qhQ i m p l i e s y ( x , y ) = ( v X ( x ) y q h Q ( y ) ) u . D e f i n e B = ( B - l ) * w i t h k e r n e l F ( y , x ) = y(x,y) assumed) ;(Y,X)

= ((

and one o b t a i n s a G-L e q u a t i o n (no t r i a n g u l a r i t y i s P P ~ ( y , ~ ) , q h ( ~ ) ) , q X ( x )=) (~ B(Y,s),A(s,x)). Ifa c o n d i -

t i o n l i k e (10.6) h o l d s so t h a t g(y,x)

= 0 f o r x < y then y i s t r i a n g u l a r

and hence so i s 6 as a V o l t e r r a t y p e i n v e r s e (no s p e c t r a l form o f B i s needed here b u t t h e w o r t h o g o n a l i t y i s used i n going from t h e s p e c t r a l f o r m o f

) ) . Moreover f r o m t h e G-L e q u a t i o n one knows t h e n t o q pP , ( x ) = ( y(x,y),q,(y)Q t h a t c(y,x) = K(y,x) i s t h e G-L k e r n e l .

y

REmARK 10.14.

Given a causal i n (10.19) t h e n (zv(t))min

= 0 f o r K,

one has e q u a l i t y i n t h e Bessel i n e q u a l i t y o f Theorem 10.12.

=

~1

and

We n o t e t h a t

w o r t h o g o n a l i t y i s used i n Theorem 10.12 and w o r t h o g o n a l i t y i s i m p l i c i t i n

Theorem 10.11 b u t b a s i c a l l y no o r t h o g o n a l i t y i s i n v o l v e d i n Theorems 10.9-10. REIIIARK 10-15- Suppose g i v e n a general .(X,t)

and l e t CosXt + J$ a ( t , s )

ORTHOGONAL POLYNOMIALS

21 5

Cosxsds be the b e s t lu approximation as i n Theorem '10.12 (so ( ~ ( h , t ) , C o s A s ) ~ = 6 ( t - s ) f a ( t , s ) = ~ ~ ( t , (sn o) t r i a n g u l a r i t y is assumed). Further l e t anti(10.15) hold ( i . e . z I T ( t , s ) = ( n ( A , t ) , C o s h s ) w = 6 ( t - s ) + z ( t , s ) w i t h c a u s a l ) so t h a t f ( x , t ) i s t h e best zw approximation as i n Theorem 10.9. Then ( B , ( t , s ) , C o s h s ) = . i r ( A , t ) a n d one has a G-L type equation z I T ( t , s ) = (

t h a t for 6 1 T ( t , ~ ) , A ( ~ , ~I)t ) follows .

T c

t , a ( t , T ) i s the G-L kernel.

REmARK 10.16- We have been u s i n g t h e analogy (10.6) of ( 6 ) in o u r development b u t one could equally well use a version of (+). T h u s f o r "polynomials" IT of t h e form (10.2) a n d orthogonal "polynomials" f as in Theorem 10.3 one considers a condition ( = ) / , T ( x , s ) f ( x , t ) d u ( A ) = 0 f o r s < t. Indeed w i t h IT a s i n (10.2) ( n ( A , s ) , f ( A , t ) ) u = ( C o s h s , f ( x , t ) ) w + id c(s,.r)( C O S h , f ( h , t ) ) u dT. Hence (10.6) implies ( m ) . Conversely i n v e r t i n g (10.2) in the form S Coshs = . i r ( h , s ) + Jo R ( s , T ) n ( X , T ) d T we obtain f o r s c t, ( C o s A s , f ( X , t ) ) w = ( I T ( h , s ) , f ( A , t ) ) , + :1 R ( s , r ) ( a ( A , r ) , f ( X , t ) ) w d . r so t h a t ( = ) implies (10.6). REmARK 10-17- The r o l e of kernel polynomial Kn(z,x) =

played here by (10.21)

RT(X,u)

=

IT

1;

p m ( z ) p n ( x )i s

f(h,t)f(u,t)dt

0

where f ( A , t ) denotes the orthogonal functions from Theorem 10.3 ( c f . [All, 2 2 C51; Ku8; Lxl]). Note here t h a t i f Qf = ( D - q ) f = -A f a s i n Corollary 10.7 one has

(10.22)

RT(A,P)

= W(f(AyT)yf(pyT))/(A2

- v2)

where W denotes t h e Wronskian. I f one defines a transformation (p s u i t a b l e ) T $,(A) = Jo q ( t ) f ( h , t ) d t then R T a c t s as a reproducing kernel ~ ( W - P ) in the space of such Indeed i t i s c l e a r t h a t ( R T ( X , u ) , f ( u y t ) ) w= f ( A , t ) and If' Analogous t o the approximation of s u i t a b l e hence ( R T ( h , ~ ) , ~ T ( u)w) = G T ( A ) . functions g ( t ) by " p a r t i a l sums" g n ( t ) = J Kn(x,t)g(z)dz one t h i n k s here of formulas of t h e type ( $ ( A ) = Jr p ( t ) f ( A , t ) d t ) $(A) - G T ( h ) = ;/ [ $ ( A ) $(p)]RT(X,~)dw(u).

We note t h a t (10.22) i s a kind of Darboux-Christoffel

re1 a t ion. The above procedures apply when dw = (2/n)dA + do w i t h say du a s u i t a b l e bounded measure. In t h i s event n ( t , s ) i n say (10.8) i s a function and everything makes sense. I f now e . g . d w = wdA w i t h l~ = c2AZm+' (c, = l/Zm m r ( m + l ) ) then we are in t h e context of t h e d i f f e r e n t i a l operator Q = Q, w i t h = (x2m+l l l,x2m+l a n d t h e orthogonal functions f ( h , t ) a r e given by rn

21 6

ROBERT CARROLL

v:(t)

= (l/cm)Jm(Xt)/(At)m (spherical functions).

connecting Cosxx and B (Cosxx) = p xQ ( y ) (BQ: D2

Q

and 6 (y,x) does n o t have t h e form & ( x - y ) 8

Q

=

i-' Q

f

+

The t r a n s m u t a t i o n k e r n e l

Q = )9, i s cjiven by (2.8) ( c f . 52). The i n v e r s e

K(y,x)

has a k e r n e l y (x,y) = ( C O S ? , X , Q~ ~ ( ( Y ) ) ~(.Cf

= AQqx Q w i t h AQ = y 2m+l

Q

t h i s i s m a n i f e s t l y a d i s t r i b u t i o n g i v e n b y (2.16).

);

There i s a g e n e r a l i z e d

G-L e q u a t i o n Q (Y,x) = ( B Q ( Y ~ F ) , A ( E , X ) )where zQ(y,x1 = A p ( x ) A q l ( ~ ) y Q ( x , ~ ) ;Ap = 1 here) and t h i s i s g i v e n i n Theorem 3.14. One c o u l d now t r y t o dupl i c a t e some o f t h e p r e v i o u s machinery i n a d i s t r i b u t i o n c o n t e x t where d i s t r i b u t i o n a l o b j e c t s as i n Theorem 3.14,

(2.16),

etc. are prototypical.

We

p r e f e r however t o r e f e r a measure w i t h growth *w A2mf1 t o Q, = P ( i n s t e a d P 2 o f P = D ) as a p o i n t o f d e p a r t u r e and r e p l a c e Coshx by p A ( x ) = ( l / c m ) J m ( A x ) Q ,

AX)-^

(see a l s o e.g.

t h e t e c h n i q u e o f t r e a t i n g random f i e l d s i n [ L x ~ ] and

Thus g i v e n an u n d e r l y i n g d i f f e r e n t i a l problem we w i l l work

Chapter 3).

w i t h t r a n s m u t a t i o n s B: P

-f

Q: p!

To begin w i t h o f course we do n o t know Q,

as orthogonal f u n c t i o n s f o r dw. b u t we assume t h e r e i s a

where t h e f a r e t o be c o n s t r u c t e d

+ f(A,t)

0 ( c f . C o r o l l a r y 10.7); then s i n c e t h e G-L equa-

t i o n s r e q u i r e d t o c o n s t r u c t t h e f i n v o l v e now a t e r m f r o m A g i v e a deeper a n a l y s i s o f t h e s i t u a t i o n .

Q

we have t o

To do t h i s we w i l l be a i d e d by de-

v e l o p i n g an i n t r i n s i c and c a n o n i c a l f o r m u l a t i o n o f t h e problem u s i n g concepts from general t r a n s m u t a t i o n t h e o r y . Thus we r e c a l l f i r s t from 51.11 and (1.9.26)

t h a t i f B: P

+

Q: p PA + p A Q (Qu =

- .

qu; Pu = ( A u ' ) ' / A p where Ap = xZm+' and say A = A A t h e n P 1 Q. Q P . t h e r e i s a r e l a t e d t r a n s m u t a t i o n i: P + Q where e.9. A5Qu = Q[A%]; Qw = w" (AQu')'/AQ

+ 4w;

6

=

f

q

-

A-'(L')' .

Then

2Q Q2 has i ( x ) = - ( m - k ) / x ). general t h e k e r n e l

of

b

= A;(Y)B(X

-+

fi

and

Thus a l t h o u g h

bwill

where K(y,x)

B(Y,x)

4will

have t h e f o r m i ( y , x )

= A>(Y)"(x-Y)

9 P

-+

f)

Q

(note f o r

one

have s i n g u l a r i t i e s i n = &(x-y)

f

k(y,x)

where

It follows t h a t

i s a "reasonable" causal k e r n e l . (10.23)

y)xmm-':

+ K(y,x)

= A-'(y)y-m-'~(y,x)xm+'

( c f . 551.11,

Q

3.8 and [C39,40,66,70,71;

F u r t h e r by V o l t e r r a i n v e r s i o n f o r B = B - l , k e r 8 = y(x,y) = Az(y) Sol]). g(x-y) + L(x,y) and we r e c a l l t h a t F(y,x) = Ap(x)Aq'(y)y(x,y) (E = B# y ) i t f o l l o w s t h a t (A = A4) k ( y , y ) '

[A'(y)K(y,y)]'

=

=

-+[q"

i$(A'/A)2

-

$(A"/A)

-

(A'/A)(mt+)/y].

A more c a r e f u l a n a l y s i s o f o u r procedure g i v e s t h e f o l l o w i n g r e s u l t . let

BP?

=

Q w i t h B(y,x)

*

$(D )B(y,x)

Thus

= k e r B g i v e n i n s p e c t r a l form f o r example so t h a t

f o r x # y. F u r t h e r l e t an expression B(y,x) = h o l d as i n (11.2) f o r example. Then o f n e c e s s i t y

= Q,(Dx)~(y,x)

-g

AQ ( y ) b ( x - y ) + K(y,x) (11.5)-(11.7)

h o l d and hence Theorem 11.3 i s v a l i d . A

about i n t e r t w i n i n g o r t r a n s m u t a t i o n BQ,

QB.

=

Nothing need be assumed

F u r t h e r , once Theorem 11.3 i s

e s t a b l i s h e d one can e s s e n t i a l l y reproduce t h e c a l c u l a t i o n s (11.5)-(11.7) f o r g ( y ) = ( B f ) ( y ) = A-'(y)f(y) + f{ A-'(y)Qm(DY)f + f{ K(y,x)[Qm(Dx)f]dx -2m-1

)xf(x)

Kx(y,O) = 0 and m K(y,x)/x + 0 as b e f o r e w i l l d o ) . ( L e t us n o t e a l s o t h a t f ( x ) = ( F ( A ) , y A ( x ) ) m ?,qf), = ( ?,-A 2q Q A )m w h i l e BQf, = say w i t h b; = v f i m p l i e s f o r m a l l y G B f =

x

-f

+

0 as x

+

0 ( t h u s e.g.

6

-

B( f , - x 2qmx )m 2.3.9,

0 and K ( y , x ) f ' ( x )

K ( y , x ) f ( x ) d x t o o b t a i n GBf = {(Dy)g = BQ,f p r o v i d e d say x 2m+l (K(y,x)

=

(r,-A2q Qx ),,,.)

=

52.10,

51.7,

c@[email protected].

g3.6,

Consequently ( c f . a l s o §2.3, e s p e c i a l l y Remark 53.5,

and 52.8 f o r r e l a t e d i n f o r m a t i o n )

Given a map B: q y

-f

q:

as i n d i c a t e d i t f o l l o w s t h a t B i s a

t r a n s m u t a t i o n , a c t i n g on s u i t a b l e o b j e c t s .

RENARK 11-5, L e t us n o t e t h a t f o r m = 11.3 becomes (*) (A'(y)K(y,y))'

=

-+ and < = 0 t h e

(1/4)[(A"/A)

-

e q u a t i o n i n Theorem

(1/2)(A'/A)2]

=

(A')"A-'/2.

T h i s r e s u l t (*) i s n o t a t v a r i a n c e w i t h r e s u l t s f o r ( A u ' ) ' / A = Qu s p e c i f y i n g A-'(y)

=

1 - K(y,y)

( c f . 53.8 and Theorems 11.6,

11.7,

and 11.10 t o f o l -

l o w ) ; one i s s i m p l y t a l k i n g about d i f f e r e n t k e r n e l s K. L e t us go n e x t t o a new d e r i v a t i o n o f a r e s u l t o f t h e t y p e j u s t a l l u d e d t o i n Remark 11.5.

We r e f e r t o s1.6 f o r some background and begin w i t h a some-

what more general o p e r a t o r Qu = ( A u ' ) ' / A - qu. For s i m p l i c i t y one c o u l d 2 but f o r A E assume A E C and use c o n s t r u c t i o n s based on (1.6.21)-(1.6.23) 1 C t h e c o n s t r u c t i o n based on (1.6.5)-(1.6.6) w i l l be r e q u i r e d . We c o n s i d e r 2 moreover a s l i g h t e x t e n s i o n i n s p e c i f y i n g q Q as t h e s o l u t i o n o f Qu = - A u A,h = A ' / A one has e.g. s a t i s f y i n g q ( 0 ) = 1 and q ' ( 0 ) = h. Then s e t t i n g

4

224

ROBERT CARROLL

where we w r i t e A 4 f o r One assumes here e.g. 0 < ~1 5 A ( x ) 5 p < m , A 1 2 C , and A + Am r a p i d l y as x + m. For A E C one can use t h e technique o f Remark 1.6.4 t o o b t a i n ( c f . a l s o Remark 1.6.1) A%(y)@,(y) 4 = exp(ihy) + [Sinh (n-Y )/XI (i'(7-1) )It@)h Q (n)dn + f m [Sink ( n - y ) I A l q ( n ) @ ~ ( l ? ) A $ ()dn ~ = exp

-

( i 5 ( n ) ) [Sinh ( n - y ) / h I @ ~ & ) l'dn + f m [Sinh ( n - y ) / h l q ( n ) @ ~ ( n ) A ~ ( n ) dn = e x p ( i h y ) - f m (~4)'(n)[CosX(n-~)@h(n) Q + Y[ S i n X ( n - ~ ) / X ] D ~(2@ ~ ( n ) l d+n Note here f o r w = A% an e q u a t i o n Qu = [ S i n h ( n - ~ ) / h '] q ( nQ) @ ~ ( n ) ~ ~ ( n ) d n . 2 2 -X u l e a d s t o w " - (q+{)w = -A w where 6 = A-%(A$)" ( c f . (11.1) - here 4 = (iu)

I

F u r t h e r ( w i t h t h e n o r m a l i z a t i o n A(0) = 1 which can always be a c h i e -q-;). ved) w(0) = u ( 0 ) = 1 and w ' ( 0 ) = $A'(O) + h = h'. Given A ' + 0 as x m the -f

asymptotic b e h a v i o r w i5(x)$:(x)

=

e x p ( i x x ) and w '

+ ft

Q

( & ) I

ihexp(ihx) prevails. (5)[Cosh(x-c)ph(c) Q

Q Q I n any case one has as usual A(y)W(@h(y),@-,(y))

mulas.

-

Similarly [Sinh(x-5)/h]

[ S i n h ( x - ~ ) / A l q ( g ) ~ ~ ( 5 ) ( p ~ ( 5 ) db iu; t we w i l l n o t use these f o r -

D , o Q ( 5 ) l d ~+ f; S h

Q

Coshx + h[Sinxx/h]

Q = - 2 i h and A (ph(y) =

c Q ( h ) a4X ( y ) + C ~ ( - X ) @ ! ~ ( ~ ) .However a d i f f e r e n t e x p r e s s i o n f o r c v i a (A(0) = 1 )

arises

W ( Q~ ~ ( OQ) , ~ - ~=( O - 2 )i i)c Q ( h ) = D ~ @ - Q ~-( omYh(o) )

(11.10) (cf.

Q

[C40,66;

Af1,Z;

Sell).

T h i s a f f e c t s t h e f o r m o f t h e s p e c t r a l measure

e t c . ( t h e d e t a i l s f o r q = 0 a r e i n d i c a t e d i n [C40,66]

and c f . Remark 3.8.9).

We g i v e now a f i r s t d e r i v a t i o n o f a f o r m u l a f o r t h e G-L k e r n e l r e l a t i v e t o Q when q = h = 0 ( c f . a l s o Theorem 11.10 l a t e r and see 53.8 f o r a d i f f e r e n t

derivation).

From t h e c o n s t r u c t i o n s based on (11.8)-(11.9)

o b t a i n s an e n t i r e f u n c t i o n @ X (,.I4

"Q qh(x)

Q ( = (ph(x)

f o r example one

here) o f exponential type x w h i l e

f o r example w i l l be a n a l y t i c i n t h e upper h a l f p l a n e and bounded by Now w r i t e i n 2 By e s t i m a t e s as i n 51.6 $ E L f o r

exp(-yImh) t h e r e ( t h e c a l c u l a t i o n s a r e i n d i c a t e d i n §1.6). say (11.8) $ =

x

4

Q (2i/x)[qh(x) -

Coshx].

r e a l and by Paley-Wiener ideas ( c f . Theorem 1.3.8 f o r example) K(x,g)exp(ihE)dg

= 0).

Since

= 2i$

K(x,c)Sinhgdg

flhSinxgK(x,c)dg

For q = h = 0 we s e t K(x,g)

[l - K(x,x)]Coshx

+

J$

IL =

( n o t e K(x,c) i s odd i n 5 and K(x,O)

= -K(x,x)Coshx n

CHZBREIII 11.6.

(0)

=

+

ft K5 (x, 0, ( l + i A ) / ( l - i X )

+

i s analy-

tic). Now g e n e r a l l y speaking t h e r e a r e s e v e r a l s t a n d a r d procedures i n l i n e a r est i m a t i o n , e.g.

smoothing, f i l t e r i n g , p r e d i c t i n g , i n t e r p o l a t i o n , e t c .

Let

us f o l l o w [Kul-221 i n d e s c r i b i n g some background s i t u a t i o n s i n a semi-heuri s t i c manner.

Thus l e t us imagine a s i g n a l Zt p e r t u r b e d by a d d i t i v e w h i t e (0 5 t 5

n o i s e Vt and o b s e r v a t i o n s Yt = Zt t Vt

T).

One can e n v i s i o n com-

p l e x v e c t o r processes e t c . b u t we w i l l t h i n k o f r e a l valued s c a l a r s f o r simp l i c i t y f r o m which development a small amount o f n o t a t i o n a l adjustment l e a d s t o t h e more general s i t u a t i o n . (3.11)

+ VtZs)

E(ZtZs + ZtVs

L e t us w r i t e

= K(t,s)

We t h i n k o f Zt and Yt as second o r d e r processes w i t h mean 0 and assume K ( t , s ) i s continuous on [O,T]

x [O,T].

(3.12)

= 6(t-s)

= EYtYs

R(t,s)

i s however a covariance.

The f u n c t i o n K need n o t be a covariance.

+ K(t,s)

Two p a r t i c u l a r cases o f i n t e r e s t i n v o l v e

Zs f o r a l l s , t so t h a t K ( t , s )

i s a covariance

(B) V t I Z s f o r t

>

(A)

V t I

s which

a l l o w s causal dependence o f Z on Y o r feedback. The problem o f smoothing p r e s c r i b e s t h e o b s e r v a t i o n s Y,,

0 5 s 5 T, and

asks f o r

lo I

(3.13)

?(tlT) =

such t h a t E \ Z t

-

H(t,s)Ysds

A

Z(t\T)I

2 be a minimum.

We t h i n k o f a H i l b e r t space Hy gen-

e r a t e d by Yt as i n Theorem 3.2 so t h a t f o r t f i x e d ? ( t l T ) i s a l i n e a r l e a s t

242

ROBERT CARROLL

squares a p p r o x i m a t i o n t o Zt and t h e p r o o f o f Theorem 3.2 g i v e s t h e necessary and s u f f i c i e n t c o n d i t i o n ( * ) 0 = E(Zt - ? ( t l T ) ) Y s f o r a l l s

[O,T].

Now w r i t i n g o u t t h e o r t h o g o n a l i t y c o n d i t i o n ( * ) we o b t a i n ( E V s V T = 6 ( ~ - s ) ) E

rT (3.14)

H(t,s)E(ZsZT+VsZT+ZsVT+VsVT)ds

+ ZtVT) =

E(ZtZT

=

.T

’0

H(t,.r) +

]

’

H(t,s)K(s,T)ds

0

(A)

above h o l d s t h e n one o b t a i n s a Fredholm e q u a t i o n f o r H T K(t,T) = H(t,T) + Jo H(t,S)K(s,T)dS. This i s o f t h e form ( K ( ~ , T ) = K(T,s)) If

CHEOREIII 3.7.

o f t e n w r i t t e n K + H + HK o r ( I - H ) ( I + K ) = I and H i s c a l l e d t h e Fredholm r e s o l v a n t o f K. When T = t we have what i s c a l l e d a f i l t e r i n g problem and one w r i t e s

?(tit)

(3.15)

t

h(t,s)Y,ds

= 0

CHEOREM 3-8- Under c o n d i t i o n s (A) o r ( B ) h ( t , s ) s a t i s f i e s f o r 0 t K(t,T) = h ( t , T ) + f0 h(t,s)K(s,T)ds.

Pfiood:

T h i s f o l l o w s immediately f r o m (3.14)

i n g t h a t i n case ( B ) f o r

:t,

T

5 t,

( w i t h h r e p l a c i n g H) upon n o t -

= E(ZtZT

K(t,.r)

T

+ ZtVT) ( i . e . EVtZT

= 0).

=

The f i l t e r i n g i n t e g r a l e q u a t i o n i s thus a c o l l e c t i o n o f Fredholm equations (indexed by t ) and i s c a l l e d a Wiener-Hopf (W-H) equation.

Such equations

a r i s e i n many areas o f mathematical p h y s i c s and t h e r e i s an e x t e n s i v e l i t erature (cf.

[Bbpl;

Kr4,7,8;

Sthl-41).

It w i l l be i n s t r u c t i v e t o s k e t c h a rough procedure f o r s o l v i n g

REflARK 3.9,

t h e e q u a t i o n o f Theorem 3.8 as f o l l o w s ( c f . [ K u ~ ] ) . able) function ( 0 5 s,t < s

t and M+(t,s)

1; M ( t , s ) f ( s ) d s .

( t ) K,

=

a)

d e f i n e i t s causal p a r t M+(t,s)

i s a (suit-

as M ( t , s ) f o r

Then w r i t e M + f ( t ) = ft M + ( t , s ) f ( s ) d s

f o r s > t.

= 0

I f M(t,s)

=

The W-H e q u a t i o n o f Theorem 3.8 can now be w r i t t e n as

(hR)+ where R

%

~(S-T)

+ K = I + K and we can t a k e h = h+ w i t h no

*

Suppose we can f a c t o r R = TT where T = T i s causal +* -1 -1 and c a u s a l l y i n v e r t i b l e (T-’ = (T )+ = T+ - see below - and T ( t , s ) = loss o f generality.

*

T ( s , t ) SO T g w i t h g, hT = KT*-’

i s anticausal).

0 and hTT

= f

g(T*)-’.

*

=

Now i f h s o l v e s

hR = K + g ( i . e .

Now hT = hT ,+

(t)

g i s anticausal).

i s causal and gT*-’

one o b t a i n s hT = (KT*-l)+ w i t h h = (KT*-l)+T-l tion. (TT*

-

t h e r e must be a f u n c t i o n Then f o r m a l l y

i s a n t i c a u s a l so

which s o l v e s t h e W-H equaOne can a l s o r e f i n e t h i s i n u s i n g R = I + K o r K + R - I so KT*-1 = *-1 -1 I ) T * - l = T - T*-’ and hence h = ( I - T T ). However i t i s easy +

LINEAR ESTIMATION

243

t o see t h a t T *-’ = I and hence (**) h = I - T -1 . The f a c t o r i z a t i o n R = TT* with T = 1, e t c . i s c a l l e d canonical and when possible i s c l e a r l y unique because of t h e causal and causally i n v e r t i b l e requirement. One sees theref o r e formally (from ( * * ) ) t h a t f i l t e r i n g and canonical f a c t o r i z a t i o n a r e equivalent ideas. One f u r t h e r deduction from t h e above remarks i s t h e so c a l l e d Siegert-Krein-Bellman i d e n t i t y . Thus l e t H be the Fredholm r e s o l vant of K defined by Theorem 3.7 so t h a t given a canonical f a c t o r i z a t i o n R = TT* w i t h h = I - T - l , I - H = ( I t K)-’ = R - l = (TT*)-1 = T*-lT-l = +

( I - h * ) ( I - h ) . Consequently one has formally H sions of t h i s w i l l appear l a t e r in more d e t a i l .

=

h* + h - h*h; o t h e r ver

RENARK 3.10. Given wide sense s t a t i o n a r y processes we w r i t e R ( t , s ) = EYtYs * = R(t-s) a n d t h e f a c t o r i z a t i o n R = TT corresponds t o t h e s p e c t r a l factori z a t i o n discussed e a r l i e r in Theorem 3.4. T h u s l e t S ( 1 ) = S y ( x ) = FR(T) ( c f . ( 3 . 1 ) ) and w r i t e t h e s p e c t r a l f a c t o r i z a t i o n as Is^[* = S w i t h s ( t ) = F-’; causal a n d c a u s a l l y i n v e r t i b l e (2 i). Let us i d e n t i f y T and s then so i n (**) one o b t a i n s PI,

(3.16)

h

=

F-’[1

- (l/ s and t h e n N

E J ( t ) J ( s ) = E V ( t ) V ( s ) + EV(t)?(s)

n

L

Y

-.s

+ E?(t)y(s)

+ E?(t)V(s).

f o r s < t we have y ( t ) l Z ( s ) f o r s < t so E?(t)?(s)

= EZ(t)Z(s).

A l s o EV(t)?(s) = E V ( t ) Z ( s )

0 f o r t > s w i t h EV(t)Y(T) = 0 f o r t >

T

-

EV(t)?(s) = 0 since EV(t)Z(s) =

holds f o r t < s g i v i n g E J ( t ) J ( s ) = E V ( t ) V ( s ) (= 0).

EIJ-VI'

=

El?\*

t)

-

(X < t )

SinA(x-t)wdA/A = [G(x+t)

G(t-x)]

An a n a l y s i s o f k e r n e l s as i n 52.11 a l l o w s us t o w r i t e B ( y , t ) = A-'(y)G(y-t)

+

Kt(y,t) and t h e k e r n e l K a r i s e s i n t h e form ( r e c a l l K(y,y)

I

Y

0 qh(y) = A-'(y)CosAy

(8.16)

= 1 - A-'(y))

+

Kn(y,n)CosAndn

0

Consequently u s i n g (8.15) one o b t a i n s (B(y,t) = 0 f o r t > y ) Y

(8.17)

B(y,t)A(t,x)dt

0 =

-

= A-'(y)[G(x+y)

G(y-x)]

+

'0

K,(y,t)[G(x+t)+G(x-t)]dt

+

Kt(y,t)[G(x+t)

-

G(t-~)]dt

The l a s t i n t e g r a l s i n (8.17) a r e ( r e c a l l K(y,O) = 0 and G(0) = 1 )

X

K(y,t)[G'(x+t)-G'

(x-t)]dt

-

0

2K(y,x)

+ K(y,y)[G(x+y)-G(~-x)]

-

e

Y

K(y,t)[G' ( x + t )

K(y,t)[G'(x+t)

-

G' (t-x)]dt

-

=

G'(Ix-tl ) I d t

0

Using ~ ( y , y ) = 1

&HE0RElll 8.6.

-

A-+(Y),

i n s e r t (8.18) i n (8.17) t o g e t (8.13).

Hence

The G-L equat on (8.13) can be d e r i v e d i n a c a n o n i c a l manner

as i n d i c a t e d .

RZmARK 8-7. Going back t o 8.5) f o r a moment we n o t e t h a t i t may be d i f f i c u l t t o r e a l i z e a 6 function e x c i t a t i o n f o r v (t,O). Y an i n p u t v ( t , O ) = f ( t ) w i t h r e a d o u t v(t,O) = g ( t ) . Y known readout f o r a 6 f u n c t i o n i n p u t . Then i n f a c t

L e t us suppose i n s t e a d L e t g 6 ( t ) be t h e un(0)

g ( t ) = 1; g 6 ( t - r )

f(- r)dr (which w i l l say i n p a r t i c u l a r t h a t once g6 i s known any o t h e r g can be computed).

Indeed i f v 6 ( t , y )

i s the s o l u t i o n o f (8.4)-(8.5)

w i t h v (t,O) Y

REFLECTION DATA

281

t

v 6 (t-T,y)f(T)dT. For y > 0 we can w r i t e then t 6 6 v t ( t , y ) = fo v t ( t - T , y ) f ( T ) d r s i n c e v ( 0 , y ) = 0 (use (8.10) w i t h a minus t 6 s i g n ) ; s i m i l a r l y v t t ( t , y ) = f0 vtt(t-T,y)f(T)dT and t h e r e f o r e ( 8 . 4 ) i s sati s f i e d f o r y > 0 ( i . e . vtt = (Av ) /A. Clearly v ( t , y ) = 0 f o r t 5 0 by conY Y s t r u c t i o n and v ( t , O ) = 10" & ( t - r ) f ( . r ) d T= f ( t ) by a l i m i t argument as y + 0. Y Now t h e problem i s t o determine g6 from ( a ) , given f and 9, and t h i s may not have a unique s o l u t i o n ( s e e [Af1,2] f o r a discussion of this p o i n t ) . For example i f t ( s ) = (Ca)(s), t: denoting the Laplace transform, then $ ( s ) =

= 6 ( t ) consider v ( t , y ) = So

A

A

A

g 6 ( s ) f ( s ) and i f f ( s ) vanishes in an unpleasant manner t h e r e will perhaps not be a unique determination of G6(s). In some instances however g can be recovered in t h e form p 6 ( t ) = C-'[ 0 and Y = 0 f o r t < 0

-

c f . Example 8.2).

t h i s e v e n t t h e s o l u t i o n o f ( 8 . 4 ) w i t h (8.7) i s v ( t , y )

In

One can

= Y(t-y).

t h i n k o f t h i s s o l u t i o n v = Y ( t - y ) as an " i n c i d e n t " o u t g o i n g wave vi which

w i l l i n f a c t be p r e s e n t f o r a l l problems ( 8 . 4 ) when A(0) = 1 ( c f . (8.23)); t h e c o r r e s p o n d i n g 'lint-ident" response d a t a G ( t ) = Y ( t ) w i l l be denoted by

+ Gi(t)

Thus l e t us t h i n k a g a i n o f decomposing G ( t ) = G r ( t )

Gi(t).

where

G r ( t ) r e f e r s t o a r e f l e c t e d displacement component a t y = 0 ( o r r e f l e c t i o n d a t a ) and t h i s corresponds t o w r i t i n g ( t > 0 ) (8.23)

1;

G(t) =

( c f . a l s o [Bbgl;

[Sinkt/k][du

Bol; S t y l ] ) .

b e f o r e ( c f . (8.13);

=

(2/7i)dk] = 1

+

:1

[Sinkt/k]do

+ Gr

Gi

=

T h i s l e a d s t o an e x p r e s s i o n f o r T(y,x)

t h u s T(y,x)

Consequently f o r x 5 y o r x

+

= ; !$

-

[[Sink(y+x)/k]

as

[Sink(y-x)/k]]du(k).

y respectively

4[G (y+x) - Gr(y-x)]

r

o r = -%[Gr(y+x)

+ Gr(x-y)l

.

-

F o r m a l l y t h e n we can w r i t e a g a i n T (y,x) = %[G;(y+x) Y a l s o t h a t G h ( t ) = J t Cosktdo(k) i s an even f u n c t i o n ) . i s n o t u n r e a l i s t i c t o suppose t h a t G I

E

It

Co o r GA piecewise continuous ( c f .

[ G o l l ) , b u t i n f a c t one can develop s t a b i l i t y e s t i m a t e s based on weaker L t y p e measurements o f t h e a p p r o x i m a t i o n t o Gh.

Using (8.23)-(8.24)

1

we ob-

t a i n Theorem 8.5 and we w i l l use t h e G-L e q u a t i o n i n t h e f o r m (8.13) f o r stability.

Thus suppose one i s g i v e n approximate d a t a G

*

0 w i t h corresponding k e r n e l K (y,x) s a t i s f y i n g (8.13)

*

-

Gr(t)

GF

Co (so

E

= 1

Gr(t)

-

A(y).

K(y,y))

and assume data G; € ( a )

E Co).

*

1 and Gr' E Lloc

We w r i t e AK(y,x)

(so

= K*(y,x)

*

E'

-

*

= 1

etc. 1

E Lloc)

K(y,x)

*

+ Gr f o r t > Set & ( t )= with G

and

r-b

( r e c a l l A '(y)

so measurement o f AK(y,x) e s s e n t i a l l y determines AA = A*(y)

From (8.13) and (8.13)* we o b t a i n ( x < y )

-

ROBERT CARROLL

284

Now it will be useful to make explicit the nature of (8.13) as a Fredholm integral equation (cf. [Cjl] for integral equations). Thus think of y as a parameter and write (x 5 y) (8.26)

C(x,s)

=

%[Gb(x+s)

h

- G~(~s-x~)l; 6Yf(x)

1

Y

=

C(x,s)f(s)ds

n

A

T(y,x)

=

%[Gr(y-x)-Gr(y+x)];

AT

=

%[E(y-x)-E(y+x)];

A6

= %[E'(S+X)-E'(

Then one can write (8.13) and (8.25) in the respective forms

?;

(8.27)

[I-& ]K

(8.28)

[6 - ty]f(x)

*

Y

Y

=

[I-C*]*K = Y

A?

f

(C; - ty)K

=

An alternative form of (8.28-) would be [I - ty]AK = AT + [6; - Cy]K* but it seems more appropriate to introduce estimates in (8.28). We recall here that the existence o f a continuous K(y,x) satisfying (8.13) or (8.27) (and * of a corresponding K (y,x)) is assured by independent considerations (and uniqueness is known). The integral equation (8.27) can be thought of in various spaces depending on the nature of 6(x,s). Thus for C(x,s) E Ltoc one has a standard theory A for 6 in L2 (with T(y,-) considered in L2 ). Similarly for 6(x,s) E Co Y (as can be posited) we have a classical theory for ey in Co (with ?(y,-) E Co). In either theory there is a Fredholm alternative (cf. [Cjl; Rbl]) etc. so we can say that x = 1 is not an eigenvalue of 6 and for any y < m y Y (I - ~ ~ 1 -exists l as an operator in L2 or CO (similarly (1 - c;)-' exists). Given &(x,s) as in (8.26) with Gb E L1 we see also that tT f will be defined Y for f E Lm so EY: Lm + Lm. Let us think of 6 working in Co generally Y (with Gb E C o ) and we will see however that stability estimates can be ob1 estimates o f the approximation of G;' to Gb. In this restained for Lloc pect let us note that if IIfllm = suplf(s)lfor 0 5 s (y then 3Y Y (8.29) l[Ci - Cylf(x)I 5 IIfllm,Y IA6(xys)lds 5 411 fll

m3Y

~[IE 0

I

( I S- X I ) 1 + I E ( S+X ) I ]dS 5 11 f11

''1

yy

0

I

E

I

( 5 ) (dS

285

REFLECTION DATA

< IIfllm I I E ~ I I 1 which means f o r o 5 x 5 y. Consequently II[C* - C ~ I ~ I I * Y COSY*Y Y L (2Y)' t h a t IIAe II = IIC - E II < IIE'II 1 where Ile: - C II r e f e r s t o t h e o p e r a t o r Yo Y Y L (2Y) Y Y Now i n o r d e r t o e s t i m a t e AK i n (8.28) l e t us g i v e an norm i n C on [O,y]. -1 and e s t i m a t e f o r (I- e*)-l, which we know t o e x i s t , i n terms o f (I- Cy) * Y Thus d e n o t i n g by L(E) t h e space o f continuous l i n e a r . e s t i m a t e s on e Y - eY o p e r a t o r s i n a Banach space E we have ( c f . [Ogl])

I f E E L(E) w i t h (I- C ) - '

CEllMlA 8.11. then ( I

-

C*)-l

e x i s t s and II(1

-

L ( E ) and II&*

E

C*)-'Il

0 ) from t h e measured response v ( t , y )

at y =

7

(A(y)v ) ( 0 z y < t Y Y ( t h i s corresponds t o x = ?

=

03,

say i n ( 8 . 1 ) ) due t o an i m p u l s i v e e x c i t a t i o n p l a c e d a t y = 0.

I t i s assumed

and A ( y ) = A- f o r y ~y ( i . e . x 2 ; ) w i t h again t h a t 0 < a 5 A(y) < 6 < 1 A E C . T h i s problem i s q u i t e d i f f e r e n t from those c o n s i d e r e d i n r e f l e c t i o n seismology, where t h e measurements a r e made a t y = 0. e.g.

The problem a r i s e s

as a subproblem i n an i n v e r s e problem f o r t h e r e c o n s t r u c t i o n of a

s p h e r i c a l l y symmetric s c a t t e r e r i n t h e t i m e domain.

Another a p p l i c a t i o n i n -

v o l v e s s t u d y i n g m a t e r i a l p r o p e r t i e s o f a l a y e r e d medium i n a water b a t h experiment; t h i s c o u l d a r i s e e.g. t i v e evaluations. (8.7), (9.1)

i n bio-medical tomography and non-destruc-

The boundary c o n d i t i o n s corresponding t o t h e problem a r e

v i z . vt(O,y) = &(y) and t h e r e a d o u t i s V(t,y)

= h(t)

TRANSMISS ION DATA

289

7

The condition A(y) = Am f o r y can a l s o be regarded as a r a d i a t i o n bounda r y condition a t y = y". Referring t o (8.10) we can w r i t e (9.2)

so t h a t

H(t)

v(y,t)

=

q XQ( y " ) G / A =

v!(y")

(9.3)

\

=

( ~ Q, ( ? ) , [ S i n x t / x ] ) ~

IT)/: H(t)SinAtdt and from (8.11)

m

lo m

G(t)Sinxtdt =

H(t)Sinxtdt

0

The function q:(y) is an even e n t i r e function of exponential type ? and t h e expression of G in terms of H in ( 9 . 3 ) can be regarded in t h e context of deconvolution ( c f . [RgZ; S t f l l ] ) . Indeed by Paley-Wiener ideas (a i s even)

A

(where

N

denotes t h e Fourier transform).

be odd extensions of G and H then

EHEOREM 9-1, The readouts G a t y

;(x)EA = 0 and

rJ

Similarly i f we take G and H t o = K" and

H at y

=

y" s a t i s f y

@

*

N

N

G = H.

EXNIPLE 9-2, We note f o r A = 1 as i n Example 8.2 we have v = Y(t-y) ( y , t >

0 ) , G ( t ) = Y ( t ) , and H ( t ) = Y ( t - ? ) = 6(t-?) * Y ( t ) . Since 6 ( t + ? ) * 6 ( t - y ) = 6 ( t ) t h e deconvolution here i s expressed by G ( t ) = 6 ( t + y ) * H ( t ) = H(t+y).

However working from Theorem 9.1 i s not too productive. rv

0 convenience) Ip,(l)

ble can w r i t e ( t a k -

a = + [ 6 ( t - 1 ) + s(-t-1)1, E = Y ( t ) - Y(-t), H = Y(t-1) - Y(-t-l), and in f a c t @ * g = F. However a natural s p l i t t i n g involving 6 ( t - 1 ) * Y ( t ) = Y(t-1) i s not v i s i b l e . Indeed cp * Y ( t ) = % [ Y ( t - 1 ) + Y ( t + l ) ] s o @ * G t H while t o deconvolute cp * r = Y(t-1) one obtains (using a Z-transform method a s i n [RgZ]) r = 2 C(-l)kt'Y(t-2k) ( k = 1 k = m ) which is a s t e p function of no apparent physical i n t e r e s t . T h u s t h e need t o s p l i t G and H simultaneously so t h a t G and H a r e involved d i r e c t l y presents some d i f f i c u l t i e s and we will proceed d i f f e r e n t l y below. ing y = 1 f o r -.,

= COSX,

-f

REIRA?W 9-3. Again taking y " = 1 f o r convenience l e t us w r i t e out t h e convolu-

t i o n in Theorem 9.1 as follows (a i s even w i t h supp @ c [-1,1], and r a r e odd), ';ict) = kjjl @(T)g(t-T)dT = @(c)i[c(t-c) + E(t+c)]dg and t h i s i s seen t o represent a c l a s s i c a l type domain of dependence s i t u a t i o n f o r the sideways Cauchy problem. The value of K ( t ) depends only on C ( c ) i n the region t-1 5 5 5 t + l and evidently one may remove the ?J in H and G f o r t > 1 .

/o'

N

-

Note H ( t ) = 0 f o r 0 5 t < 1 . One can a l s o derive t h i s s o r t of formula d i r e c t l y by working w i t h t h e sideways Cauchy problem and constructing appropr i a t e Riemann functions as before.

290

ROBERT CARROLL

L e t us use t h e t r a n s m u t a t i o n machine t o s p l i t up e v e r y t h i n g as we go along. R e c a l l t h a t ;(A)

1 / 2 n l c Q ( A ) 1 2 i s even and v pQ , ( y ) i s even i n A.

=

Also f o r

c a l c u l a t i o n i t w i l l be convenient t o remove t h e l / x f a c t o r i n (9.2). (9.5)

H ' ( t ) = (vf(y),CosAt)u

=

Thus

jOm

+f(y)eixtdx

F u r t h e r G ( h ) v f ( y ) = (1/21~)[*:(y) + *Qx(y)] where Qx(y) Q = @Qx ( y ) / c q ( - x ) i s ana l y t i c i n t h e upper h a l f p l a n e e t c . Hence s e t

1

1

m

(9.6)

Hl(t)

= (1/4n)

m

dQ,(7)eixtdA

= (1/4n)

m

*:(F)e-ihtdx

m

and i t f o l l o w s t h a t H ' ( t ) = H l ( t )

+ H1(-t).

Again we remark ( c f . Example

8.2) t h a t i n u s i n g t h e F o u r i e r t h e o r y , o r e q u i v a l e n t l y i n r e p r e s e n t i n g H by ( 9 . 2 ) and H ' by (9.5) etc.,

one a u t o m a t i c a l l y i n t r o d u c e s v a r i o u s odd and

even e x t e n s i o n s o f t h e q u a n t i t i e s G, H, e t c . ( c f . a l s o (8.11)). t h a t by formal c o n t o u r i n t e g r a l arguments Hl(t) deed by now standard arguments and p r o p e r t i e s

4

so *,(y")exp(-iAt)

Q

cexp(iA(?-t))

h a l f p l a n e Imx > 0 so f o r (9.6) i s zero.

Thus H l ( t )

7>

and f o r

7 as

-7 because

7 (and

H1(-t) i s

o f t h e r e p r e s e n t a t i o n (9.5) - i t Now from (9.6) we can w r i t e

? large

enough so t h a t

p

and 1-1 a r e

2 2 (as i n d i c a t e d above) and t h u s A(y) = Am (known) f o r y 2 But ax(y) Q w i s t h e J o s t s o l u t i o n @,(y) Q A?exp(ihy) as y + m Q

i n d i c a t e d we must have t h e n a,"(?) = A?exp(ixY).

CHEORETR 9-4, Under t h e hypotheses i n d i c a t e d , f o r (9.8)

In-

c e x p ( i h y ) f o r Imh > 0

on a l a r g e s e m i c i r c u l a r c o n t o u r i n t h e

F u r t h e r l e t us t a k e o u r r e a d o u t p o i n t

(y"= y ( z ) ) .

'L

t t h i s vanishes s t r o n g l y and t h e i n t e g r a l i n

c o n t r i b u t e s n o t h i n g t o H i f o r t > 0).

constant f o r x

0 i n (9.6) f o r t < y.

=

*:(?)

provides t h e readout H i f o r t >

simply tagging along f o r t
0;

t h i s means K ( t - T ) = 0 f o r t < 0 as d e s i r e d and moreover

= 0 f o r T > t+y.

&HE@REW 9-7, Given (9.14)

Q (-A)

m

K(t-T)

Consequently we o b t a i n expressed as above ( i n (9.13))

K(t-T)

it follows that

h o l d s which g i v e s a f i n i t e domain o f dependence r e l a t i o n between H1

and G1.

*

Now f o r t h e s t a b i l i t y q u e s t i o n we f i r s t l e t H pedance A

*

.

W r i t e AW = w

*

"oh and n o t e t h a t A5 = Am.

-

W,

AG = G

*

-

* ,G ,

G, e t c .

etc. r e f e r t o the im-

L e t us w r i t e a l s o do =

Estimates on A 5 a r e t r a n s m i t t e d t o AGr by AGr =

292

ROBERT CARROLL

Now estimates on AGr and AG; on f i n i t e i n t e r v a l s a r e going t o involve estimates on AO in L 1 ( 0 , m ) and such estimates will be d i f f i c u l t t o verify i n p r a c t i c e ( i n terms of AH1 s a y ) .

=

1: Au[SinXt/A]dh and AG;

= f t AuCosxtdx.

Hence l e t us use t h e a u t o c o r r e l a t i o n type function K ( t ) = fz H1 ( t + . r ) H 1 (-r)dr A Y of Remark 9.6 (nC = (a/2Am)G(A) = IH1 12, H1 = 0 f o r t < 7). From ( 9 . 9 ) , w i t h G ' considered even because of t h e cosine r e p r e s e n t a t i o n , we obtain

1

m

(9.15)

G ' ( t ) = (Am/a)

lfi112e-ihtdh = Am3C(t)

m

A f a c t o r of 2 ( i . e . 2Arn3C(t) i n ( 9 . 1 5 ) ) a r i s e s because of the cosine representation and must be removed when considering G' via t h e f u l l Fourier t r a n s form formulas ( c f . CC401 and e a r l i e r remarks - note the i n t e r p l a y between one and two sided d e l t a functions v i a (2/71)f: Cosxtdh = 6+ while ( 1 / a ) /f e x p ( - i h t ) d h = 2 s ) . We conclude t h a t &€Ea)R€lll 9-8, For t > 0 one has (9.15) or G ' ( t ) = Am3C(t).

REI[IARK 9-9. This i s very nice in giving a d i r e c t r e l a t i o n between H ' and

GI

so t h a t s t a b i l i t y estimates can be made d i r e c t l y via properties of H' and H without intervention of t h e s p e c t r a l measure. Unfortunately i t does not e x h i b i t the nice dependence o f G ' on only a f i n i t e range of H ' a s i n Theorem 9.7 ( b u t of course Theorem 9.7 i s n o t s u i t e d t o c a l c u l a t i o n since ~ Q ~ ( 0 ) i s not determined). Let us f a c t o r out t h e d e l t a functions i n (9.15) formally a s follows. Again work w i t h G = 1 + Gr, = 6 ( t - 7 ) + h l , e t c . f o r t > 0 and one obtains formally (note

7-t

eH1

0 and h l ( T - t ) = 0 ) rn

(9.16)

Gk(t) = hl(y+t) +

To obtain estimates now on

Y(t-7) +

h(t), h ( t ) =

t fy

E

i

and

hl(T)dT

hl(t+T)hl(T)dT E'

(hl

we assume f i r s t t h a t f o r t =

1-

h ' ) , and Gr(t) =

7

7, A2H

t fo G;(.r)d.r.

=

Then

m

(9.17)

G r ( t ) = h(t+y') +

h(t+c)h'(c)dc Y 1 2 from (9.16) ( c f . [C71,72]) since h l ( c ) h ( c ) d c = - i h 2 (c)I; =zh (y) = 0 Y i f h ( - ) = 0. I t should be no problem t o assume h and hl E L1 n Lm say and

-Jz

TRANSMISSION DATA

Ah(t+c).

293

Consequently from (9.18)

IE'I

(9.19)

5 lAhl(t+Y)I

6

f

[Ih;

m

]El

y,

-

p

K,(y,n)CosAqdr,dx

= +A~[G(?+T)

+

K2(?,t+.r)

does n o t a r i s e ) .

G(r-7) -

G(~-T)] +

co

%jo

G(t)[K2(?,t-T)

+

[C71,721) =

K2(?,T-t)

=

K2(Tyt+-c)

(9.22) becomes ( c f . (9.36) a l s o )

H(r)

= 0

- ~(y-t-~)]

+ K2(T,T-t)

where K2(c,n) = K (5,171 ( n o t e f o r

(9.24)

K(y,n)

q!(?)SinhtSinhdx

0 %[Kz(?,t-T)

y,

[$(X,y)/2i]Sinhn

c'

I = (2/n)

G(t)I(t,T,y)dt;

J + %A:[&(?-

(2/~)/:

Then f r o m (9.3) i t f o l l o w s t h a t ( c f .

00

(9.22)

A = Am a t

=

-

K2(?,t+~)ldt

Hence

294

ROBERT CARROLL

Take now

T

>

7 so

G(?-T)

=

0 and

0.

=

K2(T,t+T)

We can w r i t e t h e i n t e g r a l

term i n (9.24) i n t h e form ( i n t e g r a t i n g by p a r t s )

[+'

(9.25)

+ [-?%G

kG(t)K2(Y,t-T)dt

( t ) K2

(y,- t ) d t

=

c

%K(y,y)[G(y+~) +

G(T-~)] -

Now use K(T,Y) = 1 For

CHE0REN 9-11.

- ' :A

T >

+

(y%G'(s+T)K(y",s)ds J

F%G'(r-s)K(y,s)ds J

0

0

t o obtain T o n e has

%v c

(9.26)

H ( T ) = % [ G ( ~ + T ) + G ( T - ~ )+]

K(F,s)[G'(T-s)

-

G'(T+s)]~s

'0

L e t us n o t e i n p a s s i n g t h a t f o r t h e G-L e q u a t i o n (8.13).

T

and ?(T)

N

H(T) = %[Gr(jh)+Gr(~-y)] + 4

(9.44)

K(~,s)[G~(~-s)-G;(~+s)]ds

which f o l l o w s immediately from equations (9.24)-(9.25).

7 5 t f 37;

now G data f o r 2 ? 5 t 5 4 7 and H d a t a f o r

g5

G data f o r

T-2F5

(T

1)

27.

t5

7) {I

Take f i r s t

K(F,s)Gl(T-s)ds_= ds = - K ( ? , s ) G r ( ~ z s ) I T - ~ y + I/-2j7

P

+$-:I

z-2T)Gr(2?) (9.45)

-

-

T - 2 7K(Y,s)G;(T-s)ds

1:-~7

Ids with K2(?,S)Gr(r-s)ds = -K(?,Y)G,.(T

+

%G,(?+T)

=

t h e n we w i l l c o n s t r u c t

?. 2 7 Ln (9.44) and wr-ite

[IDT -27

K2(y,~-u)Gr(u)du.

H(T) = H(T)

-$lo

T

We assume g i v e n

?5Gr(r-7)+

+

%r $?-

,The ., e q u a t i o n

5 3 7 so

(T

K(r,~)G,l.(~-s)

-7) + K(T,

(9.44) can be w r i t t e n

K(y,s)G;(r+s)ds

-~K(T,T-~T)G~(~Y)

0,

K2(r,T-")Gr(o)du

-%K(y",y)G,(~-y)

5

=

%[

= % [ l - K ( ~ , ~ ) ] G r ( ~ -+ ~)

7T-Y K2(Ty.r+y-t)Gr(t-y")dt

We see t h a t H(T) i n v o l v e s "known" i n f o r m a t i o n w h i l e t h e r i g h t s i d e o f (9.45)

-

i n v o l v e s (unknown) G data on [0,2y] o t h e r hand i f w%take

IT-' -

Gr(T-s)ds. Gr(T-s)ds

T-Y t o

5

27 i n

a c t u a l l y G d a t a on [~-?,2?&

(9.44)-(75

T

I$y-T K2iy,s)Gr(T+s)ds

Similarly =

5

;(T)

=

-

+ IJ Kz(yh,s)Gr(T-s)ds

%Gr(T+~) +

= %[l-K(y,y)]G,(~-y)

( r e c a l l here K(y,x)

u

+

5

then f i r s t =

T

;1

(note here

On t h e

{I

K(y,s)

L(7,S) + {I

Kz(Yys)

T-s

r u n s from

5 2 7 ( c f . [C78])

K(y"s)Gk(r+s)ds

+ %K(yy2y-~)Gr(27)

4

i s odd i n x w h i l e K2(y,s)

Hence t h e r i g h t s i d e o f (9.46)

27)

K ( ~ " , Z ~ - T ) G ~ ( Z -~ )$y-T_K2(Y,s)

-7

H(T)

5

= -K(~,s)G~(T-s)

Thus (9.44) becomes f o r =

T

K(F,s)Gb(.r+s)ds

I8 K(r,sLG;(T-s)ds

-K(F,?)Gr(~-y)

27).

'v

(9.46)

T

[Ify &+: I-: I d s w i t h

G;(T+S)dS: G,(T+s)

-

i s even i n x f o r x < y).

has e x a c t l y t h e same form as t h e r i g h t s i d e

TRANSMISSION DATA

299

27,

o f (9.45). We n o t e a l s o that- i n H ( T ) where T 5 [ K(y,s)Gk(r+s)ds I % -%I0T - 2 y K(T,s)GL(T-s)ds = K ( y , t ) G L ( T + t ) d t which c o i n c i d e s i n form w i t h a t e r m i n f ; ( ~ ) . I n f a c t H and f; a r e i d e n t i c a l . Assume G i s g i v e n on [27,4?]

LEl!tilFIIA 9.19.

'v

for_ o 5 x 5 Y ) . %I$-T K(?,s)G;(T+s)ds

Write

K2(y,x)

H(T)

=

;I(T)

for

y~

T

i s even i n x f o r x

s set (10.8)

P [ N ( t ) - N(s) = m] =

t a(‘)d?lrn m! exp[-L

with P = 0 f o r m < 0 and P = 1 f o r m = 0 and t = s.

a ( ~ ) d ~ ] (m 2 0) As in [Dal], pp. 159-

160, N(t) may be considered as a random v a r i a b l e , i . e . N ( t ) =

N(t,W),

w E

i s t h e s e t of a l l functions W : t + u ( t ) = N ( t , w ) from [0,m) t o t h e non-negative i n t e g e r s , which a r e zero f o r t = 0 , continuous from t h e r i g h t , and nondecreasing, and which have only f i n i t e l y many d i s c o n t i n u i t i e s i n each f i n i t e t i n t e r v a l . T h u s P i s a p r o b a b i l i t y measure on (n,A) where A i s t h e u-algebra generated by a l l s e t s of t h e form { w ; N ( t , w ) = k3 f o r t 2 0, k a non-negative i n t e g e r (E{ 1 is t h e associated expectation operat o r ) . Finally f o r t h e process N ( t ) we define the random v a r i a b l e T ( t ) = T(t,w) as T ( t ) = 1; ( - l ) N ( T ) d T ( t t ) . Now a prototypical theorem o f the

R, where C

304

ROBERT CARROLL

type indicated i s given i n [Kcl] a s Let v ( t ) be twice continuously d i f f e r e n t i a b l e i n (-T,T). I f we define u ( t ) = E [ v ( T ( t ) ) ] then u ( t ) s a t i s f i e s (*) u " ( t ) + 2 a ( t ) u ' ( t ) = v(0) and ( B ) lim u ' ( t ) = v ' ( 0 E [ v " ( T ( t ) ) ] , 0 < t < T , with ( A ) lim u ( t ) + ast+O. CHE0REIII 10.5.

Pkoob: Since I T ( t ) l 5 t , (A) i s immediate. For each w. T i s an absolute Y continuous function of t , and OtT i s bounded from above i n absolute value By t h e c h a i n r u l e one has u ' ( t ) = by 1 , and approaches 1 a s t + 0,. (T(t))DtT(t)]. Applying t h e bounded convergence theorem ( B ) follows. Finally we observe t h a t (*) i s now equivalent t o t h e i n t e g r a l i d e n t i t y

E[vl

(10.9)

u ( t ) = v(0) + v ' ( 0 )

t exp[-ZJ;

a(s)ds]d~t

By t h e Weierstrass approximation theorem i t s u f f i c e s t o prove (10.9) f o r k v(t) = t k = 0,1, .... For k = 0 i t i s immediate. For k = 1 , u ( t ) = E [ T ( t ) ] = Jot E [ ( - l ) N ( ' ) ] d ~ . B u t E [ ( - l ) N ( T ) ] = 1 ; (-lIkP[N(.r) = k] = exp[-Z/,' a ( s ) d s ] . Thus i t remains t o prove (10.9) f o r k 2 2. I f we i n t r o k duce o k ( t ) = E[T ( t ) ] / k ! then by Fubini's theorem ( c f . [Kbl] f o r t h i s " t r i c k " which f o r k = 2 amounts t o replacing an i n t e g r a l of a s y m e t r i c i n tegrand over a square by twice t h e i n t e g r a l over half of the square) u k ( t ) = Jo t d-ckJ;k d . r k - l - - - J ~ ~ d . r l E [ ( -Nl () T 1 ) + " * N(Tk']. B u t i f T~ 5

'

E[(-l)N(T1)+

. * *

+

N('k+2)]

=

N(Tk+l 1] s i n c e N ( . C ~ + ~ )- N ( . r k t l )

E [ ( - l ) N ( T 1 )+

**.

+

N(Tk)]E[(-1)N(Tk+2)-

i s independent of N(.r,),

...,

in our previous c a l c u l a t i o n one obtains E[(-l)N('k+2) - N ( ' k + l ) ]

N(.rk).

As

=

exp[-2JTk+2 a ( s ) d s ] . Thus (replacing T ~ and + ~T ~ by + ~T and e respectively 'k+lt ~ ~ + =~lo( d-r$ t ) exp[-Zz,' a ( s ) d s ] u k ( e ) d e which i s just (10.9) f o r t h e case The r e s t follows. I v ( t ) = tk+', k > 0. Suppose now v ( t ) takes values i n a l i n e a r topological vector space F and say vtt = Av f o r some s u i t a b l e closed densely defined l i n e a r operator A w i t h w i t h v ( 0 ) = vo E D(A) and vt(0) = v1 E F s u i t a b l e ( c f . [C19,29] f o r operat o r d i f f e r e n t i a l equations). Then i n reasonable s i t u a t i o n s A will commute w i t h t h e operator E o f taking expectations so t h a t E[v"(T(t))] = E[Av(T(t))] For example i f F i s a Banach space then E[v(T(t))] would = AE[v(T(t))]. normally be a Bochner type i n t e g r a l ; f o r more general spaces one can t h i n k

RANDOM EVOLUTIONS

o f v a r i o u s weak o r s t r o n g i n t e g r a l s here.

305

Consequently we o b t a i n

&HEOREl!l 10-6- Under hypotheses o f t h e t y p e i n d i c a t e d l e t v" = Av, v ( 0 ) = vo and v ' ( 0 ) = vl. =

Then u = E [ v ( T ( t ) ) ] s a t i s f i e s u"

+

P a ( t ) u ' = Au w i t h u ( 0 )

vo and u ' ( 0 ) = vl.

T h i s k i n d o f theorem i s developed more e x t e n s i v e l y i n [RdZ] f o r example f r o m which we e x t r a c t now a few r e s u l t s .

I n t h e background here i s an e a r -

l i e r paper [Rdl] i n which t h e Ito t h e o r y o f s t o c h a s t i c i n t e g r a l s e t c . i s used t o c o n v e r t a h y p e r b o l i c e q u a t i o n f o r h ( x , t ) i n t o a p a r a b o l i c e q u a t i o n f o r H ( x , t ) = E[h(x,yt)]

where yt i s a c e r t a i n s t o c h a s t i c process.

Thus one

w i l l t o u c h upon s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n s and i n t e g r a l s l i g h t l y and

b r i e f l y here b u t we w i l l make no a t t e m p t t o g i v e a thorough d i s c u s s i o n o f t h e m a t t e r (see e.g.

[Kacl;

Kvl; Gn1,2;

Fgl; Wpl;

Icl; Idl]).

We w i l l em-

p l o y n o t a t i o n and concepts as needed and t h e n g i v e e x p l a n a t o r y comments l a t e r ; some o f t h e p r o b a b i l i s t i c ideas a r e discussed a l r e a d y i n 52.

Thus

l e t f o r convenience F be a Banach space and e.g.

A a g e n e r a t o r o f a con2 t i n u o u s semigroup ( s t r o n g l y ) ; suppose (==) ( 1 / 2 ) e ( t ) u t t + f ( t ) u t = Au. L e t

xt be a d i f f u s i o n process s a t i s f y i n g t h e s t o c h a s t i c d i f f e r e n t i a l e q u a t i o n (10.10)

dxt = f ( x t ) d t + e(xt)dbt

( x o = 0 ) where dbt i s t h e d i f f e r e n t i a l o f a s t a n d a r d Brownian m o t i o n ( t h e v a r i a n c e = 1).

2

Assume e.g. e2 + f2 5 K ( l + t ) so t h a t (10.10)

has a s o l u -

t i o n f o r a l l t 2 0. D e f i n e a d i f f u s i o n t r a n s f o r m (**) t(t) = E[u(xt)], t Now by t h e I t o c a l c u l u s du(xt) = u ( x )dxt + ( 1 / 2 ) u t t ( x t ) ( d x t ) 2 =

> 0. -

Zt

ut(xt)[f(xt)dt + e(xt)dbtl + (1/2)u ( x )e (xt)dt = Au(xt)dt + ut(xt)e(xt) Ztt t2 One assumes e.g. (***) El: ut(xt)e (xt)dt < m f o r s dbt. 0 and i t f o l lows t h a t Eu(xt) - u ( 0 ) = E l ot Au(xs)ds = lot AEu(x,)ds ( E l ot y ( w , t ) d b t = 0 general l y ) .

&HE@REN 10.7,

Hence f o r m a l l y Under t h e hypotheses i n d i c a t e d

determined by (==)-(10.10),

s a t i s f i e s $,

=A;

û

= E[u(xt)],

w i t h u and xt

and G(0) = u ( 0 ) .

EXAIIIPCE 10-8. Take f = 0 and e = J2 so xt = J2bt = b ( 2 t ) and G ( t ) = _/fu ( x ) 2 exp[-x /4t]dx/(4at)'. A r e l a t e d example a r i s e s a l s o f o r e = 42 and f = 2 k / t ( k > 0) where xt i s t h u s a Bessel process i n 2k+l dimensions g i v e n by dxt = 2 k d t / x t + J2dbt

(xo = 0).

One knows f r o m [ I d l ] ,

p. 60 t h a t ( z > 0 )

2 (10.11)

P[xt E dz] = dPt(z)

= [e-'

/4tz2ktk-4 / Z Z k r ( k+4)]dz

306

ROBERT CARROLL

= A: where ;)(t)= and hence i f utt + ( 2 k / t ) u t = Au i t f o l l o w s t h a t u(x)dPt(x) = C t -k-4 /2 2kr(k+ 0). One w r i t e s a l s o W+ = W n W., Thus one i s d e a l i n g w i t h

kFexp(iAx)dx = f k F ) .

one r e f e r s t o W functions

F

E

W,

a n a l y t i c ?or Imh >

(resp. W-)

e a s i l y t h a t n(A,t)

= U’,

E

W

I

6 (resp.

f o r example and v a r i o u s p r o p e r t i e s o f t h e

m a t r i x blocks i n U a n d a n r e established i n [ D u ~ ] . notes I m x > 0, C-

Q,

EE ,+

#

I n p a r t i c u l a r (C,

de-

I m A < 0) one says t h a t a p a i r o f n x n m a t r i x valued

e n t i r e f u n c t i o n s (E,,E-) (11.20)

One shows

Imh < 0 ) .

i s a deBranges p a i r i f

#

# 0 on ;C,

= E-E- on C; d e t E,

L: = E;~E-

d e t E- # 0 on C-;

i s i n n e r on C,

..*

Here one says Z = Z m x m i s r i n n e r i f i t i s meromorphic w i t h ZJZ:

-*

(Tat

p o i n t s o f a n a l y t i c i t y w h i l e ZJZ = J on R - f o r J ” = I one speaks o f i n n e r . 2 (resp. K2 ) t h e Hardy spaces o v e r C, (resp. C-) o f nN

L e t us denote by H

vector functions (fi(x))

=

a n a l y t i c i n C,

Ilfi(c+in)l

and sup

2”o

exp(ihx)dx w i t h F E L ).

f ( i ) (i.e.

t h e e n t r i e s , say f o r H2, f i ( A )

2

dc
0 f o r a

E

R) w i t h

To show t h a t (11.18) i s t h e c o r r e c t i n v e r s e one w r i t e s h e u r i s t i c a l -

l y , u s i n g (11.22) and Theorem 11.9

(fA(p),h) =

(11.24) =

/

(Ai*At?.h)dh

=

(F#-’bYF#-’At(h)h)dA IJ

1 ([(l/~)j

([F-lF*-’]f,A~~h)h)dl

t X #( s , h ) , X ( s , l l * ) d s ] * A t b , h ) d h

T h i s shows t h a t (11.18) extension f o r t

1

=

=

i s f o r m a l l y t h e c o r r e c t formula f o r fAE Bt and t h e

m i s immediate. For t h e l a s t statement see [ D U ~ ] . R e c a l l # # # here a l s o t h a t E E = F F, At = ( F F)-’ = F-lF#-’ , and # = * f o r A r e a l . -f

T h i s i n c i d e n t a l l y a l s o comp;etes

o u r d i s c u s s i o n o f Theorem 11.6.

The paper

[Ou8] c o n t a i n s many f u r t h e r r i c h developments r e l a t i v e t o t h e i n v e r s e spect r a l problem, s c a t t e r i n g t h e o r y f o r c a n o n i c a l e q u a t i o n s v i a t h e MarEenko equation, band extensions, entropy, L a x - P h i l l i p s s c a t t e r i n g , c h a r a c t e r i s t i c operator functions, etc.

It seems t o be t h e most complete and t h o r o u g h dis-.

c u s s i o n o f t h e m a t t e r and s p e c i a l i z e s t o o t h e r work on c a n o n i c a l equations F o r t h i s and o t h e r work on transmu-

w i t h a p p l i c a t i o n s i n many d i r e c t i o n s .

t a t i o n f o r equations w i t h o p e r a t o r c o e f f i c i e n t s , O i r a c equations, transmiss i o n l i n e problems, e t c . we r e f e r t o [Abl; K j l ; L14,9,10;

Gol; Gpl-3;

Ja1,2;

Adl; Ahl; B b s l ; C29,76;

Mc4; Rcl-3;

Sta1,Z;

Cx1,4;

W r l ; Wul; Sw1,2;

Yell.

We w i l l g i v e below a few comnents on some t r a n s m i s s i o n l i n e problems ( c f . [C76])

and i n t h i s d i r e c t i o n l e t us f i r s t d e r i v e a G-L e q u a t i o n i n t h e pre-

sent context (followin9 [ D u ~ ] ) f o r the Thus A r n ( h ) = 1 Set Hg = h

*

-

;(A)

from Theorem 11.11 and

?(A)=

F(Nf)/J2.

w i t h Xo(t,h)

= ($”,:)

(1/2)[$

-

(fl

(11.8)

*h

=

iE

( c f . a l s o Remark 2.3.8). h ( s ) e x p ( i A s ) d s = Fh.

g and ( 6 0 ) N f ( s ) i s d e f i n e d b y ( f = (fl,f2)) J Z N f ( s ) = f , ( s )

- i f 2 ( s ) f o r s > 0 w i t h &Nf(s) (11.16)

2 of

We w r i t e X ( t , s )

= fl(-s)

+ i f 2 ( - s ) f o r s < 0.

Then ( c f .

F o r example i f we have a 2 dimensional s i t u a t i o n t h e n ?(A)

if2)exp(ihs)ds + = k e r N HN = k e r

=

[t X.

; 1 [flCosAt + f 2 S i n h t ] d t w h i l e FNf//2 = (fl(-s) + i f 2 ( - s ) ) e x p ( i A s ) d s ( = F(A)). Then, r e c a l l i n g (11.8) i n t h e form

OPERATOR COEFFICIENTS

U = ( I + ?)Uo S O t h a t X 1 X # f = ( ( I + ?)Xo,f) =

31 7

+ ?)Xo as i n (11.17), one has f A ( k ) = ( X , f ) = Xo,(I + k * ) f ) = [ ( I + k*)fIw= “(1 + k*)fIA/J2.

= (I (

We w r i t e t h e Parseval formula determined by Theorem 11.11 i n t h e form (11.26)

I

m

m

( f , g ) d s = ( l / ~ ) / (fA,Amb)dA= ( 1 / 2 1 ~ ) / ([N(I+?*)flA,

(I-:)[N(I+k*)glA)dA

1

=

I

-m

-m

[N(I+k*)f,(I-H)[N(I+?*)]g)ds

=

( f , ( I + ?)N*(I - H)[N(I + ^K*)lg)ds A*

I t follows t h a t ( I + ?)“*(I - H)N](I + K ) = I which i s t h e G-L equation * in f a c t o r i z e d form. W r i t i n g N*(I- H ) N = I - N HN = I - 3C one has A

A

BHE0REIII 11-12. The G-L equation f o r K of (11.8) can be w r i t t e n as ( I + K)

*L

% ( I + ;*) = I o r ( I + l ) ( I + ?*) = SC which e x h i b i t s I + ( r e s p . I + ?*) as lower (resp. upper) t r i a n g u l a r f a c t o r s of t h e p o s i t i v e operator I - J€ ( r e l a t i v e t o a chain of projections PT: f + f ( 0 5 s 5 T ) and f 0 (s 2 T). -f

Let us show now how one can connect t h i s framework t o readout impulse responses a n d t h e s p e c t r a l representation of kernels a s i n t h e geophysical s i t u a t i o n . Take a standard model f o r transmission l i n e problems i n t h e form I = c u r r e n t , v = voltage, Z = impedance, and introduce normalized c u r r e n t and voltage V = v(x,t)Z(x)-’, I = I(x,t)Z(x)’. One s e t s a l s o r(x = ( 1 / 2 ) D,logZ(x) = Z-’Dx?’ f o r the “ r e f l e c t i v i t y ” . Then one has two equ valent 0 z v forms Dx(;) = -Dt(l/Z o ) ( l ) o r (A*) D x ( Iv) = - ( Dr D- r ) (VI ) ( 0 = D t ) - In terms

-

2 D i ) V - P ( x ) V = 0; ( D x2 - Dt)I -?-24Q(x)I = 0 where P = z’D2Z-’ = r2 - r ‘ and Q = Z ’D Z - r2 + r ’ . I f one w r i t e s WR = (V+I)/2 and WL = (V-I)/2 ( r i g h t and l e f t propagating waves) W then D ( W R ) = -(: ( D = Dt). T h i s formula leads t o numerically usewL f u l layer s t r i p p i n g techniques a s i n [ B b s l ; Lx4-61 ( c f . Remark 8.14). Let us w r i t e now (A*) i n the form

o f second order equations one obtains ( D C

ID)(,;)

W e will not deal e x p l i c i t l y here w i t h Dirac systems but r e f e r t o [ C x l ; Rcl3; L19] f o r t h i s . For (11.27) we can use t h e theory developed above f o r 0 1 Or Q = A 0 = JDX - V , J = (-, o), V = ( r o) and w i l l t r y i n p a r t i c u l a r t o provide transmutations via s p e c t r a l pairings. In p a r t i c u l a r l e t US look a t A t h e kernels I + ^K and I + L = ( I + ;)-’ based on (11.17). Thus formally go

318

ROBERT CARROLL

t o t h e s p e c t r a l formulas (11.16) and (11.18)

( n o t e we a r e d e a l i n g now w i t h

- (X(t,A), 2 - v e c t o r s X, Xo and Am i s a number) and w r i t e ( t ) G(s-t) = (l/r)II

(= ( l / ~ ) X*(t,h)X(s,h)A_dA) j: and (+) ~ G ( h - p ) / A ~ ( h )= # A COSAt) :f (X(s,A),X(s,u))ds ( = ( X (s,A),X(s,u))). Now X = ( B ) and Xo = (SinAt so we c o n s i d e r ( A ( x , A ) , C o ~ h s ) ~ ( A(x,x),SinAs)v (11.28) B(X,S) = B ( x , x ) , C ~ s A s ) ~ ( B(x,A),SinAs)w X(s,A))A-(A)dh

1

which can be w r i t t e n i n a more o r l e s s standard n o t a t i o n B(x,s) = (X(x,X),

*

X ( ~ ~ 1 ( h e)r e) t ~ a k e A r e a l w i t h dv = dh/T on 0

Edx/T on [O,-)). Xo(s,u)))v

Formally then

A(x,p)

=

and (B(x,A),(

X0( s , ~ ) ) = G ( x - p ) ~ (over (11.29)

y(s,x) =

so y ( s , x ) =

(

[

(

~(x,s),X,(s,p))

(--,a)

o r o c c a s i o n a l l y dw =

has terms (A(x,A),(

Xi(s.A),

X ~ ( S , A ) , X ~ ~ S , ~=) )B(x,u) )~ s i n c e (XE(s,X), S i m i l a r l y w r i t e do = A-(A)dA/v

(-my-)).

(

CosAs,A(x,A)

)o

(

Cos~s,B(x,h) )o

(

SinAs,A(x,A) )o

(

Sinhs,B(x,A)

and s e t

1

X o ( s , ~ ) , X * ( s , ~ ) )o and formally(y(s,x),X(x,u))

has terms

( X * ( x , ~ ) , X ( x , p ) ) ) ~ = Cosps and (Sinhs,( X * ( ~ , A ) , x ( x , p ) ) ) ~ = Sinus.

(

CosAs,

Hence n

EHEOREN 11.13. B: X o

-+

X and

S p e c t r a l k e r n e l s f o r (11.17) i n t h e form k e r B = B = I + K, y =

k e r B , B = B-’:

-+

Q = JDx

-

V and

X

-+

X o a r e determined by (11.28) and (11.

F o r m a l l y we a r e d e a l i n g w i t h t r a n s m u t a t i o n s B: JDx = Q,

29) r e s p e c t i v e l y .

a:

Q

-+

Q.,

o f (11.27), i . e . Note a l s o = A certain -J? and ?V = - V f .

We t u r n now t o t h e q u e s t i o n o f p r o d u c i n g a s o l u t i o n q ( x , t ) -?D :T

~p

= Qp = JDxp

-

Vq

where I,J,V

a r e as i n d i c a t e d t h e r e .

t, T = I,J - l = -J = J (T = t r a n s p o s e ) , i J =

i-’

amount o f e x p e r i m e n t a t i o n l e a d s one t o c o n s i d e r ( c f . [L19])

W

w

L

where w = (wl) i s i n d i c a t e d below,

(

, )o

‘L

f do, and f dg means

/I dg so

t h a t boundar? terms a r e n o t p r e s e n t upon i n t e g r a t i o n by p a r t s (see below 1 1 -;) and w r i t e w(x,t) = f o r :I dc). For w we s e t H+ = (1 l) and H- =

(1,

( l / E ) H - f ( x + t ) + ( l / P ) H + f ( x - t ) ; t h e n w s a t i s f i e s JDxw = -?Dtw w i t h w(x,O) = f f(x) = (fl). Now make some r o u t i n e c a l c u l a t i o n s u s i n g -?fl w = JD w i n t h e 2 A t 5 form -Dtwl = D w and Dtw2 = - D w and QX = x X , X = ( B ) , i n t h e form Bx 5 2 5 1 4 r B = A A and - A x - r A = AB, t o o b t a i n - 1 D p = Q(Dx)q, w h i l e

OPERATOR COEFFICIENTS

( r e c a l l w,(x,O)

= fl(x),

etc.).

31 9

Hence p r o v i d e d e v e r y t h i n g makes sense (11.

30) r e p r e s e n t s a s o l u t i o n o f - i D t q

Q(DX)q w i t h q(x,O) g i v e n by (11.31).

=

NOW from (11.31) i f we deal w i t h even e x t e n s i o n s o f t h e fi t h e n t h e second t e r m i n v(x,O)

I dg =

vanishes ( f o r

/z

([0,m) t o

dc).

(-m,m))

On t h e o t h e r

hand one can t a k e J dg as 1 ; dg i f p r o v i s i o n i s made t o have t h e "boundary" terms a t 5 = 0 v a n i s h upon i n t e g r a t i o n by p a r t s , and t h i s seems t o have It i s necessary f o r t h i s t h a t w2(t,0) = ( 1 / 2 ) [ ( f 2 ( t )

c e r t a i n advantages.

+

f 2 ( - t ) )+ ( f l ( t ) - f l ( - t ) ) ] = 0. Thus one would t a k e fl even w h i l e f2 s h o u l d be odd (we n o t e t h a t some e x t e n s i o n i s needed i n any case so t h i s i s

no problem).

Now w r i t e t h e terms i n q(x,O)

as

(

F(h)FCfl,A(x,A)

A

h2) (column v e c t o r ) one has h 0 ) and h

FCfl

=

2"

= ( hlyA)

A

t ( h2,B)

and

)w

(

F(X)

I f h = (h,,

FSf2,B(x,x) )w (Fc and Fs denote t h e F o u r i e r t y p e t r a n s f o r m s ) . A

= h,

t

h2 where h, =- (hly

(0,h2) and we want t h e n F = 1 w i t h fl and f2 chosen so t h a t A

hl and F S f 2 = h2 ( t h e p a r i t i e s , even-odd, w i l l t h e n match

x

t h a t A i s even i n f o r example).

A

A

so ( h , B ( x , X ) ) ~ A

A

=

one sees

and B odd v i a c o n n e c t i o n t o t h e second o r d e r e q u a t i o n s

Then h, = (X(x,A),hl)w

q(x,O) as q(x,O)

-

( h ,X(X,X))~

where h

'A

= hl

=

0 and we can w r i t e

A

t h

2 ( n o t e i n t h e same way

h ; , A ( x , ~ ) ) ~= 0)

(

EHEOREill 11.14,

The ( u n i q u e ) s o l u t i o n o f -?Dtq

h ( x ) can be r e p r e s e n t e d by (11.30) h2(x),B(x,x)),

=

( w i t h F = 1 and I dg = 1 ; d5) where t h e

f d e t e r m i n i n g w i s chosen so t h a t FCfl (

= O(Dx)v s a t i s f y i n g v(x,O)

A

= hl

= ( hl ( x ) ,A(X,A)),

A

F S f 2 = h2 =

and one extends fl t o be even and f2 t o be odd.

Consider now a problem f o r (11.27) w i t h i n i t i a l c o n d i t i o n s h ( x ) = (S(x),O) f o r example (one s i d e d 6 ) and t h e n t h e "impulse-response" follows.

A

F o r m a l l y hl = 6 so hl = A(0,h)

=

Hence our f o r m u l a f o r w g i v e s (**) w ( t , c )

){:;g6[

f(E-t) = ( 1 / 2 ) ( g 6 ~ ~ ~ ~ { Cosxgdc = fl(S)exp(iAg)dE

iI

1 so fl = 6 a l s o w h i l e f 2 = 0. = (1/2)(!1

Ip(x,t)

=

[

,I

(

Cosxt,A(x,x) )u

(

Sinxt,B(x,A)

)u

(-m,m)).

From (11.30) now

1

(which i s a l s o e a s i l y checked d i r e c t l y t o be a s o l u t i o n as r e q u i r e d ) . one now has a readout Ip(0,t)

i)

- i ) f ( g t t ) + (1/2)(;

( s i n c e I dc = dg and s i n c e 2J; fl(e) i n (11.32) we can i n f a c t w r i t e (**) and i n -

t e r p e r t t h e 6 f u n c t i o n s as two s i d e d a c t i n g on (11.32)

i s determined as

= G(t)

measure can be r e c o v e r e d f r o m

(00)

A m ( A ) even (see below) we have t h e n

(%

If

V ( t ) ) f o r example, t h e n t h e s p e c t r a l

G1 ( t ) = (1/r)Jm CosAtam(A)dh. -m

Given

320

ROBERT CARROLL

ME8REIII 11-15- As i n t h e geophysical s i t u a t i o n o f 598-9 t h e s p e c t r a l measure can be recovered f r o m t h e impulse response f o r (11.27) i n t h e f o r m A,(A)

=

10" G1 ( t ) C o s A t d t . REmARlc 11-16. One n o t e s t h a t (11.32) always g i v e s G 2 ( t ) = (Sinht,B(O,x))w = 0 and t h i s is due t o t h e B r e p r e s e n t a t i o n a t 0 b e i n g inadequate. T h i s i s

; 1 f ( x ) S i n A x d x where t h e r e p r e s e n t a t i o n f ( x ) fh(A)SinxxdA always produces f ( 0 ) = 0.

s i m i l a r t o ?(A)

=

L e t us r e t u r n now t o t h e 6-L e q u a t i o n o f Theorem 11.12.

(2/n)1;

=

F i r s t we t r y t o

mimic t h e procedure which works i n t h e s c a l a r case, u s i n g t h e m a t r i x k e r n e l s B and y o f (11.28)-(11.29).

Z ( X , ~ )=

(11.33)

(

T X(X,P),X~(E,P) )w = Y (E,x)

Ifwe c o n s i d e r now X(x,p) o b t a i n s (A(s,E) (11.34)

= (

A(s,c)

= ( ~ ( x , s ) , X ~ ( s , p ) ) and compose w i t h X:(c,p)

[

=

N

(

C o s s p , C o s ~ p )(Cossp,Singp) ~

(

Sinsu,Coscp)w (Sinsp,Singp )w

c ) )

-

one

with

@I

, X =~ )((8..))(Xo,Xo)m. ~ I f we w r i t e now A = 6, 'J = JC, B = 6, + K ( r e c a l l dv = dx/a) and n o t e t h a t (

v

r(,

2 s so A

0 for 5

c

x, t h e n one o b t a i n s 0 = K

and J C ) .

2s

* ~ )

Jc, 6,

'L

= ( ~(x,s),A(s,s))

g(x,s)

Xo(sy~),X:(~,v)

Note here t h a t ( ( ( B ~ ~

-

Thus d e f i n e

I n p a r t i c u l a r f r o m Am = 1

-

N

Jc

-

A

u

(K,K) as i n Theorem 11.12 ( f o r K h

^h

one has (we s e t h e r e h / s = u and

( , )U) JC(s,s) = ( X o ( s , u ) , X i ( ~ , p ) )U = ( ( h . . ) I . Now t h e elements o f 'J J C ( s , s ) a r e r e p r e s e n t e d i n sum and d i f f e r e n c e f o r m i n [Du81 f o r example and

write

we do t h i s a l s o b u t i n a d i f f e r e n t way. m

( l / a ) ~CosspCosguhdp ~

(ao)

as

t h a t hll

Gl(t)

(mm)

Thus f o r example (Cossv,Cosgp)

= ( l / Z n ) / _ f [Cos(s+s)u

= s+(t)

+ (l/n)j:

Cosxt^hdA = 6 + ( t ) + gl(t).

It follows

+ gl(s-s)I. On t h e o t h e r hand we do n o t y e t have a r e a d o u t term correspondS i m i l a r l y hZ2 = (1/2)[gl(s-c)

= (1/2)[g1(s+s)

gl(s+

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Notas de Algebra I

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Coloquio brasileiro de matematica

Apuntes De Matematica General

Les paroles de 105 chansons

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Complex Analysis in Banach Spaces (North Holland Math Studies, Vol 120: Notas De Matematica)

Real Elliptic Curves (North-Holland Mathematics Studies 54) (Notas de Matematica 81)

2001 (105)

Set Theory. Centre de Recerca Matematica Barcelona, 2003-2004

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