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

-

=

<

Rn now l e t S denote t h e space of Cm functions

(x

E

n

R ) f o r every a = ( a l ,...,a n ) and B = ( B 1

$,).

9

such

,...,

Such functions a r e c a l l e d rapidly decreasing and one says 9 k 0 in S 2 m a i f f o r any m and p , s u p I ( l + l x l ) D q k l + 0 (x E Rn) f o r IaI 5 p. This w i l l be t h e natural space f o r Fourier transforms ( s e e 53) and t h e dual space S ' , -+

t h e space of continuous l i n e a r maps S

C, i s a subspace of 27' c a l l e d t h e space o f tempered d i s t r i b u t i o n s . One shows S ' C 27' by using t h e d e n s i t y of 27 in S - s e e the references c i t e d f o r d e t a i l s ) . +

DfFZNlCl@N 2.9, For 9,lL E C:(Rn) one defines (9 * Jl)(x) = 9(x-S)Jl(S)dE (convolution). For d i s t r i b u t i o n s S,T i n U'(Rn) S * T may not be always def i n e d . However i f S E 27' and T E E ' f o r example one can define S * T E 27' by t h e r u l e (9 E U ) (2.3)

( S

*

@ T , q ( x + y ) ) = ( S x , ( T y , 9 ( x + y ) ) ) = ( T, ( S , , ~ P ( X + Y ) ) )

Ty9) = ( S

EXAMPLE 2-10, Let S

X E

Y

Y

U'(Rn) and T = Dk6

E

E l .

Then

ROBERT CARROLL

6

When t a l k i n g about convergence i n D', E l , or S ' one will always mean weak convergence (and i n f a c t t h e only l i m i t s which a r i s e w i l l involve sequences). Thus f o r example Tn + T in D' w i l l mean ( T n , v ) ( T,P) for any fixed q E D. -f

MAWCE 2.11-

Let

J, E

u(R") w t h say

J,

1 centered a t the o r i g i n , and !$ ( x ) d x f o r IP E D, a s k + m y

This shows t h a t

+

6 in

= 1.

$

c B(0,l)

Set $ k ( x )

=

=

ball of radius

kn$(kx) and then,

i$

( $ k x ) , P ( x ) ) = kn

(2.5)

2 0, supp

D'.

D'(Rn) we can define a r e g u l a r i z a t i o n T k = T * Jlk w i t h Q k a s i n Example 2.11. One can show e a s i l y ( c f . references c i t e d ) t h a t T k E C m ( R n ) and Tk T in U' a s k m. REmARK 2-12. For T

E

-f

-f

REIIARK 2.13. Let US mention here a l s o t h a t since the Lebesgue i n t e g r a l and measure theory a r e often not f a m i l i a r t o s t u d e n t s (and s c i e n t i s t s ) working w i t h d i f f e r e n t i a l equations one can nevertheless define and use L 2 theory

f o r example via distributions ( c f . [Rnl f o r example). T h u s consider Cz(0,m) and define t h e ( r e a l ) s c a l a r produce ( f , g ) = f m fgdx w i t h IIf1I2 = ( f , f ) . Let 0 q n be a Cauchy sequence in this norm topology and then f o r any t e s t function (13

I(J,,v~ -

- p m y $ ) l5

II+IIIIqn

-

o

so ($,q,,) i s Cauchy and converges t o R ( J , ) where R i s l i n e a r . That R is then a d i s t r i b u t i o n follows from t h e Banach-Steinhaus theorem f o r example ( c f . [C40] b u t we omit the d e t a i l s here ( 0 is a so c a l l e d barreled space in i t s natural topology). 2 T h u s we sayv, -+ R in D' and one can i d e n t i f y R w i t h an L function i f one 2 knows about L2; i f not we simply define L as t h e c o l l e c t i o n of d i s t r i b u tions L obtained i n this manner ( o r more generally a s the c o l l e c t i o n o f Cauchy sequences from C:(O,m) - a b s t r a c t completion). One shows e a s i l y t h a t i f two Cauchy sequences cpn and \Lln a r e equivalent ( i . e . lipn - $,I1 -+ 0 a s n m) then they determine t h e same d i s t r i b u t i o n R ( e x e r c i s e ) . Further R E S ' i s evident and one can define (R,m) = lim (pn,JIn) w i t h IIRll = lim lllpnll when 2 9, (resp. $,) a r e Cauchy sequences defining 1 (resp. m). Similarly L i s complete in norm and hence a Hilbert space; we leave t h e d e t a i l s here a s a r e l a t i v e l y easy exercise. One can obtain t h e o t h e r Lp spaces ( 1 5 p < m) by s i m i l a r procedures whereas f o r Lm one has t h e space of d i s t r i b u t i o n s f In f a c t one can say more gensuch t h a t I( f,lp ) I 5 MllcpIIL1 f o r a l l 9 E C;. e r a l l y t h a t L p i s the space of d i s t r i b u t i o n s f such t h a t I( f y 9) I 5 MlllpflLq 2 f o r a l l 9 E C: ( l / p + l / q = 1 , p > 1 ) . Let us mention a l s o t h a t i f f E L , J, E Co

qm)l = l(qn

pmll -+

-+

7

DISTRIBUTION THEORY

m

w i t h a d e t e r m i n i n g Cauchy sequence v k equality

",1

[vk(s) -

t h e n by t h e Cauchy-Schwartz i n 2 v L ( c ) I d c 1 2 5 x f 1vk - v,12dc = Xllvpk - v,tlL2 so t h a t E X

Co,

0

IX v k ( c ) d c converges u n i f o r m l y i n any f i n i t e i n t e r v a l t o a continuous func0

S i n c e - ( f , $ )= - l i M v k , $ ) = l i m ( f v k , $ ' ) = ( F , $ ' )

t i o n F(x).

t h a t F i s a p r i m i t i v e o f f, F ' = f, and F =

fX

it follows

f ( c ) d c + c i n an e v i d e n t no-

tation. L e t us make a few remarks here about t h e r e p r e s e n t a t i o n o f d i s t r i b u t i o n s i n 1 R as boundary values o f a n a l y t i c f u n c t i o n s ( c f . [Bgl; B z l ; C40; Cbl-3; H11; Odl,Z]).

These m a t t e r s a r e u s e f u l f o r e l l i p t i c t r a n s m u t a t i o n i n [C35,40].

CHEBREFII 2-14. L e t T

E

E'(R1),

The a n a l y t i c r e p r e s e n t a t i o n o r Cauchy r e p r e A

s e n t a t i o n o f T i s d e f i n e d by T ( z ) = ( l / 2 n i ) ( T t , ( l / t - z ) ) function o f z f o r z

E

and i s an a n a l y t i c

C n o t i n supp T.

vn(t)

Phoal;: F o r Imz # 0 t h e d i f f e r e n c e q u o t i e n t sequence f o r small Azn,

-

[(l/(t-z-Azn) a l y t i c i n C-R. K* so t h a t

(

converges i n E t o ( l / ( t - 2 )

(l/t-z)]/Azn,

For K = supp T p i c k a ( t )

€

27 w i t h a

=

2

)

=

E so T i s an-

E

1 on K and supp

CL

=

As z

+

xo

i n E so t h a t T ( z ) i s continuous as z

+

xo and hence a n a l y t i c a t

Tt,(l/t-z))

= ( Tt,a(t)/(t-z)

).

E

a(t)/(t-z)

R-K*,

-+

A

a(t)/(t-xo)

Since K

xo ( e x e r c i s e ) .

*

K can be t a k e n as c l o s e t o K as d e s i r e d t h e theo-

2

rem f o l l o w s . as I z (

One sees e a s i l y i n Theorem 2.14 t h a t l ? ( z ) l = 0(1z1-')

-

r e a l , ?(x+iE)

? ( x - i E ) = O ( ( X ~ - ~ ()e x e r c i s e

-

c f . [Bzl]).

-+

m

and f o r x

L e t us r e c a l l

a l s o f r o m c l a s s i c a l p a r t i a l d i f f e r e n t i a l e q u a t i o n s t h a t i f f ( t ) i s a cont i n u o u s bounded f u n c t i o n one has a harmonic r e p r e s e n t a t i o n

-m

Thus f* i s harmonic f o r Imz # 0 ( i . e . A f * = 0) and e.g. E

-+

0.

f*(t+iE)

-f

f ( t ) as

A l s o one shows e a s i l y t h a t t h e convergence i s u n i f o r m on compact

s e t s i n t.

Now l e t f ( t ) = O ( ( t 1 " )

f o r a < 0 and w r i t e ? ( z ) = ( 1 / 2 a i ) 2 Since ( l / Z n i ) [ l / ( t - z ) - l / ( t - z ) ] = ( y / n ) [ l / l t - z l 1 we see

/If ( t ) d t / ( t - z ) . t h a t fn(x+iE)

-

A

fh(x-iE) = f * ( x + i c ) and consequently as

E

-+

0, f ( x + i a )

-

A

f(x-iE)

T

E

-+

f(x).

This type o f representation f o r functions p r e v a i l s a l s o f o r

E ' i n t h e f o l l o w i n g sense.

CHE0REm 2.15,

L e t T E E ' and l e t

derivatives.

Then

lim E+O

[I [?(x+iE)

-

v

E

E be bounded a l o n g w i t h each o f i t s

?(x-is)]v(x)dx = (T,v)

8

ROBERT CARROLL

Phoo6:

-

L e t T*(z) = sgn(y)[;(z)

?(?)I

= ( ] Y ~ / T T )T( t , ( l / l t - z [

easy argument w i t h Riemann sums y i e l d s As

E

-f

0, 9 ( x + i e )

(a)

LI T * ( x + i E b ( x ) d x

.

*

Tt,ro*(x+iE)).

*

Hence IP ( x + i c )

One can a l s o check e a s i l y t h a t T ( z ) i s harmonic f o r Imz E

= (

Then an

q ( x ) u n i f o r m l y on compact s e t s and a s i m i l a r conver-

+

gence holds a l s o f o r a l l d e r i v a t i v e s o f 9 ( e x e r c i s e ) . * 9 ( x ) i n E and ( Tty9 ( x + i E ) ) + ( T,q ).

T

2 ).

+

+

0 s o from ( * )

E ' i s a l s o t h e boundary v a l u e o f a harmonic f u n c t i o n ( a c t i n g on

9 E

E

We see i n p a r t i c u l a r t h a t f o r T E E ' w i t h s u p p o r t K t h e r e

as i n d i c a t e d ) . A

i s a f u n c t i o n T ( z ) a n a l y t i c i n C-K such t h a t Theorem 2.15 h o l d s f o r 9 E D. A

A

But T i s n o t t h e o n l y such a n a l y t i c r e p r e s e n t a t i o n o f T s i n c e T + E, f o r e n t i r e , has t h e same p r o p e r t i e s ( f o r 9 E

E

U). Modulo t h i s l a c k o f unique-

ness one can a l s o p r o v e L e t T E D ' w i t h supp T = K.

CHEOREn 2.16,

a n a l y t i c i n C-K such t h a t f o r 9 E

lim

J

E+o

[f(x+ie)

-

Then t h e r e e x i s t s a f u n c t i o n f ( z )

D

f(x-is)&(x)dx

= ( Ty9)

,m

Such f a r e c a l l e d a n a l y t i c r e p r e s e n t a t i o n s o f T and we o m i t t h e p r o o f here ( c f . [Bzl] f o r example). 3,

F0URZER ANAQWIS-

The F o u r i e r i n t e g r a l and r e l a t e d t h e o r y i s one o f

t h e most s i g n i f i c a n t items i n a l l o f mathematics s i n c e i n p a r t i c u l a r i t u n i t e s b o t h p u r e and a p p l i e d mathematics i n b r e a t h t a k i n g harmony. p o i n t o f view one t h i n k s e.g. CDu2; Pel, Wo1,2]

For t h i s

o f Wiener, Kolmogorov, e t c . and books such as

a r e p a r t i c u l a r l y u s e f u l ( c f . a l s o CC19; G f l ; H l l ; S j l ;

Yal] f o r modern approaches).

We w i l l u t i l i z e F o u r i e r methods t h r o u g h o u t t h e

book i n one f o r m or a n o t h e r and g i v e here some background i n f o r m a t i o n .

Let

us d e f i n e f o r f E S(Rn) ( c f . D e f i n i t i o n 2.8)

I

a,

(3.1)

F f ( h ) = ?(A) =

f(x)e

i(X , X )

dx

-(n

where ( A , x ) =

1 X jx j*

The f o l l o w i n g formulas a r e obvious

m

(3.2)

D%(X)

=

[

il"lxclei'

Ayx)f(x)dx

-m

Now t h e i n v e r s i o n f o r m u l a f o r (3.1), has t h e form

which i s proved below i n Theorem 3.2,

FOURIER ANALYSIS

9

m

(3.4)

f ( x ) = ( I / z ~ ) ~P ( A ) e m i (

Using t h i s , w i t h (3.2)-(3.3),

dA

and t h e d e f i n i t i o n o f convergence e t c . i n De-

one proves e a s i l y

f i n i t i o n 2.8,

EHE0REm 3-1- The F o u r i e r t r a n s f o r m i s a 1-1 b i c o n t i n u o u s map S

-f

S (onto).

Now we s t a t e t h e i n v e r s i o n ( 3 . 4 ) as p a r t o f t h e f o l l o w i n g theorem For f , g E S one has (3.4) along w i t h

tHE0REm 3.2.

[

(3.5)

m

I"

I

F(A)g(A)dA =

f(x)F(x)dx

J-m

J-m

The f o r m u l a (3.5) w i l l be w r i t t e n (?,g)

= (

fY:) and i s c a l l e d t h e Parseval

r e 1a t i on.

Ptraod:

One can see e a s i l y t h a t

,1

I,

m

(3.7)

03

g(X)fv(A)e-i(A'x)dA

I

Indeed t h e l e f t s i d e i s (

I e x p ( i tA,y-x)g(x)dA)dy

&l)f(X+ddQ

=

g ( A ) e x p ( - i ( 1 , ~ () I f ( y ) e x p ( i (A,y)dy)dh =

I f(y)F(y-x)dy

=

=

I

f(y)

Consider now

I y(n)f(q+x)dn.

g(EA) i n s t e a d o f g(A) so t h a t

I, m

(3.8)

m

g ( a A ) e x p ( i (h,<)dx =

I,

E - ~

g ( z ) e x p ( i (z,E/E)dz

=

E-';(~/E)

I t f o l l o w s t h a t i n (3.7) m

(3.9)

j-

g(EA)7(oe

-i(A,x)

m

dx =

E

-

m

I,

~

:(q/E)f(x+o)dr!

=

Now one uses a w e l l known f o r m u l a ( c f . [ Y a l ] )

2 Thus, s e t t i n g g ( x ) = e x p ( - l x l / 2 ) and l e t t i n g

E

+

0 one o b t a i n s f r o m (3.9)

m

(3.11)

g ( 0 ) j"?(A)e" a

(xyx)dA = f ( x ) j-T(S)dE

2 T h i s i s (3.4) s i n c e g ( 0 ) = 1 w h i l e I G(c,)dg = ( 2 a ) n / 2 / e x p ( - 1 5 1 /2)dg = (271)n. For (3.5) we s e t x = 0 i n (3.7) w h i l e f o r (3.6) one has s i m p l y (3.12)

I

(f

*

g)(x)exp(i

A,X

dx =

i

e

I

f(x-c)g(E)d<)dx =

ROBERT CARROLL

10

(3.12)

1

=

e i (x'c)g(c)ei(x3x-c)f(x-c)dcdx

=

F(A)~(x)

Now one uses t h e Parseval formula (3.5) t o define t h e Fourier transform i n S ' by d u a l i t y ( c f . Definition 2.5 f o r the analogous porcedure f o r d i f f e r e n t i a t i o n ) . Thus, given T E S ' , consider t h e map N : q Fq + ( T , F q ) : S -+ S -+ C. By Theorem 3.1, F: S S is continuous and hence N i s continuous, as -f

-f

a composition of continuous maps, so N determines an element o f S ' , N(q) T, q ).

=

(

DEFZNZCZBN 3 - 3 , Given T

6 S'

one defines the Fourier transform FT by

EMWCE 3.4, One can check t h e following formulas e a s i l y ( T

FA

(3.15)

F(DkT) = - i x k F T

(3.16)

Dk(FT)

[m

S')

F(ixkT)

=

Let us take ? = ? i n t h e Parseval formula (3.5) so f o r f , g J i ( x ) e x p ( - i (A,x))dx with f ( x ) = ( 1 / 2 ~ ) ~and ? consequently (3.17)

S)

1

(3.14)

=

E

(q E

=n,

Fgdx

=

[

E

S F(h) =

OD

( 2 ~ ) FgdA ~

Using t h e density o f :C

o r S i n L2 one obtains immediately

2 2 CHE0REm 3-5- The Fourier transform i s an isomorphism L + L w i t h ( g

E

2

L )

Obviously a d i f f e r e n t normalization f o r F would make the correspondence g + g in (3.18) an isometry b u t we p r e f e r t h e present formulas because of (3.6) e remark t h a t S * T is n o t i n general defined f o r and (3.14) f o r example. W S,T E S ' but, when S E S ' w i t h T E Ui (a space of d i s t r i b u t i o n s we will e releave undefined), then S * T E S ' makes sense and F(S * T ) = FS FT. W mark also t h a t t h e Fourier transform can be defined f o r 2)' by using somewhat d i f f e r e n t techniques ( c f . [EaZ]).

u

Let us go now t o a t o p i c which plays an unusually important r o l e i n transmutation theory, namely t h e Paley-Wiener complex of ideas. F i r s t

11

FOURIER ANALYSIS

CHE@)REm 3-6- An e n t i r e f u n c t i o n F ( c ) , r; E C n y i s t h e F o u r i e r t r a n s f o r m F ( g ) =

JIf ( x ) e x p ( i ( x , < ) d x ,

c B(0,R) that

Pkoul;:

= t k i- i n k y o f a f u n c t i o n f E C:(Rn)

1x1 5 R I , i f and o n l y i f f o r e v e r y

= {x;

(?I

ck

IF(c)1 5 CN(l

+

N t h e r e e x i s t s CN such

Icl)-Nexp(RIImcl).

The n e c e s s i t y i s immediate f r o m t h e r e l a t i o n ( c f . ( 3 . 3 ) ) , II(-ii;k)RK ( i n t e g r a l o v e r 1x1 5 R).

F ( c ) = J DBf(x)exp(i<x,c))dx

IT)-^

f i n e (*) f ( x ) =

?(E)

w i t h supp f

fIexp(-i(x,c))F(c)dt

-

= F ( < ) and f E Cm ( e x e r c i s e

For s u f f i c i e n c y de-

(5 r e a l ) .

One sees e a s i l y t h a t

use an analogue o f (3.2) f o r t h e d i f f e r -

Now i n (*) one can use Cauchy's theorem and ( ? ) t o s h i f t t h e

entiation).

c o n t o u r o f i n t e g r a t i o n so t h a t f o r a r b i t r a r y n = a x / J x I ( a > 0 ) m

(3.19)

f(x) = (2~)-"

I.

e-i(x"-in)

F ( 5- in ) dc

Take N = n + l so t h a t m

(3.20)

If(x)/

5 (2ii)-'CNe

I f now 1x1 > 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(<)\ <

c

*

$ k ) = F(FQk).

But SUPP J/k

(l+l
by Theorem 3.6,supp

Tk

C

B(O,R+l/k). -f

m

C

S ' f o r 5 r e a l so F = T

B(O,l/k)

= FT

and hence f o r any M

Consequently I F T ~= ~

Cn(l+lr;i~-Wexp(R+l/k)(Im<\ and s i n c e Tk c supp T + supp $k and as k B(o,R).

E

E Cm ( e x e r c i s e

-

1

~

<

~

~

1

c f . Example 2.11)

B u t one checks e a s i l y t h a t supp T

i t f o l l o w s f r o m t h e above t h a t supp T

C

*

$k

12

ROBERT CARROLL

The c l a s s i c a l Paley-Wiener theorem i n v o l v e s L

2

f u n c t i o n s and we s t a t e here

some one dimensional v e r s i o n s o f t h e t y p e needed l a t e r ( t h e p r o o f s a r e l e f t as e x e r c i s e s here

-

c f . [Bsl;

Cel; Du2; P e l ; P f l ; R m l ; Yal]).

Thus f i r s t

one says t h a t a f u n c t i o n f ( z ) i s o f e x p o n e n t i a l t y p e T i f f o r each t h e r e e x i s t s M such t h a t I f ( z )

I

5 Mexp( I z I ( T t E ) ) f o r z

E C.

> 0

E

Then we have

2 F 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 T w i t h f E L on 2 t h e r e a l l i n e i f and o n l y i f f ( z ) = F ( t ) e x p ( i t z ) d t where F E L (-T,T).

&HEBREm 3.8.

JT,

f ( z ) 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 T w i t h f

6HEBREm 3.9.

f o r z r e a l i f and o n l y i f f ( z ) =

/IT F(t)exp(itz)dt

E

L'

where F(T) = F(-T) = 0

and t h e f u n c t i o n o b t a i n e d by e x t e n d i n g F t o be 0 o u t s i d e o f [-T,T]

has an

a b s o l u t e l y convergent F o u r i e r s e r i e s on any l a r g e r i n t e r v a l [-T-E,T+E]. It w i l l be u s e f u l t o i n c l u d e here some i n f o r m a t i o n about t h e F o u r i e r t r a n s -

form r e l a t i v e t o a n a l y t i c r e p r e s e n t a t i o n s e t c . as i n 52. Thus ( w o r k i n g i n n 1 1 2 R and C ) f o r f E L say w r i t e F ( x ) = f ( f , x ) = ( f , e x p ( i ~ x ) ) and l e t F ( z ) be t h e a n a l y t i c r e p r e s e n t a t i o n o f F

-

t h i s can be d e f i n e d as t h e Cauchy r e -

preresentation ?(z) = ( 1 / 2 d ) ![ F ( t ) d t / ( t - z ) ] . For f E L

mEBREiil 3.10,

Then ( c f . [ B z l ] )

one has ( z = x + i y )

w i t h F(X) = F ( f , x )

By d i r e c t i n t e g r a t i o n one has f o r y # 0 (Y(E) = Heavyside f u n c t i o n )

Ptroa6:

(e.g.

2

(7/2a) J m e x p ( i c z ) e x p f - i c t ) d 5 = l / Z s i ( t - z ) 0

f o r y > 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(<),Y(c)exp(igz))

(y > 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

<

x

<

m)

A f ( x ) ; sy(x,O) = C f ( x )

has a unique s o l u t i o n ( i n some c l a s s of functions 9 ) . To f i x ideas one thinks here of f , given on [O,m), t o be extended t o (--,a) a s an even funct i o n and A,C a r e t o a c t on this extension. The potential p ( x ) i s correspondingly extended as an even function. Then, f o r s u i t a b l e f , upon defining Bf(y) = p ( 0 , y ) i t follows t h a t QBf

=

BPf.

P m 6 : Let IL(x,y) = P(DX)v(x,y) so [ P ( D x ) - Q(Dy)15, = P(Dx)[P(Dx) - Q ( D Y ) b 0 while $(x,O) = BAf(x) = APf(x) and DyJi(x,O) = D y P ( D x ) ~ l y = O = P ( D x ) D y \ 0 ( x , 0 ) = PCf(x) = CPf(x). Consequently given uniqueness i n Cauchy problems =

of the form (4.4) f o r $, a s well a s 9 , we can w r i t e $(O,y) = BPf(y). B u t $(O,y) = P(Dx)9(x,y)lx=0 = Q ( D )p(O,y) = QBf(y) and t h e r e f o r e B i s a t r a n s Y mutation P Q. -f

Generally one will have t o f i x a domain D ( P ) so t h a t f E D ( P ) i s " s u i t a b l e " i n Theorem 4.3 and t h e extension of f (and p ) t o (-m,-) must be

REWRARK 4.4,

examined. We a r e usually not concerned here with Lp type spaces so f o r D ( P ) 2 one often takes f E C w i t h s u i t a b l e conditions a t x = 0 and perhaps x = m. In p r i n c i p l e , d i f f e r e n t extensions of f from [ 0 , m ) t o (-m,m) y i e l d d i f f e r e n t

16

ROBERT CARROLL

t r a n s m u t a t i o n s and t h i s i s discussed i n CC40; L13; L p l ; Mc3,4].

We n o t e a l -

so t h a t f o r a n o n t r i v i a l d i f f e r e n t i a l o p e r a t o r P t h e r e a r e o n l y a v e r y l i m i t e d number o f o p e r a t o r s A and C which w i l l c o m u t e w i t h

P

so some o f t h e

apparent g e n e r a l i t y o f Theorem 4.3 i s i l l u s o r y . Now i n s o l v i n g Cauchy problems o f t h e f o r m (4.4)

i t i s , perhaps s u r p r i s i n g -

We f o l l o w [Mc3,4]

l y , expedient t o use t h e method o f Riemann f u n c t i o n s .

I t w i l l be s u f f i c i e n t t o here ( c f . a l s o [C40; Ho1,2; L13; L p l ] ) . 2 D s i n c e o u r c o n s t r u c t i o n w i l l produce i n v e r s e s 73 = B - l and i f B1: w i t h B2: D2 + P t h e n BIB; 1 : P + Q ( i . e . f r o m B2D2 = PB2 we o b t a i n -1 -1 2 -1 B2 P and B1B2 P - BID B2 = QB,B;'). We w i l l b e g i n w i t h A f ( x ) = C f ( x ) = G(x) i n (4.4) (on

2

(--,m)),

P = D

, and

Q = D2

-

P

take

=

D2 + Q 2 -1 D B2 F(x) and

q with q possibly

P i c k a p o i n t (;,$) and l e t R(x,y,$,j) be t h e Riemann func2 t i o n s a t i s f y i n g DxR = Q(D )R w i t h R = 1 on t h e c h a r a c t e r i s t i c 'lines x-x^ = Y +(y-$) ( R i s c o n s t r u c t e d below). L e t D be t h e t r i a n g l e w i t h v e r t i c e s (x^-y*, 2 0), (?,;), and (i+$,O). L e t cp s a t i s f y DXv = Q(D )cp; m u l t i p l y t h i s e q u a t i o n Y by R and t h e R e q u a t i o n by IP so t h a t upon s u b t r a c t i n g one has vxxR - vRxx = complex valued.

o r e q u i v a l e n t l y Dx(cpxR - q R X ) - D (cp R - cpR ) = 0. I n t e g r a t e 9YYR - 'PRYY Y Y Y D and use Green's f o r m u l a t o o b t a i n (I' = a D = boundary D) t h i s over

(IPR -

(vXR - cpRx)dy

(4.5)

Y

r

cpR )dx = 0 Y

Using t h e a p p r o p r i a t e boundary values we have t h e n (changing (;,$)

to

and (x,Y) t o (5,111)

'x-y

EHE0REm 4.5,

'I

z

Given a Riemann f u n c t i o n R as i n d i c a t e d t h e s o l u t i o n o f DXp =

Q(Dy)cpy p ( x , O ) = F ( x ) , and cp (x,O) Y

= G(x) i s g i v e n by (4.6).

L e t us c o n s t r u c t R now f o l l o w i n g [Mc4] (assuming q E Co f o r convenience). 2 A A A Thus c o n s i d e r D R = Q(D )R and w r i t e 5 = x t y and 17 = x - y w i t h 5 = x+y and X Y n , % A 11 = x-y. Set r ( c , n ) = R(x,y,$,?) (= r(cYn,?,6)) and i n t h e r e g i o n D ' : 5

n 55 5

one has

(4.7)

4r

En + q ( ( c - q ) / Z ) r

= 0

w i t h r(t,n) = r(S,;) = I . T h i s i s a s o - c a l l e d Goursat problem f o r r and t h e r e g i o n i s shown below.

17

BASIC TRANSMUTATIONS

(4.8)

Set ( 1 / 4 ) q ( ( c - n ) / 2 ) = - f ( < , n ) a n d then ( 4 . 7 ) can be converted t o an i n t e gral equation 4

(4.9)

r(5,n)

=

1 -

1;

ds j q f ( s y t ) r ( s , t ) d t

fi

One uses successive approximations i n a standard manner i n s e t t i n g r o ( c , n ) =

1 and

rn(c,n)

(4.10) If I f ( s y t )

<

=

-

( 1;

f ( s , t ) r n b (l s , t ) d t d s

M i n D ' f o r example one finds e a s i l y ( e x e r c i s e )

(4.11) Consequent y the s e r i e s

10" rn f o r

r converges uniformly i n D ' and represents a continuous function. By ( 4 . 9 ) , r 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 in 5 and I? and, i f q E C1, then r has continuous second d e r i v a t i v e s . Hence

CHZ0REIll 4-6, The Riemann function R with continuous second p a r t i a l deriva-

t i v e s s a t i s f y i n g t h e Goursat problem ( 4 . 7 ) can be constructed a s above via 1 (4.9) when q E C . I f q E C o y (4.9) y i e l d s a generalized Riemann function r w i t h continuous f i r s t p a r t i a l d e r i v a t i v e s . By approximating (uniformly) 1 Co p o t e n t i a l s q by C p o t e n t i a l s q n the generalized Riemann function r can be approximated by twice d i f f e r e n t i a b l e Riemann functions rn so t h a t ( 4 . 6 ) can be v e r i f i e d f o r generalized Riemann functions. Pmod:

For t h e approximations r n + r = R we r e f e r t o [ M c ~ ] .

The case of principal i n t e r e s t t o us involves F = Af In this event (4.6) becomes

=

.

f and G = Cf = f ' .

18

ROBERT CARROLL

Now s e t x = 0 and d e f i n e (4.13)

K(Y,S)

-(1/2)[RS(S,0,0,Yl

=

so t h a t K w i l l be continuous.

+ Rq(S,O,O,Y)l

D e f i n e t h e n a t r a n s m u t a t i o n B by

(4.14)

Bf(y) = ~ ( O , Y ) = f ( y ) +

(4.15)

E f ( y ) = e ihy +

rY

J

K(y,S)f(S)dS -Y Q 2 Moreover, i f one takes Eh(x) t o be t h e s o l u t i o n o f Qu = -h u w i t h E?(O) = 1 9 Q 2 2 and DxEh(0) = i h t h e n q ( x , y ) = e x p ( i x x ) E h ( y ) s a t i s f i e s D q = Q(D )q = -h q X Y w i t h q(x,O) = e x p ( i x x ) and D q(x,O) = i h e x p ( i x x ) = ( e x p ( i x x ) ) ' . Using Y (4.14) we have t h e n

= K(y,x)

W r i t e now K,(y,x)

ry

K(y,c)eihcdS

-

K(y,-x)

and

EHE0REM 4.7- The o p e r a t o r s Bh and ,B w i t h continuous k e r n e l s Bh(y,x) = 6(y-x) + Kh(y,x) and ~,(y,x) = 6 ( y - x ) t K,(y,x) a r e t r a n s m u t a t i o n s D2 + Q and a r e c h a r a c t e r i z e d by qXQYh(y) = Coshy +

(4.17)

P:

,(y)

loY

Kh(y,x)CosAxdx;

= [Sinxy/h]

+

1

Y

Km(y,x)[Sinhx/A]dx

0

Pfiaab:

A f o r m u l a s i m i l a r t o (4.15) h o l d s f o r EQ & x ) and e x p ( - i h x ) so t h a t

Cosxx + h[Sinxx/x] EQ x ( y ) ] .

-t

+

~ ! , ~ ( y =) (1/2)[E!(y)

The c a l c u l a t i o n y i e l d i n g (4.17) i s t h e n r o u t i n e ( e x e r c i s e ) .

-

Now f o l l o w i n g [Mc4] l e t us determine K(y,x)

i t s e l f as a s o l u t i o n o f a Gour-

s a t problem and a t t h e same t i m e r e l a t e t h e p o t e n t i a l q t o K(x,x). t h a t K determines K t i o n (0'

-

9

o r Km f r o m (4.16).

q)E = -A E f o r E = Ex4 o f (4.15) and c o n v e r t t h i s t o an i n t e g r a l

Eh(x) Q = e ixx

+

One o b t a i n s

lox

[Sinh(x-t)/h]q(t)E:(t)dt

Put (4.15) i n (4.18) now t o g e t (4.19)

Note

Thus t a k e t h e d i f f e r e n t i a l equa-

e q u a t i o n by t h e v a r i a t i o n o f parameters technique. (4.18)

-

Eyh(y)] t (h/ZiA)[EhQ(y)

'1 -X

K(x,t)eihtdt

[SinX(x-t)/h]q(t)eihtdt

= 0

+

B A S I C TRANSMUTATIONS

I

I

t

X

+

(4.19)

19

[Sinx(x-t)/x]q(t)

K(t,S)eixSdgdt

-t

0

S+x-t eixudu

= ( 1 / 2 ) /S-x+t

One uses now r e l a t i o n s o f t h e form [Sinx(x-t)/x)eixS

t o s i m p l i f y t h e r i g h t s i d e o f (4.19) and r e p r e s e n t i t as a F o u r i e r t r a n s form (note K ( t , c ) (exercise

-

=

It\).

0 f o r 161 >

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,<,h) = m h ) (5.9)

(1/2)[f(y) - f ( - y ) l =

1

Y

0

mh(Y,c)f(<)d<

+

6

mh(Y.-'l)f(-Q)d'l

is known such generalized t r a n s l a t i o n s can be found f o r example i n t h e form Typically i f R'

24

ROBERT CARROLL

f o r s u i t a b l e F(x).

Indeed such

q

Q(Dxk=

s a t i s f y formally

Q(D )lp and Y

setting f ( X ) = V(Xy0) = (F(X)cpX,h(X)yR Q Q )A

(5.11) one has

(x,O) = h f ( x ) by (4.2). Y (one w r i t e s here q ( x , y ) = S:f(x)).

Further q(y,x)

~p

= q(x,y)

and q ( 0 , y ) = f ( y )

We n o t e t h a t one i s n o r m a l l y i n t e r e s t e d

i n x,y > 0 i n (5.10) and f o r x ) y no e x t e n s i o n o f q i s r e q u i r e d w h i l e f o r

0 5 x 5 y we can s i m p l y d e f i n e q ( x , y ) = q ( y , x ) . w r i t e f o r 0 5 x,y <

where O(x,y)

m

We n o t e a l s o

i s d e f i n e d as i n (5.8) f o r 0 ~y 5 x.

e(y,x)

=

Hence i n (5.8) we can

t h a t i f we t a k e h = 0, q even, and f even t h e n q Q

A ,h

also (5.11)) and t h i s corresponds t o Uo(y,<) i n (5.9) ( c f . (4.6)).

( x ) w i l l be even ( c f .

= -(1/2)R

0

(s,O,O,y)

b e i n g odd

Formulas o f t h e f o r m (5.10) 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 e t r e a t e d f u r t h e r i n Chapter 2. We n o t e i n passing t h a t t h e model 2 2 g e n e r a l i z e d t r a n s l a t i o n f o r Dxq = D v , q(x,O) = f ( x ) and IP (x,O) = 0 i s Y Y q(X,Y) = (1/2)[f(X+Y) + f ( X - Y ) l . Now t h e program t o a r r i v e a t t h e Parseval f o r m u l a (5.7) goes as f o l l o w s . L e t sn(x) be a sequence o f f u n c t i o n s such t h a t 6,, 2.11).

I t i s convenient h e r e t o t a k e 6,(x)

an(x) 2 0 f o r 0 < x < l / n , 0 f o r any h.

+

and Im Gn(x)dx = 1.

U

L~,,(x+Y)

+

Sn(x-y)

I;

en ( x ,Y)

1

a,

6 n ( X ) = un(X,0)

=

E l

( c f . Example

;:;1Q

=

2 l/n,

Then a l s o SA(0) = h6,,(0)

0

We w r i t e " e x p e r i m e n t a l l y " f r o m (5.10)

(1/2

6 in

= 0 f o r x = 0 and f o r x

-

=

(5.12)

w(x,y, 5 ,h) 6n ( 5 I d < ;

Rn(~h"f',h(x)d~

0

L e t Bh and B,

be determined as b e f o r e so t h a t Bh~!yh = CosXy.

a p p l y i n g Bh = I + Lh t o t h e l a s t e q u a t i o n i n (5.13) one o b t a i n s

Then f o r m a l l y

PARSEVAL FORMULAS

Now

(

i s continuous i n y b u t may grow a r b i t r a r i l y as y

Lh(y,x),6,(x))

one c o n c e n t r a t e s on an i n t e r v a l 0 5 x,y 5 u. t i o n y"(y) = 1 f o r 0 5 y 2 20 w i t h 'y (5.15)

Y ' ( Y ) [ ~ ~ ( Y )+

E v i d e n t l y R:

25

[

so

Choose a smooth c u t o f f func-

= 0 for y

Lh(y,x)Gn(x)dx1 =

+ m

2 2a+l.

c'

Then w r i t e

R:(h)Cosxydh

can be found v i a t h e i n v e r s e F o u r i e r Cosine t r a n s f o r m ( c f .

Theorem 4.1 and (5.18) below) and one w r i t e s

Note f o r 0 < x,y 5 u t h e corresponding formulas i n (5.13) i n v o l v i n g en and

w do n o t r e q u i r e an i n d e x u .

Now one shows (see t h e p r o o f o f Theorem 5.5)

t h a t UE(x,y) expressed v i a Bn and w tends t o 6 ( x - y ) so t h a t , f r o m (5.161, 2 f o r f,g E K ( u ) (5.17)

(tj(x-y),f(x)g(y))

=

J

m

f(x)g(x)dx

=

0

F u r t h e r f r o m (5.15) one has ( s i n c e 6,

-f

'R

= (2/71)[1

+

F i n a l l y one w i l l l e t u a r e p r o v i d e d below.

J 0 +

-f

6)

Y'(Y)Lh(Y~O)COs~YdYl = ( 2 / a ) [ l m

t o o b t a i n R'

+

+

CfY'(Y)Lh(Y~O)jI

R = R 4 and t h e r e m a i n i n g d e t a i l s

Thus

EHE0RElIl 5-5, L e t Q(D) = D2 - q on COY-) w i t h q E Co and l e t ~ QP ~ , ~ s( aXt i )s f y Q ( D ) q = -h2q w i t h p 4 (0) = 1 and D x p ~ , h ( 0 ) = h. D e f i n e L?,,f(A) = (f(x), pQ

(x)) for f

1,h R = RQ

Ptrood:

E

E

2Lh K say.

Then t h e r e e x i s t s RQ = (2/71)[1

Z ' such t h a t t h e Parseval f o r m u l a (5.7)

We w i l l s k e t c h t h e m i s s i n g d e t a i l s .

+ CLh(y,O)] w i t h 2 holds f o r f,g E K .

Thus f i r s t l e t us n o t e , w o r k i n g

i n t h e r e g i o n 0 ( y 5 x 5 u , x - y 5 6 5 x+y,

(w(x,y,c,h)l 5 c ( u ) say and 5 c ( u ) Im6,,(S)dc 5 c ( u ) ( r e c a l l here U i ( x , y ) = U,"(y,x) so e s t i m 0 a t e s f o r en(x,y) on 0 5 y 5 x 5 u w i l l c a r r y o v e r t o t h e r e g i o n 0 5 x,y 5 0). F u r t h e r s i n c e 6,(c) = 0 f o r 5 > 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 <

T h i s can be shown t o be e q u i v a l e n t t o

a.

( f o r f E Z ) t a k i n g r e a l values f o r

-m

< !J <

m.

One shows a l s o f o r example

i n [Mc4] as an e x e r c i s e CBRBCCARg 5-9,

L e t 16q(x)

dp i s d i s t r i b u t e d over

1

0

-

h)

2

i n Theorem 5.8.

i.e.

[O,m),

m

(5.24)

9( I h l

1

m

f(x)g(x)dx =

0

Qhf(d!J)Qhg(JlJ)dP(!J)

Then t h e measure

28

ROBERT CARROLL

-

REMARK 5-10. One can transmute P = D2 t r a n s m u t a t i o n s B1: D2

+

Q and

B2

= Bil:

q1 i n t o Q = D2 P

-+

-

q2 by composing

D2 (one can a l s o c o n s t r u c t t h e

t r a n s m u t a t i o n d i r e c t l y which w i l l be discussed l a t e r f o r more general op1 2 e r a t o r s ) . Thus e.g. i f B, % I + Kh and B2 % I + Lk one o b t a i n s f o r B = B1B2: P

-+

Q

L a t e r i n Chapter 2 we w i l l study t h e f o r m a t i o n o f s p e c t r a l p a i r i n g s t o repr e s e n t such t r a n s m u t a t i o n k e r n e l s K(y,<).

where R

P

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

Formally i n f a c t B = I + K i s

Indeed u s i n g o u r s p e c t r a l mach-

i n e r y h e u r i s t i c a l l y and f o r m a l l y one has ( s e t ;(A)

= (Pkf)(h))

S i m i l a r l y i n a f o r m a l s p i r i t ( i n t e g r a l s may sometimes be i n t e r p e r t e d as d i s tribution pairings)

so

‘v,k

(h)Rp

%

~(A-V).

Consequently, f o r m a l l y a p p l y i n g (5.27) t o q p v,k

I n t h e same way one can r e p r e s e n t t h e k e r n e l y(x,y) (5.31)

Y(x,Y)

= (q,yk(X)qA,h(Y)YR P Q

f o r B = B-l i n t h e form

Q)

We mention a l s o t h e i m p o r t a n t a d j o i n t t y p e t r a n s m u t a t i o n via

E‘ =

(5.32)

(B-’ )* o r (

v(y),Bu(y)) =

(c)

( E v(x),u(y))

E:

P

-+

Q defined

SPECTRAL THEORY IN ENERGY It i s t h e n immed a t e t h a t

= ker

?!

29

has t h e form ( c f . ( 5 . 3 1 ) )

N

(5.33)

6,

B(Y,X

=

u(x,y);

E ( Y ) = G(Y,X),f(X)) L e t us g i v e now t h e f i r s t o f

$PECCRAt CHE0RM IN CHE ENERG1J VARIABLE,

some procedures t o determine t h e s p e c t r a l measure which produce e x p l i c i t We w i l l work here f i r s t w i t h t h e e q u a t i o n

formulas.

L

Q(0)u = (Au')'/A = -A u

(6.1) where A

1 C (0,m) w i t h 0 < a

E

(i.e. A'

-+

c

A(x) 5 A <

0 as f a s t as d e s i r e d ) .

m

and A

Bp1,Z;

Sa1,4,6;

Bql-4; B v l ; Bbgl; C37,40,49,66-73,76;

Sel-3;

Stf2-6,8,9]).

Am as x

-f

"rapidly"

m

Such o p e r a t o r s a r i s e i n many p r a c t i c a l

problems ( i n geophysics, t r a n s m i s s i o n l i n e s , e t c . Bo1,2;

-+

We w r i t e q !

-

see e.g. Gi1,2;

[Bb1,2;

Bnl;

G r l ; K l l - 3 ; Ne4;

( r e s p . @'+A) 4 f o r the solutions o f

(6.1) s a t i s f y i n g (6.2)

~ Q ~ ( =0 1;) DxqA(0) 4 = 0; @p,A(x) Q % A?exp(?iAx)

as x

+

m

Q w i l l a g a i n be c a l l e d J o s t s o l u t i o n s and f o r v a r i o u s reasons t o apThe @+A p e a r l a t e r i n c o n n e c t i o n w i t h s p e c i a l f u n c t i o n s and L i e t h e o r y we w i l l c a l l t h e q QA s p h e r i c a l f u n c t i o n s .

If A E C

REmARK 6.1.

2 i n (6.1) one can s e t

J, = &u

w i t h Q ( x ) = A-4(&4)xx

and t h e n (6.3)

J,xx

- Q(x)J,

2

= - A J,

w h i l e if u ( 0 ) = 1, u ' ( 0 ) = 0 one f i n d s ( n o r m a l i z i n g t o A(0) = 1 ) (6.4)

$(O)

= 1; J , ' ( O )

= (1/2)A'(O) = h

which b r i n g s us t o t h e s i t u a t i o n o f (4.2) f o r q y y h ( w i t h q = Q ) .

L e t us

n o t e a l s o t h a t i n t h e c a l c u l a t i o n s t o f o l l o w f o r t h e Q o f (6.1) one c o u l d always add a t e r m -G(x)u t o Q(D)u w i t h say -+

<

E

Co v a n i s h i n g s u i t a b l y as

X

-; t h e n a t u r e o f t h e r e s u l t s and e s t i m a t e s would be e s s e n t i a l l y unchanged

(except f o r t h e p o s s i b l e emergence o f d i s c r e t e eigenvalues). Going now t o (6.1) one uses a v a r i a t i o n o f parameters procedure t o c o n v e r t t o i n t e g r a l equations ( s e t t i n g q = - A ' / A ) (6.5)

Q

q A ( y ) = Coshy +

Q CSinx(y-17)/h1q(rl)DrlqA(~)d~;

30

ROBERT CARROLL

One can s o l v e these equations i n t h e form 9p,(y) Q = Im9,,(X,y) c ; ~ ~ ( h , y ) w i t h p 0 = Coshy, a.

I,

= exp(ihy),

and AI2 AQ ( y ) =

0

and

m

(6.7)

Qln(A,y) =

L e t us n o t e t h a t i f A $'

'L

i h e x p ( i h y ) as y

ihA?exp(ihy)

[Sinh(~~-y)/hIq(rl)Ql~_~ (A,n)dn;

E

u

'L

e x p ( i h y ) and

m

t h e n u = A%

-f

0; hence ( 6 . 6 ) i s a c c e p t a b l e f o r n o r m a l i z a t i o n

-f

since A '

CL and one uses Remark 6.1 w i t h IL

r a t h e r t h a n t r y i n g t o work i n i'(y)

satisfies

w i t h Gh(y). Q

'L

A:'exp(ihy)

and u '

'L

The d e t a i l s o f e s t i m a t i o n

a r e g i v e n i n [C4OY67; S e l l ( c f . a l s o [ B r l ] )

and a r e sketched below as needed. Thus g i v e n t h e hypotheses on A an assumption q E L 1 (0,m) i s n o t r e s t r i c t i v e and by i t e r a t i o n (see t h e p r o o f below) one has f o r Imh 2 0

S i m i l a r l y f o r any

x

E C

The r e s u l t i n g s e r i e s converge u n i f o r m l y on compact s e t s i n t h e a p p r o p r i a t e domains and one o b t a i n s

CHE0REFR 6-2-

i s an e n t i r e f u n c t i o n o f X o f e x p o n e n t i a l t y p e y w h i l e qp,(y) Q

a Qx ( y (resp. @Q- h ( y ) ) i s a n a l y t i c f o r Imh > 0 (resp. Imh < 0) and t h e f o l l o w i n g estimates are s a t i s f i e d

Ptraad:

To check t h e estimates ( 6 . 8 ) - ( 6 . 9 )

f i r s t note t h a t ISinhxl 5

c l h l x e x p ( 1 I m x l x ) / ( l + ) A 1x1 and [ S i n X ( x - c ) \ 5 c l h l xexp( 1 I m h ) ( x - c ) ) / ( l + l h \ x ) (observe

1x1 ( x - c ) / ( l + ) x ) ( x - c ) ) 5

Ixlx/(l+lXlx)).

Hence

SPECTRAL THEORY IN ENERGY

31

1 where we take ImA L 0 (and assume say q E L ) . Similarly @i(A,y) = -/ym Cosx(q-y)q(n)@~(A,n)dn. The Cosine term can be given a crude estimate of the form ICoshxl 5 cexp((Imx1x) so that for ImA 2 0 m

(6.13)

I@i(A,y)( 5 c tellmx'(n-y'lq(n)l

(@A(h,n)ldn

5

clkle-Y1mA [lq(n)ldn Hence replacing estimates like

lAlq/(l+lA\q)

by an upper bound 1 , we get

w

(6.14)

1@2(h ,Y

I

c

el Imx I ( ' )

5c

I q(n)II@fii

( A ,111

I d d I 1I

5

rw

2 -yIrnA Iq(n)l J (q(c)ldcdn 5 c2(1/2)e-y1mx [lq(n)\dn12 c e n Further -/ym CosA(q-y)q(n)@i(A,n)dq = @~(A,Y) so (for ImA 2 0 ) m

(6.15)

]@;(A,Y)~

5 c i e I ImxI

c2 I x 1 e-YImA [lq(n)\

1 q ( n ) 1 1";

[lq(c)ldcdn

(A

, n ) 1 dn 5

5 (1/2)cZlxle -yImx [ [lq(n)Idnl2

The pattern is now clear and leads to (6.8) SO that (note l@o(h,y)l 5 exp(-ylmh) for @ = 1 an we have (6.11) for Q and Imx 5 0 (the series converges absolutely and uniformly). The estimates for @(-X,y) are virtually identical except that we work with Imx 5 0. Similar considerations apply 4 Indeed Ivo(A,y)l 5 exp(yl1mxl) with to the series for v(A,y) = vA(y). Ivpb(x,~)l 5 Ixlexp(ylImAl). Hence

Continuing we obtain

P i (xly)=

'1 Cosx(y-n)q(n)v~(x,n)dn 0

and

I P i (h,y)l

'1 exp(lIrnhI(y-n)Iq(n)ll~]exp(nIImhI)dn 5 Ihlexp(ylImhl) /oylq(n)ldn. 0 Y (6.17) (v~~(A,Y)I5 e'lmx'(y-n)Iq(~)II~i(x,~)Id~/I~I5

5

Hence

1

0

(1/2)e11mA/Y [ J;Iq(n)Idnl

2

The pattern is again clear as before and we conclude that (6.9) holds. Hence the series for qp,(y) 4 converges absolutely and uniformly on compact sets in x and (6.10) is valid. The following observations will be needed in Chapter 3 . We note that if we write (6.8) as l@n(xyy)l 5 exp(-yIma)Qn(y)cn/n! with (following (6.151, 1 5 Ixlexp(-yImh)Qn(y)cn/n!, then

\@.;I(A,Y)

IL2'(LY) - *&Y) (@ =

^cQ

ax). Q

1"1 cnQn/n!

Since

one o b t a i n s

(6.19)

IA3(x,y) - e iAy )A~'(A,Y)

-

S i m i l a r c o n s i d e r a t i o n s a p p l y t o q ( ~ , y ) and one has (6.20)

-

Iv(h,y)

I

Cosxyl 5 ce

:1 1

I

q ( n ) dn ;

~ l p ' ( ~ , y +) ASinhyl 5 ? [ A l ey'lmhl

l:lq(n)ldn

LETIIIN\ 6-3. Under t h e hypotheses i n d i c a t e d we have (6.19) and (6.201, s i m i l a r i n e q u a l i t i e s i n v o l v i n g @Q- A ( y ) .

plus

It i s perhaps somewhat more r e v e a l i n g ( i n p l a c e o f ( 6 . 5 ) - ( 6 . 6 ) ) 2 t o s t a r t from A E C (A(0) = 1) and use Remark 6.1 t o o b t a i n f o r J, = A%!,

REmARK 6.4.

h = (1/2)A'(O) = J / ' ( O ) , and Q = A-"')'

c loX

(6.21)

J , ( x ) = CosXx + h[SinAx/X]

(6.22)

A'(x)qX(x) Q

= CosXx

where (6.22) now uses o n l y A (6.23)

A?i(x)@,(x) Q

= e ihx

+

+

[SinX(x-E)/XlQ(6)J/(~)d~

(A?')'[CCISX(X-E)~~

1

Q - Sinx(x-E) h

Similarly

(and h d i s a p p e a r s ) .

E

C

-

IxW

(A')'[CosX(~-x)@!

+P ,Q dS l 6 1

+

Sinx~n-x)Dn@~]dn

One can now produce i t e r a t i o n s as i n t h e p r o o f o f Theorem 6.2 and a r r i v e a t a somewhat n e a t e r c o n s t r u c t i o n i n c e r t a i n respects.

We w i l l r e t u r n t o t h i s

l a t e r . One notes however t h a t a l t h o u g h A E C ' o n l y i s needed t o s p e c i f y q: and (PA Q by i t e r a t i o n from (6.22)-(6.23), i n o r d e r t o v e r i f y t h a t t h e d i f 2 f e r e n t i a l e q u a t i o n i s s a t i s f i e d one needs A E C . When b o t h (6.5)-(6.6) 2 and (6.22)-(6.23) a p p l y ( i . e . f o r A E C ) t h e n t h e s o l u t i o n s agree by uniqueness. The terms i n t h e @,,(-~,y) e t c . f o r

@

x

s e r i e s f o r example ( i n Theorem 6.2) s a t i s f y Gn(X,y) = r e a l so one o b t a i n s f o r X r e a l

SPECTRAL THEORY IN ENERGY

@.,(Y) -4

(6.24)

33

= @Q- x ( ~ ) ; q -QP , ( y ) = 9 Q- , ( ~ ) = v Qx ( y )

f o r any X E C w i t h G,(y) Q = @-x(y) Q ( a c t u a l l y @,(y) -9 # 0, ax Q and Q a r e l i n e a r independent w i t h

= q

x

Q

(6.25)

Q; ( y ) ) .

Further f o r

Q

A(Y)W(@.,(Y),@-,(Y)) = -2ix

(W(f,g) = f g ' - f ' g i s t h e Wronskian). Hence qhQ w i l l be a l i n e a r combinat i o n of ax Q and ,@! which by p r o p e r t i e s i n d i c a t e d i n ( 6 . 2 4 ) w i l l be q AQ ( Y ) = c Q ( x ) @Q~ ( Y ) + C Q ( - h )Q@ - ~ ( y )

(6*26)

c

( A ) = c (-A) for (where Q Q o b t a i n s ( r e c a l l A(0) = 1 )

x

real).

Using (6.25) w i t h y

+

W(qx(0),@ Q Qx ( 0 ) ) = Dx@-x(0) Q = -2ihc ( A ) ; D p xQ( O )

(6.27)

0 and (6.26) one

=

2ihc (-1)

Q Q (and thus in general C ( A ) = c (-i)). Such formulas, o r perhaps b e t t e r Q Q A(y)W(qx(y),@-,(y)) Q Q = -2ihc ( A ) , show e.g. t h a t x c (x) i s a n a l y t i c f o r Im x Q Q < 0. We c o n s i d e r now t h e p o s s i b l e vanishing of c ( x ) (= c ( A ) ) f o r Q ( D ) a s Q i n ( 6 . 1 ) (no term :(x) i s adjoined here - c f . Remark 6.1). I f C ( A )= 0 f o r A r e a l , A # 0 , t h e n c(-X) = c ( A ) = 0 and q Q x ( y ) = 0 by (6.26) which contraQ and %: one o b t a i n s d i c t s ~ Q ~ ( =0 1.) Let then ImA > 0 and from (6.1) f o r ax ( A = A

(6.28)

1

+iA)

2

DxaA Q ( O N-9A ( 0 )

- Dxz:(0)@xQ ( 0 ) = 4iA1x2 jiA(Y)I@!(Y)l

( n o t e terms with expi(x-7)y

exp(-2x2y) + 0 a s y +

=

'dY

and by Theorem 6.2 t h e

i n t e g r a l i n (6.28) makes s e n s e ) . Now by (6.27) Dx@:(0) = 0 (= DxG:(0)) means c ( - A ) = 0 and by (6.28) this can happen only i f x 1 = Rex = 0. Hence Q t h e zeros ( i f any) of x c ( - A ) i n i t s h a l f p l a n e of a n a l y t i c i t y ImA > 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 <

AIq'12dy and (6.11) holds)

34

ROBERT CARROLL

2 2 we have -A2 J " l q l Ady =

Jm

0

0

2 A l q ' I d y which i s i m p o s s i b l e .

Consequently

CHEaREm 6.5, A ( y ) W (Q~ ~ ( y ) Q, @ ~ ( y=) )2 i x c (-A) so Xc (-A) i s a n a l y t i c f o r Imx Q 4 > 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))

(<

,

)

being a d i s t r i b u t i o n p a i r i n g ) .

It follows t h a t

36

ROBERT CARROLL

Now r e c a l l t h a t A(x)W(@,@) = 2 i h c c ( - h ) i s a n a l y t i c f o r Imh > 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)

<(XI =

I

m

C(Y)A(Y)

v(h,xh(x,y)dv(A)dy

0

2 w i t h dv(A) = d x / 2 1 ~ [ c ~ ( x ).I Note here t h a t t h e i n t e g r a n d i n (6.47) has t h e form ( 1 / 2 a i ) [ ~ A 1 ( 2 A / 2 i ) { 1 where {

1

= @ ( X , Y ) [ ~ ( - ~ , X ) / ( - ~ C ~) ()

xc 4( - A ) ] f o r x > 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 - <(x) we s e e t h a t x Qu = A 2 2 2 - k x u becomes Qu = u. I t m i g h t seem a t f i r s t t h a t t h e study of Q and

ROBERT CARROLL

38

can be e a s i l y connected ( s i n c e a s u b s t i t u t i o n 5 = logx y i e l d s xDx = D C ) b u t t h e r e a r e good reasons f o r only u s i n g such a connection a f t e r c e r t a i n basic machinery i s i n place. In p a r t i c u l a r the s p e c t r a l theory i s d i f f e r e n t f o r N

4"

and n = 3 o r more parThus go t o ( 7 . 1 ) f o r t i c u l a r l y t o ( 7 . 3 ) w i t h A* = v2 - (1/4) = E(E+l) where v = E + ( 1 / 2 ) . In this section we will develop a spectral theory, w i t h a Parseval formula and and Q as we will now see.

inversion theorem, and then p u t together some formulas analogous t o ( 5 . 2 7 ) , (5.31), e t c . A p a r t i c u l a r case of g r e a t importance i n the study of special functions involves t h e Bergman-Gilbert operator and t h i s i s developed in Section 2.9 ( c f . a l s o [Bcl; C41-43,54; Cnl-7; Gjl-3; H d l ; SalO])

REMARK 7.1,

There i s a g r e a t deal of precise information i n [L17; Lrl] f o r example r e l a t i v e t o a s above which we momentarily ignore. The reason f o r this i s p a r t l y because the s p e c t r a l theory developed t h e r e uses an additiona l parameter, w h i c h a r i s e s i n describing s e l f a d j o i n t r e a l i z a t i o n s of ?,and the kind of r e s u l t s we want (which should not depend on such a parameter) would then have a form where t h e parameter was i n e x t r i c a b l y bound u p i n the REMARK 7.2,

6

formulas. r e a l . In order t o deal w i t h s p e c t r a l T h u s take ( 7 . 3 ) as indicated w i t h questions without introducing unwanted parameters ( c f . Remark 7.2) we follow

a technique of [Bbdl]. We assume as in [Bbdl] t h a t 1 ; x l q ( x ) l d x < m as well as .rowx21G(x)ldx < m and w i l l follow t h e i r notation i n this section. T h u s the J o s t solution f ( v , - k , x ) = xg(v,-k,x) 'L exp(ikx) as x m and t h e regul a r solution 9 ( v , k , x ) 'L xu 'L xv +(1/') as x + 0. We will c i t e p r o p e r t i e s of -f

the J o s t s o l u t i o n s and regular s o l u t i o n s from [Bbdl; Bel; Bdl; Dc1,2] as s i n c e we a r e primarily involved needed and do not emphasize hypotheses on

w i t h asymptotic p r o p e r t i e s and a n a l y t i c i t y p r o p e r t i e s o f f and

9

(cf. also

LCel; J d l ; Fnl; Lrl; L17; Ne1,6; Sa10-121). In c o n t r a s t t o physics where properties of a r e meaningful we a r e mainly concerned with being able t o j u s t i f y c e r t a i n mathematical developments and thus we deal w i t h p r o p e r t i e s of f and 9 ; such p r o p e r t i e s do hold of course f o r various types of hypotheses on 5 (which a r e r e a l i s t i c ) . The regular s o l u t i o n 9 ( v , k , x ) behaves l i k e v+( 1 /2 1 near x = 0 and is determined by X (7.5) When

?=

0 one has

39

SPECTRAL THEORY I N MOMENTUM

(7.6)

po(v,k,x)

= 2"r(u+l)k

-" b

x'Jv(kx)

and i n f a c t p can be determined i n terms o f such f u n c t i o n s by

(7.7)

p(u,k,~) = x

-

v+'

-

fox

E'CJ

[x'1~/2Sinv.rrl

v

(kS)J - V ( k x )

- J-,(kS)J,(kx)l~(S)p(v,k,5)dS

Under s u i t a b l e hypotheses t h e l i n e Rev = 0 w i l l be i n a r e g i o n where both p(v,k,x)

and p(-v,k,x)

usual W(p,!b)

= p$'

a r e d e f i n e d and a n a l y t i c ( c f . [ L r l ] ) .

- v'J/

and t h e n W(p(v,k,x),p(-v,k,x))

J o s t s o l u t i o n s one d e f i n e s f(v,k,x)

%

We w r i t e as

= -2v.

For t h e

e x p ( - i k x ) and corresponding t o (7.5)

and (7.7) one has t h e formulas

(7.8)

f(v,k,x)

= e

-ikx

- e k ( x - E ) ] [ q( c )

( 1 / 2 i k ) [[eik(c-x) (7.9)

f(v,k,x)

( i1w'/4)

fo(v,k,x)

%

(v, k, v)dc

+

( k5 H t ( kx 1

0, fo(v,k,x)

and f o r := (7.10)

[$[H;

= fO(v,k,X)

+ ( v2-S-,)/ c 2 ] f

-

H t ( kc 1H: ( kx

1%5 ) f ( v, k ,E )dS

e x p ( - i k x ) has t h e form

= ( ~ 1 / 2 )3-'e - $ i n ( v+ k)(kx)'H:(kx);

f o ( v y - k , x ) = IT/^) ?ie + i T ( v+4 )(kx)'HA(kx) The J o s t f u n c t i o n i s d e f i n e d as f ( v , k )

=

W(f(v,k,x),lp(v,k,x))

and thus t h e

" f r e e " Jost f u n c t i o n i s f o ( v y k ) = 2"(.rr/2)%'(~+l)k-"+~exp[-Si.rr(v-~)] (while f o ( v , - k ) = 2"(~/2)%'(v+l)k -v+4e x p [ + i ~ ( v - % ) ] . We c i t e now

L E ~ A7.3, p(v,k,X),

With i n d i c a t e d hypotheses on {, p(u,-k,x) f ( G , - k , ~ ) = f(v,k,X),

f(v,k)f(-v,-k)

f(-v,k,x)

= 4 i v k , and f o r f,-

W i t h such hypotheses p(v,k,x)

%

= f(v,k,X),

f(v,?k),

and p ' ( v , k , x )

= Ip(v,k,x),

G(c,L,x)

f(v,-k)f(-v,k)

=

-

f' = f ( ? v , k )

a r e e n t i r e i n k and a n a l y t i c i n

40

ROBERT CARROLL

f o r Rev > 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 <

( l / Z ) s + ( s - r ) and f o r s 5 r , g ( r , s ) =

=

N

(7.44)

Bf(r) = ( B ( s , r ) , f ( s ) ) ; iif(r) = (;(r,s),f(s))

“V

z: 5’

5’

N

and + c a r e transmutations with B[g(v,-k;)](r) Then B: Q + 1 - 1 g (v,-k,t-) and B[g ( v y - k y - ) l ( r ) = g ( v , - k , r ) .

=

Pfi006: The statements about B and have been proved above. To obtain t h e analogous a s s e r t i o n s f o r and consider t h e c a l c u l a t i o n s f o r I i n (7.35)again ( p 1 f, p , f 1 c, f , e t c . ) (7.37). If we interchange the r o l e s of y l a n d then one can consider m 1 ?(k,r) = vdv h(s) ( v y ~ y r ) f ( v y - k y s )d s = (7.45) rsf (v,-k) r N

[ ‘

J

[h(s)[

1

g1 ( v , - k , r ) g ( v , - k , s ) d p 1Ids

=

(1/2)h(r) =

6

h(s)F(s,r)ds

Thus z ( s , r ) = ( 1 / 2 ) 6 + ( s - r ) f o r s 2 r ( i . e . g ( r , s ) = (1/2)6+(r-s)f o r r and, a s in (7.43), from (7.30) we obtain

s)

N ( g1 ( v , - k , s ) , B ( r , s ) ) = rm g 1 ( v , - k , s ) E ( r , s ) d s = g ( v , - k , r ) -

I

(7.46)

N

To check t h e transmutation nature of and one requires gf = QB acting on s u i t a b l e o b j e c t s . Without specifying domains generally one wants t o consider action on various objects a t we simply check t h i s here formally via t h e kernels by noting “1 “1 QrB(r,s) = Q,B(r,s) and Q,B(s,r) = QrB(s,r) f o r r # s. Then u B(r,s) = s(r-s) + K(r,s) so t h a t from g ( v , - k , r ) = g 1 ( v , - k , r ) g 1 (v,-k,s)ds t h e r e r e s u l t s Nh

-

- I V

N

N

(7.47)

CD,

?cl

51

and here ( s i n c e various times) f i r s t that =

+

( l / r ) ~ ? r , r ) = (1/2)[Cl(r) -

w r i t e e.g. + r(r,s)

!:

C(~)I

+

1 s ) . T h e c a l c u l a t i o n follows [Bbdl] w i t h (note QrK(r,s) = 5sF(rys)f o r r K i n place o f b ( s e e Remarks 7.11 and 7.13) and one wants s 2”K(r,s)Dg 1 (v,-k, “ 1 ( v , - k , s ) + 0 as s + m. The resulting j u m p i n Y s ) + 0 and s 2 DSK(r,s)g K(r,s) a t s = r permits one t o w r i t e formally f o r s u i t a b l e f , Q + 8 r , s ) , f ( s ) ) = ( ; s1F ( r y s ) y f ( s ) ) = (K(r,s),:;f(s)) so ?$f = E$’f as required ( r e c a l l Q and ”1 Q have s e l f a d j o i n t form when n = 3 ) . S i m i l a r l y t ; ( B ( s , r ) , f ( s ) ) = ( Q S B ( s , r ) , N I V

N

N

N

N

f ( s ) ) =(B(s,r),Q,f(s)) or

G1%

=

%{f.

.

N

FULL L I N E SCATTERING

REIIARK 7.13-

47

The problem in u s i n g Theorem 7.10 can be seen as follows. First

of course one does not want t o mix the two s p e c t r a l measures p and p1 s i n c e i t i s much e a s i e r t o deal with transmutations expressed i n t h e form (7.44) ( c f . [C40]). More s e r i o u s l y however one simply cannot t r e a t b a s a function in view of our c a l c u l a t i o n s leading t o Theorem 7.12. T h u s e.g. B ( r , s ) = 0 f o r s > 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

FULL LINE SCATTERING

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

<

0.

p(A,f)

=

Jo

u-l/'du =

0 m

m

(8.19)

w h i l e k ( x ) = I'

One has

f(x)CosxJxdx;

f ( x ) = (1/n)

J q(x,f)A-'CosxJkdr 0

F o r a = s2, dh = 2sds, e t c . one o b t a i n s t h e s t a n d a r d Cosine i n v e r s i o n f o r mulas.

-iJx, $1 =

For

(--,a)

one has e = CosxJA, p = -x-'SinxJx,

exp(-ixJx),

=

0, e t c .

$2 = e x p ( i x A ) , 5 ' = 1/2Jx

(x

iJx, m2(x) = O ) , 5' = 0 (x < O ) ,

m,(x) >

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 <

-m

53

i(A)

M1(A)M(-A)]

M(-x) and ?,(A)

=

(ul+ivl)(u-iv)]

exp(iAx),

%

M1(-A), and t h u s MM;

=

= Si(vvl

-

uvl)

-iAx

-

A(x)eiAx

Hence

= 1/2iA.

we have f o r A r e a l e-ixx

(x

-+

-a);

p(A)u2(x)

%

e

= 1/4iAM(A)M1 (1) and A(A) = -[M(h)M1 (-A)

i n p a r t i c u l a r f r o m [Hol]

p

(x

-+

m)

+ M ( - A ) M 1 (x)]/2MM1 (A); T h i s s i t u a t i o n thus

i s a n a l y t i c f o r I m 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

FULL LINE SCATTERING

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

FULL LINE SCATTERING

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<(X ) = dw(A) = dx/ZnlMI

(8.44)

2

Thus F = 2ihM w i t h I F 1 2 = 41 I M I

less potential p (cf. [ K f l ] )

A.

t h a t MM;

=

2 agrees w i t h (8.43)-(8.44).

We observe t h a t i f A = 0 one has a t r a n s p a r e n t o r r e f l e c t i o n -

REIIARK 8.11tion of

2

and t h i s corresponds t o Re MM;

Since we know t h a t 2 i I m MM; 1 / 4 i ~o r cp(A) = M i = 1/4iXM.

= 0 by d e f i n i -

= 1/2iX by Remark 8.9 i t f o l l o w s

I n general o f course t h i s i s n o t

t r u e ( c f . Remark 8.7). L e t us summarize a few i t e m s now i n

EHE4)REm 8-12, L e t p be r e a l , even, p o s i t i v e , and continuous w i t h pexp(2Hx) 1 2 E L (0,m) and Pu = D u - pu. D e f i n e P, x , ul, u2, M, and M1 as i n (8.22), (8.23),

(8.25),

= -[MM;

+ M-M1]/2MM1

M;/Ml). pole a t

and (8.26).

and p a r e a n a l y t i c f o r ImX > - 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;

^a" = 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

<

L e t us s t i p u l a t e as a general s o r t o f conm

w i l l u s u a l l y be s u f f i c i e n t b u t t h i s can

be improved i n v a r i o u s ways. L e t us i l l u s t r a t e t h e m a t t e r w i t h some examples b e f o r e g o i n g f u r t h e r .

!%WIPCE

9.2,

I n many ways t h e p r o t o t y p i c a l s i n g u l a r o p e r a t o r can be s a i d t o

ROBERT CARROLL

60

be (usually we take here m real w i t h m > -(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 %

<'(0)xrn+'

Q

Q

i s Cm and even t h e same h o l d s f o r X.

and Vl;(x)

(m+$)C'(0)xm-'.

4

A l s o as x

t e c h n i q u e one o b t a i n s an i n t e a r a l e q u a t i o n

1

X

(9.29)

V,(x)

=

um(Xx)

+

-f

0, V,

Using t h e v a r i a t i o n o f parameters

Gm(x,x,c)x(c)VA(S)dc

Known p r o p e r t i e s o f Bessel and Hankel f u n c t i o n s ( c f . [ M b l l ) y i e l d

ROBERT CARROLL

66

(+I IU,(AX)~ 5 c[x/(l+lAlx)lmG' exp(l1mAlx) and I G (A,xy5)1 ,< C[x/(l+lXlx)l 5 C l A l - lm [lXlx/(l+lAlx)lmq' exp(lImAl(x-c)) f o r -4 < m -< 4 while lG,l [ l A ) S / ( l + l A ) 5 ) ~ - " e x p ( l I m A l (x-s)) for m >

4. One can solve (9.29) by suc-

cessive approximations i n a standard manner by s e t t i n g Vy(x) = pm(Xx) and

Vy(x) = :f G (x,x,E)X(c)V:-l(s)dc w i t h V,(x) = 1 ; V i ( x ) . From (*) we have Alx~~ f o r n 2 0 I VRA ( x ) l 5 C n ~ f " x / ( l + ~ A ~ x ) l m ~ S " e x p ( ~ I m A ~ x ) [ x / ( l + ~foXlx(5)ldEln Consequently t h e s e r i e s f o r VAconverges absolutely and uniformly on compact s e t s and gives t h e unique s o l u t i o n desired. By uniform convergence V A i s e n t i r e and

U

Hence I vQA ( x ) l 5 K(x)exp(l Imxlx) as a s s e r t e d . Next one can use S t i r l i n g ' s formula as i n Example 9.3 t o estimate c ( A ) Q given by (9.14) (note a l s o t h a t from (9.14) c ( A ) = 0 when A = i ( 2 n + p ) o r

4

i(Zn+l+a-B) unless one of these i s equal t o i m where i t would be canc e l l e d ) . Thus in a straightforward manner one obtains A =

LElllMA 9.9-

-EIs/

Given

E

> 0 there exists

K > 0 such t h a t f o r

A =

s+iq w i t h

rl

2

(p,q F 0 r e a l ) IXcQ(-A)I 5 K(l+lAl) -%(P+q);

(9.32) Further

AC

Q( - A )

i s analytic i n

I C Q ( - A ) I -1

K ( 1+ 11I )%(P+q)

w i t h zeros in t h e s e t -i[:,m).

P L C J O ~ : The estimates follow from S t i r l i n g ' s formula as indicated once

(9.14) i s known.

We note a l s o t h a t

since A W i s independent of x and p: = c aQ + c-aQ with AQ(x)W(@A,@-x) Q Q comQ QQ QQ -A and *-A e s t a b l i s h e d above (note puted from t h e asymptotic behavior of t h a t A Q exp(2px) here). Consequently Xc ( - A ) is a n a l y t i c i n 1;1. To esQ Q Q t a b l i s h (9.14) one can write ( l i m i t s as x -+ m ) lim [exp(iA-p)xp,,(x)] = 1imCc (h)exp(iAtp)xexp(iA-p)x + c ( - ~ ) e x p ( i ~ + p ) x e x p ( - i ~ - p ) x=] c ( - A ) f o r 4 4 9 ImA 0. Now use t h e representation (9.12) f o r p JQx ) with a = 4 ( p + i x ) , = +(p-iA)¶ c = % ( p + q + l ) , p,q 2 0 r e a l , and ImX = TI > 0 w i t h 0 < n < l + p/2 ( t h i s insures t h a t Reb = Re ?-,(p-ih) = +(p+q) > 0 and Re(c-b) = Re 4(1+ p+ih) = 4(1+ p - n ) > 0 ) . A standard formula f o r t h e hypergeometric 2 function (cf. [Mbl]) gives then (z Q-sh x )

b

SINGULAR OPERATORS

Z Z a r ( c ) JoItb-l ( l - t ) c - b - l -a

r

m

which i s (9.14) f o r tinuation to A

E P.

T-

as r e s t r i c t e d .

67

ZZar(c)r(-iA) dt = r ( b ) r ( c - a )

One then extends t h i s by a n a l y t i c con-

I

W e record next a few known formulas involving Jacobi functions and hyperOne can then prove geometric functions ( c f . [Ak3; Mbl; Eb2,3; F f l ; Kpl]). various r e s u l t s (e.g. Theorem 9.12) b u t more important, t h e formulas serve a s models f o r various general s i t u a t i o n s . We will show l a t e r how t o derive such formulas canonically and i n t r i n s i c a l l y i n a unified manner based on transmutation machinery ( c f . a l s o [C40,64,65]). Moreover various general r e s u l t s can then be e s t a b l i s h e d “canonically” without recourse t o special p r o p e r t i e s of hypergeometric functions. T h u s t h e models a r e chosed t o serve a s exhibition pieces f o r general r e s u l t s ( e . g . t r i a n g u l a r i t y o f transmutat i o n kernels) which a r e e s t a b l i s h e d l a t e r i n a canonical manner. Also the models e x h i b i t various typical p r o p e r t i e s o f growth and a n a l y t i c i t y which a r e used l a t e r i n t h e a b s t r a c t presentation. Thus using t h e formula F ( a , b , c , z ) = (1-z) c-a-b F(c-a,c-b,c,z) and Bateman’s i n t e g r a l ( c f . [EbZ])one has f o r y (9.35)

>

0, Rev > 0 , and Rec

> 0

r - l ( c + u ) yc+p - ( 1+y ) a tb-c+p F ( a+p ,b+p ,c+p ,- y ) = [l /r ( c ) r ( p ) ] ~ ~ x C( -1 +x)a+b-cF ’ ( a , b,c, - x ) (y-x)’-’dx

From [Ak3] f o r x

>

0 , Reu

>

0 , and Reb

0, one can w r i t e

>

Translating t h i s i n t o Jacobi functions one obtains E m N A 9-10,

For x > 0 , Rep

> 0,

and Rea

>

-1

while f o r s > 0, Rep > 0 , and ImA > -Re(a+B+l) ( r e c a l l

7aB

= 2JTcaB/r(a+1))

68

ROBERT CARROLL

a+” ’+’( Next, u s i n g (9.15)

t ) [C h2t

-

Ch2slv’-l Sh2tdt

t o g e t h e r w i t h (9.37)-(9.38),

LEIiIIRA 9-11. F o r Rea > Reg >

-+,

one has i n p a r t i c u l a r

e x p ( i x s ) = ?(-A)-’

fsm@,(t)A(s,t)dt

(Imx

> 0)

and I ‘ ( a + l ) - ’ A ( t ) ~ ~ ( t ) = IT-4 f t CosXs A(s,t)ds where L P ~ , 0 f e r t o i n d i c e s ( a , ~ )and A ( s , t ) = A ( s , t ) can be w r i t t e n i n t h e form

A , and c(-A) r e -

a,B

One can e v a l u a t e t h e i n t e g r a l i n (9.39) e x p l i c i t l y as a hypergeometric f u n c t i o n ( c f . [Kpl]).

Indeed u s i n g E u l e r ’ s i n t e g r a l ( c f . [EbZ]) w i t h a s u b s t i -

t u t i o n (Cht-Chw)/(Cht-Chs) mentioned b e f o r e (9.35), A(s,t)

(9.40)

= 5 i n (9.39),

p l u s t h e hypergeometric i d e n t i t y

one o b t a i n s

= 2 a+28+5/2 r(a+%)-1Sh2tChB-at

( C hwt-Ch2s ) “-F‘

This can be used i n p r o v i n g ( c f . [ F f l ;

(a+B, a- B, a+%, 2Cht Cht-ChS

1

Kpl])

L e t 9 Qh be t h e s p h e r i c a l f u n c t i o n tHE0RElIl 9.12. L e t x = c + i n and x E [O,m). o f Example 9.3 which i s e n t i r e i n ?, and C m i n x. F o r each n one has Kn w i t h (9.41)

Ir(a+l)-’O!)$x)

I

5 K n ( l + ( x l ) n + k ( l + x ) e x p ( In1-Rep)x

where k = 0 i f Rect > -4 and k = (+-Rea)

i f Rea <

a l s o t r u e t h a t IDypy(x)1 5 K,(l+x)”’exp( and I m h

i n the (9.42)

Consider f i r s t n = 0 and Rea > (pX

F o r p,q 2 0 r e a l i t i s

( n 1 - p ) ~ ( r e c a l l q = 2W1, p =

F u r t h e r f o r a,@ E C and c > 0 t h e r e e x i s t s 0 I @Q X ( x )I 5 Kexp[-x(ImA + Rep)].

2(a-6)).

Prruua:

-+.

-4.

K such t h a t f o r x ~c

Put t h e k e r n e l e x p r e s s i o n (9.40)

f o r m u l a o f Lemna 9.11 t o o b t a i n Ilp,(t)l

ItRe(a-B) ) t ( ShtCht)- 2Rea /ot(Ch2t-Ch2s)Rea-’ds ( c e ( I ImX -

ce

( ( I m x (+Re(a-B))t

Rea,Rea

(t)

By [Eb2] one has an e s t i m a t e 19!eayRea ( t )I 5 k(l+t)exp[-2Rea+l]t so, com(9.41) f o r n = 0. For n ~1 one uses b i n i n g t h i s w i t h (9.42), we o b t a i n

SPHERICAL TRANSFORMS known formulas ( c f . [EbZ]) (9.43)

69

such as

r(a+l)-lDtp:yB(t)

=

- % [ ( a + a + l ) 2 +A 2 ]r -1 (a+2)Sh2tp;+1'B+1

(t);

p l u s t h e d i f f e r e n t i a l e q u a t i o n and t h e n i n d u c t i o n arguments can be a p p l i e d . We o m i t t h e d e t a i l s h e r e and f o r t h e l a s t a s s e r t i o n we r e f e r t o [Kpl;C40].

PALEg-DZENER C H E O R m , SPHERICAL CRAWF0RlIl5, AND PA%EVAL F6RlllllLM FOR

10.

We g i v e f i r s t a s k e t c h o f t h e arguments i n [Kpl] used

SLNGllLAR 0PERAC0W.

t o prove Paley-Wiener t y p e theorems and t h e i n v e r s i o n f o r m u l a f o r t h e FourWe w i l l see l a t e r t h a t many o f t h e c o n s t r u c t i o n s a r e

ier-Jacobi transform.

s p e c i a l cases o f general t r a n s m u t a t i o n formulas and thus t h e t e c h n i q u e has c a n o n i c a l g e n e r a l i z a t i o n s as w e l l as i n t r i n s i c t r a n s m u t a t i o n a l meaning. Some f e a t u r e s a l s o have a group t h e o r e t i c s i g n i f i c a n c e which we w i l l d i s c u s s briefly later. [Mkl;

The i d e a s here have m o t i v a t e d some o f t h e development i n

T j l ] as w e l l as some o f o u r p r e s e n t a t i o n i n [C64,65].

We w i l l see

a l s o t h a t t h e r e a r e connections w i t h t h e more general t e c h n i q u e based on [Mc4] ( c f . o u r S e c t i o n 5 ) which we develop below f o r s i n g u l a r o p e r a t o r s (as i n [C39,4D]).

Thus l e t H denote even e n t i r e r a p i d l y decreasing f u n c t i o n s g

o f e x p o n e n t i a l t y p e so t h a t f o r any n t h e r e e x i s t s kn w i t h I g ( A ) l 5 kn D e f i n e t h e Fourier-Jacobi t r a n s -

( l + \ A l ) - " e x p ( A I I m x l ) f o r some A = A(g). form f o r f E C i by

where 9 f = [O,m)

(

f ( t ) , cQA ( t ) )as u s u a l .

Note t h a t when d e a l i n g w i t h C:

h e r e on

we mean even Cm f u n c t i o n s on R i f an e x t e n s i o n i s e v e r used. 0

One can

use formulas l i k e (9.43) and i n t e g r a t e by p a r t s t o determine t h e a n a l y t i c n

continuation o f f For

cl

=

A

a,B

i n (a,@)so t h a t f

8 = -4 one has by (9.15)

(A)

.a,B p?'-'(t)

=

i s i n f a c t e n t i r e i n (A,~,B). Cosxt w i t h

= 1 so t h a t

(10.1 ) becomes t h e Cosine t r a n s f o r m which by s t a n d a r d Paley-Wiener arguments i s a b i j e c t i o n o f C i o n t o H.

DEFINZCI0N 10.1

For f

E

C:

and Rea > -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 - ' <

Ky(l+lhl)

Rea+%

.

72

ROBERT CARROLL

P u t t i n g t o g e t h e r t h i s i n f o r m a t i o n we o b t a i n f o r t L c, ImA 2 0, Imh L -Re

ImA > -Re(a-B+l) +

(a+B+l) + y and

1 g (A

(10.12 )

y

( - 1)1 5

B(t)/zaJ

<

(1+ I A I p a + 5 (for s u i t a b l e choice o f n).

"K:, (1+ 1 A 1 ) - n e A l I m ~1 e-t(ImA+Rep

Z y ( l + l N- Z e

( A - t ) ImAe- tRep

Hence t h e i n t e g r a l i n (10.11) converges abso-

l u t e l y and i t s v a l u e does n o t depend on T- f o r T- as i n d i c a t e d . I n p a r t i c u l a r t a k e Rea > -S and IReBl < Re(a+l); t h e n Re(a-B+l) > 0 and Re(a+B+l) > 0 so one can t a k e II = 0 i n (10.11).

(recall

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

= [ r ( a + i ) / z ~ ~ ~ [ c(A ~)@:.'

p y y B

a,B

LEl!UnA 10.6-

F o r Rea >

-4and

+

5,B(

Hence

-A)@"']).

IReBl < Re(a+l),

i f g E H then

E

C i and

(ia,B);,B = g. Phaod:

From (10.11) and (10.12),

i f one l e t s

T- +

my

it follows that t > A

5

implies ( t ) = 0. A l s o Ca,,(t) i s even i n t f r o m (10.13). Now from a,B Theorem 9.12 and Lemna 10.5 one has f o r A , t 0 (note Imh = 0 here) (10.14)

iy(l+lAl)

Ig(h)D!

2Rea+l cn(l+lhl)n+k(l+t)et/Imxl-tRep

Consequently one can d i f f e r e n t i a t e

(10.13) under t h e i n t e g r a l s i g n and we have

(10.15)

< f i 1 + t ) ( l + l A l ) - 2 e -tRep -

by t a k i n g s u i t a b l y l a r g e rn.

(gY0,B)L

5 I(m(l+lx[ )-meAllrnAt

~:yB(t)/c"a,B(-A)~a,B(A)I

observe t h a t f o r F

T- >

0 and s > 0

9'a,B ( t ) E

A

in-m

?'

a,B

dxdt =

(-A)

The i n t e r c h a n g e o f i n t e g r a l s i s permissable s i n c e ( f r o m (9.40)) ( t z s > 0) w h i l e (10.12)

t h e form l g (A)@;''(t )/c" (-A) a,B Hence

in

To show t h a t g =

Ci.

@a' ( t )

~Ci,,,l(S) , ~ = (2d-4

< cexp( Rept ) ( t - s )

5a , B ( t )

I A a y B ( s , t )I

i n d i c a t e s an e s t i m a t e o f

[ 5 Kexp[-t ( I m ~ + R e p ) l (+Il A I )-'exp ( A I m A )

.

N

= c

(-A)). v'( 1 / 2 ~ )li:Tmg( a,B

Now we use Lemma 9.11 in (10.15) t o obtain A )exp( i A s ) d h . Consider now ( 10.9)- (1 0.10)

n t h e form ( c f . a l s o ( 1 0 . 3 ) )

to-invert this

1" F a , a [ i a , C l ( ~ ) C ~ ~ X ~[ di a~, C f ( h ) m

(10.17)

g(A

Phovd:

(2/7)'

=

Then f

Let a,, E C .

CHEOREN 10.7.

and f

=

E

Ciwith g =

a36

i f and only i f g

E

H

=

We know by Theorem 10.4 t h a t f

A

+

f a , B : C:

+

H i s b i j e c t i v e onto a n d

Ci,

= f if f E t > 0 , and t h u s i t i s s u f f i c i e n t t o prove (f* )" a,B a,@ ( r e c a l l C: r e f e r s t o even C: f u n c t i o n s ) . By Lemma 10.6 t h i s holds A > -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-

<

Oyly...y A = A ). Also (10.11) allows one t o determine (t) a+n ,B+n a,B f o r t > 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,B

has n o poles f o r Imh 5 0 (e.g. Rea >

(-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

<

;Q(~)F[-A) f o r A real e t c .

-f

74

ROBERT CARROLL

We r e c o r d n e x t some hypotheses and c o n c l u s i o n s f r o m [Cg3] d e a l i n g w i t h a A 2 q u i t e general s i t u a t i o n f o r equations ( c f . ( 9 . 1 ) ) Qu = (A u')'/A + p u 4 Q Q 2 4u = - A u modeled on t h e r a d i a l Laplace-Beltrami o p e r a t o r ( w i t h s i n g u l a r p o t e n t i a l ) i n a r a n k one noncompact symmetric space.

The p r o o f s a r e n o t

g i v e n here b u t t h e p r o p e r t i e s i n d i c a t e d f o r e i g e n f u n c t i o n s and r e s u l t s f o r transformations w i l l provide useful guidelines.

G e n e r a l l y here A

4

t

as

m

.1 2p > 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)( <

m

fo r any m).

Then t h e main r e s u l t s f r o m [Cg3] a r e

0

76

ROBERT CARROLL The map f

ZHE0REM 10.12.

+

g f i s a b i j e c t i o n DT

-z

H onto.

The i n v e r s i o n 9

i s g i v e n by

= 0J-l

2 where It II i s t h e L (A dx) norm ( n o t e here t h a t 2 Q 2 + - l *

xj

= iyj

and dw ( A ) = dh/

Q

F i n a l l y i n t h e c o n t e x t o f Paley-Wiener theorems l e t us r e f e r t o t h e hypotheses o f [ T j l ] o r [ F f l ] where t h e f o l l o w i n g theorem i s proved ( c f . a l s o [C40]).

F i r s t one i d e n t i f i e s a f u n c t i o n f w i t h t h e d i s t r i b u t i o n TAf (A =

A ) d e f i n e d by ( TAf,qO) = Jm f ( x ) q ( x ) A ( x ) d x f o r

Q

Q

0

v

E

D,.

L e t Jc be t h e space o f even e n t i r e s l o w l y i n c r e a s i n g f u n c t i o n s IG(x)l 5 K ( l + l h l ) Ne x p ( A I I m h [ ) f o r some N and A ) .

CHE0RE8 10.13.

o f exponential type (i.e. Then t h e map f

-+

Q f i s a 1-1 map E ' - + K onto.

Note here t h a t i d e n t i f y i n g a f u n c t i o n f w i t h TAf g i v e s QTAf = Qf which (up t o a c o n s t a n t ) i s t h e f o f Theorems 10.4,

10.7,

and 10.12.

We go now t o a

" c a n o n i c a l " d e r i v a t i o n o f Parseval formulas f o r s i n g u l a r o p e r a t o r s f o l l o w i n g t h e technique o f [Mc4] used i n S e c t i o n 5 t o o b t a i n (5.7). terms o f o p e r a t o r s o f t h e form (9.1). 2

p

Q

A

4

4 lim

= Qou +

Thus Qou = (A u ' ) ' / A Q ; ?u

Q*

and (Qo) u = [AQ(u/A ) ' ] I . We Q Q Q 2 Q does n o t have " s t r o n g " s i n g u l a r i t i e s B / X n e a r x = 0 here so t h a t

u; Qu = P u

assume

-

This pro-

and we w i l l phrase t h e m a t t e r f o r m a l l y i n

cedure was developed i n [C39,40] G(x)u where

p

=

A'/A

s p h e r i c a l f u n c t i o n s q: can be used f o r s i m p l i c i t y i n t h e e x p o s i t i o n .

Note

t h a t f u n c t i o n s o f t h e t y p e q Q f r o m 5§4,5 w i l l n o t a r i s e i n t h e s i n g u l a r AYh problem s i n c e ~ ' ( 0 1= 0 i s i m p l i c i t l y r e q u i r e d . Via (9.26) we have associV A 2 ated the operators 2, o ; and Q % Q; t h e b a s i c o p e r a t o r s t h u s have t h e P

Go

b u i l t i n (recall

p

4

= 0 f o r Qo = Q;).

Q

We w i l l use t y p i c a l a n a l y t i c i t y pro-

p e r t i e s and Paley-Wiener t y p e i n f o r m a t i o n about q: and @A!

^4o

as developed i n [ F f l ;

10.5 and 10.6 65; T j l ] ) .

-

f o r operators

K P l ] ( t h u s see Theorems 10.4 and 10.7 w i t h Lemmas

c f . a l s o Lemma 10.11, Theorem 10.12,

and [Cgl-4;

Ge1,2;

We w i l l g i v e a s k e t c h o f t h e procedure f o l l o w i n g [C39,40].

f i r s t one wants t o determine a 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 Ro f o r with

Go

as i n (9.1)

t r a n s m u t i n g D2

-+

C64, Thus

*ao

and

t h i s w i l l be known ( i f n o t i t can be "discovered" by

-

d e t a i l s o f t h i s a r e i n d i c a t e d l a t e r i n 52.2). AO

In

p a r t i c u l a r , r e f e r r i n g t o Q as P now g e n e r i c a l l y , w i t h s p h e r i c a l f u n c t i o n s P P P P P P P solutions Ex = ApqA, and q x ( x ) = C ~ ( X ) @ ~ ( X ) + ~ ~ ( - h ) @ - ~ ( x ) ,

pox, J o s t

SPHERICAL TRANSFORMS

77

one has by r e s u l t s of [ F f l ; Cg3; Ge1,2; T j l ; Kpl] as indicated i n Theorem 10.7, Remark 10.8, a n d Theorem 10.12 f o r example (10.26)

RO(A)

=

1/2nlcp(X)IC

( c f . a l s o Theorem 10.15 - typical R

P

a r e given in (9.9) and ( 9 . 2 1 ) ) .

The

^ao

transforms P and P associated with P = a r e designated as before (Pf(A) = P P fom f(x)CLA(x)dx; Pf(X) = f m f ( x k , ( x ) d x ) a n d P = P-' has t h e f o r m P F ( x ) = 0 P F(A)pA(x)dvp(h)where d v p ( A ) = cp(h)dA where C p ( A ) = R o ( h ) . We r e f e r t o ?) g e n e r i c a l l y as (with s u i t a b l e p o t e n t i a l :) and w i l l use a transmutation n A P = p Qh ) which we assume t o have been "created" B: P Q (characterized by &PA e.g. via a p a r t i a l d i f f e r e n t i a l equation technique using Riemann functions e t c . as in Sections 4-5 (such constructions a r e sketched in 511 f o r the s i n g u l a r case - one can a l s o use other techniques a s in Chapter 2 ) . Let us

I,"

-f

f i r s t give a formal g e n e r a l i z a t i o n of Lemma 5.4 as follows.

LEmmA 10-14- Let Bf(y) = ( B ( y , x ) , f ( x ) ) and Bg(x) = ( y ( x , y ) , g ( y ) ) ( 8 = B - ' ) . A n P = pQ Assume B: P + Q i s characterized by &PA i. Then (10.27)

Pmud:

*

*

PB f = q f ; @ g

=

Pg

P Formally PB*f = ( p p x ( x ) , (B ( y , x ) , f ( y ) ) )

=

P CCp,(x),B(y,x)),f(y)) =

.

( v : ( y ) , f ( y ) ) = qf. Similarly qg*g = ( 9 QA ~ y ) I ( Y ( x , y ) , g ( x )=) )( ( ~ , (QY ) , Y ( X , Y ) ) , P g ( x ) ) = h J , ( x ) , g ( x ) ) = Pg. We want t o a r r i v e a t a Parseval formula of the form ( c f . ( 5 . 7 ) ) (10.28)

(

R,Qfqg!g)A= (Ai5f,At4g); (R,Qfqg) = (L5f,L5g)

Q

Q

f o r s u i t a b l e f , g (generally of compact support - f = A f , e t c . ) . One exQ pects R = R Q t o be a d i s t r i b u t i o n i n t h e dual of some space of e n t i r e funct i o n s (e.g. R E Z ' i n Theorem 5.5) and t h e a n a l y t i c i t y p r o p e r t i e s of QfQg * * can be determined via Lemna 10.14 in terms of PB f and PB g. One will USu a l l y have some t r i a n g u l a r i t y of t h e form ~ ( y , x )= 0 f o r x > y in t h e d i s * t r i b u t i o n kernel ~ ( y , x )so t h a t B f ( x ) = ( B ( y , x ) , f ( y ) ) = 0 f o r x > u i f

/x"

f ( y ) = 0 f o r y > Q (e.g. ( B ( y , x ) f ( y ) ) = B(y,x)f(y)dy = B(y,x)f(y)dy). Similarly y(x,y) = 0 f o r y > x ( e . q . a s a Volterra i n v e r s e ) and Paley-Wie-

ner type information can be passed back and f o r t h between P and Q transforms. Since we know Ro = v P here by assumption such information i s a v a i l a b l e f o r PB*f and this i s passed t o Qf f o r use in a formula (10.28). Next, following 55 ( c f . here (5.13)) we t r y t o f i n d a generalized t r a n s l a t i o n ScSo(x) A

78

ROBERT CARROLL

(SQ(x) = 6 ( x ) / A ( x ) ) .

We o m i t i n t h i s s k e t c h any approximations t o 6

Q

9

of

t h e f o r m 6n which may be needed t o j u s t i f y t h e procedure as i n 55 ( t h i s w i l l

4

be done l a t e r ) .

Given (10.29) one m u l t i p l i e s by s u i t a b l e f,g and i n t e g r a t e s

t o o b t a i n t h e Parseval formula ( + ) (A?f,A"g) R = Sp(X)RV i n (10.28).

= (

Q

The f o r m u l a (10.29)

$p(h)R",Qf2q)x

so t h a t

serves t o determine Rw as f o l -

Set y = 0 i n (10.29) t o o b t a i n f o r m a l l y

lows. (10.30)

GQ(x) =

(

R",lpi(x) 0 )v

Operate on t h i s f o r m a l l y w i t h B t o a r r i v e a t (10.31)

BGQ(y) =

(

Rv,BPx(y)), Q

Consequently R = CpRV w i t h (.)

= ( R",P~(Y)), P

=

I n o r d e r t o make such a formula

R" = PEGQ.

more useable one i n t r o d u c e s a q a i n t h e o p e r a t o r s v

'i =

(10.32)

v

v

P

One has a t r a n s m u t a t i o n B:

+

( t h i s i s checked l a t e r ) . ~1

Q

/ x and u p =

(K(y,x),f(x) A+;,

= G[A:(y)Bf(y)]

= i[#(x)bf(xj]

I n o u r p a r t i c u l a r s i t u a t i o n w i t h A. v

t o have k e r n e l forms g(y,x) form

)

T h i s term ;(x)

aQ

Note t h a t

%//$

=

as de-

= Ap one expects B and

5 = k-'

.

= ~ ( x - Y )+ K(y,x)

here.

= A$(y);Bf

= b F [ A F ( xV) f ( x ) ]

and ;(x,y)

= 6(x-y) + i ( x , y )

has a g e n e r i c s i n g u l a r i t y o f t h e

I n t h i s e v e n t one can w r i t e Bf(y) = f ( y ) +

and &g(x) = g ( x ) + ( L ( x , y ) , g ( y )

)(I x ,y)aQ( y 1.

(10.33)

i n (9.26).

a'(y)Bap'(x) 9

= A$(y)BAi'(x)[A$x)^Pf(x)]

sired.

6 = 6" and 6 as

Q o f t h e form ( n o t e AQ = Ap)

Observe t h a t ( f o r s u i t a b l e f ) G$[L;(x)f(x)I A;(y)B;f

BR"

)

where i n p a r t i c u l a r L(x,y) =

Consequent] y

6q(x) = 6Q(x)

+

( L ( X , Y ) , ~ Q ( Y ) )= ~ Q ( x + )

;(XI;

i s w e l l d e f i n e d i n t h e case A ( y ) = y2m'1 f o r example f o r

suitable potentials

4 (cf.

i n g t o (10.33) and (.)

[C39,40;

Ge1,2]

we have f o r m a l l y

Q

and remarks l a t e r ) .

Now r e t u r n -

SPHERICAL TRANSFORMS

(10.34)

R" = PB6

Q

= P6

79

+

+

Q

( s i n c e Ap = AQ). Consequently R = :pRw = RoR" = Ro + R R" = R + R where o q 0 9 (see Remark 10.15 beR r e p r e s e n t s t h e c o n t r i b u t i o n due t o t h e p o t e n t i a l q e observe a l s o t h a t i f we t a k e t h e f o r m u l a B6 =PR" ( i n (10.31)), low). W

Q

m u l t i p l y by s u i t a b l e f ( y ) , (10.35)

and use (10.33) t h e r e r e s u l t s

(PR",f) = ( ( R " , p XP( y ) ) U , f ( y ) ) =

(R,Pf)A =

4

(

~ Q ( Y +) L ( Y ) , f ( Y ) )

=

1i m

yjo

( R V , ( f ( y ) , pPx ( y ) ) ) " = (R",Pf)"

f(y)/AQ(Y) +

=

;(Y)f(Y)dY 0

T h i s e x h i b i t s Ro and R (10.36)

(

i n t h e form ( f o r s u i t a b l e f , g o f compact s u p p o r t )

9

1i m Ro,Pf)A = Y+O f ( y ) / A Q ( y ) ;

(

Rq,Pf)x

=

jOm

We n o t e a l s o t h a t i f P f = F(X) t h e n f ( y ) = BF(y) =

( s i n c e Ap = A

Gp

again.

Q

-

some d e t a i l s a r e i n 511-12);

i(y)f(y)dy

(

P F(x),PA(y))"

and

t h i s a l s o i d e n t i f i e s Ro and

Thus i n sumnary we have f o r m a l l y e s t a b l i s h e d t h e f o l l o w i n g theo-

rem (see 5511-12 f o r f u r t h e r d e t a i l s ) CHE0REPl 10.15,

The Parseval f o r m u l a (10.28)

( f o r s u i t a b l e f,g o f compact

w i l l be a consequence o f (10.29), which i n t u r n de-

support) w i t h R =

A

where B i s a t r a n s m u t a t i o n B = B-':

t e r m i n e s R V = PB6

Q: by Eip; = p p ( h e r e P =

that

= 4(y)BA;+(x):

t h e form ;(x,y)

Q

A

+

P characterized

^oo and ^o a r e g i v e n by ( 9 . 1 ) so t h a t Ap = AQ). Given P' 6 (Pas*,i i n (9.26)) and 8 = is-' have k e r n e l s o f -f

= 6(x-y)

A

+ i ( x , y ) f o r example w i t h say L ( x )

E

l Lloc

A

(1 de-

as y + 0) i t f o l l o w s t h a t R = Ro + R q = ? J ~ + R w i t h f o r m a l l y R = $ p ( X ) I m L4( x ) QPA ( x ) d x ( g e n e r a l l y R i s a d i s t r i b u q 9 0 q t i o n ) . More p r e c i s e l y t h e a c t i o n o f R i s determined by(R ,Pf), = Im pix) 9 q 0 f ( x ) d x f o r s u i t a b l e f o f compact s u p p o r t and Ro and GP a r e a l s o i d e n t i f i e d

f i n e d as l i m c(x,y)/L;(x)&(y)

Q

through

(

Ro,Pf

= l i m f ( y ) / A Q ( y ) as y

-f

0.

P REEU?K 10-16- We n o t e e s p e c i a l l y t h a t t h e e x p r e s s i o n Im2 ( x ) P X ( x ) d x i n R

q 0 i s formal i n t h a t i t r e p r e s e n t s a d i s t r i b u t i o n ; one s h o u l d n o t t h i n k o f i m A

p o s i n g c o n d i t i o n s on L ( x ) as x "rigorously".

-f

m

i n o r d e r t o d e f i n e such an expression

We have a l r e a d y determined R,,

r i g o r o u s l y , as a d i s t r i b u t i o n ,

80

ROBERT CARROLL

v i a t h e f o r m u l a ( R , P f ) A = Im 2 ( x ) f ( x ) d x f o r s u i t a b l e f o f compact support. 9 0 We n o t e a l s o (as i n Theorem 5.6) t h a t f r o m a Parseval r e l a t i o n (10.28) one can t a k e f o r m a l l y e.g.

G(x) = 6 ( x - y ) w i t h Q i ( A ) = n QA ( y ) and produce an i n -

v e r s i o n formula (10.38)

i(y) =

(

R',Qt(x)q!(y)

)A

Theorem 10.15 shows t h e i n g r e d i e n t s which go i n t o a Parseval f o r m u l a (10.28) when i t i s d e r i v e d by what we s h a l l c a l l t h e t r a n s m u t a t i o n method.

The "ca-

n o n i c a l " f e a t u r e s a r e obvious and we see t h a t a s i d e from e s s e n t i a l l y formal c a l c u l a t i o n s t h e method hinges upon (10.29) and t h e e x i s t e n c e o f a s u i t a b l e A

L ( x ) i n (10.33)

(the condition A

Q

= Ap has a l s o been used a t s e v e r a l p l a c e s ) .

A l s o Lemma 10.14 i s e s s e n t i a l i n p l a c i n g QQ fg R d e f i n e d o v e r Pf as i n (10.35)-(10.36)

i n t h e c o r r e c t space on which

can a c t .

He w i l l proceed i n 511 t o

p u t these i n g r e d i e n t s t o g e t h e r i n a more r i g o r o u s f a s h i o n .

11- EXPLZCIC C0WERUCCIOW OF GENERALIZED CRAWLACI0bL5 AND ERAWMllEAEZObL5 f O R SINGULAR 0PERAE0W. L e t us make some comments f i r s t about g e n e r a l i z e d translation.

T h i s came up i n 55 ( c f . (5.8)-(5.13))

t h e Cauchy problem (4.4) f o r Q o r

o*

where we worked f r o m

t o produce ~ ( x , y ) = S:f(x)

satisfying

Q(Dx)q = Q(Dy)q, p(x,O) = f ( x ) , and D y ( x , O ) = 0. An e x p l i c i t c o n s t r u c t i o n was i n d i c a t e d v i a Riemann f u n c t i o n s e t c . as i n §4. I n terms o f s p e c t r a l p a i r i n g s t h e formulas (5.10)-(5.11)

are relevant.

For s i n g u l a r o p e r a t o r s

t h e m a t t e r i s s i m i l a r b u t t h e a n a l y s i s and e s t i m a t e s become somewhat more d e l i c a t e ( c f . [Bx1,2; Cpl-3;

Fi1,2;

Lg1,2;

L12,3; Cgl-4;

Mml; Ge1,2; Sol; Dgl; C27,29,30,37,40; Ho2-4; Del-4;

Lpl-31).

Pcl;

F i r s t l e t us remark t h a t

f o r s i n g u l a r o p e r a t o r s we r e s t r i c t o u r s e l v e s t o C = 0 i n (4.4) b u t f o r any A w i t h say f such t h a t ( A f ) ' ( O ) l a t i o n S:f(x)

(S

%

S(A)).

= C f ( 0 ) = 0 one o b t a i n s a g e n e r a l i z e d t r a n s -

F o r t h e moment however we w i l l o n l y be concerned

w i t h A = 1 and w i l l w r i t e q ( x , y ) = S:f(x)

P or

? we w i l l

f o r t h i s case (when d e a l i n g w i t h

w r i t e T:f(x)).

RZmARK 11.1. I n o r d e r t o produce a g e n e r a l i z e d t r a n s l a t i o n S: o f t h i s t y p e for Q =

Qi o f Example 9.2

f o r example l e t us r e c a l l f i r s t some b a s i c f a c t s

t o a d j o i n t o t h e l i s t i n Example 9.2. (9.4) w i t h C!!

= AQqA. Q

Thus we have s p h e r i c a l f u n c t i o n s

Then w i t h Ro = 4wo = 1 / 2 n / c Q I 2 =

CmA 2 2m+l

EXPLICIT CONSTRUCTIONS

where dw ( A ) =

Q

Q (X)dh

81

( w i t h Q = Q-').

( = Ro(A)dX) a n d m = Q-'

Evidently

t h e i n v e r s i o n formulas a r e b a s i c a l l y a v e r s i o n o f t h e Hankel t r a n s f o r m ( c f . [C29,30;

L191).

(11.2)

V(x,y)

0: c o n s i d e r f o r m a l l y

I n o r d e r t o produce S z f ( x ) f o r Q = = S;f(x)

( ~ X0( x ) F ( X ) , PQX ( y ) ) w

=

where f ( X ) = Q f ( X ) and I QP ~i s g i v e n by (9.4) when Q = N

Qi. C l e a r l y t h i s con-

s t r u c t i o n i s p e r f e c t l y general ( i . e . Q can be any o p e r a t o r o f t h e t y p e we have been c o n s i d e r i n g w i t h i n v e r s i o n s as i n (11.1)) and f o r m a l l y QxV = Q V, Y V(x,O) = f ( x ) , and V (x,O) = 0. V(x,y) can a l s o be w r i t t e n as Y (11.3)

S;f(x)

Y ( x , Y , ~ ) = ( PO~ ( XQ ) P ~ (Q Y ) , ~ ~ ~ ( ~ ) ) ~

= ( y(x,y,n),f(n));

We c o n t i n u e t o t a k e m > - % 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

EXPLICIT CONSTRUCTIONS

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)

-

^Oo(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

EXPLICIT CONSTRUCTIONS

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

^o

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 :$f = Q[A'f] and (?,<) = Im GQ?$dx = A A

Q

Im

A fgdx which

is t h e Parseval formula (10.28).

o Q (10.28) can be w r i t t e n when RQ (12.6)

1;

$:(A)Q%$X)gg"dA

n,

=

6QdA

4

0

Indeed Qf = Q[AQf] and

as

AQ(x)i(x)i(x)dx

tC

and i n f a c t (10.28) will determine a t r a n s p o r t of Hilbert s t r u c t u r e t o Q= QEc in general. Note here t h a t f o r iE EC ( i . e . A';€ L2 and s u p p f' i s comQ-b Q p a c t ) we have f = A,? E lE; C ( E c ) ' where = { f ; AO2f E L2; supp f compact] Q v Q, " m "v (note f o r F E EC g EE;, one has ( f , g ) = I f ( x ) g ( x ) d x = rU fgAQdx). Thus Q' 0 i n (10.28) Q a c t s on f , g E EC while Q a c t s on f",< E EG. Similarly when RQ Q 4c A A-+A 2 w" dA one has a dual notion ( E Q ) ' { f ; w Q f E L I which was useful when #(XI= nQ(X,x) was used i n t h e transform theory ( c f . [C30]). However we wQ 1 will f i n d i t more natural here t o use the Hilbert s t r u c t u r e on b A A X Q defined by "c Ac "c "c (12.6) and s e t lEQ = E (E C ( E Q ) ' ) w i t h ( f , g ) w = ( f , g ) = ( R " , ? G ) A f o r

ic

Q ,

fc

Q

Q

Q

93

CANONICAL FORMULATION

?,:

?,G)w

Im GQ ?{dX when RQ

Note t h a t we will eventually want t o c h a r a c t e r i z 2 Z C = QEc using general Paley-Wiener type theorems so Q Q that will be a space of even e n t i r e functions having c e r t a i n growth proQ p e r t i e s f o r r e a l X ( s e e [Ge1,2; C401 for t h e Bessel transform p l u s ' [ T j l ] , ((

E

=

'L

GQdX).

ic

Theorems 10.4, 10.7, 10.12, and 10.13).

In p a r t i c u l a r i f one r e s t r i c t s t h e

support of f by s u p p f C [0,13]and denotes t h e corresponding E space by E:(u) w i t h f;(o) = (1E:(u) then one has a general Hilbert s t r u c t u r e on Ec(o) "C Q and on E ( u ) . One t r e a t s EC = U E C ( a ) as a countable union of Hilbert spaces Q Q A Q a s in [Gfl] and s i m i l a r l y f o r EC = $:(u).

Q

CH€0RETil 12-3- One can c o n s t r u c t a p a t t e r n of spaces and maps r e l a t i v e t o opA

A

e r a t o r s P a c t i n g in E = Ep a n d ! Ia c t i n g i n F = F connected by t h e transmuA A @P 2 Q, B: E p F9, characterized by B p X = p: ( i n C s a y ) . Here t a t i o n B: P Ep and R a r e a s in Definition 1 2 . 2 and IE = E ' = E; = Cf; A$f E L2 1 w i t h B = F ' = c) F' = { f ; E L'}. We need consider only elements i n *C Ep = P E ~ -f

/I

-f

Q

Q

A

A

E p f o r example where E = E i s a generic notation f o r PEP as a Hi1 b e r t EC * M A A M space (e.g. take PEP = UEp as a Hilbert space) w i t h E ' = IE = E and ( f , g ) =

C

(?,j)p = (f,$)p

A

=:I A p ( x ) f ( x ) g ( x ) d x ( ? = Pf,

h

Similarly F = F = 01 QF i n generic notation. The r e l a t i o n s P = Prl q = Q-', P = P-', Q = Q- , Q B = 8, B = B-l = BQ, P* = P , Q* = m, B* = P, p* = Q, B* = PQ, B* = QP, Q- 1 = WPQ, and7p-l =

=

Pq).

QpP hold on s u i t a b l e domains.

This p a t t e r n can be arranged in diagramatic form a s i n [C40] ( c f . a l s o [C30]) b u t we simply make a few remarks here. One begins on domains EF = 2 EC, F i = Fc, EC = I f ; s u p p f i s compact and Ap'f E L } o r F C . Then we w i l l be in the context of general Paley-Wiener type theorems and PEC = for

^Ec

example will c o n s i s t of e n t i r e functions of exponential type having c e r t a i n growth p r o p e r t i e s f o r X real ( s e e below f o r t h e Bessel c a s e ) . The functions pA P ( x ) and p X Q ( x ) , being themselves e n t i r e functions of exponential type

2"

x,

FC

and normally bounded f o r A r e a l , will then a c t as m u l t i p l i e r s in or ( f o r f i x e d x ) so t h a t ! I ! and B (based on R P and R Q r e s p e c t i v e l y ) should be defined on AE c n A F c a s a natural domain. Note when R P and R Q a r e measures 4

A

A

i t obviously makes sense t o look a t p a n d P on E n F = { f ; w f E L2; G4+ E 2 Q PA L }. In any event one does not e x p e c t P and J.! t o be defined on a l l of F o r A E respectively and this i s i n accord w i t h B = P and B = Pq not being defined on a l l E o r F r e s p e c t i v e l y . Another type of domain problem involving PQ defined via R Q f o r example i s a l s o avoided by working on Fc so t h a t 3 Q f = f f o r f E Fc. As an example ( c f . a l s o Theorems 10.4, 10.7, and 10.12) we i n d i c a t e t h e framework from [Gel ,2] which was extensively t r e a t e d i n [C40].

94

ROBERT CARROLL

Then f o r Qo w i t h q $ x ) = c- 1 ( A X ) - ~ J(Ax) one d e f i n e s i n [Ge1,2] a F o u r i e r m m m Bessel t r a n s f o r m F B f ( A ) = c Q [ X ~ + ~ ~ One ] . t h i n k s o f Q d e f i n e d say on FC = 2 m+G 2 I g ; supp g compact and x-m-qg E L 1 so g = x f means f E L w i t h compact 2 2 s u p p o r t (one says f E K as i n §5). I n f a c t f o r f E K ( 5 ) , FBf(A) i s an even e n t i r e f u n c t i o n s a t i s f y i n g (12.7)

IF(A)I 5 C I A \

-m-%

exp(ol1mAl)

(lAl

large);

jOw

IF(A)12A2m+1dA <

m

2 Denote by Wm t h e s e t o f even e n t i r e f u n c t i o n s F(A) s a t i s 2 2 A sequence o f f u n c t i o n s Fn E Wm converges t o F i n W p r o v i d e d f y i n g (12.7). I F n ( A ) l 5 cnexp(o(ImA/) ( f i x e d 5 ) f o r a l l n and :1 IFn(A) - F ( A ) I 2 m2m+ldA A

DEFZNZCZBN 12.4,

2 . 2 2 i n (12.7) can v a r y f o r f E Wm ( i . e s Wm = uWm(u) i n an ob2 vious n o t a t i o n ) . The space W, can be c h a r a c t e r i z e d as t h e space o f Four2 1 ier-Bessel transforms FBf o f K The space Wm i s d e f i n e d as t h e space of

-f

0.

Note t h a t

0

.

even e n t i r e f u n c t i o n s s a t i s f y i n g ( f o r some (12.8)

5 IA1-2mexp(o(ImAl) ( 1 x 1 l a r g e ) ;

IF(A)I

1 A sequence Fn 5)

E

5)

W

f o r a ' l n and

1

Yw

converges t o F i n

0

IFn(X)

-

W

i f IFn(A)l

F(A)lA2m+'dA

-f

C

(F(A)IA2m+1dA

5 cnexp(olImAl) ( f i x e d

0.

-m-+

I n terms o f t h e Parseval formula (10.28) we have f o r f = x f, e t c . 2 Thus t h e general pro( x-m-4f,x-m-4g) = = ( R,QfQy)i = c- ( R,FBf,FBg). m2 2 2 cedure based on 55 i s t o observe t h a t FBfEWm and F B F E Wm f o r K and 1 hence (FB?)(F$) E tlm = W (W % Z i n 5 5 ) . The 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

(7,;)

F,?.

R w i l l then be l o c a t e d i n b l ' .

F i n a l l y l e t us i n d i c a t e some o f t h e a d j o i n t -

ness r e l a t i o n s s t a t e d i n Theorem 12.3 f o r completeness. DC and T E E C = C

Thus e.g.

for f E

Ec 2'

(12.9)

(Pf,$)v

P

= ( R ,Pf(A)r(A))A = ( R

Similarly f o r suitable $ (12.10)

($!$,f' ) = =

((

E

^Ec

P

, $ (A)l

w

P PA(x)f(x)dx)A =

and f ' E HC c F '

R P , eA( A ) q QA ( x ) ) , , f l ( x ) )

(RP,:(A)Qf'(A))A

= ( R p , $ ( A ) ( qQA ( x ) , f ' ( x ) ) ) A

= ($,Qf')v

The o t h e r a d j o i n t r e l a t i o n s a r e proved i n a s i m i l a r manner w i t h a c t i o n on suitable objects.

CANONICAL FORMULATION

95

The f o l l o w i n g theorem on g e n e r a l i z e d t r a n s l a t i o n w i l l be u s e f u l i n t h e t h e The p r o o f i s a o e n e r a l i z a t i o n o f an argument i n

theory a t various places.

[L12] and was developed i n [C40]. version for

Qi b u t we f e e l

t e c h n i q u e o f Theorem 12.5.

The t e c h n i q u e o f [L12] l e d t o a s p e c i a l

t h e n a t u r a l h a b i t a t i s b e t t e r i l l u s t r a t e d by t h e We r e c a l l f i r s t t h a t a g e n e r a l i z e d c o n v o l u t i o n

was d e f i n e d i n Remark 11.3 as ( f

*

g ) ( x ) = fom a(y)S:f(x)AQ(y)dy

t h e g e n e r a l i z e d t r a n s l a t i o n a s s o c i a t e d w i t h Q.

where S:

is

One o f o u r requirements i n

p u r s u i n g t h e l i n e o f p r o o f o f general Parseval formulas i n d i c a t e d i n 510 i s t o use S;

i n v a r i o u s ways as i n (10.29) f o r example.

iE I€;. Q

Then

CHEOREm 12.5.

For

= A

f",;

EC and f = A

E

lo

4

Q ?,

g = A,$

E

P i c k now fvc EC so f

Q

IEc one has

9

m

(12.11)

Pkao6:

(

For

Syf",g) = X

?€

c

Si;(x)9Y(X)AQ(x)dx

f(x)S:i(~)4~(x)dx =

= (

(Sxf,g)Q y" 7 = (r,S:$),

=

f,S:$)

EC ( c f . here a l s o remarks b e f o r e Theorem 2.2.2)

Q

let

F

= -A2; f o r m a l l y w i t h ? ( O , h ) = Q ? ( X ) and Yy(O,A) = 0 ( s i n c e Dy Y S i f ( x ) = 0 a t y = 0 ) Indeed we r e c a l l here t h a t ( c f . ( 1 1 . 3 ) ) S:i(x) =

Then G(D

(r(x,y,S),fv(S))

EC ( a s

= ( R Q , Q i ( h ) q ~ ( x ) q ~ ( y )which ) i s well defined f o r

i n Theorem 12.3 t h e q 4 A ( x ) and q h4( y ) a c t as m u l t i p l i e r s i n Note a l s o t h a t f o r y f i x e d S:f'(x)

iC f o r x,y

4

fixed).

has compact s u p p o r t i n x s i n c e U(x,y) =

S z i ( x ) a r i s e s as t h e s o l u t i o n o f t h e h y p e r b o l i c Cauchy problem GxU = CyU, U(x,O) = f ( x ) (extended t o be even), and U (x,O) = 0. Hence QSzfv(x) = Y F u r t h e r we n o t e t h a t f o r m a l l y (S;f(x),%(x))Q q(y,A) i n (12.12) makes sense.

N

and f o l l o w i n p Remark 11.3 we have RFq;(x), so t h a t i n f a c t ( 4 f y ( c ) pQe ( y ) , 6 ( X - c ) ) = Q f v ( h bQA ( y ) = F(y,A).

= ( R ~ , Q i ( h ) p4, ( y ) ( q05 ( x ) , " 0 h(X)))

Sq(y)) 4 = A(X-e)

We a l s o n o t e t h a t t h e f o r m u l a (11.9) can be proved d i r e c t l y f o r a general A

A

$(x,O,X) SYq:(x),

= vA Q ( x ) and D $(x,O,X)

Y which i s (11.9).

6

Q

(12.13)

Q

Now w ( =

?)

can be w r i t t e n w(y,A)

F ( X ) , ~ ~ ( X ) S ~ ~( uYs(i nXg ) (11.9) )

Consequently, w r i t i n g w = (

S:fv(x),AQ(x)pA(x)) Q

Y

= 0; t h i s i m p l i e s q A ( x ) p X ( y ) = IL(x,y,A)

q A ( y ) f y ( ~ ) ~ ~ ~4( x ) p =~ (( x ) )

above).

4

= qQ A ( x ~ ;o( y ) s a t i s f i e s Qx$ = Q J, w i t h

Q p a i r i n g g i v e n by R Q s i n c e J,(x,y,A)

=

(

7

we have

?(X),~~(X)S~P~(X))

=

Q ( y ) Q"f ( A ) = = pA as g e n e r a l i z e d

96

ROBERT CARROLL

Now t a k e

)u = ( R Q ,G(h)vh(x)) Q

G(A) = BG, so G(x) = N G ( x ) = (G(x),v;(x)

,E;

E

Then m u l t i p l y (12.13) by G(h) and f o r m t h e w b r a c k e t t o o b t a i n (S:?(x), To see t h a t t h i s i s (x)S:g"(x)) (which i s (12.11)). Y Q v a l i d n o t e t h a t ( G ( A ) , S X v X ( x ) ) = S:(G(~),P:(X))~ = S:t(x) s i n c e i f we s e t Y Q wn 0 v ( x , y ) = ( G(A),SXvX(x) ) u t h e n Q(DX)v = 8(Dy)v w i t h v(x,O) = ( G(x),vh(x) )u = A

Q

(x)i(x)) = (i(x),A

G(x) and D v(x,O) = 0. Hence v ( x , y ) = S:G(x). Y (12.11) a r e b a s i c a l l y a m a t t e r o f d e f i n i t i o n .

The formula (12.11) e s s e n t i a l l y d i s p l a y s S: as a s e l f a d j o i n t

REflARK 12.6-

operator i n E AQ(X

- I duality with (2 Q *o t h e n i n p a r t i c u l a r (12.13) says t h a t ( S : ) !?i(x) =

( d e f i n e d on Ec).

5) = (A

1Sxv 8 Q(X 1.

'

f,S:{)

"0

Iz

We c o n s i d e r now again P = Q A

B: P

as i n (9.1)).

I f one t h i n k s o f E

Q

Q "

(SYf",A

The r e m a i n i n g e q u a t i o n s i n

w i t h Ro =

A

3,

and general 0

known ( c f . (10.26)

A -j

Q i s t o be a 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 Bp!

o b t a i n e d v i a a PDE t e c h n i q u e f o r example b u t i n any case known, and G(y)BA?(x)

as i n (10.32)

we needed Sy6 ( x ) = 6(x-y)/AQ(x)

=

F i r s t one w r o t e down (10.29) and While t h i s i s f o r m a l l y ob-

Note t h a t f o r m a l l y 9

vious we n e e d ' a p r e c i s e v e r s i o n . v hQ( x ) s o t h a t S:SC)(x)

RQ ).

( d o = 6/AQ).

X Q

(12.14)

Q

B" =

L e t us r e c a l l t h e i n g r e d i e n t s

( w i t h A. = Ap).

going i n t o (10.28) f o r example ( R

Q = (px

( v hQ( x ) v hQ( y ) , R Q ) = P9Q16(x-C)/AQ(S

Now t o make t h i s f i r s t q u e s t i o n more p r e c i s e l e t Sn be f u n c t i o n s 6n

;1

-t

En(x)dx = 1.

6 i n say

E' ,

An

2 0,

An = 0 n e a r 0 and f o r x > 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:

*

<)(y) =

(

6:(x),A4S:i(x))

=

Consequently t o approach t h e l e f t s i d e o f (+) a f t e r (10.29 t i o n (10.28)) we can w r i t e (12.16)

and

En(x)/AQ(x) i s

=

Z

n

= ( ( SySn(x),g(x)),f(y))

x Q

( f ( y ) , ( 6~(x),AQ(x)S:f(x)))

=

(

= ( f(y),( S:6"QAQc))

f(y)(6n(x),Sz<(x)))

=

(s'*

6 p Y )

( c f . a l s o equa-

CANONICAL FORMULATION

5E

Now l e t us approximate

EC by continuous

t i n u o u s and lmdn((x)S{Gk(x)dx

Q

= (

0

((

denoting

)

El-E

duality).

ck

97

E;.

E

6n(x),~:ijk(x))

+

Then S;ik(x)

(6(X),s:ik(x))

Hence we have i n (12.16)

1.

i s con= Gk(y)

(gk = A

9" ) Q k

m

(12.17)

z kn

and f o r gk

IE (Gk Q

g in

-f

S:6:(X),qk(X)))

= ( f(y),(

5 in

+

-t

f(y)ik(y)dy =

E ) we o b t a i n t h e r i g h t s i d e o f (10.28).

Q

E m m A 12-7, The formal r e l a t i o n (12.14) makes good sense and can be estabI n p a r t i c u l a r if we EGy gk continuous, gk + g

l i s h e d as above by an argument based on Theorem 12.5. approximate d i n IEc

n - f - and k Next l e t

Q

Q

i t f o l l o w s t h a t Z:

Q'

6n/A

Q

by 6n as i n d i c a t e d t h e n f o r f,g

US

go t o t h e r i g h t s i d e o f (10.29) u s i n g t h e approximations 6; =

as above.

(10.31),

i n (12.17) tends t o t h e r i g h t s i d e o f (10.28) as

( i n t h a t order),

m

-f

E

Set t h e n R i f o r t h e corresponding o b j e c t i n

R i = PB6n

Q'

(m)

after'

The machinery has a l l been s e t up i n 910 and we w i l l

have t o show t h a t Tn =

RiV^p,qfO_g)x converges i n a s u i t a b l e manner ( c f . (+)

(

a f t e r (10.29)). n

The n a t u r a l analogue o f t h e dI;

DEFZNZCZ0N 12.8. t h e space

2;

= PE;

=PIEc where pAO

Indeed t a k i n g o p e r a t o r s

Q

in

space o f D e f i n i t i o n 12.4 i s

Go (A4 = A p ) w i t h a s u i t a b l e topology. (9.1) based on Laplace-Beltrami o p e r a t o r s as =

i n 559-10 we w i l l have Rp % $pdr w i t h i n f a c t GP = 1/2nlcp12. Then t h e na2 t u r a l analogue o f t h e W e s t i m a t e s a r i s e s as f o l l o w s . R e c a l l f r o m D e f i n i -

f^

G!

t i o n 12.2 t h a t = Pfv =m$:@[L;f] where $ i s t h e t r a n s f o r m based on = > p Thus working on E; w i t h 2 E L t h e model s i t u a t i o n i n v o l v e s $: 2 Lx + La ( i . e . E L A o r lm lF(2!pdA < m ) and t h i s i s e s t a b l i s h e d i n [ F f l ]

vF$ai

a4f

A&

?$;

0

f o r example ( a l t h o u g h s t a t e d d i f f e r e n t l y ) .

!?

1x1

Am+'

the estimate i n

f o r Ihl l a r g e , would have t h e analogue i n 5 c(~,'(a)lexp(alImXI) f o r ( X I l a r g e ( h e r e supp

i t makes sense)

f C [O,a]

%

5 cexP(olImX()/1Alm+'

IF(X)I

FP ( i.f

A!'

Since $;

and we r e f e r e.g.

large).

t o L e m a 10.5 f o r t y p i c a l b e h a v i o r o f

for

However we s i m p l y can i g n o r e such an e s t i m a t e f o r a complex and n

"C

u

=PIEc here as a space o f even e n t i r e f u n c t i o n s f = P f = "P = P f ( f = A f ) o f exponential type w i t h $ E ;: L; ( f o r h r e a l ) ( s o

deal w i t h Ep = PE;

$;$[L;?]

Cp

AP

t h a t i n p a r t i c u l a r If1 5 c$*

for

x

r e a l and l a r g e

-

c a r b i t r a r i l y small)

w i t h o u t c o n s i d e r a t i o n o f whether t h e s e p r o p e r t i e s c h a r a c t e r i z e PE;. crude e s t i m a t e w i t h supp

f" c

[O,r]

For a

we have ( c f . Theorems 9.8 and 9.12)

ROBERT CARROLL

98

P I v X ( x ) I 5 c ( x ) e x p ( x l I m X l ) w i t h c ( x ) continuous so (?(A) I ? ( x ) l e x p ( x l I m X l ) d x 5 cexp(ol1mXl). PE;

f^

converges t o

if

l$nl

<

I

5 'f c(x)A,(x) 0

f^

We w i l l say t h a t a sequence AbA

cnexp(al I m h l ) (same a ) and vp"fn

\F(x)\ 5

be t h e space o f even e n t i r e f u n c t i o n s F o f e x p o n e n t i a l t y p e ( i . e . cexp(al1mhl)) w i t h Irn IF(A)IGp(h)dA < 0

m.

W converges t o F i f I F n ( X ) l 5 cnexp(ol

We w i l l say t h a t a sequence Fn

Imxl )

( a f i x e d ) and FnCp

0

n o t e t h e n i n p a r t i c u l a r t h a t supp f i s compact f o r f want t o deal w i t h

PkuaB:

M

Assume

LZIXIIA 12.9-

?,g*

E

PW so t h a t PIK

= ApK =

> c > 0 f o r X r e a l and Ihl l a r g e . p -n* one has f g E W.

F t

7A$AA4

and f g v p = fupgvp

E

L1. 1 LA. 9

-

-f

Then W

C

2;

and

GP

§lo.

F i r s t one uses Lemma 10.13 a p p l i e d

t r i a n g u l a r i t y w i l l be discussed more c o m p l e t e l y v i a spec-

t r a l p a i r i n g s i n Chapter 2 ) . Ep

ble w i l l a l s o

assuming t r i a n g u l a r i t y o f t h e t r a n s m u t a t i o n k e r n e l s (see e.g.

Theorem 11.10

FQ so B*: lFQ

C [O,O]

K.

as i n d i c a t e d ) one A A Ac i s o f e x p o n e n t i a l t y p e f o r f,g E Ep

?;

Clearly

E

Now go t o t h e machinery sketched i n

IF;,

E

^v

has I F I z A < c l F l C P

E

1 FCP i n L A

= W.

Since F E 1?1 i s bounded on t h e r e a l a x i s ( f o r

AAA

+

P D e f i n e f i n a l l y K =IpW = { f ; f ( x ) = Im F(X)vX(x)$p(X)dX; F E W l ;

(A real).

to f

5 for

vp"f i n L,

+

-1 For an analogy t o Wm o f D e f i n i t i o n 12.4 we w i l l t a k e W = uW(0) t o

X real.

given

ic =

E

AbAn

+

*

Thus Qf = PB f and we t h i n k here a l s o o f B:

lEp L c f . Theorem 12.3).

then formally B f ( x ) =

/,"

Moreover i f f E Fc w i t h supp f

B(y,x)f(y)dy

=

f

Q

B(y,x)f(y)dy,

since

~ ( y , x ) = 0 f o r x > y, and t h u s B * f ( x ) = 0 f o r x > U. Consequently B*:FC Q "c Ac A c I : and PB*f = QfE = Ep = F = F T h e r e f o r e i n (+) a f t e r (10.29) we

E;

4

-f

9'

have ( a l s o i n Tn above b e f o r e D e f i n i t i o n 12.8) QfQg E W by Lemma 12.9 and one l o o k s f o r :pRi

= Rn = CpPgsn and

Q

R i n W'.

Here R = Ro + R

e v e n t u a l l y d e f i n e d as i n (10.36) o v e r elements P f

9

E

W.

w i l l be n

We a l s o assume l ( x )

i n (10.33) i s w e l l d e f i n e d . Thus w r i t e i n s t e a d o f (10.33) t h e approximation f o r m u l a f o r Sn E F;

4

=

EF

( c f . a l s o (10.35)) (12.18)

PRU = n

6"

Q

= 6"

Q

+

jr

L(x,y)$(y)dy

where tr a n g u l a r i t y of L i s a consequence o f assuming ~ ( y , x ) t r i a n g u l a r as Take t h e n ( y x,y) = 6 ( x - y ) + L(x,y) and a(y,x) = 6(x-y) + K(y,x)). f E lK,Pf = F E W, and a c t on t h i s f i n (12.18) t o o b t a i n ( c f . (10.35))

above

(12.19)

99

CANONICAL FORMULATION

L e t us t r e a t t h e s e terms s e p a r a t e l y . (12.20)

(6”,f)

m

(

jr

=

Q

an(x),I

f

Gn(x)A-lfdx =

Q

6

P F(h)qh(x)Cpdh) =

n

where FCP E L 1 f o l l o w s from F

E

W,

= c.

(12.21)

( R 0 ,Pf

0

Then i n (12.20) (12.22)

h)Apq,(x)^vpdhdx P

=

P F(X)~p<6n(x),q,(x)>dX

I( 6n(x),q$x)) I

=

11’ 6’(x)qA(x)dxl P

5

A o

we d e f i n e Ro ( = vp) i n W ’ v i a

As i n (10.36)-(10.37) = yjo 1i m

l,^i

’;6”

and, s i n c e I q p ( x ) I c c f o r X r e a l and x

bounded ( c f . D e f i n i t i o n 12.8) we have c i l 6”x)dx

P = P -1 , A p = A Q)

F i r s t (recal

f(y)/AQ(y) = (cP(’)Ypf)),

we s e t

( ~“(X),A-

which determines R:

Q

1f ) =

(

F(x),$P( 6n ,qA(x) P P

A

= vp(h)( 6n(x),qh(x))

= ( F(A),R;)~

i n W ’ ( s i n c e R;

5 cCp(h)).

Next we l o o k a t t h e second t e r m on th,e r i g h t s i d e i n (12.19) and w r i t e (reand l ( x ) = l i m L(x,y)/AQ(y)

c a l l L(x,y)

=

A?(x)i(x,y)Lt(y)

(12.23)

(

f ( x ) , [L(x,y)$(y)dy)

(

f(x)Ai’,f

= ( f(x)At’,[

= A?(x)ax))

i(x,y)A:(y)6n(y)dy)

=

L ~ ( x ) ~ ( x , y ) A ~ ( y ) E n d d =y )( f(x)A;’(x),$,,(x)) 0

where $,(x)

L e t us r e c o r d a g a i n ( f o r A 9 =

$(x)i(x,y)A;(y)an(y)dy.

=

Ap) t h e c o n t e n t o f (12.22) i n

1

m

(12.24)

f(x)/Ap(x)

=

Pf(h)q!(x)
0

P I f we r e s t r i c t x t o be bounded (say 0 5 x 5 a ) t h e n as above Iq,(x)l and

(0)

sup I f ( x ) / A p ( x )

I

5 cull?f$pll L1 ( 0 5 x 5 u).

i n p a r t i c u l a r t h a t i f Fn = P f n supp fn C [O,U]

(i.e.

Fn

f i n i t e i n t e r v a l [O,U~. t h e map F

+ (

f(x)A:(x),$(x))

o u t s i d e o f [O,U])

-f

0 i n L (vpdh) ( i . e .

The i n e q u a l i t y GPFn

-f

5c

shows 1 0 i n LA) with (0)

0 i n W) t h e n f n ( x ) / A p ( x ) 0 u n i f o r m l y on t h e 1 T h e r e f o r e i f + ( x ) i s any L f u n c t i o n ( o r i n -f

-f

Lie,)

i s c o n t i n u o u s ( F = P f , f/Ap E Lm s i n c e f

and determines an element R E W ‘ such t h a t $

(

,F) $

R

=

0

=

L e t us r e c a l l h e r e from D e f i n i t i o n 12.2 t h a t we a r e f(x)/Ap(x),$(x)). *C d e a l i n g w i t h c o u n t a b l e unions E: = UEC(u), = W E (u), e t c . and t h e t o p o l Qk 9 ogy i n t r o d u c e d i n D e f i n i t i o n 12.8 f o r Ep o r W i s t h e s t a n d a r d i n d u c t i v e

(

2:

l i m i t t o p o l o g y ( c f . [C19; G f l ; S j l ] .

Thus t o determine t h a t an element T

i s i n t h e dual of such a space one need o n l y check t h a t T i s continuous on

100

each

ROBERT CARROLL

~E(cI)o r bJ(u),

Conseand s e q u e n t i a l arguments s u f f i c e ( c f . a l s o §l).

q u e n t l y we can w r i t e (12.25)

(

f(x)A,'(x),$,(x))

= ( R:,F(h)

f o r some Rn E 14' s i n c e Gn w i l l q u a l i f y as a li, above. I n t h i s r e s p e c t l e t 9 us remark f i r s t (see below) t h a t $n w i l l be continuous under o u r assumptions V

A

about L, L, 1, e t c . which we e x p l i c i t l y s t a t e as

"

A

H@'@ZHHE5€$ 12-10, Assume t h e p o t e n t i a l q i n Q i s such t h a t = 2~(x)i(x,y)~:(y)

0 5 y 5 x.

= ~,(x)-e^(x,y) = 2z(x)i'(x,y)

(Ap = A

) L (x,y)

Q 1 i s continuous i n (x,y) f o r

Note f o r Ap = x2m+1 and reasonable q t h i s i s v e r i f i e d i n [Gel,

21 ( c f . [C40] f o r d e t a i l s and see a l s o Theorem 11.10). I n t h e s i t u a t i o n where Hypotheses 12.10 h o l d we c o n s i d e r Gn(x) = f x L:(x) V

L(x,y)A;'(y)b"(y)dy

=

0

ll (x,y)an(y)dy 0"

fx

and check immediately t h a t i t i s

continuous i n x ( s i n c e L (x,y) i s continuous and gn E L ' ) . T h e r e f o r e Rn i n 1 9 (12.25) above i s w e l l d e f i n e d . We can g i v e i t a formal e x p l i c i t e x p r e s s i o n ( c f . (10.34) and Theorem 10.15) b u t one s h o u l d r e c a l l Remark 10.16 i n t h i s Thus f o r m a l l y (supp f c [O,a])

connection. (12.26)

(

Rny,Pf(X)

q

:j (

J: A';

F ( 1 ) ,v ( 1 )

= ( Rn,F(X)

= ( f(x)Ai

9

I

=

m

(x,y)J"'y)

jOma;( x 1[ iy.'$x

P A p ( ~ ) 9 X ( ~ ) F ( h ) v p ( X ) d h d y d=x ,y 16" (y ) d y l dx )

which e x h i b i t s Rn f o r m a l l y , as a d i s t r i b u t i o n , b y q

REmARK 12.11.

We n o t e t h a t i t i s t e m p t i n g b u t i n c o r r e c t t o r u n t h e x i n t e The i n t e r c h a n g e o f x

g r a l i n (12.27) from 0 t o u ( s i n c e supp f c [O,u]).

and 1 i n t e g r a t i o n i s t h e n n o t j u s t i f i e d s i n c e t h e [O,u] valid a f t e r the A integration. supp f c [O,U],

F(X) = F C f =

range f o r x o n l y i s

As an example t o i l l u s t r a t e t h i s c o n s i d e r

fm

0

f(x)Cosaxdx w i t h R ( A ) = & ( A ) = (2/a)FC[1].

Then fa f ( x ) 1 dx = fm 1 ( 2 / T ) f m F(A)CosAxdh dx = fa F ( h ) ( 2 / v ) f m dh =

0

(

F(x),s(x))

0

= F(0).

0

0

i n t e r c h a n g e i n t e g r a t i o n s we o b t a i n 'f t h i s i s not correct.

0

1 Coshxdx

However i f we r u n t h e x i n t e g r a l f r o m 0 t o u and 0

f ( x ) d x = ( 2 / x ) f m F(h)[Sinhu/X]dX 0

and

Summarizing t h e above we can now s t a t e a r e f i n e d v e r s i o n o f Theorem 10.15

CANONICAL FORMULATION

$

101 A

A

Q = Q as i n (9.1) ( t h u s AQ = Ap) and l e t 8 : 4 P w i t h kernel y ( x , y ) = 6(x-y) + L(x,y) s a t i s f y Define t h e spaces E i , W, e t c . as i n Definition 1 2 . 2 and De= q!. f i n i t i o n 12.8 where Ro = i s presumed known. Then t h e Parseval formula (10.28) holds f o r f , p E IE; where R = G p R v E kr' i s defined by R = Ro + R q' which one determines by (10.36).

7lHZQ)REm 12-12, Assume Hypotheses 12.10 where A

&!

= ?lo with

A

-f

cp

$,

I t remains t o check t h e l i m i t operations Rn + R and R: Ro i n W ' A n P q A (weakly). F i r s t t h e passaoe of R: = v p ' 5 , q X ( x ) ) t o Ro = v p in W ' i s t r i v i a1 since ( F(A),R:)A = ( 6'(x),Aq1(x)f(x)) ( S ( x ) , A O 1 ( x ) f ( x ) ) = lim f ( x ) /

P400Q:

-f

-f

(There i s no need t o consider (x) = ( F ( A ) , R o ) X so :R Ro weakly i n W ' . any other topology in I d ' . ) . As f o r Rn R i n Id' weakly i t i s simplest t o q q ( c f . (12.25) and (10.36)). Recall $J,(x) = show ( fa: , i n+ ) f m L(x)f(x)dx A fo $ ( x ) L ( x , y ) A ~ ~ y ) 6 n ( y ) d iys continuous and A p ' ( x ) $ , ( x ) = A p l ( x ) J 0x (x,y) y ) . has by s n ( y ) d y = f X;(x,y)Gn(y)dy where ;(x,y) = A ~ * ( ~ ) ~ ( x , y ) A ~ (One 0 assumption t h a t < ( x , y ) = A,(x)L(x,y) i s continuous i n ( x , y ) and we know f ( x ) A p l ( x ) is continuous. In t h i s respect l e t us observe from (12.24) t h a t P f o r x bounded ( 0 5 x 5 IT) say with I q , ( x ) l L ~ C ,one has f o r f p ( x ) = f ( x ) /

A

9

-f

-f

rl

(fp(X+ X) - fp(x)l = Ifm Pf(A)Gp(h)[qA(X+X ) - q XP ( x ) ] d X [ 'J0 N IPfcpl P 0 140XldX + 2cufNmIPfGpldX. Given F pick N so t h a t I," IPf$pldX < E / ~ c , and 6 P so t h a t by uniform c o n t i n u i t y of q h ( x ) on C X E [O,N], x E [O,u]3 one has P 140Xl 5 42J"0 l P f v p l d h f o r \ A x 1 5 6. I t follows t h a t IAf p,-1 and f P i s con0 and L l ( x , y ) Ap(x) tinuous. Next r e c a l l t h a t f ( x ) = lim l?(x,y) a s y

AP(x),

-+

-+

A

L(x) = L l ( x , O ) i s then continuous. Mow f o r f fixed t h i n k o f t h e continuous 1 Lloc fApl i n Lm = ( L 1 ) ' w i t h s u p p o r t in [O,u] say and t h e continuous $,, 1 ( o r L over [O,o]). Then f o r x fixed $,(x) = f X r1(x,y)6'((y)dy = ( L l ( x Y y ) , 0 6 n ( y ) ) -+ r l ( x , O ) s i n c e d n 6 in E'. Further l + n ( x ) l Lsup I < ( x 7 y ) l I O x6 " ~ )

',"

-f

V

dy 5 s u p l r l ( x , y ) I L C ( s u p over 0 5 y 5 x ) by t h e c o n t i n u i t y of L1 i n ( x , y ) . +(x) = By bounded o r dominated convergence i n L1 i t follows t h a t $,(x) A L1 V (x,O) i n L 1 and consequently ( fh;' , G n ) -+ ( fAp' y(l (x,O) ) = ( fAp' ,ApL(x) ) = -f

(f,R).

This Page Intentionally Left Blank

1.

INCR0DUCCI0N.

We have seen i n Chapter 1 how c e r t a i n t r a n s m u t a t i o n s can

be c o n s t r u c t e d v i a Goursat problems and c h a r a c t e r i z e d v i a Cauchy problems. Some i n d i c a t i o n o f t h e c o n s t r u c t i o n o f t r a n s m u t a t i o n k e r n e l s v i a s p e c t r a l p a i r i n g s was a l s o g i v e n and we w i l l develop t h a t theme e x t e n s i v e l y i n t h i s chapter.

A l s o we w i l l d e s c r i b e a c h a r a c t e r i z a t i o n o f t r a n s m u t a t i o n s v i a

m i n i m i z i n g procedures which i s r e l a t e d t o l i n e a r l e a s t squares e s t i m a t i o n 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 (as i n Chapter 3, 5 6 ) .

One

can a l s o t h i n k o f t r a n s m u t a t i o n s as a r i s i n g f r o m 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 v i a t h e a p p r o p r i a t e Goursat problem and we w i l l show how t h e v a r i o u s approaches and ideas f i t t o g e t h e r .

F u r t h e r we w i l l g i v e an ex-

t e n s i v e development here o f general G e l f a n d - L e v i t a n (G-L) and MarEenko ( M ) equations.

The i n g r e d i e n t s ( a n a l y t i c i t y , Paley-Wiener ideas, e t c . ) needed

t o connect two l i n e a r second o r d e 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 v i a t r a n s m u t a t i o n a r e i s o l a t e d and t h e i r r o l e i n d e t e r m i n i n g p r o p e r t i e s o f t h e connection i s indicated. As a preview o f t h i s c h a p t e r we n o t e t h e f o l l o w i n g .

§ 2 discusses s p e c t r a l

p a i r i n g s f o r t r a n s m u t a t i o n k e r n e l s i n a general way and a l s o g i v e s v a r i o u s examples. (Q,

=

The p r o t o 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 s Bo: D2

D2 +

[(2m+l)/x]D)

+

Qm and B - l

Q

a r e o b t a i n e d and such t r a n s m u t a t i o n s a r e discussed

w i t h a view t o d i s c o v e r Ro.

53 i s about t h e general extended G-L e q u a t i o n I n t e r t w i n i n g i s discussed i n r e l a t i o n

and i n d i c a t e s s e v e r a l p o i n t s of view.

t o c o n n e c t i o n formulas and a p r o t o t y p i c a l G-L e q u a t i o n i s e x h i b i t e d f o r B above.

Q

§ § 4 - 5 g i v e a d i s c u s s i o n of c l a s s i c a l quantum s c a t t e r i n g t h e o r y f o l -

l o w i n g Fadeev and p r o v i d e a f i r s t e x t e n s i o n o f t h i s t e c h n i q u e t o connect t h e G-L and M equations i n terms of t r a n s m u t a t i o n ideas.

The r o l e o f a n a l y t i c i t y ,

Paley-Wiener ideas, t r i a n g u l a r i t y , e t c . i s discussed and a p p r o p r i a t e propert i e s o f k e r n e l s a r e p r o v i d e d i n a f a i r l y general s e t t i n g . new canonical i n t r i n s i c v e r s i o n of t h e G-L and

103

I n 56 we g i v e a

M connection v i a t r a n s m u t a t i o n

ROBERT CARROLL

104

f o r Fourier type operators P = D2 - q and f a i r l y general Q .

The operational

calculus f o r Fourier type eigenfunction transforms on t h e f u l l l i n e is developed and a generalized Kontorovi&Lebedev ( K - L ) inversion i s a l s o e x p l o i t ed on t h e half l i n e . Spectral i n t e g r a l s f o r t h e various kernels a r e exhibi t e d a n d t h e r o l e of G-L (resp. M) equations as lower-upper (resp. upperlower) f a c t o r i z a t i o n s of G-L data (resp. r e l a t e d PI d a t a ) appears i n a canonical manner. 57 i s about t h e c h a r a c t e r i z a t i o n of transmutation kernels via minimization procedures and one shows l a t e r in Chapter 3, 5 6 t h a t when t h e r e i s an underlying s t o c h a s t i c process this procedure i s equivalent t o e characterize l e a s t squares estimation t o determine the f i l t e r i n g kernel. W here b o t h the G-L and M equations as minimizing c r i t e r i a ( a s "Euler" equat i o n s of a v a r i a t i o n a l argument). 58 goes t o the study of transmutation f o r t h e Q type operators o f Chapter 1 , § 7 and a l s o c o n s t r u c t s kernels via Gours a t problems and successive approximations. A generalized K-L inversion N

theory i s developed (extending the c l a s s i c a l one) and various generalized t r a n s l a t i o n s , generalized convolutions, e t c . a r e discussed. 59 continues in t h e operator format and shows f i r s t how t h e Bergman-Gilbert ( B - G ) op-

5

e r a t o r a r i s e s i n s p e c t r a l form as a transmutation. Then some a p p l i c a t i o n s t o the construction of "canonical" generating functions ( f o r special functions) a r e given. General formulas f o r s p e c t r a l c h a r a c t e r i z a t i o n o f kernels and t h e construction of generating functions via residue calculus a r e g i v e n and some new applications ( e . g t o Glhittaker functions) a r e indicated. 510 involves some new work showing i n p a r t i c u l a r how one can c o n s t r u c t orthogonal functions (generalizing t h e idea of Krein f u n c t i o n s ) corresponding t o Transmutation methods a given measure of say polynomial growth on [O,m). provide t h e e s s e n t i a l i n t e r p e r t a t i o n and construction and t h e orthogonal functions are determined via a d i f f e r e n t i a l equation and a general G-L equat i o n . Some f u r t h e r i n s i g h t i n t o t h e minimization procedure of 57 i s proe wanted t o i n s e r t a secvided in the context of approximation theory. W t i o n on e l l i p t i c transmutation b u t instead r e f e r t o [C40,35]; one can show f o r example how generalized a x i a l l y symmetric potential theory (GASPT) can be constructed canonically via transmutation ideas and thus one represents GASPT a s a special case of a whole canonical family of such t h e o r i e s . General Hi1 b e r t transforms, t h e conjugate Hankel transform, Erd6lyi-Kober (EK ) operators, and i n t e r p o l a t i o n can be discussed from t h e transmutation point of view. 5l'l provides some general constructions r e l a t i n g transmutat i o n kernels t o t h e " p o t e n t i a l s " i n t h e operators from which t h e transmutat i o n s a r i s e . For example the r e l a t i o n A-'(y) = 1 - K(y,y) f o r Qu =

SPECTRAL PAIRINGS

105

(Au‘)’/A and p QX ( y ) = A-4(y)Cosxy + fy K (y,E)CoshgdE (needed i n Chapter 3, 0 5 557-8) i s obtained by a new method. SPECCRAG PAIRZNGl3 FOR GENERAflZED CRAGLACI0N AND CRANSPIllEAEl0N KERNELS. Let us t a k e operators (1.9.1) as before w i t h R Q a s i n Theorem 1.12.12 and w r i t e formally 2,

(2.1)

q6Q(A) = 1; S4(x) = S ( x ) / A (x);l!$[l](x) = 6 ( x )

Q

Q

= (

RQ,u!(x));

Similarly p[G(x-y)] = p X Q ( x ) and (A-’f(y),A-4s(x-y)) = A-’(x)f(x) in a f o r Q 4, 4 ma1 b u t j u s t i f i a b l e manner. Now l e t P and Q be qeneral operators of t h i s type based on ( A p , q p ) a n d (A ,q ) w i t h a common s p e c t r a l parameter. The Q Q main point here i s t h a t the spherical functions and J o s t s o l u t i o n s f o r both A A A 2 2 P and Q a r i s e from the equations Rp = -A p and Qp = -X p w i t h t h e same values of X . The s p e c t r a l parameter X e n t e r s in the same manner (as an eigenvalue) and since t h e “eigenfunctions” ( p0i e t c . will be defined f o r a l l 1 A

(and a n a l y t i c in r e g i o n s ) , a s long as we take s p e c t r a l i n t e g r a l s o f eigenfunctions over regions where t h e integrand has s u i t a b l e p r o p e r t i e s , t h e r e will be no problem. One need not f o r example be concerned a t a l l w i t h A 2 having i d e n t i c a l L type spectra f o r $ and Q (and generally such spectra a r e d i f f e r e n t ) . The use of the word “eigenfunction” f o r p 0i say is perhaps inappropriate s i n c e q : i s defined f o r a l l X and i s a n a l y t i c b u t we will cont i n u e t o use this terminology. Let us emphasize a l s o t h a t i n constructing transmutations via PDE techniques, Riemann f u n c t i o n s , e t c . as i n Chapter 1 , A n no i d e n t i t y of s p e c t r a f o r P a n d Q i s involved. In f a c t some of t h e main r e s u l t s (e.g. Theorems 1.5.5 a n d 1 . 1 2 . 1 2 ) involve producing a Parseval f o r mula f o r a f a i r l y general 0, with many s p e c t r a l p o s s i b i l i t i e s , i n terms of A some fixed P as base o b j e c t . We will see t h a t t h e r e s u l t i n g transmutation A A P kernel f o r say B: P -t Q : p X -t q Q can normally be expressed a s a s p e c t r a l i n t e g r a l over spec ? w i t h 8 = B” given by an i n t e g r a l over spec $ (both kernels can be considered as s o l u t i o n s of s u i t a b l e Goursat o r Cauchy probA A lems). We consider now again t h e construction of a transmutation B: P Q P characterized by &PA = p: and i n terms of PDE we a r e in the s i t u a t i o n of say Theorem 1 . 4 . 3 with operators P a n d 0 (q (x,O) = 0 i s natural because of Y fixed) s i n g u l a r i t i e s and A = 1 ) . Let us note f i r s t t h a t i f one s e t s ( f o r A P Q 2 P ( 0 ) V(X,Y) = P ~ ( X ) I P ~ ( Ythen ) ^P(Dx)p = Q ( D Y b ( = - A P ) while ~ o ( x , O ) = p X ( x ) P w i t h p (x,O) = 0. Hence formally ( t a k i n g ‘ p x ( x ) a s even i n x f o r example) Y A

-f

106

ROBERT CARROLL A

= and c o r r e s p o n d i n g l y e x t e n d i n g t h e P - f u n c t i o n s t o be even) ip(0,y) = ipA(y) Q P We can now determine t h e k e r n e l ~ ( y , x ) o f B i n terms o f s p e c t r a l (b,)(y).

p a i r i n g s and a l s o c h a r a c t e r i z e 8 = 6 - l i n t h e same way. We have been and w i l l c o n t i n u e t o be vague a t t i m e s i n s p e c i f y -

REIRARK 2.1.

i n g t h e domains o f t r a n s m u t a t i o n o p e r a t o r s such as B.

I n $ 5 5 and 12 o f

Chapter 1 one was q u i t e p r e c i s e i n d e s c r i b i n g i n v e r s i o n formulas b u t such concern w i t h domain i n general would impede t h e development o f t h e t h e o r y and b u r y e v e r y t h i n g i n a hopeless maze o f n o t a t i o n .

For example one can de-

f i n e o p e r a t o r s P, Q, B y e t c . on w i d e l y d i f f e r e n t c l a s s e s o f o b j e c t s depend-

ing on what you a r e s t u d y i n g .

F o r example we j u s t d e f i n e d B f r o m (.)

on

P

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 9 x and a l t h o u g h one can l o c a t e such o b j e c t s i n Sobolev spaces based on

(cf. [Bil;

Gf1,3;

Ral]) i t would o b v i o u s l y be P s i l l y t o w o r r y about t h i s now; one can t h i n k o f q A ( . ) i n Cp f o r example and t h e c o n s t r u c t i o n o f i n v e r s e s 8 = B - l i s o f t e n b e s t regarded i n t h a t l i g h t . Some i n d i c a t i o n o f spaces o f f u n c t i o n s where t h e t r a n s f o r m s and transmuta-

t i o n s a c t i s g i v e n l a t e r ( w i t h s u i t a b l e a t t e n t i o n t o m a t t e r s o f growth o f t h e o b j e c t f u n c t i o n f a t 0 and Now l e t T;

m

and t h e v a n i s h i n g o f f o r f ' a t 0).

be t h e g e n e r a l i z e d t r a n s l a t i o n a s s o c i a t e d w i t h

U(x,y) = T:f(x) extended t o

determined by

?(D )U; U(x,O) = f ( x ) ; Uy(x,O) = 0 ( f i s Y as an even f u n c t i o n ) . S i m i l a r l y l e t S: be t h e g e n e r a l i z -

where (+) ^P(Dx)U =

(-m,m)

A

ed t r a n s l a t i o n a s s o c i a t e d w i t h Q determined i n t h e same manner.

We w i l l

show now t h a t

(2.2)

B(Y,X)

P

Q

= ( f l A ( X ) d P X ( Y ) )" A

i s t h e k e r n e l o f t h e t r a n s m u t a t i o n B: P

A -f

Q c o n s t r u c t e d v i a Theorem 1.4.3

)" denotes t h e v - s p e c t r a l p a i r i n g as i n w i t h C = 0 and A = 1. Here ( P P 1.12.1 v i a R o r dvp ( n o t e t h a t when dvp = Cp(A)dX t h e n R 2r $(A) i s a func-

tion).

where

Thus c o n s i d e r t h e f u n c t i o n

(

,

)

denotes a s u i t a b l e d i s t r i b u t i o n p a i r i n g on (0,~). D e t a i l e d pro-

p e r t i e s o f such k e r n e l s B(Y,S) and g e n e r a l i z e d t r a n s l a t i o n s U(x,S) = TXf(S)

5

w i l l be discussed l a t e r ( c f . a l r e a d y Chapter 1) and f o r now we proceed a t t i m e s somewhat f o r m a l l y . U(x,c)) = A

A*

(

P B(Y,s),~(x,s))

5

One has 6(Dx)p = and from (2.2)

~ ( y , ~ ) ~ ? ~ U ( x=, (~ B) ( )Y . S ) , ? ~ 2 P 0 P S ~ ( y . s )= -( A nx(5),ipi(y) )" =

(

A*

Q y ~ ( y y ~T h) i.s k i n d o f c a l c u l a t i o n a r i s e s o c c a s i o n a l l y and t o j u s t i f y i t

SPECTRAL P A J R I N G S we n o t e t h a t t h e passage

107 A*

e ( y , ~ ) , ~ ~ U ( x , t~o) )( P E e ( y , ~ ) , U ( x Y ~)) i s s i m p l y P = (b;((<),q ( 5 ) ) ( a c t i n g on s u i t a b l e f u n c t i o n s ) . R e c a l l

(((E),bL(<))

(

M*

A*

A

so P Apq = ( a p q ' ) ' = Now an elementary c a l here P 4 = [ ( $ / A p ) ' A p ] l (Pqu,.A = (qp,,P )o T + A~W(P,,P~)(T) P P c u l a t i o n g i v e s f o r ( , )o = f

'?:*{

J)T

(W(f,g)

=

Poi,

-

fg'

f ' g and W(qx,qu) = 0 a t x = 0 ) .

On t h e o t h e r hand from t h e

2

A

d i f f e r e n t i a l o p e r a t o r Pu = ( A p u l ) l / A p + ppu - pu one o b t a i n s a Darboux2 T P P P P = ST(X,u). C h r i s t o f f e l t y p e formula ( A 2 - 11 )I % ( x ) q u ( x ) d x = ApW(q,,qu)(T) 0

But from Elf = I ( a c t i n g on s u i t a b l e f u n c t i o n s ) we have as i n Remark 1.11.3, P P P a P For convenience t h i n k o f R as a f u n c t i o n $ and R ( P x ( x ) , p P ( x ) )o = S ( X - u ) . 2 2 A m P n o t e t h a t sT(h,u) i s a n a l y t i c i n h , ~ . Thus &sT(h,p) + (A - p ) v f y ( x )

2

2

0

q P ( x ) d x = (A - p ) 6 ( x - p ) .

u

I t f o l l o w s t h a t i n i t s a c t i o n on s u i t a b l e o b j e c t s ?

" P

t h e o p e r a t o r sT(h,u) 0 f o r x # 1-1 , as w e l l as f o r = u, so t h a t ( P q u , P P A* P R ) = ( q ,P R ) i s confirmed i n t h i s c o n t e x t . Direct calculation using -f

x

F

i

x

t y p i c a l spherical functions w i l l also verify ;(Dx)q

the result.

Consequently

$(D )P (from above) and q(x,O) = ( e(O,c),U(x,~) ) w i t h q ( 0 , y ) = Y (a(y,c),U(O,c)) = ( B(y,S),f(c)) (U(x,y) = U(y,x) i n general here). Now as =

i n ( 2 . 1 ) B(O,S)

= ({(E),I

)v = 6 ( c ) so q(x,o)

f ( x ) ( a l s o qy(xyO) = ( B~(O.E),U(X,E)) = 0). n i t i o n o f B, and t h i s must agree w i t h q(0,y)

= (6(c),~(x,c)) = u ( ~ , o )=

Hence q ( 0 , y ) = B f ( y ) , by d e f i = ( ~ ( y , ~ ) , f ( c ) ) . It f o l l o w s

t h a t B(Y,E) g i v e n by ( 2 . 2 ) i s t h e k e r n e l o f B and we have A

KHE0REN 2-2, I f B i s t h e t r a n s m u t a t i o n B: P A

( w i t h A = 1 and C = 0, (P,Q)

%

Q determined b y Theorem 1.4.3 A

A

(P,Q)) where t h e P s p e c t r a l p a i r i n g v i s

known, t h e n B i s c h a r a c t e r i z e d by &p! g i v e n by ( 2 . 2 ) .

A -f

= q:

and t h e k e r n e l ~ ( y , x ) o f B i s

F u r t h e r one can w r i t e t h e s o l u t i o n 9 o f t h e Cauchy prob-

lem c h a r a c t e r i z i n g B i n t h e f o r m ( 2 . 3 ) where U(X,E) (2.4)

IP(X,Y) = ( v hP( x ) , f (" A ) q h4( y ) ) v ;

;(A)

= TXf(5) o r equivalently

5

= Pf(1)

Ptluod:

Only t h e l a s t a s s e r t i o n remains and ( 2 . 4 ) can be o b t a i n e d by r e a r 4 P ( a(y,s),U(x,5)) = (~x(y~y(~~(S),U(x,5)))V 4 P = ( q x ( y ) , ~ ( x , k ) )v. n ~ o w$(x,x) = ( n ~ ( c ) , ~ ( x , c ) )s a t i s f i e s PJ, = ( %(s), 4 P A* P 2 P PxU(x,s)) = (a,(c),P U ( x , s ) ) = ( P a ( s ) , U ( x , s ) ) = - h J, w i t h $ ( O , h ) = (Qx(5), 5 E x f ( 5 ) ) = ? ( A ) and $x(O,h) = 0 ( c f . ( + ) ) . Hence ( g i v e n unique s o l u t i o n s o f 2 " P P(Dx)J, = - A J , ) we have JI(x,A) = f ( h ) q x ( x ) and ( 2 . 4 ) f o l l o w s .

r a n g i n g ( 2 . 3 ) i n t h e form P(X,Y) =

L e t us r e c a l l now some t r a n s f o r m a t i o n s which mix t h e s p e c t r a l p a i r i n g and t h e g e n e r a l i z e d e i g e n f u n c t i o n . Thus s e t ( c f . 1 . 1 2 . 3 ) qF(y) = ( q x4( y ) , F ( X ) )v; P IPF(x) = ( q x ( ~ ) , F ( ~ ) ) u . Examining t h e p r o o f o f Theorem 2.2 we see t h a t g i v e n A

unique s o l u t i o n s o f ( 6 )

[Pxq

=

4Yq ,

q(x,O) = f ( x ) , qY(x,O)

=

01 (and of

108

ROBERT CARROLL

= - A 2$ ) t h e n from (2.41, v(x,O)

;;(Ox)$

P

(aA(5),1

required only that

)v =

=if(x).

f ( x ) = ( f "( x ) , v AP ( x ) ) ,

=

This

6( 5 ) and shows

COR@LLARM 2.3. Given unique s o l u t i o n s of ( 6 ) and o f Px$ = - A 2 $ w i t h a specP t r a l p a i r i n g v such t h a t ( P A ( 5 ) , l /= 6 ( S ) i t f o l l o w s t h a t P = P-' has t h e A

form P F ( x ) =

Pkooj:

(

P F(A),cpA(x) )v.

Also one can w r i t e B = @.

For t h e l a s t statement we have f r o m (2.4) and d e f i n i t i o n s above

B f b ) = v(OyY) =

(

"

Q

f(A)%(Y)

) v = !lf?i(Y) = ( @ f ) ( y ) .

It i s easy now t o c o n s t r u c t B = B - l by s i m i l a r procedures.

(2.5)

r(x,u)

Thus s e t

= (vP A ( x ) , qQ Y ) )w

and c o n s i d e r f o r V(y,n) = Syg(n)

n

(2.6)

$(XYY) =

{

v(x,n),v(Y,n))

Then as before one has a c a l c u l a t i o n ^a(Dy)$ = ( ~ ( x , n ) , ~ ~ V ( y ~ =n )( )y ( x y n ) , A

A*

A*

4 Q V(y,n)) = ( Q n y ( x , n ) y V ( y , n ) ) and from ( 2 . 5 ) Q,v(x,n) = - ( v XP( x ) , A 2RA(n)) = A n A Hence ? ( D x ) $ = Q(D ) $ w h i l e $(x,O) = (v(x,n),V(O,n))= ( y(x,n), PXv(xyd. )o = 6 ( n ) as i n ( 2 . 1 ) y we have $(O,y) =

g ( n ) ) and, g i v e n y ( 0 , n ) = ( %Q( nY) , l (y(O,n),V(y,n))

= ( 6 ( n ) , V ( y Y n ) ) = V(y,O)

= g(y).

Thus g i v e n g ( y ) extended A

to

as an even f u n c t i o n say one i s d e a l i n g w i t h a Cauchy problem Px$

(-my-) A

= Q $; $(O,Y) = g ( y j ; $,(O,Y) = 0 f o r x 2 0 ( n o t e ~ ~ ( 0 , n=) 0 so $,(O,Y) = Y Assuming unique s o l u t i o n s s e t Bg(x) = IL(0,x) = ( y ( x y n ) , g ( n ) ) and t h e n 0).

L e t g be extended t o

EHE0REllI 2.4.

A

Cauchy problem

Px$ = Q $; $(O,y)

(m)

!Yx

t i o n along w i t h

as an even f u n c t i o n and assume t h e

(-m,m)

A

=

-A2xY

= g(y);

$,(O,y)

= 0 has a unique solu-

spectral p a i r i n g w then $ i s given

Given t h e

A

by (2.6) and t h e k e r n e l o f t h e r e s u l t i n g t r a n s m u t a t i o n 8: Q i s g i v e n by t h e y(x,y)

@ ( x ) = $(O,x)

(2.7) where 6( Q) 8 =

$(XlY)

o f (2.5).

h

+.

P d e f i n e d by

One can w r i t e a l s o

= ( m y v A P( 4 v AQ( Y )) w

Given a s p e c t r a l p a i r i n g w such t h a t o n l y (C2A(n)yl Q )w = 0 Finally = 9-l where g G ( y ) = ( G ( A ) , v ~ ( y ) )w. i s known i t f o l l o w s t h a t = Qg( A).

PQ and

Pmud:

Q

BpA =

v PA .

We need check t h e d e t a i l s o n l y beyond (2.7).

o f Theorem 2.2 $(x X(~,A))~.

'Y, )

= ( y(x,n),V(y,n))

Q

"

Thus as i n t h e p r o o f

= (vA(x).(Q$(n),V(y,n)) P

C l e a r l y Q x = (c,(~),Q v(y,n)) Y Y

= (SIQh(n),Q(y,n))

=

) .*WQ

= (qA P(x),

(qnnA(n),

SPECTRAL P A I R I N G S

109

V(Y,TI) = -A 2x w i t h x(O,A) = ( a xQ ( n ) , V ( O , ~ ) ) = ( g Qx ( ~ ) , g ( n ) )= ? ( A ) and xy(O, x) = 0. Hence x ( y , h ) = < ( x ) q xQ( y ) and (2.7) f o l l o w s . F u r t h e r from (2.7)

P

and (1.12.3)

e v i d e n t l y 11,(x,O) = Rg(x) = ( q x ( ~ ) , z ( A ) ) o = P c ( x ) = (PQg)(x).

The i n v e r s i o n f o l l o w s from (2.7) by w r i t i n g P(0,y)

Q

F i n a l l y one checks t h a t R q X = (0)

q!

= g ( y ) = (r(A),q:(y))u. P O by c o n s i d e r i n a 11,(x,y) = q x ( x ) 9 p , ( y ) as i n

b e f o r e Remark 2.1.

To show t h a t 8 = B - l we have o f course t h e c h a r a c t e r i s t i c a c t i o n and &: = q ! ( e x t e n d i n g qy t o be even e t c . ) .

P = q xQ &PA

Another p o i n t o f view would

t a k e B f f r o m Theorem 2.2 and extend i t t o be even i n y ( t h i s i s o f t e n autom a t i c s i n c e terms l i k e q Then

5))

satisfies

q

=

(m)

and ( l / y ) q have a p p r o p r i a t e even p r o p e r t i e s ) . YY Y w i t h g(y) = B f ( y ) ( s i n c e a l s o qx(O,y) = ( ~ ( y , s ) , U ~ ( 0 ,

0 ) so by uniqueness q = 11, and f ( x ) = q(x,O)

= 11,(x,O) = B B f ( x ) .

Hence

BB = I and s i m i l a r l y BR = I which y i e l d s

C6ROLLARg 2.5, The t r a n s m u t a t i o n s B and €3 o f Theorems 2.2 and 2.4 a r e i n -

q-’

verse t o one a n o t h e r and

=

W Qw i t h P-’

=

QqP.

P / L U U ~ : The l a s t statements f o l l o w by w r i t i n g e.g. ’ - R )=

Pt?? = I B - l ! = (

=

p-lq-l = IPq-l. n

L e t us e x h i b i t n e x t t h e formulas a p p r o p r i a t e t o t h e t r a n s m u t a t i o n B

Q

=

(m > -%) 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<)/J6

Since Dn Yn-,-

m+% g(Jn)/Jn] n EHEBRER 2-9,

which i s t h e n a t u r a l i n v e r s e t o (2.12) f o r g = B f.

Q

For -%

<

4

m < n-5 and BQ: D2

+

9

Q,

t h e transmutation characteriz-

ed by B [Coshx] = p X ( (PXas i n ( 1 . 9 . 4 ) ) w i t h 8 =

Q

(2.15)

= (Jn/r(m+l))[Y-m-vz*

Q

vm(BQf)(Jn) = [ r ( m + l )/JIT]

(BQg)(J5)/JS

=

r(m+l

[JT/

)I

*

[Y,++ D:

B-' one has ( x 2 ,y 2

Q

Pmol;:

y,(x,y)

& r

= 2J71s nx y

r

m+l

* nmg(Jn)l

[Yn-,,+

i s even i n x )

(x2-y2);m-3/2

-m--i

F o r t h e l a s t e q u a t i o n we n o t e t h a t

(

g(Jn)dn/2Jn so f r o m (2.15) +&J"yQ(Jc,Jn)n-m-4 and t h i s g i v e s (2.16)

5,n)

(f(J~)/J5)1;

The l a t t e r e x p r e s s i o n can a l s o be w r i t t e n as ( n o t e y,(x,y) (2.16)

%

y (x,y),g(y))

4

%

%

J; y Q ( J c , J n )

.

J~1(5-17)-~-~/~/r(m+l)r(-m~)

( a f t e r a d j o i n i n g a sgnx f a c t o r ) .

W i t h a view t o s i t u a t i o n s i n t h e study o f Parseval formulas as i n § § 1 . 1 0 and 1.12 where e x p l i c i t i n f o r m a t i o n about Ro may n o t be a v a i l a b l e ( o r where Ro i s more complex) l e t us show n e x t how one can " d i s c o v e r " Ro u s i n g techniques

112

ROBERT CARROLL

s i m i l a r t o those l e a d i n g t o Theorem 10.15. T h i s approach i n v o l v e s transmu" 2 t i n g D2 = into = when P = D we w r i t e BQ: D2 and use t h e sub-

6 p;

?

-f

s c r i p t Q as i n d i c a t e d above.

I n p a r t i c u l a r we w i l l want t h e p a r t i c u l a r

Q t r a n s m u t a t i o n 6 [Cosxx] = vp,(y) and as above one expects B Q

Q

t o be a "smooth-

i n g " f u n c t i o n and y Q t o be a d i s t r i b u t i o n o f o r d e r g r e a t e r t h a n zero.

First

l e t us connect Ro t o Sy6 ( x ) as i n (10.29) where S: i s t h e g e n e r a l i z e d x Q t r a n s l a t i o n associated w i t h = $' now). S i f ( x ) w i l l have t h e general form (cf. (11.3)) s:f(x) = (y(x,y,n),f(n)) w i t h V ( X , Y , ~ ) = (Q~ ~ XQX ) ~ ~ X Y ) ,

(6

8

CQp , ( n ) ) where w

sequently

2,

i s assumed t o be o f t h e f o r m dW

w Q

(note

(

Q

Q (n),ux(n))

0

= (,;A)dh

.

here.

Con-

= 1)

6

R e c a l l a l s o here t h a t ( C lQp , ( n ) , 6 ( n ) ) Q On t h e o t h e r hand, as i n d i c a t e d a f t e r (2.1), f o r m a l l y

f o r m a l l y ( r e c a l l 6 ( n ) = S ( n ) / A (11)). =

Q

1 as i n (2.1).

Q[~(X-Y)I

fl

= ( ~(x-~),q:(y))

f iQp , ( ~ ) = ) (6(x-c),v:(E))

(2.18) (cf.

= Ip:(x)

and q [ 6 ( x - c ) / A Q ( c ) I =

( ~(X-E)/A~(E),

= vhQ(x) so t h a t f r o m (2.17) we have f o r m a l l y

sZ6Q(x) = ( ~ [ s ( x - E ) / A Q ( 5 ) ' q ~ ( y ) )=~a q [ 6 ( X - c ) / A Q l = 6 ( x - y ) / A Q ( y ) As i n t h e passage (10.29)-(+)

(10.29) a l s o ) .

(2.19)

(

f(x)g(y),s(x-y)/AQ(y))

lom lo

we have ( f , g s u i t a b l e )

= (A-'f(x),A-'g(x)) Q Q

=

m

qf&ldWQ(x)

=

Qf@iQ(i)dx =

(

Ro,gf@))h

(6 = p ) l e a d s f o r m a l l y t o t h e Parseval

Thus t h e f o r m u l a (2.18) f o r S ~ s Q ( x ) formula (2.19)

( f o r s u i t a b l e f,g)

where Ro =

A

i s specified.

Now we t r y

WQ

t o f i n d a g a i n S i 6 Q ( x ) i n t h e form ( c f . (10.29))

s i n c e dv = (2/a)dX. A-'g)

Q

Then u s i n g (2.18) we o b t a i n f o r s u i t a b l e f,g,

= (2/n)(Rv,QfQg)

so t h a t Ro

=

(Z/IT)R'

(2.20) t o o b t a i n (10.30),

i.e.

Q Q

(2.21)

Rv = Pl3 6 = Q

Q

]

6 (x) =

Q

(

Q

(as i s a l s o c l e a r from (2.17)).

Now however we can proceed as i n t h e p r o o f o f Theorem 10.15. f o l l o w , i e. 8 6 ( y ) =3PRv.

(A-'f,

Rv,qx(x) Q

Set y = 0 i n

from which (10.31) w i l l

)",

Consequently one has (13Q6Q)(x)Cosixdx

0 V

We do n o t have recourse here t o such n i c e k e r n e l s

V

K and L f o r t h e o p e r a t o r s

SPECTRAL PAIRINGS

6

=

P and

6 however

113

and c a l c u l a t i o n s involving B 6 must be handled sorneA

Q Qo

0

G:

what d i f f e r e n t l y (we r e c a l l i n p a r t i c u l a r f o r Q = = t h a t t h e kernel In t h i s d i r e c t i o n we note a l s o Theorem y of R4 i s of t h e form ( 2 . 1 6 ) ) .

Q

2.4 which s p e c i f i e s t h a t y ( x , y ) = ( C O S A X , S ~Q~ ( Y ) ) , so t h a t

Q

(2.22)

( 8 6 )(x) =

Q Q

((

Coshx,flA(y) Q ),,SQ(y))

A

EHEB)REm 2-10.

The generalized spectral function Ro for Q where B

l y determined via Ro = (2/71)PB 6

case

t o=

=

Q Q

D2

Qo the computation gives Ro = cmA q:2mt1

m calculations.

+

Go

and

=

Q

A 0

Q

can be formal-

= B-'

Q'

In the

via standard distribution

P M J O ~ : From (2.14) we have

On the other hand from ( 2 . 9 ) we should have ( c f . a l s o ( 2 . 2 2 ) )

(2.24) m

CosAxRo(A)dh = 0

0

Now (2.23) and (2.24) should agree and r e f e r r i n g t o Remark 2.8 we obtain (a = 2rn+l, m - 1 , - 2 , . . . ) , cz.fm h2m+1CosAxdi = -cmr(2m+2)Sin4a 2 f o r (2.24) m 0 t.1) Ix/-zm-2. B u t from [Mbl] one knows r(2m+2)Jn = 22m+1r(m+l)r(m+3/2)

+

2 so -cJ( 2m+2)Sin (m++) TI = -2r (m+ On t h e other hand from (2.23) 6, = -(Zm+l)r(+)/ 3/2)Sin(m+4)n/J71r(m+l). r (m+l )r(+-m) = -2(m++)r (+)r(m+3/2 )Sin (m++)n/r (m+l ) n(m++) = -2r (m+3/2)Sin (m++)x/Jd'(m++) which agrees. Hence ( r e c a l l rn > -+) ( 2 . 2 3 ) and (2.24) agree w h i 1 e 1/r(+-m)

=

r (m+3/2 ) S i n (m++)

71/71

(m++)

and t h e computation f o r Ro could have been c a r r i e d out via P73Q&Qu s i n g standard techniques. Let us make here a few f u r t h e r general comments about constructing generalized t r a n s l a t i o n s a n d transmutations. First r e l a t i v e t o formulas l i k e 1.5. 10 f o r generalized t r a n s l a t i o n l e t us w r i t e more generally (2.25)

u ( x Y y )=

(

;(h)a(A)qh(x),vh(Y) P P ,)

- ( hl f ( ~ ) c ( ~ )P x A ( x ) , x ~ ) E( Y )

where P i s some f a i r l y general operator Pu = ( A p u ' ) ' / A p - pu f o r example, P q ! and ! x s a t i s f y (*) Pq = -12q with ~ P ~ ( =0 1), DxqA(0) = 0 , x!(O) = 0 , and

114

ROBERT CARROLL

P DxxA(0) = -1 ( c f . 51.8 and n o t e t h a t o u r s t a n d a r d s i n g u l a r o p e r a t o r s a r e excluded here).

F o r convenience t a k e a l s o A measures dv = d A / 2 a l c p l 2 and dc = 2X 2 d a / a l F I and 2.6);

thus ;(A)

= (f(x),rp!(x))

and

= 1 and s p e c t r a l p a i r i n g s v i a

!. ( c f . (1.8.43) F(A) = ( f ( x ) , x , ( xP ) ) .

and a l s o sQ.4 Now f o r m a l l y

EHHE0REI 2-11,

F o r s u i t a b l e f t h e e x p r e s s i o n (2.25) r e p r e s e n t s f o r m a l l y a n P g e n e r a l i z e d t r a n s l a t i o n w i t h PxU = P U, U(x,O) = ( f ( ~ ) a ( ~ ) , r p ~ ( x ) =) "A f ( x ) , P Y UY(X'O) = (f(A)c(x),xA(x))E = Cf(x). h,

Ptraoa: (Af)'(O)

C l e a r l y U i s symmetric w i t h U(0,y) = Af(y) and Ux(O,y) = Cf(y) w h i l e Cf(0).

The c o n d i t i o n t h a t A and C commute w i t h P ( c f . Theorem A A 2 P P cp )v = ( (pf)a,rpP)\, = A P f . But = Jcop A ( x ) A 2 i 0 P f ( x ) d x and t o o b t a i n P f = -A f we can s p e c i f y D ( P ) h e r e t o c o n s i s t o f func1.4.3)

=

6

r e q u i r e s P A f = (fa,-,

t i o n s f such t h a t W(p,f)10 = r p f ' - r p ' f (

0

= 0 (e.g.

f ' ( 0 ) = 0 and compact

Note however t h a t a s i m i l a r c o n d i t i o n f o r t h e x t r a n s -

s u p p o r t would do).

form r e q u i r e s t h e n f ( 0 ) = 0 a l s o .

Thus one wants e.g.

f o f compact s u p p o r t

w i t h f ( 0 ) = f ' ( 0 ) = 0 and such t h a t t h e v a r i o u s s p e c t r a l i n t e g r a l s i n v o l v i n g 2 2 a f , -A f, c f , and -A c f make sense. One c o u l d s i m i l a r l y t a k e Q o f t h e same t y p e and c o n s t r u c t f o r -

EMARK 2-12.

ma1 t r a n s m u t a t i o n s P (2.26)

~ ( X , Y )=

Then rp(x,O)

= Af(x),

-f

(

Q v i a formulas

P Q i(A)a(A)pA(x),rpA(Y)

-

(

P

F ( ~ ) c ( ~ ) x ~ ( x ) , x : ( y )) E

pY(x,O) = C f ( x ) , and Bf(y) = rp(0,y) = ( ? ( h ) a ( A ) , p AQ( y ) )

( o t h e r terms c o u l d a l s o be i n c l u d e d ) .

REmARK 2.13,

One can a l s o use c o n s t r u c t i o n s as i n 51.4 t o o b t a i n formulas

f o r t r a n s m u t a t i o n s determined by o p e r a t o r s A and C.

L e t us n o t e f i r s t t h a t

i f P and Q a r e o p e r a t o r s o f t h e f o r m D2 + ap(x)D + b p ( x ) f o r example one 2 2 can reduce t h e e q u a t i o n [D + apD + b ]p = [ D + a (y)D + b (y)]rp t o a form P Y Q Y Q q ( c f . a l s o Remark 1.9.6). Thus s e t t i n g a p i n v o l v i n g o n l y o p e r a t o r s D2 2 2 = exp[WX a p ( ~ ) d ~one ] has D [pap] = ap[D rp + a I& + % ( a,!, + +a2)q] and i f

-

B

0 ZP we w r i t e IL = a p ( x ) a Q ( y ) v w i t h qp = bp - %a; -+ap t h e n ( 0 0 ) [Dx - qp(x)]J' = 2 [ D - q (y)lJ,. F u r t h e r , boundary c o n d i t i o n s rp(x,O) = A f ( x ) and rp (x,O) = Y Q Y C f ( x ) become ( s i n c e a ( 0 ) = 1 and ~'(0) = %aO(O)), $(x,O) = a p ( x ) A f ( x ) =

Q

AQ

~1 ( x j k a (O)Af(x) + C f ( x ) ] . Thus Y QA A D - qQ (P -+ Q) and c l e a r l y i t s u f one i s l e d t o a t r a n s m u t a t i o n D2 - qp A 2 2 " 4 f i c e s t o t a k e P = D s i n c e from t r a n s m u t a t i o n s D Q and P D2 one o b t a i n s

z f ( x ) and

J,

(x,O)

= C f ( x ) where i\Cf(x) =

B

-f

-+

A

-f

A

a t r a n s m u t a t i o n P + Q. Now f o r

= D2

-+

= D2

-

q

Q

we have a Riemann f u n c t i o n as i n 51.4 and a

SPECTRAL P A I R I N G S f o r 11, o f

f o r m u l a (1.4.6)

w i t h f r e p l a c e d by i f and g b y c^f. Assuming

(00)

i f and ^cf a r e s u i t a b l y extended we have e.g.

and Rh(x,O,O,y) (2.27)

gf(y) = i f ( y ) + Af(

X)

- %J:*

^B

based on DX$ =

= $(x,O)

%joy ?f(x)[Kl(y,x)

I:K2 (Y

9

x

K2 (Y

+

4 One can transmute P =

2

( s e t t i n g R(x,O,O,y)

= K1(y,x)

K2(xYy))

=

E’NEORElil 2-14,

115

9

+ Kl(y,-x)ldx

-

Idx

-X

D2 i n t o

$

2

D2 -

=

qo v i a a t r a n s m u t a t i o n

[D - q ( y ) ] $ w i t h even extended i n i t i a l c o n d i t i o n s A^f(x) Y A9

^B

and $ (x,O) = C f ( x ) and Y

i s g i v e n by ( 2 . 2 7 ) .

There i s a n o t h e r approach t o s o l v i n q an e q u a t i o n o f t h e f o r m ( 0 0 ) o r s i m p l y 2 2 Px$ = Dx$ = Qy$ = ( D - q ( y ) ) $ which g i v e s more immediate i n f o r m a t i o n t h a t Y t h e method based on t h e Riemann f u n c t i o n ( c f . [L13] and e s p e c i a l l y [HoZ-41). Thus l e t

D 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),

w r i t e f o r h(x,y)

(x,y),

and (x+y,O)

and

given x+y-q jo jx-y+; ( 5 If( 5 n )dcdrl y

(2.28)

Hf(x,y)

=

41 h(S,ri)f(S,v)dcdn

=

4

3ri

Y

D (variations with c o u l d a l s o be e n v i s i o n e d here f o r c e r t a i n purposes). It y 2 2 i s e a s i l y seen t h a t [Dx - D ] H f = - h f w i t h Hf(x,O) = 0 and D Hf(x,O) = 0. Y Y Hence c o n s i d e r (2.29)

$(x,Y)

+ %(x+yCf($)d$ + HJ/(x,y)

[Af(x+y) + A f ( x - y ) ]

=

J

Ef(x,y)

+

=

x-y

H$(x,Y)

2 2 2 where h(x,y) = q ( y ) . It f o l l o w s t h a t [DZ - 0 ]$ = [Ox - D ]H$ = -q$ ( i . e . Y Y 2 - q ( y ) ] $ ) and $(x,O) = A f ( x ) w i t h Dy$(xyO) = C f ( x ) . F u r t h e r i t i s Dx$ = [D: shown i n [Ho2-41 ( c f . a l s o [C29])

t h a t t h e Neumann s e r i e s f o r ( I

-

H)-’

de-

f i n e s a c o n t i n u o u s map between v a r i o u s spaces. Thus w r i t e En f o r Cn func1 2 t i o n s i n R o r R w i t h t h e Schwartz t o p o l o g y o f u n i f o r m convergence o f D‘f

5 n ( c f . [C19; S j l ] ) .

on compact s e t s f o r t h e n H: Lyoc

Eo (resp. H: En

+

Lyoc (resp. En+’ + En+’)

CNEP)RER 2.15.

q ( y ) $ w i t h $(x,O)

=

t a t i o n B: D2

-

-

If h E

and ( I

a r e continuous.

-

H)-’:

E

As a t y p i c a l theorem we s t a t e

L e t n 2 2 and A: En

€n) w i t h q ( y ) = h(x,y)

= ( I

En+’)

-f

+

D2

H)-’Ef(O,y)

E

En-’.

+ Eny C : En + En-’ ( s o E f E En f o r f E 1 2 2 Then $ = (I- H ) - E f s a t i s f i e s Dx$ = D $

Y

A f ( x ) and $y(x,O)

=

q d e f i n e d b y $(O,y) =

1”0

HnEf(O,y);

-

The corresponding transmu-

Cf(x).

= B f ( y ) can be r e p r e s e n t e d as B f ( y )

Hy(0,y)

=

41’ q(o)/$: 0

y(c,n)dCdq.

116

ROBERT CARROLL

REWRK 2-16. I n terms of c l a s s i f y i n g " a l l " t r a n s m u t a t i o n s between say P = D2 and Q = D2 - q one t h i n k s e.g. o f t h e Bh i n Theorem 1.4.7 and Theorem 1.4.9

Q = ~ o s h y+ /Y Kh(y,x)Cos xdx ( h + a ) and t h e t r a n s m u t a t i o n w i t h (*) 'Ph,h(y) 0 Bm g i v e n v i a e,(y) Q = [Sinhy/h] + Jy K (y,x)[Sinhx/h]dx ( c f . (1.4.17) w i t h O Q = p Q ). A

plus

"0." e,(O)

iQ also that f o r h + 0

The boundary c o n d i t i o n s

( 0 ) = 1 and Dxv:,,(O) = h (h # m ) h,h -- 0 and Dxeh(0) 4 = 1 exhaust t h e p o s s i b l e domains determined by

boundary c o n d i t i o n s .

(Recall

m

r e a l l y t a k i n g Cos x + h[Sinhx/x]

-f

pX,h Q

t h e t r a n s m u t a t i o n Bh i s

s i n c e Kh c o n t a i n s a f a c t o r h).

On

t h e o t h e r hand i f we t h i n k o f Bh: Coshx -+ p @ t h e n we have an i n v e r t i b l e h,h t r a n s m u t a t i o n f o r any h f m w i t h k e r n e l Kh + 6 which mixes t h e boundary conditions.

However one has a d i f f e r e n t k i n d o f t r a n s m u t a t i o n s i t u a t i o n i f i t

i s r e q u i r e d t o m i x Coshx and

!e

and p h h.

o r [Sinhx/h]

1 L ( x , t ) p h4( t ) d c Coshx = p QX ( x ) + '

To see t h i s t a k e and L = Lo) and

f o r example (p: = 'pQ

A,O

0

integrate t o get

The k e r n e l A does n o t have t h e f o r m 6(x-c) + R ( x , t ) ;

i t i s a smoothing ker-

n e l ( c f . (2.8)) and i t s i n v e r s e w i l l i n v o l v e a d e r i v a t i v e . f o r m f o r a r e l a t e d i n v e r s e l o o k a t [Sinhx/x] = eh(x) 4 + 1' 0

and d i f f e r e n t i a t e t o g e t f o r m a l l y

(2.31)

Coshx = Dxeh(x) Q + Lm(x,x)ex(x) Q

+

c

-f

see what i s happening from l o o k i n g a t s p e c t r a l i n t e g r a l s . f o r (2.30) w i t h (2.32)

By:

[Sinxx/x]

+ pf

and

5-l:

L,(x,t)eh(t)dt Q

DxL,(x,E)eh(E)dS 4

(which s h o u l d i n v e r t t h e smoothing t r a n s m u t a t i o n Coshx =

To see a t y p i c a l

p!

0

+

Q Bh(y)).

One can

Thus f o r m a l l y

[Sinhx/h] 2

;(x,y)

= ( [ S i n h y / A ] , p ~ ( ~ ) ) ~ (du = ( 2 / 1 ~ ) hdh);

g(y,x)

=

( ~ pQ ~ ( x ) , [ S i n h y / h ] ) ~ ( d w = ( 2 / r ) d h + da)

2 = ( 2 / v ) J m [Sinhy/h] Thus t a k i n g e.g. P = Q = D (so do = 0 ) one has ;(x,y) 2 0 Coshxx dx = (2/v)lm hSinhyCoshxdh % ( 2 / ~ ) 6 ' ( x - y ) w h i l e g(y,x) = IT) 0

jOmCoshx[Sinhy/X]dh

= Y(y-x)

(= 0 f o r x > y).

RzmARK 2-17. Consider now t h e t r a n s m u t a t i o n s t h a t can be connected t o a g i v e n g e n e r a l i z e d t r a n s l a t i o n U o f t h e f o r m g i v e n e.g. Thus c o n s i d e r

i n Theorem 2.11 v i a (2.25).

SPECTRAL PAIRINGS

117

f o r some k e r n e l 5 which we t a k e f o r i l l u s t r a t i v e purposes t o be say ~ ( y , c ) P Q P = ( Qx(s),ipA(y)),. Since ~ ( 0 , s ) = ( R A ( c ) , l ) , = 6 ( t ) and B ~ ( O , E )= 0 one has p(x,O) = U(0,x) = Af(x) w h i l e ip (x,O) = 0. For BUf(y) = p ( 0 , y ) one has Y

(where B i s determined by

a).

I f we want t o d u p l i c a t e t h e i n i t i a l c o n d i t i o n

and C f ( x ) = p (x,O) we a d j o i n a t e r m based on x A Q ( y ) (assumed Y Thus s e t d i r e c t l y ( t a k e A p = 1 )

A f ( x ) = p(x,O) to exist). (2.35)

~(X,Y) = (

( n o t e e.g.

(

A P f(A)a(hbh(x),q:(Y)),

-

(

P Q F ( x ) c ( ~ ~ ~ ( x ) , x ~ ( ~ ) ) ~

P P o ( Y , E ) , u ( s , ~ ) ) = ( ;(x)a(Abh(x),( ~ ( y , s ) , i p ~ ( s ) ) ; ~= (fh(A)a(A) ). I t f o l l o w s t h a t again i p ( 0 , y ) = BUf(y) = (f(A)a(x),ip:(y))y

P px(x),q:(y)) AV = ( ~ ( y , t ) , ( fa,p;(S)),)

=

(B(y,t),Af(s))

=

BAf(y).

T h i s shows i n p a r t i c u l a r

t h a t t h e same t r a n s m u t a t i o n BU ( o r g e n e r a l i z e d t r a n s l a t i o n t r a c e U(0,y) i n ( 2 . 2 5 ) ) may a r i s e from two d i f f e r e n t Cauchy problems.

T h i s should come as

no s u r p r i s e s i n c e t h e wave e q u a t i o n p r o v i d e s a l r e a d y t h e example (2.36)

i

rn

U(x,y) = U1 + U2 = ( ~ / T T ) FCfCosAxCosxydx + 0

( Z / T T ) i m F s f Sinxx Sinxy c(A)dX/A 0

w i t h U(x,O) = f ( x ) and U (x,O) = ( 2 / v ) J m c(h)FSfSinxxdh = C f ( x ) ( t a k e a l s o Y 0 say f ' ( 0 ) = 0). But U(0,y) = f ( y ) i s u n a f f e c t e d by t h e U2 t e r m a l t o g e t h e r . I n any e v e n t t h e g e n e r a l i z e d t r a n s l a t i o n U(x,y)

i n i t s " e n t i r e t y " does de-

t e r m i n e t h e Cauchy problem and t h u s t h e r e i s some p o i n t i n a s k i n g f o r t r a n s m u t a t i o n s B: P + Q determined by g e n e r a l i z e d t r a n s l a t i o n s U f o r P.

In

making t h e q u e s t i o n more p r e c i s e however one does n o t seem t o o b t a i n an i n t e r e s t i n g development. 3.

&HE GENERAL EXCENDED CELFAND-CEDICAN EQUAEION,

T h i s e q u a t i o n arose i n

s t u d y i n g t h e i n v e r s e S t u r m - L i o u v i l l e problem and i n quantum s c a t t e r i n g theor y and we w i l l g i v e some p r a c t i c a l a p p l i c a t i o n s i n Chapter 3 o f v a r i o u s such

equations.

We were l e d t o s t u d y t h e G e l f a n d - L e v i t a n (G-L) and t h e Marzenko

(M) e q u a t i o n t o g e t h e r from a u n i f i e d t r a n s m u t a t i o n a l p o i n t o f view because o f Fadeev's t r e a t m e n t i n [ F a l l f o r t h e quantum mechanical s i t u a t i o n .

We

produced v a r i o u s general forms and r e f i n e m e n t s whose c o n t e n t w i l l be developed here and i n § 5 ( c f . [C31-33,43,47-49,53,66-73,75,78,80]). Thus l e t P Q and be o f t h e f o r m (1.9.1) w i t h s p h e r i c a l f u n c t i o n s i p x , i p h , J o s t s o l u -

8

6

118

tions L e t B:

ROBERT CARROLL P

^P

aQ s p e c t r a l p a i r i n g s A?A’ Q : q PA

+

+

qy

,

(

)v and

,

(

A

(2.2).

There a r e many p o s s i b l e t r a n s m u t a t i o n s P

Theorem 1.4.3)

(v

)W

Q

6, w t), e t c . 17~

Q P = (qA(y),RA(x)

be g i v e n v i a k e r n e l e(y,x)

A

+

>v as i n

Q o f course (see e.g.

d e f i n e d as

and we c o n s i d e r i n p a r t i c u l a r a t r a n s m u t a t i o n

follows.

One determines an a d j o i n t o p e r a t o r B# by t h e r u l e ( c f . [C38,401)

(3.1)

( A Q ( Y ) ~ ( Y ) , B L J ( Y ) )= (v(Y),Aq(Y)( 6 ( Y y x ) A p 1 ( x ) A , ( X ) , u ( X ) ) )

=

~ ~ p ~ x ~ u ~ x ~ , ~ ‘ ; ~ )x) , =Y( ~A , (vx )~u (Yx )~, ~ v ( x ) ) e

where (3.2)

= k e r B i s g i v e n by

Y(X,Y) = Ag(Y)Ap’ (X)B(Y,X)

We w r i t e here (3.3) (3.4)

UE@RE!J = (B-’)#

= ( I PPA ( X ) J $ Q ( Y ) )v

rv

B’

= % a n d a s i m i l a r c a c l u l a t i o n g i v e s 8’

( ~ ~ ( x ) u ( x ) , ~ v )( x= ) (

U,A+

=

where

y ( x , y ) ~ q l ~ Q , v ) ) = (aqv(y),( iY(y,x),u(x)

))

P 0 B(Y,X) = AQ1(Y)Ap(X)Y(X,Y) = ( . O A ( X ) A q Y ))W

N

P # For general s p e c t r a l p a i r i n g s RQ and R one can d e f i n e B = B # w i t h =: ?-’ = B . The k e r n e l s a r e q i v e n by (3.2) and (3.4) and

3.1.

e v i d e n t l y B f = (IpA(y),Pf Q

)W

=

Wf. can a l s o be c h a r a c t e r i z e d as a s o l u t i o n o f

Next we show ( c f . [C39]) t h a t

a Cauchy problem as i n Theorem 1.4.3.

Thus l e t TX be t h e s e n e r a l i z e d t r a n s -

5

A

l a t i o n a s s o c i a t e d w i t h P s a t i s f y i n g T X f ( c ) = f ( c ) and D T X f ( c ) = 0 a t x = 0. 5 X S Set q ( x , y ) = ( F ( y , c ) , T E f ( E ) ) ( c f . here Theorem 2.2). From (3.4) one checks A

f o r m a l l y t h a t P(Dx)q = G(D

)q

and D q(x,O)

f ( 5 ) ) = F f ( y ) w h i l e p(x,O) Y = (g(O,c),T;f(c;)) Y

=

0. =

C l e a r l y q(0,y)

=

(g(y,c),

( ( ~P( 5 ) , l ) u y T ~ f ( =~ )A)f ( x ) .

L e t us w r i t e A f as (3.5)

A f ( x ) = ( R Q y ( f $P , ( c ) y T ~ f ( c ) ) ) A P

= ( 0 ( E ) , T t f ( E ; ) ) ( c f . here t h e p r o o f o f Theorem 1.12.5)

Then c o n s i d e r $(x,x)

A A 2 P F o r m a l l y i t f o l l o w s t h a t P(Dx)p = -1 p w h i l e p ( 0 , h ) = ( , ~ ( ~ ) , f= (f (~ A )) ) 4 P T h e r e f o r e w i t h DxJ/(O,x) = 0 and consequently $ ( x , A ) = f ( A ) q A ( x ) .

(3.6)

Af(x) =

(

RQ,Pf(h)qA(x) P )A = (f(S),(

(A(x,C),f(C)

);

A(x,S)

=

R Q , ~P( s ) qPA ( x )A) ) =

Q P P ( R , f i A ( S ) ~ A ( x )A ) =

(

P P p A ( 5 ) ~ ~ A)W ( ~ )

GELFAND-LEV ITAN EQUATION

&HE@REIII 3 - 2 - The t r a n s m u t a t i o n

119

can be c h a r a c t e r i z e d as t h e s o l u t i o n o f a

A

Cauchy problem i ) ( D x ) q = Q ( D ) q y p(x,O) = A f ( x ) , and q (x,O) = 0 where t h e Y Y o p e r a t o r A i s determined by t h e k e r n e l ( 3 . 6 ) (and one extends A f ( x ) t o be even as needed).

PrrooB:

*P

It remains t o check t h a t A commutes w i t h A

4

A

Thus f o r s u i t a b l e f, P x A f = (PxA(x,c),f(c)>; A

A*

( c f . Theorem 1.4.3).

4

-

B u t P A = PEA f o r m a l l y so P A f = A p f .

A(x,c),f(<)).

X

A*

A

A P f = (A(x,<),P

5

f(c)) = (P

I n a s i m i l a r manner we can c h a r a c t e r i z e Fj v i a a Cauchy problem.

L e t S c be

A

t h e g e n e r a l i z e d t r a n s l a t i o n a s s o c i a t e d w i t h Q and w r i t e $(x,y) S;g(n))

DJ$(O,Y)

where

7 is

=

Y Ag(y) = ( ? ( o , n ) , ~ ~ g ( O ) )= 51

considers $ ( y , ~ ) =

Then f o r m a l l y Q(Dy)$ = P"(Dx)$ w i t h

(Y(x,n),g(11))

=

= i g ( y ) we have

P Q ( R ,(oA(n),s;g(n)

))A.

( QQA ( n ) , S Y g ( n ) ) and as above ( a f t e r ( 3 . 5 ) ) { ( D y ) T = 11

= 0, and ;(O,A)

<(A).

= Q ( x ) =

Consequently ;(y,x)

v

(3.7)

For $(O,y)

= Fg(x).

Q ( ( l , c A ( o ) )v,~ig(o) )

d

$'(O,A)

= (Y(X,~),

A

g i v e n by ( 3 . 2 ) .

0 and $(x,O)

=

5

P 4 Ag(Y) = ( R ,Qg(A)qA(Y)

)A =

(~(n),( R

P

=

One

-A'G Q z(~)q~(y and )

Q

Q

, p A ( l l ) q A () A~) ) =

" " ( A ( Y 9 n ) s g ( n ) ); A ( Y , ~ ) = ( R P , f lQA ( n ) q4A ( ~ ))A = ( ~ Ql ( n ) ~ qQx ( ~ ))u

&HEOREN 3-3- The t r a n s m u t a t i o n

=

z-'

can be c h a r a c t e r i z e d as t h e s o l u t i o n " $(O,y) = Ag(y), $x (0,y) = 0 where

o f a Cauchy problem ?(Dx)$ = :(Dy)$,

A'

i s determined by t h e k e r n e l (3.7) (extended t o be even as needed). "

The o p e r a t o r s A and A a r e i n f a c t v e r y i m p o r t a n t and t h e y a r i s e i n d e r i v i n g general G-L equations as f o l l o w s .

Thus s e t (*) q Qx ( x ) = ( 6 ( x , t ) , q p ( t ) )

and o p e r a t e on t h e s e w i t h RQq A P ( y ) and RQq,(x) Qx

q PA ( y ) = (y(y,t),q:(t))

with resp-

ectively t o obtain

Q

Q P

P

(3.8)

(

RQ,q;b)qA(x)

(3.9)

(

R Q Y ~ ! ( ~ Q) q A ( x = ) ) ( R Qq A Q (x),(y(Ylt).q~(t)

(v(Y,t).(

RQ,v:(x)qA(t) 4

Note here t h a t Q [ s ( x - t ) / A AQ(x).

(3.10)

= ( R q A ( ~ ) ,6(( x , t ) , p A ( t ) ) )

)

= ( y(Y,t),6(x-t)/AQ(x)

Q

( x ) ] = q Qp , ( t ) so

)

))

=

= y(Y,x)/AQ(x)

@pA(t) Q = (

RQ,q:(t)qA(x)) Q

=

6(x-t)/

I d e n t i f y i n g (3.8) and (3.9) we o b t a i n a general G-L e q u a t i o n (

B(x,t),
Since A(y,t)Apl(t)

= A(t,y)Apl(y)

)A)

=

V(Y3x)/AQ(x)

f r o m ( 3 . 6 ) we see t h a t = ( RQyq!(y)pA(t)) P

120

ROBERT CARROLL

(3.10) has t h e form

B(x,t),AF'(t)A(y,t))

(

= (

B(x,t),Apl(y)A(t,y))

y(y,x)/

=

I n t e r c h a n g i n g x and y i n o r d e r t o have t h e v a r i a b l e s i n t h e i r cus-

AQ(x). tomary p o s i t i o n s we can s t a t e ( c f . A

L e t B: P

CHE0REI;I 3.4,

(3.4))

A

0 be our 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 &PAP = q QA

-t

w i t h k e r n e l B and i n v e r s e B = B - l having k e r n e l y. (q;(x),v!(t)

)w =

A p ( t ) ( RQ,P!(x)qA(t) P

Define A(x,t)

Ap(t)Ap'(x)A(t,x)

)A =

A,(t)

=

(as i n ( 3 . 6 ) ) . N

Then t h e G-L e q u a t i o n a s s o c i a t e d w i t h B can be w r i t t e n BA = B o r (3.11)

(

B(y,t),A(t,x)) N

W

k e r B ( B = (B

where =;

= Ap(x)Agl (Y)y(x,Y)

=

~(Y,x)

-1 #

) ). -

#

Here e v e r y t h i n g i s d e f i n e d q u i t e g e n e r a l l y , B = 73 , e t c . and t h e s p e c t r a l p a i r i n g s a r e expressed i n terms o f general R P and R Q ( c f . a l s o Remark 5.18). We can w r i t e (3.11)as an e q u a t i o n i n s p h e r i c a l f u n c t i o n s , namely v i a

C0RP)LLARg 3-8, The general G-L e q u a t i o n (3.11) can be w r i t t e n as

P and R Pq,(x) ? L e t us observe t h a t i f we o p e r a t e i n ( * ) above w i t h R Pqp,(y) r e s p e c t i v e l y t h e a n a l y s i s above can be d u p l i c a t e d t o g i v e ( c f . (

R P . PP~ ( Y Q) ~ ~ ( =x )( )B(x,t),CR P , v PA ( Y h PA ( t ) ) > = B(x,y)/Ap(y)

P ( y ) ) = (y(y,t),( R P , q QA ( t ) q 4A ( x ) ) ) . vA (3.13)

-

= ( RP,q:(t)%(y)) Q

y(x,y)

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

= A ~ ( Y ) A (~x' ) ~ ( y , x ) .

and r e c a l l from (3.2) t h a t rJ

Assume t h e hypotheses o f Theorem 3.4 w i t h 8 =

CHE0REfl 3.6. kernel

as i n (3.2).

operator

A' ( i d e n t i c a l

Define i(t,y) = t o (3.7)).

4 ( RP,q:(t)SIA(y))

s i n c e BA = B we have a l s o Now assume R P W(x)du. (3.14)

Q

Q ,

#

having

"

which can be w r i t t e n as B = BA.

g = (BA)-l

Gdh and RQ

= B

as t h e k e r n e l o f an Y

= (y(x,t),A(t,y))

hl

2-l

Then one has a " r e v e r s e " G-L e q u a t i o n o f

U

the formy(x,y)

RP , q Q A(x)

x ( t y y ) = ( R P , q QA ( t b i0( Y ) )

);

Set now i ( t , y ) = A ( y ) X ( t , y )

Q

(

I n t e r c h a n g i n g x and y again we o b t a i n

= (y(x,t),i(t,y)

B(y,x)/Ap(x)

and

(3.8)-(3.9))

"

=

A-lB = BA o r BA = i - ' B

ZdA on [0,m)

Then

(B = i B A ) .

and w r i t e W ( A ) = ;/< so t h a t dw =

I n t h i s s i t u a t i o n we n o t e f i r s t t h a t f o r m a l l y ( c f . Remark 1.11.3) (%J;)(Y)

= (

F(~,x),v!(x))

= ((

P

11 ( X ) , ~ ' (11Y )

P )u,~A(x))=

GEL FAN 0- L EV ITA N EOUAT I ON

(i.e.

P

one w r i t e s Jm

P

t o express t h e a c t i o n W = I

Q (x)pX(x)dx = 6(A-u)/$(A) 0 1 J .

on s u i t a b l e o b j e c t s ) .

121

F u r t h e r i n t h i s s i t u a t i o n c o n s i d e r A(x,<)

i n (3.6)

f o r example and w r i t e OD

(3.15)

G(x) =

W(X)p!(x)du 0

P

P

Now r e c a l l t h a t T:f(x)

= ( P f ( h ) p X ( x ) , p A ( y ) ) so w i t h PG =

W ( A ) we o b t a i n

A(x,E) = Ap(c)T:$(6)

(3.16)

&H€P)R€R 3-7. For R4

Gdx and R P

%

w”

and (3.16) h o l d s w i t h

%

Gdx w i t h W ( X ) = :/$

one has Pq! = WpX4

g i v e n by (3.15).

L e t us n o t e a n o t h e r 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

REmARK 3.8,

based d i r e c t l y on t h e Parseval f o r m u l a (1.10.28)

= BA

( c f . here [CSO;

Du7,8]).

We w i l l d i s c u s s f a c t o r i z a t i o n aspects o f t h e G-L e q u a t i o n l a t e r a f t e r est a b l i s h i n g s u i t a b l e t r i a n a u l a r i t y o f t h e o p e r a t o r s B and

$=

qf, A

g.

L e t us w r i t e ,-A

Qg, and f o r s u i t a b l e f , g one has ( A f , g ) = ( A f,A g ) = ( R , f g )

=

Q

P

n

= ( f,g)w.

Q

Q

Now p: = &p! = ( B ( y , x ) , p X ( x ) ) and i n o r d e r t o i n t r o d u c e a d j o i n t ness e a s i l y t a k e RQ ‘L 2dA o r s i m p l y use t h e n o t a t i o n ( , )w c r e a t i v e l y . Then, (3.17)

A

(

A

f,g)W =

((

Q 9 P f,l),( g,.(l, ) ) = ( ( f(y),Aq(y)( B ( Y , X ) , P ~ ( X ) W

J

I

(g(nl.AQ(n)( B ( ~ , E )P, ~ ~ ( c ) =) ) ) ~ AQ(n)8(n.E)Ap1 (S)A(x,<)dEdxdYdq

=

I

)),

f(Y)Ag(Y)8(Y,x)9(n)

A Q ( y ) f ( y ) ~ B ( Y . x ) J A’;

(S)A(X,S)

# # Now from (3.1) d e f i n e 8 = B as i n d i c a t e d t h e r e and we have ( B g,fAp) = gAQTBf) = ( ( g(n)AQ(n),B(qY6) ) , f ( E ) ) = ( ( !$Q,!3(ny<)a,’ (E)),Apf) so B# g =

(

A’;

(5)( gAQ, B ( q , S )

(3.18)

)

(f,g )u =

and consequently

I

A Q f ( Y ) j B ( Y , X ) ~A(x,c)B

1

#

a(S)dSdxdy =

( A Q ~( Y) ) [ B A B # s I (Y )dY

# T h e r e f o r e from t h e Parseval f o r m u l a we o b t a i n I = BAB and hence t h e G-L e q u a t i o n BA = (B#)-’

R m R K 3.9p l u s e.g.

= (B-’)‘

= R# =

E.

L e t us show here how a c o n n e c t i o n formula pp,(y) Q = ( 8(y,x),p~(x)) a s p e c t r a l f o r m f o r R(Y,X) l e a d s t o i n t e r t w i n i n g v i a t h e G-L

122

ROBERT CARROLL

equation.

We r e c a l l t h e c o n s t r u c t i o n i n 51.4 where a k e r n e l Kh was con-

s t r u c t e d v i a a Riemann f u n c t i o n and shown t o transmute because o f t h e Cauchy Then i n a d d i t i o n i t f o l l o w e d

problem c h a r a c t e r i z a t i o n i n Theorem 1.4.3.

t h a t Kh s a t i s f i e d a Goursat problem as i n Theorem 1.4.9 and p r o v i d e d a conn e c t i o n between Coshx and

Here we s t a r t w i t h a c o n n e c t i o n and f o r

9h,h.

s i m p l i c i t y t a k e t h e model ( q r e a l ) Qu = u"

-

q ( x ) u w i t h B: D2

+

Q: Coshx

+

A

qQ A ( y ) and B(Y,X) = k e r B = 6 ( x - y ) + k ( y , x ) where K(y,x) = 0 f o r x > 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

<8,[1x-t(

yYB + y I B i s a

+ F l ~ l " = - T ( a + l ) S i n ( ~ n a ) ( y l - a - l where l y l - '

=

0

convenient n o t a t i o n .

lx+tl-2m-2

+

1.

D2

Q:

+

Qi as above i s g i v e n by A ( t , x )

=

L e t us r e c a l l some c a l c u l a t i o n s i n 52 i n o r d e r t o s i m p l i f y t h e e x p r e s s i o n and (x!,p(x)) = ( x +B, p ( - x ) ) Since Y = (l/F(B))x!-' = ( (x-C)+6-1 , p ( - c ) ) = r ( B ) [ Y B * $ 1 ( x ) where $'(XI =

o f the A(t,x) action. we have ( ( x - ~ ) ! - ' , p ( c ) ) rp(-x).

Thus

(

hand, s e t t i n g J/(x) = V

*

= r(B)[YB

(

(x+5)!-',9(5))

~ 3 +: &"I

+

we r e c a l l t h a t ;(x)

S i m"i l a r l y i f R =

PI.

(p

(

(x+E;)!-',p(~))

G).

On t h e o t h e r

= ( (<-x)!-'

we have R = -2m-2

(

~ ( 5 ) )

B- 1 (x+c)+ ,

.nv = r(a)[v, * ;I. Consequently ( I ~ + E ; I , V P ( E ) ) = r(-2m-1 I = r(-2m-1)~-~~ * -(p~ + s i n c e ( S * T)V = S v * i.Hence ( r e c a l l Z Z z - l r ( z ) r ( z + ) / J T and r ( z ) r ( - z ) = -n/zSinaz)

p ( - s ) ) so

r(2z) =

*

I x - ~ I - ~ ~ - ~ , p ( c= )r(-2m-1)Y-2m-l )

z)

LEnDU 3-13. With t h e s t i p u l a t i o n s above, f o r 2m f i n t e g e r we have (A(E,x), p ( 5 D = BmCY-2m-l * (v + a l ( x ) = Bm[151 -2m-2 * p ] ( x ) where am = BJ(-2m-1) 2m w i l l be a c o n v e n i e n t a b b r e v i a t i o n l a t e r ( t h u s am = r(-m)/2 2r(m+l) = -IT ~ s c m v / 2 ~ ~ (m+l). +~? For 2m = i n t e g e r one can f i n d formulas f o r t h e t e r m i n v o l v i n g Jm 0

Cosxydh i n [Bbel] f o r example t o use i n p l a c e o f (*) b u t we o m i t t h i s .

Now

l o o k i n g a t Lemma 3.13 we l e t A(5,x) a c t on 5 (y,E) i n t h e same manner ( i . e . Then ( t h e f a c t o r o f

~ ~ ( y . 5%) p p ( c ) ) . (3.23)

loy

6Q(y,c)A('yx)dc

Q

4 arises

= [am/r(-2m-1)1

[+am/r(-2m-l)l[151

-2m-2 *

from B (y,-S) = B (y,S))

lo

Q

4

Ix-51 -2m-2 B ~ ( Y ~ s =) ~ S

@Q(Y,S)l(X)

2 &HE@Rm 3-14, The extended G-L e q u a t i o n f o r t h e above s i t u a t i o n D -2m-2 can be expressed as (2m C i n t e g e r ) [ % a m / r ( - 2 r n - l ) 1 [ ~ E ~

*

+

Q,

=

Q

BQ(Y.5)1(X) =

(Y)yQ(xYY).

R E ~ A R K 3-15. Note t h a t i n Theorem 2 . 9 i f one w r i t e s o u t Y-m-r/,* = f(JS)/&

the

*

nm(BQf)(Jn)

a c t i o n i s on t h e v a r i a b l e 17 i n B ~ ( J Q , J Ewhereas ;) i n Theo-

rem 3.14 we work on 5 i n 6 ( y , ~ ) .

Q

Thus i n p a r t i c u l a r Theorem 3.14 i s d i f -

f e r e n t t h a t t h e i n v e r s i o n o f Bo. L e t us w r i t e o u t Theorem 3.14 u s i n g (2.15). +c-+yQ(

J C ,411 n-"'-+

Thus f i r s t we have f o r m a l l y

[ i n / r (m+l) 1( 5-17 ) - m-3/2/r(-m-+)

and hence ( s i n c e y (x,y)

Q

GELFAND-LEVITAN EQUATION

s h o u l d be even i n x - c f . ( 2 . 1 6 ) ) y,(x,y)

125

= [EJlrsqnx y2m+1x/r(m+l)r(-m-k)1

( x 2 - y 2 ) i m - 3 / 2 w h i l e B ( y , ~ )i s i n a s i m i l a r form from (2.8). 3.14 becomes f o r m a l l y Q (AQ = y 2m+l) x sgnx ( x 2 -y2 )+ -m-3/2 =

(3.24) where

?m

=

yp[ 1 5 1

-2m-2

*

2 2 m-4

(Y - 5 )+

Hence Theorem

l(x)Y-2m

-CscmTr( -rn-k)/22m+2r(m+g)r(-2m-l) = r ( m t 1 )/Jlrr(m+%).

C(DR0CCARg 3.16,

The extended G-L e q u a t i o n o f Theorem 3.14 can be w r i t t e n i n

t h e form (3.24)

(2m # i n t e g e r ) .

REmARK 3-17, I t i s p o s s i b l e t o p l a y an i n t e r e s t i n g oame w i t h F o u r i e r t r a n s forms o f d i s t r i b u t i o n s i n o r d e r t o "check" (3.24). R e f e r r i n g t o [Bbel] f o r 2 2 x 2A+l F ( y - x )+ = A r( X+l ) y (ylsl/ formulas we n o t e f i r s t t h a t (A # + l , t 2 , . . . ) 2 2 x x++(y I s I ) and FLsgnx ( x -y )+I = iJTr( x+l )sgns ( I s I / ~ Y ) - ~ - ' J-A-+ ( Y l S l ) . Thus under F o u r i e r t r a n s f o r m (y2-x2)y-' Jlrr(m++)y2m(y1s1/2)-mJm(yls I ) and -f

sgnx (x2-y2);m-3/2 ihr(-m-k)sgns( I~l/2y)~+'J,,~(ylsl). we have a l s o ~ 1 x 1 '= - 2 S i n ( + T x ) r ( x + l ) l s l - A - l . , ~ ~ 1x sgnx] x 1 = 2i~os(klrx)r(x+l)lsl-~-'sgns. -f

x = 2iIom( x 2 -y 2 )+ 1 Consider now f o r m a l l y ( c f . S e c t i o n 2 ) F[sgnx ( x 2-y 2 )+I 2 2 2 2 1 Sinxsdx = I and F[sgnx ( x -y )+] = 2Jm x ( x -y )+Cosxsdx = J. Then some 0

m a n i p u l a t i o n o f I and J f o r s u i t a b l e A c o n f i r m s (3.24)

(the d e t a i l s are

s p e l l e d o u t i n [C40]). L e t us c o n s i d e r t h e extended G-L e q u a t i o n (B

Q

(y,t),A(t,x))

l i g h t o f Lemma 3.13 where i t was seen t h a t when Q = c o u l d be expressed t h r o u g h c o n v o l u t i o n as i n (3.23). w i t h RQ

'L

*w Q dh"

B

and

Q:

D2

-f

6 we have A(t,x)

=

Q

(y,x)

Qi t h e A(t,x)

i n the

action 4

Thus f o r general Q

= Im CosAtCosxx; 0

Q

(A)dh.

Set

as i n (3.15) W(t) = ( 2 / n ) I o tl(A)CoshtdA ( r e c a l l W(x) = w " g ( h ) / ' ; p ( h ) = ( ~ / 2 ) A

"9 JO

(1)).

Then A(t,x)

u

W(x-E)q(c)dc =

If i(x+S)cp(c)dc

Ymn

=

+ i ( x + t ) ] / 2 w i t h G(-t) = W ( t ) and e v i d e n t l y i ( x + c b ( c ) d e , Inm i(x+c)cp(c)dt = i: i ( x - c b ( c ) d c , and i(x-S)qp(c)dc. Thus f o r q even ($' = c p ) one has

= [i(x-t)

i: /:

[i(x+E) + ~(x-E)]cp(c)dc =

/:

i(x-cb(S)dc =

[w" * q I ( x ) .

CHEOREUt 3-18, The extended G-L e q u a t i o n f o r D2 and general Q w i t h RQ A

2.

dwQ

can be w r i t t e n as

REWRK 3-19, Now suppose t h e

s p e c t r a l t h e o r y i n v o l v e s a measure R

P Q ,

cpdA

A

and t h e Q s p e c t r a l t h e o r y i s determined as i n 551.10 and 1.12 v i a a Parsev a l formula tA-'f

Q

,A-'q)

Q -

= (

R,QfQg

4 rem reads f ( y ) = (RV,qf(A)cpl(y))V

)

= (RV,QfQg )

x =

so t h a t t h e expansion theo-

( q f ( x ) , q AQ V( y ) ) u ( i . e .

R

2.

RQ).

We can

126

ROBERT CARROLL

then make some c a l c u l a t i o n s based on t h e procedure used i n d e r i v i n g t h e Parseval formula which seem t o have a c e r t a i n complementary i n t e r e s t i n conn e c t i o n w i t h G-L equations (see a l s o Remark 5.18 f o r another p o i n t o f view). Thus w r i t i n g R f o r R4 and R" as above and o m i t t i n g t h e v a r i o u s l i m i t i n g procedures i n v o l v i n g R i e t c . we r e c a l l t h a t f o r m a l l y : 6(x-y)/AQ(y) = Sy6 ( x ) = x Q ( R V , qQX ( x ) Q ~ X ( y ) ) v ; 6 ( x ) = ( R " , P4~ ( x ) ) " ; SSq(y) = ( R",P!(Y))~ = BR"; R" = PB6Q. (3.26)

4

Now c o n s i d e r t h e expansion f o r m u l a above and a p p l y

B t o get formally

Bf(x) = (R",qf(h)v~(x)),~ = B[R"Qf(A)]

Here we t h i n k of t h e t r a n s f o r m p extended t o ( s u i t a b l e ) d i s t r i b u t i o n s and Hence f o r m a l l y PRf = P R"Qf as d i s t r i b u t i o n s and we w r i t e t h i s o u t i n t h e form ( b ) (C,(x),(y(x,y),

t r e a t t h e a n a l y t i c f u n c t i o n Q f ( X ) as a m u l t i p l i e r .

f ( y ) ) ) = R V ( 4a X ( y ) , f ( y ) )so Rvff(y) = ( P X P (x),y(x,y)) o t h e r hand c o n s i d e r A(E,x) i n t h e form (3.27)

A(E,x)

P

P

P

= Px[y(x,y)I.

On t h e

P

= ( R,\oA(E)aX(x))x = ( R " , P ~ ( E ) Q ~ ( x ) )=~ V

V P

P

= P[R ~ z A ( x ) l h + E = A p ( x b m PA(s)lX+x

Then w r i t e (Px P

6 'L

P: x

-f

V Q

A)

(m)

R .O.,(y)

P

= RVA ( y ) [ b A ] ( y ) = RVA (y)( ~ ( y ,

Q

Q

= PX[a4(y) Equating ( b ) and ( m ) we o b t a i n ( a f t e r a c t i n g w i t h Apl(x)( B(Y,S),A(<,X) )]. P) AQ(y)A;(x)( ~ ( y , c ) , A ( t ; , x ) ) = y ( x , y ) which i s t h e G-L e q u a t i o n (3.11)

t ; ) , q i A ( E ) ) = Ag(Y)( B(Y,E)sR"q!(E))

( r e c a l l B(Y,X)

=

= AQ(Y)( B(Y,F;),PXAp1 ( x ) A ( E , x ) )

~~(x)~i'(y)y(x,y)).

QUANClun SCACCERZNC CHEP1Rg- I n o r d e r t o p r o v i d e some background f o r our d i s c u s s i o n of G-L and M equations we g i v e some i n f o r m a t i o n here from i n 4.

verse quantum s c a t t e r i n g t h e o r y ( c f . [Adl; Consider t h e o p e r a t o r Q = D2 i s f y i n g e.g.

Im x l q ( x ) ( d x < 0

-

-.

C40; Cel; D c l ; F a l ; Nel-4,7]).

q ( x ) w i t h unknown r e a l p o t e n t i a l q ( x ) s a t 2 S o l u t i o n s o f 9p = - k q s a t i s f y i n g q(0,k) =

0 and ~ ' ( 0 , k ) = 1 a r e c a l l e d r e g u l a r s o l u t i o n s and i n deference t o t h e t r a -

d i t i o n i n p h y s i c s we develop t h i s s e c t i o n around t h e r e g u l a r s o l u t i o n . (For our purposes i n §§3,5,etc.

i t i s more n a t u r a l t o use as b a s i c i n g r e d i -

e n t s f o r G-L and M equations t h e s p h e r i c a l f u n c t i o n s P = 9 ; s a t i s f y i n g (&I = Those s o l u t i o n s @+(x) = @(x,+k) o f - A 2P w i t h ~ Q ~ ( =0 1 ) and DxqA(0) Q = 0.) 2 @ = - k @ a s y m p t o t i c t o e x p ( + i k x ) as x + m ( w i t h @: 'L + i k e x p ( + i k x ) ) w i l l be -

c a l l e d J o s t s o l u t i o n s and one shows e a s i l y t h a t W(@+,@-) -2ik.

One c a l l s F ( k ) = @(O,k) = @+(O) -F(k) w i t h

=

a+@-' - @'a +- =

t h e J o s t f u n c t i o n and ~ J ( P , @ + )

=

QUANTUM SCATTERING THEORY

(4.1)

ip(x,k) = ( l / Z i k ) [ F ( - k ) @ ( x , k )

127

- F(k)@(x,-k)]

For t h e f r e e problem ( q = 0 ) e v i d e n t l y

ip %

[Sinkx/k],

@+

-

%

e x p ( + i k x ) , and

The t h e o r y r e l i e s h e a v i l y on v a r i o u s p r o p e r t i e s o f

which a r e d e d u c i b l e f r o m w r i t i n g t h e a p p r o p r i a t e i n t e g r a l equations and making F ( k ) = 1.

estimates.

We m e r e l y i n d i c a t e t h i s here and r e f e r t o e.g.

ip

and

@+

[Cell f o r details.

Thus

(4.3)

@(x,k) = e

ikx

+

One s o l v e s t h e s e by i t e r a t i v e procedures and i n p a r t i c u l a r

LEiItiRA 4.1,

ip

i s an e n t i r e a n a l y t i c f u n c t i o n o f k of e x p o n e n t i a l t y p e x.

@(x,k) i s a n a l y t i c i n k f o r Irnk > 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-<)q(c)v(c,k)d
0

One uses here Y ’ = 6 f o r Y t h e Heavyside f u n c t i o n ( Y ( x ) = 1 f o r x > 0 w i t h Y(x) = 0 f o r x < 0 ) and we know t h a t ( 2 / n ) f m CoskaCoskpdk = 6 ( a - @ ) . ble 0

think o f t

+

x ( 0 5 5 5 x, t 5 x ) and s e t J = ( 2 / n ) J m S i n k t S i n k ( x - < ) S i n k g

dk/k = ( l / ~ ) . f - [ S i n k ( x - c ) / k ] [ C o s k ( t - c ) 0

-

0

= 1/2 ( t -+ x ) and t h e

Cosk(t+c)]dk

f i r s t t e r m i n (4.14) becomes ( 1 / 2 ) f X q ( c ) d g (assume here q E L1 near 0

-

n o t , s u i t a b l e m o d i f i c a t i o n s o f t h e p r o o f can be p r o v i d e d as i n [ C e l l ) .

On

0

t h e o t h e r hand t h e second t e r m i n (4.14) i n v o l v e s ( 0 5 n 5 t

-f

x ) J = (2/v)fom SinktSink(x-<)[Sinkg/k]dk

k(t-c)

-

Cosk(t+c)]dk,

hence t h e i n t e g r a l

fX

0

=

(l/V)fm

0

<

if

5 x; t 5 x;

[Sink(x-c)/k][Cos

which can be shown t o be 0 almost everywhere and

fc

t e r m i n (4.14) vanishes as t

-f

x.

0

The o t h e r approach t o r e c o v e r i n g t h e p o t e n t i a l v i a t h e M e q u a t i o n goes as follows; we o n l y g i v e here a b r i e f s k e t c h ( f o l l o w i n g [ C e l l ) s i n c e a more det a i l e d v e r s i o n based d i r e c t l y on t r a n s m u t a t i o n t h e o r y w i l l be g i v e n l a t e r ( f o l l o w i n g [ F a l l ) and t h i s w i l l i n t u r n be g e n e r a l i z e d c o n s i d e r a b l y . one r e f e r s t o a theorem o f [Te2]

( c f . a l s o Chapter 1, 93)

EHE0REill 4-4, A necessary and s u f f i c i e n t c o n d i t i o n t h a t F ( x ) t h e l i m i t as y

+

E

0 o f a function F(z) analytic i n y > 0 w i t h

dx = O(exp(2Ay)) i s t h a t Now l o o k a t H(x,k)

=

/,fF ( x ) e x p ( - i t x ) d x

@(x,k)

-

Thus

L

2

(-m,m)

be

/I I F ( x + i y ) l 2

= 0 f o r x < -A.

e x p ( i k x ) i n (4.3)

w i t h ( 4 . 5 ) and one o b t a i n s

QUANTUM SCATTERING THEORY

@(x,k) = e ikx

(4.16)

[

t

131

V(x,t)eiktdt n

v a l i d f o r Imk > 0 where V ( x , t )

i s LL i n t f o r each 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)

QUANTUM SCATTERING THEORY

*

*

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-<);

Since e v i d e n t l y QT,*T,g

=

T*T [Sin^kx/ 9* 0 2 T9(-k TP g )

134

=

ROBERT CARROLL

* 2 TyToD g we see t h a t T>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 ) .

QUANTUM SCATTERING THEORY

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

<

= 0 f o r m a l l y tends t o E = % and n o t e t h a t we do n o t need t o know

t h a t ZE =

-4c i n t h e arguments below.

v a l f o r m u l a f o r F o u r i e r t r a n s f o r m s we have f o r 9 ip(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)

<

By s t a n d a r d theorems on a n a l y t i c c o n t i n u a t i o n t h i s

real.

and -0- a r e a n a l y t i c c o n t i n u a t i o n s o f each o t h e r and r e p r e s e n t a

holomorphic f u n c t i o n i n C which i s p o l y n o m i a l l y bounded ( i n d e p e n d e n t l y o f x for x

c > 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,<)d<

Assume s t a n d a r d hypotheses c - ’ ( - x ) a n a l y t i c f o r Imx > 0 and polynomially bounded f o r Imx L 0. Then Q-1 c ( - A ) E S ’ and E S ’ determined Q Q by (5.21) has s u p p o r t i n [ O , w ) . CERIMA 5-11,

*

Phoad:

W e r e f e r t o Remark 5.1 f o r s t a n d a r d hypotheses and t a k e lco(-h)I-’

ROBERT CARROLL

144

p(lX1) f o r Imh 0 where p i s a polynomial (e.4. p ( l h ] ) = k ( l + l h \ ) y ) , From (5.21) now we have

<

m

q Q ( x ) = ( 1 / 4 1 ~ )j:q1(-h)e

(5.23)

-ixxdh

-

But -ixx

so f o r x

we consider (5.23) as t h e l i m i t of contour i n t e g r a l s i n t h e halfplane Imh 0. For example approximate f i r s t a l a r g e semicircular contour C by a sequence C, w i t h base l i n e s n = 6151 so t h a t the polynomial growth of c - ' ( - x ) a t m i s controlled by n > 0 i n the exponent. 0 More rigorously set l / c ( - A ) = 4 s F i and work i n S' with the Parseval f o r = x(n-ic)

< 0

0

0

h)dh Now the i n t e g r a l makes sense f o r r e a l X by standard growth f e a t u r e s of d E S and i f we take ~p E D w i t h s u p p IP C [-R,-a] then f o r n = Imh > 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-<)Vp(.Y.")d
(x-S)f(x)dxdS

Consequently one has a s a c o r o l l a r y t o Lema 5.12 -

7EHEQ)RElIl 5.13.

A

BQ = VQ EQ. N

T h i s i s of course analogous t o U = E i n t h e quantum s i t u a t i o n .

Now l e t us

145

MARCENKO EQUATION -

consider B

A

Q

N

= V

i n c o n j u n c t i o n w i t h B Wx = B

TQ

4

A

(5.28)

B

= VQXQ

4

4

t o g e t f o r example

1

W-

- W-l

r

R e c a l l Wx = PW(X)P and P = Fc here 2 = ?(A) = I c Q ( X ) I /st and 1 As i n t h e d e r i v a t i o n o f Theorem 3.18 one has W- f ( y ) = < G y , x ) ,

L e t us examine t h e o p e r a t o r (Fourier cosine transform). iw

W = W-'.

f ( x ) ) = i x f ( y ) ; p(y,x)

4' We

(Z/IT)/;

=

=

4'

w i l l w r i t e W-'(h)

W r i t e as i n Theorem 3.18

r(h)CoshxCoshydX.

A

(5.29)

W(t) = ( 1 / 2 1 ~ ) A

N

A N

even i n t.

r

u

A

-

N

4

N

N

F o r W1 use W(x-y) t o o b t a i n FWlf =

W(y-x) and k e r W2 = W(x+y).

A

+ W2 where k e r W1 = W(x-y) =

Now go t o (5.28) and w r i t e W =

A

A

A,

Note here W(A) i s even i n X and W(t) i s

and t h e n W(y,x) = W(x+y) + W(x-y).

,z e x p ( i h y )

{ ( x - y ) f ( x ) d x d y = ~(ix)f~ f ( x ) e x p ( i h x ) d x = r ( h ) F f where we w i l l t h i n k o f

J:

f(x) = 0 for x t e n as

v

Jrn -w

U Q ( x - 5 ) f ( x ) d x = Uo

V

f ( g ( x ) = g ( - x ) ) and (5.21),

Q

fs

Q

stat-

V

2 ~ F - hwe have F g Q f = FLU * f ]

c ( h ) ) F f ( n o t e FG = v

*

V

i n g t h a t +/cQ(h) =

(5.30)

Now from (5.26) w r i t -

0 ( c f . [ F a l ; C40] f o r d i s c u s s i o n ) .

<

-Q f ( c ) =

= FU F f =

4

4

= 2 ~ F - l g ) . I n (5.28) now we m u l t i p l y b y '

QF f = to =4

(+/ get

d

BQ ZQ = VQ XQ W

"Q

V

where Z Q f = UQ * f, i . e . g Q f ( c ) =

iz JI(7 ( x - c ) f ( x ) d x

=

_:I 9 ( c - x ) f ( x ) d x . T h i s

Q

" i n t u i t i v e " step w i l l lead t o a formulation very close t o t h a t o f [ F a l l ( c f . [C40] and see e s p e c i a l l y 556-7 f o r an i n t r i n s i c v e r s i o n o f a l l t h i s ) . We have t h e r e f o r e F] = I V

s i n c e FX f = F[U n

Q

Q *- f ]

=

FZ F f = (%/c ( - x ) ) F f . Q

A

4

v

4

Y

and t h e n f r o m above we o b t a i n

Q 2

r

N

(5.32) N

(W

= W-')

V

XQ W Z Q f = f

+

W ( - x - y ) f ( x ) d x = [W 4 -

*

f ) v = FW F f =

Fi

V

f ] (y)

= %i)F?.

= F

v

f = F-l[Zc v

sQ(-A)Ff.

(-h)]F(ZQf) Q

Hence

v

(-A)lFf Q-1 = F [2

T h e r e f o r e we have shown

F-ls ( - x ) F ~

Q

where s ( - A ) o r s,,(x) i s a " s c a t t e r i n g " term.

Q

Ff

*

= F - ' [ + ~ ( X ) / c p ( h ) l F ~ = F-l[Zc 'VV

f = 7 W 4 1 4

c Q (-h)][GQf]-= F-'[cQ(-h)/cQ(h)]Ff

A

4

t h a t W(t) i s even, W2f(y) = J W(x+y)f(x)dx = where f ( t ) = f ( - t ) . Hence FW2f = F(W cv 1 we have f i r s t XQW2f = F- [+/c (?,)]F&f

On t h e o t h e r hand, r e c a l l inq

(1/21~)1: s Q ( h ) e x p ( i h t ) d h so ' t h a t s Q ( - h ) =

Lz

Now s e t s ( t ) =

S ( t ) e x p ( i h t ) d t = FB; t h e n

146

ROBERT CARROLL

(5.33) (cf.

F - l s (-A)Fr = F - l F b

4

[Fal; C401

-

w r i t i n g B f ( y ) = :J 4

N

m

V

B(y+x)f(x)dx

f] = 0

one t h i n k s o f f d e f i n e d o n l y on L O , - ) ) .

Consequently,

B ( y + x ) f ( x ) d x , we have ( c f . ( 5 . 3 0 ) )

The e q u a t i o n BQ =

&HE@REIII 5.14.

*

;gQW - l

o f (5.28) becomes (5.30) o r B v

V

VQ(EQW kQ), which i n t u r n can be w r i t t e n as B

T h i s formula w i l l produce a v e r s i o n o f t h e

4

$Q= V Q I I

+

$1.

V

$Q=

M e q u a t i o n which i s q u i t e p a r a l -

l e l t o t h e quantum s i t u a t i o n o f [ F a l l as o u t l i n e d i n 54; t h e s c a t t e r i n g t e r m $3 a r i s e s i n much t h e same manner.

We c o n s i d e r i n t h a t d i r e c t i o n t h e

*Q”( c - x ) f ( x )

V

k e r n e l s i n Theorem 5.14 and n o t e f i r s t BQZQf(y) = dx) =

JJ

BQ(y,e)[i q Q ( S - x ) f ( x ) d x d c .

whereAKQ(yyx) = :J

oQ(y,i)*Q(c-x)dc

(

B (y,c),[z

Q

Consequently we have k e r B g Q = KQ and K (y,x) = 0 f o r y < x.

Q

v Q I I + S]f(y)

On t h e o t h e r

+ 1 ; f(x)J;

= J; VQ(y,c)[f(c) + :1 ti(S+x)f(x)dx]dS = Jm (y,x)f(x)dx Y Q iQ(y,c)$(c+x)dcdx. Hence f r o m Theorem 5.14 f o r y < x

while f o r y

>

hand

x

&HE@REm 5-15, The M e q u a t i o n a s s o c i a t e d w i t h Theorem 5.14 can be w r i t t e n as (5.34) w i t h (5.35) as a complement. n

I n comparing w i t h (4.31) o r (4.19) we

n o t e t h a t V ( y , ~ ) = 6 ( y - c ) + V (y,c) i n t h a t s i t u a t i o n .

Q

Q

REmARK 5-16, The data c ( A ) / c (-1) = s ( A ) e n t e r i n g Theorems 5.14 - 5.15 Q Q Q a p p a r e n t l y d i f f e r s from t h e data e n t e r i n g t h e M e q u a t i o n o f 54, namely S ( k ) = F(-k)/F(k).

We n o t e however t h a t g i v e n an o p e r a t o r D2

-

q f o r example

where e v e r y t h i n g makes sense one o b t a i n s by c a l c u l a t i o n w i t h Wronskians c Q ( ~ ) F ( A )+ cQ(-A)F(-A) = 1.

Hence s ( A )

4

+

s(x)

= l/cQ(-A)F(A).

Given

standard p r o p e r t i e s as i n Remark 5.1 and no bound s t a t e s l / c (-A)F(A) i s ana l y t i c f o r I m x > 0 and s u i t a b l y bounded t h e r e so t h a t V o ( t ) exp(iAt)dA = -(1/21~)i; sQ(A)exp(iht)dx = -S(t). 4

be t h e same as (5.34) w i t h V

Q

=

V (and VQ = 6

+

Q

= (1/21~)J1 S ( A )

Hence (4.19) w i l l a c t u a l l y Vo).

RElURK 5.17. The use o f t h e F o u r i e r technique involv.i,ng e x p ( t i x x ) on (-m,m) i s v e r y h e l p f u l h e r e i n e x p e d i t i n g c a l c u l a t i o n and producing a f o r m u l a such as (5.34)

( c f . a l s o [DuS]).

However i t s v e r y u s e f u l n e s s tends t o obscure

t h e i n t r i n s i c n a t u r e o f t h e u n d e r l y i n g t r a n s m u t a t i o n a l s t r u c t u r e and we w i l l

FOURIER TYPE OPERATORS

147

see i n t h e n e x t s e c t i o n how t o develop t h e m a t t e r c a n o n i c a l l y .

REMARK 5.18,

L e t us n o t e here another way t o l o o k a t t h e g e n e r a l i z e d G-L

equation g(y,x) =

(

~ ( y , c ) , A ( ~ , x ) ) which proves t h e a p p r o p r i a t e t r i a n g u l a r i t y

v i a t h e t h e o r y of h y p e r b o l i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s r a t h e r t h a n v i a Paley-Wiener ideas o r c o n t o u r i n t e g r a t i o n i n t h e s p e c t r a l v a r i a b l e .

The

method i s r e m i n i s c e n t of a s p e c i a l case encountered i n geophysical i n v e r s e problems ( c f . Chapter 3, 558-9). has z ( y , x ) =

(

Ifwe use (3.16)

G-L e q u a t i o n one

~ ( y , c ) , T ; i ( ~ ) ) A ~ ( x )which reminds one o f formulas d e r i v e d i n

5 5 2 - 3 t o s o l v e Cauchy problems o f t h e form

f ( y ) , px(O,y) = 0. ( 9 AP ( x ) , p AQ ( y ) ) u

i n the

=

(a)

P(Dx)p = Q ( D

lip,

y-

p(0,y) =

I n t h e p r e s e n t s i t u a t i o n w r i t e p ( x , y ) = B(y,x)/Ap(x) Then f o r m a l l y p s a t i s f i e s

y(x,y)/A,(y).

(0)

=

w i t h 9(x,y) =

( ~ ( Y , E ) , T ~ ~ ( E so ) ) t h a t P ~ ( O , Y ) = 0 and ~ ( O , Y ) = ( 1 , ~Q ~ A y )= ) 6~( y ) / A g ( y ) .

5

CH€OREIII 5-19. X(y,x)/A,(x)

= ip(x,y)

can be t h o u g h t o f as t h e impulse r e -

sponse t o an i n p u t p ( 0 , y ) = 6 ( y ) / A g ( y ) and b y domain o f dependence argurv

ments ~ ( y , x ) = 0 f o r x

(6(c),Tii(5))

= w"(x).

P m ~ u l ; : Since

(a)

<

y.

=

i s h y p e r b o l i c one can draw a domain o f i n f l u e n c e h X

Thus ~ p ( x , y ) = 0 f o r y > x. 6.

The " r e a d o u t " p ( x , O ) i s ( ~ ( O , c ) , T i g ( , ) )

The l a s t statement f o l l o w s as i n 5 2 .

&HE mARCENK0 EQ11ACION FOR FOURIER CyPE OPERACOG,

.

We go now t o t h e gen-

M e q u a t i o n as developed i n [C40,47-49,801. We w i l l u s u a l l y use t h e symbol Pu = u " - q ( x ) u t o denote a F o u r i e r t y p e o p e r a t o r as

e r a l i n t r i n s i c canonical

A

A

i n Chapter 1, 58 w h i l e 0 r e f e r s t o a " s t a n d a r d " o p e r a t o r w i t h p r o p e r t i e s as A

d e s c r i b e d i n Remark 5.1 f o r example ( a l s o Q o f t h e form u:

= (Au')'/A

-

qu

as i n Chapter 1, 56 o r Chapter 3, s8 a r e a p p r o p r i a t e and w i l l be r e f e r r e d t o A

as s t a n d a r d ) .

We assume throughout however t h a t r) i s a b s o l u t e l y c o n t i n u o u s

i n t h e sense t h a t R Q

A

A

%

dwQ = wQ(h)dh on [0,m).

We w i l l f i r s t d e r i v e a Q

v e r s i o n o f a g e n e r a l i z e d Kontoroviz-Lebedev ( K - L ) t e c h n i q u e (developed i n

-

2

2

f o r o p e r a t o r s o f t h e form c u = x u " + 2xu' + X~[I,* - G(x)]u = (I, T h i s enables one t o g i v e a %)u w i t h s p e c t r a l v a r i a b l e v - cf. 558-9).

[C43,45]

new f o r m u l a t i o n o f t h e b a s i c c o n n e c t i o n formula

%(

=

$ f r o m Theorem 5.10

and suggests some i n g r e d i e n t s f o r e x t e n d i n g t h e G-L and M c o n n e c t i o n o f 55

ROBERT CARROLL

148 A

(for P

i:

=

+@!

A 2 D ) t o a l a r g e r c l a s s of P.

F i r s t one describes a transmutation

and then f o r Fourier type operators

( c f . Chapter 1 , e8) we de-

velop and e x p l o i t t h e analogues o f t h e c l a s s i c a l transforms and e s t a b l i s h some new technique using ideas of generalzied t r a n s l a t i o n and convolution. Once t h i s machinery i s i n place t h e general idea of the G - L and M connection follows t h e l i n e s s e t f o r t h i n 55 and i n f a c t t h e procedure i s somewhat i m proved and more " i n t r i n s i c " here. Thus t h e idea of the M equation i s t o rev via B. One obtains f i r s t 'ii'=kH and then t h e general extended l a t e B and

BX" =

v

N N

% = K*, K , and a r e described i n t h e t e x t . We remark a l s o t h a t t h i s same type of M equation a r i s e s n a t u r a l l y a l s o i n a d i f f e r e n t context where one c h a r a c t e r i z e s transmutations by minimizing pro-

M equation i s

B ( X W ) where

cedures ( c f . 97) and thus we f e e l t h a t i t s " i n t r i n s i c " nature i s well established ( s e e a l s o Chapter 3 f o r connections o f t h e general machinery t o l i n e a r s t o c h a s t i c e s t i m a t i o n ) . We will a l s o develop t h e complementary r o l e of t h e G-L (resp. M ) equations as 1 ower-upper ( r e s p . upper-1 ower) f a c t o r i z a t i o n s of G-L (resp. M) data. We would a l s o l i k e t o suggest t h a t t h e "transA form calculus" of Fourier type P and t h e techniques i l l u s t r a t e d by our procedures a r e of some independent i n t e r e s t and of potential use i n various contexts. Let us t a k e t h i s opportunity t o note t h a t t h e presentation of A Theorem 2.4 f o r Fourier type P in [Hol] ( f o r Imx 2 0 ) i s somewhat misleading and our hasty reading of t h e s i t u a t i o n led t o a discussion of t h e s e ope r a t o r s a s a n example i n [C40, Chapter 2, Remark 10.12land in [C343 which was badly phrased ( a l s o the i l l u s t r a t i v e material based on this example of A Fourier type P i n [C40, Chapter 3, p p . 325-3291 and in [C68, pp. 54-57]

must be appropriately modified).

W e have accordingly c l a r i f i e d such materi a l here i n the t e x t and have indicated t h e c o r r e c t version. There is a l s o a typographical e r r o r i n [C48] in t h e expression f o r the kernel g ( y , x ) , while t h e c o r r e c t i o n p 2 + p p - on p. 50 o f [C47] was incorporated in CC801 ( i n t h e c o e f f i c i e n t of ~ ( A + uin) l i n e -2 of [C47], p. 50); these matters a r e developed c o r r e c t l y here. A

For general Q a s indicated we now f i r s t r e c a l l t h e inversion formulas

1

m

(6.1)

Qf(A) =

;(A)=

0

f ( x ) a4 A ( x ) d x ;N f ( x ) =

2

lo m

;(x)q;(x)wnO(x)dh A

where G (A)= 1 / 2 v \ c (A)[ ( E ( A ) = c ( - a ) f o r A r e a l ) . Similarly f o r P of Q 4 4 Q 2 Fourier type one h a s transforms p a n d = w i t h dv = d v p = G p ( x ) d h = dx/2nlcpl . We r e c a l l a l s o t h a t B: p Px -+ q y and ,B:- qP x Ic /c I 2 p xQ have kernels ~ ( y , x )= P O -f

FOURIER TYPE OPERATORS

149

N

(<(x),v!(Y) a l l y as

ii

and i s c h a r a c t e r i z e d more generB(Y,x) = (((x) ,P;(Y) -1 # # = (B = B v i a a formula ($(y)f(y),Cju(y)) = (+(x)u(x),Bv(x) )";

We can w r i t e f o r t h e k e r n e l s o f %(y)$'(x)&',x)

and ?x,Y)

3

= B - l and

=

= %(y)~$,~(x)~(y,x).

E-'

t h e formulas y(x,y)

).

=

One r e c a l l s t h a t by Paley-

Wiener arguments i n general ~ ( y , x ) = 0 f o r x > y and z(y,x)

=

0 f o r y > x.

P = qA. Q Now F i n a l l y we d e f i n e q PA ( x ) = a PA ( x ) / c p ( - ~ ) as b e f o r e and r e c a l l -B'kA A

we g i v e a Q v e r s i o n o f some g e n e r a l i z e d K-L t e c h n i q u e developed i n [C43-461 for

t y p e o p e r a t o r s ( c f . Chapter 1, 57 and Chapter 2, 598-9).

Thus f i r s t A

we can r e w r i t e t h e i n v e r s i o n f o r m u l a ( 6 . 1 ) as f o l l o w s . and

o^Q ( A )

=

1 / 2 T I c (A)I

Q

2

Q

Since f ( A ) , ,ph(x),

a r e even i n A one has

q - 4 i n v e r s i o n . Next one can g i v e f o r Q as f o l l o w s . Thus combining (6.1) m a l l y a new p r o o f o f t h e f o r m u l a -B Pq A = qA which g i v e s a u s e f u l f o r m o f t h e

and (6.2) one has ( w i t h obvious n o t a t i o n )

U

A

Consequently one can w r i t e ( w i t h a c t i o n i n 1~ an s u i t a b l e even f u n c t i o n s f )

U

Now go t o t h e k e r n e l expressions and s e t T ( y , x ) (y).

P

Q

= (flA(x),qX(y)

)w =

P Z?[nA(x)]

Then u s i n g (6.2)

( t h i s can a l s o be d e r i v e d d i r e c t l y as i n ( 6 . 2 ) ) . Now m u l t i p l y (6.5) by P ( x ) and i n t e g r a t e , u s i n g (6.4) f o r m a l l y ( i . e . assume t h e s ( x - p ) a c t i o n on

*u p

i n (6.4) extends t o be ~ ( X - U ) a c t i o n i n

jm*;( Y 16(

x

Q w o r k i n g on qA). Then

A-lJ

-m

ZHZ0REm 6-1. The i n v e r s i o n (6.1 (modulo i n t e r p e r t a t i o n o f (6.4)

can be w r i t t e n i n K-L f o r m as i n (6.2) and t h e r e l a t i o n E[qA] P = qA Q follows formally.

ROBERT CARROLL

150

A

n

REIIIARK 6 - 2 - Let T ( X , p ) be a s i n ( 6 . 4 ) with Q = P and observe t h a t K-L inverP P sion says ( * ) 6(x-y) = ( 1 / 2 a ) i I *l(x)f?X(y)dA a c t i n g on some f a i r l y general

Now consider T ( X , v ) a c t i n g i n A on

c l a s s of f ( c f . however [JbZ]).

(-m,m).

P

Formally f o r any function say F(X) = 1 ; f ( x ) q X ( x ) d xone would have (using

fr

(*I), Lz

P

P

P

F(x)T(l,v)dx = f ( x ) d x J r * p ( ~ ) [ ( 1 / 2 n ) i z * X ( x ) v x ( ~ ) d l l =d ~10" f ( x ) P ( x ) d x = F ( v ) so T ( A , v ) a c t s as 6 ( x - v ) on such functions (note a l s o < F ( - A ) , v T(A,u)) = F ( p ) formally here b u t t h i s will be excluded below). In particu-

*

l a r such functions F ( X ) a r e a n a l y t i c f o r ImX > 0 and i n s o f a r a s they could a l s o be represented via F ( x ) = 1 ; g ( y ) qQX ( y ) d ywe could say t h a t T ( X , p ) a c t s as ~ ( X - U ) on q QX ( y ) . Another approach here (which will show t h a t we do want F ( h ) a n a l y t i c f o r ImX > 0 and which excludes F ( - A ) )

i s t o r e c a l l t h e analyn P

t i c continuation argument of Theorem 5.10. Thus ( ~ ( y y x ) , u p X ( x =) )&:(y) 4 Q P P so *,(y) + *-x(y) = ( B ( Y , x ) , * ~ ( x ) + Q - ~ ( X ) ) = * ( y , ~ )+ *(y,-x). One w r i t e s

4

- *(y,X)

Q

-0- = -[*&(y) - *(y,-X)] and by standard continuat i o n arguments ( p l u s some estimates on polynomials) one obtains q QX ( y ) =

0, = *,(y)

=

* ( y , l ) . This means t h a t T ( h , p ) a c t s as s(A-1~)( % I S ~ ( X - U ) = two sided 6 4 P function) when a c t i n g on *A(y) over ( - m y - ) ( s i n c e 1 ; F(y,x)*p(x)dx = One wants t o d i s if *:(y)[(l/Zn)Jr q ~ ( x ) * ~ ( x ) d x ] d h= E i *rh(y)T(h,v)dA). Q Note t i n g u i s h one and two sided d e l t a functions as in [C40,67,68,71,80]. in t h i s regard t h a t ( 2 / ~ ) 1 : CosAxCosvxdx = A = 6 , ( A - v ) b u t t h i s must be w r i t t e n as A = (l/n)[: CosAxCospxdx = (1/21~)!1 [exp(ix(A+p))+ e x p ( i x ( x - ~ ~ ) ) ] dx = 62(A+u) + 6 2 ( h - p ) f o r f u l l l i n e action. Generally ( 2 / a ) f t Coslxdx = 6 1 ( ~ =) ( 1 / 2 ~ l ) l z[exp(iXx) + exp(-iix)]dx = ( l / r ) L I exp(iAx)dx = 2 6 2 ( X ) and a f a c t o r of 2 must be adjusted i n c e r t a i n formulas. In t h e above s i t u a t i o n T ( x , ~ i) s already spread

REmARK 6.3,

o u t over

(-m,m)

so we obtain

When dealing w i t h F a c t i o n i n a f u l l l i n e theory we were a b l e

i n [C47,48] t o s t i p u l a t e t h a t F(y,x) ~ ( y , x )a t s u i t a b l e times w i t h

=

0 for

-m

<

x

<

y by i d e n t i f y i n g

rv

*

(see Section 1 . 8 f o r n o t a t i o n ) . Thus take now P of Fourier type and then n i n f a c t ( 6 . 7 ) i s t h e natural formula t o use in t h e f u l l l i n e theory w i t h P Q P ; qx(y)qX(x)dw = of Fourier type. Indeed w r i t i n g a s in [C80] (*) ?(y,x) = 1 ( 1 / 2 ~ ) QQ(yb!(x)dh l~ = (1/21~)jI PM1*:(y)[z!(x) + @PX ( ~ ) l d h we , can s e t ( = )

*

= 0 in (a)f o r x+y > 0 since i n the upper half plane 1pM 0 (y)aX(x)dx h p -m 4l A p ~ M ~ * ~ ( y ) a ~i s( xa n) a l y t i c and bounded by polynomials i n X times a term exp[ - Imx (x+y)] under standard hypotheses ( c f . Lemna 5.9 and Theorem 5.10

FOUR1 ER TYPE OPERATORS

151

A

and n o t e a l s o y

0 here because i t comes f r o m Q ) .

Moreover again t h e

a c t i o n ( K ( y , x ) , q XP( x ) ) = q QX ( y ) f o l l o w s immediately from (6.7) i n t h e f u l l l i n e t h e o r y and we have

&HE@RZrn 6.4, The n a t u r a l formula f o r i n the f u l l l i n e theory w i t h Fourier t y p e *P i s ( 6 . 7 ) ( f u l l l i n e a c t i o n i n x w i t h y 2 0) and [ k ! ( x ) ] ( y ) = qQ X(y) i s a consequence t h e n o f t h e i n v e r s i o n t h e o r y f o r P on

(-m,m).

A

P X U U ~ : Standard p r o p e r t i e s f o r t h e Q q u a n t i t i e s a r e i n d i c a t e d i n Remark 4

5.1 f o r example and p r o p e r t i e s o f

p,

d i c a t e d i n Theorem 1.8.12.

etc. f o r the Fourier type P are i n -

MIL p

Q

The r e s u l t €W, = qIxf o l l o w s from t h e i n v e r s i o n P theorem 1.8.13 as rephrased below. Thus f r o m (6.7) ( = ) (;(~,X),*~(X)) = Q P P P ( 1 / 2 n ) l I pM1qX(y)(ZX(x),YI (x))dh. But from Theorem 1.8.13 w i t h u1 = and u P P P u2 = EX one has ( f o r s u i t a b l e f , F ) f ( x ) = ( 1 / 2 1 ~ ) i zp Z X ( x ) I g f ( y ) @ X ( y ) d y d x P P and F(X) = @ X ( x ) ( 1 / 2 n ) i 1 p F ( u ) Z (x)dudx so t h a t f o r m a l l y

[I

(6.8)

1-1

(1/2n) l)!(x)Zu(x)dx P

= ~(X-U)/P;

P Hence i n ( = ) ( l / 2 n ) ( XPA ( x ) y ~ u ( x ) =) (1/M1)6(h-u)/p

and

n o t e a l s o t h a t (6.7) g i v e s F(y,x) = 0 f o r y > x >

-m

&:

= q!

follows.

We

d i r e c t l y (a c o n t o u r i n -

t e g r a l argument i n t h e upper h a l f p l a n e y i e l d s t h i s s i n c e M1p/c ( - A ) i s a n a l y t i c and say p o l y n o m i a l l y bounded w h i l e @.,(y) 4 P X,(x) plr [l + n ( x , x ) / x ] e x p ( - i x x ) - c f . [Hol]).

Q

%

a i Y ( y ) e x p ( i X y ) and

A

LElnMA 6-5.

For Fourier type P

J

m P P ( 1 ~ 7 1 ) zX(x)zu(x)dx = (1/2n)

(6.9)

(A/~)@:(x)ldx = (l/pp-)6(h+u)

Prraod:

1

P zX(x)[(l/p)@-p(x)

-

2 (A/P ) 6 ( X - u )

-P

-

m

40

I t i s t h i s f o r m u l a which caused t h e m i n o r computational problem men-

To check t h e f o r m u l a t i o n e d e a r l i e r i n [C47] ( p was used i n s t e a d o f p - ) . P P P f(x)@-,(x)dX = f ( - h ) a X ( x ) d X and hence (~,G!IT)I~ f ( A ) @ , _ X ( x ) f(-X)s(X-u)dX/p = f ( - u ) / p ( i . e . ( l l 2 n ) J I @,_X(x)Xu(x)dx P -mP =

/I/I

note t h a t dh,ZV(x) P =

( l / P - 1 6 (A+u).

/I

.

Note t h a t a K-L t y p e i n v e r s i o n can a l s o be e s t a b l i s h e d from t h e 2 2 t y p e i n v e r s i o n o f 1.8.43. Thus s e t t i n g dE = :dX w i t h 2 = 2 1 dx/.irlF(X)l

R€RARK 6.6,

x!

y(X)

It f o l l o w s t h a t , one has f ( x ) = J; f"(i)x,(x)dB P with = :I f ( x ) x XP ( x ) d x . P P P f o r Z!(x) = a A ( x ) / d p ( - X ) , where M ( h ) = ( 1 / 2 i x ) F ( h ) = dp(-X) so x A = dpQX +

152

ROBERT CARROLL

P

+ d p ( - A ) D A , we have

1

m

(6.10)

f(x) = (1/2~)

F((x)E!(x)dX

m

= ker

We r e c a l l n e x t t h a t i n [C47] ;(y,x)

i , g:

-+

was w r i t t e n as

@ ,;

( r e c a l l dw = GdA = dX/2nlcQ12 here f o r $ )

and t h e subsequent c a l c u l a t i o n s were c a r r i e d o u t i n a manner c o n s i s t e n t w i t h i(y,x)

= 0 for

-m

2 0 always when r e f e r r i n g t o $). The

< x < y (note y

formula (6.11 ) does n o t a c c u r a t e l y r e f l e c t t h i s s i t u a t i o n however f o r l a r g e n e g a t i v e x and i t i s b e t t e r t o w r i t e i n t h e f u l l l i n e t h e o r y ( r e c a l l Mi/FIl c P / c i = P - A ) , ;(Y,x) = (1/2n)j: lm @ A ( y ) ~ ( A ) [ B ! ( ~ )+ @!(x)]dA.

-? eA(x)dA =

@ ~ ( Y ) [ ( C ~ / CP ~ ) @+~@!,(x)ldA (X) Then f o r x+y > 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 <

-+

m)

IJ

(l/Zn)L: [ @QA ( y ) / c p ( - X ) ] [ 6 ( X - ~ ) / ~ ] d Xs i n c e one must spread o u t t h e 6 a c t i o n t o ( - = , m ) i n t h i s s i t u a t i o n ( t h e & ( A - I J ) / $ t e r m a r i s i n g from P - P i n v e r s i o n We can spread o u t t h e 6 a c t i o n i n two ways. i n v o l v e s o n l y A,IJ on [O,m)). P P P P P P E i t h e r one can s p l i t up q A ( x ) q ( x ) i n terms o f ZX + @ A and B + @ (which W

1.1

I

J

i s done l a t e r i n another c a l c u l a t i o n , b u t works o n l y when a f u l l l i n e t h e o r y i s a v a i l a b l e ) o r we can w r i t e (6.11) as (6.13)

;(Y,X)

=

(1/4n) ~~!(X)C;:(Y) m

+

~ ~ X ( ~ J l d ~

FOURIER TYPE OPERATORS

153

Then we can where?:(y) = @QA ( y ) / c p ( - A ) has t h e same p r o p e r t i e s as *,(y). Q P P use t h e h a l f l i n e f o r m u l a ( q X ( x ) , q H ( x ) ) = 6 ( ~ - l ~ ) / $b u t remember 6 = 6 2, 1 2 s 2 here. There r e s u l t s ( $ ( y , x ) , q P ( x ) ) = :/ ~ ( Y , X ) [ C P~ @ ~ +( XcL@rU(;)ldx ) =

-

( 1 / 4 ~ ) L : (Y :[) + ~ ~ X ( ~ ) 1 6 ( h= It- ~l c)p /l 2~[%(Y) -Q + "4 *-,-(Y)]. S e t t i n g *(Y,u) g ( y , x ) ' Q ( x ) d x we o b t a i n @+ = -Q ( Y ) - *(Y,u) = 43- = -[$!Qu(~) - $(Y,-U)]

= :1

lJ

*

!J

N

and by a n a l y t i c c o n t i n u a t i o n arguments as b e f o r e we obtain%'(y)

u

= *(y,u)

=

/"%f(y)T(A,p)dX ( c f . Remark 6.2, Theorem 5.10, e t c . ) . T h i s proves t h a t cmP Q B* = i n h a l f p l a n e a c t i o n when (6.11) i s used. The p o i n t here i s a g a i n U

l

J

t h a t T(A,u)

has

4 when w o r k i n g i n X on @,(y)/cP(-x)

~ ( x - u )a c t i o n

(as w e l l as

8"

or on on q QA ( y ) ) . C l e a r l y one does n o t have t o use such arguments f o r t h e f u l l l i n e s i n c e (6.12) and ( 6 . 7 ) a r e a v a i l a b l e w i t h a f u l l l i n e t h e o r y . Maps such as

REmARK 6.9,

6 were

denied t h e s t a t u s o f t r a n s m u t a t i o n i n [Fal;

C40] s i n c e t h e o b j e c t s on which t h e y a c t a r e d i f f e r e n t from those on which say B i s a t r a n s m u t a t i o n . @!(x)dx

Ggf

=

However f o r f say o f t h e form f ( x ) =

rI

one has from (6.12) ( i ( y , x ) , f ( x ) > = @:(y)F(h)dX 12@X(y)F(X)dX 4 = b6f ( c f . a l s o Remark 3.9).

-iI

/z

F(A)

and f o r m a l l y

REmARK 6-10, Using Theorem 6.1 we can compute a l s o B(y,x) = ( . ( ( x ) , ~ ~ (Qy ) ) ~ = Ap(xlE"ql(y)] Q

( 1 / 2 n ) i I q~(y)[@;(x)/cP(-X)]dh. By c o n t o u r i n t e g r a t i o n as above t h i s shows d i r e c t l y t h a t ~ ( y , x ) = 0 f o r 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)<(x)dx

=

qqy,.)]

= '$!(Y) A

?;

refer to f o r Q we s e t Q P e.g. Mi] = c Q ( - x ) ) :(y,x) = ( 1 / 2 ~ ) [ 1 rkx(y)M1(x)p(x)xX(x)dX ( i . e . ( 6 . 7 ) ) . On P t h e o t h e r hand from Theorem 6.7 we know ( c f . Remark 6.8) ( g ( t Y r ) , Q x ( - r ) ) = From remarks above we o b t a i n t h e n (My M1, and

.rtm ;(t,T)ax(T)dT P

=

@[g(t,-)]

=

p

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

@.Q,(t).

Given @ ( f * g ) = @ ( f ) @ ( g ) f r o m above we ask now f o r a f u n c t i o n or d i s t r i b u t i o n H such t h a t @(H) = M1/Ml. Q (6.20)

H(s)

= (1/2~)

Thus

[ (M1/M1)P~xIS)dl; Q

P

=

( 1 / 2 ~ ) ~ ~ p @ ~ ( i ) M l d i /4M 1

cn

( G ( g ) = H(-E)

and r e c a l l u2(-E,)

= ul(~)).

Now i n a d d i t i o n t o b e i n g a n a l y t i c

f o r I m x > -H say one knows from [Hol; S t b l ] t h a t f o r I m l

>

0,

1x1

>

x0,

and

FOURIER TYPE OPERATORS

x E

155

P t h e r e i s a u n i f o r m e s t i m a t e u1 = @ x ( x ) = e x p ( i A x ) [ l t m(x,x)/h]

(-a,-),

A

Furthermore w i t h s t a n d a r d p r o p e r t i e s o f Q,

where m i s u n i f o r m l y bounded.

(M1/M7)p = 1/4ihMM? = 1/4ixMco(-h) should be a n a l y t i c i n t h e upper h a l f p l a n e so t h a t (6.20) makes sense f o r some reasonable c l a s s o f

= 0 f o r 5 > 0.

i n f a c t by a c o n t o u r i n t e g r a l argument H ( - S )

5.1);

EHE6RElll 6.14.

For standard

$

More d i r e c t l y we w r i t e

K(x,s) = ( 1 / 2 ~ )[ I [ c F / c i ]

P t ~ a o d : The composition F(y,s) (6.7),

(6.8),

=

* H](x) where k where SC(x,s)

Z!(s)@!(x)pdh

Ll i(y,x)SC(x,s)dx

=

f o l l o w s immediately f r o m Thus ( ~ ( y , x ) , l C ( x , s ) ) = ( 1 / 2 ~2) I-m@A(y) - Q

and (6.21).

(6.12),

Thus

we have f o r m a l l y r ( y , x ) = [ i ( y , * )

S(H) = M /MQ = cp(-X)/cD(-h). 1 1 ( s + x ) i s g i v e n by (6.21 )

( c f . Remark

pCc-Ic-1 ZPu ( ~ ) ( C P h ( ~ ) P, @ u )(d~u)d i = ( l / 2 n ) i m m @ Q, ( Y ) P [ C ~ / C ~ ]x!(s)dx P ( l / 2 n ) i a *,(Y)pMIC,(S)dX = ;(Y,X).

p(X)iI

- 6 Q

=

RENARK 6-15. We n o t e f r o m Chapter 1, S8 t h a t

-,

P Z-,(x) P u1- = Sp ( x ) = p Zxp ( x ) t AaA(x);

(6.22)

where p = 1/4ihMM1 and A = -[MMi

i-

M1M-]/2MM1

=

pSh(x) P

AZX(x) P

i-

and t h u s we do n o t have a d i r -

e c t c o u n t e r p a r t o f t h e c l a s s i c a l F o u r i e r t y p e formula i n v o l v i n g @(h)" and @(h). In any event however @(h")" = il h ( - x ) @P- x ( x ) d x = pK(x)Zx(x)dx P i-

iz

A<:

i(x)@!(x)dx

= p@(h) i- AZ(h).

REI?MRK 6-16, The c a l c u l a t i o n s l e a d i n g t o

=

B'*

H i n Theorem 6.14 were made

* o f (6.16). I n t h i s d i r P g = (1/271)11 @ ( h ) @ ( f ) @ ( g ) Z h ( x )

here m a i n l y t o e x h i b i t t h e use o f t h e c o n v o l u t i o n e c t i o n we have a l s o h

*

(f

*

g) = (h

*

f)

*

p ( h ) d h and (6.23)

*

g,h)

where we d e f i n e (;(XI

a(;).

= (1/2a)

I,

@(f)*(g)Z(h)p(A)dx

= h(-x)) Z(h) =

P lI h(x)Z,(x)dx

=

P iz h(-x)@,(x)dx

=

Thus i n ( 6 . 2 3 ) l e t us w r i t e

(note g

t f # f t 9).

D e f i n e t h e n an o p e r a t o r SC by 3B = H

t

0

f o r s u i tab1 e

i s g i v e n by (6.21) P i . e . [H t e l ( x ) = ( l / Z ~ ) i : @(H)Z(O)@!(x)pdh = ( ( 1 / 2 ~ ) j : [M /c-]pa! X)Z^ ( s 13 1 "0 e ( s ) ) d x s o (i?(y,x),e(x)) = t ' i ( y , - ) H , e ) = ($(y,-),H f- 8) = Bee.

0

( n o t e K e ( x ) = [H t e ] ( x ) = ( X ( x , s ) , e ( s ) ) w h e r e

*

X(x,s)

156

ROBERT CARROLL

The n e x t s t e p i s t o examine t h e general G-L e q u a t i o n Ap = 1 here) ;(Y,X) =

% where

where ( r e c a l l P P t" = ( v A ( t ) y v A ( x ) ) w= TxW(x)

and A ( t , x )

Here dw r e f e r s t o dw = d A / 2 r l c

(as i n (3.16)s. use B = ;AiiA-'

= (~(y,t),A(t,x)) N

A has k e r n e l

E = BA

2 QI .

P P P P A ( t , x ) = ( v x ( x ) , v x ( y ) )p = ( v x ( x ) c , ( t ) W - '

Now i n f a c t we want t o

N

(6.25)

A

( A ) ) ~ V

A

(dp = ("v/&)dh and W-l(A) = v/w).

N

The m i x t u r e o f n o t a t i o n A, W, e t c . occurs

here i n t r y i n g t o m a i n t a i n c o n t a c t w i t h [ F a l l v i a W w h i l e A has been used f r e q u e n t l y i n o u r papers. A

2

A

A

(1/4) = v/w,

D , cp

P =

Thus i n (5.29) f o r example ?(A)

=

(1/2),

= W-'(x)

.v

A

( 2 / ~ ) ,W(y,x)

vp =

= r(y,x),

etc.

=

Ic

2 QI /

I n the

s p i r i t o f (3.16) we can w r i t e a l s o ( c f . (5.29) and r e c a l l Ap = 1)

i

A

(6.26)

W(x) =

co

ru

W - ' ( x ) v ~ ( x ) d ~ ; A(t,x)

tA

= TxW(x)

0

i;mrmA 6.17,

The i n v e r s e o p e r a t o r

the kernel x ( t , x )

Pmad:

t o A o f (3.16)

(Ap = 1 ) i s determined by

g i v e n v i a (6.25)-(6.26).

Note t h a t ( r ( t , x ) , A ( x , s ) )

= ( ( v ~ ( t ) W - ' ( h ) , v S (Ps ) W ( S ) ( " XP( X ) , v p ( x ) ) ) 5

P P ( s i n c e ( v h ( x ) , v ( x ) ) = ~(A-C)/$). 5 Now t h e development i n 55 f o r t h e M e q u a t i o n i n v o l v e s t a k i n g

)

v v

= s(t-s)

2,

and a p p l y i n g another o p e r a t o r tion

$=

=

ir=

r e l a t e d t o X , so t h a t t h e o p e r a t o r M equa-

has a " n i c e " form.

i(Jc%)

B

One f e a t u r e o f t h e p r e s e n t development N

i s t o show t h a t t h e n a t u r a l "ad hoc" c h o i c e o f Jc (namely by c o n s i d e r a t i o n s o f t r i a n g u l a r i t y and " s i m p l i c i t y " ,

%=

X*), motivated

i s also the "intrinsic"

and " c a n o n i c a l " c h o i c e i n many ways (and a r i s e s a l s o i n m i n i m i z a t i o n p r o cedures as i n 5 7 ) .

One d e f i n e s t h e o p e r a t o r X

*

v i a Jc*(x,s)

= Jc(s,x) A

X(s,x)

*w i t h

g i v e n i n (6.21) and l e t us f i r s t compute t h e k e r n e l o f B = BJC d i r Y

ectly (recall

=

*

BJC* = E(JcP8 ) .

Thus

P For t h e a c t i o n <~(y,x),Z,(x)) ( w i t h ~ ( y , x ) an even f u n c t i o n v i a ~ ( y , x ) = Q P P P P B(y,x)z5(x)dx = f , B ( Y , X ) [ ~ ~ ( X ) + @ 5 ( ~ ) l =d ~ ( v A ( ~ ) , ~ A ( x ) > one has Consequently i n (6.27) one o b t a i n s 4i5M ~ , @ ( y , x vP ) ~ ~ ( x ) d x= 4igMp5(y). Q

,z

(6.28)

n

B(y,S) = ( 1 / 2 r )

m -m

[l/cQ(-A)l@~(5)@~(y)dX A

and b y c o n t o u r i n t e g r a t i o n w i t h standard p r o p e r t i e s B(y,S)

= 0 f o r 5 > 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 <

One way t o do t h i s i s t o w r i t e (l/Zr)J;

4 ( 5 ) I c Q l [*p(S) + *-u(c)]dc i s i n d i c a t e d below). Then (6.31)

so t h a t f o r m a l l y r ( t , x )

(l/ci)pll(~)@u(x)dpdc. Q P

=

158

ROBERT CARROLL

Since

@rA

= @'/c

(-A)),

+ A@, P t h i s g i v e s (6.30). We n o t e i n passing t h a t another way t o spread o u t t h e r(A,u) a c t i o n i n v o l v e s w r i t i n g r ( t , x ) as ( r e c a l l = PZ!

r(t,x) = (l/Zw)LI

iz q ( t ) $ ( x ) z ( A , u ) d A d u

l:

(1/4n)i:

=

[ qNPA A ( t ) 9q Pp ( x ) + ~ ! ( t ) ~ ~ u ( x ) l ~ ( A , u ) d A d uThen . use 6(A,u) = ( c I 2 6 -(1-LI)= 2 Q 1 21c 1 6 ( A - p ) t o g e t (6.31) and (6.30) again. N

9

2

REmARK 6-21,

Comparing (6.29) and ( 6 . 2 8 ) we see t h a t ;(x,y) = AQ(y);(y,x). V* A V A V -1v* = A B and BB = B B and f o r A = 1 t h e M e q u a t i o n w i l l Hence f o r m a l l y

%

(I

vv*

v

have t h e upper l o w e r f a c t o r i z e d form BB

* Q

= JcpJc

*

t A

XTxW(x)Jc

=

( f r o m (6.26)).

On t h e o t h e r hand t h e l o w e r upper f a c t o r i z e d form o f t h e G-L e q u a t i o n i s (for

%=

1 ) B-':

=

A o r BB*

W(A) as G-L d a t a and

%

=

T i i ( 5 ) ( c f . (3.16)).

thus seen t o be p a r a l l e l i n s t r u c t u r e . a l s o emphasized i n [Du7,8]

One can t h i n k o f

w"

%

W-l(A) as M data and t h e G-L and M equations a r e The f a c t o r i z a t i o n p o i n t o f view i s

but the c a l c u l a t i o n s there f o r the M equation

D2 .

r e l y h e a v i l y on t h e F o u r i e r t r a n s f o r m as i n 65 and t h u s a r e s p e c i a l f o r L e t us c a l c u l a t e t h e k e r n e l o f

REfllAI'K 6-22.

d i l y t h a n i n [C47,48]

P

N

(6.32)

d i r e c t l y b u t much more hanThus

P

= ( V ~ ( ~ ) , V ~ ( ~ ) W - =~ ( A ) ) ~

A(5,n)

( 1 /4v ) I m p 2 [

&*

i n o r d e r t o compare w i t h (6.31).

I CQ I '/(Mi )'I [(

(5)

+ @!

( 5 ) 1[Z!

(Q

+ @!

(n ) I

-m

T h i s decomposition spreads t h e g e n e r a l i z e d t r a n s l a t i o n over t h e whole a x i s . rv

u *

We want t o compute now?;(t,x)

= k e r 3 0 4 ~ so F ( t , x )

F i r s t one has ( c f . Lemma 6.5)

(1/2n)lI

+ (l/pp-)G(A+u)

-

P [zA(n) +

(A/p2)S(X-p) = (l/pp-)G(A+u)

N

P(s,x) = ( A ( c , n ) , ~ c ( x , n ) ) has t h e form

(t)

= (X(t,c),(A(c,n),x(x,v))).

P P @A(n)lXu(n)dn = ( l / p ) S ( X - u )

+

2

( l / p )(M-/M l m

P(s,x) = (1/4v)C,

)~(A-u)

12 P

So

[~C~~*/(M;)~I

[z!(c ) + ( ( c ) ] i m (C-/C-)+'(X)P [~(X+U)/PP- + (M;/M1 1 6 (X-TJ )/p21dudA = ( 1 / 4 ~ ) lmp 2 [ \ C g l / ( M i ) 9 +Q@ P TA JI ( ~ ) [ ( M ; / P c - ) @ ~ ( x ) + ( M ~ / P C Q ) @ ~ ~ ( X Next ) I ~ ~we -

B

-m

form ( x ( t , c ) , P ( c , x ) )

LI

1 c 12/(M;)21[

= ?(t,x)

ILZ

which g i v e s ( [ ] as i n

/c0)3@~(t)[S(h+u)/Pp-

( t ) )r ( t , x )

= (1/4n)

( l / p 2 ) (M;/M1 )S(X-u)ldudA Qm 2 = (1/4n)/ P * [ ~ C ~ ~ ~ / ( MI[ ; )] ( t ) [ ](x)dh. Now one has terms ( c times P P-" P P 2 P P 2 P PQ 2 ( 1' A ( x / I cQ [a,),( t )@A ( ) / c i 2 + @- A ( )@-A ( t )/ CQ + @A ( x )@- A ( t)/ 1 C Q 1 + p2[

+

I

I

so changing o r d e r s o f i n t e g r a t i o n i n two terms we o b t a i n r ( t , x )

=

r(t,x)

as i n (6.31). A

EHEOREIII 6-23,

A,

Given P o f F o u r i e r t y p e and C) standard as i n Remark 5.1 t h e

canonical M e q u a t i o n has t h e form

6=

*

whl*

U

X

BJc = BAJc = i(Ja\Jc ) and can be view-

# V ed as a r e l a t i o n between t h r e e t r a n s m u t a t i o n s B, B = R , and B g i v e n v i a (6.21)).

(Jc

being

T h i s can a l s o be t h o u g h t o f v i a f a c t o r i z a t i o n as i n

FOURIER TYPE OPERATORS

Remark 6.21.

I n k e r n e l form, from $(y,x)

=

159

0 f o r x > y, we o b t a i n f o r x >

y, 0 = ( g ( y , t ) , S ( t , x )

+ J ( t , x ) > w i t h S and J g i v e n i n (6.30).

(6.33)

+

Thus (x > y )

m

0 = ;(Y,X)

REClARK 6.24,

D2 one has A

For P =

g = ??

o b t a i n s (5.34) w i t h S ( t + x ) as i n 55.

REmARK 6-25.

g(y,t)[S(t,x) =

+ M(t,x)ldt

0 and M ( t , x ) = 0 i n (6.33).

Hence one

m

since S(t,x) = ( 1 / 2 r ) i m s exp[ix(x+t)]dx

I n [Ne2,3,5,7-9]

?

=

( c f . a l s o [ C t l - 3 1 ) t h e r e i s some i m p o r t a n t

work on t h r e e dimensional i n v e r s e s c a t t e r i n g i n which t h e M e q u a t i o n i s f o r mulated i n terms o f a Riemann-Hilbert problem.

The survey a r t i c l e [Ne7]

g i v e s an e x c e l l e n t d e s c r i p t i o n and we o n l y i n d i c a t e here a n o t a t i o n a l conn e c t i o n t o o u r parameters.

I t would perhaps be i n t e r e s t i n g t o t r y t o phrase

o u r M e q u a t i o n i n terms o f such Riemann-Hilbert problems b u t we have n o t P pursued t h i s . Thus i n [Ne2,3] f o r example fl = f+ ( = o u r ul) = ah and f2 = N f - (= o u r u = Z! w h i l e u1 (= u1 i n [Nel,2]) = x 1 = pul and uy = x2 = pu2 2l r ( a l s o Tr = T = T = p and RR - S12 = sZ1 = R = -A). The J o s t m a t r i x i n [Ne2,3]

which p l a y s a ( n a t u r a l l y ) c r u c i a l r o l e i n t h e t h e o r y can i n f a c t be P P Indeed one t a k e s i n [Ne2,3], g1 = p A - i x x

expressed i n terms o f c b and F. and g2 = p; + ix! P). =

and w r i t e s

(a1 ’ g 2 )

=

(fl,f2)lF

with

JI

= F/T (det

F

P

P

(i= column i n d e x ) one computes (fl,f2) = ’J P P i n terms o f p h and x A t o o b t a i n fll = fZ2= [ ( l / F ) + ( l / c b ) ]

Setting F = ((f..)) (g1,g2)lF-l

and fZ1= g12 = (6.34)

= T =

JI

=

(where cp 2 F =

[

[(l/F) - (l/ci)]. cp + F

M1

-

cP

-

F

Writing

c , - +F F cp

P =

T = 1/4ixMM1 = 1 / 2 F c i

1

2ihM). Now r e c a l l t h a t p-A = M;/M, = cp/cp and i n a Using s i m i l a r a n a l y s i s w i t h F one o b t a i n s p+A = F-/F ( c f . Remark 5.16). +

t h i s we can a l s o check t h e i m p o r t a n t r e l a t i o n QSJI? = JI- f r o m [Ne2,3] 01 Q = ( ( l o ) ) ) which determines t h e H i l b e r t problem f r o m which JI i s d e r i v e d v i a t h e s c a t t e r i n g m a t r i x S.

Here S = ((fA-:))

and t h e c a l c u l a t i o n i s

straightforward. 7-

mmimIzAmN

uA

DZRECCIUE IN CHARACEERZZZNG C R A N ~ ~ U C A C IKERNELS. ~N

We have seen how 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 (e.g. v i a PDE techniques, s p e c t r a l k e r n e l s , a c t i o n on e i g e n f u n c t i o n s , etc.).

I n t h i s s e c t i o n we w i l l i n d i c a t e another,perhaps more i n t r i n s i c ,

way t o c h a r a c t e r i z e t r a n s m u t a t i o n s v i a m i n i m i z a t i o n procedures.

There i s

a l s o some connection t o c l a s s i c a l work on orthogonal p o l y n o m i a l s ( c f .

160

ROBERT CARROLL

[Cdl-6;

C79,81;

Ghl-31 and we w i l l c o n s i d e r t h i s l a t e r ) .

More d i r e c t l y

t h e r e i s a c o n n e c t i o n t o l i n e a r s t o c h a s t i c e s t i m a t i o n which w i l l be d i s c u s sed i n Chapter 3.

I n f a c t , 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

o u r c h a r a c t e r i z a t i o n o f c e r t a i n t r a n s m u t a t i o n s v i a m i n i m i z a t i o n can be achieved v i a s t o c h a s t i c i n f o r m a t i o n and accomplishes t h e same r e s u l t i n s t o c h a s t i c geometry as l i n e a r l e a s t squares e s t i m a t i o n .

Historically i n

[ D a f l ] i t i s shown how G-L e q u a t i o n s can be o b t a i n e d by m i n i m i z i n g a c e r t a i n quadratic functional Q(t,K).

The m o t i v a t i o n t o c o n s i d e r Q(t,K) came

from a problem i n o p t i c s ( c f . [DafP])

i n v o l v i n g a feedback mechanism and

s t a t i s t i c a l a v e r a g i n g b u t no m o t i v a t i o n c o u l d be p r o v i d e d w i t h i n s c a t t e r i n g theory t o consider Q(t,K).

Thus t h e process p r o d u c i n g G-L equations ap-

peared t o s i m p l y i n v o l v e a mathematical t r i c k which was n a t u r a l l y c o n s i d e r ed t o be u n s a t i s f a c t o r y i n [ D a f l ] and t h e meaning o f such procedures seemed t o be worth p u r s u i n g f u r t h e r .

I n [C75] we p r o v i d e d an i n t e r p e r t a t i o n o f

such m i n i m i z i n g processes i n t h e c o n t e x t o f t r a n s m u t a t i o n t h e o r y which l e d e v e n t u a l l y t o m i n i m i z e a q u a d r a t i c f u n c t i o n a l e s s e n t i a l l y t h e same as Q ( t , K ) T h i s i n v o l v e s a c h a r a c t e r i z a t i o n o f t r a n s m u t a t i o n k e r n e l s themselves i n terms o f a m i n i m i z a t i o n procedure and we g i v e h e r e t h e development f o r v a r i ous s i t u a t i o n s ( c f . a l s o [C50,52,53,73,74,78]).

L e t us remark t h a t t h e r e i s

a d i s c r e t e v e r s i o n (which does n o t d i r e c t l y g e n e r a l i z e ) o f a r e l a t e d m i n i m i z a t i o n i n t h e c o n t e x t o f orthogonal polynomials, b u t w i t h o u t a c o n n e c t i o n t o Q(t,K)

n o r any e x p l i c i t l i n k t o t r a n s m u t a t i o n ( c f . [Cd4] and see a l s o

o u r t r e a t m e n t o f "orthogonal p o l y n o m i a l s " i n 110).

Although o u r c h a r a c t e r i -

z a t i o n o f transmutation kernels v i a minimization i s o f i n t e r e s t i n i t s e l f , and moreover p r o v i d e s " m o t i v a t i o n " f o r c o n s t r u c t i o n s as i n [Dadl],

there

a r e f e a t u r e s below t h e s u r f a c e (as seen i n Chapter 3 f o r example i n t h e cont e x t o f stochastic estimation).

Our procedure i s a t t i m e s f o r m a l , b u t de-

t a i l s can e v i d e n t l y be s u p p l i e d as needed, w h i l e hypotheses on c o e f f i c i e n t s , p r o p e r t i e s o f k e r n e l s , e t c . used h e r e a r e discussed elsewhere i n t h e book. I n c l a s s i c a l ( h a l f - 1 i n e ) 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 ( c f . [Fal;

C e l l and 14) one connects e i g e n f u n c t i o n s o f t h e Schrodinger o p e r a t o r 2 Q = D - q ( q r e a l h e r e ) w i t h e i g e n f u n c t i o n s o f D v i a t r i a n g u l a r transmu2

t a t i o n k e r n e l s ~ ( y , x ) = 6 ( x - y ) + K(y,x)

and we w i l l c a l l K(y.x) w i t h K(y,x)

Thus l e t q A 9 ( x ) (resp. O9A ( x ) ) be s o l u t i o n s = 0 and Q = 0 (resp. .Q,(O) Q o f (*) Qu = -A2u s a t i s f y i n g ~ Q ~ ( =0 1 ) and OxqA(0) Dx.Q:(0) = 1). We w i l l w r i t e s(A,x) f o r q QA o r 0 : and t h i n k o f c o n n e c t i n g

= 0 f o r x > y a causal k e r n e l .

s(A,x)

t o a(A,x)

= Coshx

or a(A,x)

= Sinxx/h by a f o r m u l a

MINIMIZATION

(7.1)

s(A,y)

I"

( I + K)a = a(A,y) +

=

161

K(y,x)a(A,x)dx

0

which we know t o be v a l i d f o r t h e G-L k e r n e l K = K

We can assume KO

0'

e x i s t s h e r e and o u r procedure i s designed t o c h a r a c t e r i z e i t v i a minimization.

For now l e t us t h i n k o f s = 0: and a = Sinhx/x ( c f . 54).

One knows

as b e f o r e t h a t a s s o c i a t e d t o 0 and t h e e i g e n f u n c t i o n s 0: = s i s a s p e c t r a l measure dw = dw

Q

=

which we assume here f o r convenience t o be o f t h e form dw

tdA (no bound s t a t e s ) .

Thus one can suppose e.g.

Q ( $ ) /:OA(x)O

6(x-y) ( a c t i n g on s u i t a b l e f u n c t i o n s ) and we w r i t e dw = do

:J

a(A,x)a(A,y)do

= .Q(x,y).

lo

Q (y)dw(A)

Zx + 21 d x / r

=

with

Thus

m

(7.2)

A(x,Y)

=

a(A,x)a(A,y)dw

~ ( x - Y )+

=

Nx,Y) =

(1 + Q)(X,Y)

where a = Sinhx/x (we w i l l w r i t e 1 o r I f o r t h e i d e n t i t y o p e r a t o r w i t h k e r nel 6(x-y)).

Now c o n s i d e r t h e e x p r e s s i o n (T a r b i t r a r y and f i x e d )

'0

'0

Note t h a t when K i s t h e G-L k e r n e l KO (which makes (7.1) c o r r e c t ) t h e n f o r m a l l y Z(T,K)

= 0.

We can t h i n k h e r e o f

( c a u s a l ) k e r n e l K(y,x)

kHE(?Rfm 7.1-

0, s,

i n (7.3) as unknown.

a, and dw as g i v e n and t h e I t w i l l be shown t h a t

The k e r n e l K o b t a i n e d by m i n i m i z i n g Z(T,K) o v e r a s u i t a b l e

c l a s s of admissable causal k e r n e l s s a t i s f i e s t h e G-L e q u a t i o n and r e p r e s e n t s t h e t r a n s m u t a t i o n k e r n e l KO c o n n e c t i n g s and a v i a (7.1). We proceed f o r m a l l y and r e f e r t o Chapter 1, S e c t i o n s 4-5,

Chapter 3, 98,

e t c . f o r i n f o r m a t i o n about n a t u r a l p r o p e r t i e s o f K(y,x) e t c .

Thus from

(7.3) f o r causal K

'0

'0

' 0

Now one i n t e g r a t e s i n 1, u s i n g (7.2), and t h e c o n v e n t i o n JT Q(y,y)dy = T r fl 0

f o r example t o o b t a i n ( n o t e t h a t t r a c e T r depends on T ) A

(7.5)

Z(T,K) = Z ( T ) + 2TrK + 2

I'

[YK(y,x)al(x,Ydxdy

-

162

ROBERT CARROLL

where we have w r i t t e n Z ( T ) = I.T :I

T

s(X,y)] 2dwdy which we know

-

[a(A,y)

2

{I {I K o ( ~ , x ) K o ( y , ~ )

makes sense ( i n f a c t G(T) = Jo :1 [Koa] dwdy = id [6(x-S)

+

+

s 2 ( x , < ) l d ~ d x d y = Tr[Ko(l

Here t h e t e r m r ( y , x )

*

-

O)Ko]

= (s(A,y),a(X,x))

see c a l c u l a t i o n s below a l s o ) .

i s o u r standard o b j e c t i n general WN

t r a n s m u t a t i o n t h e o r y and i n p a r t i c u l a r @(y,x) = 0 f o r x

y (i.e.

<

a n t i c a u s a l ) w i t h a 6 ( x - y ) t e r m a r i s i n g a l o n g t h e diagonal. c o n t r i b u t e s -2IJ (7.6)

K(y,y)dy = -2TrK t o (7.5).

I’

W(Y) =

K(Y,x)

0

n(x,s)g(s)dsdx =

I;

since

x ,s 1d x l ds

(Y ,x

IJ [I{

( f o r s u i t a b l e g) so t h a t T r KC? = on [x,-)

We can w r i t e now

lorn 1[I: K )a(

lom g (s

= K(-,x)

it i s

Thus t h e ;term

g(y)J{

K(y,x)n(x,y)dx]dy.

Similarly ker

K*

= Jy h(x)/X” g(y)K(y,x)dydx,

K(y,x)h(x)dxdy

and consequently

1

min (y ,c (7.7)

KK*s(Y)

* T Hence T r KK = Jo [J{ (7.8)

g(c)

=

K(y,x)K(c,x)dxdC 0

0

K(y,x)K(y,x)dx]dy

mK*g(Y) =

g(C)[r

and f i n a l l y we have

K(Y,x) ~ C ~ ( x , s ) K ( ~ , s ) d s d x l d ~

0

O*

0

Now go back It f o l l o w s t h a t T r WK = I; [I{ K(y,x)J{ Sl(x,s)K(y,s)dsdx]dy. t o (7.5) and i n s e r t t h e i n f o r m a t i o n j u s t d e r i v e d from (7.6)-(7.8) p l u s t h e N

B contribution, t o obtain

LElltmA 7-2- The e x p r e s s i o n Z(T,K)

d e f i n e d i n (7.3) can be w r i t t e n

A

Z(T,K)

(7.9)

=

Z ( T ) + T r [ K ( l + n)K* + Ks, + nK*] A

Ptlood: K(l

One o b t a i n s from (7.5),

+ n)K* +

Z(T,K)

= E(T)

KK* + KnK* w i t h T r K!2 = T r

W r i t t e n i n t h e form (7.9), Z(T,K) Q(t,K) ( o r D) i n [Daf1,2]

nK*

*

*

+ Tr[2W2 + KK + WK 1.

(note

But

d = a).

es e s s e n t i a l l y i n t h e same f o r m as t h e

and we now f o r m a l l y examine a v a r i a t i o n a l argu-

ment t o m i n i m i z e Z = Z(T,K).

Thus ( n o t e Z > 0 f r o m ( 7 . 3 ) ) we know t h e r e i s

a m i n i m i z i n g k e r n e l K = KO i n some a d d i t i v e c l a s s K o f admissable ( c a u s a l ) kernels.

n

pendent o f K) f o r L E K and = 0.

A

Then c o n s i d e r K = KO + EL i n Z(T,K) = E ( T ) + E K ( T ) (Z(T) i s indeE

a r e a l number.

T h i s l e a d s t o T r [ L ( l + n)K;]

[ ( K o ( l + n ) + n)L*] = 0 f o r L E K. k e r n e l A(y,x)

+

F o r m a l l y we s e t DEZK(T)IE=O

T r [ K o ( l + a)L*]

+

T r L n + TmL* = 2Tr

I f we w r i t e now A = K o ( l +

t h e n e v i d e n t l y k e r AL* =

:I A(y,x)L(s,x)dx

a) + a w i t h

and TrAL* =

MINIMIZATION

= :f

[I{ A(y,x)L(y,x)dx]dy.

163

The statement t h a t T r AL* = 0 f o r a l l L

w i l l be t r u e i f A(y,x) = 0 f o r x

<

E

K

y and h e u r i s t i c a l l y one concludes t h e

converse s i n c e K w i l l c o n t a i n s u f f i c i e n t l y many s u i t a b l e L.

CHE@REI 7.3, The ( u n i q u e ) 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 Ko(y,x)

+

n(y,x)

+

ry Ko(y,c)a(E,x)dc 0

= 0 f o r x < Y.

One knows t h a t t h e G-L e q u a t i o n has a unique s o l u t i o n and t h i s i s t h e t r a n s m u t a t i o n k e r n e l o f (7.1) ( c f . 54 and Chapter 3, s8).

Thus t h e G-L e q u a t i o n

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 and Theorem 7.1 i s v e r i f i e d .

REMARK 7-4- L e t us n o t e a l s o t h e f o l l o w i n g c a l c u l a t i o n which w i l l s p e c i f y ( a g a i n ) t h e minimum E o o f E(T,K)

achieved a t t h e G-L k e r n e l KO.

t h e G-L e q u a t i o n i n Theorem 7.3 we can say KO causal o p e r a t o r . (7.10)

It follows e a s i l y t h a t 1 + B

+

( 1 + B * ) ( l + K i ) = (1

which i s f o r m a l l y s e l f a d j o i n t .

+

*

R + K R = B

*

0 =

Thus g i v e n

where B i s a

(l+Ko)(l+Q) and t h u s

*

K o ) ( l + n)(l + KO)

But t h e l e f t s i d e o f (7.10) i s 1 + an a n t i -

causal o p e r a t o r so both*sides o f (7.10) must be 1 ( c f . [ D a f l ] ) .

Hence ( r e -

A

c a l l E ( T ) = Tr[Ko(l+s))Ko]) (7.11)

E o = minE(T,K)

- T r [ 2 n + 2Ko + 2K: ( s i n c e KO and B a r e causal

= g ( T ) + minE K (T) = T r [ 2 K o ( l + n ) K i

* *

+

KoR

-

cf. [Dafl]).

+

RK*] = Tr[B K0 0

+

+ Kon+ nKi]

=

K0 B] = 0

This i s t h e desired conclusion.

REMARK 7-5, The c h a r a c t e r i z a t i o n o f KO does n o t r e q u i r e t h e t r a c e argument (i.e. 1,2]

the y integral i n (7.3)).

T h i s was i n s e r t e d f o r comparison w i t h [Daf

and we w i l l show below how i t may be removed.

REMARK 7 - 6 , I n t h e same c o n t e x t o f i d e a s one can work w i t h 9: i n s t e a d o f O 4 x and k e r B = ~ ( y , x ) = 6(x-y) + K (y,x) w i t h Ko(y,x) w i t h B[Cosxx](y) = 9p,(y) Q Q0 = 0 f o r x > y. We w r i t e now a ( x , x ) = Coshx, s(x,y) = qP,(y), dw = do + ( 2 / 1 ~ ) d x , and as b e f o r e n ( x , y ) = fr a(x,x)a(A,y)du w i t h g(y,x) = <s(X,y), a(h,x)>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 <

~1

5 A(x) 5 6

<

-, A(0)

=

164

ROBERT CARROLL

Am r a p i d l y as e t c . which a r i s e i n geophysical a p p l i c a t i o n s f o r example ( c f .

= 1 f o r n o r m a l i z a t i o n ( w i t h no l o s s o f g e n e r a l i t y ) ,

x

-+

m,

A E C',

A(x)

-+

G-L equations f o r such o p e r a t o r s a r e discussed t h e r e i n

Chapter 3, § § 8 - 9 ) .

Chapter 3, 558-9 and i n Chapter 2, §12 i n d e t a i l i n a u n i f i e d manner and we

w i l l s i m p l y e x t r a c t here a few f a c t s and s t r u c t u r a l f e a t u r e s as needed ( c f . Thus ~ ( y , x ) = (px(y),Coshx)v 4

a l s o 52.10).

=

A-4(y)6(x-y)

+ K2(y.x) where

and (2i/h)[p!(y) - Cosxy] = JY K(y,x)exp(ihx)dx -Y The G-L e q u a t i o n can be w r i t t e n as ( c f . Theorem 7.3)

1 - K(y,y) = A-'(y) x ) = Kx(y,x)).

joy?o(y,c)R(E,x)dc

A

(7.12)

Ko(y,x)

+

A-'(y)G(y,x)

t

(K2(y,

= 0

A

f o r x < y. where Ko(y,x) i n Remark 7.6,

= Kz(y,x)

+

dw = du

(K(y,x)

b e i n g d e f i n e d as above) and, as

( 2 / 1 ~ ) d hw i t h n(y,x)

= (Coshy,Co~hx)~.

REmARK 7-7- We n o t e here t h a t (7.12) corresponds t o t h e G-L e q u a t i o n (O(y, ~ , ) , A ( c , x ) ) = z(y,x)

w i t h A(c,x)

=

+ R(c,x)

6(x-F,)

and f o r purposes o f r e -

c o v e r i n g A one computes w i t h an i n t e g r a t e d v e r s i o n , namely ( c f . Chapter 3, 5 58-9 and [C40,66,67,71])

(7.13)

K(Y,x) + T(y,x)

=

T(y,x)

c lo

K(y,n)Tn(n,x)dn;

=

[Sinhx/x]Coshy

do(x)

(one should n o t confuse A(y), t h e a c o u s t i c impedance, w i t h t h e o p e r a t o r A determined by A(x,Y)).

T h i s e q u a t i o n (7.13)

(and (7.12)) can a l s o be ex-

pressed i n t h e t i m e domain, which i s more r e v e a l i n g and e f f e c t i v e f o r geop h y s i c a l problems ( c f . Chapter 3, Ss8-9).

Now f o l l o w i n g t h e approach above f o r 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 s v i a m i n i m i z a t i o n we t r y t o m i n i m i z e ( a = COSAX, s = q xQ ( x ) ) (7.14)

Z(T,K)

=

=

joT;1

[(A-%a

-

A

2

s ) t (K(y,x),a(x.x))] A

dwdy

A

over a c l a s s o f a d m i s s i b l e causal (= t r i a n g u l a r ) k e r n e l s K (K(y,x) A

4

We know t h e minimum i s achieved f o r K = K = K2

= 0 for

( t h e G-L k e r n e l of

x > y). t r a n s m u t a t i o n ) and we w i l l show t h a t i n f a c t t h e " E u l e r " e q u a t i o n f o r m i n i -

m i z i n g P i s t h e G-L equation. = &(x-t)

+ R(t,x),

a:

= a(A,c))

Thus one can w r i t e ( A ( t , x )

=

(a(h,t),a(h,x)

)w

MINIMIZATION

lo jr

165

T

(7.15) -2

1;

Z =

+ 2

(AJ’a-s)‘dwdy

JF(y,x)T(y,x)dxdy +

( r e c a l l t h a t F ( y , x ) = ( ~ Qx ( y ) , C o s x x ) =w AJ’(y)6(x-y) + K(y,x) with K(y,x) = term we get a contribution -21J AJ’(y)i(y,y)dy 0 f o r x < y ) . From the 2 which cancels o u t and one obtains ( n o t e t h a t E = 10’ :/ (Al’a-s) dwdy = T A 2 A 4* lo 1 ;
(7.16)

A

E = Z + Tr[A”’(y)(h+R^KX)

A*

A

+ K(l + f l ) K A

1 A

A

( t h e t r a c e depends on T of c o u r s e ) . Now s e t K = K O + E L (KO being a miniA mum value - e.g. K2(y,x) = K(y,x) will d o ) t o obtain “Euler equations” 0

(7.17)

=

DEI(T,?

0+ E L ) ]

E=O

=

2Tr[(go(l+i2) +

A-%)L*l

This holds f o r a l l admissable L and y i e l d s formally

( L causal).

A

(7.18)

; b ( y , x ) + (K0n)(y.x) + A-’(y)Q(y,x)

which i s t h e G-L equation (7.12).

= 0

One knows t h i s has a unique s o l u t i o n so

A

T h e solution KO of t h e minimizing problem f o r I i s uniquely determined as t h e s o l u t i o n Kh(y,x) of t h e G-L equation (7.18).

&HEBR€Iil 7.8.

W e can a l s o c h a r a c t e r i z e y ( x , y )

(p

B-l) v i a a s i m i l a r minimizing procedure by simply interchanging the r o l e s of P and Q. For completeness we will sketch this. T h u s s i n c e s = A% + ( L ( x , y ) , s ) = a we minimize (7.19)

[ ;1

”

” E(T$) = Z =

[ ;1

+ 2 [i’(x)

(/l’s-a)2dvdx T

x X

= ker R

[(i’s-a)

=

2 + ( t ( x , y ) , s ( X , y ) q dwdx =

f L ( x , y ) i ( x , y ) d y d x -2

loT lox;( 1

x ,y 6 ( y ,x 1dydx

A0

[ Z(x,Y)c(x,rl)ii(Y,r7)d~dydx

+

Here we r e c a l l t h e reverse G-L equation ( c f . [C40, p . 175 and 5 3 ) (7.21)

B(Y,X)

=

(z(t,x),i(t,y))

(recall y ( x , t ) = A(t)z(t,x)).

“ The expansion of A in (7.20) can be seen

166

ROBERT CARROLL

e.g.

by matchina A-%terms i n ( 7 . 2 1 ) . Now from (7.19) ( w r i t i n g 1: 10" 2 - a ) dwdx = 2 and r e c a l l i n g t h e f o r m o f f i ( y , x ) ) e

(7.22)

n

Z = Z

We w r i t e

5

+ 2 /Tk'(x)

+

r?(x,y)z(x,y)dydx

and t h e n as b e f o r e

f o r t h e o p e r a t o r w i t h symmetric k e r n e l :(x,y)

+

=

(7.23)

&

UA*

A v

Tr[A2(Cn + PL ) + ?(Ab' +

$)?*I

m i n (x,c ) - 1 A (y)t(x,y)?(c,y)dy).

(note ker

(A%

A

A

Now s e t L = Lo + E J f o r

= f

G

m i n i m i z a t i o n and we f i n d t h e E u l e r e q u a t i o n ( c f . (7.17)) (7.24)

+ to(A'l

2Tr[(/l%

+ ;))J*]

0

=

Arguing as b e f o r e we o b t a i n f o r x > 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 =
5'

5

5

-*

N

kr

(8.4)

= sn-'KrY

- s " - ~ K ~ ; D,.[-s"-~(

-Dr[rn-3K(r,r)q(r)] Consequently one obtains ( K N

(8.5)

V

QrK = s

n-3

= (

Q,

K(rYs)6(s-r),v(s))

1=

-(rn-3K(ryr)) '6 + rn-'K(r,r)6'

,'p)

K(r,r))

[Q,K(r,s)]Y(s-r) + rn-'K6'

-

- [r2(rn-3K)'+ rn-'Kr + (n-l)rn-'K]6 V

(here K ( r , r ) K r ( r y s ) l s = r ) . Since we t r e a t K(r,s) as a distribution in I* n-3s , for P,K = s P,[K(r,s)Y(s-r)] l e t us consider the term Q,

(8.6) =

( S

n-3 s 2 [ S n-1 (KY)']I/sn-',q) = - t s n-1 (KY)',lp') = - ( K 6 + K,Y,s n-1 l p ' )

-K(r,r)rn-lv'(r) +

(

(sn-'K,Y)',q) = ( s n-3 s 2 ( sn - l K s ) l y ~ s n ,-vl ) +

+ ( rn-'K(r,r)6' Hence ( r

E

s in the 6 terms)

t

rn-'KS(ryr)6,q)

TYPE OPERATORS

[qr-PS]K -*

(8.7)

v

sn-3 [QrK-psK]Y- [r2 ( r n-3 K ) ' + r n - ' ( Kr+Ks ) + ( n - 1 ) rn-'K] 6

=

= (n-2)r"'K Since Kr + Ks = K ' and (rn-'K)' + (n-1)rnb2K = r 2 [(n-3)rn-'K + r n - ' K ' ]

+ rn-'K'

+

rn-'K'

+

2rn-1K'

173

w h i l e r 2 ( rn - 3 K ) '

+

rn-'K'

(n-1)rn-'K

+

=

rnb2(2n-4)K we o b t a i n

[Cr

(8.8)

-

-*

v

PS]K = s

n-3

-

-

[QrK-FsK]Y

2r(rn-'K)'6(s-r)

Thus t h e d e s i d e r a t a f o r t r a n s m u t a t i o n a r e (as i n ( 8 . 3 ) ) N

'v

QrK(r,s)

(8.9)

= PsK(r,s);

-2r(rn-'K(r,r))'

=

r2c

The l a t t e r c o n d i t i o n can o b v i o u s l y be expressed as r"-'K(r,r) i n agreement w i t h [Cnl-4,7]. N

Y

-

Qru = Prh

(8.10)

-"v

,w

r 2 q h + QrI = Prh

.r*"

-r2{h +

(

pG(p)dp

1;

=

Our c a l c u l a t i o n s g i v e now ( u as i n ( 8 . 2 ) )

PsK, h ) t r2
-

h

V

v

r 2 q h + (QrK.h) = P h r

+(

-

tyPSh)

&HEBREm 8-1, Under t h e c o n d i t i o n (8.9) t h e map u = Beh = h + ( t ( r , s ) , h ( s ) ) n- 3 K(r,s)Y(s-r)) i s a t r a n s m u t a t i o n Be: P + i n t r o d u c e d i n (8.2) ( z ( r , s ) = s N

2 2 2 2 Q where P = x D + (n-1)xD t k x and

rJ

#. .

5=

-

x2T(x)

(y =

2 k 4).

i n t h e r a d i a l v a r i a b l e s r,s one 2 2 2 sees t h a t Be a u t o m a t i c a l l y determines a t r a n s m u t a t i o n p (An + k ) r (An 2 s 2 s n l U + Asu where an = r An depends ,rn-l + k ( 1 - q ( r ) ) s i n c e Anu = ( r n v one o n l y on s p h e r i c a l o r a n g l e v a r i a b l e s . Hence w r i t i n g Pn = P and On =

REmARK 8.2-

has Ben: (8.11)

Given a t r a n s m u t a t i o n Be:

= $,Be Be[p

-f

-

and 2

2

+ k )u] = Be[Pnu +

(An

2

= QnBeu + C?B n eu =

2

= r [An + k (1 - q ( r ) ] B e u

I n p a r t i c u l a r i f (An sired.

+

2 k ) u = 0 t h e n Beu = 0 i s a s o l u t i o n o f (8.1) as de-

Thus t h e s t u d y o f t r a n s m u t a t i o n s f o r o p e r a t o r s

5as

i n Chapter 1 ,

87 i s i n t r i n s i c a l l y connected 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 of s e v e r a l v a r i 2 2 a b l e s (e.g. r (An + k ) ) and has a p p l i c a t i o n s n o t o n l y i n quantum s c a t t e r i n g t h e o r y b u t a l s o i n many problems i n say a c o u s t i c s c a t t e r i n g w i t h s p h e r i c a l symmetry i n p o t e n t i a l s o r o b s t a c l e s . I f one goes t o t h e c o n t e x t o f Chapter 1, 57 now ( w i t h n = 3 i n ( 8 . 1 ) here)

ROBERT CARROLL

174

we note f i r s t t h a t t h e r a d i a t i o n condition will hold in ( 8 . 1 ) - ( 8 . 2 ) from

the construction o f kernels below. T h u s u = B,h = h + frm K ( r , s ) h ( s ) d s s a t 0 when r ( h ' - ikh) + 0 s i n c e r ( u ' - iku) = r ( h ' - ikh) i s f i e s r ( u ' - iku) m 2 + r$ (Kr - ikK)hs - (h/2)irm k s q ( s ) d s (from ( 8 . 3 ) ) . For s u i t a b l e q , and K as constructed below, t h e condition a t m i s determined by t h e i s o l a t e d h term. NOW take h = g1 = f o ( v , - k , r ) / r with f o determined by (7.10) of Chap1 t e r 1 , 57 i n terms of H, ( t h e r a d i a t i o n condition i s s a t i s f i e d then). Heqce 1 f o r h = g we must have u = Beh = g as i n Theorem 1 . 7 . 1 2 a n d thus -+

EHE0REm 8.3, For s u i t a b l e ? t h e r e i s a formal equivalence between the transmutation Be of ( 8 . 2 ) f o r n = 3 and i n Theorem 1 . 7 . 1 2 (achieved via iden1 t i c a l action on the J o s t s o l u t i o n g ) and F ( r , s ) r\, 61s-r) + K ( r , s ) Y ( s - r ) .

REmARK 8.4,.

The c h a r a c t e r i z a t i o n of transmutations via action on eigenfunct i o n s i s r e l a t e d t o the following considerations. We know from Theorem 4 1.7.5 t h a t f o r s u i t a b l e f one can w r i t e p;f(v) = f ( v ) = :1 f ( s ) g ( v , - k , s ) d s and formally from ( 1 . 7 . 2 2 )

1

m

(8.12)

(

;(v),g(vy-k,r))

P

= f(r) =

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

P

0

ds

which determines G - ' ? = ( ; ( v ) y g ( v , - k y r ) ) p ( c f . a l s o Theorem 8.14 f o r the K-L version of this inversion). For now we observe t h a t (8.12) will hold f o r some reasonably l a r g e c a l s s of functions f and l e t us suppose g = Egl A 1 For s u i t a b l e f we obtain then from f ( s ) = ( f l ( v ) , = Beg as in Theorem 8.3. g1 ( v , - k , s ) ) p l (where fA, ( v ) = G 1 f ( v ) ) , B f ( r ) = ( fAl ( v ) , B g 1 ( v , - k , * ) ( p ) ) = N

A

f l ( v ) , g ( v , - k y r ) ) = Bef(r) Thus Ff = B,f f o r a reasonably l a r g e c l a s s of hr f and t h i s w i l l imply t h a t B = Be as an operator i n a well defined sense. We note a l s o t h a t t h e f i n a a s s e r t i o n of Theorem 8.3 follows formally from

(

w r i t i n g ( c f . Theorem 1.7.5 again) (8.13) (

(

g(v,-k,r),g

1

(v,-k,s))l = ( ( ? ( r , t ) , g

1

P

1 1 a ( r , t ) , ( g (v,-k,t),g (u,-k,s))

N

P

1)

(v,-k,t)),g

1

(v,-k,s))

P

1

=

= (?(r,t),&(t-s)) = ?(r,s)

V

while on the other hand ( K ( r , s ) = K(r,s)Y(s-r)) (8.14)

(g(v,-k,r),g V

1

(v,-k,s)) 1

P

1

=(

gl(v,-k,s)) 1 + ( ( K ( r , t ) , g ( v , - k , t ) ) , g P

1 1 1 Beg ,g ) 1 = ( g ( v , - k , r ) , P

1

V

(v,-k,s))

P

1

=

6(r-s)

Let u s now show how t o construct K(r,s) i n (8:2) f o r n

+

K(r,s)

= 3 following [Cnl-4,

N

Q TYPE OPERATORS

71.

Change v a r i a b l e s by 5 = (rs)'

and

ri

175

= (r/s)'

and d e f i n e M(t,n,k)

=

S K ( c n , < / r i , k ) (we w i l l suppress t h e k dependence by w r i t i n g K(r,s) and M(5, Then f o r 0 < II < 1 and 0 < < < a, M s a t i s f i e s (q = 0 f o r r > a ) 17)). 2 McI- + k 51-[l

(8.15)

-

I-

-4

a

-

q(
= (k2/2)

sq(s)ds

5

c

w i t h M(~,I-) = 0 for M(5,n)

(8.16)

> a.

T h i s can be converted i n t o t h e i n t e g r a l e q u a t i o n

1;

= (k2/2)

and one l o o k s f o r a s o l u t i o n i n t h e form M = (8.17)

Mo(<,n) = ( 1 / 2 ) Mn(S,n)

=

-

[!",

/:~t[l-t-~-q(st)]M(~~t)dtds

1;

k2n+2Mn where

sq(s)ds; IIIst[l

- t - 4- q ( ~ t ) l M ~ - ~ ( s ~ t ) d t d s

L e t I q ( r ) l 5 ; f o r 0 5 r 5 a so I M o ( ~ , n ) ] 5 a 24 M/4 and f o r 5 > 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 <

( c f . [Cn3]).

m

More s p e c i f i c a l l y assume q ( r ) = O(exp(-y

2

r ) ) f o r y > 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 <

(c/4y)exp(-yc

<

?(rs)-'exp(-yrs

as r,s

-f

-

tains K E C

2

f and l e a d s t o 2

2 2

l M ( c , r ~ ) ]5 <exp(-yc +k /4yn

2

1.

Thus

[ K(r,s)t

+ k s/4yr). I n o r d e r t o o b t a i n e x p o n e n t i a l decrease i n K 1 one r e q u i r e s t h e n t h a t k < 2yR ( r , s 2 R ) . For q E C one ob-

2

and t h e a p p r o p r i a t e

K d e r i v a t i v e s a l s o decay e x p o n e n t i a l l y f o r

k < 2yR.

REmARK 8-7- I n [Cn3] a somewhat d i f f e r e n t change o f v a r i a b l e s i s used i n d e a l i n g w i t h t h e Goursat problem (8. 3) (q = 0 f o r r > a).

Since t h i s w i l l

be a u s e f u l s u b s t i t u t i o n l a t e r i n v a r i o u s c o n t e x t s we mention i t here. Thus l e t 5 = l o g r and I- = l o g s and s e t (8.19)

M(s,I-)

=

e% ( n - 2 ) ( < + 0 ) K ( e ~ , e ~ )

( o r e q u i v a l e n t l y K(r,s)

=

(rs)-4(n-2)M(logr,logs).

Then M s a t j s f i e s

(I> 5)

176

ROBERT CARROLL

w h i l e M(5,n) = 0 f o r (5+n)/2

>

loga (with M(E,TI)

= 0 for E >

To s o l v e (8.20) one makes a f u r t h e r change o f v a r i a b l e s x = y = (~-n)/2 w i t h M(c,n) = E ( x , y ) so t h a t

-

(8.21)

Mxy

-

n i n addition). (c+n)/2 and

2 k F(x+y,x-y)E = 0; K(x,O) = (k2/2) [eZTq(er)dT;

rJ

rJ

M(x,y) = 0 f o r x > loga; M(x,y) = 0 f o r y > 0 where F(c,n)

[

co

N

(8.22)

T h i s becomes t h e n (y 5 0 )

= -[22E-e2n-e25q(e5)].

M(x,y)

eZTq(eT)dT + k

= (k2/2)

&F(~+8,~-8)~i(n,B)doda

X

Then change v a r i a b l e s a g a i n i n (8.22) back t o ( 5 , n ) ( w i t h say and 8 =

(T-LI)/~)

~1

= (T+LI)/~

t o o b t a i n , a f t e r some s i m p l i f i c a t i o n m

(8.23)

M(5,n)

= (k2/2)

(k2/2) [q%(c+O)

E [Cn3] f o r t h e d e t a i l s .

+

d-T-E

+

n+T-S

I[F(r,u)M(~,u)dvd~

-5k+r()T

c)

by successive approximations and we r e f e r t o 1 2 I f q E C one f i n d s a l s o t h a t K(r,s) E C ( s 2 r > 0 )

L e t us mention here t h a t i n [Cn1,3,7]

t r a n s m u t a t i o n Bi: = x2D2

I

?lf6-r

T h i s can be s o l v e d ( r ~>

REmARK 8.8-

e2'q(eT)dT

kk+tl)

yo

rrl -f

Q i s c o n s t r u c t e d where

+ 2xD + x2k2 - x 2 y ( x )

(c = k2q).

f o r n = 3 an i n t e r i o r

yo = x 202

+ 2x0 + x 2ko2 and

Again successive approximations

were used t o c o n s t r u c t k e r n e l s and one n o t e s t h a t ko = 0 corresponds t o t h e B-G o p e r a t o r ( c f . 59) w h i l e ko = k g i v e s an i n t e r i o r analogue o f Be i n (8.2)

1

r

(8.24)

u(r,-)

= Bi[h](r,-)

= h(r,-)

+

Ki(r,s)h(s,-)ds

0

(8.25)

QrK i( r , s )

= NO PsK i( r , s ) ;

( c a l c u l a t i o n s as b e f o r e ) .

itly for 2xD w i t h (8.26)

y= 0

c=

r Ki(r,r) = ( 1 / 2 r ) jos?(s)ds + $ r ( k %

-

k2)

It i s i n t e r e s t i n g t o d i s p l a y t h e kernels e x p l i c -

w i t h ko = 0 ( c f . [C41,42]);

x2D2 + 2xD + k2x2.

t h u s we have?'

=

yo =

x202 +

It f o l l o w s t h a t ( 0 5 5 1x1

Kd(x,S) = - ~ ~ $ ( X - E ) -[kX4(x-E#] ~J~

i (we w i l l w r i t e KO t o d e s c r i b e t h i s s i t u a t i o n which corresponds t o t h e 6-G

-0 TYPE OPERATORS

177

operator). Similarly (recall I v ( z ) = e x p ( - i i n v ) J v ( i z ) ) setting k = 0 and q = 0 (with ko replaced by k ) one inverts the map determined by the kernel x i < 5 5 x ) u + h = u + lo L o ( x , c ) u ( c ) d g where (8.26) in the form ( 0 .-d

(8.27)

Loi (x, 5 )

=

%kt5( x-4)-’11

k<’(

x- 5 1 ’(

(see here [C41,42; SalO; Lrl; Val]). These kernels will play an important role in subsequent work involving generating functions in 59 (cf.[Mdl ;Wjl]). REI!MARK 8-8. Let us make a few remarks about adjointness here and pursue the matter more completely l a t e r . Thus take n = 3, Bi as in (8.24) with ko = k vi and Ki satisfying (8.25), and Be as in ( 8 . 2 ) - ( 8 . 3 ) . Writing K ( r , s ) = Ki(r,s)Y(r-s) and k e ( r , s ) = Ke(r,s)Y(s-r) one has as above ( c f . (8.4)-(8.10)) 2 [a”r - FIEe = 5r & ( s - r ) ; [Tr s

(8.28)

y

vi

- Ps]K

+ 2 qr 6 ( r - s )

vi + K we can also write Beh y ‘yv = ( ( i i ( r , s ) , h ( s ) ) with Q r ~= P,B for = gi o r *oe. i The origin of the jumps in K or Ke involves calculations yielding

ie=

(4“ =

Upon setting k’q). g e ( r , s ) y h ( s ) )and B i h =

6

+ “e K and

=

gi

= 6

-

i

(8.29)

2r(rK ) ‘ 6 ( r - s ) = r 2 c & ( r - s ) ; - 2 r ( r K e ) ’ 6 ( s - r ) = r2G6(s-r) N A - ,

where Ki = Ki(r,r) etc. in (8.29). Now Bi and Be are transmutations P and we consider f o r example a map Be = Bi# defined by (8.30)

g

(Bef,h) =

(

f,Bih) =

h ( s ) [ r f(r)Ki(r,s)dr]ds

=

+

Q

lom +dom lor f ( r )[

fhdr

(f(s) +

S

i,

K (r,s)h(s)ds]dr =

f(r)Ki(r,s)dr,h(s))

u h , “i Then Be: Q + P has kernel y e ( s , r ) = 6 ( r - s ) + K ( r , s ) ( r s , r s ) . Writ-e ‘i i Ae ing K ( s , r ) = K ( r , s ) = K ( r , s ) Y ( r - s ) = K ( s , r ) Y ( r - s ) one has from (8.28) N Ne 2[Ps - Qr]K ( s , r ) = -s q s ( r - s ) . The corresponding calculation of the type (8.29) t o determine the j u m p involves then - 2 s ( s t e ) , = -s2ywhere “e ^Ke(s,s) = K i ( s y s ) . This leads t o K ( r , r ) = - ( 1 / 2 r ) l F sq(s)ds which i s compatible with the relation Ki(r,r) = (l/Zr)J; s<(s)ds a f t e r possible adjustment o f constants. -f

N

ie

Q

REmARK 8.10- In connection with the adjointness discussed in Remark 8 . 9 l e t

us refer t o Chapter 1 , 57 and show how we can complete somewhat the discusN : + s a t i s f i e s Egl = 9 a n d sion of B and 8 as transmutations. Thus we define now “-1 * (8.31) B =?*= (B ) :

5’ 5

h)

5’3;

178

ROBERT CARROLL

( c f . (8.30)).

Then ( r e c a l l B f ( r ) = ( B ( s , r ) , f ( s ) )

w i t h B(s,r) = 0, r > s )

co

(8.32)

( B f , h ) = (f,;h)

= (f(r),

B(s,r)h(s)ds) = ( Irf(r)B(s,r)dr,h(s)) Jr ’0 Thus t h e a d j o i n t n e s s r e l a t i o n (8.31) i s determined by k e r n e l a c t i o n as i n (8.32) and k e r B i s B(s,r) w i t h a c t i o n i n t h e r v a r i a b l e .

Now g i v e n t h e

t r a n s f o r m G d e f i n e d i n Remark 8.4 w i t h i n v e r s e determined by (8.12) (based on Theorem 1.7.5)

l e t us g i v e a f o r m a l v e r i f i c a t i o n o f t h e r e l a t i o n 1 We w r i t e o u t gg = g and Egl = g as suggested by Theorem 1.7.12. (8.33)

(

Recall B = f(r)).

1

6(rYs),g(v,-k,r))

2’

= g (v,-k,s);

(F(r,s),g

from (8.31) and d e f i n e now

1

(v,-k,s))

=

?#

=

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

so t h a t g f ( s ) =

= (

ii‘= 2-l

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

Then from (8.33) we o b t a i n f o r m a l l y

(8.34)

GBf =

Gl’if

((

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

= ( ( ;(r,s),g

1

(v,-k,s)

) , f ( r ) )=

(

= Glf;

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

= Gf

L e t f = gh i n t h e f i r s t e q u a t i o n and one has GBgh = G1gh which by t h e second e q u a t i o n must be Gh.

S i m i l a r l y i f f = & i n t h e second e q u a t i o n we ob-

’ib = Cslp which i s Glv by t h e f i r s t equation. Given a s u f f i c i eV n t l y 1 r i c h supply o f f u n c t i o n s t o which these e q u a t i o n s apply we have t h e n B =

t a i n p;

B - l and consequently by a d j o i n t n e s s

CM0REnl 8.11, Glef

=

??’.

From Theorem 1.7.12 one o b t a i n s (8.34) ( i . e . GBf = G,f = B - l and

= G f ) which l e a d s t o

We w r i t e % = i-’.

and

V

now B = 8 f o r con-

s i s t e n c y i n n o t a t i o n and n o t e t h a t B and B a r e c l e a r l y t r a n s m u t a t i o n s . Now we r e c a l l t h e K-L i n v e r s i o n f r o m 96 (Theorem 6.1 and Remarks 6.2 and 6.8) which i n f a c t i s a h y b r i d v e r s i o n o f t h e c l a s s i c a l s i t u a t i o n t o f o l l o w . We r e c a l l t h e background n o t a t i o n and machinery from

91.7.

Thus we suppose

5

now t h a t corresponding t o two o p e r a t o r s and 3’ we have d p ( v ) = ^p(v)dv and 1 2 dp ( v ) = G1(v)dv where e.g. : ( v ) = 2 i v d v / T f ( v , - k ) f ( - v , - k ) on [0,im) ( c f . (1.7.21)).

We r e f e r t o such cases as h a v i n g a b s o l u t e l y continuous spectrum = 0 f u r n i s h e s an example ( c f .

and know t h a t t h e f r e e problem t h a t fo(v,-k)

has n o zeros f o r Rev > 0 ) .

91.7 t o see

Other examples c o u l d be p r o v i d e d

( c f . here [Bdl; Bel; F u l ; R f l ; Dcl; Ne6; Sa10-131 f o r i n v o r m a t i o n on Regge poles etc.). (8.35)

Consider now t h e f o r m a l c a l c u l a t i o n based on Theorem 1.7.12 1 g (v,-k,r)

= ( ( g ,g

s

1

)

r p

,g

s

)

1

= ( g r ,( gvs ’g’)) s p

N

Q TYPE OPERATORS

179

Such a f o r m u l a i s c o n s i s t e n t w i t h a formal i d e n t i f i c a t i o n g(p,-k,s))

f*

=

? to

=

( 0 )

(g(v,-k,s),

which a l s o a r i s e s i f we t a k e t h e c o m p o s i t i o n G 0 G - l

&(v-v)/$(v)

f*

be v a l i d f o r a s u i t a b l y l a r g e c l a s s o f f u n c t i o n s

Indeed ( c f . Remark 8.4) s e t t i n g T ( v , u ) =

(

(cf. [ K i l l ) .

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

A

(8.36)

606-l? =

(

g(v,-k,s),(

f(u),g(u,-k,~))~) =

(

;(u),T(~,u))~

REmARK 8-12, As i t stands ( 0 ) i s somewhat u n s a t i s f a c t o r y because t h e hehavi o r o f g(v,-k,s) near s = 0 would r e q u i r e some d e l i c a t e c o n s i d e r a t i o n s i n o r d e r t o e s t a b l i s h a meaning f o r t h e l e f t s i d e .

We w i l l circumvent t h i s be-

l o w and p r e s e n t (*) i n a d i f f e r e n t f o r m i n c o n n e c t i o n w i t h a g e n e r a l i z e d Kontorovic-Lebedev (K-L) t h e o r y . L e t us use

h e u r i s t i c a l l y however ( f o r

( 0 )

5 and 5’)

and c o n s i d e r Bg

1

where

1

B i s g i v e n by (8.32)

( r e c a l l t h a t Egl = ( ? ( r , s ) , g (v,-k,s)) = g(v,-k,r) 1 where z ( r , s ) = ( g ( p , - k , r ) , g (p,-k,s)) 1 ) . Then w r i t e ( B ( r , s ) = 0 f o r s > r ) r,

(8.37)

Bg’ =

(

dr,s),g

1

(v,-k,s))

= ( g(u,-k,r),Tl

(v,u)

)p

= [P^/$llg(v,-k,r)

T h i s l e a d s t o t h e f o l l o w i n g f o r m a l c o n c l u s i o n based on t h e use o f Remark 8.19 f o r a p r o o f n o t i n v o l v i n g

E’: 5’

rr/

-+

Q i s c h a r a c t e r i z e d by Bg’ = ($/phl)g

Now t h e f u n c t i o n which p l a y s t h e r o l e o f t o be Wr where (**) @(v,k,r) be a n a l y t i c f o r Rev > 0).

(see

( 0 ) )

&HE(DREIII 8-13, Given a b s o l u t e l y continuous s p e c t r a f o r that B =

( 0 )

$’

it follows “1 (whereas Bg = 9 ) .

and

Q / c ( - k ) i n t h e QA t h e o r y i s g o i n g

@

k

Q

= q(v,k,r)/f(w,-k)

(which by Lemna 1.7.3 w i l l

T h i s a r i s e s i n s e v e r a l c o n t e x t s and we c o n s i d e r

f i r s t t h e i n v e r s i o n formulas o f Theorem 1.7.5 and Remark 8.4 when dp = $dw and use (1.7.11)

i n t h e form (+) f ( v , - k , x )

= -(1/2v)[f(-v,-k)p(v,k,x)

-

A

f(v,-k)q(-v,k,x)]

(we r e c a l l a l s o f ( - v , - k , x )

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

i s even i n w and f o r m a l l y

ii la

=

-(i/T)

= f(v,-k,x)).

Thus f ( w ) =

A

vf(v)@(v,k,r)dv

W

Here we use (+) and observe t h a t

im 4 vf(v)@(-v,k,r)dv

-Liw

=

im4

limwf(v)@(w,k,r)dv.

A

Consequently we have ( f ( v ) = G f ( v ) )

EHE6REm 8.14.

( G e n e r a l i z e d K-L theorem).

Given a b s o l u t e l y continuous spec-

t r u m dp = G(v)dv t h e G 4 - l i n v e r s i o n o f Remark 8 . 4 has t h e f o r m (8.38).

ROBERT CARROLL

180

EXMIPLE 8.15.

In the c a s e

r =0 we have

t h e c l a s s i c a l K-L i n v e r s i o n and we

w i l l make further comments on this l a t e r ( c f . [ J b 2 , 4 ; Kil; S a l ] ) .

In t h i s 2 = [ - v dv/

2

c a s e the "free" measure i s d p o ( v ) = ( 2 i / n ) [ v d v / f o ( v , - k ) f o ( - v , - k ) ] k r ( v t l ) r ( - v t l ) ] = -[vSinawdv/nk]

( s i n c e r ( z ) r ( l - z ) = a / S i n a z ) and

@

0

(v,k,r)

Thus w r i t i n g x-'f(s) = G ( s ) and i n p l a c e 1 of ? ( v ) s e t t i n g r ( v ) = 10" G ( s ) H v ( k s ) d s we o b t a i n from ( 8 . 3 8 ) , r G ( r ) = im 2 Wim v g ( v ) J v ( k r ) d v . The t r a n s f o r m a l s o o f t e n a p p e a r s i n terms o f HV and i n this d i r e c t i o n s i m p l y write k -+ k e x p ( i n ) above ( r e c a l l J v ( z e x p ( i n ) ) = exp ( i a v ) J V( z ) and Hv1 ( z e x p ( i n ) ) = e x p ( - i n v + l ) ) H2v ( z ) ) . Then =

(nr/2k)'exp(-4in(v-Js))Jv(kr).

I G(s)H? k s ) d s ; m

(8.39)

z(v) =

rG r )

JO

=

-;Ii

im

vE(v)Jv(kr)dv m

Using Theorem 8 . 1 4 we can g ve a more s a t i s f a c t o r y v e r s i o n o f follows.

We combine ( 8 . 3 8 ) w i t h F ( v )

Hence our r e p l a c e m e n t f o r

(0)

=

P;f(v) a s (cf [Kill f o r

( 0 )

above a s A

;+

f

-+

f)

i s t h e following a s s e r t i o n .

Based on t h e i n v e r s i o n i n Theorem 8 . 1 4 one has f o r m a l l y ( i n n terms of a c t i o n on a s u i t a b l e c l a s s o f even f u n c t i o n s f ) , ~ ( u - v )= - ( i p / n )

EHE0REfl 8.16.

; 1 g(v,-k,r) @(u,k,r)dr/r. Let us use Theorems 8 . 1 4 and 8.16 t o g i v e now a f o r m a l p r o o f o f t h e f o l l o w i n g fundamental theorem ( a more r i g o r o u s d i s c u s s i o n a p p e a r s below i n Remark e assume here the a p p l i c a b i l i t y o f the G i n v e r s i o n i n Theorem 8.14 8.18). W 1 t o ; ( v , s ) = G [ ~ ( r , s ) ] ( v ) = B [ g ( v , - k , r ) ] ( s ) = g ( v , - k , s ) ( c f . Theorem 1 . 7 . 1 2 and n o t e by Remark 8.19 below t h a t t h i s can be viewed r o u t i n e l y ) .

T1

5

EHE0REfl 8.17, Given and with absolutely continuous s p e c t r a i t follows 1 1 t h a t B[@ ( v , k , s ) / s ] ( r ) = ( 6 ( r Y s ) , @ ( v , k , s ) / s ) = @ ( v , k , r ) / r .

w i t h B(r,s) = 0 f o r s > We r e c a l l B ( r , s ) = ( g ( v , - k , r ) , g ' ( v , - k , s ) ) P 1 r and one knows from Theorem 1.7.12 t h a t ( B ( r , s ) , g ( p , - k , r ) ) = g ( p , - k , s ) which we write i n t h e form s ( v , s ) = ~ [ ~ ( r , s ) ] ( v so ) t h a t by Theorem 8 . 1 4 one h a s f o r m a l l y ( s e e Remark 8 . 1 9 ) ) Ph006:

NOW m u l t i p l y ( 8 . 4 1 ) by

1

( p , k , s ) / s and i n t e g r a t e ( u s i n g Theorem 8 . 1 6 ) t o g e t

(7 TYPE OPERATORS

181

8.18, We s k e t c h here a n o t h e r p r o o f o f Theorem 8.17 based on a t e c h n i q u e developed i n [C50,64,65]. Thus from t h e c h a r a c t e r i z a t i o n o f B i n

RE!MRK

Theorem 8.13 we w r i t e (assume no zeros o f f ( v , - k )

f o r Rev > 0 i n c o n n e c t i o n Bg' = ($/Cl)g = [ f 1 ( v , - k ) f 1 ( - v ,

w i t h h a v i n g a b s o l u t e l y continuous spectrum), - k ) / f ( v ,- k ) f ( - v ,- k 19 = [ f 1 ( v, - k ) f 1 ( - v ,- k 1/ ( - 2 v r ) 1( CP ( v ,k ,r )/ f ( v

- k 11 -

L e t us w r i t e again @ ( v , k , r ) = q ( v , k , r ) / f ( v y - k ) , [p(-v,-k,r)/f(-v,-k)]). m u l t i p l y by a f a c t o r - 2 v / f 1 ( v , - k ) f 1 (-v,-k), and w r i t e o u t t h e B a c t i o n as (8.43)

(

B(r,s),[@

1

(v,k,s)-*

1

(-v,k,s)l/s)

We r e c a l l by Lemma 1.7.3 t h a t p ( v , k , r )

= [@(v,k,r)

and f ( v , - k )

-

@(-v,k,r)l/r

are analytic i n v f o r 1 ( B(r,s),@ ( v , k , s ) / s )

Rev > 0 (and continuous up t o Rev = 0 ) and we s e t now = *(v,k,r) = 0- =

with

@+

O(-v,k,r)

= 0(v,k,r)

= *(v,k,r)

- @(v,k,r)/r.

Now (8.43) says 0,

f o r v i m a g i n a r y and 0, ( r e s p . 0-)i s a n a l y t i c f o r Rev > 0

( r e s p . Rev < 0).

By standard theorems on a n a l y t i c c o n t i n u a t i o n ( c f . 56)

0, and 0- a r e a n a l y t i c c o n t i n u a t i o n s o f each o t h e r and r e p r e s e n t an e n t i r e

f u n c t i o n 0 i n C.

Bounds can be o b t a i n e d f o l l o w i n g [Bdl;

Ne6; R f l ] e t c . and

we o n l y g i v e a q u i c k i d e a here v i a t h e e s t i m a t e s o f §1.7 ( b e f o r e (1.7.19)) based on

f

'p - q o y

%

fo, e t c .

f o r m u l a we have @ ( v , k , r ) / r ev(kr/v)v.

0 for real v

Q

It f o l l o w s t h a t @ ( v , k , r ) / r -f

t o 0 f o r real B

m.

S i m i l a r l y @(-v,k,r)/r

+ -m.

-

Thus as IvI Q0(v,k,r)/r

+

>

0, f r o m S t i r l i n g ' s

-4v -4exp(-$iriv)

i s bounded f o r Rev

2 0 and tends t o

i s bounded f o r Rev 5 0 and tends

The same c o n c l u s i o n s a l s o f o l l o w f o r *(v,k,r)

one n o t e s t h a t 6 f u n c t i o n b e h a v i o r o f R(r,s)

-

-, Rev

+exp(kin)(kr)

Consequently t h e e n t i r e f u n c t i o n 0 i s c o n s t a n t and s i n c e 0, -+

and

a t r = s i s e a s i l y accomodated. +

0 f o r real v

f o r example we have 0 = 0, which y i e l d s Theorem 8.17.

REmARK 8.19,

L e t us show t h a t t h e c h a r a c t e r i z a t i o n Bg' = ($/;l)(v)g

rem 8.13 can be o b t a i n e d w i t h o u t u s i n g

(0);

o f Theo-

o u r d i s c u s s i o n a l s o shows t h a t

(8.41) i s e s s e n t i a l l y a r o u t i n e c a l c u l a t i o n . Thus (assuming a b s o l u t e l y con1 2 tinuous spectra f o r u s i n g t h e f o r m u l a f o r dp (u) = 2 i u du/ and 1 1 1 nf (u,-k)f (-u,-k) and ( + ) f o r f we have

5')

182

ROBERT CARROLL

R e f e r r i n g back t o t h e c l a s s i c a l s i t u a t i o n ( c f . Example 8.15)

REllIARK 8-20.

we w i l l make a few more comments about t h e K-L i n v e r s i o n ( i n t h e form (8. 39)).

We r e f e r here e s p e c i a l l y t o a p e n e n t r a t i n g a n a l y s i s i n [JbZ] where i t

i s shown how t o deal w i t h what a r e o c c a s i o n a l l y p u r e l y f o r m a l expressions, by i n t r o d u c i n g convergence f a c t o r s e s s e n t i a l l y t o j u s t i f y a s h i f t i n cont o u r , and s i t u a t i o n s a r e i n d i c a t e d where t h e i n v e r s i o n f o r m u l a s do n o t work The approach i n [JbZ] s t a r t s w i t h hypotheses on G i n (8.39) and

at all.

v e r i f i e s t h e passage G

+

-f

2

G when a convergence f a c t o r exp(ev ) i s i n s e r -

t e d i n t h e v - i n t e g r a l and a l i m i t as

E +

0 i s taken.

The approach i n [ K i l l

on t h e o t h e r hand makes hypotheses on 6 ( v ) and v e r i f i e s a passage v i a (8.39).

E

+ G

+

5

One problem t h a t a r i s e s i s t h a t " n i c e " f u n c t i o n s G sometimes

give r i s e t o

which do n o t s a t i s f y t h e hypotheses i n [ K i l l f o r example so

t h a t t h e i n v e r s i o n i n t e g r a l i n (8.39) i s f o r m a l .

We r e f e r t o [Jb2;

Kill

f o r a more complete d i s c u s s i o n and here we want t o s k e t c h a procedure i n [ K i l l which shows how a f o r m u l a ~ ( u - v )= 4vexp($in(v-v))/; H1v ( k r ) J v ( k r ) d r / r based on Theorem 8.16 r e p r e s e n t s a 6 f u n c t i o n o p e r a t i o n a l l y .

The hypotheses

i n [ K i l l a l l o w one t o work w i t h f u n c t i o n s w(v) i n a s t r i p [Rev1 < 6 and t h e i m

w(v) i n v e r s i o n formulas a r e w r i t t e n as ip(r) = -4[i,vw(v)exp(~iav)Jv(kr)dv; 2 = I ; ip(r)exp(-iilrv)Hv(kr)dr/r ( n o t e t h i s has t h e same form as (8.39) i f one V

w r i t e s G ( r ) = i p ( r ) / r and G = w ( v ) e x p ( + i n v ) ) . wv and w

P

L e t us observe now t h a t i f

a r e " c y l i n d e r " f u n c t i o n s o f kx s a t i s f y i n q t h e Bessel e q u a t i o n i n

t h e f o r m (sw;)'

t

-

(l/x)[k2x2

p2]wv = 0

then, m u l t i p l y i n g t h e v e q u a t i o n

and t h e v e q u a t i o n by w and s u b t r a c t i n g , one has a f t e r an i n t e g r a t i o n 1-I bV - xw;wy]la b + ( v 2 -v 2)I a wvwvdx/x = 0. Taking Rev > \ R e v \ and u s i n g [xw;wv

by w

asymptotic properties o f w

u

=

2

J ( k x ) and wv = Hv(kx) n e a r x = 0 and x =

v

m

one o b t a i n s a k i n d o f D a r b o u x - C h r i s t o f f e l f o r m u l a m

(8.46)

xW(wv,wv)

;1

= (2i/n)e4in(v-v)

=

(v2-v2)J

Ht(kx)Jv(kx)dx/x

0

Now from (8.39) one can w r i t e ( u s i n g (8.46))

The hypotheses i n [ K i l l a r e made on ;(v)

( v i a w ( v ) ) and a l l o w one t o s h i f t

t h e c o n t o u r t o t h e r i g h t so t h a t t h e i n t e g r a l i s o v e r t h e l i n e Rev = i n t h e p i c t u r e below and p u t t i n g (8.48)

T = -ie

4 i IW

~

2lr

I [&

L v-v

t

o1

= wexp($iau) we have ( r e c a l l w i s even)

---]w(u)dp 1 u+v

=

e+i TIU

-[ 2lrl

f w(v)dv L LJ - v

,

.-

Q TYPE OPERATORS

183

-6

T h i s procedure i s j u s t i f i e d by t h e hypotheses made on w i n [ K i l l p l u s some f u r t h e r e s t i m a t e s on t h e Bessel and Hankel f u n c t i o n s which a l l o w t h e i n t e r change i n i n t e g r a t i o n i n (8.47). L e t us add here a few c o n s t r u c t i o n s based on PDE techniques f o r t h e type o p e r a t o r s ( c f . [C44]). F o l l o w i n g c o n s t r u c t i o n s i n SSl.11 and 2.1 f o r gene r a l i z e d t r a n s l a t i o n s we s e t (compare w i t h U(x,y) = (8.49)

U(r,s) =

[?(v

)/ s (v

(

$(k),p;(x)pk(y)Q

,-k,l ) l s ( v , - k , r ) g ( ~ , - k , s ) d p

A

( t h u s U(r,s)

)u )

h

= ( Cf(v)/g(v,-k,l

)I,g(.,-k,r)s(v,-k,s))

( f o r s u i t a b l e f ) and one assumes g ( v , - k , l )

P

) where f ( v ) = P;f(v)

f 0 on t h e spectrum o f

6. The

p o i n t here i s t o i s o l a t e a p l a c e o f e v a l u a t i o n , i n t h i s case r = 1 o r s = 1 A

( f o r reasons t o appear below), and t h e n t o t r a n s p o r t t h e technique f o r Q type operators.

L e t us proceed f o r m a l l y t o observe t h a t e v i d e n t l y CrU(r,s)

v

=

A

QsU(r,s) and U ( r , l ) = ( f ( v ) , g ( v , - k , r ) )

w r i t e U(r,s) = ( y ( r , s , t ) , f ( t ) ) g(v,-k,s)g(v,-k,t)) f ( t ) )w i t h r ( s , t ) a r l y D,U(r,s)Is=l

P

=

f ( r ) w i t h U(1,x) = f ( s ) .

I f we

= ( [g(v,-k,r)/g(v,-k,l)],

i t f o l l o w s f o r m a l l y t h a t DrU(rys)lr=l

= ( [Dg(v,-k,l

= Cf(s) = (r(s,t),

)/g(v,-k,l)],g(v,-k,s)g(v,-k,t))

and s i m i l -

P

= Cf(r). A c t i n g on s u i t a b l e f an easy c a l c u l a t i o n shows

t h a t C commutes w i t h holds.

P

w i t h y(r,s,t)

'ii and

a c o m p a t i b i l i t y condition Cf(1) = f ' ( 1 ) also

Consequently U(r,s) = T F f ( s ) i s a general zed t r a n s l a t i o n i n a sense

i n d i c a t e d e a r l i e r and we s t a t e t h i s as EHE@REril 8-21,

F o r s u i t a b l e f, U(r,s) = T g f ( s ) d e f ned by (8.49) i s f o r m a l l y

a g e n e r a l i z e d t r a n s l a t i o n f o r r w i t h U(1,s) where C i s g i v e n as above w i t h k e r n e l The q u e s t i o n o f whether t h e e q u a t i o n

= f(s

and DrU(l,s)

=

Cf(s)

r.

FrU = FsU p l u s

U u n i q u e l y w i l l be discussed now v i a Theorem 1.4.3.

d a t a a t r = 1 determine Thus l e t A and C be

( s u i t a b l e ) l i n e a r o p e r a t o r s and c o n s i d e r t h e Cauchy t y p e problem ( 0 0 ) -1 Qsq(r,s) = Trp(r,s); ~ ( 1 , s ) = Af(s); Drp(l,s) = C f ( s ) . We t h i n k o f 0 < r,s c

m

so f, A f , and C f a r e t o be g i v e n on (0,~)( t h i s i s discussed below) and

184

ROBERT CARROLL

the situation f o r U =

r

5’

i s d e s c r i b e d when

q

m

Q, A

=

=

1, and C i s g i v e n v i a

Now i n o r d e r t h a t one may t r a n s p l a n t t h e p r e v i o u s methods i t

as above.

i s necessary t o have a uniqueness theorem f o r s o l u t i o n s o f

n

make t h e change o f v a r i a b l e s D5, e t c . and

(0.)

a l l y here A

= x

5

= l o g r and

(om).

L e t us

= l o g s so t h a t r D r = Dn,

sDS =

can be c o n v e n i e n t l y t r e a t e d as f o l l o w s ( t a k e more gener-

,Q

y(r)],

-

r 2 D 2 + (n-1)rD + r2[k2

=

etc.).

Then w r i t e q ( r ) = ~ ( T - I ) ,n o t e t h a t 0 < r,s <

= ( r s ) l - bZnp(n,5). A

Set q ( r , s ) m

corresponds

t o - m < TI,< < m , and f o r f u n c t i o n s o f one v a r i a b l e w r i t e p ( r ) = rl-’n$(n). ^In $ which reduces t o (AA) [D 25 + exp25 Then t h e e q u a t i o n ( 0 0 ) becomes Q q = 5 ( k 2 - 4 1 ( ~ ) ) ] $ ( n , ~ )= [Dn2 + exp2n(k2 - n 4 ( n ) ) ] $ ( n , ~ ) . L e t us t h i n k o f data

6

-

f, A f , and C f g i v e n on 0 < s < -m

5

<

<

(see below).

which g i v e s r i s e t o d a t a f, A f , and C f on

m

Note t h a t t h i s i s q u i t e d i f f e r e n t f r o m p r e v i o u s

problems where data g i v e n on [ 0 , m )

c o u l d be extended i n v a r i o u s ways t o (AA) w i t h n 5 0 5 1 ) s e p a r a t e l y ; by standard t h e o r y , f o r

L e t us t h i n k o f s o l v i n q t h e Cauchy problem f o r

(-m,m).

1% r 5 1 )

and w i t h

reasonable

T-I

50

(T,

0 < r

4, unique s o l u t i o n s

t r i b u t i o n s (depending on d a t a

e x i s t i n v a r i o u s spaces o f f u n c t i o n s o r d i s -

-

c f . [C63]).

We d e f i n e a t r a n s m u t a t i o n

A A

B f ( n ) = G ( n , O ) f o r q 5 0 and f o r

6

by

5 0 r e s p e c t i v e l y which determines a t r a n s -

T-

m u t a t i o n B f o r r 2 1 and 0 < r 5 1 r e s p e c t i v e l y ; t h e two p a r t s o b v i o u s l y

6

= 5 :n- 1 The c o n n e c t i o n o f B w i t h w i l l be Sn-lW l - % n = r Qr , e t c . S i m i l a r l y w r i t e = skn- 1AS 1-%n

w i l l f i t t o g e t h e r a t r = 1.

Bsl-%n w i t h

and

A n

A

QB, Q

=

;= s%n-lCsl-%n.

ii

I t i s e a s i l y seen t h a t e v e r y t h i n g f i t s t o g e t h e r and n A A 4 -1 A1 if A and C commute w i t h Q t h e n A and E = C - (1-4n)A commute w i t h Q ( n o t e

-

D~$(o,E) = i\f(E)

&HEOREm 8.22,

A

f ( E ) = skn-’f(s),

Suppose A and C commute w i t h

t h e u n d e r l y i n g Cauchy problem ( f o r suitable f). N

B’lj’

Q (i.e.

etc.).

= {B

Hence

5’

w i t h $(0,5)

(AA)

Then B f ( r ) = q ( r , l )

and assume unique s o l u t i o n s o f A A

= z f ( t ; ) and Dn$(O,E)

Ef(5) -1 i s a well defined transmutation Q + =

a c t i n g on such f ) .

REmARK 8-23, L e t us mention a g e n e r a l i z e d t r a n s l a t i o n f o r m u l a i n c o n n e c t i o n PLI = r Z ( r n - ’ u ’ ) ’ / r n - ’ ; f o r convenience we t a k e n = 3 b u t i n 59 one sees t h a t any n can be t r e a t e d i n t h e same way. We f o l l o w Theorem

w i t h the operator

8.21 and l o o k f o r U(r,s)

s a t i s f y i n g F ( D r ) U = F ( D s ) U w i t h say U(1,s)

Denote t h e M e l l i n t r a n s f o r m by M and s e t ? ( o ) = -1 -1 JI f ( p ) = p M [ f ( ~ )=] ( 1 / 2 n i ) I p-‘-’;(o)dU = I c-im

+

cti-).

p - 1 dw. N

It

f(p)p‘dp

Since ?(Dr)r-‘-’

=

u(u+l)r-‘-’

= M[pf(p)];

f(U)p-‘-’dw

As a g e n e r a l i z e d t r a n s l a t i o n c o n s i d e r U(r,s) = i*r-‘-’

= f(s).

=

(I has l i m i t s I f ( o ) r -0-1

we see t h a t U(r,s)

sat-

h)

i s f i e s P(Dr)U = P(Ds)U and U(1,s)

= f(s) with U(r,l)

=

f ( r ) . On t h e o t h e r

5 TYPE hand DrU(r,s)lr=l -(a+l)M[pf]

OPERATORS

- i ( a + l ) f ( a ) s -a-1 dw

=

185

2 Since however M[P f ' ] = 2 we o b t a i n C f ( s ) = ( l / s ) ( s f ' ) = s f ' ( s ) . Thus one

= -(a+l)?(o)

=

Cf(s).

has t h e c o m p a t i b i l i t y C f ( 1 ) = f ' ( 1 ) as b e f o r e and consequently

CHE@RETII 6-24, U ( r , s )

=

T g f ( s ) d e f i n e d by IJ(r,s)

=

f f(a)r-a-ls-a-ldw

g e n e r a l i z e d t r a n s l a t i o n f o r F w i t h U(1,s) = f ( s ) and DrU(l,s)

is a

Cf(s) = s f ' .

=

N

EXAIIIPLE 8-29, Consider t h e operator: Qo corresponding t o 7 = 0 ( i . e . Qou = 2 2 2 x u" + 2xu' + k x u ) . The fundamental q u a n t i t i e s v0, fo, e t c . a r e g i v e n i n 1 v2 v+4 Example 9.16 and i n p a r t i c u l a r one knows t h a t Hv(k) = g o ( v y - k , l ) / t 5 n k ) i N

w i l l n o t have zeros f o r i m a g i n a r y v so o u r c o n s t r u c t i o n s a r e p e r m i t t e d ( c f .

[Cvl;

MbZ]).

F o l l o w i n g t h e model o f Q t h e o r y one expects t o have a formula o f t h e form (T!f(s),h(s)

)

= (f(s),Trh(s))

We w i l l s k e t c h a p r o o f o f

f o r s u i t a b l e f,h.

t h i s by f o l l o w i n g t h e procedure developed i n Theorem 1.12.5. U(r,s)

as i n (8.49) and c o n s i d e r

one has $ ( l , v )

=

(0)

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

from Theorem 8.21 (**) DrJ/(l,v)

#(r,v)

Take T g f ( s ) =

= ( g(v,-k,s),TLf(s)

=

(g(v,-k,s),DrU(l,s))

=

Evidently

).

= t ( v ) = P ; f ( v ) ( s i n c e U(1,s)

= f ( s ) ) and

(g(v,-k,s),Cf(s))

A

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

=

))

=

(g(v,-k,s),(ff~)G(u),g(u,-k,s)

where G ( u )

)p)

A

= Dg(v,-k,l)/g(p,-k,l)

fore),

(note ( r ( s , t ) , f ( t )

) =

(f(p),G(u)g(u,-k,s)

Now we suppose t h e i n v e r s i o n t h e o r y f o r A

i n t h e form h

(fG)(u)

€;

+

h

n -f

h to

A

be e s t a b l i s h e d f o r h i n some s u i t a b l e c l a s s o f f u n c t i o n s H . h

from be-

)p A

P;

I f we assume A

t h e n (**) y i e l d s Dr$(l ,v) = ?(v)G(v) = [Dg(v,-k,l)/g(v,-k,l

qr$ =

F u r t h e r from ( 0 ) one has (csg(v,-k,s),Trf(s)) = A2 $

(x 2

)If.

A/

(g(v,-k,s),crTgf(s)> = (g(v,-k,s)QSTrf(s)) - c f . remarks b e f o r e Theorem 2.2).

=

= v2 - k

S 2 Hence by uniqueness o f s o l u t i o n s t o Grd' = A 9 w i t h $(1) and $'(l) prescribed

(++) J/(r,v)

=

Now we w i l l need Lem-

[g(v,-k,r)/g(v,-k,l)](f(s),g(v,-k,s)>.

ma 8.26 whose p r o o f f o l l o w s .

Consider t h e f u n c t i o n R ( r , s ) = g(v,-k,r) N

g(v,-k,s)/g(v,-k,l).

Iv

E v i d e n t l y Qrn=QSn and n ( 1 , s ) = g(v,-k,s)

Dra(l , s ) = [ D g ( v , - k , l ) / g ( v , - k , l ) ] g ( v , - k , s ) . i n t h e f o r m C[g(v,-k,t)](s)

C[g(v,-k,t)]

with ( 6 )

L e t us compute CP(1 ,s) = = (g(v,-k,t),r(s,t))

= (g(v,-k,t),

) / g ( ~ > - k , l ) I ) ) = (S(v,-k,t),(g(p,-k,t), P Assume ( f o r f i x e d s ) where y ( p , s ) = G(u)g(u,-k,s).

(g(u,-k,t)g(u,-k,s),CDg(u,-k,l w

$(p,s) ) ) P

where p; a c t s and t h e n C[g(v,-k,t)](s)

I\

Y(V,S) H

=

T(v,s)

E

= [Dg(v,-k,l)/g(v,-k,l)]

I t f o l l o w s f r o m ( 6 ) t h a t Drn(l,s) = C[Q(l,-)l(s) and g i v e n s u i t g(v,-k,s). a b l e uniqueness i n t h e u n d e r l y i n g Cauchy problems as i n Theorem 8.22 we

have proved v i a Theorem 8.21

LEmA 8-26. Under t h e hypotheses i n d i c a t e d one has f o r m a l l y Q(r,s)

=

186

ROBERT CARROLL

= Tgg (v ,- k, s ) = g ( v ,- k ,r ) g ( v ,-k, s ) / g ( v ,-k, 1 )

.

Using Lemma 8.26 we o b t a i n now from (++) t h e r e l a t i o n ( = = ) ( T i f ( s ) , g ( v , - k , s ) ) r = $(r,v) = ( f ( s ) , n ( r , s ) ) = ( f(s),T,g(v,-k,s) ). Then l e t H(v) = Gh w i t h h = G-lH =

H(v),g(v,-k,s))p,

(

multiply

(.=I

by H, t a k e p brackets, and i n t e r We n o t e a l s o f r o m (-)

change o p e r a t i o n s t o o b t a i n Theorem 8.27 below. that

(

r

Tsg(v,-k,s),H(v))p

t),H(v))

P

=

((v(r,s,t),s(v,-k,t)),H(v))p

(v(r,s,t),h(t))

) =

=

= (v(r,s,t),(

EHE0REfl 8-27, With t h e hypotheses i n d i c a t e d one has ( f h(s)) = (f(s),TCh(s))

= (h

*

g(v,-k,

TCh(s).

*

h)(r) =

(

T';f(s),

f)(r).

From t h e d e f i n i t i o n o f g e n e r a l i z e d c o n v o l u t i o n i n d i c a t e d i n Theorem 8.27

Ti i n (8.49) one has now f o r s u i t a b l e f,h, ( f * h ) A h I h(s)C I [~(v)/g(~,-k,l)1g(~,-k,s)g(vy-k,r)d~1ds = I f ( v ) h ( v ) g ( v , - k , r ) d p /

and t h e c o n s t r u c t i o n o f =

M

g(v,-k,l).

Consequently, g i v e n s u i t a b l e f h / g ( v , - k , l ) For s u i t a b l e f , h w i t h

EHE0Rm 8.28,

f^ =

G f etc. (f

E

*

H etc. there follows *A

h)n = f h / g ( v , - k , l ) .

We showed i n 53 how t h e fundamental t r a n s m u t a t i o n s B and

i n the

-

theory

c o u l d be o b t a i n e d v i a p a r t i a l d i f f e r e n t i a l equations and we w i l l now use Theorem 8.22 t o s i m i l a r l y c h a r a c t e r i z e o u r 8 and

o f the

k e r n e l s a r e g i v e n i n 81.7 and e a r l i e r i n t h i s s e c t i o n . a g e n e r a l i z e d t r a n s l a t i o n o f t h e form (8.49) f o r

5'

4 theory

whose

Take U1(t,s) t o be

and c o n s i d e r f i r s t

-

y(r,

N

s) = (z(r,t),U 1(t,s)). F o r m a l l y f o r s u i t a b l e f one expects Qr; = ( T r F ( r , t ) , 1 1 -1 1 -1 1 U ( t , s ) ) = (?j$(r,t),U ( t , s ) ) = ('iS(r,t),QtU ( t , s ) ) = Os( R ( r , t ) , U (t,s)) = -1Qsq a n d ? ( r , l ) = ( x ( r , t ) , f ( t ) ) = zf(r) ( c f . remarks b e f o r e Theorem 2.2). 1 Now t h e corresponding Cauchy data i s 7 ( l , s ) = (F(l,t),U ( t , s ) ) = ( ( g(v,-k, 1 I t = z f ( s ) and we use Lemma 8.26 and Theorem 8.27 l ) , g ( v , - k , t ) ) 1, T,f(s)) 1 rp 1 1 a p p l i e d t o Ts t o w r i t e f o r m a l l y ( n o t e U ( t , s ) = U ( s , t ) e t c . ) ? ( l , s ) = r f ( s ) = ( r ( s , t ) , f ( t ) ) w i t h z ( s , t ) = ( [g(v,-k,l)/g 1 ( v , - k , l ) l g 1 (v,-k,s), 1 g (v,-k,t)) T,f(s)) t

S i m i l a r l y one has Dry(l,s) I s 1 = (Dg(v,-k,l),( Ttg ( v , - k , t ) , f ( t ) ) ) 1.

P

= ((

Dg(v,-k,l),g

1

(v,-k,t))pl,

u

P

1 = Cf(s) = t?(s,t),f(t)).

Assuming uniqueness i n t h e u n d e r l y i n g Cauchy problem we have

EHEOREm 8.29. terized via

W i t h t h e hypotheses i n d i c a t e d t h e t r a n s m u t a t i o n B i s charac1 = f o r ;(r,s) = (?(r,t),U ( t , s ) ) w i t h z ( l , s ) = A f ( s ) and

cT 5;:

-4

D$(l,s)

-

N

w

N

N

= C f ( s ) where A and C a r e g i v e n v i a k e r n e l s A and C as above.

The c h a r a c t e r i z a t i o n of 8 w i t h k e r n e l B(r,s) t h e same way.

Thus t a k e 9 ( r , s )

-1

f o l l o w s t h a t Frq = 4,q

= (B(r,t),U

can be c a r r i e d o u t i n e x a c t l y 1

(t,s))

instead o f

as b e f o r e and q ( r , l ) = ( B ( r , t ) , f ( t ) )

above.

= Bf(r).

It The

TYPE OPERATORS

187

1 1 ( t , s ) ) = Af(s) = ((g(v,-k,l),g (v,-k,t))?, I s 1 Ttg ( v , - k , t ) , f ( t ) ) ) 1 = ( A ( s , t ) , f ( t ) ) and A ( s , t ) = 'TZf(s)) = (g(v,-k,l),( P 1 1 1 ( [g(v,-k,l)/g (v,-k,l)]g (v,-k,s),g (v,-k,t) ) p . Thus A ( s , t ) has e x a c t l y t h e i n i t i a l d a t a i s ip(1,s)

same f o r m as Ah(s,t)

s)

=

p1

r e p l a c e d by p .

= (B(l,t),U

with

p1

Cf(s) = (C(s,t),f(t))

KHEBREM 8.30acterized via

r e p l a c e d by p.

Similar calculations give

where C ( s , t ) has t h e same form as ? ( s , t )

Drip(l,

but with

Thus

W i t h t h e hypotheses i n d i c a t e d t h e t r a n s m u t a t i o n B i s char1 = ?lip f o r 9 ( r , s ) = (a(r,t),U (t,s)) with ~ ( 1 , s ) = Af(s)

Trip

and D p ( 1 . s ) = C f ( s ) where A and C a r e g i v e n v i a k e r n e l s A and C as above. F o r t h e G-L complex o f ideas w r i t e g ( v , - k , r ) = r g l = (;(r,s),g 1 ( v , - k , s ) ) 1 w i t h g (v,-k,t) = B g = ( c(u,t),g(v,-k,r)). M u l t i p l y t h e f i r s t e q u a t i o n by 1 g ( v , - k , t ) and t h e second by g ( v , - k , r ) ; t h e n t a k e p b r a c k e t s and equate t o 1 1 = ( ( B(u,t),g(v,-k,u) ),g(v,-k,r)) o b t a i n ( ( z ( r , s ) , g (v,-k,s) ),g ( v , - k , t ) ) N

P

= ( a(u,t),(g(v,-k,u),g(v,-k,r))

P

) =

B(r,t).

P

Reorganizing we o b t a i n

CHEORZFII 8-31, The G-L e q u a t i o n a s s o c i a t e d w i t h 'ij and E has t h e f o r m a ( r , t ) = (F(r,s),A(s,t)) w i t h A ( s , t ) = ( g1 (v,-k,s),g 1 ( v , - k , t ) ) p . 9.

&HE BERGIIIAN-CZLBERC (B-pi)

BPERAEBA AND piENERA&ZNG FUNCKZBW.

There i s

a c o n s i d e r a b l e l i t e r a t u r e on i n t e g r a l o p e r a t o r s which t r a n s f o r m a n a l y t i c f u n c t i o n s ( o r harmonic f u n c t i o n s ) i n t o s o l u t i o n s o f e l l i p t i c equations. H i s t o r i c a l l y t h e p r i n c i p a l impetus seems t o have been Bergman's e x t e n s i v e work on t h e s u b j e c t and subsequently i m p o r t a n t c o n t r i u b t i o n s were made by numerous authors; we c i t e here o n l y t h e s u m a r y t r e a t m e n t s [Bcl;

3; Hdl; Val].

Cn1,Z;

Gjl-

I n p a r t i c u l a r c e r t a i n ( d i r e c t and i n v e r s e ) problems i n s c a t -

t e r i n g t h e o r y have been i n v e s t i g a t e d u s i n g such o p e r a t o r s ( c f . [Cn2-5,7; Gjl]).

I n t h e process o f comparing and u n i f y i n g v a r i o u s methods and p o i n t s

o f view i n t r a n s m u t a t i o n t h e o r y and a p p l i e d problems i n geophysics ( c f . [Bb1,2;

C40; S a l ] ) we were l e d t o l o o k f o r a complete t r a n s m u t a t i o n a l f o r -

m u l a t i o n f o r what we s h a l l c a l l t h e B-G o p e r a t o r . show ( c f . [C41,42]) transmutation

g:

Thus i n p a r t i c u l a r we

how t h e B-G o p e r a t o r can be c h a r a c t e r i z e d as a c e r t a i n

P"+ Qn ( d e f i n i t i o n s below) whose k e r n e l can be r e p r e s e n t e d

by a s p e c t r a l p a i r i n g o f s u i t a b l e e i g e n f u n c t i o n s o f Pn and Qn.

This places

t h e B-G o p e r a t o r i n t h e c o n t e x t o f a general t r a n s m u t a t i o n t h e o r y f o r operat o r s o f t h e f o r m Qn and i n p a r t i c u l a r t h i s a l l o w s one t o use known i n f o r m a t i o n about t h e B-G o p e r a t o r t o produce t r a n s m u t a t i o n s and c o n n e c t i o n formul a s between s p e c i a l f u n c t i o n s .

Such a t r a n s m u t a t i o n t h e o r y i s i m p o r t a n t i n

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 o f Laplace o p e r a t o r s (and t h e a s s o c i a t e d

ROBERT CARROLL

188

s c a t t e r i n g problems in [Cnl-4,6,7; Gjl] f o r example) as well as in t r e a t i n g s c a t t e r i n g problems a t fixed energy i n quantum mechanics ( c f . [C43; Cel; Dc

1 ; Bbdl; Lrl; L17; Ne6,10,11; Sa10,13] and 551.7 and 2.8). The spectral variables which a r i s e i n t h e present case correspond t o complex angular momentum variables i n quantum mechanics. There a r e a l s o some i n t e r e s t i n g connections of t h e B-G theory w i t h c e r t a i n t o p i c s in generating functions and t h i s i s p a r t i a l l y developed here ( c f . a l s o [Bbgl; C541). We r e f e r t o t h e survey a r t i c l e s c i t e d f o r general background and s t a r t here with an equation ( c f . ( 1 . 7 . 2 ) )

where (9.2)

<

depends only on angle v a r i a b l e s .

pnu

The r a d i a l p a r t times r2 i s

2 n-1 2 r [ ( r u’)l/rn--’1= r urr + ( n - l ) r u r

=

Now when a ( r e a l valued) s o l u t i o n of ( 9 . 1 ) f o r n = 2 i s expressed i n terms of t h e Bergman i n t e g r a l operator of the f i r s t k i n d one has

1, 1

(9.3)

u(x,y)

=

E(r2,t)H(x(1-t2),y(l-t2))dt/(l-t

2

)

where H i s a harmonic function RENARK 9-1, There a r e many Bergman i n t e g r a l operators d i f f e r i n g by t h e in-

troduction o f an a r b i t r a r y a n a l y t i c function ~ ( z )i n t o t h e (complex) analysis. When v = 0 t h e Bergman representation and t h e Vekua representation based on a complex Riemann function a r e i d e n t i c a l and i t i s this Bergman operator ? where cp = 0 t h a t we r e f e r t o a s the Bergman operator ( c f . [ G j 2 ] ) . 2 2 Now following [Gj2,3] we r e w r i t e ( 9 . 3 ) as u(x,y) = h + /J ~ G ( r ~ 1 -) ho ( x u , 2 1 2 2 2 yo )do where h(x,y) = Ll H ( x ( 1 - t ) , y ( l - t ) ) d t / ( l - t ) i s harmonic ( c f . a l s o N Based on t h e appropriate d i f f e r e n t i a l equation of Bergman f o r E [CnZ]). one requires G(r,?) t o s a t i s f y the Goursat type problem

G i l b e r t ’ s method of ascent then shows t h a t s o l u t i o n s ( r e g u l a r around t h e o r i g i n ) of ( 9 . 1 ) f o r n 2 can be written as ( x = ( x , , . . . , x,)) (9.5)

u(x)

=

h(x) +

lo1

u

n-’ (note G does not depend on n ) .

G ( r , l - o 2 )h(xu2 )do This will be c a l l e d t h e B-G operator.

BERGMAN-GILBERT OPERATOR

REmARK 9-2.

189

We s h a l l f i n d i t more convenient t o work from t h e G f u n c t i o n

above r a t h e r t h a n t h e f u n c t i o n ?(r

2, t )

E(r,t).

There i s a c o n n e c t i o n w i t h 2 t h e complex Riemann f u n c t i o n o f Vekua expressed v i a G ( r , l - u ) = -2zR3(z,Z, za2,0),

=

t h e l a t t e r f u n c t i o n o f n e c e s s i t y b e i n g a f u n c t i o n o f r2 = zZ ( h e r e

R 3 denotes t h e p a r t i a l d e r i v a t i v e i n t h e t h i r d argument and R = R(s,t,u,-r)

s a t i s f i e s Rst

+ (1/4)F(st)R = 0).

Now i n ( 9 . 5 ) l e t u2 = u = h +

(9.6)

p/r

and w e . o b t a i n a V o l t e r r a t y p e o p e r a t o r

jr K ( r , p ) h ( p , - ) d p ;

N

K(r,p)

=

(p/r)4n-1(l/2r)G(r,l-(p/r

0

An easy c a l c u l a t i o n based on t h e G e q u a t i o n y i e l d s

LEMl!IA 9-3, The k e r n e l ??of (9.6) s a t i s f i e s ( f o r p < r ) 2 2 " 2[ ( n - 3 ) + r F ( r ) ] K = p K~~ + ( 5 - n ) p Z .

r2Krr

N

+ (n-l)rKr

+

P

I n keeping w i t h some formulas f o r e x t e r i o r problems developed i n [CnZ] ( c f . d , a l s o 58) f o r example one i s l e d t o w r i t e K ( r , p ) = ~ " ~ K ( r , p ) i n (9.6); t h e n LEmmA 9-4- K ( r , p ) s a t i s f i e s ( f o r p < r ) r2 Krr + (n-1)pK w i t h 2 r n - 2 K ( r , r ) = -Ir F(p 2 )pdp. P

f

(n-l)rKr

+

r 2 F ( r2 ) K = p 2 K

0

We w i l l deal w i t h K below as a d i s t r i b u t i o n w i t h s u p p o r t i n t h e s e t 0 < p < r and a d i s c o n t i n u i t y a t p = r . L e t us n o t e t h a t i f T i s a d i s t r i b u t i o n and P a s u i t a b l e t e s t f u n c t i o n w i t h
%

:/

Tpdx then ( i n t h e n o t a t i o n

of ( 9 . 2 ) ) (pn-3PnT,v) = ( p n-1 ( p n-1 T1)'/pn-',p) = - ( p n-lT',p' n-3 T,Pnp). This indicates the r o l e o f the f a c t o r

= (P

)

= ( T,(pn-lp')')

i n eventually

t r a n s m u t i n g pn i n t o

Q n = r2 D2 + (n-1)rD + r 2 F ( r 2 )

(9.7)

( c f . Lemma 9.5 below).

I n w o r k i n g w i t h f o r m u l a s such as (9.6) we w i l l see

t h a t one has a t r a n s m u t a t i o n B: Pn

+

Qn i n t h e r a d i a l v a r i a b l e s

p +

r and

t h e s p h e r i c a l v a r i a b l e s g e t c a r r i e d a l o n g s i n c e r2< =a: does n o t depend on r ( c f . Remark 8.2). Thus f o r s u i t a b l e f, B [ p2A n u ] = B[Pnu +f):u] + F ( r 2 )]Bu. +n:Bu = ,'[An

LElXM 9.5, B:

pZAn

-f

= Q'Bu

I f B: Pn -t Qn i s a t r a n s m u t a t i o n t h e n B[P 2An] = r2 [An + FIB ( i . e . 2 r An + r2 F ( r2 ) ) . Thus i f Anh = 0 t h e n [An + F ( r 2 )]Bh = 0.

Thus one wants t h e m u l t i p l i c a t i o n by r2 i n o r d e r t o extend t h e t r a n s m u t a t i o n t h e o r y f r o m r a d i a l o p e r a t o r s Pu = ( r n - ' u ' ) ' / r n - ' t o c e r t a i n operators i n 2 s e v e r a l v a r i a b l e s ( i n t h i s case r A,,). We w i l l e v e n t u a l l y i d e n t i f y t h e

ROBERT CARROLL

190

operator ( 9 . 6 ) based on the B-G theory, w i t h a p a r t i c u l a r transmutation Pn 2 Qn and we remark a l s o t h a t equations of t h e type ( 9 . 1 ) with terms H(r )ru

f o r example can be reduced t o t h e form ( 9 . 1 ) . The transmutation theory f o r operators of t h e form Qn i s developed in s p e c t r a l form via J o s t s o l u t i o n s f o r Qn as in 5 5 1 . 7 and 2.8 f o r n = 3. There a r e no spherical functions t o manipulate and asymptotic estimates a r e required in the spectral pairings. However we have a special s i t u a t i o n (of considerable importance) when dealing w i t h transmutations B: Pn + Q n . The B-G operator e n t e r s the p i c t u r e and t h e s p e c t r a l pairing can be e f f e c t e d v i a t h e Mellin transform ( c f . a l s o [C35] where an analogous procedure was used i n a d i f f e r e n t c o n t e x t ) . This allows considerably more d e t a i l t o emerge and y i e l d s d i r e c t l y various i n t e r e s t i n g connections between special functions. We show how t h e Mellin transform can be used t o construct a b u t the typical transmutation kernel 6. This will not be the B-G kernel

EXAIRPI;€ 9.6.

procedure i s i n s t r u c t i v e and i s employed l a t e r when we construct t h e kernel i;the kernels B a n d a r e compared l a t e r . Take n = 3 and F(r2 ) = k 2 ( t h i s 3 corresponds t o a natural s c a t t e r i n g s i t u a t i o n ) . Thus s e t = Pn = P = 2 2 2 x D + 2xD and { = Q3 = x2D2 t 2xD -t k 2 x 2 . The equation Fp = A p has solu2 21t i o n s x7 f o r y = y, = - ( $ n - l ) i [(kn-1) + A 1' ( n = 3 here) a n d we take P 2 4 e . g . \i, ( x ) = x T , T = -4 - ( A ++) . Set a l s o v = -T-+, u = - T - I = v-+ SO 2 2 The F e q u a t i o n becomes then x p " + 2xp' t h a t T(Tt1) = u ( u + l ) = 'A = w 2 t ( k 2 x 2 - A )p = 0. S e t t i n g x = xp we have a reduced Bessel equation X I ' + ( k 2 - A 2 / x 2 )x = 0 and we w r i t e p = $(I with \i, Q ( x ) = z-'jW(z) ( z = k x ) . Here j v could be any Bessel function of order v. Now we seek a transmutation

-+.

kernel in the form of an inverse Mellin transform, which will serve a s a T Q -0-1 Q e recall that i n ,$ ( r ) ) ! J . W s p e c t r a l pairing, B ( r , p ) = ( p ,$ ( r ) ) = ( p !J dealing w i t h transmutation kernels the general theory of §§1.7 and 2.8 i n d i c a t e s t h a t C r 6 ( r r p ) = ? * B ( r , p ) i s a natural condition. Here p* i s t h e P , ru* = Pn formal a d j o i n t defined by ( P f , g ) = ( f , P g ) f o r s u i t a b l e f , g and f o r .u* 2 2 2 one has P g = ( x g ) " - ( n - l ) ( x g ) ' = x [ c j " - ( ( n - 5 ) / x ) g ' - ( ( n - 3 ) / x )g]. For N k N n-3 n-3n = 3 evidently P = P and i n general -* P [x 41 = x P'p. Let us now take v Mellin transforms ( p + U ) d i r e c t l y in the equation Q r B ( r , p ) = P p B ( r , p ) A 2 2 where Mf(o) = f ( u ) = 1; f(p)pU-'dp a n d we r e c a l l t h a t M[p D f ] = u(u+l)M[f] f o r example (see here [Zbl] f o r generalized Mellin transforms). I t i s conN

A

venient now t o take p B ( r , p ) = e ( r , p ) with e = M[e] and t o note t h a t p ( p 6 ) " Hence [ e l = p 2 e P P and under Mellin transforms one obtains = p [ p ~ " + 2BI-j. - A r* t h e equation Qre = o(o+l)e. Therefore we obtain agian $ = z-'jW(z) ( z = kr)

5

BERGMAN-GILBERT OPERATOR where

= u(of1) = v

z

-k.

191

Now one has a v a i l a b l e v a r i o u s t a b l e s o f M e l l i n

t r a n s f o r m s (e.g.

[EbZ; Oa31) and we make t h e f o l l o w i n g change o f v a r i a b l e s 2 i n o r d e r t o use t h e t a b l e s . L e t a2 = P , y = r, a = y , b = ky, ab = ky2 =

k r , a/b = l / k , e t c . From [Oa3], p.221 o r [Eb2], M[Za-'(a 2 -a 2 ) k'J%[b(a 2 -a2 ) b'11 = (2a/b)'sr(s/2)Jb+,

p. 329 i t f o l l o w s t h a t (ab). Since J, ( z ) = (1/

2 2s & = (1/2)(Tab/2) 2 ( 2 a / b ) ' S J ~ ~-5S ( a b ) r ( s / 2 ) we have M[Sin(b(a 2 -a 2 ) b')I

2Tz)-'Sinz

2

(a

-f

s ) where 0 5 a 5 a and here M [ f ] ( s )

2

J F f ( a ) a S - l d u ( c f . a l s o [Mbl],

=

p. 79 and n o t e t h a t t h e f o r m u l a on p. 49 o r [Oa3] i s i n c o r r e c t by a f a c t o r 2 2 Consequently, from (9.8), w r i t i n g S i n b ( a -a )' = Sinkr(1-p/r)',

o r r(s/2).

e t c . one o b t a i n s from t h e M e l l i n t h e o r y c+im Sinkr(l-p/r)'

,9.9)

=

kr/2)'

a - s ( 2 / k ) S / 2 r ( s / 2 ) J ~ ? i s( k r ) d s jc-iw

Now t a k e s/2 = u = -T-1 (and j v = J

i n $Q ).

-(Tt')

We then c o n s t r u c t a

transmutation kernel i n t h e form (9.10)

= (P

B(r,p)

',$

( r ) ) u= f?(k,o)p-'-'

( kr)-'Jgt5

( k r ) d T = (1/2p ) P

(thus B(r,p) = k c(k.r)PT(kr)-~J-(Tf')

(kr)do

c'( k,s)a-S(kr)-'

J?i+s/2(kr)ds) where t h e i n t e g r a l s a r e upward along some c o n t o u r c-im im

i n t h e s p l a n e as i n (9.9)

f o r example 2niE(k,s)

=

( c may v a r y ) .

(2/k)s/2(71/2)'kr(s/2)

Comparing ( 9 . 9 ) - ( 9 . 1 0 ) and we have ($' =

EHEBREm 9-7, A f o r m a l t r a n s m u t a t i o n k e r n e l 8 ( r , p ) f o r B: + s t r u c t e d i n t h e form B ( r , p ) = ( P T , $ QT ( r ) ) u = ( l / r p ) S i n k r ( l - p / r ) ' , =

(kr)

-5T+$)

2J-(

( k r ) and t h e " s p e c t r a l

"

$2

-+

cf

we t a k e o r )$:

can be conwhere $ Q T

p a i r i n g r e p r e s e n t e d by 1~ i n v o l v e s a

c o n t o u r i n t e g r a l as i n d i c a t e d w i t h c(k,T) = ( n / Z ) ~ ( k / Z ) ~ ' ( k / Z ~ i ) r ( ~ - l ) .

A s i m p l e b u t t e d i o u s c a l c u l a t i o n a l s o c o n f i r m s d i r e c t l y t h a t t h i s E(r,p) N Iv s a t i s f i e s QrB = Q 8. P

REmARK 9-8- We c o n t i n u e t o work i n t h e c o n t e x t o f Example 9.6 i n o r d e r t o show how t h e i n g r e d i e n t s f i t t o g e t h e r . [Bcl;

Cn1,2;

Gj1,3]

and thus f o r K (= (9.11 )

t h a t G(r,T)

=

= -krJ.,(kr;-5)/:-5

(where

T Q

1-02 =

l-p/r)

h e r e ) we have f r o m (9.6)

K(r,p) = - ( k/2)(p/r)'J1

while K(r,r)

One knows f o r t h i s example f r o m

[kr(l-~/r)~]/(l-~/r)~

2 -k r / 4 r e p r e s e n t s t h e d i s c o n t i n u i t y a t p = r.

I n order t o

b u i l d t h i s d i s c o n t i n u i t y i n t o t h e d i f f e r e n t i a l e q u a t i o n f o r K i n Lemma 9.3 2 (where R = k and n = 3 ) we can proceed i n s e v e r a l ways. Thus f o r example, 2 take the M e l l i n transform ( P 5 ) o f H = p K u s i n g a g a i n o u r standard = p, -f

192

ROBERT CARROLL

etc.

From [Oa3],

where L

p. 106 one has ( a

s)

-f

.

denotes t h e Lommel f u n c t i o n s lJ,V

me1 f u n c t i o n L

We r e c a l l here t h a t t h e Lomv9v 2 ( z ) s a t i s f i e s t h e e q u a t i o n z L " + z L ' + ( z 2 - V 2 ) L = z lJ+1

v,v and f o r i n d i c e s ( ~ , l ~ - l has ) t h e form Lu,u-l ( n o t e t h i s i s 1F2 and n o t 2F1). (9.13)

=

( 2 ~ + 2= S-1). culation yields

a 2'+2 -k j n a

cri

= o(o+l)^H

-

k-OzUt2. =

^H

=

Now k

= r"-'.

5rH -p 2 HP P

H

=

M[H],

H =

pK

du = -k-' (kr)-CLa+3/2,0+% ( k r )

I n p a r t i c u l a r t h i s says t h a t

and H must s a t i s f y

PR0PBSZCZbN 9.9. 2 -k2r 6(r-p).

Hence'writing

J [b(a2-a2)$] 1 2 2 (a -a 1%

serve t h a t ( c f . [ Z b l l ) M [ 6 ( r - p ) ] 6(r-p)]

( z ) = ( ~ ' + ~ / 4 ~ ) ~;2,1Jt1 F ~ ( ,-+z2) l

-k-"z-+L(z)

-' z

and an easy c a l -

= k2r'+2

Ot2

Consequently -k2r'+2 -k2r2ps(r-p)

and we ob= M[-k

2r 3

( r z p i n 6(r-p))

The €3-G k e r n e l K f o r Example 9.6 s a t i s f i e s ?&K

-

P

K =

L e t us p o i n t o u t how such a s i t u a t i o n l e a d s d i r e c t l y t o a p r o o f t h a t (9.6) i s i n f a c t a t r a n s m u t a t i o n f o r m u l a ( r e c a l l i t s o r i q i n i n v o l v e d o n l y harmonic h). Thus observe f i r s t t h a t f o r m a l l y , w i t h ( K ( r , p ) , h ( p , - ) ) a s u i t a b l e d i s tribution pairing, K(r,p),h(p,-)) = (F K(r,p),h(p,-)) - k2r2h(r,-) = 2 2 P (K,?h) - k r h and consequently ( c f . Lemma 9.5) s e t t i n g u = i h = (K,h - k 2 r2 h)

c+

(9.14)

2 2 " 2 2 2 r [A3+k I B h = r A 3 h t r k h + i + K ( r , p ) , h ( p , - ) ) + r = r2A3h t ( K(r,p),p2A3h)

2 s

K(r,p),h(p,.))

= 8 [ p 2A3h]

PR0PWICI0N 9-10. Formula (9.6) f o r o u r model problem o f Example 9.6 r e p r e " 2 2 2 r [A, + k 1. sents a t r a n s m u t a t i o n B: p A3 -f

V

We would n e x t l i k e t o compare B w i t h t h e B o f Theorem 9.7 and t h i s can be accomplished by checking t h e i r r e s p e c t i v e a c t i o n on monomials pn f o r example. n n-1 A ) = (H(r,p),p ) = H(r,n) F i r s t simply r e f e r t o (9.13) t o o b t a i n ( K ( r , p ) , p n -n -?( i . e . o = n ) . Hence f o r z = k r , i [ p n ] = r - k z ZLn+3,2,n++ ( z ) . Now r e a f t e r (9.13) w i t h Gr[rn] = n ( n + l ) r n + k2r2+n so f o r u = ~n B[p 1, Qru call

-

tri

=

n ( n + l ) r n + k2rn+'

-

k-'[n(n+l)z-'L

+ zn+']

This implies t h a t 2 where n ( n + l ) = '1 = v -k. n B(r,p),p ) we have € 3 1 ~=~ 1 =

n(nt1)u.

u must be a l i n e a r combination o f terms z-'jv(z)

On t h e o t h e r hand i f one computes B[pn] = ( n-1 ) = $(r,n) (cf. (9.9)). Thus B[pn] (pB(r,p).p 2 j v ( z ) where n ( n + l ) = v -k (we want v = [n(n+l)+&

w i l l be o f t h e form z-' = n+% s i n c e G(r,n) must

BERGMAN-GILBERT OPERATOR

be f i n i t e as r (9.10) and

+

.

0

n)

(0

193

E x p l i c i t l y we p i c k up a m u l t i p l i e r as i n d i c a t e d i n Now

= (2/k)n-'J~r(n)r-4Jn+'(kr).

B[pn]

k[pn]

must a l s o

w i l l be a m u l t i p l e o f B[pn] -n -4 To determine c" we w r i t e (L = Ln+3/2yn+5 and z = k r ) E [ p n ] = rn - k z L ( z ) W r i t i n g t h i s o u t and t a k i n g = ?(kr)-'lz [ ( - 1 )P(z/2)n+'+2P/p!r(n+p+3/2)]. ;= 2n++ k -n r ( n + 3 / 2 ) we have t h e d e s i r e d i d e n t i f i c a t i o n and t h e n one compares be f i n i t e as r

and

B[pn]

(9.15)

+

0 and hence

Ez-'Jn+%(z)

=

6[pn]

t o o b t a i n ( r ( n + % ) = J7G?-2n(2n)!/n!))

E[pn]

a(n,k)B[pn];

=

;[pn]

a(n,k) = ( 2 n + l ) ! / k 2 2n n ! ( n - l ) !

&f(EOREm 9-11. The d i f f e r e n c e between t h e t r a n s m u t a t i o n s sed t h r o u g h t h e i r a c t i o n on monomials

pn

'i and

B i s expres-

as i n d i c a t e d i n (9.15).

L e t us now g i v e a " s p e c t r a l " r e p r e s e n t a t i o n f o r t h e k e r n e l

6

analogous

of

t o Theorem 9.7. Thus we w r i t e kf = ( g ( r , p ) , f ( p ) ) so t h a t from t h e f o r m u l a v n B[P ] = cz-'Jn+%(z) (which d o e s n ' t r e q u i r e n t o be an i n t e g e r ) one g e t s (9.16)

2u+4 k -a r(a+3/2)z-'JUq5(z)

=

;[P']

Now use v a r i a b l e s as i n (9.9)-(9.10) (9.17)

-i.

g(r,cJ) = ( 1 / 2 ~ i )

= k(k,a)z-'JU,,(z)

t o obtain (u = - ~ - l )

-0-1 -4 P z Ja+4(z)$(k,o)du

where $( k,a) = ( 2 / k ) ' h r ( a + 3 / 2 ) . has k e r n e l i ( r , p )

EHEOREm 9-12. The B-G t r a n s m u t a t i o n

g i v e n by (9.17).

We want t o l o o k more c l o s e l y a t t h e d i s c o n t i n u i t y i n K i n d i c a t e d i n Lemma 9.4 r e l a t i v e t o i t s k e r n e l a c t i o n and t o t h e d i f f e r e n t i a l e q u a t i o n f o r K. Thus one has (*) u = h + (9.2) and (9.7). (9.18)

r

I ~ ~ - ~ K ( r , p ) h ( ~ , . ) dTake ~ . Qn and Pn now as i n 0 r n-3 Then w r i t i n g I = I P K(r,p)h(p,-)dp we o b t a i n 0 2

QnI = -4r hF

-

r (4n-l)h

r

Fpdp

-

0

r

0

+ ((n-l)/r)Kr

+

Fpdp + r n - ' K P ( r , r ) h

4rhr

+ F(r2)K]dp

0 NOW

assuming

P"'K

-f

0 and P

n-1 -f

0 as

P

-+

0 the l a s t integral

z

i s (by

Lemma 9.4) z = 1 ; :n-3h~2[Kpp + ( ( n - l ) / p ) K p ] d p = 1 ; h(pn-lKp)pdp = hr"' Kp(r,r) - hrr n- 1K ( r , r ) t I; (pn-lhp)pKdP. However (Kr + K p ) ( r , r ) = D K ( r , r ) - ( l - 4 n ) r 1 - n i Fpdp - %r3-nF and upon i n s e r t i n g a l l t h i s r n-3 2 i n (9.18) one g e t s (*) QnI = 10 P K(r,p)Pnhdp - r hF. The c a l c u l a t i o n s 2 r show t h a t Q'u = Pnu + r Fu = Pnh t I c~"~KP"hdp and e x h i b i t t h e way i n

=

-4Dr[r2-nf

Fpdp] =

0

194

ROBERT CARROLL

2 which t h e jump c o n d i t i o n i n K produces t h e necessary v a l u e -r hF i n (+). L e t us redo t h i s i n a d i s t r i b u t i o n c o n t e x t and o b t a i n a d e r i v a t i o n o f t h e Thus w r i t e t h e k e r n e l i n ( * ) as K ( r , ~ ) p ~ - ~

jump c o n d i t i o n i n Lemma 9.4.

where Y i s t h e Heavyside f u n c t i o n .

Y(r-p)

(9.19)

-n+3

J = r

QnI = (J,h);

+

Since

( c f . (9.18)).

)

Y(pn-'h

P P

t i o n , as

6(r-p)Dr[r

+ P n-3 P 2 [KPP

rn-'Kr(r,r)6(r-P)

( (pn-lK

An easy c a l c u l a t i o n g i v e s

) ,Yh)

P P

=

2n-4

K(r,r)] +rn-'K(r,r)6'(r-p)

+ ((n-l)/~)K~lY(r-~) hrn-'K

P

(r,r)

-

h'rn-'K(r,r)

+ (K,

t h e d e s i r e d jump c o n d i t i o n can be w r i t t e n , a f t e r some c a l c u l a -

),

( 0 )

(rn-'K)'

from which Lemma 9 . 4 f o l l o w s .

= -irF,

I;E?ltNA 9-13, The jump c o n d i t i o n i n Lemma 9.4 needed t o have Qnu = Pnh + n-3 n J p KP hdp i n (*) can be d e r i v e d as above and y i e l d s K ( r , r ) v i a ( 0 ) .

Now we want t o determine t h e d i f f e r e n t i a l e q u a t i o n f o r K as i n P r o p o s i t i o n There i s a c e r t a i n amount o f s h i f t i n g back and f o r t h between " o r d i n -

9.9.

a r y " and d i s t r i b u t i o n d e r i v a t i v e s i n t h e t e x t b u t t h e d i s t i n c t i o n should be = pn-'PT. Now c o n s i d e r c l e a r . One checks e a s i l y t h a t f o r T E D', P*[pn-'T] -* PP[p n-3 K(r,p)Y(r-p)] (p = P n ) i n n o t i n g t h a t J = Q ~ [ p n - 3 K ( r , p ) Y ( r - p ) ] from

(9.19).

Thus ( n o t e Yp = - 6 )

(9.20)

( ? * [ ~ ~ - ~ K y ] , p=)

4P

n-1 [ K Y

P

-

K61,p'

( p n-3 P n [KY],p)

= ( ( p n - l K )'Y,p)

)

P

and t h i s equals ( P " ~ ( P ' K ) Y

- r"'K

P

6

+

= ( ( ~ ~ - ~ [ K y ] ' ) ' , p= )

-

( pn-'K

P

6,p)

(the calculation involves

rn-1K6',p)

a c e r t a i n abuse o f n o t a t i o n b u t can be j u s t i f i e d ) . (9.21)

[Q: -

(Pi)*][pn-3K(r,p)Y(r-p)]

+ rn-'K(r,r)p1(r)

Hence u s i n g

= 2r[rn-2K]'6

(0)

= -r2 F 6 ( r - p )

V

PR@P05ZCZ0N 9-14, The k e r n e l K = n e l i n (*), s a t i s f i e s (9.21). -r 5-n F ( r 2 ) 6 ( r - p ) .

~ ~ - ~ K ( r , p ) Y ( r - p ) where ,

E q u i v a l e n t l y [Q:

-

K i s t h e B-G k e r -

P~I[K(r,p)Y(r-p)l =

'j d e f i n e d by E[h] = u i n (*) i s a t r a n s m u t a t i o n + F] ( r e p e a t t h e c a l c u l a t i o n s l e a d i n g t o P r o p o s i t i o n 9.10 and

Thus we know as b e f o r e t h a t p2An

-+

r2[An

use P r o p o s i t i o n 9.14). v

"

L e t us w r i t e t h e n i [ h ] = ( g ( r , p ) , h ( p , * ) ) = r'

+(

2 One has Qnra = [ r 2 D 2 + ( n - 1 ) r D + r F]r'

=

(9.22)

u =

B[pa]

= M[pE(r,p)]

p

n-3

and s e t

K(r,p)Y(r-p),p')

u(u+n-2)r'

+ Fra"

and s e t t i n g

BERGMAN-GILBERT OPERATOR

I = ( P n-3 K ( r , p ) Y ( r - p ) , p ' ) , Y(r-p)],p') = -Fr'+' + ( Consequently one has (9.23)

QnG

=

i t follows t h a t Q"1 K(r,p)Y(r-p),(pn--'[Dp']')'

[r2D2 f ( n - 1 ) r D

f

r 2F];

=

=

195 n-3 n P [K(r,p) -Fro'' + o(cr+n-Z)I.

-FrU+' f ) =

( p

u(o+n-Z)ti

W e do n o t t r y here t o deal with functions F having s i n g u l a r i t i e s a t r = 0 and assume nice behavior t h e r e . Then one expects s o l u t i o n s of (9.23) t o behave l i k e ry near r = 0 where y 2 + ( n - 2 ) y - o(u+n-2) = 0 ( c f . [C40; Cg3] Now we and 51.10 - note y ( y + n - 2 ) = o ( u + n - 2 ) ) . T h u s y = u o r y = -o-n+2. w r i t e $ QG ( r ) f o r a s o l u t i o n of (9.23) behaving l i k e r' near r = 0 and normalized i n some way ( s e e here e . g . Theorem 9.7 where we simply picked $ Q without regard t o normalization). Thus we s e l e c t $:(r) a n d ( 6 ) lim ;:(r)r-' = 1 as r 0.

s a t i s f y i n g (9.23)

-f

tHEBRElll 9-15, The "extended" B-G kernel g ( r , p ) i s characterized by t h e '4 - 'Q -0-1"Q " s p e c t r a l " formula ( T = -o-n+2 with QU = $T) B ( r , p ) = ( 1 / 2 n i ) P p $u(r)du with a s in ( 6 ) .

;:

We go next t o t h e t o p i c of generating functions and r e f e r t o [Mbl; Mdl; Mnl; Eb3; Rtl; S t k l ; Tml; Wjl-31 f o r general information. There a r e many formul a s ( p r o t o t y p i c a l l y consider e . g . (*) (xy)'(x 2-2xy)-"Ju[(x 2- 2 ~ y ) ~= ] gnJu+n(s)y'+n as in [Whl; Wjl]) and some p a r t i a l u n i f i c a t i o n has been achieved. By u n i f i c a t i o n one means t h a t t h e r e a r e c e r t a i n guiding p r i n c i ples from which various generating function formulas can be derived systema t i c a l l y . Given t h e g r e a t d i v e r s i t y of s p e c i a l functions and t h e i r o r i g i n s i t i s perhaps not s u r p r i s i n g t h a t no one d i r e c t i v e ( o r u n i f i c a t i o n ) seems

t o be known from which " a l l " generating functions a r i s e . In t h i s section we will provide another g u i d i n g p r i n c i p l e , i n c e r t a i n resoects q u i t e general, from which many generating functions can be obtained ( a l s o c e r t a i n i n t e r e s t i n g g e n e r a l i z a t i o n s a r i s e n a t u r a l l y involving e.g. s e r i e s 1 G n J v + n ( ~ ) $ v , n ( y ) with known $",,(y) which a r e not powers of y ) . The background here mo-

t i v a t i n g our i n t e r e s t was a paper [SalO] on some i n t e r p o l a t i o n formulas in quantum s c a t t e r i n g theory a t fixed energy where e.g. one obtained genera-

t i n g functions r e l a t e d t o t h e c l a s s i c a l formula of Lome1 above (namely (*) - cf. a l s o [Cel; Ne6,10,11; Sa10,13]). One can i d e n t i f y t h e generating functions i n [SalO] as transmutation kernels and provide a s t r u c t u r e f o r obt a i n i n g such generating functions via s p e c t r a l pairings involving e.g. the Mellin transform. Given t h i s o r i g i n and some knowledge about the under-

196

ROBERT CARROLL

l y i n g d i f f e r e n t i a l o p e r a t o r s one has a v a i l a b l e c e r t a i n machinery (e.g. G-L techniques) t o a s s i s t i n t h e development.

We g i v e here an i n t r o d u c t i o n t o

t h i s p o i n t o f view, along w i t h some examples suggesting f u r t h e r e x t e n s i o n s o f t h e ideas ( c f . [Bbgl;

The p e r s p e c t i v e and em-

C54] f o r o t h e r r e s u l t s ) .

along w i t h various

p h a s i s here a r e q u i t e d i f f e r e n t t h a n those o f [SalO],

c o n s t r u c t i o n s , and we achieve much g r e a t e r g e n e r a l i t y by embedding t h e ideas i n a general t r a n s m u t a t i o n machine.

One n o t e s a l s o some c o n n e c t i o n t o ap-

p l i e d problems o f a d i f f e r e n t t y p e i n [Byll; Cn8];

general t r a n s m u t a t i o n

methods c o u l d be employed i n such c o n t e x t s e x p e d i t i o u s l y . 2 We c o n s i d e r o p e r a t o r s o f t h e form (1.7.1), i . e . Qu = x u" + 2xu' + x2[k2 2 2 q ( x ) ] u and t h e e q u a t i o n (*) Cu = 1 u w i t h '1 = v = u ( o + l ) ( u = v-4 c o r N

-

-+

responds again t o complex a n g u l a r momentum). It i s convenient t o s e t 9 = 2 2 xu and t r a n s f o r m (*) i n t o (1.7.3), i . e . x 7 ' ' + x2[k2 - q ( x ) ] v = A p where we deal w i t h r e g u l a r s o l u t i o n s q(v,k,x) f(v,+k,x)

e x p ( + i k x ) as x

Q

s u i t a b l e manner

-

m

-f

Q

xu+'

as x

= xvq'

and q(v,k,x)

fo(v,-k,x)

=

For

(nkx/2)4iwYH;(kx),

2" ( Z / n ) % ( v + l ) k

-

= x??(x)

( c f . 51.7).

EMNPLE 9.16,

-v++ .v-'i

(f(v,-lk)

i

0 and J o s t s o l u t i o n s 0 at

:-+

m

Dcl; Ne6; and s1.7 f o r hypotheses).

see [Cel;

a l s o by j1"(x) t h e s o l u t i o n o f ( * ) w i t h X-"$!(X) xu+'

-f

as b e f o r e (one assumes

+

1 as x

-f

i n some Me denote

0 ( t h u s x$!?(x)

Q

see h e r e Theorems 9.15 and 9.12).

= 0 we have q 0 = 2vr(v+l)k-vx?iJv(kx),

and f o ( v , - k )

= W(fo(v,-k,x),qO(v,k,x))

=

are c a l l e d Jost functions).

2 2 We denote by P t h e o p e r a t o r 'iu = x u" + 2xu' = ( x u ' ) ' as b e f o r e and s e t 2 2 2 Qoy = x u " + 2xu' + k x u ( i . e . q = 0 i n Q o ) . F o r g e n e r a t i n g f u n c t i o n s w i t h

N

N

monomials i n y as i n (*) one i s i n t e r e s t e d i n t r a n s m u t a t i o n s B: i n t h e B-G t h e o r y ) and we c o n s i d e r f i r s t t h e t r a n s m u t a t i o n ;:(r)

w i t h k e r n e l g i v e n by Theorem 9.15,

=

g(r,p)

T-f

i: F-+q:

(as pa

-f

-~-1vQ ( 1 / 2 1 ~ i )fi p $,(r)do

( f o r n = 3).

RENARK 9-17. Take v-%)

i(r,p)

F = To so

= vO(v,k,r from Example 9.16 $u(r) "Q

= (r-'/&i) P p-'-'

see a l s o 58)

K(r,p)

=

( c f . (9.17)).

(2/k)'%(a+3/2)Jaq3(kr)do

one w r i t e s i ( r , p ) = 6 ( r - p ) + K ( r , p ) Y ( r - p )

F i r s t although

We make a few g(r,p)

= K(r,p)

must have t h e d i f f e r e n t i a l e q u a t i o n o f P r o p o s i t i o n 9.9, 6(r-p)

(9.13).

and t h i s i s c o n s i s t e n t w i t h (9.11) where On t h e o t h e r hand from [Oa3],

(2a/b)Z/2r(z/2)JZ/2(ab).

If

(9.11) and

i t i s known t h a t ( c f

-(k/Z)(p/r?J,[kr(l-p/r)']/(l-p/r)'.

f u r t h e r remarks here on t h i s .

/r and ( u =

FrK

-

f o r p < r one 2 2 K = -k r P

i s g i v e n by 2 2 ?- 2-1 da = Jo[b(a -a )']a

M[pK(r,p)]

p. 221, 2$

W i t h t h e v a r i a b l e s i n d i c a t e d above t h i s becomes

GENERATING FUNCTIONS

(z

=

-

20+1

197

an e q u i v a l e n t formula i s deduced i n [SalO] by o t h e r means). Now

M [ P ~ ( ~ , P )=] $'Q 5 ( r ) s h o u l d h o l d s o one observes t h a t i f M(v) = A

$ / ( u + + ) t h e n x-%(x)

=

= - x -3/29. Hence f o r

v

( ] / h i ) F vx = pg(r,o)

But D,[Z-~J,(Z)]

i(r,p).

one o b t a i n s g(r,p)

(x

= K(r,p)

%

(*) D p J o [ k r ( l - ~ / r ) ' ]

P)

=

-

V+l ( u s i n g (9.11)) which i s c o r r e c t f o r

mula

6 ( r - p ) + (k/2)(1-~/r)'J,[kr(l-p/r)']Y(r-p)

;=

-(r/p)'

=

-J1(z) and f r o m ( * )

one a d j o i n s a t e r m Y ( r - p ) t o Jo so t h a t ( 6 ) becomes D,[J =

=

du/(u+%) f r o m which f o l l o w s Dx[x"4']

( z ) so DZJo(z)

= -z-'J

and M($)

-0-4

LO

P < r. 5-

If

[kr(l-~/r)~]Y(r-p)l

= -(r/P)'8(r,p)

then the f o r -

6 + K i s o b t a i n e d as d e s i r e d ( r :P i n m u l t i p l i c a t i o n by & ( r - P )

M [ P ~ - ~ 6 w1 i t h M [ P ~ ] = 2 2 2 2 ( z ) s a t i s f i e s [ z DZ + 2zDZ + ( z - v )]L = r' and s i n c e L,+3/2,u+% ( z ) = L 0 &"+I 9" .-" 2 2 zV+' we o b t a i n f o r x = -k- z ZLu+l,V(~), Qrx = x - k r"'. Hence Qrx = -Qr$, " Q - Qrr5and x = d~~ 'Q - r0 as r e q u i r e d f o r e v e r y t h i n g t o f i t t o g e t h e r here

and J o ( 0 ) = 1 ) .

L e t us n o t e a l s o t h a t

M[pK]

=

N

-

( c f . also[C40]where

S t r u v e f u n c t i o n s a r i s e i n Theorem 2.12.22).0ne

t h a t t h i s c a l c u l a t i o n i s s p e c i a l f o r LLl+,,";

notes

i n general t h e M e l l i n t h e o r y

( k r ) w i t h f u n c t i o n s J,,[kr( 1 -p/r)']/ 0+3~/2,0+1-~/2 b u t t h e n i c e c o n n e c t i o n t o JI: i s m i s s i n g u n l e s s LJ = 1. It i s

r e l a t e s Lommel f u n c t i o n s L (l-p/r)p'2

t h i s example which was i n d i c a t e d i n [SalO] as a g e n e r a t i n g f u n c t i o n ( v i a i n t e r p o l a t i o n formulas e t c . ) .

To see t h i s more r e a d i l y one can s i m p l y a p p l y Thus t h e gamna f u n c t i o n has s i m p l e p o l e s

some r e s i d u e c a l c u l u s i n (9.17).

a t u+3/2 = -n ( t h e r e s i d u e a t z = -n o f r(z) b e i n g ( - l ) n / n ! ) and J-,(z) = m ( - 1 ) J,(z) ( n o t e u+% = -n-1). Hence s i n c e i ( r , P ) = K ( r , p ) f o r P < r one has f o r m a l l y , f o r a s u i t a b l e c o n t o u r c - i m (9.25)

K(r,p) = -(rP)-'

1;

-+

c+im

(kP/2)mJm(kr)/(m-l)!

From (9.11) and (9.25) we see t h a t one has a s p e c i a l case o f t h e Lommel f o r m u l a (*) where u = 1, x = k r , y = kp/2,

(kJ~/2)r(v+)/r(u+l))

Sin[kr(l-~/r)']. (kr)

(0

=

[Mbl],

l/(m-l)!

= (l/ZTi)

a(r,p)

+ p-'-'J:(r)y(o,k)dn

W r i t e t h e i n t e g r a n d I as I = P-'-'JT(Z/k)

v-4) w i t h p o l e s a t

(k/2)%[J4(kr) cf.

=

(m 2 1 ) .

Consider t h e t r a n s m u t a t i o n B o f Theorem 9.7 w i t h k e r n e l (y(5,k)

REIIARK 9.18, =

and gm-l

+

1;

p. 721.

0

=

5-35

-n so t h a t f o r m a l l y , B(r,p) =

( k ~ / 2 ) ~ Y ~ + + ( k r ) / n (! sI i n c e

J-,,+(Z)

F u r t h e r one has f o r m a l l y B[p']

= (l/rp) -5-

r T(")Jf+%

-4-

JTr

= (-1 )m+lY,+!i(z)

= y ( u , k ) $ gQ( r ) ,

m a l l y B serves as a g e n e r a t i n g f u n c t i o n f o r t h e Yn+?,.

P

-

Thus f o r -

198

ROBERT CARROLL

REMARK 9-19, The o p e r a t o r

k

= 6 + K i n Remark 9.17 has an i n -

w i t h kernel

verse g i v e n by ( c f . [C43; Gj3; L r l ; SalO; Val]) (9.26)

I1 [ k(p/r)'(

L ( r , p ) = ( k / 2 ) (1 -p/r)-'

I f one had a v a i l a b l e a s p e c t r a l t h e o r y f o r

t i o n s J/u(r) '4

;(r,p)

= 6

+ LY

(p +

r)

1-P /r)']

co i n v o l v i n g t h e r e g u l a r s o l u -

( r a t h e r than J o s t s o l u t i o n s ) t h e n one c o u l d expect t o w r i t e a

v"

spectral formula f o r

and r e l a t e i t t o a k e r n e l

'

= ker

w i t h say

=

To

i s based on t h e J o s t s o l u t i o n s However t h e s p e c t r a l t h e o r y f o r v+4 H1v ( k r ) ( v = o+*) o r i n K-L f o r m on u o ( v , k , r ) =

(B-l)*.

go = fo/r = ( T k / 2 r ) i a0(v,k,r)/r

k -v+4 = ( n / 2 k r ) *i J v ( k r ) .

= 90(v,k,r)/rfo(v,-k)

Using t h e K-L f o r -

m u l a t i o n however we can i n f a c t produce f o r m a l l y an i n v e r s i o n k e r n e l and t h e Thus

method i s g e n e r a l i z e d i n Theorem 9.23 ( c f . a l s o Remark 9.24 and [CSO]). one has f o r ? ( v ) = G O f ( v ) =

(

f(s),go(v,-k,s))

t h e K-L i n v e r s i o n ( c f . Theo-

j m

rem 8.14) f ( r ) = - ( i / T ) l i m v f ( v ) q 0 ( v , k , r ) d v ( i . e . ( fn,go(v,-k,r)) = -(i/n) ( fh, ~ * ~ ( v , k , r ) ) ~where duo = dpo = ( 2 i / n ) v 2 d v / f o ( v , - k ) f (-v,-k) =P -vSinnvdv/ 0

nk on

S y m b o l i c a l l y one w r i t e s now (-iu/n)Jr go(v,-k,r)*o(u,kyr)dr

[O,m)).

= ~(V-P)

( t h i s e x p r e s s i o n i s examined i n more d e t a i l i n Remark 9.24).

"9 s e t q O / r = J/'(r)

Now

and c o n s i d e r f o r m a l l y ( v = & )

so Q0 = ; f ( r ) / f o ( v , - k ) -1 m

(9.27)

r'( - i v / T ) q o ( v ,

;( r, p ) =

- k,p

)dv/fo( v, -k)

l i m

= u-+ one has f o r m a l l y , (;(r,p),JIT(p)) '4 = J r ;(r,p)GT(p)dp '4 i m ro ( - iv / n ) 1 ; go ( v ,- k ,P Po( u ,k, P )dp [fo (u ,- k ) / f o ( v ,- k 11( v / u dv = Li

Then f o r i m

li r'

T =

G(v-p)dv = rT so t h a t

appears t o be a k e r n e l f o r

B" = 6-l

9.24 and [C80] f o r a d i s c u s s i o n o f t h e K-L t h e o r y ) . has from (9.27) ( v = o++)

Now w r i t e ( v

-f

= (r(l-v)/ai)[J-v(kp)

-v)

(1/2Iri)(rp)

-lij m m (2/k)vr(v+l)Jv(kp)r-u-1dv be z e r o f o r p < r.

- (rp)-'(l/Zni)li;

W r i t i n g t h i s o u t one

= (1/2psi)jiI (k/2)"rUH~(kp)dv/r(v).

;(r,p)

1 c a l l t h a t r ( z ) r ( l - z ) = v/Sinnz and Hv(x) = [ J - v ( x ) Then H:(kp)/r(v)

(see here Remark

-4

-

-

Jv(x)exp(-i~v)]/iSin~v.

J v ( k p ) e x p ( - i n v ) l and

im (kr/2)vr(l-v)J-v(kp)dv

;,m

= B(p,r)

Re-

= [p-'/Zni]

( c f . Remark 9.17) which we know t o

The r e m a i n i n g t e r m i n (9.28) g i v e s f o r p < r,

( k r / 2 ) v r ( l - v ) J v ( kp)exp( - i n v ) d v .

?(r,p)

F o r m a l l y now e v a l u a t e

t h i s by r e s i d u e s a t 1 - v = -n ( v = n + l ) as (9.29)

:(r,p)

= (rp)-'

1;

(kr/2)mJm(kp)/(m-1)!

= L(r,p)

(P <

r)

=

GENERATING FUNCTIONS

p. 364 ( w i t h kLol(x,y)

From [ L r l ] ,

t h e r i g h t side of (9.29) f o r has E ( r , p )

+ L(r,p)

p <

%

r.

199

L ( r , p ) i n (9.26)),

L ( r , p ) agrees w i t h

Thus ( n o t e f r o m (9.25) and (9.29) one

= 0 f o r p > r so T ( r , p )

i s t r i a n g u l a r and

g ( r r p ) con-

t r i b u t e s a term 6 ( r - p )

ZHE8REm 9 - 2 0 , F o r o u r model o p e r a t o r o f Remark 9.17 w i t h g ( r , p ) = k e r N

P

+

co, g i v e n as 6 ( r - p ) =

i-' can

as a g e n e r a t i n g f u n c t i o n f o r p < r and t h e k e r n e l be c o n s t r u c t e d v i a K-L t h e o r y i n t h e form (9.27)-[(rp)J'/2~i][is

(kr/Z)"r(l-v)exp(-ivv)

(9.28) from which

T(r,p)

Jv(kp)dp Y(r-p).

The l a t t e r i n t e g r a l r e p r e s e n t s y ( r , p )

function f o r

p <

v

+ K ( r , p ) Y ( r - p ) v i a (9.17), one has t h e expansion

(9.25) e x p r e s s i n g K(r,p) G(r,p) f o r

v

B, B:

= 6(r-p)

as a g e n e r a t i n g

r v i a (9.29).

E v i d e n t l y one can produce g e n e r a t i n g f u n c t i o n s almost a t w i l l by t a k i n g a t a b l e o f M e l l i n t r a n s f o r m s and e x t r a c t i n g s e r i e s v i a t h e r e s i d u e c a l c u l u s . However we have a somewhat more r e f i n e d s e l e c t i o n mechanism when we i n s i s t t h a t one deal w i t h t r a n s m u t a t i o n k e r n e l s .

Indeed, i n t h e f i r s t p l a c e , work-

i n g v i a f o r m u l a s as i n Remarks 9.17,

e t c . i n t h e model case one i s ob-

t a i n i n g t r i a n g u l a r kernels g(r,p), T h i s corresponds t o $"Q" ( r )

Q ,

9.18, o(r,p),

r" so ( r / p ) '

e t c . which v a n i s h f o r p > r. = exp[olog(r/p)]

and ifp > r one

c o u l d c l o s e t h e c o n t o u r i n t e g r a l t o t h e r i g h t and o b t a i n z e r o e v a l u a t i o n ( a n a l y t i c i t y t o the r i g h t being b u i l t i n here). t r a n s m u t a t i o n k e r n e l s between o p e r a t o r s be t r i a n g u l a r ( c f . [C29,40])

and

One notes o f course t h a t o f c e r t a i n types need n o t

A second f e a t u r e

and examples a r e g i v e n below. U

i s t h a t when t h e r e i s an u n d e r l y i n g t r a n s m u t a t i o n P N

t r a l i n f o r m a t i o n a v a i l a b l e f o r Q (as w e l l as f o r

N -+

Q one may have spec-

y) which

can be used as i n

Theorem 9.20 t o develop f u r t h e r t h e i n t e g r a l r e p r e s e n t a t i o n .

I n addition

when t h e t h e o r y i s developed v i a t r a n s m u t a t i o n i d e a s one can l e a v e t h e Melv

l i n t h e o r y e n t i r e l y and work w i t h s p e c t r a l i n t e g r a l s f o r t r a n s m u t a t i o n s Q

-+

N

Q

t o produce i n t e r e s t i n g extended g e n e r a t i n g f u n c t i o n t y p e formulas, e.g.

llan$Qn+a(r)$n+B(p), "P

REmARK 9-21,

which can be u s e f u l i n v a r i o u s ways.

L e t us n o t e t h a t i n Chapter 7 o f [ C s l ] a procedure a k i n t o

t r a n s m u t a t i o n i s employed i n c o n s t r u c t i n g v a r i o u s c o n t o u r i n t e g r a l express i o n s whose k e r n e l s t h e n sometimes serve as g e n e r a t i n g f u n c t i o n s .

Thus e.g.

v

g i v e n a Bessel o p e r a t o r w r i t t e n as z2D2 + Z D + z2 = Q(Dz), c o n s i d e r solu2 t i o n s o f C(DZ)u = A u w r i t t e n i n t h e form ( C beina some general c o n t o u r ) , u ( z ) = l C K ( z , c ) v ( c ) d c where K and v can be chosen i n v a r i o u s ways as f o l lows.

L e t M(Dg)

be a d i f f e r e n t i a l o p e r a t o r w i t h f o r m a l a d j o i n t

M

*

such t h a t

200 (A)

ROBERT CARROLL

5(DZ)K(z,5)

= M(D5)K(z,5).

Then choose v such t h a t M*(D ) v

= A 2v

and C

5

so t h a t "boundary terms" via i n t e g r a t i o n by p a r t s vanish. I t follows t h a t [c(DZ) - h 2 ]u = 0 = iC [ M ( D ) - 2 ]K(z,A)v(A)dX = iC K(z,<)[M*(D ) - x 2 ] v d c 5 2 25 For example i f M ( D ) = - D 2 with ( A ) in the form z KZz t zKZ + z K + K = 0 5 5 55 one can take K(z,c) = exp[+izSinc] so t h a t ic [D2 t x 2 ] K ( z , s ) v ( c ) d ~ = 2 2 5 One takes v = exp(+icx) and ic K(z,s)[Dc + X lv(c)dr, + Ic D5[K5v - Kv5]dc. s u i t a b l e C so t h a t [ K v - Kv ] vanishes a t t h e endpojnts of C. This leads 5 5 1 t o c e r t a i n standard i n t e g r a l r e p r e s e n t a t i o n s f o r Hankel functions, HX(z) = 2

exp[-izSinc + i x c f j d ~ ; H x ( z ) = - ( 1 / n ) i L exp[-izSing + i ~ c ] d cwhere 0 + -IT -t -IT + im and L2: v+i- -+ IT + 0 + -im. Adding these one ob1 2 t a i n s ( J x = (HA + HX)/2) J x ( z ) = - ( 1 / 2 r ) i L exp(-izSing + ixg)dg where L : n + i m -+ IT + -71 + - v + i m . For = n one has only a n i n t e g r a l IT + -IT here and -(l/n)JL L1: - i m

-+

exp(izSin5) i s exhibited a s a generating function f o r t h e J n in t h e form exp(izSin5) = J n ( z ) e x p (i n c ) . Other d i f f e r e n t i a l operators a r e a l s o used in [Csl] f o r t r e a t i n g t h e Bessel functions (along other contours). The theme should be c l e a r , A g r e a t deal of f l e x i b i l i t y i n choosing contours and d i f f e r e n t i a l equations f o r K and v i s allowed in order t o study eigenfunct i o n s u h of Z ( D z ) . In c e r t a i n s i t u a t i o n s t h e kernels K will serve as gene r a t i n g functions ( i n some sense) f o r say un. The d i f f e r e n c e with transmut a t i o n constructions i s f i r s t t h e f a c t t h a t t h e contour i n t e g r a l s a r e n o t

T.

d i r e c t l y r e l a t e d t o any s p e c t r a l theory o r transform theory f o r M o r Secondly the choice of eigenfunction v X ( s ) ( a n d u,(z)) i s n o t made according t o any d i r e c t i v e and t h u s t h e procedure, although very productive, i s somewhat "disorderly" and evidently "ad hoc". 9-22. One should perhaps ask in general what i t "means" t o have a generating function. For transmutation kernels such as g ( r , p ) : pU -+ 5: i n Remark 9.17 w i t h $ Q ( r ) a n a l y t i c f o r Rev > 0 (which i s a "generic" property

REmARK

U

type operators - c f . [C43,45 and 51.7) we use J v ( k r ) Q ( k r / Z ) " / r ( v t l ) so $!(r) 2. r -% r v and p -0-lvQ iL0(r) 'L (r/p)'p-' t o g e t t r i a n g u l a r i t y as indicated by closing t h e contour to t h e r i g h t f o r p z r. On t h e o t h e r hand given for

simple poles o f ;:(r) f o r Rev < 0 as i n d i c a t e d , a (formal) generating funct i o n follows immediately by residue calculus. Hence b o t h t r i a n g u l a r i t y and t h e existence of a formal generating function a r e natural i n t h i s s i t u a t i o n . One can generalize t h e s i t u a t i o n o f Theorem 9.20 as follows. r e f e r t o a general

;itype

operator as i n ( 1 . 7 . 1 ) ; s e t

Let f ,

p,

etc.

GENERATING FUNCTIONS

( c f . ( 9 . 2 7 ) and Remark 9.24). t h a t E(r,p) = 0 f o r p > r .

We r e c a l l f ( v , - k )

(0)

rv+' f o r l a r g e v we see

%

f(v,-k,x)

[f(-v,-k)q(w,k,x)

=

-

P u t t i n g t h i s i n (9.30) one o b t a i n s

f(v,-k)p(-v,k,x)]/(-2~).

Now however ( n o t e v-+ i&. m r-0-1 q(v,k,p)dv/p

Note from Y(v,k,r) Now r e c a l l

201

+

-v-4

= -0-1)

i(p,r).

(-i/Znp)J;:

r'q(-v,k,p)dv

=

(1/21~i)

Hence f o r P < r where g ( 0 . r ) = 0 we have

i s a n a l y t i c f o r Rev > 0 so f o r m a l l y (9.32) can be evalua-

-

t e d by r e s i d u e s upon c l o s i n g c o n t o u r s t o t h e r i g h t

provided f(v,-k)

# 0

f o r Rev > 0, which i s t h e requirement f o r a b s o l u t e l y continuous spectrum. R e c a l l here f ( - v , - k ) = by those o f p(-v,k,x)

W( f (-tl,-k,x)

,q (-v,

k,x) ) w i l l have p o l e s determined

f o r Rev > 0 ( s i n c e f ( - v , - k )

i s analytic i n

ous s i t u a t i o n s a r e p o s s i b l e ( c f . [Dcl; Ne6] e t c . ) .

a n a l y t i c a t x = 0 t h e n t h e o n l y p o l e s a r e a t v = 1/2, 3/2, o t h e r s i t u a t i o n s when e.g.

Vari-

Typical'ly (simple) poles

when 1-2v = - n ( n = O , l , . . . ) and i f e.g.

can occur f o r v(-v,k,x)

tl).

...

xG(x) i s

(there are

t h e o n l y p o l e s a r e a t v = 1, 2, . . . ) .

One needs

now f o r example a f o r m u l a f o r t h e r e s i d u e i n (9.32) a t such p o l e s v i a t h e a n a l y t i c function q(-v,k,r)r(l-Zv) directly.

( c f . [NeG])

o r more s i m p l y v i a f ( - v , - k )

Thus w r i t e f o r yn = (n+1)/2 ( o t h e r values f o r vn c o u l d be hand-

l e d i n a s i m i l a r manner) (9.33)

limf ( - v , - k ) ( v - v n )

vt"

= An f-

n V

&HEB)R€:111 9.23.

t h e k e r n e l (9.30),

$:

(9.30) w i t h L(r,p)

as i n (9.32)

for

p <

N

I n t h e s p i r i t o f Theorem 9.17 w i t h B: P

r, :(r,p)

pa +

z!(r),

the kernel and

y(r,p)

T(r,p)

= 6(r-p)

-f

for

5 given

f o r m a l l y by

8 = 6-l

i s g i v e n by

+ L(r,p)Y(r-p)

i s expressed v i a t h e i n t e g r a l f o r L ( r , p )

generating f u n c t i o n L(r,p) =

1"0

rn/'

^f:

q(vn,k,p)/pf(vn,-k)

so t h a t

i n (9.33) as a where n m i g h t

r u n o n l y o v e r even o r odd i n t e g e r s i n c e r t a i n cases (we c o n s i d e r o n l y t y p i c a l s i t u a t i o n s i n p h y s i c s where vn = (n+1)/2 here).

PhaaA: From ;(r,p)

= G(p,r)

+ L ( r , p ) we want t o show t h a t i n f a c t y ( r , p )

0 f o r p > r as i n Theorem 9.17,

so t h a t o u r f o r m u l a f o r y w i l l h o l d .

=

One

notes o f course t h a t as t h e i n v e r s e o f a V o l t e r r a t y p e t r i a n g u l a r o p e r a t o r T(r,p)

w i l l n e c e s s a r i l y by t r i a n g u l a r .

E(p,r)

= ( 1 / 2 n i ) p r"-\(-v,k,p)dv/p

To see t h i s i n another way w r i t e

and f o r

p >

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 <

and one has bounds as

( i ~ ~ / ~ / 2 ) ( k r / 2 ) ' - 4 / r ( u t l ) CJ ( k r ) which i s Q

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

IJ 0 and hence i n p a r t i c u l a r t h a t

G+ = G- =

(;(r,P),v(u,k,p)/p) = (:(r,o),$:(p)) F o r m a l l y t h i s amounts t o ? ( v , p ) = p-').

REFilARK 9.25,

-

- *(r,V)

a n a l y t i c f o r Reu > 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) <;(r,p),g(U,-k,p))

v(r,p),[f(-u,-k)v(u,k,p)

r'-'/f(U,-k)

where ?I(")

a(v-vl/$

=

Then c o n s i d e r f r o m (9.30)

?(r,p),g(u,-k,p))

f(p,-k)r-'-']

= (-1/2u)( =

g(v,-k,r)g(u,-k,r)dr

(where ;:(PI = P ( ~ , ~ , P ) / Pw i t h = 6(v-u) a c t i n g on rub'.

The K-L t h e o r y g i v e s f a i r l y g e n e r a l l y a f o r m u l a 6 ( r - s ) = - ( i / n )

v*(v,k,r)g(v,-k,s)dv

( c f . however [Jb2])

and t h u s a c o m p o s i t i o n

can

m v

be w r i t t e n ( U = V-4, T = u-4)/, B(r,p)q(p,s)ds = ( - i / n ) ( l / & r i ) P [ v ( v , k , r ) / r l / j I u [ g (u,-k,s)/f (u,-k)lJ: pT-'-'dp = ( - i / n ) i i -i m *(u, k,r)ug(u,-k,s)du = a ( r - s ) ( s i n c e M;'[~(T-U)]

= p -U

and

= :/

M[p-"]

This formula a l -

pT-l-'dp).

so f o l l o w s d i r e c t l y f r o m (*) $"4u ( r ) = ( i ( r , p ) , p ' )

by composing w i t h

(-i/v)

vg (v,-k, s ) / f ( v , - k ) . One knows as b e f o r e ( * )

Another development o f G-L t y p e a r i s e s as f o l l o w s . f(v,-k,x)

=

[f(-v,-k)p(v,k,x)

- f(v,-k)v(-v,k,x)]/(-Zw).

j e c t of p r o d u c i n g an i n t e g r a l o f t h e form we c o n s i d e r ( u = V-4, $ = (9.35)

q(c,r)

Hence w i t h t h e ob-

i n a general G-L e q u a t i o n

p/r)

( i / 2 ~ ) lim$!(r)[cV-'(f(-w,-

k)/f(v,-k))

-

~ - " - ' l d w --

i( i / 2 1 ~ r ) [jzcv-'[f

(-v,-k)lp ( u , k , r ) / f

= -(i/n)

(v,-k)

/i~c'vg(u,-k,r)dv/f(w,-k)

-

p(-v,k,r)ldv = ;(S,r)

=

2 04

ROBERT CARROLL

Define then (-v-% = - u - 1 )

iTHE0)REM 9-26, From $!(r)

G-L equation ( f o r where A i s given by (9.36)

= ( g ( r Y p ) , p u )t h e r e i s a general

5

P and {) in t h e form ; ( c , r ) = 6 and given by (9.30).

( g(r,p),A(p,E))

with

RfillARK 9.27- Compare here with [Lrl], pp. 362-364 f o r

a G-L symnetric kernel F(x,y)

kr, y

= k p , e t c . ) a s follows.

6 = co.

One defines

( k L (x,y) ,L L ( r , p ) , kLlo(x,y) 21 K(r,p), x = 01 * Set Lo,(x,y) = Lol(y,x) a n d F(x,y) = Lol + (x,z)LOl ( y , z ) d z and one obtains f o r

Evidently we should our standard example Q = Qo, F(x,y) = - ( i / 2 ) J l [ i ( x y j 4 ] . have i n A above a s p e c t r a l i n t e g r a l f o r F (modulo a d e l t a function and in the appropriate v a r i a b l e s ) . Let us w r i t e t h i s out. The G-L equation of [Lrl], p. 363 is F(x,y) + L l o ( x y y ) + j f Llo(x,z)F(z,y)dz = L* ( x , y ) . Set-

ol

+ K ( r , p ) + fr K(r,s)F(s,p)ds = T h u s f o r = 6 + L and 6 = 6 + K L*(r,p) = L ( p , r ) ( L ( p , r ) = 0 for p < r). ( 6 ) w i l l agree w i t h Theorem 9.26 provided A ( s , p ) = ~ ( s - P ) + ?(s,p) = 6(s-p) + (k/2)11[k(sp)4] ( r e c a l l I 1 ( z ) i = J,(zi)). For A we have

t i n g kF(x,y)

= ?(r,p)

one has

( 6 ) ?(r,p)

pU-l6(s-p)dp = sU-', ( 1 / 2 n i ) f sU-'p-'do

The l a s t term i s ~ ( s - P ) s i n c e 6 ( s - p ) , etc.

(9.38)

Hence formally we want jrn -(sp)'(k/Z)

( k / 2 ) I l [ k ( s ~ ) 4 1=

=

2v e -inv r ( l - v ) d v

2 n i r (1 +v)

i-

This is e a s i l y p u t i n t h e Mellin framework ( ( s p ) = l / x ) i n t h e form ( k / 2 ) i m -u 2 I1 (kx-%) = - ( k / 2 n i ) L i m x ( k . 2 ) "exp(-inv)[r(l-v)/r(l+v)]d" and t h e contour integral in (9.38) y i e l d s t h e desired i d e n t i f i c a t i o n (poles a t 1-v = - n ) Now from (9.35)-(9.36) we have : ( r , p ) i(9.39) nut--'-' A(n,r) = ( l / Z T i ) iim

F ( n , r ) = - (1/2ri )

= ( ;(p,n),A(n,r))

with

dv + F ( n , r ) = s(n-r) + F ( n , r ) ;

imn'ru[f (-v,-k)/f ( v , - k)]dv i i m

Again i t i s t h e simple poles of f ( - v , - k ) f o r Rev s e r i e s representation f o r F ( n , r ) i n t h e form ( v n

> Q ,

0 t h a t give a formal

( n + 1 ) / 2 i n t y p i c a l cases)

GENERATING FUNCTIONS

( c f (9.33) f o r

?:).

205

Hence

CHEdREm 9-28, The symmetric k e r n e l A(n,r) o f t h e general G-L e q u a t i o n o f can be w r i t t e n f o r m a l l y as A ( 0 , r ) = 6 ( q - r ) + F(n,r) and Theorem 9.26 f o r

5

where

F has t h e f o r m g i v e n i n (9.39)-(9.40)

i n typical situations.

We w i l l show now t h a t t h e procedures l e a d i n g t o s p e c t r a l measures e t c . i n 51.7 a r e a l s o v a l i d when t h e a s y m p t o t i c b e h a v i o r i s modeled on t h e W h i t t a k e r f u n c t i o n s which a r i s e f r o m a Coulomb p o t e n t i a l hypotheses o f 51.7).

= a/x

(not included i n the

We o b t a i n a v e r s i o n o f t h e g e n e r a l i z e d K-L i n v e r s i o n

developed i n 58 and t h i s g i v e s a t r a n s f o r m t h e o r y f o r c e r t a i n W h i t t a k e r functions.

L e t us begin w i t h t h e W h i t t a k e r e q u a t i o n i n t h e form

uZz + [-%+ ( K / z ) + ( k - v 2 ) / z

(9.41)

2

]U

= 0

2 2 which has t h e f o r m x u " + x2[k2 - ( a / x ) ] u = x u w i t h z = Z i k x , K = - a / 2 i k , 2 = a/x and v = L++ as b e f o r e ( c f . [Cel; N e l l ) . A 2 = v -$, and u ' = ux; t h u s For s o l u t i o n s o f (9.41) we have W h i t t a k e r f u n c t i o n s (9.42)

F ( u G ~K- ,1+2U, Z ) ;

( z ) = e-4z zvq'

M K,'J

where lF1 denotes Kumner's f u n c t i o n .

One w r i t e s ( c f . " e l ] )

a/2k and V = 2nk/x = a/x, and v = L++, 50

V++K

=

K

= i n , SO n =

L + l - i n , w i t h 2v+l = 2L+2.

One has t h e n as r e g u l a r s o l u t i o n s q L = (Zik)-'-l

(9.43) (x-'-'qL(k,x) (9.44)

+

fL(k,x)

1 as x

Min,v ( Z i k x ) = (-2ik)-L-1M-in,V +

0).

(-2i kx)

For J o s t s o l u t i o n s we t a k e

= W-in,V(-2ikx);

xlim e-ikxfL(k,x)

= 1

Observe t h a t n = a/2k depends on k and s e t f i = Win,v(2ikx). ( z ) / r ( l + Z v ) and W K , v ( ~ ) a r e e n t i r e CEmA 9-29, ( c f . [ B b c l ] ) . N ~ , ~ ( Z=) M KY'J i n u with W (z) = W (z). For ( K , z ) f i x e d and l a r g v l < n/2, M (z) Q

K 3-V

K,V

K,V

zV++1 + o ( I ~ \ - ' ) I as I"\ + a. S e t t i n g W(f,g) = f g ' - fag, W ( W ~ , ~ ( ~ ) , M ~ , ~ ( z ) ) = r(1+2v)/r($+V-K); W(WK ( ~ ) , W - ~ ~ ~ ( z e x p ( k= i ~e x) p) ( + i i w ) ; W(MK . V ( z ) , ,V

( z ) ) = - 2 ~ ;and MK

M K,-V

+

( z ) = [r(2~+l)exp(+i~~)/r(v++~)]W-~,~(zexp(?in)) ,V

[r (2v+l )exp ( + i 'II ( K - v - % ) )/r ( v+%+K) l W K

,v

(z ) .

Now u s i n g these formulas w i t h z = 2 i k x and - 2 i k x = z e x p ( - i n ) one o b t a i n s

206

ROBERT CARROLL

(vL(k,x)

= (2ik)-L-1Min,v(2ikx)

W(fL,fi)

= -2ikexp(-an)

t h e zeros o f

For z = 2 i k x and z e x p ( - i T ) = - 2 i k x ,

by Lemna 9.29 so ( 6 ) FL = W(fL,vL)

and we can w r i t e now

+ FL

vP).

S i m i l a r l y F i = W(f-,v

r(Zv+l)/r(v+%+in). $-in)

=

vL

=

L 4

= (2k)

1

) = (2k)-v+'1(-i)v-%(2~+l)/r(v+

exp(TIn)[FLfi

-

Fifd]I/Zik.

One notes t h a t

occur a t t h e p o l e s o f r ( v + $ + i n ) o r V+%+in = -m (Rev < 0 ) .

F i n a l l y one w r i t e s t h e Coulomb Green's f u n c t i o n as G L ( k , r , r ' ) ( r < = min(r,r')

fL(k,r,)/Fp(k)

.

-v +% v -4

= -qL(k,r<)

and r> = m a x ( r , r ' ) as b e f o r e ) and i n w o r k i n g

w i t h t h i s i t w i l l be u s e f u l t o have fL expressed v i a (9.42) and (9.43) i n terms o f

as f o l l o w s .

We w r i t e vL(k,x) as v(-v,k,x).

( - 2 i k x ) and t h i n k o f

(7) '.k '+%(l-Zv)/r(-V+%+in). n )v ( V ,k ,x 11/2v.

= v(v,k,x)

= (-2ik)

-v-%

M-in,v + Set Fk = F(v,n) so t h a t F(-v,n) =

Then one has

(0)

fL

=

-

[F(v,n)v(-v,k,x)

F(-V,

I n keeping w i t h t h e program o f 9 1 . 7 we c o n s i d e r

I lo co

I =

(9.46)

h(r')[-Gl(kyr,r')/rr']dr'

d;v

( f o l l o w i n g [Bbdl) where

r

o f " i n f i n i t e " radius i n the

i s a semicircle

r i g h t h a l f p l a n e Rev > 0 w i t h v e r t i c a l s i d e t h e a x i s Rev = 0.

We c o n s i d e r

f i r s t t h e i n t e g r a l .h over a l a r g e s e m i c i r c l e o f r a d i u s R and l e t R We have seen t h a t

5

+-

m.

has n o zeros f o r Rev > 0 and s e t t i n g r ' = s c o n s i d e r

f i r s t r < s and t h e t e r m ( f r o m ( 9 . 4 6 ) ) J = p ~ [ v ~ ( k , r ) f ~ ( k , s ) / F ~ ( k ) ] d v . Let

J

be t h e i n t e g r a n d i n J and r e c a l l S t i r l i n g ' s formula r(z+a)

(-z)zz+a-4 f o r largzl <

TI.

Go t o [Kgl],

%

&exp(-z)

p . 316 now and t a k e x = 2 i k s and

-x = x e x p ( - i n ) as above ( t h e c h o i c e x e x p ( - i V ) i s c l e a r l y i n d i c a t e d throughout).

Take

K

= i n and we w i l l e s t i m a t e W

-K,Vb

(-x).

One s e t s z = l o g ( x / Z i v )

and from expz = ( k s / v ) we s e t 8 = (exp2z - 1 ) * = [ ( k s / v ) 2 - If'. Since I v / + we w r i t e 8 = i[l - ( k s / v ) 2 1'b 'L i - + ( k s / ~ ) ~ so i that (i-e)/(i+e) 2 Then k ( k s / v ) and 5 = ue i% i ~ l o g [ ( i - e ) / ( i + e ) ] % v i + i l o g [ 4 ( k s / v ) 2 j 5 v . from [Kgl] ( n o t e -TI+€ < argg = TI < TI-E so one r e f e r s t o '5 i n [Kgl]), -v+% v-4 v - V ~ ' . V - $ -in (-2iks) % s exp(-v)v 2 k i v (cf. also [Ofl]). Hence '-in,v f r o m - ( & ) and Lemma 9.29 one has

-

&HEP)RElIl 9.30, The a s y m p t o t i c e s t i m a t e f o r J i s IT = @(v,k i(r/s)"(rs)'

(r

<

s).

Correspondingly f o r r > s one w i l l

9 T d v a r i s i n g from (9.46) and 7 = v r e ( k , s ) f p ( k , r ) / F L

%

1

%

G EN ERAT I NG FUNCT I0NS

and 51.7 now a p p l y c o m p l e t e l y t o t h e J and J terms

The methods o f [Bbdl]

above and one can conclude t h a t I r = n i h ( r ) where (9.46)

207

a r i s i n g from 9 .

Consequently, u s i n g

(0)

9i s

the part o f I i n

( a l s o f, i s even i n

V)

EHEOEm 9-31, The completeness r e l a t i o n f o r t h e J o s t s o l u t i o n s i s 6 ( r - s ) = f J w [f,(k,r)/rl[f,(k,s)/s]dp w i t h dp = -2v 2 dv/niF(v,n)F(-v,n). We w r i t e d p ( v ) = $ ( v ) d v and s i n c e r ( Z v ) r ( l - Z v ) = n/SinZnv w i t h r ( % + v - i n )

r( -v+%+in) $-in).

=

T/Cosn(v-in) one has ;( v ) = ZkvSinZnvr(v+%+in)/TCosa( v- i n ) r ( v+

D e f i n e now a W h i t t a k e r t r a n s f o r m and i t s i n v e r s e by

1

m

n

(3.48)

Nf(v) = f ( v ) =

f(s)fR(k,s)ds/s;

rf(r) =

0

loimA

f(v)fR(k,r)$(v)dv

I n o r d e r t o use t h e g e n e r a l i z e d K-L t e c h n i q u e o f 58 now we n o t e t h a t p , n

f ( v ) , and f

R

a r e even i n v and w r i t e as b e f o r e @(v,k,r)

then using

(0)

(9.49)

r f ( r ) = (-i/n)

=

p(v,k,r)/F(v,n);

[imv?(v)@(v,k,r)dv Ljm

CHEOREm 9.32,

The g e n e r a l i z e d K-L theorem f o r t h e p r e s e n t s i t u a t i o n has t h e

form (9.49) and i n v e r t s t h e W h i t t a k e r t r a n s f o r m o f (9.48). L e t us w r i t e o u t (9.49) now i n terms of M

@( v, k , r )

=

r( v+$+in)M-i

COROCLARfl 9.33.

Kc,v

as f o l l o w s .

F i r s t note that

n, "( - 2 i k r ) / ( - 2 i k r ( 2 v + 1 ) ; t h e n one has

The W h i t t a k e r i n v e r s i o n has t h e form ( f o r s u i t a b l e f )

1

m

(9.50)

mf(v) = ? ( v ) =

- 2 i ks )ds/s;

f(s)W-in,v (v)M-i

J-2i

k r ) [ r ( v + % + i n ) / r ( Z v + l )]dv

Other transforms based on W h i t t a k e r f u n c t i o n s a r e a l s o known (see e.g. v

[Wt

2

-

q w h e r e 7; = (r u ' ) ' w i t h 11. " 2 -u-1 2 e i g e n f u n c t i o n s o f Pu = x u i n t h e f o r m u = r' o r r (A' = v -%, 0 = v-%) and Gu = (r2 u ' ) ' + r 2 [ k 2 - a / r ] u . Again s e t G:(r) = pu(k,r)/r ( u = R ) and One t h i n k s now o f a t r a n s m u t a t i o n B: P

-+

f o r m a l l y ( c f . (9.30)) (9.51)

v

B(r,p)

=

( 1 / 2 1 ~ i ).f'(2ikp)-'-1mKyu+4

We do n o t have an e x p l i c i t f o r m u l a f o r

g(r,p)

( 2 ik r )r( 2u+2 )do i n (9.51) b u t we can f o r m a l l y

r e p r e s e n t i t as a g e n e r a t i n g f u n c t i o n as f o l l o w s .

The i n t e g r a n d i s a n a l y t i c

208

ROBERT CARROLL Hence

except f o r t h e p o l e s o f r(2u+2) a t u = -1-n/2 w i t h r e s i d u e (-1)’/2n!.

&HEOREN 9-34, The t r a n s m u t a t i o n k e r n e l and f o r

(c

= 2 i k r ) g(r,p)

v i a t h e formula = Zikp, mK,%(n+l) (F)r(n/2+1 -K)/n!r(-n/Z-K). RK ,%(n + l )

Phaua:

o f (9.51) i s z e r o f o r

g(r,p)

P >

r

r i t serves as a g e n e r a t i n g f u n c t i o n f o r t h e W h i t t a k e r f u n c t i o n s

p <

= (1/2r)

10

n/2(-1)n

L e t us n o t e f i r s t t h a t ( c f . [ B b c l ] ) bi(mK,m/27m K,-m/2 ) = -Sinnm/n =

0 f o r m i n t e g r a l so t h e two f u n c t i o n s a r e l i n e a r l y dependent and i n f a c t ( z ) = r(%+m/2-~)rn~,,,/,(z). Now t h e r e s i d u e c a l c u l a t i o n ,-m/2 f o r (9.51) g i v e s B ( r , p ) = - ( l / r ) (2ikp)n/2mK,-+-n/2 ( Z i k r ) ( - l )‘/Zn! which

rf%-m/Z-K)m

K

2

t h e n y i e l d s t h e formula i n d i c a t e d .

I n terms o f convergence l e t us n o t e a l s o

(F)/(n+l)!. We see t h a t , s i n c e MK ( z ) % z’++ t h a t mK,%tn/2 (‘I = MK,$+D/2 .!J -+ m , t h e i n t e g r a n d i n (9.51) i n v o l v e s (F/:/a)“+l and hence f o r p > r as one can c l o s e t h e c o n t o u r t o t h e r i g h t t o g e t g ( r , p ) = 0.

10- 0RCH060NAt P ~ C # N ~ ~ I ~ ~AND I \ C $&RAWIU&ACZ(DN, One knows t h a t t h e r e a r e r e l a t i o n s between t h e t h e o r y o f orthogonal polynomials and methods o f i n v e r s e scattering (cf. [All;

C40,51;

Cd4,6;

Gh1,3;

Kn8; Kr9; L x l ] ) .

We work here

w i t h continuous analogues, e x t e n d i n g c o n s i d e r a b l y t h e ideas o f K r e i n f u n c t i o n s , and show how t r a n s m u t a t i o n methods p r o v i d e t h e c r u c i a l i n g r e d i e n t s and techniques f o r t h e t h e o r y .

F o r general m a t e r i a l on o r t h o g o n a l p o l y -

nomials l e t us c i t e [Akl; A l l ; Cul; F v l ; S t l l ] .

One way o f d e v e l o p i n g t h e

t h e o r y o f o r t h o g o n a l polynomials ( c f [ C u l l ) i s t o s t a r t w i t h a measure dw on an i n t e r v a l (a,b)

and d e f i n e a moment f u n c t i o n a l

b (10.1)

f(A)dw(A)

L(f) = a

The moments pn = L ( A ~ b) e i n g t h e n known one l o o k s f o r p o l y n o m i a l s pn(A) such b Various e q u i v a l e n t requirements can be u t t h a t (A) / , p n ( X ) p m ( x ) d ~ = .,6, i l i z e d . For example i t i s enough t o show (+) L(a(A)pn(A)) = 0 f o r polynomia l s ~ ( h o) f degreee m < n w h i l e L ( n ( A ) p n ( A ) ) = 0 f o r m = n. A l t e r n a t i v e l y one need o n l y show ( 6 ) L(Xmpn(A)) = Km”,

w i t h Kn # 0.

Now i n working w i t h

polynomials on t h e u n i t c i r c l e one d e a l s w i t h r e c u r s i o n r e l a t i o n s of t h e form ( c f [ A l l ] )

un+,

v o ( z ) = 1, an =

an,

= angun

+

-

b v ; v ~ =+ b~n m n + anvn w i t h say uo(z) = 1,

and an - lbnYzn> 0 ( z = e x p i e ) .

One can t r a n s f e r t h e

spectrum t o t h e r e a l a x i s h e r e and l e t n become a c o n t i n u o u s v a r i a b l e t o o b t a i n t h e e q u a t i o n s f o r K r e i n f u n c t i o n s ( c f . [Ku8; Kr91, A u

+

P, v

-+

P*)

dPdt = - i A P

-

b(t)P,;

dP,/dt

=

-b(t)P.

+

-A, b

+

-b,

Now one c o u l d d e p a r t

from t h i s p o i n t b u t we p r e f e r t o work w i t h what a r e ( e x t e n s i o n s o f ) essen-

ORTHOGONAL POLYNOMIALS

209

t i a l l y the real p a r t s of such functions ( c f . [C51; Lxl] - p r e c i s e l y we will be dealing with functions corresponding t o y+ = 2CosAt - (1/2)exp(iAt)[P + P*]

This goes back t o work in l i n e a r s t o c h a s t i c

f o r example w i t h b r e a l ) .

estimation and f i l t e r i n g a n d t h e functions b , P , a n d P, a l l have meaning in such a context ( c f . [C50,51,74; Cd4; Kr8; Dr9; Lxl] and 443.4 and 3 . 5 ) ; i n p a r t i c u l a r y+ corresponds t o a s p e c t r a l form of an even innovations process.

Q , h where p QA , h i s a soluIndeed i t was shown i n [C50,51; L x l ] t h a t Y+ % p A 2 t i o n of a d i f f e r e n t i a l equation Qu = - A u with u ( 0 ) = 1 and u ' ( 0 ) = h with l p4A I h ( y )= Coshy + {f K(y,x)CosAxdx where K(y,x) corresponds t o a f i l t e r i n g kernel (pQ f o r h = 0 i s denoted by p f a s u s u a l ) . Such a formula places A,h t h e material in t h e context of s c a t t e r i n g theory and more generally in the context o f general transmutation theory. We w i l l show now how c e r t a i n techniques in t h e theory of orthogonal polynomials have continuous analogues in the framework of transmutation theory and t h i s i n t r i n s i c connection leads t o some new r e s u l t s and points of view f o r general orthogonal functions ( c f . a l s o [Cd4,6; Gh1,3] f o r some r e l a t e d connections a t t h e d i s c r e t e level between s c a t t e r i n g theory and orthogonal polynomials - our methods a r e more general with q u i t e d i f f e r e n t focal points and deal w i t h l a r g e c l a s s e s of In p a r t i c u l a r we can allow the measures dw continuous analogues on [O,m)).

on [O,m) t o have polynomial growth and t h i s corresponds t o dealing w i t h cert a i n s i n g u l a r di f f e r e n t i a1 ooerators. Let t h e r e be given now a measure dw on to,..) which f o r the moment we assume t o be of t h e form dw = ( 2 / v ) d A + d u where do is f i n i t e . Extension of t h e theory t o measures of polynomial growth will be given l a t e r . We w i l l think of "polynomials" (of degree t ) as functions (10.2)

n(X,t)

=

CosAt +

I' 0

c(t,s)Coshsds

T h u s "polynomials" a r e e n t i r e functions o f exponential type t (and any even e n t i r e function of exponential type t has such a r e p r e s e n t a t i o n ) . The mome n t functional ( c f . ( 1 0 . 1 ) ) w i l l be given as (10.3)

L(Cosxt) =

J

m

Coshtdw(A)

=

g(t)

0

One recognizes g ( t ) as a readout impulse response in the theory of c e r t a i n geophysical inverse problems t r e a t e d by transmutation methods ( c f . [C40,66, 70-721 and Chapter 3, $ 8 ) . In f a c t i t i s convenient t o w r i t e

21 0

ROBERT CARROLL

(and extend gr t o be an even f u n c t i o n when needed). f o r "polynomials"

!

71

Now assume one l o o k s

= f such t h a t

m

(10.5)

f(h,t)f(l,s)dw(x)

= &(t-s)

0

and a c o n d i t i o n f o r o r t h o g o n a l i t y analogous t o say (+) c o u l d be

c

(10.6)

Cosxsf(h,t)dw

(the notation z(t,s) given f(A,t)

= g(t,s)

= 0

(s < t )

a r i s e s from general t r a n s m u t a t i o n t h e o r y ) .

o f t h e form (10.2) w i t h c f ( t , s )

= K(t,s)

f o r c e one o b t a i n s immediately a "G-L" equation. N

(10.7)

B(t,s) = 0 =

(

CosAt,Coshs)w

where ( C o ~ A t , C o s A s ) =~ A ( t , s ) =

R(t,S)

g

=

and g i v e n (10.6) i n

For s < t

lot

K(t,T)(

COSAS,COSXT)~~T

:1 CosxtCosxsdw can be expressed i n terms

o f t h e moment f u n c t i o n a l L o r g(.)

(10.8)

+

Indeed,

i n (10.3) as A ( t , s )

= 6(t-s) + R(t,s)

+ gr(

[ C o s x ( t + s ) + C o s A ( t - ~ ) ] d ~= l [ g r ( t + S )

It-SI)]

An i n t r i n s i c v e r s i o n o f (10.8) i s g i v e n below v i a g e n e r a l i z e d t r a n s l a t i o n .

iTHE@REl!l 10-1, Given f ( A , t )

+

= Cosxt

+ :1 K(t,s)Cosxsds w i t h (10.6) i t f o l l o w s + K(t,s)

s a t i s f i e s a G-L e q u a t i o n f o r s < t o f t h e form n ( t , s )

t h a t K(t,s)

fi K(t,T)fL(T,s)dT

= 0 with

a given

v i a t h e moment f u n c t i o n a l i n (10.8).

REMARK 10-2. W i t h f L expressed v i a (10.8) t h e G-L e q u a t i o n i n Theorem 10.1 i s i n t h e s o - c a l l e d t i m e domain v e r s i o n which i s u s e f u l i n a p p l i c a t i o n s ( c f . [C66,71,72]

and 83.8).

Such G-L e q u a t i o n s g e n e r a l l y have unique s o l u t i o n s

and can i n f a c t be s o l v e d n u m e r i c a l l y ( c f . [C40,66;

Cel; Se2,3]).

Now i n o r d e r t o c o n s t r u c t t h e a p p r o p r i a t e orthogonal f ( A , t )

L e t K ( t , s ) be 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 i n Theorem 10.1

&HE@R€lIl 10.3,

(s

one has

(*) f ( A , t )

< t ) and d e f i n e

=

Cosht + f$ K(t,y)CoshdT.

Then t h e f ( h , t )

s a t i s f y (10.5).

Pmod:

F o r m a l l y w r i t e o u t t h e l e f t s i d e o f (10.5),

C

(10.9)

f(h,t)f(x,s)dw

jOs K(s,c)A(E,t)dE

+

= I = A(t,s)

+

u s i n g (*), i n t h e form

lot

K(t,r)A(T,S)dT

jotlos K ( t , ~ ) K ( t Y ~ ) A ( ~ , r ) d 5 d Tz ( s , t ) =

+

+ a(t,s)

+

K(t,s)

S

+ K(s,t)

+

Jt K(t,T)fL(T,s)dr 0

K(s,E)a(C,t)dE

+

0

+ Jot

Jos K( t ,T ) K( t, 5 )a:d Ed=

ORTHOGONAL POLYNOMIALS

21 1

where E ( s , t ) = 5 ( t - s ) + fomin(s't)K(t,T)K(s,T)dT.

Now suppose s < t and use

Theorem 10.1 ( n o t e K ( s , t ) = 0 f o r s < t ) . One sees t h a t I = n ( t , s ) t K ( t , s ) S

Jot K(t,T)a(T,S)dT + SO K(s,c)[n(t,C) + K(t,S) + 1 ; K ( t , ~ ) s L ( ~ , t ) d ~ ] d =g 0. A s i m i l a r r e s u l t h o l d s f o r s > 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#
,..B

= ker where t & ~ ( y ) , v ( y ) % ( y ) ) = ( u(x)Ap(x),Bv(x) )). Hence f r o m t h e P P general G-L e q u a t i o n g(y,x) = ( B(Y,s),A(s,x)) w i t h A(F,x) = ( p A ( ~ ) , p x ( x )

Ap(x) = A(S,x)Ap(x) (10.24)

;(Y,X)

we must have = A:(Y)&(X-Y)

+

T(Y,X); A(C,x)

= ~(X-S)+

Ns,x)AP(x)

21 7

ORTHOGONAL POLYNOMIALS

No comparison o f measures dw and dv f o r Q and P r e s p e c t i v e l y was needed here [dh + do] = dv + <do ( r e c a l l dv = Gdh b u t p r o t o t y p i c a l l y i f (*) dw = cmh

* "+'

P P and ( ~ ~ ( ~ ) , i ? ~ =( xs )( x)- ~ s ) ) t h e n A(5,x)

w i t h v = c$2m+'

xZm+'; n(S,x) = 1 ;

[J,(h~)/(hx)~]

=

( x ~ ) ~X2m+1do. ]

[Jm(hg)/

s ( x - 5 ) t R(c,x) Thus we assume

measures o f t h e f o r m (*) and t r y t o c o n s t r u c t o r t h o g o n a l f u n c t i o n s (10.25)

f(h,t)

+

A(t)q;(t)

=

P where q X ( x ) = ( l / c m ) J m ( h x ) / ( X x ) m r e f e r s t o t h e o p e r a t o r P = Q,

and A ( t ) =

A i % ( t ) r e f e r s t o an o p e r a t o r Qu = (A u ' ) ' / A Q t q ( x ) u t o be determined w i t h AQ = AQAp.

9

I n o r d e r t o express t h e moment f u n c t i o n a l and i t s r e l a t i o n t o

n(5,x) c a n o n i c a l l y ( c f . (10.3), c i t y ) dw = :dh (10.26)

(10.4),

A 4

+ vudh and w r i t e

g ( t ) = L[n!(t)]

W(X)

=

and ( 1 0 . 8 ) ) we assume ( f o r s i m p l i -

G/u^

=

1

w n,(t)dw(h) P

=

lt;.

c

=

0

= 6 ( t ) t A p ( t ) Jrnv!)(t)&iv

Ap(t)W(t)

Then s e t q!(t)W(X)dv(h)Ap(t)

= 6(t)

=

+ Ap(t)z(t)

n

( n o t e here t h a t (10.27)

(00)

A(C,x)

= 1).

P[s(t)/~(t)] = 1 ; [d(t)/A(t)]u!(t)dt P

P

= (qA(C),qA(x)W(h) ),,AP(x)

=

Then w r i t e

T;G(S)Ap(x)

where TX denotes a c e r t a i n g e n e r a l i z e d t r a n s l a t i o n a s s o c i a t e d w i t h P ( c f . 5 P P (3.16) and (6.26)). Using (10.26) we have ( n o t e ( q h ( c . ) , y ( x ) ) v = s i x - 5 ) ) (10.28)

A(5,x)

= : 6

+

P ( q X ( 5 ) , q ~ ( ~ ) G ) v A p (=~ 6(x-S) ) + T?(5)Ap(x)

P and we w i l l r e f e r t o Z ( t ) A p ( t ) = L [ n X ( t ) ] e n t f u n c t i o n a l (ApZ

%

- s(t)

6(t) = g(t)

as t h e mom-

gr i n ( 1 0 . 4 ) )

&HEOREM 10-18. Given (10.25) w i t h and t h e k e r n e l A(<,x)

-

(AA)

:I

P q,(s)f(h,t)do(X)

=

0 f o r s < t,

determined as i n (10.28) v i a t h e moment f u n c t i o n a l

i t f o l l o w s t h a t K i n (10.25) s a t i s f i e s a G-L e q u a t i o n f o r s < t t t o f t h e f o r m ( n ( t , s ) = T S z ( s ) ) 0 = A(t)n(t,s)A,(s) + K ( t , s ) + J0 K(t,T)fi(T,s)

Z(t)Ap(t), Ap ( s ) d.r

.

Phood:

P

Simply t a k e w s c a l a r p r o d u c t s i n (10.25) w i t h q h ( s ) and use (u) t w i t h (10.28) and a ( t , s ) = T,z(s). Now t h e G-L e q u a t i o n c o n t a i n s two unknowns A ( t ) and K(t,T).

( c f . [C66,70-72,78]

I n practice

and 53.8 and 3.9) i n such a s i t u a t i o n t h e r e i s an i n -

t e g r a t e d form o f t h e G-L e q u a t i o n w h i c h does n o t c o n t a i n A e x p l i c i t l y and

218

ROBERT CARROLL

which can be used t o determine a s u i t a b l e k e r n e l from which A i s t h e n i n We w i l l proceed somewhat d i f f e r e n t l y here how-

f a c t subsequently recovered. ever.

F i r s t we w r i t e

(10.29)

A(t)i(t,s)

= A(t)?(x,t)

Then w r i t e f ( x , t ) (10.30)

= K(t,s);

;(X,t)

= pg(t)

S e t t i n g P*J, = [Ap(J,/Ap)l]'

+

Ap(s)-O.(t,s) = ;(t,s) i n (10.25)

so t h a t

jotK(t,T)q,(T)dr P

one has P*(App) = Pp and e v i d e n t l y ( n ( t , s )

( t ) P(Dt)6(t,s)

(pF(t),p!(s)c)v)

Ai*(t);(A,t))

(%

P*(DS)6(t,s).

=

=

We can w r i t e t h e G-L equa-

t i o n o f Theorem 10.18 as ( s < t ) A

(10.31)

0 = n(t,s '0

We r e c a l l now Theorem 10.6 and i n d i c a t e t h e r e l e v a n t c a l c u l a t i o n s below.

t ) and (10.31) (based on (10.30) and Theorem 10.18),

iTHHE6RER 10,19, Given

i f (10.31) has unique s o l u t i o n s t h e n f o r

[P(Dt)

-

*

;(t)]?(t,s)

=

s" =

A

2Dt?(t,t),

A

Q(Dt)K(t,s)

=

A

P (Ds)K(t,s);

i ( t , O ) = 0; D T [ ~ ( t , ~ ) / A p ( ~ ) l ( t . o=) 0.

A

Hence [ K ^ ( t , ~ ) / A ~ ( ~ ) l , ( t , 0 ) = 0 and K ( t , O ) (10.33)

P u t t i n g (10.32)-(10.33) t a i n s from (10.311, = [P(Dt)

Further note t h a t

D t K ( t y t ) = ( D t + D T ) [ A p ( t ) K ( t , ~ ) / A ~ ( ~ ) l ( t , t ) = it(t,t)+

(A6/Ap (t (; t 5 t

=t?

= 0.

n

A

-

o

Ap ( t [ (( t 3

/Ap ( T

1, ( t

Y

t

i n t o t h e above c a l c u l a t i o n s and combining one obA

= 2il(t,t)&t,s)

P*(DS)]i?(t,s)

here.

i n Theorem 10.19 v i a .?(t,s)/G(t)

t =K(t,s)

A

A

+ :1 =K(t,r)n(T,s)dT

where

By uniqueness one o b t a i n s t h e e q u a t i o n =

c(t,s).

ORTHOGONAL POLYNOMIALS

21 9

Using Theorem 10.19 and repeatina some of t h e c a l c u l a t i o n s used in i t s demo n s t r a t i o n we obtain from (10.30) ( r e c a l l D i p P ( 0 ) = 0 ) P ( D t ) i p XP ( t ) + t " P P t " Consequently JO K ( t , ? ) v , ( T ) d r + $ ( t )v , ( t ) + JO K ( t , T ) P ( D T ) v A ( T ) d T .

t(t)

EHE0REIII 10-20, Under t h e hypotheses of Theorem 10.18 one has from (10.32) * 2" a n d Theorem 10.19 [ P ( D t ) - {(t)]; = Q(0,)f = - A f .

W e r e f e r now t o a r e s u l t in 8 2 . 1 1 which aoes as follows. Suppose we a r e i n t h e s i t u a t i o n described e a r l i e r via (10.23)-(10.24) ( Q u = (A,u')'/A + q u Q w i t h AQ = A A ). Assume Kx(y,O) = 0 and lim K(y,x)/x = 0 as x + 0 ( K as i n Q P ( 1 0 . 2 3 ) ) . This c e r t a i n l y h o l d s i f e.g. m > 0 since from (10.32) (with A = As') one has AQ(t)K(t,T) = Ap(-c)i; [ - ? ( l , t ) - v , (Pt ) ] i p , ( ~P ) d v . Then (10.34)

~ ; ' ( Y , Y =) 2[A$y)K(y,y)l' = -[q

t

= G(Y) =

(1/4)(A;/AqI2 - (1/2)(A;/AQ) - (Ab/Aq)((m+')/Y)I

This i s i n f a c t a r o u t i n e c a l c u l a t i o n s i m i l a r t o the proof of Theorem 10.19

*

(one uses (**) Q,(Ox)K(y.x) =Q(D )K(y,x) f o r x < y e t c . ) . T h u s (10.34) i s Y Q i s a transmutation as i n (10.23)a condition t h a t r e s u l t s when B: Qm (10.24). (Conversely i f (10.34) holds with (**) then the kernel (10.23) determines a transmutation a c t i n g on s u i t a b l e o b j e c t s via consideration o f -f

Goursat problems e t c . - assuming s u i t a b l e behavior of Kx(y,O) e t c . ) Now we assume (10.30) a r i s e s from some d i f f e r e n t i a l operator Q so t h a t A = A-'

Q '

Let Qou = ( A u ' ) ' / A = u" + ( % / A p ) u ' t (A'/A ) u ' ( A = A A ) and we comQ A -b A Q Q Q Q P pute ( r e c a l l f = A f = +)A ~ W= CQ, + q l f ; qo = ( A ' / A )(mA)/t + A0 9 4 (1/2)(A;/AQ) - (1/4)(A'/A )'. From $ = 2 D t K ( t , t ) i n Theorem 10.19 and (10. 4 Q 34) we have then (.) q ( t ) = -q t qo. Now use Theorem 10.20 and (") w i t h -?-?2 t h e formula f o r A 'Pf t o obtain Qof t q f = A 'Pf t (q-qo)f = - A f + ($-q-q,)f 2 Q Q = - A f . T h u s we assume given dw o r d o and solve (10.31) f o r K which y i e l d s Theorem 10.19 w i t h q^ determined. Assume now we have an underlying d i f f e r e n t i a l problem f o r some 0 = Oo t q so t h a t A 'A$: i n f10.25); then from (10. A There a r e two unknowns q and A a t our disposal 34) and ( ~ m ) q = -q + q 0' Q and e . g . , taking q = 0 (we leave open various questions about o t h e r choices )

-9

A

A

A

Our connection (10.25) w i t h A = Ad5 w i l l a r i s e from an underQ lying operator Q = 0, ( A = AoAp) w i t h B: ip! + f i n (10.25) representing a Q transmutation B: P Q ( P = Q,) provided A. can be found s a t i s f y i n g q^(y) = Z;'(y,y) = qo = + ( A ' / A )(m+%)/y where 4 i s known from solving the Q Q Q 9 equation (10.31).

CHE6REIII 10.21,

-f

A-'(L5)"

ROBERT CARROLL

220

We w r i t e now (A')'A-4

Q

Q

Q Q)

=*(A'/A

Theorem 10.21 becomes

and, s e t t i n g A'

C " + ( ( 2 m + l ) / y ) C ' -$C

(10.35)

4=

A - l = C t h e equation i n

= 0

L e t us n o r m a l i z e

which i s a l i n e a r ( s i n g u l a r ) d i f f e r e n t i a l e q u a t i o n f o r C.

w i t h say C(0) = 1 and t h e n C ' ( 0 ) = 0 i s i n d i c a t e d by t h e s i n g u l a r term i n (10.35) i f we want a r e q u l a r s o l u t i o n .

F o r reasonable

$

such equations can

be s o l v e d ( c f . [C29,51,63])

C0R0LLARU 10.22.

The A

4

r e q u i r e d t o connect (10.25) t o a d i f f e r e n t i a l prob-

lem as i n d i c a t e d can be o b t a i n e d by s o l v i n g (10.35) f o r C = A!! ing f(A,t)

Phaa::

o f (10.25) a r e o r t h o g o n a l r e l a t i v e t o dw.

The c a l c u l a t i o n s f o l l o w t h e p a t t e r n o f Theorem 10.3,

9'

The r e s u l t -

u s i n g t h e G-L

e q u a t i o n (10.31) o r Theorem 10.18.

11- RELACI0G 3EtU)EEN KERNEL$ AND PflEENCIALB,

We have seen i n Chapter 1,

§ 4 f o r example how i t i s p o s s i b l e t o c o n s t r u c t t r a n s m u t a t i o n s b y s o l v i n g Cauchy problems (as i n Theorem 1.4.3) o r v i a Riemann f u n c t i o n s and Goursat problems (as i n Theorem 1.4.7). We have a l s o seen how s p e c t r a l i n t e g r a l s I n 52.7 we have

a r i s e i n v a r i o u s c o n t e x t s i n §§1.9-1.12 and §§2.2-2.10.

shown how a m i n i m i z a t i o n procedure can a l s o be used t o g i v e an " i n t r i n s i c " I n t h e p r e s e n t s e c t i o n we w i l l

characterization o f transmutation kernels.

develop f u r t h e r some o f t h e m a t e r i a l r e l a t i v e t o t h e c o n s t r u c t i o n o f k e r n e l s and i n d i c a t e some s t r u c t u r a l f e a t u r e s ; i n p a r t i c u l a r r e l a t i o n s between k e r n e l s and c o e f f i c i e n t s ( o r p o t e n t i a l s ) a r e examined. suppose we have a t r a n s m u t a t i o n B: P

I n a general s p i r i t

Q w i t h t r i a n g u l a r k e r n e l B(y,x) where 2 P and Q a r e b o t h o f t h e form Qu = (A u ' ) ' / A + pQu - q ( x ) u (p, = A'/ -f

Q

A

Q

as x

-f

a).

klim 4

(2

We w i l l express a t y p i c a l such k e r n e l i n terms o f 6 f u n c t i o n s

and f u n c t i o n s K(y,x) w i t h a d i s c o n t i n u i t y i n o r d e r t o examine what t h e prope r t y o f transmutation e n t a i l s . (1.9.26),

F i r s t l e t us r e c a l l Remark 1.6.1,

remarks b e f o r e D e f i n i t i o n 1.11.9,

(1.9.25)-

and remarks b e f o r e (2.7.12)

and

expand t h i s as f o l l o w s .

REmARK f1,1,

I n o r d e r t o a v o i d e x p l i c i t mention o f f a c t o r s l i k e p 2 above

i n Q we w i l l w r i t e g e n e r i c a l l y

&I

= (Aqu')'/AQ

+ :(X)U

(4

Q

2 - 4). Q

pQ

Fur-

t h e r we w i l l w r i t e A ( x ) = xZm+'A ( x ) where f o r convenience we t a k e A E C Q Q Q 2 and p o s i t i v e (extend A as an even f u n c t i o n i n C when necessary). A l s o assume A

Q

A (a) = Am r a p i d l y as x e t c . as i n S1.6. Q r e p l a c e s Q used a t i s o l a t e d p l a c e s elsewhere)

4,

-f

-f

03

Now we have

(6

2

KERNELS AND POTENTIALS

(11.1)

o w = w" + {w;

AgQu = Q[A+u];

Q

Q

4=

q" -

221

A-+(A+)lI

Q

Q

We know f r o m 52.2 t h a t t h e r e a r e c a n o n i c a l t r a n s m u t a t i o n s B: D A

= xZm+' f o r Q,

Q

D2

and t h u s t o s t u d y t r a n s m u t a t i o n s

8

A

2

+

Q, where

A

P

Q one can l o o k a t

+

6

?

P^

Q, so t h a t o n l y + Pn and Q , + need be considered as new. ( T h i s p o i n t o f view i s perhaps most a p p r o p r i a t e i n cases where p o = p = Pn

+

+

+

-f

AP

-

c f . Remark 11.2). Hence we w i l l examine t r a n s m u t a t i o n s B: Q, + Q where A = x and AQ = xZm+'A have t h e same o r d e r o f s i n g u l a r i t y . F o r such m Q t r a n s m u t a t i o n s we c o n s i d e r (+) 'B = k + ( y ) B ~ - ~ - % : + E v i d e n t l y Q6[xm++ 0

"+'

f ( x ) ] = 4[A'Bf]

=

Q

(,

2 Q6 B f = .2 Q B Qm f

Gn, 4.

-m-+ m +%

LQ

Ab(y)Bx [x Qmf] = @ , [ ~ ~ +where ~f] 2 2 2 = w " - [(m - k ) / x ]w ( i . e . = -(m2-k)/x ). =

$,,

i s g i v e n as b e f o r e by Qmw 2 Now 6: D + D2 + w i l l have a t r i a n g u l a r k e r n e l i ( y , x )

$,

4

-f

by v i r t u e o f t h e a n a l y s i s i n Chapter 1, 514-5 and 9-12 (assume e.g.

l?(y,x)

i s modeled on B: q ;

-f

m+4 m

q!

-

+ Q

i.e. A

q X ( x ) ) ) . Consequently 8 = k-' which i s a V o l t e r r a t y p e i n v e r s e o f x

Bf(y) =

(

B(y,x),f(x))

f ( x ) ) ] = A:'(y)f(y) (11.2)

(y) = y

i.

(

K(y,x),f(x)).

= A?(y)[ym+'f(y)

REmARK 11.2. A

Q,

Q

=

When p

Q

or

pp

,

+

(

K(y,x),xm+'

Thus

+ K(y,x);

B(y,x) = A:(y)S(x-y)

-m-Lj

For B and 8 = B - l we o b t a i n t h e n

= A:(y)i[xm+'f(x)]

+

m+4 m

q A ( y ) + (L:(y)K(y,x)x has a k e r n e l ?(x,y) = S(x-y) + i ( x , y ) q

Q X

-m-%

K(y,x) = A:(y)y

S i m i l a r l y one o b t a i n s f o r 8 = B - l = (A:(y)ixm")-'

say Q,

= S(x-y) +

= x

-

K(y,x)xm+'

-m-+--1 Lj B AQ(y)

# 0 i t i s p o s s i b l y more a p p r o p r i a t e t o r e p l a c e

by a more " c a n o n i c a l " o p e r a t o r Qm h a v i n g a p 2 term. shZmc1xAQ(x) so t h a t A ' / A

Q

Q

Q

= (2rn+l)Cothx + ( A ' / A

Q

Q

)

-+

For example i f (2m+l) = 2p

w i t h Am = shZm+'x would p r o v i d e a b e t t e r adjustment o f s p e c t r a .

adjustment i s n o t e s s e n t i a l b u t i t may be convenient a t times.

Q

then

Such an

Recall t h a t

t r a n s m u t a t i o n s a r e c o n s t r u c t e d by PDE techniques f o r example w i t h o u t r e g a r d f o r matching s p e c t r a ( c f . a l s o Remark 1.9.6).

222

ROBERT CARROLL A

Now suppose B: Q,

-+

Q i s a t r a n s m u t a t i o n w i t h , f o r example,

B:

qm h - f q Qh

and where k e r B = 6 i s g i v e n by (11.2) w i t h K(y,x) a 2 reasonable f u n c t i o n (say C ) w i t h a d i s c o n t i n u i t y a t x = y. From another (spherical functions),

= ( T ( x ) , q AQ( y ) ) "

p o i n t o f view we w i l l expect a(y,x)

dv, e t c . a r e g i v e n i n Example 1.9.2

q:,

2 and

where Am = x2m+1 and

( c f . (1.9.4)

When m

and ( 1 . 9 . 9 ) ) A*

We r e c a l l t h a t ( f o r q* r e a l ) Q II, = [A ( $ / A Q ) ' ] ' Q '* Q Q - A Q ; ~ 2~ as and Q [Rh(x)] = A [& ] - - h Q h . Hence f o r m a l l y from a s p e c t r a l

=

-+,Q,

+

GJ,

D

=

A

-

Q

AX

Now t a k e t h e e q u a t i o n by =

above (*) Q i ( D x ) 6 ( y , x ) = Q(Dy)a(y,x).

qf

with

k e r B given b y (11.2) and s p e l l o u t t h e requirements f o r t r a n s m u t a t i o n as follows.

We know t h a t Q;(Dx)K(y,x)

f o r x < y by (*) above

= 4(Dy)K(y,x)

A-+(q'

-

Q

=

A and q: = q

(B)

$(A'/A)q) w i t h (A-\)"

+

Hence (A-'q)''

(A"/A))q].

-

q((1/4)(A'/A)'

(1/2)(A"/A)

-

Q Q

)(A-$)'

(K(y,y)q(y))' (11.5)

+ KY(y,y)v(y)

On t h e o t h e r hand (11 K(y,

- (m+%)(A'/A)/x)].

+

and

(Jdy

JJ Kyy(Y,x)lp(x)dx.

[4^(Dy) + A 2 ]qh(y) Q = A-'(y)[S"

=

+ ((3/4)(A'/A)* - (1/2) + ((2m+l)/x)q' +

(A'/A)q' = A-'[q"

x ) q ( x ) d x ) ' = K ( Y , Y ) ~ ( Y ) + f6y Ky(y,x)v(x)dx

K(y,x)qy(x)dx].

+ A ' / A and ( A % ) '

A4/AQ = ( ( 2 m + l ) / x )

= A4[q"

(A'/A

J6y

+

and we w i l l w r i t e o u t t h e e x p r e s s i o n t(Dy)[Aq4(y)qT(y) Thus s e t t i n g A

K(y,x)v(x)dx)"

=

It follows t h a t

+ (l/4)(A'/Al2

((m+%)/y)(A'/A)l~(y) + ( K ( Y , Y ) ~ ( Y ) ) ' + KY(y,yb(y)

+

-

(1/2)(A"/A)

((2m+l)/y)

+

-

(A'/A))

Y

K(Y,Y)~(Y) + Now K(y,x)

jo [Qi(Dx)

= AQ5(y)i(y,x)xm+'

+

~ 2 1 K ( ~ , x k ( x ) d x= 0

w i t h t? a reasonable f u n c t i o n ( c f . [C40] and

so f o r m > - ( 1 / 2 ) a t l e a s t one expects K (y,O) = 0 and K(y,x)/x x* 0. Then from t h e d e f i n i t i o n o f Jdy [Qm(D,)K(y,x)]q(x)dx = [ X ~ ~ + ' ( K ( Y , X ) X - ~ ~ - ~ ) ~ ] ~=~ x( 2m+l X ) ~ (K(Y,X)X X -2m-1 ) x q ( ~ ) l-i K(y,x)x -2m-1

551.9-1.12) + 0 as x

Qi,

-+

(x2m+l q ' ( x ) ) (11.6)

;1

:1

+ J{ K ( y , x ) ~ - ~ ~ - ' ( x ~ ~ ' ' q ' ( x ) ) ' d xConsequently . we o b t a i n [(Qi(Dx)

+

K(Y ,Y

A 2 ) K ( ~ . x ) k ( x ) d x = Kx(y,yk(y)

1~ ' ( Y ) -

( (2m+l ) / Y ) K(Y .Y

-

b (Y

Combining (11.5) and (11.6) we have ( n o t e K'(y,y) = (KY + Kx)(y,y)) (11.7)-

0

= A-'(Y)[~^

-

+ (1/4)(A'/A)'

+

ZK'(y,y)v(y)

+

Note here a l s o t h a t K(y,y) = A-'(y)k(y,y)

(1/2)(A"/A)

-((m+%)/y)(A'/A)]q(y)

(A'/A)K(y,y)v(y)

so t h a t I?(y,y)' = (1/2)A+'(y)

KERNELS AND POTENTIALS

[2K' (y,y)

+ (A'/A)K(y,y].

223

Consequently

CHE@Rm 11-3- Given a t r a n s m u t a t i o n B: Qm

+

6 with

k e r B = ~ ( y , x ) =' ; A

6 ( x - y ) + K(y,x) as i n (11.2) (K(y,x) = 0 f o r x > 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,<)Cosxgdg

we have

K ( x , ~ ) and one has q hQ ( x )

=

5

?(x,S)CoshCdc.

-

S e l l (where 1

d i s c u s s below i n Theorem 11.7).

For A = 1 ( q = 0 ) t h e r e s u l t i n Theorem

1.4.7,

K(x,x) = A-'(x)

-

T h i s r e s u l t i s t h e form i n [C40;

which we

o b t a i n e d v i a PDE techniques and Riemann f u n c t i o n s , has t h e form

(1.4.171,

narnelyG!(x)

= ~ o s h x+

Kh(x,c)Coshgdc,

where K ~ ( ~ , S =) h

+

KERNELS AND POTENTIALS

+

i?(x,c)

-

T ( X , - E ) + hJX [?(x,n)

F u r t h e r 2DxKh(x,x)

5

(m)

=

K(x,S)(A/2i)eiA5dc

X

[SinA(x-<)/A][;(

0 ( n o t e a l s o hSinhx/x = hf:

=

-

6 1:

[Sin~(x-~)/~l~~(5)D5~~(5)ld5 =

-hSinhg + hSinAgK( 5 , c ) +

One can f o l l o w [Mc4] o r use techniques o f [C40], l a t i o n s between K(x,x)

and

CosAgdg).

w i t h (11.8) t o o b t a i n

X

(11.11)

( r i n v o l v e s a Riemann f u n c t i o n ) .

J. ,

= q ( x ) and K(x,-x)

Now f o r q = h = 0 we use

I

?(x,-n)]dn

225

4.

K5(S,n) ( x / 2 i )eixndn)]dg p. 282 e t c . t o o b t a i n r e -

Since t h e l a t t e r t e c h n i q u e i s r a t h e r c o m p l i -

c a t e d we g i v e here a new d e r i v a t i o n o f t h e r e s u l t i n [C40,66;

S e l l when q =

h = 0 ( c f . a l s o [Mcl]). Thus [ S i n x ( x - ~ ) / A ] e x p ( i x n ) = i J nn+(x-5)exp( -(x-5) iAu)du; Hence f o r g e n e r i c f ( 5 ) and [ S i n x ( x - ~ ) / h ] e x p ( - i x n ) = i j ( x - 5 ) - n e x p ( i h u ) d u (X-L)-n -i(u+x) xiu S i n x ( x - 5 ) ,ihsd5 = (11.12) f(5) f(5)dCdu '0 '0 x-25 and s i m i l a r l y 1; f(~)[Sinh(x-~)/h]exp(-iA~)d< = if," f ( 5 ) i x exp( xu)dudg k i Xx exp(ihu)/o 4 ( x -u ) f ( c ) d g d u . I n t h e same manner f o r G(C,n) = 0 when In

I"

151 ( a s f o r K o r K5) (11.13)

(11.14)

.Further i n the l a s t

Q-S

i n t e g r a l i n (11.13) we c o n s i d e r t h e f o l l o w i n g p i c t -

u r e where G(5,n) = 0 i n r e g i o n s 1 and 2 (11.15)

ROBERT CARROLL

226

the 3 r e g i o n t a k e C+T- = 2a and 5-n = 2~ and o n e o b t a i n s then f o r ( 1 1 . 1 3 ) ) I = $ XL e~x p ( i x u ) ~ ~ ( u + x ~ ~ ( x - U ) f ( c r + 8 ) G ( c r + 8 , a - B ) d u = $1X e x p ( i A u ) -X

4

A

x,u)du where we n o t e t h a t G(x,x) = G ( x , - x )

=

0.

NOW go t o ( 1 1 . 1 1 ) w i t h

Sinxg = [ e x p ( i A c ) - e x p ( - i A c ) ] / 2 i t o o b t a i n 1; K ( x , S ) e x p ( i h c ) d S = 1; Q(x,u) e x p ( i x u ) d u ( Q ( x , u ) below) s o K(x,u) = Q(x,u) and i n p a r t i c u l a r K(x,x) = Q(x,x) where i n t e g r a l s o f t h e form ( b ) will n o t c o n t r i b u t e .

Thus

EHEBREIR 11-7- Under t h e h y p o t h e s e s i n d i c a t e d one o b t a i n s (h

= q = 0 and I t f o l l o w s t h a t 1 K ( x , x ) = A-'(x). A'/A) K(x,x) = -k$ ~ ( c ) [ K ( ~ , c ) - l ] d c .

Phaod: Dropping terms o f t h e form (i), we have from ( 1 1 . 1 1 ) FK 1 ; [ S i n X ( x - c ) / A ] ~ ( g ) [ K ( g , c ) - l]ASinXcdc = -+LXX exp(iAu)[fo'(x+u)

[G(c)(K(c,t) - 1)ldedu.

%

6=

-(2i/A)

- lyx-LJ)

1

The a s s e r t i o n o f the theorem f o l l o w s .

REmARK 11-8- Suppose we s t a r t w i t h Qu = ( A u ' ) ' / A - qu f o r A Au' we c o n s i d e r bw = w" + Gw = - h2w (6 = -q-;, q' = A-'(L5)").

2

so f o r w = Further t a k e v u ( 0 ) = 1 and u ' ( 0 ) = h so w(0) = 1 w i t h w ' ( 0 ) = h = h + A ' ( 0 ) . Based on = #9,4,, w h i c h we write a s 51.4 o n e has a f o r m u l a f o r w = 9 AYh E

C

O Q

(11.16)

$!(x)

= A-'(x)CosAx

c

+ A-'(x)

Kh(x,c)CosAcdc

For q = h = 0 w i t h ( 0 0 ) 2 0 x K h ( ~ , ~=) q ( x ) + c ( x ) = q ( x ) + A-'(x)(L'(x))''. ( w i t h A-'(x) = 1 - K(x,x)) ( 1 1 . 1 6 ) s h o u l d a g r e e w i t h Theorem 11.6 so t h a t i?(x,e) = ~ ~ ( x , =t )A-+(X)KK(X,E) (hy = $ A ' ( o ) ) .

- qu a g a i n a s i n Remark 1 1 . 8 b u t - Coshx - hSinAx/A = x ($! % 9h4,h)

REIIIARK 11-9- Suppose we t a k e Qu = ( A u ' ) ' / A

r e q u i r e o n l y A E C1. The f u n c t i o n $!(x) c a n b e c o n s t r u c t e d v i a ( 1 1 . 8 ) and w i l l be e n t i r e i n A of e x p o n e n t i a l t y p e x.

Hence by Paley-Wiener i d e a s

1

X

(11.17)

$!(x)

=

CosXx + h[Sinxx/x] +

i(x,e)eiAedg

-X

(for a suitable distribution

i).

Since

x

i s even i n A , k ( y , c ) w i l l be even

i n 5 and (11.17) can be w r i t t e n ( i h ( x , c ) = h + t ( x , g ) ) , $:(x) = CosXx + 10" th(x,c)CosAgdg. Thus f o r A = 1 , ih c o i n c i d e s w i t h t h e Marc'enko kernel Kh o f Theorem 1 . 4 . 7 ( c f . remarks a f t e r Theroem 1 1 . 6 ) . For A # 1 however a l i t t l e rough a n a l y s i s (which we o m i t ) o r s i m p l y p r e v i o u s results on k e r n e l s

(from Theorems 1 1 . 6 and 1 1 . 7 f o r example) i n d i c a t e s t h a t have 6 f u n c t i o n components and we set a c c o r d i n g l y (11.18)

4

i ( x , c ) = a ( x ) [ 6 ( x + c ) + a ( x - c ) ] + K(x,S)

K'

i n (11.17) will

KERNELS AND POTENTIALS

where

i s a function.

Thus (11.17) becomes

4

(11.19)

+

+ h[Sinhx/A]

9 h ( x ) = [l+a(x)]Cosxx

227

1

X

4?(xyg)CosA
X

and a ( 0 ) = 0 i s i n d i c a t e d .

We do n o t s p e c i f y a ( x ) a p r i o r i and i n f a c t i t

can be determined from t h e a n a l y s i s which f o l l o w s .

+r

We proceed now as i n

t h e p r o o f o f Theorem 11.7 and w r i t e from (11.8) and (11.17) ( w i t h (11.19)) i(x,5)eixgdg

(11.20)

1;

I

=

S i n h ~ x - g ) [ q ~ ~ ( 5 ) - ~ D g Q~ h ( 5 ) l d=5

=

X

0

Sinh/x-S)[q[(l+a(c))Cosh5+

h(Sinhg/h) +

$1

I - A (1 +a ($) ) S i nxg + (h+a +?(5, 5 ) )Cosh E; + I

$11

?(c,q)eixndq3

- t(5)

5,

K5 ( E; ,n)ei ndnl]dS

5

We use now t h e development f o r p r o v i n g Theorem 11.7 t o o b t a i n

I = J + '/4

(11.21)

e i ~ ~ ~ ( [ ( x ++ ~~ () X - u ) ) ( l + a ) q ( i ) d ' ] d u

(lo

+( x+u) r+(x-u7

kll eixu

0 +(X+U) + 14(X-U) ) q (a+B )?( a+@,a- B ) d Bdcldu - F L e i 4(x+u) % ( X - U l X 0 q(a+B)? (a+B,a-B)dBdadu 41: eihu 5

j

I, I,

0

0

:[ h+a I +?( 5, g ) ] d ~ d u -

where J = J1 + J2 as i n d i c a t e d below.

I

rS

J1 = ( h / 2 )

n+(x-s)

15 1n - ( x - 5 )

flu

q(~)[SinA(x-c)/hl

0

dudndg = (h/4) j:ei

e

We c o n s i d e r from (11.20)

1

5 . e'xndnd5 = (h/4) 5 joxq(5 ) f ( u x s )&du

X

(11.22)

+

1

X

q(5)

0

where f can be determined by i n s p e c t i o n o f a n expanded v e r s i o n o f (11.14) which we omit.

Thus f o r 5 > x/2 one has

i5Jq-fx-E) n++fx-S)exp(ihu)dudn

=

cix-25 p+(x-s) 2E-x /U+fx-S)+ xi 'f ]exp(ihu)dndu so f(u,x,S) = -x -5 + fx-25 U-(X-E) 25-x u-(x-E) x+u, 2 ( x - 5 ) , and x-u r e s p e c t i v e l y i n t h e 3 r e g i o n s f o r u i n t e g r a t i o n ( - x y x-25),

(x-25,25-x),

and (2<-x,x)

respectively.

e r a t i o n s p r e v a i l (which we m e r c i f u l l y o m i t ) . f ( x y x , 5 ) = 0. (11.23)

J2

J2

I n p a r t i c u l a r we n o t e t h a t

Next f o r J2 we have =

A lXqn(l+a)[Sinh(x-')/x]Sinhgdg

JLX '0 NOW s e t A(x) = J , ;(S)(l+a)dg (11.24)

For E; < x / 2 s i m i l a r c o n s i d -

=

'0

and i n t e g r a t e by p a r t s i n (11.23) t o o b t a i n

= - A(x)Coshx + (1/8)i:

eixu[A'(

X+U

+ A'(

)Idu

228

ROBERT CARROLL

where A ' ( 5 ) = 4(5)[1 + a ( 5 ) l .

The f i r s t t e r m on t h e r i g h t i n t h i s l a s t ex-

p r e s s i o n r e p r e s e n t s t h e 6 f u n c t i o n c o n t r i b u t i o n i n t h e f o r m -+A(x)Cosix =

+ S(x-u)]du and hence f r o m (11.22) we o b t a i n f i r s t

-+A(x) / z e x p ( i i u ) [ 6 ( x + u )

( e q u a t i n g t h e 6 f u n c t i o n c o n t r i b u t i o n s on b o t h s i d e s ) a ( x ) = -4A(x). f r o m which f o l l o w s 1 + a ( x ) = A-'(x)

a ' = - Q ' = -%(A'/A)(l+a) = 1 and a ( 0 ) = 0 ) .

+ a(E))d5 and express

=

1 ; :(5)(1

4

=

A ' / A and A(0) = 1 ) .

w i t h a(0) = 0 ( r e c a l l

k,

M

= h + K i n (11.19),

Eh(x,c) w i t h ?(x,u) 2)] +

=

:V

+

lo '('-')

k

as i n (11.18)

Then 1+ a ( x ) = A-'(x)

and

= ( ~ ( x , ~ ) , C o s X g )w i t h B ( X , S ) = A - ' ( x ) 6 ( x - ~ )

( h / 2 ) 4 q(E)f(u,x,E)dE

%Ji'(x+u ) /?('-' [ q (atB )?( at0 ,a- 8 )

4(/i5(x+u)

( r e c a l l A(0)

Thus sumnarizing ( c f . a l s o Remark 1 1 . 5 )

&€IEQ)REm11-10. Set A(x) for

Hence

)[q(c)(l+a(c))

-

-

+ (1/4)[A'(x+u)/2)

+

+ A'(x-u)/

M

{ ( a + ~Kg ) ( a + o , a- B ) I d Bda +

(h+a'+?(~,~)){(~)ldS where f is g i v e n M

= 0.

as above w i t h f(x,x,S)

%/$ t(S,S)(A'/A)(S)d5

=

S e t t i n g u = x i n (11.25) one o b t a i n s K(x,x)

H(x) = W ' ( 0 )

-

+(A-')'(X)

-

(h/Z)logA(x) +

M

$/$ [qA-'

-

2Dx?(x,x)

and f o r general A ( A ' # 0 ) one o b t a i n s ?(x,x)

Dx[A-'

(A')'(A'/A)]dE.

= %/o

I f A = 1 one o b t a i n s K(x,x:

( X) 1 ; H ( 5 ) ( A ' /2A? ( 5 ) d51.

=

X

qdS so q =

(2A/A')(x)

t

CHAPEER 3

APP LZCACZ(DN$

I.

ZNER(DDlltEZ6N.

I n t h i s c h a p t e r we w i l l g a t h e r t o g e t h e r v a r i o u s t e c h n i q u e s

and r e s u l t s concerning t r a n s m u t a t i o n methods i n t h e areas o f l i n e a r stochast i c e s t i m a t i o n and i n v e r s e problems i n geophysics ( o t h e r m i s c e l l a n e o u s a p p l i c a t i o n s a r e i n d i c a t e d i n 910).

The remarkable s i m i l a r i t y i n p a t t e r n s and

s t r u c t u r e between formulas and methods i n these areas i s d i s p l a y e d and s t u d i e d (and these c o n t e x t s a r e c o n s i d e r e d p r o t o t y p i c a l f o r o t h e r a p p l i c a t i o n s ) . Some new r e s u l t s on geophysical i n v e r s e problems a r e i n c l u d e d 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 r o l e o f m i n i m i z a t i o n 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 i s shown t o be e q u i v a l e n t i n s t o c h a s t i c geometry t o l e a s t squares f i l t e r i n g e s t i m a t i o n .

Given t h a t many n a t u r a l processes

a r e governed by second o r d e r l i n e a r d i f f e r e n t i a l equations ( a t l e a s t t o a f i r s t a p p r o x i m a t i o n ) i t i s c e r t a i n l y no s u r p r i s e t o see t h a t c e r t a i n mathem a t i c a l p a t t e r n s and s t r u c t u r e s r e c u r f r e q u e n t l y i n p h y s i c s and a p p l i e d mathematics. above.

We w i l l weave t h e theme o f t r a n s m u t a t i o n t h r o u g h t h e t y p i c a l areas Our main i d e a i s t o emphasize

p a t t e r n s and s t r u c t u r e common t o these

areas and t o u n i f y i n a c e r t a i n way t h e t r e a t m e n t o f such t y p i c a l problems. I n f a c t , t h e use o f t r a n s m u t a t i o n methods has a l s o been h e l p f u l , f o r example as a g u i d e and t o o l i n d e v e l o p i n g new r e s u l t s about geophysical i n v e r s e problems and thus t h e t r a n s m u t a t i o n machine goes beyond mere u n i f i c a t i o n i n i t s usefulness.

The i n t e r a c t i o n w i t h i n v e r s e problems i s p a r t i c u l a r l y " i n t i m a t e "

b u t i n h i g h e r dimensions t h e study o f i n v e r s e problems i s o n l y p a r t i a l l y amenable t o t r a n s m u t a t i o n techniques and o t h e r methods a r e necessary.

As a preview o f t h e c h a p t e r we mention t h e f o l l o w i n g .

Section 2 i s a c o l l e c -

t i o n o f i d e a s and c o n s t r u c t i o n s from p r o b a b i l i t y and s t o c h a s t i c a n a l y s i s . 593-5 p r o v i d e a survey o f some o f t h e i m p o r t a n t i d e a s and c o n s t r u c t i o n s from l i n e a r stochastic estimation.

The s t r u c t u r e o f v a r i o u s i n t e g r a l equations

which a r i s e f o r t h e f l t e r i n g and smoothing k e r n e l s a r e s t u d i e d (as G-

equa-

t i o n s , W-H equations, r e s o l v a n t equations, e t c . ) and v a r i o u s r e l a t i o n s and 229

230

ROBERT CARROLL

formu as are derived which have both theoretical and computational interest (e.9. Krein-Levinson-Shur relations, Krein-Bellman-Siegert formulas, Sobolev re ations, Darboux-Christoffel formulas, etc. ) . The innovations concept is treated in some detail and underlying differential problems are obtained as in [Lx1,2] for the even and odd innovations processes. In 556-7 the spectralized innovations processes are studied via transmutation in terms of eigenfunctions q Q as in 551.4 and 1.5 and the role of minimization as Ash in 52.7 to characterize the transmutation (filtering) kernel is shown to accomplish the same thing in stochastic geometry as linear least squares estimation. There is some technical information on the representation of functions in the K-L theory (to supplement §2.8) and there is a short section on random fields following [Lx~] in order to show the connections to transmutation for singular operators. §§8-9 are on inverse problems in geophysics where transmutation methods are particularly effective and revealing for the one dimensional problem. 57 is largely on the computation of acoustic impedance as a function of travel time via reflection data using G-L methods and follows the work of the author and F. Santosa as in [C40] (with some new proofs and insight). Layer stripping methods following [Lx 4-61 are briefly discussed as is the use of transmutation for three dirnensional problems following the author and Santosa. 59 involves some new work following [C71-73,78] on recovery of the impedance profile via transmission data. Some interesting new mathematical features arise (e.g. splitting of the spectral measure) which gave connections to filtering theory (e.9. Wiener filters). The extended G-L equation in the time domain gives a model where ideas of causality and triangularity interact naturally. In 5510-11 we collect miscellaneous information on transmutation for systems and for equations with operator coefficients and mention briefly some other topics within the limitations of space guiding the preparation of this book. 2. PROBABILZQI

E€lEP)R11 AND RANDOM PRP)CE$5E5. Stochastic analysis has burgeoned

in recent years and has both used and provided many mathematical techniques. The interactions with the differential equations of physics and engineering are legion and thus it is no surprise that we will be able to indicate some interesting connections with transmutation. Actually the interaction with transmutation is quite profound since for example many of the integral equations of linear filtering and estimation theory represent transmutation formulas (in the same manner as do many - corresponding - formulas in quantum scattering theory). Indeed one can link together a great deal of the

RANDOM PROCESSES

231

machinery o f i n v e r s e s c a t t e r i n g and l i n e a r e s t i m a t i o n t h e o r y w i t h transmutat i o n ideas and techniques.

I n t h i s s e c t i o n we want t o g i v e some minimal

I t would be excessive t o a t t e m p t

background f o r t h e s t o c h a s t i c machinery.

a t u t o r i a l i n p r o b a b i l i t y t h e o r y o r i n measure t h e o r y so we g i v e a s i m p l e o u t l i n e o f some ideas i n s t o c h a s t i c processes guided by v a r i o u s " a p p l i e d " p r e s e n t a t i o n s i n e.g.

[Kacl;

Pe2; Wpl (see e s p e c i a l l y [Wpl]).

For f u r t h e r

d e t a i l s , v a r i o u s p o i n t s o f view, e x t e n s i v e e l a b o r a t i o n , e t c . we c i t e e.g. [Dul,3;

Dxl; Dagl; L x l - 3 ;

Kvl; Kul-14;

D w l ; Mhl; Lwl; Pgl; Rql-3; J d l ; Wol;

Rpl; B b i l ; Ycl; B b j l ; I c l ; M i l ; Gw1,2;

K w l ; Phl; Ydl; Fml; Kacl].

One c o n s i d e r s s t o c h a s t i c processes X t ( w ) , w E R

t E T (usually T =

(-m,m)),

and

(where n i s an a p p r o p r i a t e measure space o r p r o b a b i l i t y space).

f o r t f i x e d Xt

Then

i s a random v a r i a b l e ( i . e . measurable f u n c t i o n ) r e l a t i v e t o

some p r o b a b i l i t y d i s t r i b u t i o n dP(w) on 52.

One r e c a l l s a l s o t h e i d e a o f

o f p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n f o r a random v a r i a b l e X; t h i s i s d e t e r mined by P x ( a ) = P{w; X ( W ) 5 a }

-

f u r t h e r P X ( A ) = P{w; X ( W )

t e r m i n e s a measure on Bore1 s e t s A C pt= mt = E(Xt) = EXt = fn Xt(w)dP(w)

A } which de-

€

One d e f i n e s a mean

R based on (dP,n).

(and we n o t e t h a t EX =

L:

gdPX(c) =

fa X(w)dP(w)). We u s u a l l y assume vt = 0 and suppose f u r t h e r t h a t t h e v a r i 2 2 ance o = E I X t = E I X t I Z < m. A typical probability distribution a f u n c t i o n would be t h e z e r o mean Gaussian d i s t r i b u t i o n P(X < a ) = 1, exp (-SiS2/cr2)ds/J2~02 =

i:

dP(s) ( i d e n t i f y here

and

(-my-)

r e f e r s t o an a l g e b r a A o f " e v e n t s " (measurable s e t s ) i n

f o r example).

a, about

One

which we

w i l l m e r c i f u l l y say p r a c t i c a l l y n o t h i n g , and we denote by L t h e a l g e b r a o f Lebesgue measurable s e t s i n T.

We r e c a l l a few b a s i c ideas now and c o l l e c t

some background i n f o r m a t i o n ( n o t e E

Q

expectation).

DEFZNZEZON 2-1, Xt i s c o n t i n u o u s i n p r o b a b i l i t y a t t i f f o r each E > 0 P{IXs-Xtl 5 €1 0 as s t. I f Xt i s continuous i n p r o b a b i l i t y a t each t -f

E

-f

T t h e n one says Xt

Xt i s separable i f t h e r e

i s continuous i n p r o b a b i l i t y .

e x i s t s a c o u n t a b l e ( s e p a r a t i n g ) s e t S c T and an event A ( A = A ( s ) ) f i x e d w i t h P ( A ) = 0 such t h a t f o r any c l o s e d K c s e t s {w;

Xt(u)

subset o f A.

K; t E I n T I and { w ;

E

E

and open i n t e r v a l I t h e

K; t

E

I n S 3 d i f f e r by a

c

(8 A.

i s measurable

One knows ( c f . [ D x l l ) t h a t i f Xt i s continuous i n

p r o b a b i l i t y t h e n t h e r e i s a process 'v

P(Xt = X t )

(--,a)

i s measurable i f Xt(W) as a f u n c t i o n o f (t,w)

Xt

w i t h respect t o

X,(W)

Yt

d e f i n e d on t h e same 0 such t h a t

N

= 1 f o r each t E T,

Xt i s separable and measurable, and any

c o u n t a b l e dense s e t S c T serves as a s e p a r a t i n g s e t . We w i l l primarily be concerned w i t h complex v a l u e d "second o r d e r " processes.

ROBERT CARROLL

232

l 2 < a ) w i t h EXt = p t = 0. Such a process i s c a l l e d wide sense t s t a t i o n a r y i f i n a d d i t i o n t h e a u t o c o r r e l a t i o n ( = c o v a r i a n c e here s i n c e p =

Xt

(i.e.

EIX

i s a f u n c t i o n o f t - s a l o n e (we w i l l w r i t e R ( t , s ) = R ( t -

0 ) E(XtXS) = R(t,s)

F u r t h e r one w i l l want Xt t o be continuous i n q u a d r a t i c means o r mean s)). square continuous i n t h e sense t h a t EIXt+h - X t I 2 + 0 as h -+ 0. By t h e Markov i n e q u a l i t y

ElXI2 2

(0)

E

2

P(Ixl

-

E),

mean square c o n t i n u i t y o f Xt

p l i e s t h a t Xt i s continuous i n p r o b a b i l i t y .

We t h e n use D e f i n i t i o n 2.1 t o

One sees t h a t R(t,s)

r e p l a c e Xt by Xt i f necessary.

im-

i s nonnegative d e f i n i t e

2 i n t h e sense t h a t 1 a . t i R ( t . , t ) = E l l a.X j l 2 0 and f r o m t h e e q u a t i o n J k J k - J t R(t+h,s+^h) - R(t,s) = EXt+hXs+; - EX X = E(Xt+h - X t2 ) js+h + EXt(Xs+fi-js)

ts_ and t h e Cauchy-Schwartz i n e q u a l i t y I E X Y l 5 ( E l X l E l Y l )

mean square c o n t i n u i t y o f Xt t h a t R ( t , s )

4

i t f o l l o w s from

i s continuous i n ( t , s )

(exercise).

One denotes b y HX t h e H i l b e r t space o f random v a r i a b l e s o b t a i n e d from X t v i a sums =

ElYl

2

.

1 a.X .(w) J t,

= Y(w)

and Cauchy sequences t h e r e f r o m i n t h e norm IIYII

2

That such Cauchy sequences y i e l d random v a r i a b l e s f o l l o w s from

t h e Markov i n e q u a l i t y

g i v i n g a Cauchy sequence i n p r o b a b i l i t y , and

(o),

hence a subsequence converges almost s u r e l y ( i . e . random v a r i a b l e .

almost everywhere) t o a

F u r t h e r one can d e f i n e mean square i n t e g r a l s i n HX by

f o r say continuous f, where to = a < tl <

.. .

< tn = b, tv 5 5”

5 tv+l, and

t h e l i m i t o f these Riemann t y p e sums (presumed t o e x i s t ) i s t a k e n i n HX. b b b f ( t ) X t d t e x i s t s i f and o n l y i f f(t)f(s)

One shows(exercise) t h a t

/a

/a /a

R ( t , s ) d t d s e x i s t s as a Riemann i n t e g r a l .

-

-

One n o t e s a l s o t h a t i f we d e f i n e

U X = Xt+s t h e n EU X U X = EXt+sXt+u = R(s-u) = E X s j t so Ut determines a t s t s t u u n i t a r y o p e r a t o r on HX (ifYn Y i n HX s e t UtY = l i m UtYn). C l e a r l y UtUs -f

= Ut+sy U i ’ = U-t,

and U,

=

I so t h a t Ut i s a t r a n s l a t i o n group.

One expects now t o be a b l e t o work w i t h a F o u r i e r t r a n s f o r m i n a n a l y z i n g and d e s c r i b i n g wide sense random processes Xt as above.

F i r s t we s k e t c h a

p r o o f o f a theorem o f Bochner ( c f . [Bu2; S j l ; Wpl]).

EBE@REM 2-2. A f u n c t i o n R ( T )

E

Lm i s t h e c o v a r i a n c e f u n c t i o n o f a mean

square continuous wide sense s t a t i o n a r y process i f and o n l y i f R ( T ) = ( 1 /

21~)i: e x p ( - i w ) d F ( v ) where dF(v)

7r

F(dv) i s a f i n i t e nonnegative ( B o r e l )

measure ( c a l l e d t h e s p e c t r a l measure).

Pkaaa:

To prove i m p l i c a t i o n we can t a k e R ( t - s ) c o n t i n u o u s and nonnegative

d e f i n i t e and d e f i n e R,(T)

= (l-\xI/Zn)R(T)

for

\TI

5 2n w i t h R,(T)

= 0

RANDOM PROCESSES

elsewhere.

Then Rn

E

1

L 1 n Co and i t i s a l s o nonnegative d e f i n i t e .

m

Sn(v) =

(2.2)

233

Define

A

Rn(r)eivTdT

= Rn(v)

m

i s nonnegative d e f i n i t e ) . dT.

Now l e t f E S and de-

Then u s i n g t h e Parseval f o r m u l a we w r i t e

T)F(T)dTl = l i m

1

Hence p can be extended t o a continuous l i n e a r f u n c t i o n a l on

Co(-m,m)

t h e sup norm which means by a theorem o f F. Riesz ( c f . [ R m l ;

Rol; Mc41)

that

p(f)

ite.

=

with

/: f ( w ) d F ( v ) w i t h dF nonnegative s i n c e R i s nonnegative d e f i n -

For f E S we have then m

p(f) =

(2.4)

[

I

m

f(v)dF(v) =

m

;(-r)R(T)dT

m

Set now f n ( v ) = e x p ( - + v 2 / n ) e x p ( - i v t )

t o get

e r c i s e - cf. [Wpl] f o r t h e procedure h e r e ) .

-,2 v R ( t )

and

iz

fn(v)dF(v)

m

.+

i,

in(T)

= mexp(-+n(t-.r)2

I t f o l l o w s t h a t /:

exp(-ivt)dF(v).

(ex-

;,,(T)R(+)dT

On t h e o t h e r hand one can

m

show e a s i l y t h a t any R ( t ) d e f i n e d by ( l / 2 n ) J a e x p ( - i w t ) d F ( v ) as i n t h e statement o f Theorem 2.2 w i l l be nonnegative d e f i n i t e and continuous; i t t h e n f o l l o w s t h a t i t determines t h e c o v a r i a n c e f u n c t i o n o f some mean square continuous wide sense s t a t i o n a r y process ( c f . [Wpl] f o r d e t a i l s ) . f u n c t i o n F i s d e s c r i b e d more c o m p l e t e l y below i n any case. Now g e n e r a l l y X t may n o t have a F o u r i e r i n t e g r a l , i . e . n o t e x i s t as a mean square i n t e g r a l .

The s e t

W

[I X t e x p ( i w t ) d t

may

One can i n t r o d u c e however a s p e c t r a l

4

process X,

w i t h o r t h o g o n a l increments such t h a t

F i r s t l e t us s p e c i f y t h e s e t f u n c t i o n F i n Theorem 2.2 more p r e c i s e l y .

Thus

l e t a and b be c o n t i n u i t y p o i n t s o f F and t a k e ( c f . t h e p r o o f o f Theorem 2 A 2 2.2) f n ( v ) = (n/2n)'lb e x p ( - n ( v - v ) /2)dp w i t h f n ( T ) = e x p ( - r / z n ) [ e x p ( i T b ) a - e x p ( i T a ) l / i r . NOW (n/2a)'exp[-n(v-u) 2 /21 6 ( v - u ) as n + m so f n ( w ) -+ -f

lab ( l a b = 1 on [a,b] F[a,b)

=

and 0 elsewhere) and consequently Fab = F[a,b]

F(a,b) = F(a,b]

= lim

r:

fn(v)F(dv) = l i m

/-IFn(T)R(T)dT.

o u t t h e l a s t e x p r e s s i o n y i e l d s ( f o r c o n t i n u i t y p o i n t s o f F)

=

Writing

2 34

ROBERT CARROLL

Since t h e p o i n t s o f c o n t i n u i t y o f F a r e dense i n

l y determined by (7.6) ( e x e r c i s e - c f [Wpl]).

(--,m)

F w i l l be complete-

We n o t e t h a t (2.6) i s c o n s i s -

m

t e n t w i t h R ( T ) = (1/2n)lm e x p ( - i v T ) d F ( v ) .

Now f o r such c o n t i n u i t y p o i n t s a

and b o f F s e t f o r m a l l y

4

w i t h Xa

-f

0 i n mean square as a

-f

and, a t t h e ( p o s s i b l y ) c o u n t a b l e s e t

-m

6

o f d i s c o n t i n u i t y p o i n t s o f F, X, left.

But

Lz

Then f o r m a l l y

(2.9)

b d 1 exp(iuT)6(u-p)dudu one o n l y a c = [c,b) o r [a,d). W r i t e t h i s as

e x p ( i s ( v - v ) ) d s = 2 ~ 6 ( v - u ) so i n f

n [c,d)

o b t a i n s c o n t r i b u t i o n s on [a,b) [a$)

i s t o be mean square c o n t i n u o u s from t h e

(which may be empty) and (2.8) y i e l d s

E = (l/h)~ , R ( T ) 4

$eiuTduck

A

W r i t i n g f o r m a l l y dX,

[c,d))

=

z;; 1

FaD

A

-

XAtd

=

= (l/Zn)F([a,b)n

X,

and F ( d A ) = F([A,XtdX))

we can express

t h e i d e a o f o r t h o g o n a l increments as (2.10)

Edd,;^X,

=

where 6,u = 1 i f A = EXtXS =

(2.11)

EHE0RBll 2.3.

6,,,F(d

u

A)/2a Then f r o m ( 2 . 5 )

and 0 o t h e r w i s e .

1, :1

e - i h t e i u s EdX,dXpA

( c f . Theorem 2 . 2 )

m

=

1

2n

im

e-ih(t-S)F(dx)

= R(t-s)

A mean square continuous wide sense s t a t i o n a r y process can be A

represented i n t h e f o r m (2.5) A 2 ments w i t h E I X A [ bounded.

Phoab:

where X,

i s a process w i t h o r t h o g o n a l i n c r e -

The converse i s a l s o t r u e ( e x e r c i s e

-

c f . [Wpl]).

To p r o v e t h e

n

theorem one d e f i n e s X, (2.12)

by (2.7)

and t h e n f o r f E S i t w i l l f o l l o w t h a t

_Izf(X)d?, = ( l / 2 n ) [ z f ( t ) X t d t

( t h e i n t e g r a l s b e i n g d e f i n e d v i a l i m i t procedures from sequences f,,

in as

235

RANDOM PROCESSES

LI labd?,

Indeed from ( 2 . 7 )

below).

s t e p f u n c t i o n s fn based on c o n t i n u i t y p o i n t s one has (*) m

,:

= ( 1 / 2 i ~ ) i Ij a b ( t ) X t d t

and hence f o r

f,(h)d?, = Such f u n c t i o n s fn w i l l be u n i f o r m l y and L1 dense and f

h

( 1 / h ) L w fn(t)Xtdt. A A f o r fn + f E S w i t h fn -f f u n i f o r m l y (and w i t h t h e i n t e g r a l s i n (2.12) de-

f i n e d as mean square l i m i t s i n HX) one has E l i : f(A)d^X - i : ?(t)Xtdt12 5 + 2 E / l m l?(t)--? n ( t j l X t d t l 2, = 2 i z If(,) - fn(X)l 2 2ElLZ I f ( A ) - f , ( ~ ) I d ~ , l -m F(dx)/ZT + 2 i I lz R(t-s)[;(t) - ? n ( t ) ] [ ? ( s ) - P , ( s ) ] d t d s and t h i s vanishes A

as n +

Now t a k e f n ( h ) = e x p ( - + i 2 / n ) e x p ( - i x t o ) and f n ( t ) = (2m)'exp 2 Then i n ( * ) t h e r i q h t s i d e [ - ( n / 2 ) ( t - t o ) ] as i n t h e p r o o f o f Theorem 2.2. goes t o Xto ( t o Q to; (n/21~)'exp[-(n/Z)(t-t~) 2] + s ( t - t o ) ) while the l e f t m.

rn

exp(-ihto)dih.

s i d e o f (*) goes t o

Using t h e s p e c t r a l r e p r e s e n t a t i o n ( 2 . 5 ) one o b t a i n s a r e p r e s e n t a t i o n o f HX. Thus l e t f i r s t 4

dX,

if

,Yn

=

1;

akXtk

(tk

t k ) so f r o m ( 2 . 5 ) Yn =

Q

[I [Ia k e x p ( - i h t k )

L:

qn(h)dXh and i f Yn i s Cauchy i n HX we have E I Y n - Y m I 2 = (1/21~) 2 2 q i n L (dF) and one o b t a i n s I q n ( x ) - nm(h)l dF(h) 0. Thus nn =

-f

-f

COROLLARY 2 - 4 - One has an i s o m e t r y between H X and LL(dF/2n) g i v e n by Y =

/l q(h)d?,

and EY1Y2 = ( 1 / 2 1 ~ ) [ 1 nl(h)G2(h)dF(X). A

REIRARK 2-5. Suppose R E L1 w i t h F(dh) = R(x)dx ( c f . Theorem 2 . 2 ) . 0 for

It1

(2.13)

Set XtT =

T and c o n s i d e r GT(x) = (1/2n)

m

XtT e i x t d t

-

-m

6)

*

,:

Thus g ) = F f F g so F ( f * = F f G (G(x) = g ( - x ) ) . T = f(t-T)g(-T)dT = f o r m a l l y from 2rGT(h) = FXt one can w r i t e ( n o t e f * One r e c a l l s t h a t F ( f

[:

f (t+S

19 ( 5 1d5 1

(2.14)

4nEIGT(h)l 2 =

1,

eiXtE

The i n n e r i n t e g r a l r u n s from -T

;(A) as T

+

+

:1

eixt

1

R(t)dcdt

T - t f o r t > 0 o r f r o m -T + I t [ + T f o r t

It(.

The o u t e r i n t e g r a l can o n l y r u n from 2 aT Consequently 4 ~ r E I G ~ ( h )/2T l = LzT R(t)(l-(ltl/ZT)exp(ixt)dt + 2 m. One r e f e r s t o E I G ~ ( A ) I /ZT as energy p e r u n i t e t i m e o r

< 0 and hence has l e n g t h 2T -

-2T t o 2T.

Xl+EX:dgdt =

power and thus R ( ~ ) / ~ T iI s r e f e r r e d t o as a power spectrum sometimes w h i l e

i s c a l l e d an a u t o c o r r e l a t i o n f u n c t i o n .

-

One goes n e x t t o t h e i d e a o f w h i t e n o i s e where R ( T ) = EX t+T X t = 6 ( ~ )(one

236

ROBERT CARROLL

can a l s o m u l t i p l y by a c o n s t a n t t o g e t d i f f e r e n t s t r e n g t h s ) .

This corres-

ponds t o dF(v) = 1 dv. To t r e a t t h i s k i n d o f " n o n p h y s i c a l " s i t u a t i o n we n say t h a t a sequence Xt of mean square c o n t i n u o u s processes converges t o a 2 w h i t e n o i s e Xt i f f o r each f E L , X n ( f ) = /f f(t)X!dt i s a mean square convergent sequence ( X ( f ) n t h i s event one w r i t e s

1

+

-

and l i m EXn(f)Xn(g) =

X(f))

/I f ( t ) g ( t ) d t .

In

m

(2.16)

X(f) =

f(t)Xtdt

m

and c a l l s Xt a w h i t e n o i s e .

We can i n f a c t r e p l a c e (2.16) by a genuine

s t o c h a s t i c i n t e g r a l as f o l l o w s . (m.s.1 l i m J~t Xsds n so t h a t zb -

D e f i n e (m.s. denotes mean square) ( 6 ) Zt = = l i m xn(lab) and E ( Z ~ - Z , ) ( ? ~ - Z ~ = )

za

Consequently Zt has o r t h o g o n a l increments w i t h EdZtdZS = lab(t)lcd(t)dt. Gstdt. One can t h e n d e f i n e i n t e a r a l s f ( t ) d Z t w i t h r e s p e c t t o dZt as i n

/f

rf

4

t h e p r o o f o f Theorem 2.3 f o r dX,.

For f a s t e p f u n c t i o n we have moreover i: f ( t ) d Z t . For f E L 2 one

from (6) and t h e comnents t h e r e a f t e r X ( f ) = t a k e s l i m i t s o f s t e p f u n c t i o n s fn t o o b t a i n I f X:

ICHE0RZm 2.6,

+

Xt as i n d i c a t e d w i t h Xt a w h i t e n o i s e t h e n f o r f

E

2 L

(2.16) can be r e p r e s e n t e d r i g o r o u s l y as a s t o c h a s t i c i n t e g r a l by X ( f ) =

/f

f(t)dZt. n

One d e f i n e s now ( f o r Xt a w h i t e n o i s e ) a n o t h e r w h i t e n o i s e X, 2 h,^h E L one has

1

m

(1/2n)

(2.17)

1

m

$ ( t ) X t d t = (1/2n)

h(t)dZt =

1

m

a,

A

m

m

I

such t h a t f o r

h(v)di\,

m

=

h(v)jivdv

m

A

where dZ

V

i s d e f i n e d as b e f o r e v i a A

(2.18)

Zb

-

I -

a,

4

Z

=

-a -?a)(?d-?c)

iab(t)dZt/2r

Am

SO t h a t E ( i

s

=

(1/4n),;

-

One u s u a l l y encounters (Gaussian) w h i t e n o i s e

e t c . (EdivdZU = 8vlldv/2~).

as a formal d e r i v a t i v e o f Brownian motion. i s a Gaussian process Xt

-

i a b ( t ) i c d ( t ) d t = (1/2n)jm m lZb(v)lcd(v)dv

Thus a ( r e a l ) Brownian m o t i o n

(t 2 0 ) w i t h EXt = 0 and EXtXs

= min(t,s)

=

R(t,s)

( n o t e t h e n EIXt-X,.i2 = I t - s l b u t Xt i s n o t s t a t i o n a r y ) . It follows f o r m a l l y t h a t E$is = DtDsmin(t,s) = a ( t - s ) so itr e p r e s e n t s a w h i t e n o i s e . We remark a l s o t h a t Xt i s Gaussian means t h a t e v e r y f i n i t e l i n e a r combination Z =

1 anXtn

increments X(tl)

i s a Gaussian random v a r i a b l e and one checks e a s i l y t h a t

- X(to),

random v a r i a b l e s ( X ( t )

X(t2)

Xt).

-

X(tl),

...,

X(tn)

-

X(tn-l)

a r e independent

2 37

RANDOM PROCESSES

Now t h e n o t i o n P(a1b) means t h e ( c o n d i t i o n a l ) p r o b a b i l i t y o f a g i v e n b. Any process w i t h independent increments i s Markov ( i . e .

I X(tn-,)

xu; u = 1 ,..,n-1) = P ( X ( t n ) 5 x n has t h e p r o p e r t y E(Xt]XT; 0 5 T 5 s ) = X, i s c a l l e d a martingale.

xnml).

=

a.s.

P ( X ( t n ) 5 xn

1 X(tv)

=

A Brownian m o t i o n a l s o f o r t 2 s and such a process

More g e n e r a l l y l e t Xt be a s t o c h a s t i c process and

At an i n c r e a s i n g f a m i l y o f event a l g e b r a s such t h a t Xt

Then Xt i s a m a r t i n g a l e i f t > s i m p l i e s E ( A s ) X t

=

i s At measurable.

Xs a.s.

(limitations of

space p e r m i t t i n g we w i l l d i s c u s s m a r t i n g a l e s and p r o g r e s s i v e m e a s u r a b i l i t y more l a t e r ) . Xt on

L e t us remark t h a t one can b e g i n w i t h a ( r e a l ) Brownian m o t i o n w i t h EXtXs = ( 1 / 2 ) ( l t l + I s /

(-m,m)

p a i r o f Brownian motions on a t t = 0. t h a t dXt

[O,m),

Yt = X-t

L e t us a l s o n o t e t h a t E I X t + h %

(dt)’

-

-

I t - s l ) and t h i s determines a and Xt

Xt12

(t

0 ) , patched t o g e t h e r

= ( h l , i n d i c a t e d by w r i t i n g

and t h i s f a c t i s i m p o r t a n t i n s t u d y i n g s t o c h a s t i c i n t e g r a l s The i n t e -

i n a general sense ( b r i e f l y discussed l a t e r i f space p e r m i t s ) .

g r a l s i n t h i s s e c t i o n a r e mean square i n t e g r a l s r e l a t i v e t o s u i t a b l e Xt ( i n p a r t i c u l a r white noise i n t e g r a l s are important) w i t h the integrand i n dependent o f

more general s t o c h a s t i c i n t e g r a l s (e.g.

W;

I t o integrals) arise

i n c o n n e c t i o n w i t h t r a n s m u t a t i o n and t h i s i s i n d i c a t e d l a t e r ) . t i o n w i t h dXt

‘ I ,

(dt)’

I n connec-

above and w h i t e n o i s e i n t e g r a l s l e t us n o t e t h e f o l -

2

I t - s l one sees t h a t f o r 0 < s i 5 < t, E I X t - X s 1 2 2 + E ( X - X I + 2E(X - X ) ( X e - X s ) = ( t - s ) = E ( ( X - X ) + ( X -X ) I = E ( X - X t 5 5 s t 5 5 s t c = t-S+E-s + 2E(Xt-X5)(X - X ). Hence Xt i s a process w i t h o r t h o g o n a l i n c r e 5 s ments ( c f . ( 2 . 1 0 ) ) and one can d e f i n e i n t e g r a l s f ( t ) d X t by mean square lowing.

From E I X t - X s 1 2

=

h

convergence as i n Theorem 2.3 (where X, crements

-

was a process w i t h o r t h o g o n a l i n -

n o t e s t a t i o n a r i t y i s n o t needed i n t h i s c o n t e x t ) .

d e n t l y EdXtdXS = gstdt (t-s = (t-5) +

One has e v i -

( n o t e t h e c a l c u l a t i o n above g i v e s f o r s < 5 <

(<-TI) + (n-s))

t h e c o n c l u s i o n E = E(Xt-X5)(X

n

-Xs)

=

T-

< t

n-s -

in-

- X ) = E(X - X ) ( X - X ) = - ( T I - E ) , E(X - X ) 5 r l n 5 E T I 5 n = E(X - X ) ( X - X ) = -(q-c), SO t - s t-c+n-C+n-S + ZE - 4 ( r , - c ) ) .

deed one notes t h a t E(Xt-X5)(X (X - X s )

/z

n 5 r l r l 5 I f we s e t now Wt = X f o r o u r Brownian m o t i o n ( o r Wiener process) t h e n f o r t s u i t a b l e f, X ( f ) = 1,” f ( t ) d W t d e f i n e d as i n Theorem 2.3 w i l l determine a (Gaussian) w h i t e n o i s e X ( c f . Theorem 2.6). We n o t e t h a t w h i t e n o i s e s need n o t be Gaussian. For example t h e s t a n d a r d Poisson process P(Xt-Xs = k ) = exp[-h(t-s)]h

k

k ( t - s ) / k ! g i v e s r i s e t o EXtXS = h m i n ( t , s ) so Xt generates a

w h i t e n o i s e as b e f o r e ( c f . [ K a c l ] ) .

CZNEAR SCWHA$CZa: E$ZZIllACZ@N, We w i l l b e g i n w i t h a survey o f some techn i q u e s and ideas i n t h e areas of l i n e a r s t o c h a s t i c e s t i m a t i o n , f i l t e r i n g , 3.

238

ROBERT CARROLL

and p r e d i c t i o n .

There i s a huge l i t e r a t u r e here and we c i t e i n a general

way f o r example [ B b i l ; B b j l ; Bb11,2; Dabl; Dacl-4; Gwl-3;

Krl-9;

K r l - 2 5 ; Kvl; Kyl; Kzl-4; Kaal,2;

1-3; Lwl; L z l ; Laal; Labl; Lacl,2; Wol-4;

Bbml; Bbnl; Bbol; Dul-8;

Dwl; Dxl; Dzl; Kacl; L y l ; Lx

Ladl; Pgl; Phl; Rpl; Rql; S t h l ; S t j l ; T h i s background survey w i l l a r -

Wpl-3; Wql-3; Wsl; Ycl; Ydl; Zcl].

r i v e a t v a r i o u s i n t e g r a l e q u a t i o n s and examine t h e i r s t r u c t u r e .

The con-

n e c t i o n o f these i n t e g r a l equations t o G-L equations and u n d e r l y i n g d i f f e r e n t i a l problems i s discussed and one o b t a i n s c o n s i d e r a b l e i n s i g h t i n t o t h e i n t e r a c t i o n between s c a t t e r i n g t h e o r y , l i n e a r f i l t e r i n g and e s t i m a t i o n , and transmutation.

L e t us assume we a r e d e a l i n g w i t h second o r d e r random v a r i -

EIXtIE <

ables Xt ( i . e .

m).

(t

s t o c h a s t i c process Xt

E

L e t HX be t h e H i l b e r t space generated by t h e T) as i n 52 t o which we r e f e r f o r n o t a t i o n .

Then

g i v e n a random v a r i a b l e Y one d e f i n e s 4

DEFINICZBN 3-1, The l i n e a r l e a s t squares e s t i m a t o r Y o f Y g i v e n HX i s de2 f o r Z E HX. f i n e d as t h e element E H X s a t i s f y i n g E l ? - Y 1 2 = min E I Z - Y I

?

I

Y

CHEaREM 3.2. t E

7,

Z

HX,

E

Phao6: A

E

HX i s a l i n e a r l e a s t squares e s t i m a t o r o f Y, g i v e n Xt f o r

i f and o n l y i f ( * ) E(?-Y)Xt = 0 f o r t E E(y-Y)? = 0 ) .

EIZ-Y+Y-Y12 n

?

e q u i v a l e n t l y f o r every

a r e unique.

^v-Z E H so from ( * ) t h e r e r e s u l t s ( 7 ) E I Z - Y I 2 = ax 2 so t h e minimum i s achieved f o r Z = ^v. = E\Z-?I2 + EIY-YI

For Z A

Such e s t i m a t o r s

'i (and

E

HX,

Con4

A

v e r s e l y i f Y s a t i s f i e s t h e c o n d i t i o n i n D e f i n i t i o n 3.1 s e t Z = Y - ( E ( Y - Y ) Then Z f HX and E / Z - Y / * = E l Y - i \ * - \ E ( ? - Y ) X t \ 2 / E / X t I 2

xt!Xt/;\Xti2. = 0.

It f o l l o w s t h a t E(?-Y)Xt

( t h e l a s t i n e q u a l i t y by D e f i n i t i o n 3 . i ) .

EIY-YI

The uniqueness i s c l e a r from ( 7 ) . A

BEmARK 3-3, E v i d e n t l y we a r e d e a l i n g w i t h a H i l b e r t space p r o j e c t i o n Y A

p r o j Y on HX which i s denoted by E ( Y I H X ) =

?.

( o r H(Xt)) f o r t h e H i l b e r t space generated by Xs, A

-m

<

s 5 t, i t w i l l be o f

4

i n t e r e s t t o s t u d y f o r example Y = E ( Y ( H ( X t ) ) ,

=

G e n e r a l l y , i f one w r i t e s H X t A

c h a r a c t e r i z e d by Y E H(Xt)

A

and Y-Y IH( Xt).

As background here l e t us i n d i c a t e a t y p i c a l s i t u a t i o n a r i s i n g i n f i l t e r i n g L e t Xt and Yt be wide sense s t a t i o n a r y processes and t h e o r y ( c f . [Wpl]). assume t h e r e e x i s t bounded f u n c t i o n s Sx, Sy, and Sxy such t h a t

239

LINEAR ESTIMATION

One r e c a l l s now an i m p o r t a n t f a c t o r i z a t i o n theorem ( c f . [Wpl; Du1,2;

Pfl;

Dyl; P i l ; Kxl]) whose p r o o f w i l l however be o m i t t e d s i n c e s i m i l a r r e s u l t s

a r i s e l a t e r i n 558-9. EHE0REIII 3.4- L e t S(x) there exists

0, S

L2 w i t h

?I E

1 L , and

E

li\I2

LI

IlogS(x)ldx/(l+x2) <

$(A)= ;(-A)f o r x

= S,

e x p ( i x t ) d t ( w i t h h r e a l when S i s even) so t h a t

+ 0 for

A

f o r I m k > 0, and i n a d d i t i o n h(A)

?I

m.

Then

A

r e a l , h ( h ) = ;/

h(t)

i s bounded and a n a l y t i c

I m x > 0.

T h i s k i n d o f f a c t o r i z a t i o n o r s p l i t t i n g o f a " s p e c t r a l measure" S ( A ) i s r e l a t e d t o s p l i t t i n g expressions o f t h e f o r m dw(A) = w"(x)dx w i t h ^w = 1 / 2 n l c 9 2 (X)l ( c f . 51.6). F u r t h e r d i s c u s s i o n o f t h i s occurs a t v a r i o u s p l a c e s l a t e r i n §8-9.

Now g i v e n (3.1) l e t us t h i n k o f Xt as t h e o u t p u t o f " f i l -

t e r i n g " a w h i t e n o i s e c t o f s t r e n g t h 2n w i t h a n o n - a n t i c i p a t i v e o r causal

2.

f i l t e r having t r a n s f e r f u n c t i o n

Thus assume Zt i s t h e process w i t h o r -

thogonal increments a s s o c i a t e d w i t h ct as i n 52 and w r i t e ( c f . (2.17))

I

[

m

00

Xt =

(3.2)

h(t-s)dZs =

~(h)e-ixtd?,

m

m

-

(note h ( t - s ) = [((x)exp(-ixt)]"). A

Then f o r m a l l y i f we assume EdZtdZS =

4

stsdt w i t h EdZvdZu = 6vudv/2.ir t h e n

A

which agrees w i t h (3.1) f o r I h l dZs = Xt =

t

I ,h ( t - s ) d Z s .

2

We n o t e here i n (3.2) t h a t

= S.

Lz

h(t-s)

F o r m a l l y ( c f . (2.17) and ( 2 . 5 ) ) one can w r i t e i n (3.2)

~1t(A);xexp(-iht)dx

so t h a t

i(x)tx

= ( 1 / 2 n ) j I Xtexp(iAt)dt = dix.

It

f o l l o w s t h a t f o r m a l l y ( c f 52) (3.4)

Zt =

lot lm cSds =

t

[l/hn(hJ][j

e-ixsds]d?A

m

0

which i n d i r e c t l y (modulo a s u i t a b l e a p p l i c a t i o n o f (2.17) p r e s e n t s t h e w h i t e n o i s e ct

%

-

c f . below) r e -

ZtAas t h e r e s u l t o f f i l t e r i n g Xt by a s u i t a b l e -4

i n v e r s e F o u r i e r t r a n s f o r m o f ( l / h = H (which g e n e r a l l y would have t o be computable i n a d i s t r i b u t i o n sense).

Note t h a t l / h ^ i s a n a l y t i c f o r I m x > 0

A

so g i v e n s u i t a b l e growth we expect H t o determine a n o n a n t i c i p a t i v e o r cau-

sal f i l t e r

-

7.e.

*H

= FH w i t h H ( t ) = 0 f o r t < 0.

One o f t h e p r i n c i p a l con-

s t r u c t i o n s i n Wiener f i l t e r i n g t h e o r y now goes as f o l l o w s (we c o n t i n u e t o f o l l o w [Wpl] here and r e c a l l t h e d e f i n i t i o n o f H(Xt)

i n Remark 3.3).

240

ROBERT CARROLL

EHEORE111 3-5. L e t Xt and Yt be two wide sense s t a t i o n a r y processes s a t i s f y i n g (3.1).

Let

^h

be o b t a i n e d as i n Theorem 3.4 w i t h Zt as i n ( 3 . 2 ) and (3.4).

Then i ( Y t ( H ( X t ) )

Ptrood:

=

2 g(t-s)dZs

and g f t ) = (1/21~)c [~xy(x)/~(A)]exp(-ixt)dx

One shows e a s i l y t h a t I S x y ( X)

I S x y ( 2/ S x

1'

5 Sx(X)Sy(X)

( e x e r c i s e ) so ISXy/h* I2

and t h u s g ( t ) i n t h e statement o f t h e theorem i s w e l l de2 f i n e d as a l i m i t i n L norm. To prove t h e theorem one needs t o v e r i f y t h a t =

[L g ( t - s ) d Z x T

H(Zt),

E

H(Xt) dnd

EYtXT = E L

(3.5) for

(Sy

(t.

t The f i r s t statement i s i m n e d i a t e s i n c e e v i d e n t l y /_,g(t-s)dZ, i s t h e H i l b e r t space generated by Zs f o r

where H ( Z t )

-m

<

s

E

zt,

w h i l e from (3.4) one sees t h a t Zt E H(Xt) so H(Zt) c H ( X t ) .

Indeed g i v e n

s u i t a b l e growth o f ?I =

/_f;(

l/c

we can w r i t e (H(x) = 0 f o r x < 0 )

e x p ( - i x s ) d A = h H ( s - S ) and u s i n g e.g.

X)exp(iAS)

(2.12) as a model (3.4) y i e l d s

where we t h i n k o f t > 0 f o r convenience.

Thus Zt E H ( X t ) .

Now i n o r d e r t o

prove (3.5) one observes f i r s t t h a t by (2.17) w i t h g t ( s ) = g ( t - s )

But from (3.2) we o b t a i n t h e n E[ i I g ( t - s ) d Z S ] F T = ( 1 / 2 ~ )~ ~ ~ X Y ( h ) e x p [ - i i ( t - ~ . ) ] d =~ E Y t i T

( n o t e i n (3.1) R X y ( s - t ) = EXSTt w i t h R X y ( s - t ) = EYtYs =

(1/2n) f ? x y ( h ) e x p ( i A ( s - t ) ) d A ) .

On t h e o t h e r hand s i n c e i n f a c t X T = m

/t

g(t-S)dZg]XT = 0 when w i t h E(dZ dZ ) = 6tsds one has E[ t- s t h a t E[ ~ 1 g ( t - E ) d Z ~ ] =X ~ g(t-E)dZg]yT f o r T I t .

j:h(r-s)dZs SO

T <

t

E[C

REmARK 3-6, F r e q u e n t l y one w r i t e s now G ( X ) = fom g ( t ) e x p ( i X t ) d t ( n o t e G ( A ) i s n o t $ ( A ) ) and t h e n u s i n g ( 3 . 4 ) f o r m a l l y as dZt

I

Q

21rF-~[d?~/t?(A)] we o b t a i n

m

(3.8)

i ( Y t IH(Xt)

=

m

[G(A)/;(

x)]e-ihtd^X

A

The f i l t e r w i t h t r a n s f e r f u n c t i o n G/$ i s r e f e r r e d t o as a Wiener f i l t e r .

RENARK 3.6; (3.9)

It i s i n s t r u c t i v e t o l o o k a t a r a t i o n a l f u n c t i o n Sx(')

= k 2 [ T (a-xk)(A-6k)/ny

(h-pk)(X-pk)

LINEAR ESTIMATION

241

Then i n o r d e r t o have h as i n Theorem 3 . 4

where say I m A k > 0 and Impk > 0. one takes (3.10)

;(A)

with IAI2 = k

2

A ny(i-xk)/ny

=

(~-fj~)

( v a r i o u s A would do).

When Xt

A

can choose A so t h a t h ( t ) i s r e a l .

i s r e a l w i t h SX(A) even one

somewhat more general s i t u a t i o n can

be covered by t h e f o l l o w i n g f o r m a l procedure ( c f . [Wpl]).

i:

so t h a t t h e i n t e g r a l c o n d i t i o n i n Theorem 3.4 becomes <

Consequently

m.

lr = S

1"1

c(e)

logS(-Tane/Z)

=

an

c(e)exp(-ine)de (note

W r i t e X = -Tane/2 IlogS(-Tana/Z)lde

= lmmanexp(ine) w i t h an = ( 1 / 2 ~ )

= a-n and e x p ( i e ) = ( l - i A ) / ( l + i X ) ) .

1*hI2

When

we have l o g S ( x ) = loghh(A) + log^h(x) and one t a k e s n h ( X ) = exp[(ao/2) a-n[(l+ix)/(l-i~)]n]

( n o t e f o r A = s+iu, n > 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/<(A))]

where h designates now t h e optimal f i l t e r . In order t o compare w i t h Theorem 3.5 a n d Remark 3.6 l e t us think of Xt + Y t and Yt + Zt in Theorem 3.5 A

A

4

so t h a t we a r e concerned w i t h E = E ( Z t l H ( Y t ) ) ( c f . ( 3 . 1 5 ) ) . Thus Z ( t [ t ) i s t o correspond t o E via an i n t e g r a l of t h e form (3.15). T h i s analogy can be obtained from ( 3 . 8 ) and ( 2 . 1 2 ) by w r i t i n g f i r s t F [ G ( x ) e x p ( - i x t ) / ~ ( h ) ] = i t ( c ) so t h a t i( s ) / ~ I=T F-’[G/<](t-t). Set then F-’[G/z] = h(T) and (3.8) A t Y, e t c . ) , i = iz h(t-S)Y dc which must be compared t o (3.15) becomes ( X x 5 and (3.16) ( i . e . one wants t o i d e n t i f y h in (3.16) w i t h h o r equivalently 1 - 1/$ w i t h G/; which means G = s^ - 1 ) . Now assume ( A ) f o r s i m p l i c i t y and with R EY t Y s e t c . one has in ( 3 . 1 ) R Y Z = EYt+TZt = F -1 Syz w i t h R y Z = Then from Theorem E(Zt+., + VttT)Zt = EZt+.,Zt = K ( T ) = R - 6 r e a l and even. “ A 3.5 w i t h Syz = Syz = R - 1 = \ $ 1 2 - 1 and h = s we obtain A

A

PI,

PI,

I

m

(3.17)

g(t) = (1/2~)

m

so t h a t f o r t

[:(A)

- &I1

e-ixtdx

-

t h e r e i s no contribution from t h e l/;(x) term; we can then w r i t e u p t o a 6 function g + ( t ) = F - l [ < ( h ) ] w i t h ;(A) = G ( x ) u p t o a constant We leave t h e v e r i f i c a t i o n t h a t g + ( t ) = F-’[s”(x)] - 6 ( t ) as an exercise. > 0

244

ROBERT CARROLL FICCERZNP; AND INiXGRAI: €e?UAEIDW.

4-

We c o n t i n u e t o f o l l o w [Ku2,3,6-8,11,

and w i l l t r y t o e x h i b i t some more o f t h e i m p o r t a n t ideas

12,14-16,20,23,25]

L e t us t a k e again Y ( t ) = Z ( t )

i n l i n e a r f i l t e r i n g and e s t i m a t i o n t h e o r y . Vft)

(Y(t)

2,

+

Yt e t c . ) where V ( t ) i s w h i t e Gaussian n o i s e o f u n i t s t r e n g t h

and z e r o mean and t h e s i q n a l process Z ( t ) i s say Gaussian w i t h E Z ( t ) = 0 and E Z ( t ) V ( s ) = 0 f o r t < s.

One d e f i n e s K ( t , s ) as i n (3.11) f o r 0 5 s,t 5

T and we assume say K i s continuous i n ( t , s ) .

The i n n o v a t i o n s approach goes

back t o Kolmogorov [Kz2] and t h e i d e a i s t o r e p l a c e t h e o b s e r v a t i o n s p r o cess Y ( t ) by a s i m p l e r w h i t e n o i s e J ( t ) ( t h e i n n o v a t i o n s process c o n t a i n s t h e same s t a t i s t i c a l i n f o r m a t i o n ) .

Y ( t ) and J ( t ) w i l l be connected by a

causal and c a u s u a l l y i n v e r t i b l e f i l t e r and each o b s e r v a t i o n J ( t ) b r i n q s new We want t o determine

i n f o r m a t i o n (hence t h e name i n n o v a t i o n s ) .

A

Z(tlt)

=

A

Z ( t ) as i n (3.15) and t h e f i l t e r i n g k e r n e l h w i l l a g a i n s a t i s f y Theorem

3.8.

One d e f i n e s now p r e c i s e l y

J(t)

(4.1)

-

Y(t)

=

N

-

where Z ( t ) = Z ( t )

? ( t ) = ?(t) + V ( t )

A

t h e instantaneous e r r o r , i s t h e p o r t i o n o f Z ( t )

Z(t),

t h a t cannot be p r e d i c t e d from p a s t o b s e r v a t i o n s Y(s) and i s u n c o r r e l a t e d w i t h such Y ( s ) .

Then ( c f . [ K u ~ ] )

CHE0REfR 4-1, J ( t ) i s a w h i t e Gaussian n o i s e w i t h E J ( t ) = 0 and E J ( t ) J ( s ) =

-

Further J = ( I

6(t-s).

-

h)Y w i t h Y = ( I

h ) - l J where h = h+ i s t h e f i l -

t e r i n g k e r n e l from (3.15).

Phood:

F i r s t we show t h a t E J ( t ) J ( s ) = EV(t)V(s).

Take t > 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?\*

<

m

= Y(t)

Hence

A similar calculation As t

+

s one can l o o k

and conclude h e u r i s t i c a l l y t h a t J i s indeed w h i t e

n o i s e (a more r i g o r o u s d i s c u s s i o n appears i n [KulO]). J ( t ) = (Y-:)(t)

= E?(t)(Z-i)(s)

( r e c a l l EV(t)V(s) = & ( t - s ) ) .

E J ( t ) J ( s ) = E^i(t)Y(s) + EV(t)V(s) = E V ( t ) V ( s ) ( = 0 ) . at

Now s i n c e Z ( t )

- 1 ; h(t,s)Y(s)ds

F i n a l l y the formula

gives formally J =

t h i s i s a V o l t e r r a t y p e e q u a t i o n so t h e i n v e r s e ( I - h)-' by t h e Neumann s e r i e s 1" hn = (I - h)-'.

(I -

h)Y and

i s w e l l determined

0

REIIIARK 4-2- L e t us make a few comnents here ( f o l l o w i n g [Ku2,3])

about Kalman

Bucy f i l t e r i n g and i n n o v a t i o n s ( c f . a l s o CAm2-5; L x ~ ] ) . F i r s t one s h o u l d

FILTERING

245

work f o r convenience i n t h e c o n t e x t o f r e c o v e r i n g a Gaussian s i g n a l Z ( t ) when t h e o b s e r v a t i o n s Y ( t ) = Z ( t ) + V ( t ) a r e p o l l u t e d by n o i s e V ( t ) (assume w h i t e Gaussian n o i s e w i t h E V ( t ) V ( s ) = 6 ( s - t ) f o r s i m p l i c i t y ) . one t h i n k s o f c o n s t r u c t i n g a f i l t e r .

To do t h i s

When Y i s a v a i l a b l e from t i m e

-m

and

some c o v a r i a n c e q u a n t i t i e s a r e known one t h i n k s of a Wiener f i l t e r and t h e procedure i s c o n s i d e r a b l y s i m p l e r i f Z ( t ) i s s t a t i o n a r y .

I f however Z ( t ) i s

t h e o u t p u t o f a system d r i v e n by w h i t e n o i s e (4.2)

i ( t ) = F(t)X(t) + G(t)U(t); Z(t) = H(t)X(t); Y = Z + V

where e.g. EU(t)U(s) = Q ( t ) s ( t - s ) , E Z ( t ) V ( s ) = 0 f o r s i m p l i c i t y as i n ( A ) o f § 3 ( g e n e r i c a l l y here E V ( t ) V ( s ) = 6 ( t - s ) , e q u i v a l e n t l y EU(t)J(s) = 0 f o r s

< t

s t r u c t a Kalman-Bucy (K-B) f i l t e r .

EU(t)Y(s) = 0 f o r s < t

- or

) , and E V ( t ) V ( s ) = 0 t h e n one can con(The n o t a t i o n here i s o f t e n phrased i n

v e c t o r - m a t r i x n o t a t i o n where one r e f e r s t o c o v a r i a n c e m a t r i c e s w r i t t e n i n -'T T ' t h e form E U ( t ) - U ( s ) = Q ( t ) s ( t - s ) where U denotes transpose and i s a column v e c t o r . )

Here s t a t i o n a r i t y i s n o t c r i t i c a l i n s i m p l i f y i n g t h e c a l c u l a -

t i o n s , somewhat more general n o i s e s can be t r e a t e d , and i n f o r m a t i o n i s n o t needed f r o m t =

The connection between t h e two f i l t e r i n g methods i s

-m.

discussed i n [Am2-51 and i t i s shown i n p a r t i c u l a r t h a t f o r g i v e n ( s u i t a b l e ) c o v a r i a n c e i n f o r m a t i o n one can c o n s t r u c t o p t i m a l l y a system d r i v e n by w h i t e n o i s e which produces t h i s i n f o r m a t i o n .

For t h e moment we assume (4.2) and A

w i l l f i n d a d i f f e r e n t i a l e q u a t i o n f o r t h e e s t i m a t o r X o f t h e s t a t e s , from 4

A

which by l i n e a r i t y Z ( t ) = H ( t ) X ( t ) .

-

process J ( t ) = Y ( t )

lo

We w r i t e ( c f . Theorem 4.1)

t

A

(4.3)

F i r s t r e p l a c e Y ( t ) by i t s i n n o v a t i o n s

? ( t )= Y ( t ) - H(t):(t).

X(t) =

g(t,s)J(s)ds

and t h e o r t h o g o n a l i t y (X

lo

-

n

X ) ( t ) l J(T) for 0 5

T

< t characterizes g v i a

t

(4.4)

EX(t)J(-r) =

g(t,s)EJ(s)J(T)ds

Note here i f Y were used i n (4.3) i n s t e a d o f J we would have a W-H e q u a t i o n i n (4.4) o f t h e f o r m g i v e n i n Theorem 3.8 b u t w i t h (4.4) one o b t a i n s immed< iately for 0 -

T <

Consequently ( J = Y

t

(0)

-

g(t,z)

= EX(t)J(-r) ( r e c a l l E J ( S ) J ( T ) = 6 ( ~ - T ) ) .

4

HX)

t

(4.5)

;(t)

=

E[X(t)J(s)](Y(s)

-

H(s)i(s)]ds

from which by d i f f e r e n t i a t i o n ( K ( t ) = E X ( t ) J ( t ) )

246

ROBERT CARROLL

t

A

(4.6)

;(t)

+ E[X(t)J(t)]J(t)

E[i(t)J(s)lJ(s)ds

=

'0

=

K(t)J(t) +

'0

Now by (4.3) and

A

t h e second i n t e g r a l on t h e r i g h t s i d e i s X ( t ) and t h e

( 0 )

l a s t i n t e g r a l vanishes by t h e hypotheses a f t e r (4.2).

Consequently one ob-

t a i n s t h e Kalman-Bucy e q u a t i o n

-

? ( t )= F ( t ) ? ( t ) + K ( t ) [ Y ( t )

(4.7)

H(t);(t)]

+ K(t)H(t)[X(t) - i ( t ) ] ;

= K ( t ) V ( t ) i- F ( t ) ? ( t )

+

$(O) = 0

iTHE@REI!I 4 - 3 , Given (4.2) w i t h (4.3) and E U ( t ) J ( s ) = 0 f o r s < t (and E V ( t ) V(s) = G ( t - s ) - g e n e r i c a l l y ) one has t h e K-B e q u a t i o n (4.7) which can a l s o be A

A

w

w

4

expressed as X = FX + KV + KHX where X = X - X. One can a l s o w r i t e ( r e c a l l J = Y

-

A

w

W

Z = HX + V where X = X

-

A

X and n o t e t h a t

E Z ( t ) V ( t ) = 0 i f and o n l y i f E X ( t ) V ( t ) = 0 p r o v i d e d H ( t ) i s nondegenerate) K(t) = EX(t)J(t) = H(t)EX(t)y(t) + EX(t)V(t) =

(4.8)

H(t)E[i +

y](t)y(t)=

H(t)Ely(t)l2 = H(t)P(t,t)

D i r e c t c a l c u l a t i o n now y i e l d s f r o m (4.2) and (4.7) N

X = [F(t)

(4.9)

W r i t i n g (4.9) as

where U(t,E,)

[I;

A(T)dT

-

-

K(t)H(t)]y(t)

? = A(t)y(t)+

-

K(t)V(t)

i-

G(t)U(t)

L ( t ) w i t h y(0) = X(0) = X o one has f o r m a l l y

i s t h e a p p r o p r i a t e e v o l u t i o n o p e r a t o r ( f o r m a l l y U(t,C) 2 NOW assume E V ( t ) U ( s ) = 0, E ~ X ~= I p0, EU

cf. [cig]).

= Q ( t ) s ( t - s ) , and E X o U ( t ) = 0.

Set P ( t , t ) = E l ' j i ( t ) 1 2

and t h e n from

w r i t i n g (a,b) = Eab) (4.11)

(

f

P

=

(?,?)

+ (U(t,O)?o,U(t,O)~o)

U ( t ,c ) L ( 5 )dc, U ( t ,O

)yo)

+

(jotu ( t , 5

i-

(U(t,O)?o,

dc) +

jotu ( t 1s ) L ( s ) ds

L ( t 1d ~ ,

0

We w i l l d e r i v e now a s c a l a r form o f t h e n o n l i n e a r m a t r i x R i c c a t i e q u a t i o n N

f o r P.

Thus (X,,X,)

U

=

(Xo.Xo)

= Po,

(L(c),yo)=

(L(E,),Xo) = ( G ( S ) U ( S )

-

FILTERING

-

= 0 ( s i n c e EXoU(S)

K(s)V(E),X~)

247

= 0 and EXoV(c) = 0 i f and o n l y i f EZo

Hence t h e two m i d d l e

V ( E ) = 0 f o r nondegenerate H which h o l d s by ( A ) ) . terms i n (4.11) v a n i s h and from ( L ( S ) , L ( s ) )

-

rot

-

= (G(c)U(s)

K(E)V(S),G(S)U(S)

K ( s ) V ( s ) ) = G ( s ) G ( s ) Q ( ~ ) 6 ( c - s ) + K ( E ) K ( s ) G ( s - s ) we o b t a i n P = U ( t , O f P o U(t,c)2[G2(E)Q(E) + K2(<)1dc.

(4.12)

2 Pt = 2AP + G ( t ) Q ( t )

( r e c a l l K = HP and n o t e U ( t , t )

+

Now Ut = [F

-

KH]U = AU so f o r m a l l y

2 K ( t ) = 2FP

-

2 2 K + G Q

+

1)

=

EMOREM 4 - 4 - Given t h e hypotheses o f Theorem 4.3 p l u s E Z ( t ) V ( s ) = 0 (as i n 2 ( A ) ) , EV(t)U(s) = 0, E / X o \ = Po, EXoU(t) = 0, and EU(t)U(s) = Q ( t ) & ( t - s ) i t f o l l o w s t h a t P ( t ) = P ( t , t ) = E l ? ( t ) I 2 s a t i s f i e s (4.12) ( w i t h P ( 0 ) = Po) 2 2 2 and t h i s can be expressed a l s o i n t h e form Pt = 2FP - H P + G Q where F, G,

H appear i n ( 4 . 2 ) .

This i s a scalar form o f t h e f o l l o w i n g nonlinear = Q ( t ) A ( t - s ) as i n d i c a t e d

m a t r i x R i c c a t i e q u a t i o n ( w i t h n o t a t i o n E;(t)-GT(s) e a r l i e r ) Pt(t,t),= where P ( t , t )

=

F(t)P(t,t)

+ P(t,t)FT(t) - K(t)KT(t) + G(t)Q(t)GT(t) f o r a derivation

E X ( t ) - B T ( t ) ( c f . [Ku2,3]

-

s c a l a r d i s c u s s i o n was e x t r a c t e d

-

from which o u r

which i n v o l v e s o n l y a r e l a t i v e l y s i m p l e

extension o f notation). L e t us go now t o some f u r t h e r d i s c u s s i o n o f i n t e g r a l e q u a t i o n s and r e l a t e d formulas which a r i s e i n f i l t e r i n g t h e o r y ( c f . [Ku6,8,14,16,25;

Kr7; Bbpl;

*

Consider f i r s t t h e K r e i n - B e l l m a n - S i e g e r t (K-B-S) f o r m u l a H = h + Stjl]). * h - h h which we can r e c a s t and study more t h o r o u g h l y . Thus r e f e r r i n g t o t h e Fredholm r e s o l v a n t H i n Theorem 3.7 (Theorem 3.7 can a l s o be w r i t t e n as K = H + KH) l e t us w r i t e more g e n e r a l l y f o r K ( s , t )

symmetric and c o n t i n u o u s

I

(4.13)

K(t,s)

( n o t e H(t,s,u,T)

K(I

= =

H(t,s,u,r)

- H)

=

K(t,T)H(T,S,u,T)dT

( I - H)(I

(Bellman-Krein-Siegert)

= 0 for t or s

H(t,s,u,T)

+ K)

=

I; K

=

H+

HK = H + KH)

The f o l l o w i n g f o r m u l a s hold, where

> r)

and f o r 0 5 t < s 5 T o r 0 5 s (4.15)

I,

and f o r u = 1 (4.13) says K = H + KH o r H =

H(s,t,u,T)

+ ~ 1 - ol r ( I + K)(I

&HEOREM 4.5,

+ u

H(t,s,u,T)

<

= H(t,s,u,t)

t 5 T + H(s,t,u,s)

-

u

H(r,t,u,r)H(t,s,u,r)dr

248

ROBERT CARROLL

I n o p e r a t o r f o r m (4.15) says Uu = hu + h

Pmoa:

*

*

-

U

uh h i n an obvious n o t a t i o n ) . u u

To prove (4.14) we d i f f e r e n t i a t e (4.13) i n T t o g e t

(4.16)

-

= -uK(t,T)H(T,S,u,T)

HT(t,S,U,T)

K(tyT)HT(T,S,U,T)dT

u

Suppose now t h a t (4.17)

HT(t,S,U,T

+ g(t,s,u,T)

= -uH(t,T,U,T)H(S,T,u,T)

T

Put (4.17) i n (4.16) and use (4.13) f o r s = u/d K(t,T)g(r,s,u,T)dT s e t equal t o zero.

0 which i s t h e same as (4.13) w i t h t h e l e f t s i d e

=

(4.15) s e t N(t,s,u,T) u,T)

-

=

H(t,s,u,t)

= 0 and (4.14)

-

+ H(s,t,u,s)

which we want t o show i s zero.

uH(T,t,u,T)H(T,s,u,T)

H(t,s,u,T)

follows. T

-

= 0

which vanishes by (4.14).

-

+ H(T,t,u,T)

H(t,T,u,T)

-

0 = 0.

s 5 T and a s i m i l a r argument a p p l i e s f o r 0 5 s REIARK 4.6.

U/O

H(r,t,u,

Hence N i s c o n s t a n t

< t

Hence

N

o f H(t,s,u,T),

= 0 for 0 5 t <

5 T.

We r e c o r d here a l s o i n c o n n e c t i o n w i t h Theorem 4.5 a r e s u l t o f

[Kul6] r e l a t i n g smoothing and f i l t e r i n g ( c f . here also Theorems 3.7 and Remark 3.9). (4.18)

To prove

F i r s t one has NT = -HT(t,s,

f o r a l l T and f o r t < s = T we have by t h e symmetry i n ( t , s ) N(t,s,u,T)

-u-l i s

Hence g i v e n unique s o l u t i o n s o f (4.13) ( i . e .

n o t an eigenvalue of K) we have g(-c,s,u,T) r)H(r,s,u,r)dr

+

t o o b t a i n g(t,s,u,T)

- 3.8

Thus w i t h n o t a t i o n as i n (3.13) e t c . one can prove

?(tit)+

?(tlT) =

1

T

-

h*(t,s)[Y(s)

?(s]s)lds

t

(use h y p o t h e s i s ( A ) f o r convenience) where h * ( t , s ) (3.15). and

H

= h(s,t)

w i t h h given i n

Indeed H(t,T) s a t i s f i e s Theorem 3.7 which i s (4.13) f o r u = 1 ( K S i m i l a r l y h(t,.r) = H ( t , ~ , l , t )

a r e symmetric) so H(t,T) = tl(t,T,l,T).

W r i t i n g (4.15) now f o r u = 1 we have H(t,s) = h ( t , s ) + ( c f . Theorem 3.8). L e t t i n g t h i s a c t on Y(s) and i n t e g r a t i n g f r o m - JoT h ( r , t ) h ( r , s ) d r .

h(s,t)

0 to

T

i n s we o b t a i n (4.18).

We go n e x t t o [Ku14]

i n o r d e r t o r e c o r d some general Sobolev and Krein-Levin-

son i d e n t i t i e s d e r i v e d t h e r e .

The i d e a i s t o e x p l o i t t h e a d d i t i v e s t r u c t u r e

a v a i l a b l e i n formulas such as (4.14)-(4.15)

when K(t,s)

= K(t-s)

(Toeplitz

o p e r a t o r ) a r i s e s from a s t a t i o n a r y process (K w i l l be symmetric here so K(t-s) = K(s-t)). (4.19)

The background i d e n t i t i e s a r e t h e Sobolev i d e n t i t i e s

(Dt+D,)H(t,s,T)

= a(T,t)a(T,s)

-

a(T,T-t)a(T,T-s)

FILTERING

a(T,t) +

(4.20)

,d

a(T,u)K(u-t)du

=

249

K(T-t)

Thus t h e r e s o l v a n t H(t,s,T) o f a T o e p l i t z k e r n e l (H(t,s,T) = H(t,s,l,T)). K ( u - t ) i s n o t T o e p l i t z b u t i t s a t i s f i e s (4.19) where a ( T , t ) can be computed In f a c t a(T,t)

f r o m (4.20).

can be r e c u r s i v e l y computed u s i n g t h e so c a l l e d

K r e i n Levinson r e l a t i o n s (4.21 )

( Ds+Dt)a( t, s ) = -a ( t , t - s )a ( t,O); a ( t ,O)=K( t ,O)-

it

a ( t , u ) K( u,O)du

0

The i d e n t i t y (4.19) i s i n f a c t t h e C h r i s t o f f e l - D a r b o u x f o r m u l a f o r t h e funct i o n s a(t,s)

(which a r e e s s e n t i a l l y t h e K r e i n continuous analogues o f o r We n o t e by comparing (4.20) and (4.13)

thogonal p o l y n o m i a l s - c f . 52.10). o r Theorem 3.7 t h a t K(t,T) a(T,t)

Q

H(t,T,T)

+ :J

= H(t,T,T)

K(t,T)H(t,T,T)&G

so one has

The r e s u l t s o f [Ku14] w i l l a p p l y t o more gen-

= H(T,t,T).

e r a l k e r n e l s t h a t T o e p l i t z b u t we w i l l c o n c e n t r a t e upon t h e T o e p l i t z case. ( w i t h u = 1 ) K = H + HK

One w r i t e s now f o r i n t e g r a l o p e r a t o r s as i n (4.13) where HK = KH here by symmetry ( i . e .

K(t,T)

T ) ) and as b e f o r e I - H = (I+ K)-’ KH

Q

6 q

Q

and H(T,s,T)

K(T,t)

( s t r i c t l y HK *: f

It i s convenient t o s e t

:1 K(t,-r)H(-r,s,T)&).

= JK + JL and J(KL)

( e x e r c i s e ) J(K+L)

=

= (-IK)L

= H(s,z,

H(s,r ,T)K(.r ,t)& and (m)

J

= Dt + Ds so t h a t

+ K(JL) + K(6,

-

ST)L where

6(x-q) and one d e f i n e s

(4.22)

KGqL =

joT K(t,u)a(u-q)L(u,s)du

E v i d e n t l y JK = 0 when K i s T o e p l i t z . K = H

+

ing (I

HK t o o b t a i n (JH)(I

-

H) = ( I + K)-’)

(4.23)

JH = ( I

-

= K(t,q)L(q,s)

Now a p p l y

J

t o t h e resolvant equation

+ K) = ( I - H)JK - H(60

-

6T)K,

from which (us-

we f i n d t h a t

H)(JK)(I

-

H)

-

H(60

- sT)H

I f K i s T o e p l i t z so t h a t JK = 0 t h e r e r e s u l t s

CHE@RETO 4.7,

The Sobolev i d e n t y (4.19) f o l l o w s f r o m (4.23) f o r JK = 0 (and

(4.20) h o l d s ) .

Phood: H(T,s,T)

R e c a l l H = H(t,s,T)

= H(t,s,l,T)

w i t h HsoH = H(t,O,T)H(O,s,T).

f o r H(t.T,T)

= H(T,t,T)

H(T-s,T-t,T)

50

= a(T,t)

t h a t a(T,T-t)

and from (4.22) H6TH = H(t,T,T) Then (4.23) i s i d e n t i c a l t o (4.19)

upon n o t i n g t h a t H(t,s,T)

= H(T,T-t,T)

=

H(t,O,T)

= H(s,t,T)

( = H(O,t,T)).

=

Indeed

one can t a k e t h e r e s o l v a n t e q u a t i o n (4.13) i n t h e f o r m K ( t - s ) = H(t,s,T)

+

250

ROBERT CARROLL

T

fo K(t-T)H(T,s,T)dr

and l e t t

-+

T-t,

s

-f

T-s,

T

-f

T-c t o g e t (K i s even)

loT

K ( t - S ) = H(T-t,T-S,T) + K(t-c)H(T-S,T-S,T)dc T Thus K ( t , s ) = K(s,t) = H1 + I0 HIK(t,T)dT and K ( t , s ) = Hp + (4.24)

by uniqueness of t h e r e s o l v a n t H1 = H2. directly.

Id H2K(t,r)dT;

F i n a l l y (4.20) f o l l o w s from (4.13)

m

The p o i n t of Theorem 4.7 and (4.19) mined by a ( T , t ) = H(t,T,T)

i s t h a t H(t,s,T)

Now f o r t h e Krein-Levinson r e l a -

= H(T,t,T).

t i o n s one defines f i r s t b ( T , t ) = a(T,T-t) from (4.13)

i s completely deter-

= H(t,O,T)

H(O,t,T).

=

Then

( o r (4.20)) I

(4.25)

K(t,O)

= b(T,t)

+

ICI

I,

K(t,T)b(T,T)dT

i n o p e r a t o r n o t a t i o n we have y = ( I + K)b so b = T ( I - H)? which can be w r i t t e n as b(T,t) = K(t,O) - io H(t,u,T)

Thus s e t t i n g K = K ( t , O )

( I + K)-’t

=

K(u,O)du. Thus, s i n c e H ( t , u , t ) = a ( t , u ) , b ( t , t ) = a(t,O) = K ( t , O ) t Now f r o m (4.25) io K(u,O)a(t,u)du which i s t h e second e q u a t i o n i n (4.21). T one o b t a i n s DTb(T,t) t K(t,T)b(T,T) + fo K(t,T)DTb(T,T),dT = 0 which i n ope r a t o r n o t a t i o n i s ( I + K)bT = -K(t,T)b(T,T) (since ( I

KJTb = -HSTb = -H(t,T,T)b(T,T) (4.26)

DTb(T,t)

= -a(T,t)b(T,T)

=

-

lo

Hence bT = - ( I

-KSTb.

-

H)

Consequently

-a(T,t)a(T,O)

Next one conputes Ja = aT + at f r o m (4.20); c a l c u l a t i o n g i v e s f o r A(T,t)

=

H ) K = H).

t h u s u s i n g J K = 0 on elementary

= J a = (DT+Dt)a(T,t)

T

(4.27)

A(T,t)

+

(recall K(t) = K(-t)).

A(T,u)K(t-u)du

= -a(T,O)K(t)

Now r e f e r r i n g t o (4.25) one concludes t h a t t h e f i r s t

equation i n (4.21) i s v a l i d .

Thus

CHE0R€m 4-8- The Krein-Levinson r e l a t i o n s (4.21) f o l l o w as i n d i c a t e d . We go now t o [Ku8] f o r some i m p o r t a n t r e l a t e d m a t e r i a l .

L e t us t h i n k o f

Y ( t ) = Z ( t ) + V ( t ) a g a i n w i t h E Z ( t ) Z ( s ) = K ( t - s ) and E Y ( t ) Y ( s ) = R(t,s) 6(t-s) + K(t-s) f o r 0 < s , t IT

( c f . Remark 3.9).

=

L e t us t h i n k o f t h e un-

d e r l y i n g F o u r i e r t h e o r y o f S2 w i t h R ( T ) = ( l / Z n ) L I e x p ( - i A r ) d F ( A )

(as i n

Theorem 2.2) and Yt = If exp(-iAt)d:A (as i n ( 2 . 5 ) ) . Thus ( c f . C o r o l l a r y 2 e x p ( - i A t ) ( b y (2.10) Ed:,djiP = 6AudF(A)/2n). 2.4) Hy % L (dF/2n) and Yt We w r i t e t h e i n n o v a t i o n s process a s s o c i a t e d w i t h Y as ( c f . (3.15) and (4.1))

FILTERING

(4.28)

J(t) = Y(t)

1

251

t

-

h(t,s)Y(s)ds

0

where h s a t i s f i e s t h e W-H e q u a t i o n o f Theorem 3.8 which we w r i t e h e r e as

+ Jot h ( t , r ) K ( ? - s ) d T .

K(t-s) = h(t,s)

Now under t h e map Y ( t ) H(t,s,t). f u n c t i o n s ) where (4.29)

-

P ( x , t ) = e -ixt

%

Recall also t h e i d e n t i t y h(t,s) e x p ( - i x t ) we have J ( t )

jt H(t,s,t)e-ihsds

P(x,t)

- jotb ( t , ) e-

e - At

=

%

=

(Krein

( t - T dT

0

(s

+ t-T

and H ( t , t - r , t )

=

H(O,T,t)

= b(t,r)

=

a(t,t--c)).

The o r t h o g o n a l i t y

o f t h e i n n o v a t i o n s g i v e s t h e n ( c f . C o r o l l a r y 2.4)

I,

m

(4.30)

E J ( t ) J ( s ) = 6 ( t - s ) = (1/2n)

P(A,t)P(A,S)dF(X)

D e f i n e now t h e " r e v e r s e " f u n c t i o n s ( n o t e b ( t , T ) (4.31)

P,(h,t)

From (4.26) b t ( t , T ) (4.32)

Pt(A,t)

= e-ihti(x,t)

=

= -a(t,T)b(t,t) = -ixP(A,t)

-

1

-

i s real)

jotb(t,-r)e - i

ATdT

-b(t,t)b(t,t-T)

=

b(t,t)P,(A,t);

DtP,(A,t)

and one has d i r e c t l y = -b(t,t)P(x,t)

Consider then the kernel

I, T

(4.33)

R T ( ~ , F L=)

P(h,s)P(FL,s)dS

From (4.30) we have formally t h e reproducing p r o p e r t y ( t 5 T) (4.34)

( 1 / 2 r ) rBT(A,V)p(V,t)dF(U) -m

=

P(Ayt)

and from the r e c u r r e n c e r e l a t i o n s ( 4 . 3 2 ) one o b t a i n s a Darboux-Christoffel formula ( n o t e D t [ P ( x , t ) P ( u , t ) - P , ( h , t ) P , ( u , t ) I d i r e c t c a l cul a t i o n )

( s i n c e P(x,O)

= 1 = P,(x,O)).

=

i ( p - A ) P ( h , t ) P ( p , t ) by a

I t f o l l o w s t h a t RT i s t h e " g e n e r a t i n g func-

t i o n " o f t h e r e s o l v a n t H i n t h e sense t h a t (4.36)

RT(x,u)

= JoT Joi [ 6 ( t - s )

-

H(t,s,T)le -isxeiutdsdt

252

ROBERT CARROLL

1 one has I h(t,T),

-

and h

*

H = %

*

*

+ h - h h b e f o r e Remark 3.10 and (4.15) f o r u * * * 1 - h - h + h h = (1-h ) ( l - h ) where H % H(t,T,T), h %

Indeed r e f e r r i n g t o H = h h(.r,t).

= (1-h)[exp(-ixs)].

Since H ( t , s , t )

(

[l - H(t,s,T)],[exp(iht)exp(-ixs)])

(

P ( ~ , T ) , ( [1

-

= h(t,s)

We r e c a l l a l s o H(t,T,T)

h(~,t)],eXp(iht)

t?H€0REm 4-9, With t h e

))

=

we have from (4.29) P(X,t)

= 0 f o r t o r T > T so f o r m a l l y

= ( [l -

h~t,r)l,[P(x,~)exp(ixt)l)

= ( p(x,T),p(x,t)

=

).

hypotheses i n d i c a t e d t h e i n n o v a t i o n s J ( t )

%

P(h,t)

which generate a r e p r o d u c i n g k e r n e l RT as i n (4.33) and t h i s can be r e p r e sented as i n (4.36) o r v i a t h e D a r b o u x - C h r i s t o f f e l f o r m u l a (4.35).

5. I N N @ V A C I 0 W AND SCACCERINC, and T s i t s i k l i s [Lx1,2],

We go now t o two b e a u t i f u l papers o f Levy

which connect t h e ideas o f t h e p a s t two s e c t i o n s

d i r e c t l y t o s c a t t e r i n g ideas; we w i l l expand upon v a r i o u s r e s u l t s and cons t r u c t i o n s and make t h e f u r t h e r connections t o t r a n s m u t a t i o n t h e o r y .

We

t h e n show how t h i s c o n n e c t i o n l e a d s t o a s t o c h a s t i c i n t e r p e r t a t i o n o f t h e m i n i m i z a t i o n c h a r a c t e r i z a t i o n o f t r a n s m u t a t i o n k e r n e l s from 22.7, t h e r e i s i n f a c t an u n d e r l y i n g s t o c h a s t i c process.

when

The c o n n e c t i o n between

i n v e r s e s c a t t e r i n g techniques and l i n e a r e s t i m a t i o n problems has a l s o been noted e.g. <

m)

i n [Cacl-3;

Cd4; Du1,7,8].

Thus l e t Y ( t ) = Z ( t ) + V ( t )

(-m

< t

w i t h s t a t i o n a r y mean square continuous zero mean Gaussian s i g n a l Z ( t )

h a v i n g covariance E Z ( t ) Z ( s ) = K ( t - s ) .

L e t V ( t ) be w h i t e Gaussian n o i s e o f

u n i t i n t e n s i t y and u n c o r r e l a t e d w i t h Z ( t ) ( c f . ( A ) a f t e r (3.12)).

One

w r i t e s Y-+ ( t ) = [ Y ( t ) f Y(-y)]/2, Z-+ ( t ) = [ Z ( t ) ? Z ( - t ) 1 / 2 , and V-+ ( t ) = [ V ( t ) f V ( - t ) ] / 2 so t h a t Y, = Z+ + V+ and Y = Z + V- ( f o r say 0 5 t 5 T). E v i d e n t l y EZ+(t)Z-(s) = 0 = EV+(t)V-(s).

L e t Z = HZ be t h e H i l b e r t space

generated by Z ( t ) f o r -m < t < m = H [ Z ( t ) ; -m < t < m] and, i n t h e same nof T T t a t i o n , Hy = H [ Y ( t ) ; - T 5 t 5 T I . S i m i l a r l y one can w r i t e e.9. Hy = [ Y +-( t ) ; + 0 < t < T] so t h a t t h e r e a r e o r t h o g o n a l decompositions HZ = HZ + H i and T- +-T + T - T + Hy = H + -H; w i t h H y I H ; and H y I H Z . Consequently i n e s t i m a t i n g one Y T has e.g. E ( Z ( t ) l H y ) = E(Z+(t)I+H;) + E ( Z - ( t ) l - t l yT) . Stationarity i s l o s t i n t h e even and odd processes b u t one has i n s t e a d (5.1)

EZ+(t)Z+(S) = K+(t,S) = [K(t-S)

*

K(t+s)]/2

(i.e.

K+(t,s) i s t h e sum o f T o e p l i t z ( K ( t - s ) ) and Hankel ( K ( t + s ) ) o p e r a t o r s ) 2 and we can assume e.g. t h a t K E L on 0 5 s , t 5 T. F u r t h e r , e v i d e n t l y Now l e t us t h i n k o f f i l t e r EV+(t)V+(s) = ( 1 / 2 ) 6 ( t - s ) and EV+(t)Z+(s) = 0. i n g e s t i m a t e s ( c f . (3.15) where h i s used i n t h e r o l e o f g+) -

253

INNOVATIONS

2'(TIT) =

(5.2) We w r i t e

7* ( T I T )

I

,

E(Z ( T ) ( ' H i )

g -+ ( T , t ) Y +- ( t ) d t

=

h

A

- Z+(TIT) and f o r Z t o be t h e b e s t l e a s t = Z+(T) -

= T*(T)

+ T squares e s t i m a t o r we have as usual ? + ( T I T ) I -Hy which l e a d s t o EZ,(T)Y+(s)

lo T g+(T,t)EY+(t)Y+(s)dt. <-T obtaini for 0 - s -

=

K (T,s)

(5.3)

*

Since E Yt- ( t -) Y ? ( s ) = K-+ ( t , s )

= (1/2)g+(T,s) -

+

-

-

+

?

one

-

so K + ( t , s ) = K ( s , t ) . r here t o f i l t e r a t l e v e l t f o r t < T w i t h k e r n e l g+(t,T)

K ( t y s ) = (1/2)g+(t,s) -

(1/2)6(t-s)

joTg + ( T , t ) K + ( t , s ) d t

One notes a l s o t h a t K ( E ) = K ( - s )

(5.4)

+

S0

I t i s more n a t u r a l

so t h a t (5.3)

is

g+(t,-r)K+(T,s)dT -

One s h o u l d compare h e r e w i t h Theorem 3.8 f o r example, so f o r each t (5.4) and one w r i t e s i n an i s a Fredholm e q u a t i o n (W-H e q u a t i o n ) w i t h k e r n e l K', -1 K+- ( n o t e K+- i s a co- (% + K,)g, w i t h 9, - = (% + K+) obvious n o t a t i o n K+= v a r i a n c e and t h u s symmetric and nonnegative d e f i n i t e so 4 + K, i s s e l f a d 2 j o i n t i n v e r t i b l e ) . L e t . = D2 - Ds now and o p e r a t e on (5.4) w i t h (noting

2

t h a t DK,,

=

t

2

DK,,

DsK+(t,O)

0, K - ( t , O )

=

= 0 = K-(O,s),

etc.) t o get

W i t h hypotheses as i n d i c a t e d and W+(t) = -2Dtg,(t,t) one has 2 W-+ ( t ) ) g + w i t h D,g+(t,O) = 0 and g-(t,O) = 0. - ( t , s ) = Dsg,(t,s)

CHE0REm 5.1, CD:

-

Prraad:

We i n d i c a t e some of t h e b a s i c c a l c u l a t i o n s even though r e l a t e d com2 t Thus e.p. D t h g+(t,T)K+(T,s)dT = t + g(t,t)Kt(t,s) + gt(t,t)K(t,s) + 10 gtt(t,T)K(T,s)dT (we g'(t,t)K(t,s) 2 t t 2 omit here the s i g n ) w h i l e Ds/o g(t,T)K(T,s)dT = 10 g(t,T)D,K(T,s)dT = t g ( t , t ) K t ( t , s ) - g(tyO)Kt(O,s) - g,(t,t)K(t,s) + g,(t,O)K(O,s) + 10 g T T ( t , y ) K( -i,s)dr. Consequently p u t a t i o n s can be found i n Chapters 1-2.

*

lo t

(5.5)

%=g,(t,s)

+

v+(t,T)Ki(.,s)dT

-g+(t,O)DtK,(O,s) Now DtK+(O,s)

= K'(-s)

+

+ K'(s)

= W,(t)K,(t,s)

-

DTg,(t,O)K,(OYs) =

0 with the natural stipulation that K'(T)

be an odd f u n c t i o n ( t h i s f o l l o w s f r o m t h e F o u r i e r r e p r e s e n t a t i o n s ) and hence g i v e n t h e c o n d i t i o n s DSg+(t,O) = 0 w i t h g-(t,O) ses ( t o g e t h e r w i t h K-(O,s)

w i l l vanish.

=

= 0 from t h e hypothe-

0 ) i t f o l l o w s t h a t t h e two l a s t terms i n (5.5)

*

Hence d i v i d i n g by W ( t ) i n (5.5),

['gJWJ - - s a t i s f i e s (5.4)

254

ROBERT CARROLL

.

which has t h e unique s o l u t i o n 9., t i o n o f t h e theorem.

T h e r e f o r e .g,

-

= W g,

which i s t h e asser-

The argument i n Theorem 5.1 i s a b i t l o o s e r e g a r d i n g t h e condi-

REMARK 5.2,

t i o n Dsg+(t,O)

=

0 and g-(t,O)

= 0 b u t t h i s can be e s t a b l i s h e d w i t h F o u r i e r

techniques (see 1a t e r s e c t i o n s on geophysi c a l i n v e r s e problems and r e 1 a t e d c a l c u l a t i o n s i n Chapter 2 ) .

REMARK 5.3- I f we d e f i n e P ( t ) = E l ?

(t)I2 (cf.

( 4 . 1 1 ) ) t h e n P + ( t ) = E? ( t ) c t Z -+ ( t ) = E Z + ( t ) Z + ( t ) - E Z-+ ( t ( t ) Z-+ ( t ) = K-+ ( t , t ) - o! 9 (t,T)EY+(T)Z ( t ) d r = f _ -t =$g,(t,t) by (5.4) ( r e c a l l Z + I - H y so Z ( t ) K + ( t , t ) - fo g+(t,T)K+(T,t)dT - 4 I Z + -( t l t ) and a l s o EZ,(t)V+(t) = 0 ) . Consequently W-+ ( t ) = - 4 0 t P f ( t ) . A

t

i

+t

~

-,

A

Next one c o n s i d e r s t h e even and odd i n n o v a t i o n s J + ( t ) = Y ( t ) - Z + ( t l t ) ? which a r e w h i t e Gaussian n o i s e processes w i t h E J + ( t ) J + ( s ) = ( 1 / 2 ) 6 ( t - s ) . + T IT E v i d e n t l y -HJ = Hy i n an obvious n o t a t i o n . Let-H (t;s,T) be t h e Fredholm ? r e s o l v a n t o f t h e k e r n e l K,- as usual so t h a t (5.6) (i.e.

+

K+(t,s) = %H+(t,s,T) -

(4 +

Kf)(l

- H -+ )

=

4 - cf. (4.13),

Theorem 3.7,

etc.).

It w i l l be

useful l a t e r (and i n s t r u c t i v e now) t o r e c o r d t h e f o l l o w i n g i n f o r m a t i o n . L e t A be a z e r o mean random v a r i a b l e whose j o i n t s t a t i s t i c s w i t h Y ( t ) a r e

+

t ) , A,(t) = Gaussian and s e t A e ( t ) = EY+(t)A ( t h u s i f e.g. A = Z ( s ) ( s K+(t,s) and one assumes say A, E L2 ). E v i d e n t l y E(AIHy) T = foT a+(T,s)Y+(s) T ds + Jo a-(T,s)Y-(s)ds

and, s e t t i n g K ( T ) = A

f i l t e r i n g k e r n e l s a+(T,t) (5.7)

-

T E ( A I H y ) ( s o K(T)L'H;),

the

s a t i s f y f o r 0 5 t i T ( c f . (5.3))

A f ( t ) = %a+(T,t) +

1

I

K+(t,s)a+(T,s)ds -

0

( i n o p e r a t o r n o t a t i o n A+ = (+ + K+)a+). - -

&€tEBREI 5.4- The f i l t e r i n g k e r n e l s a,

One has t h e n r e c u r s i o n s v i a

s a t i s f y DTaf(T,t)

= -a+(T,T)g+(T,t). -

Pmraud: Take d e r i v a t i v e s i n (5.7) t o o b t a i n +DTa,(T,t) + K+(t,T)a+(T,T) + foT K,(t,s)DTa,(T,s)ds = 0. Compare t h i s e q u a t i o n w i t h (5.3), which has a unique s o l u t i o n , t o o b t a i n DTa /a (T,T) = -g,, f

t h e theorem.

Now r e c a l l t h a t (5.8)

a (T,t) f

f

which i s t h e a s s e r t i o n o f

m

-

H+) = 4 and w r i t e a, = ($ tT = 2CA (t) H (t,s,T)A (sIds.1 i c f

(4+

K+)(1 -

+

K+)-lA+ -

-

as

10

D i f f e r e n t i a t e (5.8) i n T now and r e c a l l t h e (B-K-S)

i d e n t i t y H = h* + h - h * h

255

INNOVATIONS

o r , more g e n e r a l l y and e x p l i c i t l y (4.14),

-

i n the form

DTH, ( t S ,T) = -H+- (T, t T) H, (T, s 9 T) = -9, - (TI t )g+ - (T, s

(5 9)

9

9

There r e s u l t s t h e n ( u s i n g ( 5 . 8 ) a g a i n f o r t = T) DTa,(T,t) T A+(T) + 2f0 H+(T,t,T)H+(T,s,T) A+(s)ds = -a+(T,T)H+(T,t,T)

= -2H+(t,T,T) = -a+?T,T)g+(T,t)

which i s Theoiem 5.4. -On t h e o t h e r hand one can r e g a r d (5.9) as a conseIndeed t a k e A = Z ( s ) w i t h

quence ( a s p e c i a l case i n f a c t ) o f Theorem 5.4.

A + ( t ) = K?(t,s) and p u t t h i s i n (5.7) ( f r o m which Theorem 5.4 f o l l o w s by differentiation).

The r e s u l t i n g e q u a t i o n i s i d e n t i c a l t o (5.6) upon i d e n -

so t h a t f r o m Theorem 5.4 we would have

t i f i c a t i o n o f a -+ ( t , t ) = H,(t,s,T) DTH,(t,s,T)

,( t , t )

= -H,(T,s,T)g

which i s (5.9)

( r e c a l l H+(T,s,T) -

=

g -+ ( T , s ) ) .

CHE0RZM 5 - 5 , The r e c u r s i o n s o f Theorem 5.4 i m p l y t h e B-K-S formula i n t h e form (5.9). Now i n t h e s p e c t r a l domain we have as i n S4 Y ( t )

,z

exp(-ixt)

'L

e x p ( - i x t ) d ? x ) , R ( T ) = (1/2~)iIexp(-iXT)dF(A),

(Y(t) =

R(t,s) = R(t-s) = 6 ( t - s ) A

+ K ( t - s ) , e t c . and we assume dF(x) corresponds t o a s p e c t r a l d e n s i t y R(X)dh 1 * ( Q S y ( A ) i n Remark 3.10). One assumes e.g. t h a t K ( - ) E L so K ( A ) = FK i s A

A

w e l l d e f i n e d i n a c l a s s i c a l sense and e v i d e n t l y - t h e n R ( A ) = 1 + K ( h ) . E Y ( t ) Y ( s ) = R ( t - s ) = f 1 exp(-iht)exp(i~s)Ed?xd?u d F ( x ) ( c f . (2.10)).

Note

= ( l / Z ~ ) l exp(-ix(t-s))

L e t us a l s o n o t e e x p l i c i t l y h e r e t h a t one i s t h i n k i n g n

e t c . so R ( A ) f o r example w i l l be even.

o f real Y(t), Z(t), Y+(t)

'L

Y+(t).

Cosxt and Y-

Q ,

- i S i n h t and we w i l l devote o u r a t t e n t i o n p r i m a r i l y t o

Thus t h e i n n o v a t i o n s J + ( t ) = Y+(t)

(5.10)

y+(h,t)

I n t h i s context

= CoSht

-

S

-

A

Z(tlt)

' I ,

y+(h,t)

where

g+(t,S)CoSASds

0

(similarly y-(x,t)

=

-i[Sinxt

-

t fo g-(t,s)SinAsds).

Thus one w r i t e s Y + ( t )

=

h

J

CoshtdYx and f o r m a l l y

4

S i m i l a r l y one w r i t e s J + ( t ) = J y+(x,t)dYA e t c . and we w i l l g e n e r a l l y o m i t t h e r e l a t e d formulas i n v o l v i n g = $6(t-s)

J-,

y-, e t c .

Since E J + ( t ) j + ( s ) = E J + ( t ) J + ( s )

( c f . Theorem 4.1) we o b t a i n ( c f . (4.30)) m

CHEOREIR 5 - 6 . The f u n c t i o n s y+(A,t) s a t i s f y ( 1 / 2 v ) j m y+(x,t)?+(h,s)k((x)dA &6 ( t-S

REmARK

=

). 5.7-

We c a l l a t t e n t i o n here t o t h e K r e i n f u n c t i o n s P(A,t)

o f (4.29)

ROBERT CARROLL

256

d5

A

A

(where

dYA) and a l s o t o t h e Krein-Levinson recursions (4.21) where

'L

a ( T , t ) = H(t,T,T) = H(T,t,T) ( = h ( T , t ) ) , b ( T , t ) = a(T,T-t) = H(t,O,T) = H ( O , t , T ) , and H ( t , s ) 'L H(t,s,T) i s determined by say (3.13) so t h a t A

(5.12)

Z(T(T) =

i,'

H(T,s)Y(s)ds =

,d

a(T,s)Y(s)ds

T (note t h a t Theorem 3.8 o r Theorem 3.7 give K ( T - t ) = a ( T , t ) + fo a ( T , s ) K(s-t)ds which i s ( 4 . 2 0 ) ) . The a + ( T , t ) of (5.7) play the r o l e of g ( T , t ) ? f o r A = Z + ( T ) ( c f . ( 5 . 2 ) , ( 5 . 3 ) , a n d ( 5 . 7 ) ) . Now r e f e r r i n g t o [Kr7-91 f o r example ( i f . a l s o 16121) we note t h a t Krein uses the notation L ( t , s ) 'L K ( t , s ) = K(t-s) w i t h K even and r,(t,s) 'L H(t,s,T). In order t o r e l a t e t h e

a ( t , t ) and g ( T , t ) i t will be i n s t r u c t i v e and worthwhile f i r s t t o follow t h e 2

technique suggested by [Krl] a n d f i l l i n t h e necessary d e t a i l s (we modify Consider the i n t e g r a l equation

notation as needed).

1

r

(5.13)

Q ( t , r )+

K(t-s)Q(s,r)ds = 1

r

T h i s can obviously be rewritten as ( * ) 4 Q ( t , r ) +[f K + ( t , s ) O ( s , r ) d s = 4 so Q = ( L +K+)-l/2 or Q = 1 - H+ ( c f . ( 5 . 6 ) ) which can be w r i t t e n here as Q ( t , r )= 1 - H+(t,r,r) Q, 6 ( t - r ) - g + ( r , t ) ( 1 a c t s as 6 h e r e ) . On the o t h e r hand consider $ ( t , r ) = G(t+r,2r) where G ( t , a ) = 1 r a ( t , s ) d s . Recall now Theorem 3.7 (or (4.13)) i n t h e form

ft

(5.14)

r,(s,T) =

K(S-T)

-

K(s-c)ra(S,T)dS

joa

6r[iF

K(t-s)ds K(t-s)rZr(s+r, and observe t h a t K(t-s)G(s+r,Zr)ds = 2r 2r 2r .r)ds]d.r = fo K(t+r--r)dr - JO [fo K ( t + r - S ) r Z r ( S , ~ ) d S ] d ~ = rZr(ttr,T)dT = 1 - G(t+r,2r). Hence $ ( t , r ) s a t i s f i e s (5.13) and by uniqueness we can idso t h a t 6 ( t - r ) - g + ( t , r ) = 1 - fo2rr 2 r ( t + r , s ) d s ( t h i s equae n t i f y Q and t i o n does not have a productive form however). I f we take now (5.13) in t h

:?

t h e form ( * ) following (5.13) then d i f f e r e n t i a t i n g in r one obtains i Q r ( t y r ) + fr K+(t,s)Qr(s,r)ds + K+(t,r)Q(r,r)= 0 from which by (5.4) Q r / Q ( r , r )= - g + ( r , t ) o r ( c f . a l s o [Kr7]) (5.15) Now Q =

Q r ( t , r )= - Q ( r y r ) g t ( r , t )

5 and

we will show below ( c f . [Kr7]) t h a t

257

INNOVATIONS

(5.17)

g+(r,t)

rZr(r-t,0) + rZr(r+t,0)

=

To demonstrate (5.16) one can first check easily that (*) DaG(t,a) = -ra(t, Indeed since ra(t,s) = H(t,s,a) one has from Theorem 4.5, Dara a)G(a,a). Then DaG(t,s) = -ra(t,a) + 10” ra(a,s)ra(t,a)ds (t,s) = -ra(a,t)ra(a,s), from which (*) follows immediately (note also G(a,a) = G(0,a) from the definitions - using ra(a,s) = ra(Oya-s)). Further we will show below that (5.18)

DtG(t,a)

=

G(a,a)[ra(t,a)

- ra(t,O)l

Given (5.18) we obtain (5.16) directly since from y(t,r) = G(t+r,2r) results Qr = 2Ga + Gt = -2rZr(t+r,2r)G(2r,2r) + G(2r,Lr)[rzr(t+rY2r) - rZr (t+r,O)] = -G(2r,2r)[rZr(t+r,2r) + r2,(t+r,0)] (recall rZr(t+r,2r) = rZr(r-t,0) and c(r,r) = G(2rY2r)). In order to prove (5.18) take (5.13) in the form (*) again and differentiate in t to obtain Qt(t,r) + 21; DtK+(t,s) Q(s,r)ds = 0. But 2DtK+(t,s) = K’(t-s) + K’(t+s) = DsK(t+s) - D,K(t-s) SO integrating by parts one obtains Qt(t,r) + 1 ; [K(t-s) - K(t+s)]Qs(s,r)ds = Since K-(t,s) = (l/P)[K(t-s) K(t+s)] we have Q(r,r)[K(t-r) - K(t+r)]. by (5.4) Qt/Q(rYr) = 9- or N

(5.19)

Qt(t,r)

=

Q(r3r)g-(ryt)

On the other hand setting z(t,a) = 12 ra(t,-r)d-r and integrating in T in (5.14) one has ( 0 0 ) E(s,a) = 1, K(5T)d.r - 1 , K(s-S)c(S,a)dS (note that Now differentiate ( 0 0 ) formally in s and integrate G(t,a) = 1 - g(t,a)). + 1 ,K(s-S)ES(S,a)dS = -G(a,a)[K(t-a) - K(t)] by parts to obtain (A) <,(s,a) (thus from ( 0 0 ) gs = 1 D,K(s-r)d-r - 1 DsK(s-.g,)GdS, D,K(s-<) = -D K(s-S), 5 etc.). Since ( 6 + K)ra = K by (5.14) and ( A ) ( 6 + K)[zt/G(a,a)] = K(t) But Gs = -Gs and K(t-a) one can identify G,/G(a,a) with ra(s,O) - ra(s,a). hence (5.18) follows. Further from (5.19) with Qt(t,r) = Gt(t+r,2r) and (5.18) one obtains z

Y

N

In partial sunmary we can state Settin9 rT(t,s) = H(t,s,T) etc. define G(t,a) = 1 - :1 ra(t,s) ds and c(t,r) = G(t+r,2r). Then t(t,r) satisfies (5.13), (5.15), (5.16), and (5.19) from which follows (5.17) and (5.20).

CHE0RElll 5.8.

ROBERT CARROLL

258

R€IRARK 5-9, From ( 5 . 1 7 ) and (5.20) one o b t a i n s

(5.21)

g+(r,t) c g-(r,t)

2r2r(r

=

g -+ ( r , t ) = r 2 , ( r - t , 0 )

:t , O ) ; * rZr(r+t,0)

R e c a l l i n g the i d e n t i t y a ( T , t ) = H(T,t,T) = r T ( T , t ) we can w r i t e ( a ( 2 T , $ ) = H(2T,E,2T) = H(g,ZT,ZT) = H(O,ZT-g,ZT) so e . g . rZT(T-t,O) = H(T-t,0,2T) = H(T+t,2T,ZT) (5.22)

=

a(ZT,T+t))

g ? ( T , t ) = a ( 2 T , T + t ) +. a(ZT,T-t)

T h u s i t i s e q u i v a l e n t t o compute g o r the f u n c t i o n s a(2T, ). t o (5.10) f o r y + ( x , t ) we have ( w i t h a l i t t l e c a l c u l a t i o n )

Going then

r2t S i n i t l o a ( 2 t ,T)Si n k d r

/zt

R e f e r r i n g t h e n t o P ( X , t ) i n (4.29) d e f i n e d v i a H ( Z t , s , Z t )

2t)

= exp(-2ixt)

-

=

a ( Z t , s ) ) P(h,

a ( Z t , s ) e x p ( - i x s ) d s we see t h a t there w i l l be r e l a -

t i o n s between s a y the r e a l and imaginary p a r t s o f P ( h , Z t ) and y + ( i , t ) . For completeness l e t us write this o u t further. Thus from y- = - i S i n h t + t ifo g-(t,T)SinxTdT (y- % -iAeQA = ihx!) we have t o c o n s i d e r (from ( 5 . 2 2 ) ) t (5.24) g-(t,T)SinxTdT = [a(Zt,t+-r) - a ( z t , t - T ) ] S i n i - r d r = 2t a ( 2 t , 5 )Sin h ( 5 - t 1dg = Cosx trt: ( 2 t ,5 1S i nA 5 dg - Si n h tl0 a ( 2 t ,5 ) Cos A cdg

I'

lot

0

0

Now combine ( 5 . 2 3 ) and (5.24) t o o b t a i n f o r example (5.25)

y+ + y- =

e- i h t

+

I,"

a(2t,c)eix(5-t)d5

2t We r e c a l l from ( 4 . 3 1 ) ( b ( 2 t , ~ )= a ( 2 t , 2 t - ~ ) ) P,(h,2t) = 1 - fo b ( 2 t , ~ ) exp(-ihT)dT = 1 - e x p ( - 2 i x t ) J o2 t a ( Z t , s ) e x p ( i A s ) d s . Hence from ( 5 . 2 5 ) (5.26)

[Y+ + y-1/2 = Cosxt

- ( 1 / 2 ) e i x t P,(x,t)

S i m i l a r l y y+ - y- = e x p ( i x t ) [ l + f z t a ( 2 t , c ) e x p ( - i x g ) d c ] and (5.27)

[y+

- y-]/2

=

Cosxt - ( 1 / 2 ) e

A l t e r n a t i v e l y one can write y-

=

ixt

P(X,t)

(1/2)exp(iht)[P(h,t)

- P,(x,t)l w i t h

y+ =

INNOVATIONS

2Cosxt

-

259

+ P,(~,t)l.

(1/2)exp(iht)[P(A,t)

REmARK 5-10, L e t us f o l l o w a g a i n [ L x l ] and r e o r g a n i z e some o f t h e r e s u l t s i n Theorem 5.8 i n t o a d d i t i o n a l i n f o r m a t i o n about t h e Krein-Levinson r e c u r sions (4.21).

Thus f i r s t s e t p ( T ) = a(T,O)

i n (4.21) so t h a t

I

(5.28)

p(T) = K(T)

-

A(T,s)K(s)ds 0

R e c a l l (5.22) and r e w r i t e (4.21) as (DT f Dt)a(2T,T t).

t ) = -2p(ZT)a(ZT,T

k

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

(5.29)

DTg+ + Dtg- = - 2 ~ ( 2 T ) g + ; DTg- + Dtg+ = zp(2T)g-

A l s o f r o m ( 5 . 2 2 ) and (5.28) 2p(2T) = q+(T,T)

-

g-(T,T).

Now t a k e d e r i v a -

t i v e s i n (5.29) i n T and t r e s p e c t i v e l y and s u b t r a c t t o o b t a i n Theorem 5.1

2

w i t h W+(T) = 4[p (2T) T ;(2T)] (6 means t h e d e r i v a t i v e w i t h r e s p e c t t o t h e argument). Thus p ( .) o r Wk(-) p r o v i d e two e q u i v a l e n t p a r a m e t r i z a t i o n s f o r t h e Y process. 6,

tRAN€imW&A'A&I0N AND LINEAR $TDCHAsiEIC E X I W E I 0 N .

We p i c k up f i r s t t h e

development o f S5 a f t e r Theorem 5.4 and f o l l o w [ L x l ] f o r a w h i l e .

Many o f

t h e formulas and r e l a t i o n s developed f o r l i n e a r e s t i m a t i o n have c o u n t e r p a r t s i n i n v e r s e s c a t t e r i n g t h e o r y (and r e p r e s e n t f e a t u r e s o f general transmutaSome connections a r e i n d i c a t e d as we go a l o n g and d i r e c t

t i o n theory).

l i n k s a r e e s t a b l i s h e d e.g.

i n Theorem 6.10.

F i r s t one e s t a b l i s h e s a r e s u l t

e x h i b i t i n g t h e even and odd s p e c t r a l i n n o v a t i o n s as s o l u t i o n s o f t h e n a t u r a l d i f f e r e n t i a l e q u a t i o n a s s o c i a t e d w i t h Theorem 5.1.

Thus ( c f . (5.10))

CHE0REfi 6-1. The f u n c t i o n s y,(h,t) s a t i s f y (W+ f r o m Theorem - = -2Dt[gi(t,t)] 2 2 - = -A Y,; y+(A,O) = 1; Dty+(X,O) = - 2 K ( O ) ; Y-(A,O) = 0; 5.1) EDt - W+(t)]y+ Dty-(X,O)

Phood:

= -iX.

Take e - g .

y+

g i v e n by (5.10),

d i f f e r e n t i a t e t w i c e i n t, and i n t e -

2 Dty+

2 = - A Cosht - g i ( t , t ) C o s h t + g + ( t , t ) x S i n h t g r a t e by p a r t s t o o b t a i n Dtg+(t,t)Cosht - J0t Dtg+(t,s)Coshsds 2 (where D g (t,t) means g t ( t , s ) f o r s t ) and f r o m Theorem 5.1 -J0 t Dtg+Coshsds=-JJ 2 +':[ W,(t)]g+(t,s)Coshsd~ = -D,g,(t,t)CoSht + Dsg+(t,O) - W+(t)/o t g+(t,S)COShSdS + A 2 Jot g+(t,S)CoShSdS

-

hg+(t,t)Sinht.

2

2

Dty+ = - A y+

+

Hence, s i n c e (DtfDS)g+(t,t) W+(t)y+

= gi(t,t)

w i t h W+ = - 2 g i ( t , t ) ,

( r e c a l l Dsg+(t,O) = 0 by Theorem 5.1).

Evidently

from t h e d e f i n i t i o n (5.10) y + ( h , O ) = 1 w h i l e by (5.4) and (5.1) Dty+(h,O)

-g+(O,O)

= -2K+(0,0)

= -2K(O).

=

S i m i l a r c a l c u l a t i o n s h o l d f o r y-.

=

260

ROBERT CARROLL

Now go t o (5.29)

REMARK 6.2.

( w i t h Zp(2T) = g+(T,T)

-

g-(T,T))

t o express

y+ and y- i n terms o f one another v i a

(6.1)

Dty+(h,t)

+

2 ~ ( 2 t h + ( h , t ) = -ihy-(X,t);

Dtv-(x,t)

-

2 ~ ( 2 t ) y - ( ~ , t )= - i h Y + ( h , t )

Thus ( r e c a l l g (t,O) = 0 f r o m Theorem 5.1) y i = - h S i n h t - g + ( t , t ) C o s h t Dtg+(t,s)CosAsds and t h e l a s t t e r m becomes -tIo Coshs[-D,g-(t,s) - Zp(2t)

I 0t

A r o u t i n e c a l c u l a t i o n g i v e s t h e f i r s t e q u a t i o n i n (6.1). Simg+(t,s)]ds. i l a r l y y l = -ihCosAt + i g - ( t , t ) S i n x t + if: Dtg-(t,s)Sinhsds w i t h t h e l a s t t e r m becoming if: S i n h s [ 2 ~ ( 2 t ) g - ( t , s ) equation i n (6.1) follows.

-

D,g+(t,s)]ds

It i s i n t e r e s t i n g

f r o m which t h e second

now t o w r i t e Z p ( 2 t )

=

-6/w

..

w i t h w(0) = 1 i n t h e R i c c a t i e q u a t i o n o f Remark 5.10. T h i s g i v e s e.g. w

-

W+(t)w = 0 and s i n c e w(0) = - 2 p ( O ) = -g+(O,O)

i d e n t i f y w ( t ) w i t h y+(O,t).

+ g-(0,O)

Denoting t h e Wronskian W(f,g)

= -2K(O) we can

by f g ’ - f ’ g as

usual we can combine say t h e f i r s t e q u a t i o n o f (6.1) w i t h t h e d e f i n i t i o n T h i s shows t h a t o f w t o o b t a i n - i X y - ( X , t ) = [W(y+(O,t),y+(h,t))/Y,(o,t)]. y+ and y- a r e dependent and one need o n l y s o l v e one o f t h e equations i n Theorem 6.1 ( c f . [Kr6,9]).

REMARK 6-3. The s i t u a t i o n t r e a t e d i n Theorem 5.2 ( c f . a l s o Theorem 5.3) a l s o 2 has a s p e c t r a l version. Thus i f A % IA(x)d^Yx w i t h say A E L ( i d h ) t h e esT t i m a t e E(AIHy) corresponds t o f i n d i n g t h e p r o j e c t i o n PTA o f A ( h ) on t h e sub2 ” 4 space ST = I e x p ( - i h t ) , -T 5 t 5 t l o f L (Rdh). W r i t i n g Y+(t) = 1 CoshtdY, and Y - ( t ) = -iJ SinhtdYx as b e f o r e one can conclude from t h e d e f i n i t i o n o f t h a t PTA(h) = G+(T,X) a+(T,s)

.I.

lo

with A

I

I,

a+(t,s)Coshsds; a-(T,h) = -i a-(t,s)Sinhsds A T ( n o t e t h a t f o r m a l l y E ( A I H y ) = I PTA(X)dYh). Now t r a n s f o r m t h e d i f f e r e n t i a l

(6.2)

i+(T,h)

+ $-(T,A)

=

equations i n Theorem 5.2 i n n o t i n g t h a t e.g. DT$+(T,h) = a+(T,T)CoshT + = a+(T,T)CoshT - I TD a+(T,T)g+(T,s)Coshsds = a+(T,T)

foT DTa+(T,s)Coshsds

y+(h,T).

S i m i l a r l y D $ (T,h) = -ia-(T,T)SinhT - iI0 T DTa-(T,s)Sinhsds = T - T Sumnarizing -+ ia_(T,T)Io g- (T,s)Sinxsds = a- (T,T)y-(X,T).

-ia-(T,T)SinhT

p l a y t h e same r o l e i n t h e s p e c t r a l domain as do t h e g+we see t h a t t h e y + i n t h e t i m e domain. CHE0R€TIl 6.4. v+(x,T). -

The

:+ - o f (6.2)

s a t i s f y ( a ^-+ ( O , X ) = 0 ) DT2i(T,h)

=

a+(T,T) -

TRANSMUTATION AND ESTIMATION

261

REmARK 6 - 5 - I f one w r i t e s C ( A , t ) = y,(x,t)/y+(O,t) = y(x(t),X) w i t h x ( t ) = t 2 JO ds/y+(O,s) t h e n t h e e q u a t i o n f o r y+ i n Theorem 6.1 i s t r a n s f o r m e d i n t o t h e s t r i n g e q u a t i o n o f Krein, Dym, McKean ( c f . [Kr6,9; D u l l ) . Thus y s a t 2 2 i s f i e s Dxy(x,A) + A v(x)y(x,A) = 0, y(0,x) = 1, and Dxy(O,x) = 0 where 0 5

4

x < P = x(m) and u ( x ( t ) ) = y+(O,t).

T h i s i s a so c a l l e d L i o u v i l l e t r a n s -

form and i t i s i s o s p e c t r a l i n t h e sense t h a t D,S(x)

*

= D A S (A) = ? ( A ) / I T where

2 S and S* denote t h e “ s p e c t r a l f u n c t i o n s ” o f A+ = -Dt + W+(t) and A = - * 2 2 - ( l / v ( x ) ) D x r e s p e c t i v e l y . To s p e l l t h i s o u t i e t f (resp. f ) belong t o L 2 on [ 0 , m ) (resp. L [ u d x l on [ O , L ) ) and w r i t e (+) F ( A ) = 10” f ( t ) y + ( x , t ) d t t - * w i t h F*(A) = If f * ( x ) y ( x , A ) v ( x ) d x . The s p e c t r a l f u n c t i o n s S and S a r e i n (6.3)

jm(fg)(t)dt =

lm

F+(A)G+(A)dS(x); -

m

Now from Theorem 5.4 one has dS(x) = ^R(A)dx/.ir and i n s e r t i n g t h e d e f i n i t i o n s o f LI e t c . i n (6.3) we have dS(A) = dS ( A ) .

Note h e r e e.g.

w i t h f * ( x ( t ) ) so t h a t Jf f ( x ) y ( x , x ) p ( x ) d x = F*(x) 4 2 [ f ( t ) l ~ + ( ~ , t ) l [ ~ + ( ~ , t ) / ~ + ~ ~ , t ) l ~=+ ~ ~f~( tth )+ (dAt, t/) d~ t+ ~=~ , t ~ * ** * L * * Then 1 F+G+dS = J F G dS = So f g dx = J; f ( t ) g ( t ) d t =

t o i d e n t i f y f(t)/y+(O,t) =

Jo”

F+(x).

t h a t one wants

*

Jo”

J F+G+dS.

Consider now t h e B-K-S t y p e r e l a t i o n (5.9). a v e r s i o n o f (4.15)

i n t h e form

(6.4)

= g+(t,s)

H+(t,s,T)

R e c a l l here g,(t,s) k e r n e l g+(t,s) -

(s

= -f

+ g -+ ( s , t )

-

Comparing t o (4.14) we o b t a i n

loT

g+(r,t)g+(r,s)dr -

0 f o r s > t and w r i t i n g

t ) one has (q:

g+)(t,s) -

g+ f o r t h e o p e r a t o r w i t h

(6.5)

1

-

H,- = ( 1

-

g:H1 -

*

= IT - g (r,t)g+(r,s)dr.

svt

(6.4) becomes

-

Hence

- 9), -

R e c a l l here t h e G-L e q u a t i o n f o r g+ i s (5.4) where g, = (% + K + ) - l K + w h i l e (% + K+)(I H+) = 4. Equation (6.5) can be considered as a s o r t o f nonl i n e a r G-L e q u a t i o n f o r g+.

I n t h i s same c o n t e x t we r e f e r t o t h e Sobolev

i d e n t i t i e s (4.19) which w i l l have a v e r s i o n here. t h a t DtK+(t,s)

+ D,K-(t,s)

=

0 and DtK-(t,s)

Thus one observes f i r s t

+ D,K+(t,s)

=

0.

Then

EHE0REM 6-6. I f K ( - ) i s d i f f e r e n t i a b l e t h e n t h e r e s o l v a n t H? s a t i s f i e s DtH+(t,s,T)

+

DsH-(t,s,T)

= g-(T,t)g+(T,s)

and DtH-(t,s,T)

+

D,(t,s,T)

=

262

ROBERT CARROLL

= g+(T,t)g- (T,s).

Pmol;: (6.6)

R e c a l l here ( 5 . 6 ) from which f o l l o w s

J

Dt Hf +

DtK+ =

T DtK+(t,u)H+(u,s,T)du;

0

D,K-

D H + s -

=

loT

K-(t,u)DSH-(u,s,T)du

S e t t i n g C = D H + DSH- we o b t a i n e.g. 10 T DtK+H+dU = -fo +T DuK-(t,u)H+(U,s,T)dU DUH+(u,s,T)du. H+(T,s,T)

0 =

= i C + I Dt K+H ++ J K-D9H -. But -K-(t,u)H+(u,S,T)lo Jo K-(t,u) f

= 0) 0 = i C ( t , s )

Consequently ( r e c a l l K-(t,O) T

Now H+(T,s,T)

+ fo C(u,s)K-(t,u)du.

o f s o l u t i o n s t o (5.6) one o b t a i n s C(t,s)/g+(T,s)

.

f i r s t e q u a t i o n i n t h e statement o f t h e theorem. i n a s i m i l a r way.

2

2

= g+(T,s)

-

K-(t,T)

and by uniqueness

= g-(T,t)

which i s t h e

The second e q u a t i o n f o l l o w s

2

C€t€0RERI 6.7. I f K ( - ) E C t h e n [Dt - DS]H+(t,s,T) = g+(T,t)DTg,(T,s) g+(T,s)DTgi(T,t) and H+(T,s,T) = g+(T,s) w i t h DtH+(O,s,T) = H-(O,s,T)

P/rood:

= 0.

D i f f e r e n t i a t e t h e e q u a t i o n s i n Theorem 6.6 i n t and s r e s p e c t i v e l y 2 2 (= = D t - Ds)

and s u b t r a c t t o o b t a i n e.g. (6.7)

.H+(t3s,T)

= Dtg-(T5t)gf(Tys)

-

DSg-(T,s)g+(Tyt)

Now r e f e r t o (5.29) t o r e w r i t e t h e r i g h t s i d e o f (6.7) as g+(T,s)[-2pg+(T,t) - DTg+(T3t)l

- g+(T,t)[-2Pg+(T5s) - D ~ g + ( T , s ) l = g+(T,t)DTg+(T,s) - Cl+(T,s) A s i m i l a r c a l c u l a t i o n h o l d s f o r .H- and t h i s e s t a b l i s h e s t h e

DTg+(T,t). d i f f e r e n t i a l e q u a t i o n i n Theorem 6.7.

F o r t h e r e m a i n i n g a s s e r t i o n s we know

= H,(T,s,T) and s i n c e DSKf(t,O) = K-(t,O) = 0 t h e r e s u l t s T f o l l o w from (5.6) ( t h u s 0 =dH-(t,O,T) + J0 K-(t,u)H-(u,O,T)du implies that H-(t,O,T) = H-(O,t,T) = 0 w h i l e 0 =IDsH+(t,O,T) + JoT K+(t,u)D,H,(u,O,T)du

a l r e a d y g,(T,s)

i m p l i e s DsH+(t,O,T)

= 0

%

UtH+(O,s,T)

= 0).

T h i s shows t h a t t h e s t r u c t u r e o f H,(t,s,T) i n t h e sense o f Theorem 6.7.

i s " c l o s e t o " T o e p l i t z + Hankel

Now f o l l o w i n g t h e d i s c u s s i o n o f t h e K r e i n

f u n c t i o n s P(x,s) a t t h e end o f s4 we d e f i n e ( c f . (4.36)) (6.8) (6.9)

T R+(h,u) RT -(A,p)

T

=

[6(t-s)

-Io lo '0

=

T

[ [

T'O T

[6(t-s)

Going t o (6.5) n e x t ( w i t h Cosht

-

- H+(t,s,T)]CosxtCosusdtds - H-(t,s,T)]SinxtSinpsdtds fofg+(t,s)Coshsds = y + ( A , t ) = (l*- g+)

Cosps) we can w r i t e f o r example f o r m a l l y (I- H+)(Cosus) = (1

-

g+)y+(u,-r)

TRANSMUTATION AND ESTIMATION

-

( n o t e (1

H+) = ( 1

-

*

-

g+)(l

C0sAtdtd.r = Similarly

y-(p,t)

=

g+), f r o m s

1,' -

-i[Sinut

+ T

-f

263

t ) so t h a t

y+(p,T)y+(X,T)dT = -i[ 1

Sot g - ( t , s ) S i n u s d s ]

-

g - ] ( S i n ~ s ) and

hence - ( 1 - H _ ) [ S i n u s S i n h t ] = ~ T- ( x , p ) = -JoT JOT (1 - g - ( T , t ) ) i y - ( u , T ) S i n X t foT Y - ( P , T ) Y - ( X , T ) ~ T . D e f i n e now t r a n s f o r m s

dtdT =

A

One t h i n k s here e.g.

o f f+ E -

A Sy c L Z (Rdh) where +

S:

i s spanned by {CosAtl

(resp. { S i n i t ) ) f o r 0 5 t 5 T; then t h e I y +-( X , t ) ; 0 5 t 5 T I span S.; (using (6.10))

Re-

c a l l i n g now Theorem 5.4 we have e.g.

1

m

(6.12)

RI(X,u)?+(h);(A)dX/a

= (l/n)

-m

I[

Y+(X,T)Y+(IJ,T)~T]~~~

=

[$ f,

( s )Y,

i,

f+(sh+(A,s)dsl

(IJ , ) 6 ( s-T

A

dsdT =

i,'

y,fd,s

= f, (11

T T EHE0REm 6-8, R -+ ( A , u ) g i v e n by ( 6 . 8 ) - ( 6 . 9 ) s a t i s f i e s R +-( X , P ) = lo Y +(IJ,T) y+(X,T)d-r and a c t s as a r e p r o d u c i n g k e r n e l f o r S F as i n d i c a t e d i n (6.12). T

Pnood:

The r e p r o d u c i n g p r o p e r t y f o r ST can be demonstrated as i n (6.12)

and we o m i t t h e d e t a i l s . Now observe t h a t from Theorem 6.1 t h e equations D;y,(h.t) - W+(t)y+(X,t) 2 2 2 -A v,(X,t) and DtY+(P,t) - W-+ ( t ) y-+ ( v , t ) = -v v + -(u,t) lead t o (6.13)

=

2 2 = ( A -11 ) Y + ( X , t ) y + -( P , t )

DtW(Y+(Ayt),Y,(p,t))

Hence from Theorem 6.8 we o b t a i n a D a r b o u x - C h r i s t o f f e l f o r m u l a ( c f . (4.35)) T C0RbCLA:ARM 6-9, W i t h hypotheses as i n d i c a t e d R +-( A , I J ) = W(Y+(A,T),Y+(P,T))/ 2 (A2 - IJ 1.

Pnood: One i n t e g r a t e s (6.13) from 0 t o T and notes t h a t t h e Wronskian vani s h e s a t t = 0.

=

We w i l l show n e x t how t h e c h a r a c t e r i z a t i o n o f t r a n s m u t a t i o n k e r n e l s v i a mini m i z a t i o n as i n 52.7 can be r e l a t e d t o l e a s t squares s t o c h a s t i c e s t i m a t i o n . Thus 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 r e l a t e d t o t h e d i f f e r 2 2 e n t i a l e q u a t i o n Qu = D u - qu = - A u (as i n 355-6) t h e n t h e m i n i m i z a t i o n procedure c h a r a c t e r i z i n g B: D2

-+

Q: CosXx

+

q X ( y ) (qp,(y) AQ % qX,h(y) Q

AQ

Q

y,(A,y)

264

ROBERT CARROLL

i n Theorem 6.1) can be achieved v i a s t o c h a s t i c i n f o r m a t i o n and accomplishes t h e same t h i n k i n s t o c h a s t i c geometry as l i n e a r l e a s t squares e s t i m a t i o n . We t a k e Q as i n d i c a t e d w i t h $:(O) $(X)dh on

1, Dx$:(0)

=

= h, and assume dw = dw

(which w i l l be n a t u r a l f o r t h e s t o c h a s t i c problem).

[O,m)

4 --

The

t r a n s m u t a t i o n B w i l l have t h e form

Av A4 ( y )

(6.14)

= CosXy +

I'

k(y,x)CosAxdx

0

and we know f r o m 52.7 t h a t k = kh s h o u l d a r i s e from m i n i m i z i n g (6.15)

I'

E =

-

[$:(y)

0

-

CosAy

1;

K(y,x)CosAxdx] 2 dwdy

0

where K runs o v e r a s u i t a b l e c l a s s o f admissable causal k e r n e l s . here t h a t

B has k e r n e l ~ ( y , x ) = ( $ ; ( y ) , C o ~ A x ) ~ w i t h dv

and 6 ( x - y ) = ($:(y),$:(~))~ ( A9 4 ( y ) , C o ~ X x ) ~and

PA p h ( x ) = Coshx

-+

( c f . 951.4-1.5). where W = :/$

W(A)G:(y)

IT/^);.

=

equation associated w i t h t h i s transmutation i s where (*) A(t,x) N

a(y,x)

=

d(x-y)

= ( CosAt,Cosxx)U

+

F(y,x)

(

t i o n becomes ( s (6.16)

0 =

Q

t,s

S2(t,S)

'L

x, t

Then 'L

for x

= :(y,x)

We r e c a l l a l s o t h a t

+ n(t,x) with O(t,x) = a(x,t)

= 6(t-x)

i s t h e k e r n e l f o r t h e Vol-

( f r o m t h e G-L e q u a t i o n and t h e f a c t t h a t y ( x , y ) terra inverse o f p = 6 + k).

maps

The g e n e r a l i z e d G-L

B(y,t),A(t,x))

( c f . 992.2-2.3).

and A ( t , x )

= (Z/s)dX on LO,-)

B - l = 8 has k e r n e l y ( x , y ) =

w i t h k e r n e l F ( y , x ) = y(x,y)

t h e transmutation

We r e c a l l

v

< y,

K(y,x) = 0 and t h e G-L equa-

y)

i,"

+ k(t,s) +

k(t,r)n(r,s)dr

One assumes now t h a t 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 model r e l a t e d t o Q, W, 'L

e t c . and w i l l be comparing t o t h e model f o r y+, Y+, 2K+(t,s)

+ K(t-s) w i t h k(t,s)

= K(t+s)

%

-g+(t,s).

U

e t c . SO t h a t G(t,s) Thus r e c a l l (5.4)

as

2

Also $!(t) Q y+(A,t) w i t h h = - 2 K ( O ) ( c f . Theorem 6.1 where (D - W+)y+ = 2 -A y+ so W+ = -2Dtg+(t,t) Q 4). F u r t h e r e q u a t i o n (5.10) i n t h e form (6.18)

y+(h,t)

represents p . hQ (t) =

= COSht (

-

g+(t,s)CoSASdS

jot

p(t,s),Coshs)

i n i t i a l c o n d i t i o n D,k(t,O)

= 0

'L

and s i n c e Dsp(t,s) D,g+(t,O)

= ( q44 h(t),(-ASinhs))

i n Theorem 5.1 makes sense.

from t h e Parseval r e l a t i o n o f Theorem 5.4 w r i t t e n again here as

an Next

265

TRANSMUTATION AND ESTIMATION

(6.19)

(1/21~)~ u + ( X , t ) G + ( X , s ) ~ ( x ) d h = (1/2)6(t-s) m

(Z/T);(A). R e c a l l t h a t ;(A) 2 1 ~ ) i :exp(-iXT)$(x)dX = IT)/: $(h)CosxTdh.

we see t h a t :(A)

(

i s even i n X so R ( t ) = (1/

PI,

Cosht,CosXs)u

A l s o we n o t e t h a t A(t,s)

becomes

jo CosXtCoshs(Z/T)~(x)dx

I

m

(6.20)

A(t,s)

=

m

=

IT)

[CosX(t+s) +

0

I\

t-s)]R(X)dX

R(t+S) + R ( t - S ) = 6(t+S) + 6 ( t - S ) + 2K+(t,s)

=

6 ( t - s ) ) s i n c e 6 ( t + s ) does n o t c o n t r i b u t e f o r - t , s

~1e x p ( - i h t

=

=

dC;h, Y+(t) = J Coshtd?X, Ed? d?

rz

> 0. =

PI,

6 + n(t,S)

Finally recall

(1/2a);(A)6

A1.I dX,

J+

Y + ( t ) - Z + ( t t ) y + ( A , t ) , E Z + ( t ) i + ( s ) = K-+ ( t , s ) = ( 1 / 2 ) [ K ( t - s ) f -+ - T K(t+s)], e t c . A l s o one denotes by Y y = +Hy t h e H i l b e r t space spanned by Q

Y ? ( t ) f o r 0 5 t 5 T w i t h Zf = H i t h e H i l b e r t space spanned by Z + ( t ) f o r 0 5 t <

m

( t h e s c a l a r p r o d u c t being ( f , g )

b e r t space spanned by Z ( t ) f o r E(Z-(t)lY;)

( a g a i n YT =

+ YT

=

< t <

-m

Then Z = Z+ + Z--is

Efg). m

the H i l -

+

and E ( Z ( t ) l Y T ) = E ( Z + ( t ) l Y T ) +

+ Y y and E ( Z ( t ) l Y T ) means i n s i m p l i s t i c language

t h e e x p e c t a t i o n o f Z ( t ) g i v e n Yr). F o r t h e s t o c h a s t i c problem one asks f o r t h e b e s t ( l e a s t squares) l i n e a r est i m a t i o n o f Z+- g i v e n Y; (6.21)

;+(TIT)

=

i n t h e form o f a f i l t e r i n g e s t i m a t e

E(Z,(T)IY;)

rv

= Z (T)

One wants Z+(T,T)

k C v

-

=

I

I

g+(T,t)Y,(t)dt

0

A

Z ( T I T ) t o be as s m a l l as p o s s i b l e i n a l e a s t !2

squares s e n i e so t h a t E I Z (T,T)I2 i s t o be minimal.

One t h i n k s h e r e o f

f

A

E Y;

Z,(TIT)

as t h e H i l b e r t space orthogonal p r o j e c t i o n o f Z+(T) on Y;

-

Iu

t h a t Z+(T,T)IY;.

Thus g ( T , t ) f

- +

n

serves t o l o c a t e Z+(T[T) i n YT.

t h o g o n a l i t y c o n d i t i o n can be expressed v i a EZ (T,T)Y f

(s) ? L

so

T h i s or-

= 0 for 0

5s 5T

and, changing T t o t, one o b t a i n s (6.22)

K+(t,s) =2g,(t,s)

+

which y i e l d s o u r G-L e q u a t i o n (6.17) as an e q u a t i o n f o r t h e f i l t e r i n g k e r n e l 9.,

Thus t h e i n g r e d i e n t s correspond and we c o n s i d e r n e x t t h e orthogon‘u

t

a l i t y Z+(T,T)IYT

i n t h e s p e c t r a l domain ( s e t t = T a g a i n ) .

One knows t h a t

n

EJ+(t)Y+(T) = ( 1 / 2 ) 6 ( t - T ) A

dYh one o b t a i n s f o r 0

5

T

so u s i n g Y + ( T )

5t

=

I CosxrdYx and J + ( t )

= J y+(A,t)

266

ROBERT CARROLL

(6.23)

I1

= 2

6(t-.r)

, - . A

Y,(x,t)CosuTEdY

d;

F(t,T) =

=

X 1 J -

(1/2a)2 /Iy+(X, t)CosA-r^R(A)dh = ~

-

Thus g(t,-r)

= 0 f o r T < t i s a consequence o f EJ+(t)Y+(T) = EV+(t)Y,(r)

(1/2)6(t-T)

( r e c a l l J+ = Z,

-

+

+ V+ and Z + _ L Y T ) .

=

On t h e o t h e r hand i f we con-

s i d e r E of (6.15) i n t h e p r e s e n t f o r m a t w i t h K % -G+, so t h a t one wants t o T :1 [y+(X,t) - Cosht + (g+(t,T),COsAT)] 2 dwdt o v e r some c l a s s m i n i m i z e E = lo o f causal k e r n e l s g,

t h i s amounts t o m i n i m i z i n g

u

* T Then, s e t t i n g E = I. and

T

:I

-

[d!(t)

2 CosXt] dwdt i n (6.15)

r e p l a c e s x ) one has, a f t e r i n t e g r a t i n g i n

(6.25)

E

+ 2

= E n

loT

jbK(t,-r)*(t,r)drdt

-

2

x

(where t r e p l a c e s y

( c f . 52.7)

loT

lotK(t,T)T(t,T)ddt

t

+ JOT:J It K(t,T)K(t,E)A(T,~)dTdgdt 0

As i n 52.7 t h i s l e a d s immediately t o A

E + T r [ K ( l + O)K* + W 2 +

(6.26)

E

where e.g.

T r KC&

=

*

T

= Jo I.

KO

i m i z i n g k e r n e l (e.g.

t

m*]

t

I. K(t,T)K(t,c)n(T,E)dTdCdt.

When

KO i s a min-

= t h e G-L k e r n e l k w i l l do) t h e n a s t a n d a r d v a r i a -

t i o n a l argument y i e l d s as b e f o r e EHE6REIII 6.10-

The e x p r e s s i o n E o f (6.15) can be w r i t t e n i n t h e form (6.26)

and t h e ( u n i q u e ) m i n i m i z i n g k e r n e l KO f o r E s a t i s f i e s t h e G-L e q u a t i o n (6.16)

(so t h a t KO = k ) .

-

REMARK 6-11, Note t h a t i n g o i n g from (6.25) t o (6.26) N

N

B(t,T)

= 6(t--r)

+ K(t,-r)

w i t h K(t,-r)

= 0 for

T

< t.

one used t h e f a c t t h a t This kind o f r e s u l t i s

o f course known as b e f o r e i n general t r a n s m u t a t i o n t h e o r y v i a Paley-Wiener t y p e theorems o r v a r i o u s c o n t o u r i n t e g r a l arguments ( c f . 52.5). however t h a t t h e r e s u l t (6.23),

which a r i s e s when

$!

'L

We see here

y+ i s a s s o c i a t e d t o

a s t o c h a s t i c model as i n d i c a t e d a l s o p r o v i d e s t h e r e q u i r e d i n f o r m a t i o n t o produce (6.26) y+(A,t)lY

v i a s t a t i s t i c a l considerations.

+ ( t - )=

{COSAT;

0 <

T

Note t h a t (6.23) says t h a t

< t } r e l a t i v e t o dw = (2/T)$dA;

thus i n

TRANSMUTATION AND ESTIMATION

(6.24

267

t h e m i n i m i z a t i o n k e r n e l g+ serves t o l o c a t e (y+(x,t)

-

"+

Cosxt)I Y

A+

( t - 1 n Yt = I C O S X T ; 0 5 T 5 t l . F u r t h e r t h e r e p r e s e n t a t i o n o f A(t,T) = i n t h e form n ( t , T ) = 2K+(t,-r), f o l l o w s from t h e s t o c h a s t i c 6 ( t - T + n(t,T), theory.

Thus

tHE0REm 6-12. 2K+(t,T),

Given a correspondence vA QX ( t )

(2/n)AR, f ? ( t , T ) % e t c . one can deduce t h e r e s u l t s o f Theorem 6.10 and t h e c o r r e s %

ponding G-L e q u a t i o n ( w i t h k ( t , r ) = -g+(t,T))

y+(x,t),

w

%

using o n l y stochastic i n f o r -

m a t i o n and t h e corresponding s t o c h a s t i c problem i s t h a t o f m i n i m i z i n g T 2 EJo [ J + ( t ) - Y + ( t ) + ( g + ( t , ~ ) , y + ( T ) ) ] d t o v e r a s u i t a b l e c l a s s o f causal A

k e r n e l s 9,.

T h i s serves t o l o c a t e Z + ( t l t ) = Y + ( t )

-

+

J + ( t ) i n Yt,

which i s

e x a c t l y what i s accomplished i n l i n e a r l e a s t squares e s t i m a t i o n . REmARK 6-13, L e t us emphasize h e r e t h a t one can e v i d e n t l y e s t a b l i s h v a r i o u s connections i n p a t t e r n and s t r u c t u r e between t h e s t o c h a s t i c model and t h e t r a n s m u t a t i o n t h e o r y (say v i a t h e i n t e g r a l r e p r e s e n t a t i o n s and t h e G-L equat i o n s ) b u t o u r d i s c u s s i o n shows a somewhat more d i r e c t and i n t i m a t e connect i o n between t h e t h e o r i e s by e x h i b i t i n g t h a t t h e m i n i m i z a t i o n o f (6.15) c h a r a c t e r i z i n g t h e t r a n s m u t a t i o n B " s i g n i f i e s " t h e same t h i n g i n " s t o c h a s t i c geometry" as does l i n e a r l e a s t squares e s t i m a t i o n . REmARK 6-14, L e t us g i v e a n o t h e r i n t e r p e r t a t i o n o f t h e m i n i m i z a t i o n process For s i m p l i c i t y work w i t h Q = D2

f o r I as f o l l o w s . 'P! m).

satisfying

=

Q+o

-A 2'P, ' P ! ( O )

=

-

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

= 0, w h i l e dw = ;dx 1, and DxvX(0) Q

We r e c a l l ~ ( y , x ) = 6 ( x - y ) + K(y,x)

( f o r B: D2 + Q: Coshx

on [O,

+ ~ Qp ~ ( w y i)t h h.

K r e p l a c i n g k f o r n o t a t i o n a l purposes) and z ( y , x ) = y ( x , y )

= 6(x-y)

+ K(y,

Consider now f o r m a l l y an e x p r e s s i o n (6.15) x ) ( w i t h ?(y,x) = L ( x , y ) ) . a Parseval f o r m u l a - c f . Remark 2 . 3 . 8 ) i n t h e form

lo jo T

(6.27)

Z =

m

A

If(h,y)12dwdy =

c

where f o r m a l l y f ( A , y ) = Qf(x,y)

(9

joTjy Q)

If(x,y)12dxdy

so t h a t

(via

268

ROBERT CARROLL = 0 f o r K = K).

( t h u s f(x,y)

f

Consider t h e n f o r m a l l y ,

f o r (K(~,E),L(S,X)

) =

K(y, 5 ) L ( 5, x)dS,

loT

(6.30)

j>L

+ K + KL] 2dxdy = Z

Now f o r example 1 ; { 1 L(y,x)L(y,x)dsdy

= T r LL

*

e t c . so (6.30) becomes z =

2Tr[LK* + L(KL)* + K(KL)*] + Tr[LL* + KK* + (KL)(KL)*]

= 2 T r [ ( L t LL*) t + T r LL* + Tr[K+KLL*]K*. S e t t i n g K = K + EJ and making a v a r i a 0 t i o n a l argument as b e f o r e one o b t a i n s

KL*]K*

(6.31)

0 = 2 T r ( L + LL

* * )J

* *

+ KoL J 1 + Tr[(Ko

t 2Tr[JL*Ki

t

* *

**

*

t K,LL*)J*

JKo + JLL KO] = 2 T r [(KO + L + KoL) + (KO t L + KoL)L ]J

&HE@R€m 6.15. M i n i m i z a t i o n o f Z v i a (6.27) leads t o t h e c r i t e r i o n (6.29), KO + L + KoL = 0, which c h a r a c t e r i z e s K. Ptlao6:

Set A = KO + L + KoL i n (6.31) t o o b t a i n T r A(L

A i s causal ( c f . (6.29)) and i f k e r A = A(y,x)

(5

-+

y ) = 1~in(5’Y)A(y,x)L(E,x)dx

causal).

( t h u s A(y,S)

S e t now T ( Y , ~ )= A(Y,s)

+

+

L

* * )J

one has A ( t , t )

Now = k e r AL* = 0.

i s n e i t h e r causal n o r a n t i -

#

N Y , ~so):1

T(y,S)J(y,E)dSdy

= 0

f o r J i n a s u i t a b l e c l a s s o f causal k e r n e l s , which i m p l i e s T(y,c) = 0 f o r 5 < y. Thus A ( l +*I-*) = 0 and, s i n c e (1 + L) (1 + L*)-’ = 1 + K . Consequently A = 0. rn R€illARK 6.16. m

= 1

R e c a l l here (6.13) and C o r o l l a r y 6.9.

+ K e x i s t s , so does The s i t u a t i o n when T

+

w i l l be o f i n t e r e s t i n c o n n e c t i o n w i t h v a r i o u s i n v e r s i o n formulas o f K-L

t y p e ( c f iiii2.6 and 2.8).

We assume q,

s o l u t i o n s e x i s t and w r i t e (y, (6.32)

y+(x,t)

R.

‘ I ,

p:

-

=

w

,

-t

0 rapidly a t

m

so t h a t Jost

i f . Remark 8.8 f o r h # 0 )

c Q ( x ) eix t + cQ(-A)e-iAt

= c @4 + c-aQ ).

F u r t h e r y-(X,t) = -ixO,(t) Q (y-(A,O) = 0 and Q A 4-x D~-(A,O) = - i A w h i l e 0: i s g i v e n as -x: i n 5i.8). One has t h e n f r o m (1.8.42)

as t

-+ m

(6.33)

(p:

Y_(A,t)

= (1/2)CF(h)@!A(t)

-

F(-x)@;(t)l

We r e c a l l now t h e K-L t y p e i n v e r s i o n i n t h e f o r m (assuming dw(h) = dx/28 I C Q2~) Q f ( A ) = fA( A ) = (f(x),p!(x)> w i t h (6.34)

f ( x ) = ( 1 / 2 1 ~ )P ( p ) d ! ( x ) d p 03

A

where, a c t i n g on s u i t a b l e f,

269

TRANSMUTATION AND ESTIMATION

lo co

(6.35)

f(x)

=

(1/2~1)

Q v:(x)qp(x)dx

= ~(A-u)

Now one can give expressions f o r the Wronskian W i n Corollary 6.9 i n terms of phase s h i f t ( r e c a l l y - ( x , t ) % - i x l F ( x ) l S i n ( x x + 6(x))/lAI from quantum s c a t t e r i n g theory) b u t we do b e t t e r t o work w i t h a K-L version of Corollary 6.9. The computation (6.12) will of course a l s o apply, which makes t h e present argument unnecessary in terms of r e s u l t b u t t h e discussion should be of some h e u r i s t i c value. We note t h a t i f h # 0 t h e s p e c t r a l measure has a d i f f e r e n t form ( c f . Remark 8.8) and one could perhaps develop a K-L type theory f o r such s i t u a t i o n s ; t h i s m i g h t be of use a l s o in t h e sense t h a t h = 0 does not correspond t o a s t o c h a s t i c s i t u a t i o n s i n c e h = -2K(O). In any event working w i t h h = 0 we have W(t) = W(qp,qx)(t) Q Q with W(0) = q Qp ( 0 ) 0 = 2iucQ(-p). Hence WP(O)/cQ(-u). 4 B u t one knows from (1.6.27) t h a t &,(a) Q W(0) = -2ip. On t h e o t h e r hand by (6.13), rewritten f o r q! and q xQ , we have

(6.36)

W(T)

-

W(0) = W(T) + 2ip = (u2-A2)jnTq:(t)v:(t)dt

Now as t + m evidently (A) W(t) % [ e x p ( i p t ) / c Q ( - p ) l [ a P x4( t ) - i w QA ( t ) l . Since (6.32) holds obe obtains f o r the bracket [ ] i n (A), [ 1 % i c Q ( x ) Consequently W(t) + Z i p w C i e x p ( i p t ) e x p ( i x t ) ( x - p ) - ic,(-A)exp(-ixt)(x+p). -1 cQ ( - p ) ] [ ( x - p ) ~ (A)exp(iXt) - (x+u)c ( - x ) e x p ( - i x t ) + 2pc ( - p ) e x p ( - i p t ) ] . 9 Q Q Think of moving A say a l i t t l e o f f the real a x i s (Tmh > 0 ) and consider an i n t e g r a l ( f o r s u i t a b l e f*, which we can assume t o be even) I = LE f^(u)[(W(t) 2 2 3 We w r i t e I = 1 I . where t 2 i p ) / ( p -A )]du. 1 J

1

m

(6.37)

I1 = - i c ( h ) ei h t

Q

I2

C;(p)eivtdu/

03

=

- i c (-x)e- i At

Q

(A+U )CQ(-P

Im[

11;

i?(p)eipt d P / C ~ ( - p ) ( p - x

11;

m

Let now ( f o r example) f E Cm so t h a t ? i s e n t i r e of exponential type R 0 ( c f . Theorem 1 . 3 . 6 ) ( l ? ( p ) [ I c ( l + l p [ ) - N e x p ( R I I m u ( ) )a n d take t > R w i t h ( e x p ( i p t ) l 5 e x p ( - t h u ) f o r Imp > 0. I n t e g r a t e over a l a r g e semicircular contour in the half plane Imp > 0. Under s u i t a b l e standard hypotheses 1 / c ( - p ) will be a n a l y t i c f o r Imu > 0 and bounded by a polynomial in p . I t Q 4 n follows t h a t I1 = 0 a n d I 2 = 2 n f ( A ) while I3 = 0 s i n c e f is even. There r e s u l t s ( c f . Remark 2.8.20)

2 70

ROBERT CARROLL

The manner i n which (6.35) r e p r e s e n t s a 6 f u n c t i o n a c t i n g on

EHE0REIII 6.17.

suitable functions

?

i s shown v i a t h e above c a l c u l a t i o n s f o r f

.C:

E

7- RAND0III FIELD5 AND SINCLICAR 0PERAE(DIBc We i n c l u d e some m a t e r i a l here on problems i n s t o c h a s t i c e s t i m a t i o n based on [ L x ~ ] because o f t h e i n t e r e s t i n g correspondance o f formulas and techniques w i t h those a r i s i n g i n e n t i r e l y d i f f e r e n t ways from problems i n s c a t t e r i n g t h e o r y and from general transmut a t i o n theory i t s e l f .

F o r random f i e l d s we r e f e r t o [Wpl; Ycl].

Generally

speaking a c o l l e c t i o n o f random v a r i a b l e s d e f i n e d on a common p r o b a b i l i t y s p a c e n i s c a l l e d a random f i e l d i f t h e parameter space i s m u l t i d i m e n s i o n a l (a s t o c h a s t i c process a r i s e s when t h e parameter space i s one dimensional ) . Take t h e n Xz,

z E R",

t o be a f a m i l y o f second o r d e r random v a r i a b l e s de-

E(XZI2

f i n e d on f? ( i . e .

Xz i s homogeneous i f EX, EXZXzl

w),

=

w i t h say Xz mean square continuous.

u i s independent o f z (we w i l l t a k e

= E X Z + z l l ~ z l + z t l f o r a l l z, z ' ,

z"

E

Rn.

One says

1-1 = 0 )

It f o l l o w s t h a t E X z i z I

and =

R ( z - z ' ) w i t h R nonnegative d e f i n i t e . Corresponding t o Theorem 2.2 one has n R(z) = (1/2n) JRn e x p ( - i ( v , z ) d F ( v ) and a g a i n one w i l l have a s p e c t r a l family

:,( c f .

( 2 . 5 ) and Theorem 2.3) such t h a t ( f o r d?A = ? ( d x ) )

Ion

-i( V,Z)

(7.1)

XZ =

e

xL,

r\

where-iv i s a random s e t f u n c t i o n d e f i n e d on B o r e l s e t s i n Rn w i t h ( 2 ~ ) ~ L e t G denote t h e group o f r i g i d body motions i n Rn

E?(A)?(B) = F(A n 6 ) .

( t r a n s l a t i o n and r o t a t i o n ) and d e f i n e X z t o be a homogeneous i s o t r o p i c r a n = EXt(z)Xt(zl) = R(llz-z'II ) . dom f i e l d i f f o r a l l t E G, EXZX,, p o l a r c o o r d i n a t e s one o b t a i n s i n a s t a n d a r d manner

R(T),

EHEOREIII 7.1,

0 5

T

<

m,

Introducing

i s t h e c o v a r i a n c e f u n c t i o n o f an i s o t r o p i c

homogeneous mean square continuous random f i e l d i f and o n l y i f R ( r ) = m

JO J (n-2

(Xr)FO(dX)/ ( X r ) (n-2)/2 where Fo i s a f i n i t e B o r e l measure on [0,

-) ( f a c t o r s o f 2~ e t c . a r e absorbed i n t o Fo here).

Now one can w r i t e t h e Laplace o p e r a t o r i n Rn as Au r

2

where

'L

A(Sn--')

= (r

n- 1 ur),Jrn-'

+ $,u/

can be analyzed i n terms o f eigenspaces generated by

Gegenbauer p o l y n o m i a l s C t - 2 ) / 2 ( C o s o ) .

This allows the covariance f u n c t i o n

R t o be decomposed and s t u d i e d i n terms of components

RANDOM FIELDS

(n-2)/2,

where A n m ( i r ) = J (n-2 ) / 2 + m ( i r ) / and ,,K, a n o r m a l i z i n g c o n s t a n t .

271

dm i s an eigenspace dimension,

The d e t a i l s can be found i n [Wpl; Y c l ] f o r 2 Rn and we w i l l on l y be concerned here w i t h R as i n [ L x ~ ] . Thus we go t o 2 [ L X ~ ]and l e t Y(x) = Z ( x ) + N(x), x E R , be o b s e r v a t i o n s o f a two dimen-

s i o n a l i s o t r o p i c z e r o mean Gaussian random f i e l d Z w i t h c o v a r i a n c e

-

Z ( s ) = k ( L ) where R = IIx-sII. f i e l d w i t h (*) E N ( x ) N ( s ) and t h a t k ( - )

E

= S(x-s) = 6 ( 1 ) / 2 1 ~ . L .

EZ(x)

One assumes E Z ( x ) i ( s ) = 0

Then w r i t e

L1 (i'dR).

(see here Theorem 7.1 w i t h n = 2, R = k , and Fo(dX) = ;(A)kdX and

(e)

N(x) i s a 2 dimensional w h i t e Gaussian n o i s e

a r e Hankel t r a n s f o r m s ) .

- t h u s $k

One observes now Y(x) over a d i s c DR o f

r a d i u s R c e n t e r e d a t t h e o r i g i n and we d e f i n e YR t o be t h e H i l b e r t space generated by Y(x) f o r 0 5 IIxII 5 R. b =

(7.5)

1

b(x)Y(x)dv =

DR One w r i t e s now Y ( r , e )

N(r,e)

Elements o f YR have t h e form

loR

l;ITb(r,e)Y(r,3)rdrdh

Z(r,e)

Yn(r)exp(ine),

=

= lma,Vn(r)exp(ins)

where e.g.

=

Y n ( r ) = (1/2n)lo2nY(r,e)exp(-ine)de.

One o b t a i n s t h e n e s t i m a t i o n problems f o r Y n ( r )

= Zn(r)

rn we have f o r example EZn(r)Zm(s) = EVn(r)vm(s) = 0.

f o r any n and m. < R so YR

Z n ( r ) e x p ( i n e ) , and

+ V n ( r ) and f o r n

A l s o EZn(r)Vm(s) = 0

be t h e H i l b e r t space generated by Y n ( r ) f o r 0 5 r

L e t Y:

=@-I Yi (mean square l i m i t sum) and t h e n E ( Z ( r , e ) l Y R )

(r)l$))exp(ine).

+

-

Now ( c f . (7.3)

= IYmE(Zn

w i t h dm = 1 = Km) a,

(7.6)

EZn(r)Zn(s) = kn(r,s)

Jn(Xr)Jn(Xs);(A)Xdh

= 0

A l s o E V n ( r ) i n ( s ) = &(r-s)/Z.irL ( L = I r - s l ) so t h a t Vn and Zn a r e n o t s t a t i o n ary.

However kn(r,s)

2

o r e q u i v a l e n t l y i f Z ( - ) i s mean square d i f f e r e n t i 2 2 2 a b l e one has f o r Pn(Dr) = Dr + ( l / r ) D r - (n /r ), Pn(Dr)kn(r,s) = Pn(Ds)

EHE0)REFII 7-2-

If k E C

has s t r u c t u r e g i v e n by

0 and kn(O,s) = 0 (n # 0 ) .

kn(r,s)

w i t h Drko(O,s)

PmoB:

One r e c a l l s t h a t Pn(Dr)Jn(hr)

=

t i o n follows from (7.6). =

= -A

2

Jn(hr) so t h e d i f f e r e n t i a l equa-

The i n i t i a l c o n d i t i o n s a r e a consequence o f J o ( 0 )

0 and Jn(0) = 0 f o r n = 0.

Now w r i t e t h e f i l t e r i n g e s t i m a t e s as A

(7.7)

Zn(RIR) = E(Zn(R)IYR) = 1R gn(R,s)Y,(s)ds 0

272

ROBERT CARROLL

From Tn(R,R)

(7.8

= Zn(R)

kn(R,r)

-

b e i n g orthogonal t o Yn(s) f o r 0 5 s 5 R

Z*,,(RIR)

= ( l / 2 ~ ) g n ( R y r l+

gn(R,s)kn(r,s)sds

One notes t h a t Kn d e f i n e d by t h e k e r n e l kn i s s e l f a d j o i n t and 2 nonnegative d e f i n i t e so (1/21~) + K, i s i n v e r t i b l e (assume say k n ( - ) L (0 5 r 5 R).

( r d r ) on [O,R]). 2 the L theory.

tHE0Rfin 7.3,

I t f o l l o w s t h a t (7.8) w i l l have a unique s o l u t i o n gn i n Now f o l l o w i n g t h e development i n § § 5 - 6 one has

Under t h e hypotheses o f Theorem 7.2 one has [Pn(DR)

gn(R,r) = Pn(Dr)gn(R,r); = 0 ( n # 0).

Phood:

= -20R[Rgn(R,R)1;

Wn(R)

Drgo(R,O)

=

-

Wn(R)]

0; and gn(R,O)

The p r o o f i s s i m i l a r t o Theorem 5.1 where t h e e q u a t i o n s i n Theorem

-

7.2 a r e used i n s t e a d .

I f m n = Pn(OR)

(7.9)

= (1/2a)mncin(R,r)

Wn(R)kn(R,r)

D i v i d e by Wn(R)

Pn(Dr) +

,P

one o b t a i n s kn(r,s)mngn(R,s)sds

and use t h e uniqueness f o r (7.8) t o conclude t h a t mngn(R,r)/

The boundary c o n d i t i o n s i n Theorem 7.3 r e s u l t from t h e Wn(R) = gn(R,r). c o n d i t i o n s i n Theorem 7 . 2 f o r kn. One produces i n t h i s manner f i l t e r i n g k e r n e l s gn(r,s)

for 0 5 s 5 r

we d e f i n e i n n o v a t i o n s processes b y (yn i s used i n s t e a d o f J,

R and

here because

o f t h e Bessel f u n c t i o n s Jn which a r i s e ) A

(7.10) Then Y,

yn(r) =

Yn(r)

-

Zn(rlr)

i s a w h i t e Gaussian n o i s e w i t h Eyn(r)Tn(s) = (1/2aL)S(r-s)

- s \ ) which c o n t a i n s t h e same s t a t i s t i c a l i n f o r m a t i o n as Y,

( L = Ir

( c f . 555-6). The

machinery o f 553-6 g e n e r a l l y has a v e r s i o n here f o r t h e d e t a i l s o f which we r e f e r t o [LxZ]. (7.11 )

kn(r,s)

F o r example one has a Fredholm r e s o l v a n t H,(r,s,R) = ( l / 2 ~ ) H n ( r y s y R )+

( 0 5 r,s 5 R ) w i t h H,(R,s,R)

-

( 7 12

=

gn(R,s)

0

kn(r,u)Hn(u,s,R)du

and ( c f . ( 5 . 9 ) )

DRH, (r,s 9 R ) = -Rgn (R, r )9, (R,s)

T h i s i s a Krein-Bellman-Siegert (7.13)

jR

with

I - Hn = ( I - g:)(I

t y p e r e s o l v a n t i d e n t i t y w i t h i n t e g r a l form

-

gn)

A new f e a t u r e now i s t h e f o l l o w i n g breakdown o f say Theorem 7.3 i n t o a

RANDOM FIELDS

system.

Thus f i r s t one has

EHE@REFR 7.4, = 0 and EDs

Ptoad:

If k E C

-

1

[Dr

+

l e a d s t o Theorem 7.2.

EHE6RECl 7.5,

+ ((n+l)/~)]k~+~(r,s)

0.

=

( n / R ) l g n + [Dr

= AJn-l(Ar);

[Dr-(n/r)]Jn(Ar)

-

+ ((n+l)/r)][Dr

(n/r)]

= -AJn+l(Ar)

so t h a t Theorem 7,4

On t h e o t h e r hand

For k E C

+ ((n+l)/R)lgntl

Pkood:

( n / r ) ] k n ( r y s ) + [D,

+ EDr + ( ( n + l ) / r ) l k n + l ( r y s )

(n/r)]Jn(Ar)

One notes t h a t Pn(Dr) = [Dr

-

-

t h e n [Dr

(n/s)lkn(r,s)

T h i s f o l l o w s by elementary c a l c u l a t i o n s f r o m (7.6) u s i n g

(7.14)

['JR

2 73

1

+

t h e f i l t e r i n g k e r n e l s gn(R,r)

( ( n + l ) / r ) l c ~ ~ += ~-Pn(R)gn and [Dr

= Pn(R)gn+l

Operate w i t h Or

s a t i s f i e d by gn and gn+,

and gn+l(Ryr)

-

where

pn(R)

= R[gn(R,R)

-

-

(n/r)lgn

satisfy +

[DR

gntl(R,R)l.

(n/R) and Dr + ( ( n + l ) / r ) on t h e i n t e g r a l e q u a t i o n s Using Theorem 7.4 and i n t e g r a t i n g by

and add.

p a r t s one o b t a i n s (7.15)

0 = pn(R)kn(r,R)

+

+ (l/ZT)m(R,r)

f o r m(R,r) = [OR - (n/R)]gn(R,r)

i"

kn(r,s)m(R,s)sds

0

+ [Dr + ( ( r ~ + l ) / r ) ] g ~ + ~ ( R , r ) . D i v i d e by A

and use uniqueness i n (7.8) t o o b t a i n m ( R , r ) = -pn(R)gn(R,r). s i m i l a r c a l c u l a t i o n y i e l d s t h e second e q u a t i o n above. 9

-pn(R)

Combining t h e e q u a t i o n s i n Theorem 7.5 one o b t a i n s Theorem 7.3 w i t h R) =

R i c c a t i equation t o get

pn

Remark 5.8).

from Wn ( c f .

To go now i n t o t h e s p e c t r a l domain one

r e c a l l s (7.1) and we assume Y(x)

"u

We t h i n k o f

( m ) :(A)

-V

=

(A,e) and w r i t e

exp(-(V,x))

( c f . remarks b e f o r e (5.10)). A

= ( 1 / 2 1 ~ ) + k(A).

t h e c a l c u l a t i o n here f o r two dimensions l e a d i n g t o (=).

L e t us s k e t c h

Thus f i r s t l o o k a t

(7.1) f o r k i n t h e form ( - v = ( h , e ) )

lom 2a

(7.17)

k(L) = ( 1 / 2 ~ ) *

If one w r i t e s :(A)

=

(1/21~)Jom:(A)Jo(AL)AdA let X

Y, d^Xv

Q

joei'

CoseF(AdAde)

F k t h e n F(AdAd0) = r(A)AdAdO and (7.17) becomes k ( L ) = which means ;(A)

d?", e t c . w i t h F r ( b )

= 'i;(A)/Za = ?(A)

i n (7.4).

I n (7.1) now

t h e n f o r Ilx-sll = L

2 74

ROBERT CARROLL

EYxYs = r ( L ) =

(7.18)

+ i ( p , s ) EdiL)d)i,,

e-i(v,x)

-

and one takes Ed^Xvd2 = ( 1 / 2 ~ 12)GVUF(dv) = (1/2n)2;(x)AdAde

so t h a t r(L) =

!J

Now g i v e n r = ( 1 / 2 n L ) 6 ( l ) + k we have

(1/2~1)/; y(A)Jo(AL)AdA.

so t h a t

= ~ / Z T=

^k

7= F +

+ ( 1 / 2 ~ 1 )and a measure ^rxdxde/2~1z yxdxde/(2a)

2

.

1

Thus

i n v o l v e s an i s o m e t r y between t h e H i l 2 2 b e r t space Y generated by t h e Y(x) and L (FAdAde/(2a) ) (by t h e Parseval

t h e correspondence Y ( x ) z exp(-i
lo

One has t h e n

(-L)

=

(A,e))

m

(7.19)

EY(x)Y(s) =

,e i ( v y s ) ) L 2 ( rAx d x d e / 2 ~ 1 )

Jo(AL)P(A)hdA = (e-i(vyx)

Under t h i s i s o m e t r y t h e c o e f f i c i e n t process Y n ( r ) i s mapped i n t o (1/2n) -i( v ,x )exp( - i n e )de = inJn( h r ) e x p ( - i n @ ) ( c f . comnents a f t e r (7.5 ) )

$‘exp(

since exp(-i(v,x))

e).

i n J n ( A r ) e x p ( i n ( e - @ ) ) where - v = (A,@)

=

One has t h e n

(7.20) where , ,6

c

Jn(Ar)Jn(As)F(A)AdA EYn(r)Ym(s) = ,6, 2 = (1/2a)~ooexp(i(m-n)@)d@ i s 0 f o r m # n and 6nn = 1.

can c o n s t r u c t an i s o m e t r y Y n ( r ) + Jn(xr): Yn Y,

+

yn(r,A)

(7.21 )

-+

Thus one

A

L (rAdA) and from (7.10)

with

yn(r,A)

Since Eyn(r)y,,(s) (7.22)

2

and x = ( r ,

;1

=

Jn(hr)

-

16

= (1/2aL)6(r-s)

yn( r, A)yn(s

X);hdA

gn(r,s)Jn(As)sds one has = (1 /21d)6

(r-s )

2 ClE@R€m 7.6, For n > 0 y, s a t i s f i e s [Pn(Dr) - Wn(r)]yn(r,h) = -1 yn(r,A); n n l i m 2 n!(Ar)-ny ( r , k ) = 1 as r + 0; and f o r n < 0, y n ( r y h ) = (-1) y-.,(r,h). n. Phooa: Operate on (7.21) w i t h Pn(Dr) t o o b t a i n ( r e c a l l Wn = - Z D r ( r g n ( r , r ) ) ) (7.23)

Pn(Dr)yn(r,h)

2

= -[A

-

Wn(r)lJn(Ar) + rDSgn(r,s) ls=rJn(Xr)

-

rr

s)Jn (1s)sds

As i n Theorem 7.3 one w r i t e s [Pn(Dr) = Wn(r)gn(r,s)

by Theorem 7.3.

-

P,

Put t h i s

D s ) l g n ( r , s ) = =,g,(r,s) and mngn n (7.23) and i n t e g r a t e by p a r t s

t o o b t a i n t h e f i r s t e q u a t i o n i n Theorem 7 6.

The r e s t i s obvious.

rn

RANDOM FIELDS

REmARK 7.7.

In terms of c a l c u l a t i o n one should r e a l i z e t h a t the y, can a l l

be generated from each o t h e r . (7.24)

275

AY,,+~

=

-[Dr

To see t h i s note f i r s t from Theorem 7.5 t h a t

- (n/r)

+ p n ( r ) l y n ; AY,

= [Dr

+

n+l

7 -

P~(P)]Y,,+~

Define now wn a n d u n via ( 6 ) p, = n / r - \;m/w ; p = "n+l n( n + l ) / r + G n / u n w i t h i n i t i a l conditions lirn r-'wn(r) = 1 and lim r un(r) = 1 as r 0. Then one obtains (y,

= Yn(r,A)).

-f

(7.25) and putting

XYn+, (6)

=

w(Yn~wn)/wn;AYn

= -W(Yn+l,un)

i n t o the Riccati equations (7.16) one has [ P n ( D r ) - Wn]wn =

0 and [Pnt1(D,) - Wn+l]un = 0. Thus u, and wn s a t i s f y Theorem 7.6 with = 0. From (7.25) we see t h a t y, + yntl via w, and y n + y n - l via u,. In order t o use a s t e p u p procedure t o generate a l l y n from yo via wn one shows To do t h i s note t h a t wn(r) = lim Z n n ! t h a t w ~ can + ~ be computed from w,. y n ( r , X ) / A n a s A + 0 so taking l i m i t s i n (7.25) and u s i n g the r e l a t i o n s

2

yn(s,A)wn(s)sds one obtains

W(yn,wn) = ( A / r ) $

Similarly from un(r) = l / r w n ( r ) t h e un can a l s o be generated from w,. Transforms of the form (7.25) a r e r e l a t e d t o t h e C r u m transform used i n quantum s c a t t e r i n g theory ( c f . [Cel; Cwl; Sa141). REmARK 7-8- In order t o give an inverse s c a t t e r i n g i n t e r p e r t a t i o n t o the

above s e t $ n ( r , x )

=

(rh)'yn(r,h)

so t h a t

This is t h e Schrodinger equation f o r a p a r t i c l e w i t h angular momentum n a n d 2 energy E = A . There i s a standard G-L technique f o r recovering t h e potent i a l Wn from t h e s p e c t r a l d e n s i t y r = ( l / 2~) I F ( x ) I - * where F i s the standard J o s t function ( c f . [Cel; F a l l ) . Here t h e J o s t function i s t h e same f o r a l l n whereas i n quantum physics t h e p o t e n t i a l i s constant and one c o n s t r u c t s a sequence o f J o s t functions. 8.

P;Z0PH#ZCAL ZNUZWE PR0BLEmB (REFLECCZON DAEA),

We turn next t o a more

d e t a i l e d study o f t h e operator Qu = ( A u ' ) ' / A in the context of a geophysical inverse problem ( c f . 51.6 a n d 9 2 . 1 1 ) . This operator a c t u a l l y a r i s e s i n many applied problems (e.g. in the study of transmission l i n e s ) . T h u s we

276

ROBERT CARROLL

c o n s i d e r t h e problem o f one dimensional wave p r o p a g a t i o n through a s t r a t i f i e d e l a s t i c medium and f r o m experimental i n f o r m a t i o n a t a p o i n t we a r e a b l e t o determine t h e m a t e r i a l p r o p e r t i e s through t h e medium ( o r a t l e a s t somet h i n g about t h e m a t e r i a l p r o p e r t i e s ) . i n g manner. (8.1)

The problem i s posed i n t h e f o l l o w -

The governing e q u a t i o n f o r t h e SH shear wave i s

p(x)vtt

=

(u(x)vxIx; 0 5 x <

m

where p ( x ) i s t h e d e n s i t y and u ( x ) i s t h e shear modulus which a r e unknown. The system i s a t r e s t f o r t < 0, v ( t , x )

= 0 f o r t < 0, and we i n t r o d u c e an

e x c i t a t i o n a t t h e p o i n t x = 0 o f t h e form (8.2)

vx(t,O)

= -(p(0)/P(o))?i~(t)

( t h e minus s i g n i s e x p l a i n e d a f t e r ( 8 . 7 ) ) . sumed t o be known.

Here p ( 0 ) and ~ ( 0 )can be as-

We can t h e n r e a d o f f t h e (impulse) response a t t h e same

p o i n t and c o l l e c t i n f o r m a t i o n o f t h e form (8.3)

v(t,O)

= vo(t) = G(t)

The general i n v e r s e problem t h e n i s t o determine P ( X ) and u ( x ) f o r x

>

0,

which cannot be done; however we can determine t h e "impedance" (pu)'(y) A(y) as a f u n c t i o n o f " t r a v e l t i m e " y = :/

(p/u)'dE;

=

( t h i s i s t h e standard

and n a t u r a l i n v e r s e problem here and has been s t u d i e d i n v a r i o u s ways by a number o f a u t h o r s ) .

BlagovejEenski j [Bgl] f o r m u l a t e d some complicated non-

l i n e a r i n t e g r a l e q u a t i o n s and was a b l e t o e x t r a c t some i n f o r m a t i o n about t h e governing parameters i n t h i s and more general problems ( c f . [Bgl-41). Although he was a b l e t o r e q u i r e o n l y p , E~ C1 (as we do) t h e c a l c u l a t i o n s a r e f o r m i d a b l e and t h e method i s n o t t o o r e v e a l i n g t h e o r e t i c a l l y . 2; Ne4; Sa1,2;

In [Afl,

W f l ] f o r example one s t u d i e d v a r i o u s aspects o f r e l a t e d prob-

lems i n terms o f i n v e r s e quantum s c a t t e r i n g techniques under assumptions i n 2 ~ C . The most r e l e v a n t d i s c u s s i o n i n t h i s s p i r i t f o r o u r purvolving p , E poses i s perhaps t h a t o f [Af1,2]

f o r a r e l a t e d problem b u t t h e mathematical

procedure t h e r e r e q u i r e s some m o d i f i c a t i o n s ; t h e f i n a l r e s u l t s a r e neverthe1 l e s s c o r r e c t . We w i l l r e q u i r e o n l y t h a t p , E~ C and p r o v i d e a n o n t r i v i a l and r i g o r o u s g e n e r a l i z a t i o n o f A l e k s e e v ' s t e c h n i q u e t o determine t h e spect r a l function.

Then, i n s t e a d o f u s i n g K r e i n ' s method t o e v e n t u a l l y r e c o v e r

(OF), as done by Alekseev, we produce a new v e r s i o n o f t h e G-L e q u a t i o n ap-

p r o p r i a t e t o t h i s problem.

Various techniques 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

a r e e x p l o i t e d and g e n e r a l i z e d and we r e f e r f o r background and o t h e r r e s u l t s

REFLECTION DATA

f o r r e l a t e d problems t o [Adl; 2; Gi1,2;

Bo1,2;

Lx4-6;

277

Sel-4;

Kr1,3;

Bp1,2;

Bbgl; S t m l ,

601; J a l ; B b r l ; S t f l - 9 1 . X

L e t now y ( x ) = 10 (P/u)'(S)dS (8.4)

V t t =

take t h e form

v (t,O)

= -6(t);

Y

= 0 f o r t < 0.

and v ( t , y )

that f t IA'IAldy < t i o n where A '

-+

w i t h A(0) = 1).

(8.1) becomes

(Avyly/A = TI(Dy)v

w h i l e (8.2)-(8.3) (8.5)

so t h a t , w i t h A ( y ) = (pv)'(y),

m;

v(t,O)

= G(t)

We assume

and

P

u belong t o C 1 and r e a l i s t i c a l l y

i n f a c t we w i l l be p r i m a r i l y concerned w i t h t h e s i t u a -

0 and A

+

Am r a p i d l y as y

.+

m

(we a l s o n o r m a l i z e as b e f o r e

Assume a l s o as b e f o r e 0 < a 5 A(y) 5 B <

-

f o r a l l y.

Taking F o u r i e r t r a n s f o r m s i n ( 8 . 4 ) one o b t a i n s ( c f . (1.6.1))

"v'

(8.6)

+ k2v^ = q ( y ) $ ' ; q ( y ) = - A ' / A

Here we w i l l use A and k i n t e r c h a n g a b l y s i n c e k i s customary i n p h y s i c s and

Fv

= G(k,y)

Jr v(t,y)exp(ikt)dt.

=

We w i l l c a l l r e g u l a r s o l u t i o n t h e func-

t i o n p ( k , y ) s a t i s f y i n g (8.6) w i t h p(k,O) = 1 and p ' ( k , O ) = 0 as i n (1.6.2). We w i l l c a l l J o s t s o l u t i o n s t h e f u n c t i o n s @(+k,y) s a t i s f y i n g (8.6) w i t h @(+k,y)

%

exp(*iky)A?

and @ ' ( f k , y )

%

?ikexp(kiky)A?

as y

-f

m

( c f . (1.6.2).

Equation (8.6) can now be c o n v e r t e d i n t o t h e i n t e g r a l e q u a t i o n s (1.6.5)(1.6.6)

which a r e s o l v e d by i t e r a t i o n t o y i e l d Theorem 1.6.2.

Thus we r e -

c a l l t h i s i n f o r m a t i o n i n t h e form

CHE0REm 8.1. =

Q @?,(y)

Given q

E

L1 , t h e f u n c t i o n s q ( k , y )

= p QA ( y ) ( A

can be d e f i n e d by s e r i e s as i n Theorem 1.6.2.

Imk > 0 I A 3 ( k , y ) l

5 exp(-yImk)exp[cJm

Q

k ) and @(+k,y)

One has t h e n f o r

< 0 \A$(-k,y)l Y < exp(yImk)exp[cJm l q ( ~ ) ] d n ] w h i l e @(k,y) (resp. @(-k,y)) i s a n a l y t i c f o r Y Imk > 0 ( r e s p . Imk < 0 ) . On t h e o t h e r hand p ( k , y ) i s e n t i r e w i t h I p ( k , y ) l

I e x p ( y l ImkI )exPC.#

I q ( n ) l d n ] and f o r Imk

Iq(n)ldnI.

We r e t u r n now t o t h e problem ( 8 . 4 ) - ( 8 . 5 )

and r e f e r r i n g t o [C40] f o r d e t a i l s ,

remark t h a t an e q u i v a l e n t problem a r i s e s upon r e p l a c i n g t h e impulse i n ( 8 . 5 ) by a c o n d i t i o n (8.7)

Vt(O,Y)

= 6(Y)

It i s i n f a c t somewhat more n a t u r a l t o work w i t h (8.7)

( o r w i t h an impulse

2 78

ROBERT CARROLL

i n s e r t e d d i r e c t l y i n ( 8 . 4 ) ) and we w i l l f o l l o w t h i s d i r e c t i o n ( c f . [C40;

An example w i l l p a r t i a l l y c l a r i f y t h i s equivalence and we w i l l

Stf2-41).

s i m p l y t h i n k of o u r problem subsequently as posed v i a ( 8 . 1 ) o r (8.4) w i t h = G(t).

impulse (8.7) and response v(t,O)

Take A = 1 and s t a r t w i t h i n p u t d a t a v ( 0 , t ) = - 6 ( t ) . The s o l u Y t i o n o f (8.4) i s t h e n v ( y , t ) = Y(t-y) ( f o r y , t 2 0 ) where Y denotes t h e

EHAl!lPCE 8.2.

Heavyside f u n c t i o n .

Thus v = - 6 ( t - y ) + - 6 ( t ) as y + 0 and v ( 0 , t ) = Y ( t ) . Y We n o t e t h a t t h e s o l u t i o n c o u l d a l s o a r i s e f r o m an impulse vt(y,O) = 6 ( y ) s i n c e vt = 6 ( t - y ) (6(y)

6(-y)).

3

e x p ( i A t ) d t and (8.6) becomes we must have

G

The F o u r i e r t r a n s f o r m i s FT =

+ A 2.v

YY

= 0 ( A = k).

= A ( A ) e x p ( i x y ) and w i t h i n p u t

= -exp(iAy)/iA

Now ( c f . [Bbel])

( A = 0).

-

vt = [ 6 ( t - y ) + s ( - t - y ) ] / Z .

Y(-t-y)]/Z;

t i p l y by 2 however and drop Y ( - t - y ) +

0.

F[Y(t-y)] corres-

v = -[6(t-y) - 6(-t-y)]/2; Y Working o n l y from t h e quadrant y , t 0 we mul-

o r y one i s l e d t o v = [ Y ( t - y )

6 ( y ) as t

=

More c o m p l e t e l y v i a t h e f u l l F o u r i e r the-

l/iA].

-

T(t)

*vY (0,A) = -1 i t f o l l o w s t h a t F [ Y ( t ) - Y ( - t ) ] = - 2 / i A (FY =

1 ~ 6 ( h )- l / i A ) and F 6 ( t - y ) = e x p ( i A y ) so i n some sense

ponds t o e x p ( i x y ) [ T s ( A )

if

Since v = 0 f o r t < 0

= 0 t o get v

+ - 6 ( t ) as y + 0 and vt + Y T h i s F o u r i e r p i c t u r e a l s o shows how a n a t u r a l odd and even

extension i n t o f v i s associated w i t h the s i t u a t i o n .

Moreover i n a l l prob-

lems o f t h e t y p e c o n s i d e r e d ( a r b i t r a r y A) t h e impulse response w i l l have a Y(t-y) t y p e f a c t o r

below.

-

t h e decomposition G ( t ) = 1 + G r ( t )

( t 1. 0) i s used

The f a c t o r o f 2' a r i s i n g i n v a r i o u s F o u r i e r r e p r e s e n t a t i o n s i s a l s o

c l a r i f i e d below. L e t us r e c a l l a few f a c t s about Riemann f u n c t i o n s f o l l o w i n g [C40]. g. c o n s i d e r (dw = $dA, (8.8)

S(Y,t,n)

2

= 1/2alc

4

4

12, 4

= (vA(Y)vA(n)yCosAt)o; R(y,tyn) =

Thus Rt = S and t h e s o l u t i o n o f (8.4), Vt(Y,O) (8.9)

=

vtt

Q

= Qv, w i t h v(y,O)

Q

SinAt ) 7 = f ( y ) and

g(Y) i s v(y,t) =

(

S(Y,t,n),A(n)f(n))

(up t o p o s s i b l e adjustment on y = 0 ) .

+

R(y,t,n),A(n)g(n))

Here one has S(O,t,n)

and R(O,t,n) = ( p AQ ( n ) , [ S i n A t / A ] ) w p a r t i c u l a r f o r f ( n ) = 0 and g ( n ) = 6(n)/Ao = 6 ( n ) we o b t a i n v ( y , t ) = R(y,t)

=

( v AQ ( n ) ,

( a g a i n Rt = S ) .

CosAt)w ( = ;(n,t))

(8.10)

Thus e.

and Qu = ( A u ' ) ' / A h e r e )

=

( v 0~ ( ~ ) , [ S i n A t / A ] ) ~

In

REFLECTION DATA For y = 0 one o b t a i n s t h e r e a d o u t G ( t ) =

(

279

l,[SinAt/A])W

f r o m which t h e spec-

t r a l d e n s i t y w i s determined by m

(8.11)

= ( ~ A / I T ) ~G~ ( t ) S i n A t d t

;(A)

tTHHZ6REm 8-3- The s p e c t r a l d e n s i t y :(A)

= l / 2 n l c p 1 2 can be o b t a i n e d d i r e c t l y

f r o m t h e impulse response G ( t ) v i a (8.11). Next we r e c a l l Theorem 2.11.7 which e x h i b i t s A(y) v i a a f o r m u l a A-'(y)

-

K(y,y) where K(y,x)

= 1

This is

i s t h e k e r n e l determined i n Theorem 2.11.6.

o f course a G-L t y p e k e r n e l and t h e r e w i l l be a corresponding G-L e q u a t i o n t o determine K.

T h i s was d e r i v e d i n [C40,66;

S e l l i n a r a t h e r ad hoc manner

and then a c a n o n i c a l d e r i v a t i o n was produced i n [C71,72]. t h e r e s u l t from [C40,66;

We s t a t e f i r s t

S e l l which i s u s e f u l f o r computation and t h e n we

w i l l g i v e d e t a i l s f o r t h e canonical d e r i v a t i o n .

Thus w r i t e ( + ) dw(k) =

(2/.ir)dk + do(k) and s e t [Sinkx/k]Coskydu(k)

(8.12)

T

= T(y,x);

Y

0

(y,x)

=

-1

SinkxSinkydo(k) 0

tTHEBREl?l 8-4- The a p p r o p r i a t e G-L t y p e e q u a t i o n f o r t h e d e t e r m i n a t i o n o f K(y, x ) (x

<

y ) i s g i v e n by (T b e i n g d e f i n e d by (8.12))

K(y,x) + T(y,x)

#

=

K(y,

n ITn ( n ,x )dn. T h i s G-L t y p e e q u a t i o n has a t i m e domain f o r m which i s v e r y v a l u a b l e and revealing.

Thus ( c f . ( 8 . 2 3 ) - ( 8 . 2 4 ) )

l e t us w r i t e G ( t ) = :f

+

[SinAt/A][do

(2/a)dA] = 1 + G r ( t ) ( t h e s u b s c r i p t r here r e f e r s t o r e f l e c t i o n d a t a ) . depending on whether y > x o r y

x one o b t a i n s ( c f . a l s o (8.23)-(8.24)

<

where more d e t a i l is g i v e n ) T(y,x) = [Gr(y+x)

-

Gr(y-s)]/2

It f o l l o w s t h a t Ty(y,x) = [G;(y+x) [Gr(y+x) + Gr(x-y)]/2. and t h e G-L e q u a t i o n i n Theorem 8.4 can be w r i t t e n ( x < y ) (8.13)

Then

o r T(y,x)

- G,'.(

=

ly-x1)1/2

I ~K(Y,s)CG~(X+S)-G~(~X-S~ Y

K(y,x)

+ +[Gr(y+x)-Gr(y-x)l

=

)Ids

0

ISHEBRElil 8.5,

The G-L e q u a t i o n o f Theorem 8.4 can be w r i t t e n d i r e c t l y i n

terms o f r e a d o u t data i n t h e f o r m (8.13) and g i v e n K one " s o l v e s " t h e i n v e r s e problem v i a Theorem 2.11.7

i n t h e f o r m A-'(y)

= 1

-

K(y,y).

The 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 i n Theorems 8.4 and 8.5 i n [C40,66; was l a r g e l y ad hoc i n n a t u r e .

Sell

L e t us g i v e h e r e a c a n o n i c a l d e r i v a t i o n based

on general t r a n s m u t a t i o n procedures as i n Chapter 2. L e q u a t i o n has t h e f o r m ( c f . g2.3)

Thus t h e canonical G-

280

ROBERT CARROLL

w

(8.14)

B(Y,X)

where B and =

'ii a r e

= ( B(Y,t),A(t,x))

the kernels o f transmutations D

; 1 $(A)CosAxCosAtdA

= ( CosAx,CosAt )o.

0 f o r x < y and B ( y , t )

=

2 -f

C! as usual and A ( t , x )

I n f a c t F(y,x) =

(2/.rr)J;qp(y)CosxtdA. 4

0 C O S A X , ~ ~ ( Y ) )w =

Now f o r x < y we i n t e o r a t e

i n (8.14) f o r m a l l y t o o b t a i n ( c f . (8.11)) ( B ( y . t ) , A ( t , x ) ) =

1

(

0 where

X

d(t,x) =

(8.15)

A(t,S)dc = ( C o ~ A t , [ S i n h x / h ] ) ~=

0

+ ;/

1; Sinh(x+t)GdA/;i

[G(x+t) + G ( x - t ) ]

(X > 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-'[<(s)/?(s)]. Roughly one wants a l l frequencies t o be e x c i t e d by f f o r the recovery of 9,.

8.8. Let us consider now t h e s i t u a t i o n of Remarks 1.6.1 and 1.6.4 and 92.11 where A E C2. = Q ( y ) t h e equation ( 8 . 6 ) becomes Setting A-'(fi) YY f o r $ = A% ( m ) jyy+ k2$ = Q ( y $ ( c f . ( 1 . 6 . 3 ) ) a n d t h e condition v ( k , O ) Y = 1 becomes ( 6 ) $ ' ( k , O ) - hl$(k,O) = h 2 where h l = - Z - ' ( O ) Z l ( O ) and h 2 = Z-l(O) ( Z = A-') a r e known i n terms o f P,LI evaluated a t 0. This i s a stan-

RERARK

dard type of Schrodinger equation w i t h potential Q ( y ) and conditions o f t h e type ( 6 ) can be handled by dealing w i t h "eigenfunctions" IP = 9 x Q , h as in g51.5 and 1 . 4 where I p ( k , O ) = 1 and 9 ' ( k , O ) = h l . W e w r i t e again @(?k,y) f o r t h e J o s t s o l u t i o n s of ( m ) (renormalized by @ ( t k , y ) % exp(?iky) as y +

m

f o r convenience here in r e f e r r i n a t o o t h e r papers) and one can show t h a t j ( k , y ) = h2@(-k,y)/[@'(-k,0) - h l @ ( - k , O ) ] ( c f . (1.12.10)). The function @ ( - k ) = @'(-k,O) - hl@(-k,O) has nice p r o p e r t i e s , analogous t o kc(k) and examples show t h a t @ ( - k ) = 0 a t k = 0 ( s i m i l a r t o k c ( k ) ) . In [Af1,2] t h i s i s dismissed physically a s representing a s t a t i c s i t u a t i o n which does not a r i s e i n t h e dynamical problem. This i s c o r r e c t b u t t h e pole of ? ( k , y ) a t k = 0 cannot be so l i g h t l y dismissed i n t h e subsequent mathematical a n a l y s i s . One

3

can however handle this by working w i t h (k,y) and eventually a r r i v i n g a t Y an expression f o r J / ( t , y ) analogous t o the corresponding r e s u l t obtained from (8.10). T h u s we obtain f i r s t ( c f . [C40,66; S e l l )

f o r t > 0, y 2 0 or t

0, y > 0.

Now one shows e a s i l y ( c f . (1.12.10) and

preceeding equations) t h a t cp(k,y) = [@(k)@(-k,y) - @(-k)@(k,y)]/(2ik) and consequently

282

ROBERT CARROLL

An a n a l y s i s s i m i l a r t o t h a t o f Theorem 1.6.8 g i v e s t h e s p e c t r a l f u n c t i o n f o r

2

t h e $ e i g e n f u n c t i o n problem ( = ) i n t h e f o r m (+) dw(k) = 2k dk/.rrlo(k)I2 = c ( k ) d k ( c f . a l s o [Af1,2]).

Now an i n t e g r a t i o n i n (8.20) i n y i s permissable

and we o b t a i n an analogue o f (8.10) i n t h e form m

(8.21)

J,(t,y)

-

= $,(t)

Consequently i f v(t,O) ( r e c a l l h2 = Z - l ( O ) )

h2

io

-

b(k,y)

= Z(O)$(t,O)

lI[Sinkt/kldv(k)

= g ( t ) we have i n accord w i t h [Af1,2]

g ( t ) = -1; [ S i n k t / k ] d w ( k ) .

CHEBREM 8-9- The s p e c t r a l d e n s i t y $ ( k )

2 2 2k /.rrl@(k)l f o r t h e s p e c i a l prob-

=

lem based on ( = ) can be o b t a i n e d f r o m t h e r e a d o u t g ( t ) by t h e f o r m u l a w(k) = -(2k/1~)$ g(t)Sinktdt.

A uniqueness theorem f o r o u r G-L e q u a t i o n i n Theorem 8.4 can be modeled on a procedure i n [ C e l l . W(y,x)

=

One must show t h a t t h e homogeneous e q u a t i o n

{1 W(y,q)Tn(n,x)dq

has o n l y a t r i v i a l s o l u t i o n .

Multiply

:j :j

(8.22)

n

(AA)

n

(from

by W(y,x) and i n t e g r a t e i n x t o o b t a i n , u s i n g ( * )

W(y,~)W(y,x)G(n,x)dqdx

Hence f o r any y t h e e n t i r e f u n c t i o n

=

;1

d w ( k ) [ r W ( y , ~ ) S i n k x d x ] ~= 0 0

W(y,x)Sinkxdx

( s i n c e dw > 0 ) and one can conclude t h a t W(y,x) y.

Note t h a t T

SinkxSinkqdu + ~ ( Q - x )= -G(n,x)

(8.12)) can be w r i t t e n as (*) T (n,x) = -/; t ~(n-x).

(AA)

o f k i s zero f o r k r e a l

= 0 for 0

5 x 5 y f o r each

Hence we have

CHE0Rm 8.10-

S o l u t i o n s K(y,x) o f t h e G-L e q u a t i o n i n Theorem 8.4 a r e unique.

We w i l l s k e t c h now a few r e s u l t s o f s t a b i l i t y ( f o r d e t a i l s see [C70]).

Such

i n f o r m a t i o n i s c l e a r l y o f i n t e r e s t from a p h y s i c a l p o i n t o f view and n a t u r a l l y t h e s t a b i l i t y q u e s t i o n a r i s e s i n any numerical c a l c u l a t i o n . [Stf4,5]

Symes i n

shows t h a t t h e Chudov system r e s u l t i n g f r o m t h e n o n l i n e a r G-L equa-

t i o n possesses s t a b l e s o l u t i o n s (see a l s o [ S t f 3 ] f o r a d i f f e r e n t s t a b i l i t y analysis).

Gerver i n [Gil,2]

and Bamberger-Chavent-Lailly i n [Bol,2]

were

a b l e t o whow t h a t an o p t i m i z a t i o n f o r m u l a t i o n l e a d s t o s t a b l e s o l u t i o n s f o r a w i d e r c l a s s o f impedance f u n c t i o n s .

Here we w i l l show t h a t s t a b i l i t y r e -

s u l t s can a l s o be o b t a i n e d u s i n g t h e l i n e a r G-L equation.

Our r e s u l t s a l s o

g i v e some i d e a o f how t h e s e n s i t i v i t y i s dependent on y and show t h a t small

REFLECTION DATA

283

p e r t u r b a t i o n s i n data g l e a d t o small v a r i a t i o n s i n A.

Since most numerical

schemes i n v o l v e a p p r o x i m a t i o n o f t h e i n t e g r a l s i n t h e G-L method by f i n i t e sums such s t a b i l i t y i s r e f l e c t e d i n good numerical schemes and some impres-

s i v e g r a p h i c a l d i s p l a y s have i n f a c t been o b t a i n e d ( c f . [Se2,3]).

Since i t

w i l l be convenient a t v a r i o u s p l a c e s i n t h e a n a l y s i s t o f o l l o w t o have A.

A(0)

= 1 = (pu)'(O)

=

we remark e x p l i c i t l y t h a t t h i s can be achieved by a

s c a l e change 5 = ax a t t h e b e g i n n i n g w i t h u = A., change t o have been made and t h a t A.

= 1.

Thus assume such a s c a l e

Another o b s e r v a t i o n t h a t i s help-

f u l here i s t o n o t e t h a t t h e response G ( t ) f o r A(y) = 1 can be w r i t t e n as G ( t ) = Y ( t ) ( Y = 1 f o r t > 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

<

The p r o o f i s s t r a i g h t f o r w a r d upon w r i t i n g I

(e* - e ) ]

where II (I- E)-'(i?*

- e)ll

-

- e*

= ( I

- e)[I -

",Y

-

g)-lll-l

ll~*-~~lll(I-~)~lll].

-

(I

C)-'

The e s t i m a t e comes from e s t i m a t i n g rz

and, s e t t i n g I K ( y , x ) l 5 My w i t h I I ( 1 - t )-'I1 Y IIAK(y, *)I1

(8.30)

Ell < ll(1

C)-'Il/[l

Now i n (8.28) l e t us w r i t e ( t ) llAT(y,-)llm,y

t h e a s s o c i a t e d Neumann s e r i e s . < ll~(-)ll ",2Y

< 1.

-

-

II(1

< NY[ II E U

-

"12Y

+ M IIE'II 1 Y

L (2Y)

]/[1

= Nyy one o b t a i n s

-

NyIIACyII]

*

EHEOREm 8-12, F o r y f i x e d l e t approximate d a t a Gr s a t i s f y e.g. IIE'II 1 L (2Y) < +N-' = II ( I - C ) - l I l - l / Z . < 2N [ I l ~ l l + M llE111L1(2y)] Then llAK(y,-)ll ".Y - Y ",2Y Y - Y Y where M = sup1 K(y, - ) I on C0,Yl.

Y T h i s f o l l o w s i m m e d i a t e l y from (8.30) i n n o t i n g t h a t N IIAC II 5 N IIE'II 1 Y Y1 Y L (2Y) We n o t e a l s o t h a t a Co e s t i m a t e on E ' i m p l i e s an L e s t i m a t e . Furc 1/2. t h e r f r o m K(y,y) = 1 IAAl/I[A4

+ A*'ll.

-

A-'(y)

we have lAK(y,y)I

=

\A-'(Y) -

A*-4(~)l =

Hence f o r IAAI 2 A say one has lAAl 5 I A K l $ ( l + f i )

which g i v e s a rough comparison o f IAAI and lAKI.

RZmARK 8.13,

A d i s c r e t e v e r s i o n o f t h i s s t a b i l i t y r e s u l t f o r numerical s o l u -

t i o n s o f t h e G-L e q u a t i o n i s e s t a b l i s h e d i n [SeZ]. compare t h e impedance p r o f i l e s A and A

*

*

Graphical d i s p l a y s t h e r e

corresponding t o r e f l e c t i o n data

Gr and Gr and show e x c e l l e n t s e n s i t i v i t y o f t h i s method ( c f . a l s o [Stm1,2]).

REmARK 8-14, L e t us mention h e r e some r e c e n t work o f Levy, Yagle, B r u c k s t e i n , e t . a l . i n [Lx4-6;

B b s l ] d e a l i n g w i t h s e i s m i c i n v e r s e problems (and o t h e r

problems) by " l a y e r s t r i p p i n g " methods which l e a d t o f a s t numerical procedures.

One t a k e s v as displacement and P as p r e s s u r e so t h a t t h e b a s i c equa-

L e t y be t r a v e l t i m e and A be impedance -1 Then one o b t a i n s a system w = -A Pt as b e f o r e and s e t w = vt ( v e l o c i t y ) . Y W r i t e now = A-'P, @ = A%J, p = '(M), and q = +(*-@). w i t h P = -Aw Y t' Then w i t h " r e f l e c t i v i t y " (A) r = %D l o g A ( y ) one has py + pt = - r q and q Y Y qt = - r p . One notes h e r e t h e analogy t o a l o s s l e s s t r a n s m i s s i o n l i n e probt i o n i s pvtt = -Px w i t h P =

-UV

*

X'

286

ROBERT CARROLL

lem where =

(L/C)

i s e.a.

=

Z-%

, and y

=

and

@ =

2% w i t h

i = c u r r e n t , V = v o l t a g e , Z = impedance

:J (LC) dS a g a i n r e p r e s e n t s a t r a v e l t i m e .

The i n p u t now

P(0,t) = P o 6 ( t ) and one reads say w ( 0 , t ) = w0[6(t) + 2Ej'(t)Y(t)] w i t h

Po/wo = A(0) = 1 ( t h u s t a k e wo = 1, Po = 1, and G ' = 6 ( t ) + 2?j(t)Y(t) connect t h i s w i t h (8.4)-(8.5) p(0,t)

and q ( 0 , t )

and ( 8 . 7 ) ) .

t h e n have t h e f o r m p ( 0 , t )

The downgoing and upgoing waves =

6 ( t ) + :(t)Y(t);

q(0,t) =

F u r t h e r w r i t e now p ( y , t ) = 6 ( y - t ) + v ( y , t ) Y ( t - y )

-<(t)Y(t). CJ

q(y,t)Y(t-y)

to

and q ( y , t )

=

t o o b t a i n t h e s o - c a l l e d f a s t Cholesky r e l a t i o n s ( o r downward

c o n t i nuati o n recursions ) (8.31)

v

py +

Ft

=

-r(y)%

cy - Tt

When t h e measured waves p ( O , t )

= <(t)

= - r ( y ) E r ( x ) = 2y(x,x)

and V ( 0 , t )

=

-<(t)a r e

used as i n i t i a l

d a t a i n (8.31) one o b t a i n s , upon d i s c r e t i z a t i o n , a r a p i d " l a y e r s t r i p p i n g " I f F o u r i e r t r a n s f o r m s i n t a r e taken i n say (8.31)

numerical procedure.

one has a frequency domain c o u n t e r p a r t r e f e r r e d t o as t h e Shur r e c u r s i o n s . There i s a l s o a v a r i a t i o n o f (8.31) a r i s i n g i n l i n e a r e s t i m a t i o n t h e o r y (Krein-Levinson r e c u r s i o n s ) b u t t h e procedure i s somewhat d i f f e r e n t ( c f . Theorem 4.8).

I f one c o n s i d e r s an i m p u l s i v e pressure wave o b l i q u e l y i n c i -

dent t o t h e medium t h e l a y e r s t r i p p i n g technique can a l s o be used and d a t a from two i n c i d e n t angles a l l o w s one t o r e c o v e r p and 1-1 as f u n c t i o n s o f x ( c f . [C68,69;

REmARK 8.15. [C68,69,40;

891-4;

Ne4; Sel,4,5;

Stf2-6,8-101)

We mention b r i e f l y a l s o some r e s u l t s o f t h e a u t h o r and Santosa S e l l u s i n g t r a n s m u t a t i o n methods f o r s o l v i n g some t h r e e dimen-

s i o n a l i n v e r s e problems based on a f o r m u l a t i o n o f [Bgl-4]

(whose r e s u l t s i n

t h e form o f n o n l i n e a r i n t e g r a l equations seemed i n t r a c t a b l e ) .

Our r e s u l t s

a r e a l s o a b i t u n r e a l i s t i c f o r numerical computation b u t t h e y have a number o f i n t e r e s t i n g features.

F i r s t l e t us n o t e ( c f . Remark 8.7) t h a t i f i n p u t

i n t h e one dimensional problem ( 8 . 4 ) - ( 8 . 5 ) f ( t ) t h e n (**) v ( t , y ) = J,,t v 6 ( t - r , y ) f ( T ) d T wave t o a d e l t a f u n c t i o n i n p u t .

i s taken i n t h e form v (t,O) = where v 6 i s t h e impulseY response t

6

Then G ( t ) = Jo G ( t - T ) f ( T ) d T

and i f t h e i n -

p u t f does n o t e x c i t e a l l f r e q u e n c i e s t h e r e c o v e r y o f t h e impulse response 6 G ( t ) ( o u r G ( t ) p r e v i o u s l y ) from G ( t ) may be i m p o s s i b l e .

s i o n a l problem o f [Bg1-4] h a l f space x1 = x l 0.

Now t h e 3 dimeni s a s o r t o f Lamb wave problem f o r a s t r a t i f i e d

The equations f o r displacements ui,

w i t h impulsive

stresses, a r e ( P = d e n s i t y and ( A , u) r e p r e s e n t t h e Lam6 m o d u l i ) (8.32)

REFLECTION DATA

6(x2,x3)6(t)

a t x = 0 and ui(t,x)

where

T~~

= 0.

One d e f i n e s new q u a n t i t i e s vi(t,x)

uldx2dx3

=

w i t h t r a v e l times yl(x)

=

It

=

287

= gi(t,x2,x3)

I I uidx2dx3

(p/A+Zu)'dg

= readout a t x

and w(t,x)

and y,(x)

=

I I x2

(p/u)'dE.

= :/

I t i s enough t o c o n s i d e r two vi equations i n t h e form = AT1(A.v

(8.33)

Vj,tt

where A1 =

[P(A+~U)]',

the derivative i n y (8.34)

j

j

j

A2 =

1

yj

( j = 1,2);

v.(t,O) J

and v . .(t,O)

(pu)',

JYJ

= hj(t)

( v ~ ,denotes ~

= 6(t)/Aj(0)

The w e q u a t i o n i s

here).

W t t = (A,Wy

)

j,yj

-

)y /A1 1

B(Y1)V2(t9Y2)

-

D(Y1)v2,y2(t.Y2)

where D i s known b u t n o t B and one knows (+) w,(t,O)

=

A(0)h2(t)/A,(0)

with

= j(t). The vi problems a r e t h e same as b e f o r e and one f i n d s A.(y.) J J ( j = 1,2) w i t h dy2/dy1 = A1(y,)/A2(y2) d e t e r m i n i n g y2 = y2(y1). A l s o f r o m t h e G-L e q u a t i o n kerne s K.(y.,s) e t c . one c o n s t r u c t s v Q (y ) e t c . which ? J ~j l e a d t o v . e x p l i c i t l y . T h i s a l l o w s one t o determine D(y,) above b u t B(yl)

w(t,O)

J2

2

(P/A1)([A1 - 2A21/P), i s unknown. To s o l v e (8.34) one t h i n k s f i r s t o f w1 h ( t , O ) above as g e n e r a t ng a s o l u t i o n w t o t h e homogeneous e q u a t i o n (which

=

i s t h e v1 e q u a t i o n ) and t h e n from (**) t

(8.35)

w h ( t 'Y1) = A(0)

h Hence w ( t , O )

j

h2(T)vl (t-T,Yl 1d.r

0

= h(0)[hl

*

h 2 ) ( t ) and we w r i t e w = wh t wi where wi

t h e inhomogeneous w e q u a t i o n (8.34) w i t h w;'(t,O) x(0)(hl

*

(8.36)

-

= j(t)

-

A f t e r some c a l c u l a t i o n s ( u s i n g t h e p r i n c i p l e o f no i n -

h2)(t).

coming waves f r o m (F(X) i s known

= 0 and wi(tyO)

satisfies

m)

one a r r i v e s a t t h e f o l l o w i n g i n t e g r a l e q u a t i o n f o r B

n o t e h i s used as an eigenparameter here a t t i m e s )

F(A) =

jOm f(?)*!(y)B(y)Aqdy"

= (Dx(y)/c 4 (-A) and A (y)d? = A p ( y ) d y determines where e.g. 'I'A(y) 4

4

4

7 = y(y)

o r y = y ( y ) . Here A ?I A f o r t h e v1 o r v2 e q u a t i o n s . T h i s e q u a t i o n (8.36) can a c t u a l l y be handled q u i t e e f f e c t i v e l y by t r a n s m u t a t i o n methods and mo4 t i v a t e d a c e r t a i n amount o f work on i n t e g r a l t r a n s f o r m s w i t h k e r n e l s U,(y) (cf.

[C34,40]).

(8.37)

Indeed, t a k i n g F o u r i e r t r a n s f o r m s i n (8.36) we g e t

Inm

(BA,,)(y) j~e-i"t~~(~)*!(y)dAdy W " D2 -+ Q such From general t r a n s m u t a t i o n t h e o r y t h e r e i s a t r a n s m u t a t i o n f(t) =

B(y)2(y,t)dy = (1/21~)

9:

288

that

ROBERT CARROLL

% [2exp(ihx)] Q

-4

6 (7,~)=

Q

(

= f(y). (1-

E0 has t h e form ( = ) 9(y,x) - = k e r n e l (y",'x) (y)g(x-?) + = ) w i t h zQ(Y,

Also

C o s h x , q ~ ~ ( y)w) = At5(0)A4

x) = 0 f o r x

<

7

(anticausality).

Q

Thus e.g.

(w

w

4

m

*f(y")

(8.38)

=

2 izQ(?,x)eiAxdx

A

Q P = *h(?)qh(y)

From (8.37) FG(y,.) A

Usin

(m)

J;("

B(y)E(y,T)dy

*

-)I

F[2FQ(;,

(recall y =

G(Y,t) = ~ A ~ ( Y ) [ ? Q ( ? , * )

(8.39)

=

;p(Y,

~ ( 7 ) so)

i t follows that

- ) l ( t )= 4Ap(Y)

i:

one a r r i v e s a t a V o l t e r r a t y p e i n t e g r a l e q u a t i o n

^Bp(.Y,s)FQ(?.t-S)dS =

?'((r)

where T ( T ) i s a known monotone f u n c t i o n o f

T.

B(T) +

T h i s can

be s o l v e d i n v a r i o u s c o n t e x t s by v a r i a t i o n o f V o l t e r r a techniques and y i e l d s a unique s o l u t i o n 6. sess about A1,

Then g i v e n B(yl),

p(y,)

can be determined, and sub-

w i l l be c a l c u l a b l e from t h e i n f o r m a t i o n we a l r e a d y pos-

sequently P,P,A(x)

A2, y 2 ( y 1 ) , e t c .

REmARK 8-16, R e f e r r i n g back t o t h e one dimensional problem, f r o m (8.10) one has v t ( t , y )

*

= 6(y,t) = A - l ( y ) y ( t , y ) = (qA(y),Cos?,t)w 4

with kernel 6-l =

+ L ( t , y ) . Hence G ' ( t ) = vt(t,O) = 6 ( t ) + L(t,O) ( r e T h i s i s exc a l l A(0) = 1 ) and one can w r i t e $(?,) = (2/11)[1 t FCL(t,O)].

y ( t , y ) = A'(y)&(t-y)

a c t l y t h e same as t h e r e s u l t i n Theorem 1.5.5. 9, e;Ee)PH&3ICAI: INVER5E 4RP)BLEW (CRAEMI$sle)N DAIIA).

We c o n t i n u e t h e f o r -

) /A and vt(O,y) = 6(y) and we Y Y c o n s i d e r here some problems i n v o l v i n g r e c o v e r y o f A v i a " t r a n s m i s s i o n " data

m u l a t i o n ( 8 . 4 ) w i t h (8.7) ( c f . [C71,72,73,78]).

( t h u s vtt

= (Av

Thus we c o n s i d e r t h e problem o f r e c o n s t r u c t i n g t h e

c o e f f i c i e n t A(y) i n t h e PDE (8.4),

viz. A(y)vtt

> 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 <

We n o t e

l / c (-A)

Q

Note here t h a t H l ( t )

=

+

U

suitably large v

m

2Ame- i h y jvHl(t)eihtdt

y"

Hence

=

2Ame 4 - i h y 4H 1 ( ~ )

w i l l u s u a l l y have a d e l t a f u n c t i o n component A 2 6 ( t - ? )

( c f . Example 8.2) and t h e i n t e g r a t i o n symbol i n (9.8) i s i n t e n d e d t o t a k e t h i s i n t o account.

We see t h a t t h e l o c a t i o n o f

f a c t o r i n t o t h e readout ( f o r ? s u i t a b l e r e a l one o b t a i n s

t8R0LLARU 9.5.

large).

7 only

i n t r o d u c e s a phase

Since

FQ (A)= c Q (-A) for

Under t h e hypotheses above i t f o l l o w s t h a t

;(A) =

T R A N S M I S S I O N DATA

291

" 2 l / 2 a l c Q ( h ) / 2 = ( 2 / 1 ~ ) A ~ l H ~ ( h from )l which one can r e c o v e r A by t h e methods

o f 58. R€llIIIARK 9-6, T h i s f o r m u l a seems s t r i k i n g because i t d i r e c t l y e x h i b i t s t h e s p e c t r a l measure i n terms o f t h e F o u r i e r t r a n s f o r m o f an a u t o c o r r e l a t i o n type function X ( t ) =

rf Hl(ttT)Hl(T)dT.

As i n d i c a t e d e a r l i e r t h e r e i s an

i n t i m a t e and profound c o n n e c t i o n between v i b r a t i n g s t r i n g problems and problems o f e x t r a p o l a t i o n and i n t e r p o l a t i o n f o r s t a t i o n a r y t i m e s e r i e s and t h e r e s u l t s above f i t i n t o t h a t c o n t e x t v e r y n e a t l y .

I n f a c t t h e y seem t o p r o -

v i d e a new l i n k d i r e c t l y connected w i t h t h e geophysical problem and thus perhaps w i l l l e a d t o some new d i r e c t i o n s c o n n e c t i n g t h e t r a d i t i o n a l y t i m e s e r i e s a n a l y s i s i n geophysics w i t h e x a c t techniques f o r t h e i n v e r s e problem. I n t h e same s p i r i t as t h e t r a n s i t i o n (9.5)-(9.6)

we w r i t e now ( c f . (8.10)-

(8.11) and r e c a l l q QA ( 0 ) = 1 ) m

(9.9)

+p:(0)GeiXtdh

G'(t) =

+ G1(-t)

Gl(t)

=

Lm

(9.10)

1rm$A(0)eihtdh ,

G1 ( t ) = (1/4a)

where G l ( t )

= ( 1 / 4 1 ~ ) jm*:(0)e-i"dh m

vanishes f o r t < 0 (argue as b e f o r e ) .

1

Hence as i n (9.7)

m

(9.11)

= @y(0)/cQ(-l)

+f(O)

=

Gl(t)ei"dt

0

I n general aQh ( 0 ) i s n o t known b u t i t i s r e l a t e d t o c (with the present normalization f o r -2ixcQ(x).

9 a aA)

partial

For example we have 0 v i a N;,(O) =

Q' connection

I n any event one can w r i t e

i, m

(9.12)

Gl(t)

=

H 1 ( = ) K ( t - r ) d T = ( 1 / 4 1 ~ ) p:(0)e-iAtdh/c

One n o t e s by c o n t o u r i n t e g r a t i o n t h a t f o r m a l l y since

T

27

K(t-T)

= 0 f o r t - T - 7 > 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

<

f o r t > 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

CHEBRE111 9.10,

(9.20)

b

[Ih*(t+c)llAhl(S)

*

Assume II hl It co, II hl II -, and Then l A A l on [O,Y]

a b l y small. 1I Ahll co,, ;3

f

can be

and llAhllL1 v i a IIE'II 1

L (0,ZT)

< cllAh II 1

1 L

-

(7,~);

L e t us go back t o (9.3), m u l t i p l y by (2/a)SinhT, f i r s t t h a t ( 2 / ~ ) l : SinXTSinXtdX = 6 ( t - T ) and

and i n t e g r a t e .

We n o t e

N

( c f . 58).

-

Here we r e c a l l t h a t 1

f o r 11 > y, and K(y,O)

= 0 (cf.

d n t o a r r i v e a t K(y,O) = 0 ) .

lo

K(y,y) = A-'(y),

52.11 f o r K(y,n)

H(T) =

(9.23)

J

=

1t-T.l)

(2/a) jm%[Cosllt-Tl

-

11

CosX(t+-r)]

T >

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 <

f; (where H ( T )

= 0) (9.24)

reduces t o

Indeed G ( T - ~ ) = 0 i n (9.24) w h i l e -G(y"--r)

remains,

and i n a d d i t i o n t o (9.25) m o d i f i e d below, t h e i n t e g r a l t e r m c o n t r i b u t e s

-

(9.27)

QTid(t)K2(~,t+w T)dt =

+ ~G(O)K(F,T) + 4

e

-4G(y-~)K(?,y)+

G'(s-T)K(F,s)ds

The m o d i f i c a t i o n s r e q u i r e d i n (9.25) i n v o l v e

%lom

(9.28)

G(t)K2(p,t-T)dt

= $G(O)K(?,T)

Consequently we o b t a i n from (9.24), (9.29)

0 =

A~?G(?+T)

Using again K(?,?)

tHE0REM 9-12, For

= 1 T <

7

-

- Ai5

+

%IT

G'(r-s)K(y,s)ds

0

(9.25),

G(?-T)]

(9.27),

+ %K(yz,'j;)G(y+~)

we have (8.13).

and (9.28) (G(0) = 1 ) - G(~-T)K(T,~)%

Thus

(9.24) y i e l d s t h e G-L e q u a t i o n (8.13).

REWWC 9-13, The dependence i n d i c a t e d i n Theorem 9.11 between G and H again i n v o l v e s o n l y f i n i t e i n t e r v a l s ( b u t i n a d i f f e r e n t manner t h a n i n Theorem 9.7).

We n o t e t h a t t h e G-L e q u a t i o n i n v o l v e s G on [0,27]

71 w h i l e

(9.26) p l a y s o f f H on say [?,3?]

once G i s known on [ 0 , 2 7 ] , [0,4?]

a g a i n s t G on [0,4?].

d e t e r m i n i n g K and hence A on

by s o l v i n g t h e d i r e c t problem.

for

[O,y],

K(T,-)

on [O,

O f course we know G on

One hopes t o use (9.26) and (8.13) =

(9.29) i n c o n j u n c t i o n t o develop a numerical scheme f o r example based on f i x e d p o i n t ideas and t h i s i s discussed below.

TRANSMISSION DATA

295

L e t us t h i n k o f G now as odd and G ' as even (as i s n a t u r a l from t h e s i n e and c o s i n e r e p r e s e n t a t i o n s ) and w r i t e (9.26) w i t h (9.29) as f o l l o w s . v

T

> y,

for

H(T)

y,

T

=

(integrals 0 0 for (51 >

+ %!

% [ G ( T + ~ )+ G ( T - ? ) ]

+

Now t r e a t K(7,E) as an odd f u n c t i o n i n 5 w i t h K(Y,,S) = = 0 ( v i a t h e sine representation o f K before ( 9 .

;).

7 and K(7,O)

-

u

(9.30)

G'(~+s)]ds while

+ 4, K ( ~ , s ) [ G ' ( T - s ) - G ' ( r + s ) ] d s

The by an easy c a l c u l a t i o n

22)).

-

K(~,s)[G'(T-s)

= +[G(T+~)+ G ( T - ~ ) ]

-k(y,T)

[K(Y,

*

a )

I,'

G'](T) =

-

K(y,S)G'(.r-S)dS

loY

K(k,S)G'(T+S)dS

Consequently, s e t t i n g G ( ~ , T )= ~ [ G ( T + T ) + G ( T - ~ ) ] ,we o b t a i n

H(T)

(9.31)

=

-

T

rem 9.14,

G'

( T

(T

> 0)

'

> 0 one can combine t h e e q u a t i o n s i n (9.31) t o o b t a i n

*

K ( ~ , T ) = G ( ~ , T )+ ?iK(Y,.)

REmARK 9.19.

*

G(?,T) + +K(Y,*)

CHEOREIII. 9-14, F o r

H(T)

For

GI.

L e t us g i v e a somewhat n e a t e r d e r i v a t i o n o f t h e r e s u l t i n Theo-

and more p a r t i c u l a r l y o f (9.24),

f i r s t from v(y,t)

=

as f o l l o w s ( c f . [C78]).

>u w i t h v ( 0 , t )

=

Thus

10" SinXtC(h)dh/A

1

m

=

[ S i n x t / h ] q ~ ( y ) ( 2 x / ~ ~ ) G(T)Sinxrd-cdh =

;j

0

jOm G ( T ) ( ~ / I T ) l[ o m q ~ ( y ) S i n i t S i n i r d h ] d r

We w r i t e t h e n $ ( y ,t) = ( ~ / I T q!(y)Coshtdh )J~ t h a t f r o m (9.32)

%Io

=

>"

= B(y,t)

SO

m

(9.33)

v(Y,t)

W i t h G odd one has $(y,t)

=

-Jr

G(T)$(y,t+T)dT

$(Y,t+~)ld~ G(T)$(y,t-T)dT

=

and s i m i l a r l y

i s even i n t; one o b t a i n s t h e r e f o r e ( c f . Theorem 2.3.18)

In particular f o r t > (9.35)

-

G(T)[S(Y,lt-Tl)

H(t)

=

7

+[$(y",-)

*

If we w r i t e B(y,t) = A-%(y-t)

G]

+ K2(y,t)

=

S(y,t)

now and work w i t h (9.33)

296

ROBERT CARROLL

%lo m

(9.36)

v(y,t)

=

G(T)[A-~(y)s(y-lt-TI) + K 2 ( Y , l t - r l ) l d ~

-

m

+ K2(Y,t+~)]dr

G(T)[A-'(y)G(y-t-T) 0

%I

=

%A-+(y)[G(t-y)

+ G(y+t)

-

G(y-t)]

m

+

G ( ~ ) [ K ~ ( y , l t - ~- l )K 2 ( Y , t + r ) l d r

0

T h i s i s i n f a c t e q u a t i o n (9.24) d e r i v e d r a t h e r more n e a t l y and leads t o Theorems 9.11,

9.12,

and 9.14 (a d e t a i l e d a n a l y s i s o f Theorem 9.14 appears

below)

REIRARK 9.16. T <

The emergence o f t h e G-L e q u a t i o n from ( 6 ) i n Theorem 9.14 f o r

y" can be t h o u g h t o f i n terms o f c a u s a l i t y ( c f . [ B b s l ] ) .

t h i n k o f t h e k e r n e l K(y,x)

One can a l s o ( o r K?(y,x) perhaps) i n terms o f a downward con-

t i n u a t i o n o p e r a t o r , t a k i n g d a t a on t h e s u r f a c e and p r o o a g a t i n g them downward t o depth y.

I n t h i s d i r e c t i o n one c o n s i d e r s e q u a t i o n (9.26) f o r ex-

ample o r f r o m (9.34) and B ( y , t ) = S ( y , t ) (9.37)

v(y,t)

=

+ v(O,t+y)]

%A-'(y)[v(O,t-y)

!Ijy K,(y,x)[v(O,t-x)

+

+ v(O,t+x)ldx

0

L e t us r e t u r n now t o Remark 9.13 and r e f o r m u l a t e t h e m a t e r i a l i n t o t h e f o l l o w i n g computational scheme.

The general p l a n i s t o show t h a t t h e extended

G-L e q u a t i o n o f Theorem 9.14 a l l o w s one t o map G data on [0,2?] on [0,47]

v i a an e x t e n s i o n f o r m u l a f o r K(y,x)

t h e H e q u a t i o n i n Theorem 9.14 on [y",37]

t o G data

(Theorem 9.17) and t h e n v i a

one r e t u r n s t o G d a t a on [0,27].

One t h e n hopes f o r some s o r t o f convergence upon i t e r a t i o n ( o r a f i x e d p o i n t theorem) and we s k e t c h here some r e s u l t s i n t h i s d i r e c t i o n ; one expects m i g h t have t o be t a k e n v e r y small f o r convergence. showing t h a t knowledge o f K(y,x) (x.y)

(x c y ) .

y ) = A-'(y).

for 0 5 x 5 y 5

Q = Coshy R e c a l l t h a t pp,(y)

+

f{

7

We b e g i n w i t h a formula

7 determines

K(y,x)

K(y,q)ASinhndq w i t h 1

F u r t h e r B(y,t) =(qA(y),CosAt)v Q = A-%(y)&(y-t) + Kt(y,t)

for all

-

K(y,

so

f o r x < y (K(y,O) = 0 )

Consequently Q(D ) K = DZK and we w i l l a p p l y Green's theorem ( n o t e t h e i n t e r Y change o f c o o r d i n a t e d i r e c t i o n s ) Jfa (Q - Px)dydx = Jr Pdy f (jdx i n runY n i n g v a r i a b l e s ( q , c ) f o r (y,x) t o t h e r e g i o n D = D(y) shown below

297

TRANSMISSION DATA

'

-.,

(y-x,O) UB Y = K Thus we c o n s i d e r K + ( A ' / A ) K i n t h e form -Knn + KS5 = q(n)Kn,

+nn

55

d e a l i n g w i t h K(n,y-x-n)

fr

n

+

qKndSdn.

On AB we a r e

and dg = -dn; on BC, K = K(n,n-y+x)

w i t h dn = dS;

where q ( q ) = 0 f o r n > y, so

Kndt

Kgdn = -fD

on CE, K = K(n,y+x-n)

w i t h dg = -dn; and on EA, K = K(q,n) w i t h dn = d t . fr become r e s p e c t i v e l y B (Kg - K )dn = B D Kdn C C E n En DnK = (KA - KB); fB (Kn + K )dn = J D Kdq = KC - KB; $ (K - Kn)dn = 5 A5 B n A dn = - ( K E - K C ) ; and IE ( K + K )dn = fE DnKdq = KA - KE. Since KB = K(y5 0 x,O) = 0 we o b t a i n I, = 2(KA + KC - KE) and hence

/A

Hence t h e i n t e g r a l s f o r

CHEBRERI 9-17, F o r (y,x) (9.40)

-

K(y,x) =

1

-/A -/c

7 and

x 5 y ) one has

as shown i n (9.39)

(y >

4(A'/A)Kn(n,g)dgdn

+ %[A-'C%(y-x)I

- A-'[4(x+y)ll

D The i n t e g r a t i o n o v e r D o n l y i n v o l v e s t h e r e g i o n y 5 -

7) and

o f K(y,x)

t h i s f o r m u l a a l l o w s one t o compute K(y,x) for Y

y" ( r e c a l l

for

A = Am f o r Y

27 L y 2 7 i n

terms

5y.

Take now t h e G-L e q u a t i o n f o r

T <

y i n t h e form ( c f . ( 8 . 1 3 ) )

as known f o r 0 5 x 5 y 5 2yh v i a Theorem 9.17 ( r e c a l l G + Gr - t h e d e l t a f u n c t i o n s have t h u s been removed i n Gh). Now assume

and t h i n k o f K(y,x) = 1

G(t)

i s known f o r t <

27 and s e t y = 2y i n

N

(9.42)

-E(y,x)

= K(27,x)

2Y ~ G r ( 2 y ) K ( 2 ~ , 2 y - x ) + %iG;(

-

+I

(T =

x 5 27)

27- x

-0

G;(s+x)K(Zy,s)ds

+

N

\ s - x \ ) K ( 2 Y y s ) d s = -kA2Gr 2Y++x) - ?,,;~r(s+x)K2(2~,s)ds

2y-x a r e t h e n known w h i l e t h e G terms on t h e r i g h t Changing v a r i a b l e s ( s = 2?+ t - x ) we o b t a i n t h e n a V o l t e r r a

The terms on t h e l e f t Z(7,x) a r e unknown.

-%Gr(27-x)

(9.41) t o o b t a i n

i n t e g r a l equation for G(2Ytt)

2 98

ROBERT CARROLL

(9.43)

+4

E: ( 7 , ~ ) = %A?Gr(2?+x)

Gr(2y"+t)K2(2y",2y"+t-x)dt

for 0 < x < y" by t h e G-L e q u a t i o n and t h e n by Theorem 9.17 K(2y,x) i s known f o r 0 5 x 5 27) one

LEilUilA 9-18. Given G on [0,2y"]

295 u

can f i n d G ( 0 ) f o r Gr ( G = 1 + G

r

-

(which determines K(y,x)

< 4 y by s o l v i n g t h e V o l t e r r a e q u a t i o n (9.43) f o r

O< x <-Z y ) .

We go n e x t t o t h e H e q u a t i o n from Theorem 9.14 i n t h e form =

H(T)

-

-

y"

>

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

<

=

H(T)

Hence

and H on [Y,37]

-

( w i t h K(7,x) known

~G,(T+Y) + +K(~",zT-T)G~(zT) +

5 3 7 ( r e c a l l K(y,x) i s odd i n x w h i l e

y ) ; thus H ( T ) i s known.

Then G(x) on [0,27]

can be o b t a i n e d by s o l v i n g t h e V o l t e r r a t y p e e q u a t i o n (9.45) : (9.46),

-%

namely M ( T ) = %Am G , ( T - ~ )

+ % I T K2(y",T+y"-t)Gr(t-?)dt.

Estimates o f v a r i o u s s o r t s can be e a s i l y o b t a i n e d f o r s o l v i n g t h e V o l t e r r a equations i n Lemmas 9.18 and 9.19.

22

t o r(Gr)

For a f i x e d A one maps d a t a Gr on [O,

on [0,27] and one knows t h e r e w i l l be a unique f i x e d p o i n t Gr

I f one can determine a f i x e d p o i n t Gr by an i t e r a t i v e cornputation-

= r(Gr).

a1 scheme as i n d i c a t e d , s t a r t i n g w i t h some guess Gry t h e n A can be d e t e r mined by G-L techniques. realistic; for

I t remains t o see i f t h e computational scheme i s

7 sufficiently

small one expects convergence i n some sense.

A sketch i s i n d i c a t e d o f t h e r e g i o n considered.

(9.47)

u

=

I

G(t)

\

i

/

1

,/'

t = 3y

/

I

t = 2Y

y=Y"

ROBERT CARROLL

300

10, S0mf flZXELLANE0LIS; C0PZCS. We will a c t u a l l y deal only w i t h two unrelated t o p i c s here, namely, random evolutions and some ideas involving Darboux transformations. I t seems appropriate t o mention both s u b j e c t s i n any book such as this on general transmutation theory. We will deal f i r s t with t h e Darboux transformation. T h u s we consider connection formulas between solut i o n s of d i f f e r e n t i a l equations of t h e form y " + qy = 0 which a r i s e in various physical problems ( i n p a r t i c u l a r i n t h e study of black holes) and c i t e here e.g. [Cwl; Cacl; Dd2; D11; Wx1,Z; Sa14; Ldl; Mj1,2; Adl; Lf4; Cadl; Anl]. We follow [Wx1,2] (where t h e matter a r i s e s i n the theory of black holes) and require t h a t s o l u t i o n s y and z of ( = ) Ply = y " + qy and P2z = z " + Qz = 0 be connected by a formula (+) y = az +

f o r s u i t a b l e funcT h i s almost t o o general an idea since i f y,? ( r e s p z,?) are

t i o n s a and B.

l i n e a r l y independent s o l u t i o n s of Ply

=

BZ'

0 and P2z = 0 r e s p e c t i v e l y then i t

i s always possible t o find a and B such t h a t (+) holds w i t h 7 = a7 + B?' i n addition. In f a c t i f one writes W(z,?) = z?' - z ' ? = k # 0 this will be satisfied for

(A)

w

- yz']/k and

a = [y'i'

B = [zy

ways do t h i s and we want t o examine r e l a t i o n s and t h e a , when ~ this is done. There will be t r a l theory and Darboux transformations. Now putes y ' and y" i n terms of z, z ' , and z " and

- y?]/k.

T h u s one can a l -

between t h e p o t e n t i a l s (q,Q) connections here t o isospecgiven ( = ) and (+) one comi n s e r t i n g this i n (=) we get

There is then an immediate f i r s t i n t e g r a l here of the form (note B[Q'B + 2 248' - a"] = -a[@" + 2 a ' ] o r [ Q B + a2 + ( a B I - a ' p ) ] ' = 0 ) (10.2)

x

= a

2

+

aB'

-

a'B

2 + B Q

( y , y ' ) (where y ' = a z ' + a l z + B I Z ' -BQZ = (a' - 6Q)z + (a + ~ ' ) z ' ) h a sdeterminant A = [a(a + 8') - B(a' - BQ)] and when A # 0 one can invert M i n t h e form (em) Z = [ ( a + B ' ) Y - By']/x Another r e l a t i o n which a r i s e s immediately w i t h z ' = [(8Q - a ' ) y + ay']/h. We note a l s o t h a t the map M:

(z,zl)

.+

from (+) is exp[ / ( a / ~ ) l =y B(zexp( 1 (a/@)))'from which (10.3) and z(xo) could be obtained from (0.) i f x # 0. I f f o r example xo = 0 w i t h z ( x o ) = 0 we could w r i t e (10.3) as ( 6 ) z(x) = Jf K(x,S)y(S)dS where K(x,E) = exp[-lgx ( a / B ) d d / ~ ( S ) and this has t h e appearance of a transmutational 2 A 2 formu 1a. However i f q = + v and Q = Q + 1-1 t h e s p e c t r a l parameter p

4

DARBOUX TRANS FORMAT I ON

301

w i l l u s u a l l y be bound up i n K s o t h a t we cannot t r e a t t h i s s i t u a t i o n accordI n s t e a d we have a d i f f e r e n t k i n d

i n g t o previous theory o f transmutations.

o f t r a n s m u t a t i o n a t a d i f f e r e n t i a l l e v e l as f o l l o w s .

Set

A = BDx - a - 6 ' ; B = BDx + a

(10.4)

2 2 Then e v i d e n t l y ABf = 6 f " + (Ba' - a' - 6 ' a ) f w h i l e BAf = 6 f " 2 2 a6' + 66" + a )f. Hence u s i n g (10.2) f o r f E C

-

(60,' +

2 2 2 [AB+xlf = 6 2 f " + 6 Qf;[BA+Alf = 6 f " + 6 q f

(10.5)

2 2 Consequently B[6 (D + Q ) ] = B[AB + A ] = [BA + h]B = b2(D2 + q)B and t h u s CHE@REl!I 10-1- Assume y and z a r e r e l a t e d v i a y = az n e c e s s a r i l y s a t i s f y (10.1) w i t h f i r s t i n t e g r a l L e t A and

x

+ gz'

where

a formal t r a n s m u t a t i o n a c t i n g on f

C

+

+ q] i s

62[D2

.

Now t o determine B one t a k e s t h e s i t u a t i o n o f

(A)

from (10.1).

so t h a t 6 w i l l s a t i s f y

zy

t h e d i f f e r e n t i a l e q u a t i o n determined by p r o d u c t s one c o u l d d i r e c t l y e l i m i n a t e

and

= c o n s t a n t g i v e n b y (10.2).

B be d e f i n e d as i n (10.4) and t h e n B: 62 [D 2 + Q] 2

~1

o r y?.

Alternatively

This leads t o a f o u r t h order

e q u a t i o n (assume q f Q ) (10.6)

(Q-q)BiV

-

+

(4-q)""'

+ [(Q-q)(Q+q)"

2 2 2(Q - q )6" + [ Q Q 1 - q q ' + 5 Q q ' - 5 q Q ' ] 6 '

+ (Q-qI3 + q I 2 -

4'

2

4

I6 = 0

One notes f r o m (10.1) t h a t a s o l u t i o n o f (10.6) i s enough t o c h a r a c t e r i z e t h e mapping (+) c o m p l e t e l y s i n c e (*) a = - 6 ' / 2 + (1/2)[6"' B(Q'+q')]/(q-Q).

+ 6'(3Q+q) +

The case 6 = c o n s t a n t (say B = 1 ) i s o f p a r t i c u l a r i n t e r -

e s t and we see t h a t (10.6) w i l l have a c o n s t a n t s o l u t i o n i f ( t ) (Q+q)" + 2 (Q-q) - [ ( Q ' - q ' ) ( Q ' + q ' ) / ( Q - q ) ] = 0 ( t h i s i s n o t a necessary c o n d i t i o n however).

I n t h i s s i t u a t i o n (6 = l )we have f r o m (*) a = %(q'+O')/(q-O)

f r o m (10.2) w i t h

(t), A

= a2

+

%(Q+q).

I n t h i s s i t u a t i o n t h e commutation

6

i n Theorem 10.1 can be "expanded" i n t h e f o l l o w i n g way. L e t Q = + + p 2 w i t h y = y and z = z For B = 1 w i t h ( T ) we have =

q =

11

(4-G)

. u

independent o f 1~ and B = Dx +

and

does n o t depend on

11,

11'

and

%({'+ti)/

Hence B(z ) = 11

yu connects e i g e n f u n c t i o n s as i n t h e t h e o r y o f i n t e g r a l t r a n s m u t a t i o n s .

Another p o i n t o f view h e r e would be t o observe t h a t g i v e n B i n t h i s s i t u a 2 2 2 t i o n as i n d i c a t e d , i f ( D + Q)z = - 1 ~ zU t h e n B(D + Q)zv = (D2 + q)Bzu = 1J 2 - p Bz so t h e o p e r a t o r s P, and P2 would be " i s o s p e c t r a l " i f i n f a c t Bz = !J

P

ROBERT CARROLL

302

2 l i e s i n a s u i t a b l e L t y p e space a l o n g w i t h z . I f D f c o n s t a n t one 2 2 2 2u 2 2 2 c o u l d say t h a t i f B [D + Q]zU = -LJ z t h e n B[D (D + Q)]zu = B [D + q] u 2 Bz = - P Bz so we a r e f o r m a l l y i n t h e c o n t e x t o f t r a n s m u t i n g t y p e opera-

y

11

5

P

!J

t o r s (where 6 = x ) .

However t h e p a r t i c u l a r p r o p e r t i e s (e.g.

d i t i o n s ) characterizing t h e eigenfunctions z l a t e d o r spelled out.

LJ

and y

u

= Bz

P

i n i t i a l con-

have t o be r e -

L e t us n o t e here a l s o ( c f . (10.1)-(10.2))

Summarizing some o f t h i s one can say

CHE0REm 10-2- The f u n c t i o n 6 s a t i s f i e s a f o u r t h o r d e r d i f f e r e n t i a l e q u a t i o n (10.6) and when ( i ) h o l d s so t h a t B = 1 i s a s o l u t i o n one o b t a i n s a = % ( q ' + Q ' ) / ( q - Q ) and A =

a2

then formally isospectral.

+ $(q

t

Q); f u r t h e r t h e commutation o p e r a t o r B i s

I n g e n e r a l (10.7) h o l d s and Bz

k.

= y

i s a for-

ma1 t r a n s m u t a t i o n c o n n e c t i n g ( s u i t a b l e ) e i g e n f u n c t i o n s o f P2 = L2P2 and

-P1 = 2 P1. One s h o u l d mention here,for

REmARK 10.3-

h i s t o r i c a l purposes a t least,the

procedure o f Darboux i n [ D l l ] which i s r e l a t e d t o a method o f changing pot e n t i a l s i n [Cwl;

Sa14; Cadl; L d l ; F a l l .

Thus one c o n s i d e r s y " = (Ip+h)y

t By' w i t h A and B en(*) A " + A(P-$) + 2B'(Ip+h) + b ' = 0

and z " = (J/+h)z w i t h t h e same h and assumes z = Ay t i r e i n h.

One o b t a i n s ( c f . (10.1))

and 2A' + B(q-$)

+ B"

B cannot be o f g r e a t e r degree i n h t h a n A 2 2 and one o b t a i n s a f i r s t i n t e g r a l as i n (10.2) (**) A + AB' - BA' - B Ip 2 B h = F(h).

= 0.

Here

c o n s t a n t (say B = 1 ) t h e n (**) i s a R i c c a t i e q u a t i o n 2 f o r A and one s e t s e.g. A = -y'/y"; t h e n A ' = -?"/y" + (y'/y") amd A' - A ' =

If B

(y""/y") -

9 = F(h)

+

h = k

+

h so t h a t

7'' =

[tptk+h]y

t h e y e q u a t i o n f o r a d i f f e r e n t parameter hl = k+h).

as i n d i c a t e d and c o n s i d e r z = y ' y(?'/y") + h - hl]z = [q t h -2(logy")"]z ( n o t e (logy")" = 2 - A ' = ?"/y- (y"'/y") and f r o m (*) 2A' + ( q - $ ) = 0 here - a l s o 2(y"'/y")' - y"/y" = A2 + A ' w i t h A' - A ' = p + hl S O A2 + A ' +

l e t t i n g y and

z " = [?(l/y")"

hl + 2 A '

+

h - h, = 9 + 2 A ' + h ) .

-

satisfies

Now go backwards i n

-

7 be

y"

(i.e.

t o obtain

(y"'/y")'

=

F(l/y")" = h - h l = ~ +

Consequently z s a t i s f i e s an e q u a t i o n

w i t h t h e same h as y b u t a p o t e n t i a l J/ = P

-

2(1og!)".

This procedure can

be used t o change a p o t e n t i a l i n o r d e r t o add an a d d i t i o n a l e i g e n v a l u e (bound s t a t e ) and a r i s e s i n s o l i t o n t h e o r y as w e l l as quantum s c a t t e r i n g theory (cf. also [Ial]).

REmARK 10-4. One s i t u a t i o n where formulas o f t y p e (10.3) o r

(6)

a r i s e more

DARBOUX TRANSFORMATIONS

30 3

o r l e s s n a t u r a l l y here (without binding u p a s p e c t r a l parameter i n K) i n volves one s t e p parameter s h i f t i n g in s c a l e s of special functions. Thus 2 2 2 = D2 - (m -+)/x (recall f o r example take Q, = D + ((2m+l)/x)D with Gm[xm+’u] = xm+’Qmu). Set gm = xm+%Am R where “ “ R ( p , t ) = 2?’(m+l)c-mJm(r;)

Gm

( r ; = u t ) . Based e.g. on formulas l i k e Cv-l = C; + (v/z)Cv f o r Bessel funct i o n s one has ^Rm = (2m/t)[$m-’ - im] and ;2 = -p2tkm+’/2(rn+l). Similarly R = [2r(m+l)t-2m/r(p+l)r(m-p)]Io t ^R P ( v , n ) n 2p+l ( t2 - n 2 )m-p-.l d n . ( c f . [C40,63]) fm In terms of Gm we have 6,km = -p2F? and (u) ‘Rm-’ = (l/Zrn)~~+[(m-4)/2mx] (**) i ? ” ( p , t ) = 2mJot R“m-1 ( u , T ) ( T / t ) ” ’ - % T . Here we can t h i n k of (AA) via (*) w i t h a = (m-%)/2mx and B = 1/2m w i t h y = km-’ and z = ’im so t h a t ej. 2 2 2 2 For (10.2) we obtain A = p /4m and B = (1/2m)D q = u2 - [(m-1) - k]/x

‘i” a n d

.

2 2 + (m-%)/Zmx: B ( D + Q) 8*(02 + 9 ) . The r e l a t i o n ( W ) i s a form of (10.3) or ( 6 ) expressing z = km = I Ky where K(x,4) = exp[-It ( a / ~ ) d n ] / ~= 2m -f

In t h i s case then t h e d i f f e r e n t i a l t r a n s exp[-I, (m+)dn/q] = 2m(S/x)m-’. mutation B leads t o an i n t e g r a l transmutation o f a more o r l e s s standard type

(**I.

W e go next t o a b r i e f discussion of “random e v o l u t i o n s ” following [Hfl,5; Rd1,2; G g l ; Kcl; Kb1,21. I t turns o u t t h a t formulas of the type v ( t ) = E [ u ( T ) ] where E denotes expectation and T ( t ) i s a “random time” can represent transmutations connecting s o l u t i o n s u and v of d i f f e r e n t p a r t i a l d i f e will sketch t h e f o r m a l i t i e s and give some d e t a i l s f e r e n t i a l equations. W b u t will concentrate on examples. First following [Kcl], given a non-negat i v e continuous function a ( t ) , defined on [0,m), one may a s s o c i a t e w i t h i t a Markov process, c a l l e d t h e Poisson process N ( t ) w i t h i n t e n s i t y a as f o l lows. S t a r t a t zero, i . e . N(0) = 0, and f o r t > 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;

^u

= 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+<)lf; u ( x ) x 2k exp(-x 2/t4 t ) d x . Such connection

C29; Dhl-41 b u t t h e d e r i v a t i o n

formulas a r e known f o r example from [Byl-12;

by p r o b a b i l i t y arguments i s o f i n t e r e s t . One a l s o o b t a i n s i n [RdZ] a prob a b i l i t y d e r i v a t i o n o f t h e connection T ( t ) = [r(k++)/Jd?(k)]il 1 (1-5 2 ) k-1

7" +

v(Et1d.t f o r v " = Av and

(2k/t)?'

=

AT ( w i t h t h e same Cauchy d a t a ) .

REMARK 10-9- The i d e a o f s t o c h a s t i c i n t e g r a l must be g e n e r a l i z e d beyond t h e elementary n o t i o n s of 92 when

t h e i n t e g r a n d i t s e l f i s a random v a r i a b l e .

Thus g i v e n Wt a ( r e a l ) Brownian m o t i o n one wants t o determine a meaning f o r t (om) I t ( q ) = /O q(w,s)dW, ( c f . Theorem 2.2.6). E x p l i c i t l y l e t EWt = 0, E(At)Wt+s = Wt ( m a r t i n g a l e p r o p e r t y ) , and E(At)[Wt+s - Wt] 2 = 0 2 s ( t a k e 5 = 1 f o r convenience).

Here At T >

One needs v a r i o u s hypotheses about ~ ( w , s ) o f course.

+

c A i s a r i g h t continuous f i l t r a t i o n

(A(t) = A ( t ) = n A(T) f o r

t ) , As c A t f o r s 5 t, and q ( - , t ) i s t o be measurable on A t

l e d nonanticipatory r e l a t i v e t o t h e f a m i l y At).

Generally A

t

(q

i s cal-

C A i s a fam-

( = a l g e b r a s ) and one can deal w i t h p r o g r e s s i v e l y meas u r a b l e s t o c h a s t i c processes i n t h e sense t h a t (t,w) + X t ( w ) ( 0 5 t 5 T) i s i l y o f sub 5 - f i e l d s

measurable on 1:(O,T) X AT.

Now f o r o u r f u n c t i o n s cp as above w i t h say cp t 2 measurable on 1: X A , and l e f t continuous, w h i l e fo Elq(w,s)l ds < m a.s. (almost s u r e l y ) , one can d e f i n e It(cp). =

1 q v ( w ) l AV ( t ) f o r

F i r s t take simple functions q ( w , t )

...

p a r t i t i o n s 0 5 to5 tl 5

a r e A ( t v ) measurable can approximate q ( w , t )

(Ia

tn = T say where t h e cpov

denotes t h e c h a r a c t e r i s t i c f u n c t i o n o f A").

as above by a sequence

such t h a t l i m 17 E l c p ( w , t )

<

- q n ( w , t ) l 2d t

-+

0

Ipn(w,t)

of s i m p l e f u n c t i o n s

(7 b e i n g whatever

For such s i m p l e f u n c t i o n s one d e f i n e s I t ( q ) =

needed here).

One

interval i s

1cpv(w)

-

I f qn(w,t) + q ( w , t ) as above one d e f i n e s I t ( q ) = l i m W(tv)]. 2 Note here E ( I ( c p n ) l It(qn) where t h e l i m i t i s taken i n a mean square sense. n n n n = E[q cp AvWA W] and f o r p # v , AnW i s independent o f AnW so t h a t E I I (uq nw) l '''2 = E[llp,l ' n 2 (AyW) n 21 = E1q812E(A(tn))(AnW)2 = l E l' i gn v 2l An v t = [W(tv+l)

11

1

1

/?

E l q n ( . , t ) l 2 d t ( r e c a l l E(A(A:))(AvM)n" 2 = An" y t = tu+l H - t:). S i m i l a r l y 2 2 - q n ( - , t ) I 2d t E[I(qrntn) - I(qn)l = E I I ( I ~ ~- +q~n ) I = [(cpm+n(.,t) t 0 s i n c e cpn -+ IP i n mean square. Consequently t h e d e f i n i t i o n I t ( v ) = fo q ( w ,

JT

s)dWs o f

(om)

makes sense f o r f u n c t i o n s q as i n d i c a t e d .

Now one can show t h a t such I t ( c p ) E(As)It(q)

=

-+

I s ( q ) ( c f . [Wpl;

determine a m a r t i n g a l e i n t h e sense t h a t

Kacl; I c l ; I d l ; K v l ] ) and i n p a r t i c u l a r t h i s

i s not a martingaleshows t h a t I t ( W ) = j0t WT(w)dWT(w) cannot be Wt/2 2 (Wt/2 2 e x e r c i s e ) . Thus a n c o r r e c t i o n ' ' t e r m has t o be added and t h i s l e a d s t o t h e

307

RANDOM EVOLUTIONS

I t o calculus. via d x t ( w )

One defines an I t o process r e l a t i v e t o a Brownian motion Wt at(w)dt + bt(w)dWt

=

xt(w)

(10.12)

= x0(w)

+

( ~ 4 )

as(w)ds +

jot

ib

bs(w)dWs

For s u i t a b l e nonanticipatory At measurable random processes a t ( w ) and b t ( w ) (10.12) w i l l make sense ( c f . remarks above and 52) and t h i s defines (A+). Note here t h a t (A+) i s not t h e same as d x t / d t = at(w) + bt(w)dWt/dt f o r dWt/dt a white noise ( c f . Remark 3.4.2 - f o r a good discussion of t h s point s e e [Kacl] - we will i n d i c a t e t h e d i f f e r e n c e below). Now f o r an I t o process as above one defines a s t o c h a s t i c i n t e g r a l ( i p , ( ~ ) a nonanticipa ory process a s before - q , ( w ) = ~ ( w , T ) , e t c . )

where i t i s s u f f i c i e n t t o assume t h a t I? [qT(w)aT(o)ldT<

03

a . s . and

2

IT

E l q t ( w ) b t ( w ) l d t < m , a n d one can proceed here i n t h e d e f i n i t i o n via approximating sequences cpn + q of simple functions a s before). For the I t o

formula used before i n proving Theorem 10.7 we assume now $ ( t , x ) i s continuous w i t h bounded continuous p a r t i a l d e r i v a t i v e s $t, $x, and GXx. Then t h e $ ( t , x t ( u ) ) s a t i s f i e s (*=) dyt(w) = $ t ( t , x t ) d t + 2 $,(t,xt)dxt + ( 1 / 2 ) 0 $ x x ( t , x t ) b t d t = f i t ( t , x t ) d t + $ , ( t , x t ) a t ( w ) d t + $,(t,xt)bt(w)dWt t ( 1 / 2 ) ~ ~ $ , , ( t , x , ) b2~ ( w ) d t(0’ = 1 here). To s e e t h i s h e u r i s t i c a l l y w r i t e formally $(t+dt,xttdt) = $ ( t , x t ) t Jltdt + $ dx + 2 ZX Zt 2 2 (1/2)$xxdxt + .... From dxt = a t d t + btdWt we obtain dxt2 = a d t + btdWt 2 2 Y t ZatbtdWt and s i n c e EdWt = u d t we can take f o r approximation purposes i n i n t e g r a t i o n dx: ‘L u 2b 2t d t from which (*=) follows. Finally l e t us i n d i c a t e s t o c h a s t i c process y,(w)

=

2

t h e d i f f e r e n c e between a white noise i n t e g r a l and a s t o c h a s t i c i n t e g r a l i n t h e special case yt(w) = I0t q T ( W T ( w ) ) z T ( w ) d ~and & ( w ) = lot q , ( W T ( w ) ) d W T . Formally dWT = z T ( w ) d , where W i s a Brownian motion and zT i s a white i s defined a s above a s a s t o c h a s t i c i n t e g r a l . However yt has noise and not been defined y e t s i n c e t h e integrands f o r white noise i n t e g r a l s i n 92 were not random functions. I t i s shown in [Kacl] t h a t t h e s t o c h a s t i c i n -

yt

t

2

t e g r a l version o f y t i s ( 0 = 1 here) yt(w) = 10 qT(WT(w))dWT + (1/2)u 2 DwpT(WT)d~. T h u s dyt = qt(Wt)dWt + (1/2)0 Dwpt(Wt)dt i s 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 equation corresponding t o t h e white noise equation dyt =

rt

q

(Wt 1q t .

ROBERT CARROLL

308

If. EQ?.HEZ@NSj UIEH 0PERAE0R CbEFFZCIENEti, In t h i s s e c t i o n we sketch f i r s t t h e development of transmutation operators f o r canonical equations in Hil-

bert space following [Apl; Mi1,2; Mhl].

This i s a natural background t o t h e

more d e t a i l e d work i n [Ail; Du81, t o which we r e f e r a t f i r s t f o r relevant c a l c u l a t i o n s and from which we subsequently e x t r a c t some main l i n e s . Thus l e t H be a separable Hilbert space and J an operator in H such t h a t (*) J* 2

S e t P,- = [ I 7 iJ)/2 w i t h P, + P = I and J = [iP, - i P - ] ; the P, a r e projections and we w r i t e H, = P+H. Consider (**) JDrx - Vx = f where x(r) E H , f ( r ) E H, and V(r) E BTH) =-bounded l i n e a r operators i n 1 H ( V E Lm(O,m;B(H)) and eventually a l s o V L ). One assumes e.g. x loca l l y absolutely continuous and thinks of say Bochner i n t e g r a l s ; t h u s x ( r ) = 1 x(ro) + Jr k ( s ) d s w i t h x l o c a l l y L . Assume a l s o V(r) t o be s e l f a d j o i n t r, and consider t h e operator (*) A. = JDr - V i n L2 (0,m;H) defined on D ( A o ) = = -J and J

=-I.

{absolutely continuous vector functions x with compact support such t h a t 2 Aox E L (0,m;H) and x ( 0 ) = 01. One knows e.g. t h e r e e x i s t s a J unitary ope r a t o r U(r) E B ( H ) s a t i s f y i n g (11.1)

JDrU - VU = 0; U(0) = I

*

( J unitary means U J U = UJU

*

= J o r -JUJU

*

2

w i t h comNote t h a t i f

= I ) and g E L (0,w;H)

pact support is in R ( A o ) i f and only i f :1 U * ( r ) g ( r ) d r = 0.

g = Aof = J f ' - Vf (supp f compact) then using U-l' = -U-lU'U-l with U -U-'J we have formally :/ U*g = 1; U-IJVf + 1; U*Jf' w i t h 1; U*Jf' =

*

=

= -1; (U-')'f = 1; U - l U ' U - 1f = 1: U-lJVUU-lf = -1: U-lJVf. We note t h a t i t can a l s o be assumed t h a t J V = -VJ without l o s s of g e n e r a l i t y

-1,(U*J)'f

*

since one can always pass t o an operator G AoG where G i s a unitary opera+ P-VP-IG w i t h G(0) = I ( c f . [Apl; 6121). t o r s a t i s f y i n g -JDxG = -[P,VP+ 2 I t is easy t o see t h a t A. i n ( + ) is symmetric i n Jc = L (O.m;H) ( i . e . (Aof, g ) = (f,Aog) f o r f , g E D ( A o ) ) and D(A:) = Cx E Jc which a r e l o c a l l y absolut e l y continuous w i t h Aox E X I . By standard procedures ( c f . [Dsl]) one finds

*

symmetric extensions A of A. as r e s t r i c t i o n s of A. t o domains D(A ) = { X E P P D(A:); PHx(0) = x ( 0 ) o r x ( 0 ) E PHH = 31 where H C H i s a ( - i J ) - n e u t r a l sub* space characterized by [x,yl (= i[(A:x,y) - (x,Aoy)]) = ( ( - i J ) x ( O ) , y ( O ) ) = 0 f o r x(O),y(O) E H. In order t o have a s e l f a d j o i n t extension of A. i t i s necessary and s u f f i c i e n t t h a t dim H, = d i m H- and then PH = ( 1 / 2 ) ( I t K + * * K ) = P where K: H, + H- i s an isometry onto ( P = P ) . Note a l s o JP t PJ = J ( i . e . JP = QJ = ( I - P ) J ) i n t h e s e l f a d j o i n t case w i t h K*K = P, and KK* =

P-.

In general H = PH is characterized by PJP = 0.

The operator A

P

can

OPERATOR COEFFICIENTS

309

) on D(C ) = f i u n c t i o n s w i t h a l s o be characterized a s the closure of A ( = * P P P compact support i n D ( A p ) s a t i s f y i n g i n addition Px(0) = ~ ( 0 ) ) . Further

one notes t h a t D(A*) involves y ( 0 ) E HOJPH and s e l f adjointness r e q u i r e s PH = H 0 JPH ((J*h,R) = 0 f o r ?I = P h ) . Consider now ( = ) [Ao - X]U(x,A) = 0; U(0,X) = I w i t h U(x,A) E B ( H ) and X real momentarily; thus U ' + XJU + JVU = 0. Evidently U*JU = UJU* = J ( a s f o r U = U(x,O) above). Now w r i t e

@(x,X) = U ( x , h ) P (so (11.2)

ax)

=

@

satisfies

@(f,x) = P

(m)

with @(O,h)

[- U* ( x , h ) f ( x ) d x = [

= m

P ) and s e t *

@

(x,x)f(x)dx

'0

'0

2 K2 = IL with compact supportl. This function @ ( f , X ) i s a d i r e c t i n g functional in t h e sense of Krein ( c f . [Ael; Lql; Mil]) s a f i s f y ing i n p a r t i c u l a r the condition t h a t (Ao - X ) g = f i f and only i f @(f,X) = 2 0. To check this consider f o r example f E K and set g ( r ) = U ( r , x ) J q U* ( t , l ) f ( t ) d t . Then g has compact support and (Ao - A)g = J g ' - Vg - Ag = * * * f ( s i n c e g ' = U'J I U f - UJU f = U'J I U*f - J f and hence J g ' = J U ' J I U f + f = [XU f VU]J I U*f + f = f + Xg + Vg). Now f o r g D ( K ) we need g ( 0 ) * P = Pg(0). B u t g ( 0 ) = J I r U * ( t , h ) f ( t ) d t = [PJ + QJII," U f so f o r Pg(0) = g ( 0 ) one r e q u i r e s QJ J U f = 0 = @ ( f , x ) (note JP = QJ and i n [Mil] one * * * takes @ = QJ I U f o r @ = P J U f , t h e l a t t e r instead of JP J U f - i t does f o r H valued f

E

not matter here of course).

The s p e c t r a l theory can be b u i l t u p from these

considerations as i n [Lql] and one obtains ( t a k i n g say ? = @ = QJ J U*f, (f,g) = [F(dh)?((A),G(X)) where F i s a nondecreasing l e f t continuous funct i o n e t c . We will r e f i n e t h i s below.

/_I

Now s e t U O ( x , ~ )= exp(-XJx) and n ( x , h ) = Uo-1 U - exp(XJx)U(x,A) ( t h i s i s c a l l e d a normalized matrizant i n t h e f i n i t e dimensional s i t u a t i o n - c f . [DUB; 6121). From JU' - VU = X U we have U' + JhU = -JVU = VJU (note a l s o exp(XJ)V = Vexp(-hJ)) and

a' =

lo

exp(2xJ)VJn.

Hence

X

(11.3)

+

O(X,A) = I

e2xJcV(c)Js2(c,x)dc

Jt

(note a l s o a*JR = J ) . One can s e t no = I w i t h ~ + l ( x y h=) exp(2XsJ)V(s) Jsl.(s,h)ds t o construct (n = lm n.) i n a standard manner and s e t t i n g (A) J J CL.(t,X) = 1 ; e x p ( 2 a s J ) w . ( t y s ) d s lj 2 1 ) w i t h w l ( t , s ) = V(s)J one has ( 0 ) J J t for 0 5 s 5 t w i t h ~ ~ + ~ ( =t I, s V(u)Jw.(u,u-s)du ) J

S

(11.4) (w

=

a(t,A)

1"I w J. ) .

=

I +

Note here

,d

e2xsJw(t,s)ds t

n j + l ( t y ~= )f0 e x p ( 2 h ~ J ) w ~ + ~ ( t , s =) dI$s exp(2xsJ)

31 0

ROBERT CARROLL

V(s)J$

= Jot

exp(2AgJ)wj(s,E)dgds

:J exp(ZAJ(s-t))V(s)Jw.(s,S)dtds = 1;

J t t exp(ZAJq)V(s)Jw .(s,s-q)dnds = fo exp(2XJn)I V(s)Jw.(s,s-n)dsdn. Jt rl J wj+,(tys) = is V(uJJ1" w.(u,u-s)ds and hence

1:

i

(11.5)

W(t,S)

+

JV(S)

Jd

Thus

J

=

I:

V(u)JW(u,u-s)dS

EHZtNEm 11.1- W i t h hypotheses as i n d i c a t e d one can w r i t e (11.4) w i t h (11.5) and a bound nU(t,x)ll dim

H

<

Ptraod:

m

) where

< v ( t ) e x p ( l I m h l t ) can be o b t a i n e d (as i n [ D u ~ ] when

v(t) =

e x p ( l t llV(s)llds).

Note t h a t from (11.4) U ( t , A )

= exp(-AtJ)

+

and bounds f o r llwll can be o b t a i n e d i t e r a t i v e l y v i a exp(21ImAjs)).

Jt e x p ( A ( 2 s - t ) J ) w ( t , s ) d s (0)

( n o t e llexp(2xsJ) ll

<

=

Now n o t e t h a t exp(uJ) = P+exp(ip) + P-exp(-iu) Jot exp(A(Zs-t)J)w(t,s)ds = Jot exp(ih(2s-t))P+w

and hence, f o l l o w i n g [ D u ~ ] , /$ exp(-iA(Zs-t))P-w =

+

P+w(t,(t+u)/Z)exp(iAu)du + (1/2)1: P-w(t, ( t + u ) / Z ) e x p ( - i x u ) d u = P+w ( t ( t + u )/ 2 [P+exp ( A uJ + P-exp ( -A uJ 11 + ( 1/2) ii P-w ( t ,( t + u / 2 ) (1 /2 [P+exp(-xuJ) + P - e x p ( ~ u J ) I = (1/2)_Zt w+(t,(t-u)/Z)exp(-xuJ)du + (1/2)ii (1/2)J:

w- ( t ,(t+u)/2) exp( -xuJ )du where (11.6)

W+(t,(t-U)/Z)

+ P_w(t,(t-U)/2)P-;

- P+W(t,(t-u)/Z)P,

w- (t,( t + u ) / 2 ) = P+w( t ,( t + u ) / Z ) P _ + P-w( t, (t+u)/Z)P,

EHE0REM 11.2. Under t h e hypotheses i n d i c a t e d one can w r i t e U ( t , A ) = Uo(t,h) + 1: K(tys)Uo(s,A)ds where K(t,s) = ( 1 / 2 ) [ w + ( t , ( t - s ) / 2 ) + w-(t,(t+s)/2)] where w+ (resp. w-)

*

a r e t h e J c o m u t i n g (resp. J anticomnuting) p a r t s o f w.

T h i s i s o f course a t r a n s m u t a t i o n t y p e f o r m u l a and i t can be expressed DuS]).

somewhat d i f f e r e n t l y as i n [ M i l ;

First l e t

US

i n d i c a t e a model s i t -

uation t o c l a r i f y the picture.

EXAmPLE 11-3- For i l l u s t r a t i v e purposes one can t a k e (1 c o u l d a l s o be In)

(1

-;),

'

and P - = T ( -1~ i ) s o I =

P+exp(-ihx) + P-exp(iAx) = cOsxx (Sinxx cosxx

-VJ.

Also e.g. V = (

S

r, with

r -s

r* =

r, s* = s, and t h u s JV =

*

F u r t h e r one checks e a s i l y t h a t PJP = 0 and JP = OJ w i t h P

PHI-JPH.

I n a d d i t i o n ( & ) (P

-

Q)Uo(-x,X)

= UO(x,A)(P

sume ( & ) h o l d s now i n t h e r e m a i n i n g d i s c u s s i o n .

-

Q).

= P and

We w i l l as-

It c l e a r l y extends t o 2n

OPERATOR COEFFICIENTS

31 1

dimensions and thence t o any separable H e s s e n t i a l l y as a c h o i c e o f b a s i s

-

Note a l s o t h a t H+ = (1/2)[H

vectors.

[ H t JHJ] = P+HP- + P-HP,

JHJ] = P+HP+

P-HP- w i t h H- = ( 1 / 2 )

f

g i v e s a decomposition i n t o J commuting and J a n t i -

JH+ = (1/2)[JH + HJ] = (1/2)[H - JHJIJ = H+J).

commuting p a r t s (e.g.

Now assume t h e p r o p e r t y (4) and w r i t e (++) ( c f . [ D u ~ ] ) K ( t , s ) = K(t,s)

- Q)

K(t,-S)(P

(1/2)[W(t, ( t + S ) / 2 ) + w ( t , ( t - s ) / Z ) ] P

=

w ( t , (t-s)/Z)]JQ. (11.8)

+

-

+ (1/2)J[w(t,(t+s)/2)

Then an elementary c a l c u l a t i o n y i e l d s

+

U(t,A) = U o ( t , h )

I

t, K(t,s)Uo(s,h)ds

0

L e t us n o t e i n ( M )t h a t one can w r i t e 2 ? ( t , s ) K2 = WP

-

JwJQ (w = w ( t y ( l / 2 ) ( - ) ) ) .

= K1 t , t + s )

E v i d e n t l y from P

*

= P,

and JP = QJ one has PJ = JQ and - P = JQJ. Thus JK1 = JwP -wPJ - JwJQJ = -wJQ

+ JwP and JK2

=

+

JwP

wJQ = K2J = wPJ

+ K ( t , t - s ) and *2 * Q

= Q, J

-J

=

- wJQ = -KIJ = - JwJQJ = wJQ+

We w i l l n o t check a l l t h e c a l c u l a t i o n s here b u t i t f o l l o w s now from

JwP.

[Du8; M i l ] t h a t

EHEBREIII 11-4, Under t h e hypotheses i n d i c a t e d p l u s

(4)

and smoothness on V

as needed one has (11.8) w i t h ( W ) and t h i s r e p r e s e n t s a t r a n s m u t a t i o n JD, +

JDt - V ( t ) w i t h V ( t ) = J ? ( t , t ) A

o f a "Goursat" problem JDtK

-

- t(t,t)J.

appears as t h e s o l u t i o n

Thus

n

A

A

V ( t ) K = -DSKJ w i t h K(t,O)

= 0. A

Ptaod: = -D,KJ

I JDtKUo

=

AU

A

VU = h U O + h I KUo t VUo t V f KU

o b t a i n ( u s i n g (*=)) J?(t,t)Uo -

-

V(t)K

1 KUo one o b t a i n s JDtUo + JK(t,t)Uo + A

+

4

-

A

A

and t h e n from U = Uo +

A

t)

Thus one expects (*=) JDtK

We s k e t c h some o f t h e d e t a i l s .

A

I DSkJUo =

A0

.

x 1 KUo

Since JDtUo = XUo we

+ VUo f r o m which [J^K(t, n

^K(t,t)J]Uo + ?(t,O)J

= 0 (cf.

[Du~]).

= V(t)Uo.

Various arguments t h e n produce K(t,O)J

The c a l c u l a t i o n can be f o r m u l a t e d a l s o w i t h (*=) n o t as-

sumed and one o b t a i n s [ J k ( t , t ) - ^K(t,t)J

-

V(t)]Uo + i(t,O)J + f

-

[(JDt

A

V)K + DsKJ]Uods = 0.

Riemann-Lebesgue t y p e arguments y i e l d t h e formula f o r

A

A

V i n terms o f K ( t , t )

p l u s K(t,O)J = 0 and subsequently one can deduce

REmARK 11.5- W r i t i n g A. that (I + =

( I + *K)

t)A:

-'

= Ao(I

= JDt

+

-

V and A:

t) working

(*=).m

= JDt one sees i n obvious n o t a t i o n

on s u i t a b l e o b j e c t s and we w r i t e I +

(Neumann t y p e s e r i e s f o r example) w i t h ( 1 + " L A o = A:(I

The s p e c t r a l t h e o r y can now be presented i n v a r i o u s ways.

I n [ A p l l for

ample ( c f . a l s o [Du8; Mhl; M i l ] ) one takes @ = UP w i t h R = U i ' U and s e t s

G ( x ) = l i m Q ( x , h ) as x (11.9)

G ( h ) = I+

-+

m

so by (11.4)

Im e2xsJw(m,s)ds 0

=

nm(X)=

A(h)

2

+ ?).

ex-

31 2

ROBERT CARROLL

1 L (H) a l s o ) . One can define on PH (Am) A(A) = [PG*(x)G(X!P]-' and note a l s o t h a t OIJn, = I . A Let L 2 ( H ) , H = PH, be determined by the norm IIfllA = [ ( l / ~ ) i z(A(A)f(A), 2 A f(i))HdA] ( i n f a c t L ( H ) = L2(H) s i n c e IIAII i s bounded above and below i n dependently of A ) . One has then ( s e e (11.2) f o r ?(A) = @ ( f , A ) )

where w ( m , s ) can be assumed t o make sense (assume here V

E

W i t h hypotheses a s indicated f ( x ) = (1/1~)i; @(x,A)A(A)?((X)dA

UE0REm 11.6,

f o r f E L2(H) and

fm

0

IIfl12dx = ( l / ~ ) j : (A(A)?,?)dA.

REmARK 11.7- We will i n d i c a t e here some of t h e ideas which can go i n t o a proof of this theorem; a l l d e t a i l s a r e not s p e l l e d out b u t t h e e s s e n t i a l i s present. I t i s p r o f i t a b l e here t o use the technique of [Du8; Ail] which r e f i n e s t h e approach of [Mil; Apl; Mhl] although dealing only w i t h f i n i t e dimensional s i t u a t i o n s . T h i s i s achieved i n p a r t i c u l a r by s p l i t t i n g o f f t h e space PH i n a 2n X 2n s i t u a t i o n t o deal w i t h 2 blocks of n X n matrices. T h u s in Example 11.3 t h i n k of P w i t h I n 2, 1 and U = so t h a t UP % ( AB ) Then one can work w i t h X i n H o r U i n PH equivalently. First however = X.

(i k)

l e t us record a few formulas. We r e c a l l JDxU ** J)VJst). Set now F# ( A ) = F ( A ) and one obtains t # # V;]: ,U, d s while fo (JU;) , U, = (JU)A U,,*\h # U, J Uu* + fot UA[u*Uu* + VUv,]ds ( s i n c e J" = -J #

J - U A JU1-I" = ( h - u * ) i

(11.10)

In p a r t i c u l a r f o r #

-JU J .

#

U, Up,

ds

# one has" (*) J - U (t,A)JU(t,A) = 0 and U-l = and u* = w one has a l s o

p* = A

Setting A =

W*

[J-UI JUU]/(w* -

(11.11)

t

VU = A U ( and 9 ' = exp(2Ax t A:/ Uf lJHAds = JO [JU' 1; (JU)x UIu* d s = J and (A*)* = A ) . Hence

W)

=

(S,w)U(S,w)dS

20

Consider now (11.12)

Au(A) t =

CJ -

UAJUu,]/2n(A-~*) #

One can c a l c u l a t e w i t h such o b j e c t s as reproducing kernels a n d r e l a t e the matter (following [Du8; Ail] d i r e c t l y t o t h e idea of deBranges spaces e t c . e will do this d i f f e r e n t l y below ( a s i n [ D u ~ ] ) b u t will ( c f . [Dkl; D u l l ) . W i n d i c a t e a few of the relevant formulas here h e u r i s t i c a l l y . T h u s one notes t h a t i f we s e t J = i J then s i n c e U i s a l i n e a r operator one has J U!J"UA = 0 while (11.11)' U c :Uw]/i(w*-w) 2 0. Then one can consider i n place of (11.12) N

N

(*)I

[T-

(11.13)

"t Ap(A)

-

= [J

- U,# JUu,]/2ai(X-~*) -

OPERATOR COEFFICIENTS

31 3

8

w r i t e ( c f . Example 11.3) ? = ( 1 / 2 ) ( I + % P- and = (1/2)(I % A C so t h a t i f one had a b l o c k decomposition U = ( B D) then, s e t t i n g E = A

i)

NOW

P,

iB, F = A + iB,

E F (1/2)(iE i r )

?=

C

-

F i D , F = C + i D , one has ’i;U = (l/2)(-iF v

4” -

G* = 3 w i t h ?#

-

-

Up and

p) = 6 G- = U t

(11.14)

= P.

kJu-)#* - # (GU -

kA)#(v-

(GU,,

-

F

-

-ir) and QU = Y

(we w r i t e b l o c k s t o compare w i t h t h e formulas i n [Du8] which r rrly -2 Note t h a t P - Q = J = i J , P = F, g2 = Q, PQ = 0 = GP,

i s f o l l o w e d below). and <# =

J”,

-

P + UA(P

-

Consider now ( n o t i n g t h a t (G#)* = G(w*)): -

#

-#*

w

P)A(QU - P )

-1-I

N

Q)Uu,

=

-[J

-

=

(f #-

-

U

~

optu*) ( ~ - (Ufr-p)

~- )

N

Hence w r i t i n g G,

UAJUu,].

Q

-

we have

- G#(A)Gx*(u)]/[-2.i(h-u*)]

W A ut ( 1 ) = [GZ(A)G!*(u)

Then f o r m a l l y one c o u l d c o n s i d e r c e r t a i n spaces o f v e c t o r f u n c t i o n s a scalar product

=

(

f,g)z =

(

Gt-1f,G!‘1g)L2

and x t ( A ) -t and h E H, (f(A),A,(A)h)-

7 under

i s s a i d t o be a r e p r o -

ducing kernel i f f o r f E = (f(u),h). We n o t e ,.. ## H# f i r s t t h a t s i n c e J - UJ,U, = 0 we must have G+G+ = G-G- (A = u* i n (11.14)) -1 #-1 - G-lG!-l and hence G, G+ . Consider t h e n f o r m a l l y ( t a k e u* = w and A i s r e a l so G# = G* f o r t h e

I

m

(11.15)

x

terms)

*-l-t (GT-’ f ,G, All (A ) h )dh =

)I

W

- (1 /hi

(G:-l

f ,G+( w ) h )dA/ ( A-w )

m

W

(G~-’f,G_(w)h)dh/(x-w) ( s i n c e GilGTmlG

# i n GT-’ o = p*)

(

- G*G - ( w )

= GIIG-

*-1

*

G-G-(w) = GIIGm(w)).

Now r e p l a c i n g

*

by

and G T - T one o b t a i n s by a c o n t o u r i n t e g r a l ( n o t e t h e p o s i t i o n o f f,xth>

u

= (f(u),h).

R€mAaK 11-8, L e t us r a t h e r go now t o [Du8] and use t h e n x n b l o c k s A,B, F A 2 2 2 A + iB, E = A - i B , and X = ( B ) . D e f i n e f o r s u i t a b l e f E K ( K = L ( H ) w i t h compact s u p p o r t - RZn (11.16)

A

f (A) =

%

H here and f

lom

X#(s,x)f(s)ds;

where Xo = UoP = (CoshxIn,SinAxIn) 0

?(A)

Q

(fl,f2) as a column v e c t o r )

=

as a column v e c t o r (Uo = exp(-AJx) w i t h

1 on)

- c f . Example 11.3). Note a l s o V ( R - SR, w i t h R* = R, S* = n S and f r o m (11.8) w i t h X = UP, X o = UoP one has e.g.

J

=

(-I

The i n v e r s i o n formula we w i l l o b t a i n below, f o l owing [ D u ~ ] , i s t h e n

=

314

ROBERT CARROLL

m

(11.18)

IT)

f(x) =

X(s,A)A_(A)?(h)dX -m

( = [F # ( t , ~ ) F ( t , ~ ) ] - l ) and Am = l i m At as t

w i t h At(A) = [E # ( t , ~ ) E ( t , h ) ] - ’

L e t us n o t e here i n t h e model s i t u a t i o n d e s c r i b e d i n Example 11.3, UP =

m.

A

+

*

0

( B O ), PU U*JU).

A*

Thus Am

notation (X,X)

*

B*

(o

=

A in

Q,

*

), and PU UP = E E (A=)

(note G

=

A *A + B*B ( s i n c e B*A

R and

Q,

n*s?= U*U).

o r X*X i s used i n t e r c h a n g a b l y a t times.

= A*B f r o m J =

We remark a l s o t h a t The Parseval formu-

l a a t l e v e l t i n t h e form

extends then t o t h e corresponding f o r m u l a w i t h t =

m.

One c o n s t r u c t s U e t c .

as b e f o r e and ( r e f e r r i n g t o Lou81 f o r d e t a i l s ) t h e e n t r i e s i n U(t,h)

a r e en-

t i r e f u n c t i o n s o f A o f e x p o n e n t i a l t y p e t w i t h t h e n x n b l o c k s A,B,C,D

in-

One r e f e r s t o Wm (Wiener a l g e b r a ) as t h e s e t o f m x m m a t r i x 1 f u n c t i o n s F ( A ) = c F + S F ( h ) where c F i s c o n s t a n t and kF E L~~ .(R) =

vertible.

(^kF

,f

I f cF = I Thus cF = F(o0) and kF = F - l [ F - cF]. I and bl, ( r e s p . W-) r e f e r s t o F such t h a t f - l F ( x ) = 0 f o r x I 1 c 0 (resp. x > 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 <

m

-

are

e q u i v a l e n t l y , f j ( h ) = JII~J(X)

Now w i t h a deBranges p a i r (E,,E-)

as i n d i c a t e d

one a s s o c i a t e s a deBranges space B(E+,E-) o f n x l v e c t o r valued e n t i r e func 2 t i o n s f such t h a t E;f’ E H o m 2 . One has t h e n ( c f . [DuS])

OPERATOR COEFFICIENTS

31 5

LTHE6REm 11-9, Let ( E + , E - ) be a deBranges p a i r . Then B = B(E+,E-) i s a reproducing kernel space r e l a t i v e t o t h e inner product ( f , g ) = /: (E;lf,E;lg)

- E - ( x ) E ~ ( w ) ] / [ - P I( TX - ~W * ) ] B and f o r f E B,(f,Auh)B = ( f ( w ) , h ) ) .

d h w i t h reproducing kernel AW(X) = [ E + ( ~ ) E : ( w )

(i.e. for h

E

H

Q

Cn

1,

PH,AWh

E

# REflARK 11-10- In t h e present s i t u a t i o n one w i l l have a deBranges p a i r ( F , # # # # # E ) = ( A - iB , A t iB ) ‘7r ( E + , E - ) .

P J L V O0~6 T h m m 11.9: One r e f e r s here t o [Du8; Ail] and we note t h a t the

construction (11.15) i s based on t h e present proof (only a sketch of which seems necessary h e r e ) . One can show f i r s t t h a t (C = EilE-) E;’AW(X)h = [I - C ( X ) Z * ( W ) ] E : ( W ) ~ / [ - ~ I T ~ ( ~ - U * ) ] E H 2 a n d i s perpendicular t o W z f o r E

C+.

Indeed f o r

11 E

Cn

(q

z ( ~ ) C * ( ~ ) E : h l / [ - Z n i (h-w*)ldX

and g

E:(W)h)

‘L

=

if

H

E

2

one has

iz

w

(Zg(X),[I -

( [ I - z(~)z*(X)]/[2~ri ( x - ~ ) l Z g , r ~ ) d=x

(/z[[C(X) - z(w)l/[2.i(X-~)]gdX],h) = 0 f o r w E C+ and one can extend t h e 2 2 r e s u l t t o w E R. Next one shows ( c f . [ D u ~ ] ) t h a t E;’f E H Q W 2 = H n 2 2 2 i f and only i f f E E+H n E-K and arguing as above f o r E I I A W ( X ) h with # w E C- one obtains A W ( X ) h E B f o r w E C . F i n a l l y using E+Ef = E-E-, one obtains ( c f . (11.15))

(11.21)

(

f,AWh)B= (

[E;’(h)f(x)]/[2.i(x-w)ldx,E:(w)h)

(jrn[E--l ( x ) f ( x ) l / [ 2 n i (h-w)ldX,E*(w)h)

-

(f(w),h)

=

-m

That B is a Hilbert space i s e a s i l y v e r i f i e d . # # Now ( c f . Remark 11.10) one can use ( F , E ) a s ( E + , E - ) space Bt so t h a t ( r e c a l l F#* = Fp, 1-I

(11.22)

A:(X) [X#( t,

and c f . (11.10))

- E # ( t , A ) E #*( ~ , u ) ] / ( - ~ I T ~ ( ~ - L I=* ) ]

= [F#(t,X)F#*(t,p)

1J X # * ( t

t o define a deBranges

)I

t

,LJ )l/[-~r(X-u*)I

(l/v

=

X#(s ,A ) X(s , v * ) d S

0

( r e c a l l PJP = 0 ) . Define now f* as i n (11.16) and consider f of t h e form * # Then ( 1 / ~ I) X ( s , p j ) c j f o r s < T while f = 0 f o r s 2 T. (11.23)

# ( I ) = ( l / i ~ ) X#(s,x) r

1X #*( s , p j ) c j d s

=

1 AT

0 A

2

I t follows from Theorem 11.9 t h a t IIf IIB =

(1. A T J

(Ap T (pk)cjYck) =

j

f(s)ds,ck) =

T

v.f0

’5

(h)cj

T

1kA, k 5 1-1~ j’ 5

1j,k((l/T).fOT x # (Sy1-Ik)X#*(s,1-Ij)dsg.,cJ k 2 ( f ( s ) , f ( s ) ) d s = d f l l . One shows ( c f .

=

)

‘77

BT =

1k

1j , k

10T ( x # ( S , p k )

[ D u ~ ] ) t h a t such f

31 6

ROBERT CARROLL

a r e dense w i t h s u i t a b l e a p p r o x i m a t i o n p r o p e r t i e s and t h a t t h e range o f t h e map f

A

-+

f

i s BT so t h a t

A 2 ME6REM f L 1 1 . The map f -+ f determined by (11.16) maps L (0,T) + BT 1-1 2 2 o n t o w i t h ll?llB = d l f l l L 2 and t h e i n v e r s e map i s g i v e n by (11.18) w i t h Par-

One has a l s o A m ( A ) = I

seval f o r m u l a (11.19). 1 h e L .

Pkoob:

- :(A)

(> 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+<)]. i n g t o (l/n)L:

SinhGdx.

terms (Sinps,Cosug ,,)

In f a c t f o r

?I even

4

.-+

K ( s , c ) where hij

-

= 0 f o r i = j and hll

g1(s+c)]/2,

such terms a r e zero, as a r e

and we have

%'HEtBEIII 11.17- Given h even t h e m a t r i x A(s,c) [gl(s-s)

=

+ C o s ( s - c ) ~ l ~ d u . Now w r i t e '

has t h e f o r m 6 + ( s - ~ ) I -

= [g,(s+~)

g1 b e i n g t h e r e a d o u t f r o m

We n o t e t h a t t h e formulas o f [ D u ~ ] g i v e now f o r even) 3C(t,s) = K , ( t - s ) + ? ( t + s )

(!('I- h ( u )

).

?I

+ gl(s-s)]/2

even (which i m p l i e s h i s

w i t h K , ( u ) = (o h ( u ) h(:))

Since h ( u ) = (l/Zv)L:

w i t h hZ2 =

( ~ m ) .

and Jc,(u) =

i e x p ( - i A u ) d x = (1/2n)I:

hoshudx =

OPERATOR COEFFICIENTS

=

(1/2)g1 ( u ) t h e e n t r i e s i n Jc and

2will

32 1

c o i n c i d e and o u r c o n t r i b u t i o n here

i s t o i d e n t i f y h ( u ) w i t h a r e a d o u t impulse response ( l / 2 ) g l ( u ) . though d i f f e r e n t d e l t a f u n c t i o n s were used i n f o r m i n g 6

-

Thus even

3c and 6 - % t h e h

A

r e s u l t i n g G-L equations f o r K and K a r e t h e same so one i d e n t i f i e s K w i t h K ( i . e . o p e r a t i o n a l l y t h e 6 f u n c t i o n s p l a y t h e same r o l e and l e a d t o t h e same equations).

L e t us w r i t e o u t now t h e e n t r i e s i n t h e G-L e q u a t i o n as f o l -

lows, u s i n g Theorem 11.17.

i k12h22ds, 1 kZlhllds, kll - hl, - f kllhlldS and kz2

-

h

h

-

= JK(t,t)

2;

- f

The t e r m I K H d s i n v o l v e s e n t r i e s

1 kllhllds, 1 kZ2hz2ds and we w r i t e o u t t h e equations as 0, k12 - I kl2hZ2dS = 0, kzl - f kZlhlldS = 0, and

=

k22h22ds = 0.

K ( t , t ) J so a t ( t , t ) ,

o n l y kZ2 and kll

On t h e o t h e r hand f r o m Theorem 11.4, V ( t ) Thus - kll.

0 = k21 + k12 and r = kZ2

a r e needed f o r computation o f r and we c o n s i d e r t h e n o n l y

(for 5 < x) (11.35)

0=k11(x,S)

k2z(xSS)

-

-

c c

[ g l ( ~ + S ) + g l ( ~ - S ) I / 2- %

4[g1 (x-SI-91 ( x + S ) I - 4

k l l ( x , s ) [ g 1 (s+S)+gl(s-S)Ids;

k 2 2 ( ~ , ~ ) C g(1~ - 5 )- 91 (s+E.)Ids = 0

CHEaREm 11-18. The t i m e domain G-L e q u a t i o n e n t r i e s (11.35) determine kll v i a t h e known readout i m and kZ2 (and hence r ( t ) = k Z 2 ( t , t ) - k l l ( t , t ) ) p u l s e response gl ( t ) . We see t h a t ( w i t h

?I

even) t h e i n v e r s e problem t o determine r (and hence Z )

i n a system c o n t e x t can be handled v e r y much as i n a s c a l a r c o n t e x t .

One

can a l s o t r e a t problems w i t h t r a n s m i s s i o n d a t a as r e a d o u t ( c f . 59) b u t we o m i t t h i s here ( c f . [C76]).

REmARK 11.19.

L e t us examine i n more d e t a i l t h e r e l a t i o n o f t h e canonical

system t o t h e corresponding second o r d e r problems and compare s p e c t r a l ideas. V V Thus c o n s i d e r (JDx - !)(,) = A(,) = ((vij))) i n t h e form Ix - r I = AV and - V x

-

r V = XI. Equivalently

terms o f v = V?*

and 7 = IZ-'

- X 2 7 w i t h Z ( Z - ' V ~ ) ~= - h 2 v.

(1 (VZ'),

=

-xIZ4

and (IZ-')

x = xVZ-~.

In

we have v x = - 1 Z 7 and 'I, = xvZ-l so (Z1x)x/Z = The c o n d i t i o n X(0) = ( AB ) ( 0 ) = ( o 1 ) corresponds

t o v ( 0 ) = 1 and 7(0) = 0 ( n o r m a l i z i n g w i t h Z(0) = 1) t o g e t h e r w i t h v x ( 0 ) = - h I ( O ) = 0 and 'Ix(0) = x v ( 0 ) = A . = 1 w i t h Vx(0) = - ( l / Z ) Z ' ( O )

Note however from V = Z-'v

and s i m i l a r l y I x ( 0 ) = A .

f o r v ( r e s p . 7 ) can be based on cp f o r Z - l §2.6 (assume h e r e Z c p e x p ( - i x x ) as x F(X)e-ihx]/2i

+

Zm r a p i d l y as x

-f

-f

m

and dv = d x / 2 n l c p l

as x

-f

m

m

we have V(0)

The s p e c t r a l t h e o r y

( r e s p . F f o r Z ) as i n §1.8 o r etc.).

2 while (I

w i t h du = 2 d h / n l F I 2 .

Thus Zz5v = c p e x p ( i x x ) + ihx % -AX) ? I': = [F(-h)e

We n o t e a l s o t h a t f o r V o r I

ROBERT CARROLL

322

t h e d i f f e r e n t i a l equations a r e d i f f e r e n t ( c f .

i f one wants t h e spec-

(A*));

t r a l t h e o r y f o r say V one can r e f e r t o 53.8,

Remark 3.8.8

and V ' ( 0 ) = h b u t I ( 0) = 0 w i t h I ' ( 0 ) = A ) .

Now based on p r e v i o u s h a l f l i n e

t h e o r y we have t r a n s f o r m s (u) F(A) = :/ v(x,A)dv;

/om

=

$(A)

2 .

g(x)Z(x)T(x,A)dx;

( n o t e V(0) = 1

f(x)Z-l(x)v(x,h)dx;

and g ( x ) =

f ( x ) = / , ?(A)

/0" ;(A)I(x,A)du

where dw

dA/2n1cpl i s based on t h e v e q u a t i o n ( w i t h 2-') and dp = 2 d X / \ F I 2 n i s based on t h e 'I e q u a t i o n ( w i t h w e i g h t Z). Now observe t h a t f ' = / , T v ' d v = /rF(-AZI)dv w h i l e f ( A ) = /om f Z - 1vdx = 10" f 7 dx/A = f'7dx = (-l/A)

-

(Z-1f')4.

-(l/A)/r

X Hence f ' Z - l = 10" ( f ' Z - 1 )" f d u and g i v e n t h a t f and g a r e general

/0" ( f ' Z - l f 7 d p .

f u n c t i o n s i n (u) we w i l l have a l s o ( f ' Z - l ) =

tHE@RZm 11-20. L e t dv

Hence

= dA/2nlcp12 be based on t h e v e q u a t i o n w i t h w e i g h t

2

= Z - l (and e i g e n f u n c t i o n s v) w h i l e dp = 2dX/n\FI i s based on t h e 7 equaP t i o n w i t h w e i g h t Ap = Z (and e i g e n f u n c t i o n s 7 ) . Then dv = dp.

A

Now c o n s i d e r t h e i n v e r s i o n f

R€mARK 11.21-

+

f

A +

A

f based on X = ( B ) and

Here A V = Z%, B I = f $ . I f one takes f = dw = Amdx/n on (-m,-). f A A ( o l ) t h e n we o b t a i n f l ( x ) = fl A(x,A)dw where fl = IF fl(x)A(x,A)dx. Q,

Q,

LI A

Thus s e t A = 2%

/I;f

vdw.

and we have fl =

/Om

(flZ')Z-'vdx

Consequently we can i d e n t i f y dw on

Am i s even w i t h

2

2Am(A)/n

.

=

2

1/2nlcpl ).

/f

= (flZ?i)(-m,m)

while

flfi

w i t h dv on [ 0 , m )

=

(i.e.

S i m i l a r l y by Theorem 11.20 ( 2 / n )

The way i n which fA(h)X(x,X)do d i s t i n g u i s h e s between Z/nIFl A A A A f, and f2 i s v i a t h e f a c t t h a t fl i s even a l o n g w i t h A(x,A) w h i l e f2 and

Am(A) =

B(x,A) a r e odd.

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wu YE ZB

zc ZD

WONHAM, W. 1. SIAM J o u r . C o n t r o l , 2 (1965), 347-369 2. SIAM Jour. Control , 6 ( 1 9 6 8 ) , 681-697 WILLEMS, J. 1 . IEEE Trans. AC-16 (1971), 621-634 WIMP, J . 1. Glasnik Mat., 6 (1971), 67-70 2. Proc. Edinburgh Mat. S a c . , 14 (1964), 33-40 WHITING, B. 1 . D i f f e r e n t i a l Equations, North-Holland, Amsterdam, 1984, p p . 561-570 2. DAMTP p r e p r i n t , Cambridge, 1982 YOSIDA, K. 1. Functional a n a l y s i s , S p r i n g e r , N . Y . , 1965 YOUNG, E. 1 . J o u r . Math. Anal. Appl., 16 (1966), 355-362 2. J o u r . D i f f . Eqs., 3 (1967), 522-545 3. Compos. Math., 23 (1971), 297-306 4. Ann. Mat. Pura A p p l . , 85 (1970), 357-367 5. J o u r . Math. Mech., 1 8 (1969), 1167-1175 YADRENKO, M. 1 . S p e c t r a l t h e o r y o f random f i e l d s , Kiev, 1980 YAGLOM, A . 1 . S t a t i o n a r y random f u n c t i o n s , Prentice-Hall , 1962 WOHLERS, M. 1 . IEEE Trans. CT-13 (1966), 356-444 YOULA, D. 1. IEEE Trans. CT-11 (1964), 363-372 ZEMANIAN, A . 1 . Generalized i n t e g r a l t r a n s f o r m s , I n t e r s c i e n c e , N.Y. , 1968 2. SIAM Jour. Appl. Math., 1 4 (1966), 41-59 3. Math. Proc. Cambridge P h i l . S O C . , 77 (1975), 139-143 ZADEH, A. and RAGAZZINI, J . 1 . J o u r . Appl. Physics, 21 (1950), 645-655 ZHENG, W. 1. J o u r . Math. Physics, 25 (1984), 88-90

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INDEX

Abel t r a n s f o r m , 69 a b s o l u t e l y continuous, 147 a n a l y t i c c o n t i n u a t i o n , 142, 150, 181, 202 a n a l y t i c p o t e n t i a l , 40 a n a l y t i c representation, 7 a n t i c a u s a l , 162, 291 a u t o c o r r e l a t i o n , 232 a u t o c o r r e l a t i o n f u n c t i o n , 235 Banach-Steinhaus theorem, 6, 22 band extension, 316 b a r r e l e d space, 6 Bateman i n t e g r a l , 67 B e l l m a n - K r e i n - S i e g e r t i d e n t i t y , 243, 247, 261 Bergman o p e r a t o r , 188 Bergman-Gil b e r t o p e r a t o r , 171 Bessel i n e q u a l i t y , 214 Bessel process, 305 b i n d i n g enercjies, 128 Bochner i n t e g r a l , 304 boundary values of a n a l y t i c f u n c t i o n , 3, 7 bound s t a t e , 33, 127 Brownian motion, 236 c a n o n i c a l equations, 308 c a n o n i c a l f a c t o r i z a t i o n , 243 Cauchy problem, 15, 119 Cauchy r e p r e s e n t a t i o n , 7, 12 causal, 161 causal ity, 296 causal f i l t e r , 239 Cholesky r e l a t i o n s , 286 Chudov system, 282 completeness, 42, 127 complex a n g u l a r momentum, 37 complex Riemann f u n c t i o n , 189 c o n d i t i o n a l p r o b a b i l it y , 237 continuous i n p r o b a b i l i t y , 231 continuous spectrum, 75, 127 convol u t i on, 5, c o u n t a b l e union topology, 22 Coulomb p o t e n t i a l , 205 covariance, 232 Crum t r a n s f o r m , 275 D a r b o u x - C h r i s t o f f e l formula, 107, 215, 249, 251, 263 347

348

ROBERT CARROLL

Oarboux transformation, 300 deBranges p a i r , 314 deBranges space, 314 deconvolution, 289 d e l t a function, 4 differential transmutation , 301 d i f f u s i o n transform, 305 d i s c r e t e Gelfand-Levitan equation, 211 Oirac system, 317 d i r e c t i n g f u n c t i o n a l , 309 d i s t r i b u t i on, 3 domain of dependence, 291 domain of influence, 147 downward continuation, 286, 296 d u a l i t y , 4, 5 eigenfunction, 105 Erdelyi-Kober operators, 104 entropy, 316 Euler equation, 164 Euler's i n t e g r a l , 68 Euler- Poi sson- Oarboux equation , 111 even innovations, 254 evolution operator, 246 expectation, 231 exponential type, 12 e x t e r i o r transmutation, 172 f a c t o r i z a t i o n , 121, 148, 157, 239 f i l t e r i n g , 238 f i l t r a t i o n , 306 Fourier i n t e g r a l , 8 Fourier-Jacobi transform, 69 Fourier type o p e r a t o r , 51, 147 Fredholm a l t e r n a t i v e , 289 Fredholm equation, 242 Fredholm resol vant, 242 f r e e J o s t s o l u t i o n , 39 f r e e measure, 180 fundamental system of neighborhoods, 4 Gaussian d i s t r i b u t i o n , 231 Gaussian process, 236 Gelfand-Levitan (G-L) equation, 120, 129, 187, 204, 279 generating function, 195 generalized convolution, 82, 148, 154, 186 generalized Fourier transform, 12 generalized a x i a l l y symmetric p o t e n t i a l theory, 104 generalized s p e c t r a l function, 22 generalized t r a n s l a t i o n , 23, 80, 106, 148, 153, 183 Goursat problem, 16, 18, 122, 175, 211, 311 Green's function, 16, 35, 40, 206 Hahn-Banach theorem, 1 3 Hankel operator , 252 Hankel transform, 60, 91, 109, 271 Hardy space, 314

INDEX

Harish transform, 70 harmonic representation, 7 Heavyside function, 4 Hi 1 bert problem, 159 Hi1 bert transform, 153 homogeneous, 270 hyperfunction, 3 impedance, 276, 317 impulse response, 147, 276, 317 indicia1 equation, 74 innovations, 244 interior transmutation, 177 interpolation, 195 intertwining, 21 inverse problem, 276 irregular singular point, 75 isospectral, 261 Ito calculus, 305 Ito process, 307 isotropic, 270 Jacobi functions, 61 Jost function, 56, 126, 196 Jost solution, 14, 29, 38, 60, 205 Kalman-Bucy equation, 246 Kalman-Bucy filtering, 244 Kontorovic-Lebedev (K-L) technique, 147, 149, 179, 207 kernel polynomial, 215 Krein functions, 208, 249, 251, 255 Krein-Levinson recursions, 248, 259 Kummer's function, 205 Lame moduli, 286 Laplace-Be1trami operator, 59 Laplace transform, 281 layer stripping, 285 Lax-Phillips scattering, 316 least squares estimator, 238 Lebesgue integral , 6 Legendre function, 62 Lie groups, 70 locally convex space, 4 Lornmel function, 192 Marcenko (M) equation, 136, 146, 158, 171 Markov inequality, 232 Markov process, 237 martingale, 237 matrizant, 309 measurable, 231 mean square continuous, 232 mean square integral, 232 Mehler theory, 2 Mcllin transform, 70, 140

349

350

ROBERT CARROLL

me t r izable, 5 moment f u n c t i o n a l , 209

Neumann series, 20 non -anticip ativ e f i l t e r , 239 non-destructive e v a l u a t i o n , 288 obse rv ation s , 241 odd in nov ation s , 254 one s ided d e l t a f u n c t i o n , 41, 44, 150, 292 order, 3 orthogonal increments, 233 Paley-Wiener theorem, 10 , 1 2, 71, 76 Parseval r e l a t i o n , 9 , 23, 73, 77, 314 phase s h i f t , 128 Poisson process, 237, 303 p o s i t i v e f u n c t i o n a l , 27 p o t e n t i a l , 14 Povzner-Levi tan r e p r e s e n t a t i o n , 128 power spectrum, 235 p r o b a b i l i t y d i s t r i b u t i o n , 231 q u a d r a t i c f u n c t i o n a l , 160 r a d i a t i o n boundary c o n d i t i o n , 289 Radon transform, 70 random evo lution , 303 random f i e l d , 270 random time, 303 random v a r i a b l e , 231 r a p i d l y decreas in g, 5 r e f l e c t i o n c o e f f i c i e n t , 54 Regge p oles, 40 regularization, 6 r e g u l a r s i n g u l a r p o i n t , 74 r e g u l a r g eneralized f u n c t i o n , 22 r e g u l a r s o l u t i o n , 14, 38 reproducing ker nel, 215, 251, 262, 312 r e so l vant k er nel, 35 r e v e r s e G-L equation , 120 Ri c c ati equation, 246, 260, 273, 302 Riemann fun ction , 16, 17, 82, 278 Riemann-Liouville i n t e g r a l , 132, 136 Riemann problem, 142, 159 s c a t t e r i n g matrix , 159 second o r d e r pro ces s, 231 seminorm, 4 se pa rable, 231 s h e a r modulus, 276 sideways Cauchy problem, 289 s i g n a l , 241 smoothing, 241 Sobolev i d e n t i t y , 248, 261 s o l i t o n , 302

INDEX

Sommerfeld r a d i a t i o n condition, 171 spectral d e n s i t y , 279 s p e c t r a l innovations, 259 s p e c t r a l measure, 2 7 , 37, 232 spherical functions, 14, 59 s p e c t r a l process, 233 spherical transform, 70 s p l i t t i n g , 290 s t a b i l i t y , 282, 292 standard p r o p e r t i e s , 137 s t a t i o n a r y , 232 s t o c h a s t i c d i f f e r e n t i a l equation, 305 s t o c h a s t i c geometry, 264 s t o c h a s t i c i n t e g r a l , 236, 306 s t o c h a s t i c process, 231 s t r i c t inductive l i m i t , 4 s t r i n g equation, 261 strong s i n g u l a r i t y , 59 Struve function , 7 97 support, 3 , 5 tempered d i s t r i b u t i o n , 5 t e s t function, 4 t h r e e dimensional problems, 159, 286 Toepl i t z kernel, 249 tomography, 288 t r a c e , 161 transmission c o e f f i c i e n t , 54 transmission d a t a , 288 transmission l i n e , 285, 316 transmutation, 15 t r a n s p a r a n t p o t e n t i a l , 57 t r a v e l time, 276 t r i a n q u l a r i t y theorems, 138 two sided d e l t a function, 41, 150, 292 unitary operator, 308 variance, 231 v a r i a t i o n a l argument, 162 Vekua r e p r e s e n t a t i o n , 188 Volterra operator, 20 weak convergence, 6 Hey1 f r a c t i o n a l i n t e g r a l , 136 white noise, 235 white noise equation, 307 white noise i n t e g r a l , 237 Whittaker transform, 207 wide sense, 232 Wiener algebra, 314 Wiener f i l t e r , 240 Wiener-Hopf equation, 242 Yukawa p o t e n t i a l , 40 2-transform, 289

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