Mathematical Physics

MATHEMATICAL PHYSICS

NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (121)

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

NORTH-HOLLAND-AMSTERDAM

NEW YOAK OXFORD TOKYO

152

MATHEMATICAL PHYSICS Robert CARROLL University of Illinois Urbana, Illinois, U.S.A.

NORTH-HOLLAND -AMSTERDAM

NEW YORK OXFORD TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 p.0. Box 21 1,1000 AE Amsterdam, The Netherlands Distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010, U.S.A. First edition: 1988 Second impression: 1991

LIBAARY OF CONGRESS Library of Congress Cataloging-In-PublIcatlon Data

Carroll. Robert W a y n e , 1930Mathematics p h y s i c s / Robert Carroll. p. cm. -- (North-Holland mathematics s t u d i e s ; 152) (Notas d e natenitica ; 121) Bibliography: p . Includes index. ISBN 0-444-70443-4 1. Mathematical physics. I. Title. 11. Series. 111. S e r i e s . N o t a s de natenitica ( R i o d e Janeiro. B r a z i l ) ; no. 121. O A l . N 8 6 no. 121 [ OC20 I 510 S--dcl9 88-11195 (530.1'51 CIP

ISBN: 0 444 70443 4 Q ELSEVIER SCIENCE PUBLISHERS B.V.. 1988 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 written permission of the publisher, Elsevier Science Publishers B.V. / Physical Sciences and Engineering Division, P.0. Box 103, 1000 AC Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher. No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Printed in the Netherlands

V

PREFACE

A g r e a t deal o f mathematics i s used i n studying physics, as i s w e l l known, and i t i s my b e l i e f t h a t a great deal o f physics i s used i n developing mathematics (more than i s perhaps r e a l i z e d ) .

A t one time i t seemed convenient

( f o r me a t l e a s t ) t o t h i n k o f an equation physics = geometry, b u t one might a l s o make a case f o r physics = p r o b a b i l i t y , o r physics = recursion, e t c . I t also seemed a t t r a c t i v e a t one time ( t o me) t o t h i n k o f t h e study o f phy-

s i c s (and perhaps a l s o mathematics) i n t h e context o f "recognizing God's handiwork and p r a i s i n g it". But one can a l s o ask o f course whether God had any choice i n c r e a t i o n ( c f . here [ P l ] which deals w i t h complexity, entropy, information, r e c u r s i v e games, self-reproducing machines, e t c . ) .

It i s a l s o

perhaps f i t t i n g t o t h i n k o f r e l a t i o n s between gods and c i v i l i z a t i o n s ( c f . Frequently one makes mathematical models o f a physical s i t u a t i o n [Tul]). and i f t h e model i s any good i t s mathematical study w i l l l e a d t o informat i o n o f use i n physics.

I f t h i s study can be d i r e c t e d o r guided a l s o by

physical i n t u i t i o n then so much t h e b e t t e r ; one w i l l be l o o k i n g then a t phys i c a l l y i n t e r e s t i n g features and t h e mathematical questions asked and invest i g a t e d w i l l be enriched by t h e i n t e r a c t i o n w i t h physics.

Such an i n p u t can

also a r i s e from numerical o r computer study o f a mathematical model; t h e computational a l g o r i t h m i c t h i n k i n g toward s o l v a b l e numerical problems can lead t o t h e o r e t i c a l i n s i g h t i n t o t h e model.

One i s o f course advised n o t t o

ask o n l y those questions whose answers can be computed ( b u t t h e r e may be several schools o f thought here as w e l l ) . We t r y t o provide i n general a r i c h s e l e c t i o n o f m a t e r i a l and t o i n d i c a t e as w e l l c u r r e n t areas o f i n t e r e s t and d i f f e r e n t p o i n t s o f view. We a r e esp e c i a l l y i n t e r e s t e d i n t h e i n t e r a c t i o n o f ideas from apparently d i f f e r e n t areas and t h e i r synthesis i n t h e discovery process. I n t h i s d i r e c t i o n we a l s o f e e l t h a t t h e use o f language i s enriched by knowledge of o t h e r l a n guages.

We t r y whenever p o s s i b l e t o e x h i b i t patterns and s t r u c t u r e and w i l l

vi

ROBERT CARROLL

emphasize s t r u c t u r e as p r o v i d i n g a c r a d l e f o r t h e n u t u r i n g o f t h e o r y .

We

w i l l g i v e t o t a l l y elementary i n t r o d u c t i o n s t o many areas w i t h complete det a i l s and w i l l t h e n c o n t i n u e t o develop t h e themes i n v a r i o u s ways a t v a r i ous p l a c e s i n t h e book. O c c a s i o n a l l y ( b u t r a r e l y ) we w i l l s i m p l y s t a t e a r e s u l t t h a t may be needed f o r i l l u m i n a t i o n ( w i t h r e f e r e n c e s ) and no apology seems necessary f o r o m i t t i n g t h e p r o o f .

The pace may appear t o be f a s t a t

times b u t t h e necessary d e t a i l s a r e u s u a l l y t h e r e i n t h e t e x t o r i n t h e appendices,

Once beyond t h e f i r s t c h a p t e r some o f t h e m a t e r i a l i s presented

i n a way we have found p e r s o n a l l y i n s t r u c t i v e i n l e a r n i n g and w h i c h we have used e f f e c t i v e l y i n teaching.

F o r example i n Chapter 2, 83-5, we develop a

number o f s t r u c t u r a l formulas and r e s u l t s , i n w o r k i n g o u t t h e necessary t e c h n i c a l machinery as we go along, sometimes i n a h e u r i s t i c manner.

In fact

we do n o t p r o v e t h e a b s t r a c t s p e c t r a l theorem i n H i l b e r t space f o r a s e l f a d j o i n t o p e r a t o r as such ( i t i s s t a t e d however i n §2.2),and we do n o t g i v e an " a x i o m a t i c " t r e a t m e n t o f s p e c t r a l measures, p r o j e c t i o n o p e r a t o r s , e t c . However we g i v e t h e necessary formulas, d e t a i l s , and background t o deal w i t h a l l t h e s e i d e a s and use t h e m a t e r i a l i n a way which amounts t o p r o v i n g e.g. t h e s p e c t r a l theorem a f t e r a l l .

I n f a c t i n t h i s way much more i s done,in

t h a t connections between v a r i o u s p o i n t s o f view a r e d i s p l a y e d as wel1,and one sees t h e r o l e o f t h e v a r i o u s i n g r e d i e n t s i n p r a c t i c e .

What i s a c t u a l l y

needed i s proved o r sketched more o r l e s s c o m p l e t e l y so t h a t t h e d e t a i l s can be f i l l e d i n i n any case.

The p r e s e n t a t i o n t h u s may appear somewhat

d i s j o i n t e d a t times b u t we have found i t p e d a g o g i c a l l y more s a t i s f a c t o r y t h a n a theorem-proof f o r m a t and i t has more meaning p e r s o n a l l y t o proceed

i n t h i s way.

I n t h i s s p i r i t we have o r g a n i z e d much m a t e r i a l throughout t h e

book i n a remark f o r m a t ( i n s t e a d o f theorem-proof) w i t h t h e p r o o f s o f s t a t e ments i n d i c a t e d o r c a r r i e d o u t i n t h e t e x t , a l o n g w i t h t h e general d i s c u s sion.

Exercises a r e t h e n i n t e r s p e r s e d t h r o u g h o u t t h e t e x t .

We have e x t r a c t e d m a t e r i a l from many sources w i t h ample r e f e r e n c e s .

Thus v a r i o u s ideas o f p r o o f o r p r e s e n t a t i o n , which we have found p a r t i c u l a r l y i l l u m i n a t i n g o r s t i m u l a t i n g , a r e h o p e f u l l y conveyed t o t h e r e a d e r .

I n or-

der t o i n c l u d e enough m a t e r i a l t o j u s t i f y a t i t l e as p r e t e n t i o u s as "mathem a t i c a l p h y s i c s " we have r e s o r t e d t o c e r t a i n space s a v i n g devices ( t o m i n i mize t h e number o f pages and t h e p r i c e ) .

Thus i n p a r t i c u l a r as t h e book

.,

goes on t h e r e a r e p r o g r e s s i v e l y fewer d i s p l a y e d formulas and we use t h e f o l -

*,

which a r e *, 0 , b y +, There a r e 6 d a r k symbols, used as d i s p l a y " i n d i c a t o r s " i n t h e t e x t i n t h e f o l l o w i n g o r d e r : *, A , 0 , lowing substitute.

PREFACE

6, 6,

.,**,

*A,

..., *.,

A*, A A ,

...,

Am,

vi i

..., .*, ..., .my ***,

**A,...,

... T h i s tends t o make t h e text r a t h e r dense a t times but with a l i t t l e patience and p r a c t i c e this notation is q u i t e e f f i c i e n t and useful.

**my *A*,

There is a g r e a t deal on functional a n a l y s i s i n the book, probably enough f o r a semester course i n functional a n a l y s i s , and most d e t a i l s a r e provided. In p a r t i c u l a r t h e theory of d i s t r i b u t i o n s o r generalized functions i s developed i n several ways. Although there a r e many omissions (nothing about chaos, black holes, index theory, s u p e r s t r i n g s , e t c . ) we do manage t o touch upon many t o p i c s of c u r r e n t interest (e.g. superconductivity, gauge f i e l d theory, geometric q u a n t i z a t i o n , Feynman i n t e g r a l s , quantum f i e l d theory, inverse problems, s o l i t o n theory, etc.), some of i t i n considerable d e t a i l (e.g. inverse s c a t t e r i n g and s o l i t o n t h e o r y ) . There are some ( c l e a r l y too many i n terms of o v e r a l l perspective) s e c t i o n s based on the a u t h o r ' s work and this should not be construed e n t i r e l y a s vanity ( i n p a r t i c u l a r i t allows us t o develop considerable d e t a i l in a r e a s which we know b e s t ) . The materi a l in e.g. 51.6, 1.11, 2.6, 2 . 7 provides a good model f o r discussing c e r t a i n a r e a s of research and we have employed i t s u c c e s s f u l l y i n l e c t u r e s ; t h e theory of necessary ingredients such a s s p e c t r a l measures e t c . is developed as one goes along and this seems t o make f o r meaningful pedagogy. In a sense one of t h e main c o n t r i b u t i o n s of t h e book may involve Chapter 2 where a r a t h e r f u l l discussion o f inverse s c a t t e r i n g a n d elementary sol i t o n theory i s given. There a r e a number o f new r e s u l t s and a l o t of r e c e n t m a t e r i a l . We have not spent much time on physical d e r i v a t i o n s or t h e philosophy of physics. This i s a s e r i o u s gap but one not p o s s i b l e t o bridge under t h e imposed space l i m i t a t i o n s . I t i s very productive t o l i n k mathematical development w i t h physical reasoning. For example a n i c e complex of ideas revolves around c a u s a l i t y , hyperbol i c PDE, Fourier transforms a n d Pal ey-Wiener i d e a s , s c a t t e r i n g , t r i a n g u l a r i t y of o p e r a t o r s , e t c . Similarly one has ideas o f cohomology, gauge theory, c u r r e n t s , charges, e t c . in f i e l d theory. We f e e l t h e present era t o be revolutionary i n science a n d mathematics and have t r i e d t o develop enough machinery t o help the reader storm the b a r r i cades. In the area of nonlinear PDE f o r example t h e methods of functional analysis have reached a very hybrid a b s t r a c t form,and we have preferred t o give a presentation of e a r l i e r versions of the theory,where there i s more contact w i t h t h e o r i g i n a l problems,and motivation i s more v i s i b l e . One can emphasize here t h a t i t i s wise t o s t a y reasonably c l o s e t o the source of mathematical problems i n physics i n order t o r e t a i n nourishment a n d v i t a l i t y .

viii

ROBERT CARROLL

A b s t r a c t i o n f o r i t s e l f i s o f t e n a t t r a c t i v e b u t we pursue t h i s o n l y i n t h e interest o f n u t r i e n t structure.

One should be f r e e t o use i n t u i t i o n , p i c -

tures, analogy, e t c . t o develop the a p p r o p r i a t e language f o r whatever phys i c s i s under consideration.

The r e l i g i o n o f embalming mathematics i n a x i -

omatic systems does n o t prove too p r o f i t a b l e i n mathematical physics ( a l though the reader w i l l d e t e c t vestiges o f a former f l i r t a t i o n w i t h the Muse

o f N. Bourbaki).

The book makes very modest claims.

We hope i t can be use-

f u l as a t e x t , even a more or l e s s i n t r o d u c t o r y t e x t , w h i l e s e r v i n g as a guide t o some research areas o f c u r r e n t i n t e r e s t .

There i s a l o t o f f a i r l y

s o p h i s t i c a t e d m a t e r i a l w i t h h o p e f u l l y enough r i g o r t o be b e l i e v a b l e and enough h e u r i s t i c content t o s t i m u l a t e f u r t h e r study. The author would l i k e t o thank L. Nachbin f o r adding t h i s book t o t h e Notas de Matematica series.

We would a l s o l i k e t o acknowledge the support o f

various people who made i t possible t o t r a v e l t o conferences and g i v e seminar t a l k s i n the past 3 years w h i l e t h e book was being w r i t t e n ; we mention i n p a r t i c u l a r L. Bragg, Gilbert,

J. Dettman, J. Donaldson, A. Favini,

T. Kailath, E. Magenes, P. McCoy,

and W. Zachary.

T . G i l l , R.

C. Pucci, L . Raphael, F. Santosa,

I would a l s o l i k e t o acknowledge r e l e v a n t conversation dur-

i n g t h i s p e r i o d w i t h t h e above people as w e l l as w i t h ( i n p a r t i c u l a r ) A. Arosio,

C. Baiocchi, M. Berger, M. Bernardi, A. Bruckstein, M. Cheney, D.

Colton, J. Cooper, S . Dolzycki, C. Foias, J. Goldstein, D. Isaacson, H. Kaper, T. Kappeler, D. Kaup, M. Kon, I . Lasiecka, P. Lax, T. Mazumdar, J. Neuberger, P. Newton, R. Newton, A. Pazy, H. Pollak, J. Rose, T. Seidman, G. Strang, W.

Strauss, W . Symes, P. Tondeur, G. Toth, and A . Yagle ( w i t h

apologies f o r omissions).

F i n a l l y the book i s dedicated t o my w i f e Joan.

ix

TABLE OF CONTENTS

PREFACE

V

CHWCER 1, CCASrSlCAL IDEM A I D PR0BCW

IntraZluttiun 2. Some preliminary uariatianal ideas 3, various d i f f erential eqwtians and their origins 4. Linear second arder PDE 5- Further t a p i o i n the calculus af variatians 6. Spectral theary far ardinaq differential operatars, transmutatian, and inverse problems 7. Intruductian t o classical mechanics 8. Intraductinn t a qwntwn mechanics 9, Peak problems i n PDE 10. Same nanlinear PDE 11- Ill posed prablems and regulariratian 1.

Introduction 2- Scattering thearg I (aperatar theory) 3. Scattering theory 11 (3-D) 4. Scattering theorg 111 (a medley of themes) 5. Scattering thearg IV (spectral methads i n 3-D) 6. Systems and half line prahlems 7, Refatians between patentials and spectral h t a 8- lntroductian ta salitan theory m sgstems 9. Salitans via A 10, Salikan thearg (Hamiltanian structure) 11. frame tapics i n integrable systems 1.

CHI\PCER 3-

1 2

10 16 25 35

49 57 65

74

86

99

101 108 119 137 147

168 183 192

201 211

WmE N0NCIIEAR ANAWZS: S6NE GEQHREERZC F0Rl!MCI$I 1. 2.

Intraductian Manlinear analysis

227 227

ROBERT CARROLL

X

3. 4,

5, 6, 7, 8, 9, 10*

Nnnotane nperatars enpalogical methods Convex analysis Nonlinear semigrnups and mnnatane sets Uariak ional inequa Ii t ies Quuankwn field thenry Gauge fields (physics) Gauge fields (makhematics) and geometric quantiaatian

238 252 264 272 283 286 294 301

APPENDIX A.

INCR0DUCZ10N CO CZNEAR FUICCIBNAC A N A C W I S

311

APPENDIX 3.

RCECCED C@PIC9 I N FUNCCMNAC ANACwl$

329

APPENDIX C.

INCFER0DUCCLQ)N E0 DIFFEFERENCIAI: GE0mECRy

351

REFERENCES

377

INDEX

393

T

CHAPTER 1 CLASSICAL IDEAS AND PROBLEMS

1.

ZbllR0DLIC&I0N. C l a s s i c a l l y i t was easy t o l o o k f o r "meaning" o r perhaps

"Structure" i n mathematical physics i n t h e areas i n v o l v i n g t h e c a l c u l u s o f v a r i a t i o n s (see e.g.

[ L l ] where l i t e r a r y c i t a t i o n s appear as chapter i n t r o -

ductions and cf. a l s o [Cal;Col,2;Gl;Il;Yl]).

We s h a l l use t h i s v a r i a t i o n a l

theme as a v e h i c l e t o e n t e r the s u b j e c t o f mathematical physics.

It w i l l

lead t o d e r i v a t i o n s o f many important d i f f e r e n t i a l equations and p r o v i d e i n s i g h t i n t o many physical problems v i a a m i n i m i z a t i o n ( o r b e t t e r extremal) directive.

Furthermore we w i l l be able t o d i s p l a y q u i c k l y and n a t u r a l l y

various important mathematical techniques whose f u r t h e r study has l e d t o t h e development o f whole areas o f mathematics as w e l l a s t o f r u i t f u l a p p l i c a t i o n i n physics. Thus §§2,3 and 5 w i l l deal w i t h v a r i a t i o n a l ideas and t h e o r i g i n o f some b a s i c d i f f e r e n t i a l equations.

14 discusses some fundamental methods and

r e s u l t s o f existence, uniqueness, etc. f o r c l a s s i c a l 1 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 (PDE).

96 deals w i t h some ideas o f s p e c t r a l theory, t r a n s -

mutation, and inverse theory f o r t y p i c a l o r d i n a r y d i f f e r e n t i a l equations (ODE); t h i s theme i s picked up again l a t e r i n Chapter 2 and developed exten-

s i v e l y . 917 and 8 g i v e i n t r o d u c t i o n s t o c l a s s i c a l and quantum mechanics, pres e n t i n g various p o i n t s o f view and n o t a t i o n s t o be r e f e r r e d t o f r e q u e n t l y i n o t h e r p a r t s of t h e book.

§9 introduces t h e idea o f weak problems and solu-

t i o n s i n PDE ( v a r i a t i o n a l - o p e r a t i o n a l problems) and i n d i c a t e s f i r s t some basic l i n e a r theory.

I

Then we develop the framework and s t a t e some r e s u l t s

f o r the Navier-Stokes equations, about which f u r t h e r remarks and i n d i c a t i o n s

o f proofs w i l l be given l a t e r a t various places (e.g. Ssl.10, 3.7, e t c . ) . § l o gives some f u r t h e r n o n l i n e a r problems and r e s u l t s . I n p a r t i c u l a r we a r e a b l e t o make contact w i t h some r e c e n t work on t h e Ginzburg-Landau equations and introduce ideas about s o l i t o n s , v o r t i c e s , gauge invariance, Yang-MillsHiggs .equations, e t c .

Such themes w i l l a l s o be picked up again l a t e r .

2

ROBERT CARROLL

F i n a l l y , i n $11, we g i v e some t y p i c a l r e s u l t s on ill posed problems and Tikhonov t y p e r e g u l a r i z a t i o n i n o r d e r t o i n d i c a t e an i m p o r t a n t d i r e c t i o n i n c u r r e n t research.

2. B0mE PRECImlNAR&! UARZAt10NAL IDEA$. h i s t o r i c a l int e r e s t

L e t us s t a r t w i t h a few problems o f

.

EMAIRPLE 2.1 (ErNECt'S MU)). Imagine t h a t we l o o k a t a f i s h as i n d i c a t e d

Thus t h e d i s t a n c e s b and d from t h e a i r - w a t e r i n t e r f a c e a r e known and a t c = c o n s t a n t i s known.

The p o s i t i o n o f t h e o r i g i n o f r e f r a c t i o n i s n o t known

b u t t h e v e l o c i t i e s of l i g h t i n a i r (v,)

and i n w a t e r (v,)

a r e assumed t o be

S n e l l ' s law says t h a t (*) [Sine/va] = [Sin$/vW] and t h i s i s e a s i l y

known.

v e r i f i e d experimentally ( i f the f i s h i s w i l l i n g

-

o r dead).

L e t us deduce

t h i s l a w however from F e r m a t ' s p r i n c i p l e o f l e a s t t i m e which s t a t e s t h a t t h e t i m e r e q u i r e d f o r t h e l i g h t t o pass from

A t o B s h a l l be a minimum (one as-

sumes here t h a t l i g h t t r a v e l s i n s t r a i g h t l i n e s ) .

La

We w r i t e (A) c = dTan$;

= L / v ; and tw= Lw/vw. Thus t h e a a a t i m e o f passage i s T = ta + tw= (b/va)Sece t (d/Lw)Sec$ w h i l e k = a t c =

a = bTane;

= bSece; Lw = dSec$; t

If one s o l v e s t h e second e q u a t i o n f o r Q = $ ( e ) and i n s e r t s i t i n t h e f i r s t e q u a t i o n we would o b t a i n T = T ( e ) . Then s e t t i n g T ' ( e ) 0 one would f i n d values o f e f o r which T ( e ) i s extreme (max o r min o r i n f l e c bTane

tion).

t

dTanQ.

The c a l c u l a t i o n can be shortened by d i f f e r e n t i a t i n g b o t h e q u a t i o n s

w i t h respect t o

e and e l i m i n a t i n g d$/de; t h e r e s u l t i s t h e n (*) ( e x e r c i s e ) .

EXAIIPCE 2.2 (BaACHl$e0C€WNE PR08tEm).

T h i s problem goes back t o t h e Ber-

n o u l l i b r o t h e r s and can be s o l v e d b y v a r i o u s methods ( c f . [Col-3;Yl]).

The

ingenious technique developed by E u l e r (which we p r e s e n t h e r e f o r m a l l y ) can be extended and g e n e r a l i z e d and i s amazingly p r o d u c t i v e .

Thus one imagines

a u n i f o r m b a l l w i t h a h o l e i n i t s l i d i n g under g r a v i t y w i t h o u t f r i c t i o n on a w i r e whose shape i s t o be determined so t h a t t h e t i m e o f descent f r o m A t o

B s h a l l be a minimum. (2.2)

A

VARIATIONAL IDEAS

3

2

Equating p o t e n t i a l and k i n e t i c energy one has mgy = (1/2)mv where v = ds/dt w i t h s denoting arc length. One w r i t e s ( e = d/dt, ' % d/dx) 5 = ( i2 t 2 ) 4 = i ( l + y I 2 )4 ( s i n c e j = y ' i by t h e chain r u l e ) and hence i(l+y")' = 2gy so (2.3) T = T(y) = j oX O ( d t / d x ) d x = ~ ~ o [ ( l + y ' 2 ) / 2 g y ] t d x =

We admit i n t o competition as admissable f u n c t i o n s y t h e c o l l e c t i o n A = t y E 1 C (O,xo), y ( 0 ) = 0, y ( x o ) = yo) and ask f o r y E A such t h a t T ( y ) 5 T ( t ) f o r a l l z E A (here Cn(O.xo) denotes n times continuously d i f f e r e n t i a b l e funct i o n s on ( 0 , ~ ~ ) ) . A p r i o r i such a problem w i t h general F need n o t have any s o l u t i o n y E A and such a s o l u t i o n need n o t be unique.

However i n the pre-

sent s i t u a t i o n t h e r e i s a s o l u t i o n which t u r n s o u t t o be t h e a r c o f a cycloid

- n o t r e a l l y a s u r p r i s e t o Newton f o r example.

We r e c a l l t h a t a cy-

c l o i d i s t h e path traced by a p o i n t on the circumference o f a c i r c l e when t h e c i r c l e r o l l s on a s t r a i g h t l i n e , and c y c l o i d s were o f more i n t e r e s t i n Newton's time.

Now l e t us f o l l o w Euler and assume f i r s t t h a t there i s a

minimizing f u n c t i o n y E A ( n o t necessarily unique) and f i x i t . tv E C 1(O,xo), v ( 0 ) = 0 = v ( x o ) ) and E E R (R = r e a l numbers). f i x e d and then z = y +

€9 E

so t h a t T ( y ) 5 T(z).

A

Let 6 = Pick

v

E CD

We w r i t e T(z) = T(ytEv)

A .

?(O) 5 ?(E) f o r any E (y and v are f i x e d ) . Make a p p r o p r i a t e hypotheses on F now i n (2.3) so t h a t one may d i f f e r e n t i a t e under the i n t e = T(E) and then

g r a l s i g h w i t h respect t o

E

i n the formula

X

'v

T(E) = fo oF(x,y+Eq,y'+Ev')dx

(2.4)

( c f . any reasonable book on advanced calculus f o r d i f f e r e n t i a t i o n under t h e integral sign).

Thus f o r m a l l y

rv

Now by standard c r i t e r i a f o r extreme values o f C 1 f u n c t i o n s T we want T ' ( 0 ) = 0 so (2.5) = 0 f o r X

E

= 0.

Since

v

N

E 6 was a r b i t r a r y we have

lo

0

[Fyq + F ,v']dx = 0 Y

(2.6)

v

( t h e argument o f F and F We w i l l i n (2.6) i s (x,y,y')). Y Y' sometimes r e f e r t o t h i s procedure o f reducing T ( y ) t o ?(E) and the subse-

for a l l

E

@

quent analysis as E u l e r ' s t r i c k .

v

Now f o r m a l l y

( 0 )

$OF

Y = FYI,!

U

,vldx = -loodlF ,dx X Y

However since D F t Fylyy' t F Iy" x y' Y 'Y we must a l l o w t h e f u n c t i o n y t o have another d e r i v a t i v e i n general i f ( 0 ) i s since

vanishes a t 0 and xo.

4

ROBERT CARROLL

t o be used.

This procedure can be circumvented by a technique o f du Bois

Reymond i n d i c a t e d below (which a c t u a l l y shows t h a t y" does make sense when

9 0).

FY'Y'

Thus h e u r i s t i c a l l y l e t us use

-

c0[Fy

(2.7)

DxFyl]gdx

( 0 )

and (2.6) t o o b t a i n

= 0

It w i l l f o l l o w by Lemna 2.3 below t h a t [

1=

0 i n ( 2 . 7 ) so we w i l l have t h e

Euler equations (2.8)

'

DxFyi (X,Y,Y

= Fy(X.Y,Y'

+ Note t h a t t h i s i s a second o r d e r n o n l i n e a r d i f f e r e n t i a l equation y " F YlY' For t h e brachistochrone problem w i t h F = = F ( i f Fylyl 4 0). FYlyyJ2+ F Y'X, Y [ ( l t y ' )/2gy]", a f t e r a c l e v e r change o f v a r i a b l e s ( c f . [ C O ~ ] ) , (2.8) r e duces t o t h e equation f o r a c y c l o i d ( e x e r c i s e ) . We w i l l g i v e many important examples o f Euler equations such as (2.8) ( w i t h e a s i e r c a l c u l a t i o n s ) i n t h e text.

To complete t h e present discussion we need two lemnas. X

Assume fooG(x)v(x)dx = 0 f o r a l l 9 E d where G i s assumed con-

LEmmA 2.3.

Then G z 0.

tinuous on [O,xo].

Rood: Assume G # 0 so,for some x1 E [O,xo],G(xl) G is 0 i n some i n t e r v a l I as shown

and f o r 0 < n

J.

Now choose

<

v

0 say.

By c o n t i n u i t y

G(xl) we can f i n d a s u b i n t e r v a l J C I such t h a t G(x) 2 n i n E

as i n d i c a t e d so t h a t

0 = laovGdx =

(2.10)

>

v

= 0 outside o f I and

jI vGdx :j p G d x L n2 l e n g t h J

>

v 2 n i n J.

0

J then i m p l i e s a f a l l a c y i n reasoning somewhere and by c o n t r a d i c t i o n we con-

QED

clude t h a t the lemma i s t r u e . xO

CEIIIIRA 2.4. Assume say H E Co(O,xo) and fo H(x)n(x)dx = 0 f o r any n xo) s a t i s f y i n g $on(x)dx = 0. Then H(x) = c.

Rood:

I f H = c c e r t a i n l y $oHndx

/do Hdx or

(*) #o(H - c)dx = 0.

l2oHndx = 0 and we can choose

H

Co(O,

0 and we choose c now by the r u l e cxo =

=

Then i n p a r t i c u l a r #o(H rl =

E

-

c by (*).

- c)ndx

= 0 since

2

It f o l l o w s t h a t f t o ( H - c ) dx

5

VARIATIONAL IDEAS = 0 and hence H z c.

QED

Now t h e t e c h n i q u e o f du B o i s Reymond a l l u d e d t o a f t e r I n s t e a d o f p a s s i n g f r o m (2.6) t o

I

( 0)

xO

X

(2.11)

goes a s f o l l o w s .

(0)

we w r i t e

Fy(S,y(C),y'(C))dC

0

= G(x);

= -['Gcp'dx

0

80[FYI -

Hence from (2.6) one o b t a i n s

I v F y dx

Glv'dx = 0 f o r 9 E

@

(so

0

= 9' E Co

w i t h $ o q ' d x = 0) and from L e n a 2.4 i t f o l l o w s t h a t (2.12)

FYI

-

lox

Fyd5 = 0

However f r o m (2.12) i t f o l l o w s i m 1 m e d i a t e l y (fundamental theorem o f c a l c u l u s ) t h a t F E C and ( 2 . 8 ) i s v a l i d This replaces Euler's equation (2.8).

Y'

Further i f F # 0 (Legendre c o n d i t i o n ) t h e n y " makes sense Y'Y' and belongs t o Co ( e x e r c i s e c f . [Col,3;Gl] - s i m p l y work from AFY '/Ax u s i n g (2.8)). i n any case.

-

This technique i l l u s t r a t e d t i o n s and PDE i n a b e a u t i f u

n Example 2.2 extends t o mu1 t i d i m e n s i o n a l s i t u a way.

Consider f o r example

L e t R c Rn be an open s e t w i t h a smooth enough boundary

EXAIUPLE 2.5,

r

so

t h a t t h e c l a s s i c a l Green's theorems a p p l y i n t h e form

-1

(2.13)

Auvdx =

52

J,I

DjuDjv dx

-

r

unvdo

a / a x . and un denotes t h e e x t e r i o r normal d e r i v a t i v e (we w i l l c a l l j J 1 such n r e g u l a r ) . L e t A = I u E C ( a ) , u = f on r l and c o n s i d e r t h e q u e s t i o n

where D

t o minimizing the D i r i c h l e t functional (2.14) for u

D(u) = E

A.

1,

1

(Dju)

2

dx

One would l i k e t o assume f E Co b u t even t h i s n a t u r a l h y p o t h e s i s

l e a d s t o d i f f i c u l t i e s which we i l l u s t r a t e below. F i r s t l e t us proceed f o r 1 = {cp E C (n), cp = 0 on r l . Assume D(u) has a m i n i m i z i n g m a l l y and s e t f u n c t i o n u E A , f i x it, and f o r cp E @ f i x e d c o n s i d e r v = u + w E A . Set D(v) (6)

E(E)and Jnl

t h e s t i p u l a t i o n a ( 0 ) ~ t i ( c )v i a (d/dE)ij(E)IE=O = 0 l e a d s t o 2 0. I f we assume u E C (n) t h e n an a p p l i c a t i o n o f (2.13)

DjuDjcp dx

y i e l d s (+)JR

Aucpdx = 0 f r o m which Au = 0 i n R by an argument based on

Lemma 2.3 ( e x e r c i s e ) .

Hence f o r m a l l y , m i n i m i z i n g D(u) f o r u E A amounts t o

s o l v i n g t h e D i r i c h l e t problem Au = 0 i n R w i t h u = f on

r.

To a v o i d t h e

assumption u E C2 i n p a s s i n g f r o m ( 6 ) t o ( + ) t a k e t e s t f u n c t i o n s

cp E

C,"(n)

6

ROBERT CARROLL

f o r example so t h a t ( 4 ) and (2.13) g i v e

I

(2.15) where

b u d x = (u,b) = (Au,~)= 0

n

(

,

)

denotes a d i s t r i b u t i o n b r a c k e t ( c f . Appendix B ) .

Various d i s -

t r i b u t i o n a l t y p e arguments (Weyl I s l e m a , e t c . ) can now be invoked t o a s s e r t However t h e boundary be-

t h a t Au = 0 i n s2 i n a c l a s s i c a l sense ( c f . [ J l ] ) .

To see t h i s l e t us e.9. e x t r a c t f r o m [Col] t h e 2 Take a u n i t c i r c l e R c R so t h a t

h a v i o r i s s t i l l a problem. f o l l o w i n g example.

Let

f(e)

= (1/2)a0 t

1;

anCosne t bnSinne and t r y u = ( 1 / 2 ) f o ( r ) t

1;

fn(r)

Straightforward calculation (exercise)

Cosne t gn(r)Sinne as i n [Col]. gives f o r the minimizing function (2.17)

1;

u = (1/2)ao +

w i t h min D(v) = D(u) =

1;

rn[anCosne t bnSinne]

2 2 nn(an + bn) which o f course must make sense.

We

r e c a l l here ( c f . [Col;Dl]) t h a t f o r c o n t i n u o u s f on [-7r,n] one expects t h a t 2 2 2 1 2 2 (an t bn) < w h i l e f o r f E C we have n (an t bn) < m. Thus e.g. f o r 1 f E C t h e c a l c u l a t i o n s w i l l make sense b u t f E C o i s n o t enough. Indeed a s 2 an example ( c f . [Col]) t a k e f ( e ) = ( l / n )Cos(n!e) which i s Co w i t h a

-

1

1

1;

u n i f o r m l y convergent s e r i e s r e p r e s e n t a t i o n b u t

EUmPLE 2.6.

1

maf =

1

k!/k

4

=

m.

Another i m p o r t a n t example i n t h e same s p i r i t i n v o l v e s t h e

e q u a t i o n f o r s u r f a c e s o f minimal area spanning a g i v e n "frame". Thus l e t n 2 w i t h boundary r and l e t a f u n c t i o n z = f ( x , y ) be p r e s c r i b e d on r . Con-

C R

o v e r n w i t h u = z = f on

s i d e r a s u r f a c e u = u(x,y) (2.18)

S(U) =

L e t A = { u E C1;

S

-

J n [I

2

r

whose area i s t h e n

2 4

+ uy] dA

t ux

u = z = f on

i.e. S(u) 2 S(v) f o r a l l v

r l and ask f o r a m i n i m i z i n g o b j e c t u E

A.

E A for

T h i s problem, j u s t as Example 2.5,

be s t u d i e d c a r e f u l l y b e f o r e a p r e c i s e t h e o r y can be produced. pected t h i n g s can happen ( f o r which we r e f e r e.g.

t o [Col,2;Y1]

sense and t o remarks below f o r some s p e c i f i c p a t h o l o g y ) .

must

Many unexi n a general

I f one proceeds t o

d e r i v e E u l e r equations f o r (2.18) as i n Example 2 . 5 t h e r e r e s u l t s ( e x e r c i s e ) (2.19)

2 uXx[l + u ] Y

-

~

u

2 ~ + ~u [l u +~ uX]u = ~0 YY

Here one must use i n t e g r a t i o n formulas o f t h e f o l l o w i n g t y p e ( m )

7

VARIATIONAL IDEAS (m)

In [ v ~ ( ~ / ~ u , )+F ~ ~ ( a / a u ~ ) F I =d A Jr q[(a/auX)F dy

-

(a/auy)F dx]

-

Ja d D x ( a / a u x ) F + Dy(a/auy)F]dA. W P C E 2.7. Let us i n d i c a t e here a number of simple t h i n g s t h a t can "go wrong" w i t h elementary v a r i a t i o n a l problems ( c f . [Col; 111). I t is conven-

ient t o l e t A c o n s i s t o f piecewise smooth functions = continuous functions w i t h piecewise continuous ( P C ) f i r s t d e r i v a t i v e s ( i . e . y ' is continuous except for - possibly - a f i n i t e number of f i n i t e j u m p d i s c o n t i n u i t i e s a t each of which y ' has l e f t and r i g h t sided l i m i t s ) . ( A ) In (2.18) l e t n be a u n i t c i r c l e with f = 0 so u = 0 i s minimizing w i t h S(0) = 1 ~ . Take v ( x y y ) t o represent a c i r c u l a r cone of base radius f and height l y centered a t Poy and 2H lying i n s i d e of R so t h a t S ( v f ) m ( l h )+a. Then S ( v E ) + T T b u t v 9 0 s i n c e vE(Po) = 1 f o r a l l E . (B) Look a t the D i r i c h l e t functional D ( u ) i n the form (2.14) (over a u n i t c i r c l e ) w i t h u = 0 on r so t h a t u = 0 is minimizing. Let oa be a c i r c l e of radius a < 1 i n R , centered a t the o r i g i n , < and take v = 0 in R-R,, v = l o g ( r / a ) / l o g a f o r a 2 < r < a , and v = 1 f o r r 2 2 ]Ja2 a (rdr/r2 ) = -2njloga and taking an -+ 0 one Then D(v) = [2n/(loga) a . has D(vn) + 0 b u t a t t h e o r i g i n vn = 1 f o r a l l n . These two examples show t h a t minimizing sequences may not converge t o the s o l u t i o n . ( C ) An example 2 due t o H i l b e r t involves minimizing I ( u ) = 1; t 2 / 3 ( u ' ) d t f o r u E C1 w i t h u ( 0 ) = 0 and u ( 1 ) = 1. The Euler equations a r e Dt(2t2/3u') = 0 w i t h solu+ b , f o r t > 0 a t l e a s t , so t h a t u ( t ) = t1l3 i s t h e obtion u ( t ) = vious candidate. In f a c t this u provides an absolute minimum b u t u C 1 . Thus even the Euler equations (which r e q u i r e more d i f f e r e n t i a b i l i t y ) may produce a s o l u t i o n not in the admissable c l a s s A . ( D ) A v a r i a t i o n on ( C ) due t o Weierstrass i s J ( u ) = J01 t 2 ( u ' )2 d t w i t h u ( 0 ) = 0 and u ( 1 ) = 1 ( u say absolutely continuous - c f . [Ml] and Appendix A ) . Here t h e Euler equation 2 i s D t ( 2 t u ' ) = 0 with s o l u t i o n u ( t ) = a t - ' + b so no curve passes through the points required. In f a c t t h e r e i s no s o l u t i o n in t h e c l a s s of absolut e l y continuous functions 0. To see t h i s one takes < l / n ) with un = 1 ( l / n 5 Here not only i s t h e r e no

s i n c e J ( u ) > 0 f o r such a function b u t inf J ( u ) = u n = Tan-lnt/Tan-'n o r more simply u, = n t ( 0 < t t 5 1 ) . Then J ( u n ) = l;/"t2n2dt = n2/3n3 + 0. s o l u t i o n via E u l e r ' s equation - t h e r e i s no solu-

tion a t a l l i n A .

One often speaks of a weak ( l o c a l ) minimum u E A t o a problem REIIV\RK 2.8, of minimizing T(y) = $0 F(x,y,y')dx in terms o f T(y) > T(u) f o r Y E A w i t h Ily-uU < E (I1 II denotes a C 1 norm measured i n terms of sup1 ( y - u ) ( x ) l and 11 s u p ( ( y ' - u ' ) ( x ) I f o r 0 2 x 2 x o ) . A strong ( l o c a l ) minimum u E A r e f e r s t o

8 T(y)

ROBERT CARROLL

F

T(u) f o r y

A w i t h Ily-ullo = sup

E

I (y-u)(x)l

I E on [O,x,~.

3

0, ~ ( 1 =) 1. The E u l e r ~ P C 2.9, E Consider T ( y ) = 1; ( y ' ) d t , y ( 0 ) 0 w i t h u n i q u e ( m i n i m i z i n g ) s o l u t i o n u = t. L e t IP e q u a t i o n i s D t [ 3 ( y ' ) 2] C1 w i t h ~ ( 0 =) q ( 1 ) = 0 so y = u + q i s admissable and one checks t h a t 3 2 3 2 T ( y ) = T(u) + 1; [ 3 ( q ' ) t ( q ' ) ] d t ( e x e r c i s e ) . Thus f o r 3 ( ~ ' ) + ( v ' ) 2 0 (e.g. if Iqlll 5 3 ) t h e n T ( y ) 2 T(u) so u = t i s a weak l o c a l minimum. B u t

E

f o r V, d e f i n e d by q n ( 0 ) = q n ( l )

in= - J n

0,

(n-1) ( l / n < t 5 1 ) one o b t a i n s T(u+'Pn) = - J n + 0 ( 1 ) t h e o t h e r hand Ilqnllo

.+

--

0 so t h e r e i s no s t r o n g 'local minimum a t u.

L e t us g i v e now some i m p o r t a n t examples from p h y s i c s .

EXAmPCE 2.10. first,given

+

4, =

An/ (= i n f T ( y ) ) . On

( 0 5 t 5 l / n ) , and

Thus

a system o f n p a r t i c l e s w i t h masses mi and momenta p 1. ( = rn.v.1 1 1

one forms a Lagrangian f u n c t i o n (U denotes p o t e n t i a l energy)

L = T

(2.20)

1 (1/2)mi(vi)

- u

Here one expects e.g. U = U(qi)

2

- u

=

1 (1/2)(pi12/mi

-u

where qi denote c o o r d i n a t e s .

The p r i n c i p l e

o f l e a s t a c t i o n or H a m i l t o n ' s p r i n c i p l e then says t h a t f o r to,tl g i v e n t h e t r a j e c t o r i e s o f t h e p a r t i c l e s w i l l be such as t o m i n i m i z e t h e a c t i o n (2.21) (here

;li

-

A =

L(t,qi,6i)dt LO

E u l e r ' s equations t h e n become t h e Lagrange equations

Dtqi).

T h i s i s d e r i v e d e x a c t l y as b e f o r e when t h e r e was o n l y one v a r i a b l e q ( e x e r 2 c i s e ) . One notes o f course t h a t i f T = (1/2)mq and U = U(q) then (2.22) We w i l l say a g r e a t deal more becomes F = ma i n t h e form Dt(m4) = -aU/aq. about c l a s s i c a l mechanics l a t e r from a g e o m e t r i c a l p o i n t o f view.

EUmPtE 2-11 (mAXDECC'S EQLlAel0W). The v a r i a t i o n a l f o r m u l a t i o n f o r t h i s w i l l be postponed b u t we want t h e equations recorded here a t an e a r l y s t a g e A -L One w r i t e s E f o r t h e e l e c t r i c f i e l d s t r e n g t h and H ( c f . [Fl;Lll;Sl;Tl]). f o r t h e magnetic f i e l d s t r e n g t h ( v e c t o r s ) .

The c l a s s i c a l f i e l d equations

-L

a r e Div E = 4np t o g e t h e r w i t h Curl

(2.23) where

p

t=

- ( l / c ) a $ a t ; Div

i s a charge d e n s i t y and

says t h a t D i v

+

ap/at

= 0.

G=

5is

0; Curl

a current.

ii =

2

.A

( i / c ) a ~ / a t+ ( 4 n / c ) j

The c o n t i n u i t y e q u a t i o n

We w i l l do a l l o f t h i s l a t e r i n terms o f

VARIATIONAL IDEAS

9

d i f f e r e n t i a l forms b u t f o r now l e t us make a few c l a s s i c a l comments. -

L

one takes t h e magnetic i n d u c t i o n B

a

IJH

(IJ

First

= p e r m e a b i l i t y m a t r i x ) as t h e

r e a l magnetic f i e l d s t r e n g t h and t h e n somehow one has t o choose u n i t s ( u n i t s have always been an i n p e n e t r a b l e mystery t o t h e a u t h o r and we w i l l say as l i t t l e as p o s s i b l e about them

-

see [ S 1 2 l f o r d e t a i l s ) .

In particular f o r

M a x w e l l ' s e q u a t i o n s t h e r e i s a k i n d o f h o r r o r s t o r y connected w i t h u n i t s , and we w i l l t h e r e f o r e a t v a r i o u s p l a c e s i n

f a c t o r s o f 471, e t c . (see [ S l ] )

t h i s book choose v a r i o u s e s s e n t i a l l y e q u i v a l e n t forms o f (2.23) w i t h o u t any a t t e m p t t o connect them.

Thus c o n s i d e r e.g.

A

A

-L

Curl E + ( l / c ) a @ a t = 0; D i v B = 0; D i v E = P;

(2.24)

A

-

Curl B

(i/c)aSjat = (l/c)? 2

2

2

2

o r i n t r o d u c i n g a f a c t o r o f c o n l y i n ?i (*) C u r l E + Bt = 0; D i v B = 0; Et 2 A c C u r l B = -J; D i v E = P. We w i l l u s u a l l y r e f e r t o (2.24) or (**) now as

-

A

Maxwell's equations.

The f i e l d s

f and

a r e t h e observables b u t i n s t u d y i n g

these e q u a t i o n s i t i s i m p o r t a n t t o use gauge p o t e n t i a l s (about which a g r e a t Thus, a t l e a s t l o c a l l y , one w r i t e s ( * @ )

deal w i l l be s a i d l a t e r ) . A

-

-

and E = -At

= Curlii A

Gradq. 2

W i t h t h i s c h o i c e one has o f course a u t o m a t i c a l l y D i v B -

2

= 0 and C u r l (E + At) = 0 = C u r l E

L

A

+ Bt.

It remains then o n l y t o s o l v e

(use (** ) h e r e ) (2.25) A i

hp

-

2 ( l / c )qtt = - P

-

(l/c2)itt

= -(1/c2)i

-

2 A + ( l / c )vt];

2

Dt[Div

+ G r a d [ ( l / c 2 )qt

.t

Div

21 -.A

I t i s easy t o check t h a t i f one f i n d s a s o l u t i o n (Eo,Bo)

t o (**) v i a gauge

A

p o t e n t i a l s (qo,AO) t h e n *

(2.26)

= q0 A

- xt; A

a

=

A

0

+ AX

>

g i v e s t h e same (Eo,Bo) and s a t i s f i e s t h e equations (2.25) a g a i n ( e x e r c i s e ) . These t r a n s f o r m a t i o n s ( 2 . 2 6 ) a r e c a l l e d gauge t r a n s f o r m a t i o n s and have f a r r e a c h i n g importance i n a general f o r m u l a t i o n as i n Chapter 3.

In particular 2 Div A + ( l / c )vt = 0 2 ( e x e r c i s e - p u t ( 2 . 2 6 ) i n t h i s e q u a t i o n t o o b t a i n Ax ( l / c )xtt = 2 -[Div A. + ( l / c )Dtvo] = f w i t h f known). The c o n d i t i o n (*A) determines t h e 2 so c a l l e d L o r e n t z gauge and ( 2 . 2 5 ) decouples t o g i v e (**) & ( l / c )vtt = 2 . 2 - P w i t h AA ( l / c ) A t t = - ( l / c ) J . We w i l l show l a t e r how t o express a l l 2

a

g i v e n such (qO,AO) one can choose (q,A) so t h a t

A

(*A)

-

2

-

A

-

ROBERT CARROLL

10

t h i s v i a v a r i a t i o n a l p r i n c i p l e s and s y m p l e c t i c geometry.

The n o t a t i o n i s

p u t i n t o c o n t r a v a r i a n t - c o v a r i a n t form and i n t o d i f f e r e n t i a l geometric l a n guage i n Chapter 3.

3. V A R I 0 W DIFFERENCIAL EQ11ACZQ)NS AND CHEIR 0 R I G I W .

We c o n t i n u e i n t h e Some t e c h -

s p i r i t o f § 2 t o d e r i v e v a r i o u s equations and i n d i c a t e problems.

niques o f s o l u t i o n a r e developed h e u r i s t i c a l l y and v a r i o u s mathematical machinery ( t o be e s t a b l i s h e d r i g o r o u s l y l a t e r ) w i l l be m o t i v a t e d .

E M N P L E 3.1.

L e t us use t h e v a r i a t i o n a l t e c h n i q u e o f 52 t o d e r i v e t h e equa-

t i o n o f motion f o r a v i b r a t i n g s t r i n g . P ) i s s t r e t c h e d between 0

Thus assume a s t r i n g (under t e n s i o n

5 x 5 L w i t h endpoints f i x e d ( u ( 0 , t )

= u(L,t)

=

0)

and a f t e r an i n i t i a l ( s m a l l ) displacement u(x,O) = f ( x ) t h e s t r i n g i s r e leased t o v i b r a t e (we assume ut(xyO) = i n i t i a l v e l o c i t y = 0 f o r s i m p l i c i t y ) . L 2 The k i n e t i c energy i s T = (1/2)10 putdx ( p = d e n s i t y ) and f o r small d i s p l a c e L 2 ments t h e p o t e n t i a l energy U = (1/2)J0 uuxdx a p p r o x i m a t e l y ( e x e r c i s e cf.

-

[Col]).

The l e a s t a c t i o n p r i n c i p l e o f Exercise 2.10 then asks t h a t t

(3.1)

(1/2)\

'1

L [PU;

-

2 uux]dxdt = A(u)

tn 0

should be " s t a t i o n a r y " ( o r minimal h e r e ) r e l a t i v e t o t h e admissable c l a s s A 1 = { u E C i n ( x , t ) ( o r piecewise smooth); u ( 0 , t ) = u ( L , t ) = 0; u ( x , t o ) and u(x,t,)

p r e s c r i b e d o r determined}.

Assuming P and P c o n s t a n t f o r s i m p l i c i t y 2 one o b t a i n s ( e x e r c i s e ) (*) utt - uxx = 0 ( c = u / p ) . L e t us use t h i s e q u a t i o n now t o m o t i v a t e a number o f mathematical techniques. F i r s t we observe t h a t t h i s i s a h y p e r b o l i c e q u a t i o n ( t h e wave e q u a t i o n ) w i t h "charact e r i s t i c " l i n e s x * c t = k ( t o be discussed l a t e r ) and i n f a c t t h e general 2 t G ( x - c t ) f o r F,G E C a r b i t r a r y can be p a r t i c u l a r i z e d

solution u = F(x+ct)

here t o g i v e a d ' A l e m b e r t s o l u t i o n where

7 is

(A)

u(x,t) = (1/2)[r(x+ct)

t i'(x-ct)]

t h e odd p e r i o d i c e x t e n s i o n o f f ( o f p e r i o d 2L). d

-.>

(3.2)

x

Note t h a t u immediately s a t i s f i e s ( * ) w i t h u(x,O) = f ( x ) on [O,L]

0)

0.

A t t h e end p o i n t s u ( 0 , t )

odd) and u ( L , t )

=

(1/2)[?(Ltct)

t

(1/2)[fz(ct)

t

and ut(x,

fu(-ct)] = 0 ( s i n c e

F ( L - c t ) ] = 0 (by p e r i o d i c i t y ) .

a r r i v e a t t h e same answer by s e p a r a t i o n o f v a r i a b l e s .

f" i s

Now l e t us

We t r y t o b u i l d up a

s o l u t i o n o f ( * ) i n terms o f elementary products u = X ( x ) T ( t ) which l e a d s t o 2 2 2 ( a ) X " = - A X and T" = -A c T ( i . e . X"Tc2 = XT" which can o n l y h o l d f o r 2 2 X " / X = T"/c T = k ( c o n s t a n t ) - t h a t k = -A due t o boundary c o n d i t i o n s i s

DIFFERENTIAL EQUATIONS

l e f t as an e x e r c i s e ) . t i o n ut(x,O)

11

We b u i l d i n t h e boundary c o n d i t i o n s and t h e c o n d i -

= 0 v i a X(0) = X(L) = 0 w i t h T ' ( 0 ) = 0.

This leads t o X = A n =

nn/L w i t h X = Xn = Sin(nnx/L) and Tn = C o s ( n n c t / L ) . The X e q u a t i o n i n

(0)

w i t h boundary c o n d i t i o n s X(0) = X(L) = 0 i s a S t u r m - L i o u v i l l e problem which i s s o l v a b l e o n l y f o r t h e eigenvalues

( t h e Xn a r e c a l l e d e i g e n f u n c t i o n s ) .

Now un = XnTn s a t i s f i e s (*) except f o r t h e i n i t i a l c o n d i t i o n f ( x ) = u(x,O) and t o accomplish t h i s we t r y an i n f i n i t e sum (3.3)

u(x,t)

=

lm bnXn(x)Tn(t) 1

=

1;

bnSin(nnx/L)Cos(nact/L)

E v i d e n t l y a f i n i t e sum w i l l g e n e r a l l y n o t g i v e f ( x ) = u(x,O)

so we must t r y

an i n f i n i t e sum; on t h e o t h e r hand w h i l e any f i n i t e sum s a t i s f i e s (*) p l u s u ( 0 , t ) = u ( L , t ) = 0 w i t h u (x,O) = 0 one may have convergence problems upon t d i f f e r e n t i a t i n g t h e i n f i n i t e sum. I n any e v e n t t h e r e i s no hope u n l e s s we can s a t i s f y (3.4)

f(x) =

1;

bnSin(nnx/L)

which i s c a l l e d a F o u r i e r s e r i e s ( n o t e t h a t t h e s e r i e s i s p e r i o d i c o f p e r i o d 2L and i s an odd f u n c t i o n so i t r e p r e s e n t s ?(x) on

(--,m)).

Suppose (3.4)

i s v a l i d ( i n some sense) and then, f o r m a l l y , upon n o t i n g t h a t ( e x e r c i s e ) (3.5)

f

Sin(nnx/L)Sin(mnx/L)dx = {

(L/2) f o r m = n for +

0

i t f o l l o w s t h a t ( m u l t i p l y i n g (3.4) by S i n ( m x / L ) and i n t e g r a t i n g termwise) L (3.6)

bm = ( z / L ) J

f(x)Sin(mnx/L)dx 0

2 We remark t h a t f o r ? a s i n d i c a t e d i n ( 3 . 2 ) b = O(l/m ) i s expected b u t f o r rn f o n l y PC, bm = O ( l / m ) would be normal. F u r t h e r ( r e c a l l Sin(A+B) = SinACosB N

f CosASinB) ( 3 . 3 ) and ( 3 . 6 ) l e a d t o

(3.7)

u(x,t) =

1;

bn(l/2)[Sinnn(x+ct)/L

which o f course r e p r e s e n t s ( 1 / 2 ) [ F ( x + c t ) at

(A)

again.

+ Sinnn(x-ct)/L]

+ ?(x-ct)]

and one a r r i v e s f o r m a l l y

G e n e r a l l y t h e method o f s e p a r a t i o n o f v a r i a b l e s w i l l a p p l y t o

many problems where one does n o t know a p r i o r i a s o l u t i o n l i k e

(A)

so we w i l l

want t o examine t h e method and l o o k a t t h e mathematical q u e s t i o n s i t poses for validity.

I n passing we mention t h a 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 s (3.5)

a r e a general consequence o f t h e f a c t t h a t Xn s a t i s f i e s a S t u r m - L i o u v i l l e problem and thus t h e expansion (3.4) i s a general question, namely, s t u d y

12

ROBERT CARROLL

t h e expansion of ( s u i t a b l e ) f u n c t i o n s f i n an i n f i n i t e s e r i e s o f o r t h o g o n a l eigenfunctions.

T h i s i s b e s t t r e a t e d i n t h e c o n t e x t o f H i l b e r t spaces ( o r

r i g g e d H i l b e r t spaces) and h e l p s e x p l a i n t h e need f o r H i l b e r t space t e c h niques i n mathematical physics.

REmARK 3.2,

L e t us use Example 3.1 even f u r t h e r t o m o t i v a t e c e r t a i n methods

i n v o l v i n g g e n e r a l i z e d f u n c t i o n s o f d i s t r i b u t i o n s ( c f . Appendix B).

F i r s t we

A

define the Fourier transform ( f o r n i c e functions f ) ( + ) F f ( h ) = f ( h ) =

-/I f ( x ) e x p ( i h x ) d x .

The F o u r i e r t r a n s f o r m can be extended t o a l a r g e c l a s s

o f d i s t r i b u t i o n s f E 3' f o r example (and beyond) and t h e i n v e r s i o n formula i s g i v e n by ( c f . Appendix B f o r a l l d e t a i l s h e r e ) (1/21r)/:

?(h)exp(-ihx)dh.

(m)

f ( x ) = F-'fA(x)

One d e f i n e s a c o n v o l u t i o n ( f * g ) ( x ) =

g(x-S)dg = 1: f(x-S)g(S)dS

=

LI f ( S )

f o r s u i t a b l e f,g and t h e n F ( f * g ) = FfFg.

Also

t h e 6 f u n c t i o n (which i s n o t a f u n c t i o n a t a l l b u t a measure) i s d e f i n e d by i t s a c t i o n on t e s t f u n c t i o n s

~p E

C E (C:

= Cm f u n c t i o n s w i t h compact s u p p o r t )

2 = c u x x ) as an e q u a t i o n on tt, i n x, t 1. 0, w i t h i n i t i a l d a t a u(x,O) = f ( x ) = F ( x ) on ( - m , m ) and

by t h e r u l e ( 6 , ~ =) ~ ( 0 ) . Now t h i n k o f (*) (u (-a,-)

ut(x,O)

= 0 ( t h i s i s c a l l e d a Cauchy problem).

Suppose t h a t e v e r y t h i n g i n

s i g h t has a F o u r i e r t r a n s f o r m i n x so t h a t F u(x,y) = i?(h,t). Then sat2" 2 2AX / I i s f i e s ( n o t e F f " = ( - i x ) f ) (*A) Ctt - c A u = 0; ~ ( x , o ) = f ( h ) . ConseA

= 0 ) (*.)

q u e n t l y ( s i n c e ut(h,O)

i s t h e measure d e f i n e d by ( * 6 )

$(h,t) p&t)

= ?(h)Coshct = F R u x ( t ) where p x ( t )

= (1/2)[6(x-ct)

t h i s , s i m p l y compute f o r example ( t r e a t i n g f o r purposes o f i n t e g r a t i o n

+ G(x+ct)].

To see

t h e 6 s y m b o l i c a l l y as a f u n c t i o n

- which i s e x p l a i n e d i n Appendix B )

m

(3.8)

eihx6(x-ct)dx

G(x-ct) =

A

= (eihx,6(x-ct))

= ei h c t

m

It f o l l o w s t h a t F p x ( t ) = Coshct and hence by t h e c o n v o l u t i o n theoren

(3.9)

U(x,t) = F

*

p x ( t ) = (1/2)[F(x+ct)

which agrees w i t h (3.7) o r G(x-ct) =

/I G(S-ct)F(x-c)dS

(A).

-t

F(x-c~)]

To check ( 3 . 9 ) compute f o r example F

= (s(c-ct),F(x-c))

= F(x-ct).

*

Thus we have de-

veloped a n o t h e r way t o s o l v e (*) ( t h e F o u r i e r method) which i n v o l v e s t h e use o f F o u r i e r transforms and g e n e r a l i z e d f u n c t i o n s .

T h i s method a l s o i s cap-

able o f great generalization. L e t us c o n t i n u e o u r program o f i n t r o d u c i n g v a r i o u s problems and methods b y d e s c r i b i n g some c l a s s i c a l d i f f e r e n t i a l equations o f e v o l u t i o n type.

Here

one t h i n k s o f some p h y s i c a l system e v o l v i n g i n t i m e from a g i v e n i n i t i a l

DIFFERENTIAL EQUATIONS state.

13

The Cauchy problem f o r t h e wave e q u a t i o n , or t h e f i e l d equations o f

Example 2.11,

a r e o f t h i s t y p e as a r e e.g.

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

2.10 ( i n i t i a l s t a t e s must be p r e s c r i b e d i n a s a t i s f a c t o r y manner).

First

c o n s i d e r a g a i n t h e wave e q u a t i o n .

A d i f f e r e n t i a l problem i s s a i d t o be w e l l posed i f t h e s o l u t i o n

REmARK 3.3.

-

depends ( i n some manner) c o n t i n u o u s l y on t h e boundary c o n d i t i o n s o r d a t a which f o r a p u r e e v o l u t i o n problem means t h e i n i t i a l c o n d i t i o n s . u(x,O) be

= G(x) i n t h e Cauchy problem f o r ( * )

= F ( x ) and ut(x,O)

7 anymore).

Thus l e t

( F need n o t

The ( u n i q u e ) s o l u t i o n , c a l l e d d ' A l e m b e r t s o l u t i o n , i s x+t

+ F(x-ct)] + (1/2)( hx-t

u(x,t) = (l/Z)[F(x+ct)

(3.10)

The p i c t u r e below shows how t h e s o l u t i o n a t ( x , t )

G(c)dc

depends on t h e d a t a a l o n g

The l i n e s x t c t = k a r e c a l l e d c h a r a c t e r i s t i c s h e r e and d e l i m i t t h e domains L e t a compact s e t K be g i v e n i n t h e upper

o f dependence and o f i n f l u e n c e . h a l f p l a n e and t h e compact s e t ;on shown.

t h e x a x i s be t h e r e b y determined as

Suppose t h e d a t a F and G a r e i m p e r f e c t l y known (as i s normal w i t h N

measurement e r r o r e t c . ) b u t suppose a t l e a s t t h a t f o r any such K we can f i n d

F*,G*

so t h a t s u p l F * ( x )

Then f r o m ( 3 . 1 0 ) - ( 3 . 1 1 )

-

F(x)l 5

w i t h u*

%

E

and suplG*(x)

(F*,G*)

( 1 / 2 ) 2 c T ~ = E ( l + c T ) (sup f o r ( x , t ) E K ) . on compact s e t s on

(--,-I -

Lv

G(x)l 2

(*=) s u p [ u * ( x , t )

-

E

(sup o v e r K ) . u(x,t)l

5

E

+

Thus u n i f o r m c o n t r o l o f t h e d a t a

i n s u r e s u n i f o r m c o n t r o l o f t h e s o l u t i o n on compact

s e t s i n t h e upper h a l f plane. suitably generalized

-

As a m a t t e r o f f a c t t h i s k i n d o f p r o p e r t y

-

can be used t o c h a r a c t e r i z e h y p e r b o l i c o p e r a t o r s

( c f . [ C l ;Ga;]).

EXACAmPLE 3 - 4 -

We c o n s i d e r n e x t t h e s i m p l e s t p a r a b o l i c equation, namely, t h e

heat equation

(A*)

ut = u x x (assuming t h e ( c o n s t a n t ) thermal c o n d u c t i v i t y i s

normalized by a change o f v a r i a b l e s t o be 1 ) .

T h i s c o u l d d e s c r i b e f o r ex-

ample t h e e v o l u t i o n o f temperature u i n a b a r s e t between x = 0 and x = L w i t h U(0,t)

= u ( L , t ) = 0 and u(x,O) = f ( x ) .

A s o l u t i o n by s e p a r a t i o n o f

2 v a r i a b l e s i s p o s s i b l e , f o l l o w i n g Example 3.1, and one a r r i v e s a t X " = - A X 2 and T ' = -A T w i t h X(0) = X(L) = 0. Consequently Xn = S i n ( n n x / L ) as b e f o r e

14

ROBERT CARROLL 2

= nn/L) and Tn = exp(-Ant) w i t h

(An

(3.12)

u(x,t)

2 2

1;

=

2

t/L

bne'(n

)Sin(nax/L);

f(x) =

1;

bnSin(nnx/L)

The same F o u r i e r t h e o r y a p p l i e s t o t h e expansion o f f ( c f . ( 3 . 4 ) - ( 3 . 6 ) )

but

f o r t > 0 t h e b e h a v i o r o f t h e i n f i n i t e s e r i e s f o r u i n (3.12) i s v a s t l y d i f Indeed a t f i r s t s i g h t ( 3 . 3 ) may

f e r e n t from t h a t o f t h e s e r i e s , i n (3.3).

even have t r o u b l e c o n v e r g i n g ( i f f i s say o n l y continuous w i t h no "compata2 ? a s i n (3.2) w i t h bn = O ( l / n ) b i l i t y " c o n d i t i o n s a t 0 and L ) . For f Q

u n i f o r m convergence i s assured i n (3.3) b u t a f t e r two termwise d e r i v a t i v e s one expects t r o u b l e .

On t h e o t h e r hand i n (3.12) f o r t

vergence f a c t o r exp[-n2n2t/L

2

3

> 0 one has a con-

which e a t s up p o l y n o m i a l s i n n f o r b r e a k f a s t .

One can d i f f e r e n t i a t e termwise i n x o r t a r b i t r a r i l y o f t e n i n (3.12) s i n c e t h i s o n l y b r i n g s down p o l y n o m i a l s i n n. e l l i p t i c f o r t > 0 (cf.

[Cl;Mil;Trl])

I n f a c t t h e h e a t e q u a t i o n i s hypo-

and u

€

Cm i n ( x , t ) as t h e above argu-

ment w i l l show.

REmARK 3.5. f(x)

(-m

<

Consider a g a i n t h e h e a t e q u a t i o n (A*) w i t h Cauchy data u(x,O) x <

m

c a l l e d a Cauchy and u ( L , t ) value

-

= h(t

where e.g.

f i s continuous and bounded.

=

T h i s i s again

i n i t i a l v a l u e ) problem and when c o n d i t i o n s u ( 0 , t )

= g(t)

a r e a l s o p r e s c r i b e d t h e problem i s c a l l e d a mixed i n i t i a l

boundary v a l u e problem o r a Cauchy problem w i t h boundary c o n d i t i o n s .

I n t h e p r e s e n t s t u a t i o n t h e s o l u t i o n can be w r i t t e n v i a t h e h e a t k e r n e l

f continuous and I f ( x ) l 5 c h e r e b u t some

Thus c o n s i d e r f o r s u i t a b l e f (e.g.

e x p o n e n t i a l growth o f f i s a l s o p e r m i t t e d ) One checks e a s i l y t h a t f o r t > O,Kxx

= Kt

(Am)

u(x,t) =

iz

K(x-S,t)f(S)dS

( e x e r c i s e ) and by growth s t i p u l a -

t i o n s i t i s permissable t o d i f f e r e n t i a t e under t h e i n t e g r a l s i g n i n o b t a i n u x x = ut. K(x-S,t)

-,6 ( x - s )

The p o i n t o f t h i s remark i s t o show now t h a t as t and thus u ( x , t )

c l a s s i c a l methods ( c f .

r,"

K(x-S,t)dS

(3.14)

(6(x-S),f(S))

= f(x).

i n n o t i n g f i r s t t h a t K(x-S,t)

to

0,

We do t h i s by > 0 with

= 1 f o r t > 0 ( e x e r c i s e ) and f o r any 6 > 0

1i m

K(x-S, t ) d S = 0

u n i f o r m l y f o r x E R. that J

[Jl])

-f

(Am) -+

K(x-S,t)dS

To check (3.14) (and

f o r Ix-LI

The c o n c l u s i o n i s immediate.

>

6 equals n-'/

Now, g i v e n

E

as w e l l ) s e t x-6 = (4t)'n so 2 e x p [ - l n ( l d r l f o r In( > 6 / 4 4 t .

(A&)

> 0 t h e r e e x i s t s 6 such t h a t

DIFFERENTIAL EQUATIONS

-

(f(x)

f(S)( 5

E

f o r ( x - F ( 5 26.

15

L e t M = sup ( f ( x ) ( and w r i t e ( I x - y I

I 6)

m

lu(x,t)

(3.15)

-

f(y)l =

I/

-

K(x-S,t)[f(S)

f(y)ldS( 5

m

-

I

K( X-€, ,t)2MdS 5 2MI K ( X - S t )dS t 1 x 4 126 I x-s 126 K(X-E,t)dS + E K(x-S,t)dC 0 K(X-{,t)lf(S) f ( y ) l d S 5 2M/ j I s-Y I <26 Ix-5 126 T h i s k i n d o f s i t u a t i o n (and argument) o c c u r s r e p e a t e d l y i n s o l v i n g PDE.

J

K(x-S,t)If(S)

IX-SII6

f(Y)Ids

+

9

1

-

+

Some k i n d o f k e r n e l tends t o a 6 f u n c t i o n as a parameter ( o r v a r i a b l e ) goes t o a g i v e n v a l u e (see e.g.

Example 3.7 where t h e Poisson i n t e g r a l f o r m u l a

i s discussed). The Schrodinger e q u a t i o n i s b e t t e r discussed l a t e r i n a quan-

EWAmPLE 3.6.

tum mechanical framework b u t we w i l l r e c o r d h e r e a p r e l i m i n a r y v e r s i o n ( c f . 2 [Scl;Trl]). Thus f o r a q u a n t i t y JI (wave f u n c t i o n ) such t h a t l J I l r e p r e s e n t s a p r o b a b i l i t y d e n s i t y one has (h = h/2n where h i s Planck; ( l / i ) J I t = (h/2m)JIxx a l s o as

c o n s t a n t ) (.+)

- q$

where q r e p r e s e n t s a p o t e n t i a l . One can w r i t e t h i s 2 2 ihJIt = - ( h /2m)JIxx + q$ where energy E p /2m t q, E i7iDt,

(Am)

Q

Q

p

%

-ibDx and t h e r e i s a f o r c e F = -grad q ( t h u s ( A m ) says i W t = W where

H

Q

H a m i l t o n i a n o p e r a t o r w i t h q z p o t e n t i a l energy.

T h i s e q u a t i o n has some

d e f e c t s ( i t i s n o t L o r e n t z i n v a r i a n t f o r example i n t h e f o r m ( l / i ) J I t = 3 (R/2m)A$ i n R X R ) b u t i t s t i l l has been very u s e f u l . I f one F o u r i e r t r a n s forms (A+) i n t (Dt

+

- i w ) we o b t a i n (**)

jXx - q$

($ =

= -A2$

T h i s e q u a t i o n has been s t u d i e d e x t e n s i v e l y i n

e x p ( i w t ) d t ; A 2 = 2mw/h).

quantum s c a t t e r i n g t h e o r y f o r example ( c f . [C2,3;Chl;Shl;R1] The a b s t r a c t t h e o r y o f

rI JI(x,t)

(Am)

i n the form

(0.)

and Chapter 2 ) .

ihJIt = HJI; JI(0) = JIo w i t h J I ( t )

t a k i n g values i n a H i l b e r t space f o r example has t h e s o l u t i o n (a*) $ ( t )= exp(-itH/fi)JI

EXAmPLE 3.7.

0

and one s t u d i e s t h e o p e r a t o r e x p ( - i t H / h )

i n Chapter 2.

The Laplace o p e r a t o r has come up a l r e a d y i n Examples 2.5,

e t c . and i s t h e p r o t o t y p i c a l e l l i p t i c o p e r a t o r .

2.11,

L e t us examine b r i e f l y t h e Thus one

problem a r i s i n g f r o m Example 2.5 f r o m t h e p o i n t o f view o f PDE.

has a, say bounded r e g i o n ( = open s e t t o which Green's theorems a p p l y ) R C Rn and a p r e s c r i b e d c o n t i n u o u s f u n c t i o n f g i v e n on t h e boundary r o f R. 2 E C ( 0 ) n Co($) such t h a t A u =

One poses t h e D i r i c h l e t problem o f f i n d i n g u

0 i n R and u = f on

r.

For " w i l d " s e t s R t h i s may be t h e wrong problem and 2 a t L s o l u t i o n s t a k i n g boundary values weakly o r

one i s forced t o l o o k e.g.

one imposes s t r o n g e r c o n d i t i o n s on f and asks say f o r u contiouous second d e r i v a t i v e s ) .

E

C2+a(R) ( H o l d e r

We w i l l n o t deal w i t h t h e p a t h o l o g y h e r e

16

ROBERT CARROLL

and r e f e r e.g.

t o [Gil;Lily2;Mal].

For r e l a t i v e l y n i c e n a s above t h e r e i s

a s a t i s f a c t o r y t h e o r y (some o f which i s d e s c r i b e d l a t e r ) and we s i m p l y g i v e here an example i n t h e p l a n e i n o r d e r t o e x h i b i t an e x p l i c i t s o l u t i o n v i a Thus t a k e n t o be a c i r c l e o f r a d i u s a cen-

t h e Poisson i n t e g r a l formula.

t e r e d a t t h e o r i g i n and w r i t e t h e Laplace o p e r a t o r A i n p o l a r c o o r d i n a t e s 2 2 2 so t h a t (06) [Dr + ( l / r ) D r + ( l / r )De]u = 0 w i t h u = f ( 8 ) on t h e c i r c l e r = a.

We separate v a r i a b l e s a g a i n and t r y b u i l d i n g b l o c k s R ( r ) @ ( e ) s a t i s f y i n g 2 2 (a+) [r R " + r R ' ] / R = -(8"/S) = X . By p e r i o d i c i t y 8 has t h e form Cosne o

Sinne ( A n = n ) and R=, finite at r

rn determines t h e n t h e s o l u t i o n s o f t h e R e q u a t i o n

Thus we c o n s i d e r (om) u ( r , e ) = (1/2)ao +

0.

1;

rn[anCos ne

b S i n ne] where t h e an and bn a r e determined v i a (6*) f ( e ) = a0/2 +

n [a Cos ne

+ bnSin ne].

1" a"

' n One checks e a s i l y ( e x e r c i s e ) t h a t (U) an = ( l / n a

f(e)Cos nede; bn = (l/nan)1:" f ( e ) S i n nede so t h a t 2n 2a (3.16) u ( r , e ) = ( 1 / 2 n ) \ f(e)de + ( l / n ) l y ( r / a I n [ f ( $ )

1:'

where P,(e-$) that

1;

=

ly

(r/a)'Cos

zn = l / ( l - z ) f o r

1;

(3.17)

Now t o sum t h e s e r i e s i n Po we r e c a l l

n(e-$).

121 <

1 so

1;

zn = z / ( l - z ) .

(r/a)nCosn5 = ( 1 / 2 ) 1 7 (r/a)'[ein5

(r/a)[Cosc

- (r/a)1/[1

Then, w r i t i n g 0 - 4 = 5

+

I=

- 2(r/a)~osc+ (r/al21

Combining t h i s w i t h (3.16) we have ( e x e r c i s e ) (3.18)

u(r,e) =

P(r,e,$)f($)d$; jo2=

T h i s i s t h e Poisson i n t e g r a l formula. 0 and as r

4)

+ ti(&$)

-+

2 2 2 2 P = (a -r )/[a -2arCos(e-$) + r 1211 For r < a one checks e a s i l y t h a t AP =

a an argument s i m i l a r t o t h a t i n Example 3.5 shows t h a t P(r,e, (exercire - c f . [Jl]).

4. LINEAR 9EC0ND 0RDER PDE.

I n t h i s s e c t i o n we w i l l g i v e a s k e t c h o f some

b a s i c ideas and theorems i n l i n e a r PDE.

Various i m p o r t a n t theorems w i l l be

proved by c l a s s i c a l methods; l a t e r i n Chapter 3 we w i l l use f u n c t i o n a l a n a l y t i c methods more h e a v i l y i n d e a l i n g w i t h a b s t r a c t PDE.

We w i l l c o n c e n t r a t e

on second o r d e r equations s i n c e t h e y a r e o f most f r e q u e n t occurance i n mathematical p h y s i c s ( t h e main e x c e p t i o n being f o u r t h o r d e r equations i n t h e t h e o r y o f e l a s t i c i t y and some h i g h e r o r d e r n o n l i n e a r equations i n s o l i t o n theory).

P r i o r t o say 1945 t h i s was r e a l l y a l l t h a t was known and t h e t h e o r y

PARTIAL DIFFERENTIAL EQUATIONS

c o n s i s t e d m a i n l y i n a l a r g e bag o f t r i c k s .

17

Around 1950, w i t h t h e advent o f

t h e t h e o r y o f d i s t r i b u t i o n s , and thus w i t h a language i n which t o t a l k about d i f f e r e n t i a t i o n and l i n e a r PDE,not o n l y were general and f a r r e a c h i n g theor i e s developed b u t h i g h e r o r d e r equations c o u l d be n a t u r a l l y t r e a t e d ( c f .

.

h e r e e.g

[Cl ;Col ;Frl ;G2;H1 ;M11; J1 ;Pal ;Sal ; T r l

1).

T h i s a1 1 occured i n con-

j u n c t i o n w i t h t h e development o f t h e t h e o r y o f 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 spaces (TVS) i n f u n c t i o n a l a n a l y s i s and t h e r e was a b e a u t i f u l and profound i n t e r a c t i o n (some o f t h e ensuing machinery w i l l be used a t times l a t e r i n t h i s book). action involved

Subsequent development w i t h s t r o n g g e o m e t r i c a l i n t e r -

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

Fourier i n t e g r a l operators, m i c r o l o c a l i z a t i o n , etc.

However h e r e i n t h i s

s e c t i o n we w i l l proceed c l a s s i c a l l y and t h i n k o f PDE more i n t h e c o n t e x t o f advanced c a l c u l u s ( t h e n a t u r a l h a b i t a t o f c o u r s e ) .

There a r e many e x c e l l e n t

sources o f a d d i t i o n a l m a t e r i a l and we mention o n l y [Col ;Gbl;Jl ;Mkl ;Pel ;Z1]. F i r s t we w i l l c l a s s i f y second o r d e r l i n e a r PDE i n t h e s t a n d a r d way.

For

s i m p l i c i t y t a k e 2 independent v a r i a b l e s x,y and c o n s i d e r (4.1)

Auxx t 2Buxy t Cuyy + Dux t Euy + Fu = G

where A,B,

... a r e

functions o f (x,y) alone ( n o t o f u).

The idea i s now t o

s t a r t a t some p o i n t Po = (xo,yo) and make a c o o r d i n a t e change ( x , y ) around P

0

o f Po.

(6,~)

+

i n o r d e r t o reduce (4.1) t o a " c a n o n i c a l " form i n a neighborhood

One knows ( c f .

[Pel])

t h a t f o r more t h a n two independent v a r i a b l e s

t h e r e d u c t i o n can o n l y o c c u r a t Po i n general ( n o t i n a neighborhood o f Po) b u t f o r two v a r i a b l e s t h e r e i s more f l e x i b i l i t y . Thus w r i t e C = t;(x,y) and 2 n = ~ ( x , y ) (assumed C f u n c t i o n s ) w i t h J = a ( c , n ) / a ( x , y ) = 5 Q - nXsy # 0 X Y

i n a neighborhood o f Po and then one has ux = u6Lx + u ~ , ; ~ , e t c .

...

Putting &

,

-

4

+ 2BuEn t Cunn + + Fu = G such expressions i n (4.1) we o b t a i n ( * ) x u 2 55 2. where i n p a r t i c u l a r ( A ) = AS, t 2B5 X 5 Y + CCy, B = A5 x n x + B ( E x i i y + 5 n y 1 + C S y ~ y ; C = An: + 2Bn n + Cny*. One defines now a d i s c r i m i n e n t ( a ) A = 2 .y B - A C and then 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 g i v e s ( e x e r c i s e ) N

N

tHE0REm 4-1.

2=

z2 - i? = AJ2 so t h a t sgnx

sgnA.

=

D E F L N I t L 0 N 4.2, The e q u a t i o n ( 4 . 1 ) i s s a i d t o be h y p e r b o l i c , e l l i p t i c , o r p a r a b o l i c a t Po ( o r i n a r e g i o n around Po) i f resp. A > 0, A < 0, o r A = 0. One checks e a s i l y t h a t t h e wave e q u a t i o n utt = Au i s h y p e r b o l i c , t h e Laplace e q u a t i o n Au = 0 i s e l l i p t i c , dnd t h e h e a t e q u a t i o n ut = Au i s p a r a b o l i c .

EXAIRPLE 4.3.

The T r i c o m i e q u a t i o n yuxx + u

YY

=

0 i s e l l i p t i c f o r y > 0,

18

ROBERT CARROLL T h i s e q u a t i o n served as a

h y p e r b o l i c f o r y < 0, and p a r a b o l i c f o r y = 0.

model f o r problems i n t r a n s o n i c gas dynamics and l e a d s t o a number o f i n t e r It also played a r o l e i n the

e s t i n g s t u d i e s ( c f . [C1,4;Bl;Bil;Dil;Wl,2]).

development o f t h e o r y f o r s i n g u l a r and degenerate Cauchy problems f o r which It w i l l have c h a r a c t e r i s t i c s ( d e f i n e d below) de-

we r e f e r t o [C4;Dil;Grl]. termined v i a y5,

2

+ 5; = 0 so t h a t i n t h e l o w e r h a l f p l a n e one has curves

dx/dy = td-y o r x-c = * ( 2 / 3 ) ( - ~ ) ~ /i ~l l u s t r a t e d i n (4.2)

3/2

X-c = - ( 2 / 3 ) ( - y )

S e t t i n g now x = x and t = ( 2 / 3 ) ( - ~ ) ~ " one o b t a i n s a normal f o r m (y < 0 )

+ ( 1 / 3 t ) u t which was s t u d i e d i n t h e c o n t e x t o f Euler-Poisson-

( 6 ) uxx = utt

We w i l l

Darboux (EPD) equations by W e i n s t e i n and o t h e r s ( c f . [C4;Dil]). discuss t h i s l a t e r .

Now t h e r e a r e t h r e e c a n o n i c a l forms determined b y t h e f o l l o w i n g theorem ( t h e gaps i n d i c a t e d i n t h e p r o o f a r e meant t o serve as e x e r c i s e s ) . L e t (4.1) be resp. h y p e r b o l i c , p a r a b o l i c , o r e l l i p t i c i n a

CHE0REIl 4-4.

neighborhood (NBH) R o f Po.

+

f o r m resp. u

50

%

= 0, u

55

Then 5,n can be chosen so t h a t (*) takes t h e

+

%

= 0, o r u

w i t h derivatives o f order 5 1).

55

+ u

nn

+

%

= 0

(s

denotes terms

= ? = 0. We can so t h a t assume A(xo,yo) 0 so A(x,y) # 0 i n a NBH R o f Po and we l e t A,(x,y) and 2 X2(x,y) be t h e two r e a l d i s t i n c t r o o t s o f AX2 + 2BX + C = 0 (B - AC > 0).

Roo6:

For A > 0 ( h y p e r b o l i c ) choose 5 and

r)

+

Determine t h e n 5 and n v i a (+) 5,

-2

A' = B i n n . that J = that 5

and nx

A

n

= ?i = 0 and

so t h a t

l Y 2 Y t 0 t h e n A ' > 0 and hence

a (C,r))/a(x,y)

If J

d i v i d i n g by

= X 5

'ti d= 0

so t h a t

i n ( * ) one o b t a i n s t h e canonical form.

(A1

-

A

15

r)

To see t h a t J # 0 n o t e and check t h a t s o l u t i o n s o f ( 6 ) can be found such

2 YY # 0 and n t 0 ( e x e r c i s e ) .

Next i f A = 0 i n Q we can assume n o t Y Y b o t h A and C v a n i s h s i n c e then B = 0 and we no l o n g e r have a second o r d e r

equation.

Take A(x,y)

# 0 i n a NBH a o f Po and l e t X(x,y)

i g u e r o o t o f AX2 + 2BA t C

=

-B/A be t h e un-

L e t n be a s o l u t i o n o f n x = -(B/A)ny (so

0.

.u

C = 0 i n a).

We can a g a i n f i n d

r)

such t h a t

r)

Y

# 0 and f o r 5 p i c k any func-

t i o n independent o f IT i n R ( e . g . 5 = x i s OK s i n c e J = cxny - sYnx n # 0). Y Then A ' = 0 and A' = 2' s i n c e ? = 0. Consequently 6 = 0 and, f o r 5 = x, =

A so

# 0.

D i v i d i n g by

'A' i n

( * ) we have t h e d e s i r e d form.

t i c case i s l e f t as a ( n o n t r i v i a l ) e x e r c i s e .

QED

The e l l i p -

PARTIAL DIFFERENTIAL EQUATIONS

19

C l e a r l y e l l i p t i c e q u a t i o n s have no c h a r a c t e r i s t i c s (and t h e s o l u t i o n o f e.g.

0 i s Cm i n t h e i n t e r i o r o f n ) whereas f o r h y p e r b o l i c e q u a t i o n s t h e

Au

c h a r a c t e r i s t i c s p l a y an i m p o r t a n t r o l e i n t h e t h e o r y .

I n a general sense

c h a r a c t e r i s t i c s r e p r e s e n t curves a l o n g which jump d i s c o n t i n u i t i e s o f f i r s t d e r i v a t i v e s can o c c u r o r a l o n g which s i n g u l a r i t i e s can be propagated.

"In-

i t i a l " d a t a (u and un = normal d e r i v a t i v e ) cannot be p r e s c r i b e d a r b i t r a r i l y Examples o f such b e h a v i o r w i l l be

a l o n g a c h a r a c t e r i s t i c " i n i t i a l " curve.

i l l u s t r a t e d f r o m t i m e t o t i m e as we go along.

Another p r o f o u n d d i f f e r e n c e

between h y p e r b o l i c and e l l i p t i c e q u a t i o n s concerns t h e t y p e o f problems For example we showed i n Remark 3.3 t h a t t h e Cauchy

which a r e w e l l posed.

problem f o r t h e wave e q u a t i o n i n one space dimension was w e l l posed.

Look-

i n g a t t h e Poisson i n t e g r a l formula (3.18) we see a l s o t h a t t h e D i r i c h l e t problem f o r t h e Laplace e q u a t i o n on a c i r c l e i s w e l l posed ( e x e r c i s e

-

t h a t If*

fl 5

on

E

r

-

implies Iu*

uI 5

E

i n n).

-

show

However a w e l l known ex-

ample o f Hadamard shows t h a t t h e Cauchy problem f o r t h e Laplace e q u a t i o n i s = 0 and u (x,O)

as n

Y

my

-f

= Sin(nx)/n.

gn(x) = Sin(nx)/n

f i x e d and x

#

-f

XY

0 u n i f o r m l y on t h e l i n e b u t f o r say y

o s c i l l a t e s w i l d l y w i t h am-

a square 0 5 x 5 1, 0 5 y 5 1,

Consider e.g.

= 0 i n t h e i n t e r i o r and boundary c o n d i t i o n s u(x,O)

u(0,y)

= go(y),

and u(1,y) = g l ( y )

c o r n e r s f o r c o n t i n u i t y say). hence e.g. ux(x,O)

0

>

On t h e o t h e r hand t h e D i r i c h l e t problem f o r t h e wave

m.

e q u a t i o n i s n o t w e l l posed. = fl(x),

f

f i x e d (say x = n/2 even) un(x,y)

mn

p l i t u d e going t o with u

= 0 i n , t h e h a l f p l a n e y L 0 l e t u(x,O) u YY The s o l u t i o n i s u = un = Sinh(ny)Sin(nx)/n2and

Indeed f o r uxx

n o t w e l l posed.

=

f;(x)

Since u

XY

= fo(x),

u(x,l)

( w i t h some c o m p a t a b i l i t y a t t h e

= 0 i m p l i e s ux(x,y)

=

c o n s t a n t and

we must have u x ( x , l ) = f i ( x ) = f i ( x ) .

q u e n t l y a r b i t r a r y ( c o n t i n u o u s ) boundary f u n c t i o n s f

Conse-

cannot be p r e s c r i b e d .

There a r e a few t y p i c a l c l a s s i c a l arguments which a p p l y t o t h e t h r e e p r o t o t y p i c a l e q u a t i o n s i n q u e s t i o n s o f e x i s t e n c e and uniqueness and we w i l l s k e t c h t h i s here.

F i r s t f o r t h e Laplace e q u a t i o n Au = 0 i n a " r e g u l a r " R C

Rn f o r example ( i . e . Green's theorem a p p l i e s t o R) one speaks o f a fundament a l ( o r elementary) s o l u t i o n E o f t h e e q u a t i o n i n t h e s p i r i t AE = 6 where 6 i s t h e D i r a c measure a t t h e o r i g i n .

and E

( 1 / 2 n ) l o g r f o r n = 2 where

(un = 2nn"/r(n/2)). t i o n AE = 0 becomes Err

Here tun

(B)

E ( r ) = r2-n/(2-n)wn f o r n > 2

i s t h e s u r f a c e area o f t h e u n i t sphere

Note t h a t f o r such r a d i a l f u n c t i o n s t h e Laplace equat [(n-l)/r]Er

= 0.

The r e s u l t AE = 6 o r more gen-

e r a l l y , f o r r = Ix-SI, AE = ~ ( x - S ) , w i l l ensue f r o m t h e a n a l y s i s t o f o l l o w (exercise).

Now f o r R g i v e n we e x c i s e a small b a l l

B(S,E) of r a d i u s

E

20

ROBERT CARROLL B = S(S,E)

around 5 w i t h boundary

a sphere o f r a d i u s

E

I n t h e r e g i o n nE = R-B, E ( r ) i s harmonic ( i . e . AE = 0 ) and i f Au E Co i n R

( w i t h u E Co(E)) then one o b t a i n s (4.4) t

( n o t e t h e e x t e r i o r normal n on S p o i n t s inward). Now f o r r = I x - c l = E on S , do 'L w n r n-1 , E ?, r2-n/(2-n)wn, and En 'L - ( 2 - r 1 ) r l - ~ / ( 2 - n ) w ~ . Hence t h e l a s t i n t e g r a l i n (4.4)

u(E) =

(4.5)

I

-+

EAudx

R

u(S) as

-

jr

0 ( e x e r c i s e ) and consequently

E +

[Eu,

-

Enu]do

T h i s formula f o r Au = 0 i s s l i g h t l y m i s l e a d i n g s i n c e i t seems t o suggest t h a t u and un c o u l d b o t h be p r e s c r i b e d on

r -

however we w i l l see t h a t t h e

s o l u t i o n t o t h e D i r i c h l e t problem i s unique so u a l o n e on

R.

r

determines u i n

I f we a p p l y (4.5) t o a harmonic f u n c t i o n u w i t h R a b a l l o f r a d i u s R and

c e n t e r E w i t h aB = S =

-Is

r

-

udo x-E 1 =R Here Is Eun = E ( R ) l S undo = E(R)lB Audx = 0 by Green'Is theorem a p p l i e d t o u u(5) =

(4.6)

and v

=

1.

[Eu,

i t follows that

Enu]do = (l/unRn-'

1"

We have proved t h e mean v a l u e theorem

If Au = 0 i n a r e g i o n c o n t a i n i n g B(E,R) then (4.6) h o l d s i . e . t h e v a l u e a t t h e c e n t e r 5 o f t h e sphere S equals t h e s p h e r i c a l mean value.

CKE0REill 4.5.

rHE0REm 4.6

(mAXImUJlI PRINCIPLE),

L e t R be a bounded p o l y g o n a l l y connected

open s e t and Au = 0 i n R ( u E C O ( 6 ) ) . mum values on a R =

mum o n l y on Phood:

r. I f u

r.

L e t M = max u f o r x E

terior point. u(x) = M i n

Then u a t t a i n s i t s maximum and m i n i -

# c t h e n i n f a c t i t a t t a i n s i t s maximum and m i n i -

n.

5 and suppose u ( x o )

We w i l l show u ( x ) = M f o r any x

€

= M where xo E

n i s an i n -

n and hence by c o n t i n u i t y

F i r s t we n o t e t h a t i f u ( 5 ) = M f o r E E R t h e n u z M on t h e

l a r g e s t b a l l centered a t 5 and l y i n g i n R. Indeed, l e t B(S,R) be such a b a l l and use ( 4 . 6 ) . Thus M = s p h e r i c a l average o f u over S(c,R) and hence

PARTIAL DIFFERENTIAL EQUATIONS

by c o n t i n u i t y u z

M on S(5,R) ( e x e r c i s e ) .

21

T h i s h o l d s a l s o f o r any b a l l

so u : M i n B(S,R). Now l e t y E R be a r b i t r a r y . By p o l y g o n a l l y connected we mean xo and y can be j o i n e d by a polygonal a r c a (= a

B(S,r) C B(S,R)

f i n i t e number o f s t r a i g h t l i n e segments).

...,

p o i n t s xo, xl,

x

One can f i n d a f i n i t e sequence o f

= y on a which a r e c e n t e r s o f b a l l s B(x.,R.)

n t h e p r o p e r t i e s t h a t B(x.,R.) J J

C R and x . E

J

having

J J B ( X ~ - ~ , R ~( e- x~e)r c i s e ) .

(4.7)

Since u = M i n Bo = B(x ,R ) one has u(x,) R1),

0

....

=

0

It f o l l o w s t h a t u ( y ) =

M and hence u

M i n B1

=

= B(xl,

The same argument a p p l i e s t o minima.

M.

Under t h e hypotheses o f Theorem 4.6 t h e D i r i c h l e t problem Au = CHEbREll 4.7, 2 0 i n n, u = f on r(u E Co(?t) R C (a)) has a t most one s o l u t i o n ( i t may n o t have a n y ! ) .

r,

and f on

Phoofi:

F u r t h e r g i v e n two s o l u t i o n s u* and u c o r r e s p o n d i n g t o d a t a f * if

If* -

f) 5

E

on

r

then I u *

- ul 5

E

i n R.

I f one had two s o l u t i o n s u1 and u2 f o r d a t a f t h e n u1

i s f i e s Au = 0 i n R w i t h u I n t h e second s i t u a t i o n i f It f o l l o w s t h a t

5

E

in

=

0 on

7

= u*

r.

-

u2 = u s a t -

By t h e max-min p r i n c i p l e u = @ i n

- u then

A;‘

= 0 i n R with

by Theorem 4.6.

lcl

5

E

on

;. r.

4ED

We w i l l d i s c u s s t h e D i r i c h l e t problem v i a H i l b e r t space methods l a t e r .

Now

f o r h y p e r b o l i c e q u a t i o n s we go t o t h e p r o t o t y p i c a l wave e q u a t i o n and c o n s i d e r t h e method o f s p h e r i c a l means ( c f . [C1,4;Col;Dil;Jl]). This a l s o gives us a good c o n t e x t i n w h i c h t o examine some d i s t r i b u t i o n f o r m u l a s and t o ilThus one d e f i n e s ( c f .

l u s t r a t e t h e usefulness o f d i s t r i b u t i o n techniques. Appendix

B f o r notation etc.) ux(t)

E

.

E; and A x ( t )

E

E; by

.

( A x ( t ) , 9 ( x ) ) = [n/untnl]l 9(x)dx I x Ift ( t h e s u b s c r i p t x i n p x and Ax i s n o t a p a r t i a l d e r i v a t i v e ! ) .

Then v x ( t )

(resp. A x ( t ) ) r e p r e s e n t s p h e r i c a l ( r e s p . s o l i d ) mean v a l u e o p e r a t o r s ( c f . (4.6)).

D

( p

One checks t h a t { u x ( t ) , p ( x ) )

=

( u Y ( 1 ) , 9 ( t y ) ) ( e x e r c i s e ) and (*)

( t ) , 9 ) = ( 1 1 ~ ~f) [I Yia~iaxi]dnn = (i/ontn-l) [ a ~ / a v ] d o n = ) f 4 d x = ( t / n ) ( A x ( t ) , 4 ) = ( t / n ) ( M x ( t ) , d where don = t”’

( l / w n t n-1

dQn9

ROBERT CARROLL

22

denotes t h e e x t e r i o r normal d e r i v a t i v e , and t h e i n t e g r a t i o n s i n (**) a r e r e s p e c t i v e l y o v e r I y I = 1, 1x1 = t, and 1x1 5 t. S i m i l a r l y (*A) D $ A x 2 ntl ]I v d x + [n/untn]/ vdon = ( n / t ) ( u x ( t ) - Ax(t),v) , w i t h (t),g) -[n /unt

ag/aw

i n t e g r a t i o n s o v e r 1x1 2 t and 1x1 = t r e s p e c t i v e l y .

We t h i n k o f p x ( t ) and

A x ( t ) as d i s t r i b u t i o n v a l u e d f u n c t i o n s o f t and have shown t h a t i n t h e sense o f weak v e c t o r valued d i f f e r e n t i a t i o n Dtux(t) = ( t / n ) U X ( t ) ; D t A x ( t )

(4.9)

= (n/t)h,(t)

-

Ax(t)l

v x ( t ) E Ct (El) x i n t h e sense o f s t r o n g d i f f e r b u t we d o n ' t need t h i s here. One can d i f f e r e n t i a t e

I n f a c t one can show t h a t t

e n t i a t i o n ( c f . [C4,5])

+

2

a g a i n i n t h e same s p i r i t and from ( 4 . 9 ) one o b t a i n s ( * e ) {Dt + [(n-1 ) / t l D t l p x ( t ) = Aux(t). while

Dtpx(t)

+

Note a l s o f r o m (4.8) and (4.9) t h a t p,(t) * 6 ( x ) as t + 0 * T E D; i s w e l l d e f i n e d 0. Now f o r T E D; a r b i t r a r y u,(t)

and i n f a c t D [U ( t ) * T] = D t p x ( t ) * T e t c . i n a s t r o n g o r wehk sense. t x For T E D; a r b i t r a r y u ( x , t ) = p x ( t ) * T s a t i s f i e s u t t + CHE0REm 4.8, [(n-l)/t]ut

= Au w i t h u(x,O)

= T and ut(x,O)

= 0.

Thus u s a t i s f i e s a Cauchy problem f o r a s p e c i a l case o f t h e EPD e q u a t i o n

(*&) wtt

t [(2m+l)/t]wt

Aw where 2mt1 = n-1 o r

t h i s s o l u t i o n i s unique i n D; ( c f . [C4,5]) us use t h e mean v a l u e o p e r a t o r

px(t)

- 1.

(n/2)

b u t we o m i t t h e p r o o f .

In fact Now l e t

We can use a f o r m u l a o f Wein-

f o r t h e s o l u t i o n o f ( * & ) which was v e r i f i e d i n t h e

s t e i n ( c f . [C4,5;Dil;W3])

d i s t r i b u t i o n c o n t e x t by t h e a u t h o r .

(*&) w i t h wm(0) = T and wT(0) wm( t

=

and Theorem 4.8 t o s o l v e t h e wave equa-

t i o n which corresponds t o m = -1/2 i n (*&).

(4.10)

m

=

Thus w r i t i n g wm f o r t h e s o l u t i o n o f

0 one has f o r any i n t e g e r p such m+p 1. -1/2

[r ( m t l ) t-2m/2pr (m+p+l)] [ ( 1/ t ) Dt]Pk2(m+p)wmtp( t)]

=

The f o r m u l a can be checked d i r e c t l y b u t [C4,5] It i s assumed here t h a t wm+'

p r o v i d e s a more e l e g a n t p r o o f ,

i s known and t h i s i s assured by t h e formula f o r

- n o t e wq i s known f o r q

s > q 1. -1/2 ( c f . [C4,5]

= (n/2)

-

1 by Theorem4.8)

w S ( t ) = [~r(s+i)t-2~/r(qtl)r(s-q)ll n 2qtl ( t -n ) s-q-lwq(q)dn

(4.11)

0

(again (4.11) can a l s o be v e r i f i e d d i r e c t l y and we r e f e r t o [C4,5]

f o r mean-

i n g ) . Now f o r m = -1/2 (wave e q u a t i o n ) and dimension n = 3 f o r s i m p l i c i t y we can o b t a i n t h e c l a s s i c a l Poisson s o l u t i o n by t a k i n g p = 1 so m+p = 1 / 2 = (n/2)-1 and (w' Dt(tw')

=

w'

+

= p (t)

tD

ii

t

*

T ) (*+) w-'(t)

= px(t)

*

T

t

=

[r(l/2)t/2r(3/2)](1/t)Ot(tw4)

(t2/3)Ax(t)

*

AT.

One can a l s o check

=

23

PARTIAL DIFFERENTIAL EQUATIONS

directly that if u " = Au.

= v satisfies v"

+ (2/t)v'

= Av t h e n u = ( t v ) ' s a t i s f i e s

Therefore The ( u n i q u e ) s o l u t i o n o f ( * 6 ) f o r m > -1/2 w i t h wm(0) = T E

CHEBREIII 4.9.

= 0 i s g i v e n by (4.10)-(4.11)

D; and wT(0) rem 4.8.

'G

where w 'I2-' i s known f r o m Theo-

I n p a r t i c u l a r f o r m = -1/2 one o b t a i n s s o l u t i o n s o f t h e wave equa-

tion. Now n o t e t h a t f o r

d'

= v a g a i n t h e f u n c t i o n 9 = t v a l s o s a t i s f i e s 9 " = LLP

w i t h g ( 0 ) = 0 and g t ( 0 ) = v ( 0 ) .

Hence t h e s o l u t i o n o f t h e wave e q u a t i o n i n

R3 w i t h i n i t i a l values W(0) = T E D; and Wt(0) = Dt[tvx(t)

*

TI + tvx(t)

*

S.

f u n c t i o n s now t h e s o l u t i o n W ( t ) = W(x,t) t h e d a t a S,T on t h e s u r f a c e

SE

D; i s

(*m)

W(t) =

It i s i n t e r e s t i n g t o n o t e t h a t f o r S and T

r

a t a point (x,t)

depends o n l y on

o f the intersection o f the retrograde l i g h t

cone t h r u ( x , t ) w i t h t h e i n i t i a l hyperplane t = 0

For T a f u n c t i o n xe n o t e h e r e f o r m a l l y

(cf. (4.6)).

T h i s f a c t i s a v e r s i o n o f what i s c a l l e d Huygen's p r i n c i p l e . The p i c t u r e a l s o a l l o w s us t o f o r m u l a t e an energy p r i n c i p l e . Thus f o r ( x , t ) 2 f i x e d l e t Q~ be t h e t r u n c a t e d cone i n (4.12) bounded by B ( x , t ) = { S ; t ( S - x l 2 2 2 5 ( t - T ) I , and t h e l a t e r a l s u r f a c e AT. D e f i n e < t 1, B(x,t-T) = 15; Ic-xl t h e energy o f u i n B C R3 a t t i m e t by

CHEbREm 4-10.

Suppose utt = Au ( u E C

2

i n t h e cone o f ( 4 . 1 2 ) ) .

Then t h e

energy s a t i s f i e s E( u,B( x, t-T,T) 5 E( u, B(x, t ) , O ) .

-

1

u t t ) = 2 (a/axi)[Ut(au/ There i s an obvious i d e n t i t y (A*) 2ut(Au 2 2 Given Au = utt now, i n t e g r a t e (.*) o v e r QT, a x i ) ] - [ut t (au/axi) 3,. n o t i n g t h a t t h e r i g h t s i d e i s a divergence, t o o b t a i n (u) 0 = [ 2utvi 2 2 (au/axi) - ( u t t (au/axi) )vt]da ( i n t e g r a l o v e r anT) where v = ( v 1 y v 2 y v 3 y v ) i s t h e e x t e r i o r u n i t normal t o anT. On t h e t o p v = ( O , O , O , l ) , on t h e t

Pkoo6:

1

1

1

24

ROBERT CARROLL

bottom w = ( O , O , O , - l ) ,

2

:w =

wt

so w t = 1/J2.

+ ZE(U,B(x,t),O)

0 = -zE(u,B(x,t-T),T)

(4.15)

1;

and on AT,

Hence

+

2 2 The l a s t t e k n can be w r i t t e n as (A*) 42 1 2ututvi(au/axi) vt(au/axi) 2 2 2 ut vi]do = -42 1 (vt(au/axi) utvi) ]do 5 0. Consequently E(u,B(x,

[I

I

-

[l

- 1

-

5 E(u,B(x,t),O).

t-T,T)

QEO

IfAu = utt,

tHE0REN 4.11.

t h e cone i n (4.12),

u

E

,

and u(x,O) = ut(x,O)

= 0 on t h e base o f

t h e n u E 0 i n t h i s cone. = 0 so E(u,B(x,t-T,T)

E v i d e n t l y E(u,B(x,t),O)

P4006:

C

2

Hence t h e integrand I ( a u / a x . )

2

1

= 0 f o r any T 5 t. 2 t ut = 0 a t any v a l u e o f T 5 t so u = con-

s t a n t and t h e c o n s t a n t must be 0 by c o n t i n u i t y .

QED

T h i s shows t h a t s o l u t i o n s o f Au = utt a t ( x , t ) a r e determined by t h e i r i n i t i a l data u and ut on t h e base o f t h e r e t r o g r a d e l i g h t cone a t ( x , t ) and t h a t s o l u t i o n s w i t h t h e same i n i t i a l data a r e i d e n t i c a l ( i . e . uniqueness holds).

One can e a s i l y show a l s o f r o m Theorem 4.10 t h a t i f t h e i n i t i a l d a t a

u(x,O) and ut(x,O) v a n i s h o u t s i d e o f some compact s e t then c o n s e r v a t i o n o f 3 3 energy h o l d s i n t h e f o r m E(u,R ,T) = E(u,R ,0) ( e x e r c i s e ) . For t h e h e a t e q u a t i o n (and wave e q u a t i o n ) t h e r e a r e a l s o maximum p r i n c i p l e s o f s p e c i a l forms which l e a d t o uniqueness and w e l l posedness r e s u l t s ( c f . [Jl;Spl;Prl;Zl]).

We mention here o n l y a s i m p l e one dimensional theorem f o r

t h e heat e q u a t i o n t o i l l u s t r a t e t h e m a t t e r .

CHEBREN 4.12.

L e t ut = uxx f o r 0 < x

<

Thus

L and 0 < t 5 T w i t h u continuous on

t h e c l o s e d r e c t a n g l e R: 0 5 x 5 L, 0 5 t 5 T.

Then u a t t a i n s i t s maximum

and minimum values on t h e base t = 0 o r on t h e l a t e r a l sides x = 0, x = L .

Phood:

L e t M = max u i n R and suppose t h e max o f u on t h e base and l a t e r a l

sides i s M-E ( E

^t

and

>

0.

>

0).

Let

Define w(x,t)

(x",?)

be a p o i n t where ~(2,:) = M s o 0 < x* < L 2 t ~ ( x - ? ) ~ / 4 ,L Then on t h e base and l a t -

= u(x,t)

e r a l sides w ( x , t ) 5 M-E + ~ / 4= M

-

3 4 4 w h i l e w(;,?)

w i s n o t a t t a i n e d on t h e base o r l a t e r a l s i d e s . w a t t a i n s i t s maximum so 0 < 0 if

at

(?,;)

t" <

< L and 0 <

rv

T o r wt

wt

tradiction.

2 0 if t

- wxx 2 0.

=

M so t h e maximum o f

L e t (i(,?) be a p o i n t where

t 5 T.

At

(?,y) n e c e s s a r i l y

= T and w x x 5 0 ( c o n d i t i o n f o r m a x i m a l i t y ) .

But wt

- w xx

= u

- u xx -

42L2

t The argument f o r a minimum i s s i m i l a r .

<

wt = Hence

0 which i s a conQED

25

PARTIAL DIFFERENTIAL EQUATIONS

The problem ut = uxx, u(x,O) = f ( x ) , u ( 0 , t )

C0R0CCARM 4 13u(L,t)

= g2 t ) ,

F u r t h e r if f *

0

<

f o r continuous and compatable f,gi,

-

f

t 5 T then I u *

1

5

-

\gi

E,

5

uI

E

-

g1 1 5

-

and 19;

E,

= gl(t),

and

has a t most one s o l u t i o n g2)

5 E f o r 0 5 x 5 L and

i n t h e r e g i o n so d e l i m i t e d .

The p r o o f i s l e f t as an e x e r c i s e . h e a t problem i s n o t w e l l posed.

L e t us n o t e here a l s o t h a t t h e backward To see t h i s one r e f e r s t o Example 3.4 where

we saw t h a t t h e f o r w a r d h e a t problem l e a d s t o s o l u t i o n s u ( x , t ) E Cm f o r t >

0.

Suppose we c o u l d s o l v e a backward problem u t = uxx, u ( 0 , t )

u(x,T) = f ( x ) E Co f o r 0 2 t 5 T.

=

u(L,t)

= 0,

L e t g ( x ) = u(x,O) and now s o l v e t h e f o r -

ward problem w i t h g as i n i t i a l data.

By uniqueness t h e s o l u t i o n must equal

u a g a i n and t h i s i m p l i e s f ( x ) = u(x,T)

E

Cm.

T h i s does n o t p r e c l u d e s o l v a -

b i l i t y o f t h e backward problem o f course b u t i t p u t s a s t r o n g r e s t r i c t i o n on the " i n i t i a l " data f ( x ) .

Nor does i t "prove" t h e backward problem i s n o t

w e l l posed i n some sense. for

-m

<

satisfies

x

<

m,

(A+)

However t o see t h e nonwellposedness c o n s i d e r 2 2 u ( x , t ) = ( l / o k ) e x p [ i k x + ok t]. T h i s f u n c t i o n 2 For P = - 1 t h i s i s t h e ut + puXx = 0; u(x,O) = e x p ( i k x ) / o k t

20

(A&)

.

h e a t e q u a t i o n b u t f o r p = 1 i t r e p r e s e n t s a backward h e a t e q u a t i o n ( t One sees t h e n ( f o r P = 1 ) t h a t l u ( x , O ) l 2 l u ( x , t ) l = exp(k t ) / k 2 f o r t > 0. +

-

5 1/k2

+

5. FUREHER &0PZCSC ZN &HE CACCUCW O F U A R I A E l ( D W .

0 as k

+

-f

-t).

b u t f o r any x,

We w i l l deal l a t e r w i t h

v a r i a t i o n a l ideas i n a more g e o m e t r i c a l c o n t e x t ( r e l a t e d t o inechanics f o r example) b u t f o r now l e t us g i v e some e x t e n s i o n s and r e f i n e m e n t s o f t h e development i n 53.

We a r e aiming a t general nonsmooth convex a n a l y s i s even-

t u a l l y (see Chapter 3 ) and t h e s t u d y o f n o n l i n e a r o p e r a t o r e q u a t i o n s .

This

w i l l be p i c k e d up i n Chapter 3 and t h e p r e s e n t d i s c u s s i o n i s i n p a r t h e u r i s t i c and i s based more on c l a s s i c a l a n a l y s i s .

We a l s o i n c l u d e more examples.

I n t h e c a l c u l u s o f v a r i a t i o n s one wants t o d i s t i n g u i s h between 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 an e x t r e m i z i n g f u n c t i o n (we w i l l u s u a l l y t h i n k o f t h e extreme as a minimum). s t u d y o f t h i s (see e.g.

Again t h e r e a r e good sources f o r a d e t a i l e d

[Col;Gl;Il;Tol;Yl])

and we w i l l m e r e l y p r e s e n t some

o f t h e h i g h l i g h t s ( o p t i m a l c o n t r o l t h e o r y w i l l a l s o be discussed b r i e f l y b u t not i n great d e t a i l ) .

L e t us b e g i n by c i t i n g a few more examples o f n a t u r a l

problems which can be t r e a t e d by v a r i a t i o n a l methods.

EXAFIIPCE 5.1 (ISCBPERlmE&RlC P R t B C E m ) . The o r i g i n a l problem r e q u i r e s one t o f i n d a c l o s e d c u r v e o f f i x e d c i r c u m f e r e n c e e n c l o s i n g t h e maximum area. I t 2 2 i s s i m p l e s t t o phrase t h i s as f o l l o w s . Since ds2 = dx + dy one has dx = L ( 1 - (dy/ds)*)15ds and A ( y ) = lo y ( l - (dy/ds)2)4ds i s t o be maximized f o r

26

YE

ROBERT CARROLL

C 1 ( o r piecewise smooth).

parametrized v i a x ( s )

:J

Once y ( s ) i s determined t h e c u r v e can be 2 4 [l - (dy/do) ) do. There a r e v a r i o u s o t h e r ways

o f f o r m u l a t i n g t h e problem and v a r i o u s techniques o f s o l u t i o n can be appl i e d . T h i s problem i s sometimes phrased as max A = $oydx w i t h t o ( t y ' 2 p d x = L. Another c l a s s i c a l problem i s t h e hanging c a b l e .

EMAIIIPCE 5.2.

Suppose a

c a b l e has l e n g t h L w i t h w e i g h t d e n s i t y mg ( c o n s t a n t ) and t h a t t h e supports a r e separated by a l e n g t h H < L. L e t s = a r c l e n g t h and A = Y E C1(0,L), y ( 0 ) = y ( L ) = 01. The p o t e n t i a l energy ( t o be m i n i m i z e d ) a s s o c i a t e d w i t h y L L L i s P(y) = mgIo y ( s ) d s whereas C ( y ) = Jo ( 1 ( d y / d s ) 2 p d s = Jo d x ( s ) = H. This i s sometimes phrased as m i n i m i z i n g P(y) = mglto y ( l + y ' 2 )4 dx w i t h L = $0 ( l t y ' 2 4 dx.

-

Both o f t h e s e problems a r e perhaps most e a s i l y s o l v e d u s i n g t h e method o f Lagrange m u l t i p l i e r s which we w i l l now i n t r o d u c e .

Thus ( c f . [ C o l ; G l ; S t l l )

There i s a d e l i g h t f u l d i s c u s s i o n o f Lagrange m u l t i p l i e r s , v a r -

REmARK 5.3.

i a t i o n a l techniques, e t c . i n [ S t l ]

( a t an elementary l e v e l ) some e s s e n t i a l

f e a t u r e s f r o m which we e x t r a c t here.

The book [ S t l ] develops many themes

connecting d i s c r e t e and continuous v a r i a t i o n a l t y p e problems i n a b e a u t i f u l way and some o f t h i s i n f o r m a t i o n w i l l serve as a guide and i n t r o d u c t i o n t o convex a n a l y s i s t o be t r e a t e d i n Chapter 3.

Thus one t h i n k s f i r s t o f l i n e a r

algebra, m a t r i c e s A,C,

e t c . (we o m i t arrows f o r con-

e t c . and v e c t o r s x,y,

venience i n w r i t i n g ) .

Consider t h e f o l l o w i n g problems, which a r e dual o r T complementary. F i r s t , one asks t o m i n i m i z e (A means transpose) ( * ) Q ( y ) = T -1 T T (1/2)y C y b y s u b j e c t t o t h e c o n s t r a i n t A y = f ( C p o s i t i v e d e f i n i t e and T symmetric). The dual problem i s t o maximize ( A ) - P ( x ) = - ( 1 / 2 ) ( A x - b ) C T T (Ax-b) - x f . The a s s o c i a t e d theorem i s now t h a t min Q ( y ) ( w i t h A y = f ) =

-

max(-P(x)) and t h e " o p t i m a l " dual v a r i a b l e s y,x s a t i s f y C-'y t Ax = b w i t h T A y = f (we w i l l phrase t h i n g s somewhat d i f f e r e n t l y than [ S t l ] a t times i n

A quick proof following [ S t l ] T Q ( y ) i f A y = f . To see a s s e r t s f i r s t t h a t - P 5 Q (weak d u a l i t y ) o r - P ( x )

order t o provide additional perspective). t h i s note t h a t Hence Q + P

( 0 )

Q ( y ) + P(x) = (1/2)[C-'y

0 o r - P 5 Q and P

t

+ Ax - blTC(C-'y + Ax - b].

Q = 0 i f and o n l y i f z = C-'y + Ax - b =

0 ( a t which p o i n t - P becomes maximum and Q minimum). Now d e f i n e a "LagranT T g i a n " by ( 6 ) L(x,y) = Q ( y ) + x (A y - f ) . Then f o r m a l l y 6L/6y = 0 i n v o l v e s T C-'y t Ax = b and GL/sx = 0 y i e l d s A y = f ( n o t e here e.g. 6L/6y = 0 must be i n t e r p e r t e d component wise o r as shorthand f o r a v a r i a t i o n a l d e r i v a t i v e deT -1 T -1 T termined v i a ( d / d ~ ) L ( x , y t w ) I ~ == ~0 - thus ( 1 / 2 ) y C q t ( 1 / 2 ) q C y - b q

27

CALCULUS OF VARIATIONS

t x t

T T A IP = 0 f o r IP a r b i t r a r y i m p l i e s yTC-llp

XTAT = 0 ) .

p o i n t f o r L.

-

T T T T -1 b IP t x A IP = 0 o r y C

-

bT

These " e q u i l i b r i u m " c o n d i t i o n s a c t u a l l y determine a saddle Thus ( + ) min max L = max min L (min i n y, max i n x ) and t h e

saddle p o i n t v a l u e of L i s t h e c o n s t r a i n e d minimum o f Q = n o n c o n s t r a i n e d T T Thus f i r s t l o o k a t max L = max [ Q ( y ) + x ( A y maximum o f (-P). b)] over T T x. I f A y = f t h e max i s Q ( y ) and i f A y # f t h e max i s s i n c e x can be

-

-

chosen v e r y l a r g e i n an a p p r o p r i a t e d i r e c t i o n . The min o v e r y must a v o i d T t h i s so i t can be a t t a i n e d o n l y when A y = f (a c u t e argument developed i n T Hence min max L = min Q f o r A y = f . F u r t h e r s i n c e min Q = max [Stl]). (-P) we can connect L and P and t o do t h i s c o n s i d e r ( m ) min L = min [ ( 1 / 2 ) T T yTC-'y - (b-Ax) y - x f ] o v e r y . Compute 6L/6y = 0 t o o b t a i n y = C(b-Ax) T T so t h a t p u t t i n g t h i s i n ( m ) min L = - ( l / Z ) ( b - A x ) C(b-Ax) = x f = - P ( x ) and thus (+) w i l l f o l l o w .

One has -P(x) 5 L(x,y)

<

Q ( y ) and a t t h e s a d d l e

p o i n t where C-'y t Ax = b one has e q u a l i t y . Thus again max ( - P ( x ) ) = T max min m i n max L ( x , y ) = min Q ( y ) ( f o r A y = f ) . I n these c a l c u l a L(X,Y) = X Y t i o n s t h e dual v a r i a b l e s x correspond t o Lagrange m u l t i p l i e r s ( r e c a l l 6L/6x = 0 gives the c o n s t r a i n t ) .

We r e f e r t o [ S t l ]

f o r a r i c h discussion o f t h i s

from v a r i o u s p o i n t s o f view w i t h many a p p l i c a t i o n s t o r e a l i s t i c p h y s i c a l problems. L e t us i n d i c a t e a continuous v e r s i o n o f a l l t h i s as f o l l o w s ( a g a i n f o l l o w i n g [Stl]).

One c o u l d c o n s i d e r e.g.

a b a r hanging under i t s own w e i g h t w i t h

displacement u, u ( 0 ) = 0, and s t r a i n f o r c e w = c u ' so t h a t e q u i l i b r i u m i s r e p r e s e n t e d by -Ox(cux) = f ( f = p g s a y ) . u ( 0 ) = 01 and A*

rii

- O x w i t h D(A*) = I w

E

For A C1;

.?I

O x w i t h D(A) = t u E C1;

w(1) = 01 t h i s has t h e f o r m

A*cAu = f (which forms t h e g u i d i n g p a t t e r n o f [ S t l ] ) . Now s e t Q(w) = ( 1 / 2 ) 1 2 1 2 and w(1) = 0 ( f r e e end) w h i l e P(u) = (1/2)10 c u ' = f, ( l / c ) w dx w i t h w ' io 1 1 dx - lo fudx ( w i t h u ( 0 ) = 0 ) . Then L(u,w) = lo [(1/2c)w2 t u ( w l + f ) ] d x and - ( c u ' ) ' = f ( u ( 0 ) = 0 ) w h i l e ( c f . ( 0 ) ) P ( u ) + Q(w) can be w r i t t e n 1 2 1 1 P + Q = ( l / 2 ) J o [w/c - u ' ) cdx (-w' = f and -10 f u d x = w'udx = -10 u'wdx

min P ( u )

'L

Ji

s i n c e u ( 0 ) = w(1) = 0 ) . One o b t a i n s again max ( - P ( u ) ) = max min L(u,w) = min max L(u,w) = min Q ( w ) f o r w ' + f = 0 ( e x e r c i s e ) . These examples w i l l w u serve as good models f o r m o t i v a t i n g (and u n d e r s t a n d i n g ) t h e s e c t i o n on convex a n a l y s i s i n Chapter 3 as w e l l as v a r i o u s techniques which a r i s e i n s t u d y i n g weak p r o b l e m i n d i f f e r e n t i a l e q u a t i o n s .

REmARK 5.4.

We w i l l g i v e a f u r t h e r example from [ S t l ] showing how c e r t a i n

n o n l i n e a r programming problems i n v o l v i n g c o s t o p t i m i z a t i o n a r e e s s e n t i a l l y t h e same as problems i n c l a s s i c a l mechanics ( t h e l a t t e r s u b j e c t t o be

28

ROBERT CARROLL

Again t h i s m a t e r i a l i s a l s o pre-

developed l a t e r i n c o n s i d e r a b l e d e t a i l ) .

l i m i n a r y t o t h e s e c t i o n on convex a n a l y s i s i n Chapter 3.

Thus one produces

a comodity represented by x and wants t o minimize C(x) ( c o s t ) s u b j e c t t o some admissable c o n s t r a i n t s A ( x ) ( x i s a v e c t o r ) .

L e t us t a k e x E R and C,

A t o be convex f u n c t i o n s ( i . e . C(ax t ( 1 - a ) y ) 5 a C ( x ) t (1-a)C(y) f o r 0 5 CY < 1

-

f o r s t r i c t l y convex one r e p l a c e s 5 by < ) .

l e t (**) m(b) = min C(x) f o r A ( x ) 5 b. b =

^b

such t h a t C(x*) = m ( 6 ) .

and f i n d

I n p a r t i c u l a r f o r given b

The s p e c i f i c problem now i s t o f i x As b increases m(b) must decrease

( s i n c e more x a r e admissable) and i n f a c t m(b) w i l l be a convex f u n c t i o n (exercise

-

Draw t h i s f o r C = x

cf. [Stl]).

(m(b) = min x

2 for x 2 b

2 and c o n s t r a i n t x 5 b as

= b2 f o r b < 0 and = 0 f o r b > 0 ) .

Since m i s con-

A

vex a t any p o i n t b t h e r e i s a " s u p p o r t " tangent p l a n e ( = tangent l i n e h e r e ) and t h e l i n e does n o t go above t h e graph o f m.

One can a l s o add a t e r m

?(b-^b) and t i l t t h e p i c t u r e t o make t h e tangent l i n e h o r i z o n t a l ( c f . [ S t l ] ) .

;

Then e v i d e n t l y (*A) C(G) = [C(x) t G(b-$)] and 2 0 i s in fact 4 b A(x)Lb A m in a Lagrange m u l t i p l i e r ( - y = a m / a b a t b = b ) . F u r t h e r (**) C(:) =

To see t h i s , f o r any x i n (*a) choose t h e same x i n [C(x) t ;(A(x) - ;)I. w i t h b = A(x); t h e n t h e b r a c k e t s agree and (*@) 2 (*A) s i n c e (*A)

(*A)

Consequently f o r any b and x t h e requirement A ( x ) 5

a l l o w s o t h e r choices. b implies

(**I5 A

(recall

(*A) A

A

$ 2 0).

Hence (*A) g i v e s t h e same v a l u e as 4

A

(**). Now a t x, y ( A ( x ) - b ) 2 0 n e c e s s a r i l y , and consequently, s i n c e y 2 0 Finand A(?) 5 we must have t h e Kuhn-Tucker c o n d i t i o n $(A(;) = 0. * a l l y , s e t t i n g L(x,y) = C(x) t y ( A ( x ) - b ) , one a r r i v e s a t (C and A s t r i c t l y

t,

2)

convex)

m i n,

(

*

A ( x )5b

Evidently a t y =

C(x) =

y*

max min L(X1Y) Y O x

t h e two minima agree v i a (**) and f o r o t h e r y 2 0 ( A ( x )

t)

t).

Thus t h e c o n s t r a i n e d minimum on t h e l e f t i n C(x) 2 C(x) t y ( A ( x ) (5.2) s p l i t s i n t o an unconstrained minimum o f L w i t h a parameter y f o l l o w e d

<

by a ( d u a l ) maximum o v e r y.

The o p t i m a l

^x

and

$

a l s o s a t i s f y t h e Kuhn-

Tucker c o n d i t i o n aL/ax = 0 = aC/ax t yaA/ax and $(A($)

-

^b)

=

2 and A(x) = x one has L = x2 2 , max min L = -y / 4 - by. Hence

t i c u l a r f o r our example C ( x ) = x 2x t y = 0 a t x = -y/2 and

X

Y10

x

I n par-

0.

t y(x-$) =

so

CALCULUS OF VARIATIONS max 2 ( - Y /4 YLO

-

by) = 0 a t

REmARK 5.5.

$

= 0 for

$

>

29

"2 A 0 and = b a t y = -2;

Now ( c o n t i n u i n g Remark 5.4 and f o l l o w i n g [ S t l ] )

for

<

0

(x* =^b)

l e t us l o o k a t

T h i s w i l l l e a d us i n t o t h e LeConsider t h e parabola f ( x ) = x 2 and l o o k

some o f t h e geometry behind t h i s a n a l y s i s . gendre-fenchel a t lines yx

-

(L-F) transform.

d o f s l o p e y l y i n g below t h e parabola

(5.3)

The 1 ne i s below t h e parabola when (*&) d ~ y - xx2 and t h e r i g h t s i d e y x 2 - x s l a r g e s t as a f u n c t i o n o f x when i t s d e r i v a t i v e y - 2x = 0 a t which 2 2 p o i n t x = y/2 and (*&I becomes d 2 y /4; a t d = y / 4 t h e l i n e touches t h e parabola.

The envelope o f a l l these t o u c h i n g l i n e s i s t h e parabola a g a i n

o f course ( e x e r c i s e ) and one g e t s back t o x2 by always l o o k i n g f o r t h e h i g h 2 e x t l i n e ( * 4 ) x2 = The maximum o c c u r s a t y = 2x. There i s [yx - y /4]. Y 2 t h e r e f o r e a d u a l i t y between convex f u n c t i o n s (e.g. x ) and t a n g e n t l i n e s ; one w r i t e s F * ( y ) i n s t e a d o f d ( y ) and F* i s t h e c o n j u g a t e f u n c t i o n .

Gener-

a l l y g i v e n a convex f u n c t i o n F ( x ) f o r each s l o p e y d e f i n e ( * m ) F * ( y ) = d = ma x [yx - F ( x ) ] . Then F* i s a l s o convex ( t h i s i s examined l a t e r ) aild f o r X F*(y) 2 yx - F(x) o r F(x) y x - F * ( y ) . Then maximizing o v e r y ma x F t h e maximum o f x y - F ( x ) w i l l For smooth [yx F * ( y ) ] . gives F(x) = Y 2 occur f o r y = F ' ( x ) o r x = y / 2 i n o u r example and f * ( y ) = y /4. Going back every x,y.

we can say y x 2 ing F(x) = x

-

.

- D F * ( y ) = 0 o r x - y / 2 = 0, g i v Y F* i s t h e L-F t r a n s f o r m and t h e t r a n s f o r m i s

F * ( y ) i s maximum where x The map F

-+

i n v o l u t i v e i n t h e sense t h a t F

+

F*

-t

F.

I n mechanics t h i s i s t h e map be-

Thus L = tween t h e Lagrangian L + t h e H a m i l t o n i a n H (see [ A l l and 5 1 . 7 ) . 2 . ma x 2 (1/2)mv2 - U = (1/2)mv i f U = 0 and H(p) = [pv - (1/2)mv 1; p - L ' ( v ) 2 = 0 g i v e s p = mv and H(p) = (1/2)mv2 = p /2m. G e o m e t r i c a l l y one i s p a s s i n g here from a tangent bundle t o a cotangent bundle and t h i s w i l l be c l a r i f i e d later.

REmARK 5 . 6 ,

L e t us r e c a l l a few o t h e r v e r s i o n s o f Lagrange m u l t i p l i e r s t o

supplement Remarks 5 . 3 , m i n i m i z e F(x,y,z) P

0

For example i n advanced c a l c u l u s one wants t o

s u b j e c t t o a c o n s t r a i n t G(x,y,z)

= 0.

Thus a t a minimum

say t h e d i r e c t i o n a l d e r i v a t i v e o f F i n any d i r e c t i o n tangent t o S: G = 0

a t Po must be zero. 0.

5.4.

T h i s i m p l i e s v F I S and hence VF = - V A G

S i m i l a r l y i f t h e r e a r e two c o n s t r a i n t s G(x,y,z)

o r VF + AVG =

= 0 and H(x,y,z)

=

0

30

ROBERT CARROLL

t h e n t h e s e s u r f a c e s i n t e r s e c t ( g e n e r i c a l l y ) i n a c u v e C and t h e d i r e c t i o n a l d e r i v a t i v e o f F a l o n g C m u s t b e 0. Hence OF i s i n t h e p l a n e spanned b y V G and VH o r O F = -hlvG

-

T h i s n o t i o n extends t o f u n c t i o n a l s i n a

h2vH.

s t r a i g h t f o r w a r d manner u s i n g F r e c h e t and Gateaux d e r i v a t i v e s f o r example a n d t h i s w i l l be covered l a t e r .

F o r now s i m p l y e x t e n d t h e c l a s s i c a l i d e a t o

f u n c t i o n a l s v i a E u l e r ' s " t r i c k " o f 52. 5.1-5.2

Thus f o r example t h i n k o f Examples

i n t h e form o f m i n i m i z i n g @ ( y ) =

s u b j e c t t o a con1 s t r a i n t * ( y ) = $0 G ( x , y , y ' ) d x - c = 0. Here y e A = { Y E C ( 0 ), y ( 0 ) = *,7 xo 0, y ( x o ) = yo} f o r example and we c o n s i d e r ( f o r y m i n i m i z i n g ) @ ( c , 6 ) = 1 @ (y+ s p+ 6x ) f o r v and x t e s t f u n c t i o n s i n C (O,xo) v a n i s h i n g a t 0 and xo = *(y+Ep+&)

w h i l e ;(€,ti) der that

Z ( E , ~=) 0

=

F(x,y,y')dx

$0

We need a two p a r a m e t e r f a m i l y h e r e i n o r -

0.

w i l l make sense.

T h i s w i l l a l s o be done l a t e r v i a t h e

F r e c h e t and Gateaux d e r i v a t i v e ( c f . [ C o l ; T o l ] s h o u l d hav e

t hvz =

0;

so s e t t i n g [F]

Y dx = 0 o r s i m p l y

= D

F

XY' (A*) ,F ,[]

0 at

-

E

=

6 = 0 ( w h e re

and Chapt er 3 ) .

v refers t o

Now one

~ , 6 as v a r i a b l e s )

F one a r r i v e s a t J [ F t XG] v d x = J [ F + h G ] Y Y A p p l y i n g t h i s t o Example 5.1 we o b t h [ G I y = 0.

9

t a i n ( F = y and G = ( l t y " ? )

Dx[y'/(l+y'2)!5]

= l/X.

This y i e l d s a f a m i l y

o f c i r c l e s ( e x e r c i s e ) . F o r Example 5.2, F = y ( l + y ' 2 ) 4 and G = ( l + y ' 2 y 4 so we o b t a i n (u) ( y + h ) [ ( l + y i 2 f 4 - ~ ' ~ / ( l + y ' ~ =? c] ( n o t e y ' H - H)' = y" Y' I' + y ' H ' - Hyy' - Hy,y - y ' ( H i , - H ) = 0 so y ' H - H = c - h e r e H = HY ' Y' Y Y' F + h G ). T h i s y i e l d s y + A = c C o s h ( x / c t ? ) w h i c h i s a c a t e n a r y . L e t us m e n t i o n h e r e some o f t h e c l a s s i c a l n e c e s s a r y c o n d i t i o n s

REmARK 5.7.

f o r a minimum i n t h e c a l c u l u s o f v a r i a t i o n s .

C l e a r l y t h e Euler equations

f a l l i n t o t h i s c a t e g o r y and one can c a r r y t h e a n a l y s i s o f 52 a l i t t l e f u r t h e r i n o r d e r t o o b t a i n t h e so c a l l e d L e g e ndre c o n d i t i o n .

Thus i n T ( z ) =

( c f . ( 2 . 3 ) ) l e t us s e t z = y k p as b e f o r e and expand by 2 T a y l o r ' s t h eore m ( c f . [ C o l ] ) (A*) T ( z ) = T ( y ) t ~ T , ( y , p ) t ( 1 / 2 ) ~T2(Y,p)

$0

F(x,z,z')dx

where T 1 (y , v )

= $0

[ F p + F , ~ ' ] d xas b e f o r e and Y Y X

T2(Y,v)

(5.4) The b a r i n

FYY

= ]00[Fyyp2 t 2 F y y l w '

+

FY ' Y , ~ " ] d x

e t c . means t h a t t h e a r g u m e n t y , y '

y t ~ 9and y ' = y ' t w ' f o r some 0

i s t o be r e p l a c e d by y =

We know T 1 ( y , p ) v a n i s h e s when y i s

5 8 5 E.

an e x t r e m a l and t h e n a n e c e s s a r y c o n d i t i o n f o r a minimum i s t h a t T2(.Y,v) 0 f o r a r b i t r a r y q E @ = say

now l e t s

E

+

{lp

p i e c e w i s e smooth, ~ ( 0 =) p ( x o ) = 0 1 .

0 i n ( 5 . 4 ) one wants T 2 ( y , p )

p i e c e w i s e l i n e a r f u n i t i o n as i n [ C o l ] : p = Ja[l

- (x-a)/~] for

u

1x

I f one

0 and p i c k i n g e . g . P t o be a

p =

5 U f o ; and

2

Jg[I

p = 0

+ (x-u)/u]

for

a-LI

L x 5a;

elsewhere, i t f o l l o w s t h a t

CALCULUS OF VARIATIONS

31

T 2 ( Y y q ) becomes an i n t e g r a l o v e r a-u 5 x 5 a+u w i t h q a 2 = l / u .

As

CJ

-+

0 the

o n l y c o n t r i b u t i o n comes f r o m t h e l a s t t e r m i n (5.4) and one a r r i v e s a t t h e Legendre c o n d i t i o n F (x,y,y') 0 a t x = a (a b e i n g a r b i t r a r y i n ( 0 , ~ ~ ) ) . Y'Y' RElllARK 5.8. There i s a n o t h e r l o c a l necessary c o n d i t i o n due t o W e i e r s t r a s s which can be strengthened, i n c o n n e c t i o n w i t h t h e H i l b e r t i n v a r i a n t i n t e g r a l , t o a l s o p r o v i d e s u f f i c i e n t c o n d i t i o n s f o r a s t r o n g extremum ( c f . [G1;11; The H i l b e r t i n t e g r a l w i l l a r i s e more n a t u r a l l y however i n c o n n e c t i o n

Toll).

w i t h t h e Hamilton-Jacobi e q u a t i o n l a t e r , when we deal w i t h mechanics, so we

w i l l o m i t any d i s c u s s i o n o f these m a t t e r s here.

L e t us n o t o m i t however t h e

Jacobi necessary c o n d i t i o n which i s o f a g l o b a l n a t u r e .

Thus e.g.

the arc

o f a g r e a t c i r c l e i s t h e s h o r t e s t c u r v e on a sphere c o n n e c t i n g two p o i n t s o n l y i f t h e r e a r e n o t two d i a m e t r i c a l l y opposed p o i n t s o f t h e sphere w i t h i n In particular F > 0, which i s l o c a l , cannot be a s u f f i c i e n t Y'Y' c o n d i t i o n f o r a minimum. I n o r d e r t o study t h i s t y p e o f s i t u a t i o n c o n s i d e r 2 again (A@) and w r i t e ( c f . [ G l ; I l ] ) $0 F y y l w ' d x = $0 F ' ( 9 )'/2dx = 2 YY -(1/2)$0 [DxFyyl]v dx so t h a t the arc.

V

1

-0

(5.5)

T2(y,v)

= 2S2T(y) =

[[F YY - DxFYYl l i p 2 + F y l y l v ' 2 ] d x

0

Here F e t c . a r e e v a l u a t e d a t (x,y,y') where y i s e x t r e m a l . L e t Q = ( 1 / 2 ) YY D F 3 and assume P = (1/2)FylYl 2 0. W r i t e t h e n ( 5 . 5 ) as IFYY x YY' X

-

jo [QP 0

a2T2(y) =

(5.6)

-

Dx(Pip')llpdx

where ~ ( 0 =) @ ( x o ) = 0. The Jacobi e q u a t i o n f o r t h e minimum problem a s s o c i a t e d w i t h

D E F l N I C Z 0 N 5-9.

T ( - ) i s the Euler type equation conjugate t o 0 i f and

T.

(A&)

CHE0RElll 5-10.

Qp

- Dx(PIP')

ip

= 0.

A point? i s called

$ 0

which vanishes a t 0

and assume [O,xo]

c o n t a i n s no p o i n t s

has a n o n t r i v i a l s o l u t i o n 9

One can n o r m a l i z e

'v

(A&)

here w i t h ~ ' ( 0 =) 1.

L e t P ( x ) > 0 on [O,xo]

Then a2T2(y) i s p o s i t i v e d e f i n i t e f o r 9 E Q ( i . e . ~ ( 0 *) 2 Conversely i f 6 T2(y) i s p o s i t i v e d e f i n i t e ( r e s p . n o n n e g a t i v e )

x c o n j u g a t e t o 0.

9(x0) = 0).

t h e n t h e r e a r e no p o i n t s IP E

Q

?

c o n j u g a t e t o 0 i n [O,xo]

( r e s p . i n (O,xo)

-

here

i s assumed).

The p r o o f i s sketched below f o l l o w i n g [ G l ] .

2 We know t h a t 6 T ( y ) 2 0 i s a

necessary c o n d i t i o n f o r a minimum so

t 0 R 0 L L A R g 5-11 ( J A t 0 B l N E C E ~ ~ A~ ~R N~D I E I ~ N ) .I f T(.) has a minimum a t y

32

ROBERT CARROLL

(and P ( x ) > 0 ) t h e n t h e r e a r e no p o i n t s conjugate t o 0 on ( 0 , ~ ~ ) . I f y i s an extremal f o r T ( - )

C 0 l l 0 C C A l l ~ 5.12 ( J A C 0 3 1 kllFFZCIENC C 0 N D I C I 0 N ) . w i t h Fy

-

DxFyl = 0, P ( x ) > 0, and [O,xo]

does n o t c o n t a i n p o i n t s c o n j u g a t e

t o 0 then T ( * ) has a weak minimum a t y.

2 F o l l o w i n g Legendre we n o t e t h a t f p d ( w ) = 0 f o r R o o 6 0 6 Theohem 5.10: any reasonable w and t r y t o choose w so t h a t I = PP" + Qp2 + (w 2 ) ' = 2 (Q t w')p2 + 2 ~ 9 +' i s a p e r f e c t square. Thus i f e.g. P(Q + w ' ) = w 2 2 and consequently 6 T ( y ) 2 0 ( r e c a l l P > 0 ) . Furthen I P(q' + (w/P)9)

2 t h e r (assuming w e x i s t s as i n d i c a t e d ) i f 9 i s now such t h a t 6 T ( y ) = 0 t h e n must s a t i s f y t h e Jacobi e q u a t i o n 2 P ( Q t w ' ) = w and P > 0 - r e c a l l 9

FI

(A&)

E

a).

t o g e t h e r w i t h P ' + (w/P)Ip

P u t t i n g x = 0 one f i n d s then ~ ' ( 0 )

= 0 which c o n t r a d i c t s 9 being a n o n t r i v i a l s o l u t i o n o f ( ' 6 )

lows from 9 ' + ( w / P ) v definite.

= 0).

0 (since

=

(IP : 0

also f o l -

2

It t h e r e f o r e f o l l o w s t h a t 6 T(y) i s p o s i t i v e

To see t h a t w can be found as above we use t h e f a c t t h a t t h e r e

a r e no p o i n t s

c o n j u g a t e t o 0 on [O,xo].

As usual w i t h R i c c a t i equations

one w r i t e s w = -u'P/u so t h a t t h e e q u a t i o n becomes

(A+)

- ( P u ' ) ' t Qu = 0

which i n f a c t c o i n c i d e s w i t h t h e Jacobi e q u a t i o n f o r P !

I f t h e r e a r e no

points

? conjugate

t o 0 on [O,xo]

then there i s a n o n t r i v i a l s o l u t i o n u n o t

so t h a t w e x i s t s on t h e whole i n t e r v a l .

v a n i s h i n g on [O,xo]

This l a s t p o i n t

r e q u i r e s a l s o u ( 0 ) i 0 and t h i s can be achieved by an argument based on cont i n u o u s dependence o f s o l u t i o n s o f d i f f e r e n t i a l equations on i n i t i a l data (cf. [Gl] [O,xo]

- w

-

one works on [-€,x0]

w i t h u ( - E ) = 0, u ' ( - E ) = 1 and u

= - u ' P / u i s t o apply o n l y on [O,xo]).

s e r t i o n o f Theorem 5.10. t o p y argument i n [ G l ] .

I,

The second a s s e r t i o n i s proved by a k i n d o f homoThus t h e f a m i l y

xO

(5.7)

J ( t )=

+ 0 on

T h i s proves t h e f i r s t as-

[(Fv"+ W 2 ) t

-

+ l ~ ' ~ ( 1t ) ] d x

2 x t h e l a t t e r has no conjugate p o i n t s t o 0. connects 6 T 2 ( y ) and 1 ~ 0 9 ' ~ dand Then one must show t h a t as t goes from 0 t o 1 no c o n j u g a t e p o i n t s can a r i s e . We r e f e r t o [ G l ] f o r t h e p r o o f which i s s t r a i g h t f o r w a r d b u t t e d i o u s .

Pmod

o d Ca/ru.Ueec~y 5 . 1 2 :

QED

C o r o l l a r y 5.11 i s immediate ( c f . h e r e Remark 5 . 7 -

T2(Y,9) 2 0 i s a necessary c o n d i t i o n f o r a minimum). To prove C o r o l l a r y 5.12 one f i n d s f i r s t an i n t e r v a l [O,xo+~] which c o n t a i n s no p o i n t s c o n j u g a t e t o 0 and where P > 0 (by continuous dependence as above). Consider J ( y ) = [ P v a 2 + Qp 2 I d x - a2$o 9 I 2 d x w i t h Jacobi e q u a t i o n (A#) Qp - Dx[(P-u 2 ) P I ]

$0

= 0.

Now P ( x ) 1. n > 0 on [O,xo+~] and t h e s o l u t i o n

9

of

(AM)

w i t h 9 ( 0 ) = 0,

CALCULUS

OF

33

VARIATIONS

~ ' ( 0 =) 1 depends c o n t i n u o u s l y on a ; hence f o r s u f f i c i e n t l y small a , P ( x ) a2. > 0 on [O,xo] and t h e lp above s a t i s f y i n g ( A m ) does n o t v a n i s h on (O,xo]. Hence by Theorem 5.10 J(y) i s p o s i t i v e d e f i n i t e f o r a s m a l l and t h u s t h e r e 2 [ & I 2 + Qp ]dx c/fo lpI2dx. T h i s i m p l i e s y i s

e x i s t s c such t h a t (.*)/,o minimizing since (5.8)

(A*)

-

T(y*p)

can be p u t i n t h e f o r m (T1(y,lp)

T(y) =

\

= 0)

xO [ b I 2

+ Qp2]dx +

[Slp2

+ ~ ' ~ ] d x

0

where c ( x ) , q ( x )

+

0 u n i f o r m l y on [O,xo]

as lllplll

+

0 (exercise

-

t a k e e.g.

and we assume h e r e F F ,, and 1 xyy: ; 4 F I a r e continuous i n a l l arguments t o g e t h e r ) . Now l l p l = 110 lp dS YiY 2 2 (lo l p p ' 2 d ~ ) sso i $0 lp dx < (x0/2)$0 lpI2d5 and consequently t h e l a s t t e r m i n -2 2 (5.8) i s bounded by ~ ( l + x ~ / 2 ) : / 0 l p ' dS i f 151 5 E and 111 5 E . Taking E

lllpll

= sup

t sup I l p ' ( x ) I on [O,xo]

Ilp(x)I

small enough we can make T ( y * )

REmARK 5.13.

-

.ry<

T ( y ) > 0 f o r s u f f i c i e n t l y small 1 1 ~ 1 1 ~ .QED

Many i n t e r e s t i n g problems i n v o l v i n g v a r i a t i o n a l methods a r i s e

i n o p t i m a l c o n t r o l t h e o r y ( c f . [ C l l ;Hel; I 1 ;K1 ;Lel ;Li3;Pol ;Y1 ; Z e l ] ) .

Many

problems can be phrased and s o l v e d and we c o n s i d e r a few here. Thus f i r s t t w r i t e (.A) T(y,u) = l t 1 F(t,y,);,u)dt % d y l d t ) where u i s a " c o n t r o l " v a r -

(i

iable.

u

Other f u n c t i o f a l s t o m i n i m i z e o r maximize a r e e.g.

*

(00)

T(y,u) =

i s c a l l e d an e n d p o i n t f u n c t i o n a l . crT(y,u) + B*(to,y(to),tl ,y(tl 1). Here One can have c o n s t r a i n t s G i ( t , y ( t ) , y ( t ) , u ( t ) ) = 0 o r 5 0, o r e.g. i ( t ) = The admissable y E A can have f i x e d endpoints y ( t o ) = yo \P(t,y(t),u(t)). and y ( t ) = y o r f i x e d - f r e e e n d p o i n t combinations, o r p e r i o d i c e n d p o i n t 1 1 c o n d i t i o n s , e t c . w h i l e u E 11 = some space o f admissable c o n t r o l s . Such an a b s t r a c t f o r m u l a t i o n however obscures t h e m a t t e r by n o t c a t c h i n g t h e f l a v o r Hence l e t us g i v e here some t y p i c a l problems

o r some a t t r a c t i v e examples.

which a r i s e and we w i l l go t h e a b s t r a c t t h e o r y l a t e r ( o r a t l e a s t develop t h e a b s t r a c t framework w i t h i n which c o n t r o l t h e o r y can be phrased). f i r s t l e t (06)

q(t)

Thus

= A ( t ) y ( t ) + B ( t ) u ( t ) where say (y,u) a r e column v e c t o r s

w i t h A,B b e i n g 2 X 2 m a t r i c e s . One assumes y ( 0 ) = y o (y1,y2) and and r e s t r i c t s t h e c o n t r o l s by l u i ( t ) l 5 1. One s e l e c t s a t a r g e t p o i n t ? € i n minimal t i m e by R 2 and t h e problem i s t o s t e e r t h e system f r o m y o t o (ul,u2)

choosing u s u i t a b l y .

T h i s i s c a l l e d a problem i n t i m e o p t i m a l c o n t r o l .

Let

G ( t ) be a fundamental m a t r i x s o l u t i o n ( o r e v o l u t i o n o p e r a t o r ) f o r t h e homo-

geneous e q u a t i o n G = AG, G(0) = I ( c f t i o n o f ( 0 6 ) can be w r i t t e n as (5.9)

y(t,u)

= G(t)Yo

+

t G ( t ) I o G-

[Bol;Cl;Cdl;Hol])

so t h a t t h e s o l u -

34

ROBERT CARROLL

The s e t o f a t t a i n a b i l i t y A ( t ) i s t h e c o l l e c t i o n o f p o i n t s y ( t , u ) which can be reached a t t i m e t by u s i n g a l l u E 11. I f one t a k e s e.g. t h e c o e f f i c i e n t s 1 1 o f A i n Lloc and s e t s B ( t ) u = b ( t ) u ( t ) w i t h bi E Lloc and u E Lm w i t h I u I

2

< 1 t h e n one can show e a s i l y t h a t A ( t ) i s a compact convex subset o f R ,

11 t h e t a r g e t y” = y ( t , u ) 11 such t h a t y(t*,u*) in

F u r t h e r i f we assume t h a t f o r some t = tl and u E

A ( t ) t h e n i n f a c t t h e r e i s an o p t i m a l

U*E

E

minimal t i m e t* (see h e r e [ K l ] f o r example f o r p r o o f s ) . t o c h a r a c t e r i z e u*.

L e t us see now how

One d e f i n e s t h e r e a c h a b i l i t y s e t R ( t ) = I I J G - ’ ( s ) b ( s )

u(s)ds; u E U } so t h a t A ( t ) = G ( t ) [ y o t R ( t ) ] o r R ( t ) = G - l A ( t ) closed, bounded, and convex).

-

z ( t ) = G-’(t)y

yo

E

-

is

yo ( R ( t )

C l e a r l y t* i s now t h e s m a l l e s t t i m e f o r which

R ( t ) w i t h y(t*,u*)

=

$

-

( t h u s z ( t * ) = G-;’

=

yo =

One expects t h a t t h e f i r s t c o n t a c t occurs when R ( t * ) touches z

z(t*,u*)). as shown (5.10)

Here t h e s u p p o r t i n g hyperplane t o R ( t * ) has normal rl a t z* as shown and t h u s n TA z 2 n T z f o r a l l z E R ( t * ) ( n o t e t h a t t h e v e c t o r 2 - 2 p o i n t s i n t o R ( t * ) so t* T -1 n T ( 2 - 2n) 2 0 ) . W r i t i n g t h i s o u t we o b t a i n ( a + ) lo rl G (s)b(s)[u*(s) u ( s ) ] d s 2 0.

Now r e c a l l 11 here i n v o l v e s c o n t r o l f u n c t i o n s u

Lm w i t h I u I

E

< 1.

It f o l l o w s ( e x e r c i s e ) t h a t u * ( s ) i n ( a + ) must have t h e form (am) u * ( s ) T -1 = sgn[n G ( s ) b ( s ) ] . T h i s i s a s p e c i a l case o f t h e P o n t r y a g i n maximal p r i n -

c i p l e ( c f . [Il;Kl;Pol;Yl])

which w i l l be discussed more l a t e r .

When t h e conT

t r o l s a r e g i v e n v i a B ( t ) u one o b t a i n s as above u * ( s ) = sgn[nTG-’(s)B(s)]

(as a column v e c t o r ) and we i n t r o d u c e now an a d j o i n t system as f o l l o w s .

T s i d e r t h e f u n c t i o n $ ( t ) n T G - l ( t ) (row v e c t o r ) w i t h $ ( O ) = n t i o n s a t i s f i e s (6*) b ( t ) = - $ ( t ) A ( t ) s i n c e 0 = Dtn

$6

= 6 G t lLAG =

[$

t

$A]G.

I t f o l l o w s t h a t (6.)

T

This func= $G t

u * ( t ) = sgn[$(t)B(t)],

t 5 t*, and u* i s thus determined v i a t h e a d j o i n t system.

opment o f t h e t h e o r y i n v o l v e s a H a m i l t o n i a n (6.)

.

= Dt[$(t)G(t)]

Con-

H($,y,u)

0 5 A f u r t h e r devel= $[Ay t Bu] so

t h a t t h e o p t i m a l u* s a t i s f i e s (5.11)

H($,y,u*)

=

ma x

H($,Y ,u

1

(which l o o k s more l i k e a maximal p r i n c i p l e ) .

C o n t r o l s o f t h e form (6.)

c a l l e d bang-bang c o n t r o l s and an example i s g i v e n below. t i v e example o f bang-bang c o n t r o l ( c f . [ L e l ; K l ] )

are

Thus f o r an i n t u i -

we suppose t h e problem i s

CALCULUS

OF VARIATIONS

35

0 1 0 t o s t e e r t h e system (66) $ = ( - 1 o ) y + (,)u f r o m y minimal t i m e ( l u l 5 1 ) .

E v i d e n t l y G ( t ) = (-Sint

t h o d above one o b t a i n s ( e x e r c i s e [ S i n ( t + 6 ) ] where Tan6 = -q2/n1. units apart.

t o the o r i g i n

Cost 'Int)

TI

and when u* = -1 on a c r c l e c e n t e r e d a t (-1,O).

t o a r r i v e a t y,

sgn

When u* = 1 t h e motion o c c u r s on a c i r c l e c e n t e r e d

switches

<

=

T h i s i s a bang-bang c o n t r o l w i t h s i g n

a t (1,O)

i s indicated f o r 0

= 0 in

and a p p l y i n g t h e me-

+ n2Cost]

u * ( t ) = sgn [-qlSint

y"

6 5

A typical picture

where t h e p a t h i s t r a c e d backward from t h e o r i g i n

TI

i n t i m e - t * ( c f . [Kl] f o r d e t a i l s ) .

The c o l l e c t i o n o f u n i t s e m i c i r c l e s shown i s t h e so c a l l e d s w i t c h i n g l o c u s . The f i r s t s w i t c h i s a t t = - 6 and t h e r e a f t e r an i n t e r v a l o f

TI

transpires

before the next switch.

6, SPECCRAC CHE0Rg FOR 0 D E , CRANSillLlCACI0N, AND INVERSE P R O 3 C E W .

We want t o

g i v e now some fundamental r e s u l t s f o r l i n e a r ODE and w i l l develop t h i s i n several ways.

E v e n t u a l l y we p r e f e r a p u r e l y t r a n s m u t a t i o n a l approach b u t

f o r now we w i l l interweave s e v e r a l themes (a general d i s c u s s i o n o f transmuWe w i l l deal here w i t h o p e r a t o r s o f t h e 1 4u where A E C , 0 < a 5 A 5 b < m, and A Am

t a t i o n can be found i n [C2,3]). form (*) Q(D)u = ( A u ' ) ' / A r a p i d l y as x

4=

+ m

-

+

(A(0) = 1 can a l s o be assumed and f o r convenience we t a k e

0 f o r t h e moment).

T h i s model w i l l s e r v e as a v e h i c l e t o i l l u s t r a t e and

develop t h e general t h e o r y and i t a l s o a r i s e s i n a number o f i m p o r t a n t app l i c a t i o n s i n geophysics, t r a n s m i s s i o n l i n e theory, e t c . ( c f . [C2,3,7]).

In

p a r t i c u l a r we w i l l t r e a t some a p p l i c a t i o n s i n geophysical i n v e r s e problems. 2 2 one has A4Qu = 6[A%] w h e r e ( A ) bv = D v - i v w i t h c? = A - ' ( ~ ) "

When A E C

I n p a r t i c u l a r we 2 ( a ) Qu = - 1 u and ( b ) av =

corresponds t o a 1-D Schrodinger o p e r a t o r ( c f . [ C h l ] ) .

w i l l be i n t e r e s t e d i n s o l u t i o n s o f ( v = A % ) ( 0 ) f o r t h e s o l u t i o n s o f (.a) - X 2v . We w r i t e q ! ( r e s p . Q

satisfying (6)

q QX ( 0 ) = 1; D x qQX ( 0 ) = 0; a9t X- ( x ) 'L Ai'exp('i1x)

The *?\;ill

as x

-+

c-,

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 one can c a l l t h e functions.

qx

be

spherical

For t h e c o r r e s p o n d i n g v s a t i s f y i n g (Ob) ( * ) v ( 0 ) = 1 w i t h v ' ( 0 )

= ( 1 / 2 ) A ' ( O ) = h and v i s denoted by P:,~.

Going now t o (*a) 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, and setting q = -A'/A

we o b t a i n

ROBERT CARROLL

36

(6.1)

vA Q ( y ) = CosAy +

(6.2)

A ~ ,Q( Y ) = eiAy

:j

[ S i n x ( y - n ) / h l q ( n ) D , q , (Qn ) d n

[Sin~(n-~)/hlq(n)AD~@!(n)dn

+

One can s o l v e these equations i n t h e form v X Q (y) =

1;

an(A,y)

(6.3)

with @,(A,Y)

vo

= Cosxy, @ =

vn(x,y)

0

= exp(ixy),

1;

qn(X,y) and &@!(y)

=

and

[sinx(n-~)/hlq(n)@~-~(A,n)dn; =

(’ [ s i n x ( y - n ) / x ~ q ( n ) v ~ _(A,n)dn ,

The d e t a i l s of e s t i m a t i o n a r e g i v e n i n [C2,3,7] needed.

and a r e sketched below as 1 Thus g i v e n t h e hypotheses on A an assumption q 6 L ( 0 , ~ 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 I m X 5 0 ( m ) I@,(X,Y)I 5 [ ~ ~ / n ! ] e x p ( - y I m x ) [ / ~I q ( q ) l d ~ ] ~ .S i m i l a r l y f o r any x E C (**)

YY

Ivn(A,y)l I (l/n!)exp(YlImxl)[!,, I q ( n ) l d r ~ ] ~ . 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

EHEaREIII 6.1.

qQ A ( y ) 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

a Qx ( y ) ( r e s p . + Q- x ( y ) ) i s a n a l y t i c f o r I m x > 0 ( r e s p . ImA < 0 ) and t h e f o l l o w i n g e s t i m a t e s a r e s a t i s f i e d (*A) I v 4 p , ( y ) I 5 e x p ( y l I m X l ) e x p [ # l q ( n ) \ d n l (*.)

It@ (Y) I! 2,e x p ( V I m x ) e x p [ c C I q ( n ) I d n l . Phuud:

To check t h e e s t i m a t e s (.)-(**) X l x ) and I S i n A ( x - S ) /

f i r s t n o t e t h a t ISinAxl 5 c l X l x

5 c/xlxexp( IImAl(x-c))/(l+lAlx) (note

x-S)) 5 I X l x / ( l + l x l x ) ) .

Hence

TRANSMUTAT ION

37

The p a t t e r n i s now c l e a r and l e a d s t o ( m ) so t h a t f o r

Q and

for

Imh

@ =

1 Gn

we have ( * a )

2 0 ( t h e s e r i e s converges a b s o l u t e l y and u n i f o r m l y ) .

The es-

< t i m a t e s f o r @(-h,y) a r e v i r t u a l l y i d e n t i c a l except t h a t we work w i t h Imh -

0.

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 t h e s e r i e s f o r 9(X,y)

Ivo(A,~)I (exp(yIImxI)

i Ihlexp(ylImAl).

w i t h IIP;(A,Y)I

v hQ ( y ) .

Indeed,

Hence (*m)

[q1I

=

I

{I exp( I Imhl ( y - n ) ) l q l e x p ( n l I m h l )dn 5 exp(y1 I m A l )I{ I q l d n . C o n t i n u i n g we oband I ~ i ( h , y ) l 5 {I e x P ( l I n x l ( y - n ) ) t a i n v i ( A , y ) = I$ Cosh(y-n)s(nk;(h,n)dn IqllAlexp(nlImAl)dn 5 Ihlexp(ylImAl)/{

Iqldrl.

Hence

The p a t t e r n i s a g a i n c l e a r and we conclude t h a t (*)

holds.

Hence t h e s e r -

i e s f o r v Qh converges a b s o l u t e l y and u n i f o r m l y on compact s e t s and The f o l l o w i n g o b s e r v a t i o n s w i l l be needed l a t e r . (m)

as l@,,(A,y)l

I ~ ~ @ ( E . , -Y )@ o ( h , ~ ) I I

1;

We n o t e t h a t i f we w r i t e

then

and s i m i l a r l y I A 2 ' ( A , y ) Since

holds

5 exp(-yImA)Q"(y)cn/n! w i t h ( f o l l o w i n g ( 6 . 5 ) ) [ @ A ( h , y ) l 5

IX lexp(-yImh)Qn(y)cn/n!, (6.7)

(*A)

cnQn/n! =

obtains (A*)lA3(A,y)

5 e- y ImA 1,- Q n ( y ) c n / n !

1;

1,"

-

@'(A,y)l 5 Ihlexp(-yImh) Q n ( y ) c n / n ! (@ = aQA ) . kok k k c Q / ( k + l ) ! - CQ c Q k! 5 cQexp(cQ) 5 c*Q one e x p ( i A y ) l 5 ;exp(-yImh)< l q l d r l and I A ! ' ( A , y ) -

1,-

CQ

-

1,-

ihexp(iAy)l 5 ?lAlexp(-yImh)Ja [ q [ d n . S i m i l a r considerations apply t o 9 Y and one has (-1 I v ( A , y ) - CosAyl 5 r e x p ( y l I m A [ )f{ I q l d n w i t h Ilp'(A,y) t ASinAyl 5 c l h l e x p ( y l I m h l ) / $

Iqldn.

CEtnillA 6.2. Under t h e hypotheses i n d i c a t e d we have 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 @-,(y). 4

(A*)

and

(AA),

p l u s sim-

The terms i n t h e ib s e r i e s f o r example ( i n Theorem 6.1 s a t i s f y g n ( h , y ) = @,,(-h,y)

= @Q- h ( y ) and 'p -Qi e t c . f o r A r e a l so one o b t a i n s f o r A r e a l , @,(y) -Q

( y ) = P Q- h ( y ) = .i'A(y) rl ( a c t u a l l y @l(y) -Q = * -4i ( y )

Q and

f o r any

h E

C w i t h Fl)(y) =

a r e l i n e a r l y independent w i t h (A*) F u r t h e r f o r h # 0, q!i(y)). A(y)W$(y),iP!A(y)) = - 2 i x (W(f,g) = f g ' - f ' g i s t h e Wronskian). Hence v A Q w i l l be a l i n e a r combination o f above w i l l be

(A&) q f ( y ) =

for X real).

Using

(A@)

@

@QA

Q and

4

*-A

which by p r o p e r t i e s i n d i c a t e d

c ( h ) a$ A ( y ) t c ( - h ) @Q- A ( y ) (where

Q

Q

with y

-+

0 and

(A&)

C

Q

(A)

=

c (-A)

Q

one o b t a i n s ( r e c a l l A ( 0 ) = 1 )

Q = - 2 i h c ( A ) o r D> ;i !(O) = 2 i h c ( - A ) (and thus i n W ( tQ~ ~ ( O ) , ( b _ 4 ~ ( 0= ) )Dx+-h(0) 4 4 Such formulas, o r perhaps b e t t e r A ( y ) W ( vQA ( y ) , general C ( A ) = c ( - A ) ) .

(b!(?(y)) = Q- 2 i h c Q (Qh ) ,

show e.g.

that

AC

Q( A )

i s a n a l y t i c f o r I m h < 0.

We

38

ROBERT CARROLL

c o n s i d e r now t h e p o s s i b l e v a n i s h i n g o f c ( A ) ( = c ( A ) )

I f c(A)

0 f o r h real,

9

# 0, then c ( - h )

A

which c o n t r a d i c t s v h Q ( 0 ) = 1.

= ;(A)

f o r Q(D) as i n (*).

= 0 and v Q A ( y ) T' 0 by (A&)

L e t t h e n Imh > 0 and from ( * ) f o r

-

one o b t a i n s (A = X 1 + i h 2 ) Dx@f(0)G:(O) ( n o t e terms w i t h e x p [ i ( A - T ) y ]

Ox$f(0)@f(O)

= exp(-2A2y)

+

0 as y

g!

and

4iX1A2Jr Al@fl'dy

= -+

@!

and b y Theorem 6.1

m

Now DXQA(0) Q = 0 ( = Dx%f(0)) means c (-A) = 0 and

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

Q

t h i s can happen o n l y i f A, = Rex = 0. 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 Imh

Hence t h e zeros ( i f any) o f A C ( - A )

Q

0 occur on t h e imaginary a x i s .

>

Q

such a p o i n t one would have p f = c ( A ) e A ( y ) which by (*.)

belongs t o L

Q

At 2

.

Such e i g e n f u n c t i o n s 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 ( w i t h obvious n o t a t i o n ) , g i v e n p = 2 c (A)@€ L w i t h (**), m u l t i p l y (.a) by /$ and i n t e g r a t e t o g e t ( A = i A 2 )

Q

(6.8)

A

I,

Iv12Ady =

-

I

Since D p Q ( 0 ) = 0 and A D x pQA ( y-Q )qA(y)

x2i

0 as y

-f

have -X2Jo

2Ady

CHEBREFII 6.3.

A ( y ) W (Q~ ~ ( y )Q, @ ~ ( =y ) 2iAc ) (-A)

2 J F A l p ' l dy which i s i m p o s s i b l e .

=

0 and does n o t vanish t h e r e .

>

t i o n s c ( A ) and c ( - A )

Q

9

Alv'l'dy

~x

(A

m

-f

(

+

( A i p ' ) ' v d y = -Aip'pl;

and ( * @ ) h o l d s ) we Consequently i s a n a l y t i c f o r ImA

so A C ( - A )

Q

Q

A l s o c (-1) # 0 f o r r e a l A # 0.

The f u n c -

Q

Q Q can be expressed v i a Dx@-A(0) and DxaA(0) as above.

I t 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 cq which a r e o f use i n v a r -

i o u s ways.

Thus from ( 6 . 1 ) w i t h p Qh ( y )

p(A,y),

%

- i h e x p ( - i A y ) + J{ exp[-iA(y-n)]q(n)v'(A,r~)dn. A!!exp(iAy)[iAo-p'], c (-A)

= (1/2)A:[1

Q

which equals 2iAc ( - A )

-

Q

(A+)

J{

+

makes sense.

0 and i n

5 IAlexp(nl1mAl)expJ;

= 0 since 0 : 11,(A,O).

(A+)

i f we l e t X

-r

-+

cQ(-h)

Q

One can r e p r e s e n t c ( - A ) -f

(1/2)A:

as h

( q l d c so t h a t t h e i n t e g r a l

0 so t h a t $(O,y) = Joy q(n)$(O,n)dn

(Am)

q(n)@+(n)dn] so t h a t we have

(A+)

-+

0.

The e s t i m a t e s f o r

I t f o l l o w s t h a t $(O,y)

I

CHEBREIII 6.4-

-f

( l / ~ ) p ' ( ~ , y=) $ ( h , y ) = -Sinhy

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

c (-A) f o l l o w s from ( 6 . 2 ) ; indeed Q

-f

Note a l s o from ( 6 . 1 ) ,

Cosh(y-n)q(r,)$(h,n)d~. Let A

$ ( O , O ) = 11,0

As y m , A(y)W(p,@+) by Theorem 6.3. Hence ( A + )

( l / i A ) J F exp(iAq)q(n)p'(X,n)dn].

Theorem 6.1 g i v e ip'(A,n)l in

9 @,(y) %a+, etc. p'-iXp =

Q

c (-A)

Q

b.y

(A+)

=

-f

$2/2.

= ll,oexp(J{ qdn) E

Another form f o r

-

(1/2)Ai5[1

or

(Am)

(ImA

( l / i A ) J r CosXn

0 ) and from

Consequently ( c f . Theorem 6 . 3 ) c ( - A )

Q

f o r Imh ?- 0.

REKIARK 6.5.

with

For v a r i o u s purposes one would l i k e an e s t i m a t e ( c ( - A ) l

Q

$ 0

t

f o r ImA 5 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

TRANSMUTAT I O N

(Am).

Thus i f ImA

6.1 e t c .

0 and InA

>

m

+

39

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

A i % x e x p ( i x n ) and ( r e c a l l q = - A ' / A )

?r

:$:A q(l/Z)[exp(Zixn) f o r ImA > 0 and I m x +

+ 1 I h % (1/2)A3: cQ(-h) + (l/Z)A?[l

m,

i s h e s o n l y when l o g A 5 =

-h.

(l/ix)/:

CosAnq@;dn

%

Hence q ( n ) h = -(l/2)Ai'logAm. + (1/2)A210gAm] and t h i s van-

Except f o r such i s o l a t e d cases t h e n one would

m

expect I 1 / c Q ( - x ) J 5 c f o r I m x

>

0.

L e t us emphasize here t h a t o u r development o f s p e c t r a l t h e o r y e t c . f o r t h e model o p e r a t o r (*) o r

(A)

i s designed t o show how a v a r i e t y o f methods and

ideas can be used f o r a t y p i c a l o p e r a t o r w h i l e s i m u l t a n e o u s l y d e v e l o p i n g some a p p l i c a t i o n s f o r a t y p i c a l p h y s i c a l system d e s c r i b e d by t h i s o p e r a t o r . The mathematical methods extend q u i t e g e n e r a l l y t o o t h e r s i t u a t i o n s and t h e r e a r e a l s o a l t e r n a t e methods ( f o r which we r e f e r t o [C2,3;Cdl;Mrl;La2; Jbl;Til]).

Now one can develop an expansion t h e o r y r e l a t i v e t o e i g e n f u n c -

t i o n s o f Q by s t r i c t l y t r a n s m u t a t i o n a l arguments ( c f . [C2,3]).

However we

want t o i n d i c a t e h e r e a n o t h e r method 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 . [C2,3; Dcl]).

Thus we w i l l e s t a b l i s h t h e f o l l o w i n g i n v e r s i o n .

I

m

Qf(h) =

(6.9)

f(x)A(x)q!(x)dx

=

F(x);

f(x) =

2

0

where dw(A) = l ( h ) d h = d x / 2 n l c Q ( x ) I

(thus Q = q-')

For s u i t a b l e f Fq:(x)dm(A)

=qF(x)

The t e c h n i q u e which we

d e s c r i b e now can a l s o o b v i o u s l y be a p p l i e d t o Q(D) = D2

- q^

o r Q(D)

- $

but

Consider t h e 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

we o m i t t h e d e t a i l s .

f u n c t i o n ( q ( x , x ) ?r q Qx ( x ) , @ ( x , x ) ?r a xQ ( x ) , x ( = m i n ( x , x ' ) , x > = m a x ( x , x ' ) ) 2 (@*) R(x ,x,^x) = - q ( x , x , ) @ ( x , x , ) / A ( x ) W ( q , @ ) ( r e c a l l from Theorem 6.3 t h a t A(x)W(q,@) = 2 i a c Q ( - x ) ) . :/

L e t $ E C2,

$ ( x ) [ Q ( O x ) + AZ]R(x2,x,^x)A(x)dx

;+ = ;to,

A

A

and x- = x-0 so f o r I =

one has A

(6.10)

X

li+J,(x)[Q(D,)

I=

-

*

(-1

=

$(x)A(x)Rxl;-

-

h

i L ' ( x ) A ( x ) R / &-+ + Now R i s continuous and

+ X2]R(h2,x,;)A(x)dx

xt

1'; -

J, E

R(h2,x,;o[Q(DX)

+ x2]J,[A(x)dx]

CL 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 2 A t A

s i n c e A R =~ I = ~ ( : ) A ( ~ ) [ R ~ ( x ,x , x ) - R~(~~,;-,;)I = -W(q,@)/A(x)W(q,a) ( w i t h W e v a l u a t e d a t i ) . Consequently one can make an

term gives

i d e n t i f i c a t i o n ( 0 0 ) A ( x ) [ Q ( D x ) + x 2 ]R(x 2 ,x,?) = & ( x - $ ) . S i m i l a r l y A ( x ) 2 [Q(D,) t x ]R(x2,^x,x) = 6(i-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 near 0 and

m

x21R(x2,x,y))

(e.g. =

5

E Ci(O,m))

s(x) = (A(Y)R(A

and t h e n f o r 2

.x,y),[Q(Dy)

e

= Q(D)c, ( A ( y ) 5 ( y ) [ Q ( D y )

+

x21s(y))

((

)

being a

+

40

ROBERT CARROLL

2

It f o l l o w s t h a t ( 0 6 ) C(x)/A2 = 1 , S(y)A(y)R(X ,x,Y)

distribution pairing).

dy t (1/h2)/,” e(y)A(y)R(X2,x,y)dy.

Now r e c a l l t h a t A(x)W(v,@) = 2iAcQ(-A)

i s a n a l y t i c f o r I m i > 0 w i t h a zero p o s s i b l e o n l y a t X = 0 f o r ImX 2 0 w h i l e

@!

Imx

= @ i s analytic f o r

0.

>

A l s o by Theorem 6.1 and (..),in

t h e numera-

w i l l have e x p o n e n t i a l bounds exp(y-x)ImX f o r x > y and exp 2 Consider R as a f u n c t i o n o f E = X ( E % ener-

t o r R(X2,x,y)

(x-y)ImX f o r y > x (Imx z 0 ) .

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

Except f o r a c u t on [ 0 , m )

gy).

-

[Dcl] f o r discussion

E

(cf.

t h e upper h a l f p l a n e i n X i s mapped o n t o t h e E p l a n e ) .

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 (06)

around t h i s c o n t o u r t o o b t a i n

(6.11)

m

1i m

2nic(x) =

y+”

(note generally I R / E I

%

dE

iE,;,

lo

5 ( y )R( E, x ,y dy

O(l/E3/2) a t l e a s t

-

On t h e o t h e r

c f . Remark 6 . 5 ) .

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 (6.12) i n t h e E p l a n e

( a v o i d i n g t h e c u t ) then upon i n t e g r a t i n g ( 0 6 ) around t h i s c o n t o u r we have dE j:(y)A(y)R(X2,x,r)dy

(6.13j

IEl=v

-

1‘0

t

d E l m c,(y)A(y)R(X 0

2

cAR(A 2 +iE,x,y)dy joydE lom

-ic,x,y)dy

-

= 0

Put t h i s i n (6.11) w i t h y -L m t o o b t a i n (*+) - 2 n i C ( x ) = 1 ; dEl, C ( y ) A ( y ) 2 2 Now pass t o t h e A plane, o b s e r v i n g t h e po[R(X -iE,x,y) - R(X tiE,x,y)]dy. sitions o f A

2? i E and l e t t i n g

1; v(X,x)v(X,y)do(x)dy tegrand i n

(ern)

E -L

0; we o b t a i n

(em)

w i t h dw(A) = dX/2n1cq(X)I

2

has t h e form - ( 1 / 2 n i ) [ c A ] ( 2 A / 2 i ) I

- @(X,y)/AcQ(-X)]

[@(-X,X)/(-AC~(A)) @(x,x)/xc (-A)] f o r x > y o r (-hcQ(X)) using

(A&).

CHEBREm

6.6.

Q

f o r y > x.

.

~ ( x =) :1 c ( y ) A ( y ) Note here t h a t t h e i n -

1 where

1

{

I

= v(X,y)

= YJ(A,x)[@(-A,Y)/

The e q u a t i o n (em) f o l l o w s t h e n upon

Since 5 i s an a r b i t r a r y t e s t f u n c t i o n we have proved The s p e c t r a l measure f o r t h e e i g e n f u n c t i o n s v QX ( x ) i s g i v e n by

dw(A) = C(h)dA = d x / Z n l c ( A ) I

Q

2 and t h e i n v e r s i o n ( 6 . 9 ) holds f o r s u i t a b l e f .

I n o r d e r now t o i n t r o d u c e and m o t i v a t e t h e t r a n s m u t a t i o n theme l a t e r we w i l l 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 f o l l o w i n g [C2,3,7].

From 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 something about t h e m a t e r i a l p r o p e r t i e s

TRANSMUTATION

( i n v e r s e problem).

41

The problem i s posed i n t h e f o l l o w i n g manner.

The gov-

SH shear waves i s ( 6 * ) p ( x ) v t t = [ u ( x ) v x ]x f o r 0 < where P ( X ) i s t h e d e n s i t y and ~ ( x )i s t h e shear modulus ( t h e s e a r e un-

erning equation f o r the x <

m

known).

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 vx(t,O)

= cs(t).

Then

one reads o f f t h e ( i m p u l s e ) response a t t h e same p o i n t aod c o l l e c t s i n f o r m a t i o n o f t h e form v(t,O)

=

The general i n v e r s e problem i s then t o de-

G(t).

t e r m i n e p ( x ) and u ( x ) f o r x

>

0, which cannot be done f r o m t h i s experiment;

however we can determine t h e "impedance" A(y) = (pu)'(y) " t r a v e l t i m e " y = 1 , (p/u)'dE

as a f u n c t i o n o f

( 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 1 ( c f . [C2,3] f o r r e f e r e n c e s ) . We w i l l r e q u i r e t h a t P , U E C and p r o v i d e a method here t o determine t h e s p e c t r a l measure f r o m which a v e r s i o n o f t h e G e l f a n d - L e v i t a n (G-L) e q u a t i o n i s d e r i v e d which i s a p p r o p r i a t e t o t h i s probl e m and t h i s l e a d s t o a r e c o v e r y f o r m u l a f o r A.

Various techniques o f i n -

verse 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 we r e f e r f o r background and o t h e r r e s u l t s f o r r e l a t e d problems t o [Akl ;Arl;C2,3,7;Gvl Let therefore y ( x ) =

(p/p)'(S)dS

;Chl ;Gvl ; M r l ;Pwl ;Syl

I.

so t h a t , w i t h A ( y ) = ( P L I ) ~ ( ~( )6 * ) be-

comes ( c f . ( * ) ) ( 6 . )

vtt = ( A V ~ ) ~ /=AQ(Dy)v w i t h ( 6 0 ) v (t,O) = - 6 ( t ) and Y 1 v(t,O) = G ( t ) ( a l s o v ( t , y ) = 0 f o r t < 0 ) . We assume p and IJ belong t o C

and r e a l i s t i c a l l y t h a t:1

IA'IAldy

cerned w i t h t h e s i t u a t i o n where A '

a;

0 and A

b e f o r e we n o r m a l i z e w i t h A ( 0 ) = 1 ) . f o r 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 con-

< +

+

Am r a p i d l y as y

(&A)

= (and as

5 A(y) 5 6 2. + k v one o b t a i n s ( 6 6 )

Assume as b e f o r e 0

Taking F o u r i e r t r a n s f o r m s i n

+

m

< a

=

Here we w i l l use h and k i n t e r c h a n g a b l y s i n c e k i s q(y);; q(y) = -A'/A. We w i l l c a l l customary i n p h y s i c s a n d F v = $ ( k , y ) = 1," v ( t , y ) e x p ( i k t ) d t . r e g u l a r s o l u t i o n t h e f u n c t i o n Ip(k,y) s a t i s f y i n g ( 6 6 ) w i t h p(k,O) = 1 and p ' ( k , O ) = 0 as i n ( 6 ) .

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 ( 6 6 ) w i t h @(?k,y) as y

+ m

(cf. ( 6 ) ) .

r~

exp(:tiky)A;'

and @ ' ( t k , y )

+ikexp(+iky)A-'

Equation ( 6 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

equations ( 6 . 1 ) - ( 6 . 2 ) 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 6.1 where Ip(k,y) % p Qx ( y ) and @ ( + k , y ) a Qk A ( y ) . Now go t o (&A) w i t h ( 6 0 ) and r e f e r r i n g t o [C2,3] f o r d e t a i l s , ( w e 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 ( 6 0 ) by a c o n d i t i o n ( b e ) 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 ( 6 4 ) ( o r w i t h an impulse i n s e r t e d d i r e c t l y i n ( 6 ~ ) 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 . [C23;Syl]).

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 s i m p l y t h i n k o f o u r problem subsequently as posed v i a

((A)

w i t h ( 6 4 ) and v(t,O)

=

G(t).

42

ROBERT CARROLL

The 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 ) . Y s o l u t i o n o f ( W ) i s t h e n v ( y , t ) = Y ( t - y ) ( f o r y,t L O ) where Y denotes t h e

!3AIRPI;E 6.7,

hus v = - 6 ( t - y ) -t - 6 ( t ) as y -t 0 and v ( 0 , t ) Heavyside f u n c t i o n . Y We n o t e t h a t t h e s o l u i o n c o u l d a l s o a r i s e from an impulse vt(y,O)

^v

( A = k) with

2A + X v = 0 YY Now ( c f . [ B c l ] ) F [ Y ( t ) Y(-t)]

-

= -exp i x y ) / i h ( c f . [C2,3]).

= 116(h)

-

= 6(y)

The F o u r i e r t r a n s f o r m i s

s i n c e vt = d(t-y) and d ( y ) E fi(-y). = - 2 / i h (FY

= Y(t).

A

and F b ( t - y ) = e x p ( i x y ) so i n some sense v =

l/ih)

F[Y(t-y)] corresponds t o e x p ( i h y ) [ n s ( A )

-

l/ix].

f u l l F o u r i e r t h e o r y one i s l e d t o v = [ Y ( t - y )

-

More c o m p l e t e l y v i a t h e Y(-t-y)]/2;

vy = - [ 6 ( t - y )

-

Working o n l y from t h e quadrant y , t 6 ( - t - y ) ] / 2 ; vt = [ 6 ( t - y ) + 6 ( - t - y ) ] / 2 . > 0 we m u l t i p l y by 2 however and drop Y ( - t - y ) = 0 t o g e t v = - 6 ( t ) as y + 0 Y and vt -t 6(y) as t + 0. 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 e x t e n s i o n i n t o f v i s a s s o c i a t e d w i t h t h e s i t u a t i o n .

Moreover i n

a l l problems o f t h e t y p e considered ( 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

-

t h e decomposition G ( t ) = 1

+

Gr(t) ( t L O ) i s

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

used below.

a l s o c l a r i f i e d below ( n o t e e.g.

(2/1~)lr Cosxtdx = 6 + ( t ) and (1/211)/1 exp

( - i X t ) d X = 6 ( t ) must be d i s t i n g u i s h e d , say v i a 2 6

A+).

%

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 [C2,3].

g. c o n s i d e r ( d u = Gdx, = (v,(Y)vx(n),CosAt Q Q

I: =

1/2alc

)u and R(y,t,n)

12,

and Qu = ( A u ' ) ' / A , e t c . ) (4.)

= ( v xQ( y ) v xQ ( n ) , [ S i n x t / x 1 )u.

and t h e s o l u t i o n o f ( W ) , vtt = Qv, w i t h v(y,O)

Thus e. S(y,t,n)

Thus R t = S

= f ( y ) and vt(y,O)

= g(y) i s

(+*I v(y,t) = (S(y,t,n),A(u)f(n)) + (R(y,t,n),A(n)g(n)) ( U P t o p o s s i b l e adjustment a t y = 0 ) . Here one has S(O,t,n) = ( v Qx ( n ) . C o s h t ) u and R(O,t,u) = ( v Qx ( n ) , [ S i n x t / x ] ) ~ ~ ( a g a iRt n

=

S).

I n 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 ) = ( ~Q~ ( ~ ) , [ S i n h t )/uh l

(6.14)

For y = 0 one o b t a i n s t h e readout G( t ) = ( 1 , [ S i n x t / h ] t r a l density

CHEBREFII 6.8.

I: i s

)(I)

from wh ch t h e spec-

determined d i r e c t l y

The s p e c t r a l d e n s i t y :(A)

from t h e impulse response G ( t ) v i a ;(A)

= l/ZVlc =

l2

can be obta ned d i r e c t l y

( Z A / V ) ? ~G ( t ) S i n h t d t .

and Theorem 6.1 we w r i t e $ = ( 2 i / h ) [ p xQ( x ) 2 By estimates as i n Theorem 6.1, $ E L f o r h r e a l and by Paley-

Next r e f e r r i n g t o ( 6 . 1 ) - ( 6 . 2 ) Cosxx].

'1 K(x,c)exp(ihg)dg = 2 i # K(x,c) -X Sinxgdg ( n o t e K(x,c) i s odd i n 5 and ic"(x,O) = 0 ) . Since :1 x S i n x g f ( x , c ) d c

Wiener ideas ( c f . Appendix B)

(*A) $ =

=

TRANSMUTATION ,u

N

= -K(x,x)Coshx

+ J , K5(x,5)Cosh5d5 we have

I PQ ~ ( X=) [ l

EHEBRETII 6.9. Now use (4.)

+

43

-

r(x,x)]Coshx

+

10'

5

(x,S)CoshcdE.

w i t h ( 6 . 1 ) t o o b t a i n ( c f . [C3;Mrl])

1"0 [ S i n h (x-5 ) / h ] [ q ( - x S i n x 5 + h S i n h g K ( 5 , 5

) +

r4

:J-

K5 ( 5 , r l ) ( X / 2 i )eih'dn)]d5

An a n a l y s i s o f t h i s e q u a t i o n f o l l o w i n g [C3;Mrl]

yields i n particular Iv

Under t h e hypotheses i n d i c a t e d one o b t a i n s ( q = - A ' / A )

EHE0REIII 6.10. =

(1/2)/'

0

-

q(5)[:(5,5)

l l d c and 1 - r ( x , x )

K(x,x)

= A-'(x).

We s t a t e now t h e m o d i f i e d G e l f a n d - L e v i t a n (MGL) e q u a t i o n o f [C2,3,7]

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 c a n o n i c a l Thus w r i t e ( 4 6 ) dw(k) = (2/n)dk + d o ( k ) and s e t

d e r i v a t i o n ( c f . [C3,8]).

lo m

(6.16)

T(y,x)

m

-f

T = SinkxSinkydo(k) y o 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 =

[Sinkx/k]Coskydo(k);

CHE0REIII 6.11. N K(y,x) ( x < y ) i s g i v e n by K(y,x) + T(y,x) = J{ ry

tion

-

N

K(y,q)Tq(q,x)drl

(MGL equa-

x < y).

T h i s G-L e q u a t i o n has a t i m e domain form which i s v e r y v a l u a b l e and r e v e a l ing.

Thus l e t us w r i t e G ( t ) = ;/

+ (Z/n)dX]

[SinXt/h][da

subscript r r e f e r s t o r e f l e c t i o n data).

y < x one o b t a i n s (more d e t a i l i s g i v e n l a t e r ) T(y,x)

2 o r T(y,x)

1 + Gr(t)

(the

= [Gr(y+x)

-

Gr(y-x)]/

It f o l l o w s t h a t T (y,x) = [G;(y+x) + Gr(x-y)]/2. Y and t h e MGL e q u a t i o n i n Theorem 6.11 becomes ( x < y )

= [Gr(y+x)

Gi(ly-xl)]/2

K(y,x) + ( 1 / 2 ) [ G r ( ~ + x ) - G r ( y - x ) I

-

Y, 'K(y,s)[G~(x+s)-G~(lx-sl )Ids

N

(6.17)

=

Then depending on whether y > x o r

=

0

EHE0REIII 6.12.

The MGL e q u a t i o n o f Theorem 6.11 can be w r i t t e n d i r e c t l y i n

terms o f readout d a t a i n t h e form (6.17) and g i v e n ? o n e s o l v e s t h e i n v e r s e problem v i a Theorem 6.10 i n t h e form A-'(y)

= 1

-

The d e r i v a t i o n o f t h e MGL e q u a t i o n above i n [C2,7]

K(y,y). was l a r g e l y ad hoc i n na-

t u r e and here we s k e t c h a canonical d e r i v a t i o n based on t r a n s m u t a t i o n p r o cedures as i n [C3,8]. A-'(y)Coshy

t

F i r s t from Theorem 6.9

/{ F t ( x , t ) C o s A t d t

-

vP Q,(y) =

6.10 we w r i t e (4.)

and t h e o p e r a t o r B: Cosht

n e l B ( y , t ) = A-'(y)&(y-t)

+ tt(y,t)

transmutation operator).

Then t a k e s c a l a r p r o d u c t s i n (4.)

-+

pP Q ,(y) w i t h ker-

i s c a l l e d a transmutation

D2

-+

Q (or a

w i t h Coshx i n

44

ROBERT CARROLL

(u b r a c k e t ) t o o b t a i n a canonical G-L e q u a t i o n

where

(em) r ( y , x )

i s a c t u a l l y t h e k e r n e l o f a n o t h e r t r a n s m u t a t i o n D2 = 10" ;(A)CoshxCosxtdx

and A(t,x) v xQ( y ) ) , (6.14)

= ( CosAx,Cosxt)o.

= 0 f o r x < y ( c f . [C2,3]

Q ( c f . [C2,3])

+

I n f a c t z ( y , x ) = (Cosxx,

b u t n o t e a l s o t h a t t h i s i s immediate from

by c a u s a l i t y ) and s ( y , t ) = ( 2 / ~ r ) / r px(y)CosAtdh Q ( f r o m Theorems 6.9

and 6.10 and t h e d e f i n i t i o n above o f 0 ) . formally t o obtain (B(y,t),A(t,x)) (6.18)

2A(t,x)

/"0

=

A(t,S)dS

=

=

Now f o r x

:1

( C o s A t , [ S i n x x / ~ ] ) ~=

-

Zt(y,t)[G(x+t) G(y-x)] + G ( t - x ) l d t . The l a s t i n t e g r a l s i n

= A-'(y)[G(x+y)

-

[G(x+t)

G(0) = 1 )

(MA)

I

= ?(y,t)[G(x+t)

ly

K(y,t)[G'(x+t)

-

G'(x-t)]dt

2 r ( y , x ) + F(y,y)[G(x+y) N

K(y,y) = 1

-

CHE0Rm 6.13.

A-'(y),

-

G(y-x)l

insert

(mA)

y we i n t e g r a t e i n (em

<

0 where

Consequently one o b t a i n s ( B ( y , t ) = 0 f o r t > y )

-$

=

0

(m*)

=

/[

SinA(x+t)GdA/x +

B(y,t)A(t,x)dt

+ G ( x - t ) ] d t + Jy F t ( y , t ) XN

(m*)

a r e ( r e c a l l K(y,O) = 0 and

-

+ G ( x - t ) l l t + r(y,t)[G(x+t)

-/xy

=

N

-

K(y,t)[G'(x+t)

G'(t

-

G(t-x)lli

-

x)]dt =

N

-$

-

K(y,t)[G' ( x + t )

i n (m*)

G'(lx-tl)]dt.

Using

Hence

t o g e t (6.17).

The MGL e q u a t i o n (6.17) 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 .

REmARK 6.14.

Going back t o (6.)

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 f u n c t i o n 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 f(T)dT (which w i l l say i n p a r t i c u l a r t h a t once g, be computed).

Indeed i f v * ( t , y )

L e t us suppose i n s t e a d L e t g 6 ( t ) be t h e unt g ( t ) = Jo g 6 ( t - T )

(me)

i s known any o t h e r g can

i s the s o l u t i o n o f ( 6 A ) - ( W ) w i t h v 6 ( t , 0 )

t 6 Y = 6 ( t ) c o n s i d e r v ( t , y ) = /o v ( t - T , y ) f ( T ) d T . For y > 0 we can w r i t e then t 6 6 v t ( t , y ) = JO v t ( t - r , y ) f ( T ) d T s i n c e v (0.y) = 0 (use (6.14) 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 ) = /o v t t ( t - r , y ) f ( ? ) d T and t h e r e f o r e (u) i s s a t -

isfied f o r y > 0 (i.e.

(Av ) / A ) . C l e a r l y v ( t , y ) = 0 f o r t 2 0 by y y c o n s t r u c t i o n and v (t,O) = lo G ( t - r ) f ( r ) d T = f ( t ) by a l i m i t argument as y Y * 0. Now t h e problem i s t o determine g6 from (m.), g i v e n f and g, and t h i s vtt

=

t

may n o t have a unique s o l u t i o n (see [ A k l ] f o r a d i s c u s s i o n o f t h i s p o i n t ) . For example i f $ ( s ) = ( C g ) ( s ) , 1: d e n o t i n g t h e Laplace transform, then $ ( s ) A

=

h

n

g 6 ( s ) f ( s ) and i f f ( s ) vanishes i n an unpleasant manner t h e r e w i l l perhaps

n o t be a unique d e t e r m i n a t i o n o f $ & ( s ) . t o be e x c i t e d by f f o r r e c o v e r y o f g,

Roughly one wants a l l f r e q u e n c i e s

(see 51.11 f o r f u r t h e r d i s c u s s i o n ) .

45

TRANSMUTATION

A uniqueness theorem f o r o u r MGL e q u a t i o n i n Theorem 6.11 can be modeled on a procedure i n [Chl].

W(y,x)

=

{I

One must show t h a t t h e homogeneous e q u a t i o n

W(y,n)Tn(n,x)dn

has o n l y a t r i v i a l s o l u t i o n .

(a&)

Note t h a t Tn ( f r o m

( 6 . 1 6 ) ) can be w r i t t e n as (me) T (q,x) = -1; SinkxSinkndw t 6 ( n - x ) = 6 ( n - x )

-

n

G(n,x).

(-)

!I {I

M u l t i p l y (m6) by W(y,x) W(y,n)W(y,x)G(n,x)dndx

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 ) , =

10" dw(k)[$

W(y,x)Sinkxdx]'

Hence

= 0.

f o r any y t h e e n t i r e f u n c t i o n 1; W(y,x)Sinkxdx o f k i s z e r o f o r k r e a l (dw >

0 ) and one can conclude t h a t W(y,x) = 0 f o r 0 5 x 5 y f o r each y. S o l u t i o n s K(y,x)

CHEBREI 6.15.

Hence

o f t h e MGL e q u a t i o n a r e unique.

/ A and vt(O,y) Y Y = 6 ( y ) ) and we c o n s i d e r now 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 We c o n t i n u e t h e f o r m u l a t i o n ((A)

m i s s i o n d a t a ( c f . [C3,8]).

w i t h ( b e ) ( 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 ( W ) f r o m t h e measured response v ( t , y ) a t y = 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.

due

I t i s assumed here t h a t A ( y ) =

( i . e . x 5 2) w i t h t h e o t h e r hypotheses on A unchanged. T h i s A_ f o r y L problem i s q u i t e d i f f e r e n t from t h o s e o f r e f l e c t i o n seismology where t h e measurements a r e made a t y = 0.

T h i s k i n d o f problem can a r i s e e.g.

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 o f a s p h e r i c a l l y s y m n e t r i c 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 w a t e r b a t h experiment; t h i s c o u l d a r i s e e.g.

i n bio-medical tomography and n o n - d e s t r u c t i v e e v a l u a t i o n .

The boundary c o n d i t i o n s corresponding t o t h e problem a r e (be) =

a ( y ) , and t h e r e a d o u t i s (***) v ( t , r ) = H ( t ) .

y

~y

, v i z . vt(O,y)

The c o n d i t i o n A(y) = A- f o r

can a l s o be regarded as a r a d i a t i o n boundary c o n d i t i o n a t y =

7.

Re-

f e r r i n g t o (6.14) we can w r i t e (6.19)

H ( t ) = v(?,t)

so t h a t p QA ( y 4) ~ I X= (2/n)1; .Y

= (~:(y),[SinAt/A])~

H ( t ) S i n A t d t and from Theorem 6.9 (**A) pi(;)Q

1; G ( t ) S i n x t d t = 1; H ( t ) S i n X t d t . t i o n o f exponential type

The f u n c t i o n p QA ( r ) i s an even e n t i r e func-

and t h e e x p r e s s i o n o f G i n terms o f H i n

can be regarded i n t h e c o n t e x t o f d e c o n v o l u t i o n ( c f . [ R b l j ) . Paley-Wiener ideas (@ i s even)

-

CosAtdt (where

~!(y")

@ ( t ) e x p ( i x t ) d t = $ ( A ) = 2ff 3 ( t ) -Y denotes t h e F o u r i e r t r a n s f o r m ) . S i m i l a r l y i f we t a k e c a n d = Jy-

H t o be odd e x t e n s i o n s o f G and H then $(A):"=

CHEBREm 6.16.

(**A)

Indeed b~

;''and

The readouts G a t y = 0 and H a t y =

satisfy 3

*

&

r

G =

Y

H.

L e t us now use t h e t r a n s m u t a t i o n machinery t o s p l i t up e v e r y t h i n g a s we go

46

ROBERT CARROLL

-

a l o n g ( c f . Chapter 2

e s s e n t i a l l y one r e f e r s h e r e t o " t r a n s m u t a t i o n machin-

e r y " when d e a l i n g w i t h s p e c t r a l i n t e g r a l s f o r k e r n e l s such as D and related spectral linkings). q y ( y " , i s even i n A .

Also f o r c a l c u l a t i o n

t h e 1/A f a c t o r i n (6.19). q!(y")dA. c (-A)

Q

R e c a l l n e x t t h a t :(A)

H'(t

Thus (**a)

and

= 1 / 2 n l c 0 / 2 i s even and

t w i l l be c o n v e n i e n t t o remove = (qA(y),CosAt)w Q =

lr (1/2)eiAt

F u r t h e r c ( A ) qQA ( y ) = ( l / 2 n ) [ q AQ ( y ) + qQA (Y 11 where *!(Y = @!(Y I/ i s a n a l y t i c i n t h e upper h a l f plane ( q r i s an i m p o r t a n t i n g r e d i e n t i n

-

general t r a n s m u t a t i o n t h e o r y

c f . [C2,3])

Hence s e t

m

6.20)

Hl(t) = (1/4n)I

m

*!h(y)eiAtdA

= ( 1 / 4 ~ ) 1 *?(y")e-i"dA

m

m

and i t f o l l o w s t h a t H ' ( t ) = H l ( t ) t H 1 ( - t ) .

Again we remark ( c f . Example

6.7) 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 (6.19) and H ' by ( * * a ) ,

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

extensions 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 Theorem 6 . 9 ) .

We n o t e

t h a t by f o r m a l c o n t o u r i n t e g r a l arguments H l ( t ) = 0 i n (6.20) f o r t < Indeed by now standard arguments and p r o p e r t i e s 0 so *A(Y")exp(-iht) Q

*!(y")

%

7.

c e x p ( i A 7 ) f o r ImA >

c e x p ( i h ( 7 - t ) ) 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

%

h a l f p l a n e I m A > 0 so f o r

> 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

(6.20) i s zero.

p r o v i d e s t h e readout H ' f o r t >

Thus H l ( t )

-y because

i s simply tagging along f o r t <

contributes nothing t o H' f o r t > 0). (1/2)*!(7)

(1/2)@!(7)/cQ(-A)

=

our readout p o i n t

2

=

x 2 ; and thus A ( y ) = Am f o r y ~ A?exp(iAy)

as y

+ m

%

., y

and f o r ? a s

-

(**A)

it

Now from (6.20) we can w r i t e ( * * 6 )

Jz,y H1 ( t ) e x p ( i h t ) d t .

l a r g e enough ( x

(and H 1 ( - t )

o f the representation

y') so t h a t

F u r t h e r l e t us t a k e

and P a r e c o n s t a n t f o r

p

. But a4 A ( y ) i s t h e J o s t s o l u t i o n QQA ( y )

Q

i n d i c a t e d we must have then

=

At5

exp( i$. Consequently

EHEOREm 6-17, (-A)

Under t h e hypotheses i n d i c a t e d , f o r

= Zcexp(-iA?)l:

H1 ( t ) e x p ( i A t ) d t

=

7 suitably

ZAzexp(-iAT);l

large, l / c Q

(A).

Y Note here t h a t H l ( t ) 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?&(t-?)

( c f . Example 6.7) and t h e i n t e g r a t i o n symbol i n Theorem 6.17 i s intended 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 o n l y i n t r o d u c e s a

phase f a c t o r i n t o t h e r e a d o u t ( f o r

7 suitably

large).

Since c ( A ) = c ( - A )

f o r A r e a l one o b t a i n s

COROLLARY 6.18,

Q

Under t h e hqpotheses above i t f o l l o w s t h a t $ ( A )

Q

=

1/2n1cl

= ( 2 / n ) A m l ~ 1 ( ~ ) 1from 2 which one can recover A as b e f o r e .

REmARK 6.19.

T h i s formula i n C o r o l l a r y 6.18 seems s t r i k i n g because i t

2

TRANSMUTAT I O N

47

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 t y p e f u n c t i o n JC(t) =

/IH1 (t+T)H1 (T)d?.

There i s an i n -

t i m a t e and p r o f o u n d 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 ( c f . [C3]). 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 (**@)-(6.20) we w r i t e now ( c f . (6.14) and r e c a l l ~ Q ~ ( =0 1 )) G ' ( t ) = (1/2)1

m

q!(O)weA i x t d h = G l ( t )

+ G1(-t);

W

=

0 for t

c (-A)

=

Q

i t i s related t o c

<

0 (argue as b e f o r e ) .

1 ; Gl(t)exp(iht)dt.

I n general

For example we have

Q'

Hence as i n (**&) ( 1 / 2 ) Y hQ ( 0 ) =

D@Qx ( 0 ) =

aX(O) Q is

n o t known b u t

-2ihcQ(X).

I n any case

One notes 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 K ( t - - r ) = 0 f o r t - T - y > 0; since

T

= 0 for

~7 T

t h i s means K ( t - r ) = 0 f o r t

t+G.

>

0 as d e s i r e d and moreover K ( t - - r )

<

Consequently we o b t a i n

tHE0REm 6-20, Given K ( t - r ) expressed as above i t f o l l o w s t h a t (6.22) holds 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 Hl

and G1.

Next from ( 6 . 2 1 ) w i t h G ' c o n s i d e r e d even because o f t h e Cosine r e p r e s e n t a t i o n , we o b t a i n (**+) G ' ( t ) = ( A w / n ) _ / I,;I of 2 ( i . e .

l 2 e x p ( - i x t ) d h = AwJC(t).

A factor

2AmJC(t) i n (**+)) a r i s e s because o f t h e Cosine r e p r e s e n t a t i o n and

must be removed when c o n s i d e r i n g G ' v i a t h e f u l l F o u r i e r t r a n s f o r m ( c f . [C3] and e a r l i e r remarks).

EHE0REm 6-21.

We conclude t h a t ( f o r use i n s t a b i l i t y q u e s t i o n s )

For t > 0 one has (*+)

L e t us now go back t o

(**A),

o r G ' ( t ) = AmJC(t).

m u l t i p l y by ( 2 / a ) S i n h ~ , and i n t e g r a t e . W ,e

f i r s t t h a t (2/a)Jt S i n h T S i E h t d h = 6 ( t - . r ) and (**,) ~ ' ( 7 ) = CosXy + ASinhndn =

Gosh:. rl >

Gosh? - i?(?,n)li

y"x

+ Joy zn(?',n)Coshndn

Here we r e c a l l t h a t 1

-

=

i ( y , y ) = A-'(y),

N

Jo K n ( Y , n ) C o s x ~ d ~+ +A: A = Am a t

w

y, and K(y,O)

= 0 (recall thatr(y,n)

r i v e a t K(y,O) = 0 ) .

Then from

H ( T ) = 1; G ( t ) I ( t , r , y ) d t ; [$(?-lt-Tl)

-

(**A)

=

y,

t(y,n) = 0 f o r

[$(h,y)/2i]SinAndn

(2/a)J;

i t f o l l o w s t h a t ( c f . [C3,8])

I = ( 2 / n ) / y p:(?)SinhtSinhidh

& ( ? - t - ~ ) ] and

J = (1/a)/;

note

/$ K(;Q)

[COSxlt-Tl

to ar-

(*A*)

= J

+ (1/2)AZ4

-

COSA(~+T)] &

JoY "Kn(y,n)Coshndndx % ,,

= (1/2)[K2(F,t-T)

+ K2(7,T-t)

-

K2(?,t+T)

where

K2(C,q)

48

ROBERT CARROLL

-

w

= K

rl

(5,n) ( n o t e f o r H(T)

6.23)

K2(y,t+.r)

> y,

T

does n o t a r i s e

.

Hence

becomes

(*A*)

+ G ( T - ~ )- G ( ~ - T ) ] t

= (1/2)A2[G(y+T)

-

+ (1/2)1"0 G ( t ) [ ? 2 ( y y t - T ) + &(Y,T-t)

Iu

K2(Y,t+s)]dt

d

Take now

t e r m i n (6.23) c*

-

(*AA)

ITT ( 1 / 2 )'G '( t )

-a,-

+ IT

Ki(Y,t-.r)dt

We can w r i L e t h e i n t e g r a l

i n t h e f o r m ( i n t e g r a t i n g by p a r t s )

N

[I

K2(yyt+r) = 0.

so G ( ~ - T ) = 0 and

T >

( 1 / 2 ) G ( t l r 2 ( y y r - t ) d t = (1/2)K(y,y)[G(y"+T) + {I ( l / Z ) G ' ( T - s ) K ( y , s ) d s . Now use

(1/2)G' (s+r)?(i,S)ds

+

-

G(T-~)]

F(y,y) =

l-A?

t o obtain

E5E0RETll 6.22. For T > 7 one has H ( T ) = ( 1 / 2 ) [ G ( y + ~ ) + G ( T - ~ ) ]+ ( 1 / 2 ) I{ ?(T,S)[G'(T-S) - G ' ( T + S ) ] ~ S . L e t us n o t e i n passing t h a t f o r

<

T

7 (where

H ( T ) = 0 ) (6.23) reduces t o t h e

Indeed G ( T - ~ ) = 0 i n (6.27) w h i l e - G ( ~ - T ) remains, and

MGL e q u a t i o n (6.17).

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 i n - a d d i t i o n t o (*a) -&y-T (1/ 2 ) G ( t

)z2(r,t+T)d t

:1 G'(s-T)ir(T,ss)ds.

:I

=

- (1 / 2 )G (Y-T) i?(y,,yu)+

T

(*A@)

+ (1 / 2 )

The m o d i f i c a t i o n r e q u i r e d i n (*u) involves (1/2) G'(T-s)?(F,s)ds. Consequent-

-

l y we o b t a i n from (6.23) and (*.A)

( n o t e G(0) = 1 )

- - w

w w

For

T <

7 (6.23)

-

0 = A?[G(~+T)

(*A&)

~(Y",T)-

(1/2)K(xyY)G(?++r) - G(?--.r)K(y,y)/2 + K(y,s)ds + ( 1 / 2 ) # G'((s-rl)'l?(y",s)ds. Using a g a i n

N

CHE0RETll 6.23.

(O)?(T,)

= (1/2)G(O)i?(?,~) + (l/Z)I;

G(t)z2(y",t-.i)dt

G(y-r)l +

(1 / 2 )G

( 1 / 2 ) l G'

z(Y,'f)

=

(S+T)

l-A2 we have

y i e l d s t h e MGL e q u a t i o n (6.17).

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 f r o m t h e Sine and Cosine r e p r e s e n t a t i o n s ) and w r i t e Theorem 6.22 w i t h T

>

7,

H ( T ) = (1/2)[G(r+T)

while f o r

-

T

<

Y,

G'(r+S)]ds ( i n t e g r a l s 0

with

?(7,5)=

before). G'(.c-s)ds G(T-?)],

-f

0 f o r 151 >

7).

For

+ (1/2)J Z ( T , s ) [ G ' ( r - S ) - G ' ( ~ + s ) ] d s + G ( T - ~ ) ] t (I/Z)J i'(y,s)[~l(T-s)

Now t r e a t

and

as f o l l o w s .

=

r(?,€,) as an odd f u n c t i o n i n 5 [ r ( y , - ) * G ' ] ( T ) :1 r(yN)s)

0 ( v i a t h e Sine r e p r e s e E t a t i o n as

Then by an easy c a l c u l a t i o n ( * A m ) = - JoY -K(y,s)G'(T+s)ds. Hence, s e t t i n g C(Y,T) = ( 1 / 2 ) [ G ( ~ + y ) + we o b t a i n

CHE0REm 6.24. = G(;,T)

+ G(T-;)]

= (1/2)[G(~+y)

-:(;,TI

(*A()

For

(T T

+ (l/Z)E(r,*)

REIIIARK 6.25.

> 0

-

c f . [C3] f o r f u r t h e r d e t a i l s )

> 0 one can combine t h e formulas above i n H ( T ) -

*

K(~,T)

G'.

One can 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

Theorem 6.24 and more p a r t i c u l a r l y o f (6.23) as f o l l o w s ( c f . [C3,9]). f i r s t from v(y,t)

= (q,(y),[Sinxt/A])u 9

with v(0,t) = ; 1 SinxtG(x)dx

Thus (*A+)

CLASSICAL MECHANICS

(*A+

) v ( y , t ) = :/

49

[ s i n A t / A ] p ~ ( Y ) ( Z ~ / ~ G(r)SinATdrdA )i~ = ;/

G(r)(2/n)

= [ J r v ,9( y ) S i n A t S i n x ~ d x ] d ~ . We w r i t e t h e n S ( y , t ) = ( 2 / a ) / r p,(y)CosxtdA 9 ( CosAt,p:(y))v = B(y,t) SO t h a t from (*A+ ) v ( y , t ) = (1/2)/; G ( T ) [ ~ ( Y ,( t - . r ( )

-

d ( y , t + ~ ) ] d ~ . W i t h G odd one has -it G(T)$j(y,t+r)dT = -fm0 G(T)$(y,t-r)dT

and s i m i l a r l y $ ( y , t )

i s even i n t; one o b t a i n s t h e r e f o r e v ( y , t )

= (1/2)

I f we [ d ( y , e ) * G I . I n p a r t i c u l a r f o r t > ?,H(t) = ( l / Z ) b ( Y , - ) * G I . -+ K2(y,t) = B ( y , t ) now and work w i t h (*A+) one obw r i t e B ( y , t ) = A-’&(y-t)

t a i n s (6.23) again. 7- I N P R 0 D l l C t 1 0 N PO CCAtiSICAC mECHANZCS.

Mechanics i s c e r t a i n l y one o f t h e

c o r e t o p i c s i n any study o f mathematical p h y s i c s .

We w i l l b e g i n w i t h t h e

s i m p l e s t and most i m p o r t a n t general framework and say something a b o u t c l a s s i c a l mechanics.

The b e s t r e f e r e n c e here i n o u r o p i n i o n i s [ A l l b u t we men-

t i o n a l s o [Abl ;Cal;Crl ;Go1 ;Lbl;L12;Tl].

The necessary g e o m e t r i c a l back-

ground i s i n Appendix C and we urge t h e r e a d e r t o compare t h e techniques o f c l a s s i c a l mechanics u s i n g Lagrangian and Hamil t o n i a n i d e a s w i t h o t h e r methods i n t h e book i n v o l v i n g e.g.

Lagrange m u l t i p l i e r s , a d j o i n t s t a t e s , dual

v a r i a t i o n a l problems, Legendre-Fenchel transforms, e t c .

Mechanics was i n

f a c t t h e o r i g i n o f many techniques and methods used now t h e o r e t i c a l l y i n a wide v a r i e t y o f a p p l i c a t i o n s .

We w i l l deal l a t e r i n Chapters 2-3 w i t h some

more advanced t o p i c s i n mechanics and i t w i l l be c o n v e n i e n t t o have c e r t a i n b a s i c m a t e r i a l a v a i l a b l e e a r l y i n t h e book.

We w i l l g e n e r a l l y t h i n k o f

mechanical s i t u a t i o n s where c o o r d i n a t e s qi and momenta pi = mivi ( i = 1,

...,

a r e used

3n) b u t many m a t t e r s a r e i l l u s t r a t e d e f f e c t i v e l y u s i n g s i m p l y

a 1-dimensional framework based on q and p. We c o n s i d e r k i n e t i c energy T = 2 ( 1 / 2 ) lmiv: = ( 1 / 2 ) pi/mi and p o t e n t i a l energy U = U(qi). We w i l l go d i r -

1

e c t l y t o t h e Lagrange f o r m u l a t i o n and s e t L = T - U w i t h (*) @ ( q ) = / , t l L ( t , 1 q , 4 ) d t where q E A = admissable (say C ) t r a j e c t o r i e s between q ( t o ) =Oq0 and q ( t l )

= 4,.

PHEBREIII7.1,

The r e s u l t s o f 52 g i v e immediately

A c u r v e q ( t ) i s an extremal o f @ ( q ) p r o v i d e d o t [ a ~ / a c j ] = aL/ For q % (qi), p % (pi), e t c . t h i s becomes a sys-

aq along the curve q ( t ) . tem D [aL/aq.] t 1

= aL/aqi

(pi = aL/aGi

i s c a l l e d a g e n e r a l i z e d momentum).

These equations a r e c a l l e d Lagrange’s equations ( o r Euler-Lagrange e q u a t i o n s ) and t h e f a c t t h a t motions o f t h e system c o i n c i d e w i t h such e x t r e m a l s (which a r e o f t e n m i n i m i z i n g f o r 4 ) i s r e f e r r e d t o as H a m i l t o n ’ s p r i n c i p l e o f l e a s t action.

We d i s t i n g u i s h however @ ( q ) from t h e a c t i o n i n t e g r a l S d e f i n e d by

S ( q , t ) = i L d t where y i s t h e extremal p a t h c o n n e c t i n g ( q o , t o ) t o ( q , t ) Y (see below f o r more on t h i s ) . Again q Q (qi) e t c . i s i m p l i c i t and we w i l l (A)

50

ROBERT CARROLL

forego a special n o t a t i o n

GQJ ( q i )

for this.

E v i d e n t l y o n e r e c o v e r s N e w t o n's second l a w f r o m t h e Lagrange

EYAmPtE 7.2,

Thus f o r L = ( 1 / 2 )

e q u a t i o n s when t h e f o r c e i s d e r i v e d f r o m a p o t e n t i a l .

mG2

-

-U = F o r F = ma ( i f m i s c o n s t a n t ) . C o n s i d e r q p l a n a r m o t i o n i n a c e n t r a l f o r c e f i e l d i n p o l a r c o o r d i n a t e s q1 = r

U (q ) we have Dt(m{)

n e x t e.g.

Thus U = U ( r ) and, l e t t i n g vr and ve d e n o t e u n i t v e c t o r s i n t h e

and q2 = 0 .

3

.

+ 0 rve ( e x e r c i s e ) . r Then T = (1/2)1n?~ = ( 1 / 2 ) m ( i 2 t r2G2)so t h e g e n e r a l i z e d momenta a r e p1 = r a d i a l and t a n g e n t i a l d i r e c t i o n s r e s p e c t i v e l y , r = r v =

-

mri2

2'

mi

and p2 = a L/aci2 = m r 0 . The L a grange e q u a t i o n s a r e t h e n my = 2. Ur and Dt(mr 0 ) = 0. The c o o r d i n a t e 0 = q2 i s c a l l e d c y c l i c s i n c e

aL/a;ll

2-

aL/aq2 = 0 and p2 = m r 0 i s t h e n a c o n s t a n t ( c o n s e r v a t i o n o f a n g u l a r momen2 2 ' turn). One c an w r i t e now ( 0 ) mF' = mM2/r3 - U = - V f o r V = U + ( M m/ 2r ) = r Zr e f f e c t i v e p o t e n t i a l e n e r g y ( s e t t i n g i = M / r ) . One n o t e s t h a t f o r E = T + U ( e n e r g y ) = (1/ 2 )m k 2 + V t h e c o n s e r v a t i o n o f e n e r g y f o l l o w s f r o m Et =

kVr

=

;[my +

[(2/m)(E-V)]

V ] = 0.

v r

C o n s e q u e n t l y f r o m ( 1 / 2)mk2 = E

e'

'and t h e o r b i t s can be found v i a

-

form ( 6 ) 0 = 1 [(M/r2]dr/[(2/m)(E

-

mbt'+ =

V we o b t a i n

= M/r2 = ;(dO/dr)

i n the

As an e x e r c i s e a p p l y t h i s t o

V(r))]'.

an i n v e r s e s q u a r e f o r c e ( U = - k / r ) and d e r i v e K e p l e r ' s l a w s o f p l a n e t a r y motion (cf. [All).

R e c a l l h e r e t h a t c o n i c s e c t i o n s have e q u a t i o n s r = a/

( 1 t eCos0) e t c . We r e c a l l now t h e L e g e n d r e - Fe n c h e l t r a n s f o r m o f 6 5 (Remark 5 . 4 ) and r e p h r a s e

i t h e r e i n terms o f p and v.

Thus l e t y = f ( v ) be a convex f u n c t i o n ( s a y

f " z 0 f o r f d i f f e r e n t i a b l e ) and c o n s i d e r ( v

-

Thus v ( p ) i s t h e p o i n t where pv

4)

%

f ( v ) = F(p,v)

i s maximum and one d e f i n e s

Thus Fv = 0 a t v ( p ) and f ' ( v ) = p d e t e r m i n e s v ( p ) . As i n g(p) = F(p,v(p)). 2 Remark 5 . 4 a n e as y c a l c u l a t i o n shows t h a t i f f ( v ) = mv /2 t h e n F ( p , v ) = p v 2 2 - mv /2, v ( p ) = p/m, a n d g ( p ) = p /2m. To see t h a t t h e map f g i s i n v o l u -f

g ( p ) w h i c h has an o b v i o u s g e o m e t r i c a l i n -

=

t e r p e r t a t i o n from (7.1).

I f one f i x e s v = v

G(v,p)

vp

-

t i v e one c o n s i d e r s G(v,p)

0

and v a r i e s p t h e v a l u e s o f

a r e t h e o r d i n a t e s o f t h e p o i n t s o f i n t e r s e c t i o n o f v = vo w i t h t a n -

g e n t l i n e s t o f ( v ) h a v i n g v a r i o u s s l o p e s p ( a l l o f w h i c h l i e below t h e curve).

I t f o l l o w s t h a t max G ( v , p ) = f ( v ) ( v = vo f i x e d ) and p(v,)

We l e a v e t h e d e t a i l s as an e x e r c i s e

-

c o n s i d e r e.g.

(cf. [All)

=

f'(vo)

CLASSICAL MECHANICS

51

G e o m e t r i c a l l y one i s c o n n e c t i n g h e r e t h e t a n g e n t and c o t a n g e n t spaces and t h i s i s developed below (and i n Appendix C ) . i t l y w r i t e down t h e H a m i l t o n equations.

H(p,q,t)

-

= p4

H(p) = p6

-

L(q,{,t)

(aH/aq)dq + ( a H / a t ) d t i s equal t o d[pv =

q EHEBREIR 7.3t i o n s (*)

4i

Thus dH = vdp

-

(6

-

4).

L(q,v,t)]

= v

p;

-+

Then dH = (aH/ap)dp +

when p = Lv = aL/a:

(re-

(aL/aq)dq - ( a L / a t ) d t and one o b t a i n s

b

and n o t e pv dp - L v dp = 0 ) P V P The Lagrange equations a r e e q u i v a l e n t t o t h e H a m i l t o n equa-

= aH/api;

ti

= -(aH/aqi);

i s t h e Legendre t r a n s f o r m o f L ({

4,t)

One d e f i n e s t h e H a m i l t o n i a n H =

as t h e Legendre t r a n s f o r m o f L ( q , 4 , t )

L ( 4 ) ; and here L i s assumed convex i n

c a l l (7.1) e t c . ) . (recall L

F i r s t however l e t us e x p l i c -

H~ = - L ~where H(p,q,t) -+

-

= p{

L(q,

p).

The t r a n s i t i o n t o H a m i l t o n ' s equations i s n o t j u s t an a r t i f i c e .

It allows

one t o phrase t h e t h e o r y i n t h e cotangent bundle ( o r phase space) and u t i l i z e t h e techniques o f s y m p l e c t i c geometry. c e p t u a l change i s p r o d u c t i v e .

We w i l l see l a t e r how t h i s con-

Indeed p l a c i n g o u r s e l v e s now on a m a n i f o l d

M w i t h l o c a l c o o r d i n a t e s qi l e t us s k e t c h t h e t h e o r y ( c f . Appendix C f o r ideas from d i f f e r e n t i a l geometry).

One can imagine t h e need f o r w o r k i n g on

a m a n i f o l d i f we t h i n k o f v a r i o u s c o n s t r a i n t s imposed on a dynamical system

so t h a t t h e m o t i o n i s f o r c e d t o t a k e p l a c e on some (smooth) subset M o f R3n. We denote by TM ( r e s p . T*M) t h e t a n g e n t ( r e s p . c o t a n g e n t ) bundle w i t h T M q d e n o t i n g t h e t a n g e n t space a t q (observe t h a t a c u r v e q ( t ) on M corresponds t o a tangent vector {(O) ti(0)

-

= v a t t = 0 i n t h e form v

r e c a l l v ( f ) = (V,df)

=

Dtf(q) a t t = 0).

%

1 cci(a/aqi)

with

=

ai

One notes t h a t T*M = N i s

even dimensional and f o r s i m p l i c i t y we t a k e dim T*M = 2m (where 2m

%

617).

A s y m p l e c t i c s t r u c t u r e on N i s determined by a c l o s e d nondegenerate 2-form w2

(i.e.

dw2 = 0 and f o r a l l 5 f 0 t h e r e e x i s t s

€,,TI

E TxN = TNx).

(m)

o2 =

1 dpi

p r o j e c t i o n T*M q

2 such t h a t w (€,,n) = 0

-

T*M has t h e n a t u r a l s y m p l e c t i c s t r u c t u r e determined by

A dqi. +

TI

To see t h i s ( f o l l o w i n g [ A l l ) l e t f : T*M

q and suppose 5 E Tp(T*M).

Then f,:

T(T*M)

-f

-+

M be t h e

TM takes 5 t o

52

ROBERT CARROLL

a v e c t o r S,f

coordinates evidently cise 2r

-

1 Define a 1-form o ( 5 ) = p(fJ)

tangent t o M a t q. o1 =

1 pidqi).

n o t e h e r e one can w r i t e p

1 Bj(a/aqj),

p(f&)

'L

1p j ~ j

Then u 2 = do' i s nondegenerate ( e x e r -

1 pidqiy

2r

, I ,

( i n local

5

'L

( Ipidqi)(5)).

1 a j ( a / a p .J)

+

f,E

Bj(a/aqj),

This c o n s t r u c t i o n o f

o1

is

p e c u l i a r t o T*M

-

Now t o each 5 1 2

T(T*M) one a s s o c i a t e s a 1-form u1 E T*(T*M) b y t h e r u l e (**)

y(n)

= u

E

t h e r e i s no analogous form on TM f o r example.

( n , ~ ) ,for

5

We w i l l s p e l l t h i s o u t i n l o c a l c o o r d i n a t e s 1 ) ) . Set 5 ai(a/api) + bi(a/aqi), rl % and W' 'L ajdpj + b.dq Then 5 J j'

map o1 5 (T*(T*M) T(T*M)). 5 1 2 as f o l l o w s ( n o t e w (TI) = w (~,Iw -f

-f

+ si(a/aqi),

lyi(a/api)

while w

1

(0) =

5

1 a.y

ticular if H is a

and we denote by I t h e

a l l II E T(T*M) ( w 2 as i n ( m ) ) ,

1

+ b.6

J jl

C

I n parIt f o l l o w s t h a t a = b j and b = -a. J j' j j J' 1 f u n c t i o n on T*M (any such f u n c t i o n ) t h e n t o dH us Q

on T*M one assigns a v e c t o r f i e l d 5 = IdH which i s c a l l e d Hamiltonian. d e r t h e c a l c u l a t i o n s above f o r dH =

1 (aH/api)dpi

t h e Hamiltonian v e c t o r f i e l d (ai = -(aH/aqi)

1 (-aH/aqi

)(a/aPi)

+

+ (aH/aqi)dqi

and B~ = (aH/api))

Un-

one o b t a i n s (*A)

IdH =

(aH/aPi ) ( a / a q i ) -

The v e c t o r f i e l d IdH on T*M now g i v e s r i s e t o a f l o w which we assume t o be a one parameter group o f diffeomorphisms gt: T*M t h e ODE a s s o c i a t e d w i t h IdH = 5 h e r e ( i . e .

+

qi(t)

T*M. = Si(t)

One i s s i m p l y s o l v i n g s t a r t i n g a t some

p o i n t qo and working i n l o c a l c o o r d i n a t e s ) . H i l ; L b l ] f o r a study o f such f l o w s .

We r e f e r t o [A1,ZY3;C1;Cdl;Abl; t t Thus Dtg qltS0 = IdH(q) and g i s c a l l -

ed t h e H a m i l t o n i a n phase f l o w a s s o c i a t e d w i t h t h e ( H a m i l t o n i a n ) f u n c t i o n H. t 2 t Now r e c a l l ( g )*w ( 5 , n ) = u2(gt&,g *TI) by d e f i n i t i o n s ( C f . Appendix C); thus A H a m i l t o n i a n phase f l o w preserves t h e s y m p l e c t i c s t r u c t u r e CHE0REfl 7.4. ( i . e . gt*W2 = o2 where w2 i s g i v e n l o c a l l y by ( m ) ) .

Ptvu6:

We f o l l o w [ A l l and w i l l p r o v i d e a few a d d i t i o n a l c a l c u l a t i o n s t o

make t h e p r o o f even more i n s t r u c t i v e .

L e t c be a k c e l l on N = T*M and l e t

Gc be t h e k t l c e l l swept o u t by c under t h e map gt ( 0 < t 5 t formula F ( t , x ) = g f ( x ) ( r e c a l l c i s d e f i n e d by a map f : Rk

...,

T

s a y ) by t h e

N and one o r i s a u n i t v e c t o r f o r t h e t a x i s and -f

el, ek where e i e n t s Rktl by e, ko c - G(ac) ( e x e r e., j 2 1, i s an o r i e n t e d frame i n R ) . Then a(Gc) = gTc J c i s e - a i s t h e boundary o p e r a t o r and t h e d e t a i l s a r e i n [ A l l o r sketched i n

-

Appendix C).

I n p a r t i c u l a r i f y i s a 1-chain t h e n (*@) D J T

GY

o2 = / g ~ ydH.

53

CLASSICAL MECHANICS

To see t h i s i t s u f f i c e s t o l e t Y be a 1 - c e l l f: [0,1]

N and w r i t e F ( t , s ) =

-t

w i t h 5 = aF/as and n = a F / a t b e l o n g i n g t o TN a t g t f ( s ) ( t h u s gt % 1 2 f l o w o f n ) . Then by d e f i n i t i o n s (*() 1 O* = I0 1 , o ( c , n ) d t d s . I t i s per-

gtf(s)

GY

haps i n s t r u c t i v e t o s p e l l t h i s o u t s i n c e i t r e p r e s e n t s a d i r e c t f o r m u l a t i o n o f t h e Jacobian m a g n i f i c a t i o n i n changing v a r i a b l e s . A dqi,

n

Q

1 ni(a/api)

+

Gi(a/aqi), 2

lows as above i n (7.3) t h a t (api/at)(aqi/as) 5

%

F

and ti

and 5

I (api/as)(a/api)

%

(aqi/as)

-

% A

( S , n ) = 1sini

1 a(pi,qi)/a(s,t)

%

%

4s

w

,

Thus g i v e n w

2

1 ci(a/api) + i i ( a / a q i ) - Sini 1 (api/as)(aqi/at)

Idpi it fol-

-

-

2.

( g i v e n c o o r d i n a t e s p,q(x,t)

+ (aqi/as)(a/aqi)

f o r example

r e c a l l c ( h ) = DSh(F)).

SO

ci

a t F(s,t), %

(api/as)

Thus f o r Ji = a(pi,qi)/a(s,t)

we have a change o f v a r i a b l e s f o r m u l a i n c l a s s i c a l n o t a t i o n /Jdpidqi

11 J,dtds and 1

GY

w2 =

2

1 11 Jidtds.

// w (5,n)dtds

t o more general forms w2 =

1 f 1J . .dxi

A d x . (xi

J

= pi

=

T h i s c o u l d be extended o r 4 . ) and we remark ex1

c o n t a i n s o n l y a subset o f t h e p o s s i b l e dxi A dx 2 lj* Going back now t o ( * 6 ) , s i n c e w ( c , n ) = dH(S) ( i . e . IdH % rl here, w n ( S ) = p l i c i t l y t h a t our

[I

w2(s,n) = dH(S)), one has IGy w 2 = :1 hence (*@) h o l d s .

/a Y

H so e v i d e n t l y (**) /

a 2-chain.

(7.4)

Then

O = l Gc

t dH]dt

( e x e r c i s e i n n o t a t i o n ) and

S Y

F u r t h e r i f y i s a c l o s e d c h a i n (ay = 0 ) then /

dw 2

GY

Y

F i n a l l y t o prove t h e theorem l e t c be

u 2 = 0.

= Iw 2 = [ I , - ] - 1 102 = I T .2 - 1 a Gc

dH = 0 =

g c

c

Gac

Here one has used s u c c e s s i v e l y t h e f a c t t h a t t h e boundary formula above, and (**) w i t h we a r e through.

y =

w2

g c

w2

C

i s closed, S t o k e ' s theorem, 2 2 Since f C gT*w = 1 T w g c

ac.

QED

The p r o o f c o n t a i n s a c e r t a i n amount o f i n s t r u c t i v e m a t e r i a l which has made i t l o n g e r t h a n one m i g h t expect.

R w i t h N = T*M

2,

The theorem i s i m p o r t a n t however

t h e phase f l o w preserves area (see e.g.

Q

[ A l l f o r another p r o o f ) .

The theo-

i s a so c a l l e d i n t e g r a l i n v a r i a n t o f a H a m i l t o n i a n 2 2 F u r t h e r s i n c e dH(5) = w (S,IdH) f o r 5 = i.e. iCo2 = J t w w2

.

IdH ( = n ) we o b t a i n dH(q) direction

for M =

R2 t h i s i s t h e famous theorem o f L i o u v i l l e a s s e r t i n g t h a t

rem shows a l s o t h a t phase f l o w

-

L2(n,n)

= 0 so t h a t t h e d e r i v a t i v e o f H i n t h e

i s 0 (dH(n) = n(H) e t c . ) .

o f t h e phase f l o w determined by H ( i . e .

T h i s says t h a t H i s a f i r s t i n t e g r a l H = constant along t h e f l o w ) .

This

i s o f course g o i n g t o r e p r e s e n t c o n s e r v a t i o n o f energy. Given now t h e symmplectic m a n i f o l d T*M = N and w 2 d e f i n e d by ( m ) , t o any t and t h e Poisson b r a c k e t s u i t a b l e C 1 f u n c t i o n H on N we have a phase f l o w gH o f two such f u n c t i o n s F and H i s d e f i n e d as t h e d e r i v a t i v e o f F i n t h e

ROBERT CARROLL

54

t . t d i r e c t i o n o f t h e phase f l o w gH, 1.e. ( * m ) (F,H)(x) = DtF(gH(x))lt=o = dF 2 (IdH) = w (IdH,IdF) = -(H,F ( x ) . Note from Appendix C, IdH % -H D + H D 9 P P 9 2 and (IdH,IdF) % (aH/api (aF/aqi) (aH/aqi)(aF/api) = (F,H) so t h e can-

1

-

o n i c a l equations o f m o t i o n become Dtqk = (aH/apk) = (qk,H) and Dtpk = -(aH/ E v i d e n t l y a f u n c t i o n F i s a f i r s t i n t e g r a l o f t h e phase f l o w (pk,H). aqk) t gH i f and o n l y i f (F,H) = 0. One checks e a s i l y t h a t a Jacobi i d e n t i t y h o l d s

+ ((B,C),A)

( e x e r c i s e ) (A*) ((A,B),C)

+ ((C,A),B)

= 0.

We want n e x t t o g i v e a b r i e f i n t r o d u c t i o n t o t h e Hamilton-Jacobi e q u a t i o n ( a g a i n f o l l o w i n g [ A l l ) . Thus l e t N % T*M x R 1 be an extended phase space o f dimension 2m+l and c o n s i d e r t h e 1 - f o r m w1 =

1 pidqi

-

Hdt where H = H(p,q,t)

i s a suitable function. by dw

1

(c,~)

One c o n s i d e r s a v o r t e x d i r e c t i o n E, o f w1 determined 1 = 0 f o r a l l qETN. Given dw n o n s i n g u l a r t h e d i r e c t i o n 5 i s

u n i q u e l y determined and t h e i n t e g r a l curves o f v o r t e x d i r e c t i q n s a r e c a l l e d vortex l i n e s ( o r characteristic l i n e s ) o f 1 I f y1 i s a c l o s e d c u r v e on N

.

t h e v o r t e x l i n e s emanating from y1 f o r m a v o r t e x tube.

I t i s easy t o see

t h a t if y1 and y 2 a r e two curves e n c i r c l i n g a v o r t e x tube (y, then

iYl

w1 =

Iy2w

.

Indeed by Stokes‘ theorem

I

Y

1 w1

-

-

y2 =

a,)

IY22= faayl

=

Ia dw . But on any p a i r o f v e c t o r s c,r- tangent t o t h e v o r t e x tube dw ( 5 , r l ) = 0 ( S , n l i e i n a p l a n e c o n t a i n i n g t h e v o r t e x d i r e c t i o n 5 and dwl vanishes t h e r e - e.g. s e t 5 = U E , + 6 q ) . Hence ID dw’ = 0. Now one has ( c f . [ A l l ) CHE0REIII 7 . 5 -

The v o r t e x l i n e s o f

1 pidqi

w1 =

-

Hdt have a 1-1 p r o j e c t i o n

o n t o t h e t a x i s ( p = p ( t ) , q = q ( t ) ) and they s a t i s f y t h e Hamilton equations o f (+), {i = aH/api and ii= -aH/aqi. Thus t h e v o r t e x l i n e s o f w 1 a r e t h e 1 t r a j e c t o r i e s o f t h e phase f l o w i n N T*M x R

-

P400d:

1 dpi

E v i d e n t l y dwl =

Thus ( c f .

(7.3)) for 5

=

A dqi

2 Si(a/api)

T = q AE,

i n an obvious n o t a t i o n ( n

where A T

-

.

(aH/api)dpi

+ ii(a/aqi)

[

0

A dt -

1

(aH/aqi)dqi

+ c ( a / a t ) and a s i m i l a r

-1 H 0 Hp Hp -HqOq

A dt IT

]

Now t h e rank o f A i s 2m and (qi,Gi,;) etc.). t h e v e c t o r (-H ,H , 1 ) = 5 i s an e i g e n v e c t o r w i t h eigenvalue 0. Hence i t q P Q

CLASSICAL MECHANICS

55

determines the vortex directions of W' and we see that it is also the velocity vector of the phase flow dp/dt = -aH/aq and dq/dt = aH/ap. Hence the integral curves of the canonical equations are the vortex lines of w 1 . QED Thus, by comnents above, for two curves y 1 and y 2 encircling a vortex tube generated by phase trajectories for W' = 1 pidqi - Hdt one has f w 1 = Y1 W' and W' is called the integral invariant of Poincare-Cartan (or some/ v2 times in the calculus of variations it is called the Hilbert invariant integral). If one considers curves y consisting of simultaneous states (t = constant) then / W' = f 1 pidqi and the phase flow preserves such inteY Y grals. If u is a two dimensional chain with y = a,, then J pdq = dpA Y dq and we see that the phase flow preserves the sum of (oriented) projected areas onto the (pi,qi) planes ( 1 1 dpi A dqi = f +,,I dpi A dqi). Again 9 for T*M % R2 we have Liouville's theorem. Transformations g preserving 2 are called canonical and we see that this can be expressed via (1) g*W2 = 2 or ( 3 ) f pdq = f pdq. Evidently the transformation of (2) f,, W' = J go Y 9Y T*M induced by the phase flow is canonical. Now define the action integral (cf. ( A ) and Example 2.10) by (AA) S(q,t) = Ldt where y is the extremal connecting (qo,to) to (q,t). Here by extreY ma1 one is referring to an integral curve of the canonical equations ( 0 ) or equivalently to an extremal for/ pdq - Hdt over curves for which the end Y points remain in the (q,t) space (see [All for further discussion of this). Intuitively the integral [I I - / y ](pdq - Hdt) is small o f higher order Y than the distance between y ' and y when y is a vortex direction. Alternatively, one can write in an obvious notation (6y % EIP roughly in (2.4)-(2.5) - one would write in say (2.4), T(Y+EIP)- T(y) = /fo [FYc'p + Fy,~q']dx + 0(c2) = 6T + O ( E 2 ) - cf. also [Gl]). f

we do not want extremals emanating from (qo,to) to intersect Further in (u) and this can be assured if (t-to) i s sufficiently small. If we now look in a NBH of (q,t), every point is connected to (qo,to) by a unique extrema depending differentiably on the endpoint so (u) will be well defined and one has (following [All) EHE0REIR 7.6. dS = 1 pidqi - Hdt where p = JL/aG and H = p4 - L are def ned with the aid of the terminal velocity q of y. Further S satisfies the

56

ROBERT CARROLL

Hamilton-Jacobi e q u a t i o n ( ~ m ) a S / a t

+

One l i f t s t h e e x t r e m a l s f r o m ( q , t )

Ph006:

= 0.

H(aS/aq,q,t)

space t o (p,q,t)

space T*M x R,

s e t t i n g p = aL/a4, and t h u s r e p l a c i n g t h e extremal by a phase t r a j e c t o r y (i.e.

a c h a r a c t e r i s t i c c u r v e o f pdq

i n g (qo,to) w i t h (q+eAq,t+eAt),

0 5

E: o f c h a r a c t e r i s t i c curves o f pdq

-

-

Hdt).

e5

Consider t h e e x t r e m a l s connect-

1, which g i v e r i s e t o a c o l l e c t i o n

Hdt as shown ( c f . [A1 1)

(7.7)

Here

1'hen

1

a

since B 0 =

(7.8)

[Iy'

-

However on

IZ d[pdq -

fy2 a,

+

Hdt] =

I, - fa

[pdq

p and

-

S ( q , t ) = JB [pdq

-

Hdt] = paq

as/at

-

dq = d t = 0 and on y1 and y2, pdq

aL/a6 i s an extremal o f p4

-

Hdt] =

I(P - H d t~ )

a r e Legendre t r a n s f o r m s o f each o t h e r t+At)

-

-

- H).

Hence [J

Hdt].

L e t t i n g Aq, A t

-H(as/aq,q,t)

=

4, ](pdq -

recall f o r fixed

-

Y

Mt + o(Aq,At) and hence dS = pdq = -H(p,q,t)

Hdt = L d t ( s i n c e L and H

Jyl

-

which i s

+

t h e value p =

H d t ) = S(q+Aq, 0 one o b t a i n s IB [pdq -

Hdt.

Consequently aS/aq =

QED

(A@).

It i s i n t e r e s t i n g t o n o t e t h a t t h e Cauchy problem f o r t h e Hamilton-Jacobi

e q u a t i o n (namely

p l u s S(q,to) = So(q)) can be s o l v e d u s i n g t h e canoni-

c a l equations (+) t o generate c h a r a c t e r i s t i c s . t h e c a n o n i c a l equations

= -H

and

6=

q aq y i e l d a c u r v e which maps i n t o ( q , t )

6

t o g i v e an extremal o f J L d t , L = L(q,G,t)

qo).

/Y

2

Ldt

( c a l l e d a c h a r a c t e r i s t i c from

= S (q ) 0 0

c e p t t h a t a l i e s above an arrow f r o m (qo,to)

Ja dSo

briefly,

+ Iqst L(q,G,t)dt and t h i s s a t i s f i e s qO,to To prove t h i s t h e p i c t u r e corresponding t o ( 7 . 7 ) l o o k s l i k e ( 7 . 7 ) ex-

One c o n s t r u c t s S(q,t)

(Am).

Thus ( c f . [ A l l ) ,

H w i t h q ( t o ) = qo and p ( t o ) = aso/ P space v i a t h e Legendre t r a n s f o r m p +

Ia pdq

=

-

-

I t f o l l o w s t h a t J B [pdq Hdt] So(qo+Aq) + So(qo). [So(qo) + J Y l L d t ] = S(q+Aq,t+At) - S(qo,to) which i m p l i e s t h e

= So(qo+Aq)

-

t o (qo+Aq,to) so t h a t

Hamilton-Jacobi e q u a t i o n

(Am).

One can a l s o use general methods o f charac-

t e r i s t i c s t r i p s , Monge cones, e t c . t o s o l v e

(Am)

and t h i s i s developed i n

57

QUANTUM MECHANICS [Wall f o r example ( c f . a l s o [ J l ] ) .

To s k e t c h how t h i s goes one t h i n k s o f

t h e f i r s t o r d e r n o n l i n e a r PDE St + H ( S , q , t ) = 0 = *(n,q,t,S) = no + H(n,q, q t ) where qo = t, xi = aS/aqi, and no = St ( t h u s ni 'L pi 2, aS/aqi). Now t h e standard c h a r a c t e r i s t i c s t r i p equations a r e

(7.9) (plus

q; = a*/ani

= aH/aTi;

n ! = -a*/aq 1

* = 0 - which i s a u t o m a t i c ) ;

dependent o f t f o r s i m p l i c i t y .

also

lr;

t ' = 1 = a*/anO

c f . [Jl;Wal]) i

= -aH/aqi;

s'

= n

0

t

1 niaH/ani

= -HA b u t we assume h e r e H i n L

I n any e v e n t one s e t s t =

T

( s t r p variable)

and sees t h e c a n o n i c a l e q u a t i o n s ( + ) a r i s i n g as t h e b a s i c d i r e c t onal equat i o n s f o r c h a r a c t e r i s t i c curves.

For f u r t h e r d e t a i l s see [Wal;J

an a b s t r a c t t r e a t m e n t o f Hamilton-Jacobi e q u a t i o n s see e.g. 8 . INCR0DllCUI0N U 0 QUANClllll m E C H A N W .

] and f o r

[Bdl L j l ] .

I n t h i s s e c t i o n we w i l l g ve a b r i e f

i n t r o d u c t i o n t o quantum mechanics a t t h e l e v e l o f [L13;Pkl;Scl;Shl]

f o r ex-

ample and w i l l extend a l l t h i s l a t e r b o t h g e o m e t r i c a l l y and a n a l y t i c a l l y i n t o quantum f i e l d t h e o r y e t c . ( c f . [Cgl ; C i l ;F1 ; G l l ;Gul ; I t 1 ;L14;L1 ;Mdl ;Sul]). Our aim h e r e i s t o c a p t u r e some o f t h e f l a v o r o f t h e s u b j e c t w i t h o u t b e i n g pedantic; p h y s i c a l r e a s o n i n g i s g e n e r a l l y o m i t t e d and t h e " u n d e r s t a n d i n g " i n v o l v e d i s t h e r e f o r e a t a mathematical l e v e l , a r i s i n g t h r o u g h t h e e q u a t i o n s One works w i t h wave f u n c t i o n s o r s t a t e f u n c t i o n s IL =

and t h e i r p r o p e r t i e s . IL(x,t)

f o r a s i m p l e p a r t i c l e (whatever t h a t i s ) ; x = (x1,x2,x3)

i n general

b u t a t f i r s t we w i l l use a 1-D ( = 1 - d i m e n s i o n a l ) s i t u a t i o n f o r s i m p l i c i t y . We w i l l assume t h a t b a s i c ideas i n p r o b a b i l i t y t h e o r y a r e known ( v e r y l i t t l e i s i n v o l v e d h e r e ) o r can be s t u d i e d s e p a r a t e l y ( c f . [Fel ;Do1 ;Ppl;Wol]). Thus t h e wave f u n c t i o n (which c o n t a i n s a l l p o s s i b l e i n f o r m a t i o n about t h e p a r t i c l e ) has t h e p r o p e r t y t h a t lILl

2 i s a p r o b a b i l i t y d e n s i t y i n t i e sense

t h a t t h e p r o b a b i l i t y o f t h e p a r t i c l e b e i n g i n a s e t K a t t i m e t i s IK IIL(x, t ) l2 dx (IL i s complex v a l u e d and IlJlll 2 = (IL,IL) = 1 &dx = 1 ) . Dynamical v a r i a b l e s ( f u n c t i o n s o r o p e r a t o r s i n a sense d e f i n e d below) have e x p e c t a t i o n s ( f ) = f 6fILdx r e l a t i v e t o a g i v e n IL.

For example momentum w i l l be i d e n t i -

f i e d i n c o o r d i n a t e space w i t h t h e o p e r a t o r p = ( h / i ) D x (A = h/2n, h = Planck c o n s t a n t ) and hence (*) ( p ) = ( n / i ) J &DxILdx.

Here t h e F o u r i e r i n t e g r a l

a r i s e s n a t u r a l l y i n g i v i n g a momentum r e p r e s e n t a t i o n o f p v i a here we n o r m a l l y t a k e i ( k ) =

I g(x)exp(ikx)dx

= hk.

= Fg w i t h g ( x ) = ( 1 / 2 n ) I G(k)

e x p ( - i k x ) d k and n a t u r a l l y a g r e a t deal o f fuss a r i s e s r e l a t i v e t o 2n. t h e Parseval f o r m u l a reads (g,h) tions, writing $(k,t)

=

f IL(x,t), X

=

Note First

Then f o r wave func-

I ghdx = 2nf $dk. 2

111L112 = 2 n I I $ ( k , t ) l dk.

I n o r d e r t o ac-

comodate t h e 2n most c o n v e n i e n t l y , and t o a d j u s t t o p h y s i c s usage, f o r

58

ROBERT CARROLL

quantum mechanical q u e s t i o n s we w i l l use a F o u r i e r t r a n s f o r m

-4

,v

(8.1)

Fg(k) = ( 2 ~ )

I,

g(x)e

-ikx

-4

dx =

= ( 2 ~ )

w i t h t h e corresponding adjustment t o R3. Parseval r e l a t i o n (g,h) dk = 1.

=

Thus c o n s i d e r

I

ihdx =

= h k and

Parseval formula, w i t h Dx

+

I

1,

m

-

ge

ikxdk

Then f o r T ( k , t ) = yxJl(x,t)

5Kdk = ($,;)

g i v e s IIJl1I2 =

(7) = h I k 1 T ( k , t ) l 2 d k . -Using

i k , one has (A) ( p ) = ( R / i ) / F i k $ h k

I

the

IT(k,t)l

2

(*) and t h e =

(p) as

de-

Thus t h e p r o b a b i l i t y t h a t t h e momentum 'i; = h k l i e s i n an i n t e r v a l K 2 i s IK I T ( k , t ) l dk. Next c o n s i d e r T = k i n e t i c energy which i n c l a s s i c a l phy2 2 2 2 Evids i c s i s p /2m and here should be T = (1/2m)[(Ti/i)D I = - ( h /2m)Dx.

sired.

ently ( T ) =

?=

F2/2m =

I JT$dx PI 2 k2 /2m.

=

-I

?(h2/2m)JIxxdx = -(Ti2/2m)Ix$(ik)2Fdk

= (h2k2/2m) so

C l e a r l y t h e r e have t o be r e s t r i c t i o n s on t h e f u n c t i o n s

Jl i n o r d e r t h a t pJl o r TJI make sense and as "observables" i n t h e t h e o r y we w i l l t a k e l i n e a r o p e r a t o r s A i n L2 d e f i n e d on a dense l i n e a r s e t D(A) c L2 , When t h e observable i s r e a l valued t h e o p e r a t o r A w i l l be r e q u i r e d t o be s e l f a d j o i n t ( i . e . ( A q , J l ) = (9,AJl) f o r 9,JI E D(A) = D(A*) - c f . Appendix A sometimes one w i l l deal w i t h e s s e n t i a l l y s e l f a d j o i n t o p e r a t o r s A, which means t h a t A has a u n i q u e s e l f a d j o i n t e x t e n s i o n A**).

One can check e a s i l y

t h a t p and T a r e s e l f a d j o i n t , d e f i n e d on s u i t a b l e domains (e.g. D(p) = I $ E 2 2 L , Jl' E L I , where d e r i v a t i v e s a r e always taken i n D' unless o t h e r w i s e specified).

For some purposes i t i s i m p o r t a n t t o use t h e idea o f r i g g e d H i l -

b e r t spaces t o d i s c u s s observables b u t we d e f e r t h i s here ( c f . [ B h l ] ) .

-

REmARK 8.1.

L e t us observe t h a t xp px = x ( h / i ) D x - [(h/i)Dx]x = - ( h / i ) 1 ( i n a c t i n g on any C f u n c t i o n f o r example) so p o s i t i o n and momentum do n o t

comnute as o p e r a t o r s and i n quantum mechanics t h i s i n v o l v e s t h e i m p o s s i b i l i t y o f s i m u l t a n e o u s l y measurement (Heisenberg u n c e r t a i n t y p r i n c i p l e ) . To see how t h i s works c o n s i d e r (AA) 2 = ( (A - ( A ) ) 2 ) so t h a t AA i s a s t a n d a r d

Assume ( x ) = 0 and ( p ) = 0 and c o n s i d e r 2 B t (l/fi)2(Ap)2 since now, f o r B r e a l , ( 0 ) 0 5 I , 1BxJl t $ ' I dx = 2 2 Im - m [xGJl' + ?'xJl]dx = Im - m x(lJl1 ) ' d x = 1Jl1 dx = -1 and J'Jl'dx = ?$I' 2 m - 2 dx = ( l / n ) Jlp Jldx = ( l / f i ) 2 ( A p ) 2 . It f o l l o w s t h a t ( b ) (Ax)(Ap) ~ h / 2

d e v i a t i o n i n s t a t i s t i c a l parlance. m

-/I

-


-!I

E q u a l i t y occurs f o r wave f u n c t i o n s Jl when (l/J21r/Axl)exp(-x~/2lAx~~).The u n c e r t a i n t y p r i n c i p l e here can a l s o

which i s Heisenberg u n c e r t a i n t y . 2

IJlI

=

be thought o f as a theorem about F o u r i e r i n t e g r a l s s i n c e (as above f o r T ) -I ?h2Jlxxdx = h 2( k2 ) and thus ( 6 ) says t h a t I x 2 I J l I 2 dx ( A P ) ~= ( p 2 )

I

k21$I2dk 1. 1 / 4 ( r e c a l l

I 1Jl1 2dx

= 1).

T h i s k i n d o f a n a l y s i s has been c a r -

r i e d much f u r t h e r f o r band and t i m e l i m i t e d f u n c t i o n s i n t h e c o n t e x t o f

QUANTUM MECHANICS

59

comnunications e n g i n e e r i n g and s i g n a l p r o c e s s i n g ( c f . [ D l ] ) .

2

+

U where U i s t h e o p e r a t o r determined by a ( r e a l ) p o t e n t i a l U(q) = U ( x ) and T = p2/2m as above. The SchrGdinger e q u a t i o n i n op-

Now w r i t e H = p /2m

(-in$'

e r a t o r f o r m becomes (+) ih$' = W we have a d i f f e r e n t i a l e q u a t i o n

(m)

$) so t h a t w i t h T (-12 /2m)Dx2 2 ifvbt = - ( h /2m)lLXx + U(x)$. Suppose Q =

r e p r e s e n t s a dynamical v a r i a b l e i n o p e r a t o r form and c o n s i d e r

Now f G W d x = -(h2/2m)f $,xQJdx

+ f U$Wdx

=

HQ

- QH

i s a commutator o r L i e b r a c k e t .

?DcWdx + I &UWdx

-(li2/2m)/

+

= J GHWdx so ( 8 . 2 ) becomes (**) D , C Q ) = ( ( i / h ) [ H , Q ]

QG

=

Using t h i s f o r m a l i s m one can d e r i v e

a general i n d e t e r m i n a c y r e l a t i o n e x t e n d i n g ( 6 ) .

Thus suppose [Q,R]

f o r two r e a l o r s e l f a d j o i n t dynamical v a r i a b l e s Q and R. (cf.

where [H,Q]

-iC

=

Consider f o r r e a l

( 0 ) )

W =

(8.3)

I

2 l(R+iAQ)$I dx = ((R+iAQ)$,(R+iAQ)$)

= ($,(R-iAQ)(R+

iXQ)$)

2 ( s i n c e R = R* and Q = Q*). We w r i t e h e r e e.g. (R$,R$) = ($,R*R$) = ($,R $ ) 2 = f $R $dx i n o r d e r t o c i r c u m v e n t n o t a t i o n o f t h e form 1 (R$)*R$dx ( o r worse

-

and i n c o r r e c t

-

f (%)R$dx).

I n (8.2) where H = H* ( =

i) we d i d

n o t bo-

t h e r w i t h t h i s b u t f o r R = p = ( f i / i ) D x one has p* = p ( i n t e g r a t i o n by p a r t s ) while

= -(h/i)D

= -p.

x2

Now W 2 0 i n ( 8 . 3 ) and i t s minimum occurs when W 2 2 ) - ( 1 / 4 ) ( C ) / ( Q 2 ) 2 0 and one has ( r e p l a c -

= 0 o r X = ( C ) / 2 ( Q ) so t h a t ( R

i n g R by R

-

CHEdREm 8.2 has (AR)(AQ)

( R ) and Q by Q

- (Q),

(I.INCERkAzNCy)2 (1/2)( C).

For r e a l observables R and Q w i t h [R,Q]

w i t h [R,Q]

= [R-( R),Q-(

Q)])

One says t h a t a dynamical v a r i a b l e Q i s i n an e i g e n s t a t e $ when For example i f ( H ) = E then i W t = EJ/ so $ = $,exp(-iwt),

o =

(Q)$

= i C one

=

W.

E/h.

I t w i l l be i n s t r u c t i v e t o c o n s i d e r a harmonic o s c i l l a t o r where 2 H = p2/2m + ( 1 / 2 ) k x ( k > 0 ) . The energy s t a t e a t l e v e l E corresponds t o HQ = E$ ( f r o m itWt = EJ/) o r (*A) $xx + (2m/n2)(E - ( 1 / 2 ) k x 2 ) $ = 0. T h i s can

EHmPtE 8.3.

be s o l v e d d i r e c t l y by v a r i o u s methods ( w i t h growth a t

cu

c o n t r o l l e d so t h a t

I$//= 1 ) b u t we w i l l use a d i f f e r e n t method t o determine t h e e i g e n f u n c t i o n s

60

ROBERT CARROLL

of H based on f a c t o r i z a t i o n i n t o c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s ( c f . [Gll;Pkl]). (m,/b)'q and [P,Q] (mu/%)'

Thus s e t

u1

( q = x), P = %-'[p,q]

( = c l a s s i c a l o s c i l l a t o r frequency), Q = Then JC = (1/2)(P2 + Q2)

= (k/m)'

and JC = ( f i w ) - l H .

(mh)-'p,

= -i ( e v i d e n t l y P

'L

(mwh)-'(%/i)(m,/ti)'DQ!.

and (mwh)-'(ti/i)Dx

-iD

Q

s i n c e d/dq = d/dx = (d/dQ) Now d e f i n e c r e a t i o n

and a n n i h i l a t i o n o p e r a t o r s A* and A by (**) A* = 2 '(9-iP)

so t h a t [A,A*] ercise).

= 1 and JC = A*A

+ ( 1 / 2 ) w i t h [JC,A]

and A = 2-'(Q+iP)

= -A and [JC,A*]

= A*

(ex-

Note t h a t A and A* a r e n o t H e r m i t i a n o p e r a t o r s ( H e r m i t i a n means

f o r $,x E D(A)). Now i f A$, = 0 thcnJC$o = (1/2)$0 so go i s an e i g e n f u n c t i o n o f JC. L e t Q 'L y ( q 'L x ) and s i n c e A$, = 0 means yIo0 = 2 2 -iP$, = i Dy$, = -D $ we have (*&) $,(y) = ~ ~ - l / ~ e x p (/ -2 y) ( t h e c o n s t a n t i s

(A$,x)

($,Ax)

Y O

chosen so t h a t II$ or

0

a*$, = (3/2)A*$, Thus

11 = 1 ).

Now f r o m [JC,A*]

0

and i t e r a t i n g t h i s procedure (**) JC(A*n$o)

-

= A*$, = [n+(l/2)]

i s an e i g e n f u n c t i o n f o r JC w i t h e i g e n v a l u e (n+1/2).

N o r m a l i z a t i o n i n v o l v e s t a k i n g $, w i t h A$,,

= A* one has JCA*$

= n'$n-l

and A*A$,

and t h e n A*$,,

= (A*n$o)/(n!)'

= n$,

(exercise).

= (n+l)'$n+l

One can t h i n k o f $,

as rep-

r e s e n t i n g an n p a r t i c l e o r n quantum s t a t e (each quantum o f energy %),

A* adds a quantum t o a s t a t e , i n c r e a s i n g i t s energy by a t e s a quantum.

while A annihil-

The wave f u n c t i o n s i n v o l v e Hermite polynomials and we sim-

p l y r e c o r d t h e r e s u l t s here ( c f . [ G l l ; P k l ] (n!)'4Pn(J2y)$o(y)

nu,

and

f o r details).

Thus $,(y)

=

where ( [ k ] = l a r g e s t i n t e g e r 5 k )

(Pn i s o f t e n denoted by Hn).

One can show ( e x e r c i s e

-

the discussion i n

[Gll]

i s e s p e c i a l l y n i c e ) t h a t t h e Hermite f u n c t i o n s $,(y) f o r m a complete 2 orthonormal s e t i n L (we r e f e r t o Appendix A f o r completeness i d e a s ) . One speaks o f a system, h e r e a harmonic o s c i l l a t o r , b e i n g represented mathematic a l l y by an a l g e b r a o f operators; here say p,q,H

where [p,q]

= h / i (canoni-

c a l comnutation r e l a t i o n ) serve as generators o f t h e algebra.

EMmPCE 8.4.

As a t y p i c a l quantum mechanical c a l c u l a t i o n c o n s i d e r a par-

t i c l e o f energy E i n c i d e n t from t h e l e f t i n a f o r c e f i e l d i n v o l v i n g a p o t e n t i a l U(x) o f r e c t a n g u l a r shape:

U ( x ) = 0 f o r x < -a, U(x) = Uo f o r -a <

x < a, and U ( x ) = 0 f o r x > a.

E

If

< Uo t h i s s h o u l d r e p r e s e n t a b a r r i e r

c l a s s i c a l l y which t h e p a r t i c l e c o u l d n o t pass b u t i n quantum mechanics 2 t h i n g s a r e d i f f e r e n t . Thus t h e wave f u n c t i o n $ s a t i s f i e s -(fi /2m)$,, = 2 2 2 ikx [E-U(x)]$ so f o r 1x1 > a, s e t t i n g E = h k /2m, !bXx + k $ = 0 and $ = ae

+ gexp(-ikx)

( a , g depend on x < -a o r x > a ) .

On t h e o t h e r hand f o r 1x1 < a

QUANTUM MECHANICS

-

we s e t U

0

E =

R 24k 2/2m so 9xx

=

61

^k29 and 9 = y e x p ( 2 x )

t Gexp(-^kx).

These

f u n c t i o n s must be f i t t o g e t h e r a t t h e p o i n t s x = ?a ( a l o n g w i t h t h e i r d e r i Thus i f 9 = A e x p ( - i k x ) + B e x p ( i k x ) f o r x < - a a n d 9 = F e x p ( i k x )

vatives).

f o r x > a (no waves i n c i d e n t f r o m t h e r i g h t ) some a l g e b r a ( c f . [Mol;Pkl;Rol]) g i v e s A = Fexp(Pika)[Ch2;a

-

-(1/2)i[(k/z)

(;/k)]Sh2ca]

and B = - ( i / 2 ) F

A transmission c o e f f i c i e n t T i s d e f i n e d by 1/T = l A / F I 2 and a r e f l e c t i o n c o e f f i c i e n t by R = 1-T (R = 2 2 lB/Al ). I t f o l l o w s t h a t 1/T = 1 t [Uo/4E(Uo-E)]Sh222a which measures t h e [ ( b k ) t (k/c)]Sh2$a

(Sh

'L

s i n h and Ch

transparancy o f t h e b a r r i e r .

%

cosh).

That t h e p a r t i c l e may pass through t h e b a r r i e r

i n these circumstances i s r e f e r r e d t o as a t u n n e l i n g e f f e c t .

To c o n t i n u e

t h i s theme, suppose we have a p o t e n t i a l w e l l , U(x) as above b u t w i t h Uo < 0 and Uo

<

E.

I t i s convenient t o change t h e p i c t u r e so t h a t U(x) = Uo f o r

1x1 > a, U(x) = 0 f o r 1x1 < a, and E < Uo. Then f o r 1x1 > a one has 2 -(Ti /2m)ILxx = (E-Uo)9 so 9 = Bexp(-Bx) f o r x > a and 9 = Cexp(Bx) f o r x < 2 above). I n t h e r e g i o n 1x1 < a one has -a where 6 = [(2m/h )(Uo-E)]' (= 9 = a e x p ( i k x ) + y e x p ( - i k x ) ( k as above).

It f o l l o w s upon matching s o l u t i o n s

a t x = ?a ( w i t h d e r i v a t i v e s ) t h a t t h e r e can o n l y be a s o l u t i o n when E = En where En i s a r o o t o f Tan[2a(En2m/h2)"]=2[En(Uo-En)]'/(2En-Uo)

w i t h 0 < En

Thus t h e r e i s a d i s c r e t e energy spectrum between 0 and Uo and t h e

< Uo.

r e s u l t i n g ,9,

a r e c a l l e d bound s t a t e s .

= AexpBn(atx)[exp(-iana

+_

The wave f u n c t i o n 9,

e x p ( i a n a ) ] f o r x .=-a,$,

has t h e form 9,,

= A[exp(ianx)

+_

exp

exp(-iana)] f o r x > f o r 1x1 ( a , and 9, = AexpOn(a-x)[exp(iana) an = (En2m/h2)' and B = ((Uo-E,)2m/h2)'. As a l i m i t i n g case l e t n2 Uo t o o b t a i n E = h2n2r2/4a 2m = n2h2n2/8ma2 ( n = 1,2, ...; n = 0 i s n o m i t t e d s i n c e ILo = 0 ) . The wave f u n c t i o n s become 9, = 0 f o r 1x1 2 a and 9, (-ianx)] a (where

+_

-

-f

= A[exp(nnix/Ea)

a

=

L/2 and x

pendence 9,

-f

- (-l)nexp(-nnix/2a)]

f o r 1x1 5 a.

Changing v a r i a b l e s t o

x+L/2 one o b t a i n s upon n o r m a l i z i n g and i n s e r t i n g t h e t de-

= 0 f o r x 5 0 and x

2 L w h i l e 9,

=

(2/L)'Sin(nnx/L)exp[iyn

-

iEnt/n]

for 0 x 5 L, where yn ( r e a l ) i s an a r b i t r a r y phase f a c t o r ( n o t e 2 2 2 2En = TI n IT /L m ) . 2 L e t us c o n s i d e r a general s i t u a t i o n h e r e f o r (-h /2m)J, = 2 xx2 = (E-U(x))$ ( U r e a l ) which we w r i t e i n t h e f o r m (*.) IL" + k 9 = q ( x ) 9 ( k

EMAIPLE 8.5,

2mE/h2, q = 2mU/fi2

- t h i s w i l l be p i c k e d up a g a i n i n Chapter 2 w i t h many

more d e t a i l s developed). dx < (8.5)

One assumes h e r e e.g.

q real with

and l o o k s f o r s c a t t e r i n g s o l u t i o n s o f (*.) ILl(k,x)

*

lz

2 (l+]xl )lql

i n t h e f o r m ( c f . [Chl;Fal])

exp(ikx) t s12exp(-ikx)

(x

s1 lexp( ik x )

(x

-+

I -+

-m)

-1

ROBERT CARROLL

62

(8.5)

'J'Z(k,x)

(note t h a t since q pated).

I

Q

s22exp(-i k x )

(X +

-m)

e x p ( - i k x ) + sZlexp(ikx)

(x

m)

0 rapidly at

-+

+

asymptotic s o l u t i o n s ( 8 . 5 ) a r e a n t i c i -

+m

Then e.g. JI1 r e p r e s e n t s a wave e x p ( i k x ) incoming f r o m t h e l e f t and

p r o p a g a t i n g t o t h e r i g h t ; p a r t i s r e f l e c t e d i n t h e form s 1 2 e x p ( - i k x ) and p a r t i s t r a n s m i t t e d as sllexp(ikx) The m a t r i x S ( k ) = (( s . . ( k ) ) )

(one w r i t e s a l s o sll

= T, s12 = R,

i s c a l l e d t h e s c a t t e r i n g m a t r i x and e v e n t u a l l y

1.l

( f o r t h e i n v e r s e problem) one wants t o r e c o v e r q from knowledge o f S. one d e f i n e s J o s t s o l u t i o n s fk(k,x) x

-f

m

etc.).

and l i m f - ( k , x ) e x p ( i k x )

by t h e r u l e l i m f + ( k , x ) e x p ( - i k x )

= 1 as x +

I f we t a k e

-a.

(*m)

Now

= 1 as

and c o n v e r t i t

i n t o an i n t e g r a l e q u a t i o n as i n 56 ( v a r i a t i o n o f parameters) i t f o l l o w s t h a t ' (8.6)

f+(k,x)

e ikx

=

f-(k,x)

-

im [Sink(x-t)/k]q(t)f+(k,t)dt; +

= e-ikx

1'

-m

[Sink(x-t)/k]q(t)f-(k,t)dt

These can be s o l v e d as i n 56 and e s t i m a t e s obtained.

Transmutation k e r n e l s

can then a l s o be i n t r o d u c e d b u t we o m i t them f o r now (see Chapter 2 ) . r e a l k, f + ( - k , x )

= ?+(k,x),

i 2 i k (W(f,g) = f g ' - f ' g ) .

f-(-k,x)

= i(k,x),

Thus f + ( + k , x ) and f - ( + k , x )

o f s o l u t i o n s ( f o r k # 0 ) and hence ( c . f-(k,x)

= cllft(k,x)

f,(k,x)

f o r k r e a l , and Icl2I2

form fundamental p a i r s

+ c12(f+(-k,x); = c22f-(k,x)

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 cll(k) = Eij(k)

=

= cij(k))

1 j

(8.7)

For

and W ( f + ( k , x ) , f + ( - k , x ) )

= 1

=

-c

+ Icllf.

+

22

c2lf-(-k,x) ( k ) = cZl(k), c . . ( - k ) 12 1J Various growth e s t i m a t e s and

(-k),

c

p r o p e r t i e s o f a n a l y t i c i t y can a l s o be d e r i v e d which we momentarily o m i t ( c f . 56 and Chapter 2 ) .

One can now r e l a t e t h e Jli w i t h ,f

v i a e.g. Jll = sllf+

=

and s I 2 = c 22 / c 21' S i m i 1 a r l y from J / 2 one f i n d s t h a t s Z 2 = 1/c12 and s Z 1 = c 11/ c 12. I t f o l l o w s t h a t sll 2 2 = 0, s . . ( - k ) = F . . ( k ) f o r k r e a l , and lsllI + 1s121 = s22' s l l s 2 1 s12s22 1J 1.l = ST). F u r t h e r as I k l = 1 = I s I' + I s la ( S i s a u n i t a r y m a t r i x - s 22 21 m, s12 = O ( l / l k l ) , sZ1 = O ( l / l k l ) , and sll = 1 + O ( l / l k l ) . The i n v e r s e probf-(-k,x)

t s12f-(k,x)

-

-

+

f r o m which sll

= 1/cZl

-'

lem, o f d e t e r m i n i n g t h e p o t e n t i a l from knowledge o f t h e sij

-f

( p l u s knowledge

o f t h e bound s t a t e s and n o r m a l i z i n g c o n s t a n t s ) , w i l l be s t u d i e d l a t e r ( i n f o r m a t i o n about S can be o b t a i n e d from experiments).

QUANTUM MECHANICS

63

L e t us p o i n t o u t two e q u i v a l e n t ways o f l o o k i n g a t t h e dynamics,

REIIIARK 8.6,

The former has been

namely, t h e Schrodinger and t h e Heisenberg p i c t u r e s .

d e s c r i b e d v i a t h e Schrodinger e q u a t i o n i W t = W ; t h e s t a t e J,(t,x)

evolves

i n t i m e from some i n i t i a l $(O,x)

= exp

(-iHt/'h)J,o tail).

= !b0 and f o r m a l l y one can w r i t e

-

(such o p e r a t i o n a l formulas w i l l be t r e a t e d l a t e r i n de-

= U(t)J,,

I n t h e Heisenberg p i c t u r e t h e s t a t e s remain f i x e d and t h e observables

e v o l v e i n t i m e a c c o r d i n g t o t h e r u l e ( c f . (*))

A(O),

J,

-

AH) s i n c e

H

3dA/dt = [ i H , A ( t ) l ,

( n o t e f o r m a l l y A ' = (i/R)HU*AU

o r A ( t ) = U*(t)AU(t)

(i/ti)(HA

(A*)

-

and U commute

-

U*A(i/A)HU =

Evidently the re-

see Appendix A).

l a t i o n between t h e two p o i n t s o f v i e w i s determined by (J,(t,x),AJ,(t,x) (J,o,A(t)J,o)

= (J,o,U*AUJ/o)

=

thinks o f eigenstates

sm i s

t w h i l e J, o r

as i n Example 8.3 and r e p r e s e n t s

J,

=

(UILO,Alhlo).

L e t us r e c o r d here t h e D i r a c b r a and k e t n o t a t i o n .

REmARK 8.7.

A =

r e p r e s e n t e d by ( m ] ( b r a ) .

Thus one

by I n ) ( k e t )

An o p e r a t o r a c t i o n on

J,

by A

(Jim,

i s denoted by A l n ) and ( m l A l n ) denotes a m a t r i x element 1 smASndx o r

H i s t o r i c a l l y one r e p r e s e n t e d o p e r a t o r s i n quantum mechanics v i a such A$,). matrices but t h e proper study o f l i n e a r operators i s b e t t e r c a r r i e d o u t q u i t e d i f f e r e n t l y ; we w i l l use t h e " m a t r i x " n o t a t i o n however a t t i m e s when i t doesn't lead t o trouble.

One w r i t e s e.g.

la) =

f u n c t i o n expansion r e l a t i v e t o t h e b a s i s Gn ( i . e .

1 lJ,n)(J,nl$)) ply t o write

((PI$)*

=

and e v i d e n t l y H l n ) = E,In).

I$)

f o r J, and

(PI$)-

=

($1~)).

($1

f o r Jlt

A symbol

8 ]$)(PI (%

1 I n ) ( n ( a ) f o r an eigen1 (Jln,$)ICln o r I J , ) =

J, =

Perhaps a b e t t e r n o t a t i o n i s simabove) w i t h s c a l a r p r o d u c t

v i o u s way and a l l o f these n o t a t i o n s f i t t o g e t h e r ( e x e r c i s e any q u e s t i o n ) .

1 with

I$)

=

I

(PI$)

denotes an o p e r a t o r i n an ob-

The completeness r e l a t i o n f o r J,

-

i f there i s

i s o f t e n w r i t t e n as 1 =

Sometimes one w r i t e s ( r l $ )f o r J,(r) ( c o o r d i n a t e r e p r e s e n t a t i o n ) 3 3 I r ) d r(r[J, and I I r ) d r ( r l = 1 ( t h u s t r l r ' ) = s ( r - r ' ) and

[ r )i s c a l l e d a p o s i t i o n eigenstate

-

the notation f o r

R3 s h o u l d be c l e a r ) .

3 S i m i l a r l y one d e f i n e s momentum e i g e n s t a t e s I k ) w i t h ( k l k ' ) = ( 2 7 ) 6 ( k - k ' ) ,

3

1 = ( 2 ~ ) - ~I k/ ) d k c k l , and 1 J,exp(-ikr)dr).

IJ,)

3

= ( Z T ) - ~ II k ) d k(klJ,) ( ( k l J , ) = f J , ( k )

=

I n o r d e r t o r e l a t e these r e p r e s e n t a t i o n s we t h i n k o f ( r l k )

as t h e wave f u n c t i o n f o r a system o r p a r t i c l e h a v i n g a wave number o r momentum k (such a system must be u n l o c a l i z e d i n space by t h e u n c e r t a i n t y p r i n c i p l e so ( r l k ) = c ( k ) e x p ( i k r ) = c ( k ) e x p ( i ( k , r ) ) (rlk)).

3

note ( k , r ) = ( r , k )

3

t 3

Then s i n c e ( 2 n ) s ( k - k ' ) = ( k l k ' ) = / ( k l r ) d r ( r [ k ' ) = c k c k , / d r

exp(i(k'-k,r))

(8.8)

-

we see t h a t c k can be chosen as 1 and ( r l k ) = e x p ( i ( k , r ) ) ;

( r l $ )= ( 2 ~ ) - ~( r 1l k ) d3 k ( k l $ ) = ( 2 1 ~ ) - ~ 1 d ~ k e ~ ( ~ ' ~ ) ( k ( J , )

64

ROBERT CARROLL

REmARK 8.8.

L e t us i n d i c a t e h e r e some 3-D o p e r a t o r s connected w i t h a n g u l a r

momentum and s p i n .

-

e n t s Lx = yp,

X

C l a s s i c a l a n g u l a r momentum i s L = Ly = Z P X

ZPY,

-

-

xpz, and Lz = xpy

6 and has componThe c l a s s i c a l an-

yp,.

g u l a r momentum f o r a " r o t a t o r " corresponds t o L ?, Iw, w i t h E = (1/2)Iw2 % 2 L / 2 1 ( I i s say a moment o f i n e r t i a and w an a n g u l a r v e l o c i t y ) . Now i n quan2 tum mechanics we t a k e L = q X p and t h e analogue f o r H i s H = L / 2 1 = ( 1 / 2 I )

1 LiLi

i f I makes sense (repeated i n d i c e s w i l l be sumned).

= (1/21)LiLi

1 t t l e a l g e b r a u s i n g t h e canonical q u a n t i z a t i o n r u l e s [pk,p,] w i t h [p ,q.]

= (R/i)E

y i e l d s [L ,L.]

= ifiEmjkLk

= [qk,qml

(exercise

-

ishes i f any 2 i n d i c e s a r e t h e same). o f a dumbbell o r a d i a a t o m i c m o l e c u l e

-

i t van-

For a " r o t a t o r " as above (e.g.

-

= 0

is 1 or

E~~~

k J kj m~ -1 depending on whether i j k i s an even o r odd p e r m u t a t i o n o f 123

A

think

t h e Li w i l l be p h y s i c a l

c f . [Bhl])

observables and w i t h H generate t h e a l g e b r a o f t h e system. More g e n e r a l l y 3 an o b j e c t ( p a r t i c l e ) i n R has 3 c o o r d i n a t e s qi and 3 r o t a t i o n a l degrees o f freedom.

One has a group o f r o t a t i o n s SO(3) ( o r t h o g o n a l m a t r i c e s w i t h de-

t e r m i n e n t 1 ) whose L i e a l g e b r a i s generated by

(8.9)

a1 =

[ 00 00

0 - 1 1 ; a2 = 0 1 0

( c f . Appendix C ) .

Thus

[al,a2]

[ 00

0 1 0 01 ; -1 0 0

= a 3,

a3 =

[a2,a3]

0 -1 0 1 0 0 ) 0 0

[0

= al,

and

m a t r i c e s a k ( e ) = exp(ake) generate SO(3) ( n o t e [aj,am] ak = - i y k t h e n [yl,y2]

Lk = fwk).

= i y 3 e t c . so up t o a f a c t o r o f

= a2 and t h e

[a3,al]

= ~~~~a~ and i f e.g.

ti,

yk

%

Lk, e t c , i . e .

Now one l o o k s f o r general s o l u t i o n s o f commutator r u l e s

(AA)

w i t h Jk = Jk.

T h i s w i l l n a t u r a l l y g i v e back t h e Lk b u t [Jm,Jk] = iRE mkp P i n a d d i t i o n t h e r e w i l l be s p i n o p e r a t o r s Sk s a t i s f y i n g (.A). Spin i s t h o u g h t J

o f sometimes as an i n t r i n s i c a n g u l a r momentum, n o t expressed i n terms o f pos i t i o n and momentum.

We r e f e r t o t h e p h y s i c s l i t e r a t u r e f o r t h e p h i l o s o p h y

of s p i n b u t n o t e here t h a t i t can be thought o f as a r i s i n g m a t h e m a t i c a l l y i f one r e q u i r e s a complete t h e o r y o f t h e connnutation r u l e s ( A A ) and t h e r e l a t e d e i g e n f u n c t i o n theory.

This matter i s treated i n the context o f t h e algebra

o f a n g u l a r momentum i n [ B h l ] and we f o l l o w t h a t s p i r i t (which i n f a c t has an -1 u n d e r l y i n g L i e t h e o r e t i c s i g n i f i c a n c e ) . Thus i n t r o d u c e ( A a ) H3 = 11 J3; H, = fi-l[J, + iJ2]; H- = Ti-'[J - i J 2 ] . The c o n d i t i o n JL = Jk ( o r J i = J k ) 1 i s expressed v i a H i = H3, H l = H-, and H I = H, ( n o t e s t r i c t l y we d i s t i n g u i s h between H e r m i t i a n c o n j u g a t i o n Jt and a d j o i n t s J* s t r o n g e r domain c o n d i t i o n

-

-

the l a t t e r involving a

now i t may happen t h a t Jt = J* b u t o f t e n we sim-

t generically).

p l y d o n ' t need t o worry about t h e domain and w i l l use J from

(Aa)

and (AA) one o b t a i n s (A&) [H3,H,]

=

H;, -

[H,,H-]

Now

2H3 (see here

65

QUANTUM MECHANICS

Appendix C f o r t h e L i e a l g e b r a

$=

-

so(3,C)

one can a l s o base t h e d i s c u s -

s i o n on SU(2) i n s t e a d o f S O ( 3 ) b u t we p r e f e r n o t t o ) . Set now J2 = Ti2$ 2 w i t h J? = H+H- + H3 - H3 = H-H+ + H: + H3 ($ i s a C a s i m i r o p e r a t o r s i t t i n g i n t h e c e n t e r o f t h e e n v e l o p i n g a l g e b r a E(?)

o f so(3,C)

=

r - [$,A]

= 0 for

A E E ( g ) ) . L e t fmbe a w e i g h t v e c t o r ( e i g e n v e c t o r f o r H3, H3fm = mf, with Ilf II = 1 (assumed t o e x i s t f o r some m). D e f i n e f i = H f and check t h a t m+ + m H f- = (rni1)f; ( e x e r c i s e ) . Now $ and H3 commute and H3$fm = $H3fm 3 m 2 2 = df, so t h a t = c f m and c > 0 s i n c e $ = ( l / n )J ( e x e r c i s e ) . Fur-

dfm

-

thermore in2 < c s i n c e 0 < h'2[(fm,Jlfm) -2 + (fm,J2fm)] 2 ITi-2[(fm,J 2 fm) ( f m y -2 2 2 Then s t a r t i n g from some fk (H3f; = m f k ) and a p p l y i n g J3fm)] = (c-m )llfmll

.

C

s u c c e s s i v e l y we must a r r i v e a t a l a r g e s t e i g e n v a l u e L o f H3 w i t h H + f L = 2 0. Then = (H3 t H 3 ) f E = L(L+l)f: and c = L ( L + l ) f o r t h i s L. S i m i l a r H,

$fE

l y , a p p l y i n g H- t o f: one a r r i v e s a t a s m a l l e s t e i g e n v a l u e -L o f H~ ( e x e r -

cise).

There w i l l be t h e n 2L+1 e i g e n v e c t o r s so 2L+1 must be an i n t e g e r o r

L = 0,1/2,1,3/2,

...

Thus f o r g i v e n L i n t h i s sequence one can f i n d 2L+1 L L orthonormal e i g e n v e c t o r s fm ( r e i n d e x h e r e ) spanning a space R = {f = L a f 1 (note t h a t eigenvectors o f a Hermitian operator corresponding t o L m m d i f f e r e n t eigenvalues a r e a u t o m a t i c a l l y orthogonal ). One checks e a s i l y ( e x e r c i s e ) t h a t f o r L f i x e d H+fmL = ~ ~ + ~H-f, fL k = amfk-l + ~ ~ (and H3fmL = mf,) L

1:

where am = [(L+m)(L-m+l)]&. c i b l e representation o f

The space R

serves as a space f o r an i r r e d u -

= s o ( 3 ) o r E(gu) (and J?f = L ( L + l ) f f o r any f E

2 L The number L i s c a l l e d t h e a n g u l a r momentum quantum number; L f = RL). h 2 L ( L + l ) f L (L2 % J2 = h2J? f o r L = 0,1,2 ,... - t h i s r e s t r i c t i o n on L i s necessary if L = q X p

-

c f . [Bhl]).

For L = 1/2,3/2,

... one

o p e r a t o r s and i n p a r t i c u l a r f o r L = 1 / 2 t h e m a t r i c e s u 2(fi,Jkf;)

j

arrives a t spin

w i t h elements

a r e t h e P a u l i m a t r i c e s g i v e n by

9. MEAK PRBBLEW IN PDE,

There a r e a number o f l i n e a r and n o n l i n e a r PDE

which have been s t u d i e d e x t e n s i v e l y and such s t u d i e s have o f t e n m o t i v a t e d t h e development o f whole areas o f mathematics.

We have a l r e a d y encountered

some l i n e a r problems i n § 3 , 4 b u t o n l y t h e n o n l i n e a r Hamilton-Jacobi e q u a t i o n i n 57, and we want now t o g i v e i n 59,lO some i n t r o d u c t o r y m a t e r i a l on c e r t a i n o t h e r n o n l i n e a r equations (e.g.

t h e Navier-Stokes e q u a t i o n s o f f l u i d

dynamics, n o n l i n e a r wave equations a r i s i n g i n quantum f i e l d t h e o r y , t h e Ginzburg-Landau equations o f s u p e r c o n d u c t i v i t y , e l l i p t i c equations a r i s i n g from t h e Yamabe problem i n d i f f e r e n t i a l geometry, t h e KdV and o t h e r non-

66

ROBERT CARROLL

F i r s t i t w i l l be u s e f u l ( i n

l i n e a r equations from s o l i t o n t h e o r y , e t c . ) .

f a c t v i r t u a l l y e s s e n t i a l ) t o have a v a i l a b l e some o f t h e modern f o r m u l a t i o n

o f d i f f e r e n t i a l problems i n v a r i o u s f u n c t i o n space c o n t e x t s and i n p a r t i c u l a r t h e concept and machinery o f weak s o l u t i o n s .

T h i s began i n p a r t as an

o f f s h o o t o f d i s t r i b u t i o n t h e o r y i n t h e 1950's and soon became an " i n d u s t r y " unto i t s e l f .

L e t us mention [Agl;Brl;C1;Fr2;Ftl,2;Grl;Gsl;Gwl;Hpl;Krl;Ldl

,

2;H1;H11,2;Li1-8;Mtl;Pzl;Sbl;Sh2;Trl;Vl] f o r background on methods o f f u n c A t y p i c a l s i t u a t i o n where these methods was v a l u -

t i o n a l a n a l y s i s i n PDE.

a b l e i n v o l v e s an e q u a t i o n such as Au = f i n n C R3 say w i t h u = uo on r = 2 an. The n a t u r a l problems would be ( 1 ) s o l v e Au = 0 i n R f o r u E C (a)n Co (2) w i t h u = uo E C o ( r ) on r o r ( 2 ) s o l v e Au = f E Co(n) f o r u E C 2 (n) n Co

(5) with

u = 0 on

r.

These problems make sense and can be s t u d i e d f o r n i c e

r e g i o n s b u t i f n has s p i k e s i n i t o r sharp i n t r u d i n g c o n i c a l wedges f o r example t h e r e may n o t be a s o l u t i o n . On t h e o t h e r hand one can always f i n d 2 2 weak s o l u t i o n s ( u E L (n), Au E L (n), Au i n D ' ( n ) ) , and by subsequent ana l y s i s show t h a t they a r e s u i t a b l y r e g u l a r i f t h e d a t a uo o r f a r e r e g u l a r .

2 t h e o r y here i s a t y p i c a l a p p l i c a t i o n o f f u n c t i o n a l a n a l y t i c methods

The L

and we s k e t c h i t f i r s t .

EXA1IIPCE 9.1.

The g u i d i n g p r i n c i p l e i s Green's theorem f o r n i c e o r r e g u l a r

r e g i o n s ( = r e g i o n s where Green's theorem h o l d s ! ) . r e s t r i c t i o n o f Cm(Rn) t o

'5, (*) -I, Auvdx

=

I,

Thus f o r u,v E C"(E) = DkuDkvdx

- Ir

unvdo where

un denotes t h e e x t e r i o r normal d e r i v a t i v e , Dk = a/axk, e t c . as i n (2.13). Now s e t

a(u,v) =

(A)

I

l

DkuDkvdx

+

c / uvdx and c o n s i d e r t h e problem o f s o l -

v i n g -Au + cu = f ( t h e c > 0 i s i n t r o d u c e d h e r e t o make t h i n g s t e c h n i c a l l y 2 2 1 s i m p l e r ) . We r e c a l l t h e Sobolev space H (n) = I u E L ( a ) , Dku E L ( n ) ) and 1 1 2 H,(n) = completion o f i n H (n). Take now f E L (n) = H and ask f o r u E 1

Ho(n)

Ct

= V such t h a t ( v a r i a t i o n a l D i r c h l e t problem)

( 0 )

a(u,v) = ( f , v ) f o r

, ) i s t h e H s c a l a r p r o d u c t and (( , )) w i l l denote t h e V scal a r p r o d u c t ) . I f u s a t i s f i e s t h i s t h e n we have In 1 DkuDkvdx + cI uvdx , =

all v E V ((

I n f v d x and i n p a r t i c u l a r f o r v = IP E C i t h i s means

(

-Au + cu

-

f , v ) = 0 so

-Au + cu = f i n D' w h i l e t h e c o n d i t i o n u E V i m p l i e s u = 0 on r i n some "weak" sense (see below). We see t h a t i f i n f a c t u E C 2 and (*) i s used t h e 1 Next f o r t h e v a r i a t i o n a l Neumann boundary t e r m vanishes s i n c e v E V = Ho. 1 problem l e t V = H (n) and ask again t h a t ( 0 ) hold. Then a g a i n ( t a k i n g v = IP 2 E Cm one o b t a i n s -Au + cu = f i n D ' ( n ) b u t now, i f u E C , (*) w i l l g i v e 0 1 Ir unvdo = 0 f o r a l l v E H (n). By v a r i o u s so c a l l e d t r a c e theorems v r e s tricted to

r

f i l l s up a space

('(r)

( c f . [Lil,E;Cl])

and t h u s un = 0 i n

WEAK PROBLEMS

H-+(r).

67

This sets t h e stage f o r t h e f o l l o w i n g a b s t r a c t treatment.

L e t now V C H be H i l b e r t spaces w i t h V dense i n H and c o n t i n u o u s l y embedded

(i:V+H

i s c o n t i n u o u s o r l v l H 2 kllvllV).

l i n e a r form on V X V ( i . e .

la(u,v)l

Let a(-,-)

(cllullllvll

be a c o n t i n u o u s 1-1/2

w i t h a(u,v)

l i n e a r i n u and

c o n j u g a t e l i n e a r i n v). F o r example t h i n k o f o u r Example 9.1 above w h e w H 2 1 1 = L and V = H o r Ho. We assume f u r t h e r t h a t a ( - , - ) i s " c o e r c i v e " i n t h e 1 1 sense ( 6 ) a(u,u) 1. ~ I I ~( nI oIt e~ i n o u r example f o r u E H~ o r H , a(u,u) = 2 2 IDku( + c l u I 2 ~ m i n ( 1 , c ) l l u I l ) . Now a ( - , = ) w i l l determine two l i n e a r op-

1

e r a t o r s as f o l l o w s .

First, since w

j u g a t e l i n e a r one has a(u,w)

+

V

a(u,w):

= ((c,w))and

5

C i s c o n t i n u o u s and con-

+

= Au where A i s l i n e a r and con-

t i n u o u s as an o p e r a t o r V

+ V. L i n e a r i t y i s t r i v i a l and c l e a r l y IIAull = 5 cIIuII. On t h e o t h e r hand l e t N C V be t h e s e t o f u f o r a(u,w): V + C i s continuous i n t h e t o p o l o g y o f H. Extending t h e

sup la(u,w)l/lwll which w

-+

map by c o n t i n u i t y f o r such u one has a(u,w)

A i s linear.

=

(x,w) and c l e a r l y x

n o t continuous and as an example o f A go back t o

1J

DkuDpdx + c J u i d x .

G)

= ( -Au 1 1 Ho o r H

+

.

Au where

One t h i n k s o f A as an o p e r a t o r i n H w i t h domain N = D(A);

CU,~)

so A

Take e.g. = -A

+

(A)

and w r i t e (Au,v)

v E Corn a g a i n and we have J

c and D ( A ) = { u E V;

(-A+c)u

E

A is =

Auidx = (Au, Q2 L 1 where V =

The f o l l o w i n g theorem o f L i o n s i s a v a r i a n t o f t h e so c a l l e d Lax-

M i 1gram theorem.

CHE0REI 9.2.

Given V C H H i l b e r t spaces, V dense and c o n t i n u o u s l y embedded

i n H, and a ( * , - ) a continuous, c o e r c i v e , 1-1/2 l i n e a r form on V X V i t f o l lows t h a t A: D ( A )

P4uo6:

-+

H i s 1-1 and onto.

By c o e r c i v i t y crllull

and A i s 1-1.

2

5 la(u,u)I

=

I((

Au,u))

D e f i n e t h e a d j o i n t f o r m a*(u,w)

I

5 IIAullIIull so crllull 5 IlAull

= a(w,u)

which i s a g a i n con~

t i n u o u s on V X V and c o e r c i v e w i t h say a*(u,w) = (( u,Aw))

Then A - l :

R(A)

f o r a l l u,w E V. +

= (( 3u,w))

= a(w,u)

= (( Aw,

Thus 3 = A * and A * i s 1-1 so R ( A ) i s dense.

V can be extended t o V by c o n t i n u i t y and i n f a c t i f y E V -1 yn = wn + w; t h e n wn + w and Awn = + y, yn E R ( A ) , so A

i s a r b i t r a r y l e t yn yn

+

y which i m p l i e s Aw = y ( t h e graph o f a c o n t i n u o u s o p e r a t o r i s c l o s e d ) .

Consequently R ( A ) = V.

Now s o l v i n g a(u,w)

l e n t t o s o l v i n g ( ( A u , ~ ) ) = (( Jf,w)) i s continuous).

= (f,w)

where (( Jf,w))

f o r a l l w E V i s equiva(w -* ( f , w ) :

= (f,w)

V

+

But s o l v i n g Au = J f i s accomplished v i a u = A - l J f and

uniqueness i s obvious s i n c e crllull 5 IlAull. such t h a t a(u,w) = ( f , w ) i t f o l l o w s t h a t w t h e t o p o l o g y o f H so a u t o m a t i c a l l y u

E

F i n a l l y one observes t h a t g i v e n u -+

a(u,w):

V

+

C i s continuous i n

N = D(A) and Au = f .

QED

C

68

ROBERT CARROLL

REmARK

To see t h a t D(A) w i l l be dense i n H suppose f

9.3,

0 for a l l u

(A%,$)

E

D(A).

= a*(v,$)

e x i s t s a unique v

= (AJI,v)

= (v,AJI),

IP E

D(A*) w i t h A*v = f and (u,A*v)

= H and hence v = 0 so f = 0.

D(A*), JI E D(A)) t h e r e = (AU,v)

= 0.

i s dense b y t h e r e a s o n i n g above and we show A i s c l o s e d . -+

But R ( A )

Now f o r completeness l e t us show t h a t t h i s

A* i s i n f a c t t h e a d j o i n t o f A as an unbounded o p e r a t o r i n H. w i t h un

=

Then by Theorem 9.2 a p p l i e d t o A* (determined by ( + )

- -

= a(JI,v) E

H w i t h (u,f)

E

u i n H and Aun = fn

-+

f i n H.

a d j o i n t o f A and v E D ( A ) so t h a t u t o p o l o g y w i t h (Au,v) = (u,&)

-t

Thus l e t un

Then un = A - l J f n

n e c e s s a r i l y uo = u; hence u = A - l J f E D(A) and Au = f. (Au,v)

F i r s t D(A*)

+

D(A)

E

uo i n V and

F i n a l l y l e t A be t h e

i s continuous on D ( A ) i n t h e H

( t h i s condition defines the a d j o i n t operator u

A d

A). L e t vo E D(A*) be t h e s o l u t i o n o f A*v0 = Av (R(A*) = H as b e f o r e f o r R(A)). Then (Au,v) = (u,Kv) = (u,A*vo) = (Au,vo) (by d e f i n i t i o n o f A* i n -"

(+)).

Consequently v = vo and Av

=

A*v.

L e t us show n e x t how t o f o r m u l a t e and s o l v e some weak l i n e a r e v o l u t i o n problems ( t h e p r e v i o u s d i s c u s s i o n a p p l i e s more t o e l l i p t i c problems).

EMAIIIPLE 9.4, For s i m p l i c i t y t a k e u ' t Au = f w i t h u(0,x) = u o ( x ) where e.g. A = -A t c as b e f o r e a r i s e s from a f o r m a ( - , - ) so t h a t (Au,v) = a(u,v) f o r u E D(A) ( x E R3 say).

F o r m a l l y m u l t i p l y i n g by a t e s t f u n c t i o n v ( x , t )

(with

v(x,T) = 0 ) and i n t e g r a t i n g by p a r t s one a r r i v e s a t T a(u(t),v(t))dt (u,v')dt = ( f , v ) d t t (uo,v(0)) (9.1)

loT lo

'0

n

The weak problem i s t h e n phrased as f o l l o w s . Given f E LL(H) and u E H 2 2O f i n d u E L (V) such t h a t (9.1) h o l d s f o r a l l v E L2(V) w i t h v ' E L (H) and v ( T ) = 0. We w r i t e v ( t ) e t c . i n t h i n k i n g o f v e c t o r valued f u n c t i o n s o f t 1 w i t h values i n H ( Q L2 (n)) o r V (Q H,(n) f o r example). A l l d e r i v a t i v e s a r e taken i n t h e sense o f v e c t o r valued d i s t r i b u t i o n s (see Appendix B ) .

To phrase a t y p i c a l theorem f o r ( 9 . 1 ) one f o l l o w s L i o n s [ L i l ] a g a i n ( c f . also [Cl]).

w i t h la(t,u,v)l 2 A l u I 2 2 kllull ( t h e use o f t

E

[O,T],

Thus we t a k e a f a m i l y a ( t , u , v )

5 cIIuIIIIvII and f o r some r e a l A , Re a ( t , u , u )

7-

t

Au = f changes i t t o w '

i n p a r t i c u l a r f o r o u r d i s c u s s i o n we can t a k e

a(t,u,v)

t

a here does n o t r e s t r i c t g e n e r a l i t y s i n c e a sub-

s t i t u t i o n u = wexp(at) i n u ' =

o f 1-1/2 l i n e a r forms on V X V,

a

t

(Ath)w = f e x p ( - a t )

= 0).

i s measurable and bounded f o r u,v E V f i x e d and t

One assumes t E

[O,T];

-+

also f o r

s i m p l i c i t y we assume V and H a r e separable now w i t h V C H dense and c o n t i n u o u s l y embedded ( s e p a r a b i l i t y makes m e a s u r a b i l i t y arguments e a s y ) .

Under

WEAK PROBLEMS

69

t h e s e circumstances i t makes sense t o ask f o r u as i n Examp e 9.4 s a t i s f y i n g (9.1) f o r a l l v as i n d i c a t e d and one can prove

CHMRm 9.5, E

Under t h e hypotheses i n d i c a t e d (9.1) has a un que s o l u t i o n u

Co(H) w i t h u ( 0 ) = uo. We w i l l s k e t c h t h i s modulo Theorem 9.6 w h i l e r e f e r r i n g a few t e c h -

P4006:

Thus f i r s t r e c a l l t h a t f o r H s e p a r a b l e

n i c a l d e t a i l s t o Remark 9.7 below. scalar measurability f o r t measurability o f t u,v))

-+

= (( u,A*(t)v))

i n V and hence t

+

w(t).

-+

w ( t ) (i.e.

t

-f

( w ( t ) , h ) measurable) i m p l i e s

Then from m e a s u r a b i l i t y o f t

one deduces t h a t t

-+

A*(t)v

+

a(t,u,v)

= ((A(t)

i s measurable w i t h values

2

A ( t ) u ( t ) i s measurable and i n L2(V) f o r u E L ( V ) ( ( ( A ( t )

= (( u ( t ) , A * ( t ) v ) )

). We r e f e r t o Appendix B f o r background i n f o r m a t i o n on m e a s u r a b i l i t y as needed i n t h i s s e c t i o n . I t f o l l o w s t h a t a ( t , u ( t ) , 2 2 1 v ( t ) ) E L f o r u E L (V) and v E L (V); f u r t h e r ( c f . Remark 9.7) f o r v E u(t),v))

2

2

L (V) w i t h v ' E L (H) one has v ( t ) E Co(H) and hence v ( 0 ) makes sense.

Con-

s e q u e n t l y under t h e hypotheses i n d i c a t e d t h e terms i n ( 9 . 1 ) a r e a l l w e l l defined.

2

w i t h F c H t h e space o f v E H such

Now d e f i n e H = L (V) on [O,T]

that v' E

2 L (H) and v ( T )

=

2

2

2

0 w i t h norm IIvllF = IlvllH t I v ( T ) I H .

Set

T (9.2)

E(u,v) = 0

[a(t,u(t),v(t)) - (u(t),v'(t))ldt; T L(v) = ( f ( t ) , v ( t ) ) d t + (uolv(0))

f o r u E H and v E F. L(v): F

+

(9.3)

C.

lo

Evidently u

-f

E(u,v): H + C i s c o n t i n u o u s as i s v

One can t a k e h = 0 i n t h e c o e r c i v i t y assumption and f o r v

Re E(v,v) =

joT Re a ( t , v , v ) d t

k l T Ilv1I2dt 0

+

-

+

E

F

2

( 1 / 2 ) j 0T D t l v l 2d t

( 1 / 2 ) l ~ ( 0 ) 1,nkilvll: ~

A

(k = min(l/2,k)). = L(v) f o r a l l v E

By Theorem 9.6 t h e r e i s a s o l u t i o n u

E

H such t h a t E(u,v)

F, which i s ( 9 . 1 ) .

Now f o r uniqueness, g i v e n two s o l u T T t i o n s u1 and u2 o r (9.1), u = u1 - u2 s a t i s f i e s lo a ( t , u , v ) d t = fo ( u , v ' ) d t t f o r a l l v E F ( f = uo = 0 ) . By Remark 9.7 we can w r i t e 2Re f O ( u ' , u ) d t = l u ( t ) l i (( i n O'(V');

+ lu(t)Ii

,

denotes V - V ' c o n j u g a t e l i n e a r d u a l i t y ) znd 0 = u ' + A ( t ) u hence 0 = Re Jot ( A ( t ) u , u ) d T + Re Jot (u',u)d.r = :1 Re a(t,u,u)d.r )

Jot

2

kllull d r

+

2

lu(t)lH

2

2 k/d

IIuII d r which means u : 0.

statement about Co(H) i s proved i n Remark 9.7.

eHE0REm 9.6

The l a s t

QED

(rI0Ns)- L e t H be a H i l b e r t space w i t h norm I I and F C H be a H i s continuous (F need n o t be dense

subspace w i t h norm 1I 1I such t h a t i: F

-f

70

ROBERT CARROLL

L e t E(u,v) be a 1-1/2 l i n e a r f o r m on H X F such t h a t u

n o r complete). E(u,v):

H

C i s continuous and I E ( v , v ) l 2 cllvl12 f o r v

-f

sary t h a t v

-+

l i n e a r form. Pm06:

F

-+

E(u,v)

Let v

be continuous).

-+

L(v): F

i f Kv = 0 t h e n ( v , K v ) = E(v,v)

F'

?+F

H ' then K-':

so IIK-lxII 5 (:/c)Ixl.

a map R w i t h domain

F' =

closure o f

F).

K - l on

K:

F u r t h e r K i s 1-1 s i n c e i s continuous when?has

Indeed c ~ ~ K - ~ x
I K - l x I :?IIK-lxll(xl

j - f ;, R

L ( v ) f o r a l l v E F.

= 0 which i m p l i e s v = 0 s i n c e IE(v,v)l

?C

Ifwe s e t now K(F)

the topology o f H I .

( t h u s R:

C be a continuous

E(u,v) one can w r i t e E(u,v) = ( u , K v ) where

+

H ' i s a c o n j u g a t e l i n e a r map ( H I = dual o f H ) .

cllv1I2

F ( i t i s n o t neces-

E +

Then t h e r e e x i s t s u E H such t h a t E(u,v)

By c o n t i n u i t y o f u

-f

7 in

= I(K-lx,x)l

2 1x1

Then extend K - l by c o n t i n u i t y t o

?

H ' and range i n

Evidently v

-+

= completion o f F

L ( v ) extends by c o n t i n u i t y to

A

a continuous c o n j u g a t e l i n e a r form on F which we w r i t e as L ( v ) = ( ( C L , v ) ) , A

4

t L E F ' (F' i s t h e a n t i d u a l

-

c f . Remark 9.7).

(u,Kv) = ( ( 5 , v ) ) f o r a l l v LN R x ) ) f o r x E F. L e t P: H '

E

F.

-+

Px = x f o r x

E

F" and we

Then E(u,v) = L ( v ) becomes

But t h i s means ( u , x )

((cL,K-'x))

=

= ((CL,

v

F be t h e continuous orthogonal p r o j e c t i o n so

want ( u , x ) =

((

tL,RPx

))

for x E

7.

H'

Now Q = RP:

A -f

F i s continuous ( c o n j u g a t e l i n e a r ) and hence u = Q*CL i s a s o l u t i o n ( r e -

c a l l ( ( S L , Q x ) ) = (Q*SL,x) by d e f i n i t i o n o f a d j o i n t o p e r a t o r s i n Banach QED

spaces.

REmARK 9.7,

L e t us c o l l e c t here a b i t o f t e c h n i c a l i n f o r m a t i o n .

For E a

Banach space one w r i t e s o f t e n E l f o r t h e a n t i d u a l ( i n s t e a d o f t h e d u a l ) . Thus E ' can r e f e r t o t h e space o f continuous c o n j u g a t e l i n e a r forms on E. We use t h e same n o t a t i o n and t h e meaning i s always c l e a r from t h e c o n t e x t . T h i s i s p a r t i c u l a r l y u s e f u l when we have two H i l b e r t spaces V C H, V dense and c o n t i n u o u s l y embedded i n H.

Then, i d e n t i f y i n g H = H ' by t h e Riesz theo-

rem, we w r i t e V C H C V ' f o r V ' t h e a n t i d u a l ((h,w) = (h,w) f o r h E H = H ' ) . Note H C V ' i s a continuous embedding s i n c e f o r h E H, w ~ ( h , w ) ~ / l l w l =l sup I(h,w)l/llwll

5?lhllwl/lwl (?lh(.

p l e t h e continuous c o n j u g a t e l i n e a r map v u,v) = ( A (

t

)u,v) where A ( t ) : V

-f

E

V but A(t): V

(9.1) and s e t v = y ( t ) y f o r y

It follows t h a t i n

D'

(m)

E

-+

llhllvl = s u p

V

-+

C; e v i d e n t l y a ( t ,

Note here A ( t ) u i s a c o n j u g a t e

V' i s linear.

V and

(u(t),y)'

a(t,u,v):

V,

V ' i s continuous ( s i n c e ] ( A ( t ) u , v ) l 5

cIIuIIIIvII and llA(t)uII = sup l ( A ( t ) u , v ) l / l l v l l ) . l i n e a r map on v

-f

E

Now c o n s i d e r f o r exam-

CI(0,T).

i.a ( t , u ( t ) , y )

Next we l o o k a t an e q u a t i o n Then

(f,y) o r (u(t),y)'

t

WEAK PROBLEMS + (A(t)u,y) = (f,y).

71

T h i s i s another way o f s a y i n g ( d e f i n i t i o n

+

-

c f . Appen-

A ( t ) u = f.

I n p a r t i c u l a r u ' i n (**) i s a

f u n c t i o n ( d e f i n e d a l m o s t everywhere = AE).

Note however t h a t u ' i n D ' ( H )

d i x B) t h a t i n D ' ( V ' ) (**) u '

w i l l n o t be a f u n c t i o n ; t h e d i f f e r e n c e i s t h a t u ( t ) E D ( A ( t ) ) i s n o t known

i n ( 9 . 4 ) i s n o t n e c e s s a r i l y ( A ( t ) u , y ) whereas V ' d e f i n e d on a l l V. Now g i v e n (**) one A ( t ) i s a continuous operator V t o be t r u e so a ( t , u ( t ) , y )

-f

can show e a s i l y t h a t u E Co(H) ( a f t e r p o s s i b l e m o d i f i c a t i o n on a s e t o f mea2 2 s u r e 0 ) . Thus l e t U(0,T) be t h e space o f u E L (V) w i t h u' E L ( V ' ) p r o 2 T 2 v i d e d w i t h norm Ilully = Jo [ l l u l l ~+ I l u ' I I V l ] d t . L e t e E COD, e = 1 on [O,T], and e v a n i s h i n g i n a NBH o f

-T

and T+T.

Extend u

E

U(0,T) t o

[-T,T+T]

by

symmetry around 0 and T and s e t w = eu so w E U(-T,T+T) and w = u on [O,T]. L e t vn

v

+

6 be an a p p r o x i m a t i o n t o 6 v i a say v n ( x ) = r w ( n x ) f o r

C [-l,l], v 5 0, and

Then wn = w

* vn

v

m

E

C o y supp

2

w i n L ( V ) and w,', + w ' 2 2 i n L ( V ' ) ( e x e r c i s e ) . But w;(t) E V i n f a c t and f o r n l a r g e (*.) Iwn(t)lH t t t 2 = [IIwnllv + Lr 2Re (w',w ) d c = I-T 2Re ( w ' , w > d t 5 2J-T ~~w,',llVIHwnlVd~ 5 Jvdx = 1.

n

n2

+

n

5 IIwnIIU ( 4 = U(-T,T+T)). Consequently t h e wn form a u n i f o r m Cauchy sequence i n H norm and converge u n i f o r m l y t o a continuous f u n c t i o n llw,',Ilvl]d~

( n e c e s s a r i l y wn

-+

any v E V(0,T) (9.5)

I n p a r t i c u l a r g i v e n u a s o l u t i o n o f (9.1),

w).

mined by Theorem 9.5,

we w i l l have u

Co(O,T) w i t h u ( 0 ) = uo. 2 w i t h v ( T ) = 0 one has ( r e c a l l u ' E L ( V ' ) from

loT [(u,v')

+

(u',v)]dt

€

as d e t e r Indeed f o r

(M)

(uECo(H))

= -(u(O),V(O))~

(a p r o o f can be c o n s t r u c t e d by u s i n g 2 a p p r o x i m a t i n g sequences un and vn as T T T above - e x e r c i s e ) . Then from (+t) So a ( t , u , v ) d t + Jo ( I r ' , v ) d t = JO ( f , v ) d t

so, combined w i t h (9.1), dicated.

one o b t a i n s ( u ( 0 )

-

uO,v(O)),

= 0 f o r a l l v as i n -

T h i s i m p l i e s u ( 0 ) = uo.

The machinery above (and extensions, r e f i n e m e n t s , e t c . ) became s t a n d a r d i n t h e 1950's and 1960's i n t r e a t i n g l i n e a r problems and i t s a p p l i c a t i o n t o n o n l i n e a r problems was a l s o v e r y p r o d u c t i v e i n c o n n e c t i o n w i t h monotone ope r a t o r s , v a r i a t i o n a l i n e q u a l i t i e s , n o n l i n e a r semigroups, e t c .

The emergence

o f t h e concept o f weak s o l u t i o n s was v e r y i m p o r t a n t b o t h as an i n t e r m e d i a t e e x i s t e n c e s t e p p r i o r t o p r o v i n g r e g u l a r i t y o r as an end i n i t s e l f when regul a r solutions simply d i d n o t e x i s t .

One s h o u l d n o t r e g a r d t h e absence o f

r e g u l a r o r s t r o n g s o l u t i o n s as a d e f e c t o f theory; i n areas such as f l u i d dynamics f o r example where t u r b u l e n c e can a r i s e , shock phenomena o c c u r , e t c . e x i s t e n c e and uniqueness q u e s t i o n s a r e h i g h l y n o n t r i v i a l and r e g u l a r i t y may be m i s s i n g .

L e t us i n d i c a t e h e r e a t y p i c a l weak s o l u t i o n f o r t h e Navier-

72

ROBERT CARROLL

Stokes' equations ( c a l l e d t u r b u l e n t s o l u t i o n s by Leray The Navier-Stokes equations a r e ( u

EWAIPfE 9.8.

-

ut

(9.6)

VAU +

In D.u.

with div u = 0 u(t,O) (0,T)

c1n uiDiu

= u o ( x ) on R

' A C R

-

= f

(bounded); u = 0 on B = l a t e r a l boundary o f Q = il x

= 01, H = c l o s u r e o f 9 i n L

and l e t V,

product (( u,v))

A t y p i c a l k i n d o f e v o l u t i o n problem t h a t i s

V V;

C

First l e t 9 =

Ilp

E

div 9

C;(Q)~,

1 Jn figidx HS(n)n w i t h norm II U s and s c a l a r

w i t h scalar product (f,g) =

= closure o f 9 i n

Set i n p a r t i c u l a r V1 = V so f o r s > 1, V s C

H, each space b e i n g dense i n t h e next, and w r i t e V s

C

V C H = H' C V' C

where t h e primes denote d u a l i t y ( n o t c o n j u g a t e l i n e a r d u a l i t y ) .

v E V, =

2

1 (ui,vi),,s.

=

$ etc.)

T,

grad p

t r a c t a b l e h e r e can be s t a t e d as f o l l o w s .

I I,

c f . [Li4]).

and i n i t i a l - b o u n d a r y c o n d i t i o n s o f t h e form (*@)

(cf. [LilY4;Ldl;Tel]).

and norm

-

Since

S

plr

vi E Ho(n), f o r s > 1 / 2 a t l e a s t one has a meaning f o r vi = 0 on

an ( d i v v

= 0 w i l l o f course be a u t o m a t i c ) .

One checks e a s i l y t h a t (u,v,w)

+

Set now f o r u,v E V

V x V x (V n L n ( n ) n )

b(u,v,w):

t i n u o u s ( n o t e V n L ' ( Q ) ~ = V i f n 5 4).

r

+

C i s con-

Here one has recourse t o t h e Sobo-

l e v t h e o r y ( c f . Appendix B and [Dul;Ldl;Li2,4;Sbl;Sh2,3;Tel]). Thus u E V 1 means ui E Ho(n) and hence when l / q = 1/2 - l / n ( q = 2n/n-2 - q can be a r b i t r a r y when n = 2 ) one has nu.11 < CIIU II 1 (I1 II means Lq norm) and i q i H q 2 llukllpllDkvill2IIwilln 5 cIIukllVllDkvillVllwiIln ( n o t e h e r e i f f (Dkvi)widxI g E Lq t h e n f g

E

f o r p > q since

L

n

I/R E

uk Lp and

-

f o r l / s = l / p t l / q and llfglls 5 IIfll Ilgll a l s o Lp C Lq P q i s bounded c f [Dull). Now f i r s t t r a n s f o r m (9.4)-(9.6)

-

i n t o t h e " s t r o n g " problem o f s o l v i n g (9.6) f o r u and p i n D'(Q) w i t h u ( 0 ) =

2

uo and u E L (O,T,V) given.

n Lm(O,T,H)

2

where f E L (O,T,H-l(n)n)

and uo E H a r e

The Lm c o n d i t i o n on u i s n o t necessary i n advance b u t i t a r i s e s n a t -

u r a l l y i n t h e e x i s t e n c e t h e o r y (see below).

Now we want t o g e t r i d o f t h e

unknown p somehow i n (9.6) so one n o t e s t h a t f o r 9 E 9 (gradp,v) = (gradp,q) =

-(

p,divq) = 0 i n D'(OyT)n.

by p a r t s t o o b t a i n ( 9 E 9 and

(9.8)

( u ' , ~ ) + va(u,v)

Now t a k e s c a l a r products i n (9.6) and i n t e g r a t e (

,

)

+ b(u,u,v)

T h i s leads t o t h e weak problem ( W ) :

r e f e r s t o (D'

-

D)n i n x )

= (f,lp)

2

Given f E L (O,T,V')

and uo E H f i n d

WEAK PROBLEMS

u

2 L (O,T,V)

E

Lm(O,T,H),

Ln((n)n (( u',lp) = (u,lp)'

73

u ( 0 ) = uo, such t h a t ( 9 . 8 ) h o l d s f o r a l l IP E

i n D'(0,T)).

Vn

Note t h a t t h i s f o r m u l a t i o n f o l l o w s i m -

m e d i a t e l y from (9.8) by t a k i n g l i m i t s on q i n V n L n ( ~ ) n . On t h e o t h e r hand

if u s a t i s f i e s problem ( W ) s e t u ' q ) =

E

0 i n Dl(0,T)

-

vdu t

1 uiDiu

f o r a l l IP E U by (9.8).

- f

= S i n D'(Q)n with (S,

It f o l l o w s t h a t S = -grad p f o r p

D'(Q) and (u,p) s a t i s f i e s t h e " s t r o n g " problem.

There a r e now a number o f

theorems ( w i t h v a r i o u s techniques o f p r o o f , r e q u i r i n g sometimes n 5 4 o r n =

2 ) which we w i l l c i t e h e r e w i t h o u t p r o o f s , b u t seee.g.Chapter 3 where f u r t h e r i n f o r m a t i o n w i l l be g i v e n (see [ L i 4 ; T e l ] f o r complete d e t a i l s ) . Thus IIHE0REm 9.9. There e x i s t s a s o l u t i o n o f problem W (T > 0 f i n i t e ) w i t h u ' 2 L (O,T,Vk) f o r s = n/2.

E

Uniqueness i s open i n general b u t f o r n = 2 one has For n = 2 t h e s o l u t i o n u o f problem W i s unique and a f t e r pos-

CHE0REm 9-10,

s i b l e m o d i f i c a t i o n on a s e t o f measure 0 i s continuous w i t h v a l u e s i n H and 2 0. F u r t h e r ( n = 2 ) i f f , f ' E L (O,T,V'), u(t) u i n H as t f ( 0 ) E H, 2O 1 2 If i n addition uoi E H (0)n Ho(n), uo E V, t h e n u ' E L (O,T,V) n Lm(O,T,H). -f

-f

f

E

t h e n u E L 2 ( 0 , T , H 2 ( ~ ) 2 ) w h i l e i f f E Lm(O,T,H)

L2(0,T,H)

t h e n u E Lm(O,

T, H~ (0l2 1. These r e s u l t s b e g i n t o i n d i c a t e t h e f l a v o r o f t h i s k i n d o f problem and show some o f t h e t e c h n i c a l f e a t u r e s which a r i s e .

A few f u r t h e r theorems f r o m

[ L i 4 ] which w i l l c o n t r i b u t e t o t h e p i c t u r e a r e ( r e c a l l uniqueness i s open f for n > 3) Let n

CHE0REm 9-11.

2 3 and suppose t h e s o l u t i o n o f problem W s a t i s f i e s a l s o

u

Then u ( i f i t e x i s t s ! ) i s E Ls(O,T,Lr(~)n) where 2 / s t n / r 5 1 ( r > n ) . 2 unique i n t h e c l a s s L (O,T,V) n Lm(O,T,H) n LS(0,T,Lr(n)').

Regarding s t r o n g e r s o l u t i o n s i n general one can o b t a i n some r e s u l t s i f t h e d a t a a r e assumed t o be s u f f i c i e n t l y r e g u l a r and " c o n t r o l l e d " .

Thus e.g.

2

CHE0REIR 9-12.

Let n < 4, f = 0, and uo E V n H ( R ) ~= W w i t h IIuollw s u f f i c -

i e n t l y small.

Then t h e r e e x i s t s a unique u s a t i s f y i n g problem W w i t h u ' E

L~(o,T,v) n L~(o,T,H). L e t us n o t e t h a t one can w r i t e problem W i n an a p p a r e n t l y weak-

REmARK 9.13,

e r form by u s i n g t e s t f u n c t i o n s q

E

C"(O,T,U)

f o r example and i n t e g r a t i n g

The r e s u l t i s e q u i v a l e n t t o problem W b u t we w r i t e t (u(t),q(t)) f o r comparison anyway t h e formulas ( n o t e ( * b ) I. ( u ' , d d . r

from 0 t o T i n ( 9 . 8 ) .

Q

(

Uo,Lp(o))

-

t

10

( U,q')

dr)

ROBERT CARROLL

74

(9.9)

(

u(t),v(t))

-

I

T

(uo,lp(0))

=

[b(u,v,u)

-

va(u,v)

+

( f , v ) + ( u,v') 1d.r

0

Denoting uu a s o l u t i o n t o problem W f o r (9.8)-(9.9),an

i m p o r t a n t problem i n

f l u i d dynamics i s t o ask whether uw -+ u = E u l e r f l o w as w [Kal;Tel]

-+

0.

We r e f e r t o

f o r a more o r l e s s up t o d a t e account o f t h i s and o t h e r m a t t e r s

concerning t h e Navier-Stokes equations.

The E u l e r equations a r e ( i n obvious

n o t a t i o n ) (*+) u ' t ( u g r a d ) u t grad p = f ( d i v u = 0, u = 0 on Z ) .

Here

e x i s t e n c e and uniqueness o f a smooth l o c a l s o l u t i o n a r e w e l l known.

The

q u e s t i o n o f uy

-+

u i s more d i f f i c u l t and i n p r i n c i p l e some f u r t h e r informa-

t i o n on t h i s w i l l be g i v e n l a t e r . dt

+

0 a r e e q u i v a l e n t t o uy

10. 90mE NBNCZNEAR PDE,

-+

Roughly c o n d i t i o n s such as WJ; Ilgrad uyll 2 u i n L (n) u n i f o r m l y i n t ( c f . [ K a l ] ) .

2

We g i v e f i r s t a few sample equations and remarks

concerning n o n l i n e a r phenomena.

Then as a model s i t u a t i o n which i l l u s t r a t e s

a v a r i e t y o f methods and concepts we i n d i c a t e some r e s u l t s and techniques f o r t h e Ginzburg-Landau (G-L) equations o f s u p e r c o n d u c t i v i t y .

T h i s serves t o i n -

troduce a number o f t o p i c s t o be developed l a t e r and p r o v i d e s a t a s t e o f some reasonably c u r r e n t work (see a l s o 10.10-10.12

EXANPCE 10.1.

on a t t r a c t o r s , e t c . ) .

As a s i m p l e i l l u s t r a t i o n one r e c a l l s y '

=

1 + y

2

with y(0) =

0 whole ( u n i q u e ) s o l u t i o n i s y = Tanx which becomes i n f i n i t e a t x = n/2 ( l o -

cal not global solution).

Second c o n s i d e r y ' = y 1 l 2 w i t h y ( 0 ) = 0 which has 2 s o l u t i o n s y f 0 o r y = ( x / 2 ) (nonuniqueness). The c r i t i c a l f e a t u r e here i s growth o f t h e r i g h t hand s i d e and b o t h phenomena can be encompassed i n y ' = For p

yp w i t h s o l u t i o n x+c = ~ - ~ + ' / ( l - p )( p = 1 ) .

<

1 take y ( 0 ) = 0 o r c =

0 and y = [ ( l - p ) ~ ] ' / ( ' - ~ ) i s nonunique s i n c e y : 0 i s a l s o a s o l u t i o n . For p > 1 t a k e say y ( 0 ) = 1 so c = - l / ( p - 1 ) and y = l / [ l - ( p - l ) x ] l / ( P - l ) w h i c h i s unique b u t becomes i n f i n i t e a t x = l / ( p - 1 ) . i n [Cl]

F u r t h e r d i s c u s s i o n can be found

f o r example.

We r e c a l l ( w i t h o u t p r o o f ) a few standard theorems from ODE ( c f . [Bol;Cl;Cdl] f o r proofs, d i s c u s s i o n , embellishments, e t c . )

tHE0RElll 10.2

(PZCARD-CZNDEC0FF).

r e g i o n around ( t o , y o ) w i t h I f ( t , y )

Let f(t,y)

-

be continuous i n ( t , y ) i n some

f ( t , z ) l 5 clly-zll ( L i p s c h i t z c o n d i t i o n ) .

Then t h e r e e x i s t s a unique l o c a l s o l u t i o n o f y ' = f ( t , y ) w i t h y ( t o ) = yo.

EHEBREIII 10.3

(CALIC€@-PEAN@).

Assume f ( t , y )

i s continuous i n ( t , y ) .

Then

t h e r e e x i s t s a l o c a l s o l u t i o n o f y ' = f ( t , y ) w i t h y ( t o ) = yo. We mention a u s e f u l lemma which a r i s e s i n d e a l i n g w i t h ODE ( c f . [Cl;Cal])

NONLINEAR PDE (m0WA1;c).

I ; ~ A10.4

ous.

Ifu ( t ) 5 b ( t ) +

Let a

E

L1, a

I,' a(S)u(C)dS + 1;

u ( t ) 5 b(T1e.r. Jt

(10.1)

2 0,

u

75

E

Lm, and b a b s o l u t e l y c o n t i n u -

then b ' ( s ) efst a(c )dc

We n o t e t h a t t h e r e a r e v e c t o r valued v e r s i o n s o f Theorems 10.2-10.3 o f Lemma 10.4 ( c f . [Cl;Cul] RElilARI( 10.5,

-

and a l s o

t h u s y ( t ) c o u l d t a k e values i n a Banach space).

We w i l l n o t do much i n t h i s d i r e c t i o n h e r e b u t we want t o men-

t i o n a few n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s a r i s i n g i n t h e s t u d y o f dynamic a l systems, c i r c u i t s , e t c . which a r e p r o t o t y p i c a l f o r c e r t a i n aspects o f t h e t h e o r y ( c f . [A2,3;Del;Gkl;Hrl]).

One o f t e n d e r i v e s t h e L i e n a r d equa-

-

t i o n s f r o m an e l e c t r i c a l c i r c u i t problem i n t h e f o r m (*) d x / d t = y d y / d t = - x where f r e p r e s e n t s t h e c h a r a c t e r i s t i c s o f a r e s i s t o r . i a l case f ( x ) = x 3 extensively.

f(x);

The spec-

-

x i s c a l l e d van d e r P o l ' s e q u a t i o n and has been s t u d i e d 2 Thus ( A ) d x / d t = y - x3 + x; d y l d t = - x o r + (3x - 1 ) x + x =

and t h e r e i s one n o n t r i v i a l p e r i o d i c s o l u t i o n t o which e v e r y n o n e q u i l i b r i u m s o l u t i o n tends.

Another t y p i c a l e q u a t i o n a r i s i n g e.g.

i n t h e s t u d y o f a non-

+ 6 i - Bx + l i n e a r o s c i l l a t o r i s D u f f i n g ' s equation (without f o r c i n g ) ( 0 ) 2 2 4 For 6 = 0 a Hamilton f u n c t i o n H(u,v) = v / 2 - Bu / 2 + u / 4 can be 3 = v and = BU - u ). The phase p o r % v and x T, u w i t h employed ( h e r e

x3 = 0.

x

n - .V

t r a i t f o r 6 = 0 i s i n d i c a t e d by

U

(10.3)

>

(the f i r s t f o r B

<

0 and t h e second f o r B > 0 w i t h c e n t e r s a t ~ J B ) . The L o r -

e n t z equations come up i n f l u i d c o n v e c t i o n problems i n t h e f o r m ( & ) x);

= px-y-xz;

i

= -Bz+xy

(o,p,~ >

i

= u(y-

0 ) and as f o r t h e van d e r Pol and Duf-

f i n g equations, a r e i m p o r t a n t equations i n b e g i n n i n g t h e s t u d y o f what i s c a l l e d "chaos" ( c f . [Del ;Gkl ;Hdl]). e l s o f the Volterra-Lotka type (*)

L e t us mention a l s o p r e d a t o r - p r e y mod-

x

s p e c i s models i n a more general form

= (A-By)x; (m)

i

=

= (Cx-D)y o r competing

M(x,y)x;

jl = N(x,y)y.

Equations

o f these types p l a y an i m p o r t a n t r o l e i n modeling n o n l i n e a r phenomena and t h e i r g e o m e t r i c a l and t o p o l o g i c a l s t u d y l e a d s t o a n ' i n t r i c a t e maze o f

76

ROBERT CARROLL

b i f u r c a t i o n s , c h a o t i c motion, s t r a n g e a t t r a c t o r s , f r a c t a l s , e t c .

The f a c t

t h a t any g u i d e l i n e s have emerged amidst a l l t h e pathology i s a t r i b u t e t o We w i l l make a few f u r t h e r comnents on such

t h e i n v e s t i g a t o r s i n t h i s area. m a t t e r s l a t e r from t i m e t o time. REWRI(

L e t us l i s t some o f t h e t y p i c a l n o n l i n e a r PDE which have been

10.6,

and can be s t u d i e d p r o f i t a b l y . appear l a t e r .

F u r t h e r d e t a i l s f o r s e l e c t e d equations w i l l

We have a l r e a d y mentioned t h e Hamilton-Jacobi and t h e N a v i e r -

Stokes equations ( a l o n g w i t h t h e E u l e r e q u a t i o n i n Remark 9.13). c a l equations a r i s i n g i n f l u i d dynamics a r e (KdV (10.4)

6uux + uxxx = 0 o r ut + u X + uux + uxxx = 0

ut

f

Ut

+ uux = v u

Some t y p i -

Korteweq-deVries)

%

(KdV);

(Burger's equation)

xx

( o t h e r v a r i a t i o n s on t h e s e equations a r e a l s o p o s s i b l e ) .

Burger's equation

i s a s p e c i a l 1-D Navier-Stokes e q u a t i o n and t h e r e l a t e d E u l e r e q u a t i o n ut + uux = 0 a r i s e s a l s o i n t h e study o f c o n s e r v a t i o n laws.

Other equations w i t h

o r i g i n s i n f l u i d dynamics a r e (mKdV = m o d i f i e d KdV) (10.5)

ut

- 6uuX + u

(ut ( h e r e BBM

2 6u ux + uxxx = 0 (mKdV); ut + ux + uux

?

~ + 3uYy ~ = 0~ (K-P) )

Benjamin-Bona-Mahoney

and K-P

- ux x t

= 0 (BBM);

~

Kadomtsev-Petviasvil i ).

Some i m p o r t a n t n o n l i n e a r wave equations a r e o f t e n w r i t t e n i n t h e form (** ) utt Q

(K(ux)Ix ( c f . [ M j l ] ) have t h e form

(*A)

%

=

whereas h y p e r b o l i c systems o f c o n s e r v a t i o n laws o f t e n ut + A(u)uu = 0 ( c f . [ L x l ] ) .

Some o t h e r equations a r e

( c f . [A1 1 ;Cel ;D r l ;Lml ;R2;WhlI) (10.6) utt

-

utt

- Au

2

+ m u

uxx + Sinu = 0 o r u = Xup (K-G);

(K-G WKlein-Gordon

t i o n s w i t h kup = kl

UI

tt

- Au +

2 m u = g ( S i n u ) (Sine-Gordon);

i u t + uxx + kup = 0 ( n o n l i n e a r Schrodinger)

nonlinear).

2

F r e q u e n t l y one has s t u d i e d t h e l a s t equa2 u and Xup = X l u l u ; such equations a r i s e e.g. i n quan-

tum f i e l d t h e o r y and s o l i t o n t h e o r y ( c f . [Aol;Fa3;Gll;Ll;Mdl

;Nvl;Itl]).

w i l l g i v e d e t a i l e d i n f o r m a t i o n about some o f t h e s e equations l a t e r .

We L e t us

mention a l s o t h e Ginzburg-Landau (G-L) equations a r i s i n g i n t h e t h e o r y o f s u p e r c o n d u c t i v i t y ( c f . [L15;C12]) (10.7)

-cur12A =

I$I 2A

+ (i/2k)[?vJ,

- $v$];

NONLINEAR PDE

(10.7)

4 / k 2

+ (i/k)[A.VJ, + div(J,A)] + A-AJ,

77

= IL(1

- lJ,I2)

( t h e s e w i l l be w r i t t e n d i f f e r e n t l y l a t e r ) and n o n l i n e a r e l 1 i p t i c e q u a t i o n s o f t h e form -Au + u u =

(10.8)

Au

BU';

-

[(n-2)/4

have been s t u d i e d i n c o n n e c t i o n w i t h t h e Yamabe problem i n d i f f e r e n t i a l geom e t r y ( c f . [Bel;Scl;Yal]

-

n

1,

dimension

.

The e x i s t e n c e o f a s o l u t i o n u

0 t o t h e l a t t e r e q u a t i o n i n (10.8) on a compact m a n i f o l d M w i t h s c a l a r c u r v a t u r e R > 0 was shown i n [Scl].

There a r e many o t h e r n o n l i n e a r PDE a r i s i n g

p r a c t i c a l l y everywhere i n s c i e n c e and e n g i n e e r i n g (and i n p u r e mathematics) and t h e r e a r e some l i t t l e i s l a n d s o f i n f o r m a t i o n where a " t h e o r y " can be s a i d t o e x i s t f o r c e r t a i n types o r c l a s s e s o f e q u a t i o n s . on methods, e.g.

Thus one c o n c e n t r a t e s

f i x e d p o i n t theorems, v a r i a t i o n a l i n e q u a l i t i e s , G a l e r k i n

methods, monotone o p e r a t o r t h e o r y , e t c . , which have proved u s e f u l a t one t i m e o r place, and t h e hope i s t h a t something s i m i l a r m i g h t be u s e f u l i n a g i v e n s i t u a t i o n which one m i g h t encounter.

A l t e r n a t i v e l y one i s i n s p i r e d t o d i s -

Being n o n l i n e a r i s n o t a p r o p e r t y l e n d i n g i t -

cover an e n t i r e l y new method.

s e l f d i r e c t l y t o t h e o r e t i c a l meaning o r m a n i p u l a t i o n and p r o b a b l y t h e b e s t i n t e r p e r t a t i o n was f o r m u l a t e d i n e.g.

[Pll]

by s u g g e s t i n g t h a t n o n l i n e a r an-

a l y s i s c o u l d be c o n s i d e r e d as i n f i n i t e dimensional d i f f e r e n t i a l t o p o l o g y . T h i s approach spawned a school o f " g l o b a l a n a l y s i s " which s t i l l f l o u r i s h e s i n one way o r another.

F o r c o l l e c t i o n s o f i n f o r m a t i o n and techniques on non-

l i n e a r a n a l y s i s f r o m a more c l a s s i c a l p o i n t o f view see e.g.

[Atl;Aul;Dml;

Zel,21)

REmARK 10.7.

L e t us mention some t y p i c a l r e s u l t s and methods f o r t h e G-L

equations o f Remark 10.6.

F i r s t f o r (10.7) ( w i t h n - [ ( i / k ) v + A ] + l s

t h e boundary S o f a bounded R

C

= 0 on

3 R ) perhaps t h e f i r s t general mathematical

t r e a t m e n t was g i v e n by t h e a u t h o r i n [C12] where unique weak s o l u t i o n s were 2 shown t o e x i s t f o r s u i t a b l e k < 1/42 ( k = 1/2 i s a c r i t i c a l v a l u e i n t h e theory o f superconductivity

-

we remark h e r e t h a t r e c e n t experimental d i s -

c o v e r i e s i n s u p e r c o n d u c t i v i t y may e n t a i l some changes i n t h e t h e o r y eventual

-

l y ) . The methods used i n v o l v e d s t a n d a r d use o f Sobolev i n e q u a l i t i e s , t h e use o f K o r n ' s i n e q u a l i t y t o deal w i t h e s t i m a t e s on c u r l A, and a c o n t r a c t i o n

mapping argument.

More r e c e n t l y t h e r e has been renewed i n t e r e s t i n t h e G-L

equations i n c o n n e c t i o n w i t h v o r t i c e s , b i f u r c a t i o n s , Yang-Mills-Higgs f i e l d s , etc.

L e t us s k e t c h here some o f t h e f o r m u l a t i o n and r e s u l t s f o l l o w i n g

78

ROBERT CARROLL

[Be2;Gcl ;Jal ;Tbl,2]. Take t h e 2-dimensional v e r s i o n o f (10.7) and s e t A = kA with

= 2k

2

(so A

1 i s critical).

f o l l o w here t h e n o t a t i o n i n [Tbl,2]

- we

Then (10.7) becomes (A i s r e a l

and t h e r e may be some d e v i a t i o n from t h e

c o n t r a v a r i a n t - c o v a r i a n t n o t a t i o n o f 53.8)

l:

(10.9)

[Dk

-

i A k I 2 $ = (,/2)[1$12

A

( k , j = 1,2; k # j ) . Note here c u r l A has o n l y one component B3k = (D1A2 a 2 D A )k so - c u r l A = - i D 2 B 3 + jDlB3. These equations can now be o b t a i n e d as 2 1 t h e E u l e r equations o f t h e G-L a c t i o n f u n c t i o n a l (10.10)

I, = ( 1 / 2 ) 1 ,[ldAI2

where l d A I 2 = (D1A2

(D1$2

-

-

A1$l) 2 + (D2$1

- lJll 2 12 Id 2x

+ I D A $ I 2 + (,/4)(1

R

D2A1) 2 and IDA$I 2 = I ( d

-

+ A1$2)2 + LL1 + \ 1 9l~ ) .Here ( c f .

i A ) $ I 2 = (D \L,

+ A2$2) 2 + (D2$2 - A p l ) 2 ($

=

2 Appendix C) one can t h i n k o f a t r i v i a l v e c t o r bundle E o v e r say R w i t h p r o 2 j e c t i o n IT:R x C R2 and I, a f u n c t i o n a l on C(E) B C"(E) where C(E) denotes -f

Cm, U(1) connections on E, and C"(E) denotes Cm c r o s s s e c t i o n s o f E (see 5 5 3.9-3.10

f o r connections and c u r v a t u r e ) .

Cm s e c t i o n s o f T * ( & ) a n d Cm(E)

'L

Since E i s t r i v i a l C(E)

Cm complex f u n c t i o n s on R2.

%

A

1

(R 2 )

=

Thus t h e Ai

a r e components o f t h e c o n n e c t i o n and F = D A - D A i s t h e c u r v a t u r e (dA jk j k k j F and one i n s e r t s dxi as needed); $ can be t h o u g h t o f as a H i g g ' s f i e l d 3 2 3 ( c f . Chapter 3 ) . L e t us r e c a l l h e r e t h a t i n R f o r example *dx' = dx A dx , Q

*dx2 = -dx 1 A dx3, and *dx3 = dx 1 A dx 2 w i t h **a = (-1 ) P ( ~ - P =) ~ i n

R3

so

-

Then t h a t *(dxl A dx 2 ) = dx 3 , *(dx 1 A dx 3 ) = -dx2, and *(dx2 A dx 3.) = dx'. i g i v e n a v e c t o r v = 1 v ei s e t = v.dxi w i t h v = and d e f i n e c u r l v = (*d?)- where *d? = [ ( a / a x 1 ) v 2 - w a x h )vl]dx 3 - [(a/ax 1 )v3 - (a/ax 3 )V,ldx 2 2 3 3 t [ ( a / a x )v3 - (a/ax )v21dx ( h e r e - s i m p l y i s an index l o w e r i n g o r r a i s i n g

7 1

Now t h e equations (10.9) can be w r i t t e n ( e x e r -

o p e r a t i o n as i n Appendix C ) . cise

- c f . [BeZ;Tbl])

(10.11) where DA = d

d*dA = ( i / Z ) * [ $ P

-

-

;DA$];

DA*DAJI = ( h / 2 ) * [ I J / I 2

i A i s a covariant derivative.

- 11$

S e t t i n g F = dA one can a l s o

2

w r i t e I, i n t h e form ( * i n R )

F o r t h e v o r t e x - s o l i t o n t h e o r y one t a k e s

= 1 now and p r e s c r i b e s t h e Chern

NONLINEAR PDE

79

number o f t h e l i n e bundle o f which A i s t h e c o n n e c t i o n (*.) N = ( 1 / 2 a ) ~ 2 R 1 2 A dx ). Note h e r e t h e boundary c o n d i t i o n i n dA (dA F12 = (D1A2 - D2Al)dx Q

[C12] can be w r i t t e n as n.DA$IS = 0 and f o r R q u i r e s DA$

XI

0 as

+

+

!$I

-(with

1

+

-

+

R2 i n f a c t one u s u a l l y r e -

c f . [Tbl]).

Now an i n t e g r a t i o n by

p a r t s i n (10.12) y e l d s t h e Bogomolnij f o r m u l a ( e x e r c i s e ) I, = (

(10.13)

, l

Here t h e + (resp. - ) s i g n r e f e r s t o p o s i t i v e ( r e s p . n e g a t i v e ) v o r t e x number

N.

Take N > 0 f o r s i m p l i c i t y and t h e n f r o m (**) and (10.13)

I, 2 Nn.

This

+ A1!h2 = D2$2 - A ~ $ ~D21L1 ; + ( 1 / 2 ) ( 1 J/ 1' - 1 ) = 0 ( e q u i v a l e n t l y , w i t h * 2 i n R , one has DA$ - i*DAJI = 0 and *F + ( l / Z ) ( I J / \ - 1 ) = 0 ) . The e q u a t i o n s

l o w e r bound i s r e a l i z e d i f and o n l y i f (*&) D1$l

+ A2:2

-

+;D1$2

A1!bl

= 0; F12

(*&) can be reduced t o one n o n l i n e a r second o r d e r e q u a t i o n as f o l l o w s . 6

A = A1

+

a

iA2,

-

(1/2)(D1

=

iD2),

and

two equations i n (*&) become (*+) 2a$

iff$= 0 w i t h

-

Set $ = exp f and one o b t a i n s (*.) A, = D2fl = -Afl

w i t h fl

(*&) becomes exp(if2)

+

0 as 1x1 -Af

(A*)

1 w i t h f,(e.lxl)

+ m

(from

l$l

+ (1/2)(exp(2fl)

4

s o l u t i o n A = -2ialog$.

+ Dlf2;

A 2 = -D f + D 2 f 2 ; 1 1 F12 F i n a l l y the l a s t equation i n

+

1).

-

1 ) = 0 and f o r l a r g e 1x1, $

= 2Nn + f 2 ( e + 2 r , I x I ) .

s u i t a b l e hypotneses one shows ( c f . [ T b l ] ) each p o i n t {al,...,aNl

I$I

satisfying

+

+

I t f o l l o w s t h a t f 2 must be

s i n g u l a r on some s e t and c o r r e s p o n d i n g l y $ w i l l v a n i s h on t h i s s e t . d i s c r e t e , say Z = {ai},

Set

(1/2)(D1 + i D 2 ) so t h a t t h e f i r s t

=

Under

t h a t t h e s e t Z where $ ( x ) = 0 i s

and N i s t h e s i z e o f Z.

One proves e.g.

that to

E R2N t h e r e e x i s t s unique g l o b a l Cm s o l u t i o n s t o (*&)

1 and DA$

+

v a n i s h i n g o f J/ a t a p o i n t a

0

0 as 1x1

+ m

w i t h Z = W a k I and t h e o r d e r o f

i s t h e number o f times a.

belongs t o t a

l,...,

i t i s proved t h a t weak s o l u t i o n s o f t h e G-L 3 e q u a t i o n s (10.9) ( w i t h A = l ) , s a t i s f y i n g A E C and $ E C2, a r e a l s o s o l u aNl.

Again, r e f e r r i n g t o [Tbl],

t i o n s o f t h e f i r s t o r d e r e q u a t i o n s (*&) ( t h u s c r i t i c a l p o i n t s correspond t o g l o b a l minima).

F u r t h e r one proves t h a t weak s o l u t i o n s o f (10.9) a r e r e l a -

t e d by a gauge t r a n s f o r m a t i o n t o a Cm s o l u t i o n .

REmARK 10.8.

S o l i t o n s p l a y an i n c r e a s i n g l y i m p o r t a n t r o l e i n n o n l i n e a r f i e l d

t h e o r y i n modern quantum mechanics and i n many o t h e r areas o f mathematical physics.

We make a few remarks here which a r e connected t o Remark 10.7.

Given e.g. a Klein-Gordon e q u a t i o n o f t h e t y p e utt

-

2

uxx + (1/2)m u = 0 p u t

i n a v e l o c i t y o f l i g h t t e r m c w i t h xo = c t and ( i n o r d e r t o i n t r o d u c e a

80

ROBERT CARROLL

standard p h y s i c s n o t a t i o n - c f . [ L l l ] ) w r i t e , w i t h sumnation on repeated i n 2 2 dices, (AA) DPDPq t m c (1/2)q = 0 (xP % (ct,x), xP 'L ( c t , - x ) , DP 'L a' % and (D') % ( a / a ( c t ) , - a / a x ) a/axp, D~ % all % a/axP, ( D ~ ) (a/a(ct),a/ax), Since a / a t = c ( a / a c t ) v i a a L o r e n t z m e t r i c - see here S3.8 f o r ( 0 ) e t c . ) . lJ 2 2 we have q t t - ( l / c )qxx t (1/2)m 9 = 0 and (AA) a r i s e s from a Lagrangian 2 2 2 d e n s i t y (Ae) L = (1/2)[D qDpq - (1/2)m c q 1. Take now a g a i n c = 1, and s e t P 2 2 U ( q ) = m q / 4 so t h e equations ( A A ) a r e D aL/a(D 9 ) = aL/alp = -aU/a9 ( n o t e Q

DpaL/a(Dpq) = D (D 9 ) t

t

P

-

P

D (D q ) ) . Write q = x x Cos g i v e s r i s e t o q t t

qt

and 7 ' =

(px

for simplicity

and n o t e t h a t U = 1 - p X x = -Siw (Sine-Gordon) 2 2 w h i l e UG = ( A /4)[q2 - (m2/x2)I2, w i t h Lagrange equations G: ? - 9 " = - A 2 2 [q2 - (m / A ) ] q , i s c a l l e d t h e Goldstone o r lp4 model ( c f , [Fl;Jal;Kfl]). 2 The t o t a l energy i s (A&) H = L I [$ / 2 + v l 2 / 2 t U]dx and t h e ground s t a t e t h e s t a t e w i t h l o w e s t energy o r vacumn s t a t e ) corresponds t o P O where 2 2 0 U ( l p o ) = 0 so q o = m/x and p i = -m/A f o r U = UG ( n o t e f o r U(v) = m 9 /4, 1 = 0 i s t h e ground s t a t e ) . Now one asks i f t h e r e a r e r e g u l a r s o l u t i o n s f o r (i.e.

ticular

>

+

0,

q'

-+

0

-

Finiteness o f H i n

t h e system G h a v i n g f i n i t e energy.

(A&)

requires i n par-

0, and U 0 as 1x1 so IP + I P ( + ~ )independent o f t There a r e 4 p o s s i b l e s i t u a t i o n s h a v i n g "charge" v a l -+

-+

0

and q ( + = ) = q1 o r q2. ues Q =

- p(--)

~ ( m )

2m/A (m/h,-m/A). t ) (where cog =

o f 0 (m/h,m/A),

0 (-m/h,-m/A),

-2m/x(-m,'x,m/A),

Here Q i s d e f i n e d v i a a " c u r r e n t " E~~

Lm

(A+)

J (x,t) =

-

E

P

and Ovv(x, kV

= - 1 ) so Q = ~ ( m ) v ( - m ) = Lm Dx 1 lo 2 (Jo = colD 9 = ( - 1 ) p X ) . We w i l l d i s c u s s c u r r e n t s ,

= 0 and cOl

=

-E

q(x,t)dx = Jo(x,t)dx c o n s e r v a t i o n laws, e t c . l a t e r and g i v e f u r t h e r d e t a i l s about concepts ment i o n e d here.

L e t us n o t e here DVJv = 0 o r DtJo

(which has n o t h i n g t o do w i t h dynamics).

-

-

Dx D t7 = 0 Now f i r s t we c o n s i d e r t i m e inde-

DxJl = DtDxp

pendent s o l u t i o n s w i t h f i n i t e energy so q X x = Up w i t h ( c f . (10.13)) ( A H ) H = Jm [ q o 2 / 2 t U(lp)]dx = (1/2)l: [(lp ' -+ J2U)2 i 2vtJ2U]dx > I/, J 2 l b ' d x I = -m 3 7 /q(m)J2Udg. Given q ( m ) = q ( - - ) t h e r i g h t s i d e i s (2/3)m / A and t h i s energy 9(--)

The s o l i t o n s o l u t i o n f o r 9 ' = J2U i s ~ ( x =)

i s a t t a i n e d when 9 ' = +J2U. (m/A)Tanh[m(x-a)/E]

and by L o r e n t z i n v a r i a n c e p ( x , t ) = (m/A)Tanh(ym/2)(x2 -5y = (1-8 ) ')). Next one r e -

a-Bt) i s a s o l i t o n w i t h v e l o c i t y v (B = v/c,

c a l l s a theorem o f D e r r i c k [ D k l ] which says t h a t i n 1 t D dimensions t h e r e a r e no t i m e independent s o l u t i o n s w i t h f i n i t e energy o f t h e corresponding prob2 D The t U(q)]d x f o r D 2 2 ( c f . a l s o [ K f l ] ) . [i2 t lvql lem w i t h H(p) =

LI

d i f f i c u l t y a r i s e s i n t h e t a n g e n t i a l d e r i v a t i v e ( l / r ) D e v s i n c e D,p(-,e) precludes d i f f e r e n t values

= 0

and t o a v o i d t h i s problem one can c o u p l e To see what i s g o i n g on f i r s t c o n s i d e r UG

v(+m)

t o an e l e c t r o m a g n e t i c f i e l d .

a g a i n f o r D = 2 say w i t h 9 complex and t h e r e w i l l be a continuum o f ground

NONLINEAR PDE

vo

states

(m/h)exp(ia) ( a

E

81

R ) where U ( v ) = 0.

The Lagrangian and equa-

t i o n s o f m o t i o n a r e i n v a r i a n t under t h e group U ( l ) (P t h e ground s t a t e s a r e n o t (e.g. g o = m/h

+

+

exp(iwk, w

E

R) b u t

T h i s i s c a l l e d spon-

exp(io)m/h).

taneous symnetry b r e a k i n g and w i l l l e a d t o massless (Goldstone) bosons (bosons

p a r t i c l e s o f integer spin).

%

v

Thus s e t

v1 +

=

o s c i l l a t i o n s around a ground s t a t e m/h so s e t t i n g

L becomes (.*)

LG(J,) = DP?DpJ,

e r o r d e r terms i n J,, and G2.

- UG(lJ,I)

J, =

iq2

and c o n s i d e r small

IP -

m/h

t h e Lagrangian

-

m2J,: + D J, DPJ,2 + h i g h 1-11 1 11 2 2 Thus t h e r e i s no mass t e r m f o r J,2 (% - c J , ~ ) and = D J, D'J,

i f one draws a p i c t u r e o f UG as a b o t t l e bottom o v e r (q1,q2) t h e s i t u a t i o n

i s c l a r i f i e d (exercise

-

c f . [Kfl;Mdl]).

Now add gauge p o t e n t i a l s A

P

(cf.

Remark 10.7 and c f . S3.8 f o r p o s s i b l e d i f f e r e n c e s i n i n d e x n o t a t i o n ) t o obt a i n a gauge i n v a r i a n t Lagrangian ( U ( 1 ) i s t h e group, (10.14)

LAG ( q ) = [(DP

-

-

ieAp)*(DP

-

ieAP)q

*

UG(Iql)

denotes c o n j u g a t i o n )

-

(1/4)FpvFPv]

2 2 2 2 where UG( Ilp I ) = (A / 4 ) ( 1~ I - m /A ) and Fpv = OpAv - DvAP. The gauge t r a n s f o r m a t i o n s a r e q + e x p ( i e w ( x ) ) q and A, A. ( x ) + DPw(x) where x 2, ( t , x ,x ) +

and t h e energy i s 2 d x where DQ = (V

(0.)

-

H

=

v

lI [
ieA)q, Doq = (Dt

-

z2

v

+ (@)*

P

i e A )q 0

E

1

UG(I'+'I + ( 1 / 2 ) ( E 2 + B 1 = Fol , E2 = Fo2, and B =

The ground s t a t e s a r e q o = (m/A)exp(ia) a g a i n w i t h A

= 0 and now one F,2. LJ has t i m e independent l o c a l i z e d s o l u t i o n s o f f n i t e energy. Indeed t a k i n g IP 2 @)*.Dp + UG]d 2 x w i t h say q ( r , = 0 and E = 0 one l o o k s a t H = [(1/2)B + It e ) + q ( m , e ) (= (m/x)exp(ia)), = O ( 1 / r l f E , and I B I = O(l/rl+E). f o l l o w s t h a t one must have ( e x e r c i s e - c f . [ K f l ] ) ( 0 0 ) l i m i e r A e ( r , e ) =

[D,q(-,e)]/q(-,e)

as r

+ m

(where D e q = ( l / r ) ( a / a 6 ) q -

Now one can c o n s i d e r q ( m , e ) as a map from t h e c i r c l e

ieAeq w i t h A, = A-1,). 1 -+ Sa and such maps

1 Se

can be d i v i d e d i n t o homotopy c l a s s e s depending on t h e w i n d i n g number n w i t h standard f o r m say qn(m,e) = (m/A)exp(ine). i f there are solutions w i t h Ae(r,e)

+

But by

c / r as r

+ m

(00)

t h i s i s not precluded

(and such s o l u t i o n s e x i s t ) .

We w i l l g i v e f u r t h e r d i s c u s s i o n o f Goldstone bosons, t h e H i g g ' s mechanism, s o l i t o n k i n k s and a n t i k i n k s , e t c . i n Chapters 2-3. further item following [Kfl];

For now we mention one

namely, one as a magnetic f l u x q u a n t i z a t i o n

2 Bd x a r i s i n g from t h e above d i s c u s s i o n . Thus d e f i n e magnetic f l u x as @ = I/ 2 = IIc u r l A d x = G A ds so t h a t g i v e n (o.), f o r q ( m , e ) = (m/h)exp(iu(e)), @ =

l i m I,"A,(r,e)de 2

=

(l/ie)If"

[D,q(m,e)]de/q(m,e)

= (l/e)If"

o'de = 2 N e

= nh/q where n i s t h e w i n d i n g number, h i s P l a n c k ' s c o n s t a n t , and q i s an e l -

ementary charge.

T h i s shows t h a t

@

i s quantized.

82

ROBERT CARROLL

L e t us mention t h a t t h e r e i s a n o t h e r e q u a t i o n c a l l e d t h e Ginz-

RrmARK 10.9,

burg-Landau e q u a t i o n which i s a p r o t o t y p i c a l amp1 i t u d e e v o l u t i o n equation governing t h e envelope o f s u p e r c r i t i c a l wave d i s t u r b a n c e s . It can be p u t i n 2 t h e form At AAxx = oA B l A l A where o and B a r e complex and X i s r e a l .

-

-

T h i s g i v e s r i s e t o a r i c h v a r i e t y o f phenomena i n c l u d i n g i n s t a b i l i t y , b i f u r c a t i o n s , chaos, motion on low dimensional t o r i , e t c . and we r e f e r t o [Nt1,2].

REmRK 10.10

I n c o n n e c t i o n w i t h t h e Navier-Stokes equations d e s c r i b e d b r i e f -

l y i n § 9 l e t us mention some f u r t h e r ideas based on [ F o l ] ( c f . [Cwl;Te3,4]

f o r an up t o d a t e survey and see Remarks 10.11-10.12

f o r further information)

This m a t e r i a l and t h a t o f t h e n e x t two remarks i s q u i t e t e c h n i c a l and i s onHowever we b e l i e v e t h i s exposure t o modern work w i l l whet t h e

l y sketched.

a p p e t i t e and show how a r e l a t i v e l y small amount o f t e c h n i q u e can be a p p l i e d Much o f t h e t e c h n i q u e can be l e a r n e d

t o study some f a i r l y deep problems.

Thus l e t R be an open bounded connected s e t i n

from t h e p r e s e n t book.

( n = 2 o r 3 ) w i t h boundary and

r

r

where

r'

E

C2,

w i t h R l y i n g on one s i d e o f

i s t o have a f i n i t e number o f connected components

ri.

Rn

r,

L e t V, H, and

4 be as i n Example 9.8 and A t h e F r i e d r i c h ' s e x t e n s i o n o f -A14 so t h a t 4 = and IIuII = I k u I f o r u E 4 (D(A) = 4 n H2 (n) and Au = -PAu where P: L 2

O(A')

( Q ) ~ H i s t h e orthogonal p r o j x t i o n ) . -f

One t a k e s a l s o b(u,v,w)

w i t h V ' t h e dual o f V, e t c . and we w r i t e b(u,v,w)

v)

E

V',

B ( u ) = B(u,u),

= (B(u,v),w),

and t h e nonhomogeneous problem u =

v

as i n ( 9 . 7 )

w

E

V, B(u,

on Z i n ( 9 - * @ )

i s considered where say v E H 3 j 2 ( r ) w i t h i v - v dr = 0 (H3/2(r) i s the trace 2 ri o f H (n) on r and we denote t h e space o f p as i n d i c a t e d by G 3 l 2 ( I ' ) , d r o p p i n g 2 For any 6 t h e r e i s a l i n e a r continuous map A 6 : i3l2 t h e n i n H ( R ) ~e t c . ) .

H2 such t h a t d i v A6v = 0 i n R , A6v = v on r , and I b ( v , A 6v , v ) l < Sllvlli3/2 2 IIvII ( v E V - c f . [T31]). The N-S equations (9.6)-(9-*@) then reduce t o

+

(exercise

-

c f [Fol])

w i t h w(0) = w and wo

W:

0' = uo - A69

(06) +

[O,m)

.

k

B(w) t B(w.A6v) t B(A6\p,w) = Ph = H

t vAw t

V, w = u

-

A6p

,h

= f -k vA69

-

1 ( A 6p ) .JD .JA 6p ,

Under reasonable hypotheses as i n 59 t h e c l a s s i c a l r e -

s u l t s i n d i c a t e d i n 59 can l e a d t o e x i s t e n c e o f s o l u t i o n s (@+) w E C((O,m),H) 2 n Lloc.!(O,m),V) w i t h uniqueness f o r n = 2. For n = 2 one denotes by S ( t ) t h e map wo maps. v0/2;

-f

w(t): H

+

H so t h a t S ( t ) i s a semigroup o f n o n l i n e a r continuous

Assume now v 2 vo > 0, lvl,-,3/2 set also

= A6p

.

zvo, and

S o l u t i o n s i n C(r0,Tl.V)

n = 2 i f wo E V, S(*)wo i s r e g u l a r on any [O,T]

a r e c a l l e d r e g u l a r and f o r w h i l e i f n = 3 and wo E V

( = TO(wO,p, . . . ) ) such t h a t a w as i n ( a + ) above i s r e f o r T < To. One w r i t e s R S ( t ) f o r t h e map wo + w ( t ) ( t < To).

t h e r e e x i s t s To(wo) g u l a r on [O,T]

choose 6 such t h a t 6po 5

NONLINEAR PDE

83

I t i s proved i n [ F o l ] t h a t i n f a c t f o r s u i t a b l e To, R S ( t ) w o i s a n a l y t i c on [O,To]

as a D(A) v a l u e d f u n c t i o n (D(A) i s normed by lAul which i s e q u i v a l e n t Now d e f i n e t h e D dimensional H a u s d o r f f measure o f Y C X

t o t h e graph norm).

( X a m e t r i z a b l e space) by (diamB )

(10.1 5 )

D

3

f o r c o v e r i n g s o f Y by b a l l s B . w i t h diameter B . < E . I t i s proved i n [ F o l ] J 4t h a t f o r n = 3, i f a s o l u t i o n w o f ( 0 6 ) s a t i s f i e s (N), t h e n t h e r e e x i s t s a closed set Z C [ 0 , m )

o f H a u s d o r f f dimension 5 1 / 2 such t h a t w i s an a n a l y t i c

D ( A ) valued f u n c t i o n on [ O , m ) / Z . s e t fi0

C

A l s o i f w(0)

R such t h a t ess sup I w ( x , t ) l

<

m

E

2

3 t h e r e e x i s t s a sub-

H (a)

( t E (0,T))

for all x

€

R/Ro

and

-

(0,~) w h i l e pD(Ro) 5 5/2. F i n a l l y one c a l l s a s e t X C V f u n c t i o n a l l y i n 0. For n = f o r a l l wo € X and RS(t)X = X, f o r a l l t v a r i a n t if T ( w ) = 0 0 2, g i v e n a s o l u t i o n w o f ( 0 6 ) s a t i s f y i n g (@+) t h e r e e x i s t s a f u n c t i o n a l l y i n v a r i a n t X such t h a t min I w ( t ) - w I + 0 (min f o r w € X ) as t + Moreover f o r n = 2 o r 3 t h e r e e x i s t s c ( P ~ , v ~ , ~ H ~ , ssuch ) t h a t f o r any bounded funct i o n a l l y i n v a r i a n t X C V ( i f one e x i s t s ) p D ( X ) 5 c(vO,vO,[HI ,supllwll) (sup f o r w E X ) ; e x i s t e n c e i s n o t known however f o r n = 3. These theorems a r e T

€

-.

e v i d e n t l y v e r y t e c h n i c a l and r e q u i r e a l o t o f a n a l y s i s t o prove b u t t h e y should show t h a t a l o t o f f i n e s t r u c t u r e can be d i s c o v e r e d .

F u r t h e r perspec-

t i v e can be o b t a i n e d from Remarks 10.11-10.12.

RrmARK 1O.lf.

The theme i n d i c a t e d i n Remark 10.10 has undergone tremendous

expansion and r e f i n e m e n t i n r e c e n t y e a r s and we f e e l compelled t o a t l e a s t mention a few t y p i c a l r e s u l t s and ideas.

For f u r t h e r i n f o r m a t i o n see e.g.

L e t us f i r s t e x t r a c t f r o m [Te4] some

[Cw1,2;Babl;Fo2,3,4,5;Hcl;Mgl;Te3,4].

d e f i n i t i o n s and comments r e l a t e d t o t h e Navier-Stokes e q u a t i o n s ( c f . a l s o Thus c o n s i d e r t h e N-S equations i n t h e f o r m ( 0 6 ) w i t h P = 0

Remark 10.12). o r (om) wt +

= f E H independent o f t .

2 and f ( t ) every wo i n D(A).

Aw + B(w) = f w i t h w(0) = wo (B(w)

E

'L

B(w,w)).

Take f i r s t n =

Then ( c f . S9) (om) i s w e l l posed f o r

H ( t > 0 ) w i t h s o l u t i o n w ( t ) a n a l y t i c i n t i m e and t a k i n g v a l u e s

Again S ( t ) : wo

+

w(t): H

+

D(A) i s continuous f r o m H

t h e semigroup p r o p e r t y S ( t ) S ( s ) = S ( s + t ) . H f o r S ( t ) i s d e f i n e d as b e f o r e (i.e.

+

H and has

A functionally invariant set X c

S ( t ) X = X, f o r a l l t > 0 ) and necessar-

X i s c a l l e d an a t t r a c t o r (say i n H o r D(A)) i f t h e r e e x i s t s a

i l y X C D(A).

NBH 0 o f X ( i n H o r i n D(A)) such t h a t f o r e v e r y wo E 0, S ( t ) w o + X ( i n H o r i n D(A)) as t -. It was shown i n [ F o l ] t h a t X = X(f,wo) = nS,oUt,SIS(t)wo~ -+

( c l o s u r e i n H) i s a f u n c t i o n a l l y i n v a r i a n t s e t w i t h w ( t )

+

X as t

-+

m;

X

84

ROBERT CARROLL

describes t h e "permanent" f l o w f o r f and wo given.

Now one d e f i n e s a l a r g e r f u n c t i o n a l l y i n v a r i a n t s e t ( t h e u n i v e r s a l a t t r a c t o r ) by saying t h a t a s e t A C V = D(A 4 ), bounded i n V i s absorbing i n V i f f o r every bounded B C V t h e r e e x i s t s to = t o ( B ) such t h a t S ( t ) B C A f o r t 2 to One can show t h e r e e x i s t s s u c h an

( t h i s m i g h t be b e t t e r c a l l e d S-absorbing). N

N

A ( c f . [ F o ~ ] ) . Then X = X ( f ) = ns>OUt>sIS(t)A) i s an a t t r a c t o r i n H o r V and i s t h e l a r g e s t such bounded f u n c t i o n a l l y i n v a r i a n t s e t , c a l l e d t h e u n i .v

versa1 a t t r a c t o r (X

a l l X(f,wo)).

3

m a n i f o l d and f E C"(E)2

One can p r o v e t h a t i f

then ?(f) E C"(;)';

r

= a R i s a Cm

a l s o t h e H a u s d o r f f and f r a c t a l

dimension o f X ( f ) a r e f i n i t e f o r a l l f. We remark here t h a t t h e f r a c t a l dim1i m ension o f Y = E+O l o g N E ( Y ) / l o g ( l / E ) where NE(Y) i s t h e s m a l l e s t number o f balls o f radius 5 E pD(Y) =

c o v e r i n g Y.

I n general t h e f r a c t a l dimension dM(Y) 2

H a u s d o r f f dimension and dM(Y)

i t e dimensionallity that

pD(Y) <

<

m

i s thus a f i n e r condition o f f i n -

2 2

If one s e t s e.g. G = I f l L o / v

(Lo a t y p i c a l l e n g t h i n n) so t h a t a Reynolds' number can be w r i t t e n as Re = JG i t m.

f o l l o w s t h a t d # ( X ( f ) ) 5 coG where co i s a u n i v e r s a l constant; a l s o > c G ~ ' (~c f . [Babl;Cwl;Te4]

-

n = 2 here).

pD(X(f))

For n = 3 g i v e n t h e incomplete-

ness o f t h e existence-uniqueness t h e o r y t h e r e s u l t s on a t t r a c t o r s e t c . a r e n e c e s s a r i l y somewhat l i m i t e d .

One d e f i n e s a f u n c t i o n a l l y i n v a r i a n t s e t X as

f o r n = 2 w i t h t h e p r o v i s o t h a t S ( t ) w o e x i s t s f o r a l l t > 0 when wo E X.

One

can then d e f i n e a t t r a c t o r s e t c . as before; however one does n o t know whether a t t r a c t o r s , absorbing sets, o r universal a t t r a c t o r s e x i s t .

One can a l s o g i v e

estimates on t h e f r a c t a l dimension o f a p u t a t i v e a t t r a c t o r o r f u n c t i o n a l l y i n v a r i a n t s e t r e l a t e d t o t h e Kolmogorov, Landau, L i f s c h i t z t h e o r y o f t u r b u lence ( c f . [Cwl]).

Some o f t h e i n g r e d i e n t s a r e t h e idea o f Lyapounov expon-

e n t s and t h e L i e b - T h i r r i n g i n e q u a l i t y .

REmARK 10.12,

L e t us make a few remarks based on [Fo2,3,7;Gdl

;Gjl,Z;Mgl]

which i n d i c a t e some new d i r e c t i o n s o f research r e l a t e d t o what a r e c a l l e d i n e r t i a l m a n i f o l d s (see t h e b i b 1 iography i n [Fo2,3;Mgl] ences).

f o r further refer-

The development w i l l a l s o e x h i b i t some formulas and technique o f

general i n t e r e s t and h o p e f u l l y w i l l generate i n t e r e s t i n r e a d i n g o t h e r sect i o n s o f t h e book as w e l l as t h e papers i n d i c a t e d on t h i s s p e c i f i c t o p i c . Roughly speaking one s t a r t s from t h e f a c t t h a t f o r s u i t a b l y d i s s i p a t i v e PDE t h e r e e x i s t compact g l o b a l a t t r a c t o r s A w i t h f i n i t e H a u s d o r f f and f r a c t a l dimension.

F u r t h e r some such PDE a l s o have a f i n i t e dimensional i n e r t i a l

m a n i f o l d c o n t a i n i n g t h e a t t r a c t o r and r e s t r i c t i n g t h e PDE t o t h e i n e r t i a l m a n i f o l d one o b t a i n s a system o f ODE d e t e r m i n i n g t h e l o n g t i m e b e h a v i o r o f

85

NONLINEAR PDE

We w i l l s i m p l y g i v e t h e d e f i n i t i o n s o f these terms

s o l u t i o n s t o t h e PDE.

here, p l u s some i n d i c a t i o n o f t h e equations and conceptual framework i n v o l Thus one c o n s i d e r s e q u a t i o n s o f t h e form (6*) ut + Au + R(u) = 0 where

ved.

+

R(u) = B(u,u)

Cu

- f,

u ( 0 ) = uo

E

Here A i s t o be a l i n e a r p o s i t i v e

H.

densely d e f i n e d o p e r a t o r i n a H i l b e r t space H w i t h A - l compact (D(A) w i l l be

...<

h1 <

h

l i n e a r D(A)

m

-f

j

as j

-f

(iw'' ill

H, and f E D(A')

+

v),v)

= 0,

J J

i s t o be b i l i n e a r : D(A3 x D(A)

B(u,v)

m.

D ( A s ) w i t h s c a l a r p r o d u c t (ASu,ASv) ?IAulIAvl,

L e t then A w . = X . W . w i t h 0 <

t h e graph norm i n t h e form I A u l ) .

g i v e n e.g.

One assumes e.g.

(u,~),,).

~ B ( U , V5 ) ~c ~l u l l k u l lA%1 I A v l ,

4

[ A Cul ( Z l A u I ,

-+

H, C i s

be w e l l d e f i n e d and one s e t s VZs = (6.)

(B(u,

IA'B(u,v)l 5 Such hypotheses i n s u r e

< cIA?lul I A u l ,

si -2 and ((A+c)u,u) L a l A u I

.

t h e e x i s t e n c e o f g l o b a l s o l u t i o n s o f ( 6 * ) , w r i t t e n u ( t ) = S ( t ) u o , where S F u r t h e r u ( t ) w i l l be u n i f o r m l y bounded

has t h e usual semigroup p r o p e r t i e s . i n H and i f uo

(A') B

1 2pi.

E

D(A')

( r e s p . D(A)) t h e n S ( t ) u o i s u n i f o r m l y bounded i n D I n p a r t i c u l a r a f t e r a f i n i t e t i m e S ( t ) u o E B~ i n H ( r e s p .

(resp. D ( A ) ) .

o r D ( A ) f o r s u i t a b l e u o ) where t h e B . a r e b a l l s o f r a d i i 1 The w - l i m i t s e t o f B2, namely A = u(B2) = ns10Ut,sS(t)B2 (closure i n

o r B2 i n D($)

H) i s t h e g l o b a l a t t r a c t o r (maximal a t t r a c t o r ) ; we n o t e A C Bo n B1 n B2. Now f o r t e c h n i c a l reasons t a k e = 1 for 0 5 s

5 1,

0 = 0 for

e

l e u /5

s 2 2, and

2; s e t

< p,

2 p 2 and c o n s i d e r

p =

t h e m o d i f i e d e q u a t i o n ( 6 0 ) u ' + Au t e p ( l A u l ) R ( u ) = 0. t h e NBH o f A where lAul

Then ( 6 0 )

t h e a s y m p t o t i c b e h a v i o r and bounds a r e equiva-

l e n t , and ( 6 0 ) i s e a s i e r t o work w i t h .

PN be t h e orthogonal p r o j e c t i o n H

- PN.

Then f o r any y,r,T

in

(6*)

5

One w r i t e s S ( t ) u o e t c . now f o r s o l u -

t i o n s o f ( 6 0 ) and t h e r e i s a squeezing p r o p e r t y d e s c r i b e d as f o l l o w s . 1

e

t o be a smooth r e a l v a l u e d f u n c t i o n w i t h

+

. . ,wN),

span(wl,.

Awk =

hkwk,

and Q,

Let =

w i t h /Auol 5 r, lAvol 5 r, t h e r e i s an a l t e r n a -

t i v e for t 5 T (66) IQNIS(t)uo

-

S(t)voll

LY

IP"S(t)uo

-

S(t)voll or IS(t)

A

kexp(-kahN+lt)luo - v o I f o r s u i t a b l e c o n s t a n t s depending on u0 - S ( t ) v o ] T ( c f . [ F O ~ ] f o r d e t a i l s ) . Now one l o o k s f o r an i n e r t i a l m a n i f o l d M f o r (6.)

d e f i n e d by ( 6 + ) ( A ) M i s a f i n i t e dimensional L i p s c h i t z m a n i f o l d (B)

S(t)M

cM

for all t

0 (C) d i s t ( S ( t ) u o , M )

f o r uo E B c H, B bounded. f i x e d p o i n t problem.

Qu, and c o n s i d e r (6m)

+

0 exponentially fast, uniformly

Then A c M and one reduces t h e search f o r M t o a

Indeed, w r i t e P = PN, Q = Q,,

+ Ap + PF(u)

B p ( l A u / ) R ( u ) ( n o t e PA = AP e t c . ) .

= 0;

4

p = Pu = PS(t)uo, q =

+ Aq + QF(u)

= 0 where F ( u ) =

Now one l o o k s a t maps @:PD(A)

+

QD(A) i n

a s u i t a b l e c l a s s T and e v e n t u a l l y we want q = @ ( p ) w i t h u = p + @ ( p ) . @

For

g i v e n (and po) one can i n t e g r a t e t h e p e q u a t i o n i n ( 6 ~ w ) i t h u = p + @(p).

Then f o r

CI

continuous and bounded w r i t e t;(t) = e x p [ - ( t - s ) A ]

+

86

ROBERT CARROLL

Ist e x p [ - ( t - r ) A ] u ( r ) d r ed as

'I +

--

f o r t h e unique s o l u t i o n o f

+ AC

= u r e m a i n i n g bound-

( h e r e we a r e t h i n k i n g o f t h e q e q u a t i o n i n ( 6 ~ w ) i t h A = AQ and

u = -QF(u) = -QF(p (TAc))QF(p*(p))dr s e n t i a l l y moved

-

+

Thus ~ ( 0 =)

@ ( p ) ) and p a l r e a d y determined).

and one t h i n k s now o f p = p(r,@,p0) t o 0).

exp

( n o t e t h a t one has es-

Set t h e n ( W ) T@(po) = ~ ( 0 and ) if a

T e x i s t s so t h a t ~ ( 0 =) qo then T@(pp) = qo = @ ( p o ) .

-l:

Q,

i n the class

Hence Q would be a

f i x e d p o i n t o f T and t h u s one l o o k s f o r s u i t a b l e c o n d i t i o n s on T, N, e t c . so t h a t T w i l l have a f i x e d p o i n t ( c f . [ F o ~ ] f o r d e t a i l s ) . an i n e r t i a l m a n i f o l d M v i a (p(O),q(O)) The dynamics of (6.)

M = N. (W)

+

AP

+ e

P

E M = graph

@

F i n a l l y one f i n d s

= {(po,Q(po))}

and dim

on M i s t h e n c o m p l e t e l y determined by t h e ODE

( I A ( p t @ ( p ) ) l ) P R ( p t @ ( p ) ) = 0 which i s c a l l e d an i n e r t i a l form The a n a l y s i s i n [ F o ~ ] i s somewhat c o m p l i c a t e d b u t

o f t h e equations ( 6 * ) .

a c c e s s i b l e v i a knowledge o f standard techniques developed i n t h i s book.

11. ILL: P0dED PR03CEW AND REGULARZZAtZ0N.

We have a l r e a d y encountered some

i n v e r s e problems i n 56 and 510 w h i l e some ill posed probelms were i n d i c a t e d i n e.g.

54 (backward h e a t equation, Cauchy problem f o r t h e Laplace equation,

and D i r i c h l e t problem f o r t h e wave e q u a t i o n ) .

We w i l l g i v e h e r e ( w i t h no a t -

tempt t o be e i t h e r e x h a u s t i v e o r s y s t e m a t i c ) a v a r i e t y o f o t h e r such problems and some general ideas (and theorems) about how t o deal w i t h these s i t u a t i o n s ( f o r r e f e r e n c e s see e.g. Pyl;Tkl]).

[Aul;Bll;Csl;Gz1,2;Hml

;Kul;Li9;Lvl-3;Mzl;Rml

;Sil;

L e t us emphasize t h a t i n p r a c t i c e c o n d i t i o n s o f s t a b i l i t y , con-

t i n u o u s dependence o f s o l u t i o n s on i n i t i a l o r boundary data, e x i s t e n c e and uniqueness o f s o l u t i o n s , e t c . a r e f a r more i m p o r t a n t and p r e v a l e n t than may be c u s t o m a r i l y r e a l i z e d . and must be d e a l t w i t h .

Moreover c e r t a i n i l l - p o s e d problems a r i s e n a t u r a l l y For example i n s o l v i n g an equation Ax

y where A

i s an o p e r a t o r (say a l i n e a r o p e r a t o r o r even a m a t r i x ) e r r o r s i n measurement w i l l o n l y g i v e y,

w i t h IIy-y,ll

be known t o w i t h i n an e r r o r IIA-A,II or if y

#

5 5

E E.

(some norm) and s i m i l a r l y A may o n l y But i f A i s s i n g u l a r and A,

i s not

R(A) b u t y and y, E R(A,)

one has v a r i o u s s o r t s o f problems. Anb o t h e r standard "pathology" a r i s e s i n s o l v i n g (*) y ( t ) = K(t,s)x(s)ds =

/a

I f x^(s) = x ( s ) + NSinhs then G ( t ) K X ( t ) where K i s e.g. continuous i n ( t , s ) . b = # ? ( t ) = y ( t ) + N12 K(t,s)Sinxsds. Here f o r h l a r g e l l x ( s ) - x ( s ) l l o = N (I1 I I o means sup norm on [ua,b])

b u t Ily^(t - y ( t ) l l o can be made a r b i t r a r i l y small by

t h e Riemann-Lebesgue 1emma when h i s s u f f i c i e n t l y l a r g e . d o e s n ' t depend c o n t i n u o u s l y on y.

In particular x

Another c l a s s i c a l s i t u a t i o n a r i s e s i n

1

1

F o u r i e r s e r i e s where say f ( t ) = anCosnt w h i l e f E ( t ) = (an + E/n)Cosnt so 2 % = E( (l/n ) But If-f, t ) I = ( l / n ) C o s n t l which can be made

IIf-f,l12

1

.

€11

I L L POSED PROBLEMS

a r b i t r a r i l y l a r g e (e.g.

as t

* 0).

87

T h i s example i l l u s t r a t e s t h e r o l e o f

d i f f e r e n t norms i n measuring closeness.

Generally,even when t h e we1 1 posed

problem i s known, i t may be i m p o s s i b l e t o measure t h e d a t a i n t h e c o r r e c t r e g i o n and t h e n one t r i e s t o compensate somehow, perhaps by overmeasuring d a t a elsewhere, and t h i s c r e a t e s a new t y p e o f problem.

The s t u d y o f i n v e r s e

problems u s u a l l y l e a d s t o ill posed problems f o r example.

Here g e n e r a l l y

one t r i e s t o determine t h e c o e f f i c i e n t s i n a d i f f e r e n t i a l o p e r a t o r , o r a source, o r boundary c o n d i t i o n s i n a system governed by say d i f f e r e n t i a l equat i o n s , by p e r f o r m i n g experiments on t h e system and measuring some a s p e c t o f t h e response.

For example one m i g h t d i s c o v e r t h e i n v e r s e square g r a v i t a t i o n -

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

It i s probably worth

p o i n t i n g o u t t h a t c e r t a i n n a t u r a l problems may n o t have any s o l u t i o n s .

This

i s obvious f o r e q u a t i o n s l i k e y B 2t y2 = -1 w i t h r e a l valued y b u t more s u r prising i s

EHAIPCE 11.1

(CEmg).

T h i s i s a famous example o f H. Lewy o f a d i f f e r e n t i a l

1

o p e r a t o r P(D)u = ux + i u - 2 i ( x t i y ) u t f o r which no C s o l u t i o n o f P(D)u = f Y e x i s t s i n any open s e t , f o r a l a r g e s e t o f f ( x , y , t ) E cm (x,y,t E R ) . In f a c t there are f set.

E

Cm f o r which no d i s t r i b u t i o n s o l u t i o n u e x i s t s i n any open

I f f i s a n a l y t i c o f course, by t h e Cauchy-Kowalevsky theorem, P(D)u =

f would have l o c a l a n a l y t i c s o l u t i o n s .

r e f e r t o [Hl;Sml]

We w i l l n o t go i n t o d e t a i l h e r e b u t

f o r a discussion.

I n any event one i s o f t e n f a c e d w i t h t h e i m p o s s i b i l i t y o f f i n d i n g an e x a c t s o l u t i o n t o some problem o r e q u a t i o n Ax = y and t h e r e i s an impetus t o somehow do t h e b e s t y o u can.

This i s o f course i n f l u e n c e d by t h e demands o f en-

g i n e e r i n g and a p p l i e d s c i e n c e b u t m a t h e m a t i c a l l y t h e p h i l o s o p h y i s s u r p r i s i n g l y h e a l t h y and i n t e r e s t i n g .

I n t h i s d i r e c t i o n we f i r s t c o n s i d e r Ax = y

4

R ( A ) so no s o l u t i o n i s p o s s i b l e ( t h i n k o f A as an n x m m a t r i x f o r T example). Then t r y t o f i n d x m i n i m i z i n g J ( x ) = IIAx - yl12 = (Ax-y) (Ax-y) = T T T T T T As i n Example 5.3 now x A Ax - x A y - y Ax t y y ( c f . [Bul;Gz2;Nal;Stl]). T T one c a l c u l a t e s 6J/6x = 0 t o o b t a i n A Ax = A y as t h e m i n i m i z i n g c r i t e r i o n . When (ATA)-' e x i s t s we have then a l e a s t squares s o l u t i o n x = ( A TA ) -1 A Ty.

when y

T h i s procedure i s connected w i t h l i n e a r r e g r e s s i o n i n s t a t i s t i c s f o r example. On t h e o t h e r hand (ATA)-'

may n o t e x i s t and i n t h i s case one d e a l s w i t h what

a r e c a l l e d pseudoinverses o r g e n e r a l i z e d i n v e r s e s ; t h i s amounts t o s e l e c t i n g T T < IIxII f o r a l l by some c r i t e r i o n some x = ^x f o r which (A A)x = A y (e.g. ll?d " s o l u t i o n s " x l e a d s t o t h e Moore-Penrose g e n e r a l i z e d i n v e r s e ) .

L e t us i n -

t r o d u c e now t h e i d e a o f Tikhonov r e g u l a r i z a t i o n by, means o f an example

88

ROBERT CARROLL

using generalized inverses ( c f . [Zel]

-

we w i l l g i v e more general theorems

later). S t a r t i n g w i t h Ax = y ( i n a m a t r i x - v e c t o r c o n t e x t f o r A: E F: EWAUWCE 11.2. Rn -+ Rm say) we c o n s i d e r f i r s t a p e r t u r b e d problem a c c o u n t i n g f o r say e r r o r s 2 E u c l i d i a n norm). i n measurement Ax6 = y, where lly-y611 5 6 (11 1I % I 112 = L -+

Then extend t h i s problem f u r t h e r by l o o k i n g f o r x (A)

IIAx6-y612 t yllx61I2 = S y ( x 6 )

(y >

0).

Yl6

which minimizes e.g.

Making a v a r i a t i o n a l argument w i t h

t €9 one f i n d s ( 0 ) [A*A .t Y I ] X = A*y6 ~ (A* , ~ AT h e r e ) . based on x6 = x Y Y,6 = [A*A t But A*A 2 0 and even i f (A*A)-l does n o t e x i s t we can w r i t e ( 4 ) x

J,

Now l e t P: F

yI]-'A*y6.

min IIAx-yl

2

for x

E

-+

m)

796

be t h e orthogonal p r o j e c t i o n and l e t a =

O(A) (O(A) = E here) w i t h

G

E

2 i n g IIv-yll f o r v E R(A) = R(A) here ( t h u s IlV-yl12 o f x E D ( A ) f o r which Ax =

R(A) t h e element m i n i m i z -

= a and

V

The s e t

= Py).

7 i s bounded, convex, and c l o s e d and has t h e r e -

f o r e a unique element xR o f minimal norm.

We s e t xR = Aty and c a l l At t h e

g e n e r a l i z e d i n v e r s e o r pseudo i n v e r s e (xR i s c a l l e d t h e normal s o l u t i o n ) . of

Now go back t o x

(0)

Next n o t e

and observe f i r s t t h a t IlPy-Py611 5 c6.

2

Y16

t h a t f o r any x, IIAx-y 112 HAx-Py,lI' t IIPy -y II (Pythagoras) so x 6 6 6 Y,6 minimizes $ (x,) i n (A) when y6 i s r e p l a c e d by Py6. F u r t h e r AxR = Py Y IIAx -Py 11 < IIAxR-PyII t IIPy-Py 1 I < cS and from i+ (xY,&) 5 JI ( x ) ( w i t h R 6 6 r e p l a c i n g y,) one o b t a i n s i n p a r t i c u l a r yllx 11' < IIAxR-Py)$l12Rt ~ l l x Y,6 c262 t yllx 112. Now one s c a l e s y w i t h 6 and s e t s f o r example y = 6 so Ilx

112 < !c2

t IIxd12.

I t f o l l o w s t h a t a subsequence x

Yl6 6's i n c e 1x11 5 IIxRll we must have

^x

-f

Yl6

as 6

-f

also

so Py,

~
= xR and xR i s u n i q u e l y determined because

IIAx -Pyb 5 IIAx -Py611 + IIPy -Pyll + 0 ( n o t e f r o m J, ( x < J, ( x ) f o r y Y,6 Y,6 6 2 Y Y,6j r e p l a c e d by Py6 and y = 6 we have IIAx -Py6H 5 IIAxR-Py61 t 6x: < ca2&+ 4 Y?6 R 611xRl12 -+ 0 ) . F i n a l l y , x = xR b e i n g u n i q u e l y determined, one shows e a s i l y

%'

$ ( e x e r c i s e ) . Thus t h i s procedure ( c a l l e d t h a t t h e whole "sequence" x Y9 6 Tiknonov r e g u l a r i z a t i o n ) p r o v i d e s an e x p l i c i t formula ( 0 ) f o r a sequence o f -f

objects x

Y,6

-t

xR.

The arguments have been phrased i n such a manner t h a t

one sees how t h e y can be extended and r e f i n e d t o a p p l y i n Hi1 b e r t o r Banach spaces ( w i t h s u i t a b l e hypotheses) and we w i l l do t h i s l a t e r .

Such methods

o f r e g u l a r i z a t i o n a r e v e r y powerful and u s e f u l i n many c o n t e x t s . L e t us s k e t c h now a few methods o f t r e a t i n g i m p r o p e r l y posed R?3URK 11.3. Cauchy problems as o u t l i n e d i n [Pyl] ( c f . a l s o [Bll;Ftl;Hml;Kul;Li9]). This

w i l l p r o v i d e some g u i d e l i n e s f o r f u t u r e developments. Thus t a k e f o r example ( c f . 94) (*) ut + Au = 0 i n n x [O,T], u = 0 on r x [O,T] (r = an), and u ( x , 0 ) = f ( x ) (n C Rn i s a bounded open s e t w i t h a reasonable boundary

r).

Some

89

I L L POSED PROBLEMS

o f t h e methods d e l i m i t e d i n [Pyl] a r e ( 1 ) E i g e n f u n c t i o n methods and q u a s i L e t A n and q n ( x ) be r e s p e c t i v e l y t h e nth e i g e n v a l u e and norm-

reversibility.

u + Au = 0 i n R w i t h u = 0 on

a l i z e d eigenfunction o f

r.

l

Write f ( x ) =

p n ( x ) where fn = In f ( x ) v n ( x ) d x and s e t h e u r i s t i c a l l y ( m ) u ( x , t )

2,

fn

1 fnqn(x)

e x p ( h n t ) which i n general o f course w i l l n o t converge. One a l s o t h i n k s o f 2 2 d a t a f* b e i n g mesured w i t h (**) In ( f - f * ) dx 5 a I f i t i s known e.g. t h a t 2 In u (x,T)dx 5 K2 ( t h i s i s a k i n d o f s t a b i l i z a t i o n ) t h e n f o r d a t a ? w i t h

.

1 ?:exp(2AnT)

5 K2 (and (**) f o r ? i n p l a c e o f f ) t h e f u n c t i o n G ( x , t )

1 Tn

=

pn(x)exp(A t ) ( f o r s u i t a b l e a and K) can be regarded as an approximate s o l u n t i o n o f t h e problem above. T h i s needs implementation however and f o r s t a b i l i t y one c o n s i d e r s q u a s i r e v e r s i b i l i t y v i a p e r t u r b a t i o n s o f t h e o p e r a t o r i n = 0 on r x [O,T] and F(x,O) = t h e f o r m (*A) it + A; + E 2A 2-u = 0 w i t h

=a';

f ( x ) o r perhaps (*.)

Dt[G

-

E'A^,]

+

Au = 0 w i t h

=

r

0 on

x [O,T]

and ;(x,

1

= fn I n t h e f i r s t s i t u a t i o n f o r example f o r m a l l y (*&) ;(x,t) 2 Now one l o o k s f o r a problem whose s o l u t i o n a p p r oxiqn(x)exp[-An(E A n - l ) t ] .

0) = f(x).

mates t h e s o l u t i o n o f ( + ) and we c o n s i d e r t h e problem

r

Gt + A;

= 0,

u' =

0 on

and G(x,T) = G(x,T). T h i s i s w e l l posed w i t h s o l u t i o n (*+) :(x, 2 2 2 (x)exp[-A E T + A n t ] and e v i d e n t l y lli(x,O) - f(x)l12 = fn(l n 2 n*nz exp(-AnTE ) -+O as E -+ 0 so t h i s a g a i n i s a k i n d o f approximate S o l u t i o n t o x [O,T],

t) =

1f

1

q

(+). However as E -t 0 Z ( x , T ) may n o t make sense (;(x,T) V so t h e d a t a f o r u ( x , t ) a t t = T i s n o t s p e c i f i e d as E c u l t y a r i s e s i n d e a l i n g w i t h (*a).

2,

-t

0.

-

1 fnqn(x)exp(AnT)) A similar d i f f i -

These q u e s t i o n s a r e s t u d i e d f u r t h e r i n

[Kul;Li9]). (2) Logarithmic convexity. 2 2 Then In u dx = I l u ( t ) l l

Consider ( + ) a g a i n w i t h F ( t ) =

.

(11.1)

F ' ( t ) = 21,

2 uutdx = - 2 1 uAudx = 2 1 I g r a d u l dx;

F"(t) = (11.2)

FF"

-

F"

41, gradu gradutdx

= F2 ( l o g F ) " =

( b y Cauchy-Schwartz).

2

4[/, u dxl,

= 411 utll

2 utdx

2

-

(1,

uutdx)

2

3 20

Thus l o g ( F ) i s a convex f u n c t i o n o f t and one o b t a i n s

by c o n v e x i t y ( e x e r c i s e ) F ( t ) 5 F(T) t/TF(0)l-t/T and F ( t ) > F(O)exp[tF' (O)/ 2(1-t/T)ll u(T)II 2t/T-and 1I u(t)1I2 > 1I fll 2 F(O)]. Consequently ( * m ) IIu(t)l12 5 IIfll 2 2 exp[2tllgradfll /I1 fll 3. The second i n e q u a l i t y i n d i c a t e s t h e e x p o n e n t i a l growth o f I l u ( t ) l l as t

-+

-

and t h e f i r s t i s a k i n d o f s t a b i l i t y r e s u l t on [O,T).

However n o t e t h a t if F(0) = IIfl12 i s small i t does n o t n e c e s s a r i l y f o l l o w t h a t F(T)t/TF(0)l-t/T

w i l l be small f o r t E [O,T).

These ideas have been extend-

ed and r e f i n e d c o n s i d e r a b l y i n [Bll;Ftl;Knl;Hml;Pyl]

b u t we w i l l n o t develop

ROBERT CARROLL

90

them in this book. They can be used to study uniqueness and stability questions and there are important applications to second order equations by H. Levine and others. ( 3 ) Lagrange identity method. Let vt = Av with v = 0 on r x [O,T) and then for ut t Au = 0 as in ( + ) (11.3) 0 =

Jot I,

[V(U

tL\u) t

u[vn-Av)]dxdn

v(x,n)u(x,n)l0dn t

=

R

which says 1, v(x,t)u(x,t)dx = In v(x,O)f(x)dx so for v(x,n) = u(x,2t-n) one. 2 Assume now Ilu(x,T)II 5 m and deduce obtains (A*) Ilu(t)l = JR f(x)u(x,2t)dx. that llu(T/2)1 2 mllfll together with llu(T/4)11 5 ~n~/*llfIl~/~, etc. (cf. (*.)I. The new feature here is that Lp estimates are possible from (A*) in the form say (A&) !Iu(T/2)l12 5 MfHpllu(T)Hq. REmARK 11-4- We will indicate here a few other problems following [Lvl-3;

Pyl]. Consider an inverse problem t!i - Au = $(t)f(x) in n x [O,T), u = 0 on r x [O,T), and u(x,O) = 0 where f i s unknown. This is underdetermined so assume we read off data u(x,T) = g(x) as well (and take $ = 1 for simplicity). Given normalized eigenfunctions lpn(x) of AU t AU = 0 as before one has formally (A@) u(x,t) = fnlpn(X)[l - exp(-xnt)]/xn. Hence g(x) = u(x,T) = 1 gnlpn(x) yields from (A@) (A&) f(x) = 1 ; xngnlpn(x)/(l-exp(-AnT)) provided this converges. Note that a small change in g will not necessarily yield a small change in f and this kind of inverse problem seems inherently i l l posed. Another kind of problem is the inverse problem for a Newtonian potential. Thus suppose a star shaped (relative to 0) body with unit density 6 lies in the sphere 1x1 5 R < 1 in say R 3 and on a region z of the unit sphere we can read off the external potential U(x) (aU = 0). Then the problem is to find the shape of B. If the surface of 6 is described by p = f(o,e) then

I:

where r(x,y) is the distance of x to y and y has polar coordinates p,$,e. Various stability type inequalities etc. are indicated in [Lvl] which are also related to uniqueness; we do not pursue the matter here. There are also many i l l posed problems connected with analytic continuation which we omit (cf. [Anl;Lvl-3;Pyll). On the other hand theorems on nonexistence and blowup of solutions of nonlinear first and second order evolution equations will be discussed, but only briefly for lack of space, in Chapter 3 .

We will conclude this section with another example o f Tikhonov regularization for an inverse problem in acoustic waves following the author and L.

Raphael [C6]

I L L POSED PROBLEMS

91

( c f . 56 f o r some background h e r e ) .

T h i s example i l l u s t r a t e s a

number o f i m p o r t a n t techniques and ideas r e l a t e d t o i n v e r s e and ill posed problems and b r i n g s one t o c u r r e n t r e s e a r c h i n c e r t a i n d i r e c t i o n s .

We w i l l

a l s o i n c l u d e h e r e somewhat more mathematical d e t a i l i n p r e p a r a t i o n f o r t h e developments i n Chapter 2. the details.

F i r s t we summarize t h e r e s u l t s and t h e n p r p v i d e

) /A, A = YY = 6 ( y ) , and r e a d o u t

Thus one c o n s i d e r s a 1-D s e i s m i c problem vtt

a c o u s t i c impedance, y = t r a v e l time, w i t h i n p u t v(y,O) v(0,t)

+ g ( t ) where g ( t )

= 6(t)

= ( 2 / n ) / F y(A)CosAtdA.

= (Av

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

( y ) l 5 cYIIAgll_,2y and IIAgll
tained IIAgII

(2; -+

I f o n l y e s t i m a t e s lA;l12 5 E can be ob_-2 ';i i n t h e space L ) one has an ill posed problem o f e s t i m a t i n g

which we s o l v e by use o f r e g u l a r i z a t i o n and s u m m a b i l i t y techniques. 2Y g i n Lm(0,2y) and t h e c o r r e s p o n d i n g An(y) * A(y). The Thus one f i n d s g; m,

-f

c o n n e c t i o n o f s u m m a b i l i t y and r e g u l a r i z a t i o n i d e a s w i t h t r a n s m u t a t i o n f o r mulas v i a t h e spectrum p e r m i t s one a l s o t o e x p l o r e many o t h e r q u e s t i o n s of a p p r o x i m a t i o n i n t r a n s m u t a t i o n t h e o r y and i n r e l a t e d problems o f p h y s i c s and special functions.

T h i s example t h u s serves a l s o as an i n t r o d u c t i o n t o t h a t

c i r c l e o f ideas. G e n e r a l l y we assume A 6 C

1

,A

+

A_ r a p i d l y as y

-f

m,

and 0 < a 5 A 5 B <

m

One can t h i n k o f i n p u t s ( * ) v ( 0 , t ) o r vt(y,O) o r v(y,O) w i t h Y readout (**) G ( t ) = v ( 0 , t ) o r H ( t ) = vt(O,t). The n a t u r a l i n v e r s e problem ( c f . S6).

here i s t o r e c o v e r A ( y ) from t h e readout, g i v e n say an i m p u l s i v e ( 6 f u n c t i o n ) input.

Letting G

6

o r H6 be t h e response t o a 6 f u n c t i o n i n p u t t h e problem 6 o f d e t e r m i n i n g A f r o m G o r H6 i s known t o be w e l l posed ( c f . [C2,3;Syl]). However i n p r a c t i c e t h e i n p u t may n o t be a 6 f u n c t i o n and moreover i t may 6 6 o r H , ob-

n o t e x c i t e a l l f r e q u e n c i e s so t h a t a p u t a t i v e impulse response G

t a i n e d f o r example v i a d e c o n v o l u t i o n , w i l l g e n e r a l l y be an a p p r o x i m a t i o n and t h e inaccuracy w i l l o f t e n a r i s e from s p e c t r a l " d e f i c i e n c i e s " ( c f . [Akl ;Me1 ; Pw1;Rbl;Syll).

T h i s r e c o v e r y o f G6 o r H6 has been s t u d i e d e x t e n s i v e l y , o f t e n

i n t h e c o n t e x t o f d i s c r e t e l a y e r e d media and s p i k e t r a i n a n a l y s i s , and many techniques a r e a v a i l a b l e which we do n o t d i s c u s s here. We w i l l assume h e r e 2 t h a t s p e c t r a l l y l i m i t e d ( i n L ) approximations G*& can be o b t a i n e d and we

w i l l deal w i t h t h e ill posed problem o f d e t e r m i n i n g t h e unknown G6 p o i n t wise.

One assumes e.g. G6 = 6

+

g with g

€

Co

various s i t u a t i o n s are dis-

cussed and t r e a t e d ) , and f o r any f i x e d i n t e r v a l [0,2T],

a sequence g i i s con-

s t r u c t e d (by r e g u l a r i z a t i o n and summabil i t y ) wh ch converges i n Lm(0,2T) t o g.

This i m p l i e s t h a t t h e corresponding An w i l l converge p o i n t w i s e t o t h e un-

known A on [O,T]

.

92

ROBERT CARROLL

One c o n s i d e r s (A+) vtt = Q ( D ) v = (Av ) / A a l o n g w i t h t h e a s s o c i a t e d problem 2 Y Y-u 4 f o r A E C ( A m ) utt = d ( D ) u = uyy A ?A ) " u where 4 = A-'(A')" E Co, and b Y$ u = A v ( r e c a l l k Q v = Q[A v ] from [C2,3]). v h s a t i s f i e s Qp = -A 2P w i t h

-

.

s a t i s f i e s ap' = A%h so t h a t i = ~ ( 0 =) 1 and ~ ' ( 0 =) 0. We w r i t e i h,h 2= -A P, ~ ( 0 =) 1, and i ' ( 0 ) = h = (1/2)A'(O) ( A i s normalized w i t h A(0) = 1

-

and we use h here i n s t e a d o f t h e k o f t e n found i n t h e p h y s i c s l i t e r a t u r e ) . The e i g e n f u n c t i o n expansion t h e o r y o f § 6 takes t h e form

T =:/

h = 1; h,(x)dw; (2/n)dX

+

o(A)dh

relative t o

6 for

(i

=

i A,h

AhPh(x)dx;

= :/

m

and e v i d e n t l y f = lo f v ( x ) d w where do =

f$(x)dx, etc.).

We w i l l use t h e r e s u l t s o r [ M r l ]

p r o p e r t i e s o f dw e t c . ,

here

Thus f o r now assume A E C2 i n o r -

d e r t o develop formulas and r e s u l t s . Then t h e n a t u r a l s i t u a t i o n ( c f . [C2,3; 1 M r l ] ) i n v o l v e s o E L (0,m) (o 0 as h * m ) and (a*) @ ( t ) = ;/ [l - Cosht] (&/A 2 ) = t + 10" [l - CosAt](do/h 2 ) = t + :/ (t-T)ih(T,O)dT E C3 w i t h 9'(0+) -+

+

ih:

Here t h e t r a n s m u t a t i o n s ( c f . [C2,3]) Coshx + 1 (and 9"(0 ) = - h ) . ' ;-1 have k e r n e l s ih = 6, t fh and f h = 6, + r e s p e c t i v e l y (see e. and Bh = =

ch

Then, w r i t i n g t h e

g. 9 1 . 6 f o r h a l f l i n e and f u l l l i n e d e l t a f u n c t i o n s ) . solution o f

(Am)

w i t h u(y,O)

= 6+(y) as u ( y , t )

= ($(y),Cosht)w

( n o t e
l)w = 6 + ( y ) ) we have t h e impulse response readout G6(t) = (l,Cosht)u

(11.5)

Note i n [ M r l ]

6+(t)

=

+ J;

o(h)Coshtdh = 6 + ( t ) + g ( t )

i n d e a l i n g w i t h a" and 9"'one i g n o r e s t h e 6 f u n c t i o n a t 0 so

t h a t i t i s t h e l a s t expression i n (**) which i s i n v o l v e d , i . e . (do = odh) 1 @ " ( t ) = ;/ Coshtdo = g ( t ) , and t h u s i n p a r t i c u l a r g E C when A E C2 w i t h and ih (.A) Lh(t,O) = g ( t ) . L e t us r e c a l l a l s o t h a t t h e t r a n s m u t a t i o n s ih a r e r e l a t e d t o B: ~ o s h x v ( y ) = Iph(y) and B = B - ~ by = &B and B = i h i ' .

ih

-+

Thus x

-+

(00)

~ ( y , x ) = ( ~ ( y ) , C o s h x ) = A-'(y)&+(y-x)

y, Q = (2/n)dX) and y ( x , y )

( y ) ( y = k e r 3, y

-+

x).

r e c a l l a l s o ( 0 6 ) iT;(y,x)

P

=

+

( 6 = kerB,

A-'(y)ih(y,x)

( ~ ( y ) , C o s ~ x=) ~i 5 ( y ) 6 + ( x - y ) + Ch(x,y)AL'

Using some n o t a t i o n s f r o m [ C l l ] = ~-'(y)K~(y,x), ~-'(y)

=

1

-

( c f . §§1.6,2.6)

we

K(y,y), ( i ( y ) , c o s x x )

!J

+ 6 ( y + t ) t k ( y , t ) ( t h e l a t t e r expression can be w r i t t e n a l s o as + k(y,t) when y , t > 0 ) . F u r t h e r 4 = 2D k ( y , y ) and i 5 ( y ) = 1 + 6,(y-t) # k(y,x)dx ( i h = k = i'(y)Kx(y,x) so z ( y , y ) = y-' A J0Y k ( y , x ) d x ) . The Gelfand= 6(y-t)

L e v i t a n (G-L) e q u a t i o n f o r t h e i s s i t u a t i o n a r i s e s from i ( y ) = = Coshy

+ (k(y,t),Cosht)

as ( f o r x < y )

(

ih(y,t),CosAi3

I L L POSED PROBLEMS

93

L e t us f i r s t g i v e a v a r i a t i o n o f a s t a b i l i t y r e s u l t o f t h e a u t h o r and F. Sant o s a ( c f . [C2,3]). Au

u*-u.

We phrase m a t t e r s i n terms o f a s p e c t r a l " i m p e r f e c t i o n " 1 Thus assume IIAulll 5 E ( L norm) w i t h u* % g* % A* e t c . Thus

from Ag = 1 ; (Au)Coshtdh one o b t a i n s (*+) IIAgllm 5 (11.7)

+ M(y,x)

Ak(y,x)

+ t;[Ak(y,*)I(x)

+

Now f r o m (11.6)

E,

I,'

k(Y,S)M(C,x)dS

0

=

One knows t h a t ( 1 + C )-' e x i s t s as an opY 2 an L t h e o r y can a l s o be e n v i s i o n e d ) and i n f a c t

where t y f ( x ) = J0Y A ( x , s ) f ( s ) d s .

-

e r a t o r i n Co ( o v e r [O,y]

-

t * U small a lemma i n [C2,3] says t h a t (1 + C*)-' e x i s t s w i t h l l ( 1 + Y -1 -1Y U(1 + Cy) U 5 c ( c depends on ll(1 + t ) U and !I&* - t I ; t h e - Y Y Y Y Y Y lemma i s s t r a i g h t f o r w a r d from t h e d e f i n i t i o n o f t and we l e a v e t h e p r o o f as Y an e x e r c i s e ) . Hence from (11.7) one o b t a i n s f o r 0 5 x 5 y I ( 1 + Cy*)[Ak(y,-)]

f o r Ilt Y

C;)-'u

< c

(x)l 5 IM(y,x)l

+

l M ( C , x ) l d C where I k ( y , x ) l 5 My on 0 5 x 5 Y.

My$

Now

w r i t e UfUm = s u p f ( x ) f o r 0 5 x 5 y and t h u s on 0 5 x IY, IM(y,x)l 5 ,Y IIAgll 1 IM(S,x)ldS (1/2)1{ [lAg(C+x)l + l a g and ( r e c a l l g i s even) { m, 2Y (I1 II (IC-XI )l]dC = (1/2)1'+' l A g ( n ) l d n 5 llAglll ,2y 5 2yllAgllm ?r L1 (0, X-Y Y Y 192Y 2 y ) ) . Consequently

For IIE* - t II ( < 13 I M ( x , s ) l d s 5 IIAgll < 2yllAgll ) sufY Y 1,2Y m12y f i c i e n t l y small one has an e s t i m a t e (11.8) f o r Ak(y,x), O I x 5 y, f r o m which

tHE0REm 11-5.

IAASL(Y)l 5 FyliAgllm,2y

IIAoll 1 ) .

(<

= 1 +

The p r o o f f o l l o w s from ('(y)

:1 k(y,x)dx, so AA'(y)

= 1; Ak(y,x)dx.

We imagine now IIAul12 (and hence IIAgl12) t o be g i v e n w i t h IIAol12 5

E'.

Here g

(t) = 1 ; Cosxtdu and f o r s i m p l i c i t y i n n o t a t i o n we w i l l w r i t e g ( t ) = ( 2 / n )

1 ; CoshtG(h)dh so t h a t

( m / 2 ) = Cg

=

=

$.

Then we assume lIAZI12 5

E.

One

considers ( c f . [ R a l ] )

where a w i l l be s c a l e d t o [Ral;Tkl]). i n (11.9)

E

2

(a = ke ) and g

%

T, gz

%

;El

L e t g,* be a m i n i m i z i n g f u n c t i o n f o r (11.9);

etc. (cf. here s e t t i n g g = g;

+ rp

( f o r s u i t a b l e q ) a v a r i a t i o n a l argument g i v e s v i a Parseval (llAZl12

= SAgl12(n/2)'/2)

0 = 1:

[ag,*'ip'

1; [ - g,*" + ( g : - g:)]lpdx Hence f o r m a l l y

(om)

g";

+ (g;

-

( t h e t e r m ag;'q/z

-

(l/a)g;

from which (g,*'(O) = 0 ) 0 =

g:)lp]dx

= -(l/a)g;.

t e g r a t i n g we o b t a i n (one can assume g,* and g':

= 0 under s u i t a b l e assumptions).

M u l t i p l y i n g b y CosAx and i n -+

0 a t ,,. w i t h g E ' ( 0 ) = 0 so

94

ROBERT CARROLL

g;y*'Cos

+

g;y*Sinli vanishes)

(11.10)

$;

= G;/(l

+ aX 2 ) f r o m which + aA2)]dA

g,*(t) = (2/a)jOm[i;(A)CosAt/(l

Such g;y* a r i s e i n [Ral] i n t h e c o n t e x t o f r e s o l v a n t s u m a b 1 i t y and many gen-

t:

€ e r a l i z a t i o n s a r e p o s s i b l e . Thus o u r problem now becomes: Given G; = 2 L w i t h IIA~I125 E f i n d a sequence o f g; determined by (11 10) ( w i t h say a = 2 kE ) converging p o i n t w i s e t o g = ( 2 / a ) / y ^c;(A)CosAtdA (and l o c a l l y converging 2 1 i n Lm). G e n e r a l l y one wants t o assume g E Co w i t h E L ? L2 ( t h u s g E L ) and i n p a r t i c u l a r IIg

-

g'$12 5 E ( ~ / T ) ' / ' ( i n what f o l l o w s we w i l l o c c a s i o n a l -

l y use E f o r v a r i o u s m u l t i p l e s o f E ) . As i n d i c a t e d above we w i l l have g € 1 2 1 i f A E C so an a d d i t i o n a l h y p o t h e s i s g E H (Sobolev space) w i l l be made

C

a t times f o r convenience ( d i f f e r e n t arguments a r e used l a t e r and o n l y t h e 2 h y p o t h e s i s g E Co n L i s e v e n t u a l l y used). L e t us n o t e here t h a t f o r g;

*; E

%

f i x e d and llg-g:l12 5 E we c o n s i d e r ILa(g) i n (11.9) on a c l o s e d bounded s e t 1 i n H determined by say 11g'Il; + 11g-g*1I2 < E' + c. Then f o r E and a f i x e d € 2 aa(g) i s a convex weakly l o w e r semicontinuous f u n c t i o n a l on g. B u t is

i:

weakly s e q u e n t i a l l y compact and hence a m i n i m i z i n g sequence gn (J/a(gn) 5 l P i ~ J / ~ ( g+ ) ( l / n ) say) w i l l have a weakly convergent subsequence gn

B'.

We o b t a i n J/a(g,") 5 lim_J,a(gn) so g;

in

procedure, passing t o g";

(em),

i s a m i n i m i z i n g element.

+

g,* i n

The f o r m a l

i s standard i n v a r i a t i o n a l c a l c u l u s and

can be j u s t i f i e d v i a t h e method o f du Bois-Reymond ( c f . 1 5 ) . 1 Given g E H w i t h m i n i m i z i n g elements g,* (determined by (11. 2 2 10) - f o r g: E L w i t h IIAg112 5 E ) one s e t s cl€' 5 a 5 c2€' ( o r c1 = ke ) and 2 i t f o l l o w s t h a t a subsequence g; ( n % an) can be found w i t h g;"; + g i n L 1 (and g: + g weakly i n H ) as an 0 (n + m ) .

EHE0REIR 11.6.

-+

PI LOO^:

F o l l o w i n g [Mzl] $a(g*) = aIIg:'II:

llg-g*112 < € 2 -

E'

< c w h i l e llg:

-

+ aIlg'112. 2 - g*1I2 < € 2 -

dependently o f

for E 1 l y convergent i n H ( n lets

-+

E

+ 0).

0 so g;

-+

E

+

llgt

-

L J / a ( g ) = allg81l2 2 t

:gl:

allg'll; + a/cl so Ilg*'II Hence i n p a r t i c u l a r allg;y*'11; a + c 2 E2 l l g ' l l 22 -< E'?'. Consequently llg;YIH1 5 c" ( i n 5 1 say) and one e x t r a c t s a subsequence g; + go weakE~

-f

m

corresponds t o

an

0

-

n o t e a = a ( € ) and one

F u r t h e r (an % n ) IIg* - gl12 5 llg; - g*ll + IIg* n € 2 2 g in L To i d e n t i f y go w i t h g we n o t e t h a t g;

.

-f

gl12 5

E

+

E :

go i n t h e

Schwartz space 5 ' weakly, hence s t r o n g l y s i n c e 5 ' i s Montel, w h i l e g;"; + g i n 2 L and hence i n k' ( c f . [ C l ] - a s i m p l e r argument c o u l d a l s o be found). It f o l l o w s t h a t go = g.

RrmARK 11.7,

Note t h a t g:

QED g e n e r a l l y w i l l change w i t h

E.

For each

E

one has

POSED

ILL

g:

w i t h IIAg1I2 5

f o r c e s an

-+

95

PROBLEMS

0 one must u s u a l l y change.

When we l e t

and g,* i s determined by (11.10).

E

E

-+

0 b y t h e s c a l i n g ( o r c o n v e r s e l y ) and thus g: 2 5 c one knows t h a t t h e g;

L e t us observe f u r t h e r t h a t s i n c e lg;1112

f o r m an

e q u i c o n t i n u o u s and u n i f o r m l y bounded f a m i l y on any f i n i t e i n t e r v a l (say [0, 2T1). Indeed one has e.g. Ig,*(x) g;(y)l lgGIIdS 5 (X-Y) 1 / 2 [/y-x Ig;’I 2

./y”

-

-

dEl’/2 5 ~ ( x - Y ) ” ~w h i l e 1 g i ( x )

g i ( 0 ) l 5 C”(2T)’.

By t h e A r z e l a - A s c o l i

theorem ( c f . Appendix A ) t h e r e i s t h e n a subsequence, a g a i n denoted by g,; c o n v e r g i n g u n i f o r m l y on [0,2T] t o some g1 E C0(0,2T). E v i d e n t l y g1 must be 2 2T 2 2 g s i n c e IIg* g1112,2T = lo 19; - g1 I dx and g; + g1 i n L ( a l o n g w i t h g; 2 g i n L ).

-

-+

C0R0L:I;ARU 11.8. [0,2T]

Under t h e hypotheses o f Theorem 11.6, g i v e n any i n t e r v a l

t h e r e i s a subsequence g;

g u n i f o r m l y on [0,2T].

-+

We can now a p p l y Theorem 11.5 t o o b t a i n

CHf0REN 11.9.

Assume f o r g i v e n

l I A ~ I 1 2 ( 2 / ~ ) 1 ’ 25

Assume g

E.

p l u s a growth c o n d i t i o n t i o n by (11.10) and

a

-

E

say ;A

E

one can f i n d g:

= Cg E L1 n L 2 ( g E H1

H1 w i t h 2 E L ).

be s c a l e d by cl€’

such t h a t IIAgl12 =

:% :

‘L

A

E

C2

L e t t h e g* be g i v e n v i a r e g u l a r i z a -

5 a 5 c 2 ~ ’ a Then t h e r e i s a subse-

“

quence g; such t h a t IIAgllm -+ 0 (Ag = g ; - g ) and hence lAA4(y)1 i cy Y2Y IIAgllm + 0. Y2Y REmARK 11-10. I n t e r k of computations t h e f o l l o w i n g o b s e r v a t i o n i s u s e f u l . We have

11s: -

gzll;

w h i l e a subsequence g; (n

‘L

-

52‘~’and lg:

working from some b a s i c c n -+

+

gl12 5 (*C+l)E f o r a l l g:

0, t h e whole sequence g:

g u n i f o r m l y on say [0,2T].

so t h a t i n f a c t , 2 ‘L a ) i n L n It f o l l o w s t h a t g i + g +

g (a

an) u n i f o r m l y on [0,2Tl

here).

f o r any s e l e c t i o n o f g; (see [Col] f o r d e t a i l s 2 Indeed g i v e n such a g; (+ g i n L ) one shows f i r s t t h a t g; g at -+

any to. On t h e o t h e r hand t h e s t a n d a r d p r o o f o f t h e A r z e l a - A s c o l i theorem by a diagonal process ( c f . [Col]) quence converges u n i f o r m l y .

shows t h a t any p o i n t w i s e convergent se-

Hence t h e r e i s no “ s e l e c t i o n ” process r e q u i r e d

i n Theorem 11.9 - one can s i m p l y l e t cn + 0 and choose g; v i a (11.10) w i t h say an = k E i . Then Ag = g* - g 0 u n i f o r m l y on say [0,2y] and IM4 I -t 0. n -+

L e t us c o n s i d e r now t h e f o l l o w i n g example (based on some c a l c u l a t i o n s o f A. Sarwar ( c f . r e f e r e n c e s i n [C6]).

EXAIRPCE 11.11. A(x,t)

L e t g;

= e x p ( - n t ) so A ( x , t )

= exp(-nt)Cosh(nx)

for x

<

t.

= exp(-nx)Cosh(nt)

(&*) k(y,x) + exp(-ny)Cosh(nx) + lox k(y,c)exp(-nx)Cosh(nc)dc exp(-nc)Cosh(nx)dc = 0.

f o r x > t and

The G-L e q u a t i o n (11.6) becomes ( x < y )

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

-

+ k(y,c) n ( n t 1 ) k = 0.

96

ROBERT CARROLL

2

S e t t i n g n ( n t 1 ) = p we o b t a i n k ( y , x ) = z(y)Cosh(px)

x).

Put t h i s i n (&*) w i t h x

+

( n o t e k(y,x)

i s even i n

0 t o g e t z ( y ) [ l + :1 exp(-nS)Cosh(pS)dS]

=

I t f o l l o w s t h a t z ( y ) = -[n/(pSinh(py) + nCosh(py)] and from A)& k(y,x)dx we o b t a i n A 4 ( y ) = 1 [nSing(py)/p(pSinh(py)+nCosh(py)]. 2 We n o t e a l s o t h a t ( w ) u: = ( 2 / a ) $ exp(-nt)CosAtdt = 2n/a(A2 + n ) (and 2 2 = as a = (a/2)u; = n/(A + n ) = Cg). We t h i n k here o f g*, = g; and

-exp(-ny). (y) = 1 t

-

$

p e r t u r b a t i o n o f 0; i . e . we assume A.

zE

z:

2:

= 1 w i t h cro = 0, t h e s o l u t i o n f o r which

i s (dw = (2/a)dA) uo = (CosAt,CosAy)u = 6+(y-t).

Thus k ( y , t ) = 0, A(y,x)

0, e t c . f o r t h e base problem A,

;C [ndA/(A2 + n 2 ) = dC/(C2 + 1 ) 2 so E L1 n

Now 1 ; z;(A)dA

= 1.

=

=

z:

n/2, f;;i2(h)dA = 1 ; [n2dA/(A2 + n 2 ) 2 ] = (l/n)C; 2 2 1 L , y; * 0 i n L , and ?; f 0 i n L . C l e a r l y g* = e x p ( - n t ) + 0 i n t h e space 2 n L b u t g; 76 0 i n Lm ( g i ( 0 ) = 1 and g; + 0 f o r t > 0 ) . We have a l s o kn(y,x) = -[nCosh(px)/(pSing(py) + nCosh(py)] so as n + m y p m y and ( 0 )kn(y,x) +

+

-[n/(p+n)]exp(p(x-y))

above

(u)i:(y)

1

?J

+

-

v i a IIAgtlm e s t i m a t e s . g* = g; (11.11)

g;

n/p(n+n)

y and kn(y,y)

<

+

+

-1/2.

S i m i l a r l y from

1 so An goes t o t h e r i g h t l i m i t b u t

not

For t h e r e g u l a r i z e d o b j e c t s i n (11.10) we t a k e g = 0,

~9

= exp(-nt),

0 for x

= gCn

=

;; = A;;

A

g:

= g;

= n/(A

2

2

t n ),

(2/n)lom [nCosAtdA/(A 2 +n2 ) ( l + a A 2 ) ]

=

[n/(l

4

g;,

-

an2)][(e-“/n)

-

=

dae - t/ J“]

Now i n t r o d u c e t h e s c a l i n g from Theorem 11.9 i n t h e form

a =

where from above IIul12 = c / h g i v e s 2 p l i c i t y k so t h a t a = k c / n = l / n .

Thus t a k e f o r sim-

exp(-ht/h)]

so t h a t i n f a c t g,*

n

E

i n t h e form c / h .

kc2 f o r example

Then g; = [n/(l-n)][(exp(-nt)/n) n ( t ) * 0 u n i f o r m l y on [O,m). Evidently

t h i s sequence can be used i n Theorem 11.9 and one o b t a i n s 44%(y) e s t i m a t e IIAgll

-+

2Y The l a s t example can be r e f i n e d as f o l l o w s .

EXAMPLE 11.12. where Y = ,’with

0 = k(y,x)

B < 1/2.

0 v i a an

+ yexp(-ny)Cosh(nx)

t

C$ k(y,C)yexp(-nx)Cosh(nS)dS

-

It follows t h a t 0 = kxx

where q2 = y n ( n t 1 ) .

A l i t t l e c a l c u l a t i o n gives

[l+[y/(q2-n2)][-n

i4 1 %

Take g;

=

ye

-nt

Then A i s m u l t i p l i e d by y and (&*) becomes (&*)

yexp(-nC)Cosh(nx)dS.

(11.12)

+

0.

t e-”(qSinh(qy)

[(q2

-

t k(yy5) y n ( n + l ) k and k = B(y)Cosh(qx)

+ nCosh(qy))]

n2)y/qy(q + n ) ]

+

o

Hence r e g u l a r i z a t i o n i s d e f i n i t e l y needed here and we o b t a i n

= -ye-”;

I L L POSED PROBLEMS

97

z;

so t h a t g,* + 0 u n i f o r m l y s i n c e B < 1/2. Note h e r e a l s o t h a t 2 ; :*‘dX = n Z B - l c + 0 ( B < 1/2); n ) so ;/ ECdX = n + m w h i l e 1 n + m while g$t) + 0 f o r t > 0.

= nB+l/(x2 t

however g$O)

1 1 L e t us now t a k e g E Co ( n o t C o r H ) which s h o u l d correspond r o u g h l y t o A 2 E C1 ( n o t C ). Then one i s i n t h e c o n t e x t o f vtt = Q(D ) v w i t h v ( y , t ) = Y c9X(y),CosXt)u. The same formulas h o l d f o r g e t c . ( v i z . g ( t ) = 1 ; OCoshtd~) and one can use t h e same G-L e q u a t i o n (11.6) %

k(y,t)

but the

6 operator

+ A-’(y)k(y,t),

6,(y-t)

(i.e.

itA-’k(y,t),

kh(y,t)

‘t

i s n o t used), B(y,t) = ( q X ( y ) , C o s x t ) p = A-’(y)

etc.

One can now s i m p l y t a k e t h e g z o f (11.10) as

an ad hoc formula, t h o u g h t o f perhaps as a r i s i n g i n s u n a b i l i t y t h e o r y , w i t h IA:I E L 2 , g E L 2 n Co, and Y E L 1 n L 2 2 -< E as b e f o r e and say g: E L2, A c t u a l l y t h i s use of g,* f r o m (11.10) r e p r e s e n t s an a p p r o x i m a t i o n t o g by H1

E:

.

f u n c t i o n s ;g; however t h e arguments o f Theorem 11.6 do n o t a p p l y s i n c e g 1 H The procedure o f [Ral] i n v o l v e s now t h e f o l l o w i n g program. One w r i t e s

.

as i n (11.10) (11.14)

g,*(t) = (2/a)/”[ ~ ~ C o s X t d h / ( 1 + a X 2 ) ] ; 0

g,(t)

=

(2/n)lom [ Z C o s x t d x / ( l + a i 2 ) ]

Then w r i t e a t a p o i n t o f c o n t i n u i t y o f g (hence f o r a r b i t r a r y f i x e d t )

z; -

g ( t ) l 5 l g a ( t ) - g ( t ) l + I ~ , ( t ) l where ( A ~= 7 A a ( t ) = g,*(t) [AxCoshtdX/(l+ah 2 )]. Then u s i n g e s t i m a t e s on Green’s f u n c -

-

Is,*(t) g,(t)

-

= (2/n)/;

2

t i o n s e t c . (and s c a l i n g a as b e f o r e

-

a = kE ) one shows t h a t g i v e n 6 t h e r e

) t h a t llAhl12 = IIAC;II e x i s t s ~ ( 6 such

<

E

2 -

implies Ig,*(t)

-

g ( t ) l 5 6 (see be-

low f o r Green’s f u n c t i o n s ) . L e t now

k,

be t h e Green’s f u n c t i o n f o r

Ga(x,s) = - 6 ( x - s ) .

-D2 +

x where a2 =

l/a.

= g,*(t)

g,(t)

-

-

g,(t)

(+m)

[D2

-(l/a)]

= (l/a)exp(-as)Cosh(ax)

=

for s >

Thus i n (11.14) one has =;(

Cg, Z z = Cgz) (6m) g,* = ( l / a )

g,

L e t us n o t e t h a t

= (l/a)J;

can be w r i t t e n f o r a l l x,s + exp(-(x+s)/Ja)]

so t h a t

T h i s can be w r i t t e n down e x p l i c i t l y as ( 6 + ) 6,(x,s)

( l / a ) e x p ( - a x ) C o s h ( a s ) f o r s < x and i,(x,s)

: J Ga(x,s)g:(s)ds;

(l/a)

Ga(x,s)g(s)ds.

( 0 5 x,s 5

a)

6,

i n (6+)

as ~ , ( x , s ) = (Jor/Z)[exp(-lx-sl/Ju)

and we r e c a l l t h a t two e s t i m a t e s a r e needed, one on A a ( t ) g i v e n by A a ( t ) = ( l / a ) / , G,(t,s)[gZ(s)

g ( t ) = A,(t)

-

g(s)]ds,

where we r e c a l l from (11.14) t h a t g,(t)

and one on

= (2/n)

98

:I

ROBERT CARROLL

(Cg)[CosAtdh/(l+aA2)].

Now f o l l o w i n g [Ral]

:I

k$(t,s)ds = O(a3/').

Hence

lA,(t)l 5 (l/a)llAl120(a 3/4 ) llAl120(a-1'4). Consequently f o r f o r A = g;-g, 5 C E -114 ~ < ca1/4 + 0 as a IIAl12 5 E and a s c a l e d by a = k c 2 one has lA,(t)l +

0.

T h i s e s t i m a t e i s independent o f t and can a l s o be o b t a i n e d d i r e c t l y

As f o r Aa = g a ( t ) - g ( t ) , s i n c e we a r e

from t h e s p e c t r a l formulas ( c f . [C6]).

o n l y i n t e r e s t e d i n Lm t y p e convergence on f i n i t e i n t e r v a l s p i c k t E [0,2T].

so t h a t f o r y g i v e n t h e r e exi s t s n (n independent o f t ) such t h a t [ A t 1 5 II i m p l i e s lAgl & y . I n a d d i t i o n 0 we want n = n ( a ) t o be chosen so t h a t as a + 0 (4.) exp(-Zna - 1 / 2 i a - 1 / 4 Then g i s u n i f o r m l y continuous on say [0,2T+1]

-f

2T]) Aa(x) = (l/a)I: 6a(x,s)[g(s)-g(x)]ds. I . For e s t i m a t e s c o n s i d e r ( c f . [ R a l ] )

-

+

g,(X)

2 [g(x)+n](l/a)II

g,(x)

L e t I = [ x - Q , x + ~ ] and K = [O,m) (46) ga(x) = (l/a)JI

ia(x,s)g(s)ds

Then one has f o r example

( l / a ) I K Ga(x,s)g(s)ds.

(11.15)

i a ( x , s ) d s = 1 and hence ( x E [ O ,

Now we n o t e t h a t ( 4 0 ) (l/a)I:

(see below).

i a d s + (1/a)llgl12(/K k:(x,s)ds)l/'

'> [ g ( X ) - n I ( l / a ) J ~ i a d s - ( 1 / a ) " g r 2 ( J K f ( x , s ) d S ) 1/ 2

Some c a l c u l a t i o n s which we o m i t ( c f . [C6]) y i e l d l ( l / a ) J I IK i:(x,s)ds

5 &3/2exp(-Zn/Ja).

(11.15) i n t h e form ( x

2.

Then t o e s t i m a t e A a ( t ) = g,(t)-g(t)

t E [0,2T])

5 yc

llgl12(IK k z ( t , s ) d s ) 1 / 2

112 ( t ) l

5 (y/a)II

t IIgll 2& - ' / 4 ~ x p ( - E n / J a ) .

do n o t depend on t

t h e bounds c and

ka(x,s)dsl

2,

x.

n i m p l i e s 1Ag1

< y; a l s o f o r

-1/2+ 0 (thus n

a-1/4exp(-~na e.g.,

n

CY

WP

use

+ (l/a) We n o t e e x p l i c i t l y t h a t k,(t,s)ds

Now f o r t E [0,2T],

g i v e n a r b i t r a r i l y small and ~ ( y e) x i s t s ( r e l a t i v e t o [0,2T+l]) <

5 c and

y can be

so t h a t l A t l

g i v e n ( a + 0 ) one r e q u i r e s i n a d d i t i o n t h a t

= O ( C ~ ' / ~would ) do f o r example).

Typically

w i t h T f i x e d , n i s g i v e n v i a c o n t i n u i t y and we want say n = a 1 / 4 and

= kE2 so p i c k

E

=

al"

=

n2.

T h i s g i v e s ( c f . [Ral;C6])

L e t g E co n 'L ( F E L' n L'), I I A ~ I 5 I ~E , g* = g; L', ;j* L 2 , and c o n s t r u c t g, and g,* as i n (11.14) ( w i t h a = kc 2 ). One w r i t e s Aa ( t ) = g;(t)-ga(t) and A a ( t ) = s,(t)-g(t), so t h a t I g * ( t ) - g ( t ) l i lA,(t)l + IA,(t)I. We assume E = E, 0 and t h e n lA,(t)I 2 ca!,j4 0. For t E [0,2T]

CHEBREI 11.13, E

-f

-f

choosey, + 0 w i t h n n ( y n ) (determined by u n i f o r m c o n t i n u i t y o f g on [ 0 , 2 T + l l ) Then IA,(t)l 0 u n i f o r m l y on [0,2T] r e q u i r e d a l s o t o s a t i s f y say nn = an1 / 4

.

-f

and hence llg;(t)-g(t)llm + 0 which p r o v i d e s us w i t h u n i f o r m approximations 1 ,2T t o g on [0,2T] by H f u n c t i o n s g; based on t h e p o l l u t e d d a t a 3;. The assoc i a t e d A n ( t ) then converge t o A ( t ) f o r each f i x e d t E [O,T].

99

CHAPTER 2 SCATTERING THEORY AND SOLITONS

1. ZNClEe)DLICCZBN. The idea of scattering has an obvious meaning in t h a t e.g. obstacles "scatter" incident 1ight or sound waves o r "atoms" s c a t t e r incident electromagnetic waves or other particle beams. There are two main questions which a r i s e , direct and inverse. The direct problem seeks t o determine the nature of reflected and transmitted waves when the medium or obstacle i s known. The inverse problem seeks t o determine the nature o r shape of the medium o r obstacle from knowledge of i t s "scattering" effect on incident waves. In seismic prospecting for example one attempts t o determine the nature of rock layers by t h e i r reflection of incident acoustic waves or in quantum mechanics one:?tries to determine the nature of an atomic potential from knowledge of the atoms effect on incident plane waves. In mathematical terms a number of physical inverse problems reduce t o an inverse Sturm-Liouv i l l e problem of determining the coefficients of a differential equation when sufficient spectral (frequency) information i s known. There are many questions that can be studied and we refer t o the bibliography of a r t i c l e s cited in the text for philosophy and physical background. In the f i r s t sections of this chapter we want t o bring together and organize a l i t t l e several points of view with an eye toward a semi-unified treatment of scattering, transmutation, and operator theory ( i n contexts where t h i s makes sense of course). More precisely, there are several theories involving related objects which serve very well in various contexts and occasiona l l y interact; we feel that a more profitable interaction could be motivated and achieved by indicating the relations between the methods, results, and objects in these theories, o r a t l e a s t displaying some of the related formulas. Thus we have in mind particularly, as a starting point, 1 - D scattering on (-m,m) where an operator theoretic description in Hilbert space i s described in Section 2 (whose main object seems t o be the discussion of asymptotic completeness) whereas another approach, based on Gel fand-Levitan

100

ROBERT CARROLL

(G-L) and Marzenko (M) equations and techniques i s used t o d i s c u s s and s o l v e t h e i n v e r s e problem ( d e s c r i b e d i n Sections 3-4).

Underlying both o f these

i s a t r a n s m u t a t i o n t h e o r y which serves i n many r o l e s ( c f . [C2,3,13;Chl;Lal; Fall).

Some o f t h e formulas i n t h e o p e r a t o r t h e o r e t i c approach d i s p l a y e.g.

t h e wave o p e r a t o r s i n terms o f c e r t a i n s p e c t r a l p a i r i n g s o f g e n e r a l i z e d eigenf u n c t i o n s $+ - along - ( a r i s i n g f r o m a Lippman-Schwinger e q u a t i o n ) and t h e $+, w i t h o t h e r 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 , c h a r a c t e r i z e d by d i f f e r e n t boundary o r a s y m p t o t i c c o n d i t i o n s , a r i s e a l s o i n t h e i n v e r s e t h e o r y (sometimes i n v o l v ing a spectral pairing).

I n t h e s p i r i t o f [C2,3]

( c f . a l s o 51.6) we w i l l

o f t e n t h i n k o f developments i n v o l v i n g s p e c t r a l p a i r i n g s o f general i z e d eigenf u n c t i o n s as p a r t o f t h e s t r u c t u r e o f general t r a n s m u t a t i o h t h e o r y and t h u s connections i n t h i s c o n t e x t between d i f f e r e n t approaches can be e x h i b i t e d . On t h e o t h e r hand t h e h a l f l i n e t r a n s m u t a t i o n t h e o r y f o r example can be b u i l t up almost e n t i r e l y i n terms o f s p e c t r a l p a i r i n g s o f v a r i o u s 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 so we w i l l n a t u r a l l y want t o develop t h i s theme i n v o l v i n g s p e c t r a l measures and 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 .

We w i l l a l s o d e s c r i b e some o f

t h e corresponding o p e r a t o r t h e o r e t i c and i n v e r s e t h e o r i e s i n 3-D and a t l e a s t i n d i c a t e some s p e c t r a l p a i r i n g s i n t h i s c o n t e x t as w e l l .

Another theme i s

t o c o n s i d e r t h e o p e r a t o r t h e o r e t i c c h a r a c t e r i z a t i o n o f G-L and M e q u a t i o n s i n [C3;D2]

f o r example ( c f . a l s o [C2;Fal;Kyl;Ms3])

and t r y t o r e l a t e t h i s t o

o p e r a t o r formulas o b t a i n e d from t h e H i l b e r t space p o i n t o f v i e w i n s c a t t e r i n g t h e o r y (we do n o t pursue t h i s h e r e however).

There a r e a l s o s t i l l a t

l e a s t two more p o i n t s o f view which i n t e r a c t w i t h a l l t h i s .

One a r i s e s f r o m

t h e s t u d y o f o p e r a t o r theory, f i l t e r i n g , f u n c t i o n theory, and s c a t t e r i n g f o r t r a n s m i s s i o n l i n e s e t c . ( c f . [C3,10,13,11,14;Bal;D2-4])

and t h e o t h e r i n v o l -

ves t h e theme b f L a x - P h i l l i p s s c a t t e r i n g ( c f . [C14;D2;Lx2-4]).

Some o f these

ideas a r e deveioped i n t h e t e x t . L a x - P h i l l i p s s c a t t e r i n g , which we do n o t pursue, l e a d s t o nonEuclidean wave equations as i n [Agl ;Fa2;Lx3,4;Vel],

and

i t i s p r o b a b l y here t h a t one should l o o k f o r meaning and u n i f i c a t i o n .

To e l a b o r a t e a b i t more, t h e s t u d y o f wave o p e r a t o r s and a s y m p t o t i c completeness employs m a i n l y Hi1 b e r t space methods, r e s o l v a n t formulas, s p e c t r a l decompositions o f s e l f a d j o i n t o p e r a t o r s , e t c . b u t does n o t o f t e n d i s p l a y t h e wave o p e r a t o r s W, Shl]).

i n terms o f t h e $+- i n d i c a t e d above ( c f . [ A h l ; B j l ; S j l , 2 ; 3 ( f o l l o w i n g [Chl;C3;

On t h e o t h e r hand t h e i n v e r s e t h e o r y i n R o r R

Ky1;Mrl ;Ms2,3;Lal;Dfl;Fal

;Mnl;Nw2,3;Rs1,2;Tz2;Yel,2])

uses t h e s c a t t e r i n g

and/or o t h e r f u l l l i n e g e n e r a l i z e d eigenm a t r i x i n s p e c t r a l form, t h e $+, f u n c t i o n s ( a l o n g w i t h c e r t a i n t r a n s m u t a t i o n o p e r a t o r s i n some t r e a t m e n t s ) .

SCATTERING THEORY

101

One n o t e s a l s o t h a t s o l i t o n t h e o r y uses t r a n s m u t a t i o n methods and s p e c t r a l H a l f l i n e problems on [0,

i n t e g r a l s as w e l l ( c f . [Aol;Cjl;N11-3;Nvl;Fa3]). m)

a r e t r e a t e d m a i n l y by t r a n s m u t a t i o n methods and an assortment o f 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 e i n [C2,3,10,11,13;Chl

;Bal ;Fa1 ;La1 ;Msl ; M r l ;Nw21.

Some c o n n e c t i o n s between t h e v a r i o u s e i g e n f u n c t i o n s and formulas f o r h a l f l i n e and f u l l l i n e problems a r e g i v e n i n [C2,3,10,11,13,18;Ms1-4]

and p u r e l y

f r o m an o r g a n i z a t i o n a l p o i n t o f view t h i s o f t e n leads t o new r e s u l t s and techniques f o r f u r t h e r development o f t h e theory, p l u s a deeper understandi n g o f known formulas.

I n terms o f a p p l i c a t i o n s o f course t h e s t a n d a r d con-

n e c t i o n o f i n v e r s e s c a t t e r i n g t o KdV and s o l i t o n t h e o r y i s o f c o n t i n u i n g i n t e r e s t ( c f . [Aol ; C j l ;Chl ;Msl,4;Mrl,2;

N11-3; Nvl ;Fa31) and t h i s i s developed

i n Sections 8-11 as a m a i n l i n e theme i n mathematical p h y s i c s . I n summary, t h e r e w i l l be a medley o f themes and s t r u c t u r e s i n s c a t t e r i n g theory, a l l b a s i c a l l y about t h e same t h i n g s f r o m d i f f e r e n t p o i n t s o f view. Exposure t o t h e v a r i o u s languages seems t o make t h e use o f any one more meaningful.

That t h e " u n i f i e d " examination o f such r e l a t e d ideas m i g h t be

i n t e r e s t i n g has o f course occured t o people w o r k i n g on s c a t t e r i n g t h e o r y . The m a t t e r came up e.g.

i n a c o n v e r s a t i o n w i t h P. Lax l o n g ago and my p r e -

l i m i n a r y e f f o r t s a t " u n i f i c a t i o n " l e d t o t h e t r e a t m e n t developed here. There i s some coherence, and we f e e l a l o t o f pedagogical value, b u t more i n g r e d i e n t s and o r g a n i z a t i o n a r e s t i l l needed ( c f . remarks above); t h e n one hopes t h a t a b s t r a c t i o n w i l l n o t make i t a l l u n r e c o g n i z a b l e . 2.

SCA&CERING CHE0RM.

I (0PERAC0R CHE0RM).

We begin w i t h some o p e r a t o r

t h e o r e t i c n o t i o n s and base t h e d i s c u s s i o n on [ S h l ] where any m i s s i n g d e t a i l s can be found.

As we go a l o n g a number o f r e s u l t s on s p e c t r a f o r example w i l l

s i m p l y be mentioned w i t h no a t t e m p t a t p r o o f o r understanding.

We want t o

give a f l a v o r o f t h i s information since i t i s important but our i n t e r e s t s l i e i n o t h e r d i r e c t i o n s (e.g. given i n t h a t context.

toward i n v e r s e problems) and more d e t a i l s a r e

The d i s c u s s i o n here s h o u l d p r o v i d e t h e o p e r a t o r t h e o -

r e t i c p o i n t o f view and we have t r i e d t o g i v e enough d e t a i l t o make t h i s a reasonably i l l u m i n a t i n g survey. We c o n s i d e r 1-D problems and r e c a l l p 'L 2 -ihD w h i l e H = p /2m + U = Ho + U i s a s t a n d a r d H a m i l t o n i a n and t h e SchrodX

i n g e r e q u a t i o n i s ifiij' = Hij ( c f . 51.8).

We w i l l need t o r e f e r t o t h e spec-

t r a l theorem f o r s e l f a d j o i n t o p e r a t o r s i n H i l b e r t space so we s t a t e i t b u t r e f e r t o [Ail;Rcl;Rl;Wdl]

f o r proofs.

L a t e r more d e t a i l w i l l be g i v e n f o r

s e l f adjoint realizations o f differential

expressions and t h e i n g r e d i e n t s

f o r p r o o f o f t h e s p e c t r a l theorem a r e i n d i c a t e d .

102

ROBERT CARROLL

CHE0REll 2.1.

L e t A be a s e l f a d j o i n t o p e r a t o r i n a H i l b e r t space H.

Then

t h e r e e x i s t s a f a m i l y o f ( s e l f a d j o i n t ) orthogonal p r o j e c t i o n s E ( A ) , such t h a t ( 1 ) I f A < 5 t h e n E ( A ) E ( c ) h as

E

-f

0'

( 3 ) E(A)h

-f

0 as A

--m

-f

D ( A ) i f and o n l y i n 1 A2dUE(A)hA2 (Ah,k)

=

/f Ad(E(A)h,k)

function f ( A ) =

<

= E(A) ( 2 ) For any h E H E(A+E)h

and E(A)h m

-f

( 5 ) For h

h as A E

-f

-m

E(A)

-f

f o r any h ( 4 ) u

E

D ( A ) and k E H one has

( 6 ) For f ( A ) any (say measurable) complex valued

f(A)dE(A) i s d e f i n e d

[f lf(A)12dllE(A)hi2 < /f f(A)g(A)d(E(A)u,v)

A E R,

on t h e s e t D ( f ( A ) )

= { h E H;

(7).For u E D ( f ( A ) ) and v E D(g(A)), (f(A)u,g(A) ) = (8) f ( A ) * = F(A)dE(A) ( 9 ) Ifh(A) = f ( A ) + g(A) t h e n h(A) i s an e x t e n s i o n o f f ( A ) + g(A) (10) I f h(A) = f ( A ) g ( A ) then h ( A ) i s an m

LI

e x t e n s i o n o f f ( A ) g ( A ) ( 1 1 ) I f u E O(g(A)), pointwise then fk(A)u

-+

l f k ( A ) l 5 g(A), and f k ( A )

f ( A ) u (12) I f I C (--,A]

-f

f 1)

then E ( A ) = EI(A).

We r e c a l l from §1.8 t h a t quantum mechanical observables a r e a s s o c i a t e d w t h s e l f a d j o i n t o p e r a t o r s , e.g. p (%/i)Dx, H -(I7 2/2m)Dx2 + U(x), e t c . L e t Q

Q ,

us i n d i c a t e some b a s i c f a c t s about s e l f a d j o i n t o p e r a t o r s i n H i l b e r t space w i t h o u t a l o t of f o r m a l i t y .

F o l l o w i n g [ S h l ] we assume Theorem 3.1 and use

i t t o establish various facts.

These a r e o r g a n i z e d i n a s e r i e s o f remarks.

REmARK 2.2. A E C i s i n t h e r e s o l v a n t s e t p ( A ) o f a closed (densely d e f i n e d ) o p e r a t o r A: D(A) C H H ift h e r e e x i s t s a bounded o p e r a t o r R, E L(H) such -f

t h a t (A-A)RAh = h f o r a l l h E H w h i l e RA(A-A)v = v f o r v E O(A). plement o f p ( A ) i s c a l l e d t h e spectrum a ( A ) .

R,

e r a t o r and i f A i s s e l f a d j o i n t a l l nonreal numbers a r e i n p ( A ) . t h i s l o o k a t g(A) = (z-A)-'

and use Theorem 2.1.

g(h) = (i0-h)-',

-

denote t h e i d e n t i t y ) .

xI(A)

=

cf.

[Shll).

f o r A.

E Iset

E I - t h e n g(A)(A - A ) = 1 - x ( A ) 0 I 1 and g(A)(Ao-A) = 1 where we o f t e n use 1 t o

F u r t h e r i f A.

open i n t e r v a l I c o n t a i n i n g A.

-

-

I,and g(A) = 0 f o r A

A E

w i t h (Ao-A)g(A) = 1

To see

S i m i l a r l y i f I C R i s an

= 0 then IC p(A) (exercise

open i n t e r v a l and EI = xI(A)

The com-

i s c a l l e d t h e r e s o l v a n t op-

E p(A),

w i t h xI(A)

h0 E R,

then t h e r e e x i s t s an

= 0 so t h a t p ( A ) i s open ( e x e r c i s e

There a r e v a r i o u s types o f s p e c t r a l p o i n t s a s s o c i a t e d w i t h

general o p e r a t o r s b u t f o r s e l f a d j o i n t o p e r a t o r s t h e r e i s no " r e s i d u a l spectrum" and o(A) can be decomposed i n t o d i s c r e t e spectrum Do(A) ( i s o l a t e d e i g envalues o f f i n i t e m u l t i p l i c i t y ) , continuous spectrum &(A), eigenvalues, o r eigenvalues o f

m

multiplicity.

The l a t t e r t h r e e t y p e s o f

p o i n t s c o n s t i t u t e t h e e s s e n t i a l spectrum oe(A) ( c f . below). f o r convenience t h a t t h e r e i s

l i m i t points o f We w i l l assume

no s i n g u l a r continuous spectrum ( c f . [ B j l ; R l ] )

and one shows t h a t ( c f . [ S h l ] )

MEaREIR 2.3.

L e t A be s e l f a d j o i n t .

Then A E o ( A ) i f and o n l y i f t h e r e i s

103

SCATTERING THEORY

a sequence un

E

D(A) such t h a t IIunll

1 and II(A-A)unII

0 as n

-+

-+

-.

The un i n Theorem 2 . 3 a r e c a l l e d approximate e i g e n v e c t o r s and t h i s c h a r a c t e r i z a t i o n o f o(A) i s o f t e n q u i t e useful ( t h e proof i n [Shl] i s i n s t r u c t i v e 2 2 b u t we o m i t t h i s h e r e ) . For example i f $, = cnexp[-n (x-A) / 2 ] (cn = nl"/ t h e n l l ( ~ - A ) $ ~ I+l 0 and A

v1I4)

E a(A)

f o r any r e a l A ( h e r e A n, x ) . S i m i l a r l y 2 2 (k-A) / 2 ] one has II (k-A)

f o r t h e momentum o p e r a t o r p w i t h p n ( k ) = cnexp[-n

0, b u t II (p-A)$ 1I = - f ~ l l ( k - A ) p ~ lfl o r $ = F-'pn (F h e r e uses ~/v'~IT n 2 2 Zn s y m n e t r i c a l l y ) . For H = p /2m = -(I /2m)D c o n s i d e r H, - X = (1/2m)(p t 0 x2 2 42mA)(p - J2mA) (A 2 0 ) . For ~ ( x =) cexp(-x ) w i t h / p dx 1 t a k e $,(x) = 2 2 n - 1 / 2 q ( x / n ) e x p ( i y x ) where A = 5 y /2m. Then a l i t t l e c a l c u l a t i o n shows t h a t

qn(k)ll

+

2

Il$,ll

= 1 w i t h (Ho-x)$,

0 and A

+

E

o(H0) f o r A 2 0.

There a r e e v i d e n t l y no

eigenvalues f o r Ho and A < 0 r e a l c l e a r l y l i e s i n p ( H O ) by t h e s p l i t t i n g above.

REmARK

2.4,

One wants now t o deal ( i n one l i f e t i m e ) w i t h t h e spectrum o f H

= Ho t U f o r s u i t a b l e p o t e n t i a l s U and we w i l l mention some u s e f u l c r i t e r i a

i n 1 and 3 dimensions (see [Bjl;Ka2;R1;Shl;SjlY2] enormous l i t e r a t u r e on t h i s t o p i c ) .

-

f o r a survey

t h e r e i s an

F i r s t we can d e f i n e t h e e s s e n t i a l spec-

t r u m ae(A), A s e l f a d j o i n t , v i a t h e c o n d i t i o n i n Theorem 2 . 3 , namely: oe(A) i f and o n l y i f t h e r e e x i s t s $, C D(A), ll$nll = 1, $, subsequence, and (A-A)$, a r e g i v e n i n [Shl];

x0

+

0 (oe(A) i s c l o s e d ) .

f o r example i f

E oe(A) (and c o n v e r s e l y i f A.

In particular

(A)

x0

E o ( A ) i s n o t an i s o l a t e d p o i n t t h e n

E o ( A ) i s n o t i n oe(A) then i t i s i s o l a t e d ) .

i s an e i g e n v a l u e o f f i n i t e m u l t i p l i c i t y when A.

E o(A)/oe

( e x e r c i s e ) and t h e e s s e n t i a l spectrum c o n s i s t s o f t h e c o n t i n u o u s s p e c t -

rum, t h e eigenvalues o f ues.

xo

has no convergent

Other e q u i v a l e n t c r i t e r i a

m

m u l t i p l i c i t y , and accumulation p o i n t s o f e i g e n v a l -

We w i l l exclude a p r i o r i eigenvalues o f

m

m u l t i p l i c i t y and s i n g u l a r

continuous spectrum i n general ( c f . [ B j l ; R l ; S j l , Z ] d i t i o n s and c f . a l s o Remark 2.9).

f o r d i s c u s s i o n and con-

C o n d i t i o n s can a l s o be g i v e n t o l i m i t t h e

number o f eigenvalues t o a f i n i t e s e t ( c f . [ R l ] ) .

L e t us g i v e some examples

-

o f c o n d i t i o n s on U which produce i n f o r m a t i o n about Q(A) s i t u a t i o n s and f o l l o w [ S h l ] f o r now. w i t h D(H) = D(Ho) and U ( x )

2b

Thus ( * ) I f H = H

0

we s t a y w i t h 1-D

+ U i s self adjoint

f o r a l l x t h e n (-m,b) C p(H).

(A d i s c u s s i o n

of c o n d i t i o n s on U which generate s e l f a d j o i n t H i s g i v e n l a t e r . ) Another 2 r e s u l t i s (A) Suppose U E Lloc and t + ' l U ( x ) 1 2 d x 0 as a + m ; t h e n H = Ho -+

t U i s s e l f a d j o i n t on D(H,)

and ae(Ho + U) = [0,-).

Now t h e q u e s t i o n o f

s e l f a d j o i n t r e a l i z a t i o n s o f H = Ho + U a r i s e s and t h e r e may be none o r many; i f t h e r e i s a unique s e l f a d j o i n t e x t e n s i o n o f H = Ho + U f r o m D(Ho) n D(U)

104

ROBERT CARROLL

then H i s c a l l e d e s s e n t i a l l y s e l f a d j o i n t .

Many p o t e n t i a l s do n o t g i v e r i s e

t o e s s e n t i a l l y s e l f a d j o i n t Ho + U however so one adopts t h e q u a d r a t i c f o r m Thus (Ho&,$)

approach.

+ (Up,$)

= (1/2m)(p$,p$)

and one c o n s i d e r s h(Ip,$) = (1/2m)

.C: One w i l l d e f i n e a s e l f a d j o i n t o p e r a t o r H a s s o c i a t e d w i t h h i n a w e l l determined manner and use t h i s as t h e observ(pp,p$)

f o r say q , $

E

Thus f i r s t ob-

a b l e (see [Ka2] f o r t h e r e l a t i o n s t o F r i e d r i c h s ' e x t e n s i o n s ) . serve t h a t whenever e.g.

( 0)

c(llu'1I2

2 IIuII ) 5 h ( u ) + kllul12 f o r u

t

C:

E

= h ( u ) ) then h(u,v) = l i m h(un,vn) e x i s t s whenever un -+ u and vn v 2 w i t h h(un-um) and h(vn-vm) 0 (un,vn E C o ) . T h i s determines a domain

(h(u,u) in L

-+

-+

D(h) and one says u E D(H) w i t h Hu = f i f u (f,v) f o r a l l v [Shl]

-

tions).

E

D(h).

D(h), f

E

E

L2, and h(u,v) =

T h i s o p e r a t o r H w i l l i n f a c t be s e l f a d j o i n t ( c f .

and see a l s o 51.9 f o r some analogous procedures i n d i f f e r e n t i a l equa-

H i s c a l l e d t h e form e x t e n s i o n o f Ho + U; i t c o i n c i d e s w i t h Ho + U

on D(Ho) n D(U) when t h e l a t t e r i s s e l f a d j o i n t . tence o f H v i a

(0)

A condition f o r the exis-

i n terms o f U(x) i s g i v e n by sup

U+(x) = max (U(x),O)

-

and U ( x ) = U+(x)

/xx+l

-

U-(y)dy <

where

U(x) ( c f . [ S h l ] ) . Thus

Now l e t us go t o 1-D 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 [ S h l ] f o r t h e moment. w i t h formal s o l u t i o n $ ( t ) = exp(-iHt/fi)$(O).

one c o n s i d e r s $ ' = - ( i / i l ) H $

One assumes s h o r t range p o t e n t i a l s U(x) i n H f o r example so t h a t f o r t t h e behavior i s e s s e n t i a l l y governed by exp(-iHot/h)&-(0)

with Il$(t)

-

$ (t)ll

-+

Ho. 0 as t

exp(-iHot/fi)$+(0)

with Il$(t)

-

$,(t)Il

-+

0 as t

coming and o u t g o i n g a s y m p t o t i c s t a t e s .

-+

-m

-+

-.

-+

0 as t

-+

=

and a s t a t e $ + ( t ) = These a r e c a l l e d i n -

-

Thus ( & ) llexp(-itH/n)$(O)

u, = l i m W(t)u as t Set now W e x i s t s ) where W(t) = e x p ( i t H / h ) e x p ( - i t H o / A ) and t h e n f o r m a l l y (-itHo/fi)$+(0)ll

+m

-+

Thus we imagine a s t a t e $ - ( t )

*m.

-+

exp

(when i t

fm

$(O)

= W*$?(O).

The W+- a r e c a l l e d wave o p e r a t o r s and when $ ( t ) posseses b o t h a s y m p t o t i c s t a t e s $ -+ ( t ) i t i s c a l l e d a s c a t t e r i n g s t a t e . One s e t s M+- = I u E H; l i m W(t)u e x i s t s as t -+ t-1 and R, = R(W,) = range W+. I f $ ( t )i s a s c a t t e r i n g

$(O)

s t a t e then

= W t$f ( 0 ) and-$+(O)

W;'W-$-(O)-=

S$-(O)

where S: M-

-+

M,

i s c a l l e d t h e s c a t t e r i n g o p e r a t o r and D ( S ) = M- i f and o n l y i f R- C R+ w h i l e R(S) = M+ i f and o n l y i f R,

C

R- ( e x e r c i s e ) .

A map i s c a l l e d u n i t a r y i f i t

i s an i s o m e t r y o n t o and one sees t h a t S: M- -+ M+ i s u n i t a r y i f and o n l y i f R

= R .,

There a r e v a r i o u s c r i t e r i a t o determine when M,are

n o t empty e t c .

and i n p a r t i c u l a r f o r convenience h e r e we mention ( c f . [ S h l ] )

CHEBREm 2.5.

If (l+lxl)aU(x) E L

S i m i l a r l y ifU ( x )

2

M- = L .

E

2

f o r some a > 1 / 2 then M+ = M

= L

2

= H.

L t o c and I x l " U ( x ) 5 c f o r 1x1 l a r g e ( a > 1 ) - t h e n M+ =

SCATTERING THEORY

105

+ U s e l f a d j o i n t ) i f $(O) i s an eigen-

More g e n e r a l l y (always assuming H = H,

element o f H t h e n $ ( t )has no a s y m p t o t i c s t a t e s (incoming o r o u t g o i n g ) u n l e s s

$(O) i s a l s o an eigenelement o f H, w i t h t h e same e i g e n v a l u e ( e x e r c i s e - c f . [ S h l ] and n o t e $(O) = $+(O)). Now we observe t h a t exp(isHo/h) maps M, i n t o i t s e l f and exp(isH/h) maps R+ i n t o i t s e l f w i t h ( + ) W,exp(isH,/n) -

W,. -

Indeed if e.g. fll

+

0 as t

+

Thus exp(isH,/R)u

W u

=

f t h e n llW(t)exp(isHo/h)u

( W ( t ) = exp(itH/?i)exp(-itHo/h)

--m

E M- w i t h W-exp(isH,/h)u

-

= exp(isH/h)

e x p ( i s H / h ) f l l = lIW(t-s)u

and llexp(-isHo/n)vll

= exp( i s H / * ) f

= llvll).

= exp(isHo/A)W-u.

An elementary c a l c u l a t i o n shows a l s o t h a t exp(isHo/h) maps M,- i n t o i t s e l f and exp(isH/%) maps R,

Next as i n Theorem 8.34 ( o r

i n t o i t s e l f (exercise).

d i r e c t l y from t h e Schrodinger e q u a t i o n ) one shows e a s i l y t h a t u E D(A) i f

-

F u r t h e r i f P:,- H + a r e t h e orthogonal p r o j e c t i o n s t h e n P, maps D(Ho) i n t o i t s e l f ( c o n s i d e r

and o n l y i f l i m [ e x p ( - i t H / R ) u M,

-

{iexp(-itHo/h)

u]/t

+

0 i n H as t

l ] / t l P +-u = P-+ { [ e x p ( - i t H i / h )

-

+

l]/tlu)

0.

and t h e M,- reduce Ho

w h i l e t h e R+- reduce H ( t h i s means e.g. P,Ho C HOP, o r Ho: D(Ho) n M,- + M, I 1 To see t h i s c o n s i d e r e.g. f o r u E D(Ho), HoP,u and Ho: D(Ho) n M, + M,). i h l i m ( l / t ) [ e x p ( ~ i t H o / h ) P , -u

-

= ifiP+ lim(l/t)[exp(-itHo/h)u P+u] ~

-

=

u] =

~

P,Hou.

T h i s l e a d s t o an i n t e r t w i n i n g theorem ( c f . [ S h l ] )

CHE0REm 2.6,

t h e case f o r H,

Pmod:

where Po: L2 + N(Ho) (and i f N Ho) = a, which i s 2 = -(TI /2m)Dx, Po = I = i d e n t i t y ) .

HW,Po

=2W,Ho

We can assume N(Ho) =

@

i n o u r s i t u a t i o n so Po = I and R(Ho) = L L . (1-P,)u -

= (1-P,)

The e x i s t e n c e o f t h e f i r s t l i m i t i m p l i e s t h e second l i m i t e x i s t s and W u,- E i f u E D(HW ) t h e n u E M , and W u Conversely D(H) w i t h W,Hou = HW+u. ? + E D(H)

so t h e l a s t l i m i t i n - ( 2 . 1 )

e x i s t s and hence t h e f i r s t l i m i t converges i n R, -

which i s closed, t o an element o f t h e f o r m W,f.l ] u - fll = IIW+ { ( l / t ) [ e x p ( - i t H o / h ) - l ] u - f l I I HW+u. -

Hence l l ( l / t ) [ e x p ( - i t H o / ~ ) +

0 so u E D(Ho) and W,Hou

-

=

QED

RENARK 2-7- L e t E o ( x ) and E ( A ) be t h e s p e c t r a l f a m i l y a s s o c i a t e d w i t h Ho and H.

One says f E Jc i s i n t h e continuous subspace JCc(H) if E ( A ) f i s con-

tinuous a t every A.

Such f a r e orthogonal t o eigenelements o f H and f o r H,

which has no e i g e n f u n c t i o n s one has Jcc(Ho) = L2 = Jc. an open A = UI

n

On t h e o t h e r hand g i v e n

w i t h n o n o v e r l a p p i n g i n t e r v a l s In o f l e n g t h (I,( one s e t s

106

ROBERT CARROLL

(A1 =

1 1 InIand

(A) 9

E JC i s t h e v a l u e a t t = 0 o f an incoming a s y m p t o t i c s t a t e f o r a s c a t -

f i s s a i d t o be i n t h e subspace o f a b s o l u t e c o n t i n u i t y JCac (H,) i f E o ( A ) f -+ 0 whenever (A1 -+ 0. I n f a c t f o r t h e f r e e H a m i l t o n i a n Ho, 2 Now we r e c a l l t h a t i f $ ( t ) i s a s c a t t e r i n g s t a t e JC ( H ) = JCac(Ho) = L = JC. c o then $(O) = W $ ( 0 ) w i t h $,(O) = sJ/-(O) where W,: Mi R, e t c . I n p a r t i c u l a r -+

? ?

t e r i n g s t a t e $ ( t ) i f and o n l y i f $

‘L

$ - E M- and

W-$

€

;R,

similarly

(B)

$

i s t h e v a l u e a t t = 0 o f an o u t g o i n g a s y m p t o t i c s t a t e f o r a s c a t t e r i n g s t a t e $ ( t ) i f and o n l y i f $

’L

9,

M, and $W,

E

E

.R,

The wave o p e r a t o r s a r e s a i d 2 E JCc(Ho) ( = L = JC h e r e )

t o be (weakly a s y m p t o t i c a l l y ) complete i f every 9

A s t r o n g e r ideas i s t o say t h a t t h e wave opera-

has b o t h p r o p e r t i e s (A)-(B).

t o r s a r e s t r o n g l y ( a s y m p t o t i c a l l y ) complete i f t h e y a r e complete and every $ E JCc(H) i s t h e v a l u e a t t = 0 o f a s c a t t e r i n g s t a t e .

One shows t h a t com-

pleteness i s e q u i v a l e n t t o JCc(Ho) c M,- and R, n JCc(H) = R- n JCc(H) w h i l e s t r o n g completeness i s e q u i v a l e n t t o JCc(Ho) C M,- and JCc(H) C R, ( c f . [ S h l ] 2 f u r t h e r d i s c u s s i o n i s a l s o g i v e n below). Since we t a k e JCc(Ho) = L and work 2 w i t h L = M,- as i n Theorem 2.5 one need o n l y check R., L e t us i n d i c a t e some formulas i n v o l v i n g r e s o l v a n t s R(z) = ( z -

RENARK 2.8. H)-’ = RZ.

Formally R ( z ) f = - ( i / f i ) $

exp(it/fi)(z-H)fdt

( I m z suitable

-

say

> 0 ) so t h a t ( u s i n g (1/J271 i n t h e F o u r i e r t r a n s f o r m ) m

(2.2)

R(s+ia)f = - ( i / h ) [

ei s t / h e i ( i a - H ) t / h f d t

=

U

= -i&F-’

[ ~ ( ~ , i~(ia-H)r: ) e

fl

Then by t h e Parseval formula ( z = s t i a , Imz > 0 ) (2.3)

1;

(Ro(z)f,R(z)g)ds

/0 e - 2 a T (e-iTHof,e-iTH

= 2a

g)dT =

= 2 7 1 1 e-2aT ~ ~ ( w ( T ) f ,g ) d.r

Thus f o r f E M, (2.4)

one has f o r m a l l y

(W,f,g)

=

l i m 2a lme-2aT aJ-O

im ( a/a)l;

aJ-0

(W-f,g)

=

=

( Ro ( s + i a ) f , R ( s + i a ) g )ds

S i m i l a r l y f o r Imz < 0 and f E M-, (2.5)

(w(T ) f ,9 1d-r

z = s - i a ; one uses - i z - l =

Lz

eiztdt

with

(~/TI)~I (Ro(s-ia)f,R(s-ia)g)ds

The e x i s t e n c e of t h e l i m i t s does n o t however i m p l y f

E

M, o r M-.

L e t us

SCATTERING THEORY

= R(z)W+ - f o r Imz

show now t h a t W,Ro(z)

= - ( i / l i ) l o m e i t ( 2 - H )/fiW+fdt

R(z)W,f-

h e r e ( + ) W+exp(isHo/h) =

=

-

-(i/Ti)lom W+eit(Z-Ho)/nfdt (i.e.

* 0 (recall

Indeed f o r example, f o r Imz > 0,

exp(isH/b)W,).-

(2.6)

107

R o ( z ) f E M,

and W,Ro(z)f

= W + Ro ( z ) f

A s i m i l a r argument works f o r Imz

= R(z)W+f).

,

0. Next we show f o r f E M ( m ) E(I)W+f To see t h i s we w r i t e - = W,Eo(I)f. Then = (Ro(z)fyRo(z)W:g) f o r f E M+- and g E R., f i r s t (**) (R(z)W,f,R(z)g) one shows t h a t f o r any i n t e r v a l I (*A) (a/r)JX ( R ( s + i a ) f , R ( s + i a ) g ) d s -+ (:(I) <

xJ(H).

f,g) where r ( I ) = ( 1 / 2 ) [ E ( I ) + E ( i ) ] and E(J) the l e f t side o f where Ga = a / n ( h (A E

+a

2

[I

i s fZ

Ga(s-A)d(E(A)f,g)lds = f,(A)d(E(A)f,g) ) and fa = /I Ga(s-A)ds. But f o r I = (a,b) f a ( A ) + 1

(*A)

2

Indeed by Theorem 2.1

I),f + 1 / 2 ( A = a o r b), and fa + 0 ( A

B

I)w i t h lf,(h)[

5 1 so lz fa

I n p a r t i c u l a r one notes t h a t a / : ( R ( z ) f , R ( z ) g ) d s (x)d(E(A)f:g) (E(?)f,g). a ( f , g ) f o r z = skis, a > 0. Now f r o m (**) and (*A) one has (**) (E(?)W+f, -f

-f

.%

w

f o r any i n t e r v a l I. Hence E ( I ) W-+ f = W,Eo(I)f

g ) = (ro(I)f,W:g)-

A l i t t l e f u r t h e r reasoning gives

( m

) (exercise - c f . [Shl]).

f o r f E M+. Note a l s o

from E(i)W+f=W E ( ? ) f , f E M, and ( m ), f o r J/ E JCc(Ho) one has [ E ( i ) i g E(I)]W+J/ = W,[Eo(I) - E o ( I ) ] J / = 0 so W+J/ E JCc(H) (and c o n v e r s e l y ) . T h i s leads ness.

to

t h e c o n d i t i o n R+ n JCc(H) = R--n JCc(H) i n t h e d e f i n i t i o n o f complete-

Using now ( m ) we conclude t h a t

M,

JCac(Ho) i m p l i e s R, - C JCac(H) s i n c e f o r A = UIn as i n Remark 3.7 one has E(A)W,f = WkEo(A)f + 0 as 1A1 -t 0; one 2 uses here t h e f a c t t h a t JCac(Ho) = L which-can be e s t a b l i s h e d as f o l l o w s . C

E v i d e n t l y [ R o ( z ) f ] ^ = ( z - k 2 ) - ’ F ( k ) ( F o u r i e r t r a n s f o r m ) so c a l c u l a t i n g as 2 2 2 above a/* IIRo(z)fll ds = l ? ( k ) l I / ads/[(s-k) + a 2 ]}dk and a l i t t l e a r 2= 2 In particular gument g i v e s ( E o ( I ) , f , f ) = J l;(k)l dk ( k E I c f . [ S h l ] ) . 2 2 2 ( E o ( A ) f , f ) i s c o n t i n u o u s f o r f E L so JCc(Ho) = L and i n f a c t JCac(Ho) = L

/f

-

R m R K 2.9, complete.

.

We s t i l l need some theorems t o say when t h e wave o p e r a t o r s a r e

2

(H ) = M, = L = JC ( c f . Theorem ac o We know f u r t h e r ( r e c a p i t u l a t i n g ) t h a t HW+ = WH, o (Theorem 2.6),

For H

0

2.5 f o r M+).

we can assume JC (H,) C

= JC

,

E(1)W f =-W E ( 1 ) f (Remark 2.8), R(z)W,f = W,Ro(z)f ( e q u a t i o n ( 2 . 6 ) ) , and + o ( v i a Theorem 2.5 and Remark 2.8) R, - C JCac(H). There a r e a number o f a b s t r a c t theorems i n [ S h l ] g i v i n g completeness and we o n l y s e l e c t a s i m p l e example o r two w i t h o u t g o i n g t h r o u g h t h e t e c h n i c a l d e t a i l s o f p r o o f .

/I I U ( x ) l d x

Thus i f e.g.

and H i s d e f i n e d v i a b i l i n e a r forms then t h e W, - a r e complete and R+- = JCac(H). R e c a l l s t r o n g completeness r e q u i r e s JCc(H) C R,- b u t KaCc <

m

ROBERT CARROLL

108

i n general.

JC

C

To i n s u r e s t r o n g completeness ( c f . [ S h l ] ) one can assume

f i r s t , a s b e f o e, 1 l U ( x ) l d x <

m

t o guarantee (weak) completeness

together

1 U ( x ) l p p ( x ) a d x < m f o r some a,p where P ( X ) = 1 + I x 9 a ' 1, and 1 5 p 5 2 (one c o u l d a l s o assume s i m p l y Upa E Lp f o r some a,p w i t h 1 5 p

w i t h , e.g., <

m

and a > max(p-',l-p-')).

These a r e n o t a t a l l unreasonable types o f hyWe w i l l d i s c u s s t h e q u e s t i o n o f s t r o n g and weak

potheses ( c f . a l s o [Chl]).

a s y m p t o t i c completeness f o r 3 dimensions l a t e r ( c f . [Bjl;Ka2;Rl notes here ( f r o m [Shl])

;Sjl,2]).

One

t h a t under t h e l a s t h y p o t h e s i s UP" E Lp one has Jcc

(H) = JCac(H) = R,- and t h e eigenvalues o f H a r e i s o l a t e d w i t h O,m as t h e o n l y p o s s i b l e l i m i t p o i n t s . R e c a l l t h a t s i n g u l a r spectrum i s d e f i n e d v i a JCs(H) = I I Jcc(H) We n o t e exJCac(H) and JcP H) = c l o s u r e o f t h e span o f e i g e n f n s .

.

p l i c i t l y that

s o l a t e d eigenvalues g i v e r i s e t o elements i n JCS(H) and we

n o r m a l l y wish

o exclude a p r i o r i s i n g u l a r continuous spectrum;

i n particu-

l a r eigenvalues embedded i n t h e continuous spectrum ( c f . Remark 2.10).

REmARK 2.10.

I t may be w o r t h w h i l e t o summarize here t h e v a r i o u s s p e c t r a l

decompositions ( c f . [ B j l ; R l ; S h l ] ) .

Thus f o r a g i v e n a r b i t r a r y s e l f a d j o i n t

IB Jcs, b u t t h e c l o s e d s e t s uac and u s need n o t be o p e r a t o r H i n Jf, Jc = Jc ac d i s j o i n t ( h e r e uac i s t h e spectrum o f H i n aCac e t c . ) . L e t us w r i t e u f o r P t h e p o i n t spectrum ( e i g e n v a l u e s ) and Jc f o r t h e span o f t h e e i g e n f u n c t i o n s ; P then J C c l Jc (Jc = Jcc IB JC ) where Pc = p r o j e c t i o n o n t o Jcc i s 1-P (P = p r o P P P P j e c t i o n o n t o Jc ) . uC i s t h e spectrum o f H i n Jcc and i s c l o s e d . u may n o t P P be c l o s e d and i n f a c t 0 = u (HP ) = spectrum o f H i n Jc One has u(H) = P P P P' a + 5 b u t u n uc may be nonempty. One can w r i t e Jc C Jcs, u c u s , xsL = C P P P C uc, and one d e f i n e s Jcsc as t h e complement o f Jcac i n Jcc so t h a t JC 'c, 'ac = Jc + Jc + JCs c ' The s e t o f i s o l a t e d eigenvalues ( o f f i n i t e m u l t i p l i c i t y ) p ac i s t h e d i s c r e t e spectrum ud and a/ud = u i s t h e e s s e n t i a l spectrum. Thus e u c o n t a i n s uc p l u s accumulation p o i n t s o f u p l u s eigenvalues o f m m u l t i P e p l i c i t y (an accumulation p o i n t from u m i g h t a l s o be an e i g e n v a l u e o f m mulP t ip l i c i t y )

.

3.

XA&ZERINC EHE0RU. 11 (3-D)-

We go now t o 3 dimensions and w i l l g i v e a

number o f formulas and approaches w i t h o u t concern f o r t h e d e l i c a t e q u e s t i o n s o f completeness, absence o f p o s i t i v e eigenvalues, primarily with "structural

I'

etc.

Thus we a r e concerned

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

v a n t o p e r a t o r t h e o r y and w i l l make r e l a t i v e l y s t r o n g b l a n k e t hypotheses about t h e p o t e n t i a l i n o r d e r t o guarantee n i c e behavior o f e i g e n f u n c t i o n s e t c . ( i n p a r t i c u l a r Coulomb p o t e n t i a l s , i . e . cussed).

l o n g range p o t e n t i a l s , w i l l n o t be d i s -

I n a d d i t i o n t o t h e a r t i c l e s and books mentioned e a r l i e r ( [ B j l ; K a l ;

SCATTERING THEORY

109

Rl;Shl;Sjl,E] in p a r t i c u l a r ) we want t o i n d i c a t e some f u r t h e r references, namely [Bhl ; B k l ;Am1 ;AgZ;Ahl; D11 ;Lx2-4; H1 ;H12; Ikl ;Ka4,5; Ptl ;Pql ;Rgl ;Sdl ;ShZ].

W e may occasionally s t a t e some r e s u l t s f o r n dimensions b u t mainly will conc e n t r a t e on n = 3 . A remark format i s used t o s e p a r a t e t o p i c s and many res u l t s a r e proved without being s t a t e d as theorems. Let us make a few remarks here about t h e philosophy and presentation f o r t h e next several s e c t i o n s , indeed f o r t h i s e n t i r e chapter. We will ( i n a s p i r i t already indicated in Chapter 1 ) push in some d i r e c t i o n s to give very recent r e s u l t s , b u t primari l y t h e emphasis i s on s t r u c t u r e . Thus sometimes one will say t h e same thing i n several d i f f e r e n t ways because i t seems t h a t t h e s e p a r a t e presentations a l l c o n t r i b u t e something t o t h e content o r simply because i t allows one t o develop a p a r t i c u l a r "language" in a way which can be useful l a t e r . Thus we d o n ' t aim a t p a r t i c u l a r problems or questions and mainly t r y t o provide a rich s t r u c t u r a l framework, with enough d e t a i l t o be s o l i d , within which one can discuss various t o p i c s with some understanding of what i s and was and can be. That t h e r e should appear t o be no p a r t i c u l a r d e s t i n a t i o n achieved o r targeted should not d e t e r one from t h e s c h o l a r l y goal of understanding and perhaps e x t r a c t i n g meaning of some s o r t from t h e development of a scient i f i c s u b j e c t . In any event i n a more o r l e s s introductory book we f e e l i t i s most appropriate t o e x h i b i t s t r u c t u r a l f e a t u r e s in t h e form of various types of formulas, languages, and e s p e c i a l l y connecting themes. For example t h e theme of transmutation, expressed via s p e c t r a l pairings of generalized eigenfunctions i s one such connecting d i r e c t i v e . One should of course a l s o mention t h a t along t h e way many goals a r e achieved in t e r m o f r e c o v e r i n g potent i a l s from s p e c t r a l o r experimental d a t a , describing s c a t t e r i n g phenomena, e t c . ; we just d o n ' t make a big f u s s about t h i s . To begin now l e t Ho = -A with say D ( H o ) = H L (Sobolev space) and H = -A + U with some hypotheses on U ( r e a l ) which w i l l give typical r e s u l t s with t h e l e a s t technical fuss ( s e e below). We w i l l usually take ti = 1 now and absorb t h e 1/2m f a c t o r i n t o A and U; t h i s can be accomplished e.g. by a s c a l e change in t ( f o r b ) and in x ( f o r 1 / 2 m ) .

REmARK 3.1, Let us make a few remarks about U ( t h e most u p t o d a t e survey here i s probably [ S j l ] b u t t h e r e i s too much d e t a i l f o r our purposes). In [ I k l ] f o r example one assumes U i s l o c a l l y Holder continuous except f o r a 2 f i n i t e number o f s i n g u l a r i t i e s , U E L , and t h e r e e x i s t c,R,h such t h a t I U 2+h ( x ) [ 5 c/lxl f o r 1x1 R. Then H i s lower semibounded and e s s e n t i a l l y s e l f a d j o i n t over with D ( H ) = D ( H o ) ( H i s i d e n t i f i e d w i t h whenever i t i s

Ci

110

ROBERT CARROLL

essentially s e l f adjoint).

2

(*) J [ U ( x ) /

if e.g.

S i m i l a r l y ( c f . [H12])

a-1

l d y 5 M ( i n t e g r a l o v e r l y - X I 5 R ) f o r a l l x E R3, a l l R < 1, and f i x lx-yl ed M,a ( 0 < a < 4 ) then H i s e s s e n t i a l l y s e l f a d j o i n t on C i . C o n t i n u i n g i n t h i s s p i r i t i n [ A h l ] one assumes (*) w i t h R = 1 and 0 < (A)

5 1 together w i t h

a

E L1 and g ( x ) =

For p = 1/2 and p = 1, w i t h x a r b i t r a r y , U ( x ) / l x - y I

1 [ l U ( y ) l / l x - ~ l ~ I d
I n p a r t i c u l a r (*) plus

t h i n g s happen ( l i s t e d below).

h o l d i f any o f t h e

(A)

following a r e true: (A) U E L1 n L2 ( 6 ) J I U ( x ) 1 2 ( l + l x / ) 1 t 6 d x < m ( 6 > 0) -2-E ) as 1x1 m and ( A ) holds. G e n e r a l l y ( c f . [Ag2;Ka2,4, ( C ) U(X) = O ( l X l 5 ; K d l l ) i f U ( x ) determines an Ho compact o p e r a t o r ( i . e . U: H2 + L2 i s com-f

p a c t ) t h e n H has a unique s e l f a d j o i n t r e a l i z a t i o n ( = H again f o r s i m p l i c i t y o f n o t a t i o n ) w i t h O(H)

The c o n d i t i o n i n [Ag ] i s

D(Ho) and ae(H) = ue(Ho). )2t2Ej

f o r some

[ I U(Y) I 2 / 1 X - Y I 3-p]dy < lx-Y 151 T h i s c o n d i t i o n h o l d s e.g. > 0 and some p w i t h 0 i p < 4.

E

U

if

n

(XI1 =

O ( l X l -1%) as I x

+ m

and U E Lfoc; i n p a r t i c u l a r i t i m p l i e s t h a t U

i s a s h o r t range p o t e n t i a l i n t h e sense t h a t ( @ ) f ( x ) i s a compact map H2

-+

L2 ( c f . [Ag2;Sh2]).

( 0 )

[O,m).

Going back t o [ A h l ] one has

I f U s a t i s f i e s ( * ) ( w i t h R = 1 and 0 < a 5 1 ) p l u s

i s s e l f a d j o i n t w i t h domain H2, ae(H) = ue(Ho) = ( 6 ) the equation f ( x ) =

- Ir(Ix-yl,~)U(y)f(y)dy, f

E Lm,

2

(6))

then H

K

is

# 0 r e a l , has no

-G o f Remark 3.8)

F u r t h e r (under ( * ) +

i t f o l l o w s t h a t t h e nonnegative p a r t o f os(H) = C O I .

+

(A)

and i f i n a d d i t i o n

[O,m),

n o n t r i v i a l s o l u t i o n f ( 8 r , K ) = e x p ( i ~ r ) / 4 n r- thus (A)

(x)

However t h e hYo f s h o r t range i s n o t s t r o n g enough t o exclude p o s i t i v e e gen -

values f o r example.

CHEBREIR 3.2.

(l+Jx/)ltEU(x)

F u r t h e r i t w i l l f o l l o w ( c f . [Ka2])

t h a t H i s s e l f a d j o i n t over Hz = D(H) w i t h ae(H) = pothesis

+

2

f o r k # 0 t h e r e e x i s t 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 @+(x,k) E Hloc

s a t i s f y i n g H@+ = lkI2@* along w i t h t h e Lippman-Schwinger type-equation (9 = e x p ( i ( k,x)) for

@

-

and we n o t e t h a t Lippman-Schwinger corresponds t o t h e equation

thus u s u a l l y G ( l x - y l , + l k l )

i s used i n ( 3 . 2 )

t h i s n o t a t i o n i n (3.2) i s c o n s i s t e n t w i t h [ I k l ] (3.2)

@+(X,k) = V(X,k)

-

-

-

see e.g.

[Ka4]

-

but

c f . a l s o [Aml])

G ( l x - Y l , T I k I )U(y)@+(Y,k)dY -

One shows t h a t f o r K compact ( 0 9 i s u n i f o r m l y continuous t h e r e .

4

3 K) @-+ ( x , k ) i s bounded on R x K and 2 D e f i n e now f o r f E Lac(H) = JCac(H)

@

t

-

SCATTERING THEORY

- JCac(H) One t h i n k s o f F+:

-+

111

Jc, which i s an i s o m e t r y , and FT - i s d e f i n e d by

Then Hac = HIJcac = F:MkZ F, where Mkz i s t h e m u l t i p l i c a t i o n o p e r a t o r by 2 i n Lk. F u r t h e r , d e f i n i n g - t h e F o u r i e r t r a n s f o r m on L2 = 3c by F f ( k ) = ( l / Z n ) 3 / 2 / v ( x , k ) f ( x ) d x w i t h F * F ( x ) = ( ~ / Z T ) ~ v/ (~x ,/k ) F ( k ) d k ,

one has

I k (2 Ho

=

F * M t F s o i t f o l l o w s t h a t (+) Xac = U,Jc U** U,- = F,*F. The wave o p e r a t o r s W,O?' a r e now d e f i n e d as b e f o r e , W,- = l i m e x p ( i t H / h ) e x p ( - i t H o / h ) as t ?m and f o r -+

convenience we w i l l sometimes s e t h = 1 ( v i a e.g. R e c a l l from Remark 2.7 t h a t t h e W+:

a change i n t i m e s c a l e ) .

M, R, a r e c a l l e d ( s t r o n g l y ) complete i f L2 = M+ = Jc ( H ) (which i s a s s i r e d h e r e ) w h i l e Xc(H) C R., Then g i v e n c o The s c a t t e r R, C Jcac(H) we w i l l have s t r o n g completeness ( c f . Remark 2.9). i n g o p e r a t o r i s S = WTW-

(%

-f

W;W ' -)

and a g a i n f r o m [ A h l ]

Suppose U s a t i s f i e s ( * ) ( w i t h R = 1 and 0 <

iTHE0RRn 3.3.

a

5 1 ) plus

(A)

E L2 f o r some E > 0. Then t h e wave o p e r a t o r s W, exPIUS U ( x ) ( l + I x l 1-1/2tE A i s t and a r e complete w i t h W, = U, = FfF. I f i n a d d i t i o n U E L', s e t t i n g S

I kl )

= FSF*, one has f o r kw E R"(w

.-Sn-',

(3.5)

S+(k,w,w') -

@+(y,kw)U(y)e-

(3.6)

Sf(kw) = i ( k w )

= (1/4n)/

AA

-

S - (k,-w',-w). (3.7)

=

i(k/2n)/

ST(k,w,w'),

%

+i( kw',y)

- i(k/2a )/ S, ( k ,w, -w ' (): where S+(k,-w,w') -

k

dY

S-(k,w',w)i(kw')dw'

= i(kw)

-

kw ' )dw ' S,(k,w,w')

= S+(k,w',w),

and S-(k,w,w')

=

I n f a c t S , i s t h e phase f a c t o r i n t h e a s y m p t o t i c expansion

@

f

(x,k)

= e

i(x,k)

e'iIXI

+

o(lxl-l)

+

Ikl

lyr-

S-+ ( l k l , k / l k l , X / I X I )

when t h e expansion i s l e g i t i m a t e . Thus ( 3 . 3 ) - ( 3 . 4 )

g i v e t y p i c a l s t r u c t u r a l i n f o r m a t i o n and we w i l l s k e t c h be-

low f o r m a l l y how such formulas a r i s e and a r e v e r i f i e d .

Further s t r u c t u r e

comes from [ I k l ] f o r example i n t h e f o l l o w i n g way. F i r s t v i n [ I k l ] i s @-' -1 w i t h k e r n e l G(x,y,h) s a t i s f y i n g f o r i n (3.3) and we s e t R, = -R, = ( H - A ) N

Imx

+0

(cf. [Ikl])

ROBERT CARROLL

-

We o m i t here a d i s c u s s i o n o f hypotheses and p r o p e r t i e s ( c f . [ I k l ] ) . cy

Thus

One w r i t e s R x f ( x ) = / G(x,y,X)f(y)dy and G(x,y,X) = G(y,x,A) (RX = - R x ) . now ?(k) f o r F - f ( k ) i n (3.3) and under t h e hypotheses o f [ I k l ] as i n d i c a t e d

i n Remark 3.1,

t h e r e a r e no p o s i t i v e eigenvalues.

ifv n s a t i s f i e s HP,

I t i s then proved t h a t ,

= vnqn f o r t h e n e g a t i v e eigenvalues p

( w i t h qn o r t h o n2 normalized) and f, = 1 ?,(x)f(x)dx = (f,vn) t h e n f o r f E L (.) f(x) = ( l / 2 ~ ) ~ /@~- (/ x , k ) i ( k ) d k + 1 ?,yn(x). I f Ep i s t h e r e s o l u t i o n o f t h e i d e n 2 2 t i t y f o r H w i t h P = I - Eo t h e n II Eofll = I and II fll = II E fll t II Pfll fi

1

and P f = ( l / 2 n ) 3 / 2 /

w i t h IIPfl12 = / l;(k)I2dk

one can show t h a t ((EB-Ea)f,g) %

p

( i n t e g r a l over

2

, and

- E,)f(x)

Ja <

2

= ( Z T ) - ~ / ~@ / -(x,k)f(k)dk

e t c . t o f o l l o w below).

(I).)

In particular

I k l < JB ( r e c a l l I k l

s e t t i n g U-f(x) = ( 2 ~ ) - ~ ” / @-(x,k)Ff(k)dk w i t h

F o u r i e r t r a n s f o r m ) one has W,Again S = W:W-

(EB

Jg- cf.

Ikl

Ja <

0

f(kP-(x,k)dk.

1 f^(k)i(k)dk f o r

=

o r A) and t h i s i m p l i e s (*)

R, - = PL

?d

Now PW,

UTf = U - f

= W,,

(F

U,- as i n Theorem 3.3 ( r e c a l l U,- = F -T ) . I n o r d e r t o see t h e c o n n e c t i o n between U,- and W,-

i s unitary.

=

we w i l l make some c a l c u l a t i o n s below f o l l o w i n g [ I k l ] .

RrmARK 3.4-

To see t h e r e l a t i o n between U- and W- f o r example s e t c =

( Z T ) - ~ / * and r e c a l l U - f = F+ff so t h a t UF = F T - ( f r o m (U*f,g) L e t us n o t e a l s o e x p l i c i t l y t h a t e.g. F;F-

= (f,U-g)).

= P v i a ( m ) and s i m i l a r l y U-U.

=

Now we n o t e t h a t f o r f E D(H) = D(Ho) U*Hf(x) = c l e x p ( i ( k , x ) ) 2 2 F - ( H f ) d k = c / e x p ( i ( k , x ) ) l k l F fdk and HoU*f = c / e x p ( i ( k , x ) ) l k l (FU*f)dk = 2 Now a p p l y UT t o W = l i m as t c / e x p ( i ( k , x ) ) l k l F f d k so (*A) UTH C HoU*.

F*FF*F-

-f

-m

= P.

o f exp(itH)exp(-itH,)

U*exp(itH) = exp(itH,)Uz -

=

from

-

l i m W(t) ( i . e . W U; (*A).

and n o t e t h a t = l i m U*W(t)) -

Then s i n c e (U;W(t)f,g)’

= (U;[exp(itH)

one has (H-Ho = U ( x ) ) ( * a ) (U?W(t f,g) - (U*f,g) = il: (exp( i r H o ) U ~ U ( x ) e x p ( - i t H o ) f , g ) d t . . Consider I = (exp it H o )U;U ( x )exp ( - it

i(H-Ho)exp(-itHo)]f,g) H,)f,g) (3.9)

and w r i t e

I =

e itlkl

2 F(U;U(x)e-itHof)%jdk

= c/

=

I

ei t I k

%[I @_(x,k)U(x)(e-it(Ho-Ikl

(U( x)e-i tHof ) G d k

2 )f(x))dx]dk

Now assume F g has compact s u p p o r t i n k i n o r d e r t o i n t e r c h a n g e i n t e g r a t i o n 2 o r d e r s (such f u n c t i o n s w i l l be dense i n Lk and one takes l i m i t s a t t h e end). Then J = (U*W-f,g)

- (U*f,g)

= l i m if: I(T)dT ( t

+ -m)

which we w r i t e as

SCATTERING THEORY

(F

%

F(x,k) = @-(x,k)U(x)

113

c

-

n o t e (h,F)

1 hF).

= (H-A)-'

Now f o r R,

(=

5)

and %: = ( H o - h ) - l we have f o r m a l l y (and r i g o r o u s l y when a p p l i e d t o s u i t a b l e ; e x p ( - i r ( H o - A ) ) d r = exp(-ir(Ho-,))(l/-i)(Ho-,) -1 t -t ( l / i ) ( H o objects) 1

lo

A)-'

as t

-t

-m

when

Im,

<

0 (here

%

Ikl

2

-iE).

Thus

u n i f o r m l y on compact s e t s K o f k.

F u r t h e r we n o t e t h a t (U*f,g)

= c / F g / f ( x ) @ - ( x , k ) d x d k so,

(m),

(3.13)

(U*W-f,g)

=

5+

using

(U*f,g)

-

= c / F g { l f(x)[@-(x,k)

Now t a k e l i m i t s t o g e t general g and one concludes t h a t

= (P'F-f,g)

+

UtW -

= I. Conse-

q u e n t l y one has ( r e c a l l i n g PW+ = W + and U-U? = P) U-U*W- = U- = PW- = W- and U*U - - = W*W- - = I. S i m i l a r c a l c u l a t i o n s a p p l y t o W+ and U+ and one o b t a i n s

s

=

w:w-

=

u:u-.

REmARK 3-5, The c a l c u l a t i o n s i n Remark 3.4 a r e i n some ways t h e most i n t e r e s t i n g p a r t o f theorems such as Theorems 3.2-3.3, a l " p o i n t o f view ( c f . a l s o [Ag2]).

a t l e a s t from a " s t r u c t u r -

Many o f t h e r e m a i n i n g arguments i n [ I k l ;

Ah11 i n v o l v e a p p r o x i m a t i o n arguments w i t h t h e p o t e n t i a l U(x), growth e s t i m a t e s o f v a r i o u s types, e t c .

T h i s i s a l l q u i t e necessary o f course b u t we

w i l l g e n e r a l l y f o r e g o such d e t a i l s (some comments on ( 3 . 5 ) - ( 3 . 7 )

and f u r t h e r

a n a l y t i s o f t h e connections between Hi1 b e r t space p r o j e c t i o n s E A and r e s o l vants R, and 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

@

*will

appear below).

L e t us n e x t g a t h e r some f u r t h e r background " s t r u c t u r a l " formulas f o r c l a s s i c a l o p e r a t o r t h e o r e t i c s c a t t e r i n g f o l l o w i n g [Ag2;Ahl;Aml;Kdl;La2;Ka2,4,5; Pgl ;Rql I

Ikl;

114

ROBERT CARROLL

REmARK 3.6,

L e t us f i r s t r e c a l l a few f a c t s about s p e c t r a l r e s o l u t i o n s ( c f .

Theorem 3.1 and e.g.

- R,

f o r (H-x)-’ R,(-A)

(h+A)-’

( R ~ ( A =) R ( A , A ) )

[KaZ;La2;Wdl]).

i n Remark 3.4 w i t h R, i n Appendix B.

( * r ) R(A,A) -

R(A,A)(A-B)R(A,B)

i n Remark 2.2 and i x ( A )

(x-A)-’

%

Thus l e t us r e c a l l t h e r e s o l v a n t i d e n t i t i e s (z-A)R(A,A)R(z,A); R(A,A)

R(Z,A)

X,z

(the f i r s t f o r

p ( B ) w i t h D(B) C D(A)).

Ex =

We have been u s i n g t h e n o t a t i o n

E

-

R(A,B) =

p ( A ) and t h e second f o r h E p ( A ) n

Next, g i v e n a s p e c t r a l f a m i l y E(A) as i n Theorem

2.1 f o r a s e l f a d j o i n t A one has (R(x) = R(x,A) and we can t a k e (ax,y) = lim lim b+6 ([R(t-i~)-R(t+i~)l a(x,y) h e r e ) (*+) ( [ E ( b ) - E ( a ) l f , g ) = 5.10 E C o (1/2ni)/,+& f,g)dt. T h i s i s a s t a n d a r d c o n s t r u c t i o n ( c f . [Rcl ; W d l l ) . F o l l o w i n g [Ka41 one w r i t e s a l s o ( c f . Remark 2.8

-

JCac(Ho) = L

W+ = l i m e i t H e - i t H o = l i m

(3.14)

EC

,-2~t,itH

2 E ~ m

0

W+

(3.15)

( c f . (2.5)). R(s,Ho)dA =

=

a g a i n and u s i n g P a r s e v a l ’ s theorem)

exp( - E t + i tH)exp( - i A t ) d t = [i/ ( 2 ~ r ) ’ / ~ ]H-X+ie)-l (

(~/n)l:

= EJ. lim 0

(H-X+iE)-l (HO-x-iE)-’dx

Using ( * & ) one o b t a i n s ( 5 = E,+ie)

li!

e- i t H o d t

0

One w r i t e s t h e n ( t a k i n g R(z,H) = (H-z)-’

(*. ) ( 1 / 2 n ) ” 2 1 t

2 w i t h Po = I here and h = 1 )

(A*)

( E / I T ) ~ R~ ( ~ , H ) R ( c , H ) ( H - c ) R ( 5 , H o ) d c .

W+ =

1i m

E+O

R(5,H)

(E/T)~:

But i f dE(A)

%

H one

has ( E / ~ ) R ( S , H ) R ( ~ , H )= ( ~ / n ) j dz E ( 5 ) / [ ( 5 - 5 ) ( 5 - c ) ] = ( E / ~ V ;d E ( c ) / C ( t - h ) 2 2 + E ] = 1: 5E(5-x)dE(c) = & € ( H - h ) where d E ( u ) 6 i n a standard manner. lim m Now w r i t i n g G ( 5 ) = (H-5)R(5,Ho) one has f o r m a l l y (.A) W+ = EJ.O im f;€(H-X) -f

G(5)dx =

i:

G(x+iO)dE(x)/dx where 6€(H-A)

l a t i o n s h o l d f o r W- and one has f o r m a l l y

-t

dE(A) f o r m a l l y .

(A*)

Similar calcu-

W, - = j: G(x+iO)dE(x)dx which

i s h e u r i s t i c a l l y u s e f u l i n d e v e l o p i n g t h e s t a t i o n a r y t h e o r y ( c f . [Ka4] a1 so [C4,43;Dx1]

REmARK

see

f o r connections between s p e c t r a l i z a t i o n and o p e r a t o r s ) .

L e t us r e c a l l some n o t i o n s f r o m s p e c t r a l t h e o r y f o l l o w i n g [ L a l ]

3.7.

( c f . a l s o [Rcl;Ti3]).

l i m i t point

-

-

Thus i n a c l a s s i c a l way one goes through t h e Weyl

l i m i t c i r c l e t h e o r y f o r a d i f f e r e n t i a l o p e r a t o r Qy = - y “ + qy

v i a t h e e q u a t i o n (‘6) Qy = xy w i t h y(0,A) = Sina and y ‘ ( 0 , x ) = -Cosa.

We

assume a l i m i t p o i n t s i t u a t i o n h e r e f o r s i m p l i c i t y and l e t ip(x,A) s a t i s f y (A&)

with B satisfying

t i o n 9 = B + m(A).p

E

L

(A+)

2

k e r n e l o f t h e r e s o l v a n t (R:

+

0)

Qs

=

xB and B ( 0 )

= Cosa, 0

‘(0)= Sina.

i s then produced by t h e l i m i t p o i n t t h e o r y . %

The f u n c As a

( x - Q a ) - ’ ) one takes a Green’s f u n c t i o n (Irnx

115

SCATTERING THEORY

w i t h Q(Dx)G

-

AG = -6(x-y).

One can produce a s p e c t r a l measure d p ( h ) d i r e c t -

l y from 1 i m i t i n g arguments based on t h e e i g e n v a l u e s i t u a t i o n f o r i n t e r v a l s

[O,b],

b

and we w i l l r e l a t e i t t o m ( h ) below. Thus R,f = ;/ G ( x , y , ~ ) 2 2 2 (R,f( dx 5 ( l / I I m x ( )I: I f \ dx) and one shows ( a g a i n v i a

+ m,

f(y)dy (with 1 ; limits b

f r o m e i g e n v a l u e problems on [O,b])

+ m

(3.17)

RZf =

1;

[~(X,~)F(A)/(Z-X)I~P(~) ( t h i s p e r m i t s one t o i d e n t i f y (3.17) w i t h

where F(A) = :1 f ( x ) v ( x , h ) d x ;/

Note when b < 2

Gfdy).

m

1

w i t h an e i g e n f u n c t i o n s i t u a t i o n R Z f 2r gn(x) 2 % (1/an) (0 < A, 5 A ) f o r A > o o r Pb 2 < 0 and an = Job v i d x . Note t h a t (z-Q')

1

J f(Y)vn(y)dy/an(z-hn) where p b ( h ) (A) - 1 ( l / a n2) ( A < A n 5 0 ) f o r h Q ,

RZf

1 Ipn(x)fn/an2

'L

f satisfies

W(f,q)

(A&)

= f9'

-

= f.

h

Set now Eh(x) = lo p ( x , u ) d p ( p ) w i t h f E D(Qa)

and w r i t e

f'q

+

0 at

(Am)

-.

(i.e.

g ( h ) = 10" f ( y ) E h ( y ) d y where i n a d d i t i o n

Then a l i t t l e c a l c u l a t i o n shows (**) f ( x ) = A+AA

v(x,u)v(y,u)dp(P). Ifx 2 2 and AtAAare p o i n t s o f c o n t i n u i t y o f p ( X ) t h e n EA(x,.) E L and f o r f E L , ,:q(x,A)dp(h).

Set f u r t h e r

(.A)

EA(x,y) = /A

Y

/0"

EAf(x) = E A ( x , y ) f ( y ) d y s a t i s f i e s llEAfl12 5 IIfl12 (A an i n t e r v a l ) . If f 2 E L and F ( X ) = 10" f ( x ) v ( x , h ) d x , t h e n ( e v i d e n t l y ) ( 0 0 ) E A f ( x ) = JA F(h)9(x, Q ,

Then a l m o s t More g e n e r a l l y d e f i n e ( 0 6 ) EA(x) = JA v ( x , h ) d p ( h ) . 1i m (l/o(A))/; f(x)EA(x)dx ( p u t x = 0 i n ( 0 0 ) and t a k e l i m i t s ) . There r e s u l t s a l s o (use (3.17) f o r G,

A)dp(X).

everywhere (AE) i n A ( r e l a t i v e t o dp) ( a + ) F(A) = i.e.

G =

/I9 ( x , h k ( y , h ) d p / ( z - h ) )

(3.18)

lom G(x,y,z)lom

RZEAf =

( 2-X )I~P ( )J F( P )V (Y, P )dp ( P )dY = A

EA(y,t)f(t)dtdy

J

A

( x 3 P F ( u I/

=

lom 1; [ P ( X , ~ ) P ( Y , ~ ) /

(Z-P

)I ~ (P P

/I

(note H(A)J," v ( y , h ) 9 ( y , v ) d y d p ( h ) = f; ~ ( Y , P ) ~ ( Y )=~ H Y ( u ) ) . Thus i n p a r and moreover f r o m (-) t i c u l a r (ern) ;/ RZEAfg(x)dx = fa [G(h)G(X)/(z-h)]dp (g,EAf) = Ja F(X)G(A)dp(X). The formulas above a l l o w one t o prove t h a t EA ( o r Ex) i s a s p e c t r a l f a m i l y as i n Theorem 2.1. Finally to

(&*) 1 ; g(x)EAfdx

=

2

connect p and m one r e c a l l s f i r s t from t h e Weyl t h e o r y t h a t 10" l J / l dx = -1m m ( z ) / I m z ( J / = $ ( x , z ) , t i o n s we have = $(y,z)Sina

(l/am)

f

Gpm

from (3.16)

z = u+iv). 2r

From t h e f i n i t e i n t e r v a l approxima-

(1/am)Rzpm = q m / a m ( z - h m ) and f o r x = 0, G(O,y,z)

(we assume S i n a 9 0 and i f n o t a n o t h e r argument can

116

ROBERT CARROLL

be used).

Thus by t h e Parseval f o r m u l a a p p l i e d t o G(x,y,z)

(3.19)

lob IG(x,y,z)l

2

1;

dy

i n y one has

2 2 2 bn(x)/an(Z-An) 1

2 ( f i n i t e i n t e r v a l ) . Hence passing t o l i m i t s w i t h x = 0 ( c a n c e l l i n g S i n a ) 2 (u)f; I!b(Y,z)l 2 dy = [ d p ( A ) / I z - A l 1 = -1m m(z)/Imz. T h i s l e a d s t o

/I

(3.20)

p(A) =

1i m ( l / n ) v+o

1,

[-Im m(u+iv)]du; m(z)

-

= -ctga t

(A = (ul,u,)

w i t h ui p o i n t s o f c o n t i n u i t y o f

RElWRI( 3.8,

L e t us e x t r a c t a few f u r t h e r formulas from [Rgl].

- A

P).

Thus one

deals w i t h an ( o u t g o i n g ) Green's f u n c t i o n G(r,k)

as i n Theorem 3.2, G(r,r', cv 2 k ) = - ( 1 / 4 ~ ) e x p ( i k ~ r - r ' ~ ) / ~ r -(r=' ~-G) w i t h (-A-k )G = - 6 ( r - r ' ) (we use k

f o r I k l when no c o n f u s i o n can a r i s e ) . I f we w r i t e F e ( k r ) = ( n k r / Z ) 1/ 2 t 1/2 e Jet1/2(kr) and Ge(kr) = ( n k r / 2 ) ( - 1 ) J-e-1,2 ( k r ) t h e n (He = Ge t i F e ) Fe(kr)Hi(kr') r < r' (3.21) (21+1)PL(Cose) { G(r,r',k) = (-1/4nkrr')l; Hl(kr)FL(kr') r > r' 2 I n o r d e r t o accomodate Ho = - ( h /2m)A one sometimes w r i t e s G ( r , r ' , k ) = -(2m/ 2 4nfi ) e x p ( i k l r - r ' \ ) / l r - r ' 1 . The Lippman-Schwinger e q u a t i o n ( 3 . 2 ) a r i s e s as

,I k l )

Now l e t us use o p e r a t o r n o t a t i o n a s 2 i n 51.8 ( c f . Remark 1.8.7). I n s e r t t h e (2m/fi ) f a c t o r i n G as above and 2 2 then (Ho - E)G = -1 i n o p e r a t o r form, where E Ti k /2m. We r e c a l l t h e not a t i o n s ( r l U l r ' ) = & ( r - r ' ) U ( r ) , ( r l U I J / ) = 1 ( r l U l r ' ) d 3 r ' ( r ' l J / =) f d 3 r ' ( r l U I b e f c r e ( w i t h -G( I x - y l

i n (3.2)).

Q,

r ' ) ! b ( r ' ) = f d 3 r ' U ( r ' ) s ( r - r ' ) ~ ( r ' ) = U ( r ) $ ( r ) , e t c . Thus ( k " l G l k ' ) = ( k " l 3 (H - E ) - ' I k ' ) = ( 2 ~ )s ( k " - k ' ) / ( E ' - E ) ( n o t e I k ' ) 2, t h e e i g e n s t a t e w i t h E ' = 2O 2 fi k ' /2m and ( r l k ) = e x p ( i ( k , r ) ) . Now one can w r i t e (3.22)

G(r,r',k)

I

= [ 2 m / ? 1 ~ ( 2 ~ ) d~3] k ' [ e i ( k ' y r - r ' ) / ( k 2

i

[2m/t1~(2n)~]d3k'[(rlk')( k ' l r ' ) / ( k 2 (note 1/(k2

-

k"

t

iE)

+

2 2 P(l/(k -k' )

-

- k " + i ~ ) =]

- k" + i ~ ) ]

2 2 n i s ( k - k ' ) when

E +

0

-

c f . [Rgl,

W r i t i n g t h e i n t e g r a n d as J ( r l k ' ) s ( k ' - k " ) d 3 k " ( k " l r ' ) / ( k 2 - k " 2 t i E ) p.1121). w i t h ( 2 m / n 2 ) ( 2 n ) 3 d ( k ' - k " ) / ( k 2 - k " 2 t i € )= -( k ' I ( E - H + i c ) - ' I k " ) we o b t a i n 0

(3.23)

G(r,r',k)

= (rl(E-Ho+ie)-llr')=

ZIT)-^ 1 /( rl k ' ) d3k'( k ' I ( E - H o + i c ) - ' I k") d3kY k " l r ' ) The n o t a t i o n extends t h e n t o H w i t h ( o u t g o i n g ) wave f u n c t i o n s a- f r o m (3.2)

SCATTERING THEORY

-

+

we w r i t e here

(3.24) ( q k =+exp!i(

= @-(x,k) and r e c a l l t h a t (E-Ho+iE)-l

- H0 +

= qk + ( E

$:

iE)

-1

G t o obtain

%

+ Wk

and ( 6 0 ) G+ ( r , r ' , k )

k,x))

117

= (2n)-3/d3k'

(rI(E-H+iE)-'lr')

[ ( r l J / k ) ( $ k l I r ' ) / ( E - E ' + i ~( H) ] = Ho + U(x), l / ( E - E ' + i c ) = (E-E'+iE)-', and 1 3 I d k l $ k ) ( J , k l denotes t h e completeness r e l a t i o n ) . I n t h i s notation

=

one can a l s o w r i t e qpk) =

+

[$$and A

+

%

+

as above) ( 6 6 ) A

(cpk

U- i n Remark 3.4.

+

t h o g o n a l i t y o f t h e $k as f o l l o w s . qk

+ (E - H + ic)-'lbk.

= ( w 3 fd 3 k ' ( $ ~ l ) ( $ k l ( t~h ~ a t A+\

Using t h e s e n o t a t i o n s one can show o r From (3.24) we o b t a i n

(E

1~

+

Ek) ( 6 + ) J/k =

Use now t h e f a c t t h a t U and H a r e H e r m i t i a n opera-

tors t o get (3.25)

+ I$ k+I)

( $k

=

+

k I$ k I) + ( V k l U ( E k - H - i E )

(Ip

-

(Ek ( n o t e ( A b I $ ) = (cplUA*$) t h e f i r s t t e r m i s (cp

-

+

= (qkl$k')

+

iE)-l(VJkluld'k~) +

= (cplUIA*J/)

+ ,)

k ).k (Ekl-Ek-iE)"(cpklUlikI + JlkI) = 6 ( k - k ' ) . RaRARK 3.9.

Ek'

+

-1

i n t h e n o t a t i o n o f 51.8).

From (3.24)

+ + ( c p k j ( ~ ~ - ~ o + i E ) - ' u l ~=k (,c)p k l c p k l )

= (cpklcpkl)

Put t h i s i n t o (3.25) and l e t

E

+

0 t o obtain

+

($;I

Much o f t h e f o r m a l i s m and procedures o f p o t e n t i a l s c a t t e r i n g

can be c a r r i e d o v e r t o s c a t t e r i n g by o b s t a c l e s f o r -A + U ( x ) as w e l l as f o r more general d i f f e r e n t i a l o p e r a t o r s and we w i l l s k e t c h a few f a c t s f o l l o w i n g [ S c l ] ( c f . a l s o [Rnl;Lx2,3,4;Wcl;Ctl;Cp1,2]. Thus l e t n be t h e e x t e r i o r i n 3 2 R ( f o r s i m p l i c i t y ) t o a C h y p e r s u r f a c e r and q ( x ) r e a l w i t h q ( x ) e x p ( Z a \ x ) ) bounded and u n i f o r m l y

CY

H o l d e r continuous i n R

U

r

(a > 0 and 0 <

CY

< 1).

2 L e t H = -A + q be t h e s e l f a d j o i n t o p e r a t o r w i t h domain D(H) = { g E H (n) 2 3 w i t h g = 0 on I'l and Ho = -A i n L ( R ) f o r which we have t h e s t a n d a r d Fouri e r t h e o r y based on

f = ( l / 2 n ) 3 / 2 / f e x p ( - i ( c,x))dx.

Set

H

= f AdE,

with P

m

There w i l l be eigenvalues k2 < 0 w i t h n o r m a l i z e d e i g e n f u n c t i o n s 2 j s a t i s f y i n g H p . = k .cp. a l o n g w i t h 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 cp+(x,c) s a t j J J i s f y i n g tb+ = /ci2cpk i n R, cp+ = 0 on r , and v+ = cp, - e x p ( i ( X,S ) ) s a t i s f y i n g

= lo+dE,. cp

t h e r a d i a t i o n c o n d i t i o n s as 1x1 *

o f t h e form v i = O ( 1 x I - l ) w i t h (Dr +

= fR g ( x ) c p m d x and ? ~ ' ( 5 ) = i l c l ) v , = o ( ( x I - ' ) ( r = 1x1). We s e t (2d3/'& G,(x,S)g(x)dx f o r g E L2(n). j Then fn lg! 2 dx = lS".l2 + A+ 2 J 1 IS-(C)l dS and g ( x ) = i . Jc p j + ( 2 n ) - 3 / 2 / cp+(x,S)G'(c)dS. Given a c u t o f f

1

1

f u n c t i o n I.I(x),

li E

Cm, I,

= 0 i n s i d e and near

can d e f i n e W, - = l i m e x p ( i t H ) e x p ( - i t H o )

as t

r,

and u = 1 f o r l a r g e x one

.+ f r m .

These o p e r a t o r s w i l l be

118

ROBERT CARROLL W+exp(-itHo) (W+: L2 Ai f F-: f f , F: f

independent o f , & and s a t i s f y exp(-itH)W+ moreover ( W t f ) f

A t

+

f

*

+A

= f

t h e n W+ = F:

e t c . w i t h S = WW: -

= (W-f)?

for f E

i2.

-+

(F+W+ = F), W + f = ( Z T ) - ~ / ~ @ / +(x,c)i(c)dc, = F*(FtF*)F

(recall W

= F F i n §2).

-f

-f

PL2 ( a ) ) ;

i,and

F+:

W- = FrF,

We w i l l n o t d i s c u s s

t h e d e t a i l s here. Also (FSf)(k,w) = S ( k ) F f ( k S - ) ( w ) , i s a frequency n o t a t i o n , 2 n-1 ) L2(Sn-') and (k,w) r e p r e s e n t s p o l a r c o o r d i n a t e s . w i t h S(k): L ( S -+

REmARK 3.10.

L e t us r e c o r d a few more formulas i n v o l v i n g r e s o l v a n t s and F i r s t t a k e (*+) w i t h K(z,H)

s p e c t r a l f a m i l i e s ( c f . [AgZ;Wdl]).

H-z and

pass t h e o p e r a t o r s t o g t o o b t a i n (assume a and b a r e p o i n t s o f c o n t i n u i t y ) (3.26)

(f, [E( b ) - E ( a ) ] g ) =

;1:

(1/2ni

)I

b

a

(f,

[i( t t i ~ ) - F t(- i e ) ] g ) d t

From [Ag2] one can say t h a t f o r s u i t a b l y s h o r t range p o t e n t i a l s U ( x ) t h e boundary values E f ( t ) = l i m E(z), as +Imz

-J+

R;(t)f

-

.*+

R;(t)U(x)?'(t)f

-+

0, e x i s t , and i n f a c t i ' ( t ) f

=

( h e r e F ( z ) i s t r e a t e d i n a c e r t a i n space o f opera-

t o r s which we l e a v e undefined f o r t h e mment a t l e a s t

-

t h e a s s e r t i o n about

u+

R- a r e r e f e r r e d t o as t h e l i m i t a b s o r p t i o n p r i n c i p l e which we d i s c u s s b r i e f l y i n Remark 3.11).

/,"

-

(f,[K+(t)

Thus i n (3.26) one can w r i t e (f,[Eb-Ea]f)

E-(t)f]f)dt.

= (1/2ni)

L e t F+ be t h e t r a n s f o r m s based on @+ as i n Theo-

= 1 lFif12(c)dC rem 3.2 e t c . and then one can show-that ( & m ) (f,[Eb-Ea]f) 2 ( i n t e g r a l o v e r a < 151 < b ) . A r e l a t e d k i n d o f formula comes up i n (+*)

and was n o t e x p l a i n e d t h e r e ; we g i v e a b r i e f d i s c u s s i o n here.

First a l i t t l e 1i m = E40 ( E / R )

algebra g i v e s from (3.26) as r e w r i t t e n above (+*) (f,[Eb-Ea]f)

Ib I I r ( t ? i E ) f l 1 2 d t . But l l ~ ( t + i E ) f W 2= l l F [ E ( t f i ~ ) f ] l l 2 by Parseval and one can a show t h a t (+A) F ( i i ( z ) f ) ( c ) = ( ] e l 2 z)-'?(c,z) where r ( c , z ) = (2n)-'I3

-

I

-

1 f(x)[v(x,c)

E(Z)(U(-)v( ,,c))(x)]dx

f o r w a r d as i n [AgZ, p.1881. by (+@)@,(x,c)

= exp(i(x,c))

(P = exp(i(x,c));

this i s straight-

But 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 s

@+

are given

-

"R(1c12)[U(.k(.,t)I(x) ( c f . here-(&*)). 2 2 Note t h a t i f @ = P + v s a t i s f i e s tP = 151 @ then ( H - l c l )U = -Up so t h a t

@ =

( H - ~ C . ~ ~ ) - ' [ U ~ I .It f o l l o w s t h a t ?+(c,lcl 2 ) = l i m ( E 4 0 ) ?(ey1c1 2 * i E ) = ( F + f ) ( t ) and f o r m a l l y one has ( E / T ) { I I F [ r ( t ? i c ) f l l 2d t = ( E / T ) $ b 1 ( l e i 2

v -

2 2-

-

2 -1

2

-

:,+Jc)dc.

But i n a standard way v i a t h e + E ) I f ( c , t + i E ) l dgdt = f 2 Poisson i n t e g r a l technique one sees t h a t f+(c,151 if a < 15l2 < b w i t h ^G,E(c) 0 o t h e r w i s e . Hence f o r m a l l y (f,[Eb-Ea]f) = f I F + f l2dc ( i n t e -

t )

ttE(c)

N

-f

-+

gral over a <

1c I2

< b).

T h i s can a l l be done r i g o r o u s l y as

in [AgZ]

and a

l i t t l e f u r t h e r d i s c u s s i o n i n t h e same s p i r i t w i l l e s t a b l i s h (**) ( c f . a l s o [Ikl;Ag2;Ahl]).

REMARK 3.11.

L e t us d i s c u s s b r i e f l y t h e ideas o f l i m i t i n g a b s o r p t i o n and

SCATTERING THEORY

l i m i t i n g a m p l i t u d e (see e.g.

[Rnl]).

119

Thus l e t L be a l i n e a r o p e r a t o r i n a

H i l b e r t space H and P: H + PH = E an o r t h o g o n a l p r o j e c t i o n ( t a k e f o r s i m p l i 2 c i t y H = H , L s e l f a d j o i n t , and PH = E a s u i t a b l e space o f f u n c t i o n s w i t h Consider ( + 6 ) utt + Lu = f e x p ( - i k o t ) ,

compact s u p p o r t ) .

ko > 0, f

E

H, w i t h

u ( 0 ) = u t ( 0 ) = 0. The l i m i t a m p l i t u d e p r i n c i p l e says t h a t Pv = l i m L l / T ) T 2 f0 P u ( t ) e x p ( i k , t ) d t e x i s t s and s o l v e s (**) Lv - kov = f. Since ( M )may have many s o l u t i o n s , ( 4 6 ) + t h e l i m i t i n g procedure d e t e r m i n e a s e l e c t i o n mechanism t o s e l e c t u n i q u e l y a s o l u t i o n t o ( W ) .

Another such s e l e c t i o n

mechanism, t h e l i m i t i n g a s o r p t i o n p r i n c i p l e , i n v o l v e s s o l v i n g ( W ) ( L

2 ( k o + i c ) )vE

= f,

E

> 0, and s e t t i n g Pv = l i m P v ( k o + i c ) ,

l i m i t e x i s t s ; t h i s s o l v e s (++) i n some sense. %

l i m R((ko-iE)2,L)

E

+

-

0, when t h i s

Evidently l i m (L

-

(ko+iE)2)-1

i n the s e l f adjoint situation.

4- SCAttERINP; CHE0RY- 111 (A M E W W OF CHEmES).

We go now t o some ideas and

machinery which were designed i n p a r t i c u l a r f o r t h e i n v e r s e problem (and app l i c a t i o n s i n s o l i t o n t h e o r y , e t c . ) b u t which i n t e r a c t w i t h m a t e r i a l i n 523.

We w i l l develop t h i n g s i n a Remark format a g a i n w i t h o u t p r o v i d i n g a l l o f

t h e d e t a i l s b u t w i t h enough d i s c u s s i o n , c a l c u l a t i o n s , r e f e r e n c e s , e t c . i n c l u d e d t o enable t h e i n t e r e s t e d r e a d e r t o e a s i l y f i l l i n t h e gaps.

A certain

e f f o r t i s made here t o i n d i c a t e v a r i o u s n o t a t i o n s and p o i n t s o f view so t h a t one can r e a d t h e l i t e r a t u r e and compare r e s u l t s , e t c .

I n p a r t i c u l a r we have

e x t r a c t e d m a t e r i a l f r o m [Kyl] t o p r o v i d e a p r o d u c t i v e i l l u s t r a t i o n o f t h e b r a - k e t n o t a t i o n (and o f course some o f these r e s u l t s were d i s c o v e r e d u s i n g t h i s notation

-

cf.

[Kyl]).

f o l l o w i n g [Fal;Chl], we conF i r s t f o r 1-0 s i t u a t i o n s on ( - - , m ) , 2 + SJ, = k J, f o r -m < x < m where q i s r e a l w i t h s i d e r t h e e q u a t i o n (*) -J, xx 2 say 11 ( l + l x l ) I q ( x ) l d x < m (we w i l l make whatever hypotheses on q a r e use-

REmARK 4.1.

f u l so t h a t t h e " s t r u c t u r a l " formulas f i t t o g e t h e r c o n v e n i e n t t o assume t h e r e a r e no bound s t a t e s , i . e .

-

i n particular it i s no d i s c r e t e spectrum,

a l t h o u g h t h e y can be worked i n t o t h e t h e o r y i f p r e s e n t ses on q see [Chl;Dfl;Fal,31).

-

c o n c e r n i n g hypothe-

L e t us r e c a l l from 51.8 t h a t one determines

J o s t s o l u t i o n s o f (*) v i a (4.1)

f+(k,x)

=

f-(k,x)

e ikx

-

= e-ikx

so t h a t f + ( k , x ) e x p ( + i k x )

-+

Lm [ S i n k ( x - t ) / k ] q ( t ) f + ( k , t ) d t ; +

lt

1 as x

[Sink(x-t)/k]q(t)f-(k,t)dt -+ + m .

One can " c o n s t r u c t " t h e s e f u n c t i o n s

f,- v i a i t e r a t i v e procedures as i n 51.6 and f o r r e a l ' k one has

c(f (k,x), +_

120

ROBERT CARROLL

f*(-k,x))

-

-

= + 2 i k , f+(-k,x)

f g ' here).

independent)

= f+(k,x),

v

= f-(k,x)

(W(f,g)

= f'g

+

0 we have f + ( k , x ) and f + ( - k , x ) l i n e a r l y = cllf+(k,x) + c12fi(-k,x); f + ( i , x ) = ~ ~ ~ f - ( k , +x )

One w r i t e s ( f o r k (A

-

and f - ( - k , x )

) f-(k,x)

~ ~ ~ f - ( - k , xand ) 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 ( 0 ) c ( k ) = c 2 1 ( k ) , cij(-k) = cij(k), and Ic1212 1 t lcllI

( k ) = -c22(-k), c12 L a t e r we w i l l a l s o

'4 .

w r i t e c21 = c12 = 1/T, c22 = RL/T, and cll = Rr/T. The k c . . ( k ) a r e c o n t i n u 1J ous f o r k r e a l , c12(k) i s a n a l y t i c f o r Imk > 0 w i t h ( p o s s i b l y ) a f i n i t e num-

-,

O(1) as I k l

b e r o f s i m p l e zeros k . on t h e i m a g i n a r y a x i s , kcii(k) J % 1 + a/k t o(l)/k as I k l + m .

and

a,

c 1 2 ( k ) = c21(k)

Next as i n §1.8 one c o n s i d e r s s o l u t i o n s o f (*) h a v i n g t h e a s y m p t o t i c b e h a v i o r (4.2)

{

$,(k,x)

$2(k,x)

e x p ( i k x ) t s12exp(-ikx)

x

s,. . exp( ikx ) s22exp( - ik x )

x-+-

,

-i

-+

-m

exp(-ikx) + s21exp(ikx) ~. The n o t a t i o n ( 6 ) s , ~= TL, s12 - RL, s2, = Rr, and s22 = Tr i s a l s o used and i n f a c t one has ( + ) sll s21 = sij(k),

e

s12 = c22/c21, s22 = l / c 1 2 ( = sll

I t f o l l o w s t h a t f o r r e a l k,sl,s2h 2 2 + 1s121 = 1 = +

11lc 12. 2 and lsllI

= 1 + O(l/lkl),

T = sll

= l/cPl,

t

s,2s22 = 0, s . . ( - k )

, while

and

= T), =

1J

as I k l

+ m,

s12(k)

and sll = 1 t O ( l / l k l ) . F u r t h e r T = Tr = = s22 i s a n a l y t i c f o r Imk > 0 except f o r a p o s s i b l y f i n i t e number

s21

= O(l/\kl),

o f s i m p l e p o l e s on t h e imaginary a x i s (these a r e bound s t a t e s and we w i l l assume t h e r e a r e none f o r convenience here - l a t e r when d e a l i n g w i t h s o l i t o n s e t c . t h e bound s t a t e s a r e welcomed o f course). The m a t r i x w i t h elements sij i s c a l l e d t h e s c a t t e r i n g m a t r i x S. = (( s . .)) and one has e v i d e n t l y p ) ILl(k,x 1J

= sllf+(k,x)

and $ 2 ( k , x ) = sp2f-(k,x). asymptotic behavior) (4.3)

cll(k)

c12(k) = 1

-

= (1/2ik)f:

(1/2ik)l:

L e t us r e c o r d a l s o ( f r o m ( 4 . 1 ) and

q ( t ) f - ( k , t ) e - i k t d t ; ~ ~ ~ = (( l /k2 i )k ) l I q f t e ik t d

qf-eiktdt

= c21 = 1 - ( 1 / 2 i k ) l :

qfte -iktdt

By Paley-Wiener ideas ( c f . 51.6) one can w r i t e a l s o (AR

sometimes)

( M )f,(k,x)

= exp(ikx)

Lt

+

c

A,

and AL = A

AR(x,s)exp(iks)ds and f - ( k , x )

=

exp(-ikx) + AL(x,s)exp(-i ks)ds which p r o v i d e some b a s i c t r a n s m u t a t i o n k e r n e l s AR and AL t o be s t u d i e d l a t e r . Note a l s o ReRr/IReRrI = -T 2/ I T 1 2 .

REIIIARK 4.2. (4.4)

Now t h e 1-D Lippman-Schwinger e q u a t i o n i n v e c t o r f o r m i s $(k,x)

=

IL0

-

(i/2k)f:

e k t x - y l q ( y )$ ( k ,y )dy

SCATTERING THEORY

121

ikx ,e-i kx

1

It f o l l o w s t h a t $(-k,x) = $(k,x)(co umn v e c t o r ) . a 1 f o r k r e a l and s e t t i n g H = O ) and 1 (1), so t h a t $, = e x p ( i k H x ) l , one 0 -1 has as 1x1 + m y w i t h E = x/ X I , $(k,x) = $,(k,x) + A(k,E)exp(iklxl) + o ( l ) ,

where !bo = ( e

2

-

Thus i n f a c t J, and G2 i n (4.2) a r e t h e components o f $ ( c f . [NwZ]) and $ = 1 (I)~,J,~), column v e c t o r , T, a- i n 92.3. ( E = 1 ) one I n p a r t i c u l a r as x -+ has JI1

T,

exp(ikx) + (1/2ik)exp(ikx)[/I

exp(-iky)q(y)$,(k,y)dy]

= exp(ikx)Tx

and J12 T, e x p ( - i k x ) + ( 1 / 2 i k ) e x p ( i k x ) [ / I exp(-iky)q(y)$,(k,y)dy] = exp(-ikx) + R re x p ( i k x ) w h i l e as x - m ( E = - 1 ) one has T, e x p ( i k x ) + ( 1 / 2 i k ) i z q ( y ) exp(iky)$l(k,y)dy(exp(-ikx) = e x p ( i k x ) + Rxexp(-ikx) and !b2 T, e x p ( - i k x ) + (1 / 2 i k)exp( - ik x ) / I exp( ik y ) q (y)IL2 (k,y )dy = Trexp( - ik x ) . Then one can d e f i n e -+

9(k,x) 1 1 and p ’ ( k , O ) = i k ( - l ) ; ( c f . [NwZ])

= J(k)$(k,x)

(*A)

t o be t h e s o l u t i o n o f (*) w i t h v(k,O)

=

-L

q(y)q(k,y)dy

i t f o l l o w s t h a t (**) 9 ( k , x ) = $o + ( l / k ) / t

Sink(x-y)

i s t h o u g h t o f as d e f i n i n g t h e J o s t ( m a t r i x ) f u n c t i o n 0 1 J(k). S e t t i n g Q = (1 o ) and 5 = ( T Rr) = I + A one knows t h a t kT = QkQand R.c T 0 1 a l i t t l e c a l c u l a t i o n ( e x e r c i s e ) shows t h a t JT = TPJP ( P = HQ = (-1 o ) and # # Set f ( k ) = f ( - k ) now and t h e n q# = Qq w i t h J, = d e t J = 1/T, T = T = Tl). Q#I)y i e l d s

and

(*A)

(*&) Jf-l

=

$-’QJ-’Q.

Given 9 t h e d e t e r m i n a t i o n o f J f r o m (*&)

i n v o l v e s s o l v i n g a Riemann-Hilbert (R-H) problem (see [NwZ;Gdl ;Mvl] f o r R-H problems; r e l a t i v e t o s o l i t o n t h e o r y t h e r e i s a d i s c u s s i o n i n [Fa31 and we

w i l l say more about such problems i n 511 - i n t h e p r e s e n t d i s c u s s i o n one i s assuming t h e r e a r e no bound s t a t e s ) . G e n e r a l l y t h e e n t r i e s o f J a r e n o t d i r e c t l y e x p r e s s i b l e v i a s c a t t e r i n g d a t a i n $.

(*+) J = ( l / T ) ( A

-?).

I f however q = 0 f o r x

A

h(x,y)exp(ikHy)Tdy where i n f a c t h = h(A

2-

(Dx

-

q(x))h(x,y)

0 then

F u r t h e r s i n c e 9 ( k , x ) w i l l be e n t i r e i n k o f exponen-

t i a l t y p e x (as i n 51.6) one has by Paley-Wiener ideas (*.)

-

<

2

= Dyh(x,y);

-2Dxh(x,x)

and f o r

= q(x); h(x,-x)

Ip(k,x) = expikHxl

IYI

5 1x1,

= 0.

t i o n s can be d e r i v e d as i n §1.6 f o r example by s e t t i n g (*.)

i n (*.)

Theorem 1.6.9 and [NwZ]) o r v i a e q u i v a l e n t methods i n CC3;Chl;Fal;Lal We n o t e here a l s o t h a t , s e t t i n g f ( k ) = J - l ( k ) and $-’(k)

(A*)

These equa-

= o(k),

(cf. ,2;Mrl].

one can as-

- I E L 2 (R) w i t h f a n a l y t i c f o r Imk > 0, f ( k ) -,I as I k l -+ m y and # (*&) f ( k ) = f ( - k ) = a ( k ) Q f ( k ) Q f o r k E R ( r e c a l l a g a i n we a r e d e a l i n g w i t h

sume u

t h e case o f no bound s t a t e s f o r convenience).

Ri3llARK 4.3.

We r e c a l l a g a i n t h e b r a - k e t n o t a t i o n f r o m p h y s i c s i n o r d e r t o

e s t a b l i s h c o n t a c t here w i t h [Kyl;Msl-41 Remarks 1.8.7 and 2.3.8).

( c f . [Rgl;Skl]

f o r n o t a t i o n and see

There w i l l be some h o p e f u l l y p r o d u c t i v e r e p i t i t i o n .

122

ROBERT CARROLL

2

Ho = -A i n s t e a d o f ( - 5 /2m)A.

For Hamiltonians we work w i t h e.g. speaks ( i n R3

- for

R r e p l a c e ( 2 ~ by ) ~ZIT) of k e t s

I$)

Thus one

I$)*

and b r a s

=

($1

(rlJ/) =

$ ( r ) ,( v I $ ) * = ( $ l v ) , ( r l k ) = e x p ( i k - r ) ( k - r = ( r , k ) ) , 3 3 3 6 ( r - r ' ) = ( r l r ' ) , I$) = I I r ) d r( r l $ ) , I I r ) d r(r1 = 1, ( k l k ' ) = ( Z I T ) 6 ( k -

w i t h e.g.

I$) =

k ' ) , 1 = ( 2 n r 3 J lk)d3k( k [ ,

I $(r)exp(-ik-r)dry etc. state

I$)

(2n)-3 I l k ) d 3 k ( k ( $ ) , ( k l $ ) = ( F $ ) ( k ) =

F u r t h e r n o t a t i o n i n v o l v e s r e p r e s e n t i n g an eigen-

f o r Ho v i a IHo;E) f o r example where HolHo;E)

wants t o t a k e i n t o account o t h e r o p e r a t o r s A. values a r e a, one can w r i t e IHo,Ao;E,a) a r e i m p o r t a n t as w i l l be seen below). Ao;E,alHo,Ao;F,b)

I

= 6(E-F)6(a,b)

= EIHo;E).

I f one

commuting w i t h Ho whose eigen-

f o r an e i g e n s t a t e ( t h e s e Ao,a terms Thus orthonormal i z a t i o n becomes ( \ H o ,

and completeness can be r e p r e s e n t e d by

= I ( i n t e g r a l s a r e over s u i t a b l e ranges and a r e

~HoyAo;E,a)dEda(Ho,Ao;E,a~

t o be i n t e r p e r t e d i n s u i t a b l e ways

-

i n p a r t i c u l a r one w r i t e s 6(E-Ho) =

I 6(E-Ho)dE = I ) . A l s o ( r l q l r ' ) = 6 ( r - r ' ) 3 I ( r 1 q l r ' ) d r ' ( r ' l $ ) = I d 3r 1 6 ( r - r ' ) q ( r ' ) $ ( r ' ) = q ( r ) $ ( r ) , with

1 ~Ho,Ao;E,a)da(Ho,Ao;E,a~

q(r), (rlql$> = e t c . Now r e c a l l from Remark 2.3.8 t h e Green's o p e r a t o r ( i n 3-D, w i t h t h e F o u r i e r t r a n s f o r m as above

-

F$ =

outgoing

I $ e x p ( - i k - r ) d r ) f o r Ho

= -A

(AA) G ( r , r ' , k ) = - ( 1 / 4 a ) e x p ( i k l r - r ' l ) / l r - r ' I (we sometimes abuse n o t a t i o n i n 2 u s i n g k and I k l i n t e r c h a n g a b l y when no c o n f u s i o n can a r i s e ) . Then (-A-k )G 2 = - 6 ( r - r ' ) ( o r (Ho - E)G = -1, E % k ) and i n o p e r a t o r n o t a t i o n G ( r , r ' , k ) =

.

L e t us n o t e how t h e n o t a t i o n i s u s e f u l i n p r o v i n g com-

p l e t e n e s s f o r Lippman-Schwinger (L-S) t y p e e i g e n f u n c t i o n s . w r i t e s t h e L-S e q u a t i o n i n t h e form ( c f . ( 3 . 3 ) )

I G(x,y,k)q(y)$

+ (k,y)dy

(A*)

where q 0 ( k , x ) = e x p ( i ( k , x ) ) .

Thus i n 3-D one

$+(kyx)

=

$o(k,x) +

I n o p e r a t o r form t h i s

i s (one always imagines E 0 a f t e r c a l c u l a t i o n s a r e completed - c f . ( & + ) i n + 32.3 e t c . ) ('4) $+ = $o + ( E - Ho t is)-'q$+. Now c o n s i d e r G ( r , r ' , k ) = -f

(rl(E-H+ic)-'lr'

(4.6)

)

where H = -A+q

G+ ( r , r ' , k )

Ho + q; then ( E

= ( 2 a )-3 I d3 k C ( r 1 $ ) ( $ i , / r ' ) / ( E

%

2

k )

- E' +

i ~ ) ]

-1 ' 6 ) one has (A+) $; = $o + ( E - H + i s ) q$o. We I n p a r t i c u l a r f r o m (A*) - ( k k' 3 + + know t h a t ( $ o \ $ o ) = (2n) 6 ( k - k ' ) ( F o u r i e r t h e o r y ) and t o show ( $ I$ ) = + + k + k k kl] + + ($olq(Ek-H-iE) IJ/kl) ( 2 1 ~ ) 3 i ( k - k ' )We W r i t e f i r s t ( A m ) ( $ I$ I ) = k + k+ = ( $ I $ I ) + (Ek - E~~ - i E ) - l ( $ L l q l $ k , ) ( s i n c e H and q a r e t r e a t e d as s e l f o k k k' 6 ' ( t h e f i r s t t e r m i s W k l ! b + ) = ( J , I$ ) + a d j o i n t o p e r a t o r s ) . Then from ) -1 k +kg 0 0 k ( $ o I ( E k , - H o + i ~ ) - l q /+~ k , =) ( $ k I$ k ' ) + ( E k l - E - i E ) ( $ o l q l $ k l ) which when adO o + + kk k ' ded t o ( A m ) g i v e s , when E -+ 0, ( $ k l $ k l ) = ($ol$o ) = 6 ( k - k ' ) ( 2 n ) 3 ( r i g o r can be s u p p l i e d as i n [Rgl]).

Thus ( c f . Remark 2 . 3 . 8 )

SCATTERING THEORY

123 t

The 3-D L-S 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 $ (k,x) 3 + + s a t i s f y (27)- ( $ k l $ k I ) = 6 ( k - k ' ) .

eHE0Rm 4.4, (A*)

determined by

The same k i n d o f theorem appears t o h o l d f o r t h e 1-D s i t u a t i o n

RrmARK 4.5.

o f ( 4 . 4 ) ( w i t h ( 2 a ) - l r e p l a c i n g w - ~ ) . Thus c o n s i d e r ( 4 . 4 ) and w r i t e G(x, 2 2 y,k) = - ( i / 2 k ) e x p ( i k l x - y l ) so t h a t (-D - k ) G = - 6 ( x - y ) and (Ho E)G ='-1.

-

Then (4.4) says (a*) $,(k,x)

= exp(ikx)

e x p ( - i k x ) + (E-Ho+iE)-'q$,(k,x). a p p l y w i t h $,

Q

+ (E-Ho+iE)-1q41(k,x)

and IL2(k,x) =

Hence t h e same arguments would seem t o However t h e theorem i s n o t t r u e w i t h o u t s u i t a b l e

exp(?ikx).

i n t e r p e r t a t i o n ( J da i s m i s s i n g h e r e and t h e c o r r e c t f o r m i n v o l v e s

I);

(

2a)$

($(k,x),$(k,y))dk

i n c o r p o r a t e s J da.

= 6 ( x - y ) can be proved d i r e c t l y as i n [ C l l ]

(Imx

dEda (1/

and t h i s

= -T(x)fl(x,x)f2(y,x)/2ix

for y < x

-

0 ) and we n o t e here i n o u r use o f F o u r i e r t y p e o p e r a t o r s i n [C3;17] 2 2 R2 % -A w i t h u, fl = f+ and u2 2, f2 = f-. Here (D -g+x )

>

T and Rr =

%

)

Indeed as i n Theorem 1.6.6 we c o n s i d e r a Green's f u n c -

t i o n ( f l = f+, f2 = f-, X = k ) R(x,y,x) p

/I

t h i s w i l l be discussed below. A c t u a l l y ( r e c a l l F ( k , x ) = $ ( - k , x ) )

R = s ( x - y ) and we r e c a l l W ( f l , f 2 )

Now t o use R t o determine com-

= -2ix/T.

p l e t e n e s s r e l a t i o n s one can proceed as i n Theorem 1 . 6 . 6 w i t h c o n t o u r i n t e g r a 2 p l a n e ( i t i s assumed t h a t t h e r e a r e no bound s t a t e s ) .

t i o n i n the E =

One o b t a i n s f o r m a l l y ( n o t e dE = 2Adx) 6 ( x - y ) = (1/2ni)L:

if

can be "developed" i n v a r i o u s ways.

o b t a i n f o r m a l l y t h e i n v e r s i o n o f [C3;17], f,(y,x)dx L

(p =

T).

where

namely 6 ( x - y ) = ( 1 / 2 n ) / I pfl(x,h)

On t h e o t h e r hand, one has a l s o a ( x - y ) = ( 1 / 2 n ) / r [Tf,

+ Ttl ( x , - x ) f 2 ( y , - x ) ] d x

(x,A)f2(y,x)

R(x,y,,\)ZAdX

I n p a r t i c u l a r w r i t t e n d i r e c t l y we

and f; =

f2 - Rrfl

with

With a l i t t l e c a l c u l a t i o n ( c f . a l s o Remark 4. 4 ) we see t h a t

?$;;

+ $$;:

and 6 ( x - y ) = ( 1 / 2 n ) / y ( $ ( ~ , x ) , $ ( x y ) ) d h ( c f . [Chl

CHE0REm 4-6.

One has o r t h o g o n a l i t y r e l a t i o n s i n t h e form (A)

2~)iI Tfl(x,i)f2(y,x)di x),$(A,y))dx

(p

=

( r e c a l l LL1 = Tf,

T) o r e q u i v a l e n t Y ( B ) =

Tfl

6(X-Y)

=

and $2 = T f - = T f 2 ) .

Then from J$ = q we have (bound s t a t e s can be accomodated i f p r e s e n t )

C 0 R 0 U A R y 4.7, J*(k)]-'

10" ( q ( k , x ) , d p ( E ) q ( k , y ) ) = 6 ( x - y ) where dp/dE = ( 1 / 4 n k ) [ J ( k ) f o r E = k2 > 0 '(thus (dp/dE)dE = (dp/dE)Zkdk = ( l / Z n ) ( J J * ) - ' d k ) .

(l

When q = 0 f o r x

<

REmARK 4.9,

Re). Rr 0 L e t us s k e t c h a d e r i v a t i o n o f a G-L e q u a t i o n based on C o r o l l a r y

4.7 and (*.)

f o l l o w i n g [NwZ];

then we w i l l g i v e a d i f f e r e n t approach f o l l o w -

REmARK 4.8,

0 one has ( J J * ) - l =

i n g [Kyl] i n o r d e r t o h e l p t h e r e a d e r develop s k i l l i n t h e o p e r a t o r f o r m a l ism.

T h i s w i l l t a k e t h e form o f a l o n g d i s c u s s i o n , w i t h i n t e r s p e r s e d

124

ROBERT CARROLL

remarks, r e v e a l i n g much s t r u c t u r e .

The ideas o f [ K y l ] were p i c k e d up a g a i n

i n [ F a l l and l a t e r , i n a somewhat d i f f e r e n t c o n t e x t , by t h e a u t h o r i n [C2,3, 10,20,21].

as

Thus w r i t e ( * m )

v

= (6-h)v0 = (6-hq)vo =

t e r r a t y p e o p e r a t o r and has an i n v e r s e

K:vo.

T h i s i s a Vol-

= (K:)-’ = 6-ho (we w r i t e a l s o h* q q Now i t i s c o n v e n i e n t t o work w i t h row v e c t o r s 9 , J / , e t c . so

(x,y) = h(y,x)). T t h a t J/J = v and ( J / , $ ) =

W* =

KO

v J T - l ( v J T - l ) * = V ( J J * ) - ’ ~ V *f r o m which i t f o l -

lows t h a t one should use dp = ( 1 / 4 n k ) ( J J * ) - I T = ( 1 / 4 ~ k ) ( J * ~ J ~ ) i-n’ t h e completeness formula based on row v e c t o r s

1 @odpov: = 6 = 1 vdpv*.

.

Therefore using

Now dpo = (1/4ak)and one can w r i t e

v

=

K:vo

we have f o r m a l l y (K:

=

-

K ) (@A) K:i q0dp9,* = K * - l * ! :K vodpov: = K.: But ( K * - l K:)(x,y) h: oq oq oq ’ = h:(x,y) f o r IyI < 1x1 so s u b t r a c t i n g i n (.A) one o b t a i n s (x,y) - h:(y,x) t h e G-L e q u a t i o n ( I y I < 1x1)

T h i s can t h e n be rephrased i n terms o f column v e c t o r s u s i n g dp from C o r o l l a r y 4.7 ( c f . [ N w ~ ] ) . The approach o f [Kyl] goes as f o l l o w s ( c f . a l s o [C2,3;FklY2;Fal;Wbl]).

For

s i m p l i c i t y we w i l l n o t use 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 s IHo,Ao;E,a) r a t h e r o n l y IHo,E);

however t h e o p e r a t o r n o t a t i o n

(00)

but

6(E-Ho) = i IHo,Ao;

w i l l s t i l l have t o be c l a r i f i e d ( c f . Remark 4.10). One begins by l o o k i n g f o r t r a n s m u t a t i o n o p e r a t o r s U such t h a t WU*U I and UWU* -1 -1 = I w i t h HU = UHo ( W = U U* and H i s d e f i n e d b y H = UHoU-’). Then e v i d e n t l y HU(H0 ;E) = UHoIHo;D= EU(Ho;E) SO\H;E) = UIHo;E) i s an e i g e n v e c t o r E,a)da(Ho,Ao;E,al

f o r H.

A l t e r n a t i v e l y one s t a r t s w i t h H and Ho and l o o k s f o r U w i t h W b e i n g

a measure o f t h e l a c k o f u n i t a r i t y . F u r t h e r one l o o k s f o r t r i a n g u l a r i t y U = 1 + K and U-l = 1 + L w i t h ( x l K l y ) = 0 f o r y > x and ( x l L l y ) = 0 f o r y > x. We w r i t e a g a i n H = Ho + q and one n a t u r a l l y c o n s i d e r s an o p e r a t o r L -S equat i o n o f t h e form ( c f . [Aml;Kyl;Rgl]

-

n o t e t h e r e i s some c o n f u s i o n i n v o l v i n g

o p e r a t o r s and t h e i r k e r n e l s i n h e r e n t i n t h e n o t a t i o n

-

this i s clarified i n

Remark 2.10) (4.8)

U,- = I + €lim 40 (E-HO?iE)-’qU+6(E-Ho)dE -

Here one extends

(Am)

f o l l o w i n g [Ahl;Ag2;Ikl],

-

i n t h e f o r m ( o h ) J/’

(Ah)

Uf

%

( l / 2 ~ ) ~ /e~x p/ ( - i ( k , x ) ) f ( x ) d x

= Go + (E-Ho+ie)

-1

f

qJ/ and

F*F where F denotes F o u r i e r t r a n s f o r m ( F f = f

with

q0

= e x p ( i ( k , x ) ) and F+f - =

SCATTERING THEORY

( n o t e a+ - % $-+

J ST(x,k)f(x)dx

%

$

T

i n [Ahl]).

125

Under s u i t a b l e hypotheses on

t h e p o t e n t i a l q (which we m e r c i f u l l y w i l l n o t d i s c u s s h e r e ) U n, W (wave ? t 2 o p e r a t o r s ) , where JCac = U+scU: - - (JC = L and Jcac i s t h e s t a n d a r d subspace o f a b s o l u t e c o n t i n u i t y ( c f . [Shl;Rl] and §2.2) and W+- = l i m e x p ( i t H ) e x p ( - i t H o ) as t

-+

?m

(h = 1).

s e t s A+ U, m a t r i x i s ti

i n (4.8), %

WTW-

tinguished here).

(4.9)

Thus i n a n o t h e r n o t a t i o n ( c f . [Rgl] f o r example) one

%

I + (E-Ho?iE)- 1qA+, and t h e n t h e s c a t t e r i n g -

i . e . A+ UTU- % A!At

( o p e r a t o r s and t h e i r k e r n e l s must be d i s -

One sees how

U-+ f = (2n)-3"1

i s r e l a t e d t o (4.8) v i a

(0.)

$r(k,x)ff

Thus i n o p e r a t o r n o t a t i o n ( a * ) U,

-

%

riel f o r m ) .

REmARK 4.10.

L e t us t r y t o u n r a v e l t h e n o t a t i o n .

The s t a n d a r d way o f d e a l -

i n g w i t h t h e symbol 6 ( E - H _ ) goes as f o l l o w s ( c f . [ R g l ] ) . One s e t s (om) -hi u 1 6(E-Ho) = ( E - H o t i E ) - l - ( E - H o - i E ) But s t a n d a r d formulas i n t h e t h e o r y o f

.

s e l f a d j o i n t o p e r a t o r s ( c f . [Ka2;La2;Rl]

and 5§2.2,2.3)

give formally, a t

p o i n t s o f c o n t i n u i t y o f F, 2niF(A) = JA [R(E-ie) - R(E+i€)]dE where R(E+ic) -1 = (E-H ? i E ) and F(A) denotes t h e s p e c t r a l r e s o l u t i o n f o r Ho. Thus f o r m a l l y 0 6(E-H )

+o forly-, (4.10)

%

On t h e o t h e r hand r e l a t i v e t o an e i g e n f u n c t i o n expansion

dF(E).

i.e.

(k

n,

(k,e^) - c f . [Rsl])

I

f ( x ) = ( l / 8 ~ ~ ) $ 2$'(k,s,x)l S

T+(k,s,y)f(y)dyd2Gk2dk -

o r f o r t h e F o u r i e r t r a n s f o r m ($o = e x p ( i ( k , x ) ) ) (4.11)

1

f ( x ) = ( l / 8 ~ ~ ) / ~ 2" $ o ( k y G , x ) / To(k,e^,y)f(y)dyd2
one has ( c f . [La2])

H and w i t h dE

%

f o r F t h e s p e c t r a l f u n c t i o n as above ( r e l a t i v e t o Ho o r

2kdk) (&*) (dF/dE)f

where $ denotes $'

(k/16n3)Js2$(k,~,x):

o r $o r e l a t i v e t o H o r Ho.

( E " 2 / 1 6 ~ 3 ) J 1 $ o ( k , ~ , x ) ) d 2 b ? 0 (k,;,y)l (00)

%

Hence (6.1

6(E-Ho)

n,

dFo

and t h i s s h o u l d c l a r i f y t h e n o t a t i o n

( 2 ~ ) - ~I k/ ' ) d 3 d ' 6 ( E - E ' ) ( k ' l ) .

( t h e l a s t term also

-

$(k,G,y)f(y)dyd2G

Note f o r m a l l y v i a

(4.9) o r ( a + ) (4.12)

I

le

3

1 'j l $ F ( k o y ~ ' , x )d k ' ) e -i(k = 1 I$,(k',~',x))dk'6(k'-k)kl

U+6(E-Ho) = (1/21~)

i(k"ey),d2;(

e-i( ke",c) Idy

= (E1/*/l6n3)J,

S

I$+(k,G,x))d 2.e($o(k,s,c)l -

I$'

,y

)

I(

~

6

d 2 & e -i( k2,c)

~

~

I 11 6a3

)

126

ROBERT CARROLL

We see t h a t ( 4 . 9 ) o r (@+) agrees w i t h t h e i d e a t h a t U,:- $, -+ $T and we want t o connect these w i t h ( 4 . 8 ) and ( 4 . 8 ) w i t h ( 0 6 ) more p r e c i s e l y . We remark t h a t i n [ A m l ] one s i m p l y w r i t e s t h e o p e r a t o r L-S equations as

lim I (E-Ho~iE)-lqU,dFo €4 0

(60)

U2 = I

( c o n f i r m i n g a g a i n p a r t o f ( U ) ) . Now t o pass f r o m

(4.8) t o ( 0 6 ) c o n s i d e r ( u s i n g (4.12) and hence ( 4 . 9 ) o r ( 0 6 ) )

I ( E - H o T i E ) - ' q U ,-s ( e - H o ) d E ( S o ( k ' , ~ ' , ~ ) )

(4.13)

I(

I$,(

)-I q( k / l 6r3)/ S

=

k,e^ ¶ x ) )d2e^2kdk(2 ~ ) ( k~ I -s k ) =

( E l -HOW E )-' q l GT( k',^e',x))

Consequently ( 0 4 ) f o l l o w s and t h u s (4.8) ( o r e q u i v a l e n t l y ( 6 0 ) ) p l u s (4.9) o r (*+) l e a d t o ( 0 6 ) .

Going f r o m ( 0 6 ) t o ( 4 . 8 ) i n kernel f o r m should f o l l o w

d i r e c t l y from t h e formula ( @ + ) b u t t h e c a l c u l a t i o n r e q u i r e s some manipulaNote we c o n s t r u c t $* from ( a & ) ,

t i o n o f b r a - k e t n o t a t i o n which we o m i t here. then U,

v i a (4.9

(E-HoTis)-'ql!bt)

or

(@+)¶

and (4.8) i s then a s i m p l e c a l c u l a t i o n ( t h e t e r m

-

can be r e p l a c e d by

The o p e r a t o r s U,- s a t i s f y U? U*? =

Now we r e t u r n t o [Kyl] ( c f . a l s o [Wbl]).

%*U i=

I so W

I

=

$o) ...)

G e n e r a l l y f o r HU = UHo, WU*U = I = UWU*,

e t c . , any opera-

t o r o f t h e form (4.14) w i t h [L,Ho]

U = L t P/ (E-Ho)-lqUG(e-Ho)dE = 0 w i l l s a t i s f y HU = UH.,

The o r i g i n o f t h i s formula i s u n c l e a r

( c f . however Remark 4.12) b u t t h e p r o o f i s simple, once we d e f i n e P ( l / ( E - H o ) = p r i n c i p a l v a l u e (E-Ho)-l

as ( 4 6 ) P ( l / ( E - H o ) ) = (E-Ho+iE)-'

7

Note a l s o from Remark 2.10 t h a t -nis(E-Ho) = (1/2)[(E-Ho+ic)-' and one w r i t e s a l s o ( c f . [Rgl])

- i ~ )so- t'h]a t e.g.

P(l/(E-Ho)) = (1/2)[(E-Ho+ic)-l =

(E-HotiE)-l

-

P(l/(E-Ho))

in6(E-Ho)

i.irs(E-Ho).

-

(E-Ho-i~)-l]

+ (E-H,

as i n ( b b ) .

This

formula ( 6 6 ) w i l l have an obvious sense i n a momentum r e p r e s e n t a t i o n v i a d i s t r i b u t i o n s and c o n t o u r i n t e g r a t i o n ( c f . [G2]).

Now from comments above e v i -

d e n t l y Ho6(E-Ho) = s(E-Ho)Ho = Es(E-Ho) and f r o m (4.14) a s i m p l e c a l c u l a t i o n ( m u l t i p l y on t h e r i g h t and l e f t by

1 G(E-Ho)dE

I).

=

Given t h a t e.g.

proved b u t c l a r i f i e d i n [ K y l ] )

0

-+

--).

But l i m P(E-H,))-'exp[-i(E-H,)t]

( 6 + ) UHo

-

HoU = qU ( r e c a l l

U- should have t h e form ( 4 . 1 4 ) ( n o t

then t o d e r i v e (4.8) c o n s i d e r e x p ( i H o t ) e x p

(-iHt)U- = exp(iHot)U-exp(-iHot) ( i t H ) e x p ( - i t H ) as t

Ho) g i v e s

-+

I as t

+ -m

( r e c a l l W- = U- = l i m exp

Now

= m i d ( E - H o ) as t -+

(when a p p l i e d t o

SCATTERING THEORY

127

s u i t a b l e s t a t e s ) ; an argument t h r o u g h t h e momentum r e p r e s e n t a t i o n f o r examp l e should y i e l d t h i s immediately.

I

Hence ( a p p l i e d t o s u i t a b l e s t a t e s ) (&=)

+ in/ 6(E-Ho)qU-s(E-Ho)dE which determines L f o r U- and p u t t i n g t h i s

= L

i n (4.14) one a r r i v e s a t (4.8). Now suppose one has an i n v e r t i b l e t r a n s m u t a t i o n U w i t h HU = UHo and d G f i n e Then e v i d e n t l y W - l

W by WU*U = I (W = (U*U)-').

e

= (810) for

U*-'p)

= U*-'p

= U*U and ( p l W I p ) = (pU-'

say and hence W i s p o s i t i v e s e m i d e f i n i t e (and

p o s i t i v e d e f i n i t e i n f a c t since 0 # 0 unless p = 0).

H

= UH WU* = H* = UWHoU*.

UH0U-'

0

Since 19) = UWU*lp) we have (+*) l p ) = 1 1 UIE,a),dEda(E,al

t o r as f o l l o w s .

WIF,b)odFdb(F,blU*lv)

(IE,a),

Now s i n c e W comnutes w i t h H,

gonal i n Ho and we n o t e a l s o t h a t UIE,a),

I

19) =

(+A)

REmARK 4-XX.

(

(

i t i s dia-

i s an e i g e n v e c t o r o f H.

= IE,a)

For example c o n s i d e r (Ho a c t i n g on

x l ( E - H o + i r ) - l I x ' ) = G+(x,x') -

( 2 ~ ) ~and ' ~( k " , x ' )

2

k'l)

=

=

= s(k'-k'')/(E-k'2ti,)

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

( 2 ~ i ) - ~ / ~ e x p ( k- "i (, x ' ) ) .

3 G-+ ( x , x ' ) = - ( 1 / 2 1 ~ )

and (-A-k )G

(

d3k"(xlk')( k'l(E-k'2tiE)-11k'?( k " l x ' )

k'l(E-k'2tir)-11k")

(4.17)

I

We n o t e i n passing t h a t t h i s t r i c k o f i n s e r t i n g i d e n t i t i e s i s

1 d3k'l But

=

d E i IE,a)da(alW(E)lb)odb(E,b\p).

u s e f u l i n o t h e r ways as w e l l . (4.16)

R e c a l l / IE,a)odEdaJ E,al

IHo,Ao;E,a)).

Q

and t h e s e can be i n s e r t e d anywhere. Thus

F u r t h e r WHO = HOW si.rce

One can express W through a " w e i g h t " f a c -

/

Hence ( c f . ( 4 . 6 ) )

1 d3 k ' e i(k ' , x - x ' )

2 ei i k l x - x ' / ( E - k ' + i E ) = 471 1 x - X ' 1

I

= s(x-XI).

R e t u r n i n g t o [Kyl;Wbl],

-

g i v e n t h a t qU = UHo

r e c o v e r y f o r m u l a f o r q once U i s known.

HoU one can r e g a r d t h i s as t h e

The i d e a i s t o somehow f i n d a spec-

-

t r a l f a c t o r W (E ) and t h e n u n i q u e l y determine U, and t h i s i s t h e G-L t e c h nique.

F i r s t l e t us g i v e a formula f o r fi

f-) e x p ( i t H ) e x p ( - i t H o ) so W:

%

lim (t *

e x p ( i t H o ) e x p ( - i t H ) U - l p ) = $19) ( l i m as t

$I@)=

Llv) -

-in6 E-Ho)

in!

as t

-f

s(E-Ho)qU-6(E-Ho)dElp)

-).

m)

where W,

exp(itHo)exp(-itH). m)

-+

WW: -

-

= lim ( t +

Thus l i m

and we f o l l o w (4.15) t o g e t

( s i n c e P(E-Ho)-'exp(-i(E-Ho)t)

But L i s f i x e d by ( & = ) and one o b t a i n s

2aii 6(E~Ho)qU-6(E-Ho)dE. t h a t 6(E-Ho)

UU :-

%

(+0)

5

=

+

I -

T h i s seems l i k e a s i l l y f o r m u l a b u t i f we r e a l i z e

i s g i v e n v i a (U) and U s(E-H,)

p i c k ng up t h e T m a t r i x (5 = I

+

v i a (4.12) we see t h a t one i s

2 n i -s(E-Ho)T i n [ R g l l ) .

S i m i l a r l y one

128

ROBERT CARROLL

shows t h a t ('6)

9 - l = I + 2 n i J 6(E-Ho)qU+6(E-H,)dE.

Next t h e f a c t t h a t U*U

= I i s a d i r e c t consequence o f t h e r e p r e s e n t a t i o n U* =

Fg when -

z ?

xaC=

JC

(more g e n e r a l l y JEac = U+W; - - e t c . when aC = (JCo)ac as i t does here). An ope r a t o r p r o o f o f U:U -+=- I i s g i v e n i n [Kyl] based on (4.8) e t c . ; s i n c e t h i s i s n o t enormously i n s t r u c t i v e we w i l l s k i p i t . w r i t e a general U i n (4.14) as ( W ) U = L

+

i n 1 6(E-Ho)qU6(E-H,)dE

(4.18)

U.

t

/ 1 6 ( E-H$Tq

(E-HoTiE)-'qU6(E-H0)dE;

( r e c a l l ( 6 6 ) ) t h e n e.g.

1 6(E-Ho)U"q(E-Ho-iE)-'dE)

= (It

6 ( E ' - H o ) d E ' ) = M- +

I t f o l l o w s however t h a t i f we

M+ + 1

1 (E-Ho+iE)-'qU6(E-H0)dE

( E-Ho-i

E

)-' ( E ' - H o + i E ) - '

from (4.8)

(M- + j +

M+- =

(E'-Ho+iE)-'qU

6(E-Ho)UTq(E-Ho-ic)-1dEM-

qU6 (El -Ho)dEdE'

Some c a l c u l a t i o n which we o m i t ( c f . [ K y l ] ) shows t h a t t h e l a s t 3 terms cancel

so t h a t UTU

=

REmARK 4.12.

M o r U = U-M - and s i m i l a r l y U = U+M+. L e t us comment here on t h e n a t u r e o f (4.14).

sumes any t r a n s m u t a t i o n U can be w r i t t e n i n t h i s form.

I n [Kyl] one as-

F i r s t l e t us r e c a l l

+ G (r,r',k)

= -(1/41r)exp(ikIr-r'~)/~r-r'~ s a t i s f i e s (E-Ho)Gi = 6 so t h a t k + + + + + + J G qJ, y i e l d s (E-H0)ILk = qJ,k e t c . T h i s Gk i s t h e o u t g o i n g Green's k o k k f u n c t i o n and i n general one has a g e n e r i c formula ( c f . [Rgl])

that

$+ = $

(4.19)

G(r,r',k)

=

[1/2n21r-r'

( l / 2 ~ ) ~ d! 3~k ' [ e

111C

k ' d k ' [ S i n k ' Ir-r' l / ( k 2

I f t h e c o n t o u r i s chosen t o be

+

i E

i(k ' , r - r ' )

(-my-)

/(k

-

2

2 k' )] =

-

k")]

w i t h denominator changed t o k 2

we have G+ b u t i n general (+I) G ( r , r ' , k )

-

k'

2

= -(1/4a)[Cosklr-r'I/lr-r'l

+

) (m*) ( 1 / 4 1 ~ ~ I r - r ' / ) The f i r s t t e r m i s (1; = (l/Z)L: 2 2 P l I k ' d k ' [ S i n k ' ( r - r ' I / ( k - k ' ) ] = -(1/4a)Cosk\r-r'~/~r-r'~ and t h i s i s t h e

A(k)Sinklr-r'I/lr-r'l].

term g i v i n g r i s e t o 6 i n w r i t i n g (E-Ho)G = 6 . [Sinklr-r'l/lr-r'l]

The o t h e r t e r m G2 = - ( A / ~ I T )

has no s i n g u l a r i t y so (E-Ho)G2 = 0 and i t i s determined

by boundary o r a s y m p t o t i c c o n d i t i o n s . I n p a r t i c u l a r , u s i n g (4.19) i n t h e - k ' 2+ i O ) - ' = P ( l / ( k 2 - k ' 2 ) i.rrs(k 2 - k ' 2 ) one o b t a i n s A(k) = i f o r

-

form ( k 2

G+ (and -i f o r G-).

Then depending on t h e c h o i c e o f c o n t o u r t o determine

A(k) one has a general formula (4.19) and i f we l e a v e t h e boundary o r asympt o t i c c o n d i t i o n u n s p e c i f i e d one c o u l d simply t a k e 2 Ho)-' (1/41r l r - r ' l ) P [ ~k'dk'[Sink'(r-r'I/lr-r'l] Q

Go(r,r',k)

%

P(E

-

% (PG)(ryr',k). Thus i n PG and u s i n g t h e o p e r a t o r L t o a d j u s t k 3 Thus i f J,k = W o so t h a t U % J 1 9 k I ) d d ' ( $ z ' l t h e n

(4.14) one i s u t i l i z i n g t h i s Go "boundary c o n d i t i o n s " .

@A)

%

129

SCATTERING THEORY

( v i a (4.13) and ( 4 . 1 2 ) ) (.*) N = P I (E-Ho)-'qU6(E-H0)dE

i n (4.14)

Q ,

P I (E

-

)/5.

Ho)-'q(k/l6n 3 IJlk)d24e(ILoIdE -k and ( m b ) N l $ ok I' ) = p(E"-Ho)-lql$kl,) Thus i n (4.14) (E-Ho)UI$, k 'I ) = ( E - H o ) l $ k l l ) = (E-Ho)Ll$ok " ) + q l $ k I l ) = qI$kOl) as desired.

One need o n l y show t h a t any d e s i r e d a d j u s t m e n t o f boundary c o n d i -

t i o n s can be achieved v i a L. R e t u r n i n g a g a i n t o [Kyl] we n o t e from (+@) = U+M+ we g e t M+ = L

UU :-

-

in/

(which we a l r e a d y knew).

bM- i m p l i e s 9 = M+M:'. i n g UU :=, I = U? U* and li = UM,, ,

U-M-

=

-

(4.18) t h a t t a k i n g U = U- i n U

6(E-Ho)qU-6(E-Ho)dE = 1

-

2 r i I 6qU-6 =

b so b

=

i n general i m p l i e s M+ = U+*

A l s o U+M+ = U-MNow d e f i n e W = MM';'-;

= M-'M*-'. - -

Then us-

one g e t s WU*U = I and UWU* = I.

A f t e r t h i s l o n g d i s c u s s i o n , which s h o u l d h e l p t o r e l a t e t h e techniques and n o t a t i o n o f [Kyl] t o o t h e r p o i n t s o f view and i l l u s t r a t e t h e advantages o f such methods, we go now t o t h e G-L complex o f ideas v i a t h e machinery o f Set (me) U = 1 + K and Uo = U - l = 1 1 we want t o f i n d U such t h a t WU* = U- = Uo. [Kyl].

(one c o u l d a l s o use r f o r 4 ) .

vector I q

+

L.

We t h i n k o f W as g i v e n and

L e t Q be p o s i t i o n w i t h e i g e n -

For boundary c o n d i t i o n s now one

-
r e q u i r e s ( t a k e qo f o r 0, - m , o r whatever) ( m m ) lim[
H = UH U - l , and W - l

=

H t o be Hermitian, so f r o m WU*U = I we g e t W = (U*U)-l = U-'U*-'

( H*

n o t e W commutes w i t h Ho

(U*U);

=

U*-lH0U*

=

H

=

( q l P l q ' ) = 0 f o r q ' > q. t i n g IH,A;E,a)

=

I$E)

I f K and

L

(mm)

0

and

and IHo,Ao;E,a)

i n (me) a r e t r i a n g u l a r t h e n ( a b b r e v i a =

I$:))

(***) ( q l l L E ) = ( q ( $ F ) + I9',

( q l K l q ' ) d q ' ( q ' I I L E ) and ( q l $ F ) = ( q l $ E ) + / p 9 , ( q l L l q ' ) d q ' ( q ' I $ E ) . that

UH U-'

We say now t h a t P i s t r i a n g u l a r i n t h e Q r e p r e s e n t a t i o n i f

Ho = WHoW-').

It follows

w i l l h o l d f o r t r i a n g u l a r K and L when qo i s f i n i t e ( f o r qo = -m Now w r i t e (**A) W = I + A so f r o m

some f u r t h e r c o n d i t i o n s can be imposed).

(me) and UW = U * - l we o b t a i n t h e G-L e q u a t i o n ( c f . ( 4 . 7 ) e t c . ) K + A + Kn= L*; ( q l K l q ' ) + ( q / A l q ' ) + ] ' ( q l K l q ' ? ( q " l A I q ' ) d q " 90

(4.20)

= (qlL*lq') = 0

f o r q'

c

q (note ( q ' I L l q ) * = ( q l L * l q ' ) = 0 f o r q > 9 ' ) .

We do n o t d i s c u s s

t h e method o f s o l u t i o n i n [Kyl] where one works from WU* = Uo and l a t e r asks for

u0

=

u-~.

RElllARK 4.13,

The n e x t s t e p i s t o determine W i n terms o f

5 (assume H has

o n l y a b s o l u t e l y c o n t i n u o u s spectrum); we deal h e r e o n l y w i t h t h e 1-D s i t u a t i o n f o l l o w i n g [Kyl].

To compare n o t a t i o n w i t h p a r t s o f [Kyl] we w r i t e e.g.

130

ROBERT CARROLL

$EUt(x)

(**a)

$Eut

= exp(ikx)

!bl i n (4.4).

-

A l s o as x

by

--

$EUt

(*6)

as I k (

-t

'L

= I

e x p ( i k x ) + b ( k ) e x p ( - i k x ) so

Another n o t a t i o n i n [Kyl] i s g i v e n

m.

!bk(x) = e x p ( i k x ) + / z ( x I K I x ' ) e x p ( i k x

(*u)

w i t h AL

f-(-k,x)

(in t h e sense

ky)2dk = (1/2n)[I,"

%

) d x ' so t h a t v i a say (**)

One t a k e s a l s o

K.

(*A*)

1

(x(H0,A ;E,a)

1

f o r k = JE and a = +1, so t h a t

e x p ( i a k x ) = $F(a,x)

=)I

-t

One w r i t e s b ( - k ) = b ( k ) and sometimes (**+) b ( k ) = g ( k )

b ( k ) = s12 = . lR

e x p ( - 2 i a k ) where g = 0 ( k - ' ) ILk(x)

so t h a t

( i / 2 k ) l z exp(ik1x-x' I)q(x')$FUt(x')dx'

= (1/2Jnk)

It I$E(a, 8 =))dE($L(a,

It $F(a,x)$F(a,y)dE = ( 1 / 4 n ) I ;I e x p ( i a k x ) e x p ( - i a J t e x p ( - i k x ) e x p ( i k y ) d k ] = (1/2n)

exp(ikx)exp(-iky)dk t

exp(ikx)exp(-iky)dk = 6(x-y)). t o t h i n k o f (Ho.Ao;E,a)

Note t h a t i t w i l l sometimes be convenient

as $ i ( a , x ) even though t h e x r e p r e s e n t a t i o n has n o t Note a l s o

been s p e c i f i e d ; i n case o f p o s s i b l e c o n f u s i o n we w r i t e $(a,-).

:I $ ~ ( a , x ) $ , ( a ' , x ) d x s i d e r s (xlH,A;E,a)+ (4.21)

= 6(E-E1)6(a,a') %

$ i ( a , x ) = $(a,x)

f

T

i c ) - l ( n o t e (4,21)

%

Now one con-

6(k-k')/2k).

(i/2k)Iz

1:

e

I

I dx

= +(i/Zk)exp(riklx-x'I)

where one uses ( x l y , ( E - H o ) l x ' ) Ho

(6(E-E')

$ i ( a , x ) determined v i a ( ( x l q l x ' ) = q ( x ) s ( x - x ' ) ) I(

x I I q 1x1' ) d x " $ i ( a ,x'I)

and y+(E-Ho)

i s e s s e n t i a l l y ( a & ) , up t o a m u l t i p l i e r ?

=

-

%

(E

-

$F(l,x)

=

t

( 1 / 2 4 1 ~ k ) $ ~ ( x ) ) . Thus i n p a r t i c u l a r f o r a = 1 qE % $- i n ( 0 6 ) and $; 'L $1 = $out f o r a = 1 (up t o a m u l t i p l i e r ) . Now r e f e r r i n g t o ( + a ) - ( + 6 ) i n ("0) i n 1-D t h e S(E-Ho) terms a r e g i v e n by 6(E-Ho) = 1 I $ F ( a " , - ) ) d a " ( $ F ( a " , - ) l and 9 f o r ex-

( t h e I d a " i s c r u c i a l l a t e r on f o r J I $ t ( a " , .))da"($;(a",-)l)

ample has a form ( $ ~ ( a , . ) l s l $ ~ , ( a ' , . ) ) ( o r b e t t e r ( H o , A o ; E , a ~ ~ ~ H o , A o ; E ' , a ' ~ = s(E-E')
2aiI

We w r i t e then from ( + o ) (*")

I T L ( a , x ) d x ( x l q l x l )dx'$;(a',x')

(al$(E)la')

-

= 6aal

( n o t e (r1$)= $ ( r ) = ( $ I t - ) * ) .

Similar-

One l y (*Aa) ( a l . f l ( E ) l a ' ) = 6 I + 2 a i J I ? : ( a , x ) d x ( x ( q l x ' ) d x ' i E ( at' , x ' ) . notes a l s o t h a t ($,(a,X)lql$E(b,X)) E aa= ('d'Et(a,X)lqld'o(b,x)) E = ($'o(b,x)lql$'E E t (a,x))*.

Now one assumes ( x ( q l x ' ) = 6 ( x - x 1 ) q ( x )

-t

0 as x

-+ - m

with suffi-

c i e n t r a p i d i t y (normal s c a t t e r i n g problem); o t h e r s i t u a t i o n s , c a l l e d "pathThen e.g. from (4.21) (*A&) lim $;(a,x) = lim t ox++$F(a,x) + $ F ( i l ,XI[( *1 I d ( E ) l a ) - 6+1 ,a 1 and X-t+m $E(a,x) = $E(a,x) + df(T1, x ) [ ( V 1B-l ( E l la ) - brl ,a 1. Now assume U i s such t h a t $,(x) = U $ i ( x ) behaves --; t h e n f o r ( x l H A;E,a) = ( x l U I H ,A ;E,a) (*A+) lim l i k e &x) as x o l o g i c a l " a r e t r e a t e d i n [Kyl].

+

0

L

[(xlH,A;E,a) IHo,Ao;E',a')

- ( x ~ H ~ , A ~ ; E , ~ )= ] 0.

x-t

0

-

Consider U = U-M- now w i t h (Ho,Ao;E,alM-

= 6 ( E - E ' ) ( a l ~ - ( E ) l a ' ) and s i m i l a r l y we assume a formula (Ho,

Ao;E,alW~Ho,Ao;E',a')

= 6(E-E')(alW E ) l a ' ).

leads t o m a t r i x formulas ( 1

't

Some c a l c u l a t i o n i n [Kyl]

f i r s t row and column, -1

%

second row and

SCATTERING THEORY

column, and r e c a l l W (W)la') = ((w..))

-

p22

where

1J = l/(-l~$(E)~-l),

1).

1

M-lM*- -

%

( ( P1JI . ) ) and ( a l W -(-l~$(E)~l)/(-ll$(E)l-l)y

for (a(u-(E)la') =

= 1, p i 2 = 0, p i l

pil

wll

, etc.)

131

=

= wZ2 = 1, w , ~= ( - l I $ ( E ) l l ) * ,

(-ll$(E)I

and wZ1 =

The d e r i v a t i o n i n [Kyl] seems c u r i o u s so we w i l l o b t a i n t h i s by a d i f The symbols ( k 1 I t i ( E ) / _ + l )w i l l correspond t o t h e s t a n -

ferent calculation.

It w i l l be i n s t r u c t i v e t o r e t u r n t o (+*) f o r

d a r d sij as i n d i c a t e d below.

from (.)

F i r s t we know $1 = sllf+

1-D and r e d e r i v e i t .

t h i s $1 agrees w i t h $1 i n (4.4)

'L

+ $

i n (06)

2Jnk$;

'L

b e f o r e (4.3) and Consider

i n (4.21).

now (*u) f o r say a = -1 and a ' = 1 i n t h e f o r m (4.22)

(

-1 I $ ( E ) ( l ) = - 2 a i l m m -$,(a,x)q(x)$;(a',x)dx 0

-2niJ'l'

[eikX/2Jnk]q(x)(l/2J~k)sllft(k,x)dx

From (4.3) and (+) a f t e r (4.2) (k,x)dx = sllc22

=

(-1

(*Am)

l$ll)

= (sll/2ik)lI

= c22/c21 = s12 ( n o t e here t h e n o t a t i o n 1

v i o l a t e d unless ( a l s l a ' ) = sala).

exp(ikx)q(x)f+ 'L

1, 2

[ e x p ( - i k x ) / 2 J ~ k ] q ( x ) ( s l , / 2 J ~ k ) f + ( k , x ) d x = 1 t sl1(1-cZ1) Thus t h e W m a t r i x i s (**A) = sll = s Z 2 . and wZ1 = s12 (wij 'L ( a l W l a ' ) , 1 rl, 1, 2 Now e.g.

-

(*AA)

where 1 %

we have n e a r x = -= $,(x,a) (a,x)

'L

'L

1$F(a',x)da'(a'Ip-la)

1,

$F(x,a)

=

1J

-1 i s = 1

= 1

+

+ $!(-lyx)l[

with

(*a*)

(-ll$la')

T h i s i s v a l i d n e a r x = -= and we r e c a l l h e r e $F(a,x)

-

1

= sij(-k)).

Next c o n s i d e r (**a) $,(a,x)

(*A&)

-

sll

Given t h e a s y m p t o t i c b e h a v i o r and

-

wZ2 = 1, w12 = Sl2,

-1; r e c a l l a l s o Sij(k)

are straightforward.

(*A+)

1 $;(a',x)da'(a'Ip-(a)

(( w. .)) : wll rl,

'L

(***I (ll$(l)

S i m i l a r l y from (*")

=

(*A+)

y i e l d s ( * * 6 ) $:

6-l,a11da'(a'Iv-la)

= exp(iakx)/ZJnk

=

c e x p ( i a k x ) so (*.6) t cexp(-ikx)[

says t h a t (*a+) c e x p ( i a k s ) % a y c e x p ( i k x ) t a z c e x p ( - i k x ) as x -=. From (*a+) one o b t a i n s t h e n w i t h a l i t t l e c a l c u -

3,

-f

l a t i o n (*om) ( a l p - l a ' ) =

-s

12

/s

and t h e wij

22

RrmARK 4.14, Remark 4.5.

((pi,.)):

p!'

= 1,

v!'

= 0,

u'!

= 1 / s 2 2 y and

pfl

=

can be determined f r o m t h i s .

L e t us c l a r i f y a b i t t h e i n v e r s i o n i n Theorem 4.6 r e l a t i v e t o Thus i n p a r t i c u l a r , r e f e r r i n g t o 1-D, we know t h e L-S eigen-

f u n c t i o n s $' = $

k

of

(Am)

o r ( 0 6 ) a r e o r t h o g o n a l on

(-my-)

b u t t h e f+ a r e -

n o t ( t h i s i s phrased b a d l y i n [NwZ;Fal] f o r example and s p e l l e d o u t c o r r e c t L e t us n o t e h e r e i n passing t h a t one has r e l a t i o n s between l y i n say [ C j l ] ) . t h e $1,2 (**m).

of (4.4) : (4.2), Thus (*om

and $2 = 2Jnk$;(-

$,

= $'

$1 = 2Jnk$;(l,x) ,x).

of (*6),

$i(a,x)

T

(!bE(l,x)

'L

o f (4.21),

and $k o f

$+ up t o t h e 2Jnk m u l t i p l i e r )

I n d e r i v i n g t h e formula ( A )

i n Theorem 4.6 f o r exam-

p l e v i a t h e Green s f u n c t i o n one o b t a i n e d i n f a c t (+&*) 6 ( x - y ) = (1/2n)

132

ROBERT CARROLL

/:[T(A)f+(A

, x ) f - ( h ,y) + T(-A)f+(-h , x ) f - ( - A ,y)]dh and s i n c e ;/

(-A,y)dA

i L T(A)f+(A,x)f-(h,y)dA

c u l a t i o n based on (4.23)

f+(-k,x)

(A)

Thus even though ( A ) i s

we o b t a i n ( A ) .

n o t s y m n e t r i c a l i n appearance ( i n ( x , y ) )

T-f+( -A,x)f-

t h e symmetry i s p r e s e n t v i a a c a l -

i n t h e form

-

= Tf-(k,x)

The c a l c u l a t i o n i s s i m p l y

Rrft(k,x);

- RL f - (k,x)

= Tf+(k,x)

+ T-f,(-h,x)f

Tf:fy

(*&A)

f-(-k,x)

(-h,y)

= Tftf!

+ T-[TfX

- RlfY] = ( T t T-RrR,)f:f! + T I T 1 2 f r f z - - l T I 2 R L fXfY - - _- lT12Rrf?f! -Rrf:IITf: 2 2 = 1 we have T + T R R = T s i n c e from TRF + RTI= 0 and IT1 + I R r ,L I TR,Rr = TITI’. One notes a l s o t h a t (as i n [ C l l ] ) (*do) $;: + $I$! = Tf: T-f,(-h,y) + Tf:T-f-(-h,y) = Tf:T-[Tf! - R fy] + TfXT-[Tf: - R L f - ] - T I T I 2 - IT12Rrf:f: + ITI’TfXf: - IT12 R L f -xf -yr-+-x - $1$1 y +-$;$; which i s t h e same

B -

ftf:

expansion as i n

(*&A).

L e t us now c l a r i f y some p o i n t s which a r e b a d l y s t a t e d i n some o f t h e l i t e r a ture.

F i r s t we observe t h a t u s i n g (4.23) and Theorem 4.6

(4.24)

( 1 / 2 * ) l I $1 ( h , x ) J l ( p y x ) d x = (1/2n)/f

+ R f-(X,x)]dx

( l / Z ~ ) l f T;f+(-p,x)[F-(-h,x)

To see t h i s c o n s i d e r (as i n [C3,17]) (

(1/2*)/

f+(P,x)f-(h,X)dx,p(-P))

(

Thf+(X.x)T~ft(-P,x)dx

= ~(X-P)

+ [R T-/T]6(h+p)

( 1 / 2 n ) ~ f,(-p,x)f

(x,x)dx,p(p))

= = J =

= ( 1 / T A b ( - A ) = (l/Th)( ~(X+U),P)

: (l/T-) P = (6(htp),

Note t h a t as a d i s t r i b u t i o n ( ( 1 / T p ) 6 ( h t p ) , p ( p ) ) ( p / T ) ) = y - / T so we d i s t i n g u i s h between m u l t i p l i e r s a f t e r d i s t r i b u t i o n a c t i o n

6(h+p),v(p)).

and m u l t i p l i e r s o f d i s t r i b u t i o n s . Similarly

(

I n p a r t i c u l a r Ti16(h+p) e T ( - p ) - ’ & ( h t p ) .

( 1 / 2 1 ~ ) / f+(-u,x)f-(-h,x)dx,

v(-J.I)) = ( l / T i ) v ( h ) = (l/T;)(

~ ( p ) )= (

( 1 / 2 ~ ) 1f,(p,x)f-(-x,x)dx,

Another v e r i f i c a t i o n o f ( 4 . 2 4 ) p. 306, and [Nvl], p. 19. Thus f r o m [ C j l ]

~(A-~),P(P)).

comes from t h e formulas i n [ C j l ] ,

( w i t h t h e c o n v e n t i o n Rr6(-k-q)

= RF6(k+q) and/or Rr(k)6(k+q) = R ( - q ) & ( k + q ) .

Take k = A and q = -p w i t h RrT = -RtT- t o o b t a i n [R,/\T12]6(A+p). Since Glsl = ( T 1 2 f t ( x , h ) f t ( x , - p ) m

(*&&I

I

r2 ) 6 ( A - p )

= (1/ITI

we f i n d (TI2 I = J .

-

Note

t h a t one w r i t e s (*&+) ( 1 / 2 ~ ) / f+(x,k)f,(x,q)dx ~= (l/T)G(k-q) which i n c l u d e s t h e a s s e r t i o n o f Theorem 4.6. L e t us n o t e a l s o from (4.25) ( w )( 1 / 2 r ) J- mm f - ( x , k ) f - ( x , q ) d x = ( l / T T - ) G ( k + q ) + (Rr/T 2 ) s ( k - q ) which i s a r e s u l t i n C o r o l l a r y 2 . 2 of [C17] (where A = -Rr and p = T). The formulas o f also i n [ C j l ]

133

SCATTERING THEORY

[ N v l ] a r e expressed i n a f o r m s i m i l a r t o (4.25). F i n a l l y r e t u r n i n g t o t h e L-S e i g e n f u n c t i o n s e t c . t h e r e w i l l be no problem as l o n g as we remember t h e f o r m u l a I = 1

q1 f o r a = 1 and

%

$2 f o r a

1

I.

)dEda(

Thus w i t h ZJnW;(a,x)

t e r (4.9) i n v o l v i n g I b u t f o r 1-0 o f course a = ? 1 w h i l e i n 3-D a The f o r m u l a (B) i n Theorem 4.6,

cal variables.

%

= -1 we t h i n k o f i n t e g r a l s o f t h e f o r m ( a * ) af%

spheri-

w r i t t e n as 6 ( x - y ) = (1/2n)

1 ; ($(x,x),$(x,y))dx = (1/2~)1; & [$ ;: + 6$b:]dX can be w r i t t e n (***) 6 ( x - y ) = (1/2a)/; [&;$: + $;$i]dx = l a < $ i ( a , x ) \ $ i ( a , y ) > ( r e c a l l dE = 2kdk). I n o r d e r t o o b t a i n f u r t h e r i n s i g h t i n t o and a b e t t e r understan-

REInIARK 4-15.

d i n g o f t h e o p e r a t o r s W, M+, pi, e t c . used i n [ Y y l ] and developed above, we

w i l l s k e t c h t h e approach

of

[ F a l l and v a r i a t i o n s i n [C2,3,20]).

F i r s t con-

s i d e r t h e h a l f l i n e t h e o r y w i t h i z ' = Hz, z ( 0 ) = zo, z ( t ) = e x p ( - i t H ) z o , zo I e i g e n f u n c t i o n s o f t h e p o i n t spectrum o f H, z,

= Utzo, e t c .

one w r i t e s UWU" = I (as b e f o r e ) , U*UW = I ( W =-(U*U)-'),

Thus i n [ F a l l

W(k) = l/M(k)M(-k),

(UM-')(UM-l)*

e t c . ( s o w i t h M- % M(-k) % M* % M: one can w r i t e e.g. -1 -1 L e t us r e c a l l a l s o t h a t = I, W = M ( M )*, U = UM ,, etc.).

if B

E QUW

S ( k ) = M(-k)/M(k),

%

U then

One d i f f e r e n c e h e r e w i t h t h e ma-

( n o t a t i o n o f [C2,3]).

t e r i a l from [Kyl] i s t h a t t h e domains o f H and

Ho

a r e n o t so e x p l i c i t l y i n -

v o l v e d i n [ F a l l s i n c e t h e t h e o r y i s developed v i a t h e k r e p r e s e n t a t i o n ; t h i s i s c l a r i f i e d below.

2

-D 0

t

Thus t h e model h e r e i n v o l v e s e i g e n f u n c t i o n s 0 ( x , k ) o f

2

cp

= k 0 s a t i s f y i n g 0 (0,k)

tions f(x,k)

%

e x p ( i k x ) as x

e (x,k)

(4.26)

0(x,k)

(*+A)

+

= [F(-k)f(x,k)

0 and 0 ' ( 0 , k ) = 1 a l o n g w i t h J o s t s o l u -

Thus

a.

+

= [Sinkx/k]

f ( x , k ) = eikx Then

+

=

,a -

[Sink(x-t)/k]q(t)e (t,k)dt;

[Sink(t-x)/k]q(t)f(t,k)dt F(k)f(x,-k)]/2ik

and F ( k )

M(k) (note i n

%

N

Remark 2.6.11, f-) = f'f-

+

-

F

%

2ihM2 i s u n r e l a t e d ) i s g i v e n as c ( 0 , f ) = F ( k ) ( s i n c e W ( f ,

f f - ' = 2 i k ) so t h a t F ( k ) = f ( 0 , k ) .

$ (x,k)

= e(x,k)/M(k),

$+(x,k)k

2dk so t h a t TTG

T+g = 1: g(x)$+(x,k)dx = g and (**&) T+T:

p r o j e c t i o n o n t o Jcc (H = -D st(x,k)

=

+ $ (x,-k)

convenience.

One w r i t e s f u r t h e r ( * * a ) = G(k), and T g:

= Ik and TT :+

=

(2/~)/:

= I where I

G(k) %

2 + q h e r e and s i m i l a r formulas h oCl d f o r $ - (Cx , k )

=

L e t us assume no bound s t a t e s e x i s t f o r

= (x,k)/M(-k)). The boundary c o n d i t i o n s h e r e f o r H o r Ho a r e based on Sinkx/k;

t h u s f ( 0 ) = 0 and f ' ( 0 ) = 1. Note t h a t i n general, f o l l o w i n g [C2,3,13;Mrll, 2 2 -1 i f a t r a n s m u t a t i o n Bh: D + D -q = Q has k e r n e l B~ = 6 + Kh and Bh = Bh has k e r n e l yh = 6 + Lh then f o r B h f , one wants f ( 0 ) = 1 and f ' ( 0 ) = 0 b u t w i t h

134

ROBERT CARROLL

Bhg we r e q u i r e g ( 0 ) = 1 and g ' ( 0 ) = h.

T h i s i s an unsymnetrical transmuta-

t i o n and a r i s e s because o f t h e way i n which h i s i n s e r t e d i n t o t h e k e r n e l s Kh and Lh; 6h can be symnetrized t o map Cos x t h[Sinhx/h] + p!yh i f d e s i r e d . 2 From [ F a l l we n o t e t h a t s o l u t i o n s o f i!ht = HJ, ( H = -D t q ) w i t h J,(x,O) =

$0'

( x ) have t h e form (assume no d i s c r e t e spectrum) - i k 2 t k2dk

m

(4.27)

= (2/n)l

J,'(x,t)

*(k)?(x,k)e 0

f o r example where $ i ( k ) = :/

J,i(x)J,'(x,k)dx

= * ( k ) ( t h i n k h e r e o f *(k) as 2 2 *(k)[Sinkx/k]exp(-ik t ) k dk s a t i s f i e s 2 2 = ( 2 / n ) / F *(k)[Sinkx/k]k dk and i f *(k)k E L

Now (*++) x ( x , t ) = (2/71)/:

given).

Hox = -xxx w i t h x(x,O) one has (*+.) 0 = t+*'Iim I$'(x,t) ixt

=

/om

O(x,k)

%

(IM(k)l/k)Sin(kx

(4.28) Then

d(k) =

ixt

= H

:1

-

-

x(x,t)12dx.

argM(k)) as x

+

-

R e c a l l here from

( F = M).

(*+A)

Next, g i v e n J,o s e t 2

J,,(x)$'(x,k)dx;

x' w i t h xf(x,O)

= (2/71)J'"

X'

= (2/n)/:

0

*'(k)[Sink~/k]e-'~

*'(k)[Sinkx/k]k

2dk = x:(x)

tk2dk

= TT :$',

0

=

U:q0

(To i s t h e t r a n s f o r m based on Sinkx/k analogous t o ( * @ a ) and one t

U- = U S, w r i t e s U,- % TT :o i n x or U,- % TOT+ i n k ) . Here f o r m a l l y z+ = U:zo, zt = Sz-, e t c . and U, % l i m exp(itH)exp(-itH,) as t + 2 - . One has a theorem

-

i n [ F a l l ( r e s u l t i n g from

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

(*+a))

e i g e n f u n c t i o n s o f t h e p o i n t spectrum o f H and i f i'Lt = HJ, w i t h $(x,O) = Go Thus J, = e x p ( - i t H ) $ O then (*.*I1 ; IJ,(x,t) - x ' ( x , t ) l 2dx -+ 0 as t (% z(t)) and say x t = T;[e~p(-ik~t)T,$~] = T:[exp(-ik 2 t)To(T;T,to3] (recall

+-.

-f

zt

%

l i m e x p ( i t H o ) $ = U*J,

w i t h exp(-itH,)J,,

T;[exp(-ik + O

= exp(-itHo)U:J,o

exp(-itH)bo

-

and e v i d e n t l y e x p ( - i t H ) $ o = T:[exp(-ik Hence x t = T;[exp(-ik

2 t)To$,]).

and one has ll$

-

xt1I2 = llexp(itHo)(J,-xt)l12

and exp(itH,)exp(-itH)

U:J,0112

-f

:U

t)T,$,] 2 t)To(U:$o)l

= llexp(itHo)

The boundary c o n d i 2 = T$[exp(-ik t)TtIL0] =

etc.

t i o n s a r e i m p l i c i t i n expressions l i k e exp(-itH)$, 2 "+ 2 t)$, k dk. I f one used d i f f e r e n t e i g e n f u n c t i o n s f o r (2/7r)/; $,(x,k)exp(-ik H based on s t r u c t u r a l l y d i f f e r e n t i n i t i a l values (say $(O,k) = 0 h e r e ) then e.g.

exp(-itHo)J,o

4

D(H).

T h i s however i s i n c i d e n t a l .

The p l a c e where i n i -

t i a l c a n d i t i o n s a r e used i m p l i c i t l y i n t h i s s k e t c h from [ F a l l i s s u l t s (*+.)

and (*.*)

which use (4.26),

(*+A),

in t h e r e -

etc.

RmARK 4.16.

Now c o n s i d e r t h e f u l l l i n e t h e o r y f o l l o w i n g [ F a l l . One works 2 L and JCo % JC x JC, p % $ = (pl,p2) (column v e c t o r ) , (v,;) = ;/ [(ply t (p2,?2)]dk ( k 2 % E, 2kdk % dE, and a f a c t o r o f 271 i s d i s p l a c e d here

w i t h JC

rl)

%

from t h e end o f Remark 4.14 s i n c e a symmetric F o u r i e r t r a n s f o r m w i l l be employed).

Set To:

J, .+ p : J C +

KO: p1 = (1/J2n)/;

$ e x p ( - i k x ) d x , p 2 = (1/J2n)

SCATTERING THEORY

lz

$exp(ikx)dx.

(l/JZn)/;

Then T:;

JCo

(exp(-ikx),exp(ikx))

+

I and T0T*0 = I ; n o t e T*q = 0 ( 1 / J 2 1 ~ ) [ r v,exp(ikx)dk ~ + J r v2exp

JC w i t h T;To

-+

+

lpdk =

=

Now u1 = !b2 and u2 = !bl i n [ F a l l so u;

(-ikx)dk].

'1.

=

(column v e c t o r ) i s d e f i n e d v i a components

+

( r e c a l l u1

+

u;(x,k)

=

;t(x,k)

u;(x,k)

=

5,

= T-f+

JC : J/ 0

-+

(l/JZn)/:p(x)

=

= T+T:

=

I.

(!b2,!b1)!bdx T h i s i s ach-

and one a l s o w r i t e s u - ( x , k ) = ;;(x,k = T - f + ( - k , x ) and 1 where we r e c a l l f + ( k , x ) = f + ( - k , x ) e t c . so

0-

= T-f+(-k,x)

-

and u;

= Vj2

Set t h e n

= T-f-(-k,x).

(*=A)

-

f

T(f-) = (-st 21

( -), ft

s e t (*=&) U+ = TT :o

Note from

- : 1-2 ) ( ~f+- ) -

0

+ JC

0

f- - u and T(f )+I-, , (

+

+

= U;U '-

c\,

while

-

2 -+

('1-) 92

= 9

=

T*T T*T and o + + o

?

=

-> (1/2n)J

JI

T ST* = T+T*. 0

I

0

+

R s ( k - q ) ( s i n c e RiT = -R,T-),

+ Rrs(k-q).

$4. integrals 1 J

(1/2n)

(A+*)

( 1 / 2 ~ ) J JI1 ( k , x N l (q,x)dx = (Tq/Ti)G(k+q) + and (1/2n)/

Note now t h a t (T /T-)a(k+q)

$2(k,x)$2(q,x)dx

= (T /T-)s(k+q)

q

k

: 6(k+q) so t h e e q u a t i o n (*==) can

6 ( k + q ) ) + Ss(k-q) where S appears q 5k = ( 0 v k w i t h (A*A) s(k+q) 0 The 6(k+q) terms seem t o g e t i n t h e way h e r e b u t i n f a c t t h e k-

be w r i t t e n as (*=A).

(1/2n)

('Z)($;,$i)q=

JJ'2J/1'1 '2'2'2) (here the + , ~ 1 + J / 2 ~ 2 I d k= ( 1 / 2 ~ ) ( , - ~1V$j 1 1 + @l$2p2 are i n x!). Now r e c a l l (4.25) - (*&=) i n Remark 4.14 so t h a t

J ('2)CJ1

Consequently

$1

92

= Ts(k-q),

w i t h pPf

Then (l/JZn)/ $ ( TT--ff+- - ) = ( l / J Z n ) / '('l-). JI2

i s determined v i a (*==) ('1)

J IL2(kyx)lL, (q,x)dx

in

and (4.23) (*=a)

. IL 0'

I and (*=+) S

=

(A)

and U- = T*To where T-: J c - + JC

( A ) = ( 1 / J 2 n ) / JI u;(x,JA)dx U*'U'

=

q =

-+

= Su- where S i s t h e m a t r i x w i t h T i n t h e d i a g o n a l s and e n t r i e s s.ii

f

in

and u;

$2 e t c . so t h i s corresponds t o (l/JZn)/:

= T f-(-k,x)

the (i,j) position.

?:JC

+ 9i(x)

We assume no d i s c r e t e s p e c t r a and t h e n TT :+

Then d e f i n e U+ = T*T

+ u

%

= Zhk$i(-l,x)

S i m i l a r l y T+: JC

2Jnk$i(1 , x ) ( c f . t h e end o f Remark 4.14).

+ + (q1, v 2 ) + ui(x,Jx)dx

135

-+

s c a l a r p r o d u c t comes from JE and i n v o l v e s h e r e and i n v o l v e s h e r e k

0 only

rr

so t h e r e i s no c o n t r i b u t i o n from s(k+q).

Consider S = T;15To, where an ac-

t i o n ( a ( k + q ) , e x p ( i k x ) ) = 0 s i n c e k,q 2 0 and N

(4.30)

S = ( 1 / 2 n ) l m (eiqy,e-iqy 0

T R exp(-iqx))dq )(RL Tr)(exp( i q x )

136

ROBERT CARROLL

Hence ( r e c a l l a,q 2 0 ) (A*@) ?exp(icxx) ilarly

+

x e x p ( - i a x ) = Rrexp(iay)

T = s l l = s 2 2 y Rr = sZ1 ,

Re = s 1 2 ) as

+ R / ( e x p ( - i a y ) ) and sim-

= Texp(icry)

It f o l l o w s t h a t ( r e c a l l

Texp(-iay).

-

i n (4.22)

( - 1 l$ll

(*Am)

)

(1/2n)

%

% RfS(0) % s12 and ( 1 Itill ) ( 1 / 2 n ) / exp(-iay)dexp(iax) s l l = sZ2 and s i m i l a r l y f o r t h e corresponding formulas based on

J exp(iay)dexp(iax)dy dy

T6(0)

%

‘L

t h e equations

(A*&).

F i n a l l y l e t us n e x t w r i t e o u t t h e formulas o f [ F a l l f o r t h e w e i g h t o p e r a t o r s . Thus one d e f i n e s (A1 (4.31)

UIJ’

=

(U.: JC-+ JC) and Vi: 1

/I$(x)fl

“u

A R y A2

,,” A1 ( x , y ) $ ( y ) d y ;

+

JC

-f

AL i n (**))

%

U2$ = 9 ( x )

(**+) Vl$

JC a r e g i v e n by 0

+

f,A2(x,Y)$(Y)dY

= q ; ql(k)

= (1/J2n)

( x y - k ) d x ; 1 ~ 2 ( k =)

J’(x)dx where fl = f+ and f2 = f - . t h i s c o n s i d e r e.g. (4.32)

VTToJ’= (1/2n)jom f;,fl (1/2iT)/z (UleikY)e-jkxdkdx etc.).

t a t i o n s U - s a t i s f y i n g HU = UH,

M- = U+M+,

W

e t c . w h i l e S = U;’U-,

where U,-

‘L

U = UM,,

etc.

and Nf = T;hiTo.

T:To

U = U-

Here Then

w i t h sZ1 = b/a = Rr s i n c e i n [ F a l l s12

w r i t i n g a = 1/T = c12 and b = cll our p r e v i o u s s,,,

Thus WU*U = I = UWU*,

S = M-/M,

1/MM-,

“u

U1$

Now we want t o deal w i t h transmu-

e t c . as b e f o r e .

we w r i t e f o l l o w i n g [ F a l l Ui = UN,:

=

etc.

+ a 0 a - O M i = (-b- l ) ; M = ( - ); M- = ( ’ b - ) 2 b 1 2 Oa

“+

One has N t = T;NiTo b = Rr/T,

with

4;

= (MY)* 1

sZ1 = Rry s12 = Ra = -b-/a, A

s t a t e s f o r convenience we have Wi

UiWiU?.

s2 1

Thus

and S = MY-lM;

a = 1/T, e t c . ) .

“+-I .’+*-I

= N;

lN1

i1 = ( s11521) and i2= (512 ^w2 %2h i n (*@A) o f Remark

i n [ F a l l and

+-1 -

= M2

-

’12

M2 ( n o t e here a g a i n

Assuming no bound

= (MfMf*)-l

= TW ; Tio

with I =

) ( r e c a l l o u r p r e v i o u s s12 i s

1 4.13

-

c f . a l s o 511 and [Fa3]).

SCATTERING THEORY

137

5. SCACCERINC CHE0Rg. I U (SPECCRAC MECH0D8 I N 3 4 ) .

We c o n t i n u e w i t h some

more r e c e n t developments, i n c l u d i n g a b r i e f t r e a t m e n t o f R. Newton's MarEenko method f r o m s e v e r a l p o i n t s o f view ( c f . [NwP;Rsl;Yel]).

Again a remark

f o r m a t i s used i n t h e same s p i r i t as before ( c f . a l s o [ N b l l ) . R e f e r r i n g back t o Remark 4.2 l e t us go now t o t h e M e q u a t i o n o f

REIRARK 5.1,

Newton ( c f . [ N w ~ ] ) . R e c a l l f i r s t f r o m d e f i n i t i o n s and a s y m p t o t i c p r o p e r t i e s # T # T t h a t J,' = Q8 J, ( n o t e 8#dT = b 8 = I, J/(-k,x) = J/(k,x)-, b = QSQ, e t c . ) . ( H = (;

Set y ( k , x ) = exp(-ikHx)J,(k,x)

8,

= exp(-ikHx)bexp(ikHx);

-9)

h e r e ) and t h e n y # = ;S,'Qy

thus

T

bx

(5.1)

=

1

Rrexp(2i kx) T

Rgexp( - 2 i k x ) One notes t h a t 5,

where

i s t h e m a t r i x which would a r i s e f r o m a p o t e n t i a l U(x+y).

Now i n o r d e r t o deal w i t h ( * 6 ) i n Remark 4.2 d i r e c t l y one w r i t e s i t i n t h e f o r m (*) f # Define

-

I = [u-I]Q[f-I]Q + Q[f-I]Q + 0 - 1 and t a k e s F o u r i e r t r a n s f o r m s .

(A) Z ( a ) =

(1/271)lI e x p ( i k x ) [ u ( k ) - I l d k Q ;

Then (*) becomes

[f(k)-Ildk.

( 0 )

F = 2:

*

F#Q

F(a) = (1/2n)/f

+

# QF Q

exp(-ikx)

+ ZQ. B u t f ( k ) - I i s

a n a l y t i c f o r Imk z 0 and vanishes as I k l + m t h e r e so F ( a ) = 0 f o r # Hence f o r a 2 0, F(a) = Z ( a ) Q + ( C * F ) ( a ) Q and we have Given F and Z d e f i n e d by

CHE0REM 5.2,

(A)

a <

0.

w i t h u known (and F h a v i n g t h e

p r o p e r t i e s i n d i c a t e d ) i t f o l l o w s t h a t F s a t i s f i e s t h e MarEenko (M) e q u a t i o n 2:(a)Q + :/

( 6 ) F(a) =

Z(~+B)F(B)~B f oQr

a

2 0.

I t i s more c o n v e n i e n t i n p r a c t i c e however t o use y and 8, as i n T 2 1 We n o t e t h a t bibT = bTS# = I w i t h b = QSQ ( Q = I)so Q.5' = 5- Q

REIRARK 5.3. (5.1).

Thus ( + ) J/(-k,x) = $-'QJ,(k,x),

(exercise). m a t r i x Jx

'L

y# = S i ' Q y , e t c .

There i s a J o s t

Sx e t c . and a s o l u t i o n f x t o t h e R-H problem f # = o-QfQ f o r ux

E x p l i c i t l y as b e f o r e one a r r i v e s a t a M e q u a t i o n ( e x e r c i s e - n o t e f X = y s i n c e QI = I, 8, T = sf-', e t c . ) ( m ) ~ ( a , x ) = x x ( a ) ? t 10" cX T ( a + ~ ) F ( ~ , x ) d f 3 ;E x ( " ) = ( 1 / 2 1 ~ ) j ze x p ( - i k x ) [ S ( k ) - I I Q d k ; y ( k , x ) = + x, T h i s can be e x p ( i k x ) F ( a , x ) d a ( a g a i n F 'L ( 1 / 2 n ) / [ f x - l ] e x p ( - i k x ) d k ) . ( f ( k ) = J-'(k)).

i

7

/om

broken down i n t o components and r e c o v e r y f o r m u l a s f o r q d e r i v e d (e.g. -2DxF,(0,x)

=

I t i s a l s o i n t e r e s t i n g t o proceed somewhat d i f -

2DxF2(0,x)).

f e r e n t l y ( c f . [Nw2;Kyl]).

q(x) =

Namely, c o n s i d e r f = (l/T)J, where (**) f ( - k , x )

=

(; k ) Q f( k ,x ) w i t h

(5'2)

'[ =

T -R

-Rr T

]'

'x

=

[

T -R2exp(-2ikx)

Set 5 = e x p ( - i k H x ) f = (1/T)y and t h e n

(*A)

'5

=

-Rrexp(2i k x ) T

gxQ5. Again assume no bound

138

ROBERT CARROLL

s t a t e s and one o b t a i n s a M e q u a t i o n Z(CY,X) = Z X ( a ) ? +

(5.3)

-

gx(k)

I I Q d k ; X(a,x)

;1

e ikcr

ZX(at8)X(~,x)d~Z ; X ( u ) = (1/2n)l:

-

= ( 1 / 2 ~ ) 1 1[ c ( k , x )

i]exp(-ika)dk

T h i s s p l i t s i n t o two uncoupled equations ( u s i n g a n a l y t i c i t y and asymptotic p r o p e r t i e s o f T-1 e t c . ) and f o r

CY

+ 2 x ) X 1 ( 8 , x ) d ~ ; Z2(a,x) = -k1(a-2x)

> 0 ( * a ) Zl(a,x)

-

=

-; (at2x) r

- 1;

1 ; ~ l ( a t ~ - 2 x ) H 2 ( ~ y x ) d ~where

ir(a+6

^R

m

(a) = r 9 . t

( 1 / 2 1 r ) / ~ Rrye(k)exp(ikoc)dk. (*b) -2DZ1(0,x)

Then a f t e r some m a n i p u l a t i o n one o b t a i n s a l s o We n o t e t h a t i f q ( x ) = 0 f o r x < 0

= q ( x ) = 2DxZ2(0,x).

t h i s reduces t o t h e G-L procedure ( c f . [ N w ~ ] ) . Again bound s t a t e s can be worked i n t o t h e t h e o r y b u t we o m i t t h i s . Now go t o 3-D so one considers -A9

RRnARK 5.4. equation

(*+I +(k,e,x)

where 9 (k,e,x) f u r t h e r v(k,e,x)

J/,(k,e,x)

e x p ( i ( ke,x)

=

0

=

(thus k

I k / i n say (3.2) and IL

- 9 0 (k,e,x)]

= -S-(k,e,;)

can see t h i s from ( 3 . 2 ) s h i f t e d as 1x1

+ m).

jii

and ST =

(*a)

8'

e,e')

= k(k,e',e), =

u n i t sphere S

= A(k,;,e)

-

where x = x/

XI

=

-Ixlexp(-iklxl) %

8 ' here (one

as t h e k e r n e l o f

( k / 2 a i ) A ( k ) ( t h i s i s t h e same as ( 3 . 6 ) ) .

= Zi(k,e,e')

and A(k,e,e')

QsQ where s( has k e r n e l i ( k , e , e ' ) ,

and Q f ( e ) = f ( - e ) .

8-'Q$.

-tm

For f i x e d k one t h i n k s o f A(k,e,e')

One has e v i d e n t l y A(-k,e,e')

9'

Set

or (*+) by n o t i n g t h a t t h e argument o f I x - y l g e t s

an o p e r a t o r A ( k ) and S ( k ) = I =

@-).

%

Note here from (3.5) t h a t - S (k,e',e)

and by (3.7) one expects (@-(x,k) = 9 ( k , e , x ) )

A(k,e',e)

(A*)

3 [ e x p ( i k l x - y l ) / ~ x - Y ~ I ~ ( Y ) ~ ( ~Y , ~ , Y ) ~

ke,x)) and (*=) A(k,e',e) = -(1/411) t h a t A(k,e,e) = - ( 1 / 4 1 ~ ) 1 q ( x ) y ( k , e , x ) d 3 x

(forward s c a t t e r i n g amplitude).

and

%

2 3 q9 = k 9 i n R w i t h L-S

= $(k,e,x)exp(-i(

I ~ o ( k , e ' , x ) q ( x ) I L ( k , e y ~ ) d 3 ~so

[@-(x,k)

- 1

t

A(k,-el,-e) so t h a t T T 8 has k e r n e l 8 ( k ,

=

It i s easy t o see t h a t

$*k

I

= .5$* =

One t h i n k s o f A and 8 a c t i n g on f u n c t i o n s o v e r t h e

2

as i n ( 3 . 6 ) and under reasonable hypotheses on q ( x ) ( c f . [ N w ~ ] ) one has f o r F E Lp(S2), p > 4 ( A A ) L; k211A(k)Fl12dk < cllFI12 Hence 2 2 P2 7 P' Jm II(5-1)FII dk 5 cllFII and s i n c e IIgll < cllgll i n Lp n L ( S ) i t f o l l o w s t h a t P p 2 P f o r s u i t a b l e F, if II (S-I)Fl12dk < cllFf2- so t h a t t h e F o u r i e r t r a n s f o r m o f P' ( 5 - 1 ) F w i l l make sense c l a s s i c a l l y .

-0.7

REIIIARK 5.5.

Now i n 3-D one cannot d e f i n e

v

by toundary c o n d i t i o n s ; i n s t e a d

t h e J o s t f u n c t i o n J i s determined f r o m a R-H problem f i r s t and then f i n e d v i a J. potheses

v

i s de-

Thus ( a g a i n assuming no bound s t a t e s and assuming s u i t a b l e hy-

on q ( x ) as i n [Nw2,3]) J i s determined as a s o l u t i o n o f Then J w i l l be a n a l y t i c i n C+ w i t h I I J ( k ) - Ill

QJQ$(R-H problem).

(Am) +

J#

=

0 as I k l

SCATTERING THEORY

139

m and 9 i s d e f i n e d as q ( k , x ) = J(k)$(k,x). Then Ip(-k,x) = Qp(k,x) and f o r f E Lp(S2), @(k,x) = 1 f ( e ) [ v ( k , e , x ) - $o(k,e,x)lde E Lk2 w i t h 11@11: 5

-+

t

cllfl12 Furthermore @ has a n a l y t i c c o n t i n u a t i o n s i n t o C and C-, e x p ( i k l x 1 ) P ' t @ i s bounded i n C , and e x p ( - i k l x l ) @ i s bounded i n C-. Hence @ E L 2 w i t h support i n [ - l x l , l x l l

and

Thus 9 i s e n t i r e i n k o f e x p o n e n t i a l t y p e 1x1 , b u t i t s e x i s t e n c e depends on s o l v i n g t h e R-H problem

[ N w ~ ] ) shows t h a t

(A&)

(Am)

f o r J.

2e-v[w(e.x

t

Some a n a l y s i s which we o m i t h e r e ( c f .

,e,x)

-

w(e.x-,e,x)]

c a l l e d t h e " m i r a c l e " ; t h e e q u a t i o n appears t o be i s a potential q(x)

(A&)

One can a l s o t h i n k o f

i s guaranteed ( i . e . t h e

e e

= q ( x ) and t h i s i s

dependent b u t when t h e r e dependence d i s a p p e a r s ) .

v - $ ~as a f u n c t i o n o f ke and produce a f o r m u l a (A+) 3 - J;y,~lxl h(x,y)$,(k,e,y)d y. A s p e c t r a l measure f o r t h e

Ip(k,e,x) = $,(k,e,x) 9 f u n c t i o n s can a l s o be o b t a i n e d as b e f o r e s t a r t i n g w i t h ( A = ) 6 ( x - y ) = ( 1 / 2 2nI3/ ?(k,e,x)$(k,e,y)k d dk. Assuming no bound s t a t e s one o b t a i n s 1 (q(k, = 6 ( x - y ) where (**) dp/dE = ( k / 1 6 n 3 ) [ J ( k ) J * ( k ) ] - '

x),dp(E)p(k,y)) 0).

Using

(A+)

ho (x,Y) =

I ($o( k, x ) ,d(P-PO No ( k ,Y)

3 where dpo/dE = ( k / l 6 n ) f o r E > 0. computes w v i a Radon t r a n s f o r m s uses

(A&)

RrmARK

(@A)

To o b t a i n t h e p o t e n t i a l f r o m h one f i r s t w(cr,B,x)

=

1 h ( x , y ) & ( a - e - y ) d 3y and t h e n

(we w i l l d i s c u s s Radon t r a n s f o r m a t i o n s i n more d e t a i l below). The procedure above has i n v o l v e d f i r s t d e t e r m i n i n g J v i a a R-H

5.6,

problem and t h e n u s i n g t h e G-L t y p e machinery.

It i s more d i r e c t t o use a

M method as f o l l o w s ( a g a i n assume no bound s t a t e s o c c u r ) . back t o

(E = k2 >

one o b t a i n s a l s o a G-L t y p e e q u a t i o n

(me)

JI# = $-'Q$ and s e t s y(k,e,x)

= $(k,e,x)exp(-i(

Thus one goes ke,x)).

I n the

= $(k,e,e')exp(ik(e-e',x)) same s p i r i t as b e f o r e one s e t s ( a & ) $,(k,e,e') # and t h e n Y = $,'QY. I t f o l l o w s a l s o t h a t bx a r i s e s f r o m a s h i f t e d potenn 2 t i a l q x ( z ) = q ( x + z ) . L e t 1 be t h e f u n c t i o n 1 on S and d e f i n e

(5.6)

G(a)

= i ( 2 n ) - 2 1 z kQAx(k)e kadk.

G(a,e,e')

= i(2a

-2

:I

kA

The corresponding M e q u a t i o n now i s ( a > 0 )

k,-e,e')e - i k [ a + x . ( e + e t ) l d k

140

ROBERT CARROLL

=

t G(a)? where y ( k , x )

q ( x ) t h e operators G

rf

t i c f o r Imk > 0, S;'Qy,

t

1; F ( a ) e x p ( i k a ) d a .

G#

G(a+B) and

%

e t c . ( n o t e y ( k , x ) = Y(k,e,x)

there i s a miracle equation

G ( - a - 8 ) a r e compact, y ( k , x )

%

2 11 dk <

-

Iy(k,e,x)

Under s u i t a b e hypotheses on

my

y(k,e,x)

-+

1 as

so F ( a ) = F(a,e,x),

k

t

(A&)

q ( x ) = -2e-vF(O ,e,x).

kl %

i s analy-

my

-+

lkl).

y# =

Again

We r e f e r t o "w2,

The m i r a c l e phenomena i s unpleasant

31 f o r f u r t h e r d e t a i l s and d i s c u s s i o n .

i n p r a c t i c e s i n c e i t appears u n s t a b l e t o small changes i n p o t e n t i a l e t c . and we go n e x t t o some r e c e n t v a r i a t i o n s and r e f i n e m e n t s f o r t h e i n v e r s e problem i n 3-D ( c f . a l s o [ N w ~ ] r e g a r d i n g s t a b i l i t y ) .

RrmARK 5.7, We go t o [Ye1,2] now where a method o f l a y e r s t r i p p i n g i s d i s cussed which r e l a t e s a l s o t h e Newton technique, t h e Born approximation, methods o f [Rsl],

and a t e c h n i q u e o f [l(y1].

The emphasis here i s on problems

i n a c o u s t i c s c a t t e r i n g ( s e i s m i c waves e t c . ) and one can assume t h e r e a r e no bound s t a t e s w i t h o u t l o s s o f g e n e r a l i t y .

F i r s t one d e f i n e s a g e n e r a l i z e d

Radon t r a n s f o r m f o l l o w i n g [Rsl ; Y e l l based on t h e completeness o f t h e genera l i z e d eigenfunctions $ i n [Ye1,2]

%

a- o f (*+) ( o r a+ o f ( 3 . 2 ) ) .

For v a r i o u s reasons

i t i s deemed more a p p r o p r i a t e to work w i t h u(x,k,e)

which can be r e l a t e d t o an i n c i d e n t f i e l d e x p ( - i ( ke,x)) (5.7)

u(x,k,ei)

= e

-i( ke.,x) 1

- ( 1 / 4 1 ~ ) 1[e - ik I x-y

'L

( e = ei

+(-k,e,x)

%

0 ) via

I / I x-YI 1qu(y,k,ei

One t a k e s t h e n x = I x l e s y es d e n o t i n g a s c a t t e r i n g d i r e c t i o n , and

)dy u(x,

(0.)

N

k,ei)

Q ,

e x p ( - i ( keiyx)) + [ e x p ( - i k l x l )/4nlxl]A(k,eS,ei)

where x(kyes,ei)

=

N

-J exp(i( kes,y))q(y)u(y,k,ei)dy

(*.I).

from

( t h u s A(-k,es,ei)

i s t h e same as A(k,es,ei)

Here q i s taken t o be r e a l and smooth w i t h compact s u p p o r t and

say nonnegative (so t h a t t h e r e a r e no bound s t a t e s ) . transforms i n G(x,t,ei)

%

(0.)

(F-I

s(t-ei-x)

%

(1/2 ) e x p ( i k y ) ) one o b t a i n s ( k

-I$

Note t h a t -A$ + q ( x ) $ =

+ q(x)F

REmARK 5.8,

=

-Ttt

-f

t, u

-+

T) (&*)

+ ( 1 / 4 1 ~ 1 ~ 1 ) R ( t - l x l , e ~ , ewhich ~) i s t h e t i m e response

i n t h e f a r f i e l d t o a probe G(t-ei.x) [Yell.

Taking i n v e r s e F o u r i e r

and i s c a l l e d t h e impulse response i n

k $ becomes

t h e plasma wave e q u a t i o n

(&A)

under i n v e r s e F o u r i e r t r a n s f o r m .

I n o r d e r t o d e f i n e t h e g e n e r a l i z e d Radon t r a n s f o r m one r e c a l l s n

t h e c l a s s i c a l form f i r s t , namely ( 6 . ) Rf(-c,e) = f ( ? , e ) = J f ( x ) s ( s - e - x ) d x 2 which i n t e g r a t e s f ( x ) over t h e p l a n e d e f i n e d by e.x = T ( e E S ). The c l a s s i c a l recovery formula i s (5.8)

f ( x ) = -(1/8n2)1

D?f(T=e-x,e)d 2,2e S

SCATTERING THEORY

(cf.

141

Now w i t h u as above one w r i t e s t h e i n v e r s i o n f o r m u l a from

[Dsl;Hgl]).

553-4 as

12u(x,k,e) / ;(y,k,e)f(y)dyd2ek2dk

f ( x ) = (Zn)-3Ja

(5.9)

0

( r e c a l l i(y,k,e)

s

= u(y,-k,e)

= @-(x,k) so t h i s i s p r e c i s e l y t h e

= $(k,e,x)

i n v e r s i o n we used e a r l i e r ) . Now assume f r e a l so l o Is2 u(x,k,e)i(y,k,e) 2 2 2 f ( y ) d y d ek dk = IF IS2 u(x,k,e)u(y,k,e)f(y)dyd ek dk = i ( x ) = f ( x ) and then

zm

(5.10)

/

f ( x ) = (1/16n3)/1

-

s

/ i(y,k,e)f(y)dyd2ek2dk

u(x,k,e)

= (1/2n)

One uses i n v e r s e F o u r i e r t r a n s f o r m now i n (5.10) t o g e t (G(t,x,e)

/f u ( x ,k ,e )exp ( ik t ) d k (5.11)

)/f U ( x ,k, e )exp ( - ik t )dk )

= ( 1/2n

1

f ( x ) = -(1/8n2)11

2e d t

/u"(x,t,e)D$(y,t,e)f(y)dyd

S h

Thus one has a g e n e r a l i z e d Radon t r a n s f o r m , w r i t i n g R f ( t , e ) = f ( t , e ) = J

(y ,t ,e 1f (y ) dy ,

(5.12)

f ( x ) = R-'?

=

-(1/8n2)1z

/ 2 i ( x , t , e ) D t f2"( t , e ) d

2e d t

S

When c ( x , t , e )

RrmARK 5.9, [Ye1,2]

= 6 ( t - e x ) t h i s reduces t o (5.8).

We s k e t c h now t h e " l a y e r s t r i p p i n g " approach o f Yagle-Levy (see

and c f . a l s o [Lyl;Bal;C10,11]).

= 6(t-ei-x)

+ is(x,t,ei)

RAf = ( A f , b ( r - e - x ) )

=

One t h i n k s o f a s o l u t i o n ';(x,t,ei)

o f t h e wave e q u a t i o n

((A).

Since f o r m a l l y one has

2

( f , A & ( ~ - e - x ) ) = D ( f , 6 ( ~ - e - x ) ) we have from

(set-

((A)

t i n g R i ( x , t , e ) = C(.r,t,e), e a general d i r e c t i o n - h e r e e a r i s e s f r o m R and we suppress any ei i n t h e o r i g i n o f u ) ((6) (DT 2 - Dt)C(T,t,e) 2 = R[q(x)G(x,t, One can express t h i s i n terms o f a f i r s t o r d e r system (Q = Q ( T , t , e ) )

e)].

Now t h e s c a t t e r e d f i e l d is(x,t,ei) (DT - Dt)Q = R[q'i;]. r e i n s e r t e d ) i s causal i n t h e sense t h a t i t i s 0 a t x u n t i l t h e p r o b i n g

(be)

(ei

(DT

t Dt);

wave 6(t-ei.x)

= Q;

( Y b e i n g a Heavyside f u n c t i o n ) . (6.1

i(T,t,e=ei)

i s a l s o causal when ';(x,t)

QS(r,t,e=ei)Y(t-T)

-

(**) (DT + D t ) i s ( T , t , e i )

=

ei)]

= -2Qs(T,t=r,ei)

where R[q(x)](r,ei)

S (x,t,e)Y(t-ei-x)

From t h e d e f i n i t i o n o f Q

= 6 ( t - T ) t Us(T,t,e=ei)Y(t-T).

=

+

Taking Radon t r a n s f o r m s y i e l d s ( e = ei)

A

i n ( b e ) we see t h a t Q(T,t,e=ei) i . e . Q(?,t,e=ei)

= G(t-ei-x)

reaches x; i . e . G(x,t,ei)

Qs(T,t,ei)

2,

';(x,t,ei)

and hence t a k i n g e = ei

and (DT

-

Dt)Qs(T,t,ei)

-

i n (be)

= R[qGs(x,t,

( t h e l a s t e q u a t i o n a r i s e s from

e q u a t i n g t h e c o e f f i c i e n t s o f 6 ( t - T ) i n t h e second e q u a t i o n o f ( b e ) ) . Next one notes t h a t ( u s ( x , t )

2,

rs(x,t,ei))

142

ROBERT CARROLL

x, t ) 6 ( T-ei-x)dX

/

(1/81r~)~/

s s

D2.ru(T2=e2.xye2)D 26us(~l=el.xyt,el

=

)6(T-ei.x)d2eld

2 e2dx

Here one notes t h a t t h e i n t e g r a n d i s non-zero o n l y f o r x such t h a t

"us

while

and

^u

are required only f o r

elex and

T~

e2.x.

T~ =

ei.x

T =

Virtually a l l

of t h e 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 t h e r e f o r e a r i s e s when t h e s e 3 planes coi n c i d e and t h i s a p p a r e n t l y p e r m i t s a r e c u r s i o n scheme t o be developed from

(+*) ( c f . [Ye1,2]).

Thus one s t a r t s w i t h iS(T=O,t,e=ei)

1 and Qs(T=Oyt,e=e.)

and propagates t h e equations r e c u r s i v e l y y i e l d i n g R[q(x)] A

(cf. [Yell f o r details).

Since u(T,-e)

= ;(-.rye)

recursively i n

o n l y a hemisphere o f i n c i -

dence d i r e c t i o n s ei i s needed t o t h e n compute t h e i n v e r s e Radon t r a n s f o r m and r e c o v e r q ( x ) . of R[q(x)Ts(x,t)] Gs(Tytyei)

The o n l y p o s s i b l e problem i s t h e r e c u r s i v e computation b u t g i v e n t h e d i s c u s s i o n above based on (5.13) i t i s o n l y

and ^u(T,ei)

which w i l l be needed i n computing R[qcs] and these

w i l l be known a t each s t e p from t h e r e c u r s i o n (+*).

A recipe f o r t h i s i s

g i v e n i n [ Y e l l ( c f . a l s o [Ye3,4]

where a general u n i f i e d G-L and M procedure i s developed i n 3-D i n v o l v i n g n o n l o c a l p o t e n t i a l s ) .

REmARK 5.10.

L e t us develop some more " s t r u c t u r e f o r m u l a s " f o l l o w i n g [ Y e l l

i n o r d e r t o o b t a i n t h e M e q u a t i o n o f Newton by a n o t h e r procedure ( c f . a l s o [Rsl]).

Thus l e t i +-( x , t y e i )

be a s o l u t i o n o f -A4 + q ( x ) 4 =

= o u t g o i n g ( - = incoming) r a d i a t i o n c o n d i t i o n . (+A)

N+

<-(xytyei)

= u (x,-t,-ei).

t

-f

k and

T

N+

+

a

+

By r e v e r s i n g t i m e one has

Consider t h e general Radon t r a n s f o r m R-

based on u- as i n (5.12) w i t h ( + a ) R-[;i+(x,t,ei)

J "u(y,T,e,)[u

-$,.. w i t h

(yyt,ei) - ii-(y,t,ei)]dy. k ' t o o b t a i n (+OF F R - [ f

- ~ - ( x y t Y e i ) ] ( ~ , e s )=

Apply a double F o u r i e r t r a n s f o r m + (y,k,ei) = J u-(y,k',es)[u

- u- 3 s ,

+

One can assume t h a t u and u- a r e r e l a t e d by some ( i n v e r u-(y,k,ei)]dy. + = u-(y,k,ei)S t i b l e ) s c a t t e r i n g o p e r a t o r S which we w r i t e as (++) u (y,k,ei) = 1 u-(y,k,e)S(k,e,ei)d

by (6.)

above u-(x,k,es)

(5.14)

2e.

= ;+(x,k,-es)

- i - 1 =/

FFR-[?

Now f r o m u = ( 1 / 2 n ) l u e x p ( i k t ) d k = F - ' u one has

?(y,k',-es)u

(2a)3 G(kei + k ' e s ) [ l

-

and

-

+ (y,k,ei)[l

- S-l(k)]dy

S-l(k)]

+

n o t e u , outgoing, w i l l i n v o l v e exp[- k l x l ] / l x l a t + Four e r t r a n s f o r m (F-' 'L e x p ( i k t ) / 2 n and thus (a= 2, u (y,k,ei) (cf.

(Am)

e,y)

-

we d i f f e r here i n (5.14) by

(&I3

=

from [Ye 1 ) .

m

=

Hence R - [ f

with o u r

$

= @(-k,

- G-1

=

SCATTERING THEORY 3 FF[(2a) 6(kei+k1es)(l

-

e,)

1=

Q

6 ( k + k 1 ) and FF[

S-l(k))l =

t y T

(5.15)

-

(r

-

u (x,t,ei)

N+

To see t h i s n o t e t h a t 6(kei+k'

P(t-T).

(2n)I J exp(ikt)exp(ik1r)6(ktk')(l-S-')(k)dkdk1

*

- i-1

Thus l o o k a t R-[$

( 2 ~ /exp(ikt)(l-S-')(k)exp(-ikr)dk. ) g i v e n by (+*) and l e t

143

=

=

P(t-T)

Then some c a l c u l a t i o n which we o m i t l e a d s t o

a.

u (x,t,ei)

N -

= -(1/8a 2

)Is 1 ~-(x,~,es)DtR(t-r,es,ei)r

2 d r d e s ) and f r o m (5.15) one o b t a i n s an M e q u a t i o n ( t > ei x )

(5.16)

Ts(xytyei)

=

[

2eS +

M(t-ei-x,es,ei)d

S

where M(t,es,ei)

= -(1/8n

-2ei-vGs(x,t=ei

x,ei)

2

I

)DtR(t,eS,ei)

and t h e m i r a c l e e q u a t i o n i s q ( x ) =

( c f . also (5.29)).

The c o n n e c t i o n w i t h t h e M e q u a t i o n

i n Remark 5.6 w i l l be e s t a b l i s h e d l a t e r ( c f . Remark 5.13).

REmARK 5.11.

The development i n Remark 5.10 i's somewhat c o m p l i c a t e d a t

times and t o shed some l i g h t on a l l t h i s we g i v e a n o t h e r d e r i v a t i o n o f t h e One begins w i t h t h e

M e q u a t i o n f o l l o w i n g [Rs1,2]. (5.17)

$'(k,e^,x)

so t h a t $+

'L

@-

= e ik'.x

-I

i n (3.2) and $ -

'L

L-S e q u a t i o n i n t h e form

[e'iklx-Y1/4nlx-y[]q(y)$'(k,$,y)dy Here t h e + s i g n

@+.

= (l/2n)lm f(k)exp(-ikt)dk t E v i d e n t l y (em) $-(k,s,x) = $ (-k,-$,x)

A

so t h a t f = e x p ( i k X I ) 6(t-lxl) . and $' k,G,x) = $' -k,e",x). Set now ( c f . -+

(5.18)

outgoing r a d i a t i o n m

condition since our Fourier transform i s F - ' f ( k ) (em))

+ [ e i k ' X ' / ~ x ~ ] A ( k y ~ s . ~ i ) + h(k,giYx); $+( k ,Gi x ) = e ik'i.x

si)

A

- ( 1 / 4 1 ~ ) [ e-ikes.x

=

q(x)$+(kyGi ,x)dx

U ) . T h i s i s t h e same as (em) except t h a t e x p ( - i k l x l ) i s c o n s i d e r e d (q (1/2n)exp(ikt)). outgoing there ( t h e Fourier transform being F-l I n the Q

p r e s e n t c o n t e x t (5.17) i n v o l v e s t h e same A as i n ( * w ) .

The i n v e r s i o n i s

( f o l l o w i n g (5.10)) (5.19)

f ( x ) = (1/16n3)11

[ 2$'(k,e",x)~

?'(k,g,y)f(y)dyd26k2dk

S which l e a d s t o a Radon i n v e r s i o n (5.11).

One denotes by u'(t,G,x)

verse F o u r i e r t r a n s f o r m of $' g i v e n by (5.17) (5.20)

u'= 6 ( t - G - x )

-

-

the i n -

thus ( r = I x - y I )

1 I [6(r~(t-~))/4n~~-y~]q~(y)u'(~,~,~)d~dy

144

(!b'

ROBERT CARROLL

i s extended t o n e g a t i v e k v i a (+.)

t

and one has u-(t,e",x)

= u (-t,-;,x)).

Now suppose a t l a r g e n e g a t i v e t a p r o b i n g wave uo = 6 ( t - e - x ) i s given, and t h a t q ( x ) has compact s u p p o r t . (PWE) [-A + q ( x ) ] u = -utt

The s o l u t i o n o f t h e plasma wave e q u a t i o n

f o r l a r g e n e g a t i v e t equals uo (so u i = 6 ' t h e r e )

and a f t e r s c a t t e r i n g one has u

^eS,')

-

= l i m Ixl[u(t,^ei.x)

1x1) so t h a t ( m A ) R($iyGsyt)

J exp(-ikt)A(k,e"s,;i)dk.

uo t us.

6(t-ei-x)] = -(1/4n)J

For t h e f a r f i e l d s e t (.*)

as x , t

-+

-with

T =

t-1x1

u+(t+e^s-y,6i,y)q(y)dy

T h i s a l l agrees w i t h

(0.)

-

(e"S

R(giY = x/

= (1/2n)

(6*) modulo t h e change

of s i g n a r i s i n g f r o m t h e use o f d i f f e r e n t F o u r i e r transforms.

Now t a k e a

p r o g r e s s i n g wave expansion ( Y = Heavyside f u n c t i o n and E ( s ) = s Y ( s ) ) (me) u(t,g,x) = 6(t-z.x) + F(t,e^,x) t B($,x)Y(t-$-x) t D($,x)E(t-g.x) ( F E C1 i s zero f o r t < G-x).

The " t r a n s p o r t " equations a r e determined by p u t t i n g (me)

i n t o t h e PWE and e q u a t i n g s i n g u l a r terms so t h a t one o b t a i n s

( t h i s fundamental i d e n t i t y i s a c t u a l l y t h e same as Newton's m i r a c l e ) .

Now f o l l o w i n g [Rsl] we g i v e two d e r i v a t i o n s o f t h e Newton M e q u a t i o n ( t i m e and frequency domain). (5.22)

$'(k,e",x)

F i r s t i n t h e frequency domain = JI-(k,G,x)

- (k/2ni)12A(k,;',;e)JI-(k,G',x)d~' S

+

To see t h i s n o t e t h a t t h e r e l a t i o n (-A-E)-' = (-Atq-E)-' ( c f . (5.15)). (-A-E)-lq(-A+q-E)-l f o r E = k 2 t i c can be w r i t t e n ( ~ 6 )(-Atq-k 2 t i c ) - ' = -[I-Go(+k+i~)q]-'G

0 (+k*iE)

where Go(k) i s t h e o p e r a t o r w i t h k e r n e l Go(k, I x -

y I ) = - e x p ( i k l x - y l ) / 4 n ~ x - y ~ ( c f . [Nwl;Rsl]). Apply t h i s t o q ( x ) e x p ( i k z - x ) and use (5.17) t o o b t a i n ( m + ) JI'(k,$,x) = exp(ike"-x) (-Atq-k 2 t i c ) - '

ii!

[q(x)exp(ik&?.x)] this). t

J, (k,^e,x)

the

(see here a l s o ( 6 4 ) i n 13 f o r an a l t e r n a t i v e f o r m u l a t i o n o f

S u b t r a c t i n g t h e + and - equations from ( m + ) g i v e s (H = - A t q ) - J/-(k,e^,x) = [ ( H - k 2 - i c ) - ' - (H-k2+ic)-'](qexp( i k g - x ) ) .

11;

L !l [ 1 ( i n a s u i t a b l e sense)

(ma)

But

i s ( - n i / k ) P k where Pk i s t h e s p e c t r a l pro-

j e c t i o n r e f e r r e d t o k i n s t e a d o f E (dE = 2kdk - see below and c f . (*+), ( * 6 ) , ( b m ) , e t c . i n 12.3). On t h e o t h e r hand by (5.19) - w r i t t e n as (1/8n 3 ) 1 ; Jsz - t h i s s p e c t r a l p r o j e c t i o n i s determined by t h e ISe and u s i n g t h e 9e i g e n f u n c t i o n expansion one o b t a i n s h

(5.23)

J,'(k,;,x)

-

J/-(k,$,x)

= ( - i k / 8 n 2 ) 1 2 J , - ( k y 2 ' , x ) j ?-k,G',y)qe ike -yr

S

(r

= dyd';').

Note h e r e i n (*+) o f 13 f o r example w i t h t

-2kdk one o b t a i n s ( E a i ) ( E ( A ) f , g )

= l i m JA(ARf,g)(-2kdk)

%

-k2 and d t =

so dE/dk

%

( l i m AR)

SCATTERING THEORY

(-2k/2ni). (5.24)

145

Now go t o ( 5 . 2 2 ) and w r i t e f i r s t

+

-

= $ (-k,-?,x)

$+(k,e",x)

(k/2ni)12 A ( k , 6 ' , ~ ) $ + ( - k , - ~ ' , x ) d 2 ~ ' S

+

M u l t i p l y by e x p ( - i k g - x ) and w r i t e B(k,G,x) = $ (k,C,x)exp(-ikG-x) t o g e t B(k,;,x) = B(-k,-G,x) - (k/2ni)Js A(k,s',;)exp[ik(t$'-6)-xIp(-k,-$',x)d 2.e '

-

from which ( s u b t r a c t 1 ) one has 6-1 = B--1 S e t t i n g (***) F(a,$,x)

G ' ,x)

= (l/Zn)!;

m

= ( 1 / 2 n ) L m exp[ik(a+(&;'

( k / Z n i ) J Ae

-

exp(-ikx)[B(k,s,x)

-

( k / 2 n i ) / Ae(B-1).

l ] d k and

M(a,e^,

(**A)

) - x ) ] i kA( k,$' ,$)dk one t a k e s i n v e r s e F o u r i e r

transforms i n the B equation t o obtain (5.25)

F(a,e^,x)

1;

+

(5.26)

A

A

2 e"' +

S

i2

A

4

M(a-6,e,e',x)F(-6,-g4

,x)d2g'dS

i s a n a l y t i c f o r Imk > 0 w i t h B-1 s u i t a b l y bounded

Now B(k,;,,x) F(a,G,x)

+ j2M(a,e,e',x)d

= F(-a,-e^,x)

Hence f o r

= 0.

F(a,$,x)

=

I

CY

>

SO

for a < 0

0

2 M(a,e,e' , x ) d e ' +

I" 1

S

0

A

A

M(a+G,e,e' ,x)F(6,-$'

,x)r

s

(I- = d2$'dS).

This leads t o

CHEBREIII 5.12,

One can d e r i v e t h e M e q u a t i o n (5.26) as i n d i c a t e d and t h e

miracle i s q ( x ) = -26-~F(O+,g,x).

REmARK 5-13,

- (*+)

For comparison purposes l o o k a t (5.6)

and

(**A)

-

(5.

26). Thus M(a,S,-;',x)/Zn = G(a,G',$) represents the kernel o f G ( G ( a ) : 2 2 2 2 L ( S ) + L ( S ) f o r example) and (5.26) i s a k e r n e l f o r m o f (.+).

REmARK 5-14. above.

One g i v e s a l s o i n [Rs1,2]

a t i m e domain f o r m o f t h e M e q u a t i o n

Thus t a k i n g i n v e r s e F o u r i e r t r a n s f o r m s i n (5.22) we g e t

(5.27)

u+(t,e^,x)

Now use u - ( t , s , x ) (5.28)

t = u (-t,-g,x)

usc(t,s,x)

-

= u-(t,;,x)

-

(1/2n)j S

I~u-(T,~',x)D,R(~,~',t-T)dTd'~'

w i t h u+(t,e*,x)

= u sc ( - t , - $ , x )

-

=

6 ( t - $ - x ) + uSC(t,g,x)

( 1 / 2 n ) 1 2 DtR(s,g',t-e S

I,u

m sc (T,-e',x)DtR(e^,~',ttr)dsd2he' S But by c a u s a l i t y u(t,e^,x) = 0 f o r t < 6 . x so f o r t > e - x

(5.29)

(1/2n)/

usc(t,e,x)

-

= - ( 1 / 2 ~ ) /Rt(&,e',t-e ~

( 1/ 2n ) I s *

lzl.*

S ( T ,- ^e ' ,X ) Rt ( e^,

x ) d 2, e'

' ,t + T

-

) d.r d2e"'

x ) d 2.e '

t o get

146

ROBERT CARROLL

T h i s i s t h e same as (5.26) ( c f . a l s o ( 5 . 1 6 ) ) w i t h +

-

h

-

&(a) = u

sc

= t

*

(t,e,x);

-

$ax and 4

A

M(a,e,e',x)

=

(1/2n)DaR(e",G

REmARK 5-15, Note i n pond t o

A

u (a+e-x,e,x)

F(a,e",x)

(5.30)

CL

U~'(T,-$',X).

6 = Tt $'.x.

5.26) we have F:&,-;',x) and we want t h i s t o c o r r e s sc But F(a,e^,x) - u (t,e,x) f o r a = t-e^-x so one t r i e s

Then t h e argument i s c o r r e c t f o r M

l i m i t i n (5.29)

R and we have -:'-x

%

as a

(instead o f $ l a x ; c f . [Rsl]).

The f o l l o w i n g from [ R s ~ ] serves t o u n i f y some o f t h e preceed2 i n g m a t e r i a l , One c o n s i d e r s i n 3-D ( * * a ) (A-Vw2-q+w ) $ = 0 under s u i t a b l e

R E M R K 5.16.

hypotheses on V and q.

For v a r i o u s V,q t h i s can be r e l a t e d t o v a r i o u s phy-

= exp(iw s i c a l problems. The L-S e q u a t i o n ( c f . ( 5 . 1 7 ) ) i s (*r) $'(w,;,x) $ - x ) t J G i ( q x - y ) [ q + w 2 V](y)$'(w,$,y)dy where G i ( o , z ) = -exp['+iw(z 11/41~1z I.

2

The Green's f u n c t i o n s s a t i s f y (**+) [A-w V-q+w2]G' i n (3.8) by (G here

-G i n ( 3 . 8 ) )

(**,)

G' = G'

0

= 5(x-y) and a r e g i v e n as

As i n

+ I Gi[q+w2V]G*dz. 0

(5.17)-(5.18) one has (e"' Q gSy e" Q S i ) (*A*) A(w,6',6) = - ( 1 / 4 ~ ) Ie x p [ - i w 2 ;'.y](w V+q)$+(w,$,y)dy. By r e s u l t s i n 53,5 one knows t h a t f o r t h e Schrod+ + A 2 i n g e r e q u a t i o n ( V = 0 ) (*AA) - [ 8 ~/ i w ] [ G (w,x,y) - G-(w,x,y)] = JS21L-(w,e,x)

-+

$-(q;,y)d6

( a l l formulas i n a d i s t r i b u t i o n sense).

Then w i t h some a d d i t i o n -

a l hypotheses an a n a l y s i s u s i n g F o u r i e r t r a n s f o r m and c a u s a l i t y y i e l d s

tHE0REm 5.17.

I f IT(x,Y)

I

5 Ix-yI/c,

(c-'

-2

= 1-V 2 c m ) then (*A@)

3 y ) = - ( 1 / 1 6 ~ )lf e x p ( i w ~ ) $'(qe^,x)S/'(w,~,y)de^dw ~2

(*A&)

,s

G+(O,x, 3 s ( x - y ) = - ( 1 / 1 6 ~)

lrnjS2 ( A - - q ) e ~ p [ i w ~ ( x , y ) ] $ - ( q $ , x ) ~ ~ ( w , $ , y ) d & ! (~h~e r e A-q can o p e r a t e i n x

- m

[I

o r y ) and f o r u' d e f i n e d v i a $'(w,z,x) = exp(iwt)u'(t,;,x)dt 2 (1 / 8 )~/f Js2 ( A-q)u '( t - T ( x,y) , , x ) u +(t,??,y)dGdt.

y) =

-

Thus f o r s u i t a b l e and one has

(*a*)

3

(*A+)

S(x-

-'

$'(w,z,x) = I d y $ (w,;,y)ip(y)exp(iwT(x,y)) 3 2. i ip(x) = - ( 1 / 1 6 ~ ) ( A - q ( x ) ) [ f &JJs2d e$ (w,$,x)ip'(w,<,x)

ip

define

where l ~ ( x , y ) I 5 Ix-yI/c,.

(*A,)

I f one takes now T(X,Y) = - a ( x ) + b ( y ) w i t h s u i -

3

t a b l e a,b then t h e expression becomes (**A) q ( x ) = - ( 1 / 1 6 1 ~)(A-q)!I d d S 2 d*^e$'(;J,e^,x)exp[-iwa(x)]~'(w,~) where ?''(a,<) = J d 3yip(y)exp[iwb(y )I$'(w,e*, y).

For a = b = 0 t h i s reduces t o t h e e i g e n f u n c t i o n expansion f o r t h e Schro-

d i n g e r e q u a t i o n when c = 1 . F o u r i e r expansion. c

I f i n a d d i t i o n q = 0 o f course one o b t a i n s t h e

On t h e o t h e r hand i f one t a k e s a = e ' . x / c m and b = e ' - y /

then a formula a r i s e s o f use i n i n v e r s e s c a t t e r i n a t h e o r y . Thus one has 3 2. 2 (*oo) q ( x ) = (1/1611 ) ( b q ) L f dwlS2d e $ ( w , ~ , x ) e x p [ - i w e ' . x / c m l ~ ' ( ~ , ~ , ~ ' ) 3 i Note t h a t ip ip takes where T ' ( w , $ , S ' ) = 1 d yip(y)$'(w,e,y)exp[iw;!'.y/cm].

m

-f

SYSTEMS

147

a 3-D function + a 5-D function and this is what one needs for inverse scattering. In particular using (*..) if one takes q = 0 with V E :C and c, = 1 then (9 = V ) ;-(w,$,$') = -4~w-~A(w,$,s')with (*.&) V(x) = -(1/4~ 2 )A:[ dw IS2d 2h-+ eJ,( w , - ~ , x ) e x p [ - i w ~ ' - x ] A ( w , ~ , ~ ' ) / w2 . If one uses V = 0, c 1 in (*.a) with q E Co then (9 'L q) ?-(w,$,*e') = -41~A(w,$,$l) and after some calcula2.-t A 2 tion (*a') q = (1/2a )$'-vL_/,m dw(iw)JS2 d eJ, ( w , - e , x ) e x p [ - i w ~ ' . x ] A ( w , ~ f e ' ) . Bound states can also be worked into the theory and we refer to [Rs1,2] for further details (cf. also [MNl] for interesting 3-D intertwining methods). 6. $lJsCEW AND HALF LINE PRBBLEW. A great deal of the machinery going into the study of inverse problems had its origin in Sturm-Liouville problems on a half line or in 1-D (radial) half line scattering problems in physics. A lot of information is already available in book form on this (cf. [C2,3;Chl; Lal-3;Mrll) and this author among others has emphasized the role of transmutation (or transformation) operators in this development. In fact the Russian school in general, going back to Gelfand; Levitan, Naimark, Marzenko, et. al. in the early 1950's, used these techniques extensively and one should point out their importance also in soliton theory (cf. 158-11). In recent years there has also been frequent application of transmutation theoretic techniques to systems of equations in soliton theory, transmission 1 ine problems, seismic wave problems, stochastic estimation, etc. (see e.g. [Bal;La2; Mrl;Sfl]) and the author has organized some of this systematically (mainly for second order equations) in [C3,10,11,13,14]. It seems appropriate to sketch some of this development here, mainly for systems, especially since some of the techniques and results involve concepts and methods which will be useful in 558-11 on soliton theory (cf. [C3,10,11,13,14] for additional material on second order equations).

We begin with a discussion of the integral equations of inverse scattering theory (1-D, half line mainly) and the underlying transmutations following [C10,11,13,14]. Thus one thinks of equations named Gelfand-Levitan (G-L), Marc'enko (M), Krein (K), Gopinath-Sondhi (G-S), Wiener-Hopf (W-H), etc. and there are a number of ways of deriving, relating, and regarding such equations. Without attempting to give an exhaustive bibliography we will survey some o f the material on basic integral equations and sketch some derivations, connections, points of view, etc., with the intent of providing some unification via the transmutation theme. The guiding principle will be the underlying transmutation theory for various differential operators as

148

ROBERT CARROLL

t h e o r i g i n o f t h e s p e c t r a l i n g r e d i e n t s from which one can b u i l d t h e i n t e g r a l equations.

S t r a i g h t f o r w a r d t r a n s m u t a t i o n f o r systems i s o n l y b r i e f l y d i s -

cussed h e r e and w i l l be developed more e x t e n s i v e l y i n §§8-11.

G e n e r a l l y we P B(y,x),vA(x)) =

Q form v X ( y )

r e g a r d ( s u i t a b l e ) connections o f t h e t y p i c a l = ( P (b,)(y) o f 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 v!'Q o f 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,Q (as i n t e x t ) as a t r a n s m u t a t i o n formula ( c f . 81.6).

The

r e s u l t i n g i n t e r t w i n i n g p r o p e r t y BP = QB ( a c t i n g on s u i t a b l e o b j e c t s ) o f a transmutation i s a v a i l a b l e but i t i s the basic connection o f eigenfunctions as above p l u s v a r i o u s s p e c t r a l formulas f o r t h e t r a n s m u t a t i o n k e r n e l s (e.9.

P

~ ( y , x ) = ( A A(x),q?(y) ) v ) which c o n s t i t u t e t h e h e a r t o f " t r a n s m u t a t i o n t h e o r y " as we see i t ( c f . [C2,3,11,13]). i c a l l y as a "canonical

"

I n t h i s s p i r i t t r a n s m u t a t i o n appears bas-

connection mechanism between d i f f e r e n t i a l o p e r a t o r s ;

i t s r o l e as an i m p o r t a n t i n g r e d i e n t f o r s t u d y i n g i n v e r s e problems i s one o f many byproducts.

For s i m p l i c i t y h e r e we w i l l c o n f i n e o u r s e l v e s t o one dim-

ensional problems and t h e m a t e r i a l i s p r i m a r i l y concerned w i t h h a l f l i n e problems i n geophysics, t r a n s m i s s i o n l i n e s , f i l t e r i n g theory, e t c .

L e t us

remark a l s o t h a t t r a n s m u t a t i o n ideas have been used i n v a r i o u s ways, i m p l i c i t l y or explicitly,

i n many problems, pure and a p p l i e d , r e l a t i v e t o i n t e g r a l

operators, s c a t t e r i n g t h e o r y , s p e c i a l f u n c t i o n s , e i g e n f u n c t i o n expansions, r e l a t e d equations, e t c . L e t us b e g i n w i t h some equations and n o t a t i o n .

I n terms o f t r a n s m i s s i o n

l i n e s one i s d e a l i n g w i t h a system

where I= c u r r e n t , U = voltage, Z = impedance ( w i t h Z(0) = 1), V = UZ-',

I

=

-

and r = (1/2)DxlogZ = Z'/2Z = r e f l e c t i v i t y . Thus (*) 6 V = (0: 4)V = 2 2 2 Vtt and P I = ( D - b ) I = I where r - r ' = and r t r ' = b. We w r i t e a l s o X tt 0 1o), W = QU = (Z-'U ) Z = Utt and P I = ( Z I ) / Z = Itt. One s e t s J = (-1 x x A o o), r 4 (r (-o1'0'1 ) = J, I '' = ( 0 1 1o), and H = - J (so J = HT); then t h e r e a r e canoni-

,'ZI

4

c a l equations ( c f . [C3,10,11,34;D2,3,5])

U =

(i i)

(A)

so t h a t i n f a c t (6.1) i n v o l v e s

Q(Dx)U = [JD

v ( 0 ) Q(Dx)(I)

=

-

W]U = XU where

v JDt(I). AX

It i s a l s o

o f t e n v e r y p r o d u c t i v e t o work w i t h r i g h t and l e f t p r o p a g a t i n g waves WR = (1/2)[V+I] and WL = ( 1 / 2 ) [ V - I ] s a t i s f y i n g f o r 111 = ( WW R ) ( 6 ) DxU t DtW = -W. Such equations g i v e r i s e t o l a y e r s t r i p p i n g techniqubs as i n [Bal,2;K11,2;Ly 1,2] ( c f . a l s o 55). Note a l s o t h a t i f one takes F o u r i e r t r a n s f o r m s (Dt -+ V V - i k ) i n ( a ) t h e r e r e s u l t s (+) Q(I) = -ikJ(I) so t h a t x # i k i n (A) (we w i l l see l a t e r how t o i d e n t i f y x and k ) .

Now from

(A)

one o b t a i n s ( m ) Bx

-

rB =

SYSTEMS

-

AA, -Ax

r A = AB, D,

-

r D = AC, and -Cx

149

- rC

= AD w i t h A ( 0 ) = D(0) =

( 0 ) = C(0) = 0, - A ' ( O ) = r ( 0 ) = D ' ( O ) , and B ' ( 0 ) = A = - C ' ( O ) . 4 A2 , b = BZ-', Z = CZ', and = DZ-' one o b t a i n s ax = -xZb, b,

cx =

-AZG, and

1, B

-

Setting a = -1 = A2 a, a, =

-

w i t h a(0) = z ( 0 ) = 1, b ( 0 ) = a"(0) = 0, a ' ( 0 ) = :'(O) = 2 20, and b ' ( 0 ) = - x ' ( O ) = A. Note a l s o t h a t (**) Pb = -A b, Pb = - A b y Qa = 2 = -A2yw,ith -A a, and

AZ-';

QZ

b = AO;

-

= [F-@'

i n t h e n o t a t i o n o f [C2,3]

x

+

where 2

4

(recall

'v

t h a t 2c

measures (*A) dw = dA/2nlcQ1 2 dvQ = 2dA/nlFQI 2Q(dwQ

Q

c;@-~ P

P

It was

= Z - l w i t h Ap = 2 ) .

Q

=

F p and 2cB = Fa, so f o r t h e s p e c t r a l

=

2dA/n(Fpl

and dwp = d A / 2 n I c p l 2 =

Here we r e f e r t o 51.6 f o r t h e measure dw

RfiRARK 6-1.

+

$ l e x p ( t i A x ) and @ * A % Z 2 e x p ( i i A x ) as

%

Zm r a p i d l y i s assumed and A

-+

a l s o shown i n [C3,10,14,34]

O hQ

b = q! = cPaA P

F @P ] / 2 i ; P-A

P A

Q

and f o r t h e e i g e n f u n c -

To see e.g. P = -AbZ so t h a t f r o m 1; gKZdx = 1; Z(jvAdx w i t h t h a t dv = dwp we use X m mm*l -1 m g = JO gbdwp ( c f . 91.6) one has g ' = I0 gb'dwp = J0 gZ Axdwp w h i l e on t h e

tions

one o b t a i n s dv

Q a

9-

o t h e r hand f r o m

b'

i n t h e same manner ( c f . [Chl;Dcl]).

?=

= -zx/AZ,

=

-1; Zg(g'/AZ)dx

= -(l/A)J,;

ga"'dx = ( l / A )

g ' z d x f o r s u i t a b l e g ( s i n c e z ( 0 ) = 0 ) . R e c a l l h e r e t h e i n v e r s i o n formuM W P m P m v l a s a r e e.g. g = lo g G A d x , g = J0 y A d w p , = 1; hZ-lAO:dx, and h = lo h

J,,

A0 Qdv

A

so ( s i n c e Q m

= -AS:)

=

-(l/A)

-

( Z g ' ) " above and Zg' = 1; m

m-

dvQ = 10 Ayadv

h'

Q'

(-Ag)AO!

MU

Consequently 1 ; A$a"dvQ = J0 Agadwp and t h e supply o f g

(and hence <) i s r i c h enough so t h a t one must have dwp = dvQ.

We remark

(*.) 1 = c F + c-FIndeed, w r i t e Q Q-1 Q Q' 2 F g AQ ] / 2 i w i t h Z ( Z a ' ) ' = -A a and Z(Z-'

a l s o t h a t , f r o m a Wronskian argument, a = c
aQ+

Q h

= -A2.,";

c-aQ

4-A

and

a=

[F

-

t h e n Z-'(ai;

Q

@!

A

a;)'

Note a l s o a s y m p t o t i c a l l y when 2

= +

X and (*@) f o l l o w s by a s y m p t o t i c b e h a v i o r . Zm one has from (6.2),

a ' = -AZmb and b ' =

AZila,

from which, by asymptotics, 2c = Fp and 2cp = Fa. T h i s a l s o i m p l i e s 2 Q 2 d i r e c t l y t h a t dwp = dA/2nlcpl = dv = 2dX/nIF 1 f o r example.

Q

RrmARK 6.2,

Q

Standard p r o p e r t i e s o f q,PyQ3cp.03 e t c . can be found i n [C2,3;

Chl] ( c f . a l s o 51.6).

L e t us a l s o r e c a l l t h a t one d i s t i n g u i s h e s between one

and two s i d e d d e l t a f u n c t i o n s ( o r h a l f l i n e and f u l l l i n e 6 f u n c t i o n s ) i n t h e F o u r i e r theory, v i a 6 + ( x ) = 26.

(2/a)/;

I t f o l l o w s t h a t f o r du = (2/n)dA on

6 ( x - t ) + 6(x+t) + kQ(x,t);

gp x , t )

CosAxdA = 2[(1/2n)l_z exp(iAx)dA] = [O,m)

= (B,Sin)

u

(*&) B Q = (A(x,A),CosAt)u = 6(x-t)

-

=

v

s(x+t)

t

kp(xyt);

150 V

B

Q

ROBERT CARROLL

" P = 6 ( x - t ) - 6 ( x + t ) + kQ; B = (D,Cos)

= ( C,Sin)p

where t h e f u n c t i o n s k p , Q an!

"Yp,Q

li

= 6(x-t) X Q

x ) = q and 2Dxkp(x,x) = 2Dxkp(x,x)

= p.

T h i s i s standard i n f o r m a t i o n i n t h e

h a l f l i n e t h e o r y based on §1.6 and [C2,3;Chl;Mrl;Lal] B,C,D

2

ators 0

-

satisfies for A

6

refer t o

Q

or

u(x).

$

Q

where

4

6

D2

- 4, 4

I,,

Q

Q

= A-'(k')ll

Q

( t h u@s: ;

Q

;!,

etc.).

The o r i g i n o f p r o p e r -

{ i s i n d i c a t e d below and t h e s p e c t r a l form o f t h e Ge t c . f o l l o w s from p r o p e r t i e s l i k e A = 9* QA ( y ) = ( B Q (y,x), =

n

Cosxx) i n a h a l f l i n e t h e o r y so B (y,x) = (A(y,x),CosAy) P form f o r B 9 i n ( * 6 ) i n v o l v i n g s ( x ? t ) w i l l be used below.

REmARK 6.3,

.

The expanded

L e t us say a few words on t h e t r a n s m u t a t i o n theme.

o p e r a t o r s o f t h e form Qu = (A u ' ) ' / A Q i n general A

Q

Q

and

= ($A@]

= r2 - r ' as i n d i c a t e d e a r l i e r ; n o t e a l s o i n t h i s

one has A = aZ-% = Ai(pf =

t i e s l i k e 2D k (x,x) L k e r n e l s 8';

I n f a c t s i n c e A,

a r e simply d e a l i n g w i t h h a l f l i n e Schrodinger o p e r 2 Indeed we r e c a l l t h a t i f (A q ' ) ' / A = -A 9 then $P = i

= -A2;

(pi

etc.

6 we

= Z - l we have

case f o r a =

+ 6(x+t) + kp

have t h e p r o p e r t i e s 2D k ( x , x ) = 2Dxkvg(x,

-

qu o r Qu = u "

-

We c o n s i d e r

qu ( f o r A

=

Q

1 ) where

can be modeled on t h e r a d i a l Laplace-Beltrami o p e r a t o r i n a

noncompact Riemannian symmetric space (e.g. A

= x2m+1 o r sh2m+1x).

The

t h e o r y o f t r a n s m u t a t i o n o f such o p e r a t o r s Q i s e x t e n s i v e l y developed i n [CZ,

3,10,11,14-3D,7-9,31-34]

and we o n l y r e c a l l here a few n o t i o n s needed f o r

d i s c u s s i o n o f t h e G-L and M equations.

The connections o f o p e r a t o r s P and

Q as i n d i c a t e d a r e v e r y s e n s i t i v e i n t h a t one can "see" e x p l i c i t l y t h e c o e f f i c i e n t s and t h e s p e c t r a i n terms o f t r a n s m u t a t i o n k e r n e l s and t h i s p a r t l y e x p l a i n s t h e e f f e c t i v e n e s s o f such methods i n s t u d y i n g i n v e r s e problems.

We

w i l l see t h a t t h e r e a r e many d i f f e r e n t says o f l o o k i n g a t G-L and M equat i o n s , a r i s i n g from d i f f e r e n t p o i n t s o f view, and t h e y o f t e n r e v e a l d i f f e r e n t f e a t u r e s o f these equations.

We s t a r t from o p e r a t o r s P and

Q o f t h e form

above and denote by v P Y Qt h e a p p r o p r i a t e " s p h e r i c a l f u n c t i o n s " s a t i s f y i n g Qu 2 = -A u w i t h u ( 0 ) = 1 and u ' ( 0 ) = 0 ( n o t e A ' / A w i l l have a g e n e r i c s i n g u l a r -

Q Q

i t y l / x so u'(0) = 0 i s needed and i n such s i n g u l a r s i t u a t i o n s one does n o t

have B P y Q f u n c t i o n s a t one's d i s p o s a l ) . i n d i c a t e d i n [C2,3]

Various hypotheses on A

Q

and q a r e

f o r example and we w i l l n o t go i n t o d e t a i l here ( c f .

a l s o [Arl;Cbl;Chl;Kwl;Mbl;Mrl;Lal;Sgl;Tml]). t a t i o n theme from s e v e r a l d i r e c t i o n s .

One can approach t h e transmu-

Thus f o r example i n [La3;LilO]

one

t r i e s t o f i n d i n an ad hoc manner a k e r n e l K(y,x) such t h a t f o r s u i t a b l e f u n c t i o n s f, upon s e t t i n g B f ( y ) = g ( y ) = f ( y ) +

low t h a t QBf = B ( P f ) .

$!

K(y,x)f(x)dx,

For o p e r a t o r s P and Q o f t h e form D2

one o b t a i n s ( f o r even f u n c t i o n s f w i t h f ' ( 0 ) = 0

-

-

it w i l l fol-

p and D

which i s a n a t u r a l

2

-

q

SYSTEMS

151

situat (note

0) (** An i n t e r t w i n i n g k e r n e l K as i n d i c a t e d w i l l f o r m a l l y s a t i s f y

CHEBREIII 6.4.

a Goursat t y p e problem (**) and c o n v e r s e l y a k e r n e l s a t i s f y i n g (**) w i l l f o r m a l l y i n t e r t w i n e P and Q ( a c t i n g on f u n c t i o n s f w i t h f ' ( 0 ) = 0 ) . Another way of d e t e r m i n i n g such k e r n e l s a r i s e s f r o m s o l v i n g t h e

REIMRK 6.5.

f o l l o w i n g Cauchy problem ( c f . [C2,3;ti10]). v(x,O)

= f ( x ) and vy(x,O)

Working w i t h say even f u n c t i o n s f o r y 2 0,

o f P(Dx)q = Q(Dy)q

f ( x ) f i n d a solution v(x,y)

= 0.

< x <

-m

m,

with

Then g i v e n unique s o l u t i o n s IP o f t h i s prob-

lem i n some c l a s s o f f u n c t i o n s one d e f i n e s B f ( y ) = v ( 0 , y ) .

It i s e a s i l y

seen t h a t QB = BP and some g e n e r a l i z a t i o n s o f t h i s can be found i n [C2,3]. Indeed one can deal w i t h t h e more general Cauchy problem w i t h i n i t i a l data v(x,O)

= C f ( x ) p r o v i d e d A and C a r e l i n e a r o p e r a t o r s Y L e t now do = dw r e p r e s e n t a s p e c t r a l d i s t r i b u t i o n o r ,

= A f ( x ) and IP (x,O)

comnuting w i t h P.

Q

f o r convenience, a s p e c t r a l measure f o r Q so t h a t one has i n v e r s i o n s ( c f . §1.6) (A!

f(x) = qf(A)

(*m)

= :1

Q ( x ) d x and f ( x ) = 4 f ( x ) = f; f(x)AA

We assume f o r s i m p l i c i t y t h a t do

= A@:).

t i n u o u s on

[O,m)

Q

f(h)q!(x)dw

= w dA i s a b s o l u t e l y con-

Q

Q

b u t much o f t h e t h e o r y has a v e r s i o n f o r more general s i t -

S i m i l a r l y l e t dw, r e f e r t o t h e o p e r a t o r P i n t h e same way w i t h Pf(A) = 1 ; A PA ( x ) f ( x ) d x a n d P F ( x ) = 1; F ( A b AP( x ) d w p . One can c o n s t r u c t v a r uations.

i o u s 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 as f o l l o w s .

w i t h g ( y , x ) = A:(Y)A~(X)Y(X,Y) (( , ) p Then 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 t h e

Q

Set

1 dwp and ( , )Q Q 1 dw h e r e ) . , c ~ , ~e t,c . ( o r s i mQp l y gener-

(pA P,Q

a l i z i n g Paley-Wiener theorems as i n [Cbl;Kwl;Tml])

one can show t h a t B(y,x)

0 f o r x > y and E ( y , x ) = 0 f o r x < y w i t h ( B(Y,X),P!(X)) P = p A ( x ) = BqQ ( t h u s B = k e r B, F = k e r P = @!/cp R e c a l l here a l s o t h a t 9 = cp@f+ c-@' P -A and s e t As above dw = d A / 2 n / c QAl 2 e t c .

= vA Q ( y ) = Bp,P

=

and (y(x,y),q!(y))

*!

E,

and y = k e r 3 )

f o r use l a t e r .

Q Now a s i m p l e d e r i v a t i o n o f a b a s i c G-L e q u a t i o n goes as f o l l o w s . Consider 9 P v,(y) = ( B(Y,X),P;(X)) and t a k e w - s c a l a r p r o d u c t s w i t h AA(S) so t h a t (6.4) Since ;(y,s)

Q P B(Y,S) = ( l p A ( ~ ) , A x ( S )Q) =

.-,

= 0 for

5

< y one has

(

B(y,x),A(x,S));

P

A = (qA(x),v!(S))@p(C)

152

ROBERT CARROLL

CHE0REIII 6.6.

A b a s i c G-L e q u a t i o n i s 0 = (B(y,x),A(x,5))

-

and A , g i v e n i n (6.3)

REmARK 6.7. K(y,x)

When P =

and A(x,c)

for 5 < y with 0

(6.4).

D2 -

-

p and Q = D2

+

= 6(x-S)

A(x,c)

(6

%

q one can w r i t e B(y,x)

= 6(x-y) t

6+) t o produce (A*) 0 = K(y,5)

+

A(y,c) + lo K(y,x)A(x,S)dx f o r 5 < y. T h i s i s a standard formula f r e q u e n t l y A separate encountered i n p h y s i c s but r e l a t i v e t o e i g e n f u n c t i o n s 0; and 0;. argument, which i s o m i t t e d here, shows t h a t 2 K ' ( y , y ) = q-p as i n (*+).

In

many problems i n v o l v i n g t r a n s m i s s i o n l i n e s o r a c o u s t i c waves one d e a l s w i t h 2 Qu = ( A u ' ) ' / A and P = D where e.g. 0 < a 5 A 5 B < m and A + A r a p i d l y ( c f . 51.6). I n t h i s s i t u a t i o n i t can be shown t h a t @(y,x) = A 4 ( y ) & + ( y - x ) t tx(y,x)

( c f . 51.6 and [C2,3,7,8,9])

and t h e G-L e q u a t i o n becomes

(y)A(y,c)

+ T5(y,s) + l{ Kx(y,x)A(x,5)dx

= 0 for

n o t a good f o r m u l a (because o f t h e i s o l a t e d A-'

5

( f i ) A-4

This i s a c t u a l l y

< y.

t e r m ) and i n p r a c t i c e one

uses an i n t e g r a t e d form ( c a l l e d a m o d i f i e d G-L o r MGL e q u a t i o n ) from which

A i s recovered.

T h i s w i l l be d e s c r i b e d below i n Remarks 6.29 and 6.31 ( c f .

a l s o [C3,10,11,34]

REmARK 6.8.

and 51.6}.

Another view o f (6.4) regards i t as a f a c t o r i z a t i o n problem Thus w r i t e

( c f . [C3,13,17;D2,5]). 3 i s " c a u s a l " and

r =BA i n t h e

i s anticausal.

form A = B - l g w h e r e 8 - l =

This corresponds t o a lower-upper t r i -

a n g u l a r f a c t o r i z a t i o n o f A . F u r t h e r i f we w r i t e W(h) = /v^ and i ( x ) = P Q P 1; W(h)q,(x)dv then A(x,S) = Ap(5)T$(c) where TX denotes a g e n e r a l i z e d P C One can a l s o t r a n s l a t i o n f o r P (determined by T:f(S) = (Pfq,(x),qh(5)),). P d e r i v e t h e formula A = B-':

d i r e c t l y from t h e Parseval f o r m u l a as i n [D5] A

(i

/

Thus one s i m p l y w r i t e s o u t (f,g),,, = ( A f,g) = Qf) # Q, i n terms o f vhQ = BP! t o o b t a i n I = BAB w i t h ( 0 - ' ) # = B # - l = B((AP(x)u(x), ( c f . a l s o [C3,13]).

B-'v(x))

= ( A (y)v(y),Bu(y))

REmARK 6.9.

Q

Consider a g a i n

and y ) .

relates

Q

2 = D -q and

P

2

= D -p w i t h ~ ( y , x ) = 6(x-y) t K

Since A = A p = 1 we have T ( y , x ) = (y,x) and y ( x , y ) 'L 6 ( x - y ) + L(x,y). Q y ( x , y ) and t h u s ( f o r A = 6 + A ) t h e G-L e q u a t i o n i s 6 ( x - y ) + L(x,y) = s ( x - y )

+ K(y,x) + A(y,x) + 1:

K(y,5)A(cYx)dS.

( l + L * ) = 1+A where L*(x,y) L(x,S)L(y,S)dS

=

= L(y,x)

Now ( w r i t i n g 1 f o r 6 ) c o n s i d e r (1+L)

so t h a t (LL*)(x,y)

= 1,'

(XJY = m n ( x , y ) ) and thus A(x,y) = A(y,x).

K ) ( l + L ) = 1 we have ( l + K ) ( l + A ) = ( l + L * ) o r A + K + KA = L*. (6.5)

A(Y,x)

+

K(x,S)L*(c,y)dE

K(Y,x)

+

I,' K(y,S)A(S,x)dC

= L*(Y,x)

=

Since ( l +

Written out:

L(x,y) = 0

( x < y ) . This i s t h e G-L e q u a t i o n a g a i n and t h u s A(x,y) = A(x,y) o r A(x,y)

153

SYSTEMS

= (l+L)(l+L*)(x,y)

1)

( c f . [C3,14;Lo

G-L e q u a t i o n f o r L.

which can be t h o u g h t o f as a n o n l i n e a r P = D2 w i t h q AP ( x ) = Cosrx, A(x,y) =

I n t h e event t h a t

and ( C o ~ A x , C o s A y ) ~= 6 + ( y - x ) + A(x,y) A(x,y) = ( 1 / 2 ) [ g ( x + y ) + g ( l y - x l )I, s e t t i n g x = 0 we o b t a i n g ( y ) = L(y,O) ( f r o m A = A = L ( y , x ) + L(x,y) + /{Ay L(y,c)L(x,c)dE).

T h i s i s an o b s e r v a t i o n o f T. K a i l a t h n o t e d i n Remark

4.10 o f [ C l O ]

-

M r l ] and when

q:

problem,

s t i l l holds ( L

(g

'L

(A*)

2k) w i t h

RrmARK 6.10,

c

we r e c a l l t h a t

(A@)

w^ Q

= (2/n)[l

+ FcL(y,O)]

f r o m [C2,3,26;

i s altered slightly to qQ corresponding t o a s t o c h a s t i c Ash % Lh) and g corresponds t o an a u t o c o r r e l a t i o n

= (2/1~)[1 +

;] ( c f . [C11,28]).

S i m i l a r G-L e q u a t i o n s a r i s e i n a system c o n t e x t i n much t h e

same way ( c f . [C3,10,13,14,34;D2;La2]

and t h e d i s c u s s i o n below).

There a r e

a l s o many r e l a t i o n s o f G-L equations t o general iz,ed o r t h o g o n a l polynomials, s t o c h a s t i c e s t i m a t i o n , g e n e r a t i n g f u n c t i o n s , m i n i m i z a t i o n procedures, approxi m a t i o n t h e o r y , e t c . f o r which we r e f e r t o [C2,3,11,13,15]. Now f o r t h e h a l f l i n e M e q u a t i o n we s k e t c h t h e a u t h o r ' s d e v e l -

REmARK 6.11-

based on [ F a l l ( c f . a l s o 52.7 f o r a n o t h e r ap2 proach). Fadeev worked i n t h e c o n t e x t o f quantum s c a t t e r i n g t h e o r y f o r D q ( w i t h e i g e h f u n c t i o n s O xQ) and developed a t r a n s m u t a t i o n a T approach t o t h e

opment i n [C2,3,13,17,20,21]

M e q u a t i o n ( c f . 52.4).

T h i s was p a r t i a l l y g e n e r a l i z e d by t h e a u t h o r i n [CZ,

201 and then extended c o n s i d e r a b l y i n [C3,13,17,21]

t o F o u r i e r t y p e opera-

t o r s (another, s t i l l w r e general d e r i v a t i o n , i s g i v e n l a t e r i n S2.7 [C18,35]).

'[DZ]

-

-

cf.

The t e c h n i q u e used by Fadeev and e s s e n t i a l l y u t i l i z e d a l s o i n

i s v e r y n i c e b u t i t employs t o o many p r o p e r t i e s o f t h e f u l l l i n e Four-

i e r transform.

But t h e F o u r i e r t r a n s f o r m has t o o many n i c e p r o p e r t i e s !

This

obscures t h e o p e r a t o r t h e o r e t i c a l meaning o f what i s happening and we worked w i t h t h e more general F o u r i e r t y p e o p e r a t o r s p a r t l y i n an a t t e m p t t o i s o l a t e and s t u d y such o p e r a t o r t h e o r e t i c a l s t r u c t u r a l f e a t u r e s o f t h e M e q u a t i o n . The r e s u l t j n g t h e o r y i s a c t u a l l y i n c e r t a i n r e s p e c t s much " c l e a n e r " and e a s i e r t o work w i t h and one expects some p o s s i b l e f u t u r e use o f v a r i o u s gene r a l i z e d t r a n s l a t i o n s and g e n e r a l i z e d c o n v o l u t i o n s which a r i s e n a t u r a l l y i n t h e machinery ( c f . [C3,18,35]). t h e o r y as m o t i v a t i o n ,

Thus we w i l l dispense w i t h t h e F o u r i e r

s i n c e i t i s m i s l e a d i n g , and deal d i r e c t l y w i t h F o u r i e r

t y p e o p e r a t o r s G as d e s c r i b e d below ( n o t e t h a t we do n o t need such s t r o n g hypotheses and f o r d i f f e r e n t types o f "base" o p e r a t o r G, e.g.

w i t h zero pot-

e n t i a l f o r x < 0, one can d e r i v e a c o r r e s p o n d i n g M equation; c f . a l s o S2.7 2 and comnents below). Thus t a k e G = D -g w i t h g even, r e a l , p o s i t i v e , and 1 continuous, p l u s gexp(2Plx) E L f o r some H > 0. The growth h y p o t h e s i s i s

154

ROBERT CARROLL

/I ( l t l x l 2 ) g ( x ) d x

much t o o s t r o n g and c o u l d be r e p l a c e d by say

<

-.

Also

s i n c e t h e main i n g r e d i e n t used i n t h e t h e o r y i s an i n v e r s i o n formula o f t h e t y p e ( A ) i n Theorem 2.4.6 one c o u l d dispense w i t h say g even and s i m p l y use g = 0 f o r x < 0 t o o b t a i n a r e l a t e d and s i m i l a r t h e o r y ( t h e r e would be some changes i n t h e c o n s t r u c t i o n s ) .

Note a l s o t h a t g even i s r e s p o n s i b l e f o r

some o f 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 e t c . below so one would have t o r e work t h e e n t i r e development i n , such a s i t u a t i o n (we o m i t t h i s s i n c e i t i s e s s e n t i a l l y subsumed under r e s u l t s i n §2.7). Thus f o r g even e t c . as i n d i G G 2 G c a t e d above one takes s o l u t i o n s v x and B A o f CP = -A v w i t h v J 0 ) = 1, e t c . G G as b e f o r e and s e t s x A = - O h . One w r i t e s (6.6)

u1 = f, = 2iA[M2(x)vxG u2 = f - =

-

a Gh ;

M1 ( x ) x Gx ] =

G

2iA[M2(A)q:

t Ml(h)xx]

G

= Ex

where ( c f . (**)) M 1 ( A ) = cG(-A), M 2 ( A ) = ( 1 / 2 i A ) F G ( x ) , p ( h ) = ' s l l = s22 =

1 (Q

l/4iAM2M1 = (1/2)[M;/M1

t

i s a reflection coefficient G ( c f . §2.4). One has u ( x ) = u ( - x ) = E A ( - x ) and v a r i o u s p r o p e r t i e s 1 2 o f a n a l y t i c i t y and growth a r e i n d i c a t e d i n [C3,17,21] ( i n particular p(X) i s a n a l y t i c f o r Imx > 0 ) .

The i n v e r s i o n formula i s s i m i l a r t o ( A )

M-/M

2zG = @,(x)

-Rr

= -Rp,)

n Theorem

2.4.6 and says t h a t f o r s u i t a b l e f ( c f . Remark 2.4.14) (6.7)

F(X) =

1:

f(x)@,(x)dx; G

f ( x ) = ( 1 / 2 a ) 1 1 F(A)f(x)p(A)dX = s(x-y)

G Now t h e i d e a o f t h e M e q u a t i o n i s t o r e l a t e t h e t h r e e t r a n s m u t a t i o n s B: v x Q " G V G ( d e f i n e d below). One w r i t e s + *Q (proved below), and B: ah -+ v A , B: A t h e G-L e q u a t i o n B = BA as B = 'Bz; t h e n B = and f i n a l l y BH* = ;(HAH*)=k= -f

iH,

EK where

i s a n t i c a u s a l and

^B

i s causal.

The r e s u l t i n g f a c t o r i z a t i o n o f M

V-lA

data K as K = B

B i s an upper-lower t r i a n g u l a r f a c t o r i z a t i o n o f K r e l a t e d

t o Remark 6.8 ( f u r t h e r d e t a i l s appear below). F i r s t l e t us s k e t c h a k i n d o f g e n e r a l i z e d Kontoroviz-Lebedev ( K - L ) i n v e r s i o n developed by t h e a u t h o r and based on (*.).

? ( A ) = 9 f = lo" f(x)A!(x)dx (6.8)

(*!

and

f ( x ) = ( 1 / 2 ) 1 1 f'(A)wp[cQ@p

= aQ/c-). h Q

t

Thus f o r

w

Q

= 1/2nlcQ12 we w r i t e

c,$)llX]dA = (1/2n)l:

f*( x ) * AQ( x ) d x

F o r m a l l y t h e i n v e r s i o n g i v e s T ( X , p ) = ~ ( A - P ) = ( 1 / 2 n ) f r AQA ( x )

q Qx ( x ) d x ( w i t h a c t i o n on s u i t a b l e f u n c t i o n s ) and t h i s leads t o K(y,x)

=

SYSTEMS

( 1 / 2 n ) l I A;(x)*T(y)dA be a more general P).

-

here G could w i t h B[*:(x)](y) = qP Q ( y ) ( c f . (6.3) One must be c a r e f u l i n u s i n g t h e K-L f o r m u l a s w i t h

T(A,P)a c t i n g i n P as w e l l as 16,191. B(y,x)

X and t h e d e t a i l s a r e s p e l l e d o u t i n [C3,15,

Now f o r G o f F o u r i e r t y p e we c o n s i d e r F ( y , x ) a c t i n g i n a f u l l l i n e

theory w i t h g(y,x) rJ

155

= 0 for

x < y.

<

-m

It i s then appropriate t o i d e n t i f y

w i t h t h e expression

.B(Y,x)

(6.9)

= ( 1 / 2 n ) I I pMl*X(y)c,(x)dx Q G

-

y 2 0 ) . Then i t i s immediate v i a Theorem ( Q i s a h a l f l i n e operator here G Q G 6.8 t o w r i t e ( ? ( Y , X ) , * ~ ( X ) ) = (y) ( r e c a l l M, = c i and *G = 0 / c - ) . We "U A X G now c o n s t r u c t a t r a n s m u t a t i o n B: via

*

-f

~ ( Y , x ) = ( 1 / 2 n ) I I p@y(y)Z:(X)dA

(6.10)

( t h e a c t i o n i s e v i d e n t and one sees a l s o t h ? t i ( y , x ) = 0 f o r

-m

< x

y).

I n t h e same s p i r i t one can a l s o w r i t e (6.11

1

B(Y,x) = (1/271)11 P Q~ A Y )G[ @ ~ ( X ) / C ~ I ~ ~

and e v i d e n t l y B(y,x)

One d e f i n e s now an o p e r a t o r H w i t h k e r G G n e l H(x,s) ( s + x ) g i v e n by (A&) H(x,s) = (1/21r)L1 p[c-/c-]Z (s)QX(x)dX and v G Q A i t follows that = gH ( t o see t h i s s i m p l y compose (B(y,x),H(x,s)) = g(y,s)) G G G G G One notes i n p a s s i n g t h a t @ - X ( x ) = p X X ( x ) + k D X ( x ) and B - X ( x ) = P @ ~ ( X+) G G G W,(x) ( c f . (4.23)). Next we w r i t e B = % w i t h K ( t , x ) = ( P ~ ( X ) , P ~ ( ~ ) W - ' ( A ) )

where W - l

= 0 f o r x > y.

G/w"

( c f . Remark 6.8 and n o t e h e r e AG = 1, w % w and v wG). G tA Q Then s e t t i n g i ( x ) = :1 W - l (X)qx(x)dx we have x ( t , x ) = TxW(x). The k e r n e l =

Q

A

$ ( y , s ) o f B = BH* (H*(x,s)

= H(s,x))

B ^ ( Y , ~ )= (1/2n),f:

(6.12)

(by composition

-

i s now c o n s t r u c t e d d i r e c t l y as

(l/cq)@;(Ek!(Y)dA

G G G n o t e C A ( x ) + a X ( x ) = 4ixM P ( x ) ) and by c o n t o u r i n t e g r a -

t i o n , u s i n g s t a n d a r d a n a l y t i c i t y p r o p e r t i e s ? ?B(y,s) = 0 f o r 5 > y. EHZ0REIII 6-12. i s anticausal.

The M e q u a t i o n i s Upon c o n s t r u c t i o n B

= ( 1 / 2 ~ l ) i z@:(x)[A:(y)/c-]dA

-1

Q

=

and J ( t , x ) = 6 ( t - x ) + (1/21r)L:

RMlARK 6-13, t h e case A

Q

For x , t

20

5

one can w r i t e

has k e r n e l S ( t , x ) + J ( t , x ) w i t h ( s @,(x)dA G

= BH* = g(HAH*) where Y

-

i s causal and

'i

v i a K-L t h e o r y w i t h k e r n e l ;(x,y)

k-';

= HzH* where t h e r i g h t s i d e

G c / c - ) S ( t , x ) = ( 1 / 2 n ) L I sQ(X)@,(t) Q ( XQ) @GX ( t ) @ A G X(y)dh.

one has a l s o J ( t , x )

= 1 w i t h Qu = u "

6

Hence

= (1/2n)LI @f(t)@FA(x)dh.

In

qu one can w r i t e t h e G-L e q u a t i o n o f Remark

156

ROBERT CARROLL

B-l) while the

6.8 as A = TXW(S) = 33* (3

5s'.

=

5

Hk*

HTXw^(S)H*.

M e q u a t i o n o f Theorem 6.12 i s

The f a c t o r i z a t i o n p o i n t o f view i s u s e f u l f o r com-

p u t a t i o n a l purposes when one d i s c r e t i z e s t h e problems a p p r o p r i a t e l y .

We

w i l l g i v e i n 52.7 a n o t h e r d e r i v a t i o n o f t h e r e s u l t s i n Theorem 6.12 and generalizations thereof. We go now t o systems o f t h e form ( 6 . 1 ) .

L e t us f i r s t remark t h a t i f one i s

d e a l i n g w i t h a seismic problem pvtt = -Px ( P = p r e s s u r e ) , P = pv

X'

and w =

a c o u s t i c impedance, one obvt ( v e l o c i t y ) , t h e n f o r y = t r a v e l t i m e and A -1 t a i n s w = - A Pt and P = -Awt. Then s e t t i n g = A-%P and CP = A% ! with p = Y Y (*+4)/2 and q = (*-@)/2 one o b t a i n s (A+,) p + pt = -rq and qy - qt = - r p , Y where r = (1/2)D logA ( r e f l e c t i v i t y ) . Thus we a r e i n t h e c o n t e x t o f (6) Y - i k ) (A=) where p 'L WR and q 'L WL and 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 t (D Otl D f = i k H f - r A f , where H = - j = f 2, (!)Ay and A = ( o); n o t e a l s o

*

x f

-+

(b -p),

w (WR)

w i t h WR = ( V + I ) / 2 and WL = ( V - I ) / 2 from (6) ( - W = -r ).

This i s

t h e sake as (6.2) i n [Nw2] o r (11) i n [ H w l ] except f o r a f a c t o r o f 2 ( 2 r

V , 2f

2,

y, e t c . ) .

We r e f e r t o [H21;Nw2]

2r

f o r unsupported statements i n t h i s

s e c t i o n and l e t us n o t e t h a t t h e t h e o r y o f " w 2 ] i s a l s o sketched i n 52.4.

I n f a c t we r e p e a t here ( i n expanded form and s p e c i a l t o t h e h a l f l i n e prob& 1 lem) a few equations and r e s u l t s f r o m 552.4-2.5. Thus, s e t t i n g 1 ' = ( o ) , one d e f i n e s now J o s t o b j e c t s by ( n o t e r = 0 f o r x < 0 )

f,

(6.13)

= eikHxf'

= e ikHxAy,

fr(k,x) Here f,

e ikH(x-y)

t

r ( Y ) A $ (k,y)dy;

- jOX e kH('-'

) r ( y ) A f ( k, y ) dy

(resp. f r ) d e s c r i b e s a wave which i s r i g h t o r downgoing f o r x l a r g e

(resp. l e f t o r upgoing as x = 0 ) .

I n f a c t fl ( r e s p . fr) i s t h o u g h t o f as coming i n from t h e l e f t (resp. r i g h t ) ; f, c o n t a i n s no upward t r a v e l i n g waves s i n c e i t i s downgoing as x kHx)f

+

A?' as x

= ( l / T$) ( f1z )

Thus exp(-ikHx)f, and one w r i t e s ( a * ) f,(k,O)

-m

-+

( l i m as x - +

-m)

-f

a.

0, and f r2e x p ( i k x ) 1 as x -t ( - i k x ) f o r x 5 O), e x p ( - i k x ) f r -+

-m -f

-f

Now f o r k r e a l f 0

= TAfl(k,x)

-

0

1

( a c t u a l l y fr(k,x) 1

--.

= (l/T)(/r)

= 0, fr(k,O) 2

ir/? and e x p ( i k x ) f :

i l a r l y one has f,exp(-ikx) 1 + 1 and f,exp(ikx) 2 0 2 1/T and f,exp(ikx) R /T as x -+ 0

x -+ m and e x p ( - i = l i m exp(-ikHx)fl(k,x)

w i t h l i m exp(-ikHx)fr(k,x)

W r i t t e n o u t t h i s means i n p a r t i c u l a r fr(k,O) 1 -+

... 1 ' as

-f

-+

(x

= 0 and f r2( k , x ) -+

0 as x

l / ias x -+

.+ m ) .

= 1, f r1e x p ( - i k x )

my

-+

= exp m.

f,exp(-ikx) 1

Sim-+

(x) = f r,, r y e( - k , x ) and one has i n p a r t i c u l a r (QA) f r (-k,x) RIAfr(k,x). I n a s t a n d a r d way one f i n d s a l s o t h a t fr and fL

157

SYSTEMS

a r e a n a l y t i c f o r Imk > 0, c o n t i n u o u s up t o Imk = 0, w h i l e ( f o r Imk > 0) llfLll 5 cexp(-xImk), -+

1 as I k (

7'

-

x)fL(k,x)

[Hwl],

+ m,

IIfrll 5 cexp(xImk),

kz

and

i s analytic with

!t -+

0 as I k l

-+

2

-/I?' b e l o n g t o L (R).

and e x p ( i k x ) f r ( k , x )

-.

Further exp(-ik Now f o l l o w i n g

where t h e development i s somewhat d i f f e r e n t t o t h a t i n [ N W ~ ] , we t a k e and s e t ( @ @ ) G(k,x)

(@A)

l / i i s a n a l y t i c and nonzero w i t h l / i

-

= exp(ikx)fr(k,x)

Then u s i n g s t a n d a r d t e c h -

A?'.

niques and n o t a t i o n f r o m p h y s i c s w i t h G(k,x) = ( 1 / 2 n ) I I G ( k ' , x ) / [ k ' - k - i ~ ] d k ' from

'

e x p ( - i kx)A'i

=

-

0

-(A/2n)Lm Re(k' ) f r ( k ' , x ) [ l z e x p ( - i y ( k t k ' ) ) d y ] d k ' . m

0

fr(k,x)

(6.14)

-

e

W r i t e now

so t h a t ( 0 6 ) has t h e form

= (12/2n)jm Rl(k')fr(k',x)exp(-ik'y)dk'

A(y,x)

(e+)

-

( a f t e r some c a l c u l a t i o n ) i t f o l l o w s t h a t ( @ & ) f r ( k , x )

(@A)

-ikxi' = -1-t e-ikyA(y,x)dy Then

which we p r e f e r t o t h i n k o f as t h e d e f i n i t i o n o f A i n s t e a d o f ( W ) . Set X ( x ) = ( 1 / 2 n ) ~g L~( k ) e x p ( - i k x ) d k

CHE0REI 6.14.

w i t h A d e f i n e d by (6.14).

(so Z(x) = 0 f o r x < 0 )

Then one has an M t y p e e q u a t i o n ( - x 'y

Z(c+y) = 0 f o r 5 < - y ) A(y,x)

= Z(X+Y)Al'

X

A(S,x)Z(c+y)dc.

-Al-y

zx; The r e -

covery f o r m u l a i s r ( y ) = 2A1(y,y).

RRRAaK 6-15.

We n o t e t h a t i n t e g r a l equations as i n Theorem 6.14 w i t h k e r -

n e l s 2 ( ~ + y )(Hankel o p e r a t o r s ) a r e r e f e r r e d t o as M t y p e e q u a t i o n s .

One has

a l s o from (6.14) (which i s a k i n d o f t r a n s m u t a t i o n f o r m u l a ) and

DxA(y,

(Am)

x ) = HD A(y,x) + r(x)AA(y,x) = 0. Y We t u r n n e x t t o 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 s o f t h e o b j e c t s i n t h e system t h e o r y i n terms o f t h e us n o t e f i r s t from

+ 0 )

=

(@A)

In this direction l e t etc. described e a r l i e r . 1 2 t h a t one can determine Dfr(k,O+) = - r ( O ) and D f r ( k ,

q!"

2

- i k ( r e c a l l a l s o f', = 0 and fr = e x p ( - i k x ) f o r x 2 0 which g i v e s i n i -

fR

t i a l values f o r x

g i v e i n i t i a l values f o r 0). Similar calculations f o r x > 0 for w h i l e (even though d i s c o n t i n u i t i e s o c c u r f o r t h e d e r i v a t i v e s a t 1 + x = 0 ) t h e d e r i v a t i v e s from t h e r i g h t a t x = 0 can be o b t a i n e d as Dfe(k,O )

5

= ik(l/f)

-

r ( 0 ) E L / 8 and DfL(k,O+) 2

-

= -ik(EL/f)

scramble t h i n g s r e l a t i v e t o h and k as f o l l o w s .

Next l e t us un2 1 W r i t i n g Vr,L = fr,L + f r(O)/?.

r,e

?

A

and I = 1 we know t h a t and s a t i s f y second o r d e r e q u a t i o n s r,Lz ' r , ~ fr,L 2 2 ( i B = - A B, P I = -k21, QA = - A A, QV = -k V, e t c . ) . Hence we can i d e n t i f y h

h and k a t t h i s l e v e l and express Vr,[

ponding

v pxy Q , o : ' ~ ,e t c . ,

t i e s as x

-+

m

i n terms o f A, B, C, D o r t h e c o r r e s -

t a k i n g i n t o account t h e v a r i o u s a s y m p t o t i c p r o p e r -

and t h e values a t x = 0. Any r e f e r e n c e t o F o u r i e r t r a n s f o r m A

can be abandoned a t t h i s stage.

Thus e.g. Vr

A

( r e s p . Ir)w i l l be a l i n e a r

158

ROBERT CARROLL A

A

combination o f A and C (resp. B and D ) and one expects VA and 1; t o be d i s continuous a t t h e o r i g i n i n view o f t h e c o n s t r u c t i o n o f fr ( c f . a l s o Remark 6.17).

+

We can w r i t e t h e n f o r example

^v, =

(aa)

A

One has a l s o ( 6 * ) Vl

ihe!).

ixge,] Q

Z-'[v,

+

= Z-'[(l+&)v!/f

Q

-

Xe,];

Q

"Ir= Z% -vXP

iX(l-64)O:/?J

= Z-'[ap;

t

A

and I , i s r e a l l y n o t needed here. Now one can use e q u a t i o n (*) t o P P and compare v X y b = X B X l and a = -MI: i n terms o f v;, N

represent a =

t h e a s y m p t o t i c values as x

-f

m

0

CHE0REI 6.16.

The procedure j u s t i n d i c a t e d y i e l d s 1/T = c

FP/2, :r

-

= [cQ

F-/2]/[c-

Q

Q

+ F /2],

Q

g,

and

We n o t e i n passing f r o m

REmARK 6.17. 2

There r e s u l t s

w i t h those o b t a i n e d from (a*).

= [FQ/2

(@a)

-

t h a t e.g.

-

-

Q

+ F /2 Q

c i +

=

+ c$.

c$/[FQ/2

1 fr = A-D + i(B+C) w h i l e 1 2 D x f r = ikf!. rfr and

-

= A+D i(B-C). Hence from ( a ) ( r e c a l l from ( A m ) r 2 2 1 2 2 D f = - i k f r - rf,) (&A) D x f h = i h f h - rf,. Thus x r\, k i n (U) w h i l e D x f r x r 2 1 = - i X f r - rfr and A k again. Thus we see t h a t a l t h o u g h QU = X U g i v e s A C equations f o r t h e v e c t o r s ( B ) and ( D ) d i f f e r e n t from (+), c e r t a i n complex

f

Q

l i n e a r combinations o f t h e A,B,C,D

as i n

(@a)

s a t i s f y (+).

Now i n o r d e r t o o b t a i n a s p e c t r a l v e r s i o n o f A we t r a n s f o r m t h e fr problem (Am)

to

-i)fr

(:

=

(!)A

again as i n (+) and use t h e i d e n t i f i c a t i o n X

2,

k

i n d i c a t e d i n Remark 6.17 t o w r i t e f r o m (6.14)

V (Ir) r

(6.15)

=

e

-iAx

1 (-1)

x

- /-x

,-.,

Set A +A

A (A:

+ A2),-iXy

- A

dy

1 N

= A1 and A1-A2 = A2 and one o b t a i n s a f t e r some 2 = - [ k o ( x y y ) + kO(x,y)]/2 and = [kvp(xyy) + k p ( x , y ) l / 2 .

r2

v l

R e f e r r i n g t o [ C l o , l l ] 'we a n t i c i p a t e t h e development t o f o l l o w and s t a t e Given A d e f i n e d by (6.14) one has A1(y,x)

tHE0RBIl 6.18. (1/4)[kp +

kvp

where t h e mij

-

kQ

-

kvQ]

and A2(Ylx)

= -mll(xyY)

= -mzl(x,y)

r e f e r t o t h e M m a t r i x o f [Bal;C10,11,34],

+

k"P

=

+ k

+ ] Q Q K(x,y) = -A1(y,x),

= -(1/4)[kp

and t h e M t y p e e q u a t i o n i n Theorem 6.14 i s a v e r s i o n o f t h e M e q u a t i o n o f [Bal;C10]

( c f . below) upon i d e n t i f y i n g K(x,y) = -Al(x,y)

and Z ( x ) = R ( x ) .

Phooi: To pass f r o m t h e M e q u a t i o n i n Theorem 6.14 t o t h e M t y p e e q u a t i o n o f [Bal;C10,11,34] w r i t e Al(y,x) = -J-' A2(c,x) (S+y)dS and A2(y,x) = Y Adding these equations one has -K(x,y) = Z(x+y) Jx A1(5,x)Z(S+y)dS. Z(x+y) +'-I K(x,S)Z(S+y)dC which i s t h e M e q u a t i o n o f [Bal;C10] w i t h R = -Y Z ( c f . t h e development below). QED 1 x Consider (6.14) i n t h e form fr = REMARK 6.19, mZl(x,y)exp(-iky)dy with 2 fr = e x p ( - i k x ) + JX mll(xyy)exp(-iky)dy and r e c a l l Remark 6.15 where

-

-X

SYSTEMS fr(k,x)

= Ffr(t,x)

M2*(x,t),

(Dt

+

-ik).

M 2 1 ( x y - t ) = M12(x,t),

159

= R e c a l l a l s o f r o m [Bal;C10] t h a t Mll(xy-t) and M l l = 6 ( x - t ) t mll w i t h M12 = m 1 2 ( m d u 0 c f . below). The response ( i R = M * ( & ) t o i n -

l o some Heavyside f u n c t i o n s 0 i t i a l data ( & ) i n (6) i s t h e n ( m 2 2 m ~ 2 s ( x + t ) ) and F('R) =

ll (mil m2t 16 ( x - t ) ) e x p ( - i k t ) d t

= fr(k,x)

WL

=

( c f . [Bal;ClO]).

LLI

(!R)exp(ikt)dt

The i h p u t gf ( f )

i s n o t used i n [ B a l l s i n c e d i s t u r b a n c e s a r e propagated i n t o t h e medium f r o m t h e l e f t , w h i l e fr correspontis t o a wave coming from t h e r i g h t .

However

t h i s shows t h e background f o r t h e i d e n t i f i c a t i o n s i n Theorem 6.18 and we r e f e r t o [Bal;C10,11]

WL.

and remarks below f o r f u r t h e r machinery r e l a t i v e t o W R Y

For t h e Riemann-Hilbert problem and c o r r e s p o n d i n g M e q u a t i o n t e c h n i q u e

r e l a t i v e t o t h i s h a l f l i n e problem one s i m p l y adapts t h e procedure o f §4,5 ( c f . Theorem 5.2,

Remark 5.3, and [Nw2;C1lY34]).

We go now t o t h e model equations ( 6 . 1 ) which have served so w e l l i n s t u d y i n g i n v e r s e s c a t t e r i n g by l a y e r s t r i p p i n g methods e t c . ( c f . [BalY2;K12;Lyl ,2]).

A p a r t i c u l a r l y c l e a r and i l l u m i n a t i n g e x p o s i t i o n o f t h e c o n n e c t i o n s between l a y e r s t r i p p i n g , r e l a t e d Shur a l g o r i t h m s and Krein-Levinson r e c u r s i o n s , s c a t t e r i n g , e t c . appears i n [Ba2],

I n t h e e x p o s i t i o n below, based on [C10,11,

341, we use t h e t r a n s m u t a t i o n techniques o f [C2,3,13]

t o p r o v i d e spec r a 1

formulas f o r t h e fundamental q u a n t i t i e s appearing i n t h e u n i f i e d G-L, M y K, G-S, e t c . i n t e g r a l equations o f [Bal,2] ( c f . a l s o [Bpl;Gpl;LaZ;Sa1,2] for which some s p e c t r a l i n f o r m a t i o n i s a l s o g i v e n ) .

We a l s o g i v e s p e c t r a

for-

mulas f o r v a r i o u s t r a n s m u t a t i o n k e r n e l s and G-L t y p e equations appear ng i n a systems c o n t e x t as i n [C2,3,10,11,13,14]

and emphasize a g a i n t h a t one can

view G-L, M y e t c . t y p e equations e i t h e r from t h e p o i n t o f view o f p h y s i c s where t h e y a r i s e i n s o l v i n g i n v e r s e problems o r as i m p o r t a n t s t r u c t u r a l formulas

i n v o l v i n g the connection o f d i f f e r e n t i a l operators.

The r e s u l t s

i n v o l v i n g s p e c t r a l i n g r e d i e n t s h e r e seem t o p r o v i d e 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 whole m a t t e r and c o u l d w e l l be u s e f u l f o r computations.

Our p o i n t o f

view i s t r a n s m u t a t i o n a l i n t h e sense t h a t one d e a l s w i t h c e r t a i n problems and s i t u a t i o n s i n terms o f l i n k i n g v a r i o u s u n d e r l y i n g o p e r a t o r s v i a s p e c t r a l p a i r i n g s o f 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 e x p r e s s i n g t h e r e s u l t s i n terms o f fundamental o b j e c t s f o r t h e u n d e r l y i n g o p e r a t o r s .

L e t us n o t e f i r s t t h a t

a spectral form f o r transmutation kernels r e l a t i v e t o canonical operators

Q

-

W and Qo = JDx can be expressed a s f o l l o w s ( c f . [Arl;D2,3,5;Dtl; A o c C % ( B ) = X and Q = ( o ),, (,) La2;Mrl;Sfll). One s e t s R = AJa, Q, = SinXx w i t h Q,o = SinAx and qo = (-coshx) ( n o t a t i o n o f [C14]) and R ( r e l a t i v e t o We have ( c f . [C10,34] Q,) on (-m,m) i s even w i t h 2R = dw = dwp On [ O , m ) . = JDx

(i 0")

Q

Q

160

ROBERT CARROLL

f o r more d e t a i l ). Set @ ( A , f ) = 10" f(x)@(x,A)dx and z(A,f) = :/

tHE0REm 6.20, 2,

for

@*

x

-

real

notation with

a0

(g

Similarly

i: @ ( A , f ) G ( x , x ) d A = iI @(x,x) if @(x,X)@(A,f)dA w i t h c o r r e s -

Then f ( x ) = =

and *o where Ro = l / n .

as a r o l e model, B(Y,X)

@

@(x,A)f(x)dx

and f can be m a t r i x valued).

LI @(A,f)K(x,A)dA

and f ( x ) =

ponding formulas f o r

-

(A,B)

PI,

and ? ( x , f ) .

one d e f i n e s 3 ( A , f ) K(x,f)dx

&

i.e.

W r i t i n g now i n a s t a n d a r d

(l/n)[:

=

9(y,A$o(x,A)dA,

y(x,y)

=

r:

N

@,(x,A)R@(y,A)dA, F ( y , x ) = y*(x,y) = @(y,A)60(x,A)dA one o b t a i n s t r a n s m u t a t i o n o p e r a t o r s B,E: Qo + Q w i t h 3 = B-' ( B = k e r B , y = k e r 3 , ker

i) and

+ K(y,x),

-

i t f o l l o w s t h a t BDo =

d R , etc.

=

Writing ~ ( y , x ) = 6(x-y)

= -DsK(t,s)J

w i t h W ( t ) = JK(t,t)

There i s a G-L e q u a t i o n g ( y , c ) = ( ~ ( y , x ) , A ( x , c ) )

K ( t , t ) J and K(t,O) = 0. (

M0 =

@,

f o r example, one has Q(Dt)K(t,s)

w i t h A(x,c) and

=

Ic.

N

L:

@o(x,A)R@o(c,A)dX

SinAx,SinAc)R.

h a v i n g diagonal e n t r i e s

F u r t h e r B and

There a r e s i m i l a r formulas i n t h e

(

CosAx,CosXc ) R

have t h e s t a n d a r d t r i a n g u l a r i t y .

* t h e o r y which we o m i t here

( c f . [C10,14,

rz

L e t us n o t e t h a t f o r ( f , ~ =) ~ f(A)g(A)RdA one has ( A ( x , x ) , S i n x t ) R

341).

=(B(x,h),Cosxt)R

=

0 s i n c e B i s odd and A i s even i n A.

Thus, f o r example,

4. ,

B ( x , t ) has diagonal e n t r i e s ( A(x,A),Cosxt)R and ( B(x,A),SinAt)R. On t h e V o t h e r hand, one sees e a s i l y t h a t t h e s o l u t i o n q ( x , t ) = ( I ) t o (6.1) w i t h

+

6

+

i n i t i a l d a t a (o+) i s q ( x , t ) = (( A(x,x),Cosxt)R,( t o r , where q(0,t)

(

,

Cosxtdt.

A

)R means 2R on

t o (I)(x,O) V

kHE0REtl 6.21.

5 ) + :/

t

=

G(t) =

(it) (

A

= w ).

(2R = u

[O,m)

B(x,A),SinAt)R),

The impulse response

i s t h e n g i v e n as p( G ( t ) j l and one has

+

1 ,CosAt)R = ( 2 / n ) / r AmCosAtdA from which Am = 1; G ( t )

X)

+ A(x,

The G-L e q u a t i o n i n Theorem 6.20 takes t h e form 0 = K(x,c) K(x,n)A(q,c)dn

c

for

<

x where A = 6, + A and B = 6, + K.

-

G ( t ) = 6 + ( t ) + g ( t ) ( g ( t ) even) and then r ( t ) = k 2 2 ( t , t )

(5 <

column vec-

(6.1

0 = kll(x3S)

+ [g(x+C) + g ( x - C ) I / 2

(1/2)$

+

kll(t,t)

We w r i t e with

kll(x,s)[g(s+C)

+

g ( l s - c ) ) l d s and 0 = k 2 2 ( ~ , c ) + h ( x - 5 ) - g(x+c)1/2 + ( 1 / 2 ) 4 k 2 2 ( x , s ) [ g ( l s e l ) - g(s+c)]ds. Thus f o r t h e i n v e r s e problem w i t h known g one s o l v e s t h e equations (60) f o r kll

and k22 t o g e t t h e r e f l e c t i v i t y r.

F u r t h e r , r e c a l l i n g t h e d e f i n i t i o n s o f gQ,

ip, etc. from

(*6) one can w r i t e

t h e canonical P and Q G-L equations as 0 = k ( y , x ) + [g(x+y) (1/2)#

Q

+

g(y-x)]/2

k Q ( y y t ) [ g ( x + t ) + g ( l x - t ( ) ] d t f o r x < y and 0 = kYp(x,~) + [g(X-T)

g ( x + r ) 1 / 2 + (1/2)/, k v p ( x y s ) [ g ( l s - r l ) dvp).

+

Hence

kvp

= k22 and k

9

= kl,

-

g(s+.r)]ds

for

T <

x ( r e c a l l dw

Q

-

=

and s e t t i n g L = (ks + kvp)/2 ( K r e i n k e r -

n e l ) one o b t a i n s ( r e c a l l a l s o r2 + r ' = 2D

k'

X P

(x,x);

-r'+r2= 2D k ( x , x ) ) x Q

161

SYSTEMS

CQ)R0CLAR!l 6-22, The t y p i c a l G-L e q u a t i o n f o r Theorem 6.20 l e a d s t o t h e kp)/2 and h

=

JtL ( x , s ) h ( l s - c I

+ h(x-c) +

K r e i n t y p e e q u a t i o n 0 = L(x,c) V

)ds f o r L = ( k

t

Q

g/2 ( t h i s corresponds t o WR(O,t) = 6+h and WL(O.t) = h i n t h e

development t o f o l l o w ) . W Now we go t o t h e problem ( 6 ) DxlU + D HW = W, lU = ( R), and w i l l f o l l o w [Ba I t 1 v V 1;ClOI. F i r s t note t h a t W = (1/2) - 1 ) ( 1 ) and e.gWLfor an impulse (I)(xyO) 6 = (:+) one has lU(x,O) = ( & ) ( r e c a 1 6, = 26). Various i n p u t s a r e p o s s i b l e b u t we w i l l be concerned w i t h a so...ewhat d i f f e r e n t p o i n t o f view. t h e approach o f [ B a l l i s l i k e a sideways Cauchy problem

-

In fact

p r o p a g a t i n g an i n i -

t i a l " r e a d o u t " stepwise i n t o t h e medium and f o r d i f f e r e n t values o f WR and WL a t x = 0 ( n o t t = 0 ) one g e t s G-L, M,

K, e t c . equations ( i . e .

sense t h e r e i s a u n i f i e d t h e o r y o f such i n t e g r a l equations

-

i n this

however t h i s

procedure does n o t p i c k up t h e " c l a s s i c a l " M e q u a t i o n o f Theorem 6.12.

w i l l g i v e a s p e c t r a l v e r s i o n o f t h i s here. (6.16)

(:R)(x,t) L

= M(x,t)

*

We

Thus w r i t e

W (WR)(O,t) L

One w r i t e s M = (( M..))

(i= row, j = column) and e v i d e n t l y M(0,t) = 6 1 w i t h '30 r Some c a l c u l a t i o n i n [ B a l l shows t h a t ( 6 6 ) Mll (r O)]M.

y)

DxM = IDt(-: ( x , t ) = MZ2(x,-t) and MZ1(x,t) = M

F u r t h e r t h e e n t r i e s o f M(x,t)

12 ( x , - t ) .

have s u p p o r t i n [-x,x].

We i n d i c a t e now t h e i n t e g r a l equations o b t a i n e d i n Thus one w r i t e s f i r s t ( W ) Mll(x,t)

[ B a l l ( c f . a l s o [BaZ;Bpl]). ml,(x,t)[Y(t+x)

-

Y ( t - x ) ] and MZ1(x,t) = m z 1 ( x , t ) [ Y ( x + t )

i s t h e Heavyside f u n c t i o n ( Y ( t ) = 1 f o r t

>

-

= 6(x-t)

Y ( t - x ) ] where Y

0 and Y ( t ) = 0 f o r t < 0 ) . Using

= 0 f o r t < x one o b t a i n s f r o m (6.16)

t h i s w i t h ( 6 6 ) and WR(x,t) = WL(x,t) f o r -x 2 t 5 x (6.17) WL(O,t+X) +

i-, t WR(O,t-T)mll(x,T)dT : j

WL(O,t+T)mll

+

l-xtkL(0,t+T)m21 (X,r)dT

= 0;

l,"WR(0,t-r)m2,(X,r)dT

=

(X,T)dT +

0

One assumes t h e p r o b i n g wave WR(O,t) t o have a l e a d i n g impulse and t h u s WR ( x , t ) = 6 ( t - x ) t w,(x,t)Y(t-x) takes t h e form

+

w i t h WL(x,t) = w,(x,t)Y(t-x).

Then (6.17)

162

ROBERT CARROLL w h i l e f o r -x 2 t 2 x there i s a propagating equation o f t h e form

2Dxm11(x,x)

Dxm = [JDt 0.

-

Wlm, m

(mll

m12), column v e c t o r , w i t h mll(O,O)

= m21(0,0) =

Now f o r d i f f e r e n t choices o f t h e p r o b i n g waves WR(O,t) and WL(O,t)

We c o n s i d e r ( A )

(6.18) one o b t a i n s v a r i o u s i n t e g r a l equations as f o l l o w s . wR(O,t)

= 0 and wL(O,t)

[Baly2;Ly1,2]

= R ( t ) Y ( t ) o r (B) w R ( O , t )

f o r philosophy e t c . ) .

0 = WL(O,t+X) t K(x,t) t 2

= G(x) = r - r '

where 2DxK(x,x)

= wL(O,t)

-

[C2,3;Fal;Chl]

= mll(x,t)

t

+ wL(0,t+r)] K ( X , T ) ~ T

[WR(O,t-T)

;'("')'I.

I f t h e p r o b i n g waves have t h e

form ( A ) we have a c l a s s i c a l M t y p e equation (WR(0,t-T) i s o f Hankel t y p e

= h ( t ) (see h e r e

We w r i t e now ( 6 ~ ) K(x,t)

+ m 1 2 ( x y t ) ( c f . (6+)). Then adding i n (6.18)

m21(x,t) and L ( x , t ) = mll(x,t) (6.19)

in

= 0 and t h e k e r n e l

we n o t e again how t h i s d i f f e r s from t h e M equations o f

and Theorem 6.12).

Taking (B) above as s c a t t e r i n g data and

i n t r o d u c i n g (+*) KS(x,t) = ( 1 / 2 ) [ K ( x , t )

+ K(x,-t)]

one o b t a i n s a G-L equa-

t i o n i n t h e form (6.20) (0

<

0 = KS(x,t) + [ h ( x + t ) + h ( x - t ) ] / Z

+

iox [h( I t - ~ l ) + h ( t + ~ ) ] K , ( x , ~ ) d ~

t < x ) w i t h G(x) = 4DxKS(xyx) ( n o t e here f r o m (6.18) mll(xy-x)

m21 (x,-x)

= -wL(O,O)

= 4DxKS(xyx)).

so KS(x,x) = (1/2)[K(x,x)

-

wL(O,O)] and

4

= 0 and

= 2DxK(x,x)

F i n a l l y i f one uses s c a t t e r i n g d a t a (B), r e p l a c e s t by - t i n

t h e second equation o f (6.18) and adds i t t o t h e f i r s t equation, then (6.21)

h(x-t) + L(x,t) +

- h(lt-T])L(x,T)dr

= 0

which i s c a l l e d t h e K r e i n equation ( c f . [Fal;Chl] n o t e here t h a t K S ( x y t ) = L ( x , t ) + L ( x , - t ) ,

mll(xy-x)

and C o r o l l a r y 6.22). = 0, and 2m,2(x,-x)

We =

r ( x ) so t h a t r ( x ) = -2L(x,-x). Now l e t us l o o d a t a few sideways Cauchy problems from a s p e c t r a l p o i n t o f view ( c f . a l s o [C2,3,lOy34;Sy2]). Z(Z-lUx)x say U(x,O)

=

x > 0, e t c . ) .

Take, f o r example, t h e e q u a t i o n Q(Dx)U

= 6(t)

This problem, considered as an upward Cauchy problem, was

s o l v e d i n f a c t above v i a IP i n t h e f o r m U ( x , t ) = ( CosAt,vA(x) Q ) Q (GQ =-;R on [O,.o) and r e c a l l A = aZ-' w i t h V = UZ-'). Relative t o Q with A = Z we know t h e standard t r a n s m u t a t i o n ( x , t ) = ( C o s A t , ~ ~ : ( x ) ) ~= y ( t , xQ) A Q-1 ( x ) = y(t,x)Z(x)

=

+ g ( t ) = G ( t ) ( r e a d o u t impulse response), and 6 ( x ) ( a d d i t i o n a l l y U = 0 f o r t < 0, Ut(x,O) = 0 ( = - Z Z x ) f o r

= Utty U(0,t)

Q

i s t r i a n g u l a r w i t h B ( x , t ) = 0 = U(x,t) f o r x

i s " c a u s a l " and

9

Q

>

t.

Thus U ( x , t )

( x , t ) w i l l i n f a c t p l a y t h e r o l e o f a causal Green's

SYSTEMS function.

Now l o o k a t t h e same problem sideways.

w i t h say Ux(O,t) U(x,t)

=

163

(

= 0.

= G be g i v e n

One t r i e s t o f i n d a sideways s o l u t i o n i n t h e f o r m

CosAt,F(A)vA(x)) Q

where dP = 2dA/n so t h a t U ( 0 , t )

PA

= G(t) =

(

CosAt,

Now i f t h i s i s t o r e p r e s e n t t h e s o l u t i o n

F ( A ) ) w i t h F ( A ) = FCG(t) = G ( A ) .

(Cos t.q!)x))u

L e t U(0,t)

it follows that W(A) =

A

c Q/;

(i.e.

= ;(A)

the spectral-factor

G i s e x a c t l y one w h i c h produces t h e c o r r e c t t r i a n g u l a r i t y f o r a causal s o l u tion).

We n o t e t h a t i f T ( r e s p . S ) r e f e r s t o g e n e r a l i z e d t r a n s l a t i o n r e l a -

tive to

o2

= P (resp.

Q)

t h e n TTf(T) t =

= 10" g(x)A,(x) ( Q g ( A k AQ( x ) , v $ c ) ) u general p r i n c i p l e s f r o m [C2,3]

U ( x , t ) =(Y Q ( t , S ) S t 6 ( S ) )

(6.22)

vA Q(x)dx).

P

= (yQ(tyS),6(X-S)/AQ)

f o r m a l l y s a t i s f i e s t h e Cauchy problem Q(D,)U Ut(xyO) = 0.

w i t h sxg(s) = 5 Thus i n keeping w i t h

( (~Cf)COS~T,COSht)

2

BQ(x,t)

= DtU w i t h U(x,O)

= 6 ( x ) and

On t h e o t h e r hand f o r t h e sideways Cauchy problem one c o n s i d -

e r s f o r m a l l y ( B ( x , t ) = ( q AQ( x ) , C o s A t ) = i?A(x,A),CosAt)P) t Q 1-I ( B Q ( X Y ~ ) . T T G ( ~ () = ) U(x,Y)).

(+A)

g(x,t)

The k e r n e l 8 ( x , t ) a c t s as an a n t i c a u s a l Green's f u n c t i o n s i n c e 8 ( x , t )

Q

f o r t > x (while

Q

(x,t)

= 0 f o r t < x).

t a t i o n and 8 (x,T)

Q

We n o t e t h a t

i s even i n

= (2/n)/z

G^(A)CosAtCosArdA

G i s even i n t h e c o s i n e r e p r e s e n -

t h a t (+A) g i v e s

TSO

CHEBREUI 6-23, The impulse response U ( x , t ) = (CosAt,qA(x))u Q f o r Vtt U,)

U(0,x) = 6 ( x ) , and Ut(O,x)

U(x,t)

= (I/Z)~~(X,-)

*

= 0 w i t h U(0,t)

G ( = ) = (1/2)1-'

sideways Cauchy problem w i t h V ( 0 , t ) causal s o l u t i o n U ( x , t ) tion B

Q

= 0

The c o m p o s i t i o n (+A) i s a k i n d o f

= FcG, T:G(T)

generalized convolution since f o r = (1/2)[G(tt~) t G ( l t - ~ l ) ] .

Q

=

= Z(Z-'

= G ( t ) can be expressed v i a

8 (x,T)G(t-r)dr x Q = G ( t ) and Ux(O,t)

as t h e s o l u t i o n o f a = 0.

Note t h a t a

a r i s e s as a c o n v o l u t i o n o f a noncausal Green's f u n c -

w i t h G.

Consider now i n an ad hoc manner i n (6.18) i n i t i a l c o n d i t i o n s (B) SO W,(O,t) V One has = 6 ( t ) + h ( t ) and WL(O,t) = h ( t ) and s e t h ( t ) = 0. = V To s o l v e t h e sideways Cauchy problme f o r t h i s v e c t o r so ( I ) ( o y t ) = (:). V ( T ) we c o n s i d e r f o r dv = (2/a)dA

(iIl)m

Note t h a t t h i s i s n o t t h e most g e n e r a l ' f o r m o f s o l u t i o n b u t i t s u f f i c e s h e r e ( c f . (6.26)).

Then one o b t a i n s from WR = M l l

and WL = M21

164

ROBERT CARROLL

mE0Rm

-

A),CosAt),

(C(x,A),Sinxt)

- (D(x,A),Cosxt)

CosAt)p

-

P

a r e o b t a i n e d as Mll(xy-t) 0 for

It1 >

P

+ (B(x,x),Sinht)

-

(B(x,A),Sinxt)u

( C ( x , A ) , S i r ~ x t ) ~ . M22 and M12

= M 2 2 ( x y t ) and M 2 1 ( x y - t ) = M12(x,t)

One r e f e r s now t o ( 6 4 ) t o w r i t e ( n o t e k ( x , t )

k'Q ( x , t )

= (A(x,x),

(Mll

= MEl

=

1x1.).

w h i l e c,,(x,t) t

w i t h 4M21(x,t)

P

+ (D(x,

= (A(x,h),CosAt)P

one has 4Mll(x,t)

For du = (2/lr)d

6.24.

and

+

;,,(x,t)

and k p ( x , t ) a r e even i n t

Q

k'Q ( x , t )

a r e odd i n t ) (4.)

4mll(x,t)

whereas ( 4 6 ) 4 1 n ~ ~ ( x , t )= kQ

-

kp

= k (x,t) v

-

Q v

kp t k

9'

+ kp(x,t) Hence

m21 s p e c t r a l l y v i a ( 4 0 ) and ( 4 6 ) w h i l e K, L, and KS a r e g i v e n by 2 K ( x , t ) = 2 ( m l l + m21) = k ( x , t ) + c Q ( x , t ) , One can express mll

mE0RRR 6.25, 2L(x,t)

+ m12)

= 2(mll

=

and

Q

k Q ( x , t ) + kYp(x,t),

+ K(x,-t)]

and K S ( x y t ) = [ K ( x , t )

(1/2) = (1/2)kQ(xyt). t L e t us n o t e here t h a t , s e t t i n g w ( x , t ) = J0 U ( x , ? ) d r f o r 4 as

REmARK 6.26,

we o b t a i n w ( x , t ) = ( ~ Q~ ( x ) , [ S i n A t / x ] w ) ~i t h wt(x,O)

i n Theorem 6.23, and w(0,t)

= Y(t)

+ 1; g(r)d.c.

T h i s i s t h e s o l u t i o n used i n [C2,3,7-9]

s t u d y i n g t h e one dimensional i n v e r s e problem f o r SH waves. N

tion A (x)

K i n [C3]).

, {(x)

Thus kQ(y,x) = Z%

= 2D k ( x , x )

(recall

9

-

6 = Z-'(y)tx(y,x)

= $("')'I).

in

I n that situaN

Z - l and one has ~ ( y , x ) = Z 6 ( y - x ) t Kx(y,x) w i t h K(x,x) = 1

%

( t=

(1/2)k

= 6+(x)

- Z4

and from KS =

One can a l s o compare G-L

Q x Q equations as f o l l o w s ( c f . Remark 6.29 f o r t h e MGL e q u a t i o n a s s o c i a t e d w i t h t h i s example).

From [C3] o r 51.6 t h e canonical G-L equation f o r k

t h e c o n t e x t k (y,x)

Q

= Z-4tx(y,x))

i s 0 = (6 (y,t),A(t,x))

Q

Q

(seen i n

f o r x < y where

From B =6'2 + we a l s o have 1 ; CCoshxCosAtdA = 6 ( x - t ) + A ( t , x ) . rl again f o r A = Z -4 a (a v Q~ ) A(x,A) , = Cosxx + :/ k (x,c)Cosxcdc. Thus kQ(y, Q F u r t h e r from U ( x , t ) = (a(x,x), ; kQ(y,t)A(t,x)dt = 0 (x < y). x ) + A(y,x) + 1

A(t,x)

=

Q ,

one has 6 + 2h = U ( 0 , t ) = V(0,t)

Cosxt),

V(0) =-6 + 2h w i t h I ( 0 ) = 6 ) .

= ;/

k o s x t d x ( r e c a l l Z(0) = 1 and

Hence, w r i t i n g do) = (2/n)dx t :dh

one ha

+

:1 CoSAtCoSAxGdA (1/2)1; [CoSx(t+x) + CosA(t-x)]GdA = h ( t + x ) h ( l t - x l ) since 2 h ( t ) = 1 ; SCosxtdA. T h i s y i e l d s k ( x , t ) + [ h ( x + t ) + h (

A(t,x)

=

X-

Q

tl 11 + $

[h(x+-c) + h( It-rl ) ] k Q ( X y ~ ) d= ~0 as i n (6.20). L e t us make a few o b s e r v a t i o n s here about t h e development i n

RElnARK 6.27.

[ B p l ] where G-L, M type, and G-S t y p e i n t e g r a l equations a r e d e r i v e d from t i m e domain p r i n c i p l e s .

+ hu(0,t)

= 0.

say Vx(O,t)

Thus one c o n s i d e r s utt = uxx

s ( x ) u w i t h -ux(O,t)

This corresponds t o a V problem, f o r example, w i t h s =

= -r(O)V(O,t)

- r ( O ) w i t h A(0,A)

-

= 1 so A

4

and

so - r ( O ) = - ( 1 / 2 ) Z 1 ( 0 ) = h ( r e c a l l a l s o A'(0,A) = 2 % lpQ f o r A " - 4 A = -A A ) . Such problems a r e x,h

SYSTEMS

165

t r e a t e d i n many places and we r e f e r , f o r example, t o [C2,3;Mrl]. I n [Bpl] one d e a l s w i t h a causal Green's f u n c t i o n o r impulse response G1(x,t) which i n f a c t equals i i Q ( x , t ) =

(vQ

such an i d e n t i f i c a t i o n ) .

(x),Cosxt)u

(we s e t i Q ( x , t ) =

o

i n s t e a d o f -K1 i n [ B p l ] ) .

The f u n c t i o n Kl(O,t)

o

for t <

E v i d e n t l y 8 ( x Y t ) = 6 + ( x - t ) t K,(x,t)

in

(we use K1

r e p r e s e n t s t h e measured

r e a d o u t g ( t ) . S i m i l a r l y one c o n s i d e r s an a n t i c a u s a l Green's f u n c t i o n G(x,t) + k Q ( x , t ) = 6 9( x , t ) . One knows t h a t h ( x ) = 2D k ( x , x ) = -2DxKl(xy

= 6(x-t)

X Q

gQ

x ) ( r e c a l l here

yQ) and i n c o n s i d e r i n g g Q = G f o r x

%

6Q ( x , t )

one can w r i t e a l s o

= 6(x+t)

+ 6(x-t)

t

0,

--

< t <

my

To d e r i v e i n t e g r a l

k (x,t).

Q

equations now i n [ B p l ] one uses t r a n s m u t a t i o n formulas u ( 0 , t ) = u(t,O) + J,,t K1(x,t)u(x,O)dx and f o r s u i t a b l e u, u(x,O) = u(0,x) t J,: kQ(xyt)u(O,t)dt. Using such r e l a t i o n s one d e r i v e s G-L and M t y p e equations i n [Bpl]. t i c u l a r t h e M t y p e e q u a t i o n o b t a i n e d t h e r e i s , f o r Kl(O,t)

I n par-

= g ( t ) = readout

impulse response ( W ) 0 = k Q ( x , t ) t g ( x + t ) + f-' k ( x , ~ ) g ( t + ~ ) (d t~ < x ) . On t Q t h e o t h e r hand from (6.19) w i t h wR(O,t) = 0 and wL(O,t) = R ( t ) Y ( t ) one obt a i n s ( - x < t < x ) (+m) R(t+x) t K(x,t) t

+

i s g i v e n as 2 K ( x , t ) = k ( x , t )

9

k'Q ( x , t )

JX

-t (with

R(t+T)K(x,T)dr = 0 where K ( x , t )

6

=

2DxK(x,x)

=

D

k

X Q

+ DxkvQ).

Thus we n o t e t h a t t h e problems g i v i n g r i s e t o (++) and (ern) a r e d i f f e r e n t . i s s i m p l e enough

(W)

-

,.

r e l a t i v e t o a V e q u a t i o n 6 + K1 = BQ, 6 + kQ = 6Q ,

4

e t c . and = 2D k (x,x) (= -2DXKl(x,x)). Thus -4 6 ( x , t ) = ( ~ ,Q, ~ ( x ) , C o s A t ) ~ x Q 2 where A = p Q and w = w r e f e r t o Q (Qa = A ( Z - l a ' ) ' = - A a, e t c . ) . For t h e x,h Q V e q u a t i o n ( W ) we go back t o WR = 6 and WL = w = RY so t h a t (I)(O,t) = L (A-RY and t h i s corresponds t o

-

...

A,Fexp( i x t ) ) [ -i( B,Fexp(ixt))

t i(C,Gexp(iAt))

(

(6.24)

=

where t h e p a i r i n g with G = 1 V

S

-

R

(1/2)(kQ t

(

(E = ky,)

,

)

= 1

-

6

(RY

f," R e x p ( - i x t ) d t

+ (1/2)6,(x-t)

- fi P ) *

(1/2)(ip

and I

Now t h e 1 terms g i v e c o n t r i b u t i o n s %

( k p + kvp)/2 + ( 1 / 2 ) 6 + ( x - t ) .

terms g i v e c o n t r i b u t i o n s V

Q

(1/2)(OQ

%

=

[

RY.

-

After

gQ) *

RY

-

(i

6 ( x - t ) + (1/2)[kQ+iQ]

6(x+t) + (l/2)[kQ-iQ]

6 ( x - t ) + (1/2)[kp+ip]

-6(x+t)

I n p a r t i c u l a r ( r e c a l l K = [k t

.

Thus o u r s o l u t i o n (6.24) w i t h F = 1 + R and G V R agrees e n t i r e l y w i t h t h a t o b t a i n e d v i a [ B a l l where (I) = !l)M * -' 6 = N * ( R Y ) where %

(6.25)

X >

-

We have F = 1 + R

i n v o l v e s d r = dX/2n on (--,-).

a 1 t t l e calculation the and I

1 1

t (D,Gexp(iAt))

namely, 0 = V ( x , t )

Q

-

(1/2)[kp-ip]

1

+ k v ] / 2 ) one o b t a i n s (*m) f o r V ( x , t ) = 0 when Q + R ( x + t ) + f-xt R(t+r)K(x,T)dT.

= K(x,t)

166

ROBERT CARROLL

F i n a l l y we w i l l a l s o approach t h e G-S e q u a t i o n as i n [C34] t h r o u g h t h e deThus c o n s i d e r t h e system

velopment i n [Bpl] ( c f . a l s o [Gpl;SalY2;Sol]). and QU = Utt from (*). (6.1) a l o n g w i t h P I = Itt (-m,m)

We s e t diY = (1/2a)dA on

and c o n s i d e r f o r t h e sideways Cauchy problem a s o l u t i o n i n t h e form

(6.26)

[

=

(

Famexp(iAt)

)-

+ i(Ga,exp(iAt)

4

)-

!J

T h i s a l l o w s general

I % F % J and u U % 6 % K. Suppose U0 = 0. Then ( r e c a l l U = Z% and I now I ( 0 , t ) = lo i s g i v e n w i t h U(0,t) = Z-+I) U ( x , t ) = (1/2)A ( 6 - - 6' + k' ) * I, which can a l s o be developed d i r The i d e n t i f i c a t i o n s w i t h [ B p l ] a r e p

%

Q

Q

e c t l y from (6.26). Now f o l l o w i n g [ B p l ] c o n s i d e r D t I o = 26 + h ( 1 t l ) (= Ft), t h f o r t > 0 w i t h l o ( - t )= - t o ( t ) ,and Uo = 0 ( = G ) . I ( = F) = 1 + 1 For 0

t h e corresponding U ( x , t ) one shows i n [ B p l ] t h a t U ( x , t ) = -1 f o r and hence

(m*)

-1 = ( 1 / 2 ) A 4 i Q

* I,

f o r It1 < x.

t h i s i s t h e G-S e q u a t i o n ( c f . a l s o [Gpl;SalY2;Sol]). [Bpl] one c o n s t r u c t s J, K w i t h Kt = -Z% -=A(';To = - s g n ( t ) . It f o l l o w s t h a t K = (1/2)ZQ+ f3Q *

* yo] and *A i =

+x) +

Qb"

= (1/2)Z2BQ

EHEBREIII 6.28. (1/2)K

*

I;

for It[ < x

-Z%

Q

(=A)

It1

z0

I n the notation o f

-

k"

) and J ( 0 , t ) = - F0(t

= ( 1 / 2 ) Z +Q[ T 0 ( t - x )

* io+

K = i3[(1/2)tQ

as d e s i r e d and t h u s f r o m

+

6'

13.

E v i d e n t l y Kt

one has ( c f . [C34])

(m*)

The G-S e q u a t i o n has t h e form (It1 < x ) , 1 = (1/2)Kt = K

*

[6 + (1/2)h(

Itl)] =

x

<

A f t e r some m a n i p u l a t i o n

K(x,t) + (1/2)J-\

*

lo =

K(x,r)h( I t - r l ) d r

where h i s a measured impulse response as i n d i c a t e d and K i s g i v e n by (m.1 which can be p u t i n t o t h e form K(x,t) = ;'[I + Jx t 2 2 recovery c o n d i t i o n i s Z ( x ) = K (x,x) = K (x,-x).

RENARK 6.29,

k"Q(X,r)d.r].

The s t a n d a r d

Now s p e c t r a l l y , b e f o r e d i s c u s s i n g K, i t w i l l be i n s t r u c t i v e t o

c o n s i d e r t h e k e r n e l o f t h e i n t e g r a t e d G-L e q u a t i o n o f Remark 6.26 ( c a l l e d t h e m o d i f i e d G-L e q u a t i o n o r MGL e q u a t i o n i n [Sa1,2]

-

c f . a l s o 11.6).

One

d e a l s w i t h w = ( ~ Q~ ( x ) , [ S i n A t / A ] )as ~ i n Remark 6.26 and t h e MGL e q u a t i o n i s o b t a i n e d v i a a k e r n e l K(x,E) where K 5 ( x y c ) = i 3 k (x,t;) t h e MGL e q u a t i o n i n t i m e domain f o r m ) .

5 ) i s odd i n 5 w i t h t(x,O)

J-'

Q

(see Remark 6.31 f o r

i ( x , c ) e x p ( i h c ) d 5 w i t h K(y,C) = (2/a)J;

= 1 - CosAx]/A,

I n p a r t i c u l a r K(x,x)

= 0, and s e t t i n g J, = 2 i [ q A Q(x)

[J,/2i]SinASdA.

z4,

Thus a(x,A)

K(x, =

J,

-

SYSTEMS

167 'v

CosAx = 1 ; K(x,S)ASinACdC

[?I for 5

ASdS (we w r i t e

a "normal",

s t r i c t l y one s h o u l d w r i t e

6 4

- x ) + I.

same s p i r i t .

-Z k

-

[6-

Q

+

6

k'Q 1 we

+

and f o r m a l l y t h e n ~ ( x , c ) = -K"(x,x)Y(s

Q

-

Q

+ ( 2 / n ) I y [a(x,A) - ~'CosAx][SinAg/

and K(x,t)

Q

On t h e o t h e r hand, r e c a l l i n g K(x,t)

= K(x,-x)

=

Z',

= 0

e t c . one can w r i t e K ( x , t ) = K(x,x)

k' ( x , t ) d t (so Kt t Q Z'[Y(x+t) - Y(t-x)]

Y ( t - x ) ] + Z41x

MGL e q u a t i o n K ( x , t ) =

+ JX ( x , ~ ) d ~ ] . Given ivt Q = -Z'k and one has [Kt] =

= Z%l

should w r i t e [Kt(x,t)]

t Z%inhx,SinAt)li.

f o r It1 > x w i t h K(x,x)

At]/A)

= Zk'

Now

Now we t r y t o deal w i t h K i n Theorem 6.28 i n somewhat t h e

R e c a l l Kt = -Z%

= Ca"(x,A)

[Y(x+t)

[r] -5

+ 10" [$(x,c)]Cos

not d i s t r i b u t i o n , d e r i v a t i v e here).

Z (x)kQ(x,n)dn = -K(x,x)Y(S-x)

A]dA f o r 5,x 2 0.

B* 41.=

= [l - K(x,x)]CosAx

and a(x,A)

=

-$iQ). Hence i n analogy t o

+

(?(x,A)

-

+ Z%inAx,[CosAx

the

Cos

and t h e r e i s t h u s a c e r t a i n s i m i l a r i t y i n s t r u c t u r e between t h e G-S

li

k e r n e l K and t h e k e r n e l K o f t h e MGL e q u a t i o n .

RrmARK 6-30.

L e t us n o t e a r e l a t i o n ( c f .

[C34])

between h i n t h e G-S

i n t h e Newton-Howard system t r e a t e d e a r l i e r .

e q u a t i o n and

i0 =

We f o l l o w [Sa

+ h t h e impulse response t o ?o = 6 ( x , t 0 ) one has a r e l a t i o n I ( 0 , t ) = V ( 0 , t ) + 1 ; h(t-s)U(O,x)ds f o r an a r b i t r a r y U(0,x) ( t h o u g h t o f as imposed). T h i n k now o f Z = -1 f o r x 5 0 and an i n F i r s t given

1,2;Sol;Bal].

6

coming ( f r o m t h e l e f t o r downward) wave WR = 6 ( t - x ) so t h a t W R ( O , t )

= 6(t),

WR + WL = 6 + RY and I ( 0 , t ) = W - W = 6 - RY so t h a t f o r x 5 0 (where Z = 1 and U = V w i t h I= I ) one R L has say U ( x , t ) = 6 ( t - x ) + R ( t + x ) and I ( x , t ) = 6 ( t - x ) - R ( t + x ) . L e t t i n g x = Then U ( 0 , t ) =

WL(O,t) =RY ( c f . [Ba1,2]).

0 t h e n I ( 0 , t ) = 6-R a r i s e s f r o m V ( 0 , t ) = 6+R and hence we o b t a i n

(ma)

0 =

2 R ( t ) + h ( t ) + :1 h ( t - s ) R ( s ) d s which agrees w i t h [ S a l ] ( i n [ S o l ] t h e r o l e s o f V and Ia r e i n t e r c h a n g e d ) .

One can a l s o r e l a t e t h i s t o t h e language o f

[ H w l ] as f o l l o w s ( c f . [ S a l ] ) . *

Z = 1 f o r x 5 0 so t h a t I, a

n

(-ikx)(R,/T) 6 and 6

+

and

^h

?,

*

c

= IL and V,

= exp(ikx)/f

-

= V,.

exp(-ikx)(k,/?).

,.e

0

A

f o r R = RL.

o r 0 = 2R,

t o the r i g h t with 0

Then Ve = e x p ( i k x ) / T + exp For V L ,

a t x = 0 we o b t a i n 1 = (1+6 ) / ? and 1 +

which f o l l o w s h = -2RL/(1+R,) (me)

ie, i, t r a v *e l l i n g

Think o f a

a

I?

= (1

-

t,

generating

R,)/?

from

A 0

+ h + hR,

which i s e q u i v a l e n t t o

One i n t e r e s t i n g f e a t u r e o f such e q u a t i o n s (ma) i s t o es-

t a b l i s h a r e l a t i o n between t h e G-S k e r n e l and t h e M k e r n e l R =

i,

( c f . Theo-

rem 6.14 and r e c a l l from Theorem 6.18 t h a t 8 ( x ) = F - ' i = R(x) w i t h R as i n [ B a l l and F: Dt

+

-ik).

RrmARK 6-31. L e t us n o t e h e r e a l s o a r e l a t i o n between t h e G-S e q u a t i o n and t h e K r e i n e q u a t i o n of [Ba1,2] K r e i n e q u a t i o n from [Ba1,2]

( c f . [C10,11,34]). as L ( x , t )

Thus f i r s t r e c a l l t h e K

+ I - x x H ( / t - ~ I ) L ( x , ~ ) d+~ H ( x - t )

= 0

168

ROBERT CARROLL

( - x 5 t 5 x ) . This a r i s e s from data V ( 0 , t ) = 6 + 2H and I ( 0 , t ) = 6. Now make a change i n t h e G-S f o r m u l a t i o n , i n t e r c h a n g i n g V and I e t c . W r i t i n g Z-'(x)g,(x,t)

= -Nt(x,t)

c)dc] w i t h 1 = N ( x , t ) + J-xxN(x,c)H(lt-cl)dc.

+

( l + h ) N and t h e K e q u a t i o n as 6 ( x - t )

+ L)

= 6

check t h i s s p e c t r a l l y as f o l l o w s . t ) d t = ( 1 / 2 ) I x k ( x , t ) d t = :/ N

K(x,x) = 1

We w r i t e t h i s i n t h e form 1 =

L ( x , t ) + H + J HL = 6 ( x - t ) o r ( l + h ) ( s x

It f o l l o w s t h a t 1 t J - t L ( x , t ) d t

X'

-

Q

-?

Z".

+ :1 kvp(x,

we o b t a i n t h e n as b e f o r e N ( x , t ) = Z-'[1

= N(x,x)

One knows 2L = k

k (x,t)dt = Z

Q

Hence 1 + J-xxL(x,t)dt

-12

= Z-'

u

Q

= 2-'.

+

kvp

We can a l s o and then :-J

Jo K t ( x , t ) d t = Z-%(x,x) as r e q u i r e d .

L(x, with

Note t h a t f o r

2H = h = g we can make some i d e n t i f i c a t i o n s and i n t i m e domain form t h e G-S convolution kernel H ( c f . [LaZ;Lol] 71).

'L

( 1 / 2 ) g i s t h e same as t h e kernel i n t h e MGL e q u a t i o n

and n o t e t h a t a minus s i g n a r i s e s i n t h e f o r m u l a t i o n o f [CZ,

+ r(x,t)

The MGL e q u a t i o n i s 0 = r ( x , t )

r ( x , t ) = [G

r

(X+t)

-

Gr(x-t)]/2

(t

X,

t (l/Z)g(-)

*

[(x,.)

where

t

G r ( t ) = 10 g(T)dT). We w i l l see i n subse-

7, RECACACL0PU BEt!3EEN P0CENCZAI:S AND SPECCRAI: DACA.

quent s e c t i o n s on s o l i t o n t h e o r y t h a t s p e c t r a l data, i n t h e f o r m o f t r a n s m i s s i o n and r e f l e c t i o n c o e f f i c i e n t s f o r example, p l a y s a profound t h e o r e t i c a l r o l e ( a s w e l l as a p r a c t i c a l one).

I n f a c t i t provides action-angle

v a r i a b l e s and canonical s y m p l e c t i c forms i n t h e Hamiltonian t h e o r y ( c f . 5 5 8-11).

I n a more p r a c t i c a l vein, and e s p e c i a l l y f o r general i n v e r s e prob-

lems, g i v e n measurements o f s p e c t r a l d a t a one wants t o r e c o v e r t h e p o t e n t i a l and thus t h e n o n l i n e a r map p o t e n t i a l a b l e study.

+

s p e c t r a l d a t a i s worthy o f c o n s i d e r -

Aside f r o m a l l t h a t i t i s a very i n t e r e s t i n g ma t h e m a t i c a l prob-

lem and can become v e r y d e l i c a t e .

For t h e f u 1 l i n e problem we r e f e r i n

p a r t i c u l a r t o work connected w i t h t h e i n v e r s e s c a t t e r i n g t r a n s f o r m (IST) and mention e.g.

[Aol ,2;Cjl ;Bvl;Dfl ; M r l ,2;Nll

Nvl ;Fa3;Tzl,Z;Zkl

t h i s i s sketched o r developed here and i n 58- 1.

,2].

Some o f

Corresponding r e s u l t s f o r

t h e ha1 f 1 i n e a r e more sparse a1 though d e t a i l e d connections between potent i a l s and s p e c t r a l d a t a e x i s t v i a t r a n s m u t a t i o n k e r n e l s as i n 51.6, e t c . ( c f . [C2,3;Mrl]).

2.4-2.6,

I n t h i s s e c t i o n we w i l l o r g a n i z e and r e c o r d some

r e l a t i o n s between p o t e n t i a l s and s p e c t r a l data, m a i n l y r e l a t e d t o h a l f l i n e problems, and g i v e a k i n d o f I S T f o r t h e h a l f l i n e (see [Ao3;C18,36,38;Msl, 51 and c f . a l s o [SyE;Sql]

f o r related information).

We t h i n k o f t h e maps

from p o t e n t i a l s ( o r impedance) t o s p e c t r a l d a t a b o t h as a t h i n g o f beauty and a l s o as a l o c a l i z a t i o n g i v i n g r i s e t o approximation r e s u l t s ( c f . 51.11). The development i n t h i s s e c t i o n a l s o r e v e a l s a g r e a t deal o f u n d e r l y i n g s t r u c t u r e f o r d i f f e r e n t i a l o p e r a t o r s and s c a t t e r i n g data which w i l l h e l p t o

POTENTIAL AND SPECTRUM

169

i l l u m i n a t e some o f t h e t h e o r y i n §8-11 on s o l i t o n s .

More s p e c i f i c a l l y , i n

terms o f l o c a l i z a t i o n f i r s t , we w i l l determine l o c a l i z a t i o n formulas f o r t h e main s p e c t r a l i n g r e d i e n t s a l o n g w i t h a v a r i a t i o n o f t h e IST f o r t h e h a l f I n p a r t i c u l a r we i s o l a t e a s p e c t r a l o b j e c t R 2 = F/2c- as b e i n g n a t u r -

line.

a l and s i g n i f i c a n t i n t h e t h e o r y and r e a d a b l e i n p r a c t i c e and t h i s R2 a r i s e s

i n o u r IST.

The development a l s o i n v o l v e s a s y s t e m a t i c d e r i v a t i o n o f M

e q u a t i o n s and r e c o v e r y formulas.

We n o t e a l s o t h a t c e r t a i n problems i n

f o r c e d i n t e g r a b l e systems a r e h a l f l i n e i n n a t u r e and one a n t i c i p a t e s some a p p l i c a t i o n o f t h e p r e s e n t t h e o r y i n t h a t d i r e c t i o n ( c f . [AoZ;Msl ,5;C18,35, 37,38,42]).

For background i n f o r m a t i o n on s c a t t e r i n g we r e f e r t o 54,6 and

f o r convenience we g a t h e r here again some o f t h e most needed i n f o r m a t i o n f o r t h e p r e s e n t development. Thus f o r t h e f u l l l i n e one takes a p o t e n t i a l 2 q ( x ) w i t h say ( l + l x l ) I q l d x < m and determines J o s t s o l u t i o n s o f ( * ) $ "

/I

n

-qb = -kL$ v i a (4.1),

i.e.

fl(k,t)dt

= exp(-ikx)

and f 2 ( k , x )

(A)

w r i t e s a l s o fl = ft and f 2 = f t h e r , s e t t i n g W(f,g) (4.23))

(.)

-

= fg'

fl(k,x)

SO

,f

+

t

/ R I 2 = 1 = / T I 2 4.

f o r k r e a l ) and as I k l *

-, T

=

-m,

IR2I2,

fl

+.

Fur-

= G ' i k and ( c f , =

Then ( 6 ) as x

(1/T)f +

m,

(-k, f2 %

( l / T ) e x p ( i k x ) + (R2/T)

2.

Ti

One

1 as x *

+ ( R / T ) f l ( k , x ) and f l ( k , x )

+ ( R / T ) e x p ( i k x ) and as x *

(l/T)exp(-ikx)

+

f ' g one has W ( f + ( k , x ) , f + ( - k , x ) )

(R = Rr and R2 = R-[ sometimes).

exp(-ikx) with /TI2

[Sink(x-t)/k]q(t

[Sink(x-t)/k]q(t)f2(k,t)dt.

t h a t f +-( k , x ) e x p ( + i k x )

f2(k,x) = (l/T)fl(-k,x)

x ) + R2/T)f2(k,x)

- 1,"

exp(ikx)

=

?R2 = 0 ( E ( k ) = R ( - k )

t

1 + O ( l / l k l ) , R = O ( l / l k \ ) , and R2 = 0 ( 1 /

l k l ) . We w i l l 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 so T i s a n a l y t i c f o r Imk > 0. Also ( c f . ( 4 . 3 ) ) R/T = ( 1 / 2 i k ) l I qf2(k,t)e-iktdt;

(7.1)

1/T = l - ( l / Z i k ) / :

qf2eiktdt

= 1

-

R2/T = ( 1 / 2 i k ) l I qfl(k,t)eiktdt;

(1/2ik)fm q(t)fl(k,t)e -iktdt m

which f o l l o w immediately from ( 4 . 1 ) . Next we c o n s i d e r a h a l f l i n e s i t u a t i o n and f o r completeness w i l l i n d i c a t e several p o i n t s o f view (as i n 86). =

Thus c o n s i d e r i n a system f o r m a t w = vt

v e l o c i t y and P = p r e s s u r e w i t h P = -pvx and p v

(PP)'

=

impedance and y = t r a v e l t i m e ( y ' =

be assumed) ( + ) w

Y

=

has ( c f . (.+)

=

-Px. One takes A = so thict (A(0) = 1 can

= -Awt; v = (Av ) / A ; wtt = ( A w ~ ) ~ / A Y tt Y Y S 6 a r e n o t used h e r e - A always r e f e r s t o an of

-A-'Pt;

( n o t e t h e symbols A,B,C,D impedance now).

t2

(P/!J)')

P

S e t t i n g p = (l/Z)[A-'P

+ A%]

i n S2.6) p + pt = -rAq; qy Y

-

and q = (1/2)[A-%'

- At]

qt = - r A p where rA= (1/2)Dy

one

170

ROBERT CARROLL

It w i l l be convenient i n r e l a t i n g t h i s t o t r a n s m i s s i o n l i n e s t o t a k e

logA.

Z = A - l as impedance ( w i t h r Z= -rAas r e f l e c t i v i t y ( t h i s c h o i c e Z = A - l i s made i n o r d e r t o connect n o t a t i o n h e r e t o t h a t o f 52.6 and [C10,11,341). Then we w r i t e V = UZ-'and I = I$ (U ?J w, I ?J P ) w i t h wR = ( 1 / 2 ) ( V + I ) and wL = ( l / Z ) ( V - I ) ( r i g h t and l e f t t r a v e l i n g waves). Thus wR % p and wL ?J -q and 0xwR t DtwR = -rZw L w i t h DxwL - Dt wL = -rZwR. A standard s i t u a t i o n now f o r t h e geophysical problem i s t o impose an impulse v ( 0 , t ) = - 6 ( t ) = P(0, Y = 6 ( y ) ) so t h a t an impulse response

t ) ( o r e q u i v a l e n t l y w(y,O) = vt(y,O)

w(0,t) = 6 ( t ) t 2 g ( t ) Y ( t ) i s o b t a i n e d ( 6 = 6+ here t ) = 6 t $Y and q ( 0 , t ) = - i Y . 6

+ GY and wL(O,t) = Y;

I n t h e wR,wL

- c f . §2.6).

Then p(0,

c o n t e x t t h i s becomes w R ( O , t )

F i n a l l y i n connecting t h i s n o t a t i o n t o 52.6 f o r example we w r i t e v ( 0 , t ) =

Y

t G

r

w i t h G' = 6

+ Y2:

= 6 t g = 6 t

Thus (.)

fl

= exp(ikHx)l' t

= G

G;.

Another approach t o t h e h a l f l i n e problem f o l l o w s [NwZ;Hwl] i n g (6.13).

=

which i s a standard s i t u a t i o n i n [ B a l l f o r example.

as i n 6 f o l l o w -

exp(ikH(x-y))r(y)Af,dy

and fr =

e x p ( i k H x ) A l ' - $ e x p ( i k H ( x - y ) ) r ( y ) A f r d y where r = rZ= ( 1 / 2 ) Z ' / Z as above, 1 0 1 1 0 1 ' = ( o ) , A = ( 1 o ) y H = ( o -1), fr,e = fr,[(k,x), and we t h i n k o f Z = 1 for x 2 0 .

It w i l l now be convenient t o assume r ( 0 ) = 0 ( i . e . A ' ( 0 ) = 0 = 0

0

Z ' ( 0 ) ) i n o r d e r t o have t h e t r a n s m i s s i o n and r e f l e c t i o n c o e f f i c i e n t s (T,R, and

k2 resp:)

f o r t h i s problem agree w i t h t h e corresponding f u l l l i n e co-

e f f i c i e n t s based on

= r2 - r ' = z5(Z-')"

- c f . below).

( = A-'(A')"

This

i s n o t a s e r i o u s r e s t r i c t i o n here and f o r c e r t a i n i d e n t i f i c a t i o n s i t i s n o t 3

needed a t a l l ( c f . 52.6 and [Nw2;Hwl]

f o r discussion).

r,

Note i n § 6 T, R[,

e t c . a r e used f o r t h e h a l f l i n e and t h e r e i s some d i s c u s s i o n f o l l o w i n g (6. 13) w i t h formulas i n Theorem 6.16.

For c e r t a i n i n v e s t i g a t i o n s however r ( 0 )

9 0 can be i m p o r t a n t and one can e a s i l y o b t a i n formulas r e l a t i n g R,T,R2 and 1 + e x p ( i k x ) , fl 2 + 0, i f ' + R,T,R2 f o r such s i t u a t i o n s . One has now (**) fe 0

0

0

fiexp(ikx), and if: + e x p ( - i k x ) a t m w h i l e a t -m, ?fL + e x p ( i k x ) and if:r+ 0 1 2 R2exp(-ikx) ( f r = 0 and fr = e x p ( - i k x ) f o r x 5 0 ) . F u r t h e r one has

To compare these w i t h (7.1) i s s i m p l y a m a t t e r o f r o u t i n e i n t e g r a t i o n by 2 p a r t s w i t h q = q = r - r ' and r ( 0 ) = 0; one uses a l s o ( c f . [ N w l ] and §2.6) 1 1 2 2 2 Dx{r,e = kHf r ,t - rAfr,L; Dxfr,g = ikfr,[ - rfr,e; and Dxfr,[ = -ikf,,[ -

rfr,e

.

I n t h i s d i r e c t i o n consider (from (4.3))

(*A)

cZ1 = 1 / T = 1

- (1/2ik)

POTENTIAL AND SPECTRUM 1

2

1 ; q e x p ( - i k t ) ( f e + 5 ) d t p r o v i d e d f+ = (I/~)z'/z,

t ) E1d t

=

q = r2-rl =

( - i k t ) d t = r ( 0 ) f e2 (k,O)

.

T?5(~-')11

-

1 ; r2 e x p ( - i k t ) ( $

fa1 + 52

171

Now 1 ; q e x p ( - i k t ) $1d t

+ r(O)f{(k,O) 1

f,)dt 2

+ 1 ; r2e x p ( - i k t ) ( $

-

Recall again r =

i s valid.

1 ; ( r2 - r ' ) e x p ( - i k

=

and s i m i l a r l y ; 1 qfeexp 2

5 1) d t

-

2ik1;

r e x p ( - i k t ) $ d2 t .

2

Consequently f r o m ( 7 2 ) (*@) 1/T = 1 +:1 r e x p ( - i k t ) $ ( k , t ) d t = 1/T.

ig =

calculations give

ir= Rr

RL and

0

t h a t t h e r e a r e d i f f e r e n t formulas r e l a t i n g RpT,Rr

RElRARK 7.1.

Similar

when r ( 0 ) = 0 ( c f . a l s o [Nw2] a 6 n o t e 0

0

when r ( 0 ) t 0).

and RL,T,Rr

We n o t e h e r e i n p a s s i n g t h a t t h e Zakharov-Shabat (Z-S) eigen-

-

f u n c t i o n s s a t i s f y ( c f . 19) ( * 6 ) Dxvl + isv = w 2 and Dxv2 iiy2 = wpl w i t h 1 1 9 = ('1 1 % ( 0 ) e x p ( - i s x ) as x --m ( s i m i l a r l y f o r J, = (JI, J , ~ ) , column v e c t o r , Ova JI % ( l ) e x p ( i s x ) as x -1. If we i d e n t i f y q and w w i t h -r = -rZ= rAand 1 2 2 1 w i t h k t h e n v1 % fr, v 2 % f 1 ry Q1 % yl f and J,2 % fL. -f

-f

We r e c a l l n e x t t h e h a l f l i n e f o r m a t o f 551.6,2.6,

f o r second G e n e r a l l y one makes assumptions A E C 1 , 0 < a 5

o r d e r e q u a t i o n s as i n (+).

A 5 B

<

and A

my

-f

Am r a p i d l y as y

-+ m

and [C2,3]

( w i t h A(0) = 1

A ' ( 0 ) = 0 f o r convenience as n o t e d above).

-

and h e r e we assume

Set Qu = ( A u ' ) ' / A and l e t

v Qx

(resp. 0 ,Q) be s o l u t i o n s o f Qu = - A 2 u s a t i s f y i n g (*+) Ipx(0) Q = 1, D v Q ( 0 ) = 0, Q Q Q x -4 J o s t s o l u t i o n s a r e d e f i n e d v i a a x ( x ) % Am exp 2 m. ( i x x ) as x = Now f o r A E C , which we assume f o r convenience, s e t

0 , ( 0 ) = 0, and Dxex(0) = 1 .

4

-f

&

b

-

A-'(A')" and = w" {w (w = A%) so t h a t A'Qu = ~ [ A % J ] = (note again * 2 2 q = r - r ' f o r r = r Z= -r so q = rA+ r i ) . We s e t $: = i Q= ',A' A .$' A .(jY Q e t c . which g i v e s (*.) i ! ( O ) = 1, DxGx(0) = 0, 8 $ 0 ) = 0, DO(,;) - 1, a x ( x ) = i%!(x)

recall

%

(A*)

i$Q

e x p ( i x x ) as x

?!

-f

6Q + c-6: PQX !?Q v x and O x

my

= c

e t c . ( r e c a l l h = i A ' ( O ) = 0).

and

i!

= [F$!

A

In particular

- FQi- AQ ] / 2 i h and

the spectral 2 = dX/2n(c

e i g e n f u n c t i o n expansions a r e dw measures f o r t h e 2 Q and d v 2x dX/nlF12 r e s p . Thus ( A A ) 1; i!(x)G!(y)dwQ = 6(x-y) =

Q

I

m

.QQ

J0 e x ( x )

i:(y)dvp.

Using c a l c u l a t i o n s f r o m 92.6 and [C10,11,34] we can w r i t e a g a i n 1 2 ^ 1 2 (A Z- ) (A@) = fr + fr = f2 = i X i : and Vg = fL + f = f = FGQ + 1.Q x ZiXc-e'!; t h i s l i n k s v a r i o u s n o t a t i o n s t o g e t h e r . Note h e r e a l s o ax fl and

ir

-

e

D

O

@ - Q- x ( x ) % f l ( - x y x ) ( A % k ) and from Theorem 2.6.14 w i t h R = R, T = T, e t c . (A6) 1/T = c - t (1/2)F; R/T = c - ( 1 / 2 ) f - ; and R2/T = (1/2)F - c - ( c = c F = FQy e t c . ) .

The Green's f u n c t i o n argument i n

Theorem 2.4.6

9'

gives (cf.

( 1 / 2 ~ ) / T~ f Q ( k , x ) f 2Q( k , y ) d k = 6 ( x - y ) w h i l e f r o m (A&) Q 1 we can w r i t e a l s o 2c- = (1-R2)/T and F = (1+R2)/T. One r e c a l l s a l s o from Remark 2.4.14) [C2,3]

m

(A+)

and 12.6 ( c f . a l s o (A*)) t h a t

W(fl,i)(0)

= fl(0)

= F and Fc

+ F-c-

(Am)

W($,fl)(0)

= f i ( 0 ) = 2ihc- w h i l e

= 1.

F i n a l l y l e t us make a few remarks about t h e r e l a t i o n o f impulse response t o

172

ROBERT CARROLL

T h i s was t h e key t o s o l v i n g t h e 1-D i n v e r s e

spectral quantities ( c f . j1.6).

problems i n geophysics i n [C2,3,7-91

and §1.6 v i a G-L techniques and a r i s e s

a l s o , i n v a r i o u s ways,in t h e i n t e g r a l equations o f G-L, M, G-S,

and

K types

Thus t a k i n g t h e s o l u t i o n o f (+) i n t h e form (.*)

f o r example ( c f . §2.6).

v ( y , t ) = ( ~ ,Q( y ) , [ S i n A t / h ] ) ~ o r w ( y , t ) = ( p XQ ( y ) , C o s A t ) u one has f o r t 2 0 ( g = )2: (.A) G ' = 6 + ( t ) + g ( t ) = ( l , C o s X t ) u from which ( 0 0 ) 1 / 4 ( c l 2 =

1 ; G'(t)Cos t d t .

I n t h e geophysical case moreover c - ( h ) = c (-A) i s a n a l y 2 Q Q t i c f o r I m x > 0 and t h u s knowledge o f I c I f o r h r e a l leads t o c v i a t h e Poisson-Jensen formula f o r example.

A c t u a l l y i n 51.6 and [C8] a method i s

g i v e n t o r e c o v e r c - d i r e c t l y from readout impulse response a t a p o i n t y > 0

..

Q

( t r a n s m i s s i o n d a t a ) p r o v i d e d A ( y ) = Am f o r y 'y.

I m p l i c i t i n [C8] b u t n o t

developed i s t h e f o l l o w i n g r e s u l t which d i s p l a y s F/2c- as an i m p o r t a n t spect r a l q u a n t i t y ( c f . [C36,43]

Under t h e hypotheses i n d i c a t e d ( A

t?HE0REN 7.2. A

-+

from which much o f t h i s s e c t i o n i s e x t r a c t e d ) . E

C

1

,0

5A 5 6 <

<

Am r a p i d l y i s s u f f i c i e n t h e r e ) one can w r i t e (G' = 6, + g, 6, = 2 6 )

6 ( t ) + g ( t ) = (1/2n)_/f ( F / 2 c - ) e x p ( - i h t ) d A 1- G1 ( t ) e x p ( i h t ) d t .

= Gl(t)

and

my

(06)

from which (F/2c-) =

-m

Phood:

W r i t e f r o m (@*) G ' = (v,(0),CosAt)o Q

dx and use =

f o r m ~Q ~

i n the

(A*)

cc- f o r X r e a l , f l ( 0 )

from [C2,3]

=

= (1/4~)_/: p X Q ( 0 ) [ e x p ( i X t ) / l c l 2]

( =0 c@(O) ) + c-@-(O) t o o b t a i n ( r e c a l l I c I 2 F) (a+) G ' = ( 1 / 4 ~ ) _ / : [ ( F / c - ) + (F-/c)]dh. Now

and 51.6 F and c - a r e a n a l y t i c f o r I m h > 0 w h i l e c - has no zeros

t h e r e ; a l s o F/c- w i l l be s u i t a b l y bounded so t h a t f o r t > 0,

l:

(F/c-)exp

( i x t ) d x = 0 l e a d i n g t o ( 0 6 ) f o r t > 0 as a r e p r e s e n t a t i o n o f GI, and i n Gl(t) [(F/c-)

=

= 0 f o r t < 0.

+ (F-/c)]

(00

A t t = 0 a 6 f u n c t i o n a r i s e s from (N) v i a (1/4n)

= ( 1 / 2 ) [ 1 / 2 i ~ l c 1 ~ ]= ( 1 / 2 ) [ ( 2 / n )

e x p ( i h t ) d x = 2 6 = 6, ( c f .

[C2,3,6]

and 551.6,

+

G]

1.11 f o r

where (1/2)L:

(2/n)

and f o r h a l f l i n e

and f u l l l i n e 6 f u n c t i o n s a r i s i n g f r o m Cosine and e x p o n e n t i a l r e p r e s e n t a Now w r i t e F/c- = 2 t a and F-/c = 2 + a- so t h a t = (1/2) Then upon w r i t i n g ( 0 6 ) f o r t > 0 we see t h a t i t r e p r e s e n t s ( 1 / 4 n )

t i o n s resp.). (a+.-).

Jm ( Z + a ) e x p ( - i x t ) d h = 6 + ( 1 / 4 n ) / f a e x p ( - i x t ) d x f o r t 2 0 .

-m

(1/4n)_/:~exp(-ixt)dh XtdX = (l/Z)L:

$ e x p ( - i X t ) d x = ( 1 / 4 ~ ) l z( a + a - ) e x p ( - i h t ) d h s i n c e

t)dX = 0 f o r t > 0. ( t ) = Gl(t)

Thus g ( t ) =

and f o r t > 0 t h i s i s c o n s i s t e n t w i t h g ( t ) = C;Jo;s

_/:

a-exp(-iX

I n o r d e r t o r e c o v e r 6, a t t = 0 one must c o n s i d e r G '

+ G 1 ( - t ) f o r t 0'

which produces 26 = 6,.

A p o i n t value g(0)

i s a v a i l a b l e by c o n t i n u i t y i f g ( t ) i s continfdous; i f n o t one s i m p l y i g n o r e s the matter.

QED

We r e c a l l n e x t a t y p i c a l M t y p e e q u a t i o n f o r f u l l l i n e s c a t t e r i n g ( c f . 52.4,

POTENTIAL AND SPECTRUM

173

2.6 and [ C l l ;Chl;Dfl;Fal;Nw2;C3,17;Lal;Mrl]).

2.5,

I t i s assumed through-

o u t t h a t t h e r e a r e no bound s t a t e s and t h e p o t e n t i a l s s a t i s f y s t a n d a r d con-

2.6 etc.

d i t i o n s as i n 52.4,

Thus one can determine ( i n v a r i o u s ways) a

t r a n s m u t a t i o n k e r n e l K(x,y) v i a (7.3)

fl(k,x)

K(x,Y) (K(x,y)

+

= eikx

+ 6(x-y)

Lm K(x,y)eikYdt; -

= ( 1 / 2 n ) f I [f,(k,x)

= i(x,y)

ker

'i: e x p ( i k y )

kydk

f ( k , x ) i s a transmu1 t a t i o n ) . S p e c t r a l l y g(x,y) = (1/2n)Jm- T fP ( k , x ) f 2Q( k , y ) d k i s t h e transmutaQ 4 1 2 t i o n kernel w i t h fl -+ fp ( c f . ( A + ) - we t h i n k here o f o p e r a t o r s Q = D =

where

e ikxle-i

+

6:

q and P = D2

-

p on

so i n f a c t 8(x,y) = (1/2n)L:

(-m,m))

fl(k,x)exp(-iky)dk

f o r ( 7 . 3 ) ( c f . [C3,13,17] and 52.6). An i n v e r s e t r a n s m u t a t i o n = b-' can a l s o be e a s i l y w r i t t e n down v i a t h e k e r n e l ;(y,x) = (1/2n)Im m Tpfl(k,y)f2(k, Q P

-

x ) d k and we w r i t e g ( x , y ) = 6 ( x - y ) + K(x,y) and ;(y,x) where L(y,x)

=

0 for x

<

=

6(x-y) + L(y,x)

y and K(x,y) = 0 f o r y < x ( t h e t r i a n g u l a r i t y f o l -

lows e.g. from p r o p e r t i e s o f f;" and f;" as f u n c t i o n s o f k f o r Imk > 0 and Paley-Wiener i d e a s ) . Thus i n o u r general s i t u a t i o n w i t h g: flQ -t fl, P using (4.23) we o b t a i n (y > x ) (7.4)

~ ( x - Y ) + K(x,Y) = ( 1 / 2 n ) [ l I f 1 P ( k , x ) f l Q (-k,Y)dk +

But by t r i a n g u l a r i t y i n :(y,x)

one has f o r x

P fl(-k,x)

+ Rpf~(k,y)f~(k,x)]dk.

(1/2n)L:

[RQ

-

Rqfp(k,x)fl Q ( k , y ) d k l <

y

(0.)

0 = ( 1 / 2 n ) I I [f:(k,y)

I t f o l l o w s t h a t f o r y > x (&*) K(x,y)

Rp]f!/(k,y)fp(k,x)dk.

transmutation property P i =

1:

=

Separate c a l c u l a t i o n s based on t h e

EQ f o r

example y i e l d a l s o (as i n 52.4 and 2.6)

I n t h e s p e c i a l case above ( q - p ) ( x ) = ZDxK(x,x) ( c f . [Chl ;Fa1 ;La1 ; M r l ] ) . 2 where Q = D , q = 0, T = 1, R = 0 one has f o r m a l l y ( c f . ( 7 . 1 ) )

Q

(7.5)

Q

K(x,y) = - ( 1 / 2 ~ ) 1 : Rpei k y f lP( k , x ) d k ;

p = (l/n)Dxf:

Rpe i k x fl(k,x)dx; P

R/T = ( 1 / 2 i k ) l I pf;(k,t)e-iktdt The l a s t two equations i n ( 7 . 5 ) r e p r e s e n t a f o r m o f t h e now c l a s s i c a l i n v e r se s c a t t e r i n g t r a n s f o r m ( I S T

-

c f . [Ao2;Bvl ; C j l ;N11,2]

and 552.8-2.11 ) .

In

a c e r t a i n sense i t extends t h e F o u r i e r t r a n s f o r m s i n c e f o r p small w i t h Tp %

'L

1, Rp s m a l l , fl

?r

exp(ikx), fz

( l / ~ ) l z2 i k R e x p ( Z i k x ) d k and R

?r

e x p ( - i k x ) , e t c . one has a p p r o x i m a t e l y p

%

(l/Zik)L;

p e x p ( - 2 i k x ) d x ( o r 2ikR

?J

$(2k)).

174

ROBERT CARROLL

Now one sees from ( 7 . 5 ) t h e map p

that S

2ikR/T has n i c e " F o u r i e r " p r o p e r t i e s and

S(p) i s s t u d i e d i n [Tz2]

-+

thods ( i . e .

( c f . also [Tzl])

v i a l o c a l i z a t i o n me-

Frechet d e r i v a t i v e s e t c . which a r e discussed i n § 3 . 2 ) .

a b l e (weighted Sobolev) spaces i t i s shown i n [Tz2] t h a t p

+

For s u i t -

s ( ~ ) :QN

+

s,

i s a r e a l a n a l y t i c isomorphism ( r e c a l l t h e r e a r e no bound s t a t e s h e r e by assumption).

We do n o t need t h e d e t a i l s here b u t do want t o o u t l i n e t h e

technique e n a b l i n g one t o study t h e Frechet d e r i v a t i v e map d S f o r example P (such maps a r i s e l a t e r i n s o l i t o n t h e o r y ) . Thus ( c f . Remark 7.4 below and P P fl(-k,x)f2(k,x)v(x)dx and t h i s map i s i n f a c t an i s o 13.2) (4A) d S(v) = P P P a(k)Dx[fl ( k , x ) f 2 ( - k , x ) ] o ( k ) morphism o n t o w i t h i n v e r s e (6.) ( d S)-~(CI) = P We do n o t g i v e t h e many t e c h n i c a l d e t a i l s dk where a ( k ) = T p ( k ) T p ( - k ) / 2 a i k .

LI

LI

h e r e from [Tz2] ( c f . a l s o [ T z l ] ) .

Some o f t h e t e c h n i q u e (e.g. a s y m p t o t i c

a n a l y s i s ) a r i s e s l a t e r i n §§2.8-2.11 i n s o l i t o n t h e o r y and some techniques (e.9. Frechet d e r i v a t i v e i d e a s ) come up i n d e v e l o p i n g h e r e t h e h a l f l i n e c o n t e x t . We do n o t t r y t o prove an isomorphism theorem as i n [Tz2] and t h u s w i l l n o t i n c l u d e d e t a i l s f o r t h e f u l l l i n e c o n t e x t . L o c a l i z a t i o n formulas as above i n (&A)-(&.) enable one t o study p S(p) l o c a l l y and can be used -f

i n approximation t h e o r y .

They a r e a l s o r e l a t e d t o s p e c t r a l t r a n s f o r m s o r

IST as i s i n d i c a t e d below.

Next we have

P Set S = 2ikR/T and S2 = 2ikR2/T w i t h AS = SQ - S , Ap = q-p, Then under t h e hypotheses i n d i c a t e d A ( l / T ) = - ( 1 / 2 i k ) i z ApflP( k , x ) f 2Q

CHEBREN 7.3. etc.

(k,x)dx = - ( 1 / 2 i k ) i Z Apf!fpdx

J m ApflQ ( k , x ) f 2P( - k , x ) d x .

and AS =

Q iz Apf2(k,x)f,(-k,x)dx P

w i t h AS2 =

-m

Pmod:

Use v a r i a t i o n o f parameters w i t h [D2

t

A 2 I f Q= ( q - p ) f Q t o g e t

fe = fr[1-(TP/2iA)&m f L f p p d S ] + (T P/ 2 i h ) f ; r

(7.6) f!

- p

fpfppdS;

- ( T p / 2 i A ) f t f r f f A p d c ] + (Tp/2iA)fllm P x f2f2ApdS P Q

= f;[1

Then i n s e r t t h e a s y m p t o t i c values f r o m ( 6 ) and use (4.23) t o g e t t h e theorem.

We o m i t t h e c a l c u l a t i o n s ( n o t e h = k ) .

Hence as Q

3

QED

P f o r example we a r e d e a l i n g w i t h a map o f t h e form (U) as an

approximation t o AS =

/," A p f j f i ' d k

( f i P ( k , x ) = f lP( - k , x ) ) .

Naturally, f o r -

mulas as i n Theorem 7.3 a r e more p r e c i s e and when a v a i l a b l e w i l l be p r e f e r a b l e t o a l o c a l i z a t i o n form.

REmARK 7.4. (&A)-(&@).

We s k e t c h here t h e technique l e a d i n g t o formulas o f t h e form 2 Thus f i r s t , g i v e n say f; - p f l = -k fl w i t h d f l ( v ) = Dtf,(k,p+

POTENTIAL AND SPECTRUM

175

2 pdpfl(v) = - k 2 dpfl(v) + v ( x ) + t v ) I t=O one o b t a i n s f o r m a l l y ( 6 6 ) DX d Pf l ( v ) fl(k,x) w i t h dpfl(v) 0 and D d f ( v ) + 0 as x + m. Set W = W(f2,fl) = X P l 2 i k / T and b y v a r i a t i o n o f parameters (be) d f ( v ) = ( f 2 / W ) f m f'vdt - ( f / W ) m P 1 x 1 1 flf2vdt. A s i m i l a r c a l c u l a t i o n a p p l i e s t o d f ( v ) and thence t o any P 2 q u a n t i t y expressed v i a W(f2,fl) f o r example (W 0). Thus ( 6 ~ d) f ( y ) = P 2 (f,/W)/: f$dt (f,/W)i: f2flvdt and e.g. d W(v) = d f (v)Dxfl flDxdpf2 P P 2 ( v ) + f2Dxdpfl ( v ) Dxf2dpfl ( v ) so t h a t some o r g a n i z a t i o n o f terms y i e l d s

-

+

/x

+

-

-

-

(+*I dpW(v)

-rf flf2vdt.

=

(+*I

d W ( V ) = caW/ap,v))

f;'.

S i m i l a r c a l c u l a t i o n s y i e l d ( c f . [TzZ] - we w r i t e 2 aT/ap = (T/w)flf2; aR/ap = (T/w)f2; aR2/ap = ( T / w )

The i n v e r s i o n ( 6 0 ) a r i s e s f r o m a f o r m u l a

(7.7)

= Dy[flfl m - k f 2 -mf 2 k + W(fik,fY)W(f2,f2 k - m ) / ( k 2 -m 2 ) ]

2Dy[fTf;m]f;kfi

and a s y m p t o t i c behavior, p l u s a r e l a t i o n [exp(Zi(k-m)x)/(k-m)]

as x

-+

m.

Thus

(+A)

][f

[TmT-m/2im

f;kf!$lx[fTf;m]dx

= 6(k-m).

+

ins(k-m)

One w i l l en-

c o u n t e r s i m i l a r expressions f o r some h a l f l i n e problems and i n s o l i t o n t h e One notes t h a t i n [Tz2] a r i g h t i n v e r s e f o r d R i s a s s e r t e d t o have 2 P t h e k e r n e l f o r m (l/2aim)Dx[fl(m,x)] b u t d R i s n o t o n t o i n t h e spaces conP sidered there. ory.

Now f o r h a l f l i n e problems t h e i m p o r t a n t s p e c t r a l q u a n t i t i e s i s o l a t e d i n general a r e c-,

/ c - = 3 , and F-/F = S.

F, F/2c- = R2,

I n t h i s connection

l e t us f i r s t i n d i c a t e how S and SC a r i s e i n M e q u a t i o n s f o r t h e h a l f l i n e . S i s c l a s s i c a l ( c f . [Chl;Fal]

2,3,17,20]

( c f . a l s o S2.6).

whereas SC was i n t r o d u c e d by t h e a u t h o r i n [C L e t us e x t r a c t h e r e from [C18,36]

c e r t a i n u n i f y i n g features f o r M equations.

Thus (we use

a sketch o f

+,i, e t c .

i n order

t o m a i n t a i n a c o n n e c t i o n t o t h e problems based on A and Z ) we b e g i n w i t h From t h e i o r t h o g o n a l i t y i n (u) and u s i n g (A*) one has ( W ) m * P P m P P P ~ ( x - Y ) = ( l / n i ) i m e , ( ~ ) [ f l (h,x)/FpI"dx = ( 1 / 2 n ) j m fl( x , x ) [ f l (-X,Y) - S p f l (A,y)]dh. S i m i l a r l y t h e 4 o r t h o g o n a l i t y i n ("1 g i v e s ( M )6 ( x - y ) = (1/2n) m .P P P P l mIpX(y)[fl (h,x)/c$dx = (1/2n)i: fl ( x , x ) [ f l (-X,y) + Spfl P (h,y)ldX. Now l e t g ( x , y ) : flQ + fr be as i n (7.4), B' = 6+K, E(x,y) = 0 f o r y c x, e t c . w i t h

REMRK 7.5,

inversion via

(7.8)

v'

=

fY(-X,y)

6+L, L ( y , x ) = 0 f o r x < y, e t c . + Zpfl

Q

P ( 1 , ~ ) = fl (-X,y)

L ( y , t ) [ f l P( - A , t )

+

+

P

For Z p = Sp o r

zpfl (A,Y)

-S

P

+

~pf~(x,t)ldt

P

Take x c y, m u l t i p l y ( 7 . 8 ) by fl(h,x), and i n t e g r a t e i n X t o g e t (++) (1/2n) _/f f!(A,x)[fl(-b,y) q + Epfl(X,y)]dX 9 = 0. NOW (+m) fl(X,x) P = fl(X,x) Q +

176

Im

ROBERT CARROLL

K(x,S)fl(A,S)dS Q

Q

Q 2

RQ

t o get f o r y > x

fl(h,y),

+ (1/2a)lI

0 = K(x,y)

(7.9) +

(recall (A+)

-

= T f Q (A,y)

and we p u t t h i s i n (++), u s i n g fl(-A,y) Q

,," K ( x , c ) ( 1 / 2 7 ) l z -

-

[Ep

-

RQlfp(~,x)fy(A,y)dA +

RQlfP(A,S)fp(A,Y)dhdS

We n o t e a l s o from ( + a ) - ( + & ) ( w i t h Xp = b p o r -Sp) and P t h a t f o r y > x (m*) 0 = [Ep R PI f 1 (A,x)fr(A,

here).

(A+)

with Tf2

[Ep

/I

R f l = f;,

-

T h i s w i l l a l s o h o l d o f course w i t h P r e p l a c e d by Q and hence (u)

y)dA. T(x,y)

-

= ( 1 / 2 n ) / z [Ep

an obvious a b b r e v i a t i o n .

R ]f Q (A,x)fl(x,y)dx 4 9 1 This leads t o

= ( 1 / 2 1 ~ ) i I[Zp

-

EQ]f;f~dX

in

For E p = Bp o r -Sp one has an M e q u a t i o n ( y > x ) 0 = K(x,y) +

CHE0REm 7.6,

T(x,Y) + Jxm K(x,S)T(S,y)dS.

Next one wants a formula f o r K i n Theorem 7.6 analogous t o say (&*). from (em) (me) K(x,y)

=

(1/2n)l:

-

[f,(A,x) P

f lQ( ~ , x ) ] T f Q (A,y)dA

First

and we r e -

= T f2(A,y) Q - R f Q(A,y) Q 2 t o obtain (m&) c a l l (++) f o r y > x w i t h fl(-A,y) Q 4 1 f r ( A , x ) T f Q (h,y)dA = [RQ - E p ] f ( A , x ) f ~ ( A , y ) d A . It f o l l o w s t h a t f o r

LI

lI

4 2

P Q Ep]fl(X,x)fl(h,y)dA

Q Q f2dA Q -( s i n c e l: flT P 2n6(x-y) i n ( m e ) ) . S i m i l a r l y by symmetry i f 6 t L : fl + fy we have f o r y > x 4 P ( m m ) L(x,y) = (1/2n)j: [RP - ZQ]fl(h,x)fl(h,y)dA and o f course (&*) s t i l l

y > x

(m+)

holds ( i . e .

K(x,y)

(1/2a)l:

=

[RQ

= D K(x,x)

=

(7.10)

( q - b ) ( ~ ) = (l/n)Dxl:

X

-

K(x,y) = (1/21~)j: [RQ

-DxL(x,x)

For

ME0REI 7.7.

-

).

Rp]flQ(h,y)fp(A,x)d

Since ( l / Z ) ( i - C )

we g e t f o r m a l l y

Xp =

[HQ

B p o r -Sp,

-

zp]fl(h,x)fl P

Q (A,x)dh

K(x,y) i s g i v e n by

(m+)

when y > x and

(7.10) p r o v i d e s f o r m a l l y a recovery formula f o r t h e p o t e n t i a l s . Thus tip o r -Sp p l a y s a r o l e i n t h e h a l f l i n e t h e o r y p a r t i a l l y analogous t o

Rp i n t h e f u l l l i n e (compare (7.10) and (&*) e t c . ) . We w i l l l o o k a t some now and seek t h o s e w i t h t h e most n a t u r a l con-

h a l f l i n e spectral quantities

n e c t i o n s t o t h e p o t e n t i a l (Theorem 7.2 w i l l be a guide here a l o n g w i t h r e 2 It i s convenient t o assume A E C s u l t s i n [C2,3,10,11,13,34] and 52.6). and work w i t h

6=

-

D2

4 ( 4 = A-'(k')") again. L e t us r e c a l l ( A ' ( 0 ) # [SinA(y-x)/h]~(x)iA(x)dx Q and GhQ = c f l t c-f;

= 0)

(***) v.Q A ( y ) = Coshy

t

&!).

2 i A c - a t y = 0 (we drop s u b s c r i p t s Q e t c . when no con-

Now W(i,f,)

=

f u s i o n w i l l a r i s e ) and as y (cf.

(7.1))

(**A)

+

2ixc- = i h

m,W(4,fl)

- 1;

n,

exp(ihx)[iAi

qexp(iAx)Gh(x)dx.

-

(fl

@ ] which g i v e s

Also f r o m (A) w i t h

%

177

POTENTIAL AND SPECTRUM

W(i,f,)(O) = D f l ( 0 ) ( * * a ) 2ixc- = i x - 1 ; c$oshxfl(x,x)dx. Similar r e s u l t s ; 4[Sinhx/h] hold f o r F = f l ( x , O ) where W(S',f,)(O) = -F; thus (**O F = 1 + 1 fl(A,x)dx (from (A) a g a i n ) . Me will d e r i v e general formulas f o r 1 ; A$p[dx e t c . below b u t f o r now we i n d i c a t e just t h e l o c a l i z a t i o n s t e p . Thus take e. g. C = 2ixc- with W ( i , f l ) = C. Arguing a s in Remark 7.4 one o b t a i n s (**+) & ( v ) = ( f l / C ) $ i 2 vdx - (G/C)$ fl$vdx; d f l ( v ) = -(fl/C)Jx" + f l v d x + ( a / C )

-1; i AQ ( x ) f l ( x , x ) v ( x ) d x o r a C / a q = x -2 - i f l . Similarly one has from W(s',fl) = -F (*A*) d i ( v ) = - ( f l / F ) J o e vdx + (e'/F)$ e f l v d x ; d f l ( v ) = (f1/F)Jxme'flvdx - (i/F)JXm flvdx 2 a n d (*AA) d F ( v ) = ; 1 i ; ( x ) f , ( h , x ) v ( x ) d x or a F / a q = i f , . Now although c- and F a r e individuall y natural enough t h e l o c a l i z a t i o n s involving i f , or - i f l a r e s o r t of "mixed" and one expects a b e t t e r l o c a l i z a t i o n from R2 = F/2c-, ", o r S Thus 9' -2 and consequently -2ixdc % G f i we have & [c-dc - cdc ]/c or from (**=I, (*A*) a$ / a q = (1/2ixc-2)iG. Similarly from (*AA), dF- % i f ; a n d (*A&) asQ/ Q 2 .. a q = - ( 2 i x / F ) 0 e . These have a nice "square eigenfunction" form with half l i n e eigenfunctions ( c f . 552.9 and 2.10 f o r square eigenfunctions in s o l i t o n t h e o r y ) ; however asymptotic considerations via formulas 1 ike (AA) do not seem t o give an inverse. We will i n v e s t i g a t e t h e half l i n e square eigenfunct i o n s a t another time a n d r e f e r t o [Aol,Z;Cjl;Nll ,2;Nvl;Zkl ,2] f o r background. Finally consider F/2c- = R2 with dR2 % ( 1 / 2 ) [ c - i f l + F4f1/2ix]/c-2 = (fl/4ihc-')[F; + Eixc-s']. From (A*) we have then (*A+) a R 2 / a q = (1/4ih c-')f: a n d sumnarizing we s t a t e fxm flvdx. 2

Then a s in

(W)

(**,)

dC(v)

=

2,

&HE@REIR 7.8.

by (**.),

The l o c a l i z a t i o n s f o r c - , F , 3, S , a n d R2 = F/2c- a r e given (*A@), (*A&), and (*A+).

(*AA),

We see t h a t F/2c- = R2 l o c a l i z e s l i k e R2 in (+*) and in f a c t from (A&) e t c . ( * A m ) R2 = ( l + R 2 ) / ( 1 - R 2 ) = 1 + 2 R 2 / ( 1 - R 2 ) (with = 1 These expressions will a r i s e l a t e r in our half l i n e version of 2R2/(1+R2)).

REmARK 7.9.

Ril

a spectral transform ( o r IST).

We note a l s o from Fc + F-c- = 1 t h a t R 2 +

R -2 = & = 1 / 2 / c / 2 so t h a t some range r e s t r i c t i o n on t h e o p e r a t o r dR2 may

a r i s e as f o r R in [TzZ] ( c f . Remark 7 . 4 ) ; we will examine t h i s a t another time. In any event however, in c e r t a i n r e s p e c t s R 2 seems t o be a s i g n i f i cant s p e c t r a l quantity f o r t h e half l i n e . In developing t h e IST f o r t h e half l i n e below we will want t o extend t h e Fourier Sine o r Cosine transform. In t h i s d i r e c t i o n l e t us note t h a t (*a*) + x 2,.0 0 = f 2 f ; and xGe' = ( i / 4 ) ( f22 - f;').

R m R K 7.10,

Now finding an expression f o r AR2 (or 43 o r AS) i n terms of i n t e g r a l s of

Ai

178

ROBERT CARROLL

may n o t be p o s s i b l e " e x a c t l y " (which i s o f small concern s i n c e t h e IST i t s e l f i n v o l v e s b o t h R and S = 2ikR/T).

I n t h i s d i r e c t i o n l e t us w r i t e down

some p r e l i m i n a r y formulas c o n n e c t i n g p w i t h s p e c t r a l q u a n t i t i e s . One can Q P where W(0) = 2iAcp and s t a r t w i t h Wronskian formulas f o r say W = W(GA,fl) 2 W(m) = 2 i ~ c - Thus from (D i)i: = -A2$? and (D2 b ) f lP = -A 2 flP we have 9' P Q P W(i Q ,f P )I; = IF (fi-4)GAfldx so (*@A) 2iXAc- = -IF Af6:(x)fl(x)dx (Ac- 9 cp, A Ap.1 = 6-6, e t c . ) . S i m i l a r l y W1 = W(O,,fl) ' 9 P s a t i s f i e s W1(0) = - F p and- 'W1

-

-

-

-

(a)

= -F

Q

so (*@@)

mulas below.

AF =

:I

We w i l l expand upon these f o r -

A$:(x)fl(A,x)dx. P

Summarizing here g i v e s The formulas

CHE0REm 7-11,

r e l a t e A t t o Ac- and AF.

(*@A)-(*@@)

REIIIARK 7-12, L e t us say a few works now about t r a n s m u t a t i o n k e r n e l s

Gy

-

i!

-+

i n o r d e r t o produce some a d d i t i o n a l h a l f l i n e r e c o v e r y formulas analogous

Thus one has t r a n s m u t a t i o n k e r n e l s (*a&) i ( y , x ) = ( G QA ( y ) , v* P ( x ) ) ~ Q = (;:(x),GX(y)) where ( f , g ) Q I fgdo e t c . such t h a t i:i FJ, + and ;(x,y) -Q .P Q Q v- X 9 and +: v X + v X . It i s e a s i l y shown ( c f . [C2,3;Mrl] and 582.6, 1.6, 1.11 t o (7.10).

Q

f o r d e t a i l s ) t h a t b(y,x) = 6 ( x - y ) + i((y,x)

(*@+) 2D i ( x , x ) = X

x) = 0 for x

>

4-b

= -2Dxi(x,x).

y and ;(x,y)

and ;(x,y)

= 6(x-y)

+

i(x,y) with

F u r t h e r one has t h e t r i a n g u l a r i t y i ( y ,

= 0 f o r y > x.

I n order t o obtain a spectral Set

formula f o r t? d i r e c t l y one can s u b t r a c t o f f t h e 6 f u n c t i o n as f o l l o w s . N

~ ( y , x ) = T(x,y) and n o t e t h a t K(y,x) = 0 f o r x < y w i t h a 6 f u n c t i o n c o n t r i b u t i o n a l o n g t h e diagonal x = y.

Thus f o r x i y

(*OH)

t?(y,x)

= i(y,x)

-9 B(Y,x) = 10 v X ( y ) ~ ~ ( x ) [ d o-p doQ]. S i m i l a r l y one has a k e r n e l ( ( f , g ) 2 2 P Q 1 f g d v w i t h dvQ = 2x dh/nlF( ) (*&*) i ( y , x ) = ( i : ( ~ $ ~ ( x ) ) and ~ ;(x,y)

N

*P Q-Q (OA(x),OX(y)) w i t h

Q

f o r e and we w r i t e

$

g:

'P

Oh

+;:

= 6(x-y)

and

c:

is;.

-

? ( x , y ) = lo m O ' QA ( y ) i ! ( x ) [ d v p

[C10,11,34;Chl]

and 52.6) ( * b e )

4-6

-

T,

=

One has t r i a n g u l a r i t y as be-

+ t ( y , x ) w i t h ;(x,y)

c ( x , y ) = 0 f o r y > x and t?(y,x) = 0 f o r x i(y,x)

-

m

= 6(x-y)

Hence ( x ~

+

?(x,y) where n

(*&A) ) K(y,x) = y. dvQ]. F u r t h e r as i n (*@e) one has ( c f .

= 2Dxi(x,x).

y

We summarize t h i s i n theo-

rem 7.13 and w i l l w r i t e a r e f i n e d v e r s i o n l a t e r .

CHE0REm 7.13. (*&A ) - (*&@ )

.

One can w r i t e Ap i n terms o f Ado o r Adv v i a (*em)-(*@+) o r

L e t us now r e f i n e Theorem 7.13 by u s i n g a K-L v e r s i o n 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 due t o t h e a u t h o r ( c f . [C2,3,15] (7.11)

B'(Y,x) = (1/271)l;

and e.g.

52.6).

~:(Y)[~Y(A,X)/C~I~A;

B^(Y,XI= - ( i / n ) l I

!(Y )[fY(A,x)/FpldA

Thus

POTENTIAL AND SPECTRUM (cf.

(A*)

A f u l l l i n e version of

and [C2,3]).

B"

179 (g(y,x)

c o u l d a l s o be u s e f u l so we w r i t e ( r e c a y f r o m [C2,3] c h a r a c t e r i z e d by

(*&a) ;(Y,x)

E:

fl/c-

-,f l / c -

and i ( y , x )

%

= ( 1 / 2 1 ~ ) j zTp(c,/cQ)fp(X,,y)f~(X,x)dX

= ;(x,y))

;(x,y)

-8:

by

fl/F

v^

and is

and 52.6 t h a t +

fl/F)

and C(X,Y) = ( 1 / 2 ~ ) 1 1Tp

( F /F )fQ(A,y)fi(X,x)dX. Ifone r e t a i n s a i term i n (*U) ( f i r s t e q u a t i o n ) P Q 1 = (1/2 f o r example v i a K-L r e d u c t i o n i t would a r i s e i n t h e f o r m ( * b e ) ;(y,x) IT)/: i!(x)[fl Q (A,y)/c-]dh so t h a t combining w i t h i ( y , x ) would s t i l l i n v o l v e f u r t h e r reduction.

Q

I t s h o u l d n ' t r e a l l y m a t t e r whether we use ( * t b ) o r (*&+);

e v e n t u a l l y we want e x p r e s s i o n s f o r t h e k e r n e l s i n terms o f i n t e g r a l s o f "nat u r a l " spectral quantities against t i o n o f (**+) o r (7.12)

(*&a).

i l s

Thus u s i n g

and i l s w i t h a view toward a p p l i c a w i t h (7.11) and (*&+) we o b t a i n

(A*)

~ ( Y , x ) = (l/n)lI [?!(y)G!(x)R;

+

ihG~(y)e~(x)ld~;

S i m i l a r l y , as f o r (*&+)

(obvious n o t a t i o n ) .

We see t h a t t h e s p e c t r a l q u a n t i t y R2 a g a i n a r i s e s n a t -

u r a l l y and i n t e g r a l s o f t h e f o r m CHE0REm 7.14-

,I xeidx

= 0 since

i

and

i are

even.

Thus

i,

can Under t h e hypotheses i n d i c a t e d t h e k e r n e l s i, $, and 13) i n terms o f Re = F/2c-. I n p a r t i c u l a r formally l / R 2Q - l/R;, e t c . ) (*&.) (1/2)Ab = -(l/n)DxL: X 2

be w r i t t e n v i a (7.12)-(7

1/ 2 )A6

=

( 1/ n ) D x i I

1:

The v a n i s h ng o f i n t e g r a l s

REmARK 7.15.

formulas f o r t r a n s m u t a t i o n k e r n e l s .

i P, ( x )G Q( x )AR2dX. heGdx a l s o p r o v i d e s , i n t e r e s t i n g

Thus e.g.

i(y,x) =

(1/1~)1: G:(y)G!(x)

R2dX P and z ( y , x ) = ( 1 / 1 ~ ) j : 4x(y)GX(x)R2dA. Q P Q I f we now add t h e two e q u a t i o n s i n (*&.) t h e r e r e s u l t s f o r m a l l y (*+*) Af, = P-Q 2-P-Q 2 ( l / n ) D X I I [i,v,(~R~) - X eXeX~(l/R2)]dx. I n p a r t i c u l a r formally f o r P = D P w i t h 6 = 0 and R2 = 1 (7.14)

6=

(l/r)Dxf:

[CosXxC!(l-R2) Q

-

xSin~x~!(l/R~-l)]dh

and we n o t e t h a t 1-R2 = -2R2/(1-R2) w i t h 1/R2-l = -2R2/(1+R2) ( c f . (*Am)). T h i s w i l l r e p r e s e n t one h a l f o f o u r IST f o r t h e h a l f l i n e . For t h e o t h e r

180

ROBERT CARROLL

and w r i t e o u t fl = FG t 2iAc-e'; a l o n g w i t h t h i s we h a l f we go t o (*.A)-(**.) Q P one has ( w i t h a l i t t l e c a l c u l a t i o n ) W(0) = n o t e a l s o t h a t f o r W = W(Gx,f2)

- i x with

= 2ix[c-RP/T Q pp - c Q/T P1 w h i l e f o r W(s'hQ,fg) = W one has W(0) = + FQR ]/Tp. Consequently ( A ~ I= 4-6) (*+A) 2 i x [ ( 1 / 2 ) +

W(m)

-1 w i t h W(-)

= -[F-

P 1 - (F- + FQRP ) / T p = -1: A$!(x)f2(A,P (c-R P-c ) / T ] = -IoP A$:(x)f2(A,x)dx; Q Q P Q one o b t a i n s x ) d x and we r e c a l l f2 = 4 - i x i . Expanding (*+A) and (**A)-(*.*) f o r A; = cj-b, AC- = c i - cp, e t c . ( * W ) FpI: A$!$Fdx + 2 i x c p I r A K qA Phd x =

-2ixAc-;

-I:

+

A$%'dx 4 ?p

+

A$$:dx

FP$

2ixcpi:

ixIr A%p!dx -

i h 1 , A P % dx = 1

xx

A$F!dx

= AF

(from

and (*+&)

(*.A)-(*..))

-

= 2 i x [ ( 1 / 2 ) + (cQRP cQ)/TP]; -1; A$!$!dx + P P [F- + FQR ]/T Now some c a l c u l a t i o n s which we merciQm

.

f u l l y o m i t g i v e (*++) 10 A@$!dx = -iA[TPc- + 2c-c-RP - 2c-c 1; /:A@%,' P 1; A $ F F d x =Q -1 + PTQF /2 + cP- QF + c-F R ; dx = 1 + c-F R - c-TP - c F Q P Q CJ P; P Q PQ P Q 1 ,A $ y F d x = [TpF - Fp(FQ + FQRP)]/2ix. Some f u r t h e r c a l c u l a t i o n s w i l l Q Q-P P Q Q y i e l d ( c f . ( * A m ) , (A&), e t c . ) (*+a) 1 ; Ab[Gf! + e,lpx]dx = T R2/T + R p ( l R!R:)/TQT' + (R;Q - R ; ) / T ~ T ~ = [C-(RQ - i ) / c p + 2c-c ( 1 - R~-P ) ( RP~ t RQ ~ +) Q 2 Q P 2. P - Q + R2)]/(1 P + RF)(*=*)Jr Ab[$:Gf - h exO,]dx = ( i X / ( l + R 2P ) ) 2c,CQ(RiQ - R2)(1 P

b

.

-

[cQ(R!

l)/cp

here t h a t

-

(*mA)

2c c - ( 1 - R i P ) ( 1 + R PR Q ) + 2c-c ( 1 + RP2 ) ( 1 - RpR-Q)]. Note P Q 2 2P Q 2 2 R/T = c ( l - R i ) , 1/T = c (1 + R 2 ) , and R2/T = c (R2 - 1 ) .

Also some c a l c u l a t i o n shows t h a t e.g. ( * m * ) i s s u i t a b l y a n t i s y m n e t r i c . In P 2 p a r t i c u l a r t a k e now P = D again w i t h p = 0, Rp = R2 = 0, T = 1, e t c . and P (7.15)

lom $qxSinAx/A .Q

(7.16)

1"0

For

6

-f

+ i;Coshx]dx

~ [ C o s x x- ~hSinxxi):]dx ~

= c-(RQ

Q 2

1 ) + c (R-Q 4 2

-

1)

= i h [ c - ( R 9 - 1 ) + cQ(1-R;')] Q 2

D2 t h i s l a s t e q u a t i o n approaches

interesting.

-

:I

qCos2Axdx

2.

0 which i s n o t t o o

However (7.14) and (7.15) t o g e t h e r a r e o f i n t e r e s t and we have

EHE0REm 7-16,

The formulas (*+*) and (7.14) h o l d f o r m a l l y and (*++) has

been e s t a b l i s h e d l e a d i n g t o ( * W ) and (7.16).

The formulas (7.14) and (7.15)

p r o v i d e a form o f IST f o r m a l l y f o r t h e h a l f l i n e .

Phuud:

L e t us examine t h e l a s t statement.

e t c . and R$

small TQ

For

* 1 i s approximately r e a l w i t h c

Q

+

1/2.

%

1,

4:

Cosxx,

The f i r s t e q u a t i o n

Q (7.15) becomes a p p r o x i m a t e l y a S i n e t r a n s f o r m ( * m @ ) I; 6Sin2xxdx % A(R2-1) w h i l e (7.14) i s % ( l / n ) O x l I (1-R:)CosZxxdx = ( 2 / n ) / I x(R2-1)Sin2xxdx. Q

4

Treating A(R2-l)

as odd ( v i a (*me) f o r example) t h i s l a s t e q u a t i o n i s

(4/7r)1; X ( R 2 - l )SinZxxdx which means 4(5/2) i n v e r s e 2X(R2-1)

'L

%

(2/7)J;

%

2A(R2-1 )SinAgdx w i t h

:1 4(5/2)Sin2xgdg i n agreement w i t h

(*me).

QED

POTENTIAL AND SPECTRUM

181

It i s w o r t h n o t i n g what happens i f we b e g i n w i t h a h a l f

RflllARK 7.17.

b

problem i n v o l v i n g F o u r i e r t y p e o p e r a t o r s

-

= D2

f~ extended

with

ine

t o be an

-

even f u n c t i o n (and s u i t a b l y continuous, d e c r e a s i n g a t a, e t c . c f . 52 6). = A-'(A5)" = r + ri, I f i n f a c t t h i s a r i s e s from an impedance problem w i t h

rA = (1/2)A'/A,

and h = r ( 0 ) = ( 1 / 2 ) A ' ( O ) then i n general ( u n l e s s h = we 2 d e f i n e d as s o l u t i o n s o f I% = - A 4 w i t h F(0) = 1, z ' ( 0 ) = 0, -5 x 5, , \ v j I , - n "llU n \,lUL.C " - " ""L " r\u, -

Z,g, i ) . Since

distinguish " \ v j

v,

h so

=

UllU

we w r i t e ( c f .

(A*))

a l s o t h a t 2ih;-

-

N

IIUlll

~

T

N

A

0 rapidly a t

-+

(*.&)

= 2iXc-

m

+ hF).

=

1/2<-F (R/T = F r

lar, setting =

8, T - R

4=

-

F-5- = 2FC"

( R / T ) - = -R/T,

F/E,

A% = fl a g a i n and

- Ff;

2 i x r = F-fl

R = R2 = ( 1 / 2 ) ( 8

- 1 ITI

= 1

2

= 2iX

+

-

ZF-Z-).

1~1'=

(note

fl = F r +

- s) and

T = (1/2)

Thus i n p a r t i c u -

1, $ / S =

%/%,

R + T

= S, e t c . many o f t h e c a l c u l a t i o n s o f t h e p r e v i o u s t e x t remain

v a l i d w i t h t h e main d i f f e r e n c e a r i s i n g v i a f2 =

FZ -

2iX"c-e" ( i n s t e a d o f $

-

I n p a r t i c u l a r Theorem 7.15 i s unchanged s i n c e no use o f f2 was i n v o l -

iXe).

ved i n i t s d e r i v a t i o n .

Thus (*+*)-(7.14)

For t h e o t h e r p a r t one notes t h a t W(;,,f2)(0) Q P -Fp.

Q =

F u r t h e r as i n 52.6 one has (*m+)

2 i A Z - t a n d f 2 = F Z - Z i h z - r w h i l e (-m)

(b + S )

X

1

.

one has CP =

Cf, + z - f i ,

=

.

~

Hence

(*+A)

h o l d and we have h a l f o f t h e IST. = -2iXz;

and w(e,,f,)(o) uQ P

=

becomes

-1"0 4;!(x)f2(a,x)dx P = 2iarZ-R P / T P - Ic' /Tp + '?PI; Q Q 1" ApOA(x)f2(X,x)dx *'Q P = Fp - ( f i + FQRP)/TP

(7.17)

0

/I here.

W r i t i n g t h i s o u t v i a f 2 = FG - 2ihC-g 0-4 P ..,- m .-Q-P - . we o b t a i n i n p l a c e o f (*+&) (A**) -FP$ Aplp $ dx + 2iAcpJo Aplp,e,dx - 2ih P *-Q P >-Arn .+p P tr, + (z-R' - T ~ ) / TI ; ape,;'dX + 2 i i z p / o ape,e,dx - F~ - ( ~ +a F ~ R) / There i s no need t o i n v o l v e

- ~ ~ ~ b m

Q

Now combine t h e s e equations w i t h (*+.) t o o b t a i n i n p a r t i c u l a r (A*A) Tp. m --Q.$ P P Q ,. -P - P Jo Ape ~p dx - -Tp?P[2Fp F - ( F - + F R ) / T ] and AblpxOXdx - -T Fp[-2?p A.2 p p Q Q Q + :-' - cQR / T + c /T 1. Combining these some c a l c u l a t i o n l e a d s t o (A*.)

Jr

-

(i; - iiQ)

-

+ 2-F- = 2 z :-($ - g-') F-F- - F Q P P Q P Q P Q 2 2 i n p l a c e o f (*+.). T h i s i s v i s i b l y a n t i s y m m e t r i c ( b +

+ d)8:x;

:1 !p(;d$$!

Q

= p , 6.

F P Q

- 2E-c

P Q

6) whereas

t h e check o f a n t i s y m n e t r y i n (*+m) r e q u i r e s some c a l c u l a t i o n . f u r t h e r i f one takes b = D 2 a g a i n t h e r e r e s u l t s (A*&) 10" G[F:Sinxx,'A t ~ ~ C o s A x ] d =x

?-(SQ Q 2

I ) - C (1 -

(7.14)-(7.15)

REmARK 7.18.

Q

z,)'

which agrees w i t h ( 7 . 1 5 ) .

Thus t h e h a l f l i n e IST

i s t h e same as b e f o r e . I t i s a l s o w o r t h n o t i n g a few changes t h a t o c c u r when -h =

r ( 0 ) f 0 ( r = r Z = - r A ) . Thus (7.2) h o l d s b u t one cannot i d e n t i f y R,T,R2

182

ROBERT CARROLL u

o

o

w i t h R,T,R2.

The c a l c u l a t i o n s l e a d i n g t o

l / i= c-+F/2,

-F-/2,

= F/2-c-.

irand il

in o r d e r t h a t

be necessary on

g2/T"

and

= -ik-r(0)

R/?

(A*+)

be thought o f as s o l u t i o n s t o i w

= c

2 -k w

=

v^

Thus from e a r l i e r c a l c u l a t i o n s D

(-my-).

(l-i2)/? w h i l e Dxir(Ot)

are valid with

(A()

The c o n d i t i o n r ( 0 ) = 0 i s seen t o

(0-) = - i k and D x i L ( O - ) ik A x+r and DxVL(O ) = i k ( 1 - E 2 ) / f - r ( 0 ) ( l + ( 2 ) / ? .

Such d i s c o n t i n u i t i e s i n DxVr,L would l e a d t o 6 f u n c t i o n s i n t h e second d e r i 2 v a t i v e s so t h a t = - k w would n o t h o l d . G e n e r a l l y now - r ( O ) = ( 1 / 2 ) A ' ( O )

&

= h # 0 and i ' ( 0 ) = h.

7 and

We d e f i n e n e x t , f o l l o w i n g t h e idea o f Remark 7.17, 2 2 t o be s o l u t i o n s o f & = -X IL = D -6, 6 = A-'(A')'') satisfying

(4

?I

z ( 0 ) = 1, ? ' ( O )

= 0,

g(0)

= 0, and

g'(0) =

1 (note

=

e'

b = A%).

Further

N

write

=;

(A*m)

d t Q = dA/2nl?

I

Q

2

?fl + E-fi w i t h 2iXg = F - f - Ff; ( i . e . F = F). Note t h a t 1 d i f f e r s from dw b u t dv remains unchanged. One extends

Q

Q

by CosXx and e" by SinAx/h f o r x 5 0 and d e f i n e s fl and f2 f o r a l l x by f2 = ? - i x r a n d fl = Fg + 2iX?-g, E v i d e n t l y (g,;) and (F',;') a r e 2 F = (1+ continuous a t x = 0 so fl and f2 s a t i s f y [f = -X f. S e t t i n g (A") now ("*)

R2)/T and -2;

= (1-R2)/T f o r R2,T as i n (7.1) one has a c o n s i s t e n t s e t o f

formulas.

Thus an easy c a l c u l a t i o n based on ("*)

= F-fl-Ff-

determines a s y m p t o t i c b e h a v i o r i n terms o f

identity

(4 =

It f o l l o w s t h a t ("6)

(A").

i'9).

2iXz- = -2;

1

Indeed W(d,f,)

W(Fyfl).

Hence

+

= (1-R2)/T = 2c-

a l s o 1/T =

l/r"

2iX;-

follows.

.

zfl

=

?,

+

2-f;;

+

+ hF s i n c e

= -F = - f l ( 0 )

hF/iX.

One can deduce e.g.

2iAr

F and d i c t a t e s t h e =

+ he

with f i ( 0 ) =

Hence (1+R2)/T = ( l + i 2 ) / f

hF/iX = (l-I$)/T"

+ hF/2i

= 2iXc-

= 2 i h c - and W(e",fl) (A.6)

and

=

F and

These a r e c o n s i s t e n t and R2 = [ 2 i X i 2

(AA+)

- h(l+g2)]/

[ Z i X + h ( l + i 2 ) ] which agrees w i t h a formula i n [ N w l ] modulo an adjustment o f

minus s i g n s i n

i2 ( n o t e

Z i n [Nwl]

'L

i"). Whenever w o r k i n g w i t h t h e

our

l i n e p o t e n t i a l { now one s h o u l d i n t r o d u c e Theorems 7.6-7.7

we r e p l a c e Sp by

gp =

1C"l2.

c" and

on c,9,0,

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

full in

E/'/c". S i m i l a r l y t h e l o c a l i z a t i o n s i n

Theorem 7.8 should be phrased i n terms o f rem 7.11 r e p l a c e c by

%

?,g,

and

z2 =

F/22-.

Next i n Theo-

i n Theorem 7.13 one must work w i t h dw" = dh/2n

These a l t e r a t i o n s c a r r y t o Theorem 7.14 ( c

and i n p a r t i c u l a r (7.14) i s t h e same w i t h t a i n s (7.15) w i t h these changes

i

-+

i

-+

-+

<,

ii2,9

R2_"

and R2 * R 2 .

-+

7, e t c . )

One a l s o ob-

e t c . and e v e r y t h i n g c o u l d t h e n be r e -

c a s t i n terms o f h a l f l i n e q u a n t i t i e s v i a t h e r e l a t i o n s i n d i c a t e d . Thus (A A m) F = F = (1+R2)/T = ( l + i 2 ) / ? , 2c- = ( l - $ ) / f , -2; = (1-R2)/T, 2 -; = hl

2c-

c-,

+

hF/iX, :2/? =

i, $

= $

= F/2

+

- c-,

&

he, e t c .

l/f

= F/2

+ c-, R2/T

= F/2

-

'c-, 1/T

= F/2

+

SOLITON THEORY

183

We w i l l g i v e f i r s t a b r i e f i n t r o d u c t i o n t o s o l i t o n t h e o r y w i t h s p e c i a l r e f e r e n c e t o [A11;Cel;Lml;Dal;N1;Nel;Whl]. 8. INCR0DUCCL0N C0 90CIC0N CHE0R&

We do n o t a t t e m p t t o g i v e e i t h e r a h i s t o r y o r a p h i l o s o p h i c a l d i s c u s s i o n . S u f f i c e i t t o say t h a t s o l i t o n s a r i s e as s o l i t a r y wave s o l u t i o n s f o r t h e KdV and o t h e r e q u a t i o n s and have remarkable p r o p e r t i e s ; one a l s o imagines a conn e c t i o n between such o b j e c t s and elementary p a r t i c l e s i n quantum mechanics. Other a p p l i c a t i o n s a r i s e i n plasma waves, c r y s t a l phenomena, l i g h t pulses, Thus f o r example i f we t a k e (1.10.4) i n t h e form ut - 6uux + uxxx = 0 2 one has a s o l u t i o n (*) u =(c/2)sech [ ( f i / 2 ) ( x - c t ) ] which r e p r e s e n t s a s i n g l e etc.

s o l i t o n propagating t o the r i g h t w i t h v e l o c i t y c.

The o r i g i n h e r e o f t h e

KdV e q u a t i o n i s t h e t h e o r y o f l o n g waves i n s h a l l o w w a t e r w i t h d i s p e r s i v e e f f e c t s included.

The n o n l i n e a r i t y counterbalances t h e d i s p e r s i o n t o p r o -

duce waves o f permanent form.

I n ( * ) t h e a m p l i t u d e and frequency a r e r e l a -

t e d which i s a n o t h e r f e a t u r e t o e x p e c t ( c f . [Whl] f o r p h y s i c a l d e t a i l s ) . The i m p o r t a n t " s o l i t o n " f e a t u r e s o f (*) a r e n o t y e t apparent and t h a t i n v o l v e s t h e p r e s e r v a t i o n o f f o r m (up t o a phase s h i f t ) upon i n t e r a c t i o n w i t h o t h e r such p u l s e s ( d e t a i l s l a t e r ) . s o l v e o r "understand" ut

L e t us now proceed h e u r i s t i c a l l y t o = 0 and s e t u = $$ /, - X following

- 6uux + uxxx

Gardner, Greene, Kruskal, and Miura (a n e a t e r d i s c u s s i o n w i l l f o l l o w i n r e mark 8.5). (8.1)

Thus one p o s i t s a reduced Schrodinger e q u a t i o n $xx - u l L = X $

where u = u ( x , t )

i s t o s a t i s f y t h e KdV e q u a t i o n ( t h i s s u b s t i t u t i o n

e n t i r e l y a s h o t i n t h e dark; t h e r e a r e reasons t o b e l i e v e i t m i g h t where).

s not ead some-

Thus one hopes t o f i n d u ( x , t ) as t h e p o t e n t i a l i n a f a m i l y o f i n -

verse s c a t t e r i n g problems p a r a m e t r i z e d by t and ( s u b s t i t u t i n g (8.1) i n t h e KdV e q u a t i o n ) (8.2)

x

= x ( t ) would have t o s a t i s f y

2

-$ A t + Dx[$Qx

(one assumes u

-+

-

$,QI

Q =

= 0;

0 s u i t a b l y as 1x1

-+

m,

$t +

etc.;

4xxx

-

e.g.

LI

3 ( -~ X)lLx 2 (l+lxl )luldx <

a).

There a r e several l i n e s o f r e a s o n i n g now t h a t w i l l show A t = 0 ( i s o s p e c t r a l i t y ) and l e a d t o a s o l u t i o n technique.

Q

= c$ and f o r c = - 4 i k

exploited.

3

w i t h X = -k

2

Note when it= 0 one has o b v i o u s l y t h e e q u a t i o n s Q = c$ and (8.1) can be

We mention here t h e approach o f Lax ( c f . [ L x ~ ] ) which shows t h a t

t h e KdV e q u a t i o n i s one o f many e q u a t i o n s t h a t govern t h e v a r i a t i o n o f a p o t e n t i a l i n a Schrodinger e q u a t i o n i n such a way t h a t t h e spectrum remains 2 c o n s t a n t . Thus w r i t e L$ Then (L$)t = Lt$ + L$t $xx - u$ = ( D -u)$ = A$.

184

ROBERT CARROLL

= x ~ $t

Mt = GXxt

-

-

u!bt

ut$

L$t

-

ut$ so Lt

We want a t i m e v a r -

-ut.

0 and one supposes Gt =

i a t i o n on u, and consequently on $, such t h a t A t

B$ where B i s some ( n o t n e c e s s a r i l y unique) d i f f e r e n t i a l o p e r a t o r t o be deThus g i v e Gt = B$ we o b t a i n ([L,B]

termined. At$

= LB

-

BL) LB$ = L$t = ut$ t

+ BL$ o r (-ut + [L,B])$

(8.3)

0

= X t$ =

which means t h a t B s a t i s f y i n g -ut + [L,B]

(8.4)

= 0

($t = B$)

i s a c o n d i t i o n f o r i s o s p e c t r a l e v o l u t i o n o f $. shows t h a t

B = -4D

(A)

3 + 6uD -

t 3ux

A l i t t l e e x p e r i m e n t a t i o n now

w i l l i d e n t i f y t h e KdV e q u a t i o n ut -

6uux t uxxx = 0 w i t h (8.4) ( e x e r c i s e ) . One a l s o o b t a i n s from $t = B$ ( 0 ) 3 $t = (3ux t 6uD - 4D )$. We w i l l see l a t e r how t h i s f o r m u l a t i o n i s especi a l l y p r o d u c t i v e and s i g n i f i c a n t ( c f . a l s o Remark 8.5 f o r a more p o l i s h e d version o f t h i s ) .

L e t us f i r s t examine t h e i d e a o f i s o s p e c t r a l e v o l u t i o n .

We r e c a l l t h e r e l e v a n t f a c t s about f u l l l i n e s c a t t e r i n g (see h e r e 952.1

-

and [Arl;C2,3;Chl;Fal;Lml;Kyl;Mrl]). As i n Example 1. 2 8.5 we c o n s i d e r $I' - u$ t k $ = 0 on (-m,m) and d e f i n e J o s t s o l u t i o n s f+ = 2 fl and f- = f2 as i n (1.8.6) ( q % u, X 2r -k ; however we suppress t h e t 2.7,

Example 1.8.5,

dependence i n u f o r t h e moment). a l s o (2.4.1))

As b e f o r e we d e f i n e c:;

'J

by (2.4.3)

(cf.

and one s e t s ( 6 ) RR = sZ1 = c 11/ c 12; TR = ~ 2 2= 1 / ~ 1 2 ; R L

i2

=

Using p r e v i o u s formulas we f i n d TR = TL, + R L ( - k ) T ( k ) = 0, e t c . Now fl

= + 1 /I cR 2L I1 i , RR(k)T(-k)

-

i s a n a l y t i c f o r Imk > 0 w i t h fl Paley-Wiener ideas again ( c f . formula (+) fl(k,x)

e x p ( i k x ) = O ( l / l k l ) as I k l

§1.6 and Appendix

= exp(ikx)

+

I,"

r i v e a t ( 6 ) i s i n d i c a t e d i n [Lml]. = (l/cL)btt

Another way t o a r -

Thus one imagines a s o l u t i o n t o $ "

T h i s would r e p r e s e n t e.g. some i n c i d e n t waves s c a t t e r e d

kc, c t = s , one a r r i v e s a t (+).

2.7) (8.5)

(m)

f2(k,x)

= exp(-ikx)

+

S i m i l a r l y ( c f . [Chl;Fal]

AL(x,s)exp(-iks)ds

m RR(k)eikzdk; r R ( Z ) = ( 1 / 2 n ) j -m

r(z)

- u$

where Y i s t h e

t o produce pulses going t o + a and t a k i n g F o u r i e r t r a n s f o r m s t w =

hence by

B) we have a t r a n s m u t a t i o n

AR(x,S)exp(iks)ds.

i n t h e form $ = s ( t - x / c ) + c Y ( t - x / c ) A R ( x , c t )

Heavyside f u n c t i o n .

-+ m;

= ( 1 / 2 n ) c [T(k)

-+ W,

Dt

and one d e f i n e s

r L ( z ) = ( 1 / 2 n ) Ia m RL(k)e-ikzdk;

- l]e-ikzdk

+. - i W ,

and § § Z . l

-

185

SOLITON THEORY

( r e c a l l from §2.4 e t c . T ( k )

-+

1 as I k l

-+

The zeros o f c 1 2 ( k ) = c Z 1 ( k )

m).

a r e s i m p l e and l i e on t h e upper i m a g i n a r y a x i s f o r r e a l p o t e n t i a l s and t h e s e poles o f T(k) = l/c12(k),

determine t h e bound s t a t e s ( i . e .

Imk > 0, d e t e r -

We n o t e a l s o t h a t c12 = cZ1 i s a n a l y t i c f o r Imk > 0

mine t h e bound s t a t e s ) .

and t h e f u n c t i o n s k c . . ( k ) a r e c o n t i n u o u s on t h e r e a l a x i s . Since EL(-k) = IJ RL(k) e t c . f o r r e a l p o t e n t i a l s t h e f u n c t i o n s rL and rRa r e r e a l . L e t us r e c o r d a g a i n t h e d e f i n i n g r e l a t i o n s f o r cij fm m

q(t)f-(k,t)exp(-ikt)dt;

c21 ( k ) = 1

-

-

(q

Now

(*A)

%

u ) (**) cll(k)

= (1/2ik)

(1/2ik)lI q(t)f-(k,t)exp(ikt)dt

(1/2ik)/$ q(t)f+(k,t)exp(-ikt)dt;

t)exp(ikt)dt. 1.8.5

c12(k) = 1

cZ2(k) = (l/Zik)L;

T ( k ) f 2 ( k , x ) = RR(k)fl(k,x)

+ fl(-k,x)

q(t)f+(k,

f r o m Example

and we assume f o r convenience t h a t t h e r e a r e no bound s t a t e s so

= 0 f o r z < 0.

I f we F o u r i e r t r a n s f o r m

(*A)

=

r(z)

one o b t a i n s a f t e r some c a l c u l a -

t i o n ( o m i t t e d h e r e s i n c e such e q u a t i o n s a r e a l r e a d y t r e a t e d i n §§2.1-2.7 i n more d e t a i l ) (*.)

0 = rR(x+y)

i l a r l y from T(k)fl(k,x)

+

AR(x,y) +

= f2(-k,x)

I," r R ( s + y ) A R ( x y s ) d s ( x < y ) .

Sim-

+ R L ( k ) f 2 ( k , x ) one o b t a i n s (*&) 0 =

r L ( x + y ) + AL(x,y) + rL(s+y)AL(x,s)ds ( x > y ) . These a r e o u r s t a n d a r d M equations. The i d e a a g a i n i s t h a t s p e c t r a l knowledge o f RR, R L y T ( o r some s u b s e t ) determines rRand rL and hence AR and AL ( r e c a l l we assume no bound s t a t e s f o r t h e moment).

Then as i n e1.6;

c f a l s o §§2.1-2.7) one can show

t h a t (*+) u ( x ) = -2D A ( x , x ) = 2D A ( x , x ) ( s o RR o r RL w i l l d e t e r m i n e u; r e x R X L c a l l t h a t t h e t dependence o f u and e v e r y t h i n g e l s e , except X -k2, has been momentarily suppressed).

I f t h e r e a r e now bound s t a t e s a t k = i k . s e t J

and (*. ) becomes

Thus t o determine AR (and hence AR) one needs RR(k) f o r k r e a l , t h e l o c a t i o n s k = i k . o f t h e bound s t a t e s , t h e i r number N, and t h e " n o r m a l i z i n g " c o n s t a n t s mRJ= y . c ( i k . ) = - i c l l ( i k . ) / i 1 2 ( i k j ) (c',2 = dc12/dk e v a l u a t e d a t j J11 J J k = i k . ) . The r e c o v e r y f o r m u l a (*+) has t h e same form (we r e f e r t o [Lml] J f o r t h e bound s t a t e c a l c u l a t i o n s - c f . a l s o [Chl ; F a l l ) . Now i n s e r t t h e t dependence i n u $xx(x,t) fl(k,x,t)

-

u(x,t)$(x,t) and f 2 ( k , x , t )

2

= -k $ ( x , t )

q and c o n s i d e r ( k independent o f t ) (*.) where ( 6 ) h o l d s .

and s i m i l a r l y (+)

-

(m)

t ) and AL(x,s,t).

F u r t h e r we must w r i t e c . . ( k , t )

c2,(k,t)f2(-k,x,t)

+ cZ2(k,t)f2(k,x,t)

1J

Then one has f u n c t i o n s

i n v o l v e s f u n c t i o n s AR(x,sY w i t h e.g.

fl(k,x,t)

=

and f i n a l l y ( w o r k i n g w i t h AL now)

ROBERT CARROLL

186

u ( x , t ) = 2DxAL(x,x,t) (8.8)

AL(x+Y,t)

(8.9)

AL(z,t)

where + A,-(x,y,t)

+

I,X

AL(S+y,t)AL(x,s,t)ds

= 0

(x

>

Y)

1;

= (1/21~)1: [ ~ ~ ~ / c ~ ~ ] ( k , t ) e - t~ ~ ' m d ki ( i k j , t ) e k j 2

(mjL = v j ( t ) c 2 2 ( i k j , t )

= - i ~ ~ ~ ( i k ~ , t ) / ; ~ ~t ) ( iwhere k 621(ik

t) % d ~ ~ ~ / d k j' j' e v a l u a t e d a t k = i k . i s a r e s i d u e term). The p o i n t here i s t h a t t h e t i m e J v a r i a t i o n o f A L ( Z , t ) i s v e r y simple. I n f a c t one can use t h e l i n k i n g o f u 2 and JI from t h e Lax f o r m u l a w i t h $xx - u$ = - k JI and ( o ) , namely JIt = -4JIxxx t 6 q X t 3ux$ i n t h e a s y m p t o t i c r e g i o n where u % 0 i s assumed so t h a t )Lt - 4 1 4 ~(we ~ ~o n l y need s c a t t e r i n g d a t a t o determine t h e t r a n s m i s s i o n and r e f l e c t i o n c o e f f i c i e n t s - and t h e n o r m a l i z a t i o n c o n s t a n t s as i n d i c a t e d a f t e r Q

2

(8.9)). Consider a s o l u t i o n o f yxx - uy = -k y and yt = -4yxxx o f t h e f o r m y = h(k,t)fl(k,x,t) p r o p o r t i o n a l t o fl as x + ( h i s needed s i n c e f % 1 e x p ( i k x ) cannot s a t i s f y ( 0 ) ) . E v i d e n t l y h must s a t i s f y ht = 4 i k 3 h so h ( k , t ) = h(k,0)exp(4ik3t).

Next c o n s i d e r fl

--with y

and w r i t e o u t yt = -4yxxx t o o b t a i n

2,

h(k,t)fl

%

~ ~ ~ e x p ( i tk xc 2) 2 e x p ( - i k x ) as x

+

.-.

L mJ. ( i k j , t ) hDtfl

+ htfl

+ Dt~22 =

=

LJ -8kft m.(ik.,O)e J J

3

+ D t ~ 2 1 = ( i k ) 3~ ~ ~ ( - 4 ) 3x 3 - 4 ( - i k ) c22 so D t ~ 2 1 = 0 and D t ~ 2 2 = - 8 i k c ~ ~ T) h.i s 3

= -4hD f i n v o l v e s 4 i k c21

g i v e s then (by r e l a t i o n s among t h e c .

i n d i c a t e d e a r l i e r ) cll(k,t)

= cll(k,

Ij mR. ( i k t ) = mR. ( i k O)exp(8k:t). O)exp(8ik 3t), c 1 2 ( k , t ) = c12(k,0), and 3 J j' J j' i l a r l y RL(k,t) = RL(k,O)exp(-8ik t ) and e.g. (8.11)

%(Z,t)

3

= ( 1 / 2 1 ~ ) 1 RL(k,O)e 1 -i(kzt8k t)

Sim-

I; m;(ikj,0)e-8kjt+kjz3

Thus summarizing we s t a t e ( f o r m a l l y and h e u r i s t i c a l l y ) L m.(ik.,O), and t h e p o l e l o c a t i o n s i k . f o r u(x,O) J J J have been determined, t h e subsequent temporal development o f t ( z , t ) i s d e t -

ME0REIR 8.1.

Once RL(k,O),

ermined by (8.11) and t h i s leads t o u ( x , t ) = 2DxAL(x,x,t).

RmARK 8.2.

Similarly

In o r d e r now t o a r r i v e a t an e x p l i c i t m u l t i s o l i t o n s i t u a t i o n

one wants t o deal w i t h r e f l e c t i o n l e s s p o t e n t i a l s ( i . e . RL(k) = 0 mark 8.3).

F i r s t however ( c f . [Chl;Lml;Kyl])

-

c f . Re-

we c o n s i d e r s i t u a t i o n s where

SOL ITON THEORY

187

I f t h e r e a r e no p o l e s o f RL (A*) RL(k) = c q (k-a,)/n; ( k - B . ) (m 5 n - 1 ) . J ( k ) f o r Imk > 0 t h e n i n ( 8 . 5 ) r L ( z ) = 0 f o r z < 0. I f i n a d d i t i o n T ( k ) has no poles f o r Imk > 0 (no bound s t a t e s ) t h e n (*&) a p p l i e s and AL(x,x) f o r x < 0 (so u ( x ) = 0 f o r x < 0 ) .

= 0

To f i n d u i t is e a s i e s t t h e n t o con-

R R ( k ) T ( - k ) + RL(-k)T(k) = 0 and 1 = + (RRI2 = s t r u c t RR(k) f r o m (") 2 l T I 2 + I R L I , e t c . ( c f . [Lml]) and t h e n use a s e p a r a b i l i t y t e c h n i q u e t o obt a i n AR. One has h e r e f o r T ( k ) h a v i n g n e i t h e r zeros n o r p o l e s i n Imk > 0

(

+ indicates

a 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 and R = RL o r

RR-note (1/2ni) $ [logT]dd(s-i)

= 0 and one t a k e s c o n j u g a t e s and adds t o

l o g T ( k ) = ( 1 / 2 n i ) P [ l o g T ( ~ ) ] / ( ~ - k )t o g e t ( 8 . 1 3 ) ) . More g e n e r a l l y when T ( k ) has e.g. f i r s t o r d e r zeros a i o r p o l e s B i n Imk > 0 j

(8.14)

T(k) =

n

[ ( k - a i ) ( k - B . ) / ( k - ~ i ) ( k - ~ . ) l e [(1/2ai J J

Thus suppose e.g.

( c f . [Lml]).

RL(k) = a B / ( k + i a ) ( k + i B ) ,

)J-.f% a,@>

& w ] d i 0 r e a l . Then

as above u ( x ) = 0 f o r x < 0 (we assume T has no zeros o r p o l e s f o r Imk > 0 ) . By (8.13),

u s i n g R = RL one o b t a i n s T ( k ) = k ( k + i y ) / ( k + i a ) ( k + i ~ ) where y2

a2+~' ( e x e r c i s e ) .

Then from (")

RR(k) = - a B ( k + i y ) / ( k + i a ) ( k + i B ) ( k - i y )

t h a t by (8.5) r R ( z ) = moexp(-yz) where mo = - 2 a ~ / ( y + a ) ( y + ~ )< 0. t o f i n d a s o l u t i o n now o f (*.)

=

SO

One t r i e s

i n t h e form AR(xyy) = p(x)exp(-yy) (separ-

a b l e ) and a l i t t l e c a l c u l a t i o n g i v e s ( e x e r c i s e ) moexp(-yx) + p ( x ) + p ( x ) (mo/2y)exp(-2yx) = 0 w i t h (Am) u ( x ) = -2DxAR(x,x) where exp@ = (2y/ImoI 1/2 .

2

2

2

= 2y Y(x)csch (yx++)

2 2

Another example o f t h i s t y p e i n v o l v e s RL(k) = - a /(a +k ) (a r e a l ) and i f one chooses t h e p o l e o f T ( k ) i n t h e upper h a l f p l a n e t o be a t i a = k a l s o t h e n working w i t h ?(k) = T ( k ) [ ( k - i a ) / ( k + i a ) ] and r e f e r r i n g t o (8.14) one o b t a i n s T ( k ) = k ( k + i a J Z ) / ( k 2 +a2 ). Then (") y i e l d s RR(k) and some c a l c u l a t i o n g i v e s AR(z) = r R ( z )

+

yocl,(ia)exp(-az)

( - a d z ) f o r z > 0.

( c f . ( 8 . 6 ) ) i n t h e f o r m A R ( z ) = 2aJ2exp

An assumption now o f AR(x,y) = f ( x ) e x p ( - a J 2 y )

2

leads t o

2

( e x e r c i s e ) (A&) u ( x ) = -4a Y(x)sech aJ2x which i s a t r u n c a t e d f o r m o f t h e 2 sech p o t e n t i a l o f ( * ) ( f o r t = 0; n o t e f ( x ) = -2yexp(-yx)/(l+exp(-2yx), Y

aJ2 i n AR above).

REMARK 8.3.

Going now f i n a l l y t o r e f l e c t i o n l e s s p o t e n t i a l s RL(k) = 0 (and

m u l t i s o l i t o n s o l u t i o n s ) one has r L ( z ) = 0 and, i n s e r t i n g now a t i m e dependence, we t a k e ( p o l e s a t k = i k . ) J

(A+)

AL(zyt) =

IN1 m Lj ( t ) e x p ( kJ. z )

where mL =

j

188

ROBERT CARROLL

3 m!(O)exp(-8k.t) as i n (8.10). One t r i e s f o r a s o l u t i o n o f say (8.8) i n t h e J J Set *(z) f o r m (suppressing t momentarily) ( A m ) AL(x,y) = 1 a . ( x ) e x p ( k . y ) . L J J = (m.exp(k.2)) and @(z) = ( e x p ( k . 2 ) ) (as column v e c t o r s ) so AL(x,y) = A T ( x ) J J J T @ ( y ) (A(x) ( a . ( x ) ) ) and (8.8) becomes (AL(x,y) = (x)@(y)) J

*

AT(x)[I +

(8.15)

1:

@ ( ~ ) * ~ ( s ) d s ] @ ( y+) q T ( x ) @ ( y ) = 0

where I = (( 6 . .)) and V(x) = I t

LE @ ( s ) * T ( s ) d s

i s an i n v e r t i b l e m a t r i x . I t T f o l l o w s t h a t AL(x,y) = -*T(x)V-l ( x ) @ ( y ) and AL(x,x) = -Tr[@(x)* ( x ) V - ’ ( x ) ] T 2 = -Dxlog detV ( s i n c e dV/dx = @(x)* ( x ) ) ( e x e r c i s e ) . Hence ( a * ) u ( x ) = -2Dx 2 l o g d e t V ( x ) and i n s e r t i n g now a t dependence one o b t a i n s u ( x , t ) = -2Dxlog T d e t V(x,t) where V(x,t) = I + @ ( s ) * ( s , t ) d s f o r * ( z , t ) = ( m h ( i k o)exp 3 J j’ (-8k.t)exp(k.z)). F u r t h e r i t i s e a s i l y checked t h a t t h e e n t r i e s i n V(x,t) J L J L 3 L w i l l be ( m . ( t ) = m . ( i k O ) e x p ( - 8 k . t ) ) (@A) V . . ( x , t ) = 6ij t [m.(t)exp(ki t J J j’ J !J J kj)x/(kitkj)]. I f one t a k e s a two s o l i t o n s i t u a t i o n ( N = 2 ) w i t h poles o f T ( k ) a t i k l and i k 2 ( k 2 > kl) then some r o u t i n e c a l c u l a t i o n ( c f . [Lml]) 3 3 y i e l d s , f o r y1 = klx - 4klt + 61, y2 = k2x - 4 k 2 t t 6 2 y and 6i = ( 1 / 2 ) l o g 1J

[(m, (0)/2ki )(K2-kl

11

)/(k2+kl

2 2 2 2 2 u ( x , t ) = -2(k2-kl )[(k2Csh y2+klCshyl

(8.16)

where Csh = csch, Cth = coth, and Tnh = tanh.

-

2 2 (y 1-A) -2klsech

u

)]

Now one can show ( e x e r c i s e

-

t h a t a t times l o n g b e f o r e o r l o n g a f t e r t h e 2 s o l i t o n s i n t e r a c t

c f . [Lml]) (00)

)/(k2Cthy2-klTnhyl

2k2sech 2 2 (y2+A); u

%

-2k 21 sech2 (y1+A) - 2k2sech 2 2 For kl = 1, k2 = 1.5,

(y2-A) r e s p e c t i v e l y , where A = 210g[(k2+kl)/(k2-k,)].

and 61

= 0 one has f o r example ( c f . [Lml])

REmARK 8.4, We want t o say j u s t a word here about conserved q u a n t i t i e s ( c f . [Cel;Dal;Kul;Lml;Nel;Ol] - more w i l l be s a i d l a t e r ) . To see how these a r i s e

- 6uuX + uxxx = 0 and i n t e g r a t e t o o b t a i n - uxx]dx = 0 (assuming s u i t a b l e b e h a v i o r a t f m ) .

t a k e f i r s t t h e KdV e q u a t i o n ut (06)

Hence

Dt{f

udx =

/f udx

/_fDx[3u2

S i m i l a r l y , m u l t i p l y i n g t h e KdV e q u a t i o n by u and 3 + (1/2)ux]dx 2 = 0 2 Dx[2u - uuxx i n t e g r a t e t o o b t a i n ( a + ) (1/2)Dti_1 u dx =

so

_/fu2 dx

= constant.

= constant.

[Z

Many more conserved q u a n t i t i e s can be d i s c o v e r e d i n

an ad hoc manner b u t t h e r e i s a l s o some meaning t o a l l t h i s and t h e m a t t e r

w i l l be discussed more s y s t e m a t i c a l l y below.

SOLITON THEORY

189

L e t us go n e x t t o t h e i m p o r t a n t paper [Lx5] and e x t r a c t a few b a s i c i d e a s . We work w i t h t h e KdV e q u a t i o n i n t h e form (om) ut

+

uux + uxxx = 0 and ob-

s e r v e t h a t i f v i s a n o t h e r s o l u t i o n t h e n w = u-v s a t i s f i e s ( 6 * ) wt + uwx +

wvx + wxxx = 0. M u l t i p l y by w and i n t e g r a t e by p a r t s o v e r (-a,-), assuming - 2 co w and w -+ 0 s u i t a b l y a t + m , e t c . , t o o b t a i n (6.) Dt(1/2)Lm w dx + la ( v x xx2 (1/2)ux)w dx = 0.

2 Set ( 1 / 2 ) { 1 w dx = E ( t ) and max 2vx-u,

D t E ( t ) 5 m E ( t ) from which by G r o n w a l l ' s lemma one has

= m t o obtain

(60)

E ( t ) 5 E(0)exprnt.

T h i s shows t h a t s o l u t i o n s a r e u n i q u e l y determined by t h e i r i n i t i a l values I n c i d e n t a l l y t h e e x i s t e n c e o f g l o b a l s o l u t i o n s o f (om) and r e l a t e d

u(x,O).

equations has been e s t a b l i s h e d i n v a r i o u s c o n t e x t s ( c f . [ B f l ;Ka6,7;Swl

;Te2;

Tul;Twl])

and we do n o t deal w i t h t h i s here. One notes a l s o t h a t , as b e f o r e 2 i n (*), u ( x , t ) = 3csech [ ( 1 / 2 ) J c ( x - c t ) ] i s a soliton solution o f (0.). L e t us w r i t e (om) i n t h e general form ( 6 6 ) ut = K(u) ( c f . a g a i n

REmARK 8.5, [Lx~]).

Now t h i n k o f a map u

+

Lu = L ( t ) ( L

Lu s e l f a d j o i n t i n some H i l b e r t space H.

'L

D2 + ( 1 / 6 ) u h e r e ) w i t h

event i f L(O)$(O,x) = h ( O ) $ ( O , x ) t h e n $ ( t , x ) = U(t)L(O)$(O,x)

=

Consider t h e requirement t h a t t h e

L ( t ) = Lu s h o u l d be u n i t a r i l y e q u i v a l e n t when u s a t i s f i e s ( 6 6 ) . L(t)U(t)$(O,x)

L

=

U(t)$(O,x)

= U(t)h(O)$(O,x)

I n that

s a t i s f i e s L$ =

= h(O)$(t,x)

(UU* =

u*u

=

I).Thus t h e eigenvalues h ( 0 ) would be i n t e g r a l s o f t h e e q u a t i o n ( 4 6 ) ( X ( t ) v a r i e s ) . NOW g i v e n (a+) U ( t ) - l L ( t ) U ( t )

= A ( O ) = h u ( t ) = c o n s t a n t as u ( t , x )

= L ( 0 ) independent o f t and w r i t i n g B = UtU*

we o b t a i n (6m) Lt = [B,L]

(which i s t h e same as ( 8 . 4 ) when Lt = -ut b u t h e r e Lt (6m) i s c a l l e d a Lax e q u a t i o n and (L,B)

by d i f f e r e n t i a t i n g ( 6 + ) ) . $t = Ut$(O,x)

'L

(1/6)ut).

Equation

a Lax p a i r ( n o t e (6m) i s o b t a i n e d

We n o t e a l s o t h a t w i t h $ = U$(O,x) as above (+*)

= U U*$

= B$ as i n t h e d i s c u s s i o n b e f o r e (8.3). I n the pret 2 s e n t s i t u a t i o n w i t h ( 0 . ) as t h e KdV e q u a t i o n one f i n d s t h a t L = D + ( 1 / 6 ) u 3 and B = 4[D + (1/8)uD + ( 1 / 8 ) u x ] = 4D3 + (1/2)uD + ( 1 / 2 ) u x y i e l d s [B,L] =

K(u) = ( 1 / 6 ) ( u x x x t uux) = ( 1 / 6 ) u t = Lt.

T h i s d i s c u s s i o n g i v e s a somewhat

cleaner version o f t h e h e u r i s t i c s e a r l i e r leading t o (8.4) etc.

We n o t e

a l s o t h a t f o r a f i x e d L w i t h p o t e n t i a l u ( o r u/6) t h e r e w i l l be an i n f i n i t e number o f odd o r d e r Bm such t h a t [B,,L]

= Km(u) i n v o l v e s o n l y u and i t s x

d e r i v a t i v e s ; consequently ut = Km(u) determines a h i g h e r o r d e r KdV t y p e equat i o n such t h a t t h e eigenvalues X(0) o f Lu w i t h p o t e n t i a l u a r e i n t e g r a l s .

REmARK 8.6, = 3csech

2

Again f o l l o w i n g [ L x ~ ] c o n s i d e r s o l i t a r y waves u ( x , t ) = s ( x - c t )

[ c ( x - c t ) / 2 ] which we n o t e w i l l s a t i s f y

(+A)

- c s x + s s x + s x x x = 0.

I t was d i s c o v e r e d by Gardner and Kruskal and d e r i v e d by Lax i n [ L x ~ ] t h a t

t h e wave speeds c o f such waves a r e i n t e g r a l s o f t h e m o t i o n s a t i s f y i n g c . ( u )

J

190

ROBERT CARROLL

= 4h.(u) where h i s an e i g e n v a l u e o f Lu. To see how t h i s goes t h e o r e t i c J j a l l y , f o l l o w i n g [ L x ~ ] , we c o n s i d e r ut = K(u) a g a i n and suppose ( 0 0 ) DEK(u+

E V ) I ~ == V(u)v ~ e x i s t s as a Frechet d e r i v a t i v e f o r example.

Differentiating

= K(u) i n E we o b t a i n then ( 0 6 ) vt = V(u)v where v = DEuEIE=O, uE b e i n g ‘jt a one parameter f a m i l y o f s o l u t i o n s w i t h say i n i t i a l d a t a uE(O,x) = u o ( x ) +

Ef(x).

L e t I ( u ) be an i n t e g r a l o f ut = K(u) and assume i t i s Frechet d i f -

f e r e n t i a b l e w i t h ( 0 0 ) D € I ( u + E v ) = ( G ( u ) , v ) where G

%

gradient.

I f uE i s a

one parameter f a m i l y o f s o l u t i o n s t h e n I ( u E ) i s independent o f t and hence

so i s ( G ( u ) , v ) where ut = K(u) and vt = V(u)v ( u = u ( t , x ) e t c . w i t h a c t i n g i n x,

,

)

symmetric, and D E I ( ~ E ) I E , o

= (G(u),v)

translation invariant.

)

(

,

)

f o r v = DEuE a t

Now assume s ( x - c t ) i s a s o l i t a r y wave s o l u t i o n w i t h ut

= 0). (

,

(

E

K(u) and

Then e v i d e n t l y ( G ( s ( x - c t ) ) , v ( x , t ) )

i s inde-

pendent o f t f o r v as i n d i c a t e d and s e t t i n g v ( x + c t , t ) = w ( x , t ) one o b t a i n s (G(s(x)),w(x,t))

independent o f t.

Consequently

s i n c e wt = cvx + vt = cwx + vt one has

,

(

)

,

G(s(x)),wt)

= 0 and

wt = [cD + V(s)]w ( t h e o p e r a t o r + V(s)]w) = 0 and thus ) i f c o n v e n i e n t ) ([-cD+V*(s)]G(s),w) = 0 (D* = -D). The v a l u e

V w i l l comnute w i t h t r a n s l a t i o n s ) . ((

(+a) (

(m*)

Hence (G(s),[cD

o f w a t any p a r t i c u l a r t i m e (e.g. t = 0 ) i s a r b i t r a r y and one concludes that

(W)

[-cD + V*(s)]G(s)

Next assume t h e equation ut = K(u) i s en-

= 0.

e r g y p r e s e r v i n g i n t h e sense t h a t ( u ( t ) , u ( t ) )

i s independent o f t when ut =

K(u) ( t h i s can be v e r i f i e d f o r KdV f o r example). 2(u,ut) E

(me)

V*(u)u

= 2(u,K(u))

Then ((

,

)

= (

,

)) 0 =

and p u t t i n g i n uE one o b t a i n s a f t e r d i f f e r e n t i a t i n g i n

(v,K(u)) + (u,V(u)v) = 0.

Thus s i n c e v i s a r b i t r a r y ( m b ) K(u) + 0 and f o r 0 = c s x + K ( s ) one has [cD - V*(s)]s = 0. Hence s be-

longs t o t h e n u l l space N o f cD

-

V*(s) so from

(aA)

G(s) =

KS

s i n c e dim N

This shows t h a t under t h e hypotheses i n d i c a t e d f o r I g i v e n w i t h grad I = G then a s o l i t a r y wave s ( x - c t ) s a t i s f i e s G(s) = K S where K = = 1 i s normal.

K(I,c) ( i . e . s i s an e i g e n f u n c t i o n o f G).

I f one a p p l i e s t h i s t o I ( u ) = - L = D 2 + 4 6 ) we

h ( u ) f o r Lw = hw ( i . e . w i s an e i g e n f u n c t i o n f o r X(u) obtain f i r s t

(i

3

D L = ;/6

=

v/6 when v =

DE~EIE,o)

L\;r + vw/6 = A ;

+iw.

Take s c a l a r products w i t h w and i n t e g r a t e t o g e t (wv,w)/6 = i(w,w) ( n o t e (Li,w) = (i,Lw) = A(i,w)). Normalize w so t h a t (w,w) = 1 and then i = ( 1 / 6 ) 2 2 It f o l l o w s t h a t (w ,v) = (G(u),w) which means t h a t gradh(u) = G(u) = w /6. 2 G(s) = w / 6 = K S so t h e e i g e n f u n c t i o n w o f L corresponding t o X(s) i s w = ^ C S ’ / ~ and we t a k e = 1. One then checks e x p l i c i t l y ( u s i n g -cs + s 2/ 2 + sxx =

0 etc. from

(‘A))

t h a t Lw =

f e r t o [ L x ~ ]f o r f u r t h e r discussion.

=

C S ~ / ~so/ t~h a t

c ( s ) = 4X(s).

We r e -

SOLITON THEORY

191

Consider ut + 6uux + uxxx = 0 w i t h M i u r a ' s t r a n s f o r m a t i o n ( m m ) 2 u = - v - v so t h a t (***) ut t 6uux + uxxx = -(Dx + 2 v ) ( v t - 6~ vX + vxXx). X 2 The e q u a t i o n vt - 6v vx + vxxx = 0 i s c a l l e d t h e m o d i f i e d KdV (mKdV) equa-

REmARK 8.7.

2

t i o n and we see t h a t e v e r y s o l u t i o n v o f mKdV i s mapped under (-) 2 s o l u t i o n o f KdV. I f we add t o ( m m ) t h e e q u a t i o n (**A) vt = 6v vx then (mm)

-

(**)

- vxxx

f o r m a Backlund t r a n s f o r m a t i o n (BT) between KdV and mKdV. = 0 and E(v,

Here one d e f i n e s a BT between d i f f e r e n t i a l equations D(u,x,t) = 0 as a s e t o f r e l a t i o n s i n v o l v i n g ( x , t , u ( x , t ) )

Y,T)

to a

and (Y,T,v(Y,T))

such

t h a t BT i s i n t e g r a b l e f o r v i f and o n l y i f D(u) = 0 and BT i s i n t e g r a b l e f o r

u i f and o n l y i f E ( v ) = 0 w h i l e g i v e n u ( r e s p . v ) such t h a t D(u) = 0 ( r e s p . E(v) = 0 ) BT d e f i n e s v (resp. u ) t o w i t h i n a f i n i t e s e t o f c o n s t a n t s and Such t r a n s f o r m a t i o n s w i l l be discussed l a t e r i n

E(v) = 0 (resp. D(u) = 0). more d e t a i l ( c f . §11).

E M I P L E 8.8.

Dx(u+v)/2 = a S i n [ ( u - v ) / 2 ]

and Dt(u-v)/2

= (l/a)Sin[(u+v)/2]

i s a BT t r a n s f o r m i n g t h e sine-Gordon e q u a t i o n qxt = Simp i n t o i t s e l f . The main p o i n t h e r e i s t h a t a s c a t t e r i n g problem as i n (9.1)

REmARK 8.9,

(9.2) ( i n 52.9 t o f o l l o w ) , namely, v vlt

= Avl

f o r A,B,C

+ Bv2 and v~~

= Cvl

t icvl

!x

= qv2,

v2x

-

-

i r v 2 = rvl w i t h

- Av2 l e a d s t o c o m p a t a b i l i t y e q u a t i o n s (9.3)

and an e v o l u t i o n e q u a t i o n D(u) = 0 f o r u = (q,r),given

nomial d i s p e r s i o n r e l a t i o n s f o r t h e l i n e a r i z e d problem.

say p o l y -

The e q u a t i o n s (9.1)

- (9.2) t h e n s e r v e as a BT between D(u)

= 0 and some E(v,&)

5 ) = 0 i s a p a i r o f PDE f o r v = (vl,v2)

n o t i n v o l v i n g u (see [ A o l ] f o r d i s -

cussion).

As an example o f how t h i s s i t u a t i o n l e a d s t o new i n f o r m a t i o n con-

s i d e r ( f r o m [Lml]) two S t u r m - L i o u v i l l e o p e r a t o r s (**.) ,y, and wxx =

= 0 where E(v,

(A +

J,(x,t)w

+ q(x,t)y w = A(x,t,x)y

= (1

i n which w and y a r e r e l a t e d v i a ( * 6 )

Given X~ = 0 we can be t a l k i n g about two s o l u t i o n s q , J , o f t h e KdV + y., equation. T h i s s i t u a t i o n i s a l s o r e l a t e d t o t h e Darboux t r a n s f o r m a t i o n d i s cussed i n [Lml] f o r example. c i e n t s t o o b t a i n Axx + (94)

qx

+

Now p u t ( * 6 ) A(q-J,)

and i n t e g r a t i n g one has A

-yx/F one

has

yxx = (y

2

i n t o (*a)

= 0 and 2Ax

- Ax

+

q

- q = T(t)

-

and equate c o e f f i J,

Eliminating

and l i n e a r i z i n g v i a A =

T ( t ) = 7 = c o n s t a n t we e q u a t i o n f o r x = ?. F u r t h e r

t q ) y " so s e t t i n g

i s a particular solution o f the y

= 0.

see t h a t

o f Darboux t r a n s f o r m a t i o n s ( o r t h e Crum t r a n s f o r m a t i o n ) we have 2 ( l o g ? ) " so t h e w e q u a t i o n becomes wxx = (A + q

-

and

-

T h i s i s a way o f i n -

t r o d u c i n g p o t e n t i a l changes which i s o f t e n p r o d u c t i v e .

z;

J, = q

E ( l o g 7 ) " ) w which i n v o l v e s

a p o t e n t i a l change f r o m t h e y e q u a t i o n o f -2(log;)". we i n t r o d u c e p o t e n t i a l f u n c t i o n s v i a q =

i n the s p i r i t

J,

I n t h e s p i r i t o f BT

z x so A = ( z - z ' ) / 2 i s

192

ROBERT CARROLL

a p a r t i c u l a r s o l u t i o n o f 2Ax

+v

-

J/ = 0 and A

2

-

Ax

- IP

h,

= X = -m/2 becomes

v = z X' (**+) p t p ' = m +(z - z ' ) / 2 . Now suppose 2 so t h a t (*.) zt - 3(Zx) + zxxx = 0 w i t h s u i t a b l e n o r m a l i z a t i o n (and s i m i l a r l y f o r z ' ) . Then d i f f e r e n t i a t i n g (**+) i n t and i n t e g r a t i n g i n x we o b t a i n (*A*) zt + z; = r ( z - z ' ) ( z t - z i ) d x ( a g a i n w i t h and s e t t i n g u = ztz', v = z - z ' one o b t a i n s some n o r m a l i z a t i o n ) . Using (*.) 2 zt + z; = 1 [ ( 3 / 2 ) ( v )xux - vvxxx]dx from which, a f t e r some c a l c u l a t i o n f o r p = ICI = zx and p ' =

Jlt

-

6ICICIX + ICIxxx

= 0

( n o t e h e r e from (*+)

z x x + z k X = uxx = vvx), 2 3/2)uX

The l a s t e q u a t i o n i n solution

-

2 2[P2 + PP' + ( P ' ) 2 1 vvXx + ( 1 / 2 ) v X = { - ( z - z ' ) ( z x x - ZAX)

8.18) p l u s (**+) i s a BT f o r t h e KdV e q u a t i o n .

-

Ifa

s known t h e n a n o t h e r s o l u t i o n z may be o b t a i n e d by

z' t o (*.)

s o l v i n g t h e BT which we r e w r i t e here as (8.19)

( A ) zt +

(B) p + p ' = zx

+ z;(

2

Note Dx(B) + (A) i m p l i e s Q ( z )

(**.) and Dx(A)

-

-

= 2[p 2 + pp' + ( p ' ) 2 ] = m t (1/2)(z

+ Q(z')

-

(Z-Z')(Z,~-Z'~~); 2 ' )

2

= 0 where Q ( z ) = zt

Dt(B) i m p l i e s ( z - z ' ) [ Q ( z )

-

Q ( z ' ) ] = 0.

i m p l i e s Q ( z ) = Q ( z ' ) = 0 so J/ and p s a t i s f y KdV.

-

2

3zx + zxxx i s Hence (A) + (B)

As an example o f how t o 2 2 and zt = 2p - zzxx

use t h e BT t h e o r y t a k e z ' = 0 t o f i n d z x = m + ( 1 / 2 ) z = 2mzx ( z x x = zp).

where m = -2k

2

.

T h i s leads t o

Then zx =

9. S0CI.t!0Q)NsV I A A W

q

SgXEW.

(*AA)

z = -2kTanh(kx-4k3t) f o r I z I < 2k 2 s o l i t o n f o r KdV.

i s t h e s t a n d a r d sech

We w i l l g i v e now some d i s c u s s i o n o f c e r t a i n

i m p o r t a n t f e a t u r e s o f s o l i t o n theory. i n [Aol;Cjl ;Ddl ;Fa3;N11,2;Nvl].

We f o l l o w standard source m a t e r i a l

We w i l l be p a r t l y h i s t o r i c a l i n t h e o r d e r

o f s e l e c t i n g m a t e r i a l and p a r t l y personal i n t h e e x p l i c i t choices; a l s o we

w i l l n o t be a b l e t o cover a l l o f t h e r e c e n t m a t e r i a l i n t h i s r a p i d l y moving f i e l d ( c f . however §2.11). L e t us b e g i n w i t h t h e AKNS approach ( A b l o w i t z , Kaup, Newel 1, Segur) which g e n e r a l i z e s a c o n t e x t o f Zakharov-Shabat (Z-S). F i r s t r e c a l l t h e Lax e q u a t i o n ( 6 . ) i n § 4 (Lt = [B,L]) a r i s i n g from "compatiFor L and B s u i t a b l y chosen

b i l i t y " o f qt = BJ/ and LJ/ = AJ/ when A t = 0.

(4.) i s t h e KdV e q u a t i o n f o r example b u t one must e.g. guess L and f i n d B i n o r d e r t o have t h e c o m p a t i b i l i t y be a meaningful e v o l u t i o n e q u a t i o n . We

i n d i c a t e t h e AKNS procedure now f o r t h e g e n e r a l i z e d Z-S system (9.1)

vlx

t i r v l = qv2; v2x

-

i s v 2 = rvl

193

AKNS SYSTEMS and c o n s i d e r an e v o l u t i o n o f t h e f o r m v

(9.2)

It

= Avl + Bv2; v~~ = Cvl + Dv2 = Cvl

-

Av2

One assumes A,B,C

( D = -A i n v o l v e s no l o s s i n g e n e r a l i t y ) .

a r e s c a l a r func-

t i o n s independent o f v and we n o t e t h a t r = -1 i n (9.1) y i e l d s v2xx + q ) v 2 = 0.

A

(9.3)

-

C o m p a t i b i l i t y o f (9.1) = qC-rB;

X

B +2ir,B = q -2Aq; X t

C -2icC = rt+2Ar X

and these equations can be s o l v e d f o r A,B,C t i o n equation) i s s a t i s f i e d . mention e.g.

(c2+

(9.2) r e q u i r e s ( e x e r c i s e )

i f another e q u a t i o n ( t h e e v o l u -

There a r e v a r i o u s approaches t o t h i s and we

t h e expansion o f A,B,C

i n t r u n c a t e d power s e r i e s i n

c.

Thus

2 2 EX:IURPI;E 9-1- L e t A = A. +
= a2qr/2 t a.

(and one s e t s a" = 0 ) .

I n p a r t i c u l a r t h e non-

l i n e a r Schrodinger e q u a t i o n (NLS) a r i s e s h e r e i f r = +q* (q* = 2 a2 = 2 i ; thus i q t = q x x f 2q q*. Other examples a r e g i v e n i n [Aol].

Ti)

and A2 =

L e t us w r i t e down t h e g e n e r a l i z e d Z-S

e i g e n f u n c t i o n s now and l o o k a t some o f t h e i r p r o p e r t i e s ( c f . [Aol;N12]). One s e t s r, = ( t i n i n (9.1) and determines e i g e n f u n c t i o n s by t h e p r o p e r t i e s

The Wronskian of two v e c t o r s u,v i s W(u,v) = u v - u v and e v i d e n t l y W(P, A Al 2 2 1, A $ ) = -1 w i t h W($,$) = -1. W r i t e ( A ) IP = a ( < ) $ + b(r,)$; IP = -g(c)$ + b(c.1;

!?).

One has a$ t b i = 1 and f o r s u i t a b l e r,q we can w r i t e b -a from ( 9 . 1 ) e.g. ( 0 ) ql(x,r,)exp(icx) = 1 + lt d y L i d z q ( y ) r ( z ) e x p [ 2 i r , ( ~ - z ) l

and s e t S =

(?i

= lt e x p E i r, (x-Y ) I r ( y )gl (y, r, )exp( icy )dy. 1 Corresponding formulas h o l d f o r t h e o t h e r e i g e n f u n c t i o n s . I f r , q E L one

v1 ( z , r ,)exp( ir,z ) ; v 2 (z, r, )exp( ir,x)

has e x p ( i c x ) q and exp(-ir,x)$ a n a l y t i c f o r Imr, > 0 w h i l e exp(-ir,x)?

and exp

A

( i c x ) $ a r e a n a l y t i c f o r Imr, < 0.

Irl

I t i s convenient t o assume more, e.g.

and I q l 5 c e x p ( - 2 k l x l ) i n which case e x p ( i r , x ) v , exp(-icx)!b,

and a ( < ) a r e

n

a n a l y t i c f o r r- > -k, exp(-ir,x)?, w h i l e b and

2

e x p ( i s x ) $ , and $ a r e a n a l y t i c f o r n < k,

a r e a n a l y t i c f o r - k < n < k.

Asymptotic a n a l y s i s as I s 1

-+

m

ROBERT CARROLL

194

now g i v e s formulas o f t h e t y p e v 2 e x p ( i s x ) = - r ( x ) / 2 i s i= O ( l / l c l

2

), vlexp

2

- 1/2isl: r q d y i= O ( l / l s l ), e t c . Bound s t a t e s can o c c u r when a ( < ) Ims > 0 o r $ ( c ) 0 f o r I m c < 0. A t a p o i n t where a(<,) = 0 one A A " = ck$ and a t 2k where = 0, Ip = ck$. For s u f f i c i e n t l y n i c e r,q

(isx) = 1 = 0 for

has

v

( r a p i d l y decreasing as above) one has a l s o then k;

= b ( s k ) and

.;( =

A

A

b(sk)

and moreover a ( s ) f o r example has o n l y a f i n i t e number o f zeros f o r I m s > 0. However one notes t h a t zeros o f a ( c ) a r e n o t n e c e s s a r i l y s i m p l e o r on t h e imaginary a x i s and t h e y may o c c u r on t h e r e a l l i n e (we w i l l t r y t o a v o i d this last possibility).

EXAMPLE 9.2. $1(x,c)

I f r = f q * one has e.g. $,(x,r,)

= W;(x,t),

$,(x,c)

= -vT(x,?),

'E(t),Ck

$2(x,s) = * $ ~ ( x , ~ ) ,a^(<) a(;), ;(<) = A When r = f q on t h e o t h e r hand Q1(x,s) = IL2(x,-c),

= $;(x,c),

zk = Fck, e t c .

A

4

=
*!+(x,-s), vl(x,c) = q 2 ( x , - 5 ) , v 2 ( x , c ) = -vl(x,-s), = a(-<), r\ = rck. In particular i f q i s real with r Tb(-c), sk = -sk, and ik

6(s)

=

+q a l l

o f t h e above c o n d i t i o n s h o l d and eigenvalues sk a r e e i t h e r i m a g i n a r y o r paired

(-Ck i s

C o n d i t i o n s f o r no bound s t a t e s

an e i g e n v a l u e a l o n g w i t h 5,).

a r e a l s o i n d i c a t e d i n [Aol]. a r e e n t i r e , which w i l l h o l d i f r,q * 0 f a s t e r than any

Now assume a,$,b,^b e x p o n e n t i a l as x

?-

+

( o r i f t h e y have compact s u p p o r t ) , i n o r d e r t o have a

s i m p l e d e r i v a t i o n o f t h e f o l l o w i n g r e s u l t s ( c f . [Aol]). ideas again one can w r i t e e.g.

(9.5)

IL

= (:)ei5'

(Imc > 0 and K

(K1) etc.)

Q

KO

Im exp(iss)C(Dx-Ds)Kl(x,s)

rYim

(cf. also §11)

K(x,s)eicsds;

t

By Paley-Wiener

$

= (:)e-"'

t

q i?(x,s)e-icsds

Now p u t (9.5) i n t o (9.1) t o o b t a i n e.g.

- q(x)K2(x,s)lds

/xm

-

h ( x ) i= 2K1 ( x , x ) I e x p ( i c x )

sK1(x,s)exp(iss) = 0; e x ~ ( i c s ) [ ( D ~ + D ~ ) K ~ ( x -, s r) ( x ) K 1 ( x , s ) l d s K2(x,s)exp(iss) = 0. It f o l l o w s t h a t ( c f . § § 2 . 3 - 2 . 4 )

(9.6)

-

(Dx-Ds)Kl(x,s)

w i t h (+) K1(x,x)

q(x)K2(x,s) = 0; (Dx+Ds)K2(x,s)

= - ( 1 / 2 ) q ( x ) w h i l e K1(x,s)

(6)

-

-

i=

1i m s-tm

r(x)Kl(x,S)

* 0 and K2(x,s) * 0 as s *

= 0 m.

Conversely, s o l v i n g t h e Goursat t y p e problem (9.6) w i t h (+) w i l l produce t h e h

r e q u i r e d K1, -(1/2)r(x)

K2 f o r ( 9 . 5 ) .

h

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 K and $(x,x)

( c f . [Aol]).

Next c o n s i d e r v / a =

(9.5) t o

obtain

(9.7)

v / a = (,)e1

$

t (b/a)IL extended i n t o t h e upper h a l f p l a n e and use

-isx t

q i?(x,s)e-issds

t (b/a)[(:)eicx

t

K(x,s)eicsds]

=

AKNS SYSTEMS L e t C be a contour i n I m s > 0 from - 4 0

+

195

to

m+iO

+

passing over a l l zeros

o f a ( < ) and operate on (9.7) w i t h (1/2*)JC e x p ( i s y ) d s t o g e t f o r y > x

+ :\

0 = ^K(x,y) + (Y)F(x+y)

(9.8)

K(x,s)F(s+y)ds

where ( m ) F ( x ) = ( 1 / 2 s ) I C ( b / a ) ( s ) e x p ( i s x ) d s , S(x) = ( 1 / 2 s ) i c exp(isx)ds, A

and ( 1 / 2 s ) i c [ 9 ( x , s ) / a ( 5 ) ] e x p ( i s y ) d ~ = 0 f o r y

(c"

( 1 / 2 s ) / t (^b/$)(s)exp(-isx)ds has f o r y

>

(9.9)

K(x,y)

RrmARK

9.3.

(along w i t h (9.10)

>

x.

S i m i l a r l y f o r F(x) =

a contour passing below t h e zeros o f

a")

one

x

-

1 * (o)F(x+y)

-

fXm

t?(x,s);(s+y)ds

= 0

I f a ( s ) # 0 f o r 5 r e a l and a ( < ) has i s o l a t e d simple zeros

i ) then F ( x ) = (1/2n)lf

(b/a)(c)eiSxdc

i ( x ) = (1/2a)\-

--

- ilN 1 c.ei5jx; J

(6/$)(c)e-icXdc

+ i l N C.e 1

- i s .x J

J

$(2j)/$'(tj).

and C j = I n t h e event o f slower deJ A cay o f q,r (9.10) s t i l l holds b u t t h e n o r m a l i z i n g constants c and c . a r e J J found from 7 = and c = $./a! etc. Rigorous r e s u l t s o f existence and J j J J uniqueness a r e known f o r t h e G-L-M t y p e equations (9.8) - (9.9) i n many i m where c

J

= b(s.)/a'(sj)

Cjqj

portant situations.

REmARK

9.4,

Now go back t o (9.1)

-

(9.2) and one knows from s i t u a t i o n s as

-

i n Example 9.1 t h a t t h e r e w i l l be s o l u t i o n s t o t h e c o m p a t i b i l i t y equations

B(s) *

(9.3) w i t h A ( < ) * A _ ( < ) ,

0, and C ( s )

-+

0 as 1x1 *

(we mention here

o n l y the s i t u a t i o n r = Tq* w i t h a2 = 2 i again from Example 9.1, where A = 2 2 i s i iqq*, B =-2qs i q andC=f2q*c T iq; so, w i t h q -+ 0 and qx -+ 0 a t 2x' +, A- = lim A ( < ) = 2 i s ). Set then (withv,$,a,b, e t c . time dependent) w(**) v t = v e x p ( A - t ) , $ t = $exp(-A-t), v t = vexp(-A-t), and $ t = $exp(A-t)

-

and p u t t h i s i n (9.2) t o o b t a i n e.g. (9.11)

DF =

-A-A

1

-

1 (note v 2, (,)exp(-isx) cannot s a t i s f y (9.2) so v t i s introduced). From v = 4 a $ + b $ one obtains then from (9.11) as x * m (*A) atexp(-i5x) = 0 and bt e x p ( i s x ) = -2A ( s ) b e x p ( i s x ) from which (*a) b ( s , t ) = b(s,O)exp[-2A-(c)t] and a ( s , t ) = a(s,O) time).

( t h e l a s t equation showing t h a t t h e sk are f i x e d i n

S i m i l a r l y from c

J

= $./a!

J

J

one has c . ( t ) = c exp(-2A-(sj)t, J J,O

A

cj(t) =

196

ROBERT CARROLL

A

c 3. ( t ) = ~ j y 0 e x p ( 2 A - ( c j ) t ) , nb(c,t) ;(s,O)exp(ZA-(s)t) and a*(s,t) = :(s.O). It f o l l o w s as b e f o r e ( c f . 98) t h a t (*6) F ( x , t ) = ( 1 / 2 1 ~ [Z(b/a)(t,O)exp[icx 2 A - ( t ) t ] d t - N ~ ~ , ~ e x p [ i c ~ x - ' 2 A.)t] - ( i ( w i t h a s i m i l a r expression f o r ?) and J t h i s g i v e s us some connection w i t h t h e r e s u l t s o f § 8 about t h e t i m e v a r i a -

c1

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

so K1(x,y)

PI,

Take now r = Tq* f o r i l l u s t r a t i o n and x

(i= +F*)

+F*(x+y)

and q = -2Kl(x,x)

+

becomes q ( x , t )

-

%

i n (9.9)

-(1/21~)

( t h e c . c o n t r i b u t i o n w i l l be small as x J 2 Then t h e problem tends t o a l i n e a r problem as x -+ m ( i q t = qxx + 2q q* 2 i q t - qxx w i t h s o l u t i o n s q = ( 1 / 2 1 ~ ) jaI ( k ) e x p i [ k x - ~ ( k ) t ] d k where w = -k

jm(b/a)-(e,O)exp[-2ic~-2A:(c)t]dc

-m -+

-+

m).

-

i s the dispersion r e l a t i o n

here A- = 2 i s

2

so

~ ( c =)

2iA-(-c/2)).

Next we broach t h e f a s c i n a t i n g s u b j e c t o f squared e i g e n f u n c t i o n s ( o r p r o i n ( 9 . 1 ) f o r v and m u l t i p l i e s t h e 2 equations b y q1 and q 2 r e s p e c t i v e l y t h e r e r e s u l t s (*+) (ql)x + 2icr: = 2 2 2 2 ( I ~ ~ =P w~2 ) + ~w l . Thus t h e squared e i g 29p1q2; ( P ~ -) 2~ i w 2 = e n f u n c t i o n s s a t i s f y a homogeneous f o r m o f (9.3) (*.) A x = qC - r B , B + A2 i 2 2icB = -2qA, and Cx - 2isC = 2Ar. S i m i l a r a n a l y s i s a p p l i e s t o rpl, q2, etc.

ducts o f e i g e n f u n c t i o n s ) .

I f one uses

and i n f a c t 3 s o l u t i o n s o f

(*B)

q

are

By v a r i a t i o n o f parameters one c o u l d t h e n s o l v e (9.3) (where qt and r t a r e t h e inhomogeneous terms) and i t i s n a t u r a l t o t a k e as boundary c o n d i t i o n s on t h e c o n d i t i o n s before, namely A

A,B,C

requirement a t x =

fm

-+

A - ( c ) w i t h B,C

This + q t 2 , and a3 = ('l?') qzq2 -19 (r)@l 0 = rql (9.2) as (D = ( ), Q =

2 Az - ( ~ )q( ~ ) ] @ ~ d x0 f o r @1 = qq2 f o r example). To see

( AC -A'), N = (O q)) r O t i n g ct = 0 g i l e s

p*)Nt

verse P-'

'l)

t PSXP-'

1

= (-q2

-

(AA)

-+

0 as 1x1

lz

c o n d i t i o n s (:*)

requires

m.

[(r ) -+

,

vx = i r D v t Nv; vt = Qv. = Qx t ic,[Q,D]

D i f f e r e n t i a t i 2 g and s e t -

+ [Q,N].

Set P =

('l f l ) q 2 Po2

with in-

and d e f i n e S such t h a t Q = PSP-' so t h a t Qx = PXSP-l

Vpzl -VL] PSP- PxP .

One f i n d s then from (")

-

(Am)

that

(Ad)

S =

S(-m)

t ' I- m P- NtPdx w i t h S ( - m ) = A - ( s ) ( b -:). T h i s l e a d s t o Q and hence A,B,C b u t here we o n l y look f o r c o n d i t i o n s needed f o r a s o l u t i o n t o e x i s t . Thus r e A h aexp( - i S x ) ) bexp(-iSx)) as C a l l Ip = a? + b$, $ = -$$ + bJ/ w i t h Lp % (bexp(irx) and (-aexp(iTx) T h i s leads immediately t o x -+

'

-.

a; (9.13)

2ab

Evaluating

2 8

b6

S(m) = (A&)

at

00

-(as

- bi)

and u s i n g (9.13) g i v e s

Q

AKNS SYSTEMS

-

(*+), and d e f i n i t i o n o f t h e s c a t t e r i n g data, one has a l 2 m 42 2 ) dx = [qp, + wl]dx = ab; -ab = lm (v1v2)xdj: = La ( W 2

However from (9.1), SO (A+)

197

rf

r,” (v 1 2 x , ~p

A

A

+ r$:)dx; and -a$+bb+l = L I [‘P,;, + v2GlIxdx = :2: [qp2Z2 + w l v l l d x . Def i n e ai as above ( a f t e r (A*)) and t h e n p u t t i n g (A+) i n (9.14) one a r r i v e s a t f o r i = 1,2,3

(A*)

( e x e r c se

q u i r e d f o r a s o l u t i o n of

09

ted =

A,B,C

at

* l t l ) and ( w 2

(9.15)

-

c f . [Aol]).

Thus

(A*)

i s the condition re-

9.3) t o e x i s t w i t h t h e boundary c o n d j t i o n s i n d i c a -

S i m i a r l y we c o u l d use q1 = (*?

q2 =

*5)’

?m.

(*35)’

and *3

show t h a t [(-;It

+

2A-(s)(;)Pidx

= 0

f o r i = 1,2,3. Next we use 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 s equations as f o l l o w s .

i:

From (*+)

v,v2

=

t o determine t h e e v o l u t i o n

(A*)

/,” [ q p 22 + w 21]dx

and p u t t i n g I - f =

f ( y ) d y w i t h I+f = Ix fm ( y ) d y we can w r i t e t h e r e m a i n i n g 2 e q u a t i o n s i n

(*+) as ( i = 1,2; (9.16)

( I f)g = If g )

t f = (1/2i)

Lei =

i’

[

-Dx+2q1-r - 2 r I - r D x2q1-q -2rI-q

I

I n t h e s p i r i t o f a n a l y t i c f u n c t i o n a l c a l c u l u s i f A - ( s ) i s a n a l y t i c one w r i t e s A-(s)al

Im

-m

from (9.16) and

becomes ( A m )

(A*)

It i s e a s i l y seen now t h a t

(9.17)

T 1: = ( 1 / 2 i )

[ Dx+2rI+q 2qI+q

(

u,aI-Bv)

[(-i)t*i

+ 2(i)A-ai1

/,”

dxu(x)a(x)/, B(y)v(y)dy T o f C is dyB(y)v(y)/ym d x a ( x ) u ( x ) = ( ~ I + a u , v ) and t h e a d j o i n t C

dx = 0. =

A-(C)ei

=

- 2 r I+r -Dx-2qI+r

=

I

T r 2A-(C )(q)l@idx = 0 SO T r t h a t a s u f f i c i e n t c o n d i t i o n f o r s o l v a b i l i t y i s (.A) ) + 2A-(C ) ( ) = 0. q t q One can show t h e c o n d i t i o n (*A) i s a l s o necessary under s u i t a b l e hypotheses. Then

(A*)

i n t h e f or m

(A=)

becomes ( a * )

[(-;It+

(r

-

(9.3) The n a t u r a l e v o l u t i o n equations assoc a t e d w i t h (9.1) say A _ ( < ) e n t i r e , i s and boundary c o n d i t i o n s A + A_, B,C 0 a t +my w i t h

WE0REIII 9.5.

-+

given v i a

R!ZJIARK 9.6.

(.A).

Consider (9.1) a g a i n w i t h s o l u t i o n ‘P =

one has v l e x p ( i s x )

a n a l y t i c and i t approaches 1 as

(Ti)where f o r rl

-+

-.

Ims > 0

The f u n c t i o n

198

ROBERT CARROLL

a ( < ) = W(v,lL) = l i m v l e x p ( i g x ) dependent o f t ( c f . (*A),

v,

= exp(-isxtv),

as x

etc.).

-

t o obtain a R i c c a t i equation f o r vx = u

v = v(x,t),

n

a l s o has these p r o p e r t i e s and i s i n -

+ m

Now e l i m i n a t e v 2 from (9.1) and s u b s t i t u t e (0.)

+ q ( p / q l x . Since v + o as 1c.1 + (Ims > 0 ) ( 0 6 ) u = ( 2 i c ) - ' c ) ~ i] s a permissable expansion ( e x e r c i s e ) and p u t t i n g t h i s n-1 -qr, u1 -wX, Pntl = q ( u n / q I x + uk-$!-k,.-l one can w r i t e now l o g a ( s ) = v(m) = _:I udx and hence (om )

(*+I -m

lo

p0 =

m

Lm pndx.

C , , / ( 2 i ~ ) ~ ~ + ' where )] Cn =

Since l o g a ( s ) i s indepen-

dent o f t f o r a l l c w i t h Imc > 0 t h e Cn a r e t i m e independent and these y i e l d an i n f i n i t e number o f c o n s e r v a t i o n laws. (9.18)

Co =

(-qr)dx; C3 =

I

C,

=

For example

1 (-qrx)dx;

C2 =

1 [-qrxx

2 + ( q r ) Idx;

2 2 [-qrxxx + 4q rrx + r q q x l d x

The d e r i v a t i o n does n o t use (9.2) so such i n t e g r a l s a r e c o n s t a n t s o f motion f o r any o f t h e e v o l u t i o n equations based on (9.1) o b t a i n e d v i a a r b i t r a r y We r e f e r t o [ A l o ] f o r s p e c i a l r e s u l t s r e l a t i v e t o KdV, NLS, e t c .

(9.2).

We go n e x t t o t h e q u e s t i o n o f H a m i l t o n i a n s t r u c t u r e ( c f . §§1.7, 1 .a, 3.8 f o r background and see [Aol ;N11,2;Nvl;Fa3] f o r s o l i t o n r e l a t e d t h e o r y ) . We r e c a l l f o r example i n a v i b r a t i n g s t r i n g problem one would c o n s i d e r k i n e t i c 2 2 energy T = (1/2) f mutdx and p o t e n t i a l energy V = (1/2) f cuxdx w i t h L = T-V t 2 and p = mut so T (1/2) f ( p /m)dx. The a c t i o n i n t e g r a l i s A = Itf;Ldxdt and t h e standard v a r i a t i o n a l argument m i n i m i z i n g A g i v e s 0 Dt(mut)]

d x d t which y i e l d s a wave t y p e equation f o r u.

o f [Aol] f o r example one w r i t e s f o r H(p,q,t)

f f [Ox(cux)

-

Now i n t h e n o t a t i o n

= f h(p(x,t,a),q(x,t,B)dx,

aH/

f (dH/6p)(ap /aa)dx so t h a t i f h i s a f u n c t i o n o f p and d e r i v a t i v e s pn

aa =

= D p:

t h e n (6*) (6H/6p) =

1;

(-1)'D;(ah/apn),

This involves various i n t e -

g r a t i o n s by p a r t s and s u i t a b l e " t e s t " f u n c t i o n s o f course so g e n e r a l l y we

w i l l j u s t t h i n k o f 6H/Gp f o r example as t h e Frechet d e r i v a t i v e ( c f . 53.2). Thus r e c a l l f o r i l l u s t r a t i o n D'vdx

=

(

6H/6pPv> = DEH(p+w,q,t)lE=O = f as r e q u i r e d i n (&*).

The Hamilton equations

a p / a t = 6H/6p and a p / a t = -6H/sq become then f o r H = (1/2) f ( p

4=

p/m and

EUAIRPLE 9.7. = 0 and irt m

f~ =

1 (ah/apn) 2/m + cqx)dx, 2

-Dx(cqx) (as r e q u i r e d ) .

Consider a2 = - 2 i i n (*) which becomes then i q t + qxx

-

rxx2+ 2qr2 = 0.

-ilm [qxpx + (qp) I d x .

Then

Set q = q ( x , t )

4 = 6H/6p

and

-

2 2q r

and p = r ( x , t ) w i t h H = = -6H/Gq.

Now Poisson brackets were d e f i n e d i n 51.7 b u t we do t h i s here from a some-

AKNS SYSTEMS

3.8).

what d i f f e r e n t p o i n t o f view ( c f . a l s o v a r i a b l e s (p,q)

-f

(P,Q)

and d e f i n e (A,B)

=

199

Thus one t h i n k s o f changing

/I [(GA/Gq)(SB/Sp)

-

(sA/sp)(sB/

sq)]dx.

The t r a n s f o r m a t i o n (p,q) + (P,Q) i s c a l l e d c a n o n i c a l i f ( Q ( x ) , Q ( y ) ) We t h i n k now o f = 0, and ( Q ( x ) , P ( y ) ) = s ( x - y ) ( c f . §1.7). T r equations (*A), namely ( r, + 2A ( t ) ( ) = 0, i n which i t t u r n s o u t t h a t q -q t ( q , r ) p l a y t h e r o l e o f c o n j u g a t e v a r i a b l e s (q,p), t h e map ( q , r ) -+ (Q,P) i s = 0, ( P ( x ) , P ( y ) )

c a n o n i c a l , P and Q a r e o f a c t i o n - a n g l e t y p e (H = H(P)) so aP/at = 0, and

aQ/at

= sH/sP = c o n s t a n t , and t h e i n f i n i t e s e t o f c o n s e r v a t i o n laws f o l l o w

from t h e a c t i o n - a n g l e v a r i a b l e r e p r e s e n t a t i o n ( P and Q t o be determined).

In o r d e r t o show a l l t h i s we remark f i r s t t h a t A - ( s ) i n

(*A)

can be w r i t t e n

i n terms o f t h e d i s p e r s i o n r e l a t i o n s f o r t h e l i n e a r problem. Indeed f o r x T D + m I+ -+ 0 and I: 'L ( 1 / 2 i ) ( 0 x -D:). Then as x m (*A) becomes ( 6 0 ) rt + -f

2A ( D x / 2 i ) r = 0 and -qt + 2A-(-Dx/2i)q t i o n s one o b t a i n s q = exp[i(kx-w

q

= 0.

S o l v i n g f o r p l a n e wave s o l u -

( k ) t ) ] and r = e x p ( i ( k x - w r ( k ) t ) ]

so t h a t

( v i a F o u r i e r t r a n s f o r m i n ( 6 . ) ) ( 6 6 ) A - ( s ) = .(1/2i)w ( - 2 5 ) = - ( 1 / 2 i ) w r ( 2 5 ) . q Now f o l l o w i n g [ A o l ] we show L e t A- be e n t i r e o f t h e f o r m A - ( s ) (1/2i)r: (-2c)'an w i t h an EHEBREIII 9.8. r e a l . Then t h e system (0.) i s H a m i l t o n i a n w i t h c o n j u g a t e v a r i a b l e s q and r and H(q,r) = aninCn(q,r) where Cn i s d e f i n e d by (am).

icz

Phaob:

From

(0)

= 1

r e w r i t t e n a l i t t l e one has ( b e ) q l ( x , s ) e x p ( i s x )

+ :/

q

~ ~ ( ~ , c ) e x p ( i s y ) dfrom y which (6m) 6v1(x,s)/6q(y) = e ( x - ~ k ~ ( ~ , s ) e x ~ [ i s ( ~ - x ) l where e i s t h e Heavyside f u n c t i o n . Now f o r any p i e c e w i s e d i f f e r e n t i a b l e A one can d e f i n e (**) ( 6 A / 6 q ) ( x ) = l i m ( & A ( x ) / s q ( y ) ) as y = q2(x,s)

sql(x,c)/sq(x)

and s i m i l a r l y 6 p 2 ( x , c ) / 6 r ( x )

6ql/&r = sq2/sq.

For t h e e i g e n f u n c t i o n s $:Jl,2

= 6$1,2/6q

/6r

( n o t e $1,2

t

x so i n p a r t i c u l a r

= vl(x,s)

while 0 =

one has i n t h e same manner 0

i n v o l v e s Ix

and use (**)).

Hence (*A) 192 s a ( s ) / s q ( x ) = ( 6 / 6 4 ) [ ~ 7 ~ -9 ~92$11 q2G2 and s a ( c ) / & r ( x ) = -q1JI1 where everyt h i n g i s d e f i n e d f o r Imc > 0. Now f o r r e a l 5 from (9.1), as i n (**), one

has

(**I

= 6J,

(

~

~ + 42i591$l ~ ) = ~~ ( I P ~+J v2J11); , ~ (92J/2)x -

+ v2J,l)x = 2qv7 2 J, 2 + 2w 1J,1 from which vlJ12 + v2G2 + r ~ ~ $ ~ ]( ndo yt e a = ql$2 a t m ) . I t f o l l o w s t h a t q2JI1);

and t h i s extends t o t h e upper h a l f plane.

Now l e t I s 1

r ( v JI

=

+

P

m,

~ =*

Imr

a~ -

> 0,

+

1 2m

24

[q

t o get

200

ROBERT CARROLL

From t h i s and ( W ) we o b t a n (9.21)

grad

qYr

loga =

A l s o we r e c a l l l o g a ( s ) =

1C

/(2ir,)"' from (am) w h i l e t h e e v o l u t i o n equa'T r o r ( r, + 2A-(C ) ( ) = 0 where A (5) (= ( 1 / 2 i ) ~ ( - 2 5 ) ) by as-q t q q sumption has t h e form A-(CT) = ( 1 / 2 i ) C (-2cT)nan. Hence grad C /(2i)"' q,r n = - ( 1 / 2 i ) ( C T ) n ( qr ) o r ( + 6 ) gradq,,.Cn = -(Zi)n(CT)n(i) and consequently (W) tion i s

(@A)

2 A - ( C T ) ( i ) = (l/i)l (-2)"an(C T ) n(q) r --

ili inangrad q,r C n '

T h e r e f o r e t h e ev-

o l u t i o n equations have t h e form rt = -6H/6q and qt = 6H/6r where H anCn and 6H/6r = gradrH w i t h sH/sq = grad

R?illARK 9.9.

=

il in

H.

QED q One says t h a t two f u n c t i o n s A,B o f v a r i a b l e s (p,q)

(here (r,q))

are i n involution i f (A,B) 0 ( c f . ( U ) ) . I n t h e p r e s e n t s i t u a t i o n one has f o r Cn as i n Theorem 9.8 ( r e c a l l t h e expression as i n t e g r a l s i n (9.18) etc.) 6r)

-

(+m)

0 = dCn/dt =

(6Cn/6r)(sH/6q)]dx

;:

[(sCn/Gq)qt

+ (GCn/Gr)rt]dx

=

lI [(6Cn/Gq)(6H/

I n p a r t i c u l a r t h i s would h o l d f o r H =

= (Cn,H).

Cm and by t h e Jacobi i d e n t i t y ( c f . 91.7) (Cn,Cm)

= 0.

L e t us r e c a l l a l s o

t h e n o t i o n o f complete i n t e g r a b i l i t y o f a dynamical system w i t h N degrees o f freedom; i t i s c o m p l e t e l y i n t e g r a b l e i f t h e r e e x i s t N independent constants o f motion i n i n v o l u t i o n .

I n t h a t case one can s o l v e t h e Hamilton

equations by quadrature ( c f . [ C j l ] ) .

For an i n f i n i t e dimensional system t h e

s i t u a t i o n i s more c o m p l i c a t e d s i n c e t h e number o f c o n s t a n t s r e q u i r e d i s n o t c l e a r i n general.

REmARK 9-10, L e t us w r i t e down t h e a c t i o n - a n g l e v a r i a b l e s as f o l l o w s ( c f . Assume a ( s ) and a ( 5 ) have o n l y s i m p l e zeros i n t h e i r r e s p e c t i v e [Aol]). h a l f planes o f a n a l y t i c i t y w i t h no zeros on t h e r e a l a x i s . Assume a l s o t h a t b ( 5 ) and b ( c ) can be extended o f f t h e r e a l a x i s .

Define

Given a ( < ) = 0 and cm = ( b / a ' ) ( s m ) (Imr; > 0, 1 ( m 5 N ) and 4

Am

?k = ( b / a ' ) ( t k ) (Imsk tk, and = -2il0g;~.

^Qk

etc. are also natural.

<

$(2k)

= 0 with

0, 1 k 5 N) s e t ( m * ) pm = ,,c Q, = -2ilogc,, ?k = I n §10 we w i l l see t h a t t h e a n g l e v a r i a b l e s logbm T h i s c o l l e c t i o n o f v a r i a b l e s i s denoted by S and i s

s u f f i c i e n t t o determine a l l t h e s c a t t e r i n g data v i a formulas o f t h e t y p e N N ~ o g [ ( s - ~ ~ > / ( r , - $ , ) lt ( 1 / 2 n i ) ( a ( & ) = a(s)nl (~-i,,,)/(c-Q) ( m ~ )l o g a ( s ) = l1 -Jm w [ ( l o g ( a G ) / ( ~ - ~ ) ] d(Imr, ~ > 0 ) , Wronskian r e l a t i o n s , e t c . T h i s f o r m u l a (mA)

can a l s o be expanded t o determine Cn i n terms o f l o g [ a i ( S ) ] ,

HAMILTONIAN STRUCTURE

201

-

IO~[(C-F~)/(F-~,,,)I,

and l o g [ ( 6 - t k ) / ( c - t k ) ]

t i o n t h a t (p,q) * (P,Q)

We o m i t v e r i f i c a -

( c f . [Aol]).

i s c a n o n i c a l ( c f . [ A o l ] f o r d e t a i l s and Remark 10.7)

and w r i t e down t h e H a m i l t o n i a n as ( c f . 510 f o r more d e t a i l s )

(9.23)

H =

(2/n)lI

A - ( ~ ) [ l o g [ a ~ ( ~ +) I

11N l o g [ ( E - ~ k ) / ( E - ~ k ) l ] d E

c1N log[(F-$,)/(E-cm)l

+

$ A-(s)dc (-5

4iI

To show t h a t a P / a t = 0 and a Q / a t = SH/SP one observes t h a t D t l o g [ a i ( c ) ]

= 0, A.

= 0, and Dtlogc = Dtlogb(E) = -2A-(C), Dtcm = 0, Dtlogcm = -2A-(s,), Ak 2A-(2k) ( r e c a l l h e r e from Remark 9.4 b ( s , t ) = b(c,O)exp(-ZA-(c)t) and b = A

E v i d e n t l y then SH/SP (Z/n)A ( 5 ) = -(l/n)Dtlogb boexp(2A-(c)t)). i s i n d i c a t e d i n [Aol]. c a l c u l a t i o n o f 6H/6sm and SH/S;k

= Qt w h i l e

Q,

10. 5 0 L 1 & 0 N &HE@Rg(HAIRILt0NIAN $tWetuRE). ed some f a c t s and f e a t u r e s f o l l o w i n g “121

and e m b e l l i s h t h e procedure i n

Thus f i r s t w r i t e t h e system (9.1) as

v a r i o u s ways.

vX

(10.1)

We w i l l e x t r a c t f r o m § 9 as need-

PV; P = ( - irs

=

i s1

A

and s e t

@

cl).

= (‘I

Consider v a r i a t i o n s 6 V , 6@, e t c . i n t h e f o r m (*) (6V)x

p l a c i n g V by S@(-L)].

@

But

6-r

-i6< 6q

‘ 2 ‘ 2

= PSV + ( 6 P ) V ; S P = ( (@-16V)x = -@-’PSV + @

).

Now

[ P S Vi6s + SPV] =

@-’ Q,

=

( m a t r i x A e q u f t i o n ) one o b t a i n s

@-’

-@-bX@-’ = -@-’P and

hence

X1 6 P V ( v a r i a t i o n o f parameters). (A)

Re-

+ @-’

6@ = @[J-xL@-16PPdc = -1 and hence

=

iL

where t h e l a s t t e r m i s 6@(-L).

Then we assume q,r

0

(and 6q,Sr) a r e

Q,

0 for

x 5 - L and a t x = -L, 6q, = i L e x p ( i c L ) s s w i t h 6?, = i L e x p ( - i c L ) S c f r o m (9.4) exp( -icL ) 0 ( iLexp( igL)6 s 0 iL 0 so @-lS@(-L) = ( 0 -exp( i c L ) 0 iLexp(-icL)dc) = (0 - i L bexp(-iix)) and we d e f i n e 65. Now as x +m , @ + aexp(-i<x) (bexp( is x ) -aexp( ic x ) (10.3)

I(u,v) =

(” [-Squ2v2

-m

( s o I = if ( 6 qr ) ( -uJ 2v$ 2 ) d x ) . L e t us n o t e t h a t $ = from ( A ) i n 59 w h i l e a$ + b^b with a = ql$2 s e t x = L and l e t L

-f

a

b

- a? w i t h $

q2$l,

etc.

t o o b t a n ( a f t e r some c a l c u l a t i o n )

=

b

+

6

Now i n (10.2)

ROBERT CARROLL

202

(10.4)

6a = - I ( v , $ ) + i 6 c l z (a-v1$2

+ v 2 j l ) d x ; 6g

- -I($,$)

-

-~

~ $ ~ ) d6 bx ; I(v,$) + i s c l z (vlZ2

+ $2$1)dx; 6 2 = -I(;,?)

'651:

- i 6 c l l [ Idx

(where [ ] = ( $ t $ l $ 2+ $ 2 $ l ) ) . We note now t h a t f o r two s o l u t i o n s ( ' l ) u2 and ('l) o f (9.1) ( 0 ) i(ulw2 t u2w1) = Dx[-D u w t D u w 1. In p a r t i c u l a r w2 3 1 2 3 2 1 using the asymptotic values ( 9 . 4 ) one obtains e a s i l y from t h i s ( 6 ) Dclog a ( 3 ) = - i l z [(v $ + v 2 i b l ) / a - l l d x . Further i n (10.4) f o r example one sees 1m2 t h a t 6a/6c 'I, =iLm (v1$2 + v2!b1 - a)dx so t h a t (10.4) represents 6a i n terms of 64, 6r, and 6 5 a s a " t o t a l v a r i a t i o n " . The o t h e r equations i n (10.4) have the same meaning. Now (assuming simple zeros Ck and cm) f o r s c a t t e r i n g data one takes e.g. ( c f . (9.22)) (6) b/a and ^b/if o r 5 r e a l , (L,,,,c,,,), and ( t k y t k ) ( c a l l t h i s S,). Alternatively one'can take ( m ) (S-): i / a and b/a* f o r 5 r e a l , (c,,,,~,,,), and (;k98k) where B, = l / ( b a ' ) ( c m ) and B"k = 1 / ( 6 a " l ) ( f k ) . Using 2 (10.4) one shows e a s i l y t h a t when 6 3 = 0 (*) &($/a) = ( l / a ) I ( $ , $ ) ; 6(b/a^) = (1/$2)1($,$) and some c a l c u l a t i o n , which we omit ( c f . [N12]), y i e l d s (*A) n 4 4 65, = cmIm($,$) = c, [ m m ( -6r 6 q ) ($1 + $ ) ( s m ) d x ; 6c.k = AckIk($,$) where I k denotes evaluation a t ;k (here one is i n the context of S-1. I n p a r t i c u l a r 3 , remains an eigenvalue of a ( c ) s i n c e (vm = b,,$,, a t cm) s a [ r , ( t ) , t ] = -bmIm($,$) + Further in t h e S- context (*@) ar;16cm 0 by (10.4) and (*A) ( c m = bm/a;). 6(~,,,) = (l/aA) 2 [I,,!,(+,$) - ( a " / a l ) I ($,$)I; 6 ( i k ) = - (i;/ii) m ! '"6r Ik($,$)] where e.g. I,,!,($,$) = jm(-,q)D3('%3)dx. Thus a l l of t h e v a r i a t i o n s involve inner products of ( - 6 r ) w i t h squared eigenfunctions ( c f . §9 where 6q the notation i s s l i g h t l y d i f f e r e n t ) ) and one s e t s 2 *2 j2 lL2 E- = w = ; ; ~m = Xm = ; Slk; ;k> (10.5) 2 2 $2 D3 $: rn

(~/;i)~[~i($,$)

[$iJ * =IJ;]

[

A

42

;Irn;

I"']

42

where Gk and ;k involve $ i and D 4 . evaluated a t tk. The formulas based on 3 1 data S, a r e referred t o i n t h e context of a dual inverse problem and e.g. one writes ( * 6 ) I ( v , v ) = -lI (i:)(*,ST)dx t o e x h i b i t "duality". One f i n d s then as above (10.6)

6(b/a) = ( l / a2 )I(v,v); 6 ( 6 / $ )

Then f o r S, one i s dealing ( v i a

(*&)I

=

( 1 / i 2 ) I ( $ , $ ) ; 65, = (l/cmar;12)

w i t h square eigenfunctions

HAM1LTONI A N STRUCTURE

where J

AT

AT

G:

(10.8)

L = (1/2i) -Dx-2q1+r 2rI+r

and D

A 2

e v a l u a t e d a t ;k. t h i s " d u a l i t y " n o t a t i o n i s t h a t i f one d e f i n e s ( c f . (9.16) 11

qk and X k i n v o l v i n g

203

rdy [ -Dx-2qhm 2rfXm r d y

-'qr+q]

;L

T

Ip.

5 1

-2q/xm qdy

1

The reason f o r - (9.17))

-

DX+2rrxa qdy

= (1/2i)

2r1-r I-r -Dx+2q

Dx+2rI,q

( n o t e t h a t i f we w r i t e CT = (a b, t h e n L =

(-1-:!

w h i l e C = ("

1 t) w i t h LT

-'))

t h e n e.g. (L-c)(*,*) E d= 0 and (L-c)(xm,xk) = (fl,Zk)'while ( LT- 5 ) = ( -8 C( T "T T *T ( x i , i l ) = (Qk,Q ) ((LT-s)(* ,Q ) = 0 - we n o t e t h a t CGl = cG1 i n (9.10) i s T 9 T t h e same as L Q = c* f r o m (10.7) - (10.8))..

PElllARK 10.1.

The s e t s E+ o r E- a c t u a l l y f o r m a b a s i s i n s u i t a b l e spaces and

one can expand v e c t o r s such as ( sr) o r ( :)

-Q

i n terms o f them ( c f . [Kp1,2]).

F i r s t we n o t e t h e o r t h o g o n a l i t y r e l a t i o n s f o r square e i g e n f u n c t i o n s based m T Lm u-vdx. Thus suppose Lu = cu and L v = c ' v . Then a l i t t l e c a l -

on ( u , v ) =

c u l a t i o n g i v e s (*+) I_\u(c)v(r,')dx

= [1/2i(c-c')](u2v2

-

u

~

v so ~u s i n) g ~

t h e a s y m p t o t i c s (9.4)

These a r e sometimes c a l l e d D a r b o u x - C h r i s t o f f e l t y p e formulas and a s i m p l e T c o n t o u r i n t e g r a t i o n argument f o r example y i e l d s now (*.) ( Q ( s ) , * ( e l ) ) = 2 AT -aa 6 ( 5 - 5 ' ) ( 5 , ~ 'r e a l ) . S i m i l a r c a l c u l a t i o n s y i e l d (A*) ($(L),*( 5 ' ) ) = A 2 4T na S ( 5 - 5 ' ) ( a l s o (*(c),* ( 5 ' ) ) = 0, e t c . ) and d i f f e r e n t i a t i n g t h e s e r e l a T T 2 t i o n s one o b t a i n s (u) ( D Q ( c k ) , Q ( c . ) ) = ( Q ( c k ) , D Q ( 5 . ) ) = - ( i / 2 ) a i fikj, T 5 J 5 J ( Dc*(Ck),D,* ( c j ) ) = -(i/2)aia;skj, e t c . ( c f . "121). Some f u r t h e r c a l c u l a t i o n , u s i n g (*+I i n §9 f o r example, E, o r E- as a b a s i s )

(*m)-(A*)-(AA),

e t c . now y i e l d s ( g i v e n

~

~

ROBERT CARROLL

204

+

(10.10) where

r

+ ;k6;Gk];

62kGk

-

= 2 i c cm#

2ic

(-:)

2k*A k , and

we r e f e r t o [Kp1,2]

f o r proofs. Let C (resp.

connection l e t us n o t e a few formulas ( c f . [Kp1,2]). to

+

= - ( 1 / n ) l E [(b/a)*+(^b/;)*lds

In this

t ) be

a

i n t h e 5 p l a n e passing o v e r (resp. under) t h e zeros o f assume f o r s i m p l i c i t y t h a t q , r have compact support. Then

contour a (resp

m

*

(10.11 )

= ( l / n ) l c (b/a)*d<

+

A

”

= ( l / n ) l c (;/a)

(b/a)*dc;

(l/n)J;

qTdr + ( t h e l a t t e r b e i n g c a l l e d a completeness r e l a t i o n i n p h y s i c s ) . Consider t h e s y m p l e c t i c form (A*) A =

REmARK 10.2,

onality relations n a t u r a l way

6q A 6 r w i t h o r t h o g e t c . and r e c a l l a$ + b^b = 1. One w r i t e s i n a T and 6 V T = 6 r A 6qdx = (1/2)( 6V I A ( 6 V ) where 6V =

(*=)-(A*),

i:

(A&)

Considerable c a l c u l a t i o n , which we m e r c i f u l l y

(-6r) a r e g i v e n v i a (10.10).

6s omit,

gives then

1:

(10.12)

0, a A B = - B

(a A a =

6 r A 6qdx =

+

1:

a,

etc.)

Alogb A 61og(a$)ds + 1 2 i 610gbm A 65,

1 2i610g^bk A

6Ck

I n passing one r e c a l l s t h a t (6q A 6 r ) ( v l ,v2) = has n o t been used. a l o n g w i t h 2i5,

m

lm

I(( 6 qq,,vv2l))

((,6r,v2) 6r9vl)

I but

this

Thus t h e a c t i o n a n g l e v a r i a b l e s logaa and l o g b a r i s e

and logbm e t c . ( c f . ( 9 . 2 2 ) ) and w r i t i n g / 6 q A 6rdx =

-1 6logb A & l o g a s dc +

. . . we

r e l a t e Q = -1ogb and P = loga:

(a f a c t o r of

71

i s d i f f e r e n t here from (9.22) b u t t h a t can s i m p l y be removed from t h e event u a l Hamiltonian, e.g.

(9.23),

i n order t o obtain the c o r r e c t Hamiltonian

equations). Note f o r small p o t e n t i a l s w i t h a

REmARK 10.3. and

*AT

rlr

sx)ds; r

1 and

^a

1 ( 0 ) e x p ( 2 i c x ) e t c . so one has i n (10.10) (A+) 6r

-

(l/n)iz

rf bexp(-2i<x)dS.

bexp(2icx)dc; 6q

m

rlr

Formulas f o r b and

-(l/n)Lw

*b

small p o t e n t i a l s and 6 c = 0 one has e.g.

+

1,

*T

rlr

0 -2isx -(l)e

( l / n ) _ / z bexp(2i bexp(-2icx)dc; q -(l/n) Q,

Q ,

c o u l d be o b t a i n e d by d e v e l o p i n g f o r -

mulas l i k e ( * ) i n 59 e t c . a s y m p t o t i c a l l y .

L;

+

A l t e r n a t i v e l y f r o m (10,4) 6b

Q,

I(q,$)

=

il ( 6 q ) (- v 2 f 6r

for

ql$

2

)dx

Q,

6 r e x p ( - 2 i c x ) d x so t h a t 6 r ( y / 2 ) and 26b a r e F o u r i e r transforms.

REmARK 10.4, LT as i n “121

We want t o r e c a s t t h e H a m i l t o n i a n f o r m u l a t i o n h e r e i n terms o f r a t h e r t h a t tTo f 19 (used i n [ A o l ] ) .

One c o u l d o f course

HAMILTONIAN STRUCTURE

205

choose one and f o r g e t t h e o t h e r b u t s i n c e b o t h appear a t v a r i o u s p l a c e s i n t h e l i t e r a t u r e i t may be h e l p f u l t o have them b o t h s p e l l e d o u t here. r e c a l l t h e comparison b e f o r e Remark 10.1 and r e f e r t o 59 f o r L. QT =

(l/a)('*

-1

* z ) ( c f . (9.19)) and check t h a t (.*)

Set

Thus (Am)

(LT-r,pT = (1/2i)(r)

$1

f r o m which i n p l a c e o f (9.20) one o b t a i n s f o r l a r g e

Icl,

Imr, > 0,

(*A)q@T

Q

- ( 1 / 2 i ) z i ( l / g m + l ) ( L T ) m( r ). I n f a c t t h i s has e x a c t l y t h e same form as (9. T Tq 20) w i t h L r e p l a c i n g i: ((.*) a l s o has t h e same f o r m as ( 9 . 1 9 ) ) . L e t us n o t e here t h e o r i g i n o f (.*)

o r (9.19) i n o r d e r t o see why t h i s i s c o r r e c t .

) + 2 i r , ' ~ ~=$ ~q[a -21,q l l x ~ + wlJllldy (as w e l l ( I P ~ J / -~ )2 I + r ( ' ~ ~ $ ~ ) Bl .u t ' P ~ +$ ' ~~ ~ $ =~: ;21 - [w2$2

Thus r e c a l l (*.)

i n 59 e t c . t o see t h a t e.g.

('P J,

X

as v1!b2 + v2!b1 ;1

= 2Jm

x,

9142 = a ( s i n c e $ 2 1 r(vl$l)]

[w2JI 2

+ wl!bl1dy

and a t

-m,

a g a i n ' P ~ +$ v2Jll ~ =

- &).

Hence ( ' P ~ $ +~ 2iC'P191 ) ~ and t h i s e x p l a i n s t h e c o i n c i d e n c e o f (.*)-(.A) = bp

=

-/I

q[a + 21-q(lp2J12) + w i t h (9.19)-(9.20).

[-6q('P2$2/a) + 6 r Now go t o (10.4) w i t h 65 = 0 t o o b t a i n ( 0 0 ) 6 l o g a = T l o g a = Q, ( c f . ( 9 . 2 1 ) ) . Again one w r i t e s l o g a ( ' ~ ~ $ ~ / a ) ] so d x t h a t grad ( 5 ) = -1; Fmtl/cmt1( c f . q,r (am) i n 59) where t h e (Cn = - ( 2 i ) n+l; ) ren + l ~m r p r e s e n t conserved q u a n t i t i e s , and we can proceed as b e f o r e . Thus ( L ) ( q )

cm

N

m N N

= 2 i g r a d Cmtl

and H = - 4 i c O anCntl

dispersion relation.

N

where one w r i t e s A ( < ) =

Note t h e n from A- = ( 1 / 2 i ) z i (-2r,)'an

Cn we must have 2'(-l)'aa,

= - 2 i z n so

A

-1 anr," d . ,

=

1; gncn f o r t h e and H = il a n i n

.y

= -A.

2 + 2q2q w i t h A- = 2 i s = qxx ( c f . Example 9.1 and Remark 9.4 a l o n g w i t h "121). I n t h i s case ( 0 6 ) H = m 2 -ilm [qqxx + q2q2]dx = - ( 4 / l i i ) l I 5 logaZd5 - 8 1 [(r,:-5:)/3]. Here t h e

RRRARK

-x

For r =

10.5.

-q

one has t h e NLS i q t

=

a c t i o n v a r i a b l e s a r e ( l / n ) l o g a Z , 2icm, and 2 ' 5 , w i t h a n g l e v a r i a b l e s l o g b , logbm, and logbm and Dtlogb = 6H/6(logaZ/li) = - ( 4 / i ) 5 2 = 2A- = - 2 x w i t h Dt u 2 e t c . ( n o t e i n (9.22), Q % logbm = 6H/6(2icm) = -8cm/2i = 4 i c 2 = -2A(r,,,,), m -logb, w i t h a f a c t o r o f li, so e v e r y t h i n g f i t s t o g e t h e r h e r e w i t h Remark 9 . 1 0 ) .

L e t us now work o u t t h e equations f o r a Schrodinger e q u a t i o n

REmARK 10.6.

Thus c o n s i d e r (9.1) w i t h r = -1 i n t h e s c a t t e r i n g problem f o l l o w i n g "121. 2 so t h a t v2xx + ( r , +q)v2 = 0. Take q , $ , e t c . now t o have a s y m p t o t i c b e h a v i o r at

-m

o f the form

((10.13)

and a t

m

v

Q

l e t t h e form be

206

ROBERT CARROLL

(10.14)

J,

0 icx; (,)e

%

s

j,

(i?i~)~-isx. 1 ’ lp

‘L

2isaexp(-igx) (aexp( -isx) tbexp( iex) ;

2ici;ex ( - i c x ) aexp( iex )t exp(- i5x1

6

(A

- b y$

A A

;G + bG, and ( o n l y t h e f i r s t formula i s t h e same as b e f o r e ) .

Thus

lp

= a$ t bJ,, J,

a$

A

=

= alp

J,

-

A

$t

w i t h aa

-

One shows t h a t a,lp,J,

n

bb = 1 are

4

Ims > 0 and a,lp,J, a r e a n a l y t i c f o r I m c < 0. One has a l s o (*+) a ( < ) = a ( - < ) , ;(s) = b ( - c ) , $, = -em, e t c . and i f q i s r e a l we have a(?) = a(-<), 6 ( ; ) b ( - c ) , and em = iu, w i t h nm r e a l . We assume t h e zeros em a r e analytic f o r

A

s i m p l e and do n o t l i e on t h e r e a l a x i s . t h e r e c o v e r y o f q make use o f d a t a S

The i n t e g r a l equations ( c f . §9) f o r

o r S,

as i n

(+)-(m)

which we can con-

A

dense now t o be

(am)

:S,

b/a and (cm,cm) o r S-: b/a and (c,,~,) A,

works here w i t h B,

where one

m m ( n o t e a t em, bmbm = -1).

= -l/bmar;l = b / a ‘

can compute ( c f . (*) (10.15)

A

As b e f o r e one

etc.)

6(b/a) = -(1/2ica 2

6qlp2dx; 2 cm6cm = - ( l / 2 i s m a h 2 ) f I

6w 2

dx;

2m “m

6q[Delp2,,, 2

= -(1/2icmar;l 2

-

((aJaA)

+

(1/cm))lp2mldx 2

S i m i l a r l y f o r S- one has (10.16)

”m

&(;/a)

= (1/2isa 2

)I:

6qJ,2dx; 2 8,6cm = (1/2icmah2

- ((a;/$,)

= ( 1 / 2 i c m a i ) 1 z 6q[DsJ,Em

)I: 6qG2,2,dx;

+ (l/c,)G~,Idx

2 2 2 Thus d e f i n e dual Sets O f f u n c t i o n s (A*) @ = lp2, @k = lp ( 5 ), Xk = (Delp2)ky 2 2 2 2 k 2 and = G2, qk = G ( c ), T~ = (DeJ,?)k. Then f o r L, = -(1/4)Dx - q t ( 1 / 2 ) qdy, L z = -(1/4)Dx2k - q t (1/2)R+, l l m R dy ( t h e l a s t symbol a c t s on i n t e g r a -

lt

*

/x

t i o n o b j e c t s ) , M, = -(1/4)Dx2

lim x R+m /_,dy,

-

q

one has f i r s t Ls@ = c

-

(1/2)/xa qdy, and MST = -(1/4)Dx2

2@ and M

* = c *.

- q

-

(1/2)

T T The o p e r a t o r s L, and MS

a c t on a d j o i n t s e t s o f f u n c t i o n s (6.) @T X = D $ = (Dx$E)K,X:2= (DeDxd’2)k 2 with M TsqT = 5 2 T t ( 1 / 2 ) l i m w 22( - R ) and L;aT =’ :‘aT t ( 1 / 2 ) l i m q G2(R) (here T 2 T 2 T 2 = Dxq2, *k = (Dxlp2)k, and T~ = ( D D lp ) ) . L e t us mention some formulas e x 2 k from “121 i n t h i s d i r e c t i o n . Thus i f (vl,v2) and (w,,w2) s a t i s f y (9.1) 2[v1w1 - ic(vlw2 + w i t h r = - 1 t h e n (6.1 (v2w2)xx + 4s 2 (v2w2) + 2q(v2w2)

*

*

v w )]; [vlwl 221 - 2iclp1q2 lp12 2 2 Lslp2

-

(10.17)

c

-

ic(v w

tp2 = 0.

.

= -qv w Putting v w = lp one has (66) 222 2 2 2 - ( 1 / 4 ) ( l p 2 ) x x - c lp2 - w2 + (1/2)I-qlp: = The o r t h o g o n a l i t y r e l a t i o n s w i l l be

t v2i1)],

-w221 +* 1-q

lp2;

2 2 lI lp2(c’)J,2x(s)dx

2

= 2icna 6 ( s - c ’ )

f o r e y e ’ r e a l (= 0 o t h e r w i s e ) ;

HAMILTONIAN STRUCTURE

207

The o t h e r o r t h o g o n a l i t y r e l a 2 Next one has (Q = q ; , \Ir = $:, aT = $2x,

which y i e l d s t h e f i r s t e q u a t i o n i n (10.17). t i o n s a r e proved i n a s i m i l a r way.

*T = v g x , e t c . ) (10.22)

q = - ( 2 / i n ) L I c(^b/aPdC + 4 1 ckBkak = (2/in),ff

c(b/a)ds*

-

208

ROBERT CARROLL

C l e a r l y these expressions a r e c o n s i s t e n t w i t h (10.15),

(10.16),

and t h e o r -

t h o g o n a l i t y r e l a t i o n s ; d e t a i l s o f t h e r o l e o f @,ak, e t c . as bases a r e o m i t t e d h e r e ( c f . [Kp1,2] (1/2)/"-03 [6q A

Consider n e x t ( 6 4 ) A =

f o r the relevant analytis).

[t sqdyldx where one uses t h e f i r s t

and second terms f o r 6q

i n (10.22) r e s p e c t i v e l y i n computing A.

T h i s g i v e s then ( q r e a l , R = b/a, 2 2 ( 2 ~ / ~ ) 6 1 o g ( l - I R I) A 6Argb(C)dg t 6(-2nk) A 610gbk.

1

-1:

3k = i n k ) ( 6 . ) A =

To see t h i s one computes f i r s t ( q complex) (4*) A = ;/ ( z i s / n ) s l o g ( l - ( b ~ / a ~ ) ) 2 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.g. t h e A slogbds + 1 6(2ck) A 610gbk. T 2 t e r m T = ( l / ~ ) s/ ( b / a ) a T A iL (-1/n)6($a)*T)dydx w i t h QT = and = t o obtain from ( 6 6 ) T =

p;x

= ba[(b'/b

-/I

( 2 i s a 2 / v ) s ( b / a ) A 6(^b/a)ds. 2 Thus a 6(b/a) A & ( $ / a ) = J 'L b^b[(b'/b

a few c a l c u l a t i o n s .

- a'/a)]

*

El/;)

A

b^b t 1 one has aa"(a'/a t

-

(a'/a A

$I/$)

i'/^b) - (b'/b

= b^b(b'/b t

L e t us r e c o r d

- a'/a)

A a'/a)]

2 ' 1 ; ) so b^b(b'/b

A

(is'/;

and from a$ = A

^b'/^b)

Hence J can consequently be w r i t t e n i n t h e f o r m a " a ' / b A ( a ' / a + $I/;). % a*ab'/b A a ' / a t a $ b ' / b A "a'/$ - b^ba'/a A $ ' I $ - b i b ' l b A a ' / a = - a ' / a A b'/b

-

aaa'/$ A b ' / b

-

b*ba'/a A

i(s) $'/^a A

$'/^b.

But

^a'(<)

-rf s a ' / a

a ' / a A b'/bdc (note

= a ( - c ) so

/:

b'/b]SdS = ( 2 i / n ) l- m m (sloga;)

[a'/a A b'/b +

r I sslog(l-b$a$)

= -a'(-<)

A "b/^bb^bdr; =

etc.).

LI b^bY

Hence T = ( 2 i / n )

A 61ogb sds = ( - 2 i / 1 ~ )

A 61ogbds and t h i s i n d i c a t e s t h e s p i r i t o f t h e c a l c u l a -

t i o n s (a minus s i g n s t i l l has t o be a d j u s t e d e t c . ) . F i n a l l y when q i s r e a l n 2 2 ^ 2 w i t h b = b, sk = ink, R = b/a, e t c . one has ( b ( / ( a ( = bb/a^a = ( R \ and 2 2 given 1aI2 - I b l = 1 one knows I b l so l o g b i s r e p l a c e d by i t s imaginary Next f r o m 2 (10.22) one considers an e n t i r e f u n c t i o n P o ( i ) and r e c a l l s t h a t Ms* = 5

p a r t (see Remark 10.7 f o r t h e r e l a t e d Poisson b r a c k e t s e t c . ) .

*

t o obtain formally (10.23)

DxPo(MS)* =

-41


= t ( 2 / i T ) I I Cpo(S 2 )(b/a)$2xds 2

qt + DxPo(MS)q = 0 i s t h e n 2 ( b / a ) t = 2 i c P O ( s )(b/a); ckt =

F u r t h e r ( a g a i n u s i n g (10.22)) t h e e q u a t i o n (4.)

e q u i v a l e n t t o t h e s c a t t e r i n g equations (+.) 2 -2i
/_f

HAMILTONIAN STRUCTURE

209

= Dx and 6H/6p = Dx6H/6q. Again l o g a p r o v i d e s t h e l i n k . One shows ( c f . (+A) i n 99) 61oga/sq = ~ 7 ~ $ ~ / 2 and i s a i t i s e a s i l y shown t h a t ( 6 6 ) (MS-c 2 )

-

For l a r g e s /a) 1 1 = q/2. .2 MTq/(s2)mt'. But l o g a = (1/2)10 [(9 $

2

and i f t h e d i s p e r s i o n r e l a t i o n i s -0 A ( s ) n= i c P o ( s 2 ) = lows t h a t ( m * ) that

-

( I m s > 0 ) ( + 6 ) y i e l d s (++) v2$2/a = 1 / s n so (em) 6C2mt3/6q = ( 1 / 4 i ) q q

-1" C

H = - 4 i l a2m+lC2m+3(q,r).

l o w = -(s/2~i)L;

[log(l-IRI

2

i c l ' a2m+l(c2)m

it fol-

I n s c a t t e r i n g v a r i a b l e s we know 2 2 )/(s - 5 ) I d 6 + l o g [ ( ~ - i n ~ ) / ( c + i ~ ~ ~ ) l

1

-

In and hence (ma) C2mtl = 2 l(ink)2mt1/(2m+l) (l/ri)I; S2mlog(l-IR12)dS. 3 2 3 5 particular i f = - 4 i s t h e n H = (1/2)L: (4, 2q )dx = - ( 3 2 / 5 ) 1 q k (16/n) 2 4 I: 5 l o g ( l - I R I )dS and t h i s corresponds t o t h e KdV e q u a t i o n .

-

REmARK 10.7.

-

L e t us make a few remarks on Poisson b r a c k e t s e t c . ,

t o Remark 9.10.

F i r s t t o show t h e map (p,q)

+.

w1 be s o l u t i o n s o f (9.1) f o r q1 (and u t , w2 f o r s2). 2i(52-51)[u1u2w1w2 1 2 1 2

-

u2u1w2w1] 1 2 1 2 = Dx[(u1u2 1 2

-

an i m p o r t a n t i d e n t i t y i n v o l v i n g Wronskians. in

(+A)

g o i n g back

( P y a ) i s c a n o n i c a l l e t u,, Then ( e x e r c i s e ) (m6)

u2u1)(w1w2 1 2 1 2

-

This i s

w2w1)]. 1 2

One computes grad

o f 59 and t h e n c o n s i d e r s Poisson b r a c k e t s , e.g.

a,a^,b,$

qYr ( m + ) (a(sl),b(s2)) 1i m b(sl )a(s2)x-tm

as

-Im m [ ( 6 a / s q ) ( s b / s r ) - ( 6 a / 6 r ) ( 6 b / 6 q ) l d x = [a(<, ) b ( c 2 ) exp(2i(sl-s2))]/2i(sl-s2) (si r e a l ) . Again e x p ( i k x ) / k = a i S ( k ) and one f i n d s e.g. ( s S ) (m.1 (a(c1).:(c2)) = 0; ( l o g a ( ~ ~ ) , l o g b ( ~=~ -) ()l / 2 i ( c 1 c2)) + (a/2)6(c1-c2); ( l o g s ( s l ) , l o g b ( c 2 ) ) = ( 1 / 2 i ( c 1 - s 2 ) ) + ( ~ / 2 ) 6 ( 5 ~ - 5 ~ ) .

=

iJ!

Q ,

T h i s leads t o ( P ( c 1 ) , Q ( S 2 ) ) = - 6 ( c 1 - c 2 ) f o r example f o r P = l o g a s and Q =

- ( l / n ) l ogb. L e t us i n d i c a t e some c a l c u l a t i o n s f r o m [ N v l [ f o r t h e Hamil-

REmARK 10.8.

t o n i a n f o r m u l a t i o n o f KdV t h e o r y .

Thus i n a general sense a Poisson b r a c k e t ik i s determined by (**) ( f , g ) = w (S)(af/agi)(ag/aSk) w i t h wik -w ki and -1 f , g ) , h ) t ( ( h , f ) , g ) + ( ( g , h ) , f ) = 0. Given Wik = ( w )ik one a l s o r e f e r s k Wikdgi A dS Working i n a phase space w i t h t o a d i f f e r e n t i a l form W =

1

.

1

w reduced t o a m a t r i x w i t h b l o c k s o f J = pky

s~~

= qk one o b t a i n s e q u a t i o n s

bk

can be based on e q u a t i o n s d f / d t = ( f , H ) variables (pkyqk),

(

f,g)

=

(p 2 )

down t h e d i a g o n a l and

= -aH/aqk and

( o r dSi/dt

=

6

= aH/apk.

1 wk i ka H / a S k ) .

1 (6f/6qk)(6g/6pk) - (6f/spk)(69/6qk) 2 ut = Dx(3u - u x x ) = Dx(6H/6u) where

s

~

This I n these

as b e f o r e .

Now f o r KdV one w r i t e s e.g. H(u) = 2 3 [ ( 1 / 2 ) u x t u ]dx ( n o t e o u r conventions w i t h F r e c h e t d e r i v a t i v e s e t c . a r e

/f

c o n s i s t e n t w i t h t h e n o t a t i o n Dx(6H/6u) =

-/I Dx,6(x-x')6H/6u(x')dx'

however 6H/6u(x1) a l r e a d y i n v o l v e s an i n t e g r a t i o n by p a r t s e t c . SU(X')

=

e.g. ( S , R )

2 3u

-

=

L;

uxx).

Thus one can t h i n k o f wik

(6S/su(x))Dx(6R/6u(x))dx

=

Q

-L; /f

w(x,x')

-

where i . e . 6H/

= D x 6 ( x - x ' ) and

(6S/6u(x))Dx16(x-x1)

~

-

~

21 0

ROBERT CARROLL

(6R/6u ( x ' ) )dxdx ' = J 1 Dx6 (x-x ' ) (SS/Su( x ) ) (6R/6u ( x ' ) )dxdx ' (where o f course one t a k e s D,S(x-x') = -D ' ~ ( x - x ' ) ) . Now l e t v v1 s a t i s f y -v" + w = k 2v Q ,

with

X

v

%

a e x p ( - i k x ) + b e x p ( i k x ) as x

58); here q1

%

e x p ( - i k x ) as x

(where

i

= 0 and

3

b

- cf.

= 8ik b

Thus we use a d i f f e r e n t

-m.

-f

+ m

v1

h e r e from

Remark 10.6 and, r e c a l l i n g (9.41, etc., v1 i s d i f f e r e n t f r o m 19 as w e l l . I n f a c t from 51.8 h e r e v 1 % f - = f2 w i t h a = 1/T and b = RR/T = s ~ ~ / T S. i m i l a r l y i n [Nvl] one s e t s v2(x,k)

, v,

= fl

f,

-

+ bib1

=

= l/s22 = 1/T and b = cll

(l/sll)fI

t (s12/sll)f-

= vl(x,-k)

= f- = a f + ( - k )

= !b2(xy-k) where !b2

and Jll(x,k)

+ bf+(k)

Q

+ c 12f +- so a c 22 f - + c21f:

cllft

f+ = = c12s21 = S ~ ~ / T Thus . = (l/T)G1

+ (RL/T)vl

and we r e c a l l l T j 2 +

%

= c12 %

lRI2

!b2 = = 1

w i t h RRT t RT; = 0 (so t h a t $ = R$T- = -RL/T) and 1aI2 - I b l = ( 1 / l T I 2 ) lRRl 2 / I T 1 2 = 1. I n [ N v l ] one w r i t e s r = b/a = RR and t = l / a T and we r e c a l l (v = v l y J, = J,l) (**A) a ( k ) = ( 1 / 2 i k ) [ ? v - hX]; b ( k ) = (1/2ik)[vxJ, Xm From t h i s one o b t a i n s ( a ( k ) , b ( k ' ) ) = _,I (6a/6u)Dx(sb/6u)dx = ( 1 / 2 )

iP

-

[(6a/6u)Dx(6b/6u)

(6b/6u)Dx(6a/6u)]dx

and by u s i n g (MA.) and t h e non-

homogeneous Schrodinger e q u a t i o n f o r q u a n t i t i e s 1 i k e 6v/6u one o b t a i n s ( c f . (10.4),

(+A)

i n 19, e t c . )

6a/6u = - ( 1 / 2 i k ) $ v

(**a)

= -(1/2ik)f-ft;

Note here e.g. G = s v ( y , k ) / s u ( x ) (1/2ik)!bv = ( 1 / 2 i k ) f f ( x , - k ) . 2 t u(y)G = - k3 G - s ( x - y ) v ( y , k ) w i t h G(x,y,k) = 0 f o r y (**&) -DxG

v

i s determined by a s y m p t o t i c c o n d i t i o n s a t

-m

- also

6b/6u = satisfies

<

x (since

G I + = v(x,k)). Y Y+X

Next a v a r i a t i o n on (mb) reads ( f o r fiy gi s o l u t i o n s o f t h e Schrodinger equa-

(**'I fp1Dx(f2s,) - f29px(flgl 1 it (l/k,-k2;Dx[(flf2x 2 2 91 = v , and g2 = 'p t h i s y i e l d s (gig2x - 9 2 q x ) 1 and f o r fl = f2 = ( w i t h J, exp(-ikx) a t and J, aexp(-ikx) - bexp(ikx) a t - n o t e J,l = f; f2,f- + C22f: (RL/T)-f- + ( l / T - ) f ( 1 / T - ) e x p ( - i k x ) - (RR/T)expikx)

kf2flx)

tion)

J,

%

m

%

-m

%

( a ( k ) , b ( k ' ) ) = ( 1 / 8 k k ' ) L E [a(k)a(k')e-2ik'x

- (k+k' )a( k)b( k ' )/( k-k' ) x+-m 1 i m [ - a ( k ) a ( k ' ) e - 2ik'x

(

iE eikx/k

(k-k'

+

(*A*)

-

( k t k ' ) a ( k ' ) b ( k ) e 2i ( k - k ' )'/( k - k ' ) ] b(k)b(k')e-2ikx

)a( k ' )i(k ) e - 2i

b(k)b(k')e2ikx

- (1/8kk' )

- (k-k')a(k)b(k')/(k+k')

= T i s ( k ) and t h e Riemann-Lebesgue lemma one g e t s

n(k) = (2k/n)loglaI2;

b r a c k e t s as i n (**.) one f i n d s

v

-

(k+k' I X / ( k + k l ) ]

a( k),b( k ' ) ) = (1/2)a( k ) b ( k ' ) / ( k 2 - k I 2 )

then

Y

%

%

(10.24)

Using

J,

Y

- ( a i / 4 k ) a ( k ) b ( k ) 6 ( k - k ' ).

(**m)

Setting

= argb and computing t h e necessary o t h e r

(*AA)

(n(k),n(k')) = (v(k),v(k'))

= 0 and

( v ( k ) , n ( k ' ) ) = 6 ( k - k ' ) . The d i s c r e t e spectrum i s handled s i m i l a r l y .

-

21 1

INTEGRABLE SYSTEMS

11. S0mE C0PICS IN ZNCECRAELE SWCEW. Some o f t h e more r e c e n t l i t e r a t u r e on i n t e g r a b l e systems has focused on t h e Riemann-Hi1 b e r t (R-H) problem approach ( c f . [Fa3;Mr2;Nvl]

f o r t h e R-H

and r e f e r e n c e s t h e r e and see a l s o "w2-61

We w i l l s k e t c h some of t h i s f o l l o w i n g [Fa31

problem i n s c a t t e r i n g t h e o r y ) .

and a t t h e end o f t h e s e c t i o n we d i s c u s s some t o p i c s i n v o l v i n g BacklundWe w i l l n o t d i s c u s s t h e q u a n t i z e d

gauge t r a n s f o r m a t i o n s , L i e algebras, e t c .

v e r s i o n o f s o l i t o n t h e o r y and r e f e r f o r t h a t t o [Fa51 and r e f e r e n c e s i n [Fa 31.

F o l l o w i n g now [Fa31 we use t h e n o n l i n e a r Schrodinger e q u a t i o n (NLS) f o r There w i l l o c c a s i o n a l l y be ( a d j u s t a b l e ) d i f f e r e n c e s

i l l u s t r a t i v e purposes.

i n n o t a t i o n f r o m s18-10 b u t we do n o t seek r e f u g e h e r e i n n o t a t i o n a l c o n s i s t e n c y and w i l l s t a y w i t h t h e n o t a t i o n o f [Fa31 i n o r d e r t o make e a s i e r t h e r e a d e r ' s r e f e r e n c e t o work based on t h i s n o t a t i o n .

There i s some r e p e t i t i o n

o f formulas and i d e a s f r o m 118-10 b u t t h i s i s p r o b a b l y i n s t r u c t i v e i n any Thus s t a r t i n g w i t h (*) i!bt = -J/xx + 2 ( $ 1 2 ( K = c o u p l i n g c o n s t a n t ) = Jl0(x) we c o n s i d e r o n l y s i t u a t i o n s where $ + 0 s u f f i c i e n t l y

event.

w i t h $(x,O)

r a p i d l y as 1x1

( o t h e r s i t u a t i o n s a r e t r e a t e d i n [Fa3]).

-+ m

The H a m i l t o n i a n

s t r u c t u r e a r i s e s i n (what i s by now) t h e s t a n d a r d manner ( a l t h o u g h see [Fa31 f o r t h e approach based on t h e r - m a t r i x ) . a l s ) F($,?) d z j , cnm E

=

1;

$I,$,$

J cnm(Y1,. ..,ynlzl,. E $ (N = {($,$)I

-

Thus f o r ( r e a l a n a l y t i c f u n c t i o n -

. .,zm)J/(yl

% .,

)...$(yn)$(z1)...$(zm)n dyi i s t h e "phase" space), c n m ( y I z ) = cmn(zl

y), cnm symmetric i n yi and s e p a r a t e l y symmetric i n z one d e f i n e s 6F($,$)

=

-m Im

[(aF/6$)6$

,. .

(m,n) j' where e.g.

+ (6F/6;)6;]dx

..

=

(O,O), e t c .

6F/6$(x) =

.Rz,)

1 n fcnm(x,Y1 .,Y,-~ lzl,. ,zm)$(y1 1. ..$(Y,-~ $(z1 1.. ndYidz j ' Thus generallySF/S$(x) i s a g e n e r a l i z e d f u n c t i o n . For smooth such v a r i a t i o n a l d e r i v a t i v e s t h e Poisson s t r u c t u r e i s d e f i n e d v i a (sG/s$(x)) (x),$(y)l

- (SF/s~(x))(sG/s$(x))]dx. = 0 = {?(x),$(y)I,

(A)

I F , G I = iif [(SF/S$(x))

E v i d e n t l y s$(x)/S$(y)

I$(x),$(y)l

= 6(x-y),

[J/

e t c . and A = i/f[ d F ( x )

= ib(x-y),

( 0 ) a$/at = {H,$I = i s H / s $ f o r a p r e s c r i b e d H a m i l t o n i a n f u n c t i o n H.

A d$(x)]dx g i v e s r i s e t o a s y m p l e c t i c s t r u c t u r e w i t h

and a c / a t

-i6H/6;

=

{H,?}

=

/f [\$,I2

F o r t h e NLS t h e a p p r o p r i a t e H a m i l t o n i a n i s ( 6 ) H =

+

K \J/\

4

Idx.

Now (as u s u a l ) one w i l l see t h a t (*) a r i s e s as a c o m p a t i b i l i t y c o n d i t i o n f o r t h e system

(11.1)

Fx = U(x,t,x)F;

where U = Uo

v

=

v

+ xul,

+ xvl +

U = 2O

Ft = V(x,t,h)F

JK($0; o )

x v2, vo

u3 O 0 1 0 -U1' al = (l o), a2 = (i

=

-i

1,

= JIL[;o+

i K 1J/120

u3 =

3~

(o

-

+

$a_], ,w

iJK[J/xu+

o -1),

U1 = ( l / 2 i ) ( i ):-

-

a+ = (1/2)[01

v1

=

= (1/2i)

-uo, v2

=

+ i a 2 ) , and a - =

21 2

ROBERT CARROLL

(1/2)(01 - i o 2 ) . The c o m p a t a b i l i t y c o n d i t i o n f o r (11.1) i s ( 6 ) Ut - Vx + [U,V] = 0 and f o r t h i s one r e q u i r e s (*). Now i n t h e s p i r i t o f gauge f i e l d t h e o r y e t c . ( c f . Chapter 3 ) one can r e g a r d U and V as l o c a l c o n n e c t i o n co2 2 e f f i c i e n t s i n t h e t r i v i a l v e c t o r bundle R X C ( R 2 - space-time h e r e ) and (11 . l ) i n v o l v e s a zero c o v a r i a n t d e r i v a t i v e w i t h

(6)

expressing 0 curvature.

The connection c o e f f i c i e n t s U,V a r e a f f e c t e d by a frame change G: F the f i b r e via

(m)

U

-+

GXG‘l

t

GUG-’

and V

-+

gauge t r a n s f o r m a t i o n s (as i n Chapter 3).

GtG-’

.

t GVG-’

-+

GF i n

These a r e c a l l e d

The r e p r e s e n t a t i o n o f (*) as t h e

c o m p a t a b i l i t y c o n d i t i o n (+) i s e v i d e n t l y v a l i d f o r t h e whole c l a s s o f gauge e q u i v a l e n t connections. If Next one speaks o f p a r a l l e l t r a n s p o r t induced by t h e connection (U,V). 2 y i s a c u r v e i n R f r o m ( x o y t o ) t o ( x , t ) p a r a l l e l t r a n s p o r t i s d e f i n e d by n

= exp[fy Udx t Vdt] w i t h F = A F. T h i s symbol i s d e f i n e d as f o l (*) Y Y lows. L e t y = U N y . ( a d j a c e n t segments) and s e t Ln = I t f ,,[Udx t Vdt]. Y Set AN = n N L = ” LN...L1 ( o r d e r e d ) and t h e n A = l i m AN as t h e p a r t i t i o n i s 1 n Y r e f i n e d . This i s a form o f p r o d u c t i n t e g r a l ( c f . Chapter 3 and e.g. [D12]). G(x,t)nyG-l(xo,to) One checks e a s i l y t h a t under a gauge t r a n s f o r m a t i o n A -+

and (+) i m p l i e s t h a t

nk

Y depends o n l y on ( x o , t o ) and ( x , t )

i f Fo = F ( x o y t o ) i s g i v e n one can c o n s t r u c t F

A

( n o t on y ) .

Thus

F s a t i s f y i n g (11.1).

Y O

One problem 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 ( * ) i n v o l v e s q u a s i -

REmARK 11.1.

p e r i o d i c boundary c o n d i t i o n s Jl(xtEL,t) = e x p ( i e ) $ ( x , t ) fundamental domain -L 5 x 5 L (we l e t L c a l prob em).

-

-+

and one l o o k s a t a

t o obtain the standard c l a s s i -

These c o n d i t i o n s can be expressed v i a

(*A)

U(xt2LYt,x)

Q-’

=

( e ) u ( x yt x ) Q ( e ) , V(x+2L,t,x) = Q-’VQ where Q = e x p ( i e o 3 / 2 ) . The monodromy m a t r i x T i s d e f i n e d as (**) TL(A,to) = exp/ h L U(x,to,x)dx. I f one c o n s i d e r s -L 2 a box i n R w i t h s i d e s determined by t = tly t = t2, x = L, and x = -L w i t h pos it i ve y o r i e n t e d boundary one knows t h a t A = I from above and AYl yz = A 1~ i s e a s i l y v e r i f i e d . Hence (*&) S--1 TL-lY (t2)StTL(t,) = I where S+(Aytly

,

Y n t t 2 )= exp/ V(+L,t,x)dt. y

t

E v i d e n t l y S, = Q-’(e)S-Q(e) and (*&) become;

(*+)

T,- (x,t,)Q(e) = S+(tl , ,t,)T, ( x , t l )Q(e)s;’(t, , t 9 ) so t h a t T, ( x y t ) Q ( e ) a r e con. j u g a t e f o r d i f f e r e n t values o f t. T h i s i m p l i e s i n p a r t i c u l a r (*m) TrTL(x, I

t 2 ) Q ( e ) = TrTL(x,tl)Q(e)

-

and F L ( x ) = TrTL(X,t)Q(e)

L

(independent o f t ) w i l l

be a g e n e r a t i n g f u n c t i o n f o r c o n s e r v a t i o n laws o f (*) (see below and see [Fa31 f o r d e t a i l s ) .

Now more g e n e r a l l y one d e f i n e s a t r a n s i t i o n m a t r i x (A*) T(x,y,X) = expJX U(z,x)dz ( s o T L ( x ) = T(L,-L,A)- t i s momentarily suppresY DxT(x,y,x) = U(x,x)T(x,y,x) and sed). Then one checks e a s i l y t h a t (.A) Ir\

T(x,y,x)

=

I for x

= y (which c o u l d serve as an a l t e r n a t i v e d e f i n i t i o n o f

INTEGRABLE SYSTEMS

T(x,y,A)-’

so t h a t i ( x , A ) Hence f(x,y,A)

-

(Tt

(AA)

VT).

- Elbl2

2

= 1 and

- VT

one has O$

= E(x-y,A)

U ( X , A ) + iXu3/2.

Similarly

y,A)dz

+

/y”

r,?

- VT)

= U

T(x,y)V(y)

and

= 0 ( e x e r c i s e ) so FL w i l l

= DtFL(A)

I n order t o obtain properties (AA)

f o r E(z,A) = exp(Aza3/

T(x,z,A)Uo(z)E(z-y,X)dz

( ~ m )T ( X , ~ , A ) =

and one can s o l v e

T(x,y,A) where

Next d i f f e r -

(whose s o l u t i o n i s (*+)).

= [V(L,t,X),r]

TL, aL, bL, e t c . one can w r i t e f r o m

T(x,y,A)

-

= V(x)T(x,y)

y i e l d c o n s e r v a t i o n laws as mentioned above. (A*)

= sgnrc.

= TC, w i t h C = -V(y) v i a i n i t i a l v a l u e s

Consequently (Ad) Tt(x,y)

= TL(A,t)Q(0)

o f T(x,y,A),

E

t o o b t a i n Txt = UTt + UtT and use (*) t o g e t Dx(Tt

T h i s shows a g a i n t h a t DtTrTL(A,t)Q(e)

2i)

=

= T(y,x,A),

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

T = I f o r x = y.

r

T(x,y,A)

(A*)

= 0 one has d e t T(x,y,A)

w i t h e n t i r e aL, bL s a t i s f y i n g la1 entiate

cf. [Cl])

and D T(x,y,A) = - T ( x , Y J ) U ( y , A ) . Y = 1 (exercise). We n o t e a l = uU(x,x)u where u = u1 f o r K > 0 and u = u f o r K < 0. 2 = o T ( x , y , i ) u and one can w r i t e i n p a r t i c u l a r ( c f . § § 9 - 1 0 )

T(x,z,x)T(z,y,A),

F u r t h e r s i n c e TrU(x,A)

for

-

Then v i a s t a n d a r d ODE t h e o r y ( e x e r c i s e

T).

21 3

~ { x - y , ~+)

where Uo(x) =

/y”

E(~-~,A)U~(Z)T(~,

by successive approximations ( i t e r a t i o n ) :

= E(x-y,A)

+

12x-y E(x-z,X)F(x,y,z)dz Y

This leads w i l l s a t i s f y c e r t a i n i n t e g r a l e q u a t i o n s ( c f . [Fa3]). i n powers o f l / X f o r l a r g e r e a l

e.g. t o a s y m p t o t i c expansions f o r T(x,y,A)

M a n i p u l a t i o n o f such expansions l e a d s t o expressions f o r c o n s e r v a t i o n

A.

$,$ e t c . ( c f . [Fa31 and 189-10).

laws i n terms of i n t e g r a l s o f

RrmARK 11.2.

Now l e t L

-+

m,

assuming t h e p o t e n t i a l s $,;-+

-

0 at

m

suitably

1i m

Using (11.3) one shows t h a t T+(x,A) = y - t ~ m T(x,Y, A 1E(Y ,A ) 1i m 1i m e x i s t s f o r A r e a l and w i t h r - ( x , z ) = r(x,y,z), r + ( x , z ) = yF(Y x 3 z ) 3 Y+ one has (@*) T-(x,A) = E(x,A) + r-(x,z)E(z,A)dz and T+(x,X) = E(x,A) + (e.g.

$,$E

L’).

Im r+(x,z)E(z,A)dz X

-

-

where r + ( x , z ) = u 2 f ~ ( x , z ) u 2 ( c f . [Fa31 f o r e s t i m a t e s e t c .

t h e formal procedures a r e s t r a i g h t f o r w a r d ) .

isfy

(@A)

as x * i-.

¶

The m a t r i x f u n c t i o n s T+ s a t -

dF/dx = U(x,A)F w i t h a s y m p t o t i c c o n d i t i o n s T,(x,A) This equation

(@A)

= E(x,A)-+

o(1)

p r o v i d e s an i t m e d i a t e c o n n e c t i o n t o s c a t t e r N

LF = (A/2)F where L = i a D + iJK($U- - $ u + ) . 3 x I f K > 0, L i s f o r m a l l y s e l f a d j o i n t and t h i s corresponds t o no s o l i t o n s . If 1 2 1 one w r i t e s T+(x,A) (T,,T,) (Tj - a r e columns) t h e n exp(iAx/2)T! % ( o ) + o ( 1 ) i n g t h e o r y i n t h e form

(am)

21 4

ROBERT CARROLL

2 and e x p ( - i x x / 2 ) T + +

-,

-

( 01 ) + o ( 1 ) f o r ImA 2 0, 1x1 * 1 2 % ( o ) + o ( 1 ) and exp(-iAx/2)T,

%

1 e x p ( i x x / 2 ) T+

Q

w h i l e f o r ImA 5 0, IAl 0 ( 1 ) + o ( 1 ) . Next one

d e f i n e s a reduced monodronly m a t r i x T(A) by ( 0 6 ) T(A) = l i m E(-x,A)T(x,y,A) E(y,A) as x

-+

and y *

--.

Thus-in p a r t i c u l a r T(A) = lim LE(-L,A)T(L,-L,

I:[:(

E$[:I)

A)E(-L,A) and ( W ) T(A) = ( E = sgnw). The a,b s a t i s f y l a 1 2 2 I b l = 1, a(A) = l i m exp(iAL)aL(A), b(A) = l i m bL(A) as L + -, a(A) = d e t 1 2 1 1 (T- Tt), b ( x ) = d e t (T+ T-), and a(A) i s a n a l y t i c f o r ImX > 0 w i t h a = 1 +

E

-.

The a n a l y t i c c o n t i n u a t i o n o f a ( A ) i n t o t h e l o w e r h a l f o ( 1 ) as 1A1 * p l a n e i s g i v e n by $(A) = a ( i ) . As discussed i n [C37,38] i f K > 0 a ( x ) has no zeros f o r ImA > 0 w h i l e i f

K

0 we assume t h e ( f i n i t e number o f ) zeros

<

o f a ( x ) a r e s i m p l e and do n o t l i e on t h e r e a l a x i s .

Imx

> 0 one can w r i t e f o r

(11.4)

0 (a(A . ) J

<

K

G e n e r a l l y speaking f o r

0) 2

dp] n:(A-Aj)/(A-i

a(x) = exp[(l/2ni)fm P - A

.)

J

(Poisson-Jensen) and t h i s can be extended up t o t h e r e a l l i n e by t h e Sokhots k i j - P l e m e l j formulas

(0.)

isfies

= 1aI2

+

-, y

To

hence t h e r e a l and i m a g i n a r y p a r t s a r e r e l a t e d ( 6 * ) I m r l ( A ) = -(l/v)PL:

du ( t h i s i s c a l l e d a d i s p e r s i o n r e l a t i o n i n p h y s i c s ) . x

siG(p-A).

.)/(A-x .) which f o r A r e a l s a t J J Now n ( A ) = l o g a ( A ) i s a n a l y t i c f o r ImA > 0,

by a H i l b e r t t r a n s f o r m ( c f . [C2,3]) inediate.

t

= a(A) 111 (A-A

1-(b12.

-, and

= P(l/(p-A))

N

prove (11.4) one c o n s i d e r s :(A) vanishes s u i t a b l y a t

* (l/p-A-i0)

(l/p-A)

[Ren(p)/(p-A)]

From t h i s (11.4) i s

F i n a l l v t o o b t a i n t h e t i m e e v o l u t i o n o f a.b one can t a k e l i m i t s -- i n (A&). Then V(x,A) + V ( A ) = i A 2u3/2 and mu t i p l y i n g ( ' 6 )

-t

by E(y,x) f r o m t h e r i g h t b e f o r e t a k i n g l i m i t s y (11.5)

DtT,(x,A)

= VT,

-

-

-+

+-

(A r e a ) we g e t

( i h 2/2)T+03

-

F o l l o w i n g a s i m i l a r procedure r e l a t i v e t o x one o b t a i n s (e DtT(A,t) = ( i x 2 / 2 ) [ 0 3 , T ( ~ , t ) ] which leads t o o u r standard equations ( 6 . ) at = 0 and bt 2 = - i A b ( c f . 559-10). S i m i l a r l y one o b t a i n s formulas f o r t h e d i s c r e t e spect r u m and an expression l o g a ( A )

%

i

1;

[In/,ln] w i t h In r e p r e s e n t i n g conserved

q u a n t i t i e s ( c f . [Fa31 and 59).

R m R K 11.3. Vkl;Gkbl]

We go now t o t h e R-H problem and g e n e r a l l y r e f e r t o [Mvl;Gkal;

f o r d e t a i l s o f s o l v i n g such problems.

The problem i s s i m p l y t h a t

o f t a k i n g a m a t r i x f u n c t i o n G ( x ) on t h e r e a l l i n e and r e p r e s e n t i n g i t as ( 6 6 ) G(A) = G+(A)G-(A) where G

may be a n a l y t i c a l l y c o n t i n u e d i n t o t h e upper

and l o w e r h a l f planes r e s p e c t i v e l y .

There has been a l o t o f r e c e n t s t u d y o f

such problems by Dym, Gokhberg, Krein, e t . a l . and we do n o t a t t e m p t t o g i v e

INTEGRABLE SYSTEMS

21 5

The way t h i s problem a r i s e s h e r e goes as r e f e r e n c e s (see e.g. [D2,3,5]). 1 2 1 f o l l o w s ( c f . a l s o §2.5). = (T- T), and S-(x,A) = (Tt D e f i n e (&*) S,(x,A) 2 T ) which s o l v e dF/dx = UF and e x t e n d a n a l y t i c a l l y i n t o t h e upper and l o w e r h a l f planes resp. where S+(x,A)E-l(x,A) T (x,A)

= T,(x,A)T(A)

one w r i t e s ( 6 . ) S,

= I t o ( 1 ) as = T,M,+(A)

1x1

-f

= T-M-,(A)

01.

Then u s i n g and S- = T,

= T-M -- ( A ) where ( c f . 52.4 where t h e n o t a t i o n and o r g a n i z a t i o n a r e a 1ittl e d i f f e r e n t )

M,-(A)

a Eb The reduced monodromy m a t r i x T = ( b a ) can t h e r e f o r e be w r i t t e n as T = M,M-- l = MttM:l

where S(A) = MMl;-, 3-5). that (x,A)

%*o"

1

= S+(x,A)S(A)

'la Eb/a) i s t h e s c a t t e r i n g m a t r i x ( c f . I (-b/a l / a O) f o r E = -1 i t f o l l o w s 3 - ( 0 -1 G where S* = ST ( i . e . S i s T u n i t a r y ) . Set now (*.)

=

MM ,:-

Setting Z = If o r (*A)

becomes i n terms o f S, (**) S-(x,A)

and T- = T,

= S-'

E

=

= 1 and Z = u

= S - ( ~ , h ) E - ~ ( x , h ) and G,(x,A)

= a(A)E(x,A)S;l(x,A)

( n o t e detS,

= a):

Then one can s t a t e (see [Fa31 f o r d e t a i l e d p r o p e r t i e s o f G+, - G, e t c . and a discussion o f t h e s o l u t i o n technique)

ME0REIII 11.4,

The m a t r i c e s G,(x,A)

G ( ~ , A ) where-G(x,A) 1 Eb

s o l v e t h e R-H problem G,(x,A)G-(x,A) = 1 c b e x p ( - i x x ) ) and = E ( ~ , A ) G ( A ) E - ~( x , ~ ) = ( - b e x p ( i A x ) 1

G(A) = ( - b 1 1.

P a r t l y i n t h e i n t e r e s t o f c o n n e c t i n g n o t a t i o n s we n o t e h e r e REMARK 11.5. t h e r e l a t i o n between t h e R-H problem and t h e G-L-M e q u a t i o n s ( c f . a g a i n 13ru 1 2 1 5 ) . From T- = T, one has (*.) ( l / a ) T (x,A) = T+(x,A) + rT,(x,A) (r = b/a) 2 = i u 2 f o r K < 0. We = u1 for K > 0 and and f l ( x , i ) = GT,(x,A) where 1 w r i t e T 1 (x,A.) = y . T 2 (x,A.) when a ( A . ) = 0 and residue(l/a(A))T-(x,A)lA=Aj 2 J J + J J A . ( j = 1 ,..., N) w i t h = c.T,(x,~.) (c = yj/a'(Aj)). Then g i v e n r, c J J j ji J 1 suitable properties one wants t o f i n d ,T, ,T, and ( l / a ) T w i t h t h e nece s s a r y a n a l y t i c p r o p e r t i e s . We w r i t e e.g. ( c f . (**)) (*&) T,1 :- ( 01) e x p ( - i A x / 1 2 0 0 r , ( ~ , y ) ( ~ ) e x p ( - i ~ y / 2 ) d y w i t h T, = (1 ) e x p ( i A x / 2 ) + IX rt(x,Y)(l w)exP 2) + ( i A y ) d y and p u t t h i s i n (0;) t o o b t a i n f o r y 2 x t h e f a m i l i a r (**) r,(x,y) t A(xty) t r,(x,s)A(sty)ds = 0 where A ( x ) = w(x)u- + E;(x)u+ w i t h w(x) =

<

( 1 / 4 i ~ ) [ z rexp(iAx/2)dA t ( 1 / 2 i ) l Y c j e x p ( i A . x / 2 ) . A s i m i l a r equation holds J The s o l u t i o n o f t h e f o r r - as usual and e.g. Uo(x) = u3r,(x,x)03 - r,(x,x). M t y p e equations (**) i n v o l v e s compact i n t e g r a l o p e r a t o r s w h i l e t h e s o l u t i o n o f t h e R-H problem by t h e methods o f say Gokhberg-Krein i n v o l v e WienerG;'(x,A) = I + ;/ A+(x,s) Hopf (W-H) o p e r a t o r s . Indeed one can w r i t e (*.) exp(iAs)ds and G-(x,A)

= I

t

/;@_(x,s)exp(-iAs)ds

( c f . Remark 11.3 f o r G+). -

21 6

ROBERT CARROLL

= G;'(x,X )G(x,X) f r o m Theorem At(~,s) t Q,(x,s) t ;1 A,(x,t)@(x,s-t)

F u r t h e r , 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 G-(x,h)

11.4 we o b t a i n t h e W-H e q u a t i o n (.*) d t = 0 ( s 2 0 ) whereQ,(x,s) = (-B(s-x) E B ( - S - X ) ) w i t h ~ ( s )= (1/21r)J; b(x) exp(-iXs)dx ( t h u s Q, i s known and At i s t o be determined w h i l e G(x,X) = I t [: @(x,s)exp(iXs)ds). One can t h e n w r i t e e.g. Q,- i n terms o f A, v i a @-(x,s) = @(x,-s) f ;1 A,(x,t)@(x,-s-t)dt. It i s shown i n [Fa31 how t h e G-L-M equat i o n s ( W ) can be o b t a i n e d from t h e W-H e q u a t i o n (.*) t h r o u g h a p a r t i c u l a r " r e g u l a r i z a t i o n " . We o m i t d e t a i l s h e r e and remark o n l y t h a t i n (m*) t h e 2 o p e r a t o r I + Q, can be w r i t t e n as A t K ( i n an L c o n t e x t ) where A - l i s a bounded o p e r a t o r and K i s compact; t h e n (It @ ) f= g becomes f t A - l K f = A-'g w i t h A- 1 K compact and t h i s " t r a n s f o r m " o f ( I t @)f= g i s c a l l e d regul a r i z a t i o n . One notes t h a t t h e r e i s a t r a d e o f f i n d e a l i n g w i t h W-H o r G-L -M techniques; t h e G-L-M equations a r e a s s o c i a t e d w i t h compact o p e r a t o r s ( b u t t h e t h e o r y i s g l o b a l ) w h i l e t h e W-H o p e r a t o r s a r e Fredholm (A+K = I@) With t h e t h e o r y b u t one has c e r t a i n n i c e l o c a l c h a r a c t e r i s t i c s ( c f . [Fa3]). o r g a n i z e d i n t h i s way one can now n a t u r a l l y proceed t o t h e development o f t h e H a m i l t o n i a n s t r u c t u r e v i a t h e so c a l l e d r - m a t r i x (which i s a l s o n a t u r a l i n t h e q u a n t i z e d v e r s i o n ) and we r e f e r t o [Fa3,5]

f o r this.

L e t us s k e t c h n e x t some f a s c i n a t i n g r e l a t i o n s i n v o l v i n g Backlund and gauge t r a n s f o r m a t i o n s , L i e algebras, e t c . which have been developed i n e.g. 2;Nll ;Guzl ;Sat1 -61 ( c f

. a1 so

[Botl

,

[Ao5-8; Bvl -3; Ku2; l a 1 ;Fbl ,2; Ce2; R f l ;Hcl ,2 ; W l l ] ) .

We o n l y g i v e some b r i e f i n d i c a t i o n o f t h e t h e o r y i n o r d e r t o i n d i c a t e t h e flavor.

For connections, c u r v a t u r e , and gauge t r a n s f o r m a t i o n s we r e f e r a1 so

t o Chapter 3; t h e p r e s e n t d i s c u s s i o n w i l l serve as a n i c e a d j u n c t t o 53.9Thus one views t h e Lax equations Lt = [B,L]

3.10.

o f 98 as a z e r o c u r v a t u r e

c o n d i t i o n ( c f . ( 4 ) ) . To see how t h i s goes ( f o l l o w i n g [Sat5,6]) t a k e e.g. t h e Sine-Gordon (S-G) e q u a t i o n uxt = S i n u w r i t t e n as a system ux = p and pt = Sinu.

I n x,t,u,p space, considered e v e n t u a l l y i n a bundle c o n t e x t B: (u, one t h i n k s o f d i f f e r e n t i a l forms (.A) a1 = du A d t - pdx A d t

p ) over ( x , t ) ,

and

a2 =

dp A dx + Sinudx A d t .

(x,t),p(x,t)) 1 d t and

A s e c t i o n o f t h e bundle i s a graph (x,t,u and on such a s e c t i o n a l , a2 t a k e t h e form (ma) Z1 = (ux-p)dx

Z2 = ( S i n u

-

pt)dx A d t .

One r e c a l l s now t h e Frobenius theorem o f

j i f f e r e n t i a l geometry s t a t e d i n terms o f d i f f e r e n t i a l i d e a l s ( c f . [AbZ;Wrl]). Thus c o n s i d e r forms Z ( h e r e al, a 2 ) o v e r B ( g e n e r a l l y o v e r some j e t bundle c f . Appendix C ) . {af) =

i f dI

C

L e t I ( Z ) be t h e e x t e r i o r i d e a l generated by Z ( i . e . I(Z) = I(X) i s called a d i f f e r e n t i a l ideal ai E Z; Ei E A ( B ) l ) . I and we n o t e t h a t s i n c e dal = d t A a2 and da2 = Cosu dx A a 1 one can

{I ti

A oi;

21 7

INTEGRABLE SYSTEMS

say I(Z) i s a d i f f e r e n t i a l i d e a l here.

The t h e Frobenius theorem s t a t e s

t h a t t h e r e i s a unique maximal connected i n t e g r a l m a n i f o l d f o r X o f approp r i a t e dimension a t every p o i n t o f B (where b y i n t e g r a l m a n i f o l d B one means v*u = 0 f o r u

i n Appendix C ) .

Thus e.g.

E I(Z)

-

recall

v*

%

v:

NC B

& and ( v * u , v ) = (u,v,v)

+

as

t h e forms v a n i s h on k - t u p l e s o f t a n g e n t v e c t o r s

d e t e r m i n i n g N and one i d e n t i f i e s h e r e t h e equations s o l u t i o n m a n i f o l d o f t h e S-G equation.

gl

=

z2 = 0 w i t h

the

There i s t h e n a p r o l o n g a t i o n t h e o r y The

a s s o c i a t e d w i t h I ( Z ) which we RTainly i g n o r e h e r e ( c f . [HclY2;Rfl;W11]).

i d e a ( f o l l o w i n g [Sat5,6]) i s t o add v a r i a b l e s yl,...,yn and 1-forms (me) k k k 1 n wk = dy + F dx + G d t such t h a t I w ,...,w , alYa2,...) ( h e r e I(alya2?w k ) ) k be c l o s e d under d. Here Fk, G a r e assumed t o depend o n l y on (u,p, y’) and A dx + Fkdp A d x + 1 aF k/ay j dy j A dx ( m 6 ) dwk = dFk A dx + dG k A d t = Fudu k P + Gkdu A d t + Gkdp A d t + a G k / a y j d y j A d t . B u t f r o m (me) d y j A dx = w j P u . A dx - GJdt A dx and d y j A d t = w j A d t - FJdx A d t w h i l e dp A dx = a 2 Sinudx A d t and du A d t = al + pdx A d t . The p r o l o n g a t i o n e q u a t i o n s a r e ob-

1

k

t a i n e d by s u b s t i t u t i n g t h e s e i d e n t i t i e s i n t h e e x p r e s s i o n ( W ) f o r dw and k k r e q u i r i n g t h a t dwk = 0 modulo t h e i d e a l I(alYa2,w ) generated by a1,a2, w . For t h e p r e s e n t s i t u a t i o n one o b t a i n s ( e x e r c i s e

- c f . [Sat5,6])

1

(me) Fu = 0, ‘ k j FJ(aG lay ).

G = 0, [G,F] + pGu - SinuF = 0 where [G,F] = Gj(aFk/ayJ) P P Now ( f o l l o w i n g [Sat5,6]) t o r e l a t e t h i s t o i s o s p e c t r a l e v o l u t i o n assume vec-

t o r f i e l d s F,G have been found which a r e l i n e a r i n y so F =

1 yJFk(a/ayk) J

1 yjGi(a/ayk). A Lax p a i r i s g i v e n i n terms o f t h e m a t r i c e s F = k (( F;) and G = (( G.)) by L = -Dx + F and B = G. Then t h e a s s o c i a t e d isospecJ t r a l s c a t t e r i n g problem i s g i v e n by Ly = 0 o r a y j / a x - Fiyk = 0. To see how

and G =

t h i s goes f o r t h e S-G e q u a t i o n we have ( c f . (me)) [B,L]

= [G,-Dx+F]

= [Dx,G]

- pGu = SinuFP and Lt = Fu u t + P Thus t h e S-G e q u a t i o n i n v o l v e s Lt = [B,L] as i n 88. Now t o Fppt = Fpuxt. i n t e r p e r t t h i s i n gauge t h e o r y ( c f . Chapter 3 ) l e t a x = Dx + pDu + p D +

-

= DxG

[F,G]

... and a t

- [F,G]

= Gup

+ G p + SinuF P X

X P

+ ut Du + p t Dp + ... ( t o t a l d e r i v a t i v e on t h e j e t b u n d l e obSet Dx = a x - F and Dt = a t - G so t h a t (m.) [Dx,Dt] = [ax t a i n e d f r o m B). = Dt

- Gup + pGu - SinuFP = (pt - S i n u ) FP’ P t Thus t h e c u r v a t u r e [DxyDt] o f t h e c o n n e c t i o n determined by F,G vanishes on an i n t e g r a l m a n i f o l d o f t h e S-G e q u a t i o n . Next c o n s i d e r sl(2,R) as a gauge 0 1 0 0 group, w i t h g e n e r a t o r s (**) u1 = (’ O), u2 = ( 0 -1 0 0 ), and u3 = ( 1 o ) ( c f . Appendix C). Thus [a 1 ,u 2 ] = 2u2, [ul,u3] = -2u3, and [u2,u3] = u l . Cons i d e r t h e connection determined by (**A) D, = a X + isul - qu2 - ru3 and 1)t = 0 one o b t a i n s t h e at - Aal - Ba2 - Cu3. S e t t i n g t h e c u r v a t u r e [Dx,Dt]

-

F,at

-

G I = atF

-

a x G + [F,G]

= F p

21 8

ROBERT CARROLL

standard c o m p a t a b i l i t y c o n d i t i o n (9.3) f o r AKNS systems, namely, -axA t qC - r B = 0, qt - axB - 2(qA t i s B ) = 0, and rt - a x C t 2 ( r A t i s C ) = 0. I f f o r example one l o o k s a t a NLS e q u a t i o n w i t h ( M e ) r = $, q = $, A = - 2 i s 2 i1$I2, B = 2s$ t bhX, and C = 2sq t h e n [DxyDt] (i$t + $xx + ($1 24 ) u2 t (-$ht t B = b/s, C =

sXx t

I$[

-

is,

2-

For t h e S-G e q u a t i o n one takes (*&)

$)u3.

A = a/<,

D, = a x + isul + uX(u2 - u3)/2, Dt a t - ( 1 / 4 i s ) C o s u ~ -~ ( 1 / 4 i 5 ) S i n u ( u 2 + u 3 ) w i t h [D,,D~] = ( ux t - S i n u ) ( u 2 - u3)/2.

RRRARK 11.6.

C/C,

L e t us s k e t c h b r i e f l y some p r o p o s i t i o n s f r o m [Sat51 t o phrase Thus ( c f . Chapter 3 and Appendix C f o r i d e a s

t h e s i t u a t i o n more g e n e r a l l y .

from d i f f e r e n t i a l geometry) l e t P be a p r i n c i p l e bundle f o r a L i e group G over

R 2 and (D,,D~)

be a connection on P ( i . e . a c o v a r i a n t d e r i v a t i v e h e r e ) .

L e t @ be a fundamental s o l u t i o n m a t r i x o f Dx(c)@ = 0 ( 5 i s t o be s i m p l y a = 0 t h e r e e x i s t s a s i n g l e valued parameter i n D, h e r e ) . Then i f [Dx,Dt] o f t h e equations Dx(s)@ = 0 and Dt@ = 0; CP takes values i n

s o l u t i o n @(x,t,c)

To see t h i s one w r i t e s Dx@ = 0 i n t h e f o r m ax@ =

G and i s a s e c t i o n o f P. lb w i t h U

r = L i e a l g e b r a o f G (as above) and t h i s can be viewed as a d i f -

E

Recall t h a t t h e L i e a l g e b r a i s a s s o c i a t e d w i t h

f e r e n t i a l e q u a t i o n on G.

Since D, and Dt comnute t h e f l o w s gener-

l e f t i n v a r i a n t vector f i e l d s etc.

a t e d by t h e s e o p e r a t o r s commute and @ ( x , t ) depends o n l y on t h e base p o i n t (Proposition 1).

1x1 +

m

w i t h D,

A+ as x +

+m

Assume now t h e c o e f f i c i e n t f u n c t i o n s i n D, t e n d t o 0 as 'L

aX

-

U, a t

f m

and a l g e b r a s f o r convenience). i s t s S such t h a t cp as 1x1

+

m.

Then D ~ @= 0 i m p l i e s CP

(Ut E 9 ) .

'L

exp(U,x)

w i t h A+ c o n s t a n t m a t r i c e s (one i s t h i n k i n g o f m a t r i x L i e groups 'L

Choose

exp(U,x)S(t,<)

exp(U,x)

@

'L

as x

+

-.

Then ( P r o p o s i t i o n 2 ) i f U [ V, ],

at

-m

and t h e n t h e r e ex-

Suppose f u r t h e r Dt

'L

a t - V,

= 0 the time evolution o f t h e

s c a t t e r i n g d a t a S ( t , c ) i s g i v e n by St - VtS = 0. To see t h i s assume t h e asy m p t o t i c b e h a v i o r @ 'L exp(U,x)S(t,s) i s u n i f o r m i n t and d i f f e r e n t i a b l e so t h a t ~~[exp(U,x)S(t,3)] as x

+

m,

'L

-

exp(U,x)St

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

-

SV,

V,exp(U,x)S

= 0.

= exp(U,x)(St

-

VtS)

0

'L

T h i s " e x p l a i n s " t h e success o f t h e

i n v e r s e s c a t t e r i n g method i n showing t h a t one can s o l v e f o r t h e e v o l u t i o n o f S w i t h o u t knowledge o f t h e s o l u t i o n s o f t h e a s s o c i a t e d n o n l i n e a r equation.

REIMRK 11.7. features.

We go n e x t t o [Sat3,4] Consider (*+)

D,(z,Q)

=

i n o r d e r t o e x t r a c t a few a d d i t i o n a l

ax - aJ

(z,Q) ( i n t h e s p i r i t above) where J,K a r e m a t r i x w i t h v a n i s h i n g diagonal elements. L i e a l g e b r a ;and t h a t Br i s a polynomial o r d e r r - 1 . For s u i t a b l e Br t h e c o n d i t i o n

-

Q and Dt(z,Q)

a t - zrK

-

Br

d i a g o n a l m a t r i c e s and Q ( x , t ) i s a One assumes J,K,Q t a k e values i n a i n z,Q, and d e r i v a t i v e s of Q up t o [D,,D~]

= 0 leads t o a c o m p l e t e l y

INTEGRABLE SYSTEMS

21 9

i n t e g r a b l e e q u a t i o n i n Q. Now one d e f i n e s a gauge t r a n s f o r m a t i o n G: Q 1 -+ Q2 t o be a m a t r i x valued f u n c t i o n G(x,t,z) bounded i n x on (-m,m) and a n a l y t i c

i n z such t h a t (**.) GDx(z,Ql)

= Dx(z,Q2)G and GDt z,Q,)

= Dt(z,Q2)G.

= 0 so such

= 0 i m p l i e s [Dx(z,Q2),Dt(z,Q2)]

f o l l o w s t h a t [Dx(z,Ql),Dt(z,Ql)]

It

transformations w i l l preserve s o l u t i o n s o f t h e associated nonlinear d i f f e r It can be shown ( c f . [Sat3,4;Bv3])

e n t i a l equation.

z t h e n G = @(z-C)@-',

= (al

,...,an), where

wave f u n c t i o n s a . s a t i s f y Dx(zj,Q,)Oj J = Q, + [G,J] and as x -+ +m G(x,t,z) (at

where

-m)

T

-

Go,

C = diag(zl

,...,zn),

= 0 and Dt(zj,Ql)aj -+

-

Z

i s a permutation matrix.

gauge t r a n s f o r m a t i o n G s a t i s f i e s ("*) G = z

that i f G i s linear i n

C (at

= 0.

o r G(x,t,z)

m)

and t h e

F u r t h e r Q, -1 -+ Z-TCT

It f o l l o w s f o r example t h a t t h e

-

+ Q2G

Gx = z[J,G]

GQ,

and p u t t i n g

i n t o t h i s equation leads t o a p a i r o f m a t r i x R i c c a t i

Go = @C@-',

e q u a t i o n s ~ ( A b )DxGo = [Ql,Go]

t [J,Go]Go

+ Go[J,G,].

and DxGo = [Q2,Go]

Now Backlund t r a n s f o r m a t i o n s a r e a s p e c i f i c f o r m o f gauge t r a n s f o r m a t i o n and examples based on Dx w i t h J = (; and Q = ( o- q oq ) o r (!4 :) a r e used i n

-p)

[Sat3,4]

as a b a s i s f o r some 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 r e l a t e d t o [ B v l - 3 ]

and t o o t h e r r e c e n t developments. I n 59 when d e a l i n g w i t h AKNS systems we passed r a t h e r q u i c k l y

RmARK 11.8.

o v e r t h e p r o p e r t i e s o f 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 s v , $, e t c . i n o r d e r t o s t u d y t h e squared e i g e n f u n c t i o n s .

It w i l l be i n s t r u c t i v e t o f i l l i n t h i s

gap and develop h e r e t h e completeness r e l a t i o n s f o r

v,$,

e t c . f o l l o w i n g [Ao

41. Thus r e f e r r i n g back t o 59, between (9.4) and (9.10), and t o [Ao1,2,4] u1v2 - u2v1 and f o r two f o r a d d i t i o n a l d e t a i l i f needed, r e c a l l W(u,v) s o l u t i o n s o f (9.1) w i t h d i f f e r e n t values

5 , s ' o f t h e e i g e n v a l u e one can

w r i t e (u@) D x W ( u ( s ' ) , v ( s ) ) = i ( s - c ' ) [ u 2 ( c ' ) v 1 ( ~ ) + ul(c')v2(c)1 ( c f . (01 i n l i ( u ' , v ) = L- m /L-L [ u l2 510). D e f i n e an i n n e r p r o d u c t ( u ' % u ( L ' ) e t c . ) (a&) v1 + uiv2]dx. Consider " b a s i s " v e c t o r s v,? f o r s 5 r e a l , vm = v ( s m ) , 1 5 m < N, and $

D e f i n e a d j o i n t s t a t e s ( c f . 510 where T % A k k AA 'k " k ($ ,$ ), v k = (d' ,$ 1, P k = ($2,$1) which A ) by W ) = iA 'A A 'A s a t i s f y (urn)uA - i c u l = r u 2 and u2x + i s u = qul. Then ('A&) becomes LJ i m /-LL" u l T * v d x . One r e c a l l s t h a t P[exp(+iaL)/ia] = +aS(a) ( c f . =

G(ck), 1 5 k 5

N.

LA ($2,$1i,*

(U',V)

:$

=

(10.21) and Appendix B) and i t can be shown e a s i l y t h a t = 2nas(5'-c),

($(cl),v(c)) A

=

(v(ct),G(c)) =

0,

(A@*)

($(c1),$(c))

(v(c'),v(c))

= -~Gs(c'-s),

h

To see how iaktikm, and (pk,vm) = i $ i 6 k m ( a ' * (da/d3), e t c . ) . A /-LLv(c') v ( c ) d c = l i m 1 ($ipl + J/ivZ)dx = t h i s goes c o n s i d e r e.g. (A@A)

(qk,qm) =

tE

lim/DxW($(c'),v(c))dx/i(c-c') ( l0) e x p ( i g x )

and p

P\I

= l i m [W(L)

- W(-L)]/i(c-c').

( ~ ~ ~ ~ [w h~ i l $e a) t )-a,

~p

p\r

But a t ( 01 ) e x p ( - i g x ) w i t h

-, $

%

220

ROBERT CARROLL

JI plr ( a'exp(-iEx)). exp(igx) Hence W(L) % - a ( S ) e x p [ i L ( S ' - c ) ] and W(-L) % - a ( S ' ) e x p [ i L ( s ' - s ) ] so (A@*) ( v ' , v ) = l i m a ( ~ ) e x p [ i L ( ~ ' - t ) ] / i ( c ' - c ) + l i m a ( c ' ) (S '-5 ) = 2na(S )6 ( 5 '-5 ) as d e s i r e d . The o t h e r d e r i v a t i o n s exp[-iL(S '-5 )]/i 4

Now one t a k e s again C (resp. C ) t o be contours above (resp.

are similar.

below) t h e zeros o f a (resp.

(A

(11.7)

;)a(x-y)

2) and

= (1/2n)lC v(c,xbA(c,Y)ds/a(s)

v vA

-

(5, x )GA(c ,y ) d d a (c

(1/2n Note e.g.

t h e completeness r e l a t i o n i s

i s a m a t r i x w i t h e n t r i e s vi(x)J/.(y)

as usual and t h i s n o t a A t i o n i s c o n s i s t e n t w i t h (A@&) f ( x ) = ( l / 2 n ) J c (ds/a)v(c,x)/ v ( c , y ) - f ( y ) d y - ( 1 / 2 n ) / t (dc/a)d(G,x)/ i * ( c , y ) - f ( y ) d y . To v e r i f y completeness i n L2 w i t h

J

.y

(11.7),

i n [A041 one e v a l u a t e s t h e r i g h t s i d e o f (11.7) by independent meR e c a l l t h a t t h e e x i s t e n c e and uniqueness of

thods, namely by u s i n g (9.5).

K, K i n (9.5) f o l l o w s f r o m existence-uniqueness i n t h e Goursat t y p e problem (9.6) (under s u i t a b l e hypotheses on t h e p o t e n t i a l s q , r ) . Thus one p u t s (9. 4

F i r s t however r e c a l l v = aJ/ + bJI and $ = T 0 1 bJ/,)J/ u1 where u1 = (1 o ) . T h i s becomes t h e n

5 ) i n t h e r i g h t s i d e R o f (11.7). 4

4

-a$ + bJ/ so e.g.

v XvAY

= (ajx t

a f t e r some c a l c u l a t i o n ( 0 = Heavyside f u n c t i o n ) 0 1 1 T 0 "T R = (1 O ) ~ ( Y - X )+ [ ( o ) K ( ~ 9 x 1+ (1)K (Y,x)Iole(x-Y)

(11.8)

(1 0) + I?(x,y)(O 1)]ule(y-x) kT(x,t)]u,

+ (;)(O l)ulF(x+y)

-?T(xys)F^(s+y)(l O)lul

+

+

-

=(

d s q dt6(t-s)[l?(x,s)KT(y,t)

[K(x,Y)

+ K(x,s)

+ jxw ds[KT(x,s)F(s+y)(O

( A ) ( 1 O)ul;(x+y)

O T dsC(l ) K (y,s)F(s+x)

+

-

1)

1 ^T ( 0 ) K ( Y , S ) ~ ( X + S ) I O+~

,;" d s q d t [ K( X, s ) KT(y,t )F( s + t ) - ^K( x , s ) t T ( y , t ) F^( s + t )]ul One assumes c o n d i t i o n s on t h e p o t e n t i a l s so t h a t t h e

M e q u a t i o n s (9.8)-(9.9)

A

hold f o r y

x (F,F b e i n g g i v e n by (=) i n 89) and t h e r e f o r e (11.8) reduces ( a f t e r some r e o r g a n i z a t i o n ) t o R = ;)a(y-x) as d e s i r e d . Thus (11.7), f i r s t d e r i v e d v i a o r t h o g o n a l i t y (A@*) a c t u a l l y h o l d s i n i t s a c t i o n on L2

(y

v e c t o r s v i a t h e independent v e r i f i c a t i o n (11.8) and we s t a t e t h i s as The completeness r e l a t i o n s f o r AKNS e i g e n f u n c t i o n s CHE0RRn 11.9. i s expressed v i a (11.7) ( f o r s u i t a b l e p o t e n t i a l s q,r).

v , 4, J / ,

$

RrmARK

(F0RCINC'). L e t us b r i e f l y i n d i c a t e n e x t an area o f c u r r e n t i n t e r e s t w h i c h has been e x t e n s i v e l y s t u d i e d by Kaup [ K p 3 - l l ] ( c f . a l s o [C37, 11.10

INTEGRABLE SYSTEMS 38,421).

221

We g i v e h e r e o u r f o r m u l a t i o n o f t h e m a t t e r , m a i n l y f o r t h e NLS

equation.

T h i s o n l y s e t s t h e s t a g e h e u r i s t i c a l l y and we hope t o s t i m u l a t e Kaup's r e s u l t s go much

i n t e r e s t i n f u r t h e r s t u d y o f t h e problems i n d i c a t e d .

f u r t h e r i n c e r t a i n d i r e c t i o n s and he has numerical background i n f o r m a t i o n . Thus t a k e a n o n l i n e a r e v o l u t i o n f o r say q ( x , t )

which i s " f o r c e d " b y an i n -

p u t o f t h e f o r m q ( 0 , t ) = Q ( t )( w i t h q(x,O) a l s o p r e s c r i b e d , 0 i x < =).

The

q u e s t i o n now i s n o t one o f existence-uniqueness (which can be s t u d i e d separa t e l y ) b u t r a t h e r t o use an i n v e r s e s c a t t e r i n g t e c h n i q u e i n such a way as t o determine s o l i t o n b e h a v i o r v i a s p e c t r a l i n f o r m a t i o n .

F o l l o w i n g Kaup such a = P(t)

procedure i n v o l v e s o v e r d e t e r m i n i n g t h e system by s p e c i f y i n g qx(O,t) i n o r d e r t o o b t a i n t h e t i m e e v o l u t i o n o f t h e s p e c t r a l data. lems have been s t u d i e d ( c f . [Kp3-11])

Many such prob-

b u t t h e t h e o r y i s n o t y e t complete.

Thus one l o o k s f o r t h e t i m e e v o l u t i o n s u i t a b l e s p e c t r a l d a t a (e.g. c o n s t r u c t s r e c o v e r y formulas f o r q ( x , t )

a,$)

and

Take r = q

i n terms o f such data.

=,O

f o r x < 0 i n ( 9 . 1 ) - ( 9 , 3 ) and a s i m p l e argument y i e l d s (A*+) $(O,t) = 1 0 ( b ( c y t ) ) ( n o t e v = ( 0 ) e x p ( - i c x ) and $ = ( - l ) e x p ( i s x ) f o r x 5 0 so f r o m IL = a, ( 5 , t) b - av one g e t s (A*+)). The c;assical AKNS procedure f o r d e t e r m i n i n g t i m e e v o l u t i o n o f s p e c t r a l d a t a has t o be m o d i f i e d h e r e ( c f . Remark 9.4). sume q,r,qx,rx

-+

0 at

m

as r a p i d l y as needed and s e t A,

We as-

= l i m A(x,t,c)

as x

One c o n s i d e r s as i n (9.11) t h e s o - c a l l e d t i m e dependent e i g e n f u n c t i o n s I n orJlt = $exp(-A,t),v t = qexp(A,t), it = $exp(-A+t), and = $exp(A+t). d e r t o s a t i s f y (9.2) we r e q u i r e now -f

=.

it

which i s compatable w i t h t h e a s y m p t o t i c c o n d i t i o n s (9.4) as x a t A,B,C

-+

f o r x = 0 f o r NLS w i t h Q ( t ) = q ( 0 , t ) and P ( t ) = qx(O,t)

Now l o o k

=.

(note P i s

n o t determined a p r i o r i by Q - i t must be s p e c i f i e d i n d e p e n d e n t l y - a l s o q 2 with A = i I q l + = kF f o r NLS). Thus as t y p i c a l examples t a k e (A*=) r = 2 2 2 2 i 5 , B = -q - 25q, C = i i j - 259, A, = 2 i 5 , and i q t = qxx - 2 i I q l q o r X x2 2 + 254, A, = (A&*) r = w i t h A = -i I q l t 2 i 5 , B = - i q X - 25q, C = -iiX

-TI

2 i c 2 , and i q t = qxx + 21qI2q. 2ic2, B(O,t,c) (o,t,L) (A&*)).

(11.10) (A&*)

=

= -iP

-ij9I2

t

-

One has t h e n e i t h e r

259, C ( O , t , c )

2is2, B(O,t,c)

=

iP -

-

= -if

( A U )

2541 ( i n case

259, C(O,t,c)

A(O,t,c) (A*=))

=

or

i1QI2 +

(A&.)

A

= -i? + 2 5 i ( i n case

From (11.9) a t x = 0 we have t h e n (A*=)

^bt

it = ( - i 1912

= (ilQl

2

t 4ic2);

4ic2)^b - ( i P t 259)a; at = (iP-254)G

t

-

( i P t 25Q)a; at =

(-is + 2 5 i ) G

- i I Q I 2a

+ i 1Q12a

222

ROBERT CARROLL

We n o t e t h a t t h e s i t u a t i o n (no s o l i t o n s ) w h i l e

(A&*)

(A@=)

i s s e l f a d j o i n t w i t h no zeros o f a o r

does n o t p r e c l u d e s o l i t o n s ( c f . [Aol;C38]).

a^ Our

formulas (11.10) a r e e s s e n t i a l l y t h e same t h i n g as i n [ K p l l ] b u t w i t h a d i f -

6.

I t i s perhaps worth symmetrizing e.g. f e r e n t f o r m u l a t i o n ( c f . [C37,38,42] ( A @ = ) i n (11.10) by w r i t i n g ( A & & ) ( a ) = ( B ) e x p ( 2 i c 2 t); Bt - 2 i c 2 B = i 1 Q 1 2 B

-

( i ~ + ~ r ; ~ )ata ;t 2 i s 2 a =

CHE0REm 11.11. i n t h e cases

- i l q 1 2 a + ( i i - 2a-5 ~ ) ~ .

The t i m e e v o l u t i o n o f s p e c t r a l d a t a :,a (A@m)

or

(A&*)

f o r t h e f o r c e d NLS

w i t h Q and P g i v e n independently i s determined

by t h e equations (11.10).

REmARK 11.12.

The new f e a t u r e here, discussed by Kaup and o t h e r s i n [Kp3-

111 i s t h e o v e r d e t e r m i n a t i o n .

Many models, a p p r o x i m a t i o n schemes, and num-

e r i c a l r e s u l t s a r e developed i n [Kp3-11].

The approach o f [C37,38,42]

which

i s developed here aims a t a p u r e l y mathematical r e s u l t (a f i x e d p o i n t theorem) and produces a framework which g e n e r a l i z e s n i c e l y .

There a r e many p r o -

m i s i n g f e a t u r e s and some numerical work i s i n progress.

Thus one assumes (a

suitable)

Q

i s g i v e n and assigns independently ( a s u i t a b l e ) P; t h e n q and

qx a r e computed d i r e c t l y by s p e c t r a l i z i n g t h e MarEenko k e r n e l s a t x = 0. This y i e l d s o f t h e map

($,i) = (qx(O,t),q(O,t)) m.

= N(P,q) and one asks f o r a f i x e d p o i n t

The P determined i n t h i s manner t h e n leads t o q ( x , t ) f o r x , t

> 0 v i a i n v e r s e s c a t t e r i n g and t h i s should a l l o w one t o study s o l i t o n dynamits e t c .

Requirements on P and Q should emerge from t h e i n v e s t i g a t i o n o f

The program i n d i c a t e d h e r e o f [C37,38,42]

m.

i s s t i l l i n t h e e x p l o r a t o r y stage

b u t we s k e t c h a l i t t l e more more below because o f personal i n t e r e s t and t o s t i m u l a t e f u r t h e r research. A

L e t now C (resp. C) be s u i t a b l e c o n t o u r s r e l a t i v e t o t h e zeros o f a ( r e s p .

2) as

b e f o r e and o p e r a t e on $,$ r e s p e c t i v e l y i n ( 9 . 5 ) by ( 1 / 2 n ) l c e x p ( i c y ) d c

and (1/2n)/,n e x p ( - i s y ) d c f o r y > x t o o b t a i n (11.11)

( 1 / 2 n ) l C $(c,x)eieYds

= ( A ) ~ ( y - x ) + ^K(x,y);

One knows from 19 t h a t q = -2K,(x,x) and r = so from (11.11) f o r m a l l y ( c f . Remark 11.14) ( A & + ) r ( x ) = - ( l / ~ )

( t i s suppressed momentarily). A

-2K2(x,x)

The f o r m u l a lc $ 2 ( s , x ) e x p ( i c x ) d e and q ( x ) = - ( l / n ) l ? JI1 (s,x)exp(-icx)dc. t i o n h e r e can be very general (see below) and we a r e examining o t h e r s i t u a t i o n s beyond t h e NLS. q(0,t)

= Q ( t )e, t c .

Now i n s e r t t as needed, t a k e r =

( q = 0 f o r x < O), r e c a l l

(A@+),

4

(case

and w i t h a,;

(A@=))

with

h a v i n g no

INTEGRABLE SYSTEMS zeros t a k e C =

=

(--,-I;

223

r e c a l l a l s o f o r s r e a l !bl(O,t,s)

t), i 2 ( 0 , t , c ) = S l ( 0 , t , i ) = -b(c,t), and a^(s,t) = a ( s , t ) . say (11.118) f o r example w i t h r = Ti one o b t a i n s f o r m a l l y

S"

-m

=

t q$,

more d e t a i l )

i,i

Given r =

Now w o r k i n g w i t h A

4

(A&.)

= -(l/r)

6

t h e formulas

(A&.

determine f o r m a l l y t h e

)-(A+*)

determined by i n p u t

P,q i n (11.10).

A more p r e c i s e v e r s i o n o f Theorem 11.13 i s g i v e n i n C o r o l l a r y

RElllARK 11.14,

In d e a l i n g w i t h expressions l i k e

one s h o u l d i n s e r t a

(A&=)

f a c t o r o f e x p ( - 2 i s x ) f o r example and e v e n t u a l l y l e t x t h a t q(x,t)

-6(c,

-

i n terms o f s p e c t r a l d a t a a,;

11.15 below.

=

= G ( t ) = -(l/~)l: [Qa 2is;lds.

u'(0,t)

(A+*)

EHEBREI 11-13. output

^b(c,t)

Next one has u ' = ux - - ( l / r ) J- mm [!bi-is!bl]exp(-isx)ds. B u t !bi so J/;(O,t,c.) = -i& t Qa and hence f o r m a l l y (see below f o r

E(6,t)dc.

-is$l

=

-t

The p o i n t i s

0.

has a d i s c o n t i n u i t y a t x = 0 ( u n l e s s Q ( t=) 0 ) and t h e r e p r e -

s e n t a t i o n f o r q ( x , t ) a r i s i n g from (A&+) r e a l l y i s s a y i n g t h a t (A+A) q ( x ) l'im -- l i m ( - l / r ) l : !bl ( r , x ) e x p ( - i s y ) d s = y+x ( - l / r ) i z !bl ( s , x ) e x p ( - i s x ) e x p [ i c ( x Y+' lim y)]d& =2y-tx (-2K1 (x,y). Now one knows t h a t $l(c,x)exp(-icx) = (1/2ic)q(x)

Imr

t 0(1/s ) f o r example f o r

sx) i s a n a l y t i c f o r

Imr

> 0.

> 0 and 151 l a r g e ( c f .

[Aol]),

Thus as r e q u i r e d i n K(x,y),

w h i l e !blexp(-i

f o r y < x the in-

t e g r a l vanishes by c o n t o u r i n t e g r a t i o n i n t h e upper h a l f p l a n e so t o s i m p l y

i s formally correct o f course (and can be r i g o r o u s l y c o r r e c t f o r s u i t a b l e growth e t c . ) b u t i t l e a d s f o r example a t x = 0 which may encounter d i f f i c u l t i e s i n convert o (A&.) 2 gence ( u n l e s s s t a n d a r d t i m e v a r i a t i o n nb(c,t) 2, nbo(s)exp(4is t ) i s p r e s e n t i n a way which which may n o t happen). Thus i n o r d e r t o dea w i t h (A&.) c l e a r l y e x h i b i t s i t as a f o r m u l a i n F o u r i e r ype a n a l y s i s w i t h a d i s c o n t i n u i t y a t 0 ( i . e . q ( x ) = 0 f o r x < 0 ) one must l e a v e t h e e x p ( - i s y ) t e r m i n , i n i n s e r t y = x as i n

c o u l d be m i s l e a d i n g .

(A&+)

some way, and t a k e l i m i t s . small x, JI = =

^bp

+

b

- a,;

T h i s i s most eas l y accomplished by w r i t i n g f o r

so t h a t ( c f . [ A o l ] and 19) f o r s r e a l !bl

G2 'L ^bexp(-igx)

- v 2 exp(i5x) O(x 2 )l and

(A&+)

%

sy) 'L ^bexp(-2icx) as y more r i g o r o u s l y (A+A)

t

o ( x ) ( n o t e here f r o m

O(x) s i n c e

+

i=

%

ft

i2

).

(0)

%

&, -

i n 19 cPlexp(isx)

'L

a$, 1 t

Thus i n f a c t !bl(x,s)exp(-i

x and x -t 0 and t h e n i n p l a c e o f -(l/~) lz i ( c , t ) e x p ( - 2 i s x ) d s

(A&=)

we w r i t e

= x+o 1i m

f(t,x). The f a c t o r e x p ( - 2 i s x ) o f course a l s o h e l p s f o r convergence purposes. We n o t e t h a t t h i s k i n d o f procedure c o u l d a l s o a r i s e i n F o u r i e r a n a l y s i s i n f = 0 elsewhere, w i t h F f = ( l / i s ) [ e x p ( i c E ) - 1 1 so t h a t (A+.) 1 = lim (1/2r)l: F f e x p ( - i s x ) d c . F o r x f 0 conx-to t t o u r i n t e g r a t i o n g i v e s t h e c o r r e c t answers b u t f o r x = 0, f ( O ) must be ex1i m [ e x p ( i s ~ ) - l ] d s / i c b e i n g a t b e s t ambiguous). pressed as x-to f ( x ) ((1/2a)l:

d e a l i n g w i t h e.g.

f ( x ) = 1 for 0 < x <

E,

224

Now i n

ROBERT CARROLL

(A+*)

one a n t i c i p a t e s i n a d d i t i o n t h a t a 6 f u n c t on w i l l a r i s e be-

a t x = 0. We l o o k f i r s t a t 6 : - is$, - 1 n ' 2isJ11 f o r x small. Now J12 = b2- av2 = tv2 + Thus f o r x small from ( 0 ) o f 19 a g a i n t h e ,a; f o r 5 r e a l ( c f . Remark 9.2). I 1i m -2isJ11 t e r n i n v o l v e s - 2 i c t e x p ( - 2 i s x ) b u t t h e q62 t e r m w i l l i n v o l v e cause o f t h e d i s c o n t i n u i t y i n q ( x , t )

JIi - isJI1 = qJ12

t qb2 w i t h

-

Y+X+Om

Now observe t h a t if F ( s ) = Lm PO 1i m f ( x ) e x p ( i s x ) d x then f(yx) ( 1 / 2 r ) L z F ( s ) e x p ( - i s y x ) d s + x+o f ( x ) . 1i m Hence i n d e a l i n g w i t h say lim 1 Qaexp(-igy)dc one can e q u a l l y w e l l use x+o

/IQaexp(-iA[y-x])dh

lim 1 Qaexp(-isy)ds.

%

P O 1 Qaexp(-2isx)ds i n o r d e r t o have common " F o u r i e r " f a c t o r s e x p ( - 2 i s x ) i n A

t h e a and b terms.

Now c o n s i d e r t h e adjustment by 6 f u n c t i o n s a t x = 0

needed t o ' r e g u l a r i z e "

(A+*).

^b

We n o t e f i r s t t h a t a

2,

1 as

Is1

-f

m

(Ims > 0

O(l/lsl)

f o r l a r g e 1 5 ) i s t o be expected. One a r i s e i n (A+*) because o f a n t i c i p a t e s t h e n t h a t terms @ ( x ) and/or & ( x )

say) and ( c f . [C37;42])

=

t h e c o n s t r u c t i o n s used. Indeed l o o k i n g a t (A+A) again c o n s i d e r (A+&) q ' ( x ) lim m 1i m = -2D K (x,x) = -2 is ~ + ~- L x l y+x [D x K ( X y) t DyK1(x,y)] = - ( ~ / T )C$,'(S,X) JI1 (s,x)exp(-icy)ds - ( 1 / ~ ) ~ + h(x)6,(s,x) ~ - 2isJ11( c , x ) l e x p ( - i s y ) d s . Now f o r x small we have terms (A++) I = -(l/r)lim Qaexp(-ic[y-x])ds and Y+X ( A W ) J = (1/n)lim y+x I -mm 2ic^bexp[-is(x+y)]ds = ( 1 / r ) L m 2isnbexp(-2isx)ds. This

lliA

[z

/_I

l a s t term i s the d e r i v a t i v e o f f ( t , x ) term & ( x )

=

(i/r)j_Ie x p ( - 2 i s x ) d s

in

(A+A)

( n o t e S(2x)

%

so i t c o u l d g i v e r i s e t o a (1/2r)[z

e x p ( - 2 i r x ) d s and

On t h e ( 6 ( 2 x ) , ~ ( x ) ) % J 6 ( 5 ) ~ ( 5 / 2 ) & / 2 = ( 1 / 2 ) ~ ( 0 ) so 6 ( 2 x ) % ( 1 / 2 ) 6 ( x ) ) . 1i m o t h e r hand I 2, IT)^+^ qaexp(-isy)ds and t h i s seems t o i n v o l v e a t e r n t h e r e would be a t o t a l e f f e c t o f -QS(x) -2qs(y) 'L -2Q6(x). When 9 =

i

(which c o u l d be expected); t h e s i g n i s r e a l l y n o t i m p o r t a n t s i n c e t h e manner i n which these 6 f u n c t i o n s a r e a r i s i n g here i s p a r t l y due t o t h e c o n s t r u c A

tions.

Thus one wants t o e l i m i n a t e these 6 f u n c t i o n s i n o r d e r t o o b t a i n P

and we c o n s i d e r

(Am*)

-

^P

= (1/r)lim Y-+O

Lz

Q(l-a)exp(-isy)ds t

(l/~)i$[: [2i$

I n o r d e r t o p u t e v e r y t h i n g under t h e same i n t e g r a l we i]exp(-2icx)ds. c o u l d a l s o w r i t e ( c f . (A+&)) (AmA) ^P (l/n)iz [Q(l-a) t 2 i d - i ] e x p (-2irx)ds (note t h a t

( A m A ) = (A+*)

for

4=

/_z

Q). We summarize t h i s i n A

C0R0CCARg 13.15, A r e f i n e d v e r s i o n o f Theorem 11.13 r e p r e s e n t s P and (A+A) and (AmA) r e s p e c t i v e l y .

RrmARK 11.16.

The equations

(AOm)

and

(A&*)

A

9

via

can be e a s i l y s t u d i e d by t e c h -

niques o f ODE and r e s u l t s o f [Gerl;KplO] a r e u s e f u l here. Behavior f o r A l a r g e 5 can be o b t a i n e d d i r e c t l y and i t seems t h a t standard v a r i a t i o n o f b 2 e t c . v i a terms e x p ( 4 i s t ) w i l l n o t be p r e s e n t i n a l l terms. We d e f e r det a i l s t o [C37,38,42]).

I n terms o f d e v e l o p i n g a f i x e d p o i n t theorem u s i n g

225

INTEGRABLE SYSTEMS e.g. Newton's method, Frechet d e r i v a t i v e s can be e a s i l y obtained f o r the map P + etc.

(P,:)

R a M R K 11.17. Let us i n d i c a t e here how the theory will extend when there are bound s t a t e s . Go back t o ( 9 . 5 ) where J, = ( 0, ) e x p ( i c x ) + K(x,s)exp * 1 (iss)ds and J, = (,)exp(-isx) + K(x,s)exp(-iss)ds. In deriving the M equations (9.8)-(9.9) one used contours C , C where C passesover the zeros of a i n the upper half plane e t c . . We r e c a l l t h a t $ = v / a - (b/a)J, w i t h v/a ana l y t i c above C so t h a t ( 9 . 8 ) i s obtained. This i s the o r i g i n of our contours C,? i n ( l l . l l ) - ( A & + ) and thus f o r s u i t a b l e q,r we can regard (11.11)( A & * ) a s v e r i f i e d ; t h e s p e c t r a l formulas (11 . l l ) - ( A & * ) a r e needed f o r cons i s t e n c y i n using the M equations ( 9 . 8 ) - ( 9 . 9 ) . Now evaluate e.g. ( 1 / 2 s ) A Ic J , ( s , x ) e x p ( i s y ) d s v i a a contour enclosing t h e zeros of a w i t h t h e real axi s on the bottom ( t h u s I c + = - 2 n i l r e s i d u e s ) . Since = (v/a) - (b/a) J, ( w i t h Jc ( v / a ) e x p ( i c y ) d c = 0 ) we will g e t residue c o n t r i b u t i o n s ( A = * ) 2ni 1 J,(cn,x)(b/a')(cn)exp(isny). Set cj = ( b / a ' ) ( c j ) a s in 59 and then f o r y > x (A=&) ( 1 / 2 n ) J C j ( c , x ) e x p ( i s y ) d s = - ( 1 / 2 n ) I C (b/a)J,(s,x)exp(isy)dc = K = -(l/Zn)l; ( b / a ) J , ( c , x ) e x p ( i s y ) d s + i l e x p ( i s y ) c n J , ( s n y x ) . In t h e f i n i t e sum n€ one can set x = y = 0 and r e c a l l $(sn,O) = (,n) ( a n = 0) so the residue term 1 y i e l d s (A=*) i 1(l/a,',)(o) ( r e c a l l a$ + b i = l n ) . We take r = -i now with 8 ( c ) = a(?), $ ( s ) = L ( t ) , e t c . and -4 = - 2 i \ 2 ( x y x ) gives ( A = = ) = (l/n) l i m Jm ( b / a ) J , 2 ( s y x ) e x p ( i s y ) d c . B u t 1L2 2, a e x p ( i s x ) near x = 0 so (a**) = y+x"O-7im m "= P before t o obtain now - ( ~ / I T ) ~ + ~ Lb e- x p ( - Z i e x ) d s . For -P we proceed as - ( l / n ) l i m l z (b/a)[2icJ,2(c,x) - Q J , l ( s , x ) l e x p ( i c ~ ) d+c i l ~ ~ [ 2 i s , , J -, ~&,I we re(sn,O). The residue term i s -qi 1( bh / a 'n) $ n m= -il q ( l / a , ' , ) and f o r - A c a l l a l s o Q1 bexp(-isx) so 'P ( - l / n ) l i m l m (b/a)[2icaexp(is(x+y)) - Qb 2isbexp e x p ( i s ( y - x ) ) ] d s + residues. The f i r s t term leads t o (-l/s)k$i; ( 2 i s x ) d s (giving r i s e t o a 46 adjustment a s before) and the second t o (l/rr) 6limI: (bL/a)exp(isy)dc = ( l / n ) i i z [ I [ ( l / a ) - ~ ] e x p ( i c y ) d cw h i c h should not Y"Or e q u i r e a 6 function adjustment. Thus ( c f . [C37,38,42] f o r d e t a i l s )

/xm

/,"

4

cm

;

h

-i

A

6

A

_/I

A

Q

2 For t h e NLS equation i q t = qxx + 21ql q of c a s e ( A & * ) where iTHE0REIII 11.18, r = -! with time evolution of s p e c t r a l data as in ( l l . l O ) , t h e readout i s

given by (.**) and

(**A)

^P

=

iQ$

(l/ah) + ( l / n ) i z i :

[2ic^b -

i+ Q(^a-'

-

a)]exp(-2isx)ds. Note in Corollary 11.15 one could add a term ( l / n ) (Q/P) ~ ~ ~ ~ ~ exp(-Zicx)ds t o 3 instead of 2 9 6 ( x ) and then t h e i n t e g r a l term would be ide n t i c a l to t h a t in Theorem 11.18 (when r = {, [: ( l / $ ) e x p ( - i s y ) d c = 0 f o r y > 0 follows from LZ (v/a)exp(icy)dc = 0 f o r y > x ) .

REmARK 11.19.

This Page Intentionally Left Blank

227

CHAPTER 3 SOME NONLINEAR ANALYSIS; SOME GEOMETRIC FORMALISM

This chapter w i l l be a m i x t u r e o f several themes. F i r s t , i n order t o provide some technical machinery f o r d e a l i n g w i t h n o n l i n e a r PDE

1 1NEZEe)DUCtZ"

f o r example we g i v e t h e rudiments o f c e r t a i n areas o f n o n l i n e a r analysis. One can a l s o h o p e f u l l y enjoy t h e subject f o r i t s e l f o f course. This i n cludes some f i x e d p o i n t theorems, a l i t t l e degree theory, some homotopy methods, basic m a t e r i a l on monotone and a c c r e t i v e operators and n o n l i n e a r semigroups, v a r i a t i o n a l i n e q u a l i t i e s , some convex analysis, e t c .

Then we g i v e

some i n t r o d u c t o r y m a t e r i a l on Feynman i n t e g r a l s , quantum f i e l d theory, gauge theory, geometric quantization, etc. The t o t a l package (Chapters 1-3 + Appendices A-C) should h e l p t h e reader make contact w i t h a broad spectrum o f a c t i v i t y i n mathematical physics and provide him o r her w i t h many o f t h e mathematical t o o l s needed t o work i n these areas. complete as we had o r i g i n a l l y intended.

The coverage i s n o t as

I n terms o f geometry-topology alone

l a c k o f space prevents us from discussing t h e Atiyah-Singer index theorem f o r example and we had hoped t o l e a r n something about superstrings by frami n g an i n t r o d u c t i o n t o t h i s theory. S i m i l a r l y t h e r e i s nothing about t w i s t o r geometry, very l i t t l e about r e l a t i v i t y , nothing about black holes, e t c . S i m i l a r lacunae r e l a t i v e t o nonlinear a n a l y s i s are i n d i c a t e d a t t h e end of 13.7.

Fortunately i n t h e present i n f o r m a t i o n explosion t h e r e a r e many good

sources o f i n f o r m a t i o n on a l l these t o p i c s . There are many sources o f i n f o r m a t i o n a v a i l a b l e t o 2, N0NCINEAR AIQAC&W. day on nonl inear a n a l y s i s (e.g. see [Be1 ; B r l ;AulY2;E1 ;C1 ;Ful ;F11 ;Gul ;C11;

Dml;Lul;Dnl;Atl;Pcl;Li4;Wul;Zel,2;Rkl])

and many t o p i c s t h a t one might t r y

t o cover (e.g. one t h i n k s o f successive approximations, Rayleigh-Ritz and Galerkin methods, d i f f e r e n t i a l c a l c u l u s v i a Gateaux and Frechet d e r i v a t i v e s , Morse theory and c r i t i c a l p o i n t theory i n general, homotopy and degree arguments, v a r i a t i o n a l i n e q u a l i t i e s , monotone operators, convexity, g r a d i e n t maps, steepest descent, inverse and i m p l i c i t f u n c t i o n theorems, b i f u r c a t i o n theory, s i n g u l a r p e r t u r b a t i o n and asymptotic expansions, e t c . ). One cannot

228

ROBERT CARROLL

expect everything but we will try to select some techniques and give enough information so that the reader will be able to understand and use the methods. We have already indicated some sample nonlinear problems in physics and will mention a few more typical situations here in order to indicate certain difficulties which arise ([Bell has a good selection and we extract from this at times). Some preliminary techniques and machinery are also developed in this section. We have already encountered problems of nonuniqueness and lack of global solutions (or blowup) in 51.10 (recall also the difficulties mentioned in 51.9 relative to the Navier-Stokes equations - weak and strong solutions, uniqueness, etc.). Consider now e.g. (A) y" + (1/2)Aexp(y) = 0 (c-exp(s))-+ds for h >/ with y(0) = y(1) = 0 (cf. [Bell). One has x = 2 0 where c = 1 + X-ly'(0) so y has a maximum at x1 where y(xl) = logc (note 4 2x1 and cosh 2 (~'Ac/4)/4 = c so for dyldx = A (c-exp(y))). By symnetry 1 A > 8 there is no solution ( 8 arising from the cosh equation), for A = 8 there is one solution, and for A < B there are 2 solutions. This illustrates the idea of bifurcation. (B) Another example from [Bell shows the interaction between dimension and growth. Thus consider (*) Au - u t lulau = 0 (cf. 51.10). If a 2 4/n-2 (n > 2 ,I, dimension) there is no nontrivial smooth solution on Rn with IuI 0 as x -t m while there are smooth solutions for 0 < a < 4/n-2. One notes that a solution u of (*) corresponds to a critical point of J(u) = 1 [ ( 1 / 2 ) l V ~ 1 -~ F(u)]dx where F(u) = -u 2/2 + lulau2/(a+2) EWAIIIPCE 2.1.

-+

(formally Fu = -u + lulau = f(u)). Setting (d/dE)J(u(Ex)) = 0 at E = 1 (this is contrived somewhat) one finds 0 = -((n-2)/2)/ lvul 2dx t n/ F(u)dx. On the other hand multiplying Au + f(u) = 0 by u and integrating one has 2 1 IvuI dx = 1 f(u)udx. It follows that (2n(n-2))/ F(u)dx = 1 uf(u)dx. Hence if u is a solution of (*) (2n/(n-2))/ [(ulau2/(a+2) - u 2/2]dx = 2 / [lulau2 - u Idx or [2n/(n-2)(a+2) - 111 lu/au2dx = [n/(n-2) - 111 u2dx. Hence 2n/(n-2)(a+2) - 1 > 0 or a < 4/(n-2) so in particular for a 4/n-2 we have a nonexistence result. Other behavior and problems will be illustrated by examples from time to time. Let us now go to some machinery which can be used in treating nonlinear problems. We begin with some ideas of differential calculus. DEFINICI0N 2.2. Let E and F be Banach (B) spaces and g a (single valued) map g: E -+ F (assume g is defined on all E with R(g) C F). g is continuous

if en +. e implies g(en) -t g(e); g is demicontinuous if en +. e implies g(en) g(e) weakly (i.e. (g(en),f') +. (g(e),f') for f' E F' arbitrary); g is

+.

229

NONLINEAR ANALYSIS

completely continuous i f ea + e weakly i m p l i e s g(ea) g ( e ) i n norm (ea r e see Remark 2.16); g i s u n i f o r m l y continuous i f f o r E > f e r s here t o a n e t -f

-

0 t h e r e e x i s t s 6 ( ~ such ) t h a t IIe

- ell 5

6 i m p l i e s Ug(e)

-

g(e)ll 5

g is

E;

bounded i f i t maps bounded s e t s i n t o bounded sets; g i s l o c a l l y bounded i f any e E E has a NBH i n which g i s bounded.

DEFZNZEIBN 2.3.

g: E

+

-

T such t h a t Ug(x)

i s t s a l i n e a r operator

-

e q u i v a l e n t l y [g(xo+h)

F i s Frechet ( F ) d i f f e r e n t i a b l e a t xo i f t h e r e ex-

-

g(xo)

Th]/Ilhll

+

-

g(x,)

T(x-x )II = o(llx-x,U)

or

0

0 as h + 0. T i s o b v i o u s l y un-+ g ' ( x ) maps E i n t o L(E,F)

ique ( e x e r c i s e ) and we w r i t e T = g ' ( x o ) so x

(note t h a t we want t o s p e c i f y T i s a bounded l i n e a r operator here). checks a l s o t h a t i f g i s F d i f f e r e n t i a b l e a t xo then given NBH nx-xoM 5 NBH.

E

-

such t h a t I g ( x )

Next g: E

-+

b l e ) such t h a t l i m g(xo+th)

-

a NBH o f xo).

dg ( o r dg(xo,h))

one w r i t e s

dg(xo,h)

(xo,h)

there exists a

g(xo)ll 5 ( l g ' ( x o ) l l t E ) l x - x o l f o r x i n t h i s

F i s G (Gateaux) d i f f e r e n t i a b l e ( c f . a l s o 53.5) a t xo if

t h e r e i s an operator dg: E X E

(A)

E

One

+

F (not necessarily l i n e a r i n e i t h e r varia-

g(xo)

-

0 ( f o r xo+th i n

-+

i s c a l l e d t h e Gateaux d e r i v a t i v e a t xo and

= Dtg(xo+th)lt,O.

(exercise) and from ;/

* 0 as t

tdg(x,,h)

Dt(f',g(xo+th))dt

arbitrary f ' E F' i t follows that

(0)

= adg

One checks t h a t dg(x,,ah) =

1; ( f ' , d g ( x o + t h , h ) ) d t

- g(xo)

g(xo+h)

= ;/

for

dg(x,+th,h)dt.

tH€0RBIl 2.4. I f g i s F d i f f e r e n t i a b l e a t xo then i t i s G d i f f e r e n t i a b l e . If g i s G d i f f e r e n t i a b l e a t xo, dg(xo,h) i s l i n e a r i n h ( i . e . dg(xo, ) E L(E, F ) ) , and x + dg(x,.): E + L(E,F) i s continuous i n x, then g i s F d i f f e r e n t i a b l e a t xo (dg(xo,h) = dg(xo)h = g ' ( x o ) h ) .

Pmo6:

F i r s t l e t us note t h a t t h e word map i s o f t e n used t o mean continuous

map ( s i m i l a r l y operator may imp1 i c i t l y mean continuous o p e r a t o r ) ; we w i l l t r y t o be s p e c i f i c b u t o c c a s i o n a l l y w i l l n e g l e c t t o mention continuous.

t h e f i r s t statement o f t h e theorem i s obvious. = dg(x)h where dg(x)

Ilg(x+h)

-

g(x)

E

For t h e r e s t we have dg(x,h)

L(E,F) and Ildg(x,h)II < Ildg(x)IIllhII.

- dg(x)hll

= Il/d

[dg(x+th,h)

-

Now

Hence by

dg(x,h)]dtll (/;

(a)

Ildg(x+th)

-

dg(x)Hllhlldt = o(llhll) and d g ( x ) = g ' ( x ) ( 9 ' w i l l always r e f e r t o t h e F d e r i vative).

QED

One defines h i g h e r d e r i v a t i v e s i n a n a t u r a l way and t h e r e w i l l be an analogous r e s u l t t o Theorem 2.4 connecting t h e F and G d e r i v a t i v e s . I n p a r t i c u 2 l a r (assuming both d e r i v a t i v e s e x i s t ) g " ( x ) = d g ( x ) E L(E,L(E,F)) = L2(E, F) = L(E X E,F) i s a b i l i n e a r symmetric map ( i . e . d2g(x)(e,h) = d 2g ( x ) ( h , e ) ) and dgn(x) = gn(x) E Ln(E,F) i s a symmetric n - l i n e a r map ( e x e r c i s e

-

cf.

230

ROBERT CARROLL

[Fll;Bel;Dnl;Ze2]

2 2 d g ( x ) e = Dtdg(x+te) a t t = 0 so d g

and n o t e t h a t e,g.

= DtDSg(x+te+sh) a t s = t 0 which i s t h e same as DsDtg(x+te+sh) a t 2 F u r t h e r one has a T a y l o r theorem ( c f . Appenx = t = 0 w h i c h i s d g(h,e)). (e,h)

d i x C f o r a d d i t i o n a l i n f o r m a t i o n on F d e r i v a t i v e s )

If g

CHE0REIII 2.5.

E

Cntl

( i n some NBH U o f x E E

t o F d e r i v a t i v e s ) and t h e l i n e [x,x+h]

1:

g(x+h) =

(2.1)

Rn+l

g:

E

+.

F and Cn r e f e r s

c U then

k k g ( x ) h / k ! + Rntl(xYh); (x,h) =

-

Rn+l

I '[ ( l - ~ ) ~ / / n ! ] g ~ ~ ~ ( x + s h ) h " + l d s 0

T h i s can be proved v i a c o n s i d e r a t i o n o f G ( t ) = ( y ' , g ( x + t h ) ) f o r y ' E F ' ; k k n o t e G ( t ) = ( g ' , g k ( x + t h ) h ) and we o m i t t h e d e t a i l s h e r e ( e x e r c i s e ) . Given a f u n c t i o n g(x,y)

on say

E X E one d e f i n e s p a r t i a l d e r i v a t i v e s v i a D1g(x,y)

and (6 1 g'(x,y)(e,h) = dg(x,y)(e,h) = Dtg(x+te,y+th)10 L e t us now i n d i c a t e a few general theorems a b o u t D2g(x,y)h. n o n l i n e a r maps i n B spaces and t h e n we w i l l c o n s i d e r some s p e c i a l types of e = Dtg(x+te,Y)lt,O

= Dlg(x,y)e

t

maps ( c f . [Bel;Dnl;Ze2]). tHE0REm 2.6.

F i r s t one has a c o n t r a c t i o n mapping theorem

Suppose a continuous g: B

yll ( f o r c < 1 ) i n some c l o s e d b a l l B C

B s a t i s f i e s I l g ( x ) - g(y)ll 5 CIIX E ( E Banach). Then g has a unique +.

f i x e d p o i n t xm E U. n E v i d e n t l y IIx Take any xo E B and s e t xn = g (xo! = g(xn-l). n+p-1 J n+P ntp-1 j+l c I g ( x o ) - xol 5 ~ c n / ~ l - c ~ l ~ ~ ~ x o xnll = "1, g ( x o ) - gJ(xo)ll 5 - xoll -+ 0 so x n i s Cauchy and hence xn + xm E B w i t h g(xn) = xntl +. xm. Consequently g(x,) = xm by c o n t i n u i t y o f g and i f t h e r e were a n o t h e r p o i n t

P4oo6:

In

w i t h g(?) =

we would have Uxm

-

ill = Ilg(xm)

< 1 which c o n t r a d i c t s .

+

g(?)ll

z clx,

-

?If o r c

QED Let g E C

CHEbREm 2.7 (INVERSE FLINCCI0N CHE0RElll). E

-

1

F, and suppose g ' ( x o ) i s a l i n e a r isomorphism E

i n a NBH o f xo E E, g: F onto. Then g i s a

+.

l o c a l homeomorphism o f a NBH U(xo) t o a NBH o f g ( x o ) and f o r Ily - g ( x o ) / l -1 = xn t g ' ( x o ) [y g ( x o ) ] converges

-

s u f f i c i e n t l y small t h e sequence xntl

t o t h e unique s o l u t i o n o f g ( x ) = y i n U(xo). P4006:

- yell

L e t g ( x o ) = yo and t r y t o f i n d i s s u f f i c i e n t l y small.

p

such t h a t g(xo+p) = y whenever Ily

Equivalently y

-

yo = g(xo+p)

-

g(xo) and t h u s

g(xo) - g ' ( x o ) p = o ( l l ~ l l ) . g ' ( x 0 ) p + R ( x O Y p ) = Y - yo where R = g(xo+p) Thus p = [ g ' ( x o ) ] ~ ' [ y - y o - R ( x o , p ) ] = T(p) and one can show t h a t T i s a cont r a c t i o n o f a c l o s e d b a l l B(O,E)

into itself for

E

s u f f i c i e n t l y small.

Thus

NONLINEAR ANALYSIS

(+I

('))

(cf.

-

T(P)

-

+ tP" + ( 1 - t ) p )

T(S) = g ' ( x o ) - 1 [R(xO9;)

-

-

T(;)ll

g'(xo)]($-p)dt

and IIT(p)

231

R ( x ~ , P ) I = g'(xO)-'JJ

5 cllp-~ll, c

[g'(xo

< 1, f o r p,p^

s u f f i c i e n t l y small, say p,; E B(O,E) ( t h e [ ] i n t h e i n t e g r a l i n (+) i s s m a l l ) . F u r t h e r IIT(P)II 5 IIT(p) - T(0)ll + IIT(0)II 5 cllpll + Ilg'

small f o r p , ;

1

( x o ) - (y-yo)ll 5

f o r Ily-y 1I small so T: B(O,E)

E

-+

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

B(O,E).

0 T ( p ) = p and hence g(xo+p) = y has a unique s o l u t i o n when 4y-y0// and p a r e

s u f f i c i e n t l y small.

11) and t h u s g - ' ( y ) around yo

E

F u r t h e r y depends c o n t i n u o u s l y o n p ( e x e r c i s e - c f . [Be = x i s a w e l l d e f i n e d and c o n t i n u o u s map f r o m a NBH

= T(in-l))

(Po = 0, P

a d d i t i o n gnl

CffE0REIII 2.8

E

x = xo !l w i t h g (x,)'

C

+ g'(x0)-'Cy n 111 = g'(xo) (exercise).

L e t g(x,y)

i n E X F i n t o G, g(xo,yo)

P

iteratively via

- g(xn-l 11 w h i l e i n

= x

+ P,

( I m P L I C I t Fl.NC&I0N EHE0REm).

a NBH U o f (xo,yo)

+

F u r t h e r one can c o n s t r u c t x = x,

F t o E.

QED

be a c o n t i n u o u s map o f

= 0, and assume g ( x ,y Y

O

0

)

e x i s t s and i s c o n t i n u o u s i n x w i t h g ( x , y ) a l i n e a r isomorphism F G. Then Y t h e r e e x i s t s a unique map h i n a NBH U1 o f xo, h: U1 F, such t h a t h ( x o ) = -f

-+

yo and g(x,h(x))

Phoud:

= 0 f o r x E U1.

For x n e a r xo one has g(x,y) = g ( x

where R(x, Consider t h e map y ) - R ( x , i ) = o(//y-;//) f o r (x,y) and (x,y) n e a r (xo,yo). T x ( y ) = Y - gy(xo,yo)-lg(x,y) = yo - gy-1 (X~,Y,)R(X,Y). BY t h e proof o f

{

y ) ( y - y o ) + R(x,y)

0' 0

Theorem 2.7 we see t h a t f o r x f i x e d , n e a r xo, Tx i s a c o n t r a c t i o n o f a small b a l l c e n t e r e d a t yo i n t o i t s e l f .

Hence t h e r e i s a unique f i x e d p o i n t y ( x )

o f Tx which depends c o n t i n u o u s l y on x. E v i d e n t l y y ( x ) = T x ( y ( x ) ) = y ( x ) gy-1 (xo,yo)g(x,y(x)) i m p l i e s g ( x , y ( x ) ) = 0 and y ( x o ) = yo. By uniqueness y ( x ) = h ( x ) i s t h e d e s i r e d map i n t h e theorem.

REmARK

-

QED

The i t e r a t i o n scheme i n Theorem 2.7 i s known i n o r d i n a r y c a l c u -

2.9,

l u s as Newton's method and i t can be e x t e n s i v e l y r e f i n e d ( c f . [Be1 ;At1 ; K t l ; Ze21).

I n t h e event t h a t g ' ( x o ) i n Theorem 2.7 i s a l i n e a r map o n t o ( s u r -

j e c t i v e ) b u t n o t 1-1 i t can be shown e a s i l y ( c f . [ B e l l ) t h a t g i s an open map f o r x n e a r xo ( i . e .

i t maps open s e t s t o open s e t s

showing t h a t f o r y n e a r yo t h e e q u a t i o n g(xo+p) tion x = x

0

+p

f o r s u f f i c i e n t l y small

P).

- g(xo)

-

which f o l l o w s upon = y-yo has a s o l u -

I n p r a c t i c e t o s o l v e g ( x ) = y one

o f t e n encounters s i t u a t i o n s where i t i s necessary t o use a c o n s t r u c t i o n o f approximate s o l u t i o n s o f

l i n e a r i z e d Newton t y p e problems o b t a i n e d v i a

smoothing o p e r a t i o n s and t h i s produces i t e r a t i v e l y a convergent procedure. Such (Nash-Moser-diGiorgi)

techniques a r e very powerful b u t a l s o t e c h n i c a l Y complicated and we do n o t s t u d y t h i s h e r e ( c f . [ A t l ; B e l ] ) .

232

ROBERT CARROLL

L e t us mention a " s t e e p e s t descent" method i n a s i m p l e s i t u a -

REIRAW 2.10. tion. F

E

T h i s means

Thus f i r s t we say f i s a g r a d i e n t map i f f ( x ) = gradF(x).

C1(U,R),

tition f E C

U 1

C

E, and F ' ( x ) = f ( x ) ( n o t e F ' ( x )

E

I f i n ad-

L(E,R) = E l ) .

t h e n f ' = F " corresponds t o a g e n e r a l i z e d s e l f a d j o i n t map

(d2F(x): E X E

-+

E' i s symmetric).

A typical situation i n applications

..,Dmu)dx where Dm

a r i s e s f o r m a l l y f r o m a f u n c t i o n a l J ( u ) = JA F(x,u,Du,.

...

r e f e r s t o d e r i v a t i v e s Dau w i t h la1 = m (D'u

= Dy' , : ;D

E u l e r equations r e l a t i v e t o J(u) a r e J ' ( u ) =

11.11

Dk = a / a x k ) . The = 0 where Fa

(-1)la1OaF

0

ua d e n o t i n g t h e argument p l a c e f o r C?u, and upon p o s i n g t h e matt e r i n e.g. a s u i t a b l e Sobolev space Hm one has a g r a d i e n t map i n J ' . Now suppose f = F ' and c o n s i d e r t h e problem ( m ) d x / d t = - f ( x ) ; x ( 0 ) = xo. The

= aF/aua,

idea i s t o f i n d a c r i t i c a l p o i n t t

-f

x^

f(2)

where

= 0 by l o o k i n g a t l i m x ( t ) a s

t h i s i s r e f e r r e d t o as a method o f s t e e p e s t descent.

w;

Following [ B e l l

we s k e t c h t h e p r o o f o f n

CHE0REm 2-11, L e t F E CL, F: B C H -+ R (B a c l o s e d b a l l B(xo,r) = {x; IIxxoll 5 r l i n a H i l b e r t space H), w i t h (F"(x)h,h) )allh1I2 (here (F"(x)h,h) means (9)

(

F"(x)h,h)

Then g i v e n IIF'(xo)ll/a f r t h e problem

i n H-H' d u a l i t y ) .

has a u n i q u e s o l u t i o n d e f i n e d f o r a l l t, l i m x ( t ) = xm e x i s t s , and x,

i s t h e unique minimum o f F ( x ) i n B as w e l l as t h e unique s o l u t i o n o f f ( x ) = 0 i n B.

I n addition Ilx(t)

-

xl,

O(exp(-at)).

t

Since x ( t ) = xo + lo f(x(.r))d.r = T t ( x ) w i t h Tt a c o n t r a c t i o n f o r

Pmo6:

small t we know t h e r e i s a unique l o c a l s o l u t i o n x ( t ) ( c f . Example 4.15 f o r Next l o o k a t (**) D t F ( x ( t ) ) = ( f ( x ( t ) ) , x ' ( t ) )

a discussion o f t h i s ) .

- x'(t)

(recall (

,

)

Q

(

, ))

so F ( x ( t ) ) decreases as t increases.

want now t o i n s u r e t h a t x ( t ) (extended by c o n t i n u a t i o n remains i n B and t h a t I l x ' ( t ) l l (2.2)

D2t F ( x ( t ) )

0 so f ( x ( t ) )

-+

-t

= -~(x",x') = ~(F"(x)x',x')

0.

-

We

c f . Example 4.15)

Consider ( c f .

(*)-(.))

2 2ctHx'll 2

Thus DZF 2 -2aDtF from which e v i d e n t l y D t F ( x ( t ) ) 2 -IIf(xo)ll 2e x p ( - 2 a t ) and hence I l x ' ( t ) l l 5 l l f ( x o ) l l e x p ( - a t ) .

This implies l l x ( t )

t h a t x ( t ) E B f o r a l l t under o u r hypotheses.

-

xoll 5 l l f ( x o ) l l / a so

S i m i l a r l y IIx(;)

-

-

x(t)ll 5

x ( t n ) i s a Cauchy sequence ~ ~ f ( x o ) l l [ e x p ( - a t ) / a ] f o r 0 < t 5 so when tn with x(tn) x,E B (x, i s e v i d e n t l y independent o f t h e sequence tn). Thus -f

-+

f(x,)

= 0 (Ilf(x(t))ll = llx'(t)U

(exercise) > aIIx-yl12). -

-

s(x-x,)),x-x,)~~

note

(*A)

(f(x)

-

-+

i s t h e unique such p o i n t i n

0 ) and x,

f(y),x-y)

F i n a l l y note t h a t (cf. = /O1 ( f ( X , + S(X-x,))

=

( 0 )

-

JJ ( f ' ( x t

B

t (1-t)y)(x-y),x-y)dt

and (*A)) F ( x ) - F(x,) = 10 1 (f(x, + f(x,),x-x,)ds 2 ( 1 / 2 ) a l I x - ~ J 1 ~SO

233

NONLINEAR ANALYSIS F(xoo) i s a u n i q u e minimum.

REWUN 2.12.

QED

L e t us n o t e h e u r i s t i c a l l y a d e r i v a t i o n o f t h e Lagrange m u l t i -

p l i e r r u l e f o r c o n s t r a i n e d minima (more g e n e r a l theorems w i l l be g i v e n l a t e r

-

Consider t h e advanced c a l c u l u s d i s c u s s i o n i n Remark

c f . [ L u l ;Wul;Zel]).

1.5.6 and extend i t t o t h e Banach space t h e o r y w i t h t h e F d e r i v a t i v e . i n s i m p l e s t f o r m we t r y t o m i n i m i z e J ( x ) , J: E

-+

0, G: E + R (more general c o n s t r a i n t s a r e e a s i l y handled). r e g u l a r p o i n t o f G (i.e., G'(xo):

E

-+

T(x) = (J(x),G(x)). (xo)h = 0.

Assume xo i s a

i n general, xo i s a r e g u l a r p o i n t o f G: E

E'

F i s o n t o ) ; t h u s we r e q u i r e h e r e G ' ( x o )

t a k e s a l o c a l minimum a t xo.

Thus

R, w i t h c o n s t r a i n t s G(x) =

Consider t h e f u n c t i o n T: E

-+

F if

# 0 and assume J ( x ) -f

R X R d e f i n e d by

Then a n a l y s i s o f T shows t h a t J ' ( x o ) h = 0 whenever G'

Indeed i f G ' ( x o ) h = ( G ' ( x o ) , h )

= 0 b u t J'(xo)h # 0 then T'(xo)

By t h e i n v e r s e f u n c t i o n theo-

= ( J ' ( X ~ ) ~ G ' ( X ~E )-+) :R X R would be onto.

rem ( c f . Theorem 2.7 and Remark 2.9) T i s an open map a t xo and f o r t h e r e e x i s t x,6 w i t h IIx-xoII 2

E

d i c t s xo b e i n g a l o c a l minimum.

such t h a t T ( x ) = (J(xo)-S,O)

E

given

which c o n t r a -

Thus i f N i s t h e hyperplane i n E determined 1 i n El, which i s one

by ( G ' ( x o ) , h )

= 0 i t f o l l o w s t h a t G ' ( x o ) and J ' ( x o ) E N

dimensional.

Hence t h e r e e x i s t s A such t h a t J ' ( x o ) t AG'(xo) = 0 and t h e

(Lagrange) f u n c t i o n a l F ( x ) = J ( x ) + AG(x) i s s t a t i o n a r y a t xo.

O f course

one i n t e r p e r t s N as t h e t a n g e n t space t o t h e " s u r f a c e " G(x) = 0 a t xo.

RE1N\RK 2.13 (S0lXE IRUCELCANE0l.U FACeS). L e t us r e c a l l t h a t an o p e r a t o r f: U c E -+ F i s compact i f i t i s c o n t i n u o u s and maps bounded s e t s t o r e l a t i v e l y compact s e t s ( t a k e E and F t o be B spaces

-

o r m e t r i z a b l e spaces

-

h e r e and

c f . below f o r r e l a t i o n s between compactness and s e q u e n t i a l compactness). T h i s i s g e n e r a l l y d i f f e r e n t f r o m complete c o n t i n u i t y i n D e f i n i t i o n 2.2 and a d i s c u s s i o n i s i n [ C l ] f o r example ( c f . a l s o [ B e l l and Remark 2.16). One can say t h a t f c o m p l e t e l y c o n t i n u o u s and E r e f l e x i v e i m p l i e s f i s compact. Indeed if xn i s a bounded sequence i n U convergent subsequence xn

C

E by r e f l e x i v i t y i t has a weakly Hence f ( x - ) -+ y i n F s t r o n g l y

(see Appendix A ) .

j nJ so f i s s e q u e n t i a l l y compact and hence compact ( c f . below).

pactness p r o p e r t y ( c f . [ B e l l ) i s t h a t f o r f : U C E entiable i t follows t h a t f ' ( x o ) :

E

+

-+

Another com-

F compact and d i f f e r -

F i s a compact l i n e a r o p e r a t o r .

In-

deed i f n o t l e t S be t h e u n i t sphere i n E so t h a t f ' ( x o ) S i s t h e n n o t r e l a -

-

h.)ll 2 J E (i= j ) . But f o r 8 s u f f i c i e n t l y small llf(xo+8hi) - f(xo+Bh.)R > BIlf'(x,) J (h.-h.)ll - lIf(xo+Bhi) - f ( x o ) - ~ f ' ( x o ) h i l l - l l f ( x o + B h j ) - f ( x o ) - B f ' ( x )h.ll

t i v e l y compact.

1

> BE

J

- o(le1).

P i c k hn w i t h IIhnll = 1 and

Since

E

E

such t h a t llfl(xo)(hi

i s independent o f B t h i s i m p l i e s f(xo+Bhi)

O

has no

J

2 34

ROBERT CARROLL

convergent subsequence which c o n t r a d i c t s . L e t us r e c a l l h e r e t h a t a c l o s e d subset S o f a m e t r i c space i s compact i f every c o v e r i n g o f S by open s e t s

-

c f . a l s o D e f i n i t i o n A29 and n o t e h e r e S i s compact i f and o n l y i every f a m i l y o f c l o s e d s e t s i n S has a f i n i t e subcovering (Heini-Bore1 p r o p e r t y

w i t h the f i n i t e intersection property

-

.e. f i n i t e s u b f a m i l i e s have nonvoid

- has nonvoid i n t e r s e c t i o n )

intersection

Then i n f a c t S i s compact i f and

o n l y i f i t i s s e q u e n t i a l l y compact ( i . e . every i n f i n i t e sequence i n S has a convergent subsequence i n S ) .

To see t h s, f i r s t , g i v e n S compact, i f M =

I x n l ( n = 1,2, ...) has no convergent subsequence, cover S by b a l l s B(x,E) each o f which c o n t a i n s a t most one p o i n t o f M.

The r e s u l t i n g f i n i t e sub-

c o v e r i n g i m p l i e s M i s f i n i t e which c o n t r a d i c t s .

Conversely i f S i s sequen-

t i a l l y compact (and c l o s e d ) one notes f i r s t t h a t S i s separable ( i . e . t h e r e Indeed p i c k po a r b i t r a r y i n S w i t h D =

e x i s t s a c o u n t a b l e dense s e t pn).

S ; D i s f i n i t e s i n c e i f d(p ,q ) -+ m t h e r e e x i s t a convero n gent subsequence ^qn -+ q and d(po,q) = m which i s precluded. Choose now i n -

supd(p,p

0

), p

E

d u c t i v e l y pitl d(pn,q)

such t h a t mind(pn,pitl)

( 0 5 n 5 i t l ) where di = sup

di/2

f o r q E S and 0 5 n 5 i. E v i d e n t l y do 2 dl L... and i f dn

2E

> 0

f o r a l l n t h e n no subsequence o f t h e pn i s Caucby which c o n t r a d i c t s t h e convergence o f some subsequence. d(pn,p)

<

E

Hence f o r any p E S t h e r e e x i s t s pn such t h a t

and t h e pn a r e dense.

F i n a l l y t o show S i s compact i t s u f f i c e s

t o show t h a t every c o u n t a b l e open c o v e r i n g has a f i n i t e subcovering ( e x e r cise). quence n;

Thus g i v e n S c uGi l e t xn $ unG. (xn -+

But t h e complement

x.

which c o n t r a d i c t s . Fredholm map.

E

S ) w i t h a convergent subse-

i s c l o s e d and hence x e#

a l i n e a r Fredholm map E ~ ( x =) d i m c o k e r f ' ( x ) <

-+

-1.

-+

F f o r each

F i s d e f i n e d t o be Fredholm i f f ' ( x ) i s X E

U (i.e. a(x) = dimkerf'(x) <

- ~ ( x )i s continuous

U

-+

f o r index theory). Q ,

1(-l)lalfL

We t h i n k o f more general ( e l l i p t i c t y p e ) o p e r a t o r s (**)

as i n Remark 2.10.

1 (-l)aDaAa

and

Z ( Z = i n t e g e r s ) one has i n d f ( x ) =

L e t us now g i v e some f u r t h e r d i s c u s s i o n o f o p e r a t o r s J ' ( u ) where A,

aF/aua, and la1 5

m

Given U connected and t h e f a c t t h a t t h e index

i n d f ' ( x ) = c o n s t a n t i n U ( c f . a l s o [Bsl;Dg1,2]

Au =

= S

UYGi

F i n a l l y a concept one o f t e n uses i s t h a t o f a n o n l i n e a r

Thus a map f : U c E

indf'(x) = a(x)

'rl o f UIGi

s.

= A,(x,u

,...,DSu),

A,

n o t necessarily o f the form

Such o p e r a t o r s a r e s a i d t o be i n divergence o r v a r i a -

t i o n a l form.

We assume A i s a bounded open s e t w i t h a reasonable boundary ( s o t h a t Green's theorems a p p l y i f needed) and w i l l b r i e f l y examine such op-

e r a t o r s i n t h e c o n t e x t o f Sobolev spaces W'(A). L e t us assume ( c f . [Cl;Bel; p-1 1 where E B c B ( u ) = DBu and 5 = B r l l ) (*O IAa(x,c)l 5 c [ l + lle,,,&l

NONLINEAR ANALYSIS

235

S ( u ) denotes a l l arguments f o r 161 5 s . The v a r i a t i o n a l f o r m i n W S ( A ) assoP c i a t e d w i t h A i s ( c f . 51.9)) a(u,v) = al<S(Aa(~,~ DS u),Dav> where ( ,

1

)

,...,

I _

denotes Lp-Lp' c o n j u g a t e l i n e a r d u a l i t y ( p ' = q = P/(p-1)

-

it i s often

convenient t o work w i t h c o n j u g a t e l i n e a r d u a l i t y and a n t i d u a l s V ' = c o n j u g a t e l i n e a r f u n c t i o n a l s on V and we do t h i s h e r e f o r p r a c t i c e ) .

To see t h i s

makes sense n o t e t h a t H o l d e r ' s i n e q u a l i t y g i v e s ( w i t h obvious n o t a t i o n s )

+ 1 )II vU s , p where we use t h e i n e q u a l i t y (aP ts'pp-l/p 1) < aP-' + 1 ( c f . [Hyl]). Hence (**) l a ( u , v ) l 5 g(llulls,p)llvlls,p where g( a ) i s c o n t i n u o u s and o f t h e f o r m g ( x ) = c3(xP-'

t

l ) , w i t h c3 depending o n l y on p,s, and c .

a boundary v a l u e problem (BVP) r e l a t i v e t o a(u,v)

Now t o pose

one chooses ( c f . 51.9) a

c l o s e d subspace V C Ws(A) w i t h ;'(A) C V and works i n V (boundary f u n c t i o n P P a l s c o u l d a l s o be added b u t we o m i t t h i s ) . By v i r t u e o f (*+) t h e map w + a(u,w):

V

-+

C i s c o n t i n u o u s on V and c o n j u g a t e l i n e a r ; hence t h e r e i s an e l -

ement 5 = A ( u ) E V ' ( a n t i d u a l ) such t h a t (*m) a(u,w) s i z e t h a t A: V 9 )

-+

V ' i s a n o n l i n e a r map.

which a r e p a r a l l e l t o t h o s e used i n t h e l i n e a r t h e o r y .

natural condition i n practice i s (u)-A(w),u-w)

(A*)

We empha-

= (A(u),w).

Then one imposes c o n d i t i o n s on a(.,

Re[a(u,u-w)

-

For example, a

a(w,u-w)]

0 or RdA

0, o r some v a r i a t i o n on t h i s v i a p e r t u r b a t i o n ( c f . 53.3).

Rea(u,u) > I P ( I I U I I ~ ) I I U IwI ~i t h ~ ( x ) F u r t h e r some c o n d i t i o n o f t h e form (u)

-+

as x

-+

m,

w i l l o f t e n a r i s e ( ~ ( x )need n o t be p o s i t i v e f o r small x ) ; t h i s

c o n d i t i o n corresponds t o t i n u i t y f o r A: V

-+

(Am)

R d A ( u ) , u ) ~ ~ ~ ( l l u l l ~ ) l l u l l , ,Some . t y p e o f con-

V ' i s a l s o u s u a l l y assumed.

We s h a l l show t h a t an A as

above i s e.g. demicontinuous i n W s ( A ) ( i . e . continuous from t h e s t r o n g t o p P when t h e Aa a r e e.g. c o n t i n u o u s o l o g y o f Ws t o t h e weak t o p o l o g y o f (W')'), P P i n a l l variables simultaneously ( c f . [Bel;Brl]). Thus l e t un

+

u s t r o n g l y i n W s ( A ) and n o t e t h a t A maps bounded s e t s i n t o P by (*+). Indeed ( m * ) I I A ( u ) l l W l = sup [ I ( A u , v ) l / I I v l l ]

bounded s e t s i n ( W z ) ' < g(llull ) (Ws S,P P

= W h e r e ) and g ( x ) i s bounded f o r x bounded by c o n t i n u i t y .

Since bounded s e t s i n W ' a r e r e l a t i v e l y compact s e q u e n t i a l l y i n t h e weak

t o p o l o g y (see Appendix B ) t o show A(un)

-+

A ( u ) weakly i t s u f f i c e s t o show

t h e r e i s a subsequence o f any i n f i n i t e sequence A ( u k ) o f t h e A(u,) converges weakly t o A ( u ) i n W ' ( t h e n i f A(u,)

P

which

A ( u ) weakly t h e r e must be an

i n f i n i t e subsequence A(uk) l y i n g o u t s i d e o f some NBH o f A ( u ) and t h e weakly convergent subsequence o f t h i s ensured by weak s e q u e n t i a l compactness would

236

ROBERT CARROLL

n o t converge weakly t o A(u) which would c o n t r a d i c t ) .

L e t us a l s o be e x p l i c -

i t here about t h e argument i n terms o f f u n c t i o n s and e q u i v a l e n c e c l a s s e s i n

Thus t a k e f u n c t i o n s un

Lp t y p e spaces.

in t h e above so t h a t as above

u

-+

By d e f i n i t i o n o f t h e q u o t i e n t topology, i n o r d e r t o show

A(un) i s bounded.

A i s continuous on e q u i v a l e n c e c l a s s e s i t s u f f i c i e s t o show i t i s continuous

on f u n c t i o n s .

Now choose from t h e g i v e n i n f i n i t e sequence A(uk) a subse-

quence o f uk (again denoted u k ) such t h a t u k ( x ) and Dauk(x) converge a l m o s t

.,D

uous as s p e c i f i e d , c l e a r l y A,(x,uk(x),.. almost everywhere on A .

S

L e t now ~ ( u =) 1 +

IClk(x) = v ( u k ) ( x ) w i t h 9 ( x ) = P ( u ) ( x ) . and h k ( x ) = [A,(x,ukY...,D S uk)/Jlk(x)]

on

With A,

f o r la1 5 s (see [ B o ~ ] ) .

everywhere i n A t o u ( x ) and D'u(x)

uk(x))

.. ,D

A,(x,u(x),.

-+

1 IDO"ulP-'

J,

f o r la1 5 s and s e t

bounded convergence theorem t h a t hkDau

1Ial.s

on .A,

But

i n q norm ( q = p/p-1) and i t f o l l o w s immediately from t h e Lebesgue hDau i n p norm f o r la1 5 s ( i d e n -

-+

t i f y i n g two f u n c t i o n s equal almost everywhere).

h i s

u(x))

Then J,,(X) -t J,(x) a l m o s t everywhere +. h ( x ) = (A,/J,)(x) almost everywhere

A w h i l e remaining bounded; t h i s f o l l o w s from t h e e s t i m a t e s

Jlk -+

continS

< A,(XyUk,...yDSUk~DaV)

< A,(x,u

=

,...,DS~).Da~)

Hence ( A ( u k ) , v ) = a ( u ,v) =

ll,,/<s < $k,hkDUV)

= a(u,vT = (A(u),v).

-+

1

<J,,kD

i!l

Thus':/;:)

-+

v) =

A ( u ) weak-

l y and p a s s i n g t o equivalence c l a s s e s t h i s proves

L e t A,(x,u,.

tHE0REm 2.14.

.. ,DSu)

determine a continuous f u n c t i o n i n a l l

v a r i a b l e s s i m u l t a n e o u s l y and s a t i s f y ( * 4 ) .

Then A, d e f i n e d by (*.)

v ) , i s a demicontinuous map W s ( A ) P ed s e t s .

t a k i n g bounded s e t s i n t o bound-

DEFlNltI0N 2.15.

-P

WSP(A)I,

L e t V be a r e f l e x i v e B space and l e t A: V + V '

domain D = D ( A ) C V such t h a t D i s a l i n e a r space. t o n e i f Re(A(u)-A(v),u-v)

2 0 ( > 0 ) f o r u,v

E D,

and a(u,

have a dense

A i s ( s t r i c t l y ) D-mono-

(u

v).

A i s demicontin-

uous i f un -+ u (un,u E D ) i m p l i e s A(un) -+ A ( u ) weakly. A i s hemicontinuous i f u E D, w E V, u + t n w E D (hence w E D when D i s l i n e a r ) , tn -+ 0, i m p l i e s A(u + t n w )

wo

E

+

A ( u ) weakly.

A monotone A i s maximal D-monotone i f f o r uo

V ' t h e i n e q u a l i t y Re(wo

-

A(u),uo-u)

E

D,

2 0 f o r a l l u E D i m p l i e s wo = A

A i s l o c a l l y bounded i f f o r any sequence un E D, un -+ u E D, one has (u,). A(un) bounded. In case V i s a H i l b e r t space we r e p l a c e ( , ) by ( , ) and

deal w i t h maps A: V

-+

V i n which case A i s m r e p r o p e r l y c a l l e d a c c r e t i v e

(see §3.3,3.6 f o r more d i s c u s s i o n ) . Note t h a t A m i g h t be maximal D-monotone and s t i l l a p p a r e n t l y have an extens i o n t o a D-monotone o p e r a t o r set).

'A" 3 A

with

is 3 D ( g n o t

n e c e s s a r i l y a 1i n e a r

S i m i l a r l y a D monotone A m i g h t n o t have such e x t e n s i o n s and y e t nat

237

NONLINEAR ANALYSIS

be, f o r c i b l y , maximal D-Monotone. The matter of maximal monotonicity i s b e t t e r phrased i n the context of multivalued maps ( c f . Remark 6.16). We will study monotone o p e r a t o r s i n 113.3 and 3.6 and f o r now simply make a few additional comnents t o set t h e s t a g e . In p a r t i c u l a r one t h i n k s o f perturbations of the A of Theorem 2.14 by lower order operators Ds-l (A&) Bu = 1 I 8 ISS-1 DBB8(x,u,. , u ) associated w i t h the natural forms b(u,v) = 1 ( B B ( x , u , . . . , u ) , D 8 v ) . One r e c a l l s t h a t the embedding W’(A) + P is compact f o r l / r > l / p - l / n ; in f a c t wP wrj i s compact f o r l / p wS-’(A) r - ( s - j ) / n < l / r . Then defining an o p e r a t o r 3: V -+ V ’ by ( B ( u ) , w ) = b ( u , w ) (as with A) one makes hypotheses on t h e B so t h a t 3: V V ’ will be com8 pact. For example i f one assumes estimates of the form (*&) then t h e analogue of (**I is (-1 I b ( u , v ) I 5 g ( h s - l ,p)llvlls;l ,E?‘ ^I, Let be V w i t h the ws-l topology and write b(u”,T) = (%,V) (where 3: V -+ V can a l s o be extenP ded by c o n t i n u i t y t o the c l o s u r e of v“ i n WS-l by (A+)). The argument of P h A Theorem 2.14 shows t h a t B i s demicontinuous from say V -+ V ’ and takes bounded sets i n t o bounded s e t s . Then l e t i : V +. V be the compact i n j e c t i o n . Since V i s r e f l e x i v e and separable we can v e r i f y the B complete c o n t i n u i t y of B by looking a t sequences un + u weakly ( B complete c o n t i n u i t y means complete c o n t i n u i t y on bounded s e t s - see below). Evidently B = i*
..

-+

-+

A

RmtARK 2.17. Some remarks on compactness e t c . a r e a l s o in o r d e r . F i r s t we r e s t a t e t h e d e f i n i t i o n of complete c o n t i n u i t y . T h u s a map C: E F , E and F Banach spaces, i s completely continuous i f i t i s continuous from t h e weak topology of E to t h e strong topology of F. If this holds only on bounded s e t s i n E then C will be c a l l e d B completely continuous. Note here t h a t t h e weak topology of E in general i s not metrizable and net arguments must be used. The d e f i n i t i o n s and basic f a c t s about nets a r e given in Appendix A and the added g e n e r a l i t y and f l e x i b i l i t y is well worth a few moments of ref l e c t i o n on the net concept ( c f . [Kel] f o r a complete d i s c u s s i o n ) . Now we r e c a l l t h a t a space i s c a l l e d separable i f i t has a countable dense s e t and one should check t h a t W m ( ~ ) f o r example i s separable and r e f l e x i v e f o r p += 1 P o r m. Further ( s e e e.g. [Zal]) i f E i s separable and r e f l e x i v e then E ’ i s separable and then ( c f . [ D u l l ) t h e weak topology of t h e s t r o n g l y (and weakl y ) closed u n i t ball B1 = { x E E; IIxII 5 11 i s a metric topology ( s e e below). Thus i n dealing w i t h complete c o n t i n u i t y one need only consider weak sequent i a l convergence on any ball BR = {x E E; IIxII 5 R ) C E f o r separable r e f l e x -+

2 38

ROBERT CARROLL

i v e E.

We remark t h a t i n p r a c t i c e i t i s B complete c o n t i n u i t y which i n t e r Next we have seen a l r e a d y i n

venes f r e q u e n t l y and n o t complete c o n t i n u i t y .

Remark 2.13 t h a t when E i s r e f l e x i v e a c o m p l e t e l y continuous ( o r B completeOn t h e o t h e r hand l e t T: E

l y c o n t i n u o u s ) map i s compact. and l e t e,

be a bounded weakly convergent n e t w i t h e,

w i t h Te = f and assume t h a t f,

-+

-+

F be compact L e t Tea = f,

e.

does n o t converge s t r o n g l y t o f.

Then f o r

E we can f i n d a subnet f such t h a t I l f - fll 2 E . But s i n c e T t a k e s B B bounded s e t s i n t o r e l a t i v e l y compact s e t s we know t h a t f has a convergent B f ' . Thus Te f ' # f s t r o n g l y and hence weakly. NOW, i f T i s subnet f Y Y l i n e a r , t h e n i t i s weakly continuous ( c f . Appendix A ) so Te -+ Te = f weakly Y and we have a c o n t r a d i c t i o n . I f T i s n o n l i n e a r however we seem t o need an

some

-+

-f

a d d i t i o n a l hypothesis.

Thus e.g.

i t i s B c o m p l e t e l y continuous.

above.

i f T i s compact and weakly continuous t h e n

F i n a l l y l e t us s k e t c h some f a c t s r e f e r r e d t o

F i r s t i t i s easy t o show t h a t E spearable r e f l e x i v e i m p l i e s E ' i s

separable ( c f . [Dul;Zal]).

Indeed l e t {en) be dense i n E and s e t fn = en/

IIenll so t h a t t f n l i s dense i n S1 e x i s t s fl;

E

E ' such t h a t 1lf;lIl

=

{x

1 and

E

E; IIxII = 11 C

I( f,!,,fn)l

E.

F o r e v e r y fn t h e r e

= 1 (Hahn-Banach theorem).

I f t h e l i n e a r span o f t h e fl; were n o t dense i n E ' t h e n by r e f l e x i v i t y and another v e r s i o n o f t h e Hahn-Banach theorem ( c f . Appendix B) t h e r e would be an f

E

IIf-fnll

E such t h a t IIfll = 1 and (f,f,',) = 0 f o r a l l n. >

I (f-fn)(f,l,)l =

But t h e n f o r a l l n,

1 which c o n t r a d i c t s s i n c e t h e fn a r e dense on S1.

The s e t o f f i n i t e l i n e a r combinations

anf,l,

w i t h an r a t i o n a l complex i s

t h e n dense i n E ' which shows t h a t i t i s separable.

Next t o produce a m e t r i c

(weak) t o p o l o g y on t h e u n i t b a l l B1 o f a separable r e f l e x i v e Banach space E l e t e,!, be dense i n E ' and d e f i n e l(x-y,el;)l]. than o ( E , E ' ) . be i d e n t i c a l . 3.

(Am)

d(x,y) =

1 (1/2n)[1(x-y,e;l)l/(l

+

T h i s y i e l d s a m e t r i c t o p o l o g y on B, which i s e v i d e n t l y weaker But B1 i s o ( E , E ' )

IIIBNOCONE OPERACORk.

compact and hence t h e two t o p o l o g i e s must

The t h e o r y o f monotone o p e r a t o r s ( c f . D e f i n i t i o n 2.

15) was s y s t e m a t i c a l l y developed i n connection w i t h a p p l i c a t i o n s t o PDE i n t h e 1 9 6 0 ' s and 1970's.

We w i l l make no a t t e m p t t o g i v e h i s t o r i c a l documen-

t a t i o n n o r r e f e r e n c e s t o papers b u t r e f e r t o [ C l ;Bel;Brl;Dml this.

71)

We f o l l o w here [ C l ]

;Ze3] f o r a l l

(based i n p a r t on Browder's e a r l y work i n [Br2-

and t o [Dml] f o r o u r p r e s e n t a t i o n .

We g i v e m a i n l y t y p i c a l r e s u l t s and

ideas; no a t t e m p t i s made t o g i v e b e s t p o s s i b l e theorems.

The t h e o r y has

a t t a i n e d a v e r y s o p h i s t i c a t e d a b s t r a c t form and e a r l i e r f o r m u l a t i o n s seem more i n s t r u c t i v e and u s e f u l f o r t h i s book.

MONOTONE OPERATORS

REIIMRK 3.1.

239

L e t us n o t e t h a t m o n o t o n i c i t y o f A: V

V' (V' = antidual) i s

-+

2 0 ( w h i c h corresponds t o Re(Av,v) 2 0 when

expressed v i a Re(A(u)-A(w),u-w)

I t i s c o n v e n i e n t a t t i m e s t o t h i n k o f r e a l Banach spaces and

A i s linear).

o m i t t h e symbol Re; t h e n t o go back t o t h e complex s i t u a t i o n one s i m p l y i n s e r t s Re a t t h e a p p r o p r i a t e places.

We remark a l s o t h a t f o r A as i n Theorem

2.14 a s o l u t i o n u t o Au = f i s a weak t y p e " v a r i a t i o n a l " s o l u t i o n o f t h e problem based on ( - l ~ a l D U A a ( ~ , ~ , . . . y D S u ) = g where say (g,v),z = (f,v)V-Vl.

1

I n what f o l l o w s A: D(A) C V * V ' ,

D(A) i s dense, and V i s a r e f l e x i v e Banach

space ( = B space) as i n D e f i n i t i o n 2.15 (D i s a l s o 1 i n e a r u s u a l l y ) .

CHE0Rfill 3.2.

I f A i s maximal D-monotone and l o c a l y bounded t h e n i t i s

demicontinuous.

Phood:

L e t un E D, un

uo E D.

I f A(un) P A(uo) weakly t h e n as b e f o r e t h e r e i s an i n f i n i t e subsequence c o n v e r g i n g weakly t o some wo # A(uo) (by +

B u t i f u E D i s a r b i t r a r y , Re(A,(unk)

weak s e q u e n t i a l compactness). Unk

-

u) 1. 0 and as k

Hence f o r a l l u

E

-f

m,

+ Nuo).

-

unk

D (*) R d w o

-

uo

+

-

0 and ~ ( u , , ~ )

-

A(u),uo

-

~ ( u +) wo

-

A(u),

~ ( u weakly. )

u ) 2 0 which c o n t r a d i c t s s i n c e wo

QED

C H E 0 R B 3.3.

L e t A be D-monotone and hemicontinuous; t h e n A i s maximal D-

monotone.

Phood:

Suppose (*) h o l d s f o r uo E D and a l l u E D b u t wo

-

i s dense and l i n e a r t h e r e e x i s t s v E D such t h a t R e ( w o t l y and u o + t v E D f o r 0 5 t 5 1 say.

weakly i f tn > 0, tn + 0. ( p u t u = uo+tnv i n ( * ) ) .

A(uo),v)

Now w r i t e wo

-

Since D > 0 stric-

By h e m i c o n t i n u i t y A(uo+tnv) * A(uo)

Then by assumptions Re(A(uo+tnv)

A(uo+tnv) t o o b t a i n R d A ( u o + t n v )

-

A(uo+tnv) = w0

A(uo),v) 2 R H w o

-

-

-

wo,v) 2 0

A ( u o ) + A(uo)

A(uo),v)

t h e l e f t s i d e tends t o 0 we have a c o n t r a d i c t i o n .

C0R0CLARg 3.4.

# A(uo).

> 0.

-

Since

QED

I f A i s hemicontinuous, D-monotone, and l o c a l l y bounded t h e n

i t i s demicontinuous. C l e a r l y d e m i c o n t i n u i t y i m p l i e s h e m i c o n t i n u i t y ; f u r t h e r A demicontinuous i m p l i e s A i s l o c a l l y bounded s i n c e i f un

-+

u E D, un

E

D, t h e n { A ( u n ) } i s

weakly bounded ( b e i n g weakly Cauchy) and hence by Banach-Steinhaus t A ( u n ) l i s s t r o n g l y bounded.

CHE0RElll 3.5.

I f V i s f i n i t e dimensional one has

I f V i s f i n i t e dimensional and A ( d e n s e l y d e f i n e d as i n D e f i -

n i t i o n 2.15) i s D-monotone and hemicontinuous, t h e n A i s continuous.

Phood:

Using C o r o l l a r y 3.4 i t i s enough t o show t h a t A i s l o c a l l y bounded,

240

ROBERT CARROLL

s i n c e d e m i c o n t i n u i t y i s o b v i o u s l y e q u i v a l e n t t o c o n t i n u i t y f o r V f i n i t e dimWe s h a l l show t h a t i n f a c t A monotone and dimV <

ensional.

u, un

-+

E

-

E

D consider

-

tw) =

-

u

-

0 ( t > 0 i s f i x e d h e r e ) , un

+.

u, and ailA(un)

Re(ai’A(un) -f

For w

+. m.

-1 0 5 an Re(A(un)

But a;’A(u+tw)

Without l o s s o f g e n e r a l i t y one

D, w i t h A(un) unbounded.

can assume IIA(un)II = an (3.1)

i s enough f o r

Suppose t h e c o n t r a r y ; t h e n t h e r e i s a u E D and a seq-

l o c a l boundedness. uence un

m

D i v i d i n g b y t and l e t t i n g n

D i s dense and ailA(un)

-

A(u+tw),un

-+

u

ailA(u+tw),un

m

tw)

one has l i m i n f Re(ailA(un),-w)

i s bounded t h i s h o l d s f o r any w E V.

i s bounded.

2 0.

Since

Replacing w by

-w (and by + i w i n complex spaces V ) one o b t a i n s l i m ( a ~ ’ A ( u n ) , w > =

0 for a l l

w E V which means ailA(un) 0 weakly and hence s t r o n g l y s i n c e dimV < But t h i s c o n t r a d i c t s s i n c e llailA(un)ll = 1. QED -f

-.

A v e r s i o n o f t h i s l a s t theorem was proved by Kato and we mention a l s o t h e background work o f M i n t y on monotone o p e r a t o r s and V i g i k on c o n c r e t e e l l i p t i c s i t u a t i o n s which c o n t r i b u t e d t o t h e development o f t h e t h e o r y . now some e x i s t e n c e r e s u l t s f o r n o n l i n e a r maps V

-+

We prove

V ’ d e f i n e d everywhere i n

V ( t h e s e correspond t o e l l i p t i c s i t u a t i o n s and a r e based on e a r l y work o f Browder i n t h i s a r e a ) . We g i v e f i r s t a theorem which i l l u s t r a t e s some asp e c t s o f a g e n e r a l i z e d G a l e r k i n method.

CHE0REIII 3.6.

F

-+

L e t F be a r e f l e x i v e B space, A: F -+ F ’ be hemicontinuous, B:

F ’ be B c o m p l e t e l y continuous, and f o r a l l u,v E F l e t ).(

A(v),u-v) + (3(u) m

as x

-f

m.

-

3 ( v ) , u - v ) 2 0 and

Then A maps F o n t o F ’

(q

(0)

Re(A(u)

-

Re(A(u),u) > q(IIuII)IIuII w i t h q ( x )

-+

may be n e g a t i v e f o r x s m a l l ) .

Phoo6: L e t w E F ’ be a r b i t r a r y and c o n s i d e r A1(u) = A(u) - w. I t i s easy t o see t h a t A, s a t i s f i e s t h e same c o n d i t i o n s as A (e.g. y , ( x ) = ‘p(x) - llwll) and hence i t s u f f i c e s t o show t h a t 0 E R ( A ) . L e t A be t h e d i r e c t e d s e t o f f i n i t e dimensional subspaces o f F o r d e r e d by i n c l u s i o n . For E E A l e t iE: E + F be t h e i n j e c t i o n . (To a v o i d n e t arguments one can assume V i s separa b l e r e f l e x i v e (hence V ’ separable) w i t h Vn C Vn+l a sequence o f f i n i t e dimin: Vn +. V, e t c . ensional supspaces, o r d e r e d by i n c l u s i o n , w i t h V = U V n’ Then argue as below w i t h inr e p l a c i n g iE; t h i s would correspond t o a t r a d i t i o n a l G a l e r k i n t y p e procedure. Net arguments a r e however e s s e n t i a l l y t h e same and i t seems w o r t h w h i l e t o develop some f a c i l i t y i n t h i s ( c f . [Kel;Bo4] and Appendix A).) Now i s weakly continuous (Appendix A ) and hence A E + B E

it

MONOTONE OPERATORS

E

= i$At3)iE:

241

E' i s hemicontinuous and monotone ( e x e r c i s e ) .

-+

Since E i s

f i n i t e dimensional Theorem 3.5 i m p l i e s t h a t A E t BEY and hence AEy i s continuous.

Also f o r u

E ( 6 ) Re(AEu,u) =

€

Re(

I F i E u y u ) = Re(Au,u) L v ( I I u l l ) I I u l .

Now we c i t e a lemna which i s proved i n 53.4 (needed t o f i n i s h t h e p r o o f ) .

LRRIRA 3.7, L e t A be a continuous map o f t h e f i n i t e dimensional B space E i n t o E ' such t h a t ( 0 ) h o l d s w i t h ~ ( x + ) m as x -+ Then A maps E o n t o E l .

-.

Given t h e t r u t h o f Lemna 3.7 we o b t a i n an element uE E E such t h a t AEuE = 0 and hence cp(UuIIE) 5 0 f o r each uE as E v a r i e s . l y i n E.

L e t now u

T h i s means IIuEll 5 M u n i f o r m -

F be a r b i t r a r y and c o n s i d e r a l l E

€

e. c o n t a i n i n g t h e 1 - D space Eo generated by u ) .

T h i s says t h a t t h e n e t AuE converges t o 0 weakly.

0.

€ A

c o n t a i n i n g u (i.

Then ( A u E Y u ) = (AEuE,u) = Moreover f o r such E Hence ( + ) Re

Re(AuE-Au,uE-u) = Re(AEuE,uE-u) t Re(Au,u-uE) = Re(Au,u-uE).

2 0.

[(3u-3uE,u-uE)+(Au,u-uE)]

Eo

E

Now g i v e n any Eo

E A,

w r i t e Fo = { U { u E l ;

E; E E A } ; c l e a r l y Fo C BM where BM i s t h e c l o s e d b a l l o f r a d i u s M cenNow t h e n e t uE ( a l l E) l i e s i n BM and by weak compactness has a

t e r e d a t 0.

weak c l u s t e r p o i n t uo cluster point).

E

BM (a s e t i s compact i f and o n l y i f each n e t has a

By s t a n d a r d f a c t s about n e t s (see [Kel;Dul]

and Appendix

A ) uo belongs t o t h e weak c l o s u r e o f each Fo ( c f . h e r e a l s o t h e p r o o f o f Theorem 3.22). Thus i n ( + ) l e t t i n g a subnet u E g + uo weakly we o b t a i n ( m ) > 0, s i n c e 3u o( + 3u s t r o n g l y by t h e compactness o f 3 E 0 and t h e f a c t t h a t I I u E ~ l l5 M. Moreover ( = ) h o l d s f o r any u E F s i n c e uo i s

Re(Au+3u-3uo,u-uo)

adherent t o each Fo.

Now a p p l y Theorem 3.3 t o conclude t h a t A t 3 i s maximal

F monotone and hence from (=), (A+3)uO = 3u0 o r Auo = 0.

QED

T h i s enables one i n p a r t i c u l a r t o f i n d u E V such t h a t a(u,v) all v

E

= (f,v)

for

V where f E V ' i s g i v e n a r b i t r a r i l y , whenever a ( - , - ) s a t i s f i e s (")

+ b(-,-) satisfies

Wi(A)

C V C as beP f o r e , and a,b a r e t h e v a r i a t i o n a l forms i n V a s s o c i a t e d w i t h t h e d i f f e r e n -

i n 53.2,

a(-,-)

(A*)

i n 53.2,

?(A)

t i a l o p e r a t o r s (**) and (A&) w i t h hypotheses o f t h e t y p e ( * b ) i n § 3 . 2 . Some g e n e r a l i z a t i o n s o f Theorem 3.6 i n v o l v i n g densely d e f i n e d o p e r a t o r s a r e i m p o r t a n t i n t h e t h e o r y o f n o n l i n e a r e v o l u t i o n equations and appear l a t e r ( c f . [Brly5;Cl]),

We mention f i r s t some v a r i a t i o n s and e x t e n s i o n s f o r ev-

erywhere d e f i n e d o p e r a t o r s . h y p o t h e s i s i n a f o r m a(u,v)

F i r s t i t i s possible t o r e f e r the monotonicity = (Au,v)

o r d e r d e r i v a t i v e s (see [Brly2;Dml;Cl]).

(**) a(u,v,w) u

E

ws-1

P

.

1I I_<s

t o t h e terms i n v o l v i n g o n l y t h e h i g h e s t To do t h i s one c o n s i d e r s t h e f o r m

cI

(Aa(x,u, ...,DS-l~,DS~),Da~)

where e.g.

v,w E W

S

and P T h i s i s w e l l d e f i n e d when ( * 6 ) o f 53.2 h o l d s w i t h C8 = DBu f o r =

242

ROBERT CARROLL

161 5 s-1 and 5

B

= D'v

for

161 = s; i t i s easy t o see then t h a t where g ( x ) = c ( l + x P - ' )

(*A)

f o r some c .

u,w)I I g ( l l u l l s-1 ,P )g(llvll~,p)llwllSYP (*A) i m p l i e s t h e r e i s an element S(v,u) E V ' such t h a t (**) (S(v,u),w)

la(v, Now =

S

where we assume momentarily t h a t C V C W w i t h u.v,w E V. Then P P t h e o p e r a t o r S(u,u) corresponds t o t h e A ( u ) t r e a t e d p r e v i o u s l y and as b e f o r e

a(v,u,w)

t h e map u

w i t h (BU,W) = b(u,w)

uous.

-f

S(v,u):

V

V ' w i l l be B c o m p l e t e l y c o n t i n -

-f

This leads t o t h e f o l l o w i n g simple abstract formulation f o r d i f f e r e n -

t i a l problems based on (**).

F ' i s semimonotone i f A ( u ) = S(u,u) where S: D E F I N Z C I ~ N3-8, A map A: F F X F F ' i s monotone i n t h e f i r s t argument and B c o m p l e t e l y continuous i n -+

-f

F ' i s f i n i t e l y continuous i f i t i s continuous f r o m f i n i t e dimensional subspaces E c F t o t h e weak t o p o l o g y o f F ' .

t h e second.

A map A : F

-f

EHE0REfll 3.9, L e t F be a r e f l e x i v e Banach space and A a f i n i t e l y continuous semimonotone o p e r a t o r F F ' ( A ( u ) = S(u,u)) w i t h u S(u,v) hemicontinuous, -f

s a t i s f y i n g Re(A(u),u) )'p(IIuII)IIuII

with ~ ( x )

-f

-

-f

as x

-.

-f

Then A maps F

onto F ' .

Pfiuua:

R(A) s i n c e S1(u,v) S(u,v).

=

S(u,v)

For E E A, AE =

holds a g a i n . M.

Again i t s u f f i c e s t o show 0 E

We f o l l o w t h e p r o o f o f Theorem 3.6.

Again AuE

-

y, y f i x e d , w i l l have t h e same p r o p e r t i e s as

itAiE:

E

-f

E ' i s continuous by assumptions and ( 6 )

By Lemma 3.7 we have elements uE E E w i t h AEuE = 0 and IIuEll 5 -f

0 weakly and t h e r e i s a uo E BM weakly adherent t o a l l Fo.

Given u E F and E E A, u E E, we have Re( S(u,uE)

-

S(uEYuE),u-uE) 1. 0 which

means, s i n c e S(uE,uE) = A(uE) w i t h (A(uE),u-uE) = ( A E ( u E ) , u - u E ) = 0, t h a t R e ( S ( u , u ~ ),u - u E ) > 0.

By B complete c o n t i n u i t y o f S i n t h e second argument

w i t h IIuEll 5 M we o b t a i n Re(S(u,uo),u-uo) a p p l y i n g Theorem 3 . 3 t o t h e map u

-+

2 0.

S(u,uo)

T h i s h o l d s f o r a l l u E F and

one has S(uo,uo) = A(uo) = 0.

We mention n e x t a n o t h e r t y p e o f theorem having a p p l i c a t i o n t o e l l i p t i c prob-

A t y p i c a l r e s u l t o f Browder i s c i t e d

lems which a r e n o t s t r o n g l y e l l i p t i c . g e n e r a l i z i n g a theorem o f Z a r a n t o n e l l o .

EHEBRETil 3-10, map.

L e t F be a r e f l e x i v e

B

space and T: F

Suppose ( * 6 ) I ( T u , u ) l ~ ' p ( I I u l l ) l l u l lw i t h ~ ( x )

-f

-

-

F ' a demicontinuous

-f

as x

-f

and suppose

f o r each N > 0 t h e r e i s a continuous ( s t r i c t l y ) i n c r e a s i n g f u n c t i o n $,(O)

= 0, such t h a t f o r IIuII, IIvII < N (*+) I(Tu-Tv,u-v)l

1. $N(IIu-vll)llu-vll.

Then T i s a 1-1 map o f F o n t o F ' w i t h a continuous i n v e r s e . The p r o o f i s s i m i l a r t o p r e v i o u s arguments b u t now one uses t h e Brouwer t h e orem on i n v a r i a n c e o f domain ( c f . §3.4) which i m p l i e s t h a t a b i c o n t i n u o u s

243

MONOTONE OPERATORS

1-1 map TE: E

E ' has open range (TE = i f T i E , dimE <

-+

a l s o c l o s e d by t h e c o n t i n u i t y o f T';

Since R(TE) i s

( w h i c h f o l l o w s from t h e hypotheses) we

must have R(TE) = E ' by connectedness. w i t h TEuE = 0.

m).

Hence t h e r e i s a ( u n i q u e ) uE E E

Then one argues as b e f o r e ( c f . [ C l ]

f o r details).

L e t us mention i n passing t h e a p p l i c a t i o n o f t h e s e techniques t o s i m u l t a n e o u s l y t r e a t monotone o p e r a t o r s and t h e m i n i m i z a t i o n o f convex f u n c t i o n a l s following [Brl-7]

(cf. also § 3 . 7 ) .

One c o n s i d e r s r e a l B spaces and r e a l

v a l u e d f u n c t i o n s f , g w i t h V a c l o s e d subspace o f # ( A ) , im C V C WF, and A P P bounded. L e t (A*) I ( u , v ) = I,, f ( x , v ( u ) , $ ( v ) ) d x where v ( u ) denotes t h e Dau f o r (a1 = m and $ ( v ) denotes t h e DBv f o r

IB(

5 m-1.

One wants t o make mono-

t o n i c i t y t y p e hypotheses o n l y on t h e terms o f h i g h e s t degree and t h i s i s r e -

/,

F u r t h e r l e t (")

J(u) = g(x,$(u))dx. B I f Aa denotes f (pa = Dau) and A denotes f (4, = D v ) w i t h B6 = P' B gP t h e n we assume e.g. (A*) l f ( x , v ( u ) , $ ( v ) ) l < c?1 + 1 IDaulP + ID6uIp1; I g ( x , $ ( u ) ) l 5 c [ l + 1(IBlZm-1) I D B-U I PI; l A a ( x , v ( u ) , $ ( v ) ) l 5 c sponsible f o r the formulation

(A*).

1

lDBvlD-l] ( w i t h a s i m i l a r e s t i m a t e f o r AB) and IDBuIp-']. Now ( c f . §3.2 f o r F d e r i v a t i v e s )

DEFINICZBN 3-11.

A f u n c t i o n I:V X V

+

R i s semiconvex i f (A) F o r each w E

V and c E R t h e s e t W

= I u E V; I(u,w) 5 c l i s convex (B) F o r each bounc,w ded B C V and each weakly convergent bounded n e t wa + w i n V, I(u,w,) * I(u,

w) u n i f o r m l y f o r u E B ( C ) F o r w E V f i x e d t h e map u tinuous.

I n t h i s case we c a l l x

+

+

I(u,w):

V

+

R i s con-

h ( x ) = I ( x , x ) semiconvex as w e l l .

It can be e a s i l y seen ( r e c a l l p r e v i o u s e s t i m a t e s i n 53.2 u s i n g t h e bounded

convergence theorem) t h a t under t h e above growth e s t i m a t e s , w i t h say f , g

i n t h e u,v arguments and say measurable i n x, I:V X V

-f

E

C

R i s c o n t i n u o u s and

s a t i s f i e s c o n d i t i o n s (B) and ( C ) f o r semiconvexity, w h i l e J: V

+

R i s con-

t i n u o u s and weakly c o n t i n u o u s on bounded s e t s ( h e r e use a l s o Remark 2.16 w i t h t h e f a c t t h a t V i s r e f l e x i v e and separable w h i l e t h e i n j e c t i o n Wm + P Wm- 1 i s compact). F u r t h e r because o f t h e a d d i t i o n a l growth e s t i m a t e s i n P (A*) i t r e s u l t s t h a t I ( - , - ) and J ( - ) a r e d i f f e r e n t i a b l e w i t h ( e x e r c i s e ) (Ii(U,V),W) = 1 lal=,, A6(X,v(U),$(V))@WdX;

CHE0REm 3-12. t o n e i n u.

Phood:

/, (

A,(xyv(u),~(v))Dawdx; ( I ~ ( u , v ) , w ) = 1 I B l m - 1 'A J ' ( u ) , w ) = 1 ~ ~ J,,l B ~ B ( x y i~ ( u ) ) D-B w ld x .

Suppose I ( - , - ) i s d i f f e r e n t i a b l e on V X V w i t h I i ( u , w ) mono-

Then c o n d i t i o n (A) f o r semiconvexity h o l d s .

I t s u f f i c e s t o show t h a t I(u,w)

means h ( x ) = I ( x u + ( l - x ) v , w )

- xI(u,w)

-

i s convex i n u f o r f i x e d w; t h i s (l-A)I(v,w)

1

5 0 for 0 5

x

5 1.

244

ROBERT CARROLL

0 i t w i l l s u r e l y s u f f i c e t o show t h a t h ' ( A ) i s nonde-

Since h ( 0 ) = h ( 1 ) creasing.

But e v i d e n t l y h'(A) =

hence f o r X < 5 , h ' ( 5 )

-

Ii(ucyw)

-

Ii(uA,w),us-uA) 2 0

-+

R i s determined f o r m a l l y

QED

Now when I i s g i v e n by =

(A&) I " ( U , V ) . W - Y

- I(u,w) + I(v,w) and

1; (Xu+(l-A)v,w),u-v)

h ' ( A ) = (<-A)-'(

where uX = Au+(l-A)v.

by

(

(A*)

I*

c

we see t h a t 1;:

V X V

I a l = I y I = m Aay ( x , ~ ( u ) , ~ ( v ) ) D a w D y y d x . We suppose

t h i s makes sense and f u r t h e r t h a t 1 A a y ( ~ y v ( ~ ) . ~ ( ~ )L) 0~ af o~r y la I = I y I =m Then by T a y l o r ' s theorem f o r t h e F d e r i v a t i v e ( c f . 2 § 3 . 2 ) ( I i ( u + y , v ) - I;(u,v),y) = /; Ii'(u+ty,v).y d t L 0. Hence Theorem 3.12

any r e a l e n t r i e s 5,.

would a p p l y and I would be semiconvex i n these circumstances.

We a l r e a d y

have a l a r g e c l a s s o f model problems p r o d u c i n g f u n c t i o n s I and J w i t h s u i t able properties.

& H E 0 R m 3.13.

Now a b s t r a c t l y

L e t F be a r e f l e x i v e B space and I ( -,- ) a semiconvex r e a l

f u n c t i o n on F X F ( t a k i n g f i n i t e v a l u e s ) . weakly c l o s e d bounded subset o f

F.

L e t h ( x ) = I ( x , x ) and K be a

Then h i s bounded below on K and assumes

i t s minimum t h e r e . L e t k = i n f h ( x ) f o r x E K and p i c k i n g xn E K such t h a t h ( x n ) 5 k +

Phaad:

l / n we o b t a i n a sequence w i t h h ( x n )

+

k.

By r e f l e x i v i t y K i s weakly sequen-

t i a l l y compact and a subsequence o f t h e xn ( a g a i n c a l l e d xn) converges weakly to x

€

K.

Hence I(xn,xn)

+

k.

L e t k ' > k; then f o r n l a r g e xn

But W k l Y x

i s convex and s t r o n g l y c l o s e d by

o f D e f i n i t i o n 3.11 and hence I ( x n , x ) E

W k l Y x (see D e f i n i t i o n 3.11).

( s i n c e K i s bounded) by p r o p e r t y (B)

I(x,,x) +

p r o p e r t i e s (A) and ( C ) o f D e f i n i t i o n 3.11; Appendix A).

Since xn

-+

hence W k l

x i t f o l l o w s t h a t x E Wkl,x

YX

i s weakly c l o s e d ( c f .

or I ( x , x ) 5 k ' .

k ' was a r b i t r a r y we have h ( x ) = k and hence k > - - a l s o .

QED

-

be semiconvex as above w i t h h ( x ) = I ( x , x ) e0R0CCARM 3.14, L e t I(.,.) .3xII + and F a r e f l e x i v e B space. Then h assumes a minimum on F.

Phaad: h(x)

7

L e t K = I x : IIxII 5 R I f o r R l a r g e enough so t h a t say IIxII i n f F h(x).

C0R0CCARM 3.15. IIxII

-+ m y

z

+

m

as

R implies

QED L e t I ( - , - ) be semiconvex as above w i t h I ( x , x ) = h ( x ) +

m

as

F a r e f l e x i v e B space, and J a weakly continuous f u n c t i o n on bound-

ed s e t s o f F.

Phaad:

Since

I f L = { x ; J ( x ) = c l , t h e n h assumes a minimum on L.

K = L n I x ; IIxII 5 R I i s weakly c l o s e d and bounded f o r a l l R; p i c k R

so t h a t IIxII > R i m p l i e s h ( x ) > i n f L h ( x ) .

QED

We go now t o an a b s t r a c t v e r s i o n o f t h e Lagrange m u l t i p l i e r theorem f o l l o w -

MONOTONE OPERATORS

245

i n g Browder ( c f . Remark 2.12).

ClfEQRflll 3-16, L e t h and J be two r e a l f u n c t i o n s on a B space F, w i t h b o t h d i f f e r e n t i a b l e a t xo, and J ' ( x o ) # 0. I f h has a l o c a l minimum a t xo w i t h r e s p e c t t o t h e s e t L = { x ; J ( x ) = J ( x o ) } , t h e n h ' ( x o ) = X J ' ( x o ) (some A ) . A

Pmob: L e t F

= {x; x

E

= 01 and p i c k yo

F with (J'(xo),x)

E

F such t h a t

A

I E ~and

Assume x E F, llxll = 1, w i t h J(xo) = J(xo+ex+nyo);

= 1.

(J'(xo),yo)

then i f

In1 q r e s u f f i c i e n t l y small, by t h e l o c a l minimum p r o p e r t y h ( x o ) 5 A

h(xo+sx+nyo). To r e a l i z e t h i s s i t u a t i o n , c o n s i d e r f o r x E F f i x e d w i t h IIxII = 1 (A*) J ( x o + ~ x + n y 0 ) = J ( x 0 ) + E( J ' ( X o ) , X ) + n( J'(Xo),yo) + I P ( E , ~ , x ) = J ( x o ) + n + IP(E,II,X) where I P ( E , ~ , x ) / ~ E ~++0I ~asI E , V + 0. P i c k I E I small enough, w i t h - 1 ~ 1 / 2 5 r~ 5 le1/2 so t h a t IIP(C,~,X)~5 ( ~ 1 / 2and c o n s i d e r $ ( n ) =

n

w i l l be c o n t i n u o u s i n n ( e x e r c i s e - c f . §3.2) and $ ( - I € [ / = 0 f o r some n = ~ I ( E , x ) i n t h e i n 0, w h i l e $ ( 1 ~ 1 / 2 ) > 0. Hence $(I-,) $

+IP(E,TI,X).

2) <

terval.

F o r t h i s n we have t h e n I n \ = o ( l e 1 ) and

= h(xo) c -o(lEl

E(

1-

h'(xo),x) +

r ~ (h ' ( x o ) , y o )

(Am)

h ( x o ) 5 h(xo+ex+nyo)

+ o(l~l+lnl). Hence

E(

2

h'(xo),x)

Now t h i s procedure can be c a r r i e d o u t f o r a sequence

E,

-+

0 and

a t each s t a g e en can be p o s i t i v e o r n e g a t i v e ( w i t h p o s s i b l y d i f f e r e n t c h o i We conclude t h a t ( h ' ( x o ) , x )

ces o f nn r e q u i r e d ) .

= 0 f o r t h i s , and hence

A

f o r any, x

E

F.

B u t t h e n h ( x o ) = A J ' ( x o ) must h o l d .

QED

c0R0CCAR1J 3-17, Assume F i s a r e f l e x i v e B space, I ( - , . ) i s semiconvex as i n Theorem 3.13, J i s weakly continuous on bounded sets, h ( x ) = I ( x , x ) -+ m as and L = { x ; J ( x ) = c } i s nonempty m, h and J a r e d i f f e r e n t i a b l e on F, IIxll -f

Then h ' ( x o ) = A J ' ( x O ) f o r some xo

w i t h J ' ( x ) # 0 on L.

E

L and X E R.

The p r o o f i s i m n e d i a t e f r o m C o r o l l a r y 3.15 and Theorem 3.16.

Now r e f e r r i n g

back t o Theorem 3.12 f o r example we see t h a t t h e r e i s a n a t u r a l r e l a t i o n between t h e c o n v e x i t y o f a f u n c t i o n h: F v a t i v e h ' : F +. F ' . i c a l p o i n t s o f h. D e f i n i t i o n 3.11,

+

R and t h e m o n o t o n i c i t y o f i t s d e r i -

S o l u t i o n s o f h ' x = 0 correspond t o what a r e c a l l e d c r i t On t h e o t h e r hand, even f o r semiconvex h i n t h e sense o f

t h e d i r e c t method o f t h e c a l c u l u s o f v a r i a t i o n s o r r e s u l t s

such as Theorem 3.13, a c t u a l l y g i v e extreme p o i n t s o f h ( i . e . maxima o r minima). Thus as i s p o i n t e d o u t i n [Br6], a c e r t a i n amount o f i n f o r m a t i o n i s l o s t i n p a s s i n g f r o m t h e c a l c u l u s o f v a r i a t i o n s t o t h e t h e o r y o f monotone o p e r a t o r s . I t i s i n f a c t p o s s i b l e as shown i n [Br6] t o h a n d l e b o t h t h e o r i e s by a common technique. We g i v e some i l l u s t r a t i v e r e s u l t s here. Thus l e t F be a r e f l e x i v e B space o v e r R and l e t h: F with h f

a.

L e t T: F

-f

+

(-m,m]

be a convex f u n c t i o n

F ' be hemicontinuous and monotone (cases where D(T)

246

ROBERT CARROLL

= F can again be t r e a t e d b u t we o m i t t h i s h e r e ) .

The idea i s t o f i n d uo

BR = {u; IIuII < R } , f o r some R, s a t i s f y i n g f o r a l l v E F ( a * ) ( T u o - W o , ~ - ~ o ) I f h E 0 t h e n by Theorem > h ( u o ) - h ( v ) where wo E F ' i s g i v e n i n advance.

-

If h

I f T = 0 t h e n h has a minimum a t uo.

3.3 Tuo = wo.

0 on a c l o s e d

convex s e t K, w i t h h = o f f K, t h e n f o r v € K, ( T u ~ - w ~ , v - u 2 ~ )0 which solves a problem i n monotone i n e q u a l i t i e s on convex s e t s ( c f . 553.5-3.7).

tmrmA 3.18,

Assume T: F i v e B space, h: F + (--,-]

+

F ' i s hemicontinuous and monotone, F i s a r e f l e x i s convex w i t h h # -, and w € F ' i s given. Then 0

uo s a t i s f i e s ( a * ) if and o n l y i f

Phaod: <

m

.

Assume

(.A)

(@A)

~ T v - w o , v - u o ) 2 h(uo)

and n o t e t h a t h(uo) <

-

-

h ( v ) f o r v E F.

by p i c k i n g some v f o r which h ( v )

Put t h e element vt = ( 1 - t ) u o + t w , w a r b i t r a r y , t > 0, i n

p l a c e o f v, and use t h e c o n v e x i t y o f h t o f i n d t h a t

(

in

(*A)

T v t - ~ o , ~ - ~ oL) [h(uo)

-h(w)] ( s i n c e t > 0 ) . As t + 0, Tvt + Tuo weakly i n F ' by h e m i c o n t i n u i t y and hence ( T u ~ - w ~ , w -2u h(uo) ~~ - h(w) f o r any w E F which i s ( a * ) . Conv e r s e l y i f (.*) we have

(@A)

h o l d s t h e n s i n c e (Tv,v-uo) 2 ( T u ~ , v - u ~ ) by , monotonicity

QED

imnediately.

We s h a l l assume i n what f o l l o w s t h a t h i s l o w e r semicontinuous and t h i s i n a sense i s a weaker s i t u a t i o n than t h a t which can be expected t o p r e v a i l if h ( x ) = I ( x , x ) w i t h I o n l y semiconvex.

F i r s t we r e c a l l t h a t h i s s a i d t o be

weakly l o w e r semicontinuous (LSC) on a s e t B < c, x Wc(h) = t x ; h ( x ) -

w i t h t h e xa xa)

E

- I(x,,xo)

This means xa Theorem 3.13). xo) 5 c ' .

B I i s weakly c l o s e d i n B.

Wc(h) and B bounded.

Now l e t xa

+

xo weakly

By p r o p e r t y (B) o f D e f i n i t i o n 3.11

I(xa,

0 and hence i f c ' > c i s a r b i t r a r y I(xa,xo) 5 c ' e v e n t u a l l y .

+

E

F i f f o r each r e a l c t h e s e t

W

C'rXo

e v e n t u a l l y , which i s weakly c l o s e d (see t h e p r o o f o f

s i n c e xa xo weakly and t h u s h ( x o ) = I ( x o , c',x Since c ' > c i s a r b i t r a r y we have h ( x o ) 5 c and xo E Wc(h). Thus

CHE0REN 3.19. I(x,x)

E

C

Hence xo E W

-f

I f I(. ,- ) i s semiconvex on F X F, F Banach, t h e n x

-+

h(x) =

i s weakly LSC on bounded s e t s .

We n o t e t h a t a map x

+

f(x) : F

+

R i s weakly LSC means a l s o t h a t xa

-+

x

weakly i m p l i e s l i m i n f f ( x a ) 2 f ( x ) (one should compare t h i s w i t h t h e d e f i n i t i o n above).

&HE8)REm 3.20. +

Next we have L e t F be a f i n i t e dimensional B space, T a continuous map F

F ' , and h a LSC convex f u n c t i o n , h: F

t r a r y R, t h e r e e x i s t s uo E BR w i t h (.*)

-+

(--,m).

Given wo E F ' and a r b

t r u e f o r a l l v E BR.

P ~ C J C J ~We: can assume w i t h o u t l o s s o f g e n e r a l i t y t h a t wo = 0 s i n c e T ' x = Tx

247

MONOTONE OPERATORS

-wo s a t i s f i e s t h e same hypotheses. E BR t h e r e would be a v

BR such t h a t

E

BR f i x e d t h e s e t Sv o f u

I f t h e lemna were f a l s e t h e n f o r each u

E

(exercise) while i f h(v) =

-

(

(0.)

BR s a t i s f y i n g we s e t Sv =

(0.)

Tu,v-u)

< h(u)-h(v). For v E i s open i n BRy s i n c e h i s LSC

@. By

t h e compactness o f BR we

c o u l d t h e n cover i t by a f i n i t e number o f such s e t s S v j (1 5 j 5 p).

If

J,j

i s a c o r r e s p o n d i n g c o n t i n u o u s p a r t i t i o n o f u n i t y (see Appendix B) d e f i n e

1: J,.(u)vj. 1 J,j(uj = 1 ) (u)(Tu,vj-u) < 1 ;

J,(u) = with

Evidently then

- h(J,(u)).

-

J,j(u)[h(u)

c o n v e x i t y h(J,(u)) 5 < h(u)

J,

and i s continuous.

1;

maps BR i n t o BR ( s i n c e 0 ~ $ ~ ( 5u 1) Thus f o r u

h(vj)] = h(u)

!bj(u)h(v.)

E

- 1;

BR (Tu,J,(u)-u) E

1'1

J,

j However by

Gj(u)h(vj).

and hence f o r a l l u

J

=

BR ( 0 6 ) (Tu,J,(u)-u)

But by t h e Brouwer f i x e d p o i n t theorem (see 53.4)

J,

has

a f i x e d p o i n t u i n BR and p u t t i n g u i n ( 0 6 ) we g e t 0 < 0 which i s impossible.

T h e r e f o r e Lemna 3.20 i s t r u e .

QED

L e t F be a B space, h: F + (-m,m] a convex f u n c t i o n w i t h h ( 0 ) = CmrmA 3.21. 0, wo E F ' , and l e t R be g i v e n such t h a t f o r IIuII = R t h e map. T: F + F ' s a t s a t i s f i e s (**) f o r a l l v

0

0

Paood: Set v

If uo E BR s a t i s f i e s (.*)

+ h ( u ) > 0.

i s f i e s (Tu-wo,u) IIu II < R and u

= 0 in

t h i s means IIu II < R. 0

(.*)

E

2 h(uo); by t h e hypotneses

t o o b t a i n -(Tuo-wo,uo)

Let v

E

f o r v E BR t h e n

F.

F be a r b i t r a r y and c o n s i d e r vt = ( 1 - t ) u o + t v .

T h i s t r a c e s o u t t h e l i n e f r o m uo t o v and hence f o r t small l i e s i n BR. P u t t i n g vt f o r v i n (**) one has a f t e r c a n c e l i n g t > O,( Tuo-wo,v-uo) h(uo)

- h ( v ) which i s (.*), now v a l i d f o r any v.

CE0RElIl 3-22, a g i v e n wo

E

QED

L e t T be a monotone hemicontinuous map F

f l e x i v e B space, and l e t h: F

+

F ' t h e r e e x i s t s R such t h a t

Then t h e r e e x i s t s uo

E

+

F', with F a re-

be LSC w i t h h ( 0 ) = 0.

(-m,m]

BR such t h a t (.*)

(

Tu-wo,u)

+ h(u)

>

Suppose f o r 0 f o r IIuII = R.

holds.

The p r o o f f o l l o w s p a t t e r n s e s t a b l i s h e d above i n Theorem 3.6,

3.21,

2

Lennnas 3.20,

e t c . (see [ C l ] f o r d e t a i l s ) .

Assume t h e f i r s t h y p o t h e s i s o f Theorem 3.22 and assume t h a t C0R0ttARg 3.23. as IIuIl + m y {(Tu,u) + h(u)}/llull + a. Then a s o l u t i o n o f ( a * ) e x i s t s (any w0). F o r p r o o f n o t e t h a t f o r g i v e n wo

E

F', (Tu-wo,u) + h ( u ) ,(Tu,u)

IIw Illlull and t h i s i s > 0 f o r IIuII l a r g e enough. 0

Hence t h e second hypotheses

o f Theorem 3.22 h o l d s and hence a s o l u t i o n o f (.*) u s e f u l i n f o r m a t i o n about s o l u t i o n s o f theorem ( c f . [ C l ]

CHE0REIR 3.24.

(0*)

+ h(u) -

exists.

Some f u r t h e r

i s contained i n the f o l l o w i n g

f o r proof).

Under t h e hypotheses o f Theorem 3.22 t h e s e t o f s o l u t i o n s

248

ROBERT CARROLL

S(wo) o f (.+)

(w,)

i s a c l o s e d convex s e t i n F and i f T i s s t r i c t l y monotone S

i s a single point.

We go now t o some n o n l i n e a r e v o l u t i o n equations f o l l o w i n g [Br5] ( c f . a l s o [Cl])

and f o r r e l a t e d work see [Brl;Bdl;Ka3;Tal;Li4;Mtl;Pcl;Pzl Again we

§53.6-3.7).

kill

some i l l u s t r a t i v e cases.

(cf. also

n o t g i v e t h e b e s t known r e s u l t s b u t r a t h e r p i c k The monotone o p e r a t o r aspect o f t h e t h e o r y has

a t t a i n e d a h i g h l y s o p h i s t i c a t e d f o r m i n [ B r l ] f o r example and we w i l l n o t try t o cover t h i s .

We g i v e one theorem (Theorem 3.25) t o i l l u s t r a t e some

a p p l i c a b l e a b s t r a c t t e c h n i q u e based on Browder's g e n e r a l i z e d G a l e r k i n met h o d and t h e n s k e t c h an a p p l i c a t i o n t o a n o n l i n e a r e v o l u t i o n equation. r e s u l t s on n o n l i n e a r e v o l u t i o n equations w i l l be i n §§3.6-3.7;

More

t h e technique

and p h i l o s o p h y t h e r e w i l l be somewhat d i f f e r e n t .

CHE0RETll 3-25. L e t F be a r e f l e x i v e B space, L: F * F ' a c l o s e d densely def i n e d l i n e a r map w i t h domain D = D(L), G: F .+ F ' a hemicontinuous map d e f i n ed on a l l F c a r r y i n g bounded s e t s i n t o bounded s e t s and C: F map.

Define T = L

+

F' a compact

.+.

G w i t h domain D and assume t h a t T + C i s

D monotone;

suppose f u r t h e r t h a t L* i s t h e c l o s u r e o f i t s r e s t r i c t i o n t o D(L) n D(L*) as x + Then R(T) = F ' . and t h a t Re
-.

Ptruud: As i n Theorem 3.6 i t w i l l s u f f i c e t o show 0 E R(T) and we need o n l y prove t h e theorem f o r r e a l E. L e t A be t h e d i r e c t e d s e t o f f i n i t e dimensionE* a l subspaces o f D(L) o r d e r e d by i n c l u s i o n . If E € A w i t h i n j e c t i o n iE: F t h e n d e f i n e TE = i;TiE and so on as b e f o r e and, as i n Theorem 3.6, (see E ' i s hemicontinuous and monotone, hence cona l s o Theorem 840) TE t CE: E tinuous; t h e r e f o r e , TE i s continuous w i t h ( TEu,u) 1. p(IIull )IIull and we o b t a i n uE E E such t h a t TEuE = 0 and IIuEll 5 M f o r a l l E. Now, f o r e v e r y v E E, 0 = ( T u , v ) = ( TuE,v) = ( LuE,v) t ( GuE,v). Hence i f v E D(L*) n E then ( uE, EE L*v) = -( Gu v ) and s i n c e II uEll 5 M i t f o l l o w s t h a t IIGuEll 5 N; consequently E' I( uE,L*v)l 2 NIIvll f o r a l l E E A and v E D(L*) n E. Moreover f o r any v E E -f

(W) 0

z ( (T+C)uE

- (T+C)v,uE-v)

= (CuE

- (T+C)v,uE-v) s i n c e ( T u E ,u E- v )

=

(TEuE,uE-v).

By weak compactness t h e r e e x i s t s as b e f o r e ( c f . Lemma 3.7,

Theorem 3.22,

e t c . ) uo E

each

E,

BM

= { u E F; IIuII <

and each f i n i t e s e t wl,

I( uO-uE,wi ) I

5

E.

...,w m E

MI such t h a t f o r each Eo

F', one can f i n d E E

We w i l l show now t h a t uo E D(L) and Tuo

E A,

A, E 2 Eo,

with

Suppose v

0.

E D(L) n D(L*) and l e t Eo = { v } be t h e one dimensional subspace generated by

Let

v. E

2

E

= IIvII and choose L*v E F ' as

Eo, such t h a t I(u0-uE,L*v)I 5 IIvII.

from t h e above, I(uo,L*v)l

w1 (m = 1). Then we can f i n d E E A, Since v E E n D(L*) we have then,

5 (N+l)IIvII, and t h i s h o l d s f o r a l l v

€

D(L)

17

MONOTONE OPERATORS

hence upon c l o s i n g L* on D(L) n D(L*),

D(L*);

Finally l e t v

t l y uo E D(L).

E

249

for all v

E

Consequen-

D(L*).

D(L) be g i v e n w i t h Eo = { v l .

Since C i s

compact ( c f . Remark 2.16) we can choose E E A, E 3 Eo, so t h a t f o r IICu -Cu 11 5 E , w h i l e I(Cu0,uE-v) - ( C u ,u - v ) l 5 0 0 E o (TtC)v,uo-v)/ 5 E . Then from (*+) we have (Cu,

(

E

and s i n c e

v E D.

E

> 0 i s a r b i t r a r y t h i s i m p l i e s (Cu,

E,

and

E

given

I( (T+C)v,uE-v) -

-

(T+C)v,uo-v)

2 -(2+M+llvll)

-

(T+C)v,uo-v)

0 for all

F u r t h e r s i n c e any l i n e a r L i s hemicontinuous ( c f . remarks a f t e r

Theorem B40) we know TtC i s hemicontinuous and hence by Theorem 3.3 T+C i s

D maximal monotone w i t h (T+C)uo = Cuo o r Tuo = 0. I n p a r t i c u l a r t h i s theorem a p p l i e s t o L,

1o f

i f L,

QED

= Lw, where Ls i s t h e c l o s u r e

a densely d e f i n e d l i n e a r L and Lw = L ' * where L ' = L*ID(L) n D(L*)

(since then

i' =

L;

We s h a l l a p p l y i t i n t h i s f o r m w i t h D(L) c D(L*)

= L;).

Thus l e t H

f o l l o w i n g [Br5] w i t h some m o d i f i c a t i o n s ( c f . a l s o [C1,39,40]). be a H i l b e r t space and E

C

H a dense l i n e a r subset c a r r y i n g t h e s t r u c t u r e o f

, ) be t h e H

a r e f l e x i v e B space w i t h continuous i n j e c t i o n i n t o H; l e t (

,

)E t h e E-E' d u a l i t y b r a c k e t ( c o n j u g a t e l i n e a r ) and one w r i t e s E C H C E ' where H ' i s i d e n t i f i e d w i t h H. L e t f be a map Ta X E s c a l a r p r o d u c t and

+

E ' (Ta = [r,r+a])

continuous ( 6 ) t g i v e n and u t

(

E

s a t i s f y i n g e.g.

* (f(t,u),v)E

E, I l f ( t , u ) l l E l

5 C [ I I UEI I ~ - +~ h ( t ) ] where h

0 belongs t o L u'(t)

(0.)

+

f(t,u):

E

E

+

E' i s

<

p

<

Lq w i t h h 2 0 ( l / p

-

.

o f t h e arguments a r e phrased f o r any p (1 < p < equation

-+

i s monotone on E w i t h Re( f ( t , u ) , u ) E 2 c l l u l l ~ 1 For convenience one can t a k e p 2 2 b u t most

l / q = 1 ) and (0) f ( t , . )

k ( t ) where k

the conditions (A) u

i s measurable f o r u,v E E (C) f o r 1

f(t,u(t))

=

m).

We c o n s i d e r f i r s t t h e

0 with U(T) = u E E for t 0

E

Ta and s e t F

We w i l l say t h a t u i s a weak ( o r g e n e r a l i z e d ) s o l u t i o n o f (0.) 1 i f u E Co(Ta,H) and f o r e v e r y v E C (Ta,E) w i t h v(-c+a) = 0 ( 6 * ) ,;T+a(~,~')

= LP(Ta,E).

-

d t = IrT+a(f(t,u),v)Edt

(uO,v(-c)).

The i n t e g r a l s a r e w e l l d e f i n e d and s e t -

t i n g g(u,v) = JrT+a(f ( t , u ) , v ) E d t we have ( e x e r c i s e ) (U) I g ( u , v ) l 5 cllullF 1 [ l l u l l ~ - ' + IIhll 1. Define now L: F F ' by Lu = u ' w i t h D(L) = t u E C (Ta,E); 9Indeed i t i s U ( T ) = 01 and L = L;, L i s w e l l d e f i n e d s i n c e D(L*) i s dense. 1 c l e a r t h a t L ' = -d/dt: F F ' w i t h D ( L ' ) = I u E C (Ta,E); u ( r + a ) = 01 has -+

-+

dense domain w i t h L ' operator).

C

L* ( i n what f o l l o w s L ' w i l l c o n t i n u e t o denote t h i s

We mention a l s o t h a t as b e f o r e we c o n s i d e r E

C

H c E' algebrai-

c a l l y and t o p o l o g i c a l l y w i t h H dense i n E ' and f o r p 5 2, Lp(E) c L p ( E ) ' = L q ( E ' ) on Ta s i n c e t h e n q = p/p-1 5 p and i n f a c t q 5 2 . On t h e o t h e r hand we w i l l say t h a t u

w(t) = u(t)

-

E

F i s a strong solution o f

(0.)

uo E D(LS) w i t h Lsw + Gu = 0 where Gu E F ' i s d e f i n e d by

if

250

(

ROBERT CARROLL

G u , v ) ~ = I,"?f(t,u(t)),v(t)),dt

of (U) ( c f . below).

+ A(t)u(t)

t

E

F ( G i s w e l l determined by v i r t u e

One can show t h a t t h e r e e x i s t s a unique s t r o n g = gen-

eralized solution o f

u'(t)

for v

(em)

and subsequently prove unique s o l v a b i l i t y o f (6.) First

f ( t , u ( t ) ) = 0 under s u i t a b l e hypotheses on A ( t ) .

L e t f s a t i s f y c o n d i t i o n s (A) - ( D ) above. Then G i s a c o n t i n CmmA 3.26. uous map F -+ F ' , c a r r y i n g bounded s e t s i n t o bounded sets, and G i s monotone w i t h R d G u , u ) ~ 2 cIIuII - c f o r a l l u E F (see [ C l ] f o r p r o o f ) .

CrmmA 3.27,

The s t r o n g and g e n e r a l i z e d s o l u t i o n s o f

L e t u be a s t r o n g s o l u t i o n o f

Pmod:

(em)

coincide.

(0.)

w = u-uo,Lsw + Gu = 0 ) .

(i.e.

-<

L e t wk E D(L), wk w, Lwk Lsw, and suppose v E D(L'); t h e n c l e a r l y (Lwky v ) =~ ( w ~ , L ' v )and ~ t h u s ( w ~ , L ' v )-+~ ( w , L ' v ) ~ = -JTT+a(u - u o y v ' ) E d t = +a(u, -+

v')dt

-

-f

( u o y v ( ~ ) ) . Note h e r e t h a t s i n c e v ' ( t ) E E C H C E l , i t s a c t i o n as

an element o f E ' on u ( t ) E E can be w r i t t e n as e i t h e r ( u ( t ) , v ' ( t ) )

or (u(t),

~ ' ( t ) by ) ~t h e n a t u r e o f o u r embedding H C E ' ; s i m i l a r l y ( ~ , , v ' ( t ) ) ~ = (uoy v ' ( t ) ) and ( ~ , , v ( t , ) ) ~

v ) we ~ o b t a i n (&*).

= (uoyv(to)).

any v E D ( L ' ) we have -L'*w = Gu.

Since now

(

Lwkyv)

-f

(

LSw,v) E = 4 Gu,

Conversely i f u i s a g e n e r a l i z e d s o l u t i o n so t h a t f o r (

G u , v ) ~ = (w,v' ) F =

-(

w,L'v),

then w E D ( L ' * ) w i t h Since L ' c L; we have

Thus i t s u f f i c e s t o show t h a t L o * = L.,

LS c L ' * and one need o n l y show t h e r e v e r s e i n c l u s i o n .

T h i s can be done

d i r e c t l y by a smoothing argument ( e x e r c i s e ) o r b y t h e f o l l o w i n g o b s e r v a t i o n s . I f L'*w = v E F ' = L p ( E ) ' = L q ( E ' ) t h e n f o r u E D ( L ' ) , ( w , L ' u ) ~ =

and i n p a r t i c u l a r f o r u

E

D(E) on ( r , r + a ) we o b t a i n

(

v,u)~,

-I ( w , ~ ' ) ~ d=t I ( v , u ) ~

d t ; t h i s i m p l i e s t h a t w ' = v i n D ' ( E ' ) ( c f . Remark 648) and W ( T ) = 0 i s e a s i l y seen t o be necessary as w e l l .

Hence we need o n l y show t h a t Ls i s i n

f a c t t h e o p e r a t o r d / d t w i t h domain d e l i m i t e d by t h e c o n d i t i o n s D(LS) = { u F; u ' E F ' ; U(T) = 0) where d / d t i s taken i n D ' ( E ' ) on ( T y T t a ) . T h i s can again be proved by a somewhat d i f f e r e n t smoothing argument ( e x e r c i s e Lemma 3.29 f o r a p r o t o t y p i c a l smoothing argument).

LEmmA 3-28.

E

- cf.

QED

The s t r o n g ( o r g e n e r a l i z e d ) s o l u t i o n s a r e unique and l i e i n

Co(Ta,H) w i t h

U(T)

= 0.

For w E D ( L j we Phoo6: L e t x t be 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 [ T , t ] . have (U) Re( Lw,xtw) = ReJTt (w',w)dc = (1/2)11w(t)llH2 and w E Co(Ta,H) by 2 d e f i n i t i o n o f D(L). Thus I l w ( t ) U H 5 211LwllF,IlxtwllF 5 2 1 1 L ~ l l ~ ~and U w w~ ~ w ( t ) -f

-

i s a bounded l i n e a r map from D(L) i n graph t o p o l o g y t o H w i t h a uniform bound o v e r Ta.

Extending t h e i n e q u a l i t i e s t o D(L) = D(LS) one sees t h a t any

w E D(Ls) belongs t o Co(Ta,H) and l l w ( ~ ) I =I ~1 i m l l w ( t ) l l i 5 limllLwllFIIIXtwllF = 0.

MONOTONE OPERATORS

251 N

F i n a l l y i f u and v a r e two s o l u t i o n s t h e n f o r w = u-uo, w = v-u L,w

L

= -Gu,

S

w" =

-Gv and Re( LS(w-i),xt(w-F))F

we o b t a i n

0

5 0 by

= Re( Gv-Gu,xt(v-u))

Hence l l w ( t ) - ~ ( t ) l l5 ~ 0 f o r any t by ( 6 6 ) and w =

d e f i n i t i o n o f G.

The f o l l o w i n g lemma can e v i d e n t l y be m o d i f i e d t o s e r v e f o r V c H

V a r e f l e x i v e B space ( t h u s V % E above). 2 2 L e t u E L (Ta,V) w i t h u ' E L (Ta,V') where V C H tm 3.29. b e r t spaces w i t h V dense i n H (and

'

i n D'(V')).

(u = v).

V' with

C

V ' are H i l -

C

Then u E Co(Ta,H).

2

2 L e t F(Ta) be t h e space o f u E L (Ta,V) w i t h u ' E L (Ta,V') p r o v i d e d 2 w i t h t h e norm IIul12 = I [IIuIIi + Ilu'IIVl]dc. L e t T I < T , T+a < T, and l e t

Phoo6:

be a Cm f u n c t i o n equal t o 1 on Ta and 0 i n a NBH o f 847). and

Extend u t o

f o r example by u(t+-r+a) = u(-r+a-t) Then w E F ( [ T ' , T ] )

with w =

Now by r e g u l a r i z a t i o n i n t ( c f . Remark B12) we can approximate w

u on Ta.

by a sequence o f Cm f u n c t i o n s wm v a n i s h i n g i n a NBH o f

i n F([T',T]) T.

and [r+a,T]

[T'yT]

= u(.r+t) and s e t w = eu on [T',T].

U(T-t)

and T ( c f . Theorem

T I

Indeed c o n s i d e r w m ( t ) = f w(S)lp,(t-S)dC

where

lpm

= w(mx),

lp

T'

and

E D(R),

2 lpdx = 1, w i t h supp lp C I x ; 1x1 5 1 1 and show t h a t wm -t w i n L ( V ) 2 E V and while w' w ' i n L ( V ' ) ( e x e r c i s e ) . B u t f o r these wm we have w;(t) m 2 t one can w r i t e I w m ( t ) l H = f T I2 R e ( w ' ( ~ ) , w ~ ( ~ ) ) ~= d2ce I Re(w1;1(5),wm(S))dS 5

9

0,

I

-f

' :2J

IIw~llVIIIwm(c)IIVdg5 fTI t [IIw;n(S)IIvl m2 + llwm(C,)Sv]d~. 2

where F = F([T',T].

Thus t h e c o n t i n u o u s w

continuous f u n c t i o n which means t h a t w E

Hence I w m ( t ) l H 5 IIw,,,llF

converge u n i f o r m l y i n H t o a

8 C ([T',T])

a f t e r possible modifica-

t i o n on a n e g l i g i b l e s e t and hence u E Co(Ta,H). F i n a l l y one proves ( c f . [Br5;

IIHE0REiil 3.30. 2.

QED

Cl])

L e t f s a t i s f y c o n d i t i o n s (A)

-

( D ) above w i t h uo

Then t h e r e e x i s t s a s t r o n g ( = g e n e r a l i z e d ) s o l u t i o n u o f

E

E and p 2

(0.).

The p r o o f i n v o l v e s m a i n l y t e c h n i c a l m a n i p u l a t i o n t o p u t t h e problem i n t h e c o n t e x t o f Theorem 3.25.

I n p a r t i c u l a r i n o r d e r t o a c h i e v e D(LS) C D(Ls)

one i n t r o d u c e s a w e i g h t f u n c t i o n and spreads Ta t o R, e x t e n d i n g f ( t , u ) say f(T,u)

for t

$

Ta.

as

Then an adjustment f o r m o n o t o n i c i t y must be made

and a smoothing procedure u t i l i z e d (see [Cl;Br5]

for details).

One can a l -

so extend t h e t e c h n i q u e (and use Theorem 3.25) t o c o v e r equations o f t h e form u ' + A ( t ) u + f ( t , u )

= 0, U ( T ) = uo where A ( t ) i s a s u i t a b l e f a m i l y o f

c l o s e d densely d e f i n e d l i n e a r o p e r a t o r s i n H and f i s as above. theorem i s ( c f . [Br5;C1])-

CHE0REIll 3-31,

A typical

t a k e uo = 0 f o r s i m p l i c i t y )

Let f s a t i s f y conditions (A)

-

( D ) on R w i t h 1 < p <

m

and

252

ROBERT CARROLL

l e t A ( t ) be a f a m i l y o f c l o s e d densely d e f i n e d a c c r e t i v e l i n e a r o p e r a t o r s i n H w i t h domains i n E.

L e t Mu = u ' t A(.)u

w i t h D(M) ( s u i t a b l y d e f i n e d ) Then Ms + G maps 1-1 f r o m D(MS)

dense i n F = Lp(R,E) and suppose M, = Mw.

o n t o F ' and i n p a r t i c u l a r (&+) has a unique s t r o n g s o l u t i o n o v e r R (uo = 0). The study o f a b s t r a c t e v o l u t i o n equations (6.)

u ' t A ( t ) u = f; u ( 0 ) = uo

w i t h v a r i a b l e A ( t ) has been e x t e n s i v e l y developed i n [Asl;C1,39,40;Pzl f o r example and we w i l l n o t deal w i t h t h i s here.

Ftl,2;Bwl]

\

c o n d i t i o n s D(M) dense w i t h MS =

;Lil ;

Note t h a t t h e

i n Theorem 3.31 a r e h i g h l y n o n t r i v i a l .

N o n l i n e a r v e r s i o n s o f ( b m ) w i l l be discussed f u r t h e r i n 5§3.6-3.7 f o l l o w i n g

[Bdl;Bxl;Pzl,2;Pal;Dml;Tal;Mtl;Brl].

There w i l l n a t u r a l l y be an i n t e r a c t i o n

w i t h v a r i a t i o n a l methods and convex a n a l y s i s which we s k e t c h i n 53.6-3.7 f o l l owing [ L i 3-7; E l ;Aul-3; Zel ;Dml ;Dnl ;Kbl ;Acl ;Bd2;Rkl; C11 ;Pvl ;Bw2; Bgl ;Gwl 4. E0P0C0CICAC MEtH0Dti.

1.

A l g e b r a i c t o p o l o g y i s a standard i t e m o f study t o -

day i n most u n i v e r s i t i e s ( o r should be) and i t has made enormous i n r o a d s i n t h e study o f physics.

While i t i s easy t o study homotopy t h e o r y from f i r s t

p r i n c i p l e s (which we do) i t takes some t i m e and space t o develop homology theory.

Thus i n o r d e r t o a v a i l o u r s e l v e s o c c a s i o n a l l y o f homology we w i l l

s i m p l y d e s c r i b e i t a x i o m a t i c a l l y and a s s e r t t h e e x i s t e n c e o f such t h e o r i e s i n t h e c o n t e x t s needed.

This i s n o t w i t h o u t precedent ( c f . [Cl;Sml;Hul;Gg

13) and f o r d e t a i l s i n t o p o l o g y we r e f e r t o [Sul;Ell;Bul;Hkl].

Thus f i r s t

c o n s i d e r some subcategory K o f t h e c a t e g o r y o f t o p o l o g i c a l p a i r s (X,A), X, and continuous maps f : (X,A)

-f

(Y,B) w i t h f ( A ) C B.

A C

We i n d u l g e i n t h e

l u x u r y o f u s i n g t h e concepts o f c a t e g o r y and f u n c t o r from t i m e t o t i m e ( c f .

A space X i s i d e n t i f i e d w i t h ( X , Q ) .

and Appendix C ) .

[Cl;E11]

two maps f,g:

(X,A)

+

Then g i v e n

(Y,B) we say t h a t f i s homotopic t o g, w r i t t e n f = g,

whenever t h e r e i s a map

F:

( X x I,A x I )

+

(Y,B),

I t [0,1],

f ( x ) and F ( x , l ) = g ( x ) (maps a r e assumed t o be c o n t i n u o u s ) . hand, graded A b e l i a n groups a r e c o l l e c t i o n s { G n } morphisms g i v e n as c o l l e c t i o n s

T

w i t h F(x,O) = On t h e o t h e r

o f A b e l i a n groups w i t h ho-

= { r n } , where rn: Gn -+ G,+,'d

f o r some i n t e -

ger d (which can vary f o r d i f f e r e n t {rn1).

A homology t h e o r y (HJ)

DEFZNZCI0N 4.1.

on K c o n s i s t s o f a c o v a r i a n t f u n c -

t o r H from K t o t h e c a t e g o r y C o f graded A b e l i a n groups and homomorphisms g i v e n by H(X,A)

= {H (X,A)I

9 and a n a t u r a l t r a n s f o r m a t i o n

Hq(X,A)

I f f,g:

w i t h H ( f ) o f degree 0 f o r f any morphism i n K,

a o f degree -1 o f t h e form a ( X , A ) = Iaq(X,A):

-

(A,Q) = H ( A ) } , such t h a t t h e f o l l o w i n g axioms h o l d . (A) H q-1 q l (X,A) -t (Y,B) a r e homotopic t h e n H ( f ) - = H(g): H(X,A) -P H(Y,B). (B)

-f

For any p a i r (X,A)

w i t h i n c l u s i o n maps i:A

+

X and j : (X,@)

+

(X,A)

there

253

TOPOLOGICAL METHODS

...- a (X,A) + Hq(A) - H q ( i ) i* Hq(X) - H q ( j ) + - a q (X,A) + Hq-l(A) q+l ... (C) I f U i s an open subset o f X w i t h

i s an e x a c t sequence Hq(X,A)

3

U C i n t e r i o r A = A, t h e n t h e e x c i s i o n map j : (X-U,A-U) isomorphism H ( j ) :

H(X-U,A-U)

-+

H(X,A)

*

(X,A)

induces an

(D) I f p i s a p o i n t then H ( p ) = 0 q

f o r q = 0 and Ho(p) = Z = i n t e g e r s .

S i n g u l a r homology t h e o r y i s d e f i n e d f o r a l l p a i r s and Fech homology t h e o r y i s d e f i n e d f o r say t r i a n g u l a b l e compact p a i r s (we a r e o n l y d e a l i n g w i t h co-

Z here).

efficients

I n t h e case o f p a i r s o f compact Cm m a n i f o l d s ( w h i c h a r e

known t o be t r i a n g u l a b l e ) t h e s i n g u l a r and f e c h t h e o r i e s w i l l agree. f u r t h e r s t i p u l a t e t h a t H (X,A) = 0 f o r q

63 C

consider the u n i t b a l l i d e n t i t y map lB: Bn X I

-+

+

means t h a t f,

-f

Indeed F ( x , t ) = t x f u r n i s h e s a map Bn

Bn: g ( 0 ) = 0.

Then by D e f i n i t i o n 4.1 and f u n c -

[f, = H ( f ) e t c . ) and s i n c e f g = lo, g,f,

= g,f

Now

It i s c o n t r a c t i b l e i n t h e sense t h a t t h e

Bn w i t h F(x,O) = 0 and F ( x , l ) = x.

t o r i a l i t y 1,

We can

h e r e ( c f . [Ell;Hkl;Sul]).

Bn i s homotopic t o t h e c o n s t a n t map g f , where f : Bn

-+

0: f ( x ) = 0 and g: 0

Rn.

<0

i s an isomorphism w i t h ';f

= g,

= 1.,

and hence H(0)

This

H(Bn).

By D e f i n i t i o n H(0) = Z i n dimension z e r o and t h u s H (Bn) = 0 f o r q # 0 w i t h Q

q Next one says t h a t A C X i s a r e t r a c t o f X i f t h e r e i s a con-

Ho(Bn) = Z.

t i n u o u s map r: X

-+

A c a l l e d a r e t r a c t i o n map such t h a t r i = lA where i: A

X i s t h e i n j e c t i o n (i.e. r ( x ) = x f o r x E A). f a c t f r o m t o p o l o g y (Sn = 38"'

C

Rn+'

-+

We s h a l l need t h e f o l l o w i n g

i s t h e u n i t n-sphere).

H (Sn) = 0 f o r q # 0 o r n w h i l e Ho(Sn) = Hn(Sn) = Z. q We p r e f e r t o s i m p l y assume t h i s (see e.g. [ E l l ; H k l ; S u l ] f o r p r o o f ) .

LElnmA 4.2.

now Sn were t o be a r e t r a c t o f 8"' F(x,t) = r ( ( l - t ) x ) , s t a n t map x t h a t H(Sn)

x E Sn,

w i t h r e t r a c t i o n map r : 8"'

F ( x , l ) = r ( 0 ) € Sn. Hence as above ( f o r Bn) i t would f o l l o w Z i n dimension zero which i s f a l s e by Lemma 4.2. Hence Sn i s

'+'

Any continuous map f : B

f ( x ) # x f o r a l l x E 8"'

l i e s on t h e l i n e f r o m f ( x ) t o r ( x ) . ( e x e r c i s e ) and r i s a r e t r a c t i o n 8"'

* 8"'

has a f i x e d p o i n t .

t h e n we l e t r ( x ) E Sn be such t h a t x The c o n t i n u i t y o f r i s e a s i l y checked

* Sn which i s i m p o s s i b l e .

We r e f e r a l s o t o [Gel] h e r e and n o t e t h a t i n an

-+

point.

Bn+'

i s a homeomorphism and f : K

+

QED

dimensional B space t h e

sphere IlxII = 1 i s i n f a c t a r e t r a c t i o n o f IIxII 5 1 ( c f . [ D m l ] ) . K

Then

+

tHE0REm 4.3 (BR0LUDER).

v:

Sn.

t E I , f u r n i s h e s a homotopy o f lSw i t h t h e con-

n o t a r e t r a c t o f Bn+' and t h i s leads t o

Pmad: I f

+

Suppose

Clearly i f

K i s c o n t i n u o u s t h e n f has a f i x e d

Indeed from v f v - l ( y ) = y one has f ( v - l ( y ) )

= v-l(y).

We go now t o

2 54

ROBERT CARROLL

t h e Schauder theorem i n a form which g i v e s a s l i g h t l y s t r o n g e r v e r s i o n o f Theorem 4.3 as w e l l (Remark 4.12 has a more i n t u i t i v e p r o o f o f Theorem 4.3).

CmEA l

E

4-4(IIIAZLIR), I f C c E i s a compact subset o f a B space

F(C) i s

ed convex h u l l

then i t s c l o s -

compact.

Phood:

We r e c a l l t h a t ?(C) i s t h e c l o s u r e o f t h e s e t o f p o i n t s

xi E C,

1 hi

thus g i v e n

E

= 1,

xi 2 0.

1 Aixi

= x,

It w i l l s u f f i c e t o show F ( C ) i s precompact and

z,,.

> 0 we p i c k a f i n i t e s e t

i n C such t h a t C C UB(zi,c/

..,zn

4 ) ( B ( x , r ) i s t h e b a l l o f r a d i u s r and c e n t e r x ) . L e t K be t h e convex h u l l of t h e zi; then t h e map (xi,zi) xizi: [O,1ln X {zl X X zn} -+ K w i t h -f

Xi

=

...

1

1 e x h i b i t s K as t h e continuous image o f a compact s e t and hence i n xi E C, A . as above, t h e r e i s

hixi,

Now i f x =

p a r t i c u l a r as precompact.

-1

-

1

111

-

< ~ / 4 and hence IIx Aizn(i)ll = A.(xi ~ ) t h a t Ilx a z ~ ( such ,.,i z n ( i ) 1I 1 z ~ ( ~ ) 5I I ~ / 4 . Now r(C) l i e s i n any NBH o f t h e s e t o f such p o i n t s x and

hence i n any NBH o f an ~ / 4NBH o f K.

K, j

=

1,

...,m,

r(C)

one has

-f

C uB(kj,E).

Let C

C

UB(xi,~/2),

x.11 f o r Ilx-xill f o r IIx-x.II 1

q

-

xi E C,

~ / 2 ;a . ( x ) =

1

1

E

Then l e t qi(x)

E.

-

i = 1,

... ,n

IIx-xill

f o r E/Z

= ai(x)/l

ai(x)

from

and d e f i n e (*) a i ( x )

5 IIx-xill 5 with q(x) =

i s continuous and maps C i n t o t h e convex h u l l ;(xi)

when Ilx-x.II 2

E

k. E

J

QED

which i s homeomorphic t o Bn and c o n t a i n e d i n C . r(xi)

UB(k ,E/Z), j

C i s continuous t h e n f has a f i x e d p o i n t .

Pmod:

dently

C

I f C i s a convex compact subset o f a B space E and

CHE6REGI 4.5 (3CHALIDER)f: C

In particular i f K

N

E;

=

IIx

-

ai(x) = 0

1 ni(x)xi.

Evi

o f t h e xi,

F u r t h e r s i n c e ni(x)

= 0

-

n(x)ll = I l l n q (x)(x-xi)ll 5 E . The map n o f o f 1 1 i i n t o i t s e l f has a f i x e d p o i n t y by t h e Brouwer theorem and by ( A )

n o f(y)

(A)

E,

IIx

= y f o l l o w s Ilf(y)

C with Ilf(yn)

-

ynll 5 l / n .

-

yl 5

E.

Thus we can o b t a i n a sequence yn

Then yn has a convergent subsequence yk

by compactness and by c o n t i n u i t y f ( y ) = l i m f ( y k ) = y.

C6ROCCARY 4.6.

-f

y

QED

L e t f be a continuous map o f a c l o s e d convex subset C o f a

B space E i n t o a compact subset Co C C . The p r o o f uses Lemma 4.4 t o say t h a t

Then f has a f i x e d p o i n t .

F(Co)i s

o f f t o < r ( C 0 ) has a f i x e d p o i n t by Theorem 4.5.

compact and t h e r e s t r i c t i o n There a r e many g e n e r a l i z a -

t i o n s o f t h e Schauder theorem b o t h t o l o c a l l y convex spaces, t o m u l t i v a l u e d maps, and t o c o n d i t i o n s on t h e i t e r a t e s o f a map; we c i t e e.g. 1 1 where f u r t h e r r e f e r e n c e s can be found.

D E F I N I Q I 6 N 4.7.

[Dul;ZeZ;Ks

Next ( c f . [Gnl] f o r homotopy)

L e t X and Y be t o p o l o g i c a l spaces.

A map f: X

+

Y is

2 55

TOPOLOGICAL METHODS

c a l l e d i n e s s e n t i a l i n Y i f i t i s homotopic t o a c o n s t a n t map c: X

-f

yo; i f

f i s n o t i n e s s e n t i a l t h e n i t w i l l be c a l l e d e s s e n t i a l . The case o f p r i n c i p a l i n t e r e s t i s when X = Sn i n which case a homotopy h(x, t ) c o n n e c t i n g h(0,O)

= f : Sn

-f

e x t e n s i o n f " o f f d e f i n e d on B"'. for x

E

ykgives

+

r i s e t o an

Indeed s i m p l y w r i t e f ( ( 1 - t ) x ) = h ( x , t )

Conversely i f f has an e x t e n s i o n

t E I = [0,1].

Sn,

Y and h ( - , 1 ) = c : Sn

t h e n f = c under t h e homotopy h ( x , t ) = T ( ( 1 - t ) x ) .

f":

8"'

+

Y

Thus when X = Sn one can

say t h a t f i s i n e s s e n t i a l i n Y i f and o n l y i f i t has an e x t e n s i o n F d e f i n e d on 8"'

Hence i f we t a k e Y t o be R+:'

w i t h values i n Y.

t h e n a map A: B"'

-f

t o Y must have a z e r o i n 8"' values i n Y ) .

(i.e.

R"'

- 0 )

which can be shown t o be e s s e n t i a l on Sn r e l a t i v e

R+:'

( i f n o t A i t s e l f would be an e x t e n s i o n w i t h

T h i s f a c t i s used f r e e l y f o r spheres o f r a d i u s R i n t h e f o l -

l o w i n g lemma (which i s needed f o r Lemma 3.3.7). L e t A be a continuous map o f a f i n i t e dimensional B space E i n t o

CEarmA 4.8,

E ' such t h a t Re(A(u),u) ~ I ~ ( I I u ~ I ) I I uw Ii It h q ( x )

Phood:

+ m

-f

m.

Then R ( A ) = E ' .

We can t h i n k o f E as a H i l b e r t space w i t h E = E ' and as i n t h e p r o o f

o f Theorem 3.3.6 i t s u f f i c e s t o show 0 E R(A). Re(Atu,u)

+ (l-t)llul12

= tRe(Au,u)

>

0.

{ x ; IIxII = R) and At f u r n i s h e s a homotopy i n where E+ = E

-

{O).

Choose R so l a r g e t h a t f o r

Then i f At = t A + ( 1 - t ) I , 0 < t < 1, one has f o r /lull

IIuII = R, Re(Au,u) > 0. = R,

as x

I n p a r t i c u l a r Atu # 0 on SR =

+ E

c o n n e c t i n g A w i t h I on SR

+

But on SR, I i s e s s e n t i a l i n E , o b v i o u s l y , s i n c e I i t -

s e l f has a z e r o on BR = I x ; IIxII < R1.

+ E .

t e n s i o n J o f I t o BR Extend t h i s t o 7 = 0 f o r x -f

( I f i n e s s e n t i a l t h e r e e x i s t s an ex-

Set F = I - J so F = 0 on SR and F ( x ) f x on BR. Rn+'-BR; F i s a compact map E + En+' w i t h

'+'

N

E

no f i x e d p o i n t s and t h i s i s e a s i l y shown t o be i m p o s s i b l e c o m p o s i t i o n o f homotopies A must now be e s s e n t i a l i n has a z e r o i n BR.

+ E

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

on SR and hence A

QED

We c i t e now t h e Brouwer theorem on i n v a r i a n c e o f domain which i s needed i n t h e p r o o f o f Theorem 3.3.10.

ZHE0REIIl 4.9.

L e t f: U

-f

Rn be continuous and l o c a l l y 1-1 w i t h U C Rn open.

Then f i s an open map.

Phaod: = B(xo,r)

It i s s u f f i c i e n t

t o show t h a t t o each xo E U t h e r e e x i s t s a b a l l B

such t h a t f ( B ) c o n t a i n s a b a l l w i t h c e n t e r f ( x o ) .

xo = 0 and f ( 0 )

= 0.

We can assume

Choose r such t h a t f i s 1-1 on B(O,r) and c o n s i d e r f o r

Evidently h i s continuous i n E [O,l],h(x,t) = f[x/(l+t)] - f[-tx/(l+t)]. ( x , t ) , h(x,O) = f ( x ) and h ( x , l ) = f ( x / 2 ) - f ( - x / 2 ) i s an odd f u n c t i o n . I f

t

256

ROBERT CARROLL

now h ( x , t ) = 0 f o r some ( x , t )

E

aE(0,r) X [0,1]

then x / l + t

-xt/l+t since

f i s 1-1 and t h u s x = 0 (a c o n t r a d i c t i o n s i n c e 0 4 aB). T h e r e f o r e d(f,B(O, r ) , y ) = d(h(x,l),B(O,r),O) # 0 f o r e v e r y y E B(0,s) (some s ) and t h i s i m Here d r e f e r s t o t h e t o p o l o g i c a l degree which i s

p l i e s B(0,s) C f ( B ( 0 , r ) ) .

QED

examined be1 ow.

DEFINXEI0N 4-10, The degree i s d e f i n e d as a f u n c t i o n d: { ( f , A J ) , open, f: y

-+

Rn continuous, y E R'/lf(aA)}

( 6 ) d(f,A,y)

E A

= d(f,Al,y)

open s e t s such t h a t y t

E

[0,1]

when h:

and y ( t ) $ h(aA,t)

#

f(iT/A,

X [0,1]

+

R s a t i s f y i n g ( A ) d(id.,A,y)

-f

+ d(f,A2,y)

A C Rn

whenever A , ,

A2

c

= 1,

A are d i s j o i n t

i s independent o f

U A2) ( C ) d ( h ( x , t ) , A , y ( t ) )

Rn i s continuous, y: [ O , l ]

-+

Rn i s continuous,

f o r t E [0,1].

T h i s d e f i n i t i o n i s o f course useless u n t i l we see what i t i s we want here. The i d e a o f degree goes back t o Brouwer and was extended by Leray-Schauder; r o u g h l y deg f ( f : A

-+

Rn as above) i s an i n t e g e r determined by f l a Awhich

when nonzero i m p l i e s t h a t f has a z e r o i n A .

The degree s h o u l d a l s o be a 1 H e u r i s t i c a l l y f o r say f E C

homotopy i n v a r i a n t (as i n D e f i n i t i o n 4.10).

we g e t an a l g e b r a i c c o u n t o f t h e number o f s o l u t i o n s o f f ( x ) = 0 i n A prov i d e d f ( x ) # 0 on

ai

(see below).

Then we say xo

€

A i s a regular point i f

d f ( x o ) = f ' ( x o ) i s n o n s i n g u l a r ; o t h e r w i s e xo i s c r i t i c a l .

On t h e o t h e r hand

yo i s c a l l e d a r e g u l a r v a l u e o f f i f f - ' ( y 0 )

c o n t a i n s no c r i t i c a l p o i n t s o f

f; o t h e r w i s e yo i s c a l l e d a c r i t i c a l value.

One knows by S a r d ' s theorem

Now

t h a t t h e s e t o f c r i t i c a l values o f f has measure zero ( c f . [Sml;Bel]). given f

E

1 C , f:

-t

R',

A open, bounded, and connected Kan open connected

s e t by d e f i n i t i o n cannot be w r i t t e n as t h e d i s j o i n t union o f 2 nonempty open sets

-

A i s l o c a l l y connected i f f o r e v e r y p o i n t xo E A and open U 3 xo

t h e r e e x i s t s a connected open V, xo

E

V c U

-

one knows a l s o t h a t any 2

-

p o i n t s o f a compact connected and l o c a l l y connected m e t r i c space S - A say can be j o i n e d by an a r c i n Sl],and

X;

f(x)

f ( x ) # yo on a z , t h e s e t f - ' ( y 0 ) = { x

yo} i s f i n i t e when yo i s a r e g u l a r value.

imp1 i c i t f u n c t i o n theorem ( c f . Theorems 3.2.7-3.2.8)

T h i s f o l l o w s from t h e which decrees t h a t f - l

( y o ) must be d i s c r e t e and thus cannot have a l i m i t p o i n t i n

i.

Now d e f i n e

t h e degree o f f a t yo b y

Thus d ( y ) i s an i n t e g e r ( p o s i t i v e , n e g a t i v e , o r z e r o ) . nonzero one knows f - ' ( y )

However i f i t i s

i s n o t empty. One can extend t h i s d e f i n i t i o n by 1 approximation t o f u n c t i o n s f E C a t p o i n t s where d f ( x ) i s s i n g u l a r and t o

TOPOLOGICAL METHODS functions f

E

Co.

However a n e a t e r way i s d i s p l a y e d i n [ S m l ]

which we i n d i c a t e here ( c f . a l s o [Br8;Ze2]). L e t yo E R n / f ( a i )

DEFINlCI0N 4-11.

257

f o r example

Thus

and P = q ( y ) d y ( d y = dyl

. . . 4 dyn

a Cm n - f o r m on Rn w i t h compact s u p p o r t K C R n / f ( a i ) such t h a t yo

l

The degree o f f a t yo can t h e n be d e f i n e d as d(f,A,yo)

P = 1.

be

E

K and

=

JAv o f .

Such d i f f e r e n t i a l forms P w i l l be c a l l e d admissable f o r (yo,f). To show t h a t t h i s i s w e l l d e f i n e d ( i . e .

independent o f P) one shows f i r s t

t h a t i f v = 9 d y i s a Cm n - f o r m on Rn w i t h compact s u p p o r t K such t h a t 1 v = 0 t h e n t h e r e e x i s t s an ( n - 1 ) - f o r m w w i t h v = dw and supp w C K ( e x e r c i s e -

see [Sml]). =

Then i f P and

v with

p-n

I,,

P

= 0 etc.

v

n a r e b o t h admisdable i n D e f i n i t i o n 4.11 one has - J n o f = J dw o f

Set v = dw so t h a t J P o f

J a i w o f = 0 s i n c e supp w C K i m p l i e s w = 0 on f ( a n ) . o f = J A n o f and d i s w e l l d e f i n e d . It i s now a b a s i c a l l y rou-

J d(w o f )

Hence

J =

t i n e m a t t e r t o show t h a t d(f,A,yo)

We w i l l s k e t c h some o f t h i s and r e f e r t o [ S m l ]

t e properties.

t a i l s ( c f . a l s o [Br8;Dml;Ze2]).

(think o f C

f o r f u l l de-

Thus

The degree o f D e f i n i t i o n 4.11 has t h e f o l l o w i n g p r o p e r t i e s

tBE0RRn 4-12. 1

i n D e f i n i t i o n 4.11 has a l l t h e a p p r o p r i a -

f u n c t i o n s f o r 1 - 9 ) : ( 1 ) I f Iyl-yoI

i s small d(f,A,yl)

yo) ( 2 ) I f yo i s a r e g u l a r v a l u e f o r f t h e n d.(f,A,yo)

and i n p a r t i c u l a r d(f,A,yo)

= 0 i f yo

ft i s a continuous 1-parameter f a m i l y

t f i x e d and yo

'#

ft(A)

f o r t E [0,1]

= d(f,A,

= d(yo) (from (4.1))

f ( i ) ( 3 ) (Homotopy i n v a r i a n c e ) If 1 X [O,l] Rn which i s C on A f o r +

t h e n d(ftyA,yo)

i s independent o f t

( 4 ) I f f = g on aiz and yo B f(ai;) = g(ai;) then d(f,A,yo) = d(g,A,yo) ( 5 ) I f Ai C A , A . n A = @ and yo $ f ( i / u A i ) t h e n d(f,Aiyyo) = 0 f o r a l l but a fin1 j i t e number o f i and d(f,A,yo) = 1 d(f,Ai,yo) (6) (Excision) I f Q C i i s c l o s e d and yo 4 f ( Q )t h e n d(f,A,yo) = d(f,A/Q,yo) ( 7 ) L e t A and be bounded 1 - n open s e t s o f dimension n and m r e s p e c t i v e l y w i t h f E C (A,R ) and f E Rm). I f yo E R n / f ( a x ) and yo E Rm/?(ai) t h e n d ( f X A X 7, (yo,F0)) =

C1(f,

7,

- 4 -

d(f,A,yo)d(f,A,yo) for x

E

0) i f 0 Rn,

ai ( i . e . 9 g(ai)

( 8 ) I f f ( x ) and g ( x ) never p o i n t i n o p p o s i t e d i r e c t i o n s f(x)

+ hg(x)

4 0 f o r X 2 0, x E a x ) t h e n d(f,A,O) = d(g,A, g E C(V,Rn), U,V bounded open s e t s i n

( 9 ) L e t f E C(U,V),

and V . be t h e s e t o f open connected subsets of

J

a r e d i s j o i n t compact subsets o f V.

V / f ( a n ) , whose c l o s u r e s

Then i f zo E Rn/(g o f)(aG),d(g

o f,U,

zo) = 1 d(f,U,Vj)d(g,Vj,zo) w i t h a f i n i t e sum ( h e r e d(f,U,v) i s c o n s t a n t f o r v E Vj). One shows t h a t t h e p r o p e r t i e s 1 - 9 a r e v a l i d f o r c o n t i n u o u s funct i o n s by u n i f o r m a p p r o x i m a t i o n o f a continuous f by C1 f u n c t i o n s fn. Then one has (10) I f f and g a r e continuous, yo f$ f ( a x ) ,

and f = g on

ai

then

2 58

ROBERT CARROLL

= d(g,A,yo)

d(f,A,yo)

(11) Ifv

C(a?,Rn) and yo $

E

v(aK)

pends o n l y on t h e homotopy c l a s s o f IP ( h e r e d(f,A,yo) continuous e x t e n s i o n f o f tf

v

-

to A

t h e n d(v,A,yo)

de-

i s t h e same f o r any

indeed i f g i s any o t h e r e x t e n s i o n ft =

+ ( 1 - y ) g can be used w i t h (3)). (1) i s t r i v i a l ( e x e r c i s e ) and

P R U O ~ : See here [ S m l ] f o r m i s s i n g d e t a i l s . f o r (2) i f f-’(y0)

w i t h Ni a NBH o f xi on which f i s a homeomorphism,

= {xi}

Ni n N . = 0, and N = nf(Ni), J

c o n s i d e r f o r suppu c N ( p a d m i s s a b l e f o r y$N)

( c f . ( 4 . 1 ) and Appendix C).

For ( 3 ) l e t Y = { f t ( x ) ,

$ Y and Y i s compact.

Then yo

n Y =

E

ax,

0 5 t 5 11.

L e t u be admissable f o r yo and f w i t h suppp

= IA u o f t i s continuous i n t and hence c o n s t a n t

Then d(ft,Ayyo)

0.

x

For ( 4 ) one a p p l i e s ( 3 ) t o t h e f a m i l y t f + ( 1 - t ) g .

s i n c e d i s an i n t e g e r .

( 5 ) 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 ( e x e r c i s e ) and ( 6 ) f o l l o w s from ( 5 ) We l e a v e ( 7 ) as an e x e r c i s e w h i l e ( 8 ) f o l l o w s f r o m c o n s i d e r a -

immediately.

t i o n o f t h e homotopy t f ( x ) + ( I - t ) g ( x ) . t i o n (exercise). we t a k e fn defined.

-f

F i n a l l y (9) i s a r o u t i n e calcula-

f, + f E Co(/l,Rn)), For t h e approximations

f u n i f o r m l y on/1.‘

E

t h e a p p r o x i m a t i n g sequence one c o n s i d e r s a homotopy tfn + ( 1 - t ) g E

C’,

g,

-f

C1(A,Rn)), is

To see t h i s i s independent o f

= l i m d(fnyA,yo).

Then d(f,A,yo)

f,

For n l a r g e yo q! f n ( a i ) and d(fn,A,yo)

f; t h e c o n c l u s i o n i s v i r t u a l l y immediate ( e x e r c i s e ) .

9

where g, The v e r i f i -

c a t i o n o f ( 1 ) - ( 9 ) now f o r continuous maps i s easy ( e x e r c i s e ) and p r o p e r t i e s ( 1 0 ) - ( 1 1 ) a r e a l s o immediate.

QED

I n o r d e r t o extend degree t h e o r y t o reasonable c l a s s o f maps.

CEIIBIIA 4.13.

If

dimensional spaces one has t o f i n d a

We r e f e r back t o Theorem 4.5 now and check

K c B i s a c l o s e d bounded subset o f a

B i s compact then T i s a u n i f o r m l i m i t ( i . e . maps TE (dimR(TE) <

Phooa:

Since

T(K)

-

B space B and T: K

compact maps a r e assumed c o n t i n u o u s ) .

i s compact f o r

E

> 0 it

can be covered by open s e t s Ni

( 1 5 i 5 j(&)) w i t h c e n t e r s xi and we t a k e a p a r t i t i o n o f u n i t y suppGi c Niy 847).

and

Set TE(x)

-+

i n norm) o f f i n i t e dimensional

1 d ~ ~ ( x= )1 f o r x E T(K) ( c f . Lemma = 1 gi(T(x))xi and e v i d e n t l y T E ( x )

-f

54.4,

$iy

$i

2 0,

Theorems 4.5 and

T ( x ) u n i f o r m l y i n x.

By Theorem 4.5 we know t h a t a compact map o f a c l o s e d convex bounded s e t D B y B Banach, i n t o i t s e l f has a f i x e d p o i n t . Now l e t U C B be bounded and

C

TOPOLOGICAL METHODS open, B Banach, and T = I

-

K:

u’ + B

259

where K i s compact.

F i r s t n o t e t h a t T(aG) i s c l o s e d s i n c e i f T(xn) = xn K(xn)

+

k (subsequence) and xn

K(x) = k and y = x-K(x). set

E

< d/2.

Let

-+

k+y = x

4

F o r yo

E

aq

L e t yo E B/T(afi).

- K(xn)

by c l o s u r e

-

-f

y we have

b u t by c o n t i n u i t y = d > 0 and we

T ( a i ) then dist(yo,T(a@)

KE be a f i n i t e dimensional E-approximation o f K as i n Lem-

ma 4.13 w i t h range i n a f i n i t e dimensional space VE c o n t a i n i n g yo.

-

KE e v i d e n t l y TE(x)

fined.

One s e t s d(T,U,yo)

= I

# yo f o r x

E

au and

hence d(TE,VE n E,yo)

usapproximation

= d(TE,VE n U,yo) and a l i t t l e c a l c u l a t i o n ,

i n g homotopy again, shows t h a t t h i s does n o t depend on t h e ( c f . [Bel;Sml]

F o r TE i s de-

E

A l l o f t h e p r o p e r t i e s i n Theorem 4.12 ex-

and Theorem 4.25).

t e n d now t o t h e B space s i t u a t i o n f o r T = I - K and v i a ( 1 0 ) - ( 1 1 ) one need onl y deal w i t h homotopy c l a s s e s o f maps T:

d(T,U,y,)

-

( c f . h e r e Theorem 4.25).

yo) = 1 f o r yo E U ( c f .

REmARK 4.14,

(4.1));

ac

+

R/IyoI i n o r d e r t o d e t e r m i n e

L e t us a l s o n o t e e x p l i c i t l y t h a t d(I,U,

a l s o d I-K,U,yo)

= d( I-K-yo,U,O).

L e t us s k e t c h a p r o o f o f t h e Brouwer f i x e d p o i n t theorem (Then+l We must show t h e r e i s no r e t r a c t i o n r: B

orem 4.3) u s i n g degree ideas. .+

Sn.

0

Suppose t h e r e were and n o t e t h a

o f degree d(r,BntlO) i s impossible.

RElllRRK 4.15,

= d(I,Bntl,O)

r(dBntl)

so t h a t by p r o p e r t y 4 C Sn which

The s t u d y o f o p e r a t o r s T = I - K as above i s c a l l e d Leray-Schau-

d e r t h e o r y and has many a p p l i c a t i o n s . veloped so f a r i n 112-4. rem 2.6),

4

T h i s i m p l i e s 0 E r(Bntl)

= 1

L e t us summarize t h e main p o i n t s de-

Thus F and G d e r i v a t i v e s , c o n t r a c t i o n maps (Theo-

i n v e r s e f u n c t i o n theorem (Theorem 2.7) which used a Newton t y p e

a p p r o x i m a t i o n technique, descent (Theorem 2.9),

imp1 i c i t f u n c t i o n theorem (Theorem 2.8),

monotone o p e r a t o r t h e o r y i n 53.3,

steepest

homotopy arguments

and i n v a r i a n c e o f domain from §3.4 were used i n p r o v i n g r e s u l t s i n §3.3, as w e l l as t h e Brouwer f i x e d p o i n t theorem, and a p p l i c a t i o n s were i n d i c a t e d i n §3.3 t o d i f f e r e n t i a l problems.

I n §3.4 we proved t h e Brouwer and Schauder

f i x e d p o i n t theorems a l r e a d y a l o n g w i t h i n v a r i a n c e o f domain, and have developed some homotopy and degree ideas.

Further applications a r e i n order.

EMRIPLE 4.16, Consider Au t f(x,u,Du) = 0, x E A , and u = 0 on an w i t h say For A t h e r e a l i z a t i o n o f A c o r r e s p o n d i n g t o 0 O i r i c h l e t boundary f E Cm. c o n d i t i o n s A - l w i l l make sense as a compact o p e r a t o r i n s u i t a b l e spaces and o u r e q u a t i o n becomes u

- KF(u)

= 0 ( F ( u ) = f(x,u,Du)).

F o r s u i t a b l e spaces

one t h i n k s e.g. o f t h e c l a s s i c a l Schauder e s t i m a t e s ( c f . [Nil;ZeE;B2;Gil; S m l ] ) and spaces o f t h e form C”@(A) o r 2 say UuU < c w h i l e u E C2”(A) n W (A) 1,B P

W 1( A ) ; t h e n a p r i o r i e s t i m a t e s g i v e p w i t h a compact i n j e c t i o n o f t h i s

260

ROBERT CARROLL

space i n t o C l Y B f o r example (C2ya denotes Holder continuous second d e r i v a More s i m p l y ( f o l l o w i n g [Sml])

t i v e s etc.).

i n t h e p r e s e n t s i t u a t i o n use l u

It m -< 1, IIKulll 5 c where II Ill i s a C 1 norm (I1 II c o u l d a l s o be used - c f . 1,B For reasonable f, n o t growing t o o r a p i d l y , t h e a p r i o r i e s t i m [Gil;Ze2]). a t e s w i l l i n v o l v e ( 0 ) IIuII < c ( f o r u = KFu - I I U I I ~5, ~c can a l s o be used) 1 1 and one c o n s i d e r s a b a l l U = {u; IIulll 5 l t c l i n t h e B space C ( A ) ( w i t h u = 0 on 3;). L e t Tu = u-KFu and by ( a ) Tu # 0 f o r Ilulll = l t c ( i . e . u E aG). Thus d(T,U,O) i s d e f i n e d and one c o n s i d e r s Tt(u) = u - tKF(u) (0 5 t 5 1).

By t h e homotopy p r o p e r t y o f degree d(T,U,O)

= d(I,U,O)

= 1.

Hence Tu = 0

1 has a s o l u t i o n i n U i n C ( A ) ( c f . a l s o Theorem 4.25) and a l i t t l e f u r t h e r work w i l l e s t a b l i s h smoothness. Note t h a t t h e a p r i o r i e s t i m a t e i s essent i a l here f o r e x i s t e n c e .

EW\WI;E 4-17,

As an example o f t h e use o f f i x e d p o i n t theorems d i r e c t l y

consider f i r s t y ' = f(x,y)

w i t h y ( 0 ) = yo o r y ( x ) = yo

T x ( y ) where f ( s , y ) E F, y E F ( F Banach).

J , f(s,y(s))ds

t

=

I n t h e p r o o f o f Theorem 2.11 we

had a s i m i l a r s i t u a t i o n ( w i t h f ( s , y ( s ) ) r e p l a c e d by f ( y ( s ) ) E H = H i l b e r t 1 and f E C ) and remarked t h a t f o r x small Tx was o b v i o u s l y a c o n t r a c t i o n so l o c a l l y t h e r e e x i s t s a unique y ( x ) w i t h y ( x ) = T x ( y ) .

T h i s remark i s based

on working w i t h T x ( y ) i n Co(H) (Co(B) would i n v o l v e t h e same arguments f o r 1 B bounded) and n o t i n g t h a t IITx(y) yoIIH 5 :f l l f ( y ( s ) ) l l d s . When f E C one

-

-

has by D e f i n i t i o n 2.3 and Theorem 2.4 I f ( y ( s ) )

f(yo)l 5 [llf'(yo)l

-

IIy(s)-yoll f o r s small and hence f o r x small enough IITx(y) f(s,y)

i s say compact on [O,a]

X F

-+

F and e.g.

llf(s,y)ll

yoll

t

1.

€1 Now i f

5 c ( l + / l y / / ) then we

can use degree t h e o r y f o r example t o o b t a i n a s o l u t i o n o f y ( x ) = T x ( y ) on (no s h r i n k i n g o f t h e i n t e r v a l i s needed

[O,a]

-

t h e bound

t

theorem would g i v e a unique s o l u t i o n on a s m a l l e r i n t e r v a l ) . Tx(y) f o r y

E

Co(F) w i t h Y ( S ) E B = B ( y o y b ) = {y; Ily(S)-yon

c o n t r a c t i o n map F i r s t the set

5 b l and 0 5

S

i x l i e s i n a t r a n s l a t e by yo o f t h e c l o s e d convex h u l l o f t h e r e l a t i v e l y

X B

compact image C o f f: [O,x]

F times x (B = B(0,b)

-+

-

n o t e h e r e :f

f(s,

y ( s ) ) d s i s a l i m i t o f Riemann sums ~ ~ f ( s ~ , y ( s ~ ) [ ( s ~ - s ~where - ~ ) / si-l x]

?(c)

si

5

Referring t o Thus T x ( y ) C which i s compact by Lemma 4.4. 5 si). t h e Arzela-Ascol i theorem (Appendix A , Theorem A33) we see by e q u i c o n t i n u i t y t h a t t h e image under T,[T(y)](x) i s r e l a t i v e l y compact ( 0 5 x ( y ) (y = ATy i n Co(F); 0 c1 t

CJ:

= Tx(y), o f Co(B) = Mb i n Co(F)

5 a).

x 5 1)

(sup norm)

F u r t h e r g i v e n a s o l u t i o n o f y ( x ) = ATx one has Ily(x)ll (IIyoII

+ 1: c ( l + l l y ( s ) l l ) d s 5

l y ( s ) I d s from which Ily(x)ll 5 c l e x p ( a c ) 5 c2 by Lemna 1.10.4.

r > c2 so (1- T)y # 0 f o r any y ( x ) w i t h Ily(x)ll = r anywhere on [O,a]

Choose (i.e.

TOPOLOGICAL METHODS

sup y ( x )

= r).

Hence d(1-T,Br(0),O)

a s o l u t i o n ( c f . Theorem 4.25).

261

= 1 and ( I - T ) y = 0 has

= d(I,Br(0),O)

L e t us n o t e here t h a t i f one had worked w i t h

t h e Schauder f i x e d p o i n t theorem i t would have been necessary i n general t o find a set A

C

Co(F) such t h a t T(A) C A.

s h r i n k i n g t h e i n t e r v a l i n general.

T h i s c o u l d be o b t a i n e d o n l y by

One n o t e s h e r e (and i n Theorem 2.11)

t h a t a l o c a l s o l u t i o n f o r 0 5 x 5 a (a < a ) which remains bounded as x

-+

a

w i l l n o r m a l l y have a l i m i t y ( a ) which can then be used as an i n i t i a l v a l u e f o r a c o n t i n u a t i o n y ( x ) f o r a 5 x 5 B ( B < a ) . For t h i s one needs o n l y t h a t 1 e.g. Ilf(s,y)ll ~ ~ p ( s ) $ ( I I y l lw) i t h $ ( c ) < m f o r 5 < m and say v E Lloc ( c f . [ C 13).

I n Theorem 2.11 we were a b l e t o c o n t i n u e t h e s o l u t i o n f o r a l l t be-

cause o f t h e hypotheses on f which l e t t o bounds on x ( t ) e t c . One can a l s o use f i x e d p o i n t ideas f o r a s e m i l i n e a r h y p e r b o l -

EXAntPCE 4.18.

f o r example ( u can a l s o be i n c l u d e d by Y Thus f o r t h e c h a r a c t e r i s t i c i n i t i a l v a l u e 1 1 problem u = f, u(x,O) = v ( x ) E C on [O,a], u(0,y) = $(y) E C on [O,b] XY ( ~ ( 0 )= $(O), f continuous, w i t h If(x,y,u,O)l 5 M ( l + l u l ) and If(x,y,u,v) i c problem such as u

XY

= f(x,y,u,uX)

expanding t h e equations below).

-

5

f(x,y,u,v)I

(4.3)

C I V - V I one

writes

U(X,Y) = P ( X ) + $ ( Y ) V(X,Y) = 9 ' ( x ) +

- ~ ( 0 +)

I,";1

f(5,~,U(c,~),v(S,~)d5d~;

I Y f(x,~,u(x,~),v(x,~)d~ 0

Then s o l v e t h e second e q u a t i o n v = Tu by c o n t r a c t i o n , o b t a i n a p r i o r i e s t i m a t e s v i a t h e Gronwall lemma (Lemma 1.10.4),

and use Schauder's f i x e d p o i n t

theorem f o r t h e f i r s t e q u a t i o n w i t h v = Tu ( c o n t i n u a t i o n s as i n Remark 4.17 may a l s o be needed b u t we o m i t t h i s ) . There a r e many d i f f e r e n t t y p e s o f i n t e g r a l o p e r a t o r s t o which

EXAIIIPCE 4.19.

v a r i o u s n o n l i n e a r techniques can be a p p l i e d ( c f . [Ksl,2;Mtl ;Ze2]). g. an i n t e g r a l o p e r a t o r ( K u ) ( t ) = J A K(t,s,u(s)ds son o p e r a t o r .

Thus e.

i s r e f e r r e d t o as a Ury-

Here one t h i n k s o f a f u n c t i o n K ( t , - , - ) :

A X Cn

-f

Cn f o r exam-

p l e ( t a k e n = 1 here and t i s a parameter); f o r f u n c t i o n s u ( - ) : A * Cn one Cn ( K i s c a l l e d a N e m y t s k i j o r s u b s t i t u -

writes (Ku)(t,-)

= K(t,-,u(.)):

t i o n operator).

Various hypotheses a r e p l a c e d on K so t h a t e.g. measurable

functions u

-+

A

-+

measurable f u n c t i o n s K u ( t , - ) .

(t)= JA G(t,s)f(s,u(s))ds

An o p e r a t o r o f t h e form (Ku)

i s c a l l e d a Hammerstein o p e r a t o r and t h e s e a r i s e

f r e q u e n t l y i n d i f f e r e n t i a l e q u a t i o n s ( c f . Example 7.16). a r i s e s from say u '

+

Au = f ( t , u )

w i t h u ( 0 ) = uo E E.

Another o p e r a t o r

Thus ( c f . Appendix B)

i f -A generates a s t r o n g l y c o n t i n u o u s semigroup i n a B space E, G ( t ) =

262

ROBERT CARROLL t h e n ( 6 ) u ( t ) = G(t)uo

exp(-At),

equation o f Volterra type.

G(t-s)f(s,u(s))ds

f

which i s an i n t e g r a l

There a r e many theorems f o r a l l o f these t y p e s

o f o p e r a t o r s and we w i l l n o t even a t t e m p t t o g i v e references. The f o l l o w i n g g e n e r a l i z a t i o n o f a r e s u l t o f Dolph-Minty i s w o r t h c i t i n g ; t h e p r o o f i s i n s t r u c t i v e b u t we o m i t i t h e r e i n r e f e r r i n g t o [ C l ]

(it involves

monotone o p e r a t o r techniques as i n 53.3). L e t H be a H i l b e r t space,

tHE0REIII 4.20.

K E L(H) be monotone, and F: H 2 0 f o r IlxII

be hemicontinuous, bounded, and monotone, w i t h Re(F(x),x)

+

H

M.

>

Then t h e e q u a t i o n y t KFy = 0 has a s o l u t i o n .

REmARK 4-21, L e t us say a few works about p r o p e r maps ( c f . [Bel;Ze2;Dml]). As n o t e d i n [Ze?] t h e r e i s a sense i n which uniqueness i m p l i e s e x i s t e n c e ( g o i n g back t o Schauder).

Thus i f

A i s an n X n m a t r i x and Ax

most one s o l u t i o n t h e n r a n k A = n and detA # 0 so t h a t x = A-’y tion.

= y has a t

i s a solu-

One expects t h i s s i t u a t i o n t o p r e v a i l a l s o f o r l i n e a r compact p e r t u r -

b a t i o n s o f t h e i d e n t i t y which w i l l be Fredholm o p e r a t o r s o f i n d e x 0 ( c f . G e n e r a l l y one wants t o know when a (conRemark 2.13 and see e.g. [Sh3]). t i n u o u s ) o p e r a t o r f: E -+ F (E,F Banach) has open range so t h a t f ( x o ) = yo i m p l i e s f ( x ) = y has a s o l u t i o n f o r y in some NBH o f yo.

Recall here t h e

open mapping theorem f o r l i n e a r o p e r a t o r s (Theorem A45) which says t h a t f

i s open i f i t i s o n t o F ( t h e n f ( U ) i s open f o r U open).

Similarly for fin-

i t e dimensional spaces we have i n v a r i a n c e o f domain (Theorem 4.9) so t h a t f ( U ) i s open f o r U open when f i s l o c a l l y 1-1.

A r e l a t e d question i s o f

course whether x depends c o n t i n u o u s l y on y = f ( x ) ( s t a b i l i t y ) .

One c o u l d

d i s c o u r s e a t l e n g t h i n t h i s area o f ideas b u t we w i l l t r y t o e x t r a c t a few key ideas and theorems.

Thus f i r s t one d e f i n e s a continuous map f:

1 (E,F Banach) t o be p r o p e r i f f- ( K ) i s compact f o r K compact. f o r a p r o p e r map t h e s e t o f s o l u t i o n s S = { x E Y

E; f ( x )

E

+

F

In particular

= y l i s compact.

Next (see [ B e l l f o r m i s s i n g d e t a i l s )

CHE0REll 4.22.

The f o l l o w i n g a r e e q u i v a l e n t . ( 1 ) f i s proper ( 2 ) f i s a c l o -

sed map ( i . e .

f ( C ) i s c l o s e d f o r C c l o s e d ) and S i s compact f o r y f i x e d Y ( 3 ) I f E and F a r e f i n i t e dimensional then f has t h e p r o p e r t y I l f ( x ) l l + m when llxll + m. Phuod:

-

F o r ( 1 ) i m p l i e s ( 2 ) we need o n l y show f i s closed.

and l e t y,

= f ( x n ) + y , xn E C.

compact and hence a subsequence x, f ( x ) = y.

L e t C be c l o s e d

Since Y = {yn} i s compact, u = f - ’ ( Y ) E u converges t o x E C.

For ( 2 ) i m p l i e s (1) one t a k e s a compact subset

is

By c o n t i n u i t y

K

C F with f-l(K)

TOPOLOGICAL METHODS

263

= D and covers D w i t h c l o s e d s e t s D, h a v i n g t h e f i n i t e i n t e r s e c t i o n p r o p e r t y ( c f . Remark 3.2.13). A l i t t l e argument ( c f . [ B e l l ) shows t h a t "0, = @ which i m p l i e s D i s compact.

( 3 ) i m p l i e s ( 2 ) and c o n v e r s e l y f o r E,F f i n i t e

dimensional i s l e f t as an e x e r c i s e .

QED

Under hypotheses o f t h e f o r m made i n §§3.2-3.2 f o r o p e r a t o r s Au = 1 S la1 5s (-l)lulDaAa(x,u, 0 u), A: Ws -+ W-' one can show t h a t A i s p r o p e r ( c f . P P Now f o r g l o b a l r e s u l t s one r e c a l l s f i r s t f r o m Theorem 2.7 t h a t i f [Bell). g ' i s 1-1 o n t o t h e n g: E F i s a l o c a l homeomorphism. The e x t e n s i o n as a

...,

-+

g l o b a l theorem i s ( c f . [ B e l l f o r p r o o f ) L e t f: E

CHE0RElil 4.23.

-+

F be continuous.

Then f i s a homeomorphism i f and

o n l y f i s a l o c a l homeomorphism and i s p r o p e r . We conclude t h i s s e c t i o n w i t h a few more a p p l i c a t i o n s o f degree t h e o r y , properness, e t c . t o a r r i v e a t an i n v a r i a n c e o f domain theorem f o r B spaces; t h e p r e s e n t a t i o n f o l l o w s [ B e l l and we p r o v i d e some f u r t h e r d e t a i l s e s t a b l i s h i n g t h e degree p r o p e r t i e s o f maps IiK i n B spaces which were o m i t t e d a f t e r Theorem 4.12.

One can deal w i t h continuous maps f:

E

-f

F f o r which a

I-K where K i s compact - b u t we w i l l F - c f . [Bel;Dml;Ze2] ( v a r i o u s o p e r a t o r s a r e admissable, E n o t d w e l l on t h i s h e r e ) . R e c a l l f r o m D e f i n i t i o n 4.5 t h a t f: Sn Y i s indegree f u n c t i o n i s d e f i n e d as i s t h e case f o r f

=

-f

-f

e s s e n t i a l i f i t i s homotopic t o a c o n s t a n t and i n t h i s event f extends t o a map 8"' 8"' B"'.

-+

-f

Y; t h i s p r o p e r t y c h a r a c t e r i z e s i n e s s e n t i a l i t y .

Rn+'

i s e s s e n t i a l as a map Sn

-f

Rn+'/{OI

Further i f f:

t h e n f must have a 0 i n

E v i d e n t l y e s s e n t i a l i t y i s a homotopy i n v a r i a n t and one extends t h e s e

ideas t o B spaces by w o r k i n g w i t h compact p e r t u r b a t i o n s o f a f i x e d c o n t i n uous map f (e.g. f = I ) . Thus t a k e f = I f o r s i m p l i c i t y ( w i t h E = F ) , S C be t h e s e t o f E c l o s e d ( E Banach), and C a component o f E/S. L e t CI(S,E) -+ E compact, and CY(S,E) c CI(S,E) t h e subset where g # 0 on S . The map g E CY(S,E) i s i n e s s e n t i a l w i t h r e s p e c t t o C i f g has an

maps g = I+K, K: E extension

9".

C y ( C u S,E)

-

otherwise g i s essential.

Thus t o show t h a t a

g i v e n g E C I ( C U S,E) has a z e r o i n C one need o n l y show t h a t g i s essent i a l w i t h r e s p e c t t o C. One shows i n a s t a n d a r d manner ( c f . [ B e l l ) t h a t e s s e n t i a l i t y (and i n e s s e n t i a l i t y ) o f maps g i n CY(S,E) what i s c a l l e d compact homotopy where h ( x , t ) : g(x,t) = I t h(x,t), gl.

gl(x)

=

I + h(x,l),

i s i n v a r i a n t under

S X [O,l]

and g o ( x ) = I

-+

E i s compact,

+ h(x,O) l i n k s go and

As one expects now (see [ B e l l f o r p r o o f )

CHE0REfil 4.24,

L e t D be a convex bounded domain i n

E

(domain = open connec-

264

ROBERT CARROLL

ted set).

Then f , g

E

CY(ai,E)

a r e compactly homotopic i f and o n l y i f d ( f ,

Further i f f E CY(ai,E) then f i s essential r e l a t i v e t o D D,O) = d(g,D,O). i f and o n l y i f d(f,D,O) # 0. I n p a r t i c u l a r i f d(f,D,O) 4 0 t h e n f ( x ) = 0 has a s o l u t i o n i n D.

REmARK 4.25.

There a r e many deep and b e a u t i f u l r e s u l t s i n n o n l i n e a r a n a l y -

s i s which can be developed u s i n g degree t h e o r y ( c f . [Bel;Br8;Dml

;Sml;Ze2]).

We have o n l y t r i e d t o g i v e t h e f l a v o r o f (some) degree arguments here. us c i t e however a few f u r t h e r r e s u l t s .

I f D C E i s a bounded domain and f = I+K,

EHE0REflI 4.26,

l o c a l homeomorphism w i t h d(f,D,p)

Let

First K compact, i s a

= 21 t h e n f ( x ) = p has e x a c t l y one s o l u -

t i o n i n D.

f ( x ) = p l i s d i s c r e t e and f i n i t e . and by ( 5 ) i n Theorem 4 . 1 2 , ~ t l =

1

6

= t x E E; P Cover S w i t h d i s j o i n t open s e t s 0 C D P P d(f,OX,p). One then shows v i a homotopy

P J L V V ~ : f w i l l be p r o p e r on a bounded

( e x e r c i s e ) so t h e s e t S

t h a t f o r any p a t h p ( t ) i n D w i t h s u i t a b l e 0 Op(t),f(p(t)))

i s constant ( c f . [ B e l l

Hence d(f,Ox,p)

i s c o n s t a n t and t h e r e can o n l y be one x ( n o t e x~Ox,OxnOy=O). Thus under t h e hypotheses o f Theorem 4.26 i f f ( D ) n f ( a 6 ) = @

REmARK 4-27. then f : D

-

3 p ( t ) the function d(f, P(t) homotopy and connectedness a r e used).

+

f ( D ) i s 1-1.

To see t h i s n o t e f ( D ) C E / f ( a i ) so d(f,D,p')

d e f i n e d f o r a l l p ' E f ( D ) and i s c o n s t a n t ( = 21).

is

I n view o f t h i s one asks

whether a 1-1 map f o f a bounded open s e t D o n t o f ( D ) i s a homeomorphism ( i n v a r i a n c e o f domain).

Such a theorem ( i . e .

showing f i s open) i s t r u e f o r

f = I+K f o r example and we r e f e r t o [ B e l l f o r d e t a i l s .

f i x e d p o i n t index i(T,G)

= d(1-T,G,O)

T a r e compact w i t h ( I - T ) x d(1-S,G,O)

+ 0 and ( I - S ) x

f e r e n t ( c f . Theorem 4.25). to < 1.

# 0 on

= t(1-T)x + (1-t)(I-S)x

t h i s cannot be a compact homotopy on <

Assume S and

ac.

aG

Hence H(t,x)

=

x

#

Then i f d(1-T,G,O)

t h e r e i s a X < 0 such t h a t ( I - T ) x = h(1-S)x f o r some x

deed i f one s e t s H(t,s)

0

Next one d e f i n e s a

(G a bounded open s e t ) .

-

E

at.

In-

[tTx + (1-t)Sx] then

modulo 0 s i n c e t h e degrees a r e d i f =

0 f o r some xo,t0,

Hence ( I - T ) x o = - [ ( l - t o ) / t o ] ( I - S ) x o .

xo

E

aE,

and

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

t o eigenvalue problems see [Bel;Dml;ZeZ]).

5.

C0)NVEX ANAL&315.

thods i n §§1.5,1.7,3.3,

We have a l r e a d y g i v e n some d i s c u s s i o n o f c o n v e x i t y mee t c . and w i l l t r y now t o be somewhat more s y s t e m a t i c .

We r e c a l l i n p a r t i c u l a r t h e Legendre-Fenchel t r a n s f o r m a t i o n f r o m Chapter 1 and t h e r e l a t i o n o f c o n v e x i t y t o monotone o p e r a t o r s i n 53.3.

Convex a n a l y -

s i s as a s u b j e c t i n i t s e l f has been developed i n p a r t i c u l a r i n e.g.

[Aul-3;

CONVEX ANALYSIS

We w i l l f o l l o w h e r e [ E l ]

E1;Cel;RklI.

265

i n p a r t i c u l a r t o d i s p l a y some o f t h e

b a s i c ideas i n convex a n a l y s i s and t h e p r o o f s o f some i m p o r t a n t f a c t s w i l l be g i v e n ( o r a t l e a s t sketched).

The i d e a i s t o g i v e a t a s t e o f t h e s u b j e c t .

Some o f t h e methods and p r o o f s a r e a l r e a d y p r e s e n t e d i n 93.3 i n complete det a i l , w i t h perhaps s l i g h t l y d i f f e r e n t t e r m i n o l o g y a t times, and t h i s can s e r v e as a f o u n d a t i o n ( c f . 93.7 f o r examples o f v a r i a t i o n a l i n e q u a l i t i e s ) .

REmARK 5.1.

Take E t o be a

We c o l l e c t h e r e some ideas and d e f i n i t i o n s .

TVS o v e r R ( g e n e r a l l y a LCS and f r e q u e n t l y a B o r H space). convex i f h x t ( 1 - x ) y E A whenever x,y E A ( 0 < A < 1 ) .

A set A c E i s

The convex h u l l co

( A ) o f A i s t h e i n t e r s e c t i o n o f a l l convex s e t s c o n t a i n i n g A ( t h u s co(A) =

{l Xixi,

xi E A,

1 Xi

N

= 11 and

G(A) = r ( A ) ) .

We r e f e r t o t h e appendices

f o r t h e Hahn-Banach (H-B) theorem and r e l a t e d ideas.

I n view o f t h e iden-

t i t y o f weak and s t r o n g c l o s u r e o f convex s e t s we know t h a t i f a sequence xn .+

x weakly t h e r e e x i s t s a sequence o f convex combinations yn =

i n norm f o r E Banach ( e x e r c i s e

- c f . [El]).

A f u n c t i o n f: A

(F =

vex, i s convex i f f [ h x t ( 1 - h ) y ] 5 h f ( x ) t ( 1 - h ) f ( y ) dently i f f: E

-+

R

{+-I).

U

i s c a l l e d t h e e f f e c t i v e domain o f f .

m)

Given A C E t h e i n d i c a t o r f u n c t i o n o f A i s d e f i n e d by x A ( x ) = 0 f o r x and x A ( x ) =

m

for x

4

-03

and n o t i d e n t i c a l l y =

One d e f i n e s t h e e p i g r a p h e p i ( f ) f o r a f u n c t i o n f: E E X R; f ( x ) 5 a } .

E

o f f.

Thus t h e p r o j e c t i o n o f e p i ( f ) on E i s dom(f).

r e c a l l s t h a t f: E

-+

@

'i as

m.

the set epi(f) =

One checks e a s i l y ( e x Next one

i s c a l l e d l o w e r semicontinuous (LSC) i f Cx E E; f ( x ) E

R and l i m i n f f ( x ) 2 f ( x o ) as x

semicontinuous (USC) i f - f i s LSC ( s o x A i s LSC

-

-f

i s convex i f and o n l y i f e p i ( f ) i s convex.

< a1 i s closed f o r a l l a A i s closed

A

T h i s i s t h e s e t o f p o i n t s l y i n g above t h e graph

{(x,a)

-+

E

C l e a r l y A i s convex i f and o n l y i f x A i s convex.

A.

A convex f i s c a l l e d p r o p e r i f i t i s nowhere

e r c i s e ) t h a t f: E

Evi-

i s convex t h e s e t s I x ; f ( x ) 5 a 1 and { x ; f ( x ) < a1 a r e

The s e t dom(f) = I x ; f ( x ) <

convex.

R

-+

N In Akxk -+ x E, A C E con-

resp. open).

o n l y i f e p i ( f ) i s closed.

- resp.

xo.

-+

USC

More g e n e r a l l y ( e x e r c i s e ) f : E

f i s upper

- i f and o n l y i f

-+

a

i s LSC i f and

Various o t h e r p r o p e r t i e s a r e i n d i c a t e d i n [ E l ;

A u l ] f o r example and w i l l be developed as needed.

RElllARK 5.2. L e t o(E,E') and u ( E ' , E ) be t h e s t a n d a r d weak t o p o l o g i e s on E R. F o r e ' E E l and a E R t h e conand E ' ( c f . Appendix A ) and l e t f: E -+

tinuous ( a f f i n e ) function x a l l x E E, a ,(x,e')

-

-+

(x,e')

f(x) or a

f ( x ) l ( c f . Remark 1.5.4 and 91.7). c o n j u g a t e o r p o l a r f u n c t i o n o f f.

F

-

a

i s l e s s t h a n f i f and o n l y i f f o r

f * ( e ' ) where (*) f * ( e ' ) = The f u n c t i o n f*: E '

-+

t:F

I( x , e ' ) -

fi i s c a l l e d t h e

One now r e p e a t s t h i s process as b e f o r e

266

ROBERT CARROLL

t o determine f**: E

E) v i a

f**(x)

(A)

R

-f

=

( n o t El'

"P

-

-f

{(x,e')

-

we use t h e weak t o p o l o g y h e r e so El'

f*(e')1.

One d e f i n e s a

r

regularization

g o f f as t h e p o i n t w i s e supremum o f c o n t i n u o u s a f f i n e f u n c t i o n s x = L a ( x ) ( L E E ' so L ( x ) = ( L , x ) )

-

(exercise

cf. [El])

which a r e l e s s t h a n f .

%

L(x) + a

-+

It i s e a s i l y seen

t h a t i f t h e r e e x i s t s a continuous a f f i n e f u n c t i o n l e s s

t h a n f then e p i ( g ) = CO e p i ( f ) .

I t f o l l o w s t h a t f** i s t h e r - r e g u l a r i z a t i o n

g o f f and i f i n f a c t f i s a p o i n t w i s e sup o f continuous a f f i n e f u n c t i o n s

( f E r ( E ) ) t h e n f** = f ( n o t e f*

f

Then n e c e s s a r i l y ( L , x ) + a = f ( x ) so L a ( y ) = ( L , y ) + a =

La(x) = f ( x ) . (L,x)

r(E')).

One says a continuous a f f i n e f u n c t i o n La 5 f i s e x a c t a t x

REmARK 5.3.

+ f ( x ) 5 f ( y ) o r (L,y)

(L,y-x)

E

- f(x).

-

f(y)

One says now t h a t f :

E

-

i(L,x) +

f(x).

Hence by (*) f * ( L

i s subdifferentiable a t x E E

i t has a continuous a f f i n e m i n o r a n t which i s e x a c t a t x ( t h u s Ln above). The " s l o p e " L E E ' o f such an La i s c a l l e d a s u b g r a d i e n t o f f a t x and t h e c o l l e c t i o n o f such s u b g r a d i e n t s i s t h e s u b d i f f e r e n t i a l a f ( x ) .

5 f * * ( x ) 5 f ( x ) from L a ( x )

= f ( x ) one has La(x) = f**

then a f

( x ) and hence if a f ( x ) # 0 then f ( x ) = f * * ( x ) and i f f ( x ) = f * * ( x ) ( x ) = af**(x).

a f

E

+ f ( x ) 5 f ( y ) f o r a l l y E E.

( x ) i f and o n l y i f f ( x ) i s f i n i t e and ( y - x , L ) F u r t h e r s i n c e L,(x)

Thus L

F i n a l l y f ( x ) = min f ( y ) f o r y

E

E i f and o n l y i f 0

E

af(x)

(from f ( x ) + ( y - x , L ) 2- f ( y ) e t c . above). 11) t h a t L

E

We remark a l s o ( e x e r c i s e - c f . [ E a f ( x ) i f and o n l y i f f ( x ) + f * ( L ) = ( x , L ) w h i l e a f ( x ) i s con-

vex and a(E',E)

closed i n E l .

We r e c a l l n e x t from § 3 . 2 t h a t t h e l i n e a r Gateaux d i f f e r e n t i a l f ' ( x ) i s c h a r a c t e r i z e d by ( 0 ) ( f ' ( x ) , y ) = 1 i m [ f ( x + x y ) - f ( x ) ] / x = d f ( x , y ) and here we s p e c i f y t h a t d f ( x , y ) i s t o be l i n e a r ( d f ( x , y ) = ( f ' ( x ) , y ) ) b u t

RRllARK 5.4.

c o n t i n u i t y of x

-+

f ' ( x ) i s n o t assumed as i n Theorem 3.2.4 ( c f . a l s o [Aul]).

L e t us notenow t h a t f o r a convex f u n c t i o n f, i f f ' e x i s t s a t x then f ' ( x ) = a f ( x ) . Indeed ( c f . Theorem 3.3.12) f i r s t f ( x + x ( y - x ) ) 5 ( 1 - x ) f ( x ) + x f ( y ) by by c o n v e x i t y so ( f ' ( x ) , y - x )

5 l i m [ ( l - x ) f ( x ) + xf(y) - f ( x ) ] / x

= f(y)

- f(x)

5 f(x+w) - f ( x ) . Hence f ' ( x ) E a f ( x ) . On t h e o t h e r hand f o r L E a f ( x ) one must have )I( w,L) 5 f(x+hw) - f ( x ) which i m p l i e s (w,L) 5 ( w,

or (f'(x),w) f'(x)).

Therefore ( w , f ' ( x ) - L ) '0

f o r a l l w which i m p l i e s f ' ( x ) = L.

a f ( x ) reduces t o a s i n g l e p o i n t L = f ' ( x ) . ideas ( c f . [ E l ] )

One shows a l s o u s i n g epigraph

t h a t i f f i s continuous a t x and has a unique a f ( x ) t h e n

at x af(x) = f'(x).

Thus e s s e n t i a l l y l i n e a r G d i f f e r e n t i a b i l i t y i s t h e same

as uniqueness o f a f ( x ) . entiable

Thus

Next ( c f . Theorem 3.3.12)

recall that f o r G differ-

as above c o n v e x i t y o f f ( x ) i s e q u i v a l e n t t o ( f ' ( x ) , y - x )

2 f(y)

-

CONVEX ANALYSIS f ( x ) and hence f ' ( x ) w i l l be a monotone map.

-

f(u2)

f(ul)

-

t h e form t u 1 - u 2 , f ' ( u 1 )

REmARK 5.5.

-

5 f(ul)

and (ul-u2,f'(u2))

267

5

Here f r o m (u2-ul,f'(ul))

f ( u 2 ) one o b t a i n s m o n o t o n i c i t y i n

LO.

f'(u2))

L e t us sumnarize some r e s u l t s whose p r o o f s a r e c o n t a i n e d i n §3.

B space w i t h C C

3 ( o r can be e x t r a c t e d e a s i l y ) .

Take E t o be a r e f l e x i v e

E a nonempty c l o s e d convex s e t .

L e t f be convex LSC and proper.

The prob-

lem o f concern f i r s t i s t o f i n d u e C such t h a t f ( u ) = i n f f ( v ) ( v e C ) . E q u i v a l e n t l y i f one d e f i n e s ?(v) = f ( v ) f o r v t h e n m i n i m i z e i ( v ) o v e r E. i s corecive over C ( i . e .

C and ? ( v ) =

m

for v

#

C

Assume a l s o e i t h e r t h a t C i s bounded o r t h a t f

f(u)

-+

m

if u

E

C and IIuII

-+

-).

Then t h e r e e x i s t s

a t l e a s t one u e C m i n i m i z i n g f ( v ) (which i s unique i f f i s s t r i c t l y convex).

F u r t h e r t h e f o l l o w i n g 3 c o n d i t i o n s a r e e q u i v a l e n t when f i s G d i f f e r -

e n t i a b l e w i t h x * f ' ( x ) continuous ( 1 ) f ( u ) = i n f f ( v ) ( v

€

C) ( 2 ) ( f ' ( u ) ,

v - u ) 1. 0 f o r a l l v e C ( 3 ) ( f ' ( v ) , v - u )

1. 0 f o r a l l v E C.

t h i s e q u i v a l e n c e ( c f . 53.3 and [ E l ] ) .

Thus g i v e n ( 1 ) i t f o l l o w s t h a t f ( u )

< f [ ( l - x ) u + xv] and hence [ f ( u + x ( v - u ) ]

-

f(u)]/x

gives (3).

-

f'(u),v-u)

(f'(u+x(w-u)),w-u)

we g e t ( 2 ) a t

=

(3) recall

( 1 - A ) U + xw, w E C, t o g e t ( f '

Conversely g i v e n ( 3 ) t a k e v '0

For ( 2 )

( c f . Remark 5.4) so adding t h i s w i t h ( 2 )

'0

[ ( 1 - x ) u + xw],w-u)

> 0.

2 0 and ( f ' ( u ) , v - u )

The converse ( 2 ) i m p l i e s ( 1 ) f o l l o w s as i n Remark 5.4. f i r s t that c f ' ( v )

L e t us s k e t c h

( a f t e r d i v i d i n g by

x

= 0.

x

> 0 ) and by c o n t i n u i t y o f

x

+

I n e q u a l i t i e s such as ( 2 ) - ( 3 ) a r e

c a l l e d v a r i a t i o n a l i n e q u a l i t i e s and more g e n e r a l l y ( a s i n 53.3) one wants t o s o l v e v a r i a t i o n a l i n e q u a l i t i e s o f t h e form ( 6 ) ( f i ( u ) , v - u )

> 0 and ( f i ( v ) , v - u ) + f 2 ( v ) - f 2 ( u ) f = fl+f2,

+ f2(v) - f2(u)

> 0 which a r e e q u i v a l e n t t o ( 1 ) - ( 3 ) when

fly f2 LSC convex, and fl i s c o n t i n u o u s l y G d i f f e r e n t i a b l e ( n o t e

here t h e s i m i l a r i t y t o e.g.

(o*)

i n 53.3).

Hence t h e r e w i l l be theorems

r e l a t i v e t o ( 6 ) analogous t o those i n 53.3 (see 53.7 f o r v a r i a t i o n a l i n e q . ) .

REmARK

5.6,

L e t us say a few words abour p r i m a l and dual problems f o l l o w i n g

Thus one s t a r t s w i t h a problem ( P ) F i n d i n f f ( v ) ( v E E say and f o r a [El]. s o l u t i o n v = u one w r i t e s i n f P = f ( u ) - e v e n t u a l l y one wants t o work o v e r a convex s e t C, v E C, and we d e f e r t h e r o l e o f C t o a p p l i c a t i o n s l a t e r ) . Next t h i n k o f a f a m i l y o f p e r t u r b e d problems ( P ) F i n d i n f v ( v , p ) ( v E E) where P v: E X F - + R , v(v,O) = f ( v ) , and F i s a TVS. T y p i c a l examples f o r v a r e g i v e n below ( c f . Example 5.8, discussed i n Remark 5.11. to

v

as i n (*).

e t c . ) and t h e seeming ad hoc n a t u r e o f t h i s i s

L e t now

v*:

E' X F'

I n p a r t i c u l a r t h e problem (P*)

-+

R

be t h e c o n j u g a t e f u n c t i o n

F i n d sup [-v*(O,p')]

for p'

E F ' i s c a l l e d t h e dual problem t o ( P ) and one w r i t e s sup P* = -v*(O,p*)

for

268

ROBERT CARROLL

I n general

any s o l u t i o n p ' = p*.

-

problems P and P* ( e x e r c i s e such t h a t f ( u o ) <

-

[El;Aul]

5 inf P

n o n t r i v i a l means e.g.

~ ( 0 , p ' ) - q(u,O)

Remark 1.5.4,

r e f e r back t o [ S t l ] ,

< sup P*

and we n o t e t h a t q*(O,p')

E and p E F so q*(O,p') e.g.

-m

R

t h a t t h e r e e x i s t s uo

o r -v*(O,p')

- q(u,p]

zlp(u,O)).

for u L e t us

E

I n p a r t i c u l a r t o i s o l a t e a n i c e class o f

Then f o r q E r o ( E X F ) s e t h ( p ) = i n f P -+

f o r nontrivial

and §1.7 f o r m o t i v a t i o n a l m a t e r i a l and t o

f o r further details.

i t w i l l f o l l o w t h a t h: F

m

= sup[( p , p ' )

problems one f i r s t denotes by ro(EXF) o r r o ( E ) t h e minus i-.

<

i s convex ( e x e r c i s e

P

r

space o f Remark 5.2

= i n f q(u,p)

(u

E

E) and

- n o t e q i s convex as a

4

p o i n t w i s e sup o f continuous a f f i n e f u n c t i o n s ) ; however h

r o ( F ) i n general.

F u r t h e r h * ( p ' ) = q*(O,p') s i n c e h * ( p ' ) = sup [( p , p ' ) - h ( p ) ] = sup [C p , p ' ) inf - U q ( u , p ) I = sup sup [( p,p') - q ( u , p ) ] = q*(O,p') ( n o t e - i n f J = s u p ( - J ) ) . Now one says P i s normal i f h ( 0 ) i s f i n i t e and h i s LSC a t 0. It i s e a s i l y seen ( e x e r c i s e

-

c f . [ E l ] ) t h a t i f P i s normal so i s P* and i n f P

sup P*

Next one says P i s s t a b l e i f h ( 0 ) i s f i n i t e and h i s s u b d i f f e r -

i s finite.

e n t i a b l e a t 0.

Then P i s s t a b l e i f and o n l y i f F i s normal and P* has a t To see t h i s n o t e i f P i s s t a b l e h ( 0 ) i s f i n i t e and ah

l e a s t one s o l u t i o n .

( 0 ) i s nonempty so a h ( 0 ) = ah**(O) i s nonempty ( c f . Remark 5.3) and h ( 0 ) = h**(O) from Remark 5.3 so P i s normal ( e x e r c i s e t h e LSC r e g u l a r i z a t i o n o f h

- n o t e h** 5 1 5 h w i t h ?;

l a r g e s t LSC m i n o r a n t o f h ) .

Continuing i n

t h i s v e i n one can show ( c f . [EL]) t h a t t h e f o l l o w i n g a r e e q u i v a l e n t : ( 1 ) P and P* a r e normal and have s o l u t i o n s ( 2 ) P and P* a r e s t a b l e ( 3 ) P i s s t a b l e .

RENARK 5.7,

One can summarize some o f t h e preceeding by s a y i n g ( c f . [ E l ] ) t h a t i f P and P* have s o l u t i o n s w i t h i n f P supP* f i n i t e then a l l s o l u t i o n s u o f P and p* o f P* a r e l i n k e d by (+) q(u,O)

+ q*(O,p*)

= 0 o r (O,p*)

E aq

Conversely g i v e n ( + ) t h e n u and p* a r e s o l u t i o n s o f P and P* w i t h (u,O). i n f P = supP*. To see t h i s n o t e i n p a r t i c u l a r t h a t i n f P = q(u,O) = supP* = -q*(O,p*) 3 that L

and q(u,O) + q*(O,p*) E

= q;(p')

-+

fi v i a

(m)

-L(u,p')

R

and ifq i s convex t h e n a l s o L

- f i s convex).

Now

P'

-+

- q ( u , p ) ] so t h a t P One shows ( c f . [ E l ] - t h e de-

L ( u , p ' ) i s a concave USC f u n c t j o n F ' u

*

Next one d e f i n e s a

= sup [( p , p ' )

( u i s a parameter h e r e ) .

t a i l s a r e r o u t i n e h e r e ) t h a t Lu: p ' -+

= 0 ( r e c a l l f r o m Remark 5.3

a f ( x ) i f and o n l y i f f ( x ) + f * ( L ) = ( x , L ) ) .

Lagrangian L: E X F ' -L(u,p')

= ( (u,O),(O,p*))

-+

L ( u , p ' ) i s convex ( f i s concave if

269

CONVEX ANALYSIS

= sup [( p , p ' )

from which f o l l o w s +*(O,p')

U

- v(u,p)]

=

i!fLi n( uf , p ' )

becomes (P*) F i n d sup lem P* (namely f i n d sup [-p*(O,p'p]) P' o v e r p ' - where L % Lv). S i m i l a r l y , assuming v E r o ( E X F ) , r ( F ) and hence

(5.2)

.

v;*

=

( c f . Remark 5.3).

(pU

= v;*(P)

Ip(U,p)

':!

=

"! L(u,p') P

f r o m which v ( u , O ) = p ' ) ( f o r p E r,(EXF)

o r only

vu

vu: p

-+

v(u,p) E

It follows t h a t

-

[(P,P')

and prob-

L ( u , p ' ) (sup

v;(P')I

= ';P[(P,P')

+

and problem P becomes P: F i n d E r ( F ) f o r a l l u i s enough).

L(U,P')I

i n f sup

L(u,

i s a saddle p o i n t o f L means t h a t L(G,p') -< L(U,p*)

Now be d e f i n i t i o n (ij,p*)

-< L(u,p*)

f o r a l l u,p'. From Remark 5.6 we r e c a l l f o r general problems t h a t sup i n f L(U,P') s sup^* 2 i n f P always, o r v(u,o) or [-v*(o,p')l 5 P i n f sup L(u,p'). Hence when (Gyp*) i s a s a d d l e p o i n t o f L one has L(U,p*) PI = i n f L ( u , ' * ) = +*(O,p*) = L ( i , p ' ) = v(U,O) ( i n f o v e r u ) . Thus v(U,O) P L e t us n o t e t h a t t h i s says t v*(O,p*) = 0 and i n f P = supP* as above.

ilf

"!

sup inf P'

L(u,p')

= sup P* = sup[+*(o,p'))

Again one r e f e r s back t o Remarks 1.5.3 and 1.5.4 which appear now as s p e c i a l cases o f t h e general framework now a v a i l a b l e ( c f . a l s o Remark 5.10). F o l l o w i n g [ E l ] l e t us e x h i b i t a t y p i c a l s i t u a t i o n o f a c a l c u MAmPtE 5.8. l u s o f v a r i a t i o n s f l a v o r . Thus suppose f ( u ) = J(u,Au) where A E L(E,F) w i t h v(u,p)

= J(u,Au-p)

L e t J* E r ( E ' X F ' ) be t h e c o n j u g a t e f u n c t i o n w i t h v*(O,p')

see below).

- J(u,Au-p)].

sup [ ( p , p ' ) UYP

u)

-

(p'q)

-

-

f o r example (A w i l l o f t e n be a d i f f e r e n t i a l o p e r a t o r

J(u,q)]

S e t t i n g q = Au-p one has v*(O,p')

= sup

[(A*p',

U,P

= J*(A*p',-p').

=

Hence t h e problem P* i s t o f i n d

i!lFl

and h e r e we have a v i s i b l e c o n n e c t i o n between P* and P, inf One can a p p l y t h e r e s u l t s o f t h e preceeding r e namely t o f i n d uEE J(u,Au). [-J*(A*p',-p')]

marks now, assuming e.g. J i s convex, e t c . ( c f . [ E l ] ) . e s t i s t h e s i t u a t i o n where G(Au)] and P*: F i n d J(U,dG) t J*(/\*p*,-p*)

F ( G ) + F*(A*p*) A*p* E

= F(u)

ilf[ F ( u )

+ G(Au) w i t h P: F i n d

+

[-F*(A*p') - G * ( - p ' ) ] . The e x t r e m a l i t y c o n d i t i o n P = 0 can be decoupled i n t o ( e x e r c i s e - c f . [ E l ] ) (**)

- (A*p*,U)

a F ( G ) and -p*

J(u,Au)

O f particular inter-

E

= 0 and G(Ali)

+ G*(-p*)

+ (p*,AG)

= 0 which means

aG(AG).

2

Take now a r e a l problem as i n s1.9. Given f E L ( A ) ( A C Rn 1 1 bounded) f i n d u E H o ( ~ ) such t h a t (*A) a(u,v) = ( f , v ) f o r a l l v E Ho where

EXA1IIPI;E 5.9.

270

ROBERT CARROLL

Inl (Diu,Div)

( t h u s we have t h e weak D i r i c h l e t problem). One can 4 1 1 use t h e norm lluJl = a(u,u) on Ho and w r i t e ( f , u ) = ( f , u ) i n H0-H-' d u a l i t y 1 ( r e c a l l t h e Poincar6 t y p e i n e q u a l i t i e s IIul12 5 cIIuII r e l a t i v e t o Ho - c f . [ L i 1 2 n 21). Now t a k e E = Ho, F = L ( A ) , A 'L grad w i t h A* 'L - d i v , E' = H - l , F ( u ) = - ( f , u ) , and G(p) = ( 1 / 2 ) ~ I~p I 2 dx w i t h G(Au) = (1/2)JA Igradu12dx. One a(u,v)

=

knows i n a general sense by c a l c u l u s o f v a r i a t i o n s techniques i n 551.2, t h a t t h e unique s o l u t i o n

-

mum o f (1/2)a(u,u)

(f,u)

1.5

( o b t a i n e d i n 91.9) w i l l a c h i e v e t h e mini1 2 i n Ho ( i . e . P: F i n d i n f [ ( 1 / 2 ) I A l g r a d u l dk - (f,

of

(*A)

-

otherNow e v i d e n t l y F * ( u ' ) = sup ( u ' + f , u ) = 0 i f u ' + f = 0 and = u)]). 2 wise (sup o v e r u ) w h i l e ( c f . below) G * ( p ' j = ( 1 / 2 ) J A I p ' I dx so P*: F i n d

[-':'

"?

( - d i v p ' + f , u ) - ( 1 / 2 ) ~ , , I P ' ~ ~ ~ XOne ] . can e l i m i n a t e t h e p ' f o r P which F*l/\*p') = m ( s i n c e sup ir! p ' i s d e s i r e d and a - s i g n appears) and t h e n 2 ( * e ) P* Isup [ - ( 1 / 2 ) l A I p ' I dx] (sup o v e r p ' w i t h - d i v p ' + f = 0 ) . The exi s -(f,ii) + s:p ( - d i v p * + f , u ) 2 which i s t r i v i a l l y s a t i s f i e d and (*&) ( 1 / 2 ) I A I g r a d i l dx + ( 1 / 2 )

tremal c o n d i t i o n (**) a t a s o l u t i o n (Ci,p*) = (-divp*,i)

2 I p * l dx = -IA p*gradGdx which means t h a t p* = -gradfi almost everywhere Thus t h e s o l u t i o n ii o f P and p* o f P* a r e l i n k e d by p* = -gradii. Note (AE).

J

A

also that formally divp* = f

'~r

This k i n d o f a n a l y s i s i s e a s i l y ex-

-Ail = f .

I

tended t o a n o n l i n e a r D i r i c h l e t problem ( f E La

-

1 Di(JDi~Ja-2Di~)

l/a

+

l / a ' = 1)

I,

(**) f =

w i t h ( c f . 53.3) (*m) P 'L i n f [ ( l / a ) l y JDiuladx 01 Here G(p) ( l / a ) l J Ipiladx and G*(p') 'L (1/

JA fudx] ( i n f over u E W ).

Q

2

The p r i n c i p l e behind t h i s d u a l i t y goes as f o l l o w s ( c f . [E F i r s t check t h a t i f 9 ( t ) = ( l / a ) ( t l a t h e n \ p * ( t ) = ( l / a ' ) l t l a ' . Take

a')l J Ip;I

11).

,

dx.

more g e n e r a l l y 9 t o be a s u i t a b l e even f u n c t i o n w i t h c o n j u g a t e q * and s e t F(u)

q(IIull); then g ( u ' ) = +o*(llu'll) i s c o n j u g a t e . sup sup [ ( u , ~ ' ) - ~ ( ( I I U I I ) ]= t)O [(u,u')

F * ( u ' ) = sp:

~ ( t )= ]sup[tllu'II

-

To see t h i s w r i t e

-

p ( t ) ] (sup i n t E R ) = q * ( l l u ' l l ) .

~(IIuII)]=

:$

(A*)

[tllu'll

-

I n any case a p p l y t h e

s i m p l e s i t u a t i o n 9 = ( l / a ) ~l p l a d x t o t h e i n d i v i d u a l pi t o o b t a i n G and G*.

REmARK 5.10.

L e t us make a few more comments about saddle p o i n t s and d u a l -

i t y and expand t h e d i s c u s s i o n o f Remark 5.7 (see a l s o Remark 5.11 and [Mcl;

We t h i n k o f L ( u , p ' ) d e f i n e d on A X B C E X F (E,F' r e f l e x i v e B spainf sup i n f L ( u , p ' ) 5 L ( v , p ' ) f o l l o w s i n a general way L(U,P' ) P' u < L ( v , p ' ) and hence ( A A ) L(u,p') 5 L ( u , p ' ) ( c f . Remark - P P P 5.7). Given a saddle p o i n t (li,p*) w i t h L ( i , p ' ) 2 L(U,p*) 5 L(u,P*) one has L ( i , p ' ) = L(U,P*) = i n f L(u,p*). But L ( u , p ' ) L:'! L(U,p') and P ~ ( u , p ' ) so i n f sup ~ ( u , p ' ) 5 SUP i n f L(U, i n f ~ ( u , p * ) ( i n f over u) 5 P p ' ) ( i n f i n u, sup i n p ' ) which t o g e t h e r w i t h (AA) i m p l i e s sup i n f L ( u , p ' )

Rwl]).

ces) and f r o m

s'y

"!

"Y

'nuf

i!f

iCf

i:f':!

suy

CONVEX ANALYSIS = i n f sup L ( u , p ' ) .

min.

271

When a sup o r i n f i s a t t a i n e d one o f t e n w r i t e s max o r

Now t y p i c a l s i t u a t i o n s i n v o l v e A and B convex, bounded, closed, and

nonempty w i t h p '

L(u,p')

+

concave USC and u

+

L(u,p')

convex LSC.

One

shows t h e n t h a t t h e s e t o f s a d d l e p o i n t s i s convex and o f t h e form A. (exercise

-

Further i f p '

cf. [El]).

L ( u , p ' ) and u

+

-+

L(u,p')

X Bo

are G-diff-

e r e n t i a b l e then (Gyp*) i s a s a d d l e p o i n t o f L i f and o n l y i f (A*) ( ( a L / a u )

2 0 and

(U,p*),u-U)

(

5 0 (see [ E l ] f o r a more general

(aL/ap')(i,p*),p'-p*)

-

version t h e p r o o f i s s t r a i g h t f o r w a r d ) . A t y p i c a l e x i s t e n c e r e s u l t now says t h a t t h e r e e x i s t s a t l e a s t one s a d d l e p o i n t under t h e c o n d i t i o n s i n d i c a t e d above. venience u

+

L e t us s k e t c h a p r o o f o f t h i s f o l l o w i n g [ E l ] assuming f o r conL(u,p')

i s s t r i c t l y convex.

Since E and F ' a r e r e f l e x i v e , A

and B a r e compact f o r t h e weak t o p o l o g i e s o f E and F ' ( A l a o g l u theorem Appendix A).

Further the semicontinuity properties f o r p '

L ( u , p ' ) w i l l h o l d f o r t h e weak t o p o l o g y ( e x e r c i s e

+

-

+

-

L ( u , p ' ) and u

c f . Remark 5.1).

Now

-+ L ( u , p ' ) b e i n g weakly LSC i s bounded below on A and a t t a i n s i t s minimum f ( p ' ) a t a unique p o i n t U ( p ' ) (by s t r i c t c o n v e x i t y ) . Then f ( p ' ) min L ( u , p ' ) = L ( U ( p ' ) , p ' ) . = Since f ( p ' ) i s t h e n t h e l o w e r bound o f con-

for p' E B y u

U

cave weakly USC f u n c t i o n s i t w i l l be concave and weakly USC, hence bounded above, and w i l l a t t a i n i t s upper bound a t some p o i n t p* SO t h a t f ( p * ) = max max min L ( u , p ' ) ( w i t h (A&) f ( p * ) 5 L(u,p*)). Next by c o n c a v i t y f(P') = L(u,(l-A)p*+ Ap') 2 (1-h)L(u,p*) + AL(u,p') and t a k i n g u = UA = i [ ( l - A ) p * + ~ p ' ]one has

+

(A*)

f(p*) 2 f((1-A)p*+Ap')

2 (l-A)L(cA,p*) f ( p * ) 1. L ( U A , p ' ) .

= L(GA,(l-A)p*+Ap')

A L ( U ~ , ~ ' ) .Since L(UA,p*) 2 f ( p * ) t h i s i m p l i e s

(Am)

0 and from UA E

A

(weakly s e q u e n t i a l l y compact) e x t r a c t a weak-

l y convergent subsequence Uk

+

6.

Now l e t A n

-+

p ' ( o r t h e sequence x k ) .

Then U = u(p*) and t h i s i s independent o f

Indeed L(U,,(l-A)p*+Ap')

hence by c o n c a v i t y i n p', (l-A)L(C$,p*) which by USC and LSC L ( i , p * )

5 l i m i n f L(G,,p*)

T h e r e f o r e U = U(p*).

<

<

+ AL(EA,p')

L(u,(l-A)p*+Ap')

5 L(u,(l-A)P*+AP')

and from

5 l i m sup L ( u , ( l - h ) p * + X p ' ) Now r e t u r n t o ( A m ) and pass

L(u,p*) ( A = A k + 0 ) . t o l i m i t s A k -+ 0 t o o b t a i n (@*) f ( p * ) 2 L ( G , p ' ) f o r a l l p '

E

B.

The p o i n t

(Gyp*) i s now t h e saddle which i s c h a r a c t e r i z e d v i a (**) and (A&) as f o l l o w s . Ift h e r e e x i s t s (ij,p*) and a such t h a t L ( U , p ' ) 5 " f o r a l l p ' and L(u,p*) L i n f S U P L ( u , p l ) = suy CL f o r a l l u t h e n (Gyp*) i s a s a d d l e p o i n t w i t h CL = P' P inf L ( u , p ' ) ( n o t e a = L ( i , p * ) and L ( U , p ' ) 5 L(G,p*) 5 L ( u , p * ) ) . U

REmARK 5.11. L(u,p')

Given a problem P

Q,

i n f F ( u ) one wants t o w r i t e F ( u ) = sup

i n o r d e r t o make an immediate and d i r e c t c o n n e c t i o n t o saddle prob-

lems ( c f . Remark 5.6 where t h e p e r t u r b a t i o n IP a r i s e s i n a seemingly ad hoc

272

way).

ROBERT CARROLL

E, then F1(u) =

-

F = F, t F2 w i t h F1 convex, LSC, and p r o p e r on - FC(u')], and hence F ( u ) = [ ( u , u ' ) t F2(u)

I n p a r t i c u l a r i f e.g.

s:y

[(u,u')

F j i ( u ' ) ] (so L ( u , p ' )

( u , p ' ) t F2(u)

event we t h i n k o f L ( u , p ' )

-

':!

FC(p'),

p'

u', B = E l ) .

I n any

on A X 6 C E X F ' (E,F r e f l e x i v e B spaces, A,B

c l o s e d convex) as g i v e n somehow and we d e f i n e f o r u E E and p E F, q(u,p) m

if u

#

A, w i t h ( c f . (m))

(@A)

q(u,p)

hypotheses on L as i n Remark 5.10,

=

"?P

[(p,p')

t L(u,p')].

one e a s i l y checks ( c f . [ E l ] )

t h a t (u,p) i n f sup

+ q ( u , p ) i s convex and LSC on E X F and t h e problem o f d e t e m i n i n g

L(u,p')

= i n f q ( u , O ) l e a d s t o two e q u i v a l e n t dual problems

=

Then, w i t h

P'

1 ) F i n d sup i n f

L ( u , p ' ) (sup over p ' , i n f o v e r u ) ( 2 ) F i n d su? [-q*(O,p')]. P equivalence o f ( 1 ) and ( 2 ) one has q*(O,p') = sup [( P,p') inf UYP [ - L ( u , P ' ) ] SO -q*(O,p') = L(u,P'). Thus SUPC-V*(O,P')I

To see t h e

U

REmARK 5.12.

We have seen how v a r i a t i o n a l i n e q u a l i t i e s such as ( 2 ) - ( 3 ) i n

Remark 5.5 a r i s e a b s t r a c t l y and we w i l l g i v e examples and a p p l i c a t i o n s a l o n g w i t h f u r t h e r t h e o r y i n 53.7.

There a r e many Such a p p l i c a t i o n s ; huge groups

o f r e a l i s t i c p h y s i c a l problems can be phrased as v a r i a t i o n a l i n e q u a l i t i e s ( o r q u a s i - v a r i a t i o n a l i n e q u a l i t i e s ) and we r e f e r t o [Acl ;Bgl ; B r l ;Bw2;Dvl ;K b l ;Gwl ;Li3-7;Zel ;Bd2].

6- N0NCZNEAR SrEmZGR0llP.5 AND A0N0C0NE SrECSr.

The ideas o f n o n l i n e a r semigroup

and m u l t i v a l u e d monotone o r a c c r e t i v e o p e r a t o r s p r o v i d e d a s t i m u l a t i n g chapt e r i n n o n l i n e a r f u n c t i o n a l a n a l y s i s i n t h e 1960's and 1970's. some m a t e r i a l a v a i l a b l e i n book o r l e c t u r e n o t e f o r m (see e.g. Dpl;Mtl;Au3;Pz2]) appear.

There i s [Brl;Dml;Cxl;

and s e v e r a l o t h e r books a r e i n p r e p a r a t i o n o r about t o

We w i l l o n l y g i v e a b r i e f i n t r o d u c t i o n t o t h e s u b j e c t f o l l o w i n g

[ P z ~ ] (and my notes from some l e c t u r e s o f Pazy a t t h e U n i v e r s i t y o f Maryland

i n 1972-73).

The c o n n e c t i o n o f t h i s m a t e r i a l t o convex a n a l y s i s and v a r i a -

t i o n a l i n e q u a l i t i e s w i l l a l s o be i n d i c a t e d ( c f . 53.7 as w e l l ) . Thus l e t E be a B space ( r e a l f o r convenience) and A an o p e r a t o r w i t h domain E D(A) C E and values i n 2 ( = t h e c o l l e c t i o n o f subsets o f E). We w i l l see t h a t such o p e r a t o r s a r e n o t o n l y n a t u r a l b u t a b s o l u t e l y necessary i n a p p l i cations.

One wants now t o s t u d y i n i t i a l v a l u e problems (*) u '

t

Au 3 f ,

u ( 0 ) = uo where u ' i s a s t r o n g v e c t o r valued d e r i v a t i v e and we g e n e r a l l y r e -

q u i r e u ' e x i s t s A E w i t h u a b s o l u t e l y continuous ( i n p a r t i c u l a r u = l u ' d ~ + u ~ ) .

DEFZNZCI0N 6.1,

A i s accretive i f llx-?+x(y-~)ll > IIx-x"llfor

x

> 0 and a l l (x,

y ) , (:,$) i n G(A) = { ( x , y ) ; y E Ax}. We mention a l s o t h a t a map A: O ( A ) C E' E - 9 2 i s monotone if ( x - y , x ' - y ' ) 2 0 f o r a l l x,y E D(A), x ' E Ax, y ' E Ay.

NONLINEAR SEMI GROUPS

273

L e t us n o t e t h a t when A i s a c c r e t i v e t h e e q u a t i o n x+hy = z f o r ( x , y ) (i.e.

(l+AA)x 3

z ) has a t most one s o l u t i o n .

Indeed i f xi+Ayi

=

E

G(A)

z t h e n IIxl

-

-x2 + h(yl y2)11 2- IIx 1 - x 21I i m p l i e s x1 = x2 and hence y1 = y2. Thus i f z E R(l+r\A) t h e r e e x i s t s a u n i q u e (x,y) E G(A) w i t h x t h y = z and hence ( 1 +

i s s i n g l e valued. I n f a c t (1 + A A ) - ' i s nonexpansive s i n c e i f x1 + hyl = z1 and x2 + hy2 = z t h e n Ilx - x + x ( y y )ll = IIz -z II > IIx - x II = l l ( 1 t hA)-'

1 (l+hA)- z211.

-

xA)-lzl

2 1 2 1-2 1 2 - 1 2 The argument can be r e v e r s e d so t h a t i f (l+AA)-'

nonexpansive t h e n A i s a c c r e t i v e .

is

Now one d e f i n e s a k i n d o f s c a l a r p r o d u c t

( c a l l e d s e m i - i n n e r p r o d u c t ) on E so t h a t one can handle t h e i d e a o f a c c r e t i v e i n a manner s i m i l a r t o t h e s i t u a t i o n i n H i l b e r t space. first

(A)

z2;Dml]) (0)

[x,y],

= [IIx+Ayll

that A 1i m

[x,y]

=

-

(A>

IIxll]/A

0).

Thus d e f i n e

One checks ( e x e r c i s e

-

c f . [P

i s nondecreasing and t h i s l e a d s t o t h e d e f i n i t i o n inf [x,ylA = [x,yIA. It i s r o u t i n e t o prove ( c f . Cpz2-j)

+

[x,y-JA

( 1 ) [ , 3: E X E + R i s USC ( 2 ) [crx,By] = IBI[x,y] ( 3 ) PR0P05fCf0N 6.2. [x,y] 1. 0 i f and o n l y i f Ilx+Xyll 2 IIxII f o r a l l A > 0 ( 4 ) [x,y+z] 5 [x,y] + [x,zl

(5) [x.ax+yI

= a l x l + [x,yl

[O,y] = llyll ( 8 ) [x,y] - [x,z] t i v e i f and o n l y i f [x-;,y-;]

RrmARK

(6) -[x,-yI

5 lly-zll.

5

[x,YI (7) ~ [ x I IIyll ,Y and I ~ + eE i s a c c r e -

It f o l l o w s t h a t A: E

2 0 f o r every (x,y),

(i,;) i n

G(A).

For a ( r e a l ) H i l b e r t space E = H one has [ x , y l A = ( l / A ) [ l l x + 2 nxn ] / ( i i X + ~ y n + tixi) and [X,y] = ( ~ , ~ ) / I I ~ I I For . E = LP(A), 1 < p 1i m m y one can e v a l u a t e i n a s t r a i g h t f o r w a r d way ( l / A ) [ ( I A lu+hvlpds)l'p (JA l u I p d s ) l / P 1 - [u,v] t o o b t a i n [u,v] = I I u l l " ~ I A v ( s ) ) u I p - ' s g n u d s ( e x e r iyn2

6.3,

-

-

The s e m i - i n n e r p r o d u c t a s s o c i a t e d w i t h

cise).

[XJ]

i s ( y , x ) + = /lx/l[x,y]

(cf. [Dmll). Now a map J: E {x'

J(x)

E

3

E';

2

t'

i s c a l l e d a d u a l i t y map ( r e l a t i v e t o a gauge 9 ) when

(x',x)

= IIx'IIIIxII;

IIx'II = 9 ( l I x l l ) l .

Here 9 s h o u l d be a con-

t i n u o u s and i n c r e a s i n g f u n c t i o n w i t h 9 ( 0 ) = 0 and 9 ( r ) [Brl;Pcl;Dml;Bdl]). F(X)

= {XI E

(x',x)

The most common

E'; ( x ' , x )

9

+

m

as r *

m

(cf.

i n v o l v e s 9 ( r ) = r and we w r i t e t h e n

2

= 11x11~ = 11x811I .

I n [pz21 one uses J(X) = t x ' E E ' ;

= IIxII when IIx'II = 1 1 b u t t h i s i s n o t perhaps b e s t c a l l e d a d u a l i t y

map; i t i s more a p p r o p r i a t e t o n o t e s i m p l y (as i n [ P z ~ ] ) t h a t F ( x ) = IIxII J ( x ) where F

2,

9 ( r ) = r i s a d u a l i t y map.

Then i f x ' E J ( x ) s e t y ' = IIxIIx'

so ( y ' , x ) = IIxII 2 and Ily'II = IIxII. Now when E ' i s s t r i c t l y convex, one sees immediately t h a t F and hence J i s s i n g l e valued (see Remark 6.16 f o r p r o o f a space E ' i s s t r i c t l y convex i f t h e u n i t sphere does n o t c o n t a i n l i n e segments

-

i . e . IIxII = llyll = 1 and x = y i m p l i e s llAx+(l-A)yll < 1 f o r A

E

(0,l))

and we r e c a l l a r e s u l t o f Asplund which says t h a t any r e f l e x i v e B space E

-

274

ROBERT CARROLL

can be p r o v i d e d w i t h an e q u i v a l e n t norm under which E i s s t r i c t l y convex w h i l e E ' i s a l s o s t r i c t l y convex under t h e new dual norm.

EHE0REN 6.4.

[x,y]

Let x '

Ptuud:

< IIxtAyll

-

= max ( x ' , y )

for x'

E

J(x).

J ( x ) and A > 0 so ( x ' , x + A y ) = IIxIl

E

IIxII whence ( x ' , y ) ( [ x , y ] .

E J ( x ) such t h a t ( x ' , y )

= [x,y].

Ife.g. E

+

= allxlI

115'11 5 1 then 5 '

E

E ' w i t h t h e same norm.

B[x,y]

Define a l i n e a r functional 5 '

( t h u s ( 5 ' , x ) = IIxII and(C',y)

= [x,~]).

V ' and by Hahn-Banach w i l l have an e x t e n s i o n x ' To see 115'11 5 1 n o t e f i r s t [x,By]

B by P r o p o s i t i o n 6.2 ( e x e r c i s e ) .

E'

2 B[x,y]

Hence (E',crx+By) 5 allxII + [x,By]

By] 5 IIax+Byll ( a g a i n u s i n g Prop. 6.2). By11 o r II€,'II 5 1 and x ' E

Thus A ( x ' , y )

A(x',y).

F o r t h i s l e t x,y be l i n e a r l y independent

and V C E t h e subspace generated by I x , y l . on V by (c',ax+By)

+

One needs t h e r e f o r e o n l y t o f i n d x '

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

f o r any = [x,ax+

5 IIax+

(C,',ax+By)

e x t e n d i n g 5 ' w i t h IIx'II 5 1 e x i s t s s a t i s f y i n g

9 ED

( x ' , x ) = IlxII and ( x ' , y ) = [x,y].

REMRK 6.5, From Theorem 6.4 i t f o l l o w s immediately t h a t [x,y] '> 0 if and o n l y if t h e r e e x i s t s x ' E J ( x ) w i t h ( x ' , x ) 2 0, and moreover t h e f o l l o w i n g ( 1 ) A i s a c c r e t i v e ( 2 ) (l+AA)-'

c o n d i t i o n s a r e now seen t o be e q u i v a l e n t . i s nonexpansive f o r A >

(x*,y^)

For every (x,y),

o

o

(3) [x-;,y-jr]

f o r (x,y),

i n G(A) t h e r e e x i s t s x '

(;,;I

i n G(A) ( 4 )

J ( x - i ) such t h a t

E

(

x',y-;)

A c t u a l l y ( 4 ) w i l l h o l d f o r a l l x ' E J ( x - $ ) as w i l l be seen i n t h e

> 0.

course o f subsequent developments.

-

cise

One a l s o shows i n a r o u t i n e way ( e x e r -

c f . [ P z ~ ] ) t h a t i f A i s a c c r e t i v e then

fi

i s accretive

an a c c r e t i v e A i s c l o s e d i f and o n l y i f R(1tA) i s closed.

(A

G(A)) and

%

We remark t h a t

c a l c u l a t i o n s w i t h m u l t i v a l u e d o p e r a t o r s a r e modeled on standard o p e r a t o r c a l c u l a t i o n s ; one s i m p l y works w i t h p a i r s (x,y) E G ( A ) ( i . e . y

E

A ( x ) ) . NOW

d e f i n e A t o be s - a c c r e t i v e i f ( x ' , y - y ) 5 0 f o r a l l x ' E J ( x - ? ) and a l l ( x , y ) and

(x*,;)

tive.

E v i d e n t l y when J i s s i n g l e valued s - a c c r e t i v e

E G(A).

f

accre-

A number o f s i m p l e r e s u l t s i n v o l v i n g s - a c c r e t i v e o p e r a t o r s f o l l o w .

EHE0REN 6.6.

( 1 ) I f A i s a c c r e t i v e and

t i v e . ( 2 ) I f S: E

-f

E is

B

i s s-accretive then A t B i s accre-

nonexpansive then 1-S i s s - a c c r e t i v e ( 3 ) I f S ( v )

i s nonexpansive d e f i n e D(A) = I x E then A i s s - a c c r e t i v e ( 4 ) IfA: E

-f

E; E

l i m [x-S(p)x]/p

= Ax e x i s t s ( P

t

* 0 )I;

i s a c c r e t i v e and continuous ( c o n t i n u -

ous i m p l i e s s i n g l e v a l u e d ) then A i s s - a c c r e t i v e .

Pkood:

We sketch t h e p r o o f o f a few p o i n t s and l e a v e t h e r e s t as an e x e r -

cise (cf. [ P z ~ ] ) .

~XII - ( x ' , s x - s i )

For ( 2 ) l e t S be nonexpansive and x ' E J(x-;);

2 o since ( x',sx-sx') < I I S ~ - S ~ ^ I
IIX-~I.

But

(

then Ilxx',x-x^)

=

NONLINEAR SEMIGROUPS

-

Ilx-ill so ( x ' , x - ? )

(x',Sx

-

275

S?) 2 0 f o r a l l x ' E J(x-(0

and t h i s i m p l i e s

For ( 1 ) t a k e ( x y ), (x^ ,y^ ) E G(A) and (x1,y2), (ily 1' 1 ,1 ,1 By d e f i n i t i o n (x,y +y ) and (X^,'y1+y2) E G(AtB). There e x i s t s

1-S i s s - a c c r e t i v e .

$2) E G(B).

'A

2 0 and t h e n ( x',Y~-;~)

x ' E J(x,-Xnl)

such t h a t

s-accretive.

Hence t h e r e e x i s t s x ' E J(xl-;,)

.y^2)) 2

(

x',yl-yl)

0.

such t h a t

2 0 since B i s (

x',y1+y2

-

(ilt

QED

Consider now t h e i n i t i a l v a l u e problem (IVP) (*) and d i s c r e t i z e [O,T]

... <

0 = to < tl <

A s o l u t i o n o f t h i s w i l l be any p i e c e w i s e c o n s t a n t

f u n c t i o n v ( t ) on [O,T]

+ Avl 3 0 o r v1 + hoAvl

t h e n (vl-vo)/ho

d i s c r e t i z a t i o n (i.e.

u I 0

<

= ( l + h o A ) - l v 0 and

E

C([O,T],E)

<

E

w i t h T-tN <

E )

where IIvo

-

i s called a mild solution o f the IVP i f

> 0 t h e I V P has an €-approximate s o l u t i o n vE w i t h l l u ( t ) - v " ( t ) l l

E

f o r t E [O,tN,].

CHEBREIII 6.7. t

Set v,

T h i s process can t h e n be i t -

3 vo.

maxLlit = max(ti-ti-l)

A function u

E.

f o r any

Indeed l e t hi

One c a l l s €-approximate s o l u t i o n o f u'+Au 3 0 a s o l u t i o n vE o f an

erated.

E

s a t i s f y i n g v ( t ) = vi on (timl,ti] and v ( 0 ) = vo where

If A i s accretive there exists a solution,

and r e c a l l ( l + h A ) - l i s w e l l d e f i n e d .

= ti-ti-l

E

Consider ( 6 ) [ ( ~ ~ - v ~ - ~ ) / ( t ~ + - tAvi ~ -3 ~ )0 ]

tN< T.

w i t h i n i t i a l v a l u e vo. vi s a t i s f i e s ( 6 ) .

with

L e t A be a c c r e t i v e i n E and uo

E

fi).I f

f o r every

E

>

0 u'

Au 3 0 , u ( 0 ) = uo, has a €-approximate s o l u t i o n vE then t h e r e i s a unique

m i l d s o l u t i o n t o which t h e vE converge. This i s proved i n [PzZ] v i a some t e c h n i c a l lemmas which we w i l l s t a t e as The main such lemma (Lemma 6.8) can be proved by r e -

needed b u t n o t prove.

c u r s i o n i n a more o r l e s s s t r a i g h t f o r w a r d manner. 4

LEIRIRA 6.8. s e t ti =

i l1 hk,

A

0; hi > 0; h j > 0; L,M 2 0; 1 5 i

L e t ;ij

'

tj =

1;

A

hk,Ato = 0 =

^to.

Suppose aio

N; 1 5 M + Lti,

j

5 N; and

< M + a oj M + Then aij

+ [hi/(hi+:j)]Eiyj-l. L t j y and (+) aij :A,[h^j((hi+hj)]ai-l,j [(ti-tj)2+ h t i + ht.]'L where h = maxhi and ^h = maxh J j* L e t v and v^ be s o l u t i o n s o f d i s c r e t i z a t i o n problems ( 6 ) f o r ( * ) L€l!W 6.9. w i t h 1 5 i 5 N and 1 5 j 5 one has llv(t)-;(s)ll 3h "2

+ h t + *hs]%yll

N^.

Then f o r any a p p r o p r i a t e t , s and (x,y) E G(A) 2 + 3h 2 + 2 Ilv(O)-xll + llvh(O)-xII + [ ( t - s ) + Z(h+;)lt-sl

(ti-ti-l = hi,

4

4

t.-t. = J J-1 One r e f e r s t h i s t o Lemma 6.8 v i a aij

Phood: w i l l hold.

ij, h =

maxhi, A

^h

= maxi.).

A

A

4

= llv(ti)-v(t.)ll '

J since then (+)

To see t h i s n o t e t h a t i f (v-w)/S + Av 3 0 and (v-w)/Gh + A$ 3 0

l l ( 6 / ( 6 + ~ ) ) l l ~ - w l l . Indeed 0 5 [ V - ; , ( V - W ) / ~ - ( ~ t h e n IIv-$l 5 ( ~ / ( 6 + ~ ) ) l l w - ~ +

276

ROBERT CARROLL

-

[II &V+Wll llv-;ll]/$

+

II v-;ll]/s

A

A

llv-;ll]/:

from which t h e r e s u l t f o l l o w s .

a r e e a s i l y v e r i f i e d w i t h e.g.

+

-

A

[IIv-v-v+wll

-

= [Ilw-vll

-

Ilv-^vlI]/s+[ll~-~ll

The o t h e r hypotheses o f Lemna 6 . 8

A c r u c i a l s t e p h e r e i s t o n o t e t h a t by a c c r e t [v(ti-l)-v(ti)]/hi E A v ( t i ) one has Ilv(ti)-xll 5 Rv(ti)-x+h + h.llyll. It i s hi - y l l l and hence Ilv(ti)-xll 5 llv(ti-l)-xll 1 b i t r a r y (x,y) E G ( A ) i s i n t r o d u c e d and t h i s l e n d s f l e x i b i l ll$(O)-xll.

ates.

M

= ~ ~ v ( t i ) - ~ ( 0 )L~ =~ ,llyu, and

aio

= Ilv(O)-xll

vness and because [(v(ti-l)-v(ti))/ h e r e t h a t an a r ty t o the estim-

QED

The r e s t i s s t r a i g h t f o r w a r d .

REmARK 6.10, We s k e t c h now t h e p r o o f o f Theorem 6.7. L e t vE and vn w i t h 0 < n 5 E be E and n approximate s o l u t i o n s w i t h i n i t i a l v a l u e uo so by Lemma 6.9 f o r h,nh < E , (x,y) E G(A), ( m ) l l v E ( t ) - v n ( s ) l l 5 lIvE(O)-xll + ~ ~ v ' ( O ) - X ~ ~ 2 2 + [ ( t - s ) + ~ E +T 6~ + 2 ~ T ] % y l l 5 IIvE(0)-xll + IIvn(0)-xll + ~ l ; - s ~ l l y l +l [ 6 ~ (T+~)]'I1yll ( t h e l a s t i n e q u a l i t y i s an easy e x e r c i s e and we n o t e i n p a s s i n g

vn

that

i s a l s o an

t h a t IIvE(0)-xoll 5 < -

E

+

0

(d6 < 3).

E

and ~ l v ' ( 0 ) - x o ~ 5 ~

E

One knows by assumption now

(xo=uo i n ( * ) ) from which IlvE(0)-xll

+

Ilx-x 11. Hence ( m ) becomes Ilv'(t)-v'(s)ll 5 P I t - s I l l y l l + 3 [ ~ ( T + ~ ) ] ' l l y l l f o r xo E D(A) say and ( x , y ) E G ( A )

+

E

5 2~ + 211x-xoII + 3 [ ~ ( T + ~ ) ] ' l l y l l and one i n f [ 2 r + 211x-xoll + 3[r(T+r)]'llyll] ( i n f o v e r (x,y) E G

For t = s lIv'(t)-v'(t)ll

d e f i n e s (**) $ ( r ) = (A)).

approximate s o l u t i o n ) .

Ilvn(0)-xll 5

IIx-x II;

2~ + 211x-xoII

E

Now $ ( r )

211x-xoll 5 4 4 .

-,0

as r

0 s i n c e f o r E f i x e d we can choose x E D ( A ) w i t h Take t h e n any y E Ax and f i x llyll so $(r)depends o n l y on r

and can be made <

E

-+

say.

Hence l l v E ( t ) - v ' ( t ) l l

5 $(E), 0

< 0

5

and v E ( t )

E,

s o some u ( t ) .

i s t h e r e f o r e a Cauchy n e t which converges ( u n i f o r m l y on [O,T)

+ IlvE(t)

To see t h a t u i s continuous we c o n s i d e r l l u ( t ) - u ( s ) l l 5 l l v " ( t ) - u ( t ) l l -vn(s)ll

+

llv'(s)-u(s)S.

o b t a i n s (*A) l l u ( t ) - u ( s ) l l 5 211x-xoll p ( r ) = i n f [rllyll + 211x-xoll] have p ( r )

+

0 one If one s e t s

Using t h e above i n e q u a l i t i e s and l e t t i n g

+

It-slllyll f o r (x,y)

G(A).

( i n f o v e r (x,y) E G(A)) then f o r xo

0 when r * 0 ( a s b e f o r e f o r p ( r ) ) .

E

-+

D(A)

we

Hence f o r I t - s l < r I l u ( t ) -

u(s)ll < P r ) and u i s u n i f o r m l y continuous on [O,T),

we have a m i l d s o l u t i o n .

E

E

We n o t e a l s o t h a t i f vE

-+

so e x t e n d i n g t o [O,T]

u and v'

+

determine

2 m i l d so u t i o n s based on p o s s i b l y d i f f e r e n t i n i t i a l values t h e n passing t o limits i n

( m

) we g e t I l u ( t ) - $ ( t ) I I 5 IIU(O)-XI~+ ~ ~ f i ( o ) - x lfl o r x

c o n t i n u i t y o f t h e norm t h i s holds a l s o f o r x E one o b t a i n s ( * a ) l l u ( t ) - j ( t ) l

REmARK (*A),

6-11.

D(A)

E

D(A).

BY

and choosing x = c(0)

5 ~ ~ u ( O ) - ~ ( Owhich ) ~ ~ , proves uniqueness.

QED

The c o n c l u s i o n o f Theorem 6.7 s h o u l d be supplemented w i t h ( m ) ,

and ( * a ) o f Remark 6.10.

m - a c c r e t i v e if R ( l + A ) = E.

One says now t h a t an a c c r e t i v e A: E

We w i l l show t h a t i n t h i s event R ( l + l A )

-+

E

is

= E for

277

NONLINEAR SEMIGROUPS

a l l A > 0 and A i s t h e n sometimes c a l l e d h y p e r a c c r e t i v e . t h a t a monotone map A: E

+

We remark a l s o

E ' (assume E' i s s t r i c t l y convex f o r s i m p l i c i t y )

i s c a l l e d hypermaximal monotone i f A+AF i s o n t o f o r a l l o f d u a l i t y d e f i n e d a f t e r Remark 6.2

-

x

0 ( F i s t h e map

>

c f . h e r e Remark 6.16,

D e f i n i t i o n 2.15,

and 53 f o r f u r t h e r i n f o r m a t i o n on maximal m o n o t o n i c i t y and c f . [Dml;Brl]

-

comparisons

i n f a c t f o r r e f l e x i v e spaces A: D c E

monotone i f and o n l y i f i t i s D maximal monotone).

+

To see t h a t R(l+xA) =

when R ( l + A ) = E'one checks f i r s t t h a t ( * 6 ) ( J x ~ , ( ~ - J x ~E) G /~ ( A) f o r and any x.

= (ytAn-y)/A =

rl E

show

E for

=

E(l+pA)

given

z E E l e t Tx

Ilx-y#.

= y so x = y + x ~w i t h

Set Jxx = ( l r X A ) - ' x

n

x

>

E 0

Then ( l - J x x ) / x

E Ay.

x

R(l+xA) = E f o r some

Ay. Assume now e.g.

for

E ' i s hypermaximal

and we w i l l

x / 2 ( t h i s w i l l p r o v e t h e d e s i r e d c o n c l u s i o n ) . Thus (X/p)z + ( l - A / p ) J x x f o r x E E, so ITx-Tyll < l(~~-x)/p]

p >

=

For I ( ~ - A ) / L I I < 1 ( o r

1.1 >

h/2) t h e r e e x i s t s a f i x e d p o i n t o f T so

t h a t x = ( X / ~ . I ) Z + ( l - h / u ) J A x o r z = J A x + 1 . 1 ( x - J ~ x ) / x . Thus t = 5+1.1~w i t h n E

AS (by ( * 6 ) ) and t h i s means z

E

R(l+pA).

As a c o r o l l a r y o f a l l t h i s one

can say now t h a t i f A i s m - a c c r e t i v e t h e n t h e e q u a t i o n u'+Au 3 0 , u ( 0 ) = uo

E

D(A),

has a unique s o l u t i o n ( m i l d ) on

Indeed i t i s s u f f i c i e n t t o

[O,m).

show t h e r e e x i s t s e-approximate s o l u t i o n s and one l o o k s a t ( 6 ) i n t h e f o r m u(ti)

t hiA(u(ti)

3 u(ti-l).

Then one o n l y needs (l+hiA)u(ti)

3 u(ti-l) t o I f o n l y R(1-t

develop an e-approximate s o l u t i o n , w h i c h h o l d s by R(l+hA) = E.

AA)

3

D(A)

w i t h A a c c r e t i v e one can i n f a c t c o n s t r u c t €-approximate s o l u = (1+

t i o n s v i a u(ti)

RrmARK 6-12.

D(A)

when uo E

and produce a m i l d s o l u t i o n .

L e t u be t h e m i l d s o l u t i o n o f u'+Au 3 0 , u ( 0 ) = xo

s e t u ( t ) = S(t)x,

where i n f a c t S ( t ) :

D(A) + D(A)

For A m - a c c r e t i v e S ( t ) :

+

S(t)x

0

i s continuous

- S(t)yll 5 IIx-yll ( f r o m

D(F)+ D(A)

i s c a l l e d a (non-

l i n e a r ) semigroup o f continuous c o n t r a c t i o n s ( w i t h g e n e r a t o r -A n i t i o n B.33).

f o r uo

E

n ) t Aui 3 0, and ui

=

Ax.

see

( U ~ - U ~ - ~ ) / ( ~ /

[ ~ + ( t / n l A l - ~ u ~ - I~n. t h i s s p i r i t we can r e o r g a n i z e

t h e preceeding r e s u l t s i n a semigroup c o n t e x t . E

c f . Oefi-

fi)and we

t h a t t h i s i s r e a l l y an € - a p p r o x i m a t e s o l u t i o n w i t h ti = i t / n ,

E

-

One r e c a l l s a l s o a theorem o f C r a n d a l l - L i g g e t which produces

s o l u t i o n s o f t h e I V P v i a u ( t ) = l i m [l+(t/n)A]-"uo

if f o r xn

D(A) and

s i n c e u = l i m vE w i t h vE

S has t h e p r o p e r t i e s ( 1 ) S ( 0 ) = I ( 2 ) t ( t ) E D(A). ( 3 ) S ( t t s ) x o = S ( t ) S ( s ) x o (by uniqueness) ( 4 ) I I S ( t ) x

( = ) i n Remark 6.10).

E

D ( A ) and yn E Axn w i t h xn

+

x and yn

F i r s t one says A i s c l o s e d -+

y one has x E D(A) and y

Then one shows ( e x e r c i s e ) t h a t an m - a c c r e t i v e A i s c l o s e d .

Further

f o r A m - a c c r e t i v e t h e preceeding theorems i m p l y ( c f . [ P z ~ ] ) ( 1 ) A generates k a semigroup o f c o n t r a c t i o n s on D(A) ( 2 ) l i m J x = S ( t ) x f o r x E 0%) where

278

ROBERT CARROLL

-

A -+ 0 and h k -+ t ( 3 ) IJ:,nx

S(t)xll 5 211x-Cll + ( t / J n ) l l d

for

(S,n)

E G(A).

One says t h a t u i s a s t r o n g s o l u t i o n o f t h e I V P (*+) u'+Au 3

REClARK 6-13.

0, u ( 0 ) = x, if u i s L i p s c h i t z continuous and d i f f e r e n t i a b l e , u ( t ) E D(A) AE, and u s a t i s f i e s (*+) ( L i p s c h i t z means l l u ( t ) - u ( s ) l l 5 c l t - s l ) . One n o t e s It i s a l s o

f i r s t t h a t for E r e f l e x i v e Lipschitz implies d i f f e r e n t i a b i l i t y . worth p o i n t i n g o u t t h a t i f u ' ( t ) e x i s t s so does Dtllu(t)ll = [u,u'].

-

l i m [Ilu(t+h)ll

llu(t)ll]/h = lim[llu(t)+hu'(t)ll

-

I l u ( t ) l l ] / h = [u,ul].

Indeed

NOW ev-

i d e n t l y s t r o n g s o l u t i o n s o f t h e I V P a r e unique s i n c e g i v e n 2 such s o l u t i o n s a t a p o i n t o f d i f f e r e n t i a b i l i t y we have D t l l u ( t ) - v ( t ) l l - u l E Au and - v l

E

Av so [ u - v , - u I + v ' ]

= -[u-v,-u'+v'].

2 0 and Dtllu-vll 5 0.

But

Now IIu-vII i s

a l s o continuous s i n c e i t i s L i p s c h i t z and hence l l u ( t ) - v ( t ) l l 5 l ~ u ( O ) - v ( O ) l ~ .

Of course one can a l s o i n v o k e t h e uniqueness theorem f o r m i l d s o l u t i o n s once one proves s t r o n g i m p l i e s m i l d (which we l e a v e as an e x e r c i s e ) . We w i l l i n d i c a t e a few examples, r e f e r r i n g t o [PzZ] f o r ad-

EXANPCE 6.14.

Consider f i r s t Au = Dxv(u)

d i t i o n a l d e t a i l s ( c f . a l s o [Bxl;Gml;Au3;Dml]). where

v

i s a continuous s t r i c t l y i n c r e a s i n g f u n c t i o n R

= R and t r y t o s o l v e ( * a )

+

R , v ( 0 ) = 0, v ( R )

ut+Au = 0 ( t 2 0 , 0 < x < 1 ) ; u ( 0 , t ) = 0 ( t 2 0);

u(x,O) = u o ( x ) . Here t a k e e.g. D(A) = I u E Co, u ( 0 ) = 0, v ( u ) a b s o l u t e l y 1 c o n t i n u o u s } . One can show t h a t A i s m - a c c r e t i v e i n L ( 0 , l ) = E ( c f . [ P z ~ ] ) Now one says u i s a weak s o l u 1 t i o n o f ( * a ) , i n t h e form ut + Dxv(u) = 0, i f ti E C((O,T), L ( 0 , l ) ) and (A*) :1 1 ; [unt + v(u)n,]dxdt + 101 u(x,O)n(x,O)dx 0 f o r a l l rl E C m w i t h

and hence

has a unique m i l d s o l u t i o n .

(*a)

compact s u p p o r t i n [0,1)

It can be checked t h a t a m i l d s o l u t i o n i s

X [O,T).

a weak s o l u t i o n and t h i s r e l a t e s t h e i d e a o f m i l d s o l u t i o n t o " c l d s s i c a l " ideas i n POE.

Another example from [ P z ~ ] i n v o l v e s

( 0 < x < 1, t > 0 ) ; u(x,O) = u o ( x ) ; u ( 0 , t )

=

(fi)

ut

u ( 1 , t ) = 0.

-

[u'(u)lXx = 0

Suppose again u'

i s a s t r i c t l y i n c r e a s i n g f u n c t i o n , ~ ( 0 =) 0, and v(R) = R.

Take E = L1(O,l)

and D ( A ) = t u

(

E; v ( u ) and vp(u),

E

r

(A&)

= a A ) (-)

S t i l l another example i s ( A C Rny

has a unique m i l d s o l u t i o n .

- au

ut

u E ) E; ~u ( 0~ ) =

Again one shows A i s m - a c c r e t i v e ( c f . [ P z ~ ] )

u ( 1 ) = 01 w i t h Au = -v(u),,. and thus

a b s o l u t e l y continuous; ~

+ B ( U ) 3 0; u ( x , t ) =

o

(X E

r,

t > 0 ) ; U ( X , O ) = u,(x)

ii(A),

where B i s a maximal monotone graph w i t h ~ ( 0 =) 0. Set D ( A ) = I u E 1 1 -Au + R ( u ) E L ( A ) } C L ( A ) and one can show t h a t A i s m - a c c r e t i v e ( [ P z ~ ] ) . We make f i n a l l y a few remarks about t h e problem (A&) u ' + Au 1 By d i s c r e t i z a t i o n one o b t a i n s 3 f, u ( 0 ) = uo E D(A) f o r f E L (0,T;E).

RmARK 6.15. (v.-v

i-1 requires 1

)/(ti-ti-l)

1;

ti L-

+ Avi = fiy v ( 0 ) = v o y and f o r an E - d i s c r e t i z a t i o n one

Ilf(s)

I

-

fillds

< E.

One deals w i t h €-approximate s o l u t i o n s

2 79

NONLINEAR SEMIGROUPS

and m i l d s o l u t i o n s as before and f o r A m - a c c r e t i v e t h e r e e x i s t s a unique mild solution (cf. [Pz~]).

R m R K 6-16, L e t us mention e x p l i c i t l y t h a t t h e monotone o p e r a t o r s d e a l t w i t h i n §§3.2,3.3 were s i n g l e valued b u t t h e r e i s a p a r a l l e l t h e o r y f o r mult i v a l u e d monotone maps ( c f . o p e r a t o r s A: D ( A ) C E E

Ay.

-+

[Brl;Dml;Bdl

,Z;Dpl;Pcl;Tal;Zel]).

2 E ' a r e determined by

(

Thus monotone

2 0 for x'

x-y,x'-y')

E

A monotone A i s maximal monotone i f G ( A ) i s a maximal monotone s e t i t i s n o t t h e p r o p e r r e s t r i c t i o n o f a monotone s e t i n E X E l ) .

(i.e.

Ax, y '

n o t e t h a t i n D e f i n i t i o n 3.2.15

L e t us

i f one says f o r A b e i n g D maximal monotone

t h a t G ( A ) s h a l l n o t be t h e p r o p e r r e s t r i c t i o n o f a monotone s e t w i t h domain

D t h e n ( t a k e V r e a l ) ( w ~ - A u , ~ ~ - u2) 0 f o r (u,Au) i m p l i e s wo

E

wo = Auo.

G(A),

E

(uo,wo) E D X V '

I f A i s r e q u i r e d then t o be s i n g l e valued we would have

Au,.

It w i l l be i n s t r u c t i v e h e r e t o p r o v e a few f a c t s f o r m u l t i v a l u e d

A i n o r d e r t o enhance t h e r e s u l t s o f §3.3.

Thus ( f o l l o w i n g rBdl;Dml;Tal])

l e t us f i r s t s k e t c h a p r o o f o f some f a c t s ( c f . Remark 6.11) r e l a t i v e t o t h e

Z E ' say b e i n g maximal monotone and R(A+xF) = E ' ( r e c a l l F = UxUJ and E i s t o be r e f l e x i v e w i t h E and E ' s t r i c t l y convex). Thus f i r s t assume R(A+xF) = E' f o r some x and suppose A i s n o t maximal monotone so t h a t t h e r e e x i s t s ( x ,y ) E E X E ' n o t i n G(A) w i t h (x-xo,y-yo) 0 f o r a l l (x,y) ideas o f A: E

.+

0

E

G(A).

0

B u t t h e r e e x i s t s (xl,yl)

5 0.

(xl-xo,F(xl)-F(xo))

E

G(A) such t h a t XFxl+yl

(A+)

0

0

so (A+)

B u t i n f a c t F i s monotone s i n c e f o r x ' E F ( x ) and

y ' E F ( y ) one has ( x ' - y ' , x - y ) = ( x ' , x ) t IIy'IIIIyll - Ilx'IIllyll - Ily'IIIIxll = (IIX'II It f o l l o w s from

= AFx +y

+ (y',y) -

-

IIy'Il)(llxll

t h a t x1 = xo so ( x ,y ) = ( x 0

0

-

(x',y)

-

( y ' , x ) L IIxlIIIIxII

-

llyll) = (Ilxll

llyll)2 > 0.

y ) E G(A) which c o n t r a -

1' 1

d i c t s ( r e c a l l a l s o F i s s i n g l e valued i n t h e p r e s e n t s i t u a t i o n ) .

On t h e

o t h e r hand t o show A maximal monotone i m p l i e s E ' = R(A+XF) l e t us check f i r s t t h a t F i s c o e r c i v e and e.g. hemicontinuous (hence demicontinuous by Theorems 3.2 and 3.3 s i n c e F i s o b v i o u s l y bounded v i a IIF(x)ll = IIxll). Thus 2 ( c f . [ T a l l ) i f f,g E F ( u ) t h e n ( f , u ) = ( g , u ) = IIul12 = IIfl12 = IIgll w h i l e f o r

0

<

X < 1

(

(l-h)f+Ag,u)

= IIul12 and l l ( 1 - X ) f + ~ g I < IIuII s i n c e Ilfll

But by m o n o t o n i c i t y o f F, l l ( 1 - ~ ) f + h g l l = sup

(

(l-h)f+xg,u)/llull

= IIgll = IIuII. >

IIuII so ll(1-

h)f+xgll = IIuII = Ilfll = Ilgll and t h i s i m p l i e s f = g when E ' i s s t r i c t l y convex. T h i s proves F i s s i n g l e v a l u e d f o r E ' s t r i c t l y convex.

x

X)u+xv] and as verges w h i l e IIf,II

.+

A.

Ilf,

1I 0

Il(1-h )u+x vII. 0

= F[(1-

t h e r i g h t s i d e o f ( f A , ( l - h ) u + X v ) = ll(l-x)u+Xvl12 con-

= l l ( l - h ) u + ~ v l l i s bounded.

n e t f s .+ f weakly and consequently ( f , ( l - A IIfn

Next p u t f,

0

By t h e A l a o g l u theorem a sub-

0 )u+Xov) =

I l ( l - ~ o ) u + ~ o v l 1so 2 that

B u t by d e f i n i t i o n s IIfll 5 IIf

II so IIfll = IIf 1I 10 xo

280

ROBERT CARROLL

and hence f E F[(1-Xo)utxov] y i e l d s f,

-+

must equal f,

s i n c e IIF(x)H = IIxl,F

0

-

f weakly ( e x e r c i s e

.

A l i t t l e f u r t h e r argument

c f . arguments i n S3.3).

i s bounded and coercive.

We n o t e a l s o t h a t

Now one can p r o v e a r e s u l t

s a y i n g t h a t ifA i s maximal monotone and B i s e.g. monotone, s i n g l e valued, hemicontinuous, bounded, and c o e r c i v e t h e n t h e r e e x i s t s xo such t h a t (Am) (

u - x o y B x o t v ) 1. 0 f o r a l l (u,v)

trary y

0

E

G(A); a p p l i e d t o Bx = AF(x)

0

etc. f o r r e l a t e d situations).

§3.3, e.g.

The r e s u l t

ment below and we f i r s t n o t e t h a t F: E +

(Fx,x-y),

x weakly and l i m sup for all y

E

(

+

(Am)

+

+

3.3.6,

w i l l f o l l o w f r o m some argu-

5 0 i m p l i e s l i m i n f ( Fxa,xa-y) 2

Fxa,xa-x)

E ( E = D(F) here).

x weakly ( o r xu

Theorems 3.3,3,

E ' i s pseudomonotone which means This n o t i o n w i l l a r i s e again i n To see t h a t F i s pseudomono-

t h e s t u d y o f v a r i a t i o n a l i n e q u a l i t i e s i n §3.7. t o n e l e t xn

yo f o r a r b i -

E Axo o r yo E R ( A t x F ) which w i l l prove t h a t A max-

we g e t y -,F(x0)

i m 1 monotone i m p l i e s R(A+xF) = E ' (cf.

t h a t xu

-

x as a n e t

- e i t h e r argument i s t h e same);

2 ( Fx,xn-x) so l i m i n f ( Fxn,xn-x) 2 0 and hence 2 0. For y E E = D(F) a r b i t r a r y , a g a i n by m o n o t o n i c i t y (*A) l i m i n f ( Fxn,xn-y) 2 l i m ( Fy,xn-y) = ( Fy, x-y). L e t w = (1-A)x+Ay and p u t t h i s i n (.A) i n place o f y t o obtain l i m by m o n o t o n i c i t y lim

(a*)

inf

(

(

(

Fxn,xn-x)

Fxn,xn-x)

= 0 when l i m sup ( Fxn,xn-x)

(Fw,x-y)

f r o m which l e t t i n g A

i n f (Fx.,x.-y) 1

1

Hence u s i n g (@*) l i m i n f

L ( Fw,h(x-y)).

Fxi,xi-x+X(x-y))

+

0, l i m i n f

[( Fxiyxi-x)

= lim inf

t

(

(

(

Fxiyx-y)

2

2 ( Fx,x-y). Hence l i m 2 ( Fx,x-y) as r e q u i r e d t o

Fxi,x-y)

Fxiyx-y)]

show F pseudomonotone. Now t o prove

(Am)

one may f i r s t assume w i t h o u t loss o f g e n e r a l i t y t h a t 0

E

D ( A ) and we can assume B i s pseudomonotone, c o e r c i v e , bounded, and demicon tinuous.

F o l l o w i n g [Dml;Tal;Bdl]

and work w i t h u

E

8,(0)

f o r example assume f i r s t dimE <

= t u ; lull < nl.

my

E = R",

Indeed i f t h e r e s u l t i s t r u e on

1. 0 and B w i l l be c o e r c i v e and Since 0 E D(A), -(xn,Bxn) 2 0 and

Bn(0) one f i n d s xn such t h a t (u-xn,Bxn+v) continuous i n Rn ( c f . Theorem 3.3.5).

c o e r c i v i t y i n t h e form (Bu,u) ~ q ( I I u l l ) l l u l l w i t h q ( r )

+

m

as r

implies

+ m

l x n l 5 c ( n o t e q ( r ) may be n e g a t i v e f o r small r ) . Hence one e x t r a c t s a subsequence xk + x w i t h B ( x k ) + B(x) and (u-x,Bx+v) > 0 f o r any u. Thus t a k e n E K C R compact and convex; i f t h e theorem i s f a l s e one has K = UtUZIEAX

z

01.

There must be a f i n i t e subcovering so K = UKi L e t pi be a p a r w i t h Ki o f t h e form Ki = UzIEAz{ x E K; ( zi-x,z!+Bx) < 01. {x

E

K; ( z - x , z ' + B x )

<

t i t i o n o f u n i t y w i t h supp qi C Ki and i s a continuous map K (00)

(f(x)-x,C

+

1 qi

1

= 1 on K.

Then f ( x ) =

K which has a f i x e d p o i n t xo E K.

qi(X)(Z;+BX))

=

(1

VJ

1vi(x)zi

But f o r x E K

. ( X ) ( Z . - X ) , ~v ~ ( x ) ( z ; + B x ) ) = J J

1~ i 9 . i

NONLINEAR SEMIGROUPS

281

(z.-x,zt+Bx). Now if i = j and qi(x) # 0 t h e n x E Ki and (zi-x,z;tBx) < 0. J I f i # j and q.tp.(x) t 0 t h e n x E Ki n K. and tz.-x,z;+Bx) t (z.-x,z!+Bx) = 1 J J 1 J tz.-x,z!+Bx) + ( z . - x , z ! - z ! ) < 0. Hence ( f ( x ) - x , $ q i ( x ) z { t B x ) < 0 on K J J 1 1 J which i s a c o n t r a d i c t i o n a t x = xo. F i n a l l y l e t dim E = and c o n s i d e r t h e s e t A o f f i n i t e dimensional subspaces o r d e r e d by i n c l u s i o n (as i n 53.3). For F E A l e t iF: F

-+

E

be t h e i n j e c t i o n w i t h BF = I p B i F .

maximal m n o t o n e w i t h D(A) = E and t h i n k o f AF = I p A i F .

We r e c a l l A i s One f i n d s x F E F

such t h a t ( 0 6 ) ( UF-XFSVF~BFXF)2 0 f o r (uF,vF) E G(AF). Again by c o e r c i v i t y IIxFll 5 c and s i n c e E i s r e f l e x i v e t h e r e i s a subnet xa * x weakly w i t h Bxa y weakly ( r e c a l l B i s bounded). I t f o l l o w s t h a t f o r (u,v) E G(A) ( 0 + ) l i m sup (xa,Bx,) ( ( u - x , ~ ) + t u , y ) ( e x e r c i s e ) . Now t o show (om) l i m i n f < 0 assume i t n o t so; t h e n i t would f o l l o w f r o m ( o + ) t h a t ( u - x , (xa-xyBxa) v ) -+ ( u , y ) - ( x , y ) > 0 f o r (u,v) E G(A) which i m p l i e s (x,-y) E G(A) s i n c e +

A i s maximal.

Hence s e t t i n g u = x i n

had been assumed. inf

(

Hence

(0.)

Hence

(

one a r r i v e s a t (om) whose d e n i a l

i s v a l i d and hence by pseudomonotonicity l i m

Bxa,x - x ) 1. ( B x , x - u ) f o r any u.

xa-u).

(0+)

I n a d d i t i o n by ( 0 6 )

(

v,u-x,)

> ( Bx,

2 ( v , u - x ) - l i m i n f ( Bxa,xa-u) 2 0, which i s

v+Bx,u-x)

(A=).

This development g i v e s a f u r t h e r exposure t o mu1 t i v a l u e d a n a l y s i s and comp l e t e s t h e p r o o f o f t h e e q u i v a l e n c e o f A maximal monotone and R(A+hF) = E ' .

7, UARIACZ0NAL INEqllALICIE$.

We t u r n now t o v a r i a t i o n a l i n e q u a l i t i e s o f t h e

f o r m ( 6 ) i n 53.5 f o r example and w i l l g i v e examples and theorems f r o m [Bw2; Bgl ;Acl ;Bd2;Dvl; Kbl ;Gwl ; B r l ;Pcl ;L i 3 - 7 ; Zel]. t h e need f o r m u l t i v a l u e d maps as i n 53.6.

The examples w i l l a1 so i n d i c a t e We w i l l see t h a t t h e use o f v a r -

i a t i o n a l i n e q u a l i t i e s a l l o w s one t o c o n c e n t r a t e i n one i n e q u a l i t y a l l o f t h e i n t r i n s i c f e a t u r e s o f c e r t a i n p h y s i c a l phenomena such as g o v e r n i n g equations, boundary c o n d i t i o n s , jump c o n d i t i o n s , e t c .

We n o t e a l s o t h a t a l t h o u g h l i n -

e a r d i f f e r e n t i a l equations o f t e n a r i s e i n d e s c r i b i n g t h e phenomena, t h e p r problems a r e n o n l i n e a r because o f t h e c o n s t r a i n t s , e t c .

We w i l l n o t d w e l l

on r e g u l a r i t y questions, i n o r d e r t o emphasize s t r u c t u r e , and remark t h a t t h e s t u d y o f g e n e r a l i z e d s o l u t i o n s , r e l a x e d problems, e t c . i s i m p o r t a n t and necessary i n s t u d y i n g v a r i a t i o n a l problems o f a l l t y p e s ( c f . [Acl ;Aul ,2]). We have a l s o a b b r e v i a t e d t h i s s e c t i o n due t o space l i m i t a t i o n s ( c f . Remark 7.7).

L e t us t r y t o g a i n a l i t t l e p e r s p e c t i v e f i r s t by f o r m u l a t i n g some

problems connected t o frameworks developed a1 ready.

REmARK 7.1.

Consider t h e problem ( c f . [Bdl;Bw2;Dvl;Li4;Pcl])

w i t h A bounded and

-> 0;

r

= a h reasonably smooth (*) ut

and uun = 0 on E =

r

X [O,T]

- Au

for

= f(x,t);

w i t h u(x,O) = u o ( x ) i n A .

A c Rn

X E

u > 0; un

Here un

%

ex-

282

ROBERT CARROLL

t e r i o r normal d e r i v a t i v e and we t h i n k o f an L 2 c o n t e x t .

The c o n d i t i o n uun

r

i n t o 2 p a r t s rl and r 2 ( n o t known) on which u = 1 2 2 0 and u = 0 resp. Take now V = Lp(O,T; H ( A ) ) , H = L (0,T;L ( A ) ) , and V ' = "1 1 vu.vvdx i n H and Lq(O,T;H ( A ) ' ) ( p 2 2, l / p t l / q = 1 ) . D e f i n e a(u,v) = lA 1 The problem (*) f o r uo l e t K = { v E V; v 0 o n = ] be a convex cone i n H 2 E L , f E H, e t c . i s e q u i v a l e n t t o t h e problem: Given uo E L2 and f E H f i n d T T f i n d u E K such t h a t u ( 0 ) = uoy u ' E V ' , and (A) lo ( u ' - f , v - u ) d t + 10 a(u, 0 d i v i d e s t h e boundary

.

E v i d e n t l y any s o l u t i o n o f (*) s a t i s f i e s

v - u ) d t 2 0 f o r a l l v E K.

f o r t h e converse assume u i s a s o l u t i o n o f t o obtain f i r s t (u'-f,p)

t a(u,q)

(A)

and s e t v = u

= 0 so t h a t u s a t i s f i e s

(A)

and

f o r + # E C,"

k p

(*) i n a d i s t r i b u -

t i o n sense and o n l y t h e boundary c o n d i t i o n s need be checked. F i r s t f o r any T T w E K t a k e v = u+w i n (A) t o o b t a i n ( 0 ) lo ( u ' - f , w ) d t + lo a(u,w)dt 1. 0. T Taking w = -u one has lo ( u ' - f , u ) d t t ld a ( u , u ) d t & 0 from which i n f a c t , T T u s i n g ( 0 ) (6) lo ( u ' - f , u ) d t + lo a(u,u)dt = 0. Since (A) f o l l o w s from ( 0 ) and (6) t h e y a r e e q u i v a l e n t t o n o t e t h a t u E H1 i m p l i e s u E

(A).

i'on r

Now t o check t h e boundary c o n d i t i o n s we and un E H-'I

so uun i s w e l l d e f i n e d

By Green's f o r m u l a f r o m ( * ) ( n o t e E =

( c f . [Li2]).

T

d t component) (+) JE vundo = lo [ ( u ' - f , v ) one sees t h a t

(A)

r

X [O,T]

t a(u,v)]dt

-

and do has a

and u s i n g

( 0 )

i m p l i e s J7 uundo = 0 w i t h l7 vundo 2 0 f o r a l l v I

Hence uun = 0, u 1. 0 v i a u

E

and (6)

E

K.

-

K, and un 2 0 v i a l.;. vundo 2 0 f o r v E K where

v 5 0.

We n o t e t h a t i f one d e f i n e s t h e LSC convex f u n c t i o n IP by ~ ( v =) (1/ 2 From Re2 ) l A l v v l dx t h e n p i s G - d i f f e r e n t i a b l e w i t h ( D P ( u ) ¶ v ) = a(u,v). mark 5.4 b ( u ) = I P ' ( U ) = a p ( ~ )and one has ( p ' ( u ) , v - u ) c l p ( v ) T T (A) has t h e form ( m ) lo ( u ' - f , u - v ) d t 5 JO [IP(V) - v ( u ) l d t .

REmARK

7.2.

-~

( u so ) that

L e t us i n d i c a t e t h e connection o f some ideas h e r e f o l l o w i n g es-

p e c i a l l y [Li4].

One notes a l s o t h a t t h e problems and r e s u l t s o f 53.3 a r e i n

p a r t concerned w i t h v a r i a t i o n a l i n e q u a l i t i e s and we w i l l p r e s e n t some d i f f e r e n t p o i n t s o f view here.

I n p a r t i c u l a r pseudomonotone o p e r a t o r s a r e n a t -

u r a l i n c e r t a i n contexts i n v o l v i n g v a r i a t i o n a l i n e q u a l i t i e s . Thus l e t A: V

t y p i c a l " e l l i p t i c " problem. and f E V ' be given. f o r a l l v E K. vdx, Au =

-f

V',

Then f i n d u E K such t h a t (*)

One t h i n k s here e.g. o f a(u,v)

-1 D.(a..D.u) 1 1J J

+ aou, Au

xi) 2 0 on T , uau/avA = 0 on f i r s t K i s bounded; A: K

C

V

= f, u

r, etc. -,V ' i s

Consider a

K C V a c l o s e d convex s e t ,

(A(u),v-u) ~ ( f , v - u )

1

lAa..D.uDivdx 1J

2 0 on r , au/avA

J

=

t JA aou

1 a i jD.uCos(n, J

i n t h e s p i r i t o f Remark 7.1.

Suppose

bounded and pseudomonotone.

Then t h e r e

e x i s t s u E K s a t i s f y i n g (**). Note here t h a t i f K i s n o t bounded one w i l l assume A c o e r c i v e ( i . e . t h e r e e x i s t s vo such t h a t

(*A)

(A(v),v-vo)/IIvll

+

-

VARIATIONAL INEQUAL I T 1ES

as IIvII

283

i n which case t h e same r e s u l t w i l l h o l d ( c f . 53.3 f o r t h e r o l e

-+ m )

o f c o e r c i v i t y i n p r o d u c i n g boundedness o f approximate s o l u t i o n s

-

a1 so n o t e

t h a t some a u t h o r s r e q u i r e pseudomonotone o p e r a t o r s t o be bounded as p a r t o f We t a k e h e r e V s e p a r a b l e ( f o l l o w i n g [ L i 4 ] )

the definition).

h i b i t a c l a s s i c a l G a l e r k i n t y p e method. and c l o s e d o f dimension

5 my w i t h

UK,,,

K,,, C

Thus l e t

i n o r d e r t o ex-

K+,l

C

K, K,,

convex

dense i n K and one l o o k s f o r urn E ,K,,

such t h a t (**) (A(um),v-um) ) ( f , v - u

) f o r a l l v E .,K ,, To produce t h i s urn m l e t Vm be an m-dimensional H i l b e r t space c o n t a i n i n g I$, w i t h s c a l a r p r o d u c t

[

,1(

[~1g,w],

=

[

ng

For , I,,,).

Vmy

E

TI

E

g E V', w

E

.,K,,

i s continuous on Vm so ( g , w ) =

(g,w)

(check t h a t Vm

L(V',V,)

(**) becomes (*&) [nA(u,(,v-u,] urn] f o r a l l v

+

2 [vf,v-u

L e t Pm: Vm

-+

m

-+

V i s c o n t i n u o u s here).

] o r [um,v-um] 2 [u,+nf-nA(u,),v-

Km be t h e c a n o n i c a l p r o j e c t i o n and ( * 6 )

i s e q u i v a l e n t t o urn = Pm(um+ f - A(um)) ( e x e r c i s e can show v

-f

Pm(v+ f- A ( v ) ) :

Brouwer f i x e d p o i n t theorem. t h e c o n t i n u i t y o f A: ,K,,

-+

K,

-+

draw a p i c t u r e ) .

I f we

To show c o n t i n u i t y i t i s s u f f i c i e n t t o show

V ' (weak t o p o l o g y

-

we n o t e t h a t " a l l " t o p o l o g i e s

u i n K so A m i s bounded i n V ' and we can e x t r a c t a subsequence A(uk) y weakly. Thus l e t un

-+

-f

Then l i m sup ( A ( u k ) , u k - u ) (A(u),u-v)

5 0 and hence l i m i n f ( A ( u k ) , u k - v )

which i m p l i e s ( y - A ( u ) , u - v )

L 0 for all v

c o n t i n u i t y i s e s t a b l i s h e d , and urn e x i s t s . t h e u,

-

Km i s continuous t h e n urn e x i s t s by t h e

a r e e q u i v a l e n t on f i n i t e dimensional spaces). (u,)

Then

E

V.

Now s i n c e Km

=

2

(y,u-v)

Hence y = A(u), C

K w i t h K bounded

a r e bounded ( a l o n g w i t h A(um) i n V ' ) so one e x t r a c t s a subsequence

u -+ u i n V weakly w i t h u E K s i n c e K i s weakly closed. Since UKm i s dense j i n K f o r any such u E K t h e r e e x i s t s uo E UK, such t h a t IIu-uollv 5 6 . Then i ( f , u -u ) f o r j l a r g e enough by (*a) (uo E , , ,K f o r some M ) j o and s i n c e ( A ( u . ) u - u ) 5 C E one has l i m sup ( A ( u . ) u . - u ) = l i m sup [ ( A ( u . ) , J O J J J Hence l i m i n f ( A ( u . ) , u . - v ) U.-U ) + (A(U.),Uo-U)] < CE + ( f , u - u o ) < CIE. J O J J J > ( A ( u ) , u - v ) f o r a l l v E V. But i f v E UKm f o r j l a r g e enough ( A ( u . ) , u . - v ) J J < ( f , u - v ) f r o m (*a) as b e f o r e so l i m i n f ( A ( u . ) , u - v ) z ( f , u - v ) and hence j J j ( A ( u ) , u - v ) I ( f , u - v ) f o r a l l v E UK., Since UK, i s dense i n K we g e t (**). (A(uj),uj-uo)

REMARK 7.3,

An a l t e r n a t i v e way o f f o r m u l a t i n g and t r e a t i n g Remark 7.2 (*)

i s based on t h e epigraph idea ( c f . 53.5).

x K( v )

= 0 i f v E K.

Thus l e t x K ( v ) =

m

if v

$

One assumes A i s d e f i n e d on a l l V ( n o t j u s t on K

p r a c t i c e t h i s i s n o t u s u a l l y a problem) and then (*) i n g u E V such t h a t ( A ( u ) - f , v - u )

K and

-

in

i s equivalent t o find-

+ xK(v) - xK(u) 2 0 f o r a l l v

E

V (note a l -

so t h a t x K i s convex, proper, and LSC). More g e n e r a l l y as i n 13.3 one has a V ' and a p r o p e r convex LSC f u n c t i o n 9 and d e a l s

s u i t a b l e o p e r a t o r A: V

-f

284

ROBERT CARROLL

w i t h t h e problem (*+) F i n d u for all v

V.

E

E

+ v(v)

V such t h a t ( A ( u ) - f , v - u )

- v(u)

0

I f A i s bounded and pseudomonotone w i t h v as i n d i c a t e d w h i l e m and [(A(u),u-vo) + lp(u)]/IIuII + 00 when IIuII + V ' g i v e n t h e r e e x i s t s a s o l u t i o n o f (*+). T h i s i s proved i n

t h e r e e x i s t s vo w i t h lp(vo) < m

then f o r f

[Li4],

E

f o l l o w i n g MOSCO, v i a epigraph ideas (see e s p e c i a l l y [ A c l ] f o r e p i N

graph methods). = (A(v),O)

for

= V X R and K = e p i

The i d e a here i s t o s e t

7=

, u

( v , ~ )E V.

-+..,

'> 0 f o r a l l

such t h a t ( A ( u ) - f , v - u )

6

.u

(f,-l) E V ' (exercise

- note

with

i(T)

Then A i s pseudomonotone and (**) i s equiva-,.d

lent t o finding

v

ry

,u

E

where f =

N

= ( u , ~ )E K involves a '>v(u)).

This i s a l N

most e q u i v a l e n t t o t h e problem o f Remark 7.4 b u t n o t q u i t e s i n c e K i s n o t bounded and t h e r e i s n o t c o e r c i v i t y o f t h e form m a n i p u l a t i o n u s i n g t h e p a r t i c u l a r form o f

REmARK 7.4,

(*A).

7 suffices

A l i t t l e technical ( c f . [Li4]).

Recall now from 53.5 t h e i d e a o f s u b g r a d i e n t e t c . so t h a t f o r

a proper convex f u n c t i o n ~ ( u on ) V, alp(u) C V ' i s t h e s e t o f x such t h a t ) ~ ( v -) ~ ( u '>(x,v-u) i s now e q u i v a l e n t t o A(u)

-

f + av(u).

(i.e. (*m)

V

alp:

-t

2").

The problem (**) o f Remark 7.3

F i n d u E V such t h a t - ( A ( u )

- f ) E alp(u) o r

0 E

I n p a r t i c u l a r one sees how m u l t i v a l u e d o p e r a t o r s a r i s e

n a t u r a l l y i n t h e study o f v a r i a t i o n a l i n e q u a l i t i e s .

REmARK 7.5,

L e t us l o o k a t some c o n c r e t e examples from [Bw2;Bd2;Dvl;Kbl].

Thus f i r s t c o n s i d e r " o b s t a c l e " problems i n 2 dimensions f o r example. Thus 2 d e s c r i b e d say by y = $ ( x ) , continuous, and 0 5 x

g i v e n an o b s t a c l e A c R

< L, one wants t o connect x = 0 and x = L by an e l a s t i c s t r i n g passing be-

l o w A which cannot p e n e n t r a t e A.

Thus u ( x ) z $ ( x ) and a t y p i c a l example i s

Thus u ( x ) i $ ( x ) , u ( 0 ) = u ( L ) = 0, u" > 0, and u ( x ) < $ ( x ) i m p l i e s u " = 0. Note t h a t t h e c o n d i t i o n (u-$)u" = 0 d e s c r i b e s t h i s l a s t s i t u a t i o n . L e t K = 1 L 2 Iv,v(O) = v ( L ) = 0, v ~ $ i n1 say Ho w i t h energy E ( v ) = (1/2)J0 v ' dx and o f course one must r e q u i r e $ ( x )

0 a t 0 and L e t c .

We c o u l d ask now f o r u E

K such t h a t E(u) < E(v) f o r a l l v E K o r e q u i v a l e n t l y one asks f o r u E K Lsuch t h a t (A*) I. u ' ( x ) [ u ' ( x ) - v ' ( x ) ] d x 5 O f o r a l l v E K. The e q u i v a l e n c e

i s an easy e x e r c i s e f o l l o w i n g p r e v i o u s p r a c t i c e . A h i g h e r dimensional v e r 3 v i a A(u) = -Au t hu f o r exam1 p l e and a(u,v) = A vu-vvdx + f A Auvdx. L e t yo be t h e t r a c e map H ( A ) -t I?' s i o n o f t h i s problem can be phrased i n say R

( r ) and

take $

E

H'(A)

w i t h yo$ '> 0 and K = I v E HA, v i $ i n A } .

By prev-

i o u s r e s u l t s we know t h e r e e x i s t s a unique s o l u t i o n o f t h e problem (u) Find

VARIATIONAL INEQUALITIES

u

K such t h a t a(u,u-v)

E

5 JA f ( u - v ) d x f o r a l l v E K, f o r f

Some a n a l y s i s as above g i v e s t h e n (-Au+Au-f).(u-$) -Au

t

Au

= f when u <

REmARK 7.6.

285

E

L

2

given.

= 0 i n A which means t h a t

JI.

Another c l a s s i c a l problem i n v o l v e s t h e f l o w o f f l u i d s a c r o s s

porous media.

Thus one has a permeable dam w i t h w a t e r l e v e l a t t h e l e f t

h i g h e r t h a n a t t h e r i g h t and A i s t h e wet p o r t i o n o f t h e dam w i t h (unknown) boundary 'P as shown

(7.

Y'

r)\

y = 'Pb)

I X

I > One can f o r m u l a t e t h e problem now i n v a r i o u s ways ( c f . [BwZ;Kbl])

s i m p l y s t a t e one f o r m u l a t i o n .

and we

Thus l e t p be p r e s s u r e (atmospheric p r e s s u r e

= 0) so p 2 0 i s r e q u i r e d and s e t w(x,y)

(0,c)

D e f i n e D = (0,c) X Y so p = 0 on and i t i s c o n v e n i e n t t o r e s t r i c t a t t e n t i o n t o D* = 1 Let K = I v E H X (O,y*) where y* i s l a r g e enough so t h a t A C D*.

(D*),

v = g on aD*,

(0,m)

= Jm p ( x , t ) d t .

o/n

where g i s d e f i n e d below) and K, 2 2 Here g i s d e f i n e d on aD* by g(x,O) = [y,-y2](x-c)/2c t

= I v E K; v

2 + y2/2;

2 0 i n D*I.

g(0,y)

=

rY* Y

(yl-t) d t , g(c,y) = I Y * ( y 2 - t ) ' d t , and g(x,c) = 0. Then one l o o k s f o r w E Y Kt such t h a t lD*vw.v(v-w)dxdy 2 ID ( - l ) (* v - w ) d x d y f o r a l l v E K+. Given a

s o l u t i o n o f t h i s we have p = -D w and A = { ( x , y ) E D*; w > 01 ( c f . [ B w ~ ] Y f o r d e t a i l s ) . One s h o u l d n o t e t h a t v a r i a t i o n a l i n e q u a l i t i e s a r e an e f f i c a c i o u s way t o deal w i t h many f r e e boundary problems ( c f . [Li4;Bw2;Kbl]).

REmARK 7.7,

We had o r i g i n a l l y i n t e n d e d t o s k e t c h i n t h i s s e c t i o n some t e c h -

niques i n v o l v i n g p e n a l t y and r e g u l a r i z a t i o n methods from [ L i 4 ] f o r example as we1 1 as t h e existence-uniqueness p r o o f s f o r t h e Navier-Stokes theorems c i t e d i n 51.10 ( f o l l o w i n g [ L i 4 ]

-

c f . a l s o [Fol-6;Te1,3,4]).

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

Lack o f space

S i m i l a r l y we t h i n k i t would be appro-

p r i a t e t o t r e a t a b s t r a c t Hamilton-Jacobi equations ( f o l l o w i n g [Ljl;Bd3]), v a r i a t i o n a l convergence ( d ' a p r e s [ A c l ] ) ,

compensated compactness ( c f . [Mhl]),

homogenization, e t c . t o c i t e b u t a few areas o f c u r r e n t i n t e r e s t and a g a i n must work w i t h i n page l i m i t a t i o n s .

Another t o p i c would be t h e c o n s i d e r a t i o n

o f e l l i p t i c problems near t h e c r i t i c a l Sobolev exponent (where compactness f a i l s ) and f o r t h i s we r e f e r t o [ B x l ] .

A number o f i n t e r e s t i n g r e s u l t s on

n o n l i n e a r wave equations can be found i n [ G f l ;Str1-3;Fr3;Bfl-3;Ghl

Ka6,7; Te2;Swl; Tcl ;Swl-3;Wil-3]

;Oal ;Kcl;

f o r example, a1 ong w i t h n o n l i n e a r e v o l u t i o n

286

ROBERT CARROLL

r e s u l t s f o r o t h e r equations, b u t we w i l l n o t have space t o even s k e t c h any o f t h i s , a l t h o u g h i t has s i g n i f i c a n t impact on t h e study o f quantum f i e l d theory, s o l i t o n theory, e t c .

We a l s o o m i t any d i s c u s s i o n o f t h e t r a n s i t i o n

t o t u r b u l e n c e and chaos i n d e a l i n g w i t h n o n l i n e a r e v o l u t i o n equations ( c f . [ B i r l ; N t l - 3 1 f o r example). 6. QUANenm FtELD eHE0R&J. We w i l l s k e t c h here some o f t h e machinery which

a r i s e s i n quantum f i e l d t h e o r y and t h e s t u d y o f gauge f i e l d s .

No a t t e m p t i s

made t o d e s c r i b e t h e p h y s i c s b u t i t i s p o s s i b l e t o g e t an i d e a o f wh,at i s going on m a t h e m a t i c a l l y by s i m p l y w r i t i n g o u t t h e equations, v a r i a t i o n a l principles, etc.

F o r r e f e r e n c e s we mention [ L l ;Nkl ;Gul ;Hal ;J a l ;G11 ; C i 1 ;L p l ;

J k l ;H t l ;Bql ;Sxl ;F1 ;Fa3; F y l ;I t 1 ;Sul ;Re1 ; T t l ;Q1;Cgl ;A1 1 ;By1 ; D j l ;Fa4;Fdl ;Gxl ;

A number o f t h e mathematical problems i n v o l v e e.g. a n a l y s i s

B l e l ;Mdl ; R j l ] .

o f c r i t i c a l Sobolev i n d i c e s where standard compactness r e s u l t s f a i l ( c f . [Tbl,2;Ul]),

o r t h e s t u d y o f t o p o l o g i c a l - g e o m e t r i c questions, e t c . which

can be e a s i l y understood a t l e a s t , even i f s o l u t i o n s t o t h e problems a r e d i f f i c u l t ( i f available).

We f e e l t h a t enough background d i s c u s s i o n o f v a r -

i a t i o n a l ideas, Lagrange and Hamilton equations,

e t c . has been g i v e n a l r e a d y

so t h a t we can s i m p l y w r i t e down Lagrangians, a c t i o n i n t e g r a l s , e t c . , i n f l a t o r curved spaces, w i t h o u t a l o t o f formal j u s t i f i c a t i o n .

Geometric

ideas such as connections, c u r v a t u r e , e t c . and d i f f e r e n t i a l forms a r e a l s o s i m p l y used as needed; t h e d e f i n i t i o n s a r e e i t h e r g i v e n i n t h e t e x t o r can be found i n Appendix C.

Although t h e p h y s i c i s t s approach t o mathematics may

seem c a v a l i e r a t times (and o c c a s i o n a l l y l e a d s t o e r r o r ) s t i l l t h e exposure t o t h i s k i n d o f c r e a t i v e h e u r i s t i c mathematical t h i n k i n g i s w e l l w o r t h an e r r o r o r two; one can w o r r y about r i g o r once t h e r e i s some substance w o r t h w o r r y i n g about. L e t us r e c a l l from § § 1 . 7 - 1 . 8 t h e c l a s s i c a l f c r m a l i s m ( s e t h e r e I H , F I = -(H, F) =

1 (aH/api)(aF/aqi) -

IH,pkI and Dtqk = IH,qk},

(aH/aqi)(aF/api)

f o r t h e Poisson b r a c k e t ) Dtpk =

a l o n g w i t h t h e o p e r a t o r form h d A / d t = i [ H , A ( t ) ] ,

A(0) = A (we w i l l t a k e h = 1 and/or c = 1 a t t i m e s h e r e ) . procedure r e q u i r e s a l s o [p,q]

= -i (Q I p , q l = 1 ) and

The q u a n t i z a t i o n

iJ't = HJ/ f o r s t a t e s J,

i D w h i l e pt = i[H,p] and qt = i[H,q]). ( r e c a l l as o p e r a t o r s p Q, - i D and q q P We r e c a l l a l s o Example 1.8.3 where t h e q u a n t i z a t i o n procedure f o r a harmonic Q,

o s c i l l a t o r leads n a t u r a l l y t o c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s . a f i e l d t h e o r y one r e p l a c e s t h e sum

1 by

Now i n

an i n t e g r a l , j u s t as i n t h e case o f

continuous e l a s t i c s,ystems i n t h e c a l c u l u s o f v a r i a t i o n s , o n l y now t h e r e i s an a d d i t i o n a l q u a n t i z a t i o n i n v o l v e d .

Thus e.g.

f o r a f r e e Klein-Gordon (K-

287

QUANTUM FIELD THEORY

G)

f i e l d one has

2 2 (Dt-d+m )IP= 0 c o r r e s p o n d i n g t o a Lagrangian L0 w i t h den-

s i t y to ( c f . §1.10) (*)

C,(v,av)

( h e r e x'

D ~ D ~DU,

%

= (ct,x),

x

2

= (ct,-x),

= (i3j =~ (a/ax,), )

(a/a(ct),-a/ax)

2

-

= ( 1 / 2 ) [ 1 D'qDUlp

Lo(lp,a9)

gii

(1,-1,-1,-1),

'L

D~

2 2 m IP 3 w i t h Lo =

d3x

'=

u = DUD o r

(a/a(ct),a/ax)

%

f

ru

1

(ap)

=

1

(a/ax'), = ( l / c I D t - A, g gij = 6ij ( h e r e gA' = gA,), = gpvxVy etc. cf. [Lll]). R e c a l l a l s o t h a t one r e f e r s t o v e c t o r s vJ as c o n t r a v a r k b i jJ, i a n t and v j = gjkv as c o v a r i a n t , l e n g t h s a r e 1 [ g . .x x ] ds f o r a c u r v e

-

'

1

1 g'"(af/ax ) eIJ I J d f s ( a f / a ( c t ) e o i z1 ( c f . 51.10 and Appendix C - 9:, TxM TEM, g = 1 g..dx Id x j , i j i 1J gx(v)w = gx(v,w), 1 v Dx 1 g . . v dx .(gi = gji) o r 9:, vi 1 gijvJ = viy g;:' w = c widxi 1 wJDx =l J w P, w 1 = giJwj, and i n p a r t i c u l a r 1 D v f dx" 1 (1 g'"Dvf)DP as i n d i c a t e d ) . One sometimes w r i t e s 6q = 1 D!Jqdx' and

x ( s ) (sum on repeated i n d i c e s ) , D f =

3 (af/axi)ei

%

-f

*

1

-+

-+

*

-+

-+

( c f . 51.10) i s t h e g r a d i e n t o p e r a t o r on f u n c t i o n s t o 1-forms

we see t h a t D

U

( c f . §1.10) i s t h e g r a d i e n t from f u n c t i o n s t o t a n g e o t v e c t o r s . Note 3 2 p,p' = p: - C1 pi, and n o t a t i o n a l l y we can w r i t e e.g. D' =

w h i l e 0'

1

a l s o t h a t e.g.

1 gU"Dv also

9

= (l/c2)D2 - A. I n p h y s i c s one w r i t e s ' t It w i l l be c o n v e n i e n t t o t a k e c 1 w i t h xo

= gxDP so t h a t D DY = D'D

,i

= Diq

and q'

= t and x = ( t , x )

i 'I = D q.

=

( t h e r e s h o u l d be no c o n f u s i o n here i n d i s t i n g u i s h i n g 3-D

o r 4-0 x ) f r o m now on; a l s o t a k e h = 1 and t h e a c t i o n f u n c t i o n a l w i l l be (A)

4

S = 1 L d t = J t d x.

t i o n s o f t h e f i e l d as

A now s t a n d a r d v a r i a t i o n a l argument g i v e s t h e equa-

( 0 )

( a / a x ' ) [ a ~ / a ( D ~ p ) ] = aL/aq and one d e f i n e s a con-

Then TI ( t o q ) by ( & ) n ( x , t ) = a L ( q , a q ) / a + where i 'L Dtq. H = 1 H(n,q)d3x where H = nq - c i s a H a m i l t c n i a n d e n s i t y . I f t h e r e

j u g a t e momentum (0)

a r e a number o f f i e l d s q , one sums o r i n t e g r a t e s o v e r a and na = a C ( q , a q , ) / a(aoqff).

A s t a n d a r d n o t a t i o n a l s o expresses

at/a

(aoqa)as s L / s ( a O q a ) ( c f

T h i s i s formal so f a r and one must t a k e i n t o account e.g.,

83.2).

orderings, l o c a l properties, etc.

operator

The q u a n t i z a t i o n s t e p i s e a s i l y expressed

f o r m a l l y v i a (m) [q(x,t),lp(x',t)l = 0; [ T ( x , t ) , n ( x ' , t ) l = 0; [ n ( x , t ) , q ( x ' , t ) ] = - i 6 3 ( x - x ' ) where fi3 i s a D i r a c measure i n R 3 and t h e dynamical equat i o n s a r e f o r m a l l y (**) i,(x,t)

= i[H,qff]

= na and ;,(x,t)

It

= i[H,n,].

seems t o be e a s i e r and more i n s t r u c t i v e f o r m a l l y (and u s e f u l as i n 53.2) t o t h i n k o f Poisson b r a c k e t s however and w r i t e (e.g.

aL/aGff = ac/a(aoq,), ~ ~ ) ( a c / a ~ , ) l I, , 1 i C , 1, e t c . ) t,

= 6~/6+,

-+

{n,(x.t).n,(y,t)} aH/aT ; CL

{F,P;j =

=

(*A)

1 [(aF/ana)(aP;/aq,)

{n,(xyt),q,(y,t)l

= { ~ ~ ( ~ , t ) ~ q , ( Y , t =) l 0; aoq,(xlt)

a 0n, ( x , t ) = {H,T,} = -aH/aqa. TI = a c o / a i = i and Ho = ml,

(*) we have

-

3

L = J Ld x, H =

3

1 n,ia - (aF/a

= 6 (x-Y);

= {HYq,l = For t h e K-G f i e l d w i t h to g i v e n by 2 2 2 co = ( 1 / 2 ) [ n + I v 3 v I 2 + m IP 1. I n = n,(X¶t)

2 p a r t i c u l a r aHo/an = n and aHo/aq = m 9 ; one has a L0/ a q = -aHo/aq = Dtn =

288

2 -m q

ROBERT CARROLL

+ 1,3 Diq2

and Dtq = -aHo/an =

so D2q = -m 2v + h p .

IT

t

I n a d d i t i o n t o t h e K-G Lagrangian to o f (*) l e t us mention ex-

REmARK 8.1.

p l i c i t l y a l s o t h e q 4 s e l f i n t e r a c t i n g t h e o r y based on t = to - (A/4!)v

t

t h e Sine-Gordon (S-G) Lagrangian ( c f . 552.9-2.11)

A)[Cos(JAq/m) R m R K 8.2.

-

and

+ (m4/

1 1 ( t h i s l a t t e r 1; l e a d s t o a good f i e l d t h e o r y i n 2-0, n o t 4 ) .

L e t us i n d i c a t e a s t a n d a r d q u a n t i z a t i o n o f t h e K-G f i e l d v i a

F o u r i e r t h e o r y ( c f . a l s o 51.8). form (nq +

= ( 1 / 2 ) 1 D’qD’q

4

m2v

=

Thus w r i t e a H e r m i t i a n s c a l a r f i e l d i n t h e 3

0 ) v ( x , t ) = / d k [ a ( k ) f k ( x ) + a*(k)f;:x)]

e x p ( - i k . x ) / [ ( 2 1 ~ ) ~ 2 ~ , ] ’ , wk = (k2+m2)’, t c a l l e d a i n physics, ko w k Y and x Q

and k - x = w k t %

-

where f k ( x ) =

( k , x ) (a* i s u s u a l l y

- t h e n o t a t i o n i s standard, as

(t,x)

a r e o t h e r s , and we w i l l t r y t o be c l e a r and c o n s i s t e n t i n o u r use o f sym-

- however we w i l l o c c a s i o n a l l y use d i f f e r e n t n o t a t i o n s ) . One w r i t e s a(t)L$b(t) = abt - a b and i t f o l l o w s t h a t ( e x e r c i s e - c f . [ B q l ; I t l ] ) (*@) a ( k ) = i/ d3xf;(x,t)a Lq ( x , t ) ( n o t e / f;(x,t)ia:fk(x,t)d 3x -- 63 ( k - k ) and bols

3O

-

/ f k ( x , t ) i g f (x,t)d x = 0 a ( k ) i s t i m e independent). Then e.g. [a(k),a* ( k ) ] = / / ogk d xd 3y[fi(xYt)a”dp(xYt)Yfk(YYt)a:~(YYt)] = i/ d 3 X f { ( X ~ t ) c f k ( X , t ) = 63(k-k) and s i m i l a r l y [ a ( k ) , a ( k ) ]

= [a*(k),a*(k)]

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

= 0 ( t h e a ( k ) and v ( x , t )

-

one uses h e r e ( m ) s i n c e 2 2 a 0v = IT). I n t h i s t e r m i n o l o g y we have a l s o ( H = ( 1 / 2 ) ( n 2 + 103v12 + m v ) ) 3 3 I t i s i n s t r u c t i v e here t o d i s (*6) H = / Hd x = ( 1 / 2 ) / d kwk[a*a + aa*]. C r e t i Z e / d 3k

AVky 63 (k-;)

Q

(1/2)wk(aiak + aka;),

m

1 Hky Wk

= (Ikl2+m2)’,

= &k$, e t c . ( c f .

[ai,a;]

Hk

=

Example 1.8.3

One t h i n k s o f energy e i g e n f u n c t i o n s as pro-

The ground s t a t e i s

1 wk/2 (which w i l l be fuss - c f . [ B q l ; I t l ; G l l ] ) .

energy E =

without a l o t o f

H =

v k ( n k ) , Hkvk = wk(nk++)vky and v k = (l/nk!)’(a*)nkqk

%

( 0 ) (nk = O y l y 2,...). i t has

Gk;/AVk,

ak = JAVka(k),

f o r a p a r a l l e l development). d u c t s ‘IPk where v k

Q

vo

‘IPk(0) w i t h a k p k ( 0 ) = 0 and

s i m p l y removed f r o m t h e t h e o r y We do n o t g i v e h e r e a l o n g d i s -

cussion b u t simply r e c o r d some formulas i n d i c a t i n g what i s g o i n g on.

Thus

f i r s t ( f o r general background) v i a c o n s i d e r a t i o n s o f symmetry and conservat i o n laws ( c f . [ B q l ; G l l ; I t l ] ) one has an energy-momentum v e c t o r (H,P) where 3 3 P Q - / 1 ~ 0 ~ x9 d ( 1 / 2 ) / d k k(a*a+aa*) Q ( 1 / 2 ) k(aCak + aka;) here w i t h

1

Q

P kv k (,kn ) = k ( n k f 4 ) q k ( n k ) e t c . a n d , s e t t i n g N~ = a;akywith =

1k

Nk ( P = (P’),

1 wk/2 one 1 wk/2 = 1 wka;ak

o f energy

H

-

k = (k’)).

Nkqk = nkqk and P’

Now t o remove t h e vacuum e x p e c t a t i o n v a l u e

s t i p u l a t e s a z e r o energy ground s t a t e w i t h = (1/2):aLak

+ aka;:

(*+I H’ =

where : : denotes t h e so c a l l e d

Wick o r d e r i n g where a n n i h i l a t i o n o p e r a t o r s ak appear t o t h e r i g h t o f c r e a t i o n o p e r a t o r s a;

(see [ G l l ]

f o r a good d i s c u s s i o n o f t h i s

- g e n e r a l l y if

QUANTUM FIELD THEORY

Q = ( A t A*)/42 one has :Qn: = 2-”‘1

289

(;)A*jAn-J).

Note h e r e t h a t wkagakvk

= Wkakgk and observe t h a t c l a s s i c a l l y , where t h e commutators a r e a l l zero,

t h e r e i s no z e r o p o i n t energy.

H

c o n t i n u o u s case v i a

=

This n o t a t i o n i s then t r a n s f e r r e d t o t h e

J d3 k w k a * ( k l a l k ) e t c .

Somewhat more g e n e r a l l y i f

one wants t o deal w i t h p a r t i c l e s and a n t i p a r t i c l e s a f i e l d i s used (charged s c a l a r f i e l d

3 d X : S * I I + vv*vv +

w*v:

- v1

where n =

v 2 as i n

and

v*

appropriate quantization.

’ (F”)

O

=

( r e c a l l i n g E = -vv

o

(F’”

- At, B

=

v

X E = -B

- aAV/axu).

= aA!’/axv

= -1 ( i

-

E Ox By EX -Bz 0‘

v E

t’

row, w = column)

0, v X B = Et,

=

Dp(DwAv) = 0 and D‘F’”

Note f o r gij

2 l ) ,one w r i t e s Fuv

(FL) =

(8.2)

(11 =

E

p a r t i c u l a r (*.) aFuv/axv = 0 o r nA’ =

(cf. [Bql;Itl;Gll]).

E E ’ - E Ox By -BZ -Ex -B Ox By -E: B i -Bx Ox

i

F’”

=

We r e c a l l f i r s t t h e c l a s s i c a l f r e e Maxwell equa-

t i o n s ( c f . Example 1.2.11 and §1.10) and w r i t e

and gii

H

We go n e x t t o t h e e l e c t r o m a g n e t i c f i e l d and w i l l i n d i c a t e t h e

REmRRK 8.3.

(8.1)

= (vl+iv2)/J2

say) w i t h e.g.

(*A)

= (v1-iv2)/J2

v

= gij

etc.)

In

+ DV

t D’F”’

= 0 (i= j ) , goo = 1,

and FPv = -gVaFg w i t h

= gu,Ft

-iy

We w i l l phrase a l l t h i s below i n terms o f d i f f e r e n t i a l forms, c o n n e c t i o n s , c u r v a t u r e , e t c . b u t i t w i l l be i n s t r u c t i v e t o have v a r i o u s p o i n t s o f view. 2 Now t h e c l a s s i c a l Lagrangian 1; = -(1/4)FpvFpv = ( 1 / 2 ) ( E 2 - B ) ( E = I E I , etc.) leads t o tion.

ac/aio

IT =’

= 0 and t h i s i s n o t s a t i s f a c t o r y f o r q u a n t i z a -

=

v

’ = -A’

( n o t e (A’)

%

(Ao,-A)

Du

while

t h e n a s u i t a b l e Lagrangian d e n s i t y i s aL/ai

’w i t h x 0 ( n o( tt,ex ) )

I f one uses a L o r e n t z gauge as i n Example 1.2.11 so D A’

t h i s means d i v A + v t = 0 s i n c e A’

(A*)

and (A’)

’ -(l/Z)(D =

a

= a/ax’

c

=

‘L

(Ao,A)).

=

=

A )(DVAp) w i t h

V ’

IT’ =

We r e f e r h e r e t o Re-

mark 9.3 f o r f u r t h e r d i s c u s s i o n ) . L e t us now p u t t h e Maxwell t h e o r y i n t h e language o f d i f f e r e n -

RfiltARK 8.4.

t i a l forms ( c f . Appendix C ) . (again c = 1 )

(AA)

Thus f o l l o w i n g [Gul;Wtl;Fsl;Cul]

one w r i t e s

B dz A dx - Bzdx A dy + Exdt A dx + Y Then t h e equations V X E = -Bt and 0.B = 0 a r e e q u i -

F = -Bxdy A dz

-

E d t A dy + E,dt A dz. Y v a l e n t t o dF = 0. The equations v X B

Et and v.E = 0 ( i n t h e absence o f

c u r r e n t s and charges) can be w r i t t e n as 6 F = 0 ( o r 6 F = -u0J when v X B =

Et +

poJ).

We r e c a l l h e r e t h a t 6 = *d* where

(Am)

*(dx A dy) = -dz A d t ,

290

ROBERT CARROLL

*(dy A dz)

-dx A d t , *(dx A dz) = -dy A d t , *(dx A d t ) = dy A dz, *(dy A d t ) = -dx A dz, and *(dz A d t ) = dx A dy. The L o r e n t z gauge w i t h d i v A t qt : :

Since dF = 0 we w r i t e ( a t l e a s t l o c a l l y ) F = dA

= 0 l o o k s now as f o l l o w s .

( A = a 1-form) and any A t d f = A ' g i v e s t h e same F. dS which i s

IY

( n o t e d i v S = -65, 5 =

Lorentz gauge i n v o l v e s 6A' = 0 o r 6 A have SdA' = -AA' =

nA' =

= 0 and coupled w i t h SF = 0 we

Note a l s o t h a t f o r A ' =

0.

1 AidxHone

has S A ' =

We w i l l see l a t e r i n c o n n e c t i o n w i t h g e o m e t r i c quan-

as d e s i r e d .

-D,,[A')'

1 viei, - Af

R e c a l l now -A = Sd t i 5 = 1 vidx , e t c . ) . Then t h e

t i z a t i o n how one deals w i t h A as a c o n n e c t i o n and F as a c u r v a t u r e i n , s u i t a b l e f i b r e bundles ( c f . a l s o [Gul;G11;Blel;Cul;Tr1;Sxl]).

REII~ARK 8.5, As a n o t h e r i n t r o d u c t o r y p o i n t o f view we want now t o develop a l i t t l e t h e use o f a c t i o n i n t e g r a l s and t h e Feynman p a t h i n t e g r a l . T h i s p o i n t o f view has become fundamental i n many i n v e s t i g a t i o n s and t h e r e a r e many a p p l i c a t i o n s ( c f . [Bql ;Nkl ;J a l ;B11 ;F1 ;Fyl ;Sul ; A l l ;L1 ;Re1 ;Mml ;Lcl ,2]). We w i l l g i v e f i r s t a p a r t i a l l y h e u r i s t i c d e r i v a t i o n f o l l o w i n g [Fyl;Sul;Rel]. 2 2 2 Thus s t a r t w i t h ih!bt = W w i t h H = TtV = p /2m + V -h Dx/2m t V(x). We

-

t h i n k o f a propagator o r Green's f u n c t i o n G s a t i s f y i n g as an o p e r a t o r (H ihDt)G(t,to)

o r i n c o o r d i n a t e f o r m (Hx

= -itiS(t-to)I

-ihS(x-y)6(t-to).

-

ihDt)G(x,t,y,tO)

One w r i t e s i n s t a n d a r d n o t a t i o n G(x,t,y,to)

G(t,to) = o(t-to)exp[-iH(t-to)/h]

= (xlG(t,to)l

F o r m a l l y o f course

y ) and t h e s t a t e s a r e t o e v o l v e v i a # ( t ) = G(t,to)!b(to). (46)

=

where 0 i s a Heavyside f u n c t i o n . Now

it/n

f o r m a l l y and h e u r i s t i c a l l y ( f o l l o w i n g [ S u l ] ) we w r i t e X =

.

and s e t G(x,

t,y) = (xlexp[-h(T+V)/N]. .exp[-h(T+V)/N]ly) w i t h exp[-X(T+V)/N = exp(-XT/N) 2 2 2 2 exp(-XV/N) + O ( A /N ). One assumes t h a t t h e O ( X /N ) t e r m i s w e l l behaved

when a p p l i e d t o s t a t e s , etc.,

and f o r reasonable p o t e n t i a l s t h i s can be j u s N Next one wants t o r e p l a c e t h e t e r m (A+) (exp[-h(TtV)/N]) by [exp

tified.

(-XT/N)exp(-XV/N)]

N

and i n t h i s d i r e c t i o n one notes t h a t [l + (x+yn)/nIn

e x p ( x ) as l o n g as yn (A+)

-f

one can w r i t e e.g.

(-XT/N)exp(-XV/N)

-

0 when n

-f

-

(exercise

- c f . [Sul]).

-+

To work w i t h

[ e ~ p ( - X T / N ) e x p ( - h V / N ) ] ~ - [exp(-X(TtV)/NIN = [exp

e~p(-x(T+V)/N)][exp(-x(T+V)/N)]~-~ t

... t

[exp(-XT/N)

exp(-XV/N)IN-' [exp(-xT/N)exp(-XV/N) - exp[-X(T+V)/N]] and t h e N [ ] terms 2 2 a r e O(X /N ) . With some r e f i n e m e n t t h i s l i n e o f reasoning can be developed t o prove t h e T r o t t e r p r o d u c t f o r m u l a ( c f . Appendix B f o r semigroups and see

- c f . a l s o 53.6).

[Sul;Dwl;Pz]

f o r proof

tHE@REIn 8.6

L e t A and B be l i n e a r o p e r a t o r s i n a B space E such t h a t A, B,

exp(At), Qt and A+B a r e generators o f c o n t r a c t i o n semigroups Pt t - lim t / n t / n nxx. exp[(A+B)t]. Then f o r x E E,R x - n- ( P Q ) Q

and Rt

Q

Q

exp(Bt),

QUANTUM FIELD THEORY From t h i s one w r i t e s t h e n G(x,t,y)

= lim

291

( X I [exp(-xT/N)exp(-xV/N)]

N I y ) and

= I between terms ( j = l,...,N-1) one o b t a i n s ( c f . j §§2.3-2.4) ( A m ) G(x,t,y) = l i m 1 dxl...dXN-lf-l ( xjtl lexp(-xT/N)exp(-xV/N) I x . ) (xo = y and x = xN). Now V i s a m u l t i p l i c a t i o n o p e r a t o r so exp(-xV/N)I J x . ) = I x . ) exp(-xV/N) and t o t r e a t t h e o p e r a t o r exp(-xT/N) one i n s e r t s momJ J entum s t a t e s 1 = 1 d p l p ) ( p I w i t h ( P I E ) (2rh)-'exp(-ipg/h) so t h a t ( n l e x p ( - x T / N ) I c ) = J dp( n l e x p ( - x T / N ) l p ) ( P I S ) = (1/2nh)/: dpexp(-xp 2/2mN)exp[ip(n-

inserting 1 dx.Ix.)(x

J

J

The i n t e g r a l s can be e v a l u a t e d v i a /: exp(-ay 2+by)dy = J(n/a)exp(b 2/ 2a) ( e x e r c i s e ) and hence ( nlexp(-xT/N) 15) = (mN/2aAh 2 )4exp[-mN(n-E) 2/2Ah2].

c)/h].

P u t t i n g a l l t h i s together then y i e l d s (8.3)

G(x,t,y)

m(x.tl-x.)

,+

= 1i m

Letting ( E i = exp[-XV(xj)/N]). t h e p a t h i n t e g r a l f o r G as

(Aj"

= xy,-xj).

2 ) N/2nN-1e-[

dxl...dxN-l(mN/2sxh

3

Now t h e [

E

2hh2

2

N

lEJ N

= t / N = t i x / i N and r e o r g a n i z i n g one o b t a i n s

t e r m i s an a p p r o x i m a t i o n t o t h e Riemann i n -

t e g r a l 10 d ~ [ ( 1 / 2 ) m X ~- V ( x ) ] d r o v e r a p a t h

X(T)

t

j o i n i n g y = x 0 and x = x N One t h i n k s o f sumning

and o f course t h i s r e p r e s e n t s an a c t i o n S = Jo LdT.

o v e r a l l p o s s i b l e ( c o n t i n u o u s ) broken l i n e paths c o n n e c t i n g y and x; hence by a p p r o x i m a t i o n one i n t e r p e r t s t h e i n t e g r a l (@*) G(x,t,y) (-))/h]

= c

1 exp[iS(x

as a sum o v e r a l l p o s s i b l e c o n t i n u o u s paths x j o i n i n g y and x ( h e r e

c = ( r n / 2 a i f i ~ ) ~+/ ~m as N rigorous theory).

-f

m

which was one o f t h e problems i n d e v e l o p i n g a

One extends t h e s e c o n s i d e r a t i o n s t o 3-D and t o a Lagran-

- V(x) i n a more o r l e s s s t r a i g h t f o r w a r d way, e x c e p t t h a t t h e v e c t o r c o n t r i b u t i o n ( i e / n c ) 1(xjtl - x . ) A ( x ) must i n v o l v e

g i a n L = ( 1 / 2 ) m l i I 2 + (e/c);-A

e v a l u a t i o n o f A(x) a t x = ( 1 / 2 ) ( x . + x ) o r e.g. ( 1 / 2 ) f A ( x j ) + A(xjtl)] must J j+l be used. T h i s f e a t u r e i s standard i n t h e t h e o r y o f Brownian m o t i o n and t h e I t o i n t e g r a l and w i l l n o t be discussed h e r e ( c f . [Sul;Wgl]). [Sul ;Fyl ; G l l ;Nkl ;Bql ; I t 1 ;Mml ;Lcl,2]

We r e f e r t o

f o r t h e p h i l o s o p h y o f t h e Feynman i n t e -

g r a l i n c o n n e c t i o n w i t h d i f f r a c t i o n , quantum mechanical paths, p r o b a b i l i t y , e t c . and remark h e r e o n l y t h a t s t a t i o n a r y phase arguments l e a d one t o conc l u d e t h a t t h e main 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 occurs when 6 s 6x = 0 and t h i s leads one t o e v a l u a t i o n o v e r t h e c l a s s i c a l p a t h determined by t h e Lagrange equations.

The i n t e g r a l (@*) i s o f t e n w r i t t e n as

1 exp[iS(x)/h]D(x)

where t h e magic D ( x ) i s used s y m b o l i c a l l y as some s o r t o f

" t h i n g " i n p a t h space.

(@A)

G x,t,y)

=

The r e a d e r s h o u l d n o t e t h a t we have f a r t o o much

2 92

ROBERT CARROLL

r e s p e c t f o r t h e p a t h i n t e g r a l and i t s consequences t o be mocking about D ( x ) b u t i t s use and i n t e r p e r t a t i o n a r e s t i l l perhaps i n an e x p l o r a t o r y stage; f o r a good t r e a t m e n t o f t h e f u n c t i o n a l i n t e g r a t i o n p o i n t o f v i e w see e.g. The mathematical p h i l o s o p h y has i n v o l v e d i n s e r t i n g convergence

[ G l l ;Mml].

f a c t o r s , w o r k i n g t h r o u g h imaginary t i m e

and u s i n g a n a l y t i c c o n t i n u a t i o n ,

e t c . and t h e p a t h i n t e g r a l may be one o f t h e most i m p o r t a n t and p r o d u c t i v e n o n r i g o r o u s o b j e c t s y e t conceived i n mathematical physics; however c o n s i d e r ,2]). a b l e r i g o r can be p r o v i d e d ( c f . [Gll;Mml;Lc1,2;All 2 -2 one w r i t e s T -t - i r i n (.*) w i t h h = 1, j , + - x , and t formally

(0.)

t V(x))ds].

In particular if -f

- i t t h e n one has

n dx(s)exp[-J; (mi2/2 G(x,-it,y) = ( x l e x p ( - t H ) l y ) = J W(Y 9x9 t 1 t 2 Here t h e t e r m exp[-Jo m i ds/2] w i l l serve as a convergence f a c -

t o r and w i t h some r e o r g a n i z a t i o n , u s i n g t h e T r o t t e r p r o d u c t formula again, one o b t a i n s t h e Feynman-Kac formula ( 0 6 ) ( x l e x p ( - t H ) l y ) = J exp[-Jot V(x)ds] dWt

YX

where dWt

YX

denotes c o n d i t i o n a l Wiener measure i n t h e p a t h space W(y,x,

t ) (see h e r e [ G l l ]

REmARK 8-7.

f o r details).

L e t us now see how t h i s approach l o o k s i n f i e l d t h e o r y ( c f . Thus t a k e S(tl,t2,[v])

[Fa4;Rel;Nkl;Itl;NxlI).

=

c2

d4xI;(~,av) where [v]

c o u l d be any c o l l e c t i o n o f l o c a l f i e l d s and we f o l l o w [Rel] i n o r d e r t o g i v e 4 a s k e t c h o f what i s g o i n g on. F i r s t as i n ( 0 ) e t c . (@+) 6 s = 1 d x[aL/av -

+ ID dop(ac/a(apv))sv where Is i s a s u i t a b l e s u r f a c e i n t e -

ap[ar/a(apv)]]6v gral.

Given

[ar/a(apv)]

6q =

0 on u one o b t a i n s t h e E u l e r equations

(a=)

aL/av

-

a

’

One s h o u l d remark h e r e t h a t v a r i o u s c o n s e r v a t i o n

= 6 S / S q = 0.

laws a r i s e from t h e i n v a r i a n c e o f L under L o r e n t z t r a n s f o r m a t i o n s + t r a n s l a t i o n s ( t h e Poincar;

group); a l t e r n a t i v e l y one s t i p u l a t e s t h a t t h e a c t i o n be

i n v a r i a n t under such t r a n s f o r m a t i o n s o f c o o r d i n a t e s and f i e l d s .

Thus c o o r -

denotes a L o r e n t z t r a n s f o r m a t i o n ) ’ 2 6x’a f up t o o r d e r (Sx) w h i l e f u n c t i o n s change a c c o r d i n g t o ’ ( h e r e s,f denotes t h e f u n c t i o n a l change a t x - i . e . t h e form o f f i s p o s s i b d i n a t e changes a r e x’

-f

a’

+ Atx”

(A’

6f =

l y c o o r d i n a t e o r frame dependent).

4 4 Then 6(d x ) = d xJ6x’ P

’

’

t

( J a c o b i a n ) and

st = 6 t + 6x’a 1: = 6xpa 1; + ( a t / a L p ) ~ ~ +9 (at/a(aup))60apv w h i l e s a q = 0 O p4 [ ~ ~ , a ~+ ] av 6 q = a 6 p ( e x e r c i s e ) . We can w r i t e t h e n ( 6 * ) 6s = J d x[SC P O 4 P O 4 + aPsxpc] = 1 d x[capsxp t 6xpa c + a [ ( a ~ / a ( a ~ q ) ) ~= ~ 1~ d~ ]xau[c6xp ] + ( a t / a ( a q ) ) 6 0 v ] (where (-) has been used). T h i s can a l s o be w r i t t e n as 6s = J d4 x p ,[[tg; - ( a ~ / a ( a g ) ) a ~ ~ ] +s x( ~a c / a ( a p ~ ] ( g i = 6;). Ifone

’

’

imagines now some g l o b a l (x-independent) t r a n s f o r m a t i o n fixp = (SxP/Swa)Swa and 6q = ( b / 6 w a ) 6 w a t h e n SS = - J d 4 xa j’6wa where -j: = [tg: - ( a C / a ( a p v ) ) p a apq]6xP/6wa t (aC/a(a v ) ) ( s q / s w a ) i s c a l l e d a c u r r e n t d e n s i t y ; thus when SS

’

QUANTUM FIELD THEORY

293

a j' = 0.

T h i s i s a k i n d o f E. Noether u a theorem f o r c l a s s i c a l f i e l d t h e o r y r e l a t i n g a c o n s e r v a t i o n l a w t o i n v a r i a n c e t o f t h e a c t i o n . I n p a r t i c u l a r ( c f . Remark 1.10.8) c o n s i d e r ((A) . . 0 = L * dxo = 0 f o r a l l 5wa one concludes t h a t

.

t l

Zm d3xa j' = It2 dxoao[z d 3 x j i t /tt'dxo/z d3xaijd The l a s t t e r m w i l l vanP a I i s h i f boundary c o n d i t i o n s a r e s u i i a b l e and hence t h e charges ( 6 0 ) Q"z ( T ) = 3 .o -Jm m d XJ,(T,X) a r e t i m e independent ( i . e . f r o m ((A) Qa(T1) = Q a ( T 2 ) ) . Hence -m

6 s = 0 i m p l i e s e x i s t e n c e o f conserved charges.

It w i l l be h e l p f u l t o i n t r o d u c e a few a d d i t i o n a l ideas.

RfiRARK 8.8.

we c o n s i d e r t h e p a t h i n t e g r a l i n phase space f o l l o w i n g [Sul;Fa4]

First

since i n

p a r t i c u l a r t h i s l e a d s t o many i n t e r e s t i n g r e s u l t s d i r e c t l y and i s f r e q u e n t l y used i n t h e Russian l i t e r a t u r e .

As n o t e d i n [ S u l ] i t i s p o s s i b l e t o make

e r r o r s b u t we w i l l n o t d i s c u s s t h e p h i l o s o p h y o f paths e t c . here. Thus as N b e f o r e one w r i t e s down G(x,t,y) = l i m <XI [exp(-AT/N)exp(-AV/N)] l y > and i n s e r t s now a l t e r n a t i v e l y r e s o l u t i o n s o f t h e i d e n t i t y u s i n g c o o r d i n a t e and = lim

momentum e i g e n s t a t e s t o o b t a i n ( ( 6 ) G(x,t,y)

( p2€/2m) I pN) ( pNI exp(VE) I xN-l) -it/hN, xo = y, and xN = x. (

pi(exp(VE)lxi-l)

. . .( x1 I exp( p 2 ~ / 2 m )I pl) Now

(

1 [$-'dp (

xi/exp(p2€/2m)lpi)=

= exp(V(xi-l)~)(pilxi-l)

dx.]dp& x l e x p P ' p1 I exp(Vc) ly) where E = e x p ( p : ~ / 2 m ) ( x ~ / p ~ and )

w h i l e ( x l p ) = (21~fi)-'exp(ipx/li)

It f o l l o w s t h a t gives a 6 function normalization J d x ( p l d ( x 1 p ' ) = 6(p-p'). im N-'dpid~i]dpN(2rrh)-Nexp[( i / f i ) J o t ( p i H)d.r] where ( b e ) G(x,t,y) % Jhno 2 1; ( p i - H1d.r % ( t / N ) C o [ ~ ~ ( x ~ - x ~ - ~ ) / ( t pj/2m / N ) - V(xj_,)l.

-

RRRARK 8.9, S(t,T,x)

(S =

t u d e yT + xt.

= (xt1yT) = J D(x)exp(i/h) L e t us w r i t e (@A) as (6.) G(x,t,y,T) t Cd-r) where ( x t l y T ) i s w r i t t e n t o denote t r a n s i t i o n a m p l i -

/T

One expects o f course by c o n s t r u c t i o n t h a t (+*) G(x,t,y,T)

jm G(x,t,z,~)G(z,r,y,T)dz -m

and $ ( x , t )

press G v i a Remark 8.8 as G(x,t,y,T)

=

Z l

G(x,t,y.T)$(y,T)dy.

=

We a l s o ext

= J [dpdq/2~]exp[(i/h)/T

(pG-H)d.r]

(sometimes h e r e D ( x ) i s w r i t t e n as ~ [ d q ] w i t h an u n s p e c i f i e d n o r m a l i z i n g constant

K

and sometimes s i m p l y as [dq]).

I n f i e l d t h e o r y one i s concerned

w i t h c a l c u l a t i n g Green's f u n c t i o n s d e f i n e d by G(x l,...,xn)

= (OITv(xl)

q ( x n ) l 0 ) f r o m which S m a t r i x elements can be o b t a i n e d ( c f . 53.9).

...

Here T

denotes a t i m e o r d e r e d p r o d u c t where t h e f a c t o r s a r e o r d e r e d so t h a t l a t e r times s t a n d t o t h e l e f t o f e a r l i e r times.

We w i l l n o t d w e l l h e r e on t h e

p h i l o s o p h y o f p e r t u r b a t i o n t h e o r y , t i m e o r d e r e d products, Feynman diagrams, Feynman propagators, e t c . ( c f . [It l ;G11; F y l ;Nsl ;Bql ;Cgl ;L1; Fa4;Ael ;Bog1 b u t we w i l l e x h i b i t some i n t e r e s t i n g formulas and p o i n t s o f view.

1)

L e t us

f o l l o w [Cgl] here t o s k e t c h t h e c o n s t r u c t i o n o f a g e n e r a t i n g f u n c t i o n W(J) f o r such Green's f u n c t i o n s ( c f . a l s o [ A e l ] ) .

Thus r e c a l l f i r s t ( c f . Remark

2 94

ROBERT CARROLL

1.8.6) t h e Heisenberg dynamical v a r i a b l e s w i t h t i m e independent s t a t e vect o r s l a ) and t h e Schrodinger t i m e dependent s t a t e v e c t o r s l a , t ) ' = e x p ( - i t H ) l a ) (TI = 1 ) . Consider e.g. G(tl,t2) = (OITQH ( t , ) Q H ( t 2 ) 1 0 ) = I dqdq'(01 q ' , t ' ) ( q ' , t ' ( T Q H (t,)Q H( t 2 ) 1 q , t 1 0 ) where ( O l q , t ) = v o ( q , t )

i s t h e ground s t a t e w a v e f u n c t i o n (and QH

%

= vo(q)exp(-iEot)

Heisenberg r e p r e s e n t a t i o n o f Q

2,

S

q - i . e . Q I q ) = 914)). Now f o r t ' > tl > t2 > t we can w r i t e (.A) (q',t'l T QH ( t l ) Q H ( t 2 ) l q , t ) = ( q'lexp(-iH(t'-tl))QSexp(-iH(tl-t2))Q Se x p ( - i H ( t 2 - t ) ) l q 2 ) = J (q'Iexp(-iH(t'-tl))(q,)(ql

I Q s e x p ( - i H ( t l - t 2 ) ) l q 2 ) ( q 2 ( q Se x p ( - i H ( t 2 -

(QH = e x p ( i H t ) Q S e x p ( - i h t ) o r A ( t ) = U*AU i n Remark 1.8.4). Then u s i n g t h e p a t h i n t e g r a l we o b t a i n (one t h i n k s o f tl and t2 as break p o i n t s t))1q)dqldq2

i n the paths) [pt

- H(t,q)]]

(

q',t'lTQH(tl)QH(t2)lq,t)

=

I [ d p d q / 2 ~ ] q ( t ~)q(t2)eXp[iI;'d.r

and e v i d e n t l y t h e same r e s u l t h o l d s f o r t2 > tl.

G(tl,t2) = J [dpds/2.k0(s',t)v~(q,t)q(tl )q(t2)exp[it'd.r[p4 where we have added a " c u r r e n t " Jq t o 1; ( J = 0 o u t o f [T,T']); one can change H

-+

H-Jq.

One can remove t h e

which we r e f e r t o [Cgl;Ael]

Hence ( + a )

- H

+ Jqll equivalently

vo,v,* terms by an argument f o r

and t h e r e r e s u l t s f o r m a l l y (qi = q ( t i ) )

The c o n s t r u c t i o n extends t o G ( t l,...,tn) = ( O I T q ( t l ) ...q( t n ) I O ) and t h e s e c a l l a l l be generated f o r m a l l y v i a

w i t h G ( t l,...,tn) = (-i)'6'W(J)/6J(tl)

...6J(tn)/J,0

( n o t e one takes func-

t i o n a l d e r i v a t i v e s and t h e n l a b e l s tl, t2, e t c . ) .

Here W(J)

'L

(010) ( f o r H

-Jq) and W(0) = 1.

The formula l o o k s somewhat more " r i g o r o u s " i f one goes T,,) 'L in G(-iTl ,..., -kn) i n which case t o imaginary t i m e and w r i t e s S ( T ,..., ~

S ( T1

,. . . ,T,

= lim

.

GnWijE ( J ) / 6 J ( T ) . . .6J ( T~ ) I J= where ( c f [Cgl ;Ae 13 ) (A6 ) GE( J ) 1 [ d q ] e ~ p [ J ~ ~ ' d s ( - m ; ~ / 2- V(q) + J ( s ) q ) ] ( f o r 1: = m42/2 - V where con-

vergence can be e n v i s i o n e d more e a s i l y ) . normalization f a c t o r

4

= (q',t'lq,t)

Here one o m i t s t h e J independent

f r o m (8.6) s i n c e i n f i e l d t h e o r y one i s

o f t e n i n t e r e s t e d i n "connected" Green's f u n c t i o n s where (1/WE(J))6'WE(J)/ 6J(<)...6J(?n)IJ=o

= G(zl,...,Xn)

f a c t o r i s removed from WE and f u n c t i o n ( c f . Remark 9.6

-

9. GAUGE FZECD.5 (PHg3ZCS).

(WE(J) = W(J)

iE i s considered

-

c f . §3.9).

Hence t h i s

as t h e fundamental g e n e r a t i n g

->

x = (it,x)). We f i r s t c o n t i n u e t h e d i s c u s s i o n o f 58 a l i t t l e

i n c e r t a i n d i r e c t i o n s i n o r d e r t o have a v a i l a b l e a few more ideas and cons t r u c t i o n s from c l a s s i c a l quantum f i e l d t h e o r y .

GAUGE FIELDS

RmARK 9.1.

295

L e t us f i r s t extend t h e g e n e r a t i n g f u n c t i o n o f Reamrk 8.9 t o a Thus we have f i e l d s v and 4 / d x(na v H + Jv)]

f i e l d s i t u a t i o n f o l l o w i n g [Cgl;Acl;Fa4;Itl;Nsl]. 'TI

say and one d e f i n e s (*) Z(J) = W(J) = 1 [dpdn]exp[i

= 1 [&]exp[i

/d4x(t t &)I.

replacing x

(it,x)

J(x)

u

= (t,x)

(A)

ZE(J) = i E ( J ) = I

We d e f i n e t h e n ( G = G ( ~ l y . . . y x n ) y

(x)]).

0

-

This i s r e l a t e d t o a Euclidean version

I

?lJ

=

[dv]exp(ld4x"[L(v(~)) +

N

r

y

WE = WE(J))

Based on Feynman t h e o r y t h i s i s c a l l e d a connected Green's f u n c t i o n and t h e f o r example. For t h e s t a n d a r d 2 2 i l l u s t r a t i v e example t ( v ) = t o ( v ) + tl(v), cO(v) = ( 1 / 2 ) a A d v - (1/2)m v , and t l ( v ) = -X/4!v 4 one o b t a i n s ( 0 ) ZE(J) = 1 [ d v ] e ~ p [ - / d ~ Z [ ( a ~ v ) ~+/ 2 p h i l o s o p h y i s discussed i n [Cgl ; I t l ; B o g l ; R y l ]

2

+ m2v 2 /2

-

Jv]]. I f we drop t h e E s u b s c r i p t f o r t h e Euclidean s i t u a t i o n and w r i t e Zo(J) = / [dv]expJ d4 x"(Co+Jv) ( n o t e to - ( 1 / 2 ) ( a v ) 2 ( 1 / 2 ) ( v 3 v ) 2 - m2v 2 /2 i n 2 c o o r d i n a t e s ) then f o r m a l l y ( b ) Z ( J ) = e x p [ J0d 4 y (1/2)(V3v)

Q ,

( a l i t t l e s t r a n g e perhaps b u t f o r m a l l y an obvious n o t a t i o n ) . 2 2 2 Replace now - ( a o v ) 2 - (v3v) i n ( 0 ) by v ( a T + v3)v ( d i v e r g e n c e theorem f o r 2 4 - 4s u i t a b l e v , A3 % v 3 ) and w r i t e ( + ) Z,(J) = J [dv]exp[-(1/2)id xd yK(2,y) 4t 1 d zJy] where K(;,?) = ~~(;-y)(-a' - v32 + m2 ) . T h i s can be t r e a t e d as t1(6/6J)1Zo(J)

an

m

dimensional Gaussian i n t e g r a l , namely as a l i m i t i n g f o r m o f (.)

...r h N e x p [ - ( 1 / 2 ) l

viKijvj

1

/ dvl

Jkvk] ( l / d e t K ) e x p [ ( l / Z ) I Ji(K-l)ijJj] as N 4 and one w r i t e s t h e n (**) Zo(J) = e x p [ ( l / 2 ) i d x" 4 Working t h r o u g h momd 4 ~ J ( ~ ) A ( ~ , ~ ) Jwhere ( ~ ) ] / d4yK(2,y")A(7,?) = 6 (x-z). +

Q

( c f . [Fyl;Itl;Cgl;Nsl])

-t

C

I

U

entum space i t can be shown t h a t ( e x e r c i s e - c f . [ F y l ; B q l ; I t l f o r v a r i o u s 4 2 w = / [ d k e x p ( i k . ( X " _ ? ) ) / ( 2 ~ ) ~ ( r+~ m ) ] ( k = ( i k o ,

d e r i v a t i o n s ) (*A) A(;,?)

z0

N

= T = i t and ko = i k o we o b t a i n t h e Feynman propagator (*@) Letting = i A F ( x - y ) = i/ ( d4 k l ( 2 n ) 4 ) e x p [ - i k . ( x - y ) ] / ( k 2 m2 + i E ) ( c f . a l s o

k)). );A -,:(

-

2 t h e i c t e r m corresponds t o a damping f a c t o r i c v / 2 i n t h e Lagran2 g i a n and one can t a k e +k i n t h e e x p o n e n t i a l ) . Note h e r e x - p % t E - x - p , k 2 2 Q k ki, e t c . ( t h e r e l a t i o n s between 3-D and 4-D terms a r e o b v i o u s ) . 552.3-2.4;

0

-1

RRilARK 9.2. equation.

For completeness we s h o u l d say something h e r e about t h e D i r a c Thus ( c f . [ B q l ; I t l ] )

( * b ) i$t = (-ia.v

=

aiak

+ akaio=

t i o n now i s Y

{:

apylJ.

c l a s s i c a l l y one l o o k s a t a wave e q u a t i o n

+ Bm)$ = H9 where 9 i s a 4 - v e c t o r wave f u n c t i o n ,

{ai,ak}

2

0 f o r i # k, Iai,B} = 0, and a: = B = I. A s t a n d a r d n o t a i J v = B, Y = Bai ( i = 1,2,3), {-rl ,Y I = 2guv, *'Y = y0ylJyo, and

Then ( * b ) becomes (*+) (iv'alJ

-

m)$ : ( i y

- m)$

The K-G e q u a t i o n i s o b t a i n e d by m u l t i p l y i n g by ( i y + m).

= 0 (c = h = 1).

One w r i t e s a l s o

2 96

ROBERT CARROLL

0

(9.2)

y

where t h e

5

=

(note

53

=

I 0 i -I); y

i

1;

= ( O-5 i

0

i a r e the Pauli spin matrices i i s o f t e n as

5

o r -ti).

T~

ai

= (5i

ai

5

1

(A

0 -i

0 1

= (l

o),

o), and

a2 = (i

We r e f e r t o [ I t l ; B q l ]

f o r the philos-

ophy o f t h e D i r a c e q u a t i o n and w i l l o n l y mention a few f e a t u r e s here.

Y (*+) and

-

G(i$

ia.$y’

(*m):

-

P

+

I$

= 0,

m ) $ l ( x l ) = 0).

-+

11 t o

such t h a t

?(A)

$ ‘ ( X I ) = % ( x ) t h e n t h e equations

5 = $*ro, a r e

covariant (i.e.

i n particular

An a p p r o p r i a t e Lagrangian here i s

o b t a i n (*+) and ( * m ) .

a t p i and ii = a t l a i t so H

1: =

(A*)

f t d4x independently w i t h r e s p e c t

m)$ and one v a r i e s t h e a c t i o n S =

t o $ and =

and i f $

= yo?*yo,

(iy”(a/ax’’)

4 matrix

= f1 then there e x i s t s a 4 x

First

= A t xu ,

one notes t h a t if e.g. A i s a L o r e n t z t r a n s f o r m a t i o n x ’ = Ax, x ” A:

“1,

The a p p r o p r i a t e c o n j u g a t e f i e l d s a r e

+ m]$ ( j = 1,2,3). j Now t o i n t r o d u c e gauge f i e l d s we’ proceed f i r s t f r o m a p h y s i c s

RrmARK 9.3.

f d3x$[-iyja

=

p o i n t o f view as i n [Cgl;Fa4;Itl;Rel;Ql;Mdl]. 11 t h e e l e c t r o m a g n e t i c s i t u a t i o n where

v

=

We r e c a l l from Example 1.2.

vo - x t

+ vx a r i s e

and A = A.

( ( I P ~ , A ~o)r (q,A) a r e c a l l e d gauge p o t e n t i a l s and g i v e t h e same E and B); we r e c a l l a l s o t h e L o r e n t z gauge where d i v A + q t = 0 ( c = 1 h e r e ) . w r i t e here A’ axlax

v

%

= (q,A)

(xt,-V3x)).

-

w i t h gauge t r a n s f o r m a t i o n s A’ + A p

L e t us

apx (where a’”x

Now c o n s i d e r a c l a s s i c a l wave f u n c t i o n

9 satisfying a 9 i s then

Schrodinger e q u a t i o n ( 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 f i e l d s and Products o f t h e form $*A$

called 9).

9

+

e

e x p ( i e ) $ and one asks about

volve derivatives o f

\t. and

=

a r e unchanged under a phase r o t a t i o n e ( x ) here.

these t r a n s f o r m v i a

Operators A g e n e r a l l y i n -

all$

+

a

$ ’ = exp(ie(x))[aP$

!J

+ i(a,e)$]. I f now one r e p l a c e s a by a gauge c o v a r i a n t d e r i v a t i v e D = a !J ” + ieA ( e b e i n g a charge) and i f A t r a n s f o r m s v i a A A ’ = A’ - ( l / e ) a p e !J 1-1 P U then DU$ exp(ie)Dp$ and 9*Dp$ i s i n v a r i a n t . Thus t h e e l e c t r o m a g n e t i c -+

-+

c o u p l i n g i s i n t i m a t e l y connected t o l o c a l phase i n v a r i a n c e .

Note here t h e

Lagrangian 1: o f ( * b ) i n Remark 9.2 ( a p p l i e d t o f i e l d s ) i s to =:(if = ?(iy’aP

J’A’

-

m)q and t h i s i s now r e p l a c e d by

where J’ = wy’q

(”)

1; =

?(iypD

’

-

i s a (conserved) e l e c t r o m a g n e t i c c u r r e n t .

g i a n 1: i s i n v a r i a n t under A

-+

”lJ

A

- a e

and

c+-+

exp(ie)p.

grangian f o r quantum electrodynamics (QED) i s t h e n 4)FPvFpv ( c f . Remark 8.3).

(Aa)

-

m)q

m ) q = to -

The Lagran-

A “complete” La-

t = to

-

J’A

’-

(l/

One can a l s o imagine o t h e r gauge i n v a r i a n t La-

grangians b u t t h i s one seems d i s t i n g u i s h e d by l e a d i n g t o a r e n o r m a l i z a b l e t h e o r y (a s u b j e c t we w i l l m e r c i f u l l y o m i t ) . t h e o r y based on U ( l ) (= phase r o t a t i o n s ) .

QED i s c a l l e d an A b e l i a n gauge Now c o n s i d e r gauge groups more

GAUGE FIELDS

I n p a r t i c u l a r c o n s i d e r S U ( 2 ) which was

c o m p l i c a t e d t h a n phase r o t a t i o n s .

used i n t h e o r i g i n a l Yang-Mills t h e o r y . t i o n s JI

+

e x p ( i ~ . a / 2 ) J I where

T =

2 97

i n e r e f e r s here t o isospin rota-

u = ( T ~ , T ~ , Ta ~ r e) t h e P a u l i m a t r i c e s o f

T-cx = 1 T . C Y . and JI ( i n t h e sim1 1' JI p l e s t v e r s i o n ) w i l l have a f o r m such as ( ' ) where JIi a r e complex f u n c t i o n s

Remark 9.2,

i s a v e c t o r (a1,a2,a3)

CY

with

$2

( 2 component s p i n o r s ) .

For t h e 4-D t h e o r y one o f t e n d e a l s w i t h composite

s p i n o r s JI = (')

( b i - s p i n o r s ) where

e q u a t i o n (iy'la,

-

X

a r e 2 component s p i n o r s and a D i r a c

m)JI = 0 becomes (A&) i v t = mp -iu.v+;

= (01,02,u3)).

(U

v,x

Now S U ( 2 )

o f 2 x 2 m a t r i c e s A = (( a . .)) 'J w r i t e A = a + i u - a where all 4 2 = a - i a 3 w i t h a + a2 = 1 ( I 4 a gauge t r a n s f o r m a t i o n JI + JI'

ixt

-mx

=

-iu-v

v

3

r o t a t i o n s i n R 3 ( c f . Appendix C) and c o n s i s t s

%

w i t h detA = 1 and A*A = AA* = I. One can = a4+ia 3; a12 = ial+a2,

and t h e u

aZ1 = ial-a2,

form a b a s i s o f SU(2)).

and aZ2 We c o n s i d e r

= GJI on 2-component s p i n o r s w i t h G = e x p ( i T . a

( x ) / 2 ) so t h a t aUJI G(a J I ) + (aPG)JI. Set D = a + i g B = 13 + i g B where 1 bllP - b 3!J BP = (1/2)T-bl, = (1/2)~':' = ( 1 / 2 ) ( ( bij)) where P P %b12 = bP-1bp, b21 3 P 1 2 = bP+ibP, and bZ2 = -bP ( p = 0,1,2,3). = G(DPJI) One wants then DPJI * D;JIl

.Y

-f

(ap

which means

+ (a,G)JI + igB;(GJI) LJ T h i s r e q u i r e s igB;(GJI) = igG(BUJI)

+ i g B ' ) I L ' = G(a,JI)

G(aPJI) + igG(BPJI).

" a r b i t r a r y " JI, and hence g)(aPG)G-' -(l/g)a out

1-1

(A+)

(A+)

B;

= G[BP

( c f . a l s o Remark 9.4).

= G(aP + i g B

-

!J

(a,,G)JI,

+ (i/g)G-'(aPG)]G-'

)JI

=

t o hold f o r

= GBPG-l

+

(i/

Note f o r G = e x p ( i e ) one o b t a i n s B ' !J - BP = e). It i s i n s t r u c t i v e t o work

e which i s t h e c o n d i t i o n i n QED ( g on an i n f i n i t e s i m a l l e v e l

t h i s t o [Ql;Itl;Cgl].

(%

G = 1 + (i/2)r.a)

and we r e f e r f o r

W i t h o u t d e l v i n g i n t o t h e p h i l o s o p h y t o o much a t t h i s

t i m e we s i m p l y remark now t h a t an a p p r o p r i a t e Yang-Mills Lagrangian f o r t h e i n t e r a c t i o n o f a D i r a c b i s p i n o r w i t h a Y-M f i e l d w i l l be (JI

L

(Am)

(cjkm

-

= $(iyPDP

m)JI

-

(1/2)TrFUVFuV where "F:

= avb!

-

(') X

aPb:

as above)

+ gsjkmbibt

i s t h e L e v i C i v i t a symbol which i s 1 f o r even p e r m u t a t i o n s o f t h e i n -

d i c e s , -1 f o r odd permutations, and 0 o t h e r w i s e ) . ngth tensor i s F

= (l/ig)[Dv,DP]

= avBP

-

Note h e r e t h e f i e l d s t r e -

aPBu + ig[Bv,BP]

and observe

t h a t yPDP(:)

= y'yg)

REmARK 9.4.

It i s probably worth w r i t i n g o u t the s i t u a t i o n f o r scalar

(D

Q

DP) makes sense.

fields

v

v

= e x p ( i T - a ( x ) ) v where T = ( T . ) (row v e c t o r ) .

v'

= (pi)

(column v e c t o r , i = 1,2,3)

w i t h Y-M gauge t r a n s f o r m a t i o n s

Here T. generates i s o J s p i n r o t a t i o n s about t h e j a x i s and s a t i s f i e s t h e S U ( 2 ) a l g e b r a r u l e s [T j' Tk] = i E j k m P ( n o t e h e r e [u./2,uk/2] = isjkmum/2 where u = r j ) . Now DP = J j I a t i g T - b and a s u i t a b l e Lagrangian can be based on i: = ( l / 2 ) ( D ! J v ) - ( D P v )

-

+

P

J

!J

(1/4)TrFPvFP"

-

V ( v - v ) ( g i v e n a p o t e n t i a l V , e.g. V ( v - v )

= (1/2)mp-p

+

2 98

ROBERT CARROLL

I n t h i s connection one notes a l s o t h a t f o r quarks qf =

(A/~!)(v.v)~).

-

( q a f ) (column v e c t o r 1,2,3,

a = 0,1,2,3

= m2 = m3 = m, f = f l a v o r = l y . . . y n y a = c o l o r =

ml

and each qaf i s a D i r a c b i s p i n o r ) one has an SU(3) t h e o r y

w i t h gauge t r a n s f o r m a t i o n s (**) q ' ( x ) = Gq = l o r (A = (Ak) a row v e c t o r , k = 1,...,8, matrices

-

see e.g.

[Lpl;Cgl]).

exp[(l/2)giA.A(x)]q(x)

A k = generators

i n co-

o f SU(3) a r e 3 x 3

The c o l o r gauge f i e l d based on (a*) i s

ii

Eu

= gA (A / 2 ) = (g/2)A . A and t h e t r a n s f o r m r u l e i s = G[gP + i G - l ( a u G ) ] uk k u G-' as b e f o r e ( n o t e a l s o from GG-' = I one has (auG)G-l t Ga ( G - l ) = 0 so u

-

B; = GKuG-l

iGa (G-l) VAd

again DP =

t iB

(au

1.1

)

G[EP - i a u ] G - l ) . : (a I t ig ) and f u u

u

The c o v a r i a n t d e r i v a t i v e ,is

=

= q(iyPDu

- m)q

-

(1/4)TrF'"FPV

(summed o v e r f l a v o r s ) ( c f . below f o r t h e f o r m a t i o n o f such Lagrangians).

REmARK 9.5,

L e t us p o i n t o u t here t h a t i n l o o k i n g a t t h e above formulas f o r

c o v a r i a n t d e r i v a t i v e s and Lagrangians e t c . one sees immediate i n t e r p e r t a t i o n i n terms o f c l a s s i c a l d i f f e r e n t i a l geometry.

Covariant d e r i v a t i v e s a r e o f

-

course based on t h e i d e a o f connections ( c f . 53.10) so t h a t e.g. D

lJ

=

aP

t

igP where

trie components o f : i

(g/2)A .A o r

=

u

u

iiP

= gT-bu o r Bu = (g/2).r-bP i d e n t i f y

A:r

w i t h C h r i s t o f f e l symbols

0-4 w h i l e u,w r u n over v a r i o u s s e t s

-

terms l i k e

(here A i s a Dirac index

we a r e t h u s t h i n k i n g o f geometry i n

t h e i n t e r n a l charge space o r c o l o r space, o r whereever).

The Riemann c u r -

v a t u r e t e n s o r i s d e f i n e d v i a RV = a r" - a r" t Y A r" - r Ar" and i n pa8 a PR R ua u 8 Xu ua A6 t h e case o f = (g/2).r.bu f o r example t h i s w i l l correspond t o ( c f . [Cgl]) w

-

aaiBB

u a iia -

t o T r F a B FI R

.

REmARK 9.6,

[iga,iii,]

= ( i g / 2 ) ~ %and ~ Tr(.r.Fa,)(T-FaB)

i s proportional

T h i s w i l l be e x p l i c a t e d i n 510 i n a c o o r d i n a t e f r e e manner. The n e x t s t e p i s t o q u a n t i z e t h e Y-M f i e l d s and one must r e -

gard some o f t h e development h e r e as h e u r i s t i c ( b u t h o p e f u l l y s t i m u l a t i n g ) . We saw how t h e p a t h i n t e g r a l f o r m u l a t i o n i n Remark 8.9 l e a d s t o t h e connect e d Green's f u n c t i o n s G(X"l Remark 9.1). =

1 [&d7i]exp[i/

N

v i a the generating functional ( c f . also

To be more e x p l i c i t l e t us r e c a l l (*) i n t h e f o r m Z ( J ) = i ( J ) 4 4 d x(doq H ( n , q ) t Jp)] = 1 [Clp]exp[i/ d x(C t Jp)] w i t h

-

N

WE(J)

,.. . ,xn)

G(;l,...lgi)

g i v e n by (A) and

via

(0).

We a l s o r e c a l l from Remark 8.

3 t h a t i n t h e e l e c t r o m a g n e t i c f i e l d one uses f i e l d v a r i a b l e s A

ac/ai\

Fi

(=

-AP

f o r the f o f

(A*)

i n § 8 ) ; one s e t s a l s o

IT

P

=

P.

with

ar/aAP.

'TI

=

For QED

i n t h e U(1) gauge f o r m u l a t i o n as i n Remark 9.3 one has an 1; as i n (A@), name l y 1:

=7(ivPau -

m)q

-

(1/4)FuvFu" (Fuv = auAu

-

apAv).

For gauge t h e o r y

however i t w i l l be necessary now t o impose a gauge f i x i n g c o n d i t i o n and work w i t h s t a t e s IL where e.g. a P A IIL) = 0 ( L o r e n t z gauge f o r example); t h i s amP

ounts t o removing redundant degrees o f freedom and w i l l be r e f l e c t e d i n t h e

GAUGE FIELDS

299

p a t h i n t e g r a l f o r m u l a t i o n ( c f . [Cql ;Fa4;Rel ; D , i l ] ) .

There a r e g l o b a l o b s t r u -

-

c t i o n s t o g l o b a l gauge f i x i n g ( G r i b o v a m b i g u i t y

c f . [Svl;Jkl])

and t h i s i s

b r i e f l y discussed l a t e r . Note f o r t h e QED gauge w i t h 9 = 0 one has e.g. no = o and v 3 e n = o (c = - ( I / ~ ) ( ~ F ~ ~ F O + ' F..F i j SO 71P = ai:/a(aoAu) = F ). FPv = - ( 1 / 4 ) 1 Fv;

Thus t a k e now SU(2) Y-M f i e l d s w i t h i: = -1(J1 / 4 ) T r F (a = 1,2,3

Zo(J)

(9.3)

(a'Aav-avAau)

[dAUle

= =

- a Aa

a Aa

say) where Fa''=

P4v x(i:otJ$

iJ

t

Id4xA:(gPVa2-auav)A:;

gEabci'Ac P

" Y1;

d4xc0 =

I

Z(J) =

V

-

p P V

and w r i t e 1 /d4x[a

[dAp]e

Aa-a A a )

u v

v u

i/ d4x(LtJPAP)

Now one would l i k e t o d u p l i c a t e t h e procedure o f ( = ) i n Remark 9.1 i n v o l v i n g detK, A, e t c . b u t t h i s w o n ' t work because e.g. have an i n v e r s e ( n o t e e.g. t i o n operator

-

t o o many f i e l d s ;

c f . [Cgl]).

K

PV

KPV = g P v a 2

Kv i s p r o p o r t i o n a l t o K

PA

-

a a

so Kuvuii::

does n o t a projec-

The gauge freedom has i n v o l v e d i n t e g r a t i n g o v e r

i n p a r t i c u l a r o r b i t s which a r e gauge e q u i v a l e n t and have

f u n c t i o n a l volume must now be f a c t o r e d o u t .

Some examples a r e g i v e n i n [Cg

13 t o m o t i v a t e t h e procedure b u t we s i m p l y i n d i c a t e h e r e t h e general method. Thus e.g.

t h e a c t i o n i s i n v a r i a n t under A

A i - . r / 2 = U[Au..r/2

t

U

+

A' where U = e x p ( - i ~ . e / 2 ) w i t h P

( c f . G i n Remark 9.3,

(l/ig)U-laPU]U-l

G

n,

exp(i7.42)).

The a c t i o n i s c o n s t a n t on t h e o r b i t o f t h e gauge group SU(2) formed by a l l

Ae and one wants t o r e s t r i c t t h e p a t h i n t e g r a l t o a " h y p e r s u r f a c e " which

P I f f a ( A p ) = 0, a = 1,2,3, i s such a hyperi n t e r s e c t s each o r b i t o n l y once. 8 s u r f a c e then f a ( A P ) = 0 s h o u l d have a u n i q u e s o l u t i o n e f o r A g i v e n and P

Since gauge f i x i n g i s g e n e r a l l y o n l y pos-

t h i s i s a gauge f i x i n g c o n d i t i o n .

s i b l e l o c a l l y t h i s procedure i s o n l y h e u r i s t i c ; however i t i s p r o d u c t i v e i n terms o f p e r t u r b a t i o n t h e o r y (see [ J k l ] f o r a good d i s c u s s i o n o f gauge f i x ing).

The d i s c u s s i o n i n [ S v l ] shows t h a t i f t h e c o n d i t i o n s a t m amount t o 4 U t m l f o r example then t o p o l o g i c a l o b s t r u c t i o n s e x i s t .

s t u d y i n g M = S4 = R

The a p p r o p r i a t e i n t e g r a l i n SU(2) i s w r i t t e n [de] = D e f i n e now (**) A[';],

where (Mf)ab

= 6f,/6eb

=

so t h a t Af[A,]

1 [de]6[fa(A;)]

( n o t e detMf # 0 s i n c e fa(A:)

s o l u t i o n and 6 h e r e r e f e r s t o a 6 f u n c t i o n

-

-

= de'

w i l l provide e x p l i c i t f o r -

l e f t i n v a r i a n t Haar measure f o r which [Vil;Hg2] mulas).

3 n1 dea ( d ( 8 0 ' )

e.g.

= detMf

i s t o have a unique formally J des(g(e)) =

J (ae/ag)dgs(g) = ( a e / a g ) l g = O ) .

One checks e a s i l y t h a t A(A';), i s gauge i n 4 variant (exercise cf. [ C g l ] ) and (am) J [dAP]exp(i/ d x C ) = J [de][dAP] J [de][dAP]A (A ) 6 ( f a ( A P ) ) e x p ( i / d 4 xC) ( n o t e Af(AP)6(fa(A:))exp(i/ d4xL)

-

f C

4 Af(AP) and e x p ( i / d x C ) a r e i n v a r i a n t under Au i s independent o f 8 w i t h / [de] t h e

m

-f

AP).

The i n t e g r a n d i n

(0.)

volume one wishes t o f a c t o r out; one

300

ROBERT CARROLL

a r r i v e s a t t h e Fadeev-Popov ansatz i n v o l v i n g t h e c o n s i d e r a t i o n o f ( 0 6 ) Zf

( J ) = I[dAu]detMfs(fa(Au))exp[iI

d4x(C

+ J,A')].

Thus [dAA]

a term detMf6(fa(Au)) i n o r d e r t o e l i m i n a t e t h e

i s m o d i f i e d by

orbital integral.

Further The f u r -

d e t a i l s and d i s c u s s i o n can be found i n [Cgl ;Ga4;Djl;Nsl;Ryl;Pdl].

t h e r a n a l y s i s o f detMf i n terms o f Feynman diagrams e t c . l e a d s t o t h e Fadeev-Popov ghost f i e l d s and i n t h i s r e s p e c t f o r Mf = 1 + L one w r i t e s detMf = exp(Tr[logMf]) = exp[TrL + ( 1 / 2 ) T r L 2 + + ( l / n ) T r L n + ...I = e x p [ I d4 x

...

4

4 Laa(X,X) + ( 1 / 2 ) l d xd YLab(XYY)Lba

RFmARK 9.7.

...I

I n Remark 1.10.7 and 1 10.8 f o l l o w i n g [ T b l ; J a l ]

remarks on Y-M-H f o l l o w i n g [Sagl;Tb2;

L e t us make a few a d d i t i o n a

;Gfl].

Ja1;Eal;Ghl

we discussed

f e l d s i n connection w i t h t h e G-L equa-

some f e a t u r e s o f Yang-Mills-Higgs tions.

+

Y,X)

I n p a r t i c u l a r we g r a d u a l l y i n j e c t more and more machinery

i n t o t h e t h e o r y i n p r e p a r a t i o n f o r t h e general f o r m u l a t i o n i n 53-10. Thus 4 t h e idea now i s t o work o v e r Minkowski space M = R w i t h rl i j = -gij = (-1,

l,l,l)here and t o f i r s t l o o k f o r l o c a l e x i s t e n c e o f s o l u t i o n s o f t h e e v o l u t i o n problem w i t h Cauchy t y p e d a t a s p e c i f i e d a t say t = 0 ( c = 1 - t h e temp o r a l gauge A'

= 0 i s used i n [Eal;Ghl]

f o r example a l t h o u g h f o r c e r t a i n

l o c a l e s t i m a t e s a gauge t r a n s f o r m a t i o n i s u s e f u l ) . group w i t h L i e a l g e b r a v

g

+

5 and

g l ( p , R ) ) so t h a t [ea,eb]

L e t G be a compact L i e

l e t 8 be a r e a l m a t r i x r e p r e s e n t a t i o n o f

fabcec

=

( t h e ea a r e g e n e r a t o r s ) .

(0:

The Y-M po-

valued l - f o r m over M o f t h e form A = A a e dx' = Audx' w i t h u a e )dx' A d x V = F dx' A dx" where F = a A - a A + PV a P" 'V 1!" V l J The n o t a t i o n h e r e and i n §3.10 i s e q u i v a l e n t t o p r e v i o u s n o t a t i o n

tential i s a

c u r v a t u r e F = (Fa

'

[A ,Av].

-

b u t we w i l l n o t always a t t e m p t t o a d j u s t

s i g n s f o r example s i n c e d i f f e r e n t

authors use d i f f e r e n t conventions; anyone who i s s e r i o u s l y concerned can On a f l a t t = to s p a c e l i k e h y p e r s u r f a c e i and F = F . . d x d x j ( i , j = 1,2,3). The

e a s i l y produce a u n i f i e d n o t a t i o n . i n M one w r i t e s A = Ayeadxi = Aidxi

'J

Higgs f i e l d i s a v e c t o r valued f u n c t i o n 9 on t o r space

E

%

M

w i t h values i n t h e r e a l vec-

Rp which serves as a r e p r e s e n t a t i o n space f o r t h e m a t r i c e s Ba.

The c o v a r i a n t d e r i v a t i v e i s D

q = a,p+

u

Auq

( t h e use o f t h i s c o v a r i a n t d e r i -

v a t i v e i s o f t e n r e f e r r e d t o as minimal c o u p l i n g o f A t o 9 ) and i f U i s a smooth G v a l u e d f u n c t i o n on M i t generates gauge t r a n s f o r m a t i o n s o f (A,@

' f e r e n c e a r i s e s from t h e U - l and

(Dug)' =

Tr[-(1/4)FpVFu"]

U(Duv).

-

-' +

' i factors there).

v i a (a+) 9 ' = Up, A ' = UApU

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

-

the sign d i f -

(0.)

F L Y = UFpv

For a gauge i n v a r i a n t Lagrangian one has ( 6 * ) 1; =

(1/2)(Dp9).(D'v)

v a r i a n t polynomial ( i . e .

Ua Uml ( c f . Remarks 9.3-9.4

-

P(9) where P(9) i s assumed t o be an i n -

P(Uq) = P ( 9 ) ) o f degree 5 4 ; T r denotes here (nega-

GAUGE FIELDS

301

a t i v e ) t r a c e o v e r t h e m a t r i c e s B a where e a i s chosen t o be r e a l a n t i s y m m e t r i c and fabci s c o m p l e t e l y a n t i s y m m e t r i c ((DUq)-(D'q)

w i t h Treaeb = 6ab,

= (Du

V ) ~ ( D ' ~ f) o~r some i n v a r i a n t " c o n t r a c t i o n " i n E = Rp). The c o r r e s p o n d i n g 3 H a m i l t o n i a n i s (6.) H = I d x [ n - n / 2 + Tr(EiEi/2 + F . . F . . / 4 ) + (Diq)-(Div)/2 1J 1J + P ( v ) + Tr(AoC)] where Ei = E:ea = atAi - aiAo + [A 0' A i] = Foi, n = a tq + A q = Doq, and C = Caoa = -a . E . + [E.,A.] - (n-eaq)ea. The i n i t i a l v a l u e 0 J J J J c o n s t r a i n t i s C = 0 and (Ei,n) a r e momenta c o n j u g a t e t o (Ai,q). Most o f t h e terms i n H s h o u l d make sense by now ( c f . a l s o 510); we remark ( c f . [ E a l ] ) t h a t t h e energy momentum t e n s o r i s (6.) (D'v)*(D"q)

-

(1/2)n'"(Dav).(Daq)

-

T'"

n'"P(q)

o f t h e equations o f m o t i o c ( 6 6 ) VvF'"

-

= Tr[FPaF:

= -((D'q)

+

(1/4)n'"FaBFaB]

= 0 as a consequence

where aVT'" -eaq)ea

and (D,(D'P))~

=

aP/avk ( h e r e vyFaB = a F + [ A ,F I ) . Note a l s o t h e B i a n c h i i d e n t i t y ( 6 + ) Y aB y aB + v B Fy a = 0. Now one w r i t e s o u t t h e Hamilton e q u a t i o n s i n t h e vyFaB VaFBy +

temporal gauge An = 0 as (9.4)

Dt

Ei

-aj[Ai,:j]-[A.,F.

=

9 n

J

1

.]-((Dilp)-@a~)Oa 1J

For t e c h n i c a l reasons one now s p l i t s E = Eidx i v i a Ei = EiT + EiL ( t r a n s v e r s e = divergence f r e e and l o n g i t u d i n a l = c u r l f r e e ) where

aiEi

T

= 0 and

E

ijk

L c L j Ek = 0 and t h e n EL i s r e p l a c e d by a smoother o b j e c t EC such t h a t Ei = Ei

when t h e c o n s t r a i n t aiE: 11 f o r details).

=

aiEi

= [E.A.]

-

J .J

(n.eaq)e

a

i s s a t i s f i e d (see [Ea

The e v o l u t i o n equations ( 9 . 4 ) can t h e n be w r i t t e n as ( 6 = )

d u / d t = Au + J(u) (where u = (Ai Ei q n ) as a column v e c t o r - h e r e Au i s t h e T The o p e r a t o r A same as t h e f i r s t t e r m i n (9.4) w i t h Ei r e p l a c e d by Ei). 2 generates a 1-parameter group on (HS+l x H s ) where H, i s t h e a p p r o p r i a t e Sobolev space and f o r s 2 1 one can e s t a b l i s h l o c a l e x i s t e n c e and uniqueness o f s o l u t i o n s t o t h e i n t e g r a l e q u a t i o n (+*) u ( t ) = e x p [ A ( t - t ) ] u ( t o ) + 0

ItQ d s e x p [ A ( t - s ) ] J ( u ( s ) ) .

F u r t h e r t e c h n i c a l argument l e a d s t o g l o b a l s o l u -

t i o n s and we r e f e r t o [Eal;Ghl;Sagl;Gfl]

f o r theorems i n t h i s s p i r i t .

p r a c t i c e one i s o f t e n i n t e r e s t e d i n t h e s t a t i c Y-M-H s i t u a t i o n on R ample where (AM) i s independent o f time, A. [1qI2

-

1 1 ( c f . [Jal;Tb2]

= 0, and e.g.

P(v)

%

3

In f o r ex-

(x/8)

and 51.10.

10. GAUGE F I E L D S (WCHEmACICS) AND GE0mECRIC qllANCIZACI0N. We go now t o a more mathematical f o r m u l a t i o n o f t h e gauge i n v a r i a n c e ideas u s i n g t h e l a n guage o f f i b r e bundles, connections, c u r v a t u r e , e t c . ( c f . [Gy2;Trl ;Sxl ; J a l ; Tbl-3;Eal;Fdl;Ul

;Boul;Ghl ;Pbl ; B l e l ;Gy2;Thl ;Burl ;Gul ;Dhl ; D j l ;Ael ; K h l ] ) .

All

o f t h e necessary g e o m e t r i c a l d e f i n i t i o n s and ideas a r e presented i n t h e t e x t

302

ROBERT CARROLL

There w i l l be no space t o g i v e i n f o r m a t i o n about i m p o r t -

o r i n Appendix C.

a n t a p p l i c a t i o n s o f index t h e o r y and cohomology f o r example t o mathematical ;C1; J k l ;Gul ; H t l ;Wol ; S r l ; S S l ]

physics ( c f . [Gyl,2;Bsl

RElllARK 10.1.

f o r this).

The general f o r m a t f o r gauge f i e l d t h e o r y i n v o l v e s a p r i n c i -

4

p a l f i b r e bundle 7 : P -+ M w i t h s t r u c t u r e group G ( c f . Remark 10.2 - M = R w i t h Minkowski m e t r i c f o r example and G = SU(2) f o r Y-M t h e o r y i s t y p i c a l c f . [ B l e l ;Burl;Gul;Gy2;Ttl;Dhl;Djl;Aell). feomorphism y: P G.

+

-

A gauge t r a n s f o r m a t i o n i s a d i f -

P such t h a t y ( u g ) = y ( u ) g and noy =

TI

f o r u E P and g E

The s e t o f gauge t r a n s f o r m a t i o n s w i t h t h e composition l a w o f maps' f o r m

t h e gauge group Y. t i a b l e map

T:

P

+

Generally a " p a r t i c l e " f i e l d i s a s u f f i c i e n t l y d i f f e r e n F such t h a t T(pg) = g-'T(p);

here F i s a d i f f e r e n t i a b l e

m a n i f o l d on which G a c t s f r o m t h e l e f t and i n p r a c t i c e F i s o f t e n Rp.

Q be t h e s e t o f such f i e l d s and one assumes i n p h y s i c s t h a t

Let

for y

T I TOY

E

w i l l be c a l l e d t h e space o f p h y s i c a l c o n f i g u r a t i o n s . Now k k one d e f i n e s a Lagrangian on Q v i a a map L: J (P,F) + R where J i s t h e kk k Y.

Hence Q/Y =

@

j e t bundle ( c f . Appendix C ) and i t i s r e q u i r e d t h a t L ( j ( f ) ) = L ( j ( f ) ) f o r P P9 k p E P and g E G. This i m p l i e s L ( j ( f ) ) i s c o n s t a n t a l o n g f i b r e s so i t can P k k be considered a f u n c t i o n on M and we w i l l want a l s o L ( j ( f ) ) = L ( j ( f o y ) ) 1 P P f o r y E Y. In p r a c t i c e one w i l l d e f i n e L on J (P,F) and i n t h i s s i t u a t i o n 1 a coordinate free d e f i n i t i o n i s possible v i a j ( f ) = (p,f(p),df(p)) so f o r P n ( p ) = x, L ( j p1 ( f ) ) ( x ) = L o ( f ) ( x ) where Lo: @ + R ( e x e r c i s e - c f . [ B l e l ; B u r 11 f o r details).

REmARK 10.2.

The q u e s t i o n o f gauge i n v a r i a n c e i s considered i n Rem.10.8.

L e t us d e f i n e a p r i n c i p a l f i b r e bundle P ( o v e r M w i t h L i e

group G ) as f o l l o w s . late a triple

F i r s t i n o r d e r t o connect n o t a t i o n t o [ B l e l ] we s t i p u -

TI^

= (P,M,G) w i t h n: P M t h e p r o j e c t i o n and G a L i e group For each g E G t h e r e i s t o be a diffeomorphism R * P + P g' (Rg(p) = pg) such t h a t p(g1g2) = (pg1)g2 and pe = p. One r e q u i r e s t h e map P X G P t o be e.g. C" and i f pg = p then g = e ( f r e e a c t i o n ) . The (C") -1 map TI: P + M i s o n t o and n - ' ( n ( p ) ) = Ipg; g E G ; 7 (x)) i s c a l l e d t h e f i b r e -+

( c f . Appendix C ) .

+

over x and TI-'(x) i s d i f f e o m o r p h i c t o G ( b u t t h e r e i s no n a t u r a l group s t r u c t u r e on T I - ' ( x ) ) .

F i n a l l y f o r each x E M t h e r e e x i s t s an open U 3 x and a

diffeomorphism TU: .-'(U)

-+

su(pg) = s u ( p ) g (su: n-'(U) ( o r a c h o i c e o f gauge).

U X G o f t h e form TU(p) = (n(p),s,(p)) -f

G).

where

T h i s TU i s c a l l e d a l o c a l t r i v i a l i z a t i o n

T h i s d e f i n i t i o n can be souped up i n terms o f t h e

c o n s t r u c t i o n s i n Appendix C b u t we o m i t d e t a i l s ( c f . [Cl;Tdl;Khl;Ttl]). us a l s o f o r m a l l y mention t h e t r a n s i t i o n f u n c t i o n s guv: U n V t r i v i a l i z a t i o n s Tu and TV around x

E

My x

E

U

-t

Let

G for. 2 l o c a l

n V, d e f i n e d v i a guv(x)

=

303

GAUGE FIELD THEORY sU(p)sv(p)-'.

These have t h e p r o p e r t i e s ( 1 ) guu(y) = e ( 2 ) gvu(y) = g;i(y)

( 3 ) guv(y)gvw(y)gwu(y) = e . Local s e c t i o n s o f IT^ a r e Cm maps U : M + P such TI o o = 1 l o c a l l y . E v i d e n t l y t h e r e i s a n a t u r a l correspondence l o c a l

that

s e c t i o n s and l o c a l t r i v i a l i z a t i o n s . One can d e f i n e connections i n v a r i o u s ways and we f o l l o w [ B l

REilARK 10.3.

e l ] i n g i v i n g several equivalent d e f i n i t i o n s ( t h e

5

i s demonstrated i n [Bl

e l ] and some r e l a t i o n s w i l l become c l e a r h e r e v i a t h e d i s c u s s i o n and use o f t h e concepts). t a i n e d i n e.g.

M o t i v a t i o n a l d i s c u s s i o n and g e o m e t r i c a l i n t u i t i o n can be ob[Khl ;Blel;Gg2;Ttl;Bsl;Ae1;Spil;Dzl].

f i b r e bundle nG: P

-f

M

Thus g i v e n a p r i n c i p a l

(dimM = n ) ( 1 ) A c o n n e c t i o n a s s i g n s t o e v e r y p

E

P a

subspace H C T (P) ( h o r i z o n t a l subspace) such t h a t T ( P ) = H 8 V ( d i r e c t P P P P P One assumes H depends smoothly on p sum) where V = { x E Tp(P); aX, = 01. P P i n t h e sense t h a t t h e r e w i l l be l o c a l "frames" o f n v e c t o r f i e l d s spanning H ( n o t e 8 does n o t i n v o l v e o r t h o g o n a l i t y n e c e s s a r i l y ) ( 2 ) A c o n n e c t i o n i s P a q v a l u e d 1 - f o r m w d e f i n e d on P (<= L i e a l g e b r a o f G) such t h a t ( i ) F o r A c o n e d e f i n e s a ( v e r t i c a l ) v e c t o r f i e l d A* v i a A* = Dt(pexp(tA))It=O (see P Then w(A*) = A i s r e q u i r e d . ( i i ) L e t Ad : be d e f i n e d Appendix C ) . p&. as t h e homomorphism o f g induced by Adg: G + G: g ' + 9;'g-l ( i . e . A d = (Ad g g)*; Adg i s sometimes c a l l e d I and a good d i s c u s s i o n o f t h i s map can be g found i n [Hg2,3;Tdl;Wrl] - c f . a l s o L e m a 10.4 below and l e t us n o t e t h a t

E

< + 9"

?+

+ Ad . G + GL(c) g i v e s a homomorphism Ad* = ad: g l ( 5 ) where (adA) g' One r e q u i r e s t h a t w ( R * X ) = Ad -1 w (X) f o r a l l g E G, p E (B) = [A,B]). Pg g 9 P P, and X E T ( P ) ( i . e . R*w = Ah w). We n o t e i n comparing t h e s e two d e f i P 9 g-1 n i t i o n s t h a t i f w i s a c o n n e c t i o n 1 - f o r m t h e n H = { X E Tp(P); w (X) = 01. P P P F o r completeness we s t a t e

Ad: g

CmmA

10.4.

momorphism.

Pttaod:

Then q * :

$+

+

H a l o c a l ( a n a l y t i c ) ho-

i s a L i e a l g e b r a homomorphism.

L e t 5 E Ge so t h a t 9,s

recall that = (TT )

L e t G and H be L i e groups w i t h q : G

= & ( 5 ) = Tq(5) E He (q* = (q*),

i s i d e n t i f i e d w i t h Ge as i n Theorem C37).

h e r e and we

Now l e t X w i t h X

9

5 be t h e l e f t i n v a r i a n t v e c t o r f i e l d generated by 5 ( T i s t h e t a n -

g e g e n t space f u n c t o r as i n Appendix C). ( i . e . Ip(gp) = q ( g ) \ o ( p ) ) we have (*) T ( T ~ ( o~ q) ) e s = dTq(g)q*5 = Yq(g) f i e l d generated by 9,s.

Then s i n c e P o

T

=

9 T9(g) = ( T v ) ~ ( T T ~ ) &T(P = o

o IP l o c a l l y

5 = (v,) 9 X 9 T g e where Y i s t h e l e f t i n v a r i a n t v e c t o r

Since ( i n an obvious n o t a t i o n ) & [ X , X ]

=

[Y,Y],

an argument s i m i l a r t o t h a t b e f o r e Theorem C37, we have 1p,[5,5]

=

[~p,S,v~Sl.

by

A t h i r d equivalent d e f i n i t i o n o f connection ( c f . [ B l e l ] f o r d e t a i l s ) assigns

304

ROBERT CARROLL

t o each t r i v i a l i z a t i o n ( o r gauge c h o i c e ) TU: IT-'(U)

-+

valued 1-

U X G a

above t h e n one r e q u i r e s (A) ) ) f o r a l l Y, E Tx(M) (we here l e f t t r a n s l a t i o n ( c f . Appendix C ) and Adg-l(U,V)wU(Yx) = g ~ ~ w u ( Y x ) g u so v

(t,'becomes-1i n a more

"casual" n o t a t i o n v a l i d f o r m a t r i x groups ( e ) wv = gUVdgUVt guvwuguv.

The

1-forms wu a r e c a l l e d gauge p o t e n t i a l s i n p h y s i c s and we w i l l g i v e examples below r e l a t i n g t h i s t e r m i n o l o g y t o t h e c l a s s i c a l p h y s i c s language.

REmARK 10.5.

We go n e x t t o t h e c u r v a t u r e o f a connection.

L e t Ak(N,g) be One

t h e s e t o f g valued k-forms on N (N a m a n i f o l d and g a L i e a l g e b r a ) . defines a bracket f o r 9

and

E Ai

by (6) [9,J,](Xl,...,X

J, E A'

1 (-1 )"['+' ( x u ( l ) . . ¶ x u (i1) ~ ~ ( x u ( i t l - .,xu( Y

t a t i o n s o f (1,2

,... ,itj) and

i l y t h a t [J,,v]

= (-l)ij[9yJ,ly

x],9]

= 0

(x

9.

t h e XI

i + j) = (l/i!j!) u runs o v e r permu-

a r e v e c t o r f i e l d s on N.

One checks eas-

(l)iK~b,JI1,~l t (-l)kj[[x,~l,JI1

k

A ) and d[v,J/]

E

i+ j ))Iwhere

= [dp,J,]

+ (-l)i[9ydJ,].

t

(-1)ji"JIy

Now g i v e n a connec-

one w r i t e s X E T ( P ) as X = XH t X v where k P k H I f 9 E A (P,g) d e f i n e 9 H E A (P,g) by (XH = X )

t i o n 1-form w on nG = (P,M,G) n*(Xv) = 0 and w(XH) = 0. H H VI (X1,. ,Xk) = 9(X1 ,. . ,Xk).

u-

..

.

(P,g)

The e x t e r i o r c o v a r i a n t d e r i v a t i v e o f 9 i s (*) 1 2 E A (P,g) i s ( m ) !Jw = DWw E A

The c u r v a t u r e o f w

D 9 - (dp)H E Aktl(P,g).

(Ow i s a l s o c a l l e d t h e f i e l d s t r e n g t h ) .

= !Jw = dw t (1/2)[w,w].

f o r Y,Z E Tp(P)

To see t h i s we n o t e f i r s t .that (1/2)[w,w](Y,Z)

- [w(Z),w(Y)])

(1/2)([w(Y),w(Z)]

(*A)

Then one shows t h a t (**) DWw

= [w(Y),w(Z)]

=

so i t i s necessary t o p r o v e

dw(YH,ZH) = dw(Y,Z) + [w(Y),w(Z)].

I f Y and Z a r e h o r -

i z o n t a l t h i s i s immediate s i n c e w(Y) = w(Z) = 0 and Y = Y H y Z = ZH.

If Y

and Z a r e v e r t i c a l we can suppose Y = A* and Z = B* f o r A,B E ( c f . 10.1). P P Then however dw(Y,Z) = A*w(B*) - B*w(A*) - w([A*,B*]) = -w([A*,B*]) = -w([A,B]*)

-IA,B]

=

and n o t e e.g. zero).

=

-[w(A*),w(B*)]

= -[w(Y),w(Z)]

(exercise

Hence b o t h s i d e s o f =

?P

(*A)

where

vanish.

? is

F i n a l l y i f Y i s v e r t i c a l and Z i s

the horizontal l i f t o f a vertical f i e l d

P X on M ( c f . below) and Y = A*. Then dw(Y,Z) = A*w(?) P = 0 s i n c e w ( r ) = 0, w(A*) = A i s c o n s t a n t , and [A*,?] (*A)

v a n i s h and (**) i s proved.

- Tw(A*) - w([A*,?]) = 0 ( c f . below).

Thus

We remark t h a t i f X i s a v e r -

t i c a l f i e l d on M t h e r e e x i s t s a unique v e c t o r f i e l d -4

cf. [Blel]

t h a t a v e c t o r f i e l d A* a c t i n g on a c o n s t a n t w(B*) = B w i l l be

horizontal take Z

both sides o f

-

y on

P (horizontal l i f t )

Y

such t h a t w(X) = 0 and n*(X ) = X f o r p E P; t h i s i s a l g e b r a i c a l l y c l e a r P .(PI from t h e f a c t t h a t IT*: ( M ) must be an isomorphism and smoothness HP -+ T d P ) can be e a s i l y shown ( e x e r c i s e - c f . [ B l e l ] ) . The v a n i s h i n g o f [A*,?]

305

GAUGE FIELD THEORY

-

Dtvt-1 .(X)

,u

f o l l o w s from [A*,X]

=

= 0 where v t = p e x p ( t A ) i s t h e 1-parameter

group of diffeomorphisms generated by A* ( e x e r c i s e

-

c f . P r o p o s i t i o n C29).

RRIIARK 10.6. The B i a n c h i i d e n t i t y ( o r homogeneous f i e l d e q u a t i o n s ) i s g i v e n w w by D R = 0 where w i s a c o n n e c t i o n f o r m w i t h c u r v a t u r e SLw; i n f a c t dQw = To see t h i s n o t e t h a t s i n c e w vanishes on h o r i z o n t a l v e c t o r s DwQw

[f2w,w].

= 0 f o l l o w s from dnw = [Rw,w]

[w,w])

-

= (1/2)[dw,w]

( n o t e [[w,w],w]

(recall

= 0).

Dwv

Now dQw = d(dw + ( 1 / 2 )

=

= [ d w , ~ ] = [dw + ( ~ / ~ ) [ w , w ] , w ] = [fZw,w]

(1/2)[w,dw]

We can express a l l t h e s e q u a n t i t i e s l o c a l l y i n terms

of t h e gauge p o t e n t i a l s wu = ufiw ( t a k e h e r e u t o be a l o c a l s e c t i o n , =

1, w i t h corresponding t r i v i a l i z a t i o n T U ( u ( x ) g ) = ( x , g ) ) .

a?

gauge p o t e n t i a l s a r e a s s o c i a t e d t o f i e l d s t r e n g t h s

flu

= dwU

(l/?)[w,w]) 1 E A (N,g), where

v

o u

nu =

~$2’ and ( * a )

+ (1/2)[wu,wu]. Indeed by r u l e s a l r e a d y e s t a b l i s h e d RU = ufi(dw + = d(4fiw) + (1/2)[a~w,ufiw]. I n terms o f m a t r i x groups G, w i t h v J, E AJ(N,g)

as b e f o r e , one has

(*&I [v,$]

= q A $

-

(-l)ij$

A

v

and J, a r e t h o u g h t o f as m a t r i c e s o f R v a l u e d forms and q A J, i s mat-

r i x multiplication with the entries multiplied via

(exercise

I t f o l l o w s t h a t (*+) Qw = dw + w A w, RU = dwU + wu A wu, dnU =

IT

Thus t h e wu =

$A

-

wu

wu

A

%

54,

-

c f . [Blel]).

= g;$luguv,

and

(Bianchi).

EMtAmPtE 10.7. Given M = Minkowski space w i t h G = U ( 1 ) = { e x p ( i e ) , e E R 1 and g = U ( 1 ) = { i a ; a E RI, l e t w be a c o n n e c t i o n 1-form on P and uu a l o c a l 1 s e c t i o n . Then wu = ufiw % - i A E A (U,R) corresponds t o t h e v e c t o r p o t e n t i a l li 2 and t h e f i e l d s t r e n g t h % Fu = -dAU E A (U,R). Given a n o t h e r l o c a l s e c t i o n

u,, one shows e a s i l y t h a t Fu = Fv ( n o t e i n p a s s i n g t h a t u ( x ) = u u ( x ) g u v ( x )

exercise).

Indeed f r o m

( 0 )

wv

=

gUVdgUV -1

giidgUV + w

=

-

since

+ g -1 uv~uguv U Then g;:guv = I i m p l i e s d(gUV)gUV + giVdgUV = 0 o r dg; = -2 -1 -2 -gUVdgUV. Hence -d(guvdguv) = gUVdgUV A dgUV = 0 and hence dwU = dwV o r Fu = Fv. T h i s says t h a t f o r A b e l i a n G one expects t h e Fu t o determine a w e l l

U(1) i s A b e l i a n .

d e f i n e d f i e l d s t r e n g t h on M.

Now c o n s i d e r a n o n a b e l i a n G (e.g.

w i t h f i e l d s t r e n g t h s nu as i n Remark 10.6. eral

Q,

#

5.

Again

54,

=

gu$,v2,

G = SU(2)) b u t i n gen-

Hence one t h i n k s o f t h e f i e l d s t r e n g t h as t h e c u r v a t u r e slw

which l i v e s on P ( n o t M ) .

REmARK 10.8,

We w i l l make a few more comments now about t h e geometric formu-

l a t i o n i n o r d e r t o g i v e a l l t h e o b j e c t s i n Remark 10.1 an i d e n t i t y i n t h i s context.

I f G a c t s on F ( F i s o f t e n a v e c t o r space) f r o m t h e l e f t one forms

a r i g h t G space s t r u c t u r e on P X F (P

Q

(P,M,G))

by w r i t i n g ( p , f ) - g = (p.g,

g - l f ) and P XG F = ( P X G ) / G i s a v e c t o r bundle w i t h p r o j e c t i o n nF: P XG F +.

M d e f i n e d by vF((p,f),G)

= ~ ( p = ) x.

The f i b r e s IT;’(x)

a r e homeomorphic

306

ROBERT CARROLL

t o F (under q ( f ) = (p,f).G: maps

P

T:

F

We denote by C(P,F)

n-'(x)).

-+

F such t h a t T(Pg) = g"T(p)

-+

-

o f s e c t i o n s of P XG F

exercise

-

sT(pG) = ( p , ~ ( p ) ) - G where x

-

(c(P,F)

i s isomorpi.lic t o t h e space

n o t e t h e s e c t i o n sT One says f : P

pG).

t h e space o f

T

-+

i s determined by

P i s an automorphism i f

+

i t i s a diffeomorphism such t h a t f ( p g ) = f ( p ) g ; t h i s induces a diffeomorph-

ism

F:

M

gn(p))= n ( f ( p ) ) .

M g i v e n by

+

f" =

automorphism f such t h a t

%

= Ad

Adg w i t h (Ad),

-

we again denote by

Y

the

If one l e t s G a c t on i t s e l f v i a g - g ' = I g g '

s e t o f gauge t r a n s f o r m a t i o n s ) . = gg'g-l ( I g

A gauge t r a n s f o r m a t i o n o f P i s an

lM ( c f . Remark 10.1

9

e t c . ) an ( a n t i ) i s o m o r p h i s m C(P,G)

fT: P

2

Y

P: f ( p ) = p z ( p ) ( n o t e e.y. f T (pg) = pgT(pg) = p g g - l T ( p ) g = pT(p)g = f T ( p ) g ) . There a r e many c a u t i o n s t o observe here i n t h e geometry and n o t a t i o n ( c f . [ B l e l ; B s l ] ) and we w i l l n o t

can be w r i t t e n down v i a

C(P,G)

T E

n,

-+

We remark t h a t i f f E

t r y t o develop t h i s here.

Y

and w i s a connection 1 -

-

form then f * w i s a l s o a connection 1 - f o r m ( s t r a i g h t f o r w a r d one w r i t e s C f o r t h e space o f c o n n e c t i o n 1-forms on P. P i s C(P,i) where t h e r e p r e s e n t a t i o n G g

+

Ad

2

Y

(we o m i t t h e d e t a i l s

down an e x p o n e n t i a l map Exp: C(P,T) C(P,;)

-+

GL(c) i s t h e a d j o i n t r e p r e s e n t a t i o n

Then i n a c e r t a i n sense ( c f . [ B l e l ] ) C(P,c)

g' t h e L i e a l g e b r a o f C(P,G)

%

-+

-f

c f . [ B l e l l ) and

The gauge a l g e b r a o f

-

can be considered as one can s i m p l y w r i t e

C(P,G) v i a Exp(H)p = expH(p) f o r H

E

Now g o i n g back t o Remark 10.1 l e t P

and check t h e necessary f a c t s ) .

and l e t G a c t on F v i a GL(F); P XG F can be d e f i n e d v i a maps T : P 1 1 F as b e f o r e . Consider Lagrangians d e f i n e d on J (P,F) ( J (P,F) % I ( p , f , e ) , (P,M,G)

e:Tp(P)

+

F l i n e a r } ) here w i t h t h e n a t u r a l m a n i f o l d s t r u c t u r e ( e x e r c i s e -

c f . Appendix L ) .

We say L: J 1 (P,F)

o ( R g - l ) * ) = L(p,f,e)

(9-

w e l l d e f i n e d Lo: C(P,F)

-f

n,

-+

e

R i s a Lagrangian i f L(pg,g-l-f,g-l-

G a c t i o n on F) and i n t h i s s i t u a t i o n t h e r e i s a

C"(M) g i v e n f o r x E M, J/

E

C(P,F),

and p

E

P with

n ( p ) = x by c 0 ( J / ) ( x ) = t(p,J/(p),dJ/p).

To see t h i s one must show L(p,J/(p), Since J/ o R = g-'-J/ we dJ/ ) i s independent o f t h e c h o i c e o f p E n - ' ( x ) . P 9 o (R )* = g-l.dJ/ P o r d\llPg = g-'*dllp o (Rg-l), so C(pg,J/(pg),dipg) have dJ/ = L(pg,Egl.

J/(pg,g-l-d$p

o ( R g - l ) * ) = L(p,Q(p),d$p)

as r e q u i r e d .

e a s i l y ( c f . [ B l e l ] ) t h a t G i n v a r i a n t L ( i . e . C(p,g-f,g.e) n e c e s s a r i l y g i v e r i s e t o gauge i n v a r i a n t E

Y -

f: P

+

P, J / : P

-f

c0

( i . e . Lo($) = L o ( ( f

F, ( f - l ) * J / ( p ) = J / ( f - ' ( p ) ) ) .

One sees

= L(p,f,e))

-1

do n o t

)*$) f o r f

To remedy t h i s one i n -

troduces connections as i n physics and i n o u r p r e s e n t geometrical c o n t e x t t h e r e l e v a n t theorem i s

&HEQ)REIII 10.9,

L e t L: J(P,V)

f u n c t i o n L: C(P,V)

X C

-+

-+

R be a G i n v a r i a n t Lagrangian and d e f i n e a

C"(M) by L($,w) = L(p,J/(p),DwJ/,).

d e f i n e d and gauge i n v a r i a n t i n t h e sense t h a t L ( f V , f * w )

Then L i s w e l l = L(ll,w)

for f

E

Y.

307

GAUGE FIELD THEORY

The p r o o f i s e s s e n t i a l l y 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 and we r e f e r t o [ B l e l ] f o r details.

T h i s d i s c u s s i o n has p l a c e d some elementary gauge f i e l d i d e a s

i n a g e o m e t r i c a l c o n t e x t f r o m which one can develop a m a t h e m a t i c a l l y r i g o r o u r t h e o r y t o supplement t h e p h y s i c s ( c f . [Blel;Bsl;Gy2]). t o a v o i d becoming e n t a n g l e d i n n o t a t i o n and f o r m a l i s m d i f f e r e n t i a l geometry

-

-

One s h o u l d t r y a c o n s t a n t danger i n

b u t t h e reward i s g r e a t s t r u c t u r a l beauty and i n -

s i g h t o f which we hope a t a s t e has been conveyed here.

Such s t r u c t u r e i s

becoming i n c r e a s i n g l y i m p o r t a n t i n f o r m u l a t i n g many p h y s i c a l t h e o r i e s today. There i s today ( a g a i n ) a f r u i t f u l and f a s c i n a t i n g i n t e r a c t i o n between mathem a t i c i a n s and p h y s i c i s t s and perhaps t h e danger o f o v e r h a s t y f o r m a l i z a t i o n

w i l l be circumvented.

Other g e o m e t r i c a l machinery based on s y m p l e c t i c geom-

e t r y a r i s e s i n what i s c a l l e d geometric q u a n t i z a t i o n and we w i l l t r y t o g i v e a b r i e f t a s t e o f t h a t as w e l l ( c f . [Gul;Srl;SS1;Wol;Soul]).

L e t us p o i n t

o u t however t h a t t h e r e w i l l be no a t t e m p t t o say a n y t h i n g about s t r i n g s and s u p e r s t r i n g s ( c f . [Gxl]),

t w i s t o r t h e o r y ( c f . [ P l l ; C c l ;Gyl,2]),

etc.

Suf-

f i c e i t t o say t h e r e i s a g r e a t deal o f marvelous i n t e r a c t i o n between phys i c s and geometry, a l g e b r a i c t o p o l o g y , a l g e b r a i c geometry, e t c .

REmARK 10.10.

L e t us g i v e a few ideas about geometric q u a n t i z a t i o n f o l l o w -

i n g [Wol;Gul;Srl;SSl;Soul

;Htl].

We w i l l t r y t o a v o i d e x c e s s i v e g e o m e t r i c a l

machinery.

H e u r i s t i c a l l y ( f o l l o w i n g [Wol]) c o n s i d e r c l a s s i c a l phase space In = T*M w i t h volume element E~ = dpl A dpn A dq 1 A dqn (w = dpi A dqi i s 2 t h e s y m p l e c t i c form). L e t H = L (In) be t h e obvious H i l b e r t space w i t h (v,

1

J / ) = ( 1 / 2 1 r ) ~ lI ~~

(h = 1 ) and one wants t o l o o k f o r observables as a c o l l e c t i o n o f s e l f a d j o i n t o r H e r m i t i a n o p e r a t o r s on H ( t h e elements o f H c o r $ E ~

respond t o s t a t e s ) . E

H via

~p +

Thus c l a s s i c a l observables f E C m ( N ) a c t on s u i t a b l e

-iXp where

v

X f i s t h e v e c t o r f i e l d determined by Xf J w + d f =

0 ( c f . 51.7 and Appendix C ) .

Thus i n l o c a l c o o r d i n a t e s Xf = (af/ap,)(a/aqa)

-(af/aqa)(a/ap,) 'L I d f i n 51.7 o r Appendix C ( r e c a l l I: dH + IdH, I: w: 5 1 2 where w ( Q ) = w ( Q , C ) i n 51.7, - { F , H I = (F,H) = (aH/api)(aF/aqi) - (aH/ 5 2 aqi)(aF/api) = w (IdH,IdF), e t c . ) . Note t h a t a f a c t o r o f 2 occurs i n t h e 2 d e f i n i t i o n o f I i n [Wol] ( i . e . 2w i s used) and t h i s changes { , I by a f a c -f

1

t o r o f 2, e t c .

We can s i m p l y i g n o r e t h i s w i t h o u r n o t a t i o n .

L e t us r e c a l l

h e r e a l s o t h e L i e d e r i v a t i v e i n Remark C.28 and t h e formula Lx = i ( X ) o d t d o i ( X ) ( i ( X ) w = x J w and i ( X ) d f = ( d f , X ) = X ( f ) ) .

Xf J w = - d f ( e x e r c i s e dp,)

A dqa

- 1 dp,

-

n o t e f r o m Appendix C X f J

A X f -1 dq,).

1 dp,

Then one checks t h a t A dqa =

Then e v i d e n t l y L(Xf)w = X f J dw

1 (Xf +

J

d(Xf J w)

308

=

-

ROBERT CARROLL

d ( d f ) = 0 and t h e f l o w qf generated by X f determines l o c a l 1-parameter t pf % g ha = af/ap,, and

-

f a m i l i e s o f canonical t r a n s f o r m a t i o n s ( c f . 51.7

ba

= -af/aqa w i t h f

Q

t e r m i n o l o g y o f [Wol] and w2 = de where 0 Now back t o q u a n t i z a t i o n . A

s a t i s f y (*.)

4

2

Note a l s o I f , g l = Xf(g) = w (X

H, e t c . ) .

The o p e r a t o r s

=I

p,dq".

iYf^ corresponding

A

= - i k where k = t f , g l and t h e corresponding f

do t h i s ( e x e r c i s e ) .

4

Adding a f u n c t i o n i n t h e form f

s a t i s f a c t o r y s i n c e t h e b r a c k e t c o n d i t i o n (*') -i[Xf(q)

(A*) ? ( q ) =

will

-iXf

However i t i s n o t a s a t i s f a c t o r y q u a n t i z a t i o n s i n c e Xf

= 0 whenever d f = 0.

takes

-

t o g , f should

A

[f,g]

X ) i n the 9

f'

-

i(Xf

_I e ) q ]

+

Q

Xf + f i s also n o t

w i l l n o t hold.

However i f one

w2 = de - e b e i n g r e -

f q (where

f e r r e d t o as a s y m p l e c t i c p o t e n t i a l ) then t h e c o n s t a n t s w i l l a c t by m u l t i -

2

plication i n L

(m)

and (*m) holds ( e x e r c i s e ) .

T h i s c o n s t r u c t i o n however de-

pends on t h e c h o i c e o f e and f cannot perhaps be determined g l o b a l l y . i n t r o d u c e gauge t r a n s f o r m a t i o n s : When e i s r e p l a c e d by e l = Then w i t h

changes 9 by q ' = e x p ( i u ) q .

f^

f"

as i n

(A*)

e + du

exp(iu)?(q) =

Hence

one f^(ql)

and

i s w e l l defined, independently o f t h e c h o i c e o f gauge p o t e n t i a l . The i n t u i t i v e background has now been sketched i n Remark 10.

REWRK 10.11.

10 b u t more must s t i l l be done, f i r s t t o remove t h e a m b i g u i t y i n phase, and l a t e r t o r e s t r i c t H ( o r t h e admissable wave f u n c t i o n s ) .

We w i l l o n l y ad-

dress t h e f i r s t p o i n t h e r e ( f o l l o w i n g [Wol]) and t h e key l i e s i n t h e expression

(A*)

which can be i n t e r p e r t e d v i a a H e r m i t i a n l i n e bundle w i t h a con-

n e c t i o n v y p o t e n t i a l 8, and c u r v a t u r e ( 1 / 2 ) Q = w

2 ( e x i s t e n c e i s discussed

To see what t h i s means l e t B be a l i n e bundle over

below).

(a v e c t o r

bundle w i t h t y p i c a l f i b e r C) and l e t (

, ) be a H e r m i t i a n s t r u c t u r e H: B

R: b

as a f u n c t i o n o f b ( c f . Appendix C ) .

(b,b) which i s smooth ( i . e .

-f

C")

-+

A connection v on B i s d e f i n e d t o be a map vx on C"(B) = C" s e c t i o n s o f B ( X E V(m) = complex v e c t o r f i e l d s on m) such t h a t

-

gvy ( f , g E CF(m)

complex valued f u n c t i o n s

x ( f ) s + f v X s ( X E v(m), s E c"(B),

( X E V(m),

s,sl

E

C"(B)).

f E c"(m))

-

(AA)

( 1 ) vfx+gY = f o x

and X,Y E V(m)) ( 2 ) v x ( f s ) =

( 3 ) vx(s+sa) = vx(s) + vx(s')

Note t h a t t h i s i s r e l a t e d

t o t h e c l a s s i c a l co-

v a r i a n t d e r i v a t i v e v by v ( s ) = Va/axv(s) B dx' and V x ( s ) = ( V ( s ) , X ) g. [GyZ]). (A@)

X(s,sl)

(see e.

The connection i s c o m p a t i b l e w i t h t h e H e r m i t i a n s t r u c t u r e when = ( v x s y s ' ) + ( s , v x s I ) f o r X E V(m) and s , s ' E C"(B).

v a t u r e o f V i s t h e complex 2-form

v ] -

+

)s (X,Y

E V(m),

s

on

m

The c u r -

d e f i n e d by ( ' 6 ) 2n(X,Y)s = i ( [ v x ,

Cm(B)).

Note t h a t one c o u l d r e f e r everyy [X,YI t h i n g h e r e back t o a p r i n c i p a l bundle f o m a t w i t h say G = U ( l ) b u t t h i s V

E

would s i m p l y be an e x e r c i s e i n n o t a t i o n which we p r e f e r t o o m i t .

Further-

309

GAUGE FIELD THEORY

more t h e p r e s e n t approach g i v e s an i n t r o d u c t i o n t o some c l a s s i c a l g e o m e t r i c Now l e t (U,$)

p o i n t s o f view.

so J/

be a l o c a l t r i v i a l i z a t i o n ($: U X C

i n D e f i n i t i o n C.13) and w r i t e s = $ ( - , l )

% T ';

be t h e 1 - f o r m on U d e f i n e d by (X J 6 ) s = i v x s ( X E V(U)). Vxs' = [ X ( f )

(A+)

-

o f Remark 10.10 we see t h a t

(A*)

meaning i n terms o f

e

Such a B i s c a l -

i ( X J ~ ) f ] sand p u t t i n g t h i s i n

(1/2)s2 = dB ( t h i s i s t h e o r i g i n o f t h e f a c t o r o f p a r i n g now w i t h

Let

I f s ' = f s i s any o t h e r smooth s e c t i o n one checks

l e d a potential l-form. easily that

n-'(U),

+.

( u n i t section).

QJ

(A&)

yields

2 mentioned above). (A*)

Com-

has a g e o m e t r i c a l

B = c o n n e c t i o n p o t e n t i a l and c u r v a t u r e ( 1 / 2 ) n

%

w

2

.

Now t h e q u e s t i o n i s whether one can produce a l i n e bundle B o f t h e t y p e i n d i c a t e d and t h i s i s answered by W e i l ' s i n t e g r a l i t y c o n d i t i o n ( c f . [Wol]). T h i s can be phrased as f o l l o w s . smooth c u r v e w i t h t a n g e n t T.

Suppose B e x i s t s and l e t y: [0,1]

+

m

be a

A s e c t i o n s ' o v e r y i s s a i d t o be p a r a l l e l i f

VTs' = 0 which means l o c a l l y ( i n say U ) t h a t t h e r e i s a d i f f e r e n t i a l equation

(Am)

T(v) = i ( T

i s assured).

J B)V

w h e r e g s = s ' ( t h u s l o c a l existence-uniqueness

Suppose now y ( 0 ) = y ( 1 ) w i t h y = aX,(X,cU ) ; t h e n f r o m

s ' ( y ( 1 ) ) = e x p ( i l B )s'(y(O)) = e x p ( i /

Y

y divides a closed oriented

that exp(ilZ

a) =

2 surface

=, R C

)s'(y(O))

(Am)

by S t o k e ' s theorem.

If

i n t o 2' p a r t s El and C2 one sees

1 ( c f . [Wol]) and hence

IZ

R = 2nn.

This should h o l d f o r

2 - c y c l e s e t c . and one a r r i v e s a t t h e i n t e g r a l i t y c o n d i t i o n : The Chern c l a s s 2 o f B ( w i t h c o n n e c t i o n V ) :cohomology c l a s s o f R i n H (W,R) s h o u l d be " i n t e g r a l " ( c f . [Wol;Ssl]

and see a l s o [Gy2;Ttl;Vml;Dyl]).

i n f a c t necessary and s u f f i c i e n t .

This c o n d i t i o n i s

Such a bundle B i s c a l l e d a p r e q u a n t i z a -

t i o n bundle b u t much more i s needed s t i l l and we r e f e r t o [ G a l ; S r l ; S s l ; H t l ; Wol] f o r a l l t h a t .

This Page Intentionally Left Blank

31 1

APPENDIX A INTRODUCTION TO L I N E A R FUNCTIONAL ANALYSIS

We w i l l o u t l i n e h e r e some b a s i c ideas o f f u n c t i o n a l a n a l y s i s and prove t h e It i s n o t much h a r d e r t o cover l o c a l l y convex spaces,

essential things.

r a t h e r than o n l y Banach spaces, so we do t h i s ( f o l l o w i n g [ C l ] ) . The i n c r e a sed p e r s p e c t i v e and i n s i g h t seem w e l l w o r t h any e x t r a e f f o r t and i n f a c t some r e s u l t s f o r Banach spaces a r e proved more n a t u r a l l y i n t h e general cont e x t i n a way which b e t t e r d i s p l a y s t h e i r meaning and s t r u c t u r e . DEFINICI0N A.1,

A ( l e f t ) v e c t o r space F o v e r a f i e l d K ( K = R o r C h e r e ) i s

a c o l l e c t i o n o f elements s a t i s f y i n g ( A ) F i s an a b e l i a n group under There i s a map (a,x) BX,

-+

ax: K X F

-+

t

(B)

F such t h a t a(x+y) = axtay, (a+B)x = a x +

and a(Bx) = (aB)x (C) The m u l t i p l i c a t i v e . i d e n t i t y 1 o f K a c t s as an i d -

e n t i t y i n F i n t h e sense t h a t l x = x f o r a l l x DEFZNZCI0N A.2,

E

A t o p o l o g i c a l v e c t o r space (TVS

F.

-

we suppress t h e word l e f t

from now on) i s a v e c t o r space F endowed w i t h a topology s a t i s f y i n g ( A ) (a,x) ax: K X F + F i s continuous (B) (x,y) x -x: F F i s continuous.

-+

-+

-+

xty: F X F

-+

F i s continuous ( C )

-+

In a d d i t i o n we s h a l l assume t h a t a l l TVS a r e Hausdorff spaces u n l e s s o t h e r w i t h x # y, t h e r e a r e open

wise s t a t e d ( i n a Hausdorff space f o r any x,y,

-

sets

see below

DEFINICI0N A.3,

-

U,V w i t h x E U and y

E

V such t h a t U n V = @ (@ = empty).

A seminorm i n a v e c t o r space F o v e r K i s a f u n c t i o n p: F

R satisfying (A) p(xty) 5 p(x)

t p ( y ) (B) p(ax) = I a I p ( x ) . p ( x ) = 0 i m p l i e s x = 0 then p i s c a l l e d a norm.

*

If i n addition

We r e c a l l now t h a t a t o p o l o g y on a c o l l e c t i o n o f o b j e c t s F i s d e f i n e d by t h e p r e s c r i p t i o n o f what a r e t o be open sets, where, t o q u a l i f y as a f a m i l y o f open sets, a f a m i l y 0 o f subsets o f F must s a t i s f y (*) Ma E 0 (any index set

-

E

0)

(A)

W nE 0 ( f i n i t e i n t e r s e c t i o n

-

F

E

0).

Then one d e f i n e s a

neighborhood NBH o f a p o i n t t o be any s e t c o n t a i n i n g an open s e t c o n t a i n i n g t h e p o i n t and we denote by N ( x ) t h e f a m i l y o f NBH o f x.

It i s elementary t o

312

ROBERT CARROLL

v e r i f y t h a t N(x) s a t i s f i e s the following four rules.

DEFlNWI0i-i A.4.

Neighborhood axioms: (A) Every s e t i n F c o n t a i n i n g a s e t i n

N ( x ) i t s e l f belongs t o N ( x ) . ( B ) Ni section)(C) x

E

that for a l l y

N if N

E

E

N ( x ) i m p l i e s nNi

E

E

~ ( x )( f i n i t e i n t e r -

N ( x ) ( 0 ) If V E N ( x ) t h e r e e x i s t s W

E

N ( x ) such

W, V E N ( y ) .

Here (A), (B), ( C ) a r e obvious and ( D ) f o l l o w s upon t a k i n g W t o be an open s e t c o n t a i n i n g x and c o n t a i n e d i n V and o b s e r v i n g t h a t a s e t i s open i f and o n l y i f i t i s a NBH o f each o f i t s p o i n t s . e r a l t o p o l o g y (see [ B o ~ ] ) t h a t i f each x N(x) s a t i s f y i n g (A)

-

E

It i s a standard theorem o f gen-

F has a s s o c i a t e d t o i t a f a m i l y

( D ) then t h e r e i s a unique t o p o l o g y on F such t h a t

N ( x ) i s p r e c i s e l y t h e f a m i l y o f NBH o f x f o r t h i s t o p o l o g y .

I n f a c t one can

d e s c r i b e t h i s t o p o l o g y by p i c k i n g as open s e t s those s e t s 6 such t h a t f o r each x E 0 one has 0

E

N(x) (exercise).

We c a l l a f a m i l y ~ ( x c) N ( x ) a

base o f N ( x ) o r a fundamental system o f NBH (FSN) a t x i f every tains a set B

€

B(x).

NE

N ( x ) con-

Then N ( x ) i s recovered as t h e f a m i l y o f s e t s c o n t a i n -

i n g a set i n B(x). DEFZNZCIBN A.5,

I f F and G a r e t o p o l o g i c a l spaces, then a map f: F + G

continuous a t x i f f o r any

W.

WE

N(f(x)) there i s a V

E

is

N ( x ) such t h a t f ( V ) c

I n view o f t h e NBH s t r u c t u r e f o r TVS [ t h a t i s , N ( x ) = x + N(O)],

one

need o n l y check c o n t i n u i t y a t 0 f o r l i n e a r maps between TVS. It i s e a s i l y checked t h a t t h i s i s e q u i v a l e n t t o t h e requirement t h a t f - ’ ( U )

be open f o r any open U ( e x e r c i s e ) and t h e c o n t i n u i t y p r o p e r t i e s o f D e f i n i t i o n A.2 f o l l o w immediately from t h e d e f i n i t i o n n ( x ) = x + N ( 0 ) . DEFINICZ0N A.6.

A LCS ( 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 a TVS

whose t o p o l o g y i s d e f i n e d by a f a m i l y o f seminorms pa. F; pa(x) 5

E},

then F i s Hausdorff. -+

= {x E

t h e n a FSN B ( 0 ) i s taken t o be t h e f a m i l y o f f i n i t e i n t e r -

s e c t i o n s o f t h e VJE), e a r map f: E

Thus i f V,(E)

w h i l e i f , f o r any x

A morphism E

-+

E

F, t h e r e i s a pa w i t h pa(x) # 0

F o f TVS ( o r LCS) i s a continuous l i n -

F; f i s a homomorphism i f i t i s a l s o open ( i . e . f ( 0 ) i s open

f o r 0 open) and an isomorphism i f i t i s 1-1 and b i c o n t i n u o u s . DEFZNZCZQN A.7. r i c d: F X F

-+

A m e t r i c space i s a s e t F w i t h a d i s t a n c e f u n c t i o n o r metR s a t i s f y i n g ( A ) d(x,y) 2 0 ( B ) d ( x , y ) = 0 i f and o n l y i f x =

( D ) d(x,z) 5 d(x,y) t d(y,z). A TVS i s c a l l e d m e t r i z a b l e i f t h e r e i s a t l e a s t one m e t r i c d e f i n e d on i t such t h a t t h e s e t s Vn =

y ( C ) d(x,y) Iy

E

= d(y,x)

F; d ( x , y ) 5 l / n l form a FSN a t x.

APPENDIX A

31 3

A net i n F i s a p a i r ( S ,> ) , where S i s a f u n c t i o n w i t h v a l ues i n F and > d i r e c t s t h e domain A o f S ( t h a t i s , CL > 6, B 2 implies a > Y; a L a ; and a , E~ A i m p l i e s t h e r e e x i s t s y E A w i t h y ) a and y 2 B ) . A

D E F I N I C Z 0 N A.8,

denoted a l s o b y Sa o r {Sa}, i s e v e n t u a l l y i n a s e t B i f and o n l y

net (S,)),

i f t h e r e e x i s t s B such t h a t a > B implies S

E

B.

A n e t converges t o x i f

and o n l y i f i t i s e v e n t u a l l y i n e v e r y NBH o f x. Since t h e c l o s u r e

Ti

o f B c o n s i s t s o f a l l p o i n t s x such t h a t e v e r y NBH o f x

i n t e r s e c t s B, we see t h a t i f x t o x.

E

B,

t h e n t h e r e i s a n e t Sa i n B c o n v e r g i n g

Indeed, d i r e c t t h e NBHs U o f x by i n c l u s i o n and p i c k a p o i n t Su i n

each U n B.

I n a m e t r i c TVS F t h e t o p o l o g y i s d e f i n e d by a c o u n t a b l e FSN

a t 0 formed o f s e t s Vn = {q E F; d(lp ,0) < l/n}. i f t h e r e i s a sequence i n B converging t o x.

Hence x

E

BC

F i f and o n l y

Thus, i n general, B i s c l o s e d

i f and o n l y i f no n e t i n B ( o r sequence i f F i s m e t r i c ) converges t o a p o i n t o f F-B.

T h i s means t h a t c l o s e d s e t s a r e c o m p l e t e l y d e s c r i b e d by convergence

p r o p e r t i e s , and s i n c e c l o s e d s e t s a r e t h e complements o f open, s e t s we can c o m p l e t e l y d e s c r i b e t h e t o p o l o g y i n terms o f convergence (see [Kel ;Bo4;Dul] f o r f u r t h e r discussion).

DEFINZCDDN A.9,

A n e t Sa i n a TVS F i s a Cauchy n e t i f , g i v e n any V E N ( O ) , - S, E V f o r a , 2~ B . A space F i s complete i f

t h e r e i s a B such t h a t S,

every Cauchy n e t converges; i f F i s m e t r i c (or m e t r i z a b l e ) i t i s complete i f e v e r y Cauchy sequence converges.

A complete m e t r i z a b l e LCS i s c a l l e d a Fre-

c h e t space and a complete normed space i s c a l l e d a Banach space. I f F i s a TVS over C t h e ( t o p o l o g i c a l ) dual o f F i s t h e v e c t o r space o f a l l continuous l i n e a r maps f ' : F * C.

The a c t i o n o f f ' on f w i l l be w r i t t e n

( f ' , f ) = ( f , f ' ) = f ' ( f ) . Evidently, ( A f ' , f )

= A( f ' , f ) and ( f ' + g ' , f )

= ( f',

f ) +(g',f).

We go n e x t t o some v e r s i o n s o f t h e Hahn-Banach complex o f ideas.

EHE0REm A.10

(HAHN-BANACH).

L e t p ( - ) be a c o n t i n u o u s seminorm i n a (complex)

LCS F and suppose g i v e n a l i n e a r f o r m L d e f i n e d on a c l o s e d l i n e a r subspace G C F such t h a t I L ( z ) l 5 p ( z ) f o r z E G. Then L may be extended t o a cont i n u o u s l i n e a r form u on a l l F w i t h l u ( x ) l

I p(x) f o r x

€

F.

P h o o ~ : We prove t h e theorem f o r r e a l spaces i n remarking t h a t t h e complex case can t h e n be o b t a i n e d by an elementary c a l c u l a t i o n ( c f . t h e d i s c u s s i o n below).

For L d e f i n e d as above on any c l o s e d l i n e a r subspace G

C

F and xo $k

G, we w i 1 show t h a t L can be extended t o t h e l i n e a r space G ' spanned by xo

and G wh l e preservi.ng t h e p r o p e r t i e s i n d i c a t e d .

Then by Z o r n ' s lemma o r

t r a n s f i n t e i n d u c t i o n one can extend t o a l l o f F.

Thus f o r any z,z' E G,

314

ROBERT CARROLL

n o t e t h a t L ( z ' ) - L ( z ) = L ( z ' - z ) 5 p[(z'+xo)+(-z-x,)]

5 p ( z ' + x o ) + p(-z-xo) SO 5 p ( z ' t x o ) - L ( z ' ) . T h i s h o l d s f o r any z,z' E G and hence one has sup [-p(-z-x,)-L(z)] 5 i n f [ p ( z ' + x o ) - L ( z ' ) ] (sup o v e r z , i n f o v e r 2 ' ) .

-p(-z-xo)-L(z)

L e t Y be a number between these extremes and d e f i n e L on G ' by L(axo+z) = t L(z) f o r

z

E

To show t h a t L ( y ) 5 p ( y ) on GI, f i r s t l e t a > 0; t h e n

G.

s i n c e z / a E G we have y 5 p ( z / a t x o ) - L ( z / a ) ; axo)-L(z);

-s, s

ay 5 ap(z/a+xo)-aL(z/a)

and L(z+axo) = a y + L ( z ) (p(z+axo).

> 0; t h e n z / a = -z/s

+ L(z/s) 5

and -p(z/s-xo)

E

y;

f o r x i n some NBH V o f 0 i n F.

choose a FSN which i s symmetric ( i . e .

x

-p(x-sx0) + L ( z ) 5

Since p i s now c o n t i n u o u s a t 0, i t

ys; and y a t L ( z ) = L(z+axo) 5 p ( z + a x o ) .

follows that p(x) 5

= p(z+

On t h e o t h e r hand, l e t a =

E

But we can always

V i m p l i e s -x

E

Then

V ) i n a LCS.

suppose t h e e x t e n s i o n o f L t o have been c a r r i e d o u t t o a l l F w i t h L ( x ) 2 p ( x ) on F. Therefore,

IL(x) I

5 E f o l l o w s a l s o now - L ( x ) = L ( - x ) c p ( - x ) 5 E. f o r x r5 V and L i s continuous. Also t h i s shows t h a t

From L ( x ) ( p ( x ) IL(x)

(p(x)

C0R0ttARg A.11.

I

5

E

since -L(x) = L(-x) 5 p(-x) = p(x).

QED

For any continuous seminorm p on a LCS F t h e r e i s a u E F'

( F ' = dual o f F ) w i t h l u ( z )

I

(p(z)

and u ( x o ) = p ( x o ) , xo a r b i t r a r y , can be

p r e s c r i b e d i n advance.

Phaod:

{axel)

D e f i n e L on { x o l ( = subspace spanned by xo =

.p(x0) and extend L.

REmARK A.12.

by L(ax,)

=

QED

Thus i n p a r t i c u l a r , i n a LCS F, where continuous seminorms a l -

ways e x i s t t h e r e a r e a u t o m a t i c a l l y n o n t r i v i a l elements o f F ' , and t h i s i s a good reason t o work i n LCS. We say now t h a t a s e t B i s d i s c e d i f

AX E

B whenever x E B and 1x1 5 1.

t h e r , i f F i s a TVS o v e r R, then B i s convex i f x,y E B f o r 0 < A < 1.

E

B implies

AX t

Fur-

(1-x)y

I f F i s a TVS o v e r C, l e t Fo be t h e same space c o n s i d e r -

ed as a TVS o v e r R ( m u l t i p l i c a t i o n b y i, then, i s t r e a t e d as an automorphism o f Fo and n o t as a d i a l a t i o n ) .

Then B

C

F i s convex i f i t i s convex i n Fo.

I n a complex TVS E a complex hyperplane determined b y f ( x ) = a t i g i s t h e i n t e r s e c t i o n o f two r e a l hyperplanes Re f ( x ) = a and Re f ( i x ) = - B ( n o t e t h a t

I m F ( x ) = -Re f ( i x ) ) .

Conversely, i f

Ho i s a r e a l homogeneous hyperplane de-

termined b y g ( x ) = 0, g a r e a l l i n e a r f o r m on E, t h e n Ho n iH, homogeneous hyperplane determined by g ( x ) correspond t o continuous forms ( f - l ( O )

-

ig(ix)

0.

i s a complex

Closed hyperplanes

i s c l o s e d when f i s c o n t i n u o u s ) .

We

s t a t e n e x t a geometrical v e r s i o n o f t h e Hahn-Banach theorem w i t h o u t p r o o f (see [ B o ~ ]

- a p r o o f f o r normed spaces i s sketched below i n Theorem A.16).

31 5

APPENDIX A

We remark here a l s o t h a t a TVS F i s o f t e n s a i d t o be l o c a l l y convex i f N ( x ) in F

0

has a FSN c o n s i s t i n g o f convex s e t s .

Then i t can be shown ( c f . [ B o ~ ] In

f o r d e t a i l s ) t h a t seminorms can be found which determine t h e t o p o l o g y .

t h i s d i r e c t i o n we c o n s i d e r s e t s B which a r e convex, d i s c e d ( o r symmetric as above), a b s o r b i n g ( i . e . point.

x E

pB

for

1p1

2 p o ) , and have 0 as an i n t e r i o r

Then d e f i n e t h e gauge o r Minkowski f u n c t i o n a l o f B by

i n f p f o r x E pB and 0 < p . a seminorm ( e x e r c i s e ) .

= i x E F; p ( x )

It follows t h a t

Note p ( x ) i n

( 0 )

( 0 )

p(x) =

5 1 1 and p i s

i s d e f i n e d f o r B o n l y convex and

The p r o p e r t y o f b e i n g d i s c e d t r a n s l a t e s i n -

a b s o r b i n g w i t h 0 E i n t e r i o r B.

t o p(ax) = I a I p ( x ) b u t i s n o t needed t o d e f i n e p ( x ) f o r B. I f E i s a TVS o v e r R and A i s a convex open nonempty s e t i n E

CHE0RElll A.13,

with M a linear variety (i.e.

t r a n s l a t e o f a l i n e a r subspace) n o t i n t e r s e c -

t i n g A, t h e n t h e r e i s a c l o s e d hyperplane H c o n t a i n i n g

M and n o t i n t e r s e c -

t i n g A. I n p a r t i c u l a r ( c f . [ B o ~ ] ) , if A i s convex open and nonempty, and B i s convex nonempty w i t h A n B = Since 0

4

C,

t h e n C = A-B i s convex open and nonempty ( e x e r c i s e ) .

by Theorem A.13 t h e r e i s a l i n e a r c o n t i n u o u s f o r m f # 0 on E

such t h a t f ( x ) =

@,

>

0 i n C.

Set a

For x E A, y E B one has then f ( x ) > f ( y ) .

i n f f ( x ) f o r x E A and n o t e t h a t a i s f i n i t e w i t h f ( x ) ? a f o r x E A and

f ( x ) 5 a f o r x E B.

Thus A and B l i e i n t h e h a l f - s p a c e s f ( x )

< a determined by t h e c l o s e d hyperplane f ( x ) = a.

now A # @ be c l o s e d and convex and x W o f 0 such t h a t A+W n x+W =

$

A.

l y (exercise).

Then t h e r e i s a convex open NBH

(exercise).

open, by t h e above t h e r e i s a hyperplane

2 a and f ( x )

As an a p p l i c a t i o n , l e t

Since A+W and x+W a r e convex and

H s e p a r a t i n g them - i n f a c t s t r i c t -

We conclude Every c l o s e d convex s e t A

CHE0REm A.14-

C

E i s the intersection o f the

closed half-spaces containing it.

A v a r i a t i o n on t h i s which i s o f t e n u s e f u l i s g ven by L e t G be a c l o s e d l i n e a r subspace o f a Banach space E and xo E

LEilllllA A.15,

E an element n o t i n G.

Then t h e r e i s a uo E E

w i t h ( u o , x o ) = 1 and

(

uo,z)

= 0 f o r a l l z E G.

Pmag: L e t d

= i n f IIxo-zII f o r z

E G.

C l e a r l y d > 0, s i n c e o t h e r w i s e xo

would be a l i m i t of elements o f G and hence would belong t o G.

z

E G,

Ilx

0

-211

2 d.

xo and G by t h e r u l e L(axo + z ) = a. s i n c e - z / a E G.

Thus f o r a l l

D e f i n e a l i n e a r form L on t h e l i n e a r space spanned by Then IIax0+zII = lallxo+z/al1 2

Thus / L ( a x o + z ) l = IuI 2 IIaxo+zll/d.

laid,

Then by Theorem A . l l

31 6

ROBERT CARROLL

we can extend L t o a continuous l i n e a r f u n c t i o n uo w i t h Thus

(

uo,xo) = 1 and

(

I( uo,x)I

cllxfl/d.

Also IIuoll 2 l / d .

uo,z) = 0 f o r z E G.

QED

L e t E be a r e a l normed space and A a convex open (nonempty)

CHE0Rm A.16,

M be a l i n e a r v a r i e t y n o t i n t e r s e c t i n g A. Then t h e r e e x i s t s a c l o s e d hyperplane H 3 M and n o t i n t e r s e c t i n g A ( i . e . t h e r e e x i s t s e ' E E ' and c E R such t h a t ( e y e ' ) c f o r e E M and ( e y e ' ) < c f o r e E A ) . set.

Let

Let 0

P4006:

E

A w i t h o u t l o s s o f g e n e r a l i t y and l e t G be t h e subspace o f E

generated by M ( n o t e A i s a convex open (NBH) o f 0 and hence absorbing). Then M i s a hyperplane i n G, 0

f

M, so t h e r e e x i s t s a l i n e a r f u n c t i o n a l f

on G such t h a t M = { x ; f ( x ) = 11. t i o n a l ) o f A d e f i n e d by f ( a x ) = a 2 p(ax) f o r x

(0)

E

Thus I f ( x ) l 5 p ( x ) f o r x

E

u on E w i t h l u ( x ) l 2 p ( x ) .

L e t p be t h e gauge ( o r Minkowski f u n c -

so t h a t f ( x ) = 1 5 p ( x ) f o r x

M and

E

M.

Evidently

a > 0 w h i l e f o r a < 0, f ( a x ) 5 0 5 p ( a x ) .

G and one extends f t o a continuous l i n e a r f o r m

Let H = I x

6

E; u ( x ) = 11 be t h e corresponding

c l o s e d hyperplane i n E which does n o t i n t e r s e c t A ( u ( x ) < 1 f o r x E A).QED

DEFZNZEI0N A.17,

A s e t B i n a TVS i s bounded i f i t i s absorbed by any NBH

o f 0 o r e q u i v a l e n t l y by any V i n a FSN o f 0. for

B absorbed by V means B

C

XV

I A ~ 2 x0.

DEFZNZCION A . M .

F and G a r e s a i d t o be i n d u a l i t y i f t h e r e i s a b i l i n e a r

form ( , ) on F X G w i t h t h e p r o p e r t i e s (A) For any x # 0 i n F t h e r e i s a y E G such t h a t ( x , y ) # 0 ( 6 ) For any y # 0 i n G t h e r e i s an x 6 F such t h a t (

x,y)

#

0.

If F i s a LCS and F ' = G i t s dual, t h e n t h e n a t u r a l a c t i o n o f F ' on F g i v e s a bracket

(

,

)

as d e s c r i b e d e a r l i e r .

t o assert t h a t i f x

E

Here one uses t h e Hahn-Banach theorem

G, x # 0, t h e r e i s a seminorm p w i t h p ( x ) f 0 ( a l l

# 0 (see C o r o l l a r y

spaces a r e H a u s d o r f f ) and hence an x ' E F ' w i t h (x,x') A.ll).

We s h a l l assume a l l spaces a r e LCS from now on.

DEFINZEIBN

A.19.

L e t F ' be dual t o F, F a LCS; t h e n t h e s t r o n g topology on

F ' i s t h e t o p o l o g y o f u n i f o r m convergence on bounded s e t s o f F.

S E F ' converges t o S E F ' means (S,,b)

-f

(S,b) u n i f o r m l y f o r b

Thus a n e t E

B with B

a bounded ' s e t i n F. I t i s e a s i l y checked ( e x e r c i s e ) t h a t t h e c l o s e d convex d i s c e d envelope of a

bounded s e t i s bounded and hence a FSN o f 0 i n F ' f o r t h e s t r o n g t o p o l o g y i s formed o f t h e p o l a r s o f c l o s e d bounded d i s c e d s e t s B of a d i s c e d B C F i s d e f i n e d by Bo = { x '

E

F';

C

F.

The p o l a r Eo

I ( x ' , x ) l 5 1 f o r x E B}.

C

F' In

31 7

APPENDIX A

general, f o r a r b i t r a r y B C F, one d e f i n e s Bo = I x ' E F ' : R e ( x ' , x ) 5 1 f o r x E

BI; t h e n o t i o n s a r e e q u i v a l e n t f o r d i s c e d B.

3

fro.

5 implies

Bo

= { f i n i t e sums 1 h . b J J' I f F i s a normed space, t h e n t h e s t r o n g t o p o l o g y i n F'

r B i s t h e convex d i s c e d envelope o f B; r B

b . E By

J

Note t h a t B C

1 Ihjl

5 11.

i s t h e s t a n d a r d norm t o p o l o g y determined, f o r example by ( 6 ) IIx'II = sup 1(x, x ' ) I (sup f o r # x l l = 1).

One s h o u l d check t h a t i f F i s a Banach space, t h e n so i s F' w i t h t h e norm ( 6 ) ( e x e r c i s e ) .

DEFINICL0N A.20.

The weak t o p o l o g y on F ' , denoted by u(F',F),

s e s t ( = weakest) t o p o l o g y on F ' making a l l t h e l i n e a r maps x '

C continuous ( x E F ) .

i s t h e coar-+

(x,x'

F'

):

+

Thus i n s t a n d a r d n o t a t i o n F C CF = IIF C and o(F',F)

i s t h e induced t o p o l o g y o f t h e p r o d u c t w i t h a FSN c o n s i s t i n g o f f i n i t e i n tersections o f sets I x '

E

F';

l(x',x)l

5

E}.

I t i s e a s i l y seen ( e x e r c i s e ) t h a t t h e weak o r s t r o n g t o p o l o g i e s on F " make

F ' i n t o a H a u s d o r f f space.

A b a r r e l i n a TVS i s a closed, convex, disced, a b s o r b i n g

DEFZNZCI0N A.21,

A TVS i s tonne16 o r b a r r e l e d i f e v e r y b a r r e l i s a NBH

s e t (a French j o k e ) .

R e c a l l t h a t a s e t V i s a b s o r b i n g i f i t absorbs any p o i n t x.

o f 0.

A B a i r e space i s a t o p o l o g i c a l space such t h a t e v e r y coun-

DEFINZCI0N $1.22,

t a b l e union o f c l o s e d s e t s w i t h o u t i n t e r i o r p o i n t s i t s e l f has no i n t e r i o r point.

CHEBREIII A.23. P400d:

A complete m e t r i z a b l e space i s B a i r e .

L e t E be a complete m e t r i z a b l e space w i t h d i s t a n c e f u n c t i o n d ( x , y ) .

Suppose E = UE,

En closed, w i t h no En c o n t a i n i n g a nonvoid open s e t .

El # E and CE1 = complement o f El open b a l l B1 = B

P ~ , E ~=)

does n o t c o n t a i n B(p1fi1/2),

I x E E; d(x,pl)

5 ~~1 w i t h

1/2.

<

E~

Since E2

t h e nonvoid open s e t CE2 n B ( P ~ , E ~ c/ o~n)t a i n s

an open b a l l B2 = B ( P ~ , E ~w )i t h

E~

< 1/4.

We o b t a i n by i n d u c t i o n a sequence

Bn = B ( P ~ , E ~o )f open b a l l s w i t h t h e p r o p e r t i e s 0 <

E~

<

1/2n,

Bn+l

~ ~ / 2 )and , Bn n En = a~ ( n = 1,2, . . . ) . But f o r n < m we have d(pn,p pntl) t h e p, d(p,,p)

-+ d(pntl

,P,+~)

.. .

i.

d ( ~ , - ~ ,pm) 5 1/2"'

+

... +

1/2m < 1/2'.

f o r m a Cauchy sequence and t h e r e f o r e converge t o a p o i n t p.

5 d(pn,pm) + d(pm,p) 5 ~,,/2

p E Bn f o r e v e r y n. UEn ;

i.

th is contradicts

Then

i s open; consequently, CE1 c o n t a i n s an

t d(pm,p)

tends t o ~

C B(p,,

)

< d(pn.

Thus Since

~ i t/ f o2 l l o w s t h a t

Hence p l i e s i n none o f t h e s e t s En and t h e r e f o r e p $?

.

QED

The use o f t h i s concept w i l l appear now as we discuss t h e Banach-Steinhaus

31 8

ROBERT CARROLL

complex o f ideas. tHE0REIII A.24,

Pmod:

F i r s t we prove

Every B a i r e LCS i s b a r r e l e d . Since B i s absorbing we can w r i t e F = UnB. Since

L e t B be a b a r r e l ,

If x 0 E B i s i n t e r -

F i s B a i r e some nB, and hence By has an i n t e r i o r p o i n t . i o r and xo = 0 we a r e through.

I f xo

+ 0,

Consider t h e map g d e f i n e d by g ( z ) = (1/2)z By c o n v e x i t y g ( V ( x o ) )

[ztx,]).

n o t e t h a t -xo

€

B by discedness

L e t V(xo) be a nbh o f xo i n B and z E V(xo).

and hence O E B by c o n v e x i t y .

-

(1/2)x0 ( o r g ( z ) + xo = ( 1 / 2 )

B, g ( x o ) = 0,and g(V(xo))

C

N(0).

C

QED

Every Frechet o r Banach space i s b a r r e l e d .

C0R0LCARM A.25,

We want now t o make a s i m p l e b u t i m p o r t a n t o b s e r v a t i o n about convergence. Let t

-+

f(t): R

-+

F be a map and F a t o p o l o g i c a l space.

f ( - ) has a l i m i t L as, say, t t h a t f ( t n )-+ i.

Indeed, t a k e L = 0 and l e t t,

be a n e t t,

+

0 w i t h f(t,)+

L e t Vn be a decreasing sequence o f NBHs o f t h e f o r m Vn = I t E R;I

0.

l/nl.

There i s a NBH W o f 0 i n F such t h a t f(t,)

Therefore, we can p i c k tnE Vn such t h a t f ( t n ) i n g from t h e t,. assumption.

w

Then we a s s e r t t h a t

0, i f f o r any sequence tn + 0 i t i s t r u e

+

-+

f(w):A

#

tl

i s n o t e v e n t u a l l y i n W.

W, n = 1,2,.

..,

t h e tn com-

0, b u t f ( t n )+ 0, which i s i m p o s s i b l e under o u r

Then tn

-+

Thus ( g e n e r a l i z i n g t h i s a l i t t l e ) whenever we have f u n c t i o n s -+

F, where A i s a m e t r i c space and F a t o p o l o g i c a l space, we can

r e f e r the discussion o f l i m i t s t o t h a t o f sequential l i m i t s . DEFZNIEZ0N A.26-

L e t E and F be LCS and c o n s i d e r t h e v e c t o r space L(E,F) o f

continuous l i n e a r maps E

-+

o f s e t s N(B,V) =

Iu

The s t r o n g t o p o l o g y i n L(E,F) has a FSN composed

E L(E,F);

vex d i s c e d s e t s i n E.

L e t V r u n over a FSN o f 0 i n F composed o f

F.

convex d i s c e d c l o s e d s e t s .

u(B)

C

V l , where B runs o v e r a l l bounded con-

The weak t o p o l o g y o r t o p o l o g y o f s i m p l e convergence

has a FSN composed o f f i n i t e i n t e r s e c t i o n s o f s e t s N(x,V) = t u E L(E,F); u(x) c V},

where x E E.

Consider now what i t means t o be bounded i n L(E,F)

f o r t h e weak o r s t r o n g

t o p o l o g i e s and n o t e o f course t h a t D ' = L(D,C) i s a s p e c i a l case w i t h V o f t h e f o r m V = { z E C;IzI 5 € 1 .

A set

H

L(E,F) i s s t r o n g l y bounded i f f o r

C

any convex d i s c e d bounded B C E and any V as i n D e f i n i t i o n A.26 t h e r e e x i s t s i0 such t h a t

H c AN(B,V)

then H C L(E,F) over y '

E

f o r 111

5 Ao.

Note a l s o t h a t AN(B,V)

i s bounded means t h a t f o r

H) o r e q u i v a l e n t l y B

C

1x1

Aoy

(6)

= N(BdV), and

Uy'(B) C A V ( u n i o n

~ n y ' - ~ ( V( i)n t e r s e c t i o n o v e r y ' E H ) .

i t i s r e q u i r e d t h a t C l y ' - l ( V ) f o r y ' E H absorb e v e r y bounded s e t .

H c L(E,F)

Thus

Similarly

i s weakly bounded ifand o n l y i f f o r V as above t h e s e t n y ' - ' ( V )

31 9

APPENDIX A

y ' E H, absorbs e v e r y p o i n t y E E.

A s e t H C L(E,F) i s e q u i c o n t i n u o u s i f f o r any V i n a FSN DEFINZCI0N A.27, o f 0 o f convex d i s c e d c l o s e d s e t s i n F t h e s e t n y ' - l ( V ) , y ' E H, i s a NBH o f E q u i v a l e n t l y , t h e r e i s a NBH U E N(0) i n E w i t h y ' ( U )

0 i n E.

C

V f o r y'EH.

It i s obvious t h a t i f H C L(E,F) i s equicontinuous, t h e n i t i s b o t h s t r o n g l y and weakly bounded. L e t E and F be LCS w i t h E b a r r e l e d .

CHE0REIR A.28,

ed s e t H c L(E,F) Let V

Ro06:

E

N ( 0 ) i n F be convex, disced, and closed.

T absorbs e v e r y p o i n t

Then T = n y ' - ' ( V )

S i n c e H i s weakly ( o r s i m p l y ) bound-

( y ' E H) i s closed, convex, and d i s c e d .

ed,

Then e v e r y weakly bound-

i s b o t h s t r o n g l y bounded and equicontinuous.

y E E and hence T i s a b a r r e l .

Since E i s b a r r e l e d

T i s a NBH o r 0 i n E and hence H i s e q u i c o n t i n u o u s (and consequently s t r o n g QED

l y bounded).

I f V i s a convex d i s c e d NBH o f 0 i n a LCS F, t h e n a s e t A A

D E F I N I C I B N A.29,

c F i s s a i d t o be small o f o r d e r V i f f o r any x,y E A one has x-y E V.

set B

C

F i s precompact i f f o r any V as above t h e r e i s a f i n i t e c o v e r i n g o f

B by s e t s xi t V, small o f o r d e r V.

A i s compact i f i t i s precompact and

complete; A i s r e l a t i v e l y compact i f

A' i s

compact ( c f . a l s o Remark 3.2.13

f o r compactness). I f H C L(E,F)

C H E 0 R m A.30.

i s equicontinuous,

E,F b e i n g LCS, t h e n t h e weak

t o p o l o g y on H i s e q u i v a l e n t t o t h e t o p o l o g y o f u n i f o r m convergence on p r e compact s e t s o f E.

Pmub: E

E).

Suppose Ua E H converges weakly t o 0 ( i . e .

Ua(x)

+

0 i n F f o r any x

We want t o show t h a t i f B i s precompact, t h e n g i v e n V a convex d i s c e d

c l o s e d NBH o f 0 i n F we can make U,(B)

C

V f o r a "large".

But since H i s

equicontinuous, t h e r e i s a convex d i s c e d NBH W o f 0 i n E such t h a t U(W)

By precompactness t h e r e a r e a f i n i t e number o f xi such t h a t

f o r a l l U E H. t h e s e t s xi

t

W c o v e r B.

we can make Ua(xi) Consider now H from H

+

C

F (i.e.

ous maps H

+

V/2

C

F.

t o p o l o g y and thus

C

Then Ua(B)

V/2 f o r a l l xi

-L(E,F) x(U)

C

equicontinuous.

?€

Hence U,(B)

For x E E, l e t

C V.

be t h e map U

QED +

U(x)

U ( x ) i s continuous when H has t h e weak

It i s an easy e x e r c i s e t o show t h a t H i s e q u i -

C(H,F).

continuous i f and o n l y i f x

-+

t Ua(W) and by weak c o n t i n u i t y

L e t C(H,F) be t h e space o f c o n t i n u -

= U(x), x E E).

Observe t h a t U

UUa(xi)

i f a i s large.

+

y: E

+

C(H,F)

t h e t o p o l o g y o f u n i f o r m convergence on

i s c o n t i n u o u s when C(H,F) has

H (cf. [Boll).

Then observe t h a t

320

ROBERT CARROLL

i s equicontinuous ( c l o s u r e

i n FE

- i n t h e weak t o p o l o g y ) .

= nEF and V i s g i v e n as above i n F, then,

-+

U weakly

by e q u i c o n t i n u i t y , s i n c e Ua

N ( 0 ) i n E such t h a t F(Ua) = Ua(x) C V f o r a l l Ua and x Then U(x) = l i m U a ( x ) E V f o r a l l x E W, s i n c e V i s closed. Hence E W. c x(H) C V f o r x E W, and t h i s means t h a t x -+ ^ji: E C(T,F) i s continuous o r E

H, t h e r e i s a

Indeed i f Ua

WE

-+

that

L

i s equicontinuous.

It i s moreover obvious t h a t

!, c o n s i s t s o f l i n e a r

maps ( s i m p l y extend t h e l i n e a r r e l a t i o n s by c o n t i n u i t y ) . L e t E be b a r r e l e d and Un

CHE0REI A.31 (BANACH-XEINHAU.5).

uence converging s i m p l y t o a map Uo: E x E E).

Pkood:

Then Uo E L(E,F) and Un

-t

-t

F (i.e.

E

L(E,F) a seq-

Uo(x) = l i m Un(x) f o r each

Uo u n i f o r m l y o n precompact s e t s i n E.

F i r s t observe t h a t a Cauchy sequence { x n l i n any LCS G i s bounded.

Indeed i f W i s a convex d i s c e d NBH o f 0 i n G t h e n f o r n,m 2 N (some N) we have x -x E W; hence xn E xN + W f o r n N. The f i n i t e s e t x1 ,. . .,xN ben m longs t o some aW, s i n c e W i s absorbing, and hence any xn belongs t o aW + W C

(a+l)W.

bounded. chy.

Thus { x n l i s absorbed by any such W i n a FSN and i s consequently Hence t h e s e t H = { U n l i s weakly bounded, s i n c e Un i s weakly Cau-

By Theorem A.28 H i s equicontinuous and by Theorem A.30 t h e weak topo-

l o g y on H i s e q u i v a l e n t t o t h e t o p o l o g y o f u n i f o r m convergence on precomp a c t s e t s o f E. We n o t e t h a t i f t h e n Ua(x)

-+

F u r t h e r by t h e preceding d i s c u s s i o n

?is

a dense subset o f E and H

0 for x E

? implies

Ua(y)

-f

C

(weak)

b i t r a r y and p i c k x E

C

0 f o r any y E E (Ua E H).

V/2 f o r a l l U

so t h a t y - x E W.

E

WE

-t

0.

QED

Indeed

N(0) i n E, conLet y

H.

Then Ua(y-x) + Ua(x)

V f o r a l a r g e enough and consequently Ua(y)

C

€

E be a r -

V/2 + V / 2 c

Hence

L e t E be b a r r e l e d and Ua E L(E,F) be a s i m p l y bounded n e t

CQ)R0ttAR&l A.32.

which converges s i m p l y on a dense s e t L(E,F) and Ua

L(E,F).

L(E,F) i s equicontinuous

g i v e n V E N(0) i n F, convex disced, closed, t h e r e i s a vex, disced, closed, such t h a t U(W)

C

-+

C E t o a map Uo:

E

-f

F.

Then Uo E

Uo u n i f o r m l y on precompact subsets o f E.

&HE0REI A.33 (ASC0tI-ARZECA). L e t E and F be H a u s d o r f f spaces w i t h E l o c a l l y compact. Then H C C(E,F) i s r e l a t i v e l y compact f o r t h e t o p o l o g y o f u n i f o r m convergence on Compact subsets o f E i f and o n l y i f H i s e q u i c o n t i n u ous and f o r a l l x E E t h e s e t H(x) = { h ( x ) ; h E H I i s r e l a t i v e l y compact. We s h a l l o n l y need t h i s theorem when E c Rn i s compact and F i s a Banach space and t h e p r o o f i s s t r a i g h t f o r w a r d , a l t h o u g h l e n g t h y , so we s i m p l y r e f e r t o [BolY2;C1],

o r o t h e r standard r e f e r e n c e s on a n a l y s i s .

APPENDIX A

321

L e t E,F be ( H a u s d o r f f ) LCS, F complete, and f a l i n e a r map

CHE0REIII A.34.

d e f i n e d on a dense l i n e a r subspace A C E w i t h values i n F.

-

I f f i s continu-

ous o n A, t h e n i t can be extended by c o n t i n u i t y t o a unique l i n e a r c o n t i n u ous map f: E

-+

F.

Pmu6: L e t xa E A, xa

+

x E E; t h e n {xu) i s a Cauchy n e t i n A.

Then e v i -

i s Cauchy ( e x e r c i s e ) and hence converges t o a u n i q u e l i m i t i n

d e n t l y If(x,)l

F; x b e i n g g i v e n i t i s easy t o see t h a t t h i s l i m i t i s independent o f t h e n e t x

+

Thus f o r any x E E, l i m f ( y ) = g ( x ) e x i s t s as y

x.

now V ' be a c l o s e d NBH o f g ( x ) .

-+

x, y

E

A.

Let

Then t h e r e i s an open NBH V o f x such t h a t

f ( V n A) C V', since f ( y ) g ( x ) as y + x, y E A. But V i s a NBH o f each o f i t s p o i n t s and hence f o r e v e r y z E V we have g ( z ) = l i m f ( y ) as y * z and y -+

E

Hence g ( z ) E f ( V n A ) C V ' , which says t h a t g i s c o n t i n u o u s a t x.

V n A.

Since one can t a k e a FSN c o n s i s t i n g o f c l o s e d s e t s , we a r e through. L e t now E and F be LCS and c o n s i d e r a map and p i c k f '

E

F'; then e

-f

u:

E

+

F. Suppose u i s c o n t i n u o u s

E * C i s continuous.

(u(e),f'):

QED

Hence i t d e t e r -

mines an elerrent 5 o f E ' and we can w r i t e ( e , S ) = ( u ( e ) , f ' ) . By l i n e a r i t y t t 5 = u ( f ' ) f o r u t h e l i n e a r transpose F ' + E ' . Examples o f such maps appear f o r example i n d i s c u s s i n g d i f f e r e n t i a t i o n i n D' and F o u r i e r t r a n s f o r m in

5'. C l e a r l y i f e

(u(e),f'

-+

0 weakly ( i . e . f o r u ( E , E ' ) ) then ( e , S )

-+

0 and hence

0 f o r any f ' E F ' .

Thus u i s c o n t i n u o u s f r o m o ( E , E ' ) t o o(F,F'). t 0 weakly t h e n ( e , u ( f ' ) ) * 0 f o r any e E E and

) +

On t h e o t h e r hand, i f f ' + t hence u i s c o n t i n u o u s from u( F ' ,F)

+

u( E ' ,E).

Now suppose f o r e v e r y bound-

ed s e t AC E t h e r e i s a bounded B C F such t h a t tu(Bo)

C

A';

then f '

-+

0

+

0 s t r o n g l y i n E ' ( c f . D e f i n i t i o n A.19) and

tu would be s t r o n g l y continuous.

But u(A) i s bounded i n F s t r o n g l y as t h e

strongly i n F ' implies t u ( f ' )

image o f a bounded s e t by a l i n e a r continuous map. one has o b v i o u s l y u ( A ) ~ = ~ ~ - ' (o Ar ~tu(Bo) ) C A'.

CHE0REIR A.35. t o o(F,F')

and

L e t u: E + F be continuous. Then u i s c o n t i n u o u s f r o m o(E,E') t u i s b o t h s t r o n g l y continuous and c o n t i n u o u s f r o m u(F',F) t o

Next observe t h a t i f u: E e + (u(e),f' E

)

E ' such t h a t Iq(e)

E

-+

F i s continuous f o r u(E,E') and u(F,F') t h e n IP:

i s continuous f o r u(E,E').

seminorms f o r o(E,E') given

We have proved

( u l i n e a r here).

o(E',E)

ei

Hence t a k e B = u(A) and

I

i c max I(e,e;)

Hence t h e r e a r e a f i n i t e number o f

I

(see D e f i n i t i o n A.20).

a r e d e s c r i b e d by q ( e ) = max I ( e , e i )

1,

Indeed

i = 1,2,...,

and

we know by c o n t i n u i t y t h e r e i s such a q and a 6 so t h a t q ( e ) 5 6 i m -

p l i e s Iq(e)

I

E.

P i c k then, f o r any e w i t h q ( e )

B 0,

X = 6/q(e),

and we

322

ROBERT CARROLL

have l q ( h e ) l 5

E

or Iq(e)l 5

= (E/6)q(e);

1x1 I q ( e ) l

t r a r i l y large t o obtain Iq(he)l = ~ / above. 6

5

E

f o r q(e) = 0 p i c k Ihl a r b i o r q ( e ) = 0.

Pick then c =

Thus f o r t h i s s e t o f e; one knows t h a t ( e , e i ) = 0, i = 1, ...,n, By l i n e a r a l g e b r a t h i s means t h e r e i s an element e ' =

i m p l i e s q ( e ) = 0.

1 aiei

such t h a t q ( e ) ( e , e ' ) f o r a l l e E E ( i . e . I e ; f L = Ee;3 a l g e b r a i c I a l l y where f o r subspaces G we w r i t e Go = G and { e i l i s t h e space spanned by t h e ei). tu:

F'

E' ltnear.

-+

LEllllllA A.36. o(F,F')

E'

By E,E'

d u a l i t y e ' i s unique and we w r i t e again e l = t u ( f ' ) f o r Thus we have

Any l i n e a r map u: E

-+

F which i s continuous f o r u ( E , E ' ) and t t u: F '

can be r e p r e s e n t e d i n t h e f o r m ( u ( e ) , f ' ) = ( e , u ( f ' ) ) and

-+

i s continuous f o r o ( F ' , F ) and a(E',E).

DEFINZCI0N A.37.

The Mackey t o p o l o g y T(E,E') on E i s t h e t o p o l o g y o f u n i -

form convergence on a l l convex d i s c e d o(E',E)

compact s e t s o f E l .

We c i t e t h e f o l l o w i n g p a r t i c u l a r case o f a theorem o f Mackey and Arens w i t h o u t p r o o f (see [ B o ~ ] ) .

CHE0REm A.38.

A t o p o l o g y T on E i s compatable w i t h t h e E - E ' d u a l i t y ( i . e .

y i e l d s E ' a g a i n as dlral) i f and o n l y i f

T i s a t o p o l o g y o f u n i f o r m conver-

gence on s e t s c o v e r i n g E ' which a r e convex d i s c e d and a(E',E) compact.

In

p a r t i c u l a r a topology T on E y i e l d s E l as dual i f and o n l y i f o(E,E:) C T

c T(E,E'). A few a d d i t i o n a l remarks about t h e Mackey t o p o l o g y w i l l be i n s t r u c t i v e even i f n o t used.

Thus l e t u: E

F be continuous f o r o(E,E') and u ( F , F ' ) and

-+

l e t W = KO be a NBH o f 0 i n F f o r T ( F , F ' ) ( K b e i n g o ( F ' , F ) compact, convex, t and d i s c e d ) . Since u i s continuous f o r o(F',F) and o(E',E) by Lemma A.36 t we know C = u(K) i s convex, disced, and u(E',E) compact. Hence Co i s a NBH -1 0 o f 0 i n E f o r T(E,E') and c l e a r l y Co = u ( K ) ( a l l p o l a r s a r e b e i n g taken between E and E ' and t h e a s s e r t i o n can be v e r i f i e d as i n t h e p r o o f o f Theorem A.35).

UfE0REm A.39,

Consequently

I f u: E

-+

F i s l i n e a r continuous f o r o(E,E')

and o ( F , F ' ) t h e n

i t i s continuous f o r T(E,E') and T ( F , F ' ) . Now r e f e r back t o Theorem A.16. convex s e t s places us i n

Eo,

I f E i s complex, c o n s i d e r a t i o n o f c l o s e d

and continuous r e a l l i n e a r forms g determined

by c l o s e d r e a l hyperplanes a r e i n 1-1 correspondence w i t h complex c o n t i n u o u s l i n e a r forms as i n d i c a t e d i n Remark A.12.

CHE0REIII A.40.

We conclude immediately

The c l o s e d convex s e t s i n E a r e t h e same f o r a l l t o p o l o g i e s

323

APPENDIX A

compatable w i t h t h e E - E ' d u a l i t y . I n f a c t more i s t r u e and a theorem o f Mackey says t h a t bounded s e t s i n E a r e t h e same f o r a l l t o p o l o g i e s compatable w i t h E-E' d u a l i t y .

We p r o v e n e x t an

i m p o r t a n t lemma known as t h e theorem o f b i p o l a r s ( a l l p o l a r s i n E - E l ) .

If A

C E m A.41.

E, E and LCS, t h e n Aoo i s t h e c l o s e d convex envelope f o r

C

u(E,E') o f A and 0 ( i . e .

Phao6:

Aoo =

{I hixi,

hi

2 0,

I A.

= 1, xi

1

E

U

I f B i s t h e convex envelope o f A and 0, c l e a r l y Bo = A',

A,

C l e a r l y Aoo 3

o n l y c o n s i d e r A convex c o n t a i n i n g 0.

4

A

and Re ( a , y )

c l u d e t h a t Aoo =

> 1.

Consequently y

A.

E

A'

and a

E

E, d

4 i,

strictly. Thus Re(x,y)

H i t has an e q u a t i o n Re(x,y) = 1 f o r some y E E l .

< 1 for x E

01.

so we need

and i f a

t h e r e i s a c l o s e d hyperplane H ( r e a l ) which separates a and Since 0

A

$ Aoo;

we con-

QED

Next observe, by Theorem A . 2 8 and t h e Tikhonov theorem, a s s e r t i n g t h a t t h e p r o d u c t o f compact s e t s i n a p r o d u c t CE i s compact ( c f . [Kel;Bo4]),

that

s i m p l y bounded s e t s B i n t h e dual o f a b a r r e l e d space E a r e e q u i c o n t i n u o u s and r e l a t i v e l y compact f o r o(E',E);

i n fact,

these t h r e e p r o p e r t i e s a r e e q u i -

On t h e o t h e r hand i t i s e v i d e n t t h a t B C E ' i s e q u i c o n t i n u o u s i f

valent.

and o n l y i f Bo

C

E i s a NBH o f 0 ( i . e .

i f and o n l y i f B C Vo f o r V a NBH o f

But t h i s says t h a t t h e Mackey t o p o l o g y T ( E , E ' ) on E i s c h a r a c t e r -

0 i n E).

i z e d by t h e p r o p e r t y t h a t convex d i s c e d s e t s i n E ' a r e u(E',E) compact i f and o n l y i f they a r e e q u i c o n t i n u o u s .

Hence i f E i s b a r r e l e d t h e n i t s i n i -

t i a l t o p o l o g y must be T ( E , E ' ) . I f u: E

tHE0REm A.42.

-f

F i s c o n t i n u o u s f o r u ( E , E ' ) and o ( F , F ' ) w i t h E and

F b a r r e l e d t h e n u i s s t r o n g l y continuous. From Theorem A . 2 8 and t h e Tikhonov theorem we have a l s o

tHE0REm A.43

The s t r o n g l y c l o s e d u n i t b a l l i n t h e dual E ' o f a

(ACA0W.I).

Banach space E i s weakly compact. Now f o r e E E, e; Banach.

Thus e

0 i n E ' s t r o n g l y i m p l i e s (e,e;)

+-

6

E El' and

element

A -+

e: E

c a l l e d semireflexive.

0 so e determines an

-+

= 0 i m p l i e s ( e y e ' ) = 0 f o r a l l e l so e = 0 by Hahn+-

E" i s an embedding and i f i t i s o n t o E" t h e n E i s

I f i n a d d i t i o n t h e s t r o n g t o p o l o g y o f El' i s equiva-

l e n t t o the topology o f E then E i s c a l l e d r e f l e x i v e .

When E i s Banach we

s e t E" = ( E l ) ' w i t h norm determined by t h e r u l e IIe"ll = sup ~ ( e " , e ' ) ~ / l l e ' t l . Then E" i s a Banach space ( j u s t as E ' was). +

(e,e'

):

E'

-t

L e t e E E and observe t h a t e '

C i s o b v i o u s l y norm continuous, s i n c e

I(

eye'

)I

5 IIellIle'II.

We

324

ROBERT CARROLL

e^ =

then i d e n t i f y e w i t h an element

F u r t h e r i: E

i ( e ) E El'.

El' i s an i s o -

+

m e t r i c isomorphism i n t o ( n o t e I l i ( e ) l = I l e l ) and E i s r e f l e x i v e when i i s onWe emphasize t h a t r e f l e x i v i t y i s a p r o p e r t y o f E and i; E may be

t o El'.

i s o m e t r i c a l l y isomorphic t o El' w i t

o u t E being r e f l e x i v e .

The f o l l o w i n g

theorem i s o f t e n u s e f u l .

A bounded s e t i n a r e f l e x i v e Banach space E i s weakly sequen-

MEQREm A.44, t i a l l y compact.

L e t yn be a bounded sequence i n E, IIynll 5 k.

Phaod:

l i n e a r span determined by t h e yn.

L e t G be t h e c l o s e d

Then G i s separable and r e f l e x i v e ( e x e r I

c i s e ) and, a r g u i n g as i n Remark 3.5.16,

G i s separable.

L e t y;

be dense i n

G ' and from t h e bounded sequence ( y ' y ) s e l e c t a convergent subsequence 1 1' P 2p,i) f r o m t h e se( y i ,yPyi ). S i m i l a r l y s e l e c t a convergent subsequence ( y ' ,y 1 ' 1 and so on. Then t h e sequence xi g i v e n by xi = y P y i has t h e quence ( y 2yyp,i)

p r o p e r t y t h a t 1im(y;,xi)

e x i s t s f o r n = 1,2

= lim(i(xi),y;)

.:...

By u s i n g

C o r o l l a r y A.32 we can conclude t h a t t h e r e i s an element y " i n G" such t h a t

l i m (y',xi)

f o r a l l y' E GI.

= (y",y')

with ( y ' , x i ) 4 y ' , y )

Thus by r e f l e x i v i t y we have a y But any z ' E E ' determines a y '

for a l l y' E G'.

such t h a t ( z ' , y ) = ( y ' , y ) f o r y E G and, s i n c e xi xi)

+

E

G

E

E

G'

G, i t f o l l o w s t h a t ( z ' ,

( z ' , y ) f o r z ' E E l , which proves t h a t xi converges weakly t o y.

QED

We prove n e x t a v e r s i o n o f t h e Banach theorem o f homomorphisms o r open mapp i n g theorem.

CHEQREm A.45.

L e t E and F be complete m e t r i z a b l e TVS.

continuous map u: E

Phoob:

Then e v e r y l i n e a r

F, o n t o F, i s a homomorphism ( i . e .

-t

i s open as w e l l ) .

It s u f f i c e s t o p r o v e t h a t t h e image under u o f any NBH V o f 0 i n E

Now (x,y)

i s a NBH o f 0 i n F.

-f

x-y i s a continuous f u n c t i o n o f x and y, so

-

t h e r e e x i s t s a NBH W o f 0 such t h a t W x E E belongs t o nW f o r l a r g e n.

Since x/n -+ 0 as n

W c V.

-+

any But

m,

Thus E = UnW and F = u ( E ) = unu(W).

F i s a B a i r e space (Theorem A.23) and so one o f t h e nu(W) c o n t ai n s a nonv o i d open s e t .

Further, since y

c o n t a i n s a nonvoid open s e t W ' .

- W'.

But s i n c e y

since W '

- W'

-+

radius

Let

E.

E~

-

Thus u(V)

E

u(W)

-

- -

u(W) 3 u(W)

-

-

u(W)

3

W'

W ' i s open, and

i t i s open and c o n t a i n s

Hence t h e c l o s u r e o f t h e image o f V c o n t a i n s a

> 0 l e t BE, B i be open b a l l s w i t h c e n t e r a t 0 i n E,F o f

> 0 be a r b i t r a r y and

t h e r e i s a sequence

3

- W ' ) i s t h e u n i o n o f open s e t s

= U(a

Now f o r

n y i s a homeomorphism i n F, u(W) i t s e l f

a - y i s a homeomorphism t h e s e t a

0; hence i t i s a NBH o f 0. NBH o f 0.

+

qiy

i = O,l,...,

1;

with

Then from t h e above

E~

<

E

qi

-+

0 such t h a t u(B

0'

E;

)

3

BA-. I

APPENDIX A

325

we w i l l show t h a t t h e r e i s an x E BZEe w i t h u ( x ) = y.

I f y E B'

Indeed,

n0

s e t t i n g i = 0 above we f i n d f i r s t an xo E BEosuch t h a t lly-u(xo)ll < nl.

B'

Since y - u ( x o )

we n e x t t a k e i = 1 t o f i n d x1 E BE w i t h IIy-u(xo)-u(xl)I

'I1

C o n t i n u i n g we o b t a i n a sequence xn w i t h xn

< q2.

'I,,+~. L e t z m

<

x II <

m

E

s e r i e s xo

lim

(

++

~

+~

E

+

= xo +

... +

E ~ .

~

...

B

I

- ~(1,"xi)" ... +

and IIy

Ell

I I X ~ ++ ~

+ xm so t h a t f o r m > n, IIzm-znll

T h i s shows t h a t zn i s a Cauchy sequence and t h a t t h e

... converges

x1 +

... + E

E

t o an element x such t h a t IIxII = l i m IIznll <

~ ~But. u i s c o n t i n u o u s and hence y = u ( x ) . Thus maps o n t o a s e t u(B ) c o n t a i n i n g an open b a l l BAo i n F. ~ <) 2

any open b a l l B 2% 2EO Consequently t h e image o f a NBH o f 0 i n E under u c o n t a i n s a NBH o f 0 i n F. Thus i f u i s 1-1 f r o m E o n t o F, then c o n t i n u i t y ( p l u s l i n e a r i t y ) y i e l d s an isomorphism o f TVS.

L e t E and F be complete m e t r i z a b l e TVS.

C0R0CtARlj A.46. u: E

+

A u s e f u l c o r o l l a r y i s t h e s o c a l l e d c l o s e d graph theorem. Then a l i n e a r map

F ( i n t o ) i s continuous i f and o n l y i f i t s graph G(u)

C

E X F is

closed. P4006:

The n e c e s s i t y i s obvious ( e x e r c i s e ) .

On t h e o t h e r hand, i f G(u) i s

c l o s e d then i t i s m e t r i z a b l e and complete s i n c e E X F has t h e s e p r o p e r t i e s . We mention i n passing a theorem which i s f r e q u e n t l y u s e f u l (see [ C l ]

for a

p r o o f i n r e f l e x i v e spaces and f o r r e f e r e n c e s t o t h e l i t e r a t u r e ) . L e t A: E

tHE0REm A.47.

E and F Banach spaces.

-t

F be a c l o s e d densely d e f i n e d l i n e a r o p e r a t o r w i t h

Then t h e f o l l o w i n g a r e e q u i v a l e n t : ( A ) R(A) i s

c l o s e d i n F (B) R(A*) i s c l o s e d i n E ' (C) A i s open = If; ( f , f ' ) = 0 f o r a l l f ' E N(A*)I ( F ) R(A*) E

=

(D) A* i s open ( E ) R ( A )

{el; (e',e)

= 0 for all e

N(A)l.

L e t us make a few comnents here about H i l b e r t space.

D E F I N I t I 0 N A.48,

L e t H be a v e c t o r space and (

, ) a sesquilinear, p o s i t i v e

d e f i n i t e , nondegenerate, H e r m i t i a n form on H ( c a l l e d a s c a l a r p r o d u c t ) . Thus ( x , y ) = (y,;),

(x,x)

1. 0, ( x , x )

= 0 i f and o n l y i f x = 0,

(ax,y)

= u(x,y),

( x , w ) = E(x,y), ( x + x ' , Y ) = (x,Y) + ( x ' , ~ ) , and (x,y+Y') = ( x , Y ) + ( x . ~ ' ) . Then H i s c a l l e d a p r e H i l b e r t space, and i f i t i s complete i t i s a H i l b e r t space ( t h e t o p o l o g y i s determined by t h e norm IIxl12 = ( x , x ) ) .

RENARK A.49,

There i s a c a n o n i c a l i s o m e t r i c a n t i i s o m o r p h i s m between H and

H ' when H i s H i l b e r t (see [ B o ~ ] ) .

Indeed, f i r s t n o t e t h a t t h e Cauchy-Bunya-

kovsky-Schwartz i n e q u a l i t y (we w i l l c a l l i t Cauchy-Schwartz) llyll f o l l o w s i m n e d i a t e l y f r o m t h e axioms.

I (x,y)l

5 IIxII

Consider then t h e c o n t i n u o u s

326

ROBERT CARROLL

l i n e a r map ux: y

y,x)

-+

f o r x fixed.

We w r i t e (y,x) = u x ( y ) =

and check t h a t @ ' ( a x ) = a e ' ( x ) w i t h e ' ( x + z ) = e ' ( x )

te'(x),y)

8' i s a n t i l i n e a r , 8 : H

-+

H I , and e ' ( x )

t

(

ux,y) =

e'(z).

Thus

In

0 means (x,x) = 0 o r x = 0.

f a c t 8' maps o n t o H ' by a c l a s s i c a l r e s u l t o f F. Riesz (proved below). norm i n H ' i s now IIux(I = sup l u x ( y ) I = sup I ( y , x ) l Cauchy-Schwartz i n e q u a l i t y IIuxll > Ilxll -

we have 1I u II = IIxII X

.

f o r IIyll = 1.

The

By t h e

Ilxll and s i n c e t a k i n g t h e sup i m p l i e s IIuxll

Hence 8 ' i s an isometry.

A f u r t h e r r e s u l t we

s h a l l need (and prove below) i s t h a t i f F C H i s a c l o s e d subspace o f H, t h e n t h e r e i s a continuous orthogonal p r o j e c t i o n P: H IIxII,

F

= P(H),

and Px

-f

F s a t i s f y i n g IIPxll 5

= x f o r x E F, w h i l e K = P-l(O) i s a c l o s e d subspace

o f H, w i t h H = K 8 F ( o r t h o g o n a l d i r e c t sum where o r t h o g o n a l i t y o f x and y means (x,y)

= 0).

These f a c t s and t h e Riesz theorem a r e easy consequences

o f t h e f o l l o w i n g d i s c u s s i o n ( f o r s p e c i a l i z e d t e x t s on H i l b e r t space see e.g.

[ A i 1 ;Mu 11) We sketch b r i e f l y some i n f o r m a t i o n l e a d i n g t o t h e c o n c l u s i o n s j u s t mentioned.

2

F i r s t i t i s elementary t o v e r i f y t h e p a r a l l e l o g r a m l a w II ( 1 / 2 ) ( x t y ) l l t 2 ll(1/2)(x-y)l12 = (1/2)[llxIl + llyl12]. Now l e t F C H be a c l o s e d subspace and l e t x E H be a r b i t r a r y . We s h a l l produce a unique y E F such t h a t IIx-yll = i n f IIx-zII f o r z E F.

L e t d = i n f IIx-zII f o r z E

F and p i c k zn

E F such t h a t

IIx-z 112 < d2 t l / n . Then z ' = ( 1 / 2 ) ( z +z ) E F so IIx-z'II ' d and hence ( m ) n 2" II (1/2)(zn-zm)l12 = (1/2)[11x-zn112 + IIX-ZJl 3 11 ( 1 / 2 ) ( x - z n + x - z m ) i l ~ 5 ( 1 / 2 )

-

(2d2 t l / n

t

l/m)

-

d2 5 ( l / Z ) ( l / n + l / m ) .

and converges t o a unique y

E

F.

Hence zn i s a Cauchy sequence

One w r i t e s y = Px and v e r i f i e s e a s i l y t h a t

P i s a continuous l i n e a r o p e r a t o r ( e x e r c i s e ) .

Further i f z

F i s arbitrary,

E

X < 1, t h e n y t X(Z-y) E F so Ily t X(z-y)l12 2 IIy-xl12. T h i s can be w r i t 2 t e n Ily-xl12 + X lz-yI12 + 2XRe(y-x,z-y) 2 Ily-xI12 which becomes 2Re(y-x,z-y) 0

<

-xllz-yll

2

.

5 0.

Since h i s a r b i t r a r i l y s m a l l , one has Re(x-y,z-y)

z = y + az' f o r z'

E

F arbitrary.

We o b t a i n Re ( x - y , z ' )

t h e orthogonal decomposition x = ( x - y )

t

y.

= 0.

Now p i c k

T h i s leads t o

F i n a l l y t o show t h a t x

-f

ux i s

o n t o H ' , l e t x ' E H' and c o n s i d e r t h e c l o s e d hyperplane F = x ' - l ( O ) C H. The orthogonal complement L o f F i s a l i n e and i f b E L, b form y

-+

(y,b) vanishes on F.

( y , x ' ) = (y,hb)

REmARK A.50. where (x,y)

for a l l y

E

+ 0,

Hence t h e r e e x i s t s a s c a l a r X

+

the l i n e a r

0 such t h a t

H, which i m p l i e s x ' = uXb.

Since i t i s o f t e n much e a s i e r t o argue i n r e a l H i l b e r t spaces = (y,x)

and (x,y)

i s l i n e a r i n b o t h v a r i a b l e s we i n c l u d e here a

few f a c t s r e l a t i n g r e a l and complex s c a l a r p r o d u c t s and l i n e a r forms ( c f . a l s o Remark A.12).

The f o l l o w i n g p o i n t o f view i s developed i n [Bbl].

APPENDIX A

327

F i r s t one can d e f i n e a H i l b e r t space v i a t h e p a r a l l e l o g r a m l a w as f o l l o w s .

A H i l b e r t space i s a Banach space ( r e a l o r complex) whose norm s a t i s f i e s t h e 2 i d e n t i t y (**) IIx+yl12 + Ix-y1I2 = 211xll + 211yl12 f o r a l l v e c t o r s x and y. I n a r e a l H i l b e r t space t h e i n n e r p r o d u c t o f x and y , denoted (x,y), i s de2 f i n e d by t h e f o r m u l a (*A) (x,y) = (1/4)[llx+y1I2 IIx-yll 3. I n a complex H i l -

-

b e r t space t h e i n n e r p r o d u c t o f x andy ( i n t h a t o r d e r ) , denoted by ( x l y ) , i s d e f i n e d by t h e r u l e (**) ( x l y ) = (1/4)[11~+y11~- IIx-yl12 + iIIx+iyl12 - i l l x 2 - i y l 3. Then one checks e a s i l y t h a t ( , ) and ( [ ) s a t i s f y a l l o f t h e r e q u i r e d p r o p e r t i e s o f a s c a l a r p r o d u c t i n D e f i n i t i o n A.48.

L e t us see how

one can prove some s i m p l e f a c t s about complex spaces f r o m t h e c o r r e s p o n d i n g Thus e.g. suppose we have proved I ( x , y ) I 5 I I x l l I y l 2 f o r r e a l spaces v i a ( f o r IIxII = llyll = 1 ) (*A) p l u s 4 1 ( x , y ) [ 5 IIx+yl12 + IIx-yll 2 2 = 211x11 + 211yll = 4. Then i n t h e complex case, we may w r i t e I ( x l y ) [ = u ( x l y ) f a c t s about r e a l spaces.

f o r suitable = (x,y)

+

(u(

=

1.

Then ( u x l y ) = I ( x l y ) I i s r e a l ; t h e r e f o r e (*&) ( x l y )

i ( x , i y ) y i e l d s ( u x l y ) = (px,y).

c i t i n g t h e r e a l case j u s t proved,

Since IIpxlI = llyll = 1 we have,

I(xly)l = I(uxly)l

= I(ux,y)I

'1.

Theo-

rems about l i n e a r forms can be proved u s i n g t h e r e l a t i o n s i n Remark A.12. Thus e.g.

i f g i s a r e a l l i n e a r form t h e n f ( x ) = g ( x )

complex l i n e a r f o r m such t h a t Ref = g. g ( i x ) = - h ( x ) and h ( i x ) = g ( x ) .

-

i g ( i x ) determines a

One w r i t e s f ( x ) = g ( x ) + i h ( x ) w i t h

This Page Intentionally Left Blank

329

APPENDIX

B

SELECTED TOPICS I N FUNCTIONAL ANALYSIS

We w i l l c o v e r h e r e some t o p i c s i n f u n c t i o n a l a n a l y s i s o f s p e c i a l i n t e r e s t . F i r s t we g i v e a b r i e f i n t r o d u c t i o n t o d i s t r i b u t i o n t h e o r y i n n o t i n g t h a t a more t o p o l o g i c a l t r e a t m e n t i s e a s i l y a c c e s s i b l e (and sketched) w i t h t h e machinery o f Appendix A ( c f . [ C l ] ) .

I n f a c t t h e d i s t r i b u t i o n spaces were a

model f o r t h e development o f t h e general machinery f o r LCS.

However t h e

main d i r e c t i v e was PDE and a g r e a t deal o f t h e progress i n s t u d y i n g PDE o v e r t h e l a s t 35 y e a r s o r so has been due t o t h e development and s y s t e m a t i c use of t h e t h e o r y o f d i s t r i b u t i o n s (and i t s e x t e n s i o n s t o boundary values o f a n a l y t i c f u n c t i o n s , h y p e r f u n c t i o n s , e t c . ).

There a r e many t r e a t m e n t s o f

t h i s t h e o r y a v a i l a b l e ( c f . [Bml;C1;Frl;G2;H1;Hv1;Jol;Ncl;Sal;Yol]).

It 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 h o n e s t l y w i t h o u t t h e general theor y o f LCS and almost immediately b e g i n t o use d i s t r i b u t i o n s and t h e r e l a t e d

T h i s i s t h e approach we w i l l adopt h e r e and f o r o u r purposes we w i l l o f t e n c o n f i n e o u r a t t e n t i o n t o R1 = R. Thus F o u r i e r theory.

DEFINZCIBN 3.1,

Let R

be Rn o r an open s e t i n Rn.

D e f i n e C i as t h e v e c t o r

space o f Cm f u n c t i o n s i n R w i t h compact s u p p o r t ( s u p p o r t cp = supp 9 i s t h e s m a l l e s t c l o s e d s e t o u t s i d e o f which l i n e a r map T: C i +

cp

5

0).

A distribution T i n R i s a

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 con-

s t a n t s C and k (depending on K) w i t h (*) I T ( 9 ) l = I(T,cp)l 2 C I s u p l D a ~ I , la1 5 k ( q E C i ( K ) = { q E C i ; supp 9 c K I , Da9 = DY p D : , Dk = a/axk,

...

a = (al

,...,an),

order

< k.

la1 =

I ak).

I f k i s t h e same f o r a l l K one says T i s o f

One w r i t e s ~ ' ( n f) o r t h e v e c t o r space o f such d i s t r i b u t i o n s T.

T h i s can be s t a t e d i n terms o f s e q u e n t i a l c o n t i n u i t y as f o l l o w s .

Given a

compact s e t K c R l e t DK be t h e space o f Corn f u n c t i o n s i n R w i t h

support i n

K. One places a t o p o l o g y on D~ by s p e c i f y i n g t h a t a sequence o f 9 j E D~ converges t o 0 p r o v i d e d sup)Da9.1 -+ 0 u n i f o r m l y on K f o r each f i x e d a . E v i J d e n t l y i f T s a t i s f i e s t h e c o n d i t i o n s o f D e f i n i t i o n B . l t h e n (T,q ) -+ 0 when j -+ 0 i n D ( i . e . T: DK -+ C i s c o n t i n u o u s ) . On t h e o t h e r hand i f (*) i n K

330

ROBERT CARROLL A

D e f i n i t i o n B . l does n o t h o l d f o r some K = K, w h i l e ( T , v . ) + 0 whenever IP

J

+.

j

0 i n D K a r b i t r a r y , then, f o r any j , t a k i n g C = k = j i n D e f i n i t i o n B . l , we have

I( T,lpj I

(T,qj)

1suplDa'pjI

> j

(la1

5 j ) f o r some 9 j E D C One can assume IDalp.( 5 l / j f o r la( 5 j ( i . e . v j +. 0

= 1 (by l i n e a r i t y ) and then

i n Dt when we l e t j r u n ) a l t h o u g h

(

T,'p

J

.)

J

+

0.

T h i s c o n t r a d i c t s and hence

we can s t a t e

A l i n e a r map T: C i

tHE0REm 8.2.

+.

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

o n l y i f T i s a continuous l i n e a r map DK

C f o r every K C R compact.

By Theorem 8.2 i n o r d e r t o t e s t whether o r n o t a s p e c i f i c ob-

REmARK 3.3, j e c t (e.g.

+

a d e l t d o b j e c t d e f i n e d by ( 6 , ~ )= ~ ( 0 ) )i s a d i s t r i b u t i o n one E

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

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

I n p r a c t i c e t h i s i s a l l we need. DK. a t o p o l o g y on D = C i , 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 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 t h a t a l i n e a r map T: D t i n u o u s i f and o n l y i f T: D K m i n i n g " sequence K

C

n

Kn+l

+

+.

F, F a LCS, i s con-

F i s continuous f o r each Kn i n any " d e t e r -

o f compact s e t s which exhaust R ( i . e .

R =

UK,).

T h i s a l l o w s one t o s p e c i f y d i s t r i b u t i o n s T E D'(n) as continuous l i n e a r maps T: D

+

D 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

C when

f o r t h e d u a l i t y n o t a t i o n D-D'. Let R = R

EXAAIRPCE 3.4. clearly

( 6,q

j

)

=

'p

.(O)

J

1 +

'p + 0 i n D be a g e n e r i c sequence. Then j K 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 . For any f E

and l e t

E v i d e n t l y ( f , q ) + 0 so f = /f f ( x ) ' p ( x ) d x f o r 'p E .C: L1oc j determines a d i s t r i b u t i o n . I n p a r t i c u l a r one d e f i n e s t h e Heavyside funcdefine

(f,'p)

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 D i s c o n s t r u c t e d v i a a t o p o l o g y

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 objects i n Dl. -(T,DkP):

D

More p r e c i s e l y l e t T +

D

+

C.

E

D' and c o n s i d e r t h e map M:

C l e a r l y M i s l i n e a r and Dk: DK

-f

'p .+

Dkv

+

DK I s continuous;

hence ( g i v e n t h a t t h e t o p o l o g y o f DK i s i n f a c t t h e t o p o l o g y induced by D) by Remark 8.3,

Dk: D + D i s continuous.

Since T: D

+

C i s c o n t i n u o u s by de-

f i n i t i o n , M i s continuous and hence determines an element i n D ' ( R ) -(T,Dk'p)).

=

T h i s leads t o

DEFZNltI0N 8-5- Given T T,Dkq ).

q ) = -(

(M(lp)

E

D' one d e f i n e s DkT by t h e formula

(IP E D) (DkT,

APPENDIX B

Given T = f

E)tAmPI;E 3.6.

E

1 C (0)we see t h a t D e f i n i t i o n 8 . 5 reduces t o t h e

standard f o r m u l a o f i n t e g r a t i o n by p a r t s . one has DY = 6 s i n c e (DY,qP= -(Y,v') DEFINZtI0N 8.7-

331

A p p l i e d t o T = Y o f Example 8.4

= -/rv'(x)dx = q(0) = ( 6 , ~ ) .

L e t E denote C"(n) w i t h t h e t o p o l o g y o f u n i f o r m convergence T h i s w i l l be a m e t r i z a b l e

on compact s e t s o f f u n c t i o n s and a l l d e r i v a t i v e s .

space ( t h e t o p o l o g y i s d e f i n e d by a c o u n t a b l e number o f seminorms) and conI f K C Kn+l w i t h R = UKn i s n a d e t e r m i n i n g sequence o f compact s e t s t h e n a sequence v k +. 0 i n E means

vergence can always be r e f e r r e d t o sequences. t h a t f o r any p and n f i x e d , supIDaqkI

-f

0 for x

Kn and la1 5 p.

E

The dual

space E ' ( = t h e space o f continuous l i n e a r maps E * C) i s i n f a c t t h e space o f d i s t r i b u t i o n s T w i t h compact s u p p o r t (we o m i t t h e p r o o f o f t h i s b u t i t i s routine

-

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 i n an

open s e t A C n i f ( T , q ) = 0 f o r a l l q E C i ( A ) .

The complement i n R o f t h e

u n i o n o f a l l such A where T = 0 i s c a l l e d supp T. DEFINt&I0N 8.8.

For R = Rn now l e t

5 denote t h e space o f Cm f u n c t i o n s IP n

(x E R ) f o r every a =

and B = ( B 1' B,). Such f u n c t i o n s a r e c a l l e d r a p i d l y decreasing and one says v k 0 2 m a i n 5 i f f o r any m and p f i x e d , sup I ( l + l x l ) D v k l 0 ( x E Rn) f o r la1 5 p.

such t h a t sup ( x B D a v ( x ) ( <

m

(al,...,an)

...,

-f

-f

T h i s w i l l be t h e n a t u r a l space f o r F o u r i e r t r a n s f o r m s (see below) and t h e dual space t i ' , t h e space o f continuous l i n e a r maps

D' 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 . t h e d e n s i t y o f D i n ti ( c f . l o c . c i t . ) . DEFINItI0N 3.9. (convolution). fined.

5

-f

F o r q y $ E Cl(Rn) one d e f i n e s (9 For d i s t r i b u t i o n s S,T i n D ' ( R n ) ,

i s a subspace o f

C,

One shows

5' C D' by u s i n g

*

$)(x) =

S

*

/z

v(x-S)$(C)dS

T may n o t be always de-

However i f S E D' and T E E ' f o r example one can d e f i n e S

*

T E D'

E D) ( S * T,q) = (S IT , q ( x t y ) ) = ( Sx,( T , q ( x + y ) ) ) = ( T X Y Y Y' S x y p ( x t y ) ) ) ( t e n s o r p r o d u c t s Ia r e d e f i n e d l a t e r and a r e n o t needed here).

by t h e r u l e ( q (

Let S

EMmPtE 3.10,

E

D ' ( R n ) and T = Dk& E E ' .

q ( x t y ) ) ) = -(S,(6y,Dkq(~+y)))

-(S,D

v(x)) = k

When t a l k i n g about convergence i n D',

(

Then ( S DkS,q).

*

T,q) = ( S , ( D 6 x k y' Hence S * Dk6 = D k S.

El, o r 5' one w i l l always mean weak

convergence (and i n f a c t t h e o n l y l i m i t s which a r i s e w i l l i n v o l v e sequences). Thus f o r example Tn

More general ideas a r e sketched l a t e r . mean (Tn,v)

-f

( T , q ) f o r any f i x e d q

E

D.

-f

T i n D' w i l l

S t r o n g convergence i n D' e t c . w i l l

o f t e n be p r e s e n t b u t n o t needed f o r t h e a p p l i c a t i o n s o f d i s t r i b u t i o n s t o mathematical p h y s i c s i n t h i s book. EMmPLE 3-11,

L e t $ E D(Rn) w i t h say $ L 0, supp $

C

B(0,l) = b a l l o f radius

332

ROBERT CARROLL

Set $,(x)

1 c e n t e r e d a t t h e o r i g i n , and f $ ( x ) d x = 1. for

v

E

D, as k

-+

($k(X)#(x))

m,

= (6,~). T h i s shows t h a t g k

One can show e a s i l y t h a t Tk E Cm(Rn)

Jlk as i n Example B.11. -f

L e t D,

and Tk

*

~l~ w i t h

+

T in

L e t us s k e t c h t h e c o n s t r u c t i o n o f D and D' from " f i r s t p r i n c i -

Thus l e t ,K,, C

be a d e t e r m i n i n g sequence as b e f o r e w i t h

D and d e f i n e seminorms i n D, by N

%

+. v(0)

m.

REmARK 3.13, ples".

= f$(F,)v(S/k)dS

D'.

For T E DI(R") we can d e f i n e a r e g u l a r i z a t i o n Tk = T

RENARK 3.12.

D' as k

= knI$(kx)v(x)dx

6 in

-f

= kn$(kx) and then,

Km

p = ( p l,...,pn) w i t h V(K,,E,S)

1 pi

here.

For I p I =

{v

D,,,; Np(v) 5

=

E

E

P

(v) =

sup IDpvI ( x E K),

UK,

= R.

where

V ( K , E ) = {v E Dm; Np(v) 5 € 1 P m f o r I p l 5 s). E v i d e n t l y V(K,,,,l/n,s) = set

(A)

nV (K , l / n ) f o r I p I 5 s, and t h i s p a r t i c u l a r s e t o f f i n i t e i n t e r s e c t i o n s f o r ~m n = 1,2 and s = 0,1, ... s u f f i c e s t o d e f i n e a FSN B(0) i n D, as i n Appen-

,...

d i x A.

The c o n s t r u c t i o n y i e l d s , w i t h a r o u t i n e argument f o r completeness,

t h e r e s u l t t h a t Dm i s a Frechet space ( e x e r c i s e ) .

L e t now D = C i w i t h im:

We s h a l l p u t on D t h e f i n e s t ( = s t r o n g e s t ) l o c a l l y

Dm -+ D t h e i n j e c t i o n .

a r e continuous ( s t r o n g e s t means t h e convex t o p o l o g y such t h a t a l l t h e im l a r g e s t c o l l e c t i o n o f open s e t s o r NBHs). plicitly.

L e t Vm

€

ced envelope i n D o f taining B =

UV .,

We c o n s t r u c t t h e NBH system ex-

N(0) i n Dm as above and w r i t e r(UV,) UV ;,

f o r t h e convex d i s -

t h i s i s t h e s m a l l e s t convex d i s c e d s e t i n D con-

L e t A be t h e c o l l e c t i o n o f such r ( u V m ) and t a k e i t as a

FSN f o r a t o p o l o g y T on D.

Since Vm C r(UVm) w i t h i m ( V m ) = Vm we see t h a t

i i s continuous. On t h e o t h e r hand i f im i s continuous f o r a l o c a l l y conm vex t o p o l o g y T ' on D and V i s a convex d i s c e d NBH o f 0 f o r T ' then V 3 r ( u ( v n D,)) with v n E N ( O ) i n D, s i n c e i i l ( v ) = v n D ~ . Hence v E N ( O ) for

T and T i s s t r o n g e r than T I .

The s e t s B = r(UVm) a r e a b s o r b i n g s i n c e t h e y a r e taken as NBHs o f 0 and ( Y , x ) Y X i s continuous; hence t h e t o p o l o g y -f

T i s l o c a l l y convex and indeed t h e f i n e s t l o c a l l y convex t o p o l o g y on D such t h a t t h e im a r e a l l continuous.

D w i t h t h i s topology i s c a l l e d t h e s t r i c t

i n d u c t i v e l i m i t o f t h e Dm and one checks e a s i l y t h a t i t i s independent o f t h e d e t e r m i n i n g sequence I(,.

The c o n n e c t i o n o f t h i s w i t h o u r p r e v i o u s d i s -

cussion r e v o l v e s around.

CHEBREIII B.14.

A l i n e a r map f : D

each r e s t r i c t i o n fm= flDm:Dm

-+

-+

F, F a LCS, i s continuous i f and o n l y i f

F i s continuous.

To prove t h i s simply n o t e t h a t f i s continuous i f and o n l y i f f - l ( V ) E N(0) i n D f o r V E 3 ( O ) i n F. Take V convex and d i s c e d so t h a t f- 1 ( V ) i s a l s o and

3 33

APPENDIX B t h e n f- 1 ( V )

E

N ( 0 ) i n D i f and o n l y i f f- 1 (V) n

D,,,E

N(0) i n

I),

which i s

e q u i v a l e n t t o s a y i n g fm i s continuous.

We now d e f i n e t h e Schwartz d i s t r i b u t i o n s as before, D' = c o n t i n u o u s l i n e a r maps D + C.

We remark t h a t o t h e r e q u i v a l e n t d e s c r i p t i o n s o f t h e t o p o l o g y T i n D a r e poss i b l e (see [Sal]).

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

Thus l e t { a } = { a o =+,

L. Schwartz.

be a sequence o f open sets,

fily...l

c an, such t h a t any compact Kc Rn i s e v e n t u a l l y c o n t a i n e d i n t h e nn. Let E,,

-+

ward

{E}

be a decreasing sequence o f p o s i t i v e numbers t ~ =l ~ E ~ ,wE ith ~ , . . .

0, and tml = {mo,m l , . . . l a sequence o f p o s i t i v e i n t e g e r s i n c r e a s i n g t o m.

L e t V ( { m } , { ~ I , { f i l ) be t h e s e t o f f u n c t i o n s

D such t h a t f o r x $

p E

As I n } , t ~ l tml , vary, t h e V ( { m l , { E l , t n l )

nn, IOPV(x)I 5

if I p I 5 mn. form a FSN f o r t h e t o p o l o g y T ( e x e r c i s e ) .

One s h o u l d a l s o n o t e e x p l i c i t l y

t h a t t h e t o p o l o g y induced by T on Dm i s e q u i v a l e n t t o t h e o r i g i n a l t o p o l o g y o f Dm (see below where we complete t h e d i s c u s s i o n r e l a t i v e t o D e f i n i t i o n B.

5). Thus l o o k a t Dk as a map D -+ D. It w i l l be continuous i f a l l r e s t r i c t i o n s t o a Dm a r e c o n t i n u o u s from Dm + D. But Dk maps Dm -+ Dm and D i n duces on D,

t h e o r i g i n a l D,

t o p o l o g y (say Tm).

t o p o l o g y induced by D on Dm.

Tm. E,S)

Since im: Dm -+ D i s continuous, we know T;n c

Conversely, p i c k some Vm = V(K,,E,S) Consequently TmcT.,'

ME0REN 3.15.

Dk: D

+

and d e f i n e f o r example V

= V(Kp, P Then V = r ( u V ) E N(0) i n D and V n Dm = P But Dk: Dm D i s continuous and we have m

f o r a l l p ( n o t e Vp c Vp+l).

V ' = Vm. m

To see t h i s l e t TI;I be t h e

-+

D i s a continuous l i n e a r map.

We t h e n d e f i n e DkT f o r T E D' as i n D e f i n i t i o n B.5.

R e c a l l now by C o r o l l a r y

A.25 t h a t a Frechet space i s b a r r e l e d so Dm i s b a r r e l e d . CHE0REN B.16.

Phood:

Hence

D i s barreled.

L e t B be a b a r r e l i n D.

tions i t i s clear that iil(B) i n t h e Frechet space Dm.

Then i f i,: Dm

+

D a r e t h e canonical i n j e c -

i s a b a r r e l i n Dm.

Hence i - ' ( B ) i s a NBH o f 0 m But by o u r c o n s t r u c t i o n o f NBHs o f 0 i n t h e induc-

t i v e l i m i t t o p o l o g y on D, we see t h a t B 2 r ( U i i 1 ( B ) )

E

N(0) i n D.

QED

C l e a r l y compact s e t s i n D ( o r i n any LCS) a r e bounded, and we s h a l l t h a t conv e r s e l y , c l o s e d bounded s e t s i n D a r e compact.

T h i s remarkable p r o p e r t y

(never t r u e i n an i n f i n i t e dimensional Banach space) makes c e r t a i n powerful arguments p o s s i b l e i n D and D'.

To see t h i s , f i r s t l e t B C D be bounded and

r e c a l l t h e a l t e r n a t i v e d e s c r i p t i o n o f t h e t o p o l o g y o f D a f t e r Theorem 8.14. Suppose t h e supports o f t h e f u n c t i o n s i n B were n o t a l l c o n t a i n e d i n some f o r some f i x e d m.

Then we c o u l d f i n d a sequence xn,

$

l x n l 2 n, and a sequence

334

qn E

ROBERT CARROLL

B w i t h qn(xn)

+ 0.

n But then p i c k any {m) = Imo,m l,...l, ilk= { x E R ;

1x1 5 k } f o r k 2 1, fi0 = 9, and choose Icl = { q k ( x k ) / k l .

xV(Iml,{e),Inl)

= Iq E

t a i n B f o r any f i x e d A .

D;

lDprpl

f o r I p I 5 mk and x

5

It i s obvious t h a t E ilk)does

n o t con-

Consequently, B c Dm f o r some m and e v i d e n t l y B

w i l l be bounded t h e r e ( r e c a l l t h a t D induces on Dm t h e t o p o l o g y Tm). CHE0Rm 3.17.

Hence

B c D i s bounded i f and o n l y i f B c Dm f o r some m and i s

bounded t h e r e . Now l o o k a t B c Dm bounded.

Given any

E,S,

t h e r e i s a A.

such t h a t B c

f o r I A l 2 Ao. I f s = 1, I D p q I 2 I A ~ Ef o r I p I 5 s = 1, and hence by A s c o l i - A r z e l a (Theorem A.33) B i s r e l a t i v e l y compact i n C(K,,,,C) = Co(K,,).

AV(K,,,,E,S)

Indeed bounded d e r i v a t i v e s i m p l y e q u i c o n t i n u i t y and a l s o t h e boundedness o f B ( x ) ( h e r e E = ,,K,,

F = C i n Theorem A.33).

S i m i l a r l y i f Cs(l(,)

denotes t h e

space o f s-times c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s on Km ( w i t h t h e t o p o l o g y o f u n i f o r m convergence o f f u n c t i o n s and a l l d e r i v a t i v e s o f o r d e r :s ) then B i s r e l a t i v e l y compact i n any C'-'(K,,) by a r e p e t i t i o n o f t h e above reasoning ( a p p l i e d f i r s t t o t h e ( s - 1 ) - o r d e r d e r i v a t i v e s i n Co(K,,),

etc.).

We l e a v e t h e remaining d e t a i l s as an e x e r c i s e i n r e c a l l i n g t h a t a c r i t e r i o n f o r compactness o f a s e t K i n a m e t r i c space i s t h a t every i n f i n i t e sequence i n K s h o u l d have a convergent subsequence. CHEOREIR B.I.8. DEFINICl0N 3.19,

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

Bounded s e t s i n D o r E a r e r e l a t i v e l y compact.

A ( H a u s d o r f f ) LCS which i s b a r r e l e d and i n which every

bounded s e t i s r e l a t i v e l y compact i s c a l l e d a Montel space. CHE0RElTl 3.20.

D and E a r e Montel spaces.

Ifa sequence o f d i s t r i b u t i o n s T E D' i s weakly convergent j then i t i s s t r o n g l y convergent.

CHE0REIII 3-21.

P4ool;:

By Theorem A.31 T. i s u n i f o r m l y convergent on precompact s e t s o f D. J But bounded s e t s i n D a r e r e l a t i v e l y compact, hence precompact (see [ B o ~ ] )

and consequently T. converges s t r o n g l y . QED J CHE0REIII 3.22. D i f f e r e n t i a t i o n Dk: D' D' i s continuous i n t h e s t r o n g t o p -+

o l o g y (and i s i n f a c t a homomorphism).

Ptoul;: R e c a l l t h a t DkT i s t h e map

q

-+

-(T,Dkq):

D

-+

C.

L e t B C D be bound-

ed; then u s i n g Theorem 8.17 i t i s easy t o see t h a t Dk(B) i s bounded i n D ( o r s i m p l y n o t e t h a t t h e l i n e a r continuous image o f a bounded s e t i s bounded). But, i n an obvious n o t a t i o n , (DkTa,B) = -(Ta,DkB) bounded s e t s then so does DkTa.

and i f Ta

-+

0 u n i f o r m l y on

We l e a v e t h e r e s t as an e x e r c i s e .

QED.

APPENDIX B

REMARK B.23.

335

L e t us mention here a l s o t h a t s i n c e t h e Lebesgue i n t e g r a l and

measure t h e o r y a r e o f t e n n o t 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 ) w o r k i n g w i t h d i f f e r e n t i a l e q u a t i o n s and p h y s i c s one can n e v e r t h e l e s s d e f i n e and use

2

L

t h e o r y f o r example v i a d i s t r i b u t i o n s ( c f . [ R c l ] ) .

and d e f i n e t h e ( r e a l ) s c a l a r p r o d u c t ( f , g ) = ;/

f g d x w i t h IIfl12 = ( f , f ) .

Let

be a Cauchy sequence i n t h i s norm t o p o l o g y and t h e n f o r any t e s t f u n c t i o n

9,

JI

Thus c o n s i d e r Ci(0,m)

I($,vn

E Ci,

- (P,)I

= l(9n-vm,$)l ~ I I J / U l l ~ p ~ -* v ~0l l so

converges t o L ( $ ) where L i s l i n e a r .

($,vn) i s Cauchy and

That L i s t h e n a d i s t r i b u t i o n f o l l o w s

from t h e Banach-Steinhaus complex o f i d e a s (see Theorems A.31-A.33 that

v,,

-+

and n o t e

D i s b a r r e l e d by Theorem B.16 - c f . a l s o Theorem B.26). Thus we say 2 L i n D' and one can i d e n t i f y L w i t h an L f u n c t i o n i f one knows about

2 L2; i f n o t we s i m p l y d e f i n e L as t h e c o l l e c t i o n o f d i s t r i b u t i o n s L o b t a i n e d i n t h i s manner ( o r more g e n e r a l l y as t h e c o l l e c t i o n o f e q u i v a l e n c e c l a s s e s

- abstract

o f Cauchy sequences f r o m C: i f two Cauchy sequences 9,

and ,$,

completion).

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

are equivalent (i.e.

llgn-J/nll

t h e n t h e y d e t e r m i n e t h e same d i s t r i b u t i o n L ( e x e r c i s e ) . f i n e (L,M)

lim

=

(9

,J/ ) n m

-f

0 as n

+

m)

F u r t h e r one can de-

w i t h IILII = l i m lllpnll e t c . when qn (resp. $,)

are

We l e a v e as an e x e r c i s e t h e p r o o f

Cauchy sequences d e f i n i n g L (resp. M).

2 t h a t L d e f i n e d i n t h i s way i s complete ( c f . [ R c l ] ) . One can o b t a i n 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 More space o f d i s t r i b u t i o n s f such t h a t I( f,lp)l 5 MlllpllL1 f o r a l l lp E C,". g e n e r a l l y we can d e f i n e Lp as t h e space o f d i s t r i b u t i o n s f such t h a t

< MIIvIILq

for all

q~ E

C,"

( l / p + l / q = 1).

I( f,lp)I

L e t us mention a l s o t h a t if f E L

2

w i t h a d e t e r m i n i n g Cauchy sequence v k E C T t h e n by t h e Cauchy-Schwartz i n 2 2 2 equality [ v k ( c ) ~p,(~)]dcI 5 x/f Ipk-qml dc = xllq k-9 m11 L 2 so t h a t

-

:I/

/f 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 c o n t i n u o u s func-

t i o n F(x). that

-1im ( 9

Since -(f,J/)=

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

k

,J/) =

l i m (/qk,J/') = (F,J/') it follows

+ c i n an e v i d e n t way.

= f, and F = JX f ( S ) d c

L e t us a l s o make a few c l a s s i c a l remarks about t h e Lebesgue i n t e g r a l ( c f . Thus f i r s t f o r t h e Riemann i n t e g r a l o f a c o n t i n u o u s f u n c t i o n f b one w r i t e s f(x)dx = l i m f(ci)Aix where Aix = xi - xi-l, i = 1,

[Szl;Til]). on [a,b]

...,n,

/a

1

xo = a, xn = b, and ciE Aix.

t i o n s t h e y a x i s by A.y = y .1 1 o f x such t h a t f ( x ) E Aiy. Ln =

1 qim(Ei)

where m(Ei)

F o r t h e Lebesgue i n t e g r a l one p a r t i -

- y i - 1 ( i = 1, ..., n ) and takes Ei f o r t h e s e t Then p i c k qi E Aiy a r b i t r a r y and f o r m t h e sum

= measure Ei.

I f i n f a c t Ei

j o i n t i n t e r v a l s one sees e a s i l y t h a t L = l i m i t Ln ( n monotone decreasing upper sum

1 yim(Ei)

-f

i s a union o f d i s m)

e x i s t s (note the

and t h e monotone i n c r e a s i n g l o w e r

336

ROBERT CARROLL

1 yi-lm(Ei)

0 ) . More g e n e r a l l y m(Ei) must be d e f i n e d o r one can use o t h e r d e f i n i t i o n s o f t h e i n t e g r a l . For exsum

d i f f e r by l e s s t h a t max lAiy(

ample l e t us say a s e t E has measure <

E

-+

i f E can be c o n t a i n e d i n a f i n i t e E has measure 0 i f such E.

o r c o u n t a b l e s e t of i n t e r v a l s o f t o t a l l e n g t h < a c o v e r i n g can be found f o r any measurable i f f o r any

E

Now l e t f 2 0 on [a,b]

E.

and say t h a t f i s

0 i t can be c o n v e r t e d t o a continuous f u n c t i o n by L e t f 2 0 be measurable and E.

>

changing i t s values on a s e t o f measure < 0 l e t fn

be such a continuous f u n c t i o n . Assume t h e fn b can be chosen such t h a t t h e Riemann i n t e g r a l s f n ( x ) d x have a comnon bound b and d e f i n e GLB ra f n ( x ) d x as t h e Lebesgue i n t e g r a l o f f ( c f . CBo3,Rzll f o r for

E,

-f

Co[a,b]

E

/a

f u r t h e r d i s c u s s i o n and d e t a i l s

-

c f . a l s o Remark 8.24 below).

A few f a c t s and d e f i n i t i o n s about v e c t o r valued f u n c t i o n s w i l l

REmARK 8.24,

The space K i s Co(R) as an i n d u c t i v e l i m i t o f Co(Km) ( c f . 0 i.e. K = U C w i t h t h e f i n e s t l o c a l l y convex t o p o l o g y such

a l s o be u s e f u l . Remark 8.13

-

(h)

t h a t t h e i n j e c t i o n s Co(Km)

-f

K a r e continuous, Co(K,,,) = continuous f u n c t i o n s The dual K ' i s t h e space

having s u p p o r t i n Km w i t h t h e sup norm t o p o l o g y ) . o f (Radon) measures.

Denote Rn o r any open n

C

Rn by E and d e f i n e K(E) i n

t h e same way; i f one has a compact E use Co(E) i n s t e a d w i t h u n i f o r m convergence and measures a r e elements o f t h e d u a l . F i s a LCS over R; l e t f : E

-+

then f

(f(x),z'

E

K(E,F)).

Then x

-+

Let

u

E

K ' and z ' E F ' where

F be continuous w i t h compact s u p p o r t (we say ):

E

p o r t and one s e t s @ ( z ' ) = / ( f ( x ) , z ' ) d u .

+

R i s continuous w i t h compact sup-

T h i s d e f i n e s a l i n e a r form on F '

and hence an element o f F ' * ( * denotes a l g e b r a i c d u a l ) which we c a l l u ( f ) o r / fdu so t h a t ( 1 f d u , z '

)

= / ( f ( x ) , z ' ) du.

If h: E

F the topology o(F,F').

-+

[0,1]

We n o t e t h a t o ( F ' * , F ' )

induces on

i s continuous w i t h compact support,

h = 1 on K = supp f, and C i s t h e weakly c l o s e d convex envelope o f f ( E ) i n F ' * then 1 fdu

E

u(h)C

C

F ' * f o r any p o s i t i v e

u.

I f t h e c l o s e d convex en-

velope i n F o f a compact s e t i s compact t h e n p ( f ) E

F

and t h i s holds when-

ever F i s complete f o r example o r i n t h e weak t o p o l o g y o f t h e dual o f a Banach space. Now f o r v E K(E) and 1 5 p < m , w r i t e Np(v) = ( I E I v ( x ) l Pdu ) l / P * For h 2 0 l o w e r semicontinuous ( i . e .

/*

hdp = sup / vdu f o r

l o w e r semicontinuous.

v 5

h.

l i m i n f h ( x ) 2 h ( x o ) as x

For any g

0, /*gdu

=

inf

/*

-+

xo) define

hdp f o r h 2 g

L e t F be a Banach space now and Fp(E,F) t h e space o f

F, d e f i n e d everywhere on E, such t h a t N ( f ) = ( / * l l f ( x ) l l P P dv)"' < m. With N as serninorm Fp(E,F) i s complete b u t i t i s n o t H a u s d o r f f . P L e t Np(E,F) be t h e adherence o f 0 i n Fp(E,F) and Lp(E,F) t h e c l o s u r e o f

a l l functions E

-f

K(E,F) i n Fp(E,F);

t h e n Lp(E,F) = Lp(E,F)/Np(E,F).

We know i f f E K(E,F)

337

APPENDIX B

1 fdu

and F i s Banach t h e n

I I / fdull 5 / I I f ( x ) l l d p = N l ( f ) .

E F; f u r t h e r

Hence 1 f + I fdu: K(E,F) -+ F i s c o n t i n u o u s when K(E,F) has t h e t o p o l o g y o f 1: (E,F). 1 Since K(E,F) i s dense we can extend t h i s map t o t (E,F) t o d e f i n e 1 f d p E F.

Since any r e a l measure can be w r i t t e n 1-1 =

I.I

+

- u-

with

p'

F

0 one can speak

o f i n t e g r a t i n g w i t h r e s p e c t t o any measure; complex measures can a l s o be

A f u n c t i o n f:

handled r o u t i n e l y .

z' E

f o r any

F' x

+

(f(x),z')

E-+ F

i s s a i d t o be s c a l a r l y i n t e g r a b l e i f

i s integrable.

I f f o r example F i s r e f l e x i v e

Banach and f ( K ) i s bounded f o r K compact t h e n a s c a l a r l y i n t e g r a b l e f s a t i s f i e s l fdp

F.

E

s e t K c E and F', x

+

+

F i s u-measurable i f f o r any compact

0 t h e r e i s a compact K ' c K w i t h

E

t i n u o u s on K ' .

z' E

F o r F Banach, f : E

p(K

nCK') 5

and f con-

E

E q u i v a l e n t l y f i s u-measurable i f and o n l y i f ( A ) f o r e v e r y

(f(x),z'

)

i s measurable and (B) f o r any compact K c E t h e r e i s

a denumerable s e t H c F such t h a t f ( x )

for

E

K

almost a l l x E

p

(thus F

We r e f e r t o [ B o ~ ] f o r f u r t h e r i n -

separable a u t o m a t i c a l l y s a t i s f i e s ( B ) ) .

f o r m a t i o n on measures, almost everywhere (a.e o r AE) t e r m i n o l o g y , e t c . There i s a m e a s u r a b i l i t y gap h e r e i n p r e s e n t a t i o n b u t anyone who has gone t h i s f a r can e a s i l y p i c k up some elementary measure t h e o r y f r o m say [ B o ~ ; L e t us a l s o r e c a l l i n passing t h a t a Banach valued f u n c t i o n f :

M1,E;Till. R

+

E i s c a l l e d a b s o l u t e l y continuous on say [O,T]

exists

such t h a t

&(E)

(an,Bn) n (a,,,,~,,,)

and f ( x )

-

= @

1 Ilf(5,)

-

f o r m # n.

f ( a ) = la f ' (x t)dt.

f(an)//

5

I f e.g.

E

i f f o r each

-

5 6 ( ~ )and R t h e n one knows f ' e x i s t s AE

whenever

E =

1 /an

0 there

E

an(

I f E is r e f l e x i v e t h i s same r e s u l t h o l d s b u t

i f E i s n o t r e f l e x i v e an a b s o l u t e l y continuous f may n o t be d i f f e r e n t i a b l e

anywhere ( c f . [ B d l ] ) .

L e t us a l s o mention t h e Lebesgue dominated conver-

If 1 5 p <

gence theorem.

m,

fn -t f AE,

( f n ( x ) ( 5 ( g ( x ) l A E f o r some g E Lp,

then f E Lp and I I f - f II + 0. Another u s e f u l f a c t i s t h a t i f fn f i n Lp n P + f AE. (1 5 p < m ) t h e n t h e r e e x i s t s a subsequence f nk 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 j s one o f t h e most s i g n i f i c a n t items -f

i n a l l o f mathematics.

We g i v e h e r e some b a s i c i n f o r m a t i o n . Iv

for f E

$(R")

( 0 )

Ff(h) = f ( x ) =

[I f ( x ) e x p ( i ( x , x ) ) d x

The f o l l o w i n g formulas a r e obvious ( 6 ) Du?(x) =

~1exp(i(X,x))D'f(x)dx.

(-i)lalhB?((h) =

(0,and

US

define

1 X.x J j*

jm -m iIalxaexp(itX,x))f(x)dx;

Now t h e i n v e r s i o n f o r m u l a f o r

which i s proved below, has t h e form (+) f ( x ) = Using t h i s , w i t h

Let

where ( h , x ) =

ZIT)-^^^

(01,

F(x)exp(-i(x,x))dx

t h e d e f i n i t i o n o f convergence e t c . i n D e f i n i t i o n

B.8, one proves e a s i l y

ZHE(DREFI B.25.

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 5

Now we prove t h e i n v e r s i o n ( + ) as p a r t o f t h e f o l l o w i n g theorem.

+

ti (Onto

338

ROBERT CARROLL

For f , g

rHE0REIR 3-25. dx; ( f

*

gjv(A) =

E

si one has (*) a l o n g w i t h

?(X)t(X).

(m)

[I q A ) g ( h ) d h [z fc =

The f i r s t formula i n ( m ) i s w r i t t e n ( T , g ) = ( f ,

n,

g ) and i s c a l l e d t h e Parseval r e l a t i o n .

Phoo6:

One can see e a s i l y t h a t (**)

/f g(A)?(A)exp(-i(A,x))dA

f(x+n)dn.

Indeed t h e l e f t s i d e i s J g ( A ) e x p ( - i t h , x ) ) [ /

= f f(y)[/

exp(i(A,y-x))g(A)dAldy

exp(-i(A,x))dA

= E-'~(E;/E). =

E-~/:

f(y)exp(i(~,y)dy]dA

/_fg(EA)exp(i(h,c))dA

I t f o l l o w s t h a t i n (**),

La 2

;(Q/c)f(x+n)dn

=

Cong(z)

E-~[:

f g(ci)?(h)

(*A)

c ( c ) f ( x + E 5 ) d c . Now one uses a

[Yol]) f e x p ( - l x l /2)exp(i(A,x))dx 2 Thus, s e t t i n g g ( x ) = exp(-1x1 / 2 ) and l e t t i n g

w e l l known formula ( c f .

(-lAI2/2). from (*A) g(0)Lf ?(A)exp(-i(A,x))dA

[z z(n)

f f ( y ) F ( y - x ) d y = f ?f(n)f(n+x)dn.

s i d e r now g(Eh) i n s t e a d o f g(X) so t h a t exp(i(z,c/E))dz

=

= (21~)"'exp E

* 0 one o b t a i n s

This i s (*) s i n c e g ( 0 )

= f ( x ) l z ;S'(c)dc.

2

= 1 w h i l e 1 g ( c ) d c = ( 2 1 ~ ) ~ ' exp(-IE;I ~f /2)dS = ( 2 ~ ) ~ For . the f i r s t part o f (m)

we s e t x = 0 i n (**) w h i l e f o r t h e second p a r t one has s i m p l y f ( f * g ) ( x )

exp(i(A,x))dx = J exp(i(A,x))[/ (i(X,x-c ) ) f ( x - c ) d c d x =

f ( x - c ) g ( C ) d c ] d x = f f exp(i(A,E,))g(c)exp

?(X)g(A).

Now one uses t h e Parseval formula

si

9' * (T,Fp) : si + si

t o define the Fourier transform i n

Thus, g i v e n T E si', c o n s i d e r t h e map N: IP

by d u a l i t y .

*

(m)

+

Fp

5 i s continuous and hence N i s continuous, as a composition o f continuous maps, so N determines an element o f 5 ' . C.

By Theorem 8.8 F:

Given T E si' one d e f i n e s t h e F o u r i e r t r a n s f o r m FT by (IP E

DEFINlCI0N 3.26.

EMIIIPLE 3.27.

-+

One can check t h e f o l l o w i n g formulas e a s i l y ( T E si ): F6 = 1;

F(DkT) = -iAkFT; Dk(FT) = F ( i x k T ) .

f" =

i n t h e Parseval f o r m u l a ( m ) so f o r f , g E s, F(h m w i t h f ( x ) = ( 1 / 2 ~ ) ~and F consequently lm Fgdx = ( 2 2 Using t h e d e n s i t y o f C," o r 8 i n L one o b t a i n s immediately

L e t us t a k e

--,.,

exp(-i(A,x))dx

EHE0RERl 3.28.

The F o u r i e r t r a n s f o r m i s an isomorphism L 2 + L2 w i t h ( g E L 2 )

l"j2dh = (2a)'J:

Jm -m

lg12dx.

Obviously a d i f f e r e n t n o r m a l i z a t i o n f o r F would make t h e correspondence g

*

i n Theorem 8.28 an i s o m e t r y b u t we p r e f e r t h e p r e s e n t formulas because o f (m)

and Example 8.27 f o r example.

We remark t h a t S

*

T i s n o t i n general

d e f i n e d f o r S,T E 8 ' b u t , when S E 3' w i t h T E 0 i (a space o f d i s t r i b u t i o n s we w i l l n o t pursue f u r t h e r ) then S

*

T E 3 ' makes sense and F(S*T)

FSFT.

We remark a l s o t h a t t h e F o u r i e r t r a n s f o r m can be d e f i n e d f o r D' by u s i n g

APPENDIX B

339

somewhat d i f f e r e n t techniques. L e t us go now t o a t o p i c which p l a y s an u n u s u a l l y i m p o r t a n t r o l e i n many areas, namely t h e Paley-Wiener complex o f ideas. An e n t i r e f u n c t i o n F ( c ) ,

CHE0REIR 3.29. F(s) =

[f

f(x)exp(i(x,s))dx,

supp f C B(0,R) = I x ;

cN

S~ = Ck +

1x1 5 R),

c

First

i s the Fourier transform

E Cn,

i n k y of a f u n c t i o n f E C,"(Rn) w i t h

i f and o n l y i f f o r e v e r y N t h e r e e x i s t s a

such t h a t ( * ) I F ( < ) \ 5 cN(1 + (<1)-Nexp(RlIm51).

Ph006:

The n e c e s s i t y i s immediate from t h e r e l a t i o n ( c f . ( b ) ) , I"(-ick)'k

F ( c ) = 1 D B f ( x ) e x p ( i ( x , s ) ) d x ( i n t e g r a l o v e r 1x1 5 R). For s u f f i c i e n c y de-n m Lm exp(-i(x,E))F(C)dE ( 5 r e a l ) . One sees e a s i l y t h a t

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

36) =

-

F ( 6 ) and f E Cm ( e x e r c i s e

use an analogue o f ( 6 ) f o r t h e d i f f e r e n -

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

tiation).

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 rl = a x / I x I

/_fexp(-i(x,c-in))F(c,-in)dC.

Take N = n + l so t h a t \ f ( x ) \ 5 (2n)-'CN

e x p ( R l q l - ( x , ~ ) ) [ ~ (1+151)-Ndc.

0 ((x,n) = alx1 w h i l e

R)q]

(a > 0 ) f ( x ) = ( Z T T ) - ~

I f now 1x1

= Ra).

R we l e t a

7

+ m

t o obtain f ( x ) =

QED

Hence supp f C B(0,R).

Cf(E0REm 3-30, An e n t i r e f u n c t i o n F ( c ) 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 and o n l y if f o r some c o n s t a n t s R, N, and C one has (***) I F ( c ) l 5 C ( l +

I 5 I INexp(R I Imc I 1. Pmod:

For 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 J/ E C," equal t o 1

on K w i t h supp J/ C K '

2

K.

Then one sees e a s i l y t h a t f o r any 9 E E , ( T , l p )

Since T E D' by D e f i n i t i o n B . l t h e r e e x i s t c o n s t a n t s c

= ( J / T , q ) = (T,qJ/).

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

1sup

lDax [

(la1 5 k ) f o r

x

E

D ~ , . B u t x = I ~ J / E D ~f ,o r a n y v E E s o I(T,lp)I 5 c l s u p ( D a ( 9 J , ) ( c c ' 1 sup I D a q I (la1 5 k , x E K ' ) where c ' depends on c and J / . We can a l s o t a k e K' C B(0,R)

f o r some R.

t a i n s (***). some T

E

3'.

has F T k = F(T

Now t a k e 9 = e x p ( i ( x , ~ ) ) so

(

T , v ) = FT and one ob-

For s u f f i c i e n c y we know F E 3 ' f o r 5 r e a l so F =

7=

FT f o r

L e t LLk be an approximate i d e n t i t y as i n Example B . l l and one

*

J/k) = F(FJ/k).

But supp J/k

[ F k ( C ) l 5 CM(1+1c1)-Mexp(lIm61/k). I:/)N-Mexp(R+l/k)lImcl

B(O,l/k)

and hence f o r any

3 m

M

t

and s i n c e Tk E Cm ( c f . Remark B.12) by Theorem 8.29 But one checks e a s i l y t h a t supp (T*J/k)

supp Tk C B(O,R+l/k). Supp $'k and as k

C

COnSeqUentIy l F T k l = I F r k l 5 CM(1

i t f o l l o w s t h a t supp T

C

B(0,R).

C

supp T +

QED

2 The c l a s s i c a l Paley-Wiener theorem i n v o l v e s L 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 of 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 . [Chl;Btl;Dl;Ppl;Rdl;Yol]).

Thus f i r s t one says t h a t

34 0

ROBERT CARROLL

a f u n c t i o n f ( z ) i s of e x p o n e n t i a l t y p e T i f f o r each such t h a t I f ( z ) l 5 M e x p ( l z l ( T + ~ ) f o r z E C.

0 there exists M Then we have E

>

n

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 LL o n 2 F ( t ) e x p ( i t z ) d t where F E L (-T,T). t h e r e a l l i n e i f and o n l y i f f ( z ) = 1 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 E L tJHE0REIR 3.32. T F ( t ) e x p ( i t z ) d t where F(T) = F(-T) = 0 f o r z r e a l i f and o n l y i f f ( z ) =

CHE@REIII 8.31,

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 i n t e r v a l [-T-E,T+E]. Theorem 8.31 can be proved by d i r e c t e x t e n s i o n f r o m C; formula and a l i t t l e t h o u g h t

-

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

Theorem B.32 i s harder.

some i n f o r m a t i o n about l i n e a r semigroups which w i l l be u s e f u l a t v a r i o u s times ( c f .

[Cl;Dul;Ftl;Bzl;Hpl;Ka2;Tal;Yol]

A basic

for further details).

m o t i v a t i o n i s t o s o l v e t h e Cauchy problem f o r t h e simple case ( * a ) u ' + Au = 0, U ( T ) = u

0

where A i s a c o n s t a n t unbounded o p e r a t o r i n

A family S(t)

DEfZNltI@N 3-33,

a

6anach space.

L ( F ) ( L ( F ) = bounded l i n e a r o p e r a t o r s on

E

2t

t h e Banach space F) w i l l be c a l l e d a s t r o n g l y continuous semigroup ( 0

-) i f S(t+T) = S ( t ) S ( T ) , S(0) = I, and t

-+

<

The

S ( t ) x E Co(F) f o r any x E F.

i n f i n i t e s i m a l g e n e r a t o r -A o f S ( t ) i s t h e l i n e a r o p e r a t o r d e f i n e d by -Ay =

l i m [S(t)y

-

y]/t,

CHE6RElII 3.34.

t

-+

O+, on those y

E

D(A) f o r which t h e l i m i t e x i s t s .

D(A) i s dense, A i s closed, and f o r t > 0, d S ( t ) y / d t =

- A S ( t ) y = - S ( t ) A y when y E D(A) ( h e r e d / d t denotes o r d i n a r y v e c t o r valued d i f f e r e n t i a t i o n as a l i m i t o f d i f f e r e n c e q u o t i e n t s ) .

x

Phoud: y,

If y, = 10 S ( t ) y d t , then e v i d e n t l y x-'yx

= (l/t)/k

-

F.

a r e dense i n

W r i t i n g -At

-

[S(S+t)y

= [S(t)

-

-+

y as

x

-f

I ] / t we have as t

S(S)yIdE; = (l/t)I$+t S(S)ydS

-

0, and hence such +

0, (*&) -kty,

( l / t ) f b S(S)ydS

-,S(h)y Fur-

Since l i m Atyh e x i s t s , y, E D(A), which i s consequently dense.

y.

t h e r , f o r t > 0, A > 0, and y S ( t ) y ] = -S(t)A,y

-f

D(A) c o n s i d e r -A,S(t)y

€

x

- S ( t ) A y as

-+

-AS(t)y = d+S(t)ydt = -S(t)Ay.

Similarly, x-l[S(t)y

Ay,

-f

-f

d - S ( t ) y / d t = - S ( t ) A y as h

=

y, and AY,,

w.

-+

-AI:

S(S)ydS.

( * r ) we

E v i d e n t l y , AS(c)yn = S(c)Ayn

-f

w as t

-f

will

obtain

(**I

S(t)y

-

y =

Then t o show A i s c l o s e d l e t yn E D ( A ) , yn

(llS(~,)Il 5 c t h e r e by Banach-Steinhaus), = ( l / t ) $ S(S)wdg

S(t-A)y] = -S(t-A)

which means t h a t d S ( t ) y / d t e x i s t s as

Also, i n t e g r a t i n g and u s i n g

-1: AS(S)ydS

-

0, by s t r o n g c o n t i n u i t y ( s i n c e S ( t - A )

remain bounded by Banach-Steinhaus), described.

-

= h-l[S(t+x)y

0, which i m p l i e s S ( t ) y E D(A) w i t h

0.

-+

+

S(5)w u n i f o r m l y on [ O , t ]

and hence from (**) Aty = l i m Atyn

Thus y E D(A) w i t h Ay = w.

QED

APPENDIX B

341

Thus i f -A i s t h e i n f i n i t e s i m a l g e n e r a t o r o f a s t r o n g l y c o n t i n u o u s semigroup S ( t ) , i t f o l l o w s t h a t f o r any uo

E

D(A) t h e Cauchy problem (*a) has a s o l u -

t i o n u = S(t)uo, which i s i n f a c t s t r o n g l y d i f f e r e n t i a b l e f o r t > 0 w i t h u ' = -S(t)Auo continuous t h e r e .

tions u

E

This s o l u t i o n i s unique ( i n t h e class o f func1 E C ( F ) f o r t > 0 w i t h y ' t Ay = 0 and

C 1 (F) f o r t > 0 ) s i n c e i f y

c

y ( 0 ) = uo t h e n c o n s i d e r f o r 0 <

= f(c).

< t the function S(t-c)y(c)

Upon

d i f f e r e n t i a t i n g we o b t a i n ( d / d E ) f ( c ) = S(t-E)y' + S ( t - t ; ) A y ( c ) = 0 ( e x e r c i s e ) and hence f ( t ) = y ( t ) = f ( 0 ) = S(t)uo = u ( t ) . Now t h e f u n c t i o n h

+

i h ( A ) = ( x I t A ) - l = Rh(-A) where i t makes sense i s c a l I f IlS(t)ll (m

l e d t h e r e s o l v a n t o f -A.

on [O,S]

(by Banach-Steinhaus) t h e n

w r i t i n g any t as t = ns t t;, 0 2 5 < 6, one has llS(t)U 5 U S ( 6 ) l l n l l S ( ~ ) I I 5

mntl

5 mexp(wt) where w

wo = i n f h ( t ) / t f o r t

>

= (logm)/s.

0.

Then s e t h ( t ) = l o g l l S ( t ) l l and c o n s i d e r

We have h(tl+t2)

= logllS(tltt2)ll

5 logllS(tl)ll

wo i s e i t h e r f i n i t e o r --. z h ( t o ) / t o 5 wo + E . Then f o r ( n - l ) t o < t < n t o one has t h e e s t i m a t e h ( t ) / t 5 [ ( n - l ) h ( t o ) + h ( t - ( n - l ) t o ) ] / t 5 h ( t o ) / t o + [logm t w t o ] / t , which i m p l i e s l i m sup h ( t ) / t IlS(t,)II

= h(tl)

If finite,

< w t E. 0 (exercise).

t h(t2) with h ( t )

given

E

5wt

t logm and

> 0, t h e r e i s a to > 0 such t h a t wo

Hence wo = l i m h ( t ) / t as t

-+

-, and t h e case wo

=

--

i s similar

The number wo i s c a l l e d t h e t y p e o f S ( t ) and by s u b a d d i t i v i t y

f o r any w1 > wo,

IIs(t)II 2 m e x p ( w l t )

f o r some

m (exercise).

CHE0Rfm 8-35. I f S ( t ) i s a s t r o n g l y continuous semigroup o f t y p e wo w i t h i n f i n i t e s i m a l g e n e r a t o r -A then f o r Reh > wo, ?((A) i s d e f i n e d and i s g i v e n by t h e formula (Riemann i n t e g r a l ) EA(A)y = ir e x p ( - A t ) S ( t ) y d t . Pmvd:

For Reh > w1 > wo w i t h

exp[-(o-wl)t]

x

= u

f i T

one has \ e x p ( - x t ) \ l l S ( t ) y l l 5mlyn

L1, and t h e i n t e g r a l makes sense, d e f i n i n g a bounded operaTo show t h a t Eh(A)y € D ( A ) c o n s i d e r m/u-w € L ( F ) w i t h IIE,(A)II tor 1' f o r a > 0, ( * m ) A a i x ( A ) y = - ( l / a ) i T e x p ( - h t ) [ S ( t + a ) y - S ( t ) y l d t = -(l/cx) exp( - A t ) S ( t ) y d t t (1 / a ) e x p ( h a ) / : exp( - A t ) S ( t ) y d t . Hence [exp( xa )-l]I," E

kh(A)

A

6

a "A

(A)y

-f

-XR",(A)y

t y as u

-f

0, and hence $(A)y E D ( A ) w i t h Aih(A)y =

F u r t h e r , f o r y E D(A) b o t h e x p ( - A t ) S ( t ) y and A e x p ( - A t ) S ( t ) y

-XR,(A)y t y. a r e c o n t i n u o u s and i n t e g r a b l e ; hence s i n c e A i s closed, i t can be c a r r i e d under t h e (Riemann) i n t e g r a l s i g n i n Theorem 8.35 t o o b t a i n ARVh(A)y = K,(A)Ay

(use here t h e d e f i n i t i o n o f t h e Riemann i n t e g r a l as a l i m i t o f sums)

Thus we have shown (A+XI)gh(A)y = y and f o r y E D ( A ) ,

[Afih(A)

t h I ] y = y which determines Gh[A)

as ( A + X I ) - l .

Rv,(A)(A+XI)y

=

QED

We c i t e n e x t t h e P h i l 1 ips-Miyadera e x t e n s i o n o f t h e H i l l e - Y o s i d a theorem.

342

ROBERT CARROLL

It i s o f f r e q u e n t use i n app i c a t i o n s o f semigroup t h e o r y t o s o l v i n g t h e

b u t w i l l n o t be needed i n f u l l s t r e n g t h here; we s k e t c h

Cauchy problem (*.)

i n s t e a d a p r o o f o f t h e c o r o l a r y which i s used more o f t e n .

A necessary and s u f f i c i e n t c o n d i t i o n f o r a c l o s e d l i n e a r

CHEBREIII 3.36.

o p e r a t o r -A w i t h dense domain t o generate a s t r o n g l y continuous semigroup S ( t ) i s that there exist

m

> 0 and w such t h a t Il(X1

C@RBCCARY B.37

+ A)-'II

5 m/(A-w)n

for

I n t h i s event IlS(t)ll 5 mexp(wt).

> w and i n t e g e r s n.

a l l real

A necessary and s u f f i c i e n t c o n d i t i o n f o r a

(HIftE-YBdIDA).

c l o s e d densely defined l i n e a r o p e r a t o r -A t o generate a s t r o n g l y continuous

5 l / X f o r X > 0.

semigroup o f c o n t r a c t i o n o p e r a t o r s i s t h a t 1I (xI+A)-lll

Ptooh:

F i r s t n o t e t h a t I I ( I + c Y A ) - ~ I5 I 1 for

f o r t 1. 0. E

F ) as t

-f

0 (exercise).

[Sm(t-s)Sn(s)xJds t h a t Sn(t)x

2 0 and s e t S n ( t ) = ( l + t A / n ) - n

Then IISn(t)ll 5 1, so t h a t t h e S n ( t ) a r e u n i f o r m l y bounded. f o r t > 0 and S n ( t ) x

thermore one has S;l(t) = -A(I+tA/n)-"'

(x

CY

-

Consider now [ S n ( t )

= l i m J:-'[-S,'(t-s)Sn(s)x

S,(t)x

[I+(~/n)A]-~-'xds.

= l i m I:-'[(s/n)

2

-

-

+

Fur-

Sn(0)x = x

Sm(t)]x = l i m j i E ( d / d s )

+ Sm(t-s)S;l(s)x]ds. It follows (t-~)/m]A~[I+((t-s)/m)A]-~-'

Then i f x E D(A ) we have an easy e s t i m a t e s i n c e i n p a r -

t i c u l a r t h e r e s o l v a n t o f A commutes w i t h A; one has t h e r e l a t i o n S n ( t ) x -m-1 [I t (s/n)A]-"' A'xds. S,(t)x = Jot [ ( s / n ) - ( t - s ) / m l [ I + ( ( t - s ) / m ) A l f o l l o w s t h a t IISn(t)x - S,(t)xll 2 11A2xllJ~[ ( s / n ) + (t-s)/m]ds = ( t 2/ 2 )

It

2 ( l / n + l/m)llA xII.

Hence t h e S n ( t ) x form a Cauchy sequence, and l i m S n ( t ) x 2 However, D(A ) i s dense i n 2 o u r Banach space F ( e x e r c i s e ) ; n o t e f o r example t h a t D ( A ) = (A+hI)-lD(A)

e x i s t s u n i f o r m l y i n t i n any f i n i t e i n t e r v a l .

f o r h > O a n d (A+XI)-l has range D ( A ) which i s dense.

By v i r t u e o f t h e u n i -

form boundedness o f t h e S n ( t ) i t f o l l o w s t h a t S ( t ) x = l i m S n ( t ) x = l i m ( I + t A / n ) - n x e x i s t s f o r any x E F ( c f . Theorem A.31).

I t i s easy now t o

check t h a t S ( t ) has t h e d e s i r e d p r o p e r t i e s f o r a s t r o n g l y c o n t i n u o u s semigroup and we l e a v e t h i s as an e x e r c i s e . As an a p p l i c a t i o n c o n s i d e r A = -A

+

BED

c d e f i n e d i n H = L L ( Q ) by D i r i c h l e t o r

Neumann c o n d i t i o n s ( c f . §1.9). Then i n e i t h e r case Re((A+h)u,u) = (ctReA) 2 I l u l l ~ t J l v u l dx and Theorem 1.9.2 i m p l i e s t h a t A+h i s 1-1 o n t o H f o r Rex 2 + c > 0. From Re((A+h)u,u) 2 (c+Reh)Null we o b t a i n furthermore II(A+X)ull 2 >

(c+Rex)llull and hence II(A+h)-lI1

< l/(c+Reh).

Consequently f o r c = 0 Coro-

l l a r y 6.37 i m p l i e s t h a t -A generates a s t r o n g l y continuous semigroup o f cont r a c t i o n o p e r a t o r s , s o l v i n g s t r o n g l y t h e r e f o r e t h e Cauchy problem f o r u ' Au = 0 w i t h D i r i c h l e t o r Neumann boundary c o n d i t i o n s .

APPENDIX 6

343

To r e l a t e t h e semigroup ideas t o monotone o p e r a t o r s as i n

3.3 we d e f i n e a

l i n e a r monotone o p e r a t o r A t o be maximal l i n e a r , o r s i m p l y maximal, i f i t i s n o t t h e p r o p e r r e s t r i c t i o n o f a n o t h e r monotone l i n e a r o p e r a t o r .

We empha-

s i z e t h a t t h i s i s n o t t h e same as s a y i n g t h a t A i s maximal D ( A ) monotone i n t h e sense o f D e f i n i t i o n 3.2.15.

R e c a l l a l s o t h a t monotone o p e r a t o r s i n

H i l b e r t spaces a r e c a l l e d a c c r e t i v e ( c f . 13.3).

ZHE0REIII 3-38, - A i s t h e i n f i n i t e s i m a l g e n e r a t o r o f a s t r o n g l y c o n t i n u o u s semigroup o f c o n t r a c t i o n o p e r a t o r s i n t h e H i l b e r t space H i f and o n l y i f A i s a ( c l o s e d ) maximal a c c r e t i v e l i n e a r o p e r a t o r w i t h dense domain. For p r o o f we n o t e f i r s t t h a t i f S ( t ) i s a c o n t r a c t i o n semigroup w i t h generat o r -A t h e n f o r y

E

D ( A ) and A

-f

0,

(A*)

2

0 L ( l / ~ ) { i l S ( ~ ) y -l l ~IIyll 1 =

- (YJY). Hence A

([S(A)Y-Y]/A,S(A)Y) + ( Y , [ S ( ~ ) Y - Y I / A ) + - ( A Y , Y )

t i v e and D(A) i s dense by Theorem 6.34 w i t h A closed.

i s accre-

For t h e r e m a i n i n g de-

t a i l s we r e f e r t o 13.3 and remarks below.

RrmARK 8-39, A maximal a c c r e t i v e l i n e a r o p e r a t o r i n a H i l b e r t space need n o t be closed, b u t a densely d e f i n e d maximal a c c r e t i v e o p e r a t o r A i s closed.

w, and u E D ( A ) t h e n r d e f i n e d on D(K) = D(A);A E C } by T(V+AU) = Av + A W i s an a c c r e t i v e e x t e n s i o n o f A,

Indeed, i f un E D(A), un .(v+Au;v

E

+

u, Aun

-f

which c o n t r a d i c t s ; however, i f u E D(A) t h e n by Theorem 3.3.3 Au = w s i n c e any l i n e a r A i s a u t o m a t i c a l l y hemicontinuous (see t h e p r o o f o f C o r o l l a r y 6. 41 below f o r r e l e v a n t d e t a i l s and see Remark 3.6.16

f o r a d i s c u s s i o n o f max-

imal m o n o t o n i c i t y i n t h e c o n t e x t o f m u l t i v a l u e d maps).

S i m i l a r l y a closed

maximal a c c r e t i v e o p e r a t o r i s densely d e f i n e d ( e x e r c i s e ) . We p r o v e now a v a r i a t i o n on Theorem 3.3.6

due t o Browder which can be used

i n c i d e n t a l l y t o complete t h e p r o o f o f Theorem 6.38.

CH€@REm 8.40.

F be a r e f l e x i v e Banach space and A: F

Let

hemicontinuous o p e r a t o r (D = D(A) -2

s i o n A t o any

3

D

f o r u E D with q(x)

Pmvd:

(E n o t + m

C

-f

F ' a D-monotone

F ) which has no proper monotone extenIfRe ( A ( u ) , u ) ~ q ( l l u I )IIull I

necessarily linear).

as x

-f

m,

then R(A) = F ' .

By Theorem 3.3.3 A i s maximal D-monotone.

R e f e r r i n g f o r comparison

t o t h e p r o o f o f Theorem 3.3.6 l e t w E F ' be a r b i t r a r y and A be t h e f a m i l y o f f i n i t e dimensional subspaces o f D o r d e r e d by i n c l u s i o n . F i s t h e i n j e c t i o n and we d e f i n e again A E = i f A i E .

F o r E E A, iE: E +

Then as b e f o r e , AE i s E-

monotone and hemicontinuous and hence continuous ( b y Thecrem 3.3.5) Re(AEu,u) ~ q ( l l u l ~ ) U u fl lo r

E'

U E

E ( c f . §3.3).

and we p i c k uE E E such t h a t AEuE = i f w .

with

By L e m a 3.3.7 AE maps E o n t o Then q(lI uil

)IluEII 5 Re

(

AEuE,uL)

344

ROBERT CARROLL

) = Re (w,u ) < IIwllIIuEll. Consequently, v(IIuEll) 5 IlwII and hence E E E Ilu II < M f o r some M independent o f E. L e t BM = { u E F;IIull 5 M I and then by E weak compactness ( c f . Theorem A.43) t h e r e e x i s t s uo E BM such t h a t f o r each

= Re ( i*w,u

Eo E A, each f i n i t e s e t vl,

I( u,-uE,vj

w i t h E 3 Eo and

...vm i n F ' , and each ) I 5 f o r 1 5 j 5 m. E

Eo E A c o n t a i n i n g v and l e t E E A w i t h Eo

E

> 0, one can f i n d E E A

Now f o r any v E D p i c k some

E; t h e n (AA) Re (Av-w,v-uE) = Re (Av-AuE,v-u ) > 0 s i n c e AEuE = i * w means AiEuE AuE = w. But choosing E E v1 = Av-w above t h e r e e x i s t s E 3 Eo w i t h l( uo-uE,Av-w)( 5 E . Hence Re ( A v -w,v-uo) -w,v-u

O

from (")

-E

)

and, s i n c e

C

i s a r b i t r a r y , we conclude t h a t Re ( A v -

E

Then i f uo E D one deduces t h a t Auo = w s i n c e A i s maximal D-

5 0.

But uo must be i n D, s i n c e if n o t we c o u l d d e f i n e a monotone op-

monotone.

N

N

erator

2

A w i t h D(K) = D(A) u uo, Au

0

QED

c o n t r a d i c t s t h e hypotheses. Now suppose A: V

+

This

w, and Au = Au f o r u E D ( A ) .

=

V ' i s a l i n e a r monotone o p e r a t o r w i t h dense ( l i n e a r ) do-

main D = D(A) i n a r e f l e x i v e Banach space V.

It i s a u t o m a t i c a l l y hemicon-

tinuous, s i n c e from u E 0, w E V, u+tnw E D (which i m p l i e s w E D),

it fol-

lows t h a t A(u+tnw) = Au + t n A w Au, s t r o n g l y i n f a c t ( c f . D e f i n i t i o n 3.2. 2 1 5 ) . I f Re ( A u , u ) 2 cIIuII f o r example then f o r any w E V ' Theorem B.40 -f

y i e l d s uo E V such t h a t Re (w-Av,uo-v)

5 0 for a l l v

be maximal D(A)-monotone by Theorem 3.3.3

Hence i f uo

t o n e o p e r a t o r s a r e D maximal monotone). But again u monotone.

0

further A w i l l

E D(A);

( t h u s densely d e f i n e d l i n e a r mono€

D ( A ) t h e n Auo = w.

must be i n D(A) i f we assume f o r example t h a t A i s maximal N

Indeed, i f uo $ D(A),

d e f i n e A 3 A on D(X) = Iu+auo;u

E

D(A);a E

N

Cl by A(u+auo) (

,

= Au

+

aw.

Then s e t t i n g v = -u/a E D(A) and r e c a l l i n g t h a t N

i s c o n j u g a t e l i n e a r d u a l i t y , t h e r e r e s u l t s Re (A(u+auo),u+auo)

)

Re (wo-Av,uo-v)

C0R0LCARM 3.41.

5 0 f o r a l l u E D(A).

=

2

T h i s c o n t r a d i c t s and we have proved

L e t V be a r e f l e x i v e Banach space and A: V

-f

d e f i n e d maximal monotone l i n e a r o p e r a t o r s a t i s f y i n g Re ( A u , u )

R(A)

= la1

V ' a densely

1. cIIuII 2 . Then

V'.

I n p a r t i c u l a r i f A i s maximal a c c r e t i v e i n a H i l b e r t space H and h > 0 t h e n 2 Moreover, r e f e r r i n g back t o t h e p r o o f

A+h s a t i s f i e s Re((A+h)u,u) 2 XIIuII o f Theorem 8.40,

.

l e t A = A + X w i t h AEuE =

itw,

uo as b e f o r e ; t h e n (")

r e w r i t t e n as Re((A+h)v-w,v-uE) = Re((A+h)v-(A+h)uE,v-uE) v

E

D(A) ( s i n c e as b e f o r e AEuE =

o f course

it

i t w means AiEuE = AuE =

can be

XIIv-u E112 f o r a l l (A+h)uE = w

-

here

i s taken w i t h r e s p e c t t o t h e s c a l a r product i n H ) . Then by 2 1Iv-u 112 (upon s e l e c t i n g a subnet

weak l o w e r s e m i c o n t i n u i t y l i m i n f IIv-uEll

0

uEa converging weakly t o u o ) and, as before, Re((A+h)v-w,v-uEa)

can be made

APPENDIX a r b i t r a r i l y c l o s e t o Re( (A+A)v-w,v-uo). -w,v-u

> Allv-u 112 > 0. 0)0 -

B

345

Hence one concludes t h a t Re( (A+h)v

Now a r g u i n g as before, if uo E D(A) = D(A+A) t h e n

(A+A)uo = w, s i n c e A+A i s maximal O(A)-monotone.

4

On t h e o t h e r hand, i f uo N

cy

D(A),

c o n s t r u c t (A+h)-

A + h as above ( b u t now (A+A)uo = w, e t c . ) and N

XIIv-u It2 w i l l i m p l y t h a t A i s an

observe t h a t t h e n Re((A+x)(v-uo),v-uO)

0

a c c r e t i v e l i n e a r e x t e n s i o n o f A which i s excluded.

Hence we have proved

t h a t i f A i s maximal a c c r e t i v e .(and densely d e f i n e d ) t h e n R(A+A) = H f o r any A > 0; moreover,

2 hllul12 we a l s o o b t a i n II(A+A)-lII

from Re((A+h)u,u)

Then, u s i n g C o r o l l a r y B.37 and Remark B.39,

5 l/A.

i t f o l l o w s t h a t -A generates a

s t r o n g l y continuous semigroup o f c o n t r a c t i o n s , which proves one h a l f o f Theorem B.38. To complete t h e p r o o f o f Theorem B.38 i t i s o n l y necessary t o show t h a t t h e c l o s e d densely d e f i n e d a c c r e t i v e o p e r a t o r A o f

(A*)

i s maximal a c c r e t i v e .

Now we know by C o r o l l a r y 8.37 t h a t A+A i s 1-1 o n t o H f o r A > 0.

Then assume

N

A i s a maximal a c c r e t i v e e x t e n s i o n o f A (whose e x i s t e n c e f o l l o w s f r o m a v e r s i o n o f Z o r n ' s lemma f o r example

- exercise - c f . [Kel]).

From what we have

j u s t proved x + h i s 1-1 o n t o H f o r a l l A > 0 ( C o r o l l a r y 8.41). C

x, one has (Z+A)(A+A)-'x

= x so A+:

maps D ( A ) 1-1 o n t o H.

But s i n c e A Hence D(A) =

N

D ( A ) and A i s i n f a c t maximal a c c r e t i v e .

Theorem 8.38 i s t h u s c o m p l e t e l y

proved, a l b e i t somewhat c i r c u i t o u s l y , and we have i n c i d e n t a l l y proved

A densely d e f i n e d l i n e a r a c c r e t i v e o p e r a t o r i n a H i l b e r t space

LEarmA B.42.

H i s maximal a c c r e t i v e i f and o n l y i f R(A+hI) = H f o r a l l A > 0.

L e t us mention here a few f a c t s about Sobolev spaces f o r r e f e r e n c e a t v a r i ous p l a c e s i n t h e book (see [Adl;Mgl]

f o r further details).

bp(R) i s t h e space o f ( e q u i v a l e n c e c l a s s e s o f ) f u n c t i o n s P u E L P ( n ) such t h a t D"u E L P ( n ) f o r la1 2 m. The norm i s d e f i n e d by IIuI1 m,P = Jn I D " ~ l ~ d x 1 (sum ~ ' ~ o v e r la1 ( m ) . Wm(n) i s a r e f l e x i v e Banach space P i n Wm f o r p # 1,- ( e x e r c i s e ) . Wm(n)o i s d e f i n e d t o be t h e c l o s u r e o f P P' Now we s h a l l c a l l a bounded r e g i o n R C Rn v e r y r e g u l a r i f i t s boundary r i s

DEFZNZCZ0N %.43.

[I

Cy

a Cm compact (n-1)-dimensional m a n i f o l d w i t h R l y i n g on one s i d e o f

r.

We

say t h a t R s a t i s f i e s t h e cone c o n d i t i o n i f t h e r e i s a f i x e d cone K such t h a t a t any p o i n t p E i n a.

r

one can p l a c e t h e v e r t e x a t p and have K-p l i e e n t i r e l y

E v i d e n t l y a v e r y r e g u l a r r e g i o n s a t i s f i e s t h e cone c o n d i t i o n .

CHE0REIII 3-44.

L e t n C Rn be a bounded open s e t s a t i s f y i n g t h e cone c o n d i -

t i o n and Rs t h e i n t e r s e c t i o n o f R w i t h any t r a n s l a t e o f RS ( s 5 n; m and s integers).

Then @(R) P

C

L q ( R S ) a l g e b r a i c a l l y and t o p o l o g i c a l l y ( i . e .

346

ROBERT CARROLL

continuous i n j e c t i o n ) f o r n > mp, n-mp < s, q 5 sp/(n-mp).

If n i s (e.g.)

holds f o r n-m 5 s.

mp, n-mp < n-1, q 5 (n-l)p/(n-mp)

For p = 1 t h i s

v e r y r e g u l a r , then $(n) C Lq(T) f o r n > ( t h e sense i n which "values" on ns o r

r

a r e determined i s i n d i c a t e d below). Thus i n p a r t i c u l a r one has Wm(n) C Lnp/(n-mp)(n) and W"(n) C L(n-l)P/(n-mp) P1 ( r ) w h i l e f o r n = 3 we have W, (n) C L 6 (n) and W,1(n) C Lp4 ( r ) . We s h a l l c a l l such theorems Sobolev t y p e embedding theorems.

One should a l s o r e c a l l t h e

elementary f a c t s ( e x e r c i s e ) t h a t f o r bounded n, LP(n) C Lq(n) f o r p 2 q, a l g e b r a i c a l l y and t o p o l o g i c a l l y , and f u r t h e r i f f E Lp, g E Lq, t h e n f g E Ls f o r 1/s = l / p

LP norm and p,q,s compact f o r l / p

5 IIfll IIgll

(see [ D u l l ) ; here II II denotes t h e P . P q is 2 1. We r e c a l l a l s o t h a t t h e embedding WS(n) + W:(n) P ( s - j ) / n < l / r ( c f . [Adl;Myl]).

l / q w i t h Ilfgll,

t

-

To d e s c r i b e t h e boundary values (and values on l o w e r dimensional s l i c e s ) i n -

r

d i c a t e d i n Theorem 8.44 we s h a l l d i s c u s s t h e t r a c e you on E Wm(s?). D e f i n e D(fi) t o be t h e r e s t r i c t i o n o f D(Rn) t o

P r e g u l a r D ( 6 ) i s dense i n Wm(n) ( c f . [ l i 2 ] ) . P where n i s t h e u n i t i n t e r i o r normal and u E

When n i s v e r y

D e f i n e y . u = a j u / a n J on J

r,

D(6). Then y . can be t h o u g h t o f J 1, m > 0, and extends by c o n t i n -

as a map i n t o Wm-J-l'p(r) f o r example, p > P u i t y t o a l l Wm(n). I n f a c t one has ( c f . [ L i z ] ) P QXE0REfll 3-45, For m > 0 an i n t e g e r , p > 1, t h e map y : u ul: $(a)

o f a function u

c.

Iyou,ylu,

-f

...,

nWm-J-l/p(r) i s continuous and onto. Thus Ily .uII i n ' h - 1 /PI P J t h e boundary space i n d i c a t e d i s bounded by c1IuII i n Wm S i m i l a r l y given g j P' E Wm-j-l'p(r) t h e r e e x i s t s J E Wm(n) such t h a t y . u = g . and IIuII i n Wi(n) 5 P P J J c~ Ilg.11 (norm g i n w m - j - l / P , sum f o r o 5 j 5 [m-l/p]). J j p For p = 2 one w r i t e s Hm(n) = W!(n) f o r example and we c i t e a n o t h e r embedding -+

theorem o f Sobolev t y p e which i s f r e q u e n t l y u s e f u l

tHE0REB B.46,

I f R i s v e r y r e g u l a r (bounded) t h e n Hm(R) C C

c a l l y and t o p o l o g i c a l l y f o r 2k wn():

c

k (6) a l g e b r a i -

2m-n ( k and n a r e i n t e g e r s ) ; s i m i l a r l y

i f j < Zm-n/p.

c

We w i l l have occasion t o use f u n c t i o n s 9 = 1 i n a NBH o f a g i v e n compact K with

J, E

and K

C:

f o r example.

To c o n s t r u c t such f u n c t i o n s l e t

t h e compact s e t o f a l l p o i n t s a t d i s t a n c e -<

t h i s discussion).

Let

x,

from K

(E

Rn be g i v e n

i s fixed i n

be 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 KE ( i . e . X, = 1

on KE, x E = 0 elsewhere) and c o n s i d e r J, = cp, s t a n d a r d approximations t o 6 and l / m < e a s i l y checked t h a t supp

E

K C

J, C

KEtlIm

E).

* x,

(where cpm i s one o f o u r

E v i d e n t l y J, E Cm, and i t i s

w i t h J, = 1 i n K,-l/m

(simply look a t

347

APPENDIX B I K v,(x-S)dS P

aE0REIII 3.47.

and a t 1 xE(x-S)vm(S)dS). The f o l l o w i n g i s immediate. 1515‘h C Rn w i t h 2 K2€ f o r some E , Given two compact s e t s K C

one can f i n d 9

E

C i w i t h 9 = 1 on a compact NBH o f K and 9 = 0 on Cc.

In

p a r t i c u l a r i f K i s compact and V an open NBH o f K, t h e n t h e r e e x i s t s 9 E C; w i t h 9 = 1 on K and 9 = 0 o u t s i d e o f V . One can use t h i s lemna t o show t h a t g i v e n an open c o v e r i n g o f an open R Rn by open s e t s fii ( R = Wi) t h e r e e x i s t f u n c t i o n s 9i supp

ai

c

qi’ 0 5 $i(x)

2 1 , and on any compact K

(depending on K) o f t h e ibi a r e n o t z e r o w h i l e

-

Hl]

C;(Rn)),

E

C

such t h a t

C R o n l y a f i n i t e number

19i(x)

= 1 i n R (cf.

[Sal;

t h i s i s c a l l e d a p a r t i t i o n o f u n i t y and t h e r e i s a more complete d i s -

c).

c u s s i o n i n Appendix

REUIARK 3-46, L e t us make a few c o n e n t s about v e c t o r v a l u e d d i s t r i b u t i o n s . Thus l e t F and G be v e c t o r spaces ( t h e c o n s t r u c t i o n h e r e i s a l g e b r a i c ) and d e f i n e B(F,G;C)

t o be t h e space o f a l l b i l i n e a r forms F X G

-+

Thus b

C.

E

B(F,G;C) means b(f,+f2,g) = b(fl,g) + b(f2,g), b(f,gl+g2) = b(f,gl) + b(f, S i m i l a r l y , B(F,G;H) denotes g2), b(af,g) = ab(f,g), and b ( f , a g ) = a b ( f , g ) . H i f H i s a v e c t o r space. Then l e t A =/1(F,G) be t h e b i l i n e a r maps F X G -+

s e t o f a l l formal l i n e a r combinations

1 a J. ( f .J, g . J)

w i t h a E C and ( f g ) j j yj F X G. D e f i n e a v e c t o r space s t r u c t u r e on A by a l a . ( f . , g . ) = aaj(fj,gj) J J J and aj(fj:gj) t bk(fk,gk) = (amt$,)(fmygm) where t h e m i n d e x s e t i n -

1

1

1

1

c l u d e s t h e J and k i n d i c e s .

L e t A.

be t h e s u b v e c t o r space c o n s i s t i n g o f

f i n i t e l i n e a r combinations, w i t h c o e f f i c i e n t s i n C, o f elements o f t h e form

- (fld(fa) - a(f,g).

(f1+f2,g) and

DEFlNI&I0N 8-49,

(f2’9)Y

-

(f,g,+g2)

(f,g1)

-

(f42)’

(af,g)

-

.(flS)Y

The t e n s o r p r o d u c t F IG i s d e f i n e d t o be t h e q u o t i e n t

v e c t o r space A(F,G)/Ao(F, F ) .

If b

E

B(F,G;C)

we can extend b t o be a l i n e a r map b: A

-+

C by t h e r u l e

1

we can f a c t o r Since b vanishes t h e n on A b(1 a.(f.,g.)) = a.b(f.,g.). J J J J J J 0: i t t h r o u g h t h e q u o t i e n t J = A/Ao t o determine a l i n e a r map b: F IG C (if -+

=

v(z),

= b(z)).

b(f,g)

where

+ A/A

. o Conversely i f b: F

i s t h e c a n o n i c a l map, v ( f , g )

IBI

G

-+

t o be a b i l i n e a r map F X G

between B(F,G;C) K);

v: A

and L ( F IG;C)

= f

Ig, t h e n b(;)

C i s l i n e a r we d e f i n e b ( f , g ) = 6 ( f I g ) = -+

C.

Thus one has a 1-1 correspondence

(where L(H,K)

h e r e denotes l i n e a r maps H

-+

t h i s i s e a s i l y seen t o be an ( a l g e b r a i c ) i s o r o r p h i s m o f v e c t o r spaces.

We remark t h a t t h e t e n s o r p r o d u c t F IG i s , i n f a c t , c h a r a c t e r i z e d by a u n i versal f a c t o r i z a t i o n property.

There i s a (unique up t o isomorphism) v e c t o r

348

ROBERT CARROLL

space J = F I G and a b i l i n e a r map v : F X G -+ J such t h a t v ( F X G ) generates J ( i . e . J i s t h e s e t o f f i n i t e l i n e a r combinations o f elements v ( f , g ) ) and i f K i s any v e c t o r space w i t h b: F X G a l i n e a r map b ' : J

K a b i l i n e a r map t h e n t h e r e e x i s t s

-+

b' o

K such t h a t b

+

v

(cf. [Cl]).

Next l e t us r e c a l l t h a t i f E i s a TVS t h e r e e x i s t s a unique (up t o isomor-

E* such

phism) complete TVS denoted by n

E i s isomorphic as a TVS t o a

that

There a r e v a r i o u s n a t u r a l t o p o l o g i e s on F IG f o r LCS

dense subspace o f E.

F and G due t o Grothendieck [ G t l ] .

* p r o p e r t y t h a t L(F InG;C)

The n t o p o l o g y i s c h a r a c t e r i z e d by t h e

= B(F,G;C)

where L and

maps; we w i l l n o t have occasion t o use t h e

IT

B r e f e r now t o continuous The

t o p o l o g y as such.

E

topol-

ogy on F IG i s t h e t o p o l o g y o f u n i f o r m convergence on p r o d u c t s o f (convex, d i s c e d ) equicontinuous s e t s i n F ' X G ' ( c f . Appendix A f o r equicontinuous sets). and g ' E

Thus a n e t ea

B when A

-+

0 i n F BEG i f <e,,(f',g')>+

0 uniformly f o r f ' E A

C F ' and B C G I a r e convex, disced, equicontinuous sets; A

the completion i n t h e

E

t o p o l o g y i s t h e n denoted by F BEG.

topology i s stronger than the

T

E

t o p o l o g y and when A

4

space t h e canonical i n j e c t i o n E I n F every LCS F.

I n general t h e

E i s a so c a l l e d n u c l e a r

-+

E I E F i s an isomorphism i n t o , f o r

Most o f t h e d i s t r i b u t i o n spaces a r e n u c l e a r b u t we w i l l n o t

Now f o r S E D; and T E D' one can i d e n t i f y Y S I T w i t h t h e d i s t r i b u t i o n W E D' d e f i n e d by (W,v(x,y)) = ( Sxy( Ty,lp(x,y))) x ,Y i n t h e con(we r e p e a t some f a c t s below = ( Ty,( Sx,v(x,y))) f o r IP E D

e x p l i c i t l y use t h i s p r o p e r t y .

t e x t o f tensor products).

D' i n t o (Dx ID ) * Y

L(Dx

Y

X *Y

Here one n o t e s t h a t t h e r e i s a n a t u r a l map D; I

I Dy;C)

d e f i n e d by e x t e n d i n g t h e map

IT

(

S IT,v

I $)

ID )*. But W can e a s i l y Y Y be shown t o be continuous D C and s i n c e Dx I D i s c l e a r l y dense i n XYJ( Y we have W = Sx B T i n D which g i v e s an a l g e b r a i c map D;( I D' i n t o Dx,Y Y X,Y Y . The Schwartz k e r n e l theorem, which we do n o t need, s t a t e s i n f a c t ,Y t h a t D; ID' = D ' = D;(Di) a l g e b r a i c a l l y and t o p o l o g i c a l l y ( f o r s u i t a b l y Y X,Y d e f i n e d t o p o l o g i e s ) . Using t h e t e n s o r p r o d u c t one can now d e f i n e t h e con= (

S,v X T,$) and thus W a c t s l i k e S,

i n (D,

-+

volution S

*

( S IT,v(x+y))

T, when i t makes sense ( c f . D e f i n i t i o n B.9) by ( S for

v

E

D and s u i t a b l e d i s t r i b u t i o n s S and T.

seen t h a t when S E E' and T E D' t h e n S d e f i n e D ' ( F ) = L(D,F),

*

*

T,v)

=

It i s e a s i l y

T E D' i s w e l l d e f i n e d . Next we

F a LCS a l g e b r a i c a l l y ; no t o p o l o g y i s needed h e r e b u t h

D' neF. L e t Dk = a/axk and A d e f i n e Dk i n D ( F ) as t h e e x t e n s i o n by c o n t i n u i t y o f Dk I I i n D' I,F. This i s continuous n t h e E t o p o l o g y s i n c e if U c D" = D and V C F ' a r e equicontinUOUS, v E U f ' E v, t h e n C c DkT, f,v If ' ) = -( 1 To 5 f,DkV 181 f ' ) i n fact it i s

n s t r u c t i v e t o t h i n k o f D'(F)

APPENDIX B

349

and DkU i s equicontinuous with U. Clearly one has ( lDkT, f,,q B fl) = ( DkTa,d( fa,f' ) = ( 1 (f,,f' ) DkTa,9) and this leads to the determination o f DkT, T E D'(F), by either o f the rules (DkT,f9 = -(T,Dkq) in F or (DkT, f ' ) = D6T.f') in D'.

This Page Intentionally Left Blank

351

APPENDIX C INTRODUCTION TO DIFFERENTIAL GEOMETRY

We begin with some ideas from differential geometry in a classical spirit and refer to [Al;Acl;Azl ;Blel;Bthl ;Cl;Cel;Gy2;Hg2,3;Spil;Stel] for background. We do not attempt to be exhaustive and will often give several definitions for the same object to illustrate different points of view. Let M be a Hausdorff space, U C M open, M = UU,, and pa : a, U Rn a homeomorphism of Ua onto an open set Ua C Rn (n fixed). The pair is called a chart. Suppose for each a,8 the map p B o p i l:ya(Ua n UB) * pB(Ua n U ) is Cm (resp. real analytic). By differentiable one means here 8 that the range coordinates are differentiable functions of the domain coordinates. Further, let us require that the collection of charts (Ua,pa) be a maximal collection satisfying these conditions (this can always be achieved by extension if not initially true). This (maximal) collection of charts is then called a maximal atlas and M equipped with such an atlas is called a Cm (resp. real analytic) n-dimensional manifold. DEFLNL&I0N C.1. -f

),Y.U*(

Note that M in Definition C.l is automatically locally compact (since it is locally homeomorphic to Rn), and frequently it is convenient to assume that M is paracompact or countable at infinity (and hence automatically paracomWe shall, in fact, always assume that M is countable at pact - see [Bo~]). infinity, which means M = u s , Km compact; further by taking unions and so on there is no loss of generality in assuming Km C Km+l. Then (exercise) M is paracompact, which means that any open covering (Wa) of M has an open locally finite refinement. Thus, there is an open covering (V,) such that each V 8 C Wa for some a and each x E M has a NBH V such that V n V = 9 exB cept for a finite number of B. In this situation each compact K will also In particular, we can now assume only intersect a finite number of the V B' there is a locally finite covering (Ui) of M by a countable family of relatively compact open domains of local coordinates (a Ua arising in a chart (Ua,qa) is called a domain of local coordinates andqadescribes the coordin-

352

ROBERT CARROLL

a t e functions).

Indeed, f i r s t cover M by r e l a t i v e l y compact open domains o f

l o c a l c o o r d i n a t e s Vg;

t h i s can be done because i f x

E

U, and U, w i t h

c Ua,

is a r e l a t i v e l y compact open NBH o f x (such U e x i s t s i n c e M i s l o c a l l y homeomorphic t o Rn a t x v i a a chart.

$

v,), then

'P,~U

= 9;

i s a homeomorphism and (U.9;)

Then p i c k a l o c a l l y f i n i t e r e f i n e m e n t (U,)

i n t e r s e c t s o n l y a f i n i t e number o f t h e ;U,

t h e U,

o f (V ) such t h a t each B a r e then c o u n t a b l e and

We w i l l say t h a t a f u n c t i o n f on M belongs t o C",

we can index them as Ui. L~o,,

e t c . i f f o r e v e r y c h a r t (U,9,)

etc.,

where U,

every

v

the functions f o

9 ':

I\r

E

C"(ca), L p ( c ) ,

( r e c a l l t h a t g E Lyoc(n), n open, ifv g

q,(U,)

E

LP(n) f o r

Unless o t h e r w i s e s t a t e d f u n c t i o n s on M w i l l be r e a l ; how-

C:(n)).

E

is

e v e r i t i s obvious t h a t most arguments apply as w e l l t o complex f u n c t i o n s . We w i l l o n l y deal e x p l i c i t l y w i t h t h e Cm case and o t h e r s i t u a t i o n s can be handled by changing s u i t a b l e words, e.g. Cm

+

real analytic, etc.

By a v a r -

i e t y i n Rn one means t h e s e t o f common zeros i n Rn o f some c o l l e c t i o n o f f u n c t i o n s fl

,. . .,fa.

L e t M and N be C" m a n i f o l d s w i t h a t l a s e s { ( U , , q , ) } and DEFZNZCZaN C.2, L e t f: M + N be a map such t h a t whenever f ( U a ) c W t h e map J, o t ( W J, ) I . B B f3 'El f o 9, : q,(U,) + J, ( W ) ( i n t o ) i s a Cm map. Then f i s c a l l e d a C" morphism B B , 1 i n t h e c a t e g o r y rn o f C manifolds, and i f f i s o n t o N w i t h Cm i n v e r s e f t h e n f i s c a l l e d a diffeomorphism o r an isomorphism i n For c a t e g o r i e s see 53.4 and D e f i n i t i o n s C.17-C.19. f: M

-f

N i n a category

rn

m.

We say t h a t a morphism

i s an isomorphism i f t h e r e i s a morphism f ' : N

such t h a t f o f ' and f ' o f a r e t h e i d e n t i t i e s .

+

M

There a r e now v a r i o u s e q u i -

v a l e n t ways t o d e f i n e t h e tangent and cotangent spaces T (M) and T*(M) a t a P P p o i n t p E M. One u s u a l l y d e f i n e s one o r t h e o t h e r f i r s t (by v a r i o u s methods) and then t h e o t h e r i s d e f i n e d by d u a l i t y .

We s h a l l g i v e several v e r s i o n s o f

t h i s i n t h e f o l l o w i n g and show t h e i r equivalence.

F i r s t i n p a s s i n g we r e -

c o r d a lemma which i s u s e f u l i n c o n s t r u c t i n g v a r i o u s o b j e c t s on a Cm manif o l d ( c f . a l s o Theorem B.47). Let

CEllImA C.3.

a function

J,

KC

E C"(M)

M be compact and V an open NBH o f K.

such t h a t

J,

= 1 on K and

To prove t h i s f i r s t observe t h a t i f Vi E M with

vi

J,

= 0 o u t s i d e o f V.

i s a r e l a t i v e l y compact open NBH o f x

c o n t a i n e d i n a domain o f l o c a l c o o r d i n a t e s Ui,

another open NBH W . o f x w i t h 1

i n g o f K by such NBHs Wi

-

compacts Ki = Wi

ii C Vi

( i = l,..,,N)

w i t h Ki c Vi

C

vi

Then t h e r e e x i s t s

n e c e s s a r i l y compact.

t h e n one can f i n d

A f i n i t e cover-

produces a f i n i t e c o v e r i n g o f K by

c Ui.

Then t r a n s p o r t each Ki t o Rn v i a

353

APPENDIX C

-Wi

vi and a p p l y Theorem 6.47 t o o b t a i n f u n c t i o n s p o r t ) such t h a t J,i =

1

-

= 1 on Ki

(l-$,)(l-J,2)--.(l-J,N)

= E

and Jli

C”(M)

E Cm(M)

( w i t h compact sup= 0 o u t s i d e o f Vi. The f u n c t i o n J, J,i

K

i s one on

The r e m a i n i n g d e t a i l s a r e easy ( t h e f u n c t i o n b e f o r e r e a c h i n g t h e boundary o f V so t h e Vi

J,

-

and z e r o o u t s i d e o f UVi. s h o u l d be damped down t o z e r o

s h o u l d be chosed w i t h

an open c o v e r i n g Ui o f M as above one can f i n d f u n c t i o n s )Li

Ti

C Ui n

T h i s lemma can a l s o be deduced as a s p e c i a l case o f t h e f a c t t h a t g i v e n

V).

Jli 2 0 , and

C Ui,

1 $i

= 1 (partition o f unity

-

J,i

E

Cm w i t h supp

c f . [Spil;Stel]).

I t f o l l o w s now i n p a r t i c u l a r t h a t i f V i s an open subset o f M w i t h p

i f f E C“(V) that

7=

E

then t h e r e i s an open NBH N o f p and a f u n c t i o n ?E C”(M)

V and such

f i n N ( t a k e N t o be a r e l a t i v e l y compact open NBH o f p w i t h

V, W open, p i c k

C W

T=

C

$f).

T h i s f a c t a l l o w s us t o phrase c e r t a i n arguments based on f u n c t i o n s Cm

J,

as i n Lemna C.3 r e l a t i v e t o

and W, and s e t

and

i n some ( v a r y i n g ) NBH o f p i n terms o f Cm(M).

1

Now ( c f . [ C e l l ) l e t C ( p ) be

t h e s e t o f equivalence c l a s s e s o f c o n t i n u o u s l y d i f f e r e n t i a b l e f i l n c t i o n s i n some open NBH o f p; t h e NBH Df depends on t h e f u n c t i o n f, and f i s i d e n t i f i e d w i t h g i f t h e y a r e equal i n some NBH o f p. We w r i t e f f o r t h e f u n c t i o n and 1 1 i t s c l a s s and r e c a l l t h a t C ( p ) i s c a l l e d t h e s e t o f germs o f C f u n c t i o n s a t p ( t h u s c l a s s f = germ f ) . By t h e above remarks, i f f E C 1 ( p ) , t h e r e i s 1 a function C (M) w i t h ? = f i n an open NBH o f p, so we c o u l d e q u a l l y w e l l

7E

base t h e d i s c u s s i o n on g l o b a l f u n c t i o n s . We now d e f i n e an e q u i v a l e n c e r e l a 1 t i o n R i n C ( p ) ( c o n s i d e r e d as a v e c t o r space o f e r R w i t h mf+b’g d e f i n e d and 1 C on Df n D ) by s a y i n g f : g i f a l l f i r s t p a r t i a l d e r i v a t i v e s o f f - g a r e 9 0 a t p. To s p e l l o u t t h e d e t a i l s , one means t h a t i f (Ua,qa) i s any c h a r t a t Evidently C Rn, then a / a x i ( ( f - g ) o pa -1 ) = 0 a t q,(p). p w i t h qa(Ua) = t h e d e r i v a t i v e s depend on t h e q a , b u t t h e i r v a n i s h i n g i s independent o f coo r d i n a t e systems because o f t h e c h a i n r u l e ( c o n s i d e r e.g.

o

(q6

o

q;’)]

etc.).

a/axi[(

(f-g) o

qil)

I t i s e a s i l y seen t h a t t h i s r e l a t i o n i s w e l l d e f i n e d

on germs and i s an equivalence c o m p a t i b l e w i t h t h e v e c t o r space s t r u c t u r e o f f i 4 1 1 C ( p ) ( i . e . f z g, f : g i m p l i e s a f + 6 = ag+Bi). Thus C ( p ) / R i s a v e c t o r space which we c a l l T*(M) ( c o t a n g e n t space) and we denote t h e c a n o n i c a l ( a l l P 1 f = d f . T h i s p o i n t o f view g e b r a i c ) homomorphism C ( p ) -t C (p)/R by f -f

w i l l be expanded l a t e r when we c o n s i d e r j e t s .

Next we w i l l show t h a t T;(M)

i s an n-dimensional v e c t o r space w i t h b a s i s dx l,...,dxn t e m q a a t p as above where xi = pripa

i n a c o o r d i n a t e sys-

( p r denoting p r o j e c t i o n ) .

A basis i s

d e f i n e d as usual t o be a s e t o f l i n e a r l y independent elements bi spanning Ti(M) i n t h e sense t h a t any v e c t o r b E T*(M) has a unique expansion P

1 a.b J j’

354

ROBERT CARROLL

1 Aixi :0 i n C ( p ) which means t h a t a t Suppose t h a t Aidxi = 0. Then f ~p,(p), x j = a / a x . ( f o 9, -1 ) - 0 and consequently t h e dxi a r e l i n e a r l y i n d e J pendent (iff = x = p r . q then f o p i 1 i s t h e f u n c t i o n x + x j on q,(U,)). j ~a~ -1 ( i . e . X i = a/axi(f o On t h e o t h e r hand l e t f E C ( p ) and xi = ( a f / a x . ) )

1

1

-

evaluated a t q,(p)

f E

xixi

1 hidxi,

and d f =

1 (af/ax.)

dxi. ! P o f coordinates

1 P

t h i s a b b r e v i a t e d n o t a t i o n w i l l be used f r e q u e n t l y ) ; t h e n which means t h a t t h e dxi span T*(M) and d f = P i s c o m p l e t e l y determined r e l a t i v e t o any c h o i c e

Thus T;(M) pa

a t p, where t h e 9,

simply p r o v i d e a b a s i s i n which t o de-

s c r i b e T;(M).

(M) = T;(M)* ( * denotes a l g e b r d i c dual over R and we w i l l P L e t a c o o r d i n a t e system q a a t p be selected, w i t h dxi have T*(M) = Tp(M)*). P t h e a s s o c i a t e d b a s i s o f T*(M) and l e t v . ( a ) be a standard dual b a s i s f o r P J Tp(M). Thus t h e v . ( a ) a r e determined by ( v . ( a ) , d x i ) = 6 . . (Kronecker d e l t a ) J J 1J and any v E T (M) can be w r i t t e n v = pivi(a) r e l a t i v e t o t h i s b a s i s . Now 1 p i f f E C ( p ) we can d e f i n e an a c t i o n o f v E T (M) on f by t h e composition f P df +
We now d e f i n e T

1

(l

-f

1

(1

...,

j = 1, n, as b e i n g a b a s i s f o r j’ Change c o o r d i n a t e s and we w r i t e v = u.(a/axj). P J o f b a s i s formulas i n T and T* w i l l be g i v e n below. Thus any v E T d e t e r P P P 1 mines a map ( c a l l e d a d e r i v a t i o n ) , a g a i n denoted by v, C ( p ) R, s a t i s f y i n g

we can now t h i n k o f t h e symbols a/ax

T (M) r e l a t i v e t o t h e

1

(pa

-f

v(fg) = f(p)v(g)

+

v ( f ) g ( p ) ( e x e r c i s e ) and t h i s f o r m u l a i s t h e b a s i s f o r an

alternative definition o f T

P

(M).

Summarizing

1 1 L e t C ( p ) be t h e v e c t o r space o f germs o f C f u n c t i o n s a t p DEFZNICI8N C.4. E M and R t h e equivalence r e l a t i o n d e f i n e d by f z g i f a l l t h e f i r s t p a r t i a l 1 d e r i v a t i v e s o f f - g v a n i s h a t p. Then C (p)/R = T*(M) i s an n-dimensional P v e c t o r space w i t h b a s i s dxi ( 1 2 i 5 n ) r e l a t i v e t o a c o o r d i n a t e system

(M) i s d e f i n e d as T*(M)* and can be i d e n t i f i e d w i t h t h e n-dimensionP P a l v e c t o r space having a ( d u a l ) b a s i s composed o f symbols a/axi ( r e l a t i v e t o a t p.

T

a c o o r d i n a t e system qa). 1 ( r e s p . q ) r e f e r t o x ( r e s p . y ) c o o r d i n a t e s . For any f E C ( p ) B we have d f = ( a f / a x . ) dxi = ( a f p y . ) dy.. By t h e c h a i n r u l e ( A ) a/axi 1 P-1 3 P J ( f o q i l ) = a / a x i [ ( f o q B o (9 o C” )I = a / a y . ( f o qgl)ay./axi (note R a J J t h a t q6qh1 maps x y ) . Thus ( * ) (af/axi)p = (af/ayjip(ayj/axi!p. The L e t now q,

1

1

1

-f

I:

i s called m a t r i x ( j = row index, i = column i n d e x ) ( 6 ) J ( x ) = (( ayj/axi)) Y t h e Jacobian o f t h e map $ a B = q R q i ’ and s i n c e $ a B i s a diffeomorphism d e t J

APPENDIX C

355

#

0 on va(Ua r~ Up). To see t h i s l o o k a t t h e map yi = y i ( x ( y ) ) t o o b t a i n 6 j k = 1 (ayi/axj)(axj/ayk) o r J ( x ) J x ( y ) = I and a s i m i l a r f o r m u l a h o l d s upY on i n t e r c h a n g i n g x and y. Thus J x ( y ) = J ( x ) - ’ and d e t J ( x ) # 0. Hence Y Y (which a l s o f o l l o w s d i r e c t l y as i n ( a ) . ( 9 ) ( a f / a y ) = 1 (axi/ay ) ( a f l a x . ) k P k P 1 P By d e f i n i t i o n o f d f we have f u r t h e r ( m ) dy = (ayj/axk)pdxk. T h e r e f o r e we j = can w r i t e ( e x e r c i s e ) (*) a/ayk = 1 ( a x . / a y ) a / a x . and i f 1 pi(a/axi) J k P J 1 h j ( a / a y j ) t h e n (*A) h j = 1 ( a ~ ~ / a x ~ ) The ~ p ~l a .s t two formulas f o l l o w by

1

d u a l i t y and a r e s t a n d a r d r e s u l t s o f l i n e a r a l g e b r a . L e t TIM) be t h e d i s j o i n t u n i o n UT (M) and T*(M) = uT*(M). P P These s e t s a r e c a l l e d t h e t a n g e n t and c o t a n g e n t bundles o f M r e s p e c t i v e l y

D E F I N I C I B N C.5.

( t h e y w i l l be seen below t o be Cm m a n i f o l d s w i t h a n a t u r a l bundle s t r u c t u r e ) . The change o f b a s i s f o r m u l a s a t a p o i n t p E M a r e g i v e n by c o e f f i c i e n t changes i n d i c a t e d by

(+) and

(m)

and (**) w i t h

(*A).

L e t now V be a f i n i t e dimensional v e c t o r space o f dimension n o v e r C o r R. To d e f i n e t h e e x t e r i o r p r o d u c t V A V one can use s e v e r a l e q u i v a l e n t t e c h n i ques.

For example ( c f . [ B o ~ ] ) V A V = V

I

V/J where J i s t h e v e c t o r space

i n V IV c o n s i s t i n g o f f i n i t e l i n e a r combinations w i t h c o e f f i c i e n t s i n C o r

R

o f elements o f t h e f o r m x

p r o d u c t o f v e c t o r spaces). p o i n t o f view.

P(m)

Thus l e t

I

y + y Ix (see D e f i n i t i o n B.49 f o r t h e t e n s o r

We s h a l l however adopt a somewhat d i f f e r e n t

P(m)

be t h e p e r m u t a t i o n group on m symbols and f o r

... Ixm)

...

Then extend t h i s I I xu(,,,). o(1) o p e r a t i o n i n t h e obvious way by l i n e a r i t y t o I m V = V I IV (m t i m e s ) . o E

define u(xl I

= x

...

An element

T E I m V w i l l be c a l l e d symmetric

a l t e r n a t i n g i f oT = (sgno)T f o r a l l u

1 (sgn

u ) u (sum

over u

E

E

P(m)) as a map

i f oT = T f o r a l l u E P(M) and

D e f i n e ( * a ) Am = ( l / m ! )

P(m).

E V

+

ImV.

Then observe t h a t AmT

i s always a l t e r n a t i n g and AmT = T i f T i s a l t e r n a t i n g ( e x e r c i s e ) . We d e f i n e AmV t o be t h e v e c t o r space o f a l t e r n a t i n g elem-

D E F I N I & I 0 N C.6. e n t s i n ~VI.

REMARK C.7,

One can a l s o d e f i n e AmV as ImV/kerAm b u t t h e n o t a t i o n w i l l be

s i m p l e r i n v a r i o u s formulas i f we use D e f i n i t i o n C.6 and t h i n k o f AmV ImV.

F u r t h e r l e t us say t h a t a l i n e a r map J,: I m V

alternating i f

J,

o u = (sgno)JI f o r a l l u

E

P(m).

+

C

W, W a v e c t o r space, i s

Then AmV as a q u o t i e n t i s

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 t h a t i f $ i s any a l t e r n a t i n g l i n e a r map I m V W, t h e n t h e r e i s a u n i q u e l y determined l i n e a r map A

q : AmV

6

n o Am where A:,,

ImV

-+

* W such t h a t J, = In t h i s

I m V / k e r Am i s t h e c a n o n i c a l map ( e x e r c i s e ) . 4

f o r m u l a t i o n one w r i t e s c u s t o m a r i l y Am(vl

I

... I vm)

=

-+

v1 A

... A

vm f o r

356

ROBERT CARROLL

a r b i t r a r y vi

E

However, we a r e g o i n g t o d e f i n e now ( u s i n g D e f i n i t i o n C.6)

V.

Iv ) f o r u E ArV and v E A S V which w i l l i n -

( * 6 ) u A v = ((r+s)!/r!s!)Arts(u

t r o d u c e a convenient numerical f a c t o r i n t o t h e formulas.

... A

Thus i n p a r t i c u l a r

...

I vm). I n any event, i f La(ImV,W) i s t h e vecvm = m!A ( v I m 1 t o r space o f a l t e r n a t i n g l i n e a r maps I m V -+ W, i t f o l l o w s immediately t h e n v1 A

t h a t La(ImV,W) i s isomorphic as a v e c t o r space t o L(A"V,W) pondence

JI

rl

-+

under t h e c o r r e s -

i n d i c a t e d above.

k Iv i s a b a s i s f o r V t h e n t h e n elements v I i, ik k A c o n s t i t u t e a b a s i s f o r IV ( e x e r c i s e ) . Hence elements o f t h e form v il k Indeed n o t e t h a t f o r T A v w i t h il < < ikspan t h e v e c t o r space A V. Now i f vi

( i = 1,.

...

.,n)

...

..

bk

I V,

6

Ak(uT) = ( 1 k ! )

which i m p l i e s e.g.

1 (Sgnr)ToT

two o f i t s terms a r e interchanged.

(L)

CHE0RElI C.6.

1(Sgnro)(Sgna)roT = (sgno)Ak(T) ... A v i k i s m u l t i p l i e d by -1 i f

= (l/k!)

t h a t an element v

The elements v il A k b a s i s f o r A V whenever t h e vi ( i = 1

i,

A

... A

vik,

,. . . ,n)

... <

il <

P R O O ~ : Only l i n e a r independence remains t o be shown. A...Av

v

im

il

,..., i kwhere )

= 0 ( ( i ) = (il

ik, constitute a

a r e a b a s i s f o r V.

i, <

... <

1a( i 1

Thus suppose

ik.P i c k some p a r t i c u -

be t h e r e m a i n i n g i n t e g e r s i n t h e s e t l a r ( i ) i n t h i s sum and l e t iktl,...,i n ,n) w i t h o u t r e p i t i t i o n . M u l t i p l y by v . A . . . A v , and then a l l terms (1, Ik+r in A v . w i l l i n v o l v e a repeated element and except vi A v. A v. A

...

...

I

1k+r

1k

...

lh

hence must vanish (by remarks above an i n t e r c h a n g e o f t h e repeated element would r e v e r s e s i g n ) .

m...

s i n c e An(vi I

I

I

Hence a(i)vil

... A v i

= 0.

But v

..

A . Avin i, v . ) i n v o l v e s t h e l i n e a r l y independent terms o(vi A

1n = 0 f o r any ( i ) .

# I

I

0

...

QED (i) k The p r o o f a l s o shows t h a t A V = 0 f o r k > n s i n c e repeated elements would k occur i n t h e b a s i s v e c t o r s . Now we can c o n s i d e r A T*(M) w i t h AoT* = R and P P 1 P i c k i n g a b a s i s o f T* corresponds t o s e l e c t i n g l o c a l c o o r d i n a t e s A T* = T*. P P P (pa a t p and u s i n g t h e dxi (i= 1,. ,n) as a b a s i s . Then (modulo r e d u c t i o n vim).

Hence a

t o i n d i c e s j, < dyiK

... <

..

j,) t h e change o f b a s i s formula f o r a t e r m dy

follows d i r e c t l y from

(m).

.. A

As f o r c o e f f i c i e n t changes we r e c a l l f i r s t

(see e.g.

[ B o ~ ] ) t h a t t h e standard t e n s o r law f o r an e q u a l i t y

...I dy ik

=

1 b ( j )dx j l

A . il

1 a(i)dyi

I

lay. )b ' j m J~ 1k ( j ) w i t h summation on repeated i n d i c e s ( e x e r c i s e ) . Then e v i d e n t l y under t h i s k k r u l e t h e ensuing i d e n t i t y i n I T* would map i n t o an i d e n t i t y i n A T* ( i t i s P P e a s i l y seen t h a t t h e necessary antisymmetry p r o p e r t i e s o f t h e a ( i ) and b ( j )

I...@ dx

i s a ( i ) = (axjl/ayi,

r e q u i r e d o f a l t e r n a t i n g forms would be preserved).

)...(ax,

F i n a l l y one c o u l d pass

357

APPENDIX C k t o a b a s i s expansion i n A T* P' a(i)dyi, A (cf. [Bo~]). If <

... <

ik s

j'1 <

(sum o v e r j

<

1 ... < . .. <

T h i s can be sumnarized i n t h e f o l l o w i n g way

... A

dyik

=

1 b(j)dxj,

jk, t h e n ( e x e r c i s e ) (*+) a(i) j k ) ;(( axj/ayi))

1 index, p = column index); dyi,

= (( axj,/ayi,))

... A

1 det

=

A

... A dxj k

1det

w i t h il

((axj/ayi))b(j)

(1 5 p,m 5 k; m = row

...A d x j,' J D e f i n e A T*(M) as t h e d i s j o i n t u n i o n UA T*(M) (a b u n d l e DEFINI&I0N C.9. k P s t r u c t u r e w i l l be i n t r o d u c e d l a t e r ) . L e t rk: A T*(M) + M be t h e p r o j e c t i o n k map o f AkT$M) + p. A s e c t i o n o f A T*(M) i s d e f i n e d t o be a map s : M + k A T*(M) such t h a t r k o s = i d e n t i t y and w i l l be c a l l e d a d i f f e r e n t i a l form A

dyik

=

(( ayi/ax.)) dxj,A

k

k

o f degree k (no c o n t i n u i t y o r d i f f e r e n t i a b i l i t y o f s i s r e q u i r e d a t t h e momThus a d i f f e r e n t i a l form o f degree k i s a f u n c t i o n s on M such t h a t k s ( p ) E A T*(M) f o r each p and t h e change o f b a s i s and c o e f f i c i e n t formulas P a r e g i v e n by (*+). ent).

Next i f gi E T*, xi E T we d e f i n e , w i t h a s l i g h t abuse o f n o t a t i o n (see P P' D e f i n i t i o n 8.49) ( * m ) (gl I Igk)(xl, xk) = (Ig.,Ixi) = ( (gl I

...,

...

I... Ix k ) ) = n(gi,xi).

I3 gk),(xl

...

1

S i m i l a r l y g ( x ly...yxk) i s defined k any g E I T* by e x t e n d i n g ( * m ) l i n e a r l y . Thus IT* can be regarded as P P k space o f k - l i n e a r maps ( o v e r R ) o f n T -+ R (where nk means T X X P P k times, and a map i s c a l l e d m u l t i l i n e a r i f i t i s s e p a r a t e l y l i n e a r i n

...

argument).

for the Tpy each

Indeed, by e x t e n d i n g o u r p r e v i o u s c h a r a c t e r i z a t i o n o f t e n s o r

p r o d u c t s i n an obvious way ( c f . D e f i n i t i o n 8.49) t h i s corresponds t o s a y i n g k k t h a t I T* i s t h e a l g e b r a i c dual o f IT and here ( * m ) e s t a b l i s h e s t h i s duP k P a l i t y ( t o see t h a t we g e t a l l o f (I T )* s i m p l y check t h e dimension). P k S i m i l a r l y A T* can be c h a r a c t e r i z e d as t h e v e c t o r space o f a l t e r n a t i n g k 'k l i n e a r maps n T -t R. Here an a l t e r n a t i n g m u l t i l i n e a r map i s d e f i n e d t o P be a map ( x ,,..., x k ) + R which vanishes whenever two o f t h e xi c o i n c i d e ; k such a map determines an a l t e r n a t i n g l i n e a r map IT -t R and c o n v e r s e l y (exk P k e r c i s e ) . Hence A T* w i l l appear as t h e a l g e b r a i c dual o f A T ( c f . Remark P P C . 7 ) and t h e d u a l i t y can be e x h i b i t e d by u s i n g t h e ( ) p a i r i n g (*.) bek k k i( tween I T and I T* ( r e c a l l t h a t we c o n s i d e r A T* C IT* and t h e a c t i o n o f k P P P P In w E A T* o f (xl ,..., x k ) i s t h u s e x a c t l y i t s a c t i o n as a t e n s o r on I x i ) . P k p a r t i c u l a r i f w E IT* and xi E T P P (C.l)

... Ix k )

Akw(x l , . . . ~) = ( A k w , x1 I k

(l/k!)~sgnT(w,T-l(~xi)) = (l/k!)lsgndw,x *1

( l / k ! ) z s g n c r w(x 01

,..., x

) OK

= (l/k!)lsgnr(rw,Ixi)

I... Ix

k'

)

=

=

358

ROBERT CARROLL

I r T * and v E I S T * , Arts(u Iv ) P P x )sgno where t h e x ) = ( r ! s ! / ( r + s ) ! ) 1 u(xi, x )v(x. (xl r+s iP J' js r t s i n t o i n c r e a s i n g sequences il, i and sum i s o v e r p a r t i t i o n s o f 1, r ,xrts) = 1 sgnou(x jl ,js. Consequently from (*6) u A v (xl,. il " * " ) which d e f i n e s t h e wedge p r o d u c t u A v EArtsT; C I Xi,)V(Xj1 9 * * * "j s rtsT; by i t s a c t i o n on ( x l,...,x r t s ). We n o t e i n passing t h a t f o r gi E T i , xi E T , t h e r u l e g1 A A gk(x1 x k ) = k!Ak(gl B Igk) ( c f . Remark C.7) P A gk(xl x k ) = 1 sgno ( g B Igo(k),xl I y i e l d s (A*) g1 A a(1) 8 Xk) = 1 sgno ( go(l),xl )...( g u ( k ) , x k ) = d e t ( g i , x . ) which r e v e a l s t h e deJ terminant i n i t s natural habitat. Then one can show ( e x e r c i s e ) t h a t f o r u

,...,

,...,

...,

,.. .

... ...

E

,...,

...,

..

,..., ,...,

...

...

...

Now t o d e f i n e t h e idea o f t a n g e n t v e c t o r t o a c u r v e we f i r s t c o n s i d e r two c" N ( f can be C 1 here, i . e . t h e c o o r d i n m a n i f o l d s M and N and a Cm map f : M -+

ates o f f ( x ) a r e C chart

(Ua,qa)

1

f u n c t i o n s o f t h e i r arguments xi).

and f ( p )

E

N belong t o a c h a r t ( W 6 , J , @ )

Let p E M belong t o a w i t h f(Ua)

We w i l l o c c a s i o n a l l y abuse n o t a t i o n by w r i t i n g p

(Ua,va) i n s t e a d o f p

E

i n order t o simultaneously e x h i b i t t h e l o c a l coordinates

Let v E T

pa.

an a r b i t r a r y tangent v e c t o r a t p; t h e n we d e f i n e i t s image f,v t h e r u l e f,v(g) f,v

W6 (such a

C

g i v e n i n advance).

s i t u a t i o n can o b v i o u s l y always be r e a l i z e d f o r any (WB,J,B)

T

E

E

Ua

be

P

f ( P ) by = v ( g o f ) , where v ( g o f ) i s d e f i n e d by (*) and c l e a r l y

i s w e l l d e f i n e d by d u a l i t y ( n o t e t h a t dg = 0 a t f ( p ) i m p l i e s d ( g o f )

= 0 a t p).

I n p a r t i c u l a r i f v = a/axi

we want t o express f,(a/axi)

i n terms

o f t h e b a s i s elements spy. a t f ( p ) . To do t h i s consider ( ? = J, fp;') (AA) -1 a/ax+i 0 0 ( $ p a )Ip,(p) = 1a/ayk(g )$b(f(p))(afk/axi)px(p) as f,(a/ax.) ( c f . (A)). Hence u s i n g a p r e v i o u s a b b r e v i a t i o n we r e w r i t e (u) 1 = 1 (aFk/axi)(a/ayk) and t h e m a t r i x (( aik/axi)), k = row index, i = column

J,;?

$il

A

B

index, i s c a l l e d a Jacobian ( c f . (6) where f i s t h e i d e n t i t y and M The map f,

=

N).

i s a l s o scjmetimes denoted by d f .

I n p a r t i c u l a r c o n s i d e r a map q: J ( i . e . q i s a curve).

-+

M where J

R i s say an open i n t e r v a l

C

f o r to E J f i x e d can be de-

Thus a g e n e r a t o r o f Tt,(J)

noted by d / d t and we c o n s i d e r q,(d/dt)

= q'(to) E T

Thus d / d t ( h ( q ( t ) ) ) q(t,)' But i f = q ' ( t o ) ( h ) a t to and q ' ( t o ) i s c a l l e d t h e t a n g e n t t o q ( - ) a t q ( t o ) . and d q k / d t = vk (where ;I = qk a r e l o c a l c o o r d i n a t e s a t q ( t o ) g i v e n by (W,$) J, 0

q ) then ( d / d t ) h ( q ( t ) )

=

1 (ah/aqk)(dik/dt)

=

1 vk(ah/aqk)

which shows

t h a t q ' ( t o ) = 1 v k ( a / a q k ) . Conversely t h i s f o r m u l a c o u l d be used t o d e f i n e the tangent vector t o a curve q ( t ) .

DEFZNZCZ0N C.10.

If f: M

Tf(p)(N) by t h e r u l e f,v(g)

-f

N is a C

1 map then i t d e f i n e s a map f,:

= v ( g o f ) where v E T (M), f,v

P

E

Tp(M)

Tf(p)(N),

-+

and

359

APPENDIX C

1

g E C (f(p)) i s arbitrary.

The e f f e c t on b a s i s v e c t o r s i s shown i n (")

i f M = J i s an open i n t e r v a l , w i t h t t a n g e n t v e c t o r f ' ( t o )= d/dt E T

1 vk(a/aYk)

+

and

i s the

f ( t ) a c u r v e i n N, f,(d/dt)

t o f ( t ) a t to ( h e r e vk = d y k / d t a t to,

(J), and t h e y k ( t ) a r e t h e c o o r d i n a t e s o f f ( t ) ) .

t o

One s h o u l d n o t e t h a t f, does n o t g e n e r a l l y p r o v i d e a map o f v e c t o r f i e l d s ( = s e c t i o n s o f t h e t a n g e n t b u n d l e s ) s i n c e f ( p ) may equal f ( p ' ) f o r p # p ' . Now t h e wedge p r o d u c t d e f i n e d above has t h e f o l l o w i n g a d d i t i o n a l p r o p e r t i e s ( u t v ) A w = u .A w t v A w, u A ( v t w ) = u A v + u A W, u A w = v A u, au A v u A av = a ( u A v ) , and ( u A v ) A w = u A ( v A w). (4.)

If u

DEFINICI0N C.11,

... c ... A

jk, t h e n du

E

E

1 b ( j ) d x j , A ... A 1 1 (ab ( J. 1/axi)dxi

A k T* i s g i v e n by u =

Akt'T*

P i s d e f i n e d by du =

P d x . (sums o v e r 1 5 i 5 n and j, < Jk

...

dxjK,

j, <

A j, < j k ) . A f o r m i s c l o s e d i f du

A dx

= 0 and e x a c t i f u = dv f o r some form v.

d ( u A v ) = du A v +

The f o l l o w i n g f a c t s a r e e a s i l y proved ( e x e r c i s e ) :

A l s o i t i s c l e a r t h a t d(u+v) = du t dv.

( - l ) r s u A dv; d2 = 0.

t h a t i f f: M

-t

N is C

1

as b e f o r e and p E (Uc,,lp,),

W , then one can d e f i n e a map f*: T;(pl(N)

-t

f(p)

T;(M)

E

(W8,d'B),

Next observe f(Uc,)

C

by t h e r u l e f * ( d g ) =

8 v E d ( g o f ) . T h i s i s e v i d e n t l y w e l l d e f i n e d . Moreover f o r dg E T;(p)(N), Tp(M), ( f * ( d g ) , v ) = ( d ( g o f ) , v ) = v ( g o f ) = f,v(g) = (dg,f,v) which exk h i b i t s f * and f, as a d j o i n t maps. We can t h e n extend f* t o forms u E A T*

a t f ( p ) by t h e f o r m u l a (*4) f * u ( v

,,...,v k )

= u(f,vl,..

.,f,vk)

f o r vi E TP (see (C.1)). R e c a l l here t h a t f*u i s c o m p l e t e l y determined by i t s a c t i o n as k a k - l i n e a r map on A T P' 1 CHE0REIR C.12, I f f : M + N i s a C map t h e n i t determines a map f*: T*f( p ) (N) ( N ) . F u r t h e r f* i s ad+ Tfp(M) by t h e r u l e f * ( d g ) = d(g o f ) f o r dg E T* k f(p) k (N) A T;(M) (k 2 0 ) d e t e r j o i n t t o f, and i t extends t o a map f*: A T* f(P) mined by t h e f o r m u l a (A&) f o r k 1. 1 and, when u i s a f u n c t i o n , by t h e r u l e f * u ( p ) = u ( f ( p ) ) . F u r t h e r d f * = f * d and f * ( u A v ) = f * u A f*v. -f

L e t us n o t e i n p a s s i n g a few p a r t i c u l a r formulas which a r e u s e f u l i n v a r i o u s §1.7). L e t dxi generate T* and vi 'L a/axi P F o l l o w i n g t h e general f o r m u l a s above one has i n p a r t i c u l a r

places (see e.g.

(C.2)

(dxi A dx.)(vl,v2) J

=

1

(

dxi,vl)

(dxi,v2)

(

generate T

P'

dxj,v,)

(dx.,~)

J 21 A n o t a t i o n used i n §1.7 f o l l o w i n g [ A l l i n v o l v e s wA(S) = ( A , S ) = 1 AiSi so w1 = 1 Aidxi and f u r t h e r w i = Aldx2 A dx3 + A2dx3 A dxl t A3dxl A dx2 w i t h = d i v A dxl A dx2 A dx3. Consequently i f A = g r a d f, w: = 1 (af/axi)dxi

2

360

Q

ROBERT CARROLL

1 df, and i n general dwA = d ( 1 Aidxi)

1 1 (aAi/axj)dxj

=

m

=

2 wcurlA.

m

o f Cm m a n i f o l d s ( D e f i n i t i o n C.2) means diffeomorphism. There a r e s e v e r a l

We c o n s i d e r now m a n i f o l d s i n t h e c a t e g o r y and r e c a l l t h a t isomorphism i n

A dxi

approaches t o v e c t o r bundles b u t we g i v e o n l y one s e t o f d e f i n i t i o n s here. P r i n c i p l e bundles a r e discussed i n §3.10. L e t E,M E

DEFINIC10N C.13.

m

and n : E

-+

Assume t h e

M be a morphism o n t o M.

f i b e r n - l ( x ) (complete i n v e r s e image) has t h e s t r u c t u r e o f an n-dimensional r e a l o r complex v e c t o r space f o r each x E M ( n <

i s t o be f i x e d f o r a l l x ) . -1 Assume t h e r e i s an open c o v e r i n g { V k ) o f M and maps r k : n (V,) Vk X F where F i s a f i x e d copy o f Rn o r Cn c a l l e d t h e t y p i c a l f i b e r such t h a t (A) m

-+

T~

i s an isomorphism i n

m

(B)

T

~

n~- ' ( x: )

-+

F i s an isomorphism o f v e c t o r

spaces and ( C ) t h e f o l l o w i n g diagram commutes -1

(C.3)

TI

n

Evidently

Tkx

'k

(vk)

V k X F

b

-\v k /

i s the restriction o f

= projection

Tk

to

"-'(XI.

Then extend ( V k d k ) t o be

a maximal f a m i l y h a v i n g these p r o p e r t i e s and c a l l t h e r e s u l t i n g o b j e c t n = (E,M,F,a)

a v e c t o r bundle (VB).

A s e c t i o n s o f a VB i s a map s : M

-+

E such

that a o s = identity. The p r o d u c t m a n i f o l d V k X F i s d e f i n e d as f o l l o w s .

RElMRK C.14.

F is a

t r i v i a l m a n i f o l d ( i . e . one c o o r d i n a t e system w i l l s u f f i c e ) and we c o n s i d e r

Vk as an open subset o f M w i t h t h e induced c h a r t s ; then extend b o t h f a m i l i e s Consider t o be maximal and taken t o be i n t h e form (UCr,vCr) and (WB,$B). Vk X F w i t h c h a r t s (U,XWgyqax$B)

and extend these a g a i n t o be a maximal fam-

i l y t o o b t a i n Vk X F as a C a m a n i f o l d .

S i m i l a r l y n - l ( V k ) i s a m a n i f o l d as

The ( V k , ~ k ) a r e c a l l e d t r i -1 ) : Vi n V j L(F) i s x note F i s f i n i t e dimensional).

an open subset o f E w i t h t h e induced s t r u c t u r e . v i a l i z a t i o n s and we remark t h a t t h e map x a u t o m a t i c a l l y Cm ( e x e r c i s e

-

-+

(T.T j i

-+

L e t n: E + M and 7 1 ' : E ' M ' E Y%. A VB morphism n i n t h e c a t e g o r y Y3 o f v e c t o r bundles i s a p a i r o f m a n i f o l d morphisms m:

DEFINICZ0N C.15, M ' and e: E

-+

-+

E ' such t h a t t h e f o l l o w i n g diagram commutes

-+

a'

M

3

APPENDIX C

361

E' a c o n t i n u o u s l i n e a r map ( E x = TI -1 ( x ) = IT etc.). Vecm(x) X' t o r bundles o v e r a f i x e d M w i l l be denoted by UB(M) and morphisms i n t h i s

w i t h ex: Ex

-+

c a t e g o r y w i l l be assumed t o i n v o l v e m = i d e n t i t y .

R m A R K C.16.

Since o u r m a n i f o l d s a r e always c o u n t a b l e a t i n f i n i t y and para-

compact we can always f i n d a c o u n t a b l e l o c a l l y f i n i t e f a m i l y (U,)

o f relativ-

e l y compact open domains o f l o c a l c o o r d i n a t e s which a r e s i m u l t a n e o u s l y domains o f t r i v i a l i z a t i o n s ( c f . D e f i n i t i o n C.2).

One need o n l y observe t h a t

t h e r e s t r i c t i o n o f a t r i v i a l i z a t i o n i s a g a i n a t r i v i a l i z a t i o n and use a p r e vious construction. Since t h e use o f c a t e g o r y t e r m i n o l o g y has been n o t i n f r e q u e n t h e r e we w i l l g i v e t h e necessary d e f i n i t i o n s .

One t h i n k s o f t o p o l o g i c a l spaces and con-

t i n u o u s maps, groups and homomorphisms, Cm m a n i f o l d s and morphisms as i n D e f i n i t i o n C.2,

etc.

DEFZNZCI0N C.17. A c a t e g o r y i s a c l a s s o f o b j e c t s A,B, ...such t h a t f o r any two o b j e c t s A,B t h e r e i s a s e t ( p o s s i b l y v o i d ) Hom(A,B) o f morphisms A -+ B s a t i s f y i n g ( A ) Hom(A,B) and Hom(C,D) a r e d i s j o i n t u n l e s s A = C and B = D i n which case t h e y a r e equal (B) To each t r i p l e (A,B,C) composition from Hom(A,B)

X Hom(B,C)

-+

Hom(A,C):

= y(6a) whenever t h e compositions make sense ( C )

t h e r e i s a map c a l l e d

( a , ~ )

-+

60, such t h a t

(y6)a

For each o b j e c t A t h e r e i s

an element lA E Hom(A,A) ( n e c e s s a r i l y unique) such t h a t

lAa =

a and 81A = B

whenever t h e compositions make sense.

D E F I N I C I 0 N C.18-

A c o v a r i a n t f u n c t o r T: A

-+

8, A and B b e i n g c a t e g o r i e s , i s

a "map" which a s s o c i a t e s an o b j e c t T(A) E B t o each A E A and a morphism and T(a6) = T(B) t o each a E Hom(A,B) such t h a t T ( l A ) = 1 T(A) T(a)T(6) whenever t h e map a6 i s d e f i n e d . A c o n t r a v a r i a n t f u n c t o r S r e v e r s e s

T ( a ) : T(A)

-t

arrows ( i . e .

S ( a ) : S(B)

-+

S(A) and S ( a 6 ) = s ( ~ ) S ( a ) ) .

D E F I N I C I 0 N C-19- The f u n c t o r s o f t h e same v a r i a n c e f r o m A

-t

B f o r m t h e ob-

j e c t s o f t h e f u n c t o r c a t e g o r y F(A,B) and t h e morphisms i n t h i s c a t e g o r y a r e c a l l e d natural transformations. t u r a l transformation S

-+

They a r e d e f i n e d by s a y i n g t h a t t i s a na-

T i f t h e r e i s a c o l l e c t i o n o f morphisms tA:S(A)

T(A) such t h a t t h e f o l l o w i n g diagrams commute ( a E Hom(A,B))

-+

362

ROBERT CARROLL

The most n a t u r a l examples o f v e c t o r bundles a r e perhaps t h e t a n g e n t and cotangent bundles o f a m a n i f o l d M ( c f . D e f i n i t i o n C.5).

To i n t r o d u c e t h e bun-

d l e s t r u c t u r e we s h a l l r e d e f i n e them now i n a s l i g h t l y d i f f e r e n t way.

We r e -

c a l l here t h a t t h e Frechet d e r i v a t i v e was d e f i n e d i n D e f i n i t i o n 3 . 2 . 3 .

Thus

i f f: U

F i s a continuous map o f an open subset U o f a Banach space E i n t o

-+

a Banach space F then f i s Frechet d i f f e r e n t i a b l e a t xo E U i f t h e r e e x i s t s a continuous l i n e a r map A: E = 0 as x

-+

xo.

-f

F such t h a t l i m [ l l f ( x ) - f ( x o ) - x ( x - x o ) l l ] / l l x - x o l l

We w r i t e A = f ' ( x o ) = D f ( x o ) and f ' becomes a map U

-+

L(E,F)

i f f i s d i f f e r e n t i a b l e a t every p o i n t o f U; s i m i l a r l y , f " becomes a map U L(E,L(E,F)) -f

L(E,F)

tives.

If x

e t c . and t h e d e r i v a t i v e s a r e u n i q u e l y defined.

-f

-f

Df(x): U

i s continuous we say f E C 1 on U and s i m i l a r l y f o r h i g h e r d e r i v a W r i t i n g L ~ ( E , F ) f o r symmetric m - l i n e a r maps

d"~ -f

F ( i . e . m-linear

maps u n a f f e c t e d by i n t e r c h a n g i n g two arguments) i t i s e a s i l y seen t h a t D m f

(x,)

E

L ~ ( E , F ) ( c f . 93.2).

L e t now U C El and V C E2 be open s e t s and suppose f: U X V ous.

Consider (x,y)

E

-f

F i s continu-

U X V and denote f o r example by Dlf(x,y)

t h e Frechet

d e r i v a t i v e a t (x,y) o f f ( - , y ) w i t h r e s p e c t t o t h e f i r s t argument (when i t exists).

Then ( e x e r c i s e ) Df(x,y) e x i s t s and i s continuous i f and o n l y i f

both Dlf(x,y)and

D2f(x,y) e x i s t and a r e continuous and i n t h i s event one has

= 1 Dif(x,y)vi f o r vi E E. The D i f a r e c a l l e d p a r t i a l d e r i Df(x,y)(vl,v2) v a t i v e s and f o r F Ei = R we o b t a i n t h e customary n o t i o n o f p a r t i a l d e r i v a -

tive.

The c o o r d i n a t e w i s e continuous d i f f e r e n t i a b i l i t y o f say

on a man-

y ~ ~ y ~ a '

i f o l d M t h u s i s e q u i v a l e n t t o t h e e x i s t e n c e and c o n t i n u i t y o f (IP y ~ - ' ) ' i n f a c t equals t h e Jacobian J ( x ) ( e x e r c i s e t h e same t h i n g t o say e.g. v,.":'

-

Ba

which

see ( 6 ) ) . S i m i l a r l y i t means

E Cm u s i n g e i t h e r Frechet d i f f e r e n t i a b i l i t y

o r t h e usual c o o r d i n a t e w i s e v e r s i o n .

We r e f e r t o 53.2 f o r a d d i t i o n a l i n f o r -

mation about Frechet d e r i v a t i v e s (e.g. T a y l o r ' s theorem, e t c . ) .

Using t h e

Frechet d e r i v a t i v e we can g i v e a n o t h e r d e f i n i t i o n o f T(M) where t h e bundle s t r u c t u r e i s apparent ( c f . [ L g l ] f o r t h i s p o i n t o f v i e w ) .

DEFLNZCZ0N C.20.

For x € M we c o n s i d e r t r i p l e s (U,y~,u) where ( U , ~ J ) i s a

c h a r t a t x and u i s an element o f t h e v e c t o r space Rn i n which v(U) = We say (U,y~,u)

5

lies.

i f ( & y ~ - ~ ) '=u w (where ( $ y ~ - ' ) ' can o f course be de-

(W,$,w)

f i n e d i n some NBH o f ~ ( x ) ) . This i s c l e a r l y an equivalence r e l a t i o n by t h e

I-' c h a i n r u l e ( e x e r c i s e ) and i n p a r t i c u l a r one has t h e f a m i l i a r [ ( $ v - ~ ) ' V(X) -1 The s e t o f equivalence c l a s s e s o f such t r i p l e s a t x i s de= (V$ f i n e d t o be Tx(M) and each c h a r t ( U , ~ J ) a t x determines a b i j e c t i o n Tx(M) -f

Rn ( o n t o ) by (U,y~,u)-

-+

u.

One g i v e s Tx(M) t h e t r a n s p o r t e d t o p o l o g i c a l

APPENDIX C

363

v e c t o r space s t r u c t u r e f r o m Rn which i s e v i d e n t l y independent o f t h e c h a r t (U,y) s e l e c t e d t o e x h i b i t t h e b i j e c t i o n ( r e c a l l t h a t TVS s t r u c t u r e i n v o l v e s o n l y convergence and 1 i n e a r s t r u c t u r e and does n o t i n v o l v e c o o r d i n a t e v a l u e s ) . Again s e t T(M) = UTx(M) w i t h n t h e p r o j e c t i o n Tx(M) c h a r t on M we have a n a t u r a l b i j e c t i o n T ~ ~ ( U , P ~ U =) * (x,u)

T

~

:n-'

(U)

-+

-+

I f (U,v) i s a

x.

U X Rn determined by

( o r = u, depending on c o n t e x t ) f o r x E U.

T(M) by s i m p l y t r a n s p o r t i n g t h e Cm s t r u c t u r e o f U X Rn t o n

o

(n-'(U),(vXl)

makes sense s i n c e i f (U,v), ~,][(vXl)

~ ~ 1= -xuw: l

o

f i n i t i o n C . l ) xuw(x,u)

( U ) by a c h a r t This

(W,9) a r e c h a r t s on M a t x t h e n t h e map [ ( $ X l ) o

v ( U n W ) X Rn

-+

9(U n W) X Rn i s g i v e n by ( c f . Deand t h i s i s an isomorphism o f

= ((9v-1)(x),(Jnp-1)'u)

W i t h t h i s C" s t r u c t u r e on T(M) (and on n-'(U))

t r i v i a l manifolds.

a r e c l e a r l y isomorphisms i n

m

the

'I,,

as w e l l (see D e f i n i t i o n C.2) and axiom (A) o f

Since n i s l o c a l l y t h e c o m p o s i t i o n o f morphisms

D e f i n i t i o n C.13 h o l d s .

p

o

= n i t i s a l s o a morphism and T ( M ) i s a v e c t o r bundle.

T ,,

L e t now f : M

-+

M ' be a morphism i n

f, = T f determines a morphism T(M) functor. -+

-1

and t h e n e x t e n d i n g t o a maximal f a m i l y on T(M).

7),

(B)

Axioms

We d e f i n e c h a r t s on

and (C) o f D e f i n i t i o n C.13 a r e t h e n c l e a r l y s a t i s f i e d .

JI

B

( W ) i s C" B

wL4

T ( M ' ) i n U B and t h a t T:

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

C").

- cf.

(see D e f i n i t i o n C.2).

-+

V3 i s a f u n c t o r .

-f

-+

UB i s a

t h e map ILBfv,': E

va(Ua)

Tx(M) we d e f i n e t h e n

= (f(x),(WBy98y($Bfv~')'u).)

I t f o l l o w s t h a t T f i s a morphism T(M)

i s s i m p l y t h e map (x,u)

m

m

T h i s i s w e l l d e f i n e d (ex-

and e v i d e n t l y Tf(x,(Ua,va,u)')

[Cl])

o T f o :;T

s e q u e n t l y T:

I f (U,,va,u)'

= (WBy9By(9,fvi1)'u)- i n Tf(x)(M).

= Ti(f(x)y(%fvL1)'u). 7

(i.e.

-+

Thus r e c a l l f i r s t t h a t whenever f ( U a ) C W,

TXf(Ua,va,u)' ercise

m

(f(x),(i,fv;')'u)

-f

and f

T(M') s i n c e E

Con-

C".

To s p e l l o u t t h e correspondence w i t h o u r (V,9,v)

at a

as x c o o r d i n a t e s .

Then

p r e v i o u s t r e a t m e n t o f t h e tangent space we suppose (U,v,u) p o i n t p E M and t h i n k o f 9 as y c o o r d i n a t e s and

v

5

-+ y and Dy.(x)u = 1 D.y.(x)ui. W r i t i n g ( v . ) and (ui) as column vecJ -1 1 J J t o r s we o b t a i n ( v j ) = (9v ) ' u = (( D . y . ( x ) ) ) (ui) (i= column i n d e x and j =

k - ' :x

row i n d e x ) .

T h i s corresponds t o

F u r t h e r n o t e t h a t if f: M -+

7

a:

1 J

(*A)

and r e f e r s t o c o e f f i c i e n t changes.

R and if we i d e n t i f y Tp(R) w i t h R t h e n T x f : Tx(M) -1 R can be w r i t t e n i n t h e f o r m T x f ( v x ) = ( f v a ) ' 7 a x v x f o r a t r i v i a l i z a t i o n : T(U,)

-+

df =

Ua X Rn.

1 Dk(f

T h i s o f course corresponds t o t h i n k i n g d i r e c t l y o f Txf

1 (af/axk)dxk

w i t h components vi

+

=

1Dk(f

o v-')dxk

s i n c e i t s a c t i o n on a v e c t o r

T

~

( r e l a t i v e t o a dual b a s i s f o r t h e d x k ) i s expressed by

o v a l ) v k = D ( f o v a ' ) ( v l....,vn).

F u r t h e r i n f o r m a t i o n about v e c t o r bundles and t h e i r morphisms can be found i n

~

v

~

364

ROBERT CARROLL

[Cl;Hbl;Lgl;Pll].

We conclude t h i s t o p i c w i t h a b r i e f d i s c u s s i o n o f j e t bun-

d l e s ( c f . [Cl;Hbl;Pll]

f o r further details).

Let q

DEFZNICL0N C.21.

E

M and I

be t h e s e t o f f u n c t i o n s i n Cm(M) v a n i s h i n g

q

a t q ( r e a l o r complex f u n c t i o n s ) ; I i s an i d e a l i n Cm(M) considered as a k r i n g o r algebra. L e t Zq(n) = Iktl* : m ( ~ ) f o r any VB IT as above and d e f i n e t h e v e c t o r space o f k j e t s o f IT a t q as J k ( T ) = C"(n)/Zi. The canonical map f q k i s s a i d t o map f i n t o i t s k j e t a t q. R e c a l l t h a t + j k ( f ) q : C"(T) J (IT) 2 q I f o r example means t h e s e t o f f i n i t e sums 1 figi w i t h fi, gi E I ( i d e a l q q product), e t c . -f

I n order t o describe j,(f)

as s i m p l y t h e c o l l e c t i o n o f p a r t i a l d e r i v a t i v e s q o f f a t q o f o r d e r l a \ 5 k we p r o v i d e some formulas now which a l s o serve as good e x e r c i s e s i n h a n d l i n g Frechet d e r i v a t i v e s , t a n g e n t v e c t o r s , e t c . ( c f . [Cl;P11]

f o r more d e t a i l s ) .

There i s a unique l i n e a r map d k * Ik LE(T(M) ,C) such t h a t i f q' 9 9 k a r e v e c t o r f i e l d s on M ( i . e . Cm s e c t i o n s o f T(M)) and f E I t h e n k k q ...,vq) = (v'v 2...v f ) ( q ) and t h e sequence 0 + Ik+'Ik L;(T(M),C) k q + 0 i s e x a c t (vi means vi(q) and t h e n e x t t o l a s t map i s d ). 9 q P I L U U ~ : We r e c a l l t h a t a sequence i s e x a c t when t h e image o f one map i s t h e

tElMitA C.22. k

-f

vl,...,v k 1 dqf(vq,

-f

kernel o f t h e next.

-f

Now u s i n g r e a l valued f u n c t i o n s f o r s i m p l i c i t y f o r k =

0 we have 0 + I1 + Io = C"(M) R 0 which i s o b v i o u s l y e x a c t and f o r k = 1 q use T a y l o r ' s theorem t o conclude t h a t 0 + I: Iq1 + we s e t d 1 ( f ) = qd f and q 9 L e t us T* = LS(T , R ) + 0 i s e x a c t ( c f . 53.2 f o r T a y l o r ' s theorem e t c . ) . q 1 q c l a r i f y some n o t a t i o n i n t h i s connection. Thus suppose t h a t ei % a/axi i s a -f

-f

-f

basis o f T

w i t h e t PI, dxi a dual b a s i s o f T* We can t h i n k o f q : U + E C Rn 9 q' determining l o c a l coordinates a t q w i t h q ( q ) = 0 f o r s i m p l i c i t y . Setting ? -1 k(l/a!)DaF(0)xa + = f o q , T a y l o r ' s f o r m u l a becomes e.g. F ( x ) = lal

1

i('), Gcx) = 1 I a l = k + l Ra ( 0 ) x a

where t h e Da denote t h e u s u a l p a r t i a l d e r i v a -

t i v e s and Ra(0) i s t h e usual remainder t e r m o b t a i n e d from T a y l o r ' s theorem. We can w r i t e now Dlalr(0){e'?il 1 t h i n k i n g o f vectors v =

1 xiei

= D"fU(0) which we denote a l s o by D a f ( q ) and

i n Rn we g e t D ~ " ~ ? ( O ) V=~ 1 ~ I( l a l ! / a ! ) x a D a

( l / a ! )D"f(q) 7(0). I n what f o l l o w s we s h a l l o c c a s i o n a l l y w r i t e f = 1 la1 5 k xa + g w i t h an obvious abuse o f n o t a t i o n . F u r t h e r w i t h these n o t a t i o n s we have t h e r u l e ( d f , v ) = D?(O),v ( c f . D e f i n i t i o n C.4 and subsequent e q u a t i o n s ) . 1 1 1 Thus d = d maps I o n t o T* i n an obvious way w h i l e i f f E I w i t h d f = 0 i t q 9 q q f o l l o w s t h a t D"T(0) = 0 f o r la1 5 1 and hence by T a y l o r ' s theorem f"= ;?=

1 \ a l = 2Ra ( 0 ) x "

or f

E

I'

q'

The general case now f o l l o w s e a s i l y by i n d u c t i o n

365

APPENDIX C

(cf. [Cl;pll]

QED

for details).

k k k k = k ! S (w) where S ( w ) Now observe t h a t i f f E I = I and d f = w t h e n d,f q i i s t h e element o f L;(T(M) ,R) qd e f i n e d qby (v’ ..,vk) nw(v ). Indeed i f k q q’j v’,...,~ a r e Cm v e c t o r f i e l d s t h e n vlfk = k ( v f ) Qfk-’ and 9v2v1fk = k ( k - 1 ) 1

-+

(v’f)(v2f)fk-* + k(v2v’f)fk-’ k ( k - l ) ( v ’ f ) ( v 2 f ) f k - * + h where h E Ik-’ etc. 1 k q I n d u c t i v e l y one o b t a i n s i n p a r t i c u l a r vk...vlfk = k ! ( v f ) . . . ( v f ) + g where 1 1 k k 1 k g E I q . Consequently d k f k ( v k v ) = k!w(v )...w(v ) = k ! S (w)(v q’ * . YVq). q q’ q q q I;= C.23. There i s a unique map d k * Zk-’ -+ L i ( T ( M ) q y ~ q )such t h a t i f f E k q’ q k Z k - ’ ( r ) and h E C”(IT*) t h e n d ( h o f ) = h ( q ) d f ( h o f i s t h e element o f Cm 9 q m q I f g E C (M,C) s a t i s f i e s g ( q ) = 0 w i t h (M,C) d e f i n e d by q + h ( q ) ( f ( q ) ) ) . k k k = w and i f f E Cm(v) s a t i s f i e s f ( q ) = y t h e n d ( g f ) = k ! S (w) Iy. The dgq q sequence o zk + zk-l + L ~ ( T ( M ) ~ +, ~o ~i )s e x a c t . q q k A d d i t i o n a l d e t a i l s f o r t h e p r o o f can be found i n [Cl;P11]. Now s i n c e Z (IT) q k C Z k - l ( ~ ) t h e map f j k ( f ) q :C”(IT) + J IT)^ induces a l i n e a r map j ( f ) + 9 k k-1 k-1 k k q with kernel Z ( T ) ~ / Z IT)^ ( e x e r c i s e - we a r e jk-l(f)q: J IT)^ J w o r k i n g w i t h v e c t o r spaces). Using Lemna C.23 we have an e x a c t sequence 0 L E ( T ( M ) q y ~ q )+ J k IT)^ + J k - l ( ~ ) q + 0 where t h e i n j e c t i o n i: L i + Jk i s c h a r -

...,

+

-+

-+

IT)^

+

-+

k a c t e r i z e d by t h e r e l a t i o n i ( S (w) Iy ) = j k ( ( l / k ! ) g k f )

( f o r any g E C”(M,C) q = w, and any f E Cm(n) w i t h f ( q ) = y ) . These seq-

s a t i s f y i n g g ( q ) = 0, dg k q k L i ( T ( M ) IT ) ( e x e r c i s e ) . uences i m p l y t h a t J (IT) i s isomorphic t o q 9’ s, F u r t h e r g i v e n ei a b a s i s f o r T(M) as b e f o r e t h e n y - + ty/(e ) l l k l = m i s an q isomorphism o f L i ( T ( M ) IT ) w i t h @ IT which means t h a t J (TI) i s i s o 9 9’ q Ial=m q morphic t o @ a l < k IT^.

I

_

L e t Jk(,)

DEfINZIIZ0N C.24, M

-+

J

k

(IT)

= uJk(,)

( q E M) and f o r f E C”(IT) q Given a t r i v i a l i z a t i o n f o r IT,

by j k ( f ) ( q ) = j k ( f ) q .

U X F, w i t h l o c a l c o o r d i n a t e s 9 : U k k o f J (IT) o v e r U, T ~ :J (IT)(U) -f

f o r Jk(,)\U).

For a f u n c t i o n f

E

-+

?C

al

(IT

T:

IT-’(U)

+

Rn, we c o n s t r u c t a t r i v i a l i z a t i o n k F, as f o l l o w s (we w r i t e J ( I T ) ( U )

xm@\alzk

C

define j k ( f ) :

we w r i t e

f^

=

‘I

o f o 9-l

E

Cm(U,F)

k ) describes t h e desired t r i v i a l i z a I _ k t i o n i n terms o f a l l p a r t i a l d e r i v a t i v e s o f o r d e r 5 k . J (IT) i s t h e n e a s i l y k seen t o be a VB and j, w i l l a l s o be t h o u g h t o f as a map C“(IT) C”(J ( I T ) ) .

and t h e map j k ( f )

+

(q,tDaf(v(q))l

<

-+

I n p a r t i c u l a r one can show t h a t 0

-+

Li(T(M),n)

-+

J

k

(IT)

-+

Jk-l(n)

+

0 i s an

exact sequence o f v e c t o r bundles where L ~ ( T ( M ) , I T ) has a s u i t a b l e bundle structure (cf. [Cl;Pll]).

There i s a l o t o f f u r t h e r i n t e r e s t i n g and u s e f u l

i n f o r m a t i o n on j e t bundles and d i f f e r e n t i a l o p e r a t o r s i n [ P l l ;Ccl;Cl] example.

for

The machinery was i m p o r t a n t i n t h e development o f t h e A t i y a h -

366

ROBERT CARROLL

Singer index theorem and i s a l s o n a t u r a l f o r s t u d i e s o f general d i f f e r e n t i a l systems ( c f . [Qil;SteE;Swel]).

We mention i n passing a s o r t o f n e a t f a c t ,

namely t h e n a t u r a l equivalence o f Hom and CmL i n ( c f . [Cl;P11]) L e t T be a f u n c t i o n on M such t h a t T ( q )

CmrmA C.25.

essary and s u f f i c i e n t c o n d i t i o n t h a t T q

-+

T(q)jk(f)

REmARK

E

q

C"(5)

f o r each f

E

E

E

Cm(L(Jk ( R ) , c )

k L ( J (n) , r l q ) . = Horn(

F

( r ) , ~ i )s t h a t

Cm(r).

L e t us now address some s p e c i f i c needs i n t h e t e x t .

C.26,

A nec-

Thus f i r s t

i n J1.7 f o r example one d e a l s i n a v e r y c u r s o r y manner w i t h i n t e g r a l s o v e r s i n g u l a r c h a i n s and c e l l s e t c . on a m a n i f o l d M.

The necessary background i s

i n [ A l l o r i n any book i n v o l v i n g i n t e g r a t i o n on m a n i f o l d s and s i n g u l a r homo l o g y ( c f . [Dzl;Fsl;Nzl;Rhl;Wtl]);

o u r excuse f o r n o t t r e a t i n g t h i s more ex-

t e n s i v e l y h e r e i s ( i n a d d i t i o n t o space l i m i t a t i o n s ) t h e d e s i r e t o s t e e r t h e reader t o [ A l l .

Thus v e r y b r i e f l y k - c e l l s on M a r e t r i p l e s (D,f,0) = u where k M i s a smooth map o f a k-dimensional polyhedron D i n R t o M and 0 i s k I f w E A = k-forms on M one d e f i n e s I u w = t o r e p r e s e n t an o r i e n t a t i o n .

f: D

+

1

ID f*w. Chains a r e f i n i t e c o l l e c t i o n s C = miui o f c e l l s w i t h m u l t i p l i c i t y k f a c t o r s mi and g i v e n D C R as above w i t h a D = Di (Di s u i t a b l y o r i e n t e d -

1

18,

we o m i t d i s c u s s i o n o f t h i s h e r e ) one d e f i n e s ao = where Si 'L (Di,fiyOi). k Similarly a C = miaoi. One says w E A i s c l o s e d i f dw = 0 and v i a Stokes'

1

theorem JaC w = 1 dw so we see t h a t IaC w = 0 i f w i s closed. C i s a c h a i n C w i t h a C = 0 so e v i d e n t l y I c dw = 0 f o r any c y c l e .

A c y c l e on M A form w i s

Ifone c o n s i d e r s t h e deRham complex 0 R Ao * A k 0 (map i s d) then H (M,R) = k e r dk/ I m dk-l i s c a l l e d t h e k - t h

c a l l e d e x a c t i f w = dv. -+

... -+An

-+

-f

1

-f

deRham cohomology group (dk = d a t l e v e l k ) . A v e r y b e a u t i f u l s e t o f t h e o k corresponds t o t h e r e a l s i n g u l a r o r Cech cohomo ogy o f M k and moreover each cohomology c l a s s i n H w i l l have a unique harmon c r e p r e rems shows t h a t H

s e n t a t i v e (Hodge t h e o r y ) ;

i n a d d i t i o n t h i s t h e o r y can be embedded

A t i y a h - S i n g e r complex o f ideas ( c f . [Bsl;Ggl;Gyl;Pll;Rhl]).

n the

L o c a l l y over

n i c e ( t o p o l o g i c a l l y s i m p l e ) r e g i o n s t h e deRham complex i s however e x a c t and t h i s i s t h e c o n t e n t o f t h e c l a s s i c a l Poincar;

lemma.

CEIR1RA C.27 (P01NCARf). L e t U be a s t a r l i k e s e t i n Rn and l e t w k w i t h dw = 0; then t h e r e e x i s t s ak-' w i t h dak-' = wk = w.

Pkaod:

D e f i n e a map H: Am + A

... A d x 1m . , Hw(x)

fdxi,

A

... A

dxim

=

4

Now

be g i v e n

m-1 such t h a t dH t Hd = I as f o l l o w s . (-l)p-l(lJ t m - l f ( t x ) d t ) x dxi A

( d x j means t h i s t e r m i s m i s s i n g ) .

t h e n Hw(x) i s inchanged.

k

iP

If permuted by

I

T,

f

For w = A

... A d x i P A -+

sgn-rf,

367

APPENDIX C

A dx

i,

1;

d(Hw)(x) =

(C.6) A

4

... Adxi

A P

(-l)p-’[x

... A d x

1 f .Jd x J.

A dx

. ..

(C.7) x dx

j

i,

A

... A

dx

I

A

j

+ 1 ; tm-lf(tx)dtdx. ]

lP

... A

d x 1m . t

A

d x . A dx A,.. A dx A fp J i, iP

.. A dxi,

im

1 m 1 Jo t f ( t x ) l o dxi,

... A dx i m

so (dw i s an m+l form) Hdw(x) =

1 Jo1

d x j A dxi

tm

A

which i m p l i e s

dHw t Hdw = m[J, t m - l f ( t x ) d t ] d x A

tmf j ( t x ) d t d x

... A dx im + 1 1 (-l)p[li tmfj(tx)dt]xip

f . ( t x ) d t x . A dxi A J n J A dx A A dxi, iP

...

i,

c1n JOl

m = m[Ji t m - l f ( t x ) d t d x i

tx)dtx B u t dw =

iP

= f , [m tm- ’f

A

... A

dxi

t

m

i,

A

... A

1 tmfjxj]dtdx

i,

dx A

im

+

... A

1 [J,l dxi

m

tmfj(tx)dt] =

= w

Thus i f dw = 0 one has dHw = w and w i s e x a c t (on n i c e s t a r l i k e r e g i o n s ) . QED

RrmARK C.28,

S e c t i o n 1.7 a l s o r e q u i r e s o c c a s i o n a l l y t h e use o f i n t e r i o r

p r o d u c t s and L i e d e r i v a t i v e s . Thus t h e i n t e r i o r p r o d u c t i s determined by k k (A+) ( i x w ) ( C 1 , E ~ - =~ w) (X,sl, ..., ( i x = i ( X ) ) and t h e L i e d e r i v a 1 t i v e o f a 1 - f o r m i s ( A m ) (Lxw ) ( E ) = Dtw (g*c)lt=O where gt i s t h e f l o w d e t -

~ ~ - ~ j

....

ermined by X (see remarks b e f o r e Theorem C.10 and a l s o t h e d i s c u s s i o n below on L i e t h e o r y ) .

R e c a l l t h a t t h e L i e b r a c k e t o f v e c t o r f i e l d s E , ~ I i s (6m)

[E,nIi = E.arl./ax - n.agi/ax One can d e f i n e a L i e d e r i v a t i v e o f J 1 j J j* t h i n g ” and i n p a r t i c u l a r f o r a v e c t o r Y ( a * ) ( L Y) = Dt(g;tYgtp)It=O X P [gitY t - Y ] / t as t + 0 ( i t i s i n s t r u c t i v e t o draw a p i c t u r e here, S P P t - the we o m i t - c f . [Wrl]). One c o u l d a l s o work w i t h LxY = Dtg,Ylt,O

“any= lim

which -t g*

c o n t e x t i s a p o s i t i o n a l device.

PR0P053dZtZ0N C.29,

LxY = [X,Y].

-t D e f i n e now H ( t , u ) = D Y t ( f o g )It=o. ( L X Y ) ( f ) = D,[gitY t ]fl -t u t $=Ot t 9 P = f ( g ( h ( g ( p ) ) ) = ( f o g ) (h ( g ( p ) ) where Y generates t h e f l o w hU. Set now Then Y t ( f o g - t ) = DUH(t,u)lu=O and ( L X Y ) f = DtDuH(t,u)lu,t,O. S P By t h e c h a i n r u l e one has K(t,u,s) = f ( g s ( h u ( g t p ) ) ) so H(t,u) = K ( t , u , - t ) . 2 2 2 u t a H / a t h (0,O) = [a K/axlax2 - a K/ax3ax2](0,0,0). B u t K(t,u,O) = f ( h ( g p))

Pkood:

so (aK/au)(t,O,O) = Y t f = ( Y f ) ( g t p ) = ( Y f ) o g t ( p ) and (a*K/atau)(O,O,O) g p s u X Y f . A l s o K ( O , u , s ) = f ( g ( h p ) ) SO (aK/as)(O,u,O) = Xf(h‘p) = X f o hu(p) 2 and (a K/auas)(O,O,O) = Y X f . T h i s i m p l i e s L X Y f = (XY-YX)f = [X,Y]f. QED

=

368

ROBERT CARROLL

RrmARK

(cnntinued).

C.28

Note n e x t t h a t

L X d f = d L X f on f u n c t i o n s f

(@A)

To see t h i s n o t e t h a t s i n c e L X d f and dLXf a r e one forms

(where L X f = X f ) .

we w i l l l e t them a c t on Y and show i d e n t i t y i n Y ( L X f ) = Y D t ( f o gt)ItZe w h i l e from (Am) t (g,Y,f)lt,O = D t Y ( f o g )ItZ3 = Y D t ( f o r u l e h*f = f o h, (h,Y,f) = ( Y , f o h)).

i(X).

To s e t t h e stage one w r i t e s ( c f .

(*A).

Thus d ( L X f ) ( Y ) =

L X ( d f ) ( Y ) = Dt(df,g,Y)lt=O t = Dt t g )It=o as r e q u i r e d ( n o t e here t h e Next one shows L x = I ( X ) o d + d o

(A+))

i ( X ) w = X -I w f o r t h e i n t e r i o r

A l s o i t i s convenient t o speak o f " a n t i d e r i v a t i o n " 8 i n AT*, namely k ou A v t (-1) u A Ov ( k = deg u - 0 i s c a l l e d a d e r k i v a t i o n i f t h e r e i s no f a c t o r ( - 1 ) ). Thus d i s an a n t i d e r i v a t i o n and so i s i ( X ) w h i l e L x i s a d e r i v a t i o n . One shows e a s i l y t h a t i f a ( s u p p o r t d i m i n -

product.

8 : Ak -+ A k + l Y O(u A v ) =

i s h i n g ) map D s a t i s f i e s D ( f f ) = (Df ) f + fi D f 2 and D ( f w ) = (Df)w t f D w 1 1 2 s ( f E A', w E A ) w i t h D: Ao + A and Al1 As'' ( s even) then i t has a unique

1

e x t e n s i o n t o a d e r i v a t i o n on AT* ( e x e r c i s e

-

see e.g.

[Lsl]).

But Lx and

i ( X ) o d t d o i ( X ) a r e d e r i v a t i o n s a g r e e i n g on f,f2 and f w which i m p l i e s

t h e y a r e equal on any f o r m ( e x e r c i s e ) .

Note h e r e i f f E A',

= X ( f ) w h i l e i ( X ) d f = (df,X) = X ( f ) .

LXf = (X,f) = dLXxi

s i n c e LXd = dLX.

d(X,xi)

= dLXxi

i ( X ) f = 0 and

On b a s i s elements, LXdxi

On t h e o t h e r hand [ i ( X ) d

t di(X)]dxi

so L x = i ( X ) o d t d o i ( X ) on AT*.

= di(X)dxi

As examples l e t us c i t e

1 aidxi, dw = 1 dai A dxi, X J dw = 1 ( X J dai) A dxi - 1 d a p X J dxi = 1 1 5.(aai/ax.)dxi - g.(aa./axi)dxiy X J w = 1 aiciy J J J J t 1 a.(ag./axi)dxiy Lxw = i ( X ) o d t d o i ( x ) d(X J w) = 1 g.(aa./axi)dxi J J J J 1 cj(aaj/axi)dxi t 1 a.(ag./axi)dxi. J J (0.)

1 Cia/axiy

X =

RENARK C.30. and 3.

=

w =

=

Symplectic geometry a r i s e s in 51.7 as w e l l as i n Chapters 2

We make no a t t e m p t a t s y s t e m a t i c e x p o s i t i o n b u t s i m p l y i n d i c a t e a

few formulas which a r e needed i n t h e t e x t (see [Al;Gul;Lbl;Ol;Dzl] tional information

f o r addi-

- some f u r t h e r development and formulas a l s o appear i n t h e

t e x t o f Chapters 1-3).

We work w i t h t h e s y m p l e c t i c s t r u c t u r e on T*M d e t e r -

>:

1

dpi A dqi. The 1 - f o r m w1 = pidqi i s mined by t h e canonical 2-form w2 = 2 a s s o c i a t e d t o w by t h e r u l e o f a s s i g n i n g a 1-form w1 t o 5 E T v i a w 1 ( n ) = 2 5 1 P 5 w (n,S) ( n E Tp). One denotes by I t h e map I : T* + T * w5 + 5 . L e t now H P P' be a f u n c t i o n so dH E T*; IdH i s c a l l e d a H a m i l t o n i a n v e c t o r f i e l d . Note 2 here dH = (aH/api)dpi t (aH/aqi)dqi so (dH,n) = w (n,IdH); i n p a r t i c u l a r

1

for Q =

(C.8)

C

c.(a/api) 1

t ni(a/aqi)

2 w (rl,IdH) = ti 1Idpi,IdH)

1 (dpi

"i (dqi,IdH)

1

one has (dH,n) =

A dqi)(n,IdH)

=

>: ci(aH/api)

+ ni(aH/aqi),

369

APPENDIX C

T h e r e f o r e IdH =

1 [-(aH/aqi)a/api

mulas and IdH i s c a l l e d (dH)

-

#

+ (aH/api)a/aq$ g i v e s agreement i n t h e f o r I 2r #: T* -t Tp). P IdH: 1 [(aH/ap.)dpi t (aH/aqi)dqiJ

i n [ C u l l f o r example ( i . e .

Then i n l o c a l c o o r d i n a t e s t h e map I: dH

-+

'0)

1

[-(aH/aqi)a/api + (aH/api)a/aqi has t h e form (;! ( w o r k i n g on 0 1 c o o r d i n a t e s i n p,q o r d e r t h i s corresponds t o an a c t i o n o f t h e form ( - 1 o)

H ) = (H -H ), t h e l a t t e r two v e c t o r s b e i n g column v e c t o r s ) . P P q R a R K C.31. We w i l l make a few remarks h e r e about Riemannian t y p e geometry (H

q

and r e f e r t o §3.10 f o r ideas o f connections and c u r v a t u r e ( f o r r e f e r e n c e s see e.g.

[Bthl;Cul;Dzl;Spil

;Sxl;Ttl]).

We r e f e r t o e a r l i e r remarks a b o u t

t e n s o r p r o d u c t s f o r background and t h i n k i n g o f V = T (M) one d e f i n e s elemi P i e n t s g = g . . e I eJ as c o v a r i a n t t e n s o r s o f r a n k 2. Here ei ( r e s p . e ) i s 1J i a b a s i s o f V ( r e s p . V * ) so ei a/axi and e 2r dxi. E v i d e n t l y gij = g(ei, Q

e . ) and g i s c a l l e d a m e t r i c i f i t i s symmetric (gij = g . . ) and nondegenerJ J' ate. We w i l l o n l y be concerned w i t h L o r e n t z m e t r i c s where gij = (1,-1,-1,

-1) i n a s u i t a b l e b a s i s o r i n Riemannian m e t r i c s where gij 2 0. always t h a t t h e r e i s a smooth dependence x

-f

One assumes

g ( x ) w i t h l o c a l i z a t i o n s as i n -

.

The i n v e r s e o f t h e m a t r i x (( g . .)) i s denoted by (( s'')) 1J an i n d e x r a i s i n g and l o w e r i n g c o n v e n t i o n which goes as f o l l o w s .

Given a

m e t r i c g on M as above, f o r x E M,g(x)

T:(M)

dicated.

induces a map Gx: Tx(M)

-f

One has via

I n p a r t i c u l a r ( u s i n g s u p e r s c r i p t n o t a t i o n on g(x)). Gx(v)w = gx(v,w) (gx i j i t h e n Gx(v) = v a r i a b l e s xi now) i f v = via/axi dx = g . .v dx (sumnation on k i k lJ repeated i n d i c e s ) s i n c e Gx(v)(a/ax ) = g ( v a/ax',a/ax ) = gikvi w h i l e 9 . . . . vJdxi(a/ax k ) = gijvJ6i = g v j = gjkvj. Thus i n terms o f components vi'J kj vi = g. .vJ and G l o w e r s i n d i c e s (G, i n v e r s e r a i s e s i n d i c e s ) . Here t h e i n lJ -1 , x ij i v e r s e map Gx i s determined v i a (( g )) = (( gij))-' w i t h GX'(w) = gijw.a/ax i i i i ij J = w a / a x f o r w = widx L e t us d e f i n e h e r e a l s o t h e ( i . e . w = g wj). Q

Hodge

*

o p e r a t o r (used e.g.

in

1.10).

Thus l e t t h e o r i e n t e d volume element

...

. ..

...

be rl = ( l / n ! ) n i, in dxi' A A dxin = 1gI4dx1 A A dxn where 1g/ = l d e t (( gij)) I and d e f i n e oil * * ' i n = sgn d e t (( gij)) g'"' g .i n j n rl j,. . j n . Any k form can be w r i t t e n as a = ( l / k ! ) a i , a(i)

(")

= a(e. ' I

*a(vktl,.

,...,e l. k ) . . ,vn)n

g i v e n by (*a).

for e

'L a/axiK). iK = a A Gx(vk+l) A

. ..

...

. dxil Now *: Ak ...lh

A -+

.. .

... A

.

d x l K (where

An-k i s d e f i n e d by

A Gx(vn) and t h e components a r e

-.

(l/k!)a. ,,jl *3ng (cf. [Ttl]). IK+I . in JI jk ih+, j K C I * * i g iA, - , ez j n A e 3 , *e 1 = I n p a r t i c u l a r i n R3 w i t h s t a n d a r d m e t r i c g one has *1 = e 2 3 3 k(n-k) t s e A e , *(e2 A e ) = e l , e t c . w i t h ** = 1. G e n e r a l l y ** = ( - 1 )

. .

=

on k-forms where s i s t h e index o f g ( i . e . orthonormal b a s i s f o r which g(ei,ei)

= -1

s i s t h e number o f v e c t o r s i n an

-

orthonormal means g(ei,ei)

=&l

370

ROBERT CARROLL

one = 0 f o r i # j ) . For Minkowski space w i t h g % (1,-1,-1,-1) 1 1 2 3 0 2 (Q = e0 A e A e2 A e3), *eo = -e A e A e , * e l = -e A e A 3 1 - 2 e etc., *(eo A e - e A e 3 , * ( e 2 A e 3 ) = -eO A e 1 , etc., *(eo A e 1 A e 2 3 1 2 k+l = -e , * ( e A e A e3) = -eo, e t c . and ** = ( - 1 ) We remark f i n a l l y t h a t n(k+l)+s+l*d*a f o r a E A k and t h e 6: Ak Ak-' i s d e f i n e d by ( 0 6 ) 6a = ( - 1 ) 3 Laplace-Beltrami o p e r a t o r i s A = 6d + d6: Ak -+ A k . I n R w i t h s = 0, 6a = 4 (-1)3(k+1)+'*d*a = (-1)3k*d*a w h i l e i n R w i t h s = 1 t h e -1 f a c t o r i s 1 =

and g(eiye.)

has *1 = -nJ

.

-+

i i (-l)4(k+1)+2. Using t h e index r a i s i n g o r l o w e r i n g convention vidx v i i- i 3 i 2 a/ax i n t h e form vie v ei one has i n R 5 v e 1. - + ?= v.ei *reA 1 -t 6 T = -*d*?E Ao and one w r i t e s d i v 5 = -65. S i m i l a r l y one can d e f i n e t h e c u r l as c u r l 5 = (*d?)lJ. ++

-+

++

Iv

R?3MRK C.32,

We g i v e n e x t a few comments and examples about L i e groups f o l -

I t i s a shame t o g i v e so l i t t l e o f a

l o w i n g [C41;Hg2,3;01;Tdl;Vrl;Wrl].

s u b j e c t o f unique beauty through t h e i n t e r p l a y o f geometry, algebra, t o p o l ogy, and a n a l y s i s b u t t h e r e f e r e n c e s i n d i c a t e d cover t h i s .

EXMPtE C.33. maps Rn

GL(n,R)

(resp. GL(n,C)

* Rn (resp. Cn

-+

Cn);

i s t h e group o f a l l n o n s i n g u l a r l i n e a r

i t can be thought o f a l t e r n a t i v e l y as t h e

group o f a l l i n v e r t i b l e n x n m a t r i c e s w i t h r e a l (resp. complex) c o e f f i c i e n t s . TM- 1 L e t O(n) be t h e group o f r e a l orthogonal n x n m a t r i c e s M ( i . e . M = T where M means t h e transpose o f M ) and l e t U(n) be t h e group o f u n i t a r y n x n = TM-l

complex m a t r i c e s ( i . e . One w r i t e s SL(n,R) (resp. GL(n,C))

where

( r e s p . SL(n,C))

denotes t h e complex c o n j u g a t e o f M).

f o r t h e group o f m a t r i c e s i n GL(n,R)

w i t h d e t e r m i n a n t equal t o one and then we s e t SO(n) = O(n)

n SL(n,R) w i t h W ( n ) = U(n) n SL(n,C). 2 l e a v i n g t h e f o r m x1 +

Next l e t SO(p,q) 2 + xp2 - xp+l

be t h e s e t o f mat-

...

- ... -

invarx 2 P+q i a n t and n o t e t h a t SO(3,l) i s t h e L o r e n t z group (we use t h i s i n t h e form xo 2 2 2 x1 x2 x3 i n t h e t e x t ) . We d e f i n e a l s o i n passing SP(n,R) ( r e s p . Sp(n,

r i c e s i n SL(p+q,R)

-

-

-

C ) ) t o be t h e s e t o f m a t r i c e s i n GL(2n,R)

.. . +

t h e form x1 A x ~ t+ ~

(resp. GL(2n,C)) l e a v i n g i n v a r i a n t

xn A x~~ ( r e s p . z, A z

n+l

+

... +

zn A z ~ ~ )A l .l

o f these groups w i l l be seen t o be L i e groups.

REmARK C.34. a6

-

BY =

Consider e.g.

SL(2,C)

1 (hence SL(2,C) i s a 3

-

c o n s i s t i n g o f 2 x 2 m a t r i c e s (" B, w i t h '!64 complex dimensional s u r f a c e i n C ) .

-

Since a , ~ , y , 6 cannot a l l v a n i s h t o g e t h e r one can express f o r example y as a f u n c t i o n o f a,B,6 when B # 0 and t h u s t h r e e o f t h e q u a n t i t i e s a , ~ , y , 6 w i l l s u f f i c e as c o o r d i n a t e s i n t h e NBH o f any p o i n t o f SL(2,C). sing that there i s a 2

-+

We n o t e i n pas-

1 mapping o f SL(2,C) o n t o SO(3,l) ( x and -x

t h a t SL(2,C) i s l o c a l l y isomorphic t o SO(3,l);

+

y ) so

t h e corresponding L i e algebras

APPENDIX C

sl(2,C)

and s o ( 3 , l )

371

a r e isomorphic (see below)

I n f a c t t h e L i e algebra sl(2, C ) c o n s i s t s of 2 x 2 complex m a t r i c e s w i t h t r a c e 0 f o r which a b a s i s i s X = 0 1 0 0 (o o); Y = (1 o); H = Note a l s o t h e expressions exptX = (: along

(b -01).

w i t h t h e r e l a t e d expressions exptY = (: connect sl(2,C)

p)

i)

and exptH = ( e x p t e 0x p ( - t ) ) which

and SL(2,C).

A

DEFINICI0N C.35.

t o p o l o g i c a l group i s a group G w i t h a t o p o l o g y c o m p a t i b l e

w i t h t h e group s t r u c t u r e ( t h i s means t h a t (x,y)

uous).

A

coordinates). a

xy-l: G X G

-+

G i s contin-

L i e group i s a group G which i s a r e a l ( o r complex) a n a l y t i c mani-

f o l d w i t h t h e map (x,y) (U

-+

X U ,vCr X %

-t

xy-l: G X G

-+

G a n a l y t i c (when expressed i n l o c a l

Here t h e p r o d u c t m a n i f o l d G X G i s d e f i n e d i n terms o f c h a r t s Y

J

~

)

i n t h e obvious way.

One can e a s i l y v e r i f y t h a t t h e groups o f Example C.33 a r e L i e groups. One 2 matrix

uses h e r e t h e t o p o l o g y and a n a l y t i c s t r u c t u r e determined by t h e n

components i n GL(n,R) o r GL(n,C) as l o c a l c o o r d i n a t e s ; t h i s i s t r a n s m i . t t e d t o subgroups i n t h e obvious way.

Since t h e r e s t r i c t i o n MM* = I (M* =

T or i

TM) f o r c e s t h e c o e f f i c i e n t s o f M t o l i e i n a bounded r e g i o n i n n2 dimensional r e a l o r complex Euclidean space t h e groups O(n), SO(n), U(n seen t o be c l o s e d compact subgroups o f GL(n,R) o r GL(n,C); SL(n,R),

, and SU(n) a r e SO(n), SU(n),

and U(n) a r e a l l connected w h i l e SO(n) i s t h e connected

SL(n,C),

component o f t h e i d e n t i t y i n O(n).

We mention a l s o t h e P o i n c a r g group con-

s i s t i n g o f t h e s t a n d a r d s e m i d i r e c t p r o d u c t o f SO(3,l) w i t h t h e t r a n s l a t i o n s in R

4 (inhomogeneous L o r e n t z group).

L E SO(3,l)

and a a r e a l 4 - v e c t o r where t h e p r o d u c t r u l e i s d e f i n e d by (L,a)

(L"L,

(i,b) =

Thus G c o n s i s t s o f p a i r s (L,a) w i t h

a + Lb).

Then M = G/H (H = SO(3,l))

i s c a l l e d Minkowski space.

We have discussed v e c t o r f i e l d s on a m a n i f o l d M e a r l i e r ; t h e y f o r m a module o v e r t h e r i n g Cm(M) fX(g). i n C"(M)

under t h e r u l e s ( X + Y ) ( f ) = X ( f ) t Y ( f ) and ( f X ) ( g ) =

F u r t h e r i t i s e a s i l y v e r i f i e d t h a t [X,Y]

= XY

-

YX i s a d e r i v a t i o n

whenever S and Y a r e v e c t o r f i e l d s and t h a t t h e f o l owing Jacobi

i d e n t i t y h o l d s ( e x e r c i s e ) (me) [X,[Y,Z]]

+ [Y,[Z,X]]

+ [Z,[X

Y]]

= 0.

Thus

t h e s e t o f v e c t o r f i e l d s on M forms a L i e a l g e b r a where t h i s i s d e f i n e d f o r f i n i t e dimensional s i t u a t i o n s by

A

DEFINIC10N C.36,

L i e a l g e b r a i s a ( f i n i t e d i m e n s i o n a l ) r e 1 o r complex

v e c t o r space F w i t h a b i l i n e a r o p e r a t i o n (X,Y) w i t h [X,X]

= 0.

A

morphism h: F

-+

= h[f,f']

L e t now G be a L i e group w i t h

T

*

g

such t h a t (a*) h o l d s

E o f r e a l o r complex L i e a l g e b r a s i s a

l i n e a r map such t h a t [ h f , h f ' ]

P'

3 [X,Y]

-+

f o r f , f ' E F. pg: G

+

G t h e l e f t t r a n s l a t i o n by p.

372

ROBERT CARROLL

i s an a n a l y t i c d i f f e o m o r p h i s m ( e x e r c i s e ) . P ( T ~ ) * = Trp: Tx(G) -+ Tpx(G) i s denoted a l s o by dT i n some formu P X i s a vector field, written also g X * M + T(M), t h e n g -+ ( d r ad 9' determines another v e c t o r f i e l d X on G. We say t h a t X i s l e f t i It i s e v i d e n t t h a t

T

-+

The map as and i f 'v

) X = x 9 7 P9 variant i f

u

Now g i v e n 5 E Te(G) ( e % i d e n t i t y ) one can c o n s t r u c t a unique xPg = xPg. l e f t i n v a r i a n t (C") v e c t o r f i e l d X on M such t h a t Xe = 5 . Indeed d e f i n e X

P

= (TT ) 5 and n o t i n g t h a t t h e diagram

P e

G

(C.9)

5 3P

commutes we have, by f u n c t o r i a l p r o p e r t i e s o f t h e t a n g e n t space f u n c t o r T and p r e v i o u s d e s c r i p t i o n s above) (TT ) (TT ) 5 = 9 P P e ) 5 which i s l e f t i n v a r i a n c e . To show uniqueness l e t Y be l e f t i n v a r (TT 9P e i a n t w i t h Ye = 5 which i m p l i e s (TT -1) Y = 5 ; b u t by f u n c t o r i a l p r o p e r t i e s P P P It remains t o check t h e smooth(TT ) (TT -1) = i d e n t i t y and hence Y = Xp. P e P P P Now ( r e c a l l T f = f, ness, i . e . we want t o show X f E C"(G) when f E C"(G). (see [ClY41;Lg1;Wrl]

and (f,v)(g) = v ( g o f ) ) ( X f ) ( p ) = (Xp,df) = X p ( f l p (TTp)e5(f)p = S ( f 0 and t h i s can be p u t i n t h e form (em) ( X f ) ( p ) = D t f ( p y ( t ) ) l t = O where y Tp)e i s a smooth curve, i . e . a C" map J G w i t h J an open i n t e r v a l c o n t a i n i n g -+

t h e o r i g i n , such t h a t y ( 0 ) = e and y ' ( 0 ) = y*(Dt) a n a l y t i c , depending on c o n t e x t ) .

= 5 (smooth may a l s o mean

Indeed g i v e n such a

y,

one has (&*) D t ( f o

o~ y),(Dt)(f)p = (TTp)(Y*Dt)(f)p = c ( f o T ~ ) ~ Consequently : from (&*) X i s e a s i l y seen t o be smooth as r e q u i r e d ( t h e e x i s t e n c e o f such

-tP

0

v),

=

(

T

curves y i s covered below).

We observe n e x t t h a t i f X and Y a r e l e f t i n v a r -

i a n t v e c t o r f i e l d s t h e n so i s [X,Y].

To see t h i s suppose f i r s t t h a t IP i s a

u

map G

-+

o

X(f o

cp =

N

G and w r i t e & ( X p ) = XcpcPz i n which case f o r f cp).

I t f o l l o w s t h a t X ( y f ) o IP = X ( y f o

cp)

E

C"(G)

one has ( X f )

= X ( Y ( f o I P ) ) and

U U

hence &[X,Y]

= [X,Y].

Therefore t a k i n g

i n v a r i a n t w i t h X and Y which proves

&HE@REIII C.37.

L e t G be a L i e group.

cp

T

P

we see t h a t [X,Y]

Then t h e l e f t i n v a r i a n t (C")

is left

vector

f i e l d s on G form a L i e a l g e b r a ? w h i c h induces a L i e a l g e b r a s t r u c t u r e on Te(G) under t h e r u l e [X,Y], (C")

= [5,n]

where X,Y a r e t h e unique l e f t i n v a r i a n t

v e c t o r f i e l d s a s s o c i a t e d w i t h 5 , €~ Te(G).

c i s c a l l e d the L i e algebra

o f G and we always i d e n t i f y i t w i t h Te(G) as i n d i c a t e d . EXNIIPCE C.38.

We mention here a few examples o f p a i r s (G,:);

the v e r i f i c a -

t i o n w i l l f o l l o w immediately once t h e e x p o n e n t i a l map i s d e f i n e d below.

Thus

APPENDIX C

373

( A ) G = GL(n,R) and =;

L(Rn) = g l ( R n ) = a l l l i n e a r maps Rn

SL(n,C)

= a l l n x n m a t r i c e s w i t h t r a c e 0 ( C ) G = U(n) and

and

= sl(n,C)

Ti =

N

g = u ( n ) = a l l skew h e r m i t i a n m a t r i c e s ( i . e . M t

T

N

g = so(n) = a l l skew symmetric m a t r i c e s ( i . e . M t

and

';3 =

so(p,q)

= m a t r i c e s o f t h e form M =

-+

R" (B) G =

0 ) (D) G = SO(n) and

(E)

M = 0)

G = SO(p,q)

x z ) where X1 and X3 a r e skew

($I

x 2 .'3

symmetric o f o r d e r p and q r e s p e c t i v e l y and X2 i s a r b i t r a r y ( F ) G = Sp(n,R) X = sp(n,R) = m a t r i c e s o f t h e f o r m M = (" T ~ where ~ ) X2 and X3 a r e symand

x,

m e t r i c n x n m a t r i c e s and X1 i s a r b i t r a r y .

I

L e t J be an open i n t e r v a l c o n t a i n i n g 0 and l e t G be a L i e

RRN\RK C.39,

An a n a l y t i c homomorphism (when d e f i n e d )

group.

q:

J

-+

G i s called a local

one parameter subgroup o f G ( n o t e t h a t y ( 0 ) = e); i f J = R t h e word l o c a l i s I f X i s a (C")

suppressed.

f o r X a t p i s a smooth map

v e c t o r f i e l d on G t h e n a l o c a l i n t e g r a l c u r v e

J

y:

.+

G w i t h y ( 0 ) = p and y ' ( t ) = y * ( D t l t

=

.

I n t e g r a l curves e x i s t l o c a l l y as i n d i c a t e d below. XY(t) To show t h a t l o c a l i n t e g r a l curves e x i s t l e t (U,lp) be a c h a r t a t p; t h e n i t i s easy t o see t h a t a v e c t o r f i e l d X on G can be w r i t t e n l o c a l l y as X =

1 Xia/axi

w i t h Xi

E

C"(U) and t h e e q u a t i o n y ' ( t )

a system o f o r d i n a r y d i f f e r e n t i a l equations. serve t h a t i f f

E

( f o q-')(lp(Y(t)))

Cm(G) t h e n y * ( D t ) t f =

equations a r i s i n g from Y

dxi/dt

= Xi(x,

( v ) , .

X

can be w r i t t e n as

Y(t)

= (Dt)tf(y(t)).

F ( l p ( y ( t ) ) ) so D t f ( y ( t ) ) =

1 (af/axi)(dxi/dt)

w r i t e s as

=

= Xi(t)

Thus x i ( y ( t ) )

B u t now f ( y ( t ) ) =

1 (afu/axi)(axi/at)

.. , x n ( - ) ) ,

X

Y(t) i = 1,.

which one

Consequently t h e

a c c o r d i n g t o o u r conventions.

y'(t) =

and ob-

can be expressed i n t h e f o r m ((A) Iv . . ,n, where Xi = Xi o IP -1 and an i n i Now by s t a n d a r d theorems i n ODE t h i s

t i a l condition y ( 0 ) = p i s prescribed.

Iv

system always has unique l o c a l s o l u t i o n s ; n o t e t h a t t h e C" f u n c t i o n s Xi

sat-

i s f y a L i p s c h i t z c o n d i t i o n f o r example and t h u s t h e so c a l l e d P i c a r d - L i n d e l o f f theorem a p p l i e s . (om)

I n p a r t i c u l a r , i n o r d e r t o produce t h e curves y o f

one c o u l d use any g l o b a l Cm v e c t o r f i e l d X w i t h Xe = 5 and l e t

y

be a

l o c a l i n t e g r a l c u r v e f o r X.

That such v e c t o r f i e l d s e x i s t i s e a s i l y seen.

Indeed t a k e a t r i v i a l i z a t i o n

T ~ :T(U)

and w r i t e

E

T,

6 = e

(recall 5

p i c k i n g open s e t s V c

.+

Te(G)).

c W c w c U with

-

w i t h (U,lp) a c h a r t a t e E G, -1 Then d e f i n e X = T f o r p E U and, U X Rn,

compact, l e t JI

w h i l e JI = 0 o u t s i d e o f W; t h e v e c t o r f i e l d X =

x

N

UP E

C"(G)

be 1 on

v

t h e n f u l f i l l s o u r needs.

X = (T P N ) 6 ) and check as an e x e r c i s e t h a t X i s i n f a c t a n a l y t i c ( i . e . t h e Xi i n P e an e x p r e s s i o n X = 1 xi(a/axi), Xi = xi 0 I p - l , a r e a n a l y t i c ) . L e t v t ( p ) be t h e unique l o c a l i n t e g r a l c u r v e a t p ( i . e . lpo(p) = p). By uniqueness one

Now l e t X be a l e f t i n v a r i a n t v e c t o r f i e l d on G w i t h Xe = 5 ( i . e . T

-

374

ROBERT CARROLL

has v S ( v t ( p ) )

= vstt(p)

w i t h vp_t(vt(p))

= p, whenever e v e r y t h i n g makes sense,

and again by standard theorems about ODE ( c f . [ C d l ] )

( t , p ) - + v t ( p ) i s Cm i n

f o r example w h i l e p + q t ( p ) i s a diffeomorphism o f some open NBH of say e o n t o i t s image. For s u i t a b l e t,p,g we can w r i t e i n

some open NBH of (0,e)

( 9 ) 5 (TT X = ( T o~ ~ p ~ ( g ) )and ' (9, o ~ ~ ( 9 ) =) 'X 'IP P g g o v t = v t o T by uniqueness, s i n c e t h e i n i t i a l c o n d i t i o n s P P Hence g i v e n q t ( e ) defined i n some i n t e r v a l J we can d e f i n e

an obvious n o t a t i o n t h i s means t h a t a r e t h e same.

'I

q t ( p ) f o r any p by q t ( p ) =

T~

o vt(e).

T h i s means i n p a r t i c u l a r t h a t t

-+

p t ( e ) i s a l o c a l a n a l y t i c homomorphism s i n c e one has t h e e q u a t i o n vstt(e) v s ( v t ( e ) ) = r v t ( e ) o v,(e) when t h e

'iii

are analytic

= pt(eks(e)

- c f . [Cdl]).

=

(we g e t a n a l y t i c s o l u t i o n s o f (6.) F u r t h e r we may now extend t h e d e f i n i -

t i o n o f v t ( e ) f o r example t o a l l values o f t

R u s i n g t h e group p r o p e r t y .

E

Hence t h e f o l l o w i n g d e f i n i t i o n makes sense. L e t v t ( e ) be t h e ( u n i q u e ) one parameter subgroup o f G gen-

DEFINIE10N C.40,

Then q l ( e )

e r a t e d by t h e l e f t i n v a r i a n t v e c t o r f i e l d X. exp Xe ( o r exp X) and t h e map Xe map.

W r i t i n g JI ( e ) = q

st s = 1, v t ( e ) = exp(tXe). S

Y

-f

exp Xe: g

( e ) one has JI;(e)

-f

i s d e f i n e d t o be

G i s c a l l e d t h e exponential = t X e so t h a t ,

= tp;(e)

setting

We remark t h a t when G = GL(n,R) f o r example t h e e x p o n e n t i a l map i s t h e usual exponent o f a m a t r i x ( r e c a l l g = L ( R " ) ) . subgroups of GL(n,R) above.

and t h i s p o i n t o f view serves t o c l a r i f y Example C.38

One should observe t h a t t h e e x p o n e n t i a l map does n o t always map o n t o

G even when G i s connected.

SL(2,R)

A s t a n d a r d counterexample i s p r o v i d e d by G =

( c f . [Tdl]).

EMmPLE C.41.

Consider t h e r o t a t i o n group G = SO(3) and i n

[ 00 10 -10 1; 00 0

(C.10)

Similar s i t u a t i o n s occur f o r L i e

a1 =

a2 =

[

0 0 1 0 0 0 1 ; a3 = -1 0 0

[

5=

so(3) s e t

0 -1 0 1 0 01 0 0 0

Then r o t a t i o n s o f a n g l e e around t h e xk a x i s i n R 3 a r e d e s c r i b e d by m a t r i c e s ak(e) =

Thus t h e c t k ( 6 ) a r e one

expake and e v i d e n t l y a k = (d/de)ak(e)l,,O.

parameter subgroups o f G and we have f o r example

We n o t e i n passing a l s o t h a t t h e m u l t i p l i c a t i o n t a b l e f o r S O ( 3 ) i s determined by [al,a21 = a 3 ; [a2,a31 = al; [a3,",] = a2. For t h e Lorentz group SO(3,l) we add a f o u r t h row and f o u r t h column o f zeros t o t h e "k o f (C.14) and den o t e them a g a i n by "k E so(3,l);

t h e n elements B~

E

s l ( 3 , l ) a r e d e f i n e d by

APPENDIX C

375

B1 = 1 i n t h e ( 1 , 4 ) and ( 4 , l ) p o s i t i o n s , B2 = 1 i n t h e ( 2 , 4 ) and ( 4 , 2 ) p o s i -

t i o n s , B3 = 1 i n t h e ( 3 , 4 ) and ( 4 , 3 ) p o s i t i o n s , and t h e o t h e r e n t r i e s a r e 0. One s e t s a , ( @ ) = expakg w i t h

Bk(0) =

expBke so t h a t t h e a k ( e ) a r e as b e f o r e

w i t h an a d d i t i o n a l 1 i n t h e ( 4 , 4 ) p o s i t i o n (and 0 i n o t h e r f o u r t h row and column p o s i t i o n s ) w h i l e e.g. Coshe 0 0 Sinhe )

This Page Intentionally Left Blank

377

REFERENCES

ABBREVIATIONS: LMN = L e c t u r e Notes i n Mathematics; LNP = L e c t u r e Notes i n Physics; AMS = America1 Mathematical S o c i e t y ; S I A M = S o c i e t y f o r I n d u s t r i a l and Appl i e d Mathematics; North-Hol l a n d = North-Hol l a n d Pub1 i s h e r s , Amsterdam; S p r i n g e r = S p r i n g e r Verlag, N.Y., etc.; DAN = Doklady Akad. Nauk SSSR; UMN = Uspekhi Mat. Nauk; JMP = J o u r n a l o f Mathematical Physics; CMP = Comnunicat i o n s i n Mathematical Physics; JMAA = Journal of Mathematical A n a l y s i s and A p p l i c a t i o n s ; JDE = J o u r n a l o f D i f f e r e n t i a l Equations. \

A

AB

AC AD AE AG

AH A1 AK AL

AM

AN A0

AR AS

ARNOLD, V. [l]Mathematical Methods o f C l a s s i c a l Mechanics, S p r i n g e r , 1978. - [2] O r d i n a r y D i f f e r e n t i a l Equations, MIT Press, 1981. - [3] Geometrical Methods i n t h e Theory o f O r d i n a r y D i f f e r e n t i a l Equat i o n s , S p r i n g e r , 1983. ABRAHAM, R. and MARSDEN, J. [l]Foundations o f Mechanics, Addison-Wesl e y , 1978. - [2] and RATIU, T., M a n i f o l d s , Tensor A n a l y s i s , and A p p l i c a t i o n s , Addison-Wesley, 1983 ATTOUCH, H. [l]V a r i a t i o n a l Convergence f o r F u n c t i o n s and Operators, P i t man, London, 1984 ADAMS, R. [l]Sobolev Spaces, Academic, N.Y., 1975 ABERS, E. and LEE, B. Physics Reports, North-Holland, 1 (1973), 1-141 AGMON, S. [l]L e c t u r e s on E l l i p t i c Boundary Value Problems, Van Nostrand, 1965. - [2] Ann. Scuola Norm. Sup. Pisa, 2 (1979), 151-218. - [3] Comn. Pure Appl . Math. , 39 (1986), S3-Sl6 ALSHOLM, P. and SCHMIDT, G. [l]Arch. Rat. Mech. Anal., 40 (1971),281-311 AKHIEZER, N. [l]and GLAZMAN, I.,Theory o f L i n e a r Operators i n H i l b e r t Space,Pitman, 1981. - [2] C a l c u l u s o f V a r i a t i o n s , B l a i s d e l l , 1962 I z d . Nauka, MOSCOW, ALEKSEEV, A. [l]Some Methods and A l g o r i t h m s 1967, pp. 9-84 ALBEVERIO, S. and HOEGH-KROHN, R. [l]Mathematical Theory o f Feynman Path I n t e g r a l s , S p r i n g e r LNM 523, 1976. - [2] and BLANCHARD, P., CMP, 83 (1982), 49-76 and Trends Appl. Pure Math. Mech., Pitman, 1981, pp. 1-22 AMREIN, W., JAUCH, J., and SINHA, K. [l]S c a t t e r i n g Theory i n Quantum Mechanics, Benjamin, 1977 ANGER, G. ( E d i t o r ) [l] I n v e r s e and I m p r o p e r l y Posed Problems i n D i f f e r e n t i a l Equations, Akad. Verlag, B e r l i n , 1979 ABLOWITZ, M. [l]and SEGUR, H., S o l i t o n s and t h e I n v e r s e S c a t t e r i n g Transrorm, SIAM, 1981. - [2] S t u d i e s Appl. Math., 58 (1978), 17-94. [3] and SEGUR, H., JMP, 1 6 (1975), 1054-1056. - [4] and KAUP. D, NEWELL, A., and SEGUR, H., S t u d i e s Appl. Math., 53 (1974), 249-315. - [5] and BEALS, R. and TENENBLAT, K., S t u d i e s Appl. Math., 74 (19 86), 177-203. - [6] and NACHMAN, A., S t u d i e s Appl. Math., 71 (19841, 243-250 and 251-262. - [7] and NACHMAN, A., Physica 18D (1986), 223 -241. - [8] and NACHMAN, A. and FOKAS, A., L e c t . Appl. Math., 23 (1_986), 217-222. AGRONOVIC, Z. and MAREENKO, V . [l]The I n v e r s e Problem o f S c a t t e r i n g Theory, Gordon-Breach, N.Y., 1963 AROSIO, A. [l]Ann. Math. Pura Appl., 218 (19831, 173-218

...,

ROBERT CARROLL

3 78

AT AU

AZ B

BA BB BC BD

BE BF

ALTMAN, M. [l]A U n i f i e d Theory o f N o n l i n e a r Operator and E v o l u t i o n Equations w i t h A p p l i c a t i o n s , Dekker, N.Y., 1986 AUBIN, J. [l] and EKELAND, I.,A p p l i e d N o n l i n e a r Analysis, Wiley, N.Y., 1984. - [2] A p p l i e d F u n c t i o n a l A n a l y s i s , Wiley, N.Y., 1979. - [3] and CELLINA, A., D i f f e r e n t i a l I n c l u s i o n s , Springer, 1984. AUSLANDER, L. and MACKENZIE, R. [l]I n t r o d u c t i o n t o D i f f e r e n t i a b l e Manif o l d s , McGraw H i l l , N.Y., 1963 BERS, L. [l]Mathematical Aspects o f Subsonic and Transonic Gas Dynami c s , Wiley, N.Y., 1958. - [2] and SCHECHTER, M., P a r t i a l D i f f e r e n t i a l Equations, I n t e r s c i e n c e , N.Y., 1964, pp. 131-299 BRUCKSTEIN, A. and KAILATH, T. [l]and LEVY, B., S I A M Jour. Appl. Math., 45 (1985), 312-335. - [2] S I A M Review, 29 (1987), 359-389. BERBERIAN, S. [l]L e c t u r e s i n F u n c t i o n a l A n a l y s i s and O p t i m i z a t i o n Theory, Springer, 1974 BRYtKOV, Y. and PRUDNIKOV, A. [l] I n t e g r a l Transforms o f G e n e r a l i z e d Functions, MOSCOW, 1977 BARBU, V. [l]N o n l i n e a r Semigroups and D i f f e r e n t i a l Equations i n Banach Spaces, Noordhoff, Leyden, 1976. - [2] Optimal C o n t r o l o f V a r i a t i o n a l I n e q u a l i t i e s , Pitman, 1984. - [3] and DAPRATO, G., Hamilt o n Jacobi Equations i n H i l b e r t Space, Pitman, 1983. - [4] and PRECUPANU, T., C o n v i x i t y and O p t i m i z a t i o n i n Banach Spaces, Reid e l , Dordrecht , 1986 BERGER, M. [l]N o n l i n e a r i t y and F u n c t i o n a l A n a l y s i s , Academic, 1977. 121 Contem. Math. AMS, 54 (1986), 1-8 BONA, J. [l]and WINTHER, R., S I A M Jour. Math. Anal., 14 (1983), 10561106. - [2] and SOUGANIDIS, P. and STRAUSS, W., S t a b i l i t y and I n t o appear. - [3] and SCHONBECK, s t a b i l i t y o f S o l i t a r y Waves M., Proc. Roy. SOC. Edinburgh, l O l A (1985), 207-226 BENSOUSSAN, A. and LIONS, J. [l]A p p l i c a t i o n s o f V a r i a t i o n a l Inequal it i e s i n S t o c h a s t i c C o n t r o l Theory, North-Holland, 1982 BOHM, A. [l] Quantum Mechanics, Springer, 1979 BITSADZE, A. [l] Equations o f Mixed Type, Moscow, 1959 BERTHIER, [l] S p e c t r a l Theory and Wave Operators f o r t h e Schrodinger Equation, Pitman, 1982 BAUMGARTEL, H., and WOLLENBERG, M. [l]Mathematical Theories o f S c a t t e r ing, Birkhauser, 1983 BLOOM, F. [l] I11 Posed Problems f o r I n t e g r o d i f f e r e n t i a l Equations i n Mechanics and Electromagnetic Theory, SIAM, 1981 BREMERMAN, H. [l]D i s t r i b u t i o n s , Complex V a r i a b l e s , and F o u r i e r Transforms, Addison-Wesley, 1965 BEN ISRAEL, A. and GREVILLE, T. [l] Generalized Inverses, Wiley, 1974 BOURBAKI, N. [l] Fonctions d ' u n e V a r i a b l e RCelle, Chap. 1-10, Hermann, Topologiques, Chap. 1-5, P a r i s , 1951. - [2] Espaces V e c t o r i e l s Chap. 1-7, Hermann, Hermann, P a r i s , 1953-55. - [3] Int,&yation, P a r i s , 1952-59. - [4] Topologie Generale, Chap. 1-10, Hermann, P a r i s , 1961. - [5] A l g k b r e M u l t i l i n e a i r e , Hermann, P a r i s , 1958. BURRIDGE, R. [l] Wave Motion, 2 (1980), 305-323 BJORKEN, J. and DRELL, S. [l]R e l a t i v i s t i c Quantum Mechanics and Relat i v i s t i c Quantum F i e l d s , McGraw H i l l , N.Y., 1964-65 BROWDER, F. [l]N o n l i n e a r Operators and N o n l i n e a r Equations o f E v o l u t i o n i n Banach Spaces, AMS, 1976. - [2] B u l l . AMS, 69 (1963), 862-874; Trans. AMS, 117 (1965), 530-550. - [3] B u l l . AMS, 70 (1964), 551553; B u l l . AMS, 71 (1965), 780-785; B u l l . AMS, 72 (1966), 89-95. - [4] I l l i n o i s Jour. Math., 9 (1965), 608-616 and 617-622. - [5] Annals Math., 80 (1964), 485-523 and 82 (1965), 51-87. - [6] Arch. Rat. Mech. Anal., 20 (1965), 251-258; Proc. N a t l . Acad. Sci., 56

...,

BG

BH BI BJ

BK BL BM BN BO

BP BQ BR

.

REFERENCES

BS BT BU BV

BW BX

BY BZ BEC BLE BOG BTH

BUR BOT BIR BAB C

379

(1966), 419-425 and 1080-1086; B u l l . AMS, 71 (1965), 176-183. 171 Proc. N a t l . Acad. S c i . , 53 (1965), 1272-1276 and 54 (1965), 1041-1044; Arch. Rat. Mech. Anal., 21 (1966), 259-269; B u l l . AMS, 73 (1967), 322-327 and 470-476. - [8] Proc. Symp. Pure Math., AMS, 1986, pp. 203-226. BOOS, B. and BLEECKER, D. [l]Topology and A n a l y s i s , S p r i n g e r , 1985 BOAS, R. [l]E n t i r e Functions, Academic, 1954 BOTT, R. and TU, L. [l]D i f f e r e n t i a l Forms i n A l g e b r a i c Topology, S p r i n ger, 1982. BEALS, R. [l]and COIFMAN, R., Comm. Pure Appl. Math., 37 (1984), 39-90 and 38 (1985), 29-42. - [2] and COIFMAN, R., S c a t t e r i n g and Inv e r s e S c a t t e r i n g f o r F i r s t Order Systems, 11, t o appear. - [3] and TENENBLAT, K. , I n v e r s e S c a t t e r i n g and t h e Backlund Transform. BAIOCCHI, C. [l]Rend. Sem. Mat. Padova, 35 (1965), 380-417 and 36 (19 66), 80-121. - [2] and CAPELO, A., V a r i a t i o n a l and Q u a s i v a r i a t i o n a1 I n e q u a l i t i e s , Wiley, 1984 BREZIS, H. [l]Ope'rateurs Maximaux Monotones, North-Holland, 1973. [2] D i r e c t i o n s i n PDE, Academic, 1987, pp. 17-36 BOGOLIUBOV, N. and SHIRKOV, 0. [l]I n t r o d u c t i o n t o t h e Theory o f Quant i z e d F i e l d s , I n t e r s c i e n c e , 1959 BERENS, H. and BUTZER, P. [l]Semigroups o f Operators and Approximation, Springer, 1967 BECKER, P., BOHM, M., and JOOS, H. [l]Gauge Theories o f S t r o h g and E l ectroweak I n t e r a c t i o n , Wiley, 1984 BLEECKER, D. [l]Gauge Theory and V a r i a t i o n a l P r i n c i p l e s , Addison-Wesl e y , 1981 BONGAARTS, P. [l]C W I S y l l a b i , Amsterdam, 1985, pp. 1-70 BOOTHBY, W. [l]An I n t r o d u c t i o n t o D i f f e r e n t i a b l e M a n i f o l d s and Riemann i a n Geometry, Academic, 1975 BURR, J. and DEKERF, E. [l]C W I S y l l a b i , Amsterdam, 1985, 91-113 BOITI, M. [l]and TU, G., Nuovo Cimento, 718 (1982), 253-264. - 12) and KONOPELEENKO, B. , and PEMPINELLI, F. , I n v e r s e Prob., 1 (1985), 3333- 56 BIRNIR, B. [l] Physica 19D (1986), 238-254 BABIt, A. and V I S I K , M. [l]Pitman Research Notes 122, 1985, pp. 11-34 CARROLL, R. [l]A b s t r a c t Methods i n PDE, Harper-Row, 1969. - [2] Transm u t a t i o n , S c a t t e r i n g Theory, and S p e c i a l Functions, North-Holland, 1982. - [3] Transmutation Theory and A p p l i c a t i o n s , North-Holland, 1985. - [4] and SHOWALTER, R., S i n g u l a r and Degenerate Cauchy Problems, Academic, 1976. - [5] Ann. Math. Pura Appl., 56 (1961), 1-31. - [6] and RAPHAEL, L., S I A M Jour. Appl. Math., t o appear. [7] and SANTOSA, F . , Math. Meth. Appl. Sci., 3 (1981), 145-171. [8] and SANTOSA, F., Conf. I n v e r s e S c a t t e r i n g , SIAM, 1983, pp. 230-244. - [9] and SANTOSA, F., Jour. Acous. SOC. Amer., 76 (1984), 935-941. - [ l o ] A p p l i c a b l e Anal., 22 (1986), 21-43. - [ll]Oakland Conf. PDE, Pitman, 1987, pp. 1-38. - [12] and GLICK, A., Arch. Rat. Mech. Anal., 16 (1964), 373-384. - [13] Acta Applicandae Math., 6 (1986), 109-184. - [14] S p r i n g e r LNM 1223, 1986, pp. 25-36. [15] and DELIC, G., Math. Meth. Appl. Sci., t o appear. - [16] and BERKOPEC, T., Funkc. Ekvac., 29 (1986), 281-298. - 1171 and DOLZYCKI, S., A p p l i c a b l e Anal., 19 (1985), 189-200. - [18] A p p l i c a b l e Anal., 26 (1987), 61-85. - [19] Math. Meth. Appl. S c i . , 6 (1984), 467-495. - [20] Rocky Mount. Jour. Math., 12 (1982), 393-427. [21] A p p l i c a b l e Anal., 18 (1984), 39-54. - [22] JMAA 92 (1983), 410-426. - [23] Proc. Roy. SOC. Edinburgh, 91A (1982), 315-334. [24] Comm. PDE, 6 (1981), 1407-1427. - [25] Anais Acad. B r a s i l ,

380

CA

CB

cc

CD CE

CF CG CH CI

CJ

CL CM CN

co CP CR

cs

CT

cu cw

cx

ROBERT CARROLL Cienc., 54 (1982), 271-280. - [26] Rend. Accad. L i n c e i , 72 (1982), 65-70. - [27] Jour. Math. Pures Appl., 63 (1984), 1-14. - [28] A p p l i c a b l e Anal., 17 (1984), 217-226. - [29] and DOLZYCKI, S . , JMP, 25 (1984), 91-93. - [30] B o l l . Un. Mat. I t a l . , 58 (1986), 465-486. - [31] and SANTOSA, F., Math. Meth. Appl. Sci., 4 (1982), 33-73. - [32] Osaka Jour. Math., 1 9 (1982), 815-831 and 833-861. - [33] A p p l i c a b l e Anal., 16 (1983), 85-90. - [34] and DOLZYCKI, S . , A p p l i c a b l e Anal. , 23 (1986), 185-208. - [35] Forced N o n l i n e a r Schrodinger Equations, i n p r e p a r a t i o n . - [36] CR Roy. SOC. Canada, 9 (1987), 237-242. - [37] Proc. Conf. D i f f . Eqs., Ohio Univ., 1987, t o appear. - [38] Proc. Symp. Howard Univ., 1987, t o appear. [39] and COOPER, J., Math. Annalen, 188 (1970), 143-164. - [401 and STATE, E., Canad. Jour. Math., 23 (1971), 611-626. - [41] I n t r o d u c t i o n t o L i e Theory, Lectures, Univ. I l l i n o i s , 1970. - [42] I n v e r s e S c a t t e r i n g and Forced N o n l i n e a r Systems, t o appear. - [431 and WANG, C., Canad. Jour. Math., 17 (1965), 245-256. CARATHEODORY, C. [l]V a r i a t i o n s r e c h n u n g und Partielledifferentialgleichungen E r s t e r Ordnung, L e i p z i g , 1956. CHEBLI, H. [l]Jour. Math. Pures t p p l . , 58 (1979), 1-19. CARTAN, H. and SCHWARTZ, L. [l]Seminaire, P a r i s , 1963-64. CODDINGTON, E. and LEVINSON, N. [l]ODE, McGraw-Hill, N.y., 1956 CHERN, S. [l]Lectures, Univ. Chicago, 1959. - [2] and PENG, C., Manuscripts Math. , 28 (1979), 207-217. CANNON, J. [l]The 1-D Heat Equation, Addison-Wesley, 1984 CHANG, T. and L I , L. [l]Gauge Theory o f Elementary P a r t i c l e s , Oxford, 1984 CHADAN, K. and SABATIER, P. [l]I n v e r s e Problems i n Quantum S c a t t e r i n g Theory, Springer, 1977 CHAICHNAN, M. and NELIPA, N. [l] I n t r o d u c t i o n t o Gauge F i e l d Theories, Springer, 1984 CALOGERO, F. and DEGASPERIS, A. [l]S p e c t r a l Transform and S o l i t o n s , I, North-Hol land, 1982 CLARKE, F. [l] O p t i m i z a t i o n and Nonsmooth A n a l y s i s , Wiley, 1983 CESARI, L. [I] O p t i m i z a t i o n Theory and A p p l i c a t i o n s , Springer, 1983 CONTI, R. and SANSONE, G. [l]N o n l i n e a r D i f f e r e n t i a l Equations, Permagon, N.Y., 1984 COURANT, R. [l]and HILBERT, D., Methods o f Mathematical Physics, 1,2, W i l e y - I n t e r s c i e n c e , N.Y., 1953-1962. - [2] D i r i c h l e t ' s P r i n c i p l e , I n t e r s c i e n c e , 1950. - [3] D i f f e r e n t i a l and I n t e g r a l Calculus, I n t e r s c i e n c e , 1936-37. COOPER, J. [l]JMAA, 49 (1975), 130-153; JMAA, 36 (1971), 151-171; JDE, 9 (1971), 453-495. - [2] and STRAUSS, W., JDE, 52 (1984), 175-203; Jour. F n l . Anal., 47 (1982), 180-229. CORBEN, H. and STEHLE, P. [l]C l a s s i c a l Mechanics, Wiley, 1950. CARASSO, H. and STONE, A. [l]I m p r o p e r l y Posed Boundary Value Problems, Pitman, 1975 COLTON, D. [l]and KRESS, R., I n t e g r a l Equation Methods i n S c a t t e r i n g Theory, Wiley, 1983. - [2] and MONK, P. and SANTOSA, F., Oakland Conf. PDE, Pitman, 1987, pp. 39-73. CURTIS, W. and MILLER, F. [l]D i f f e r e n t i a l M a n i f o l d s and T h e o r e t i c a l Physics, Academic, 1985 CONSTANTIN, P. [l]and FOIAS, C., Corn. Pure Appl. Math., 38 (1985), 127. - [2] and FOIAS, C. and TEMAM, R., Memoir 314, AMS, 1985. [3] and FOIAS, C., and TEMAM, R., and MANLEY, O., J o u r . F l u i d Mech., 150 (1985), 427-440 CRANDALL, M. [l]Proc. Symp. Pure Math., AMS, 1986, pp. 305-337

REFERENCES

CZ D

DC

DE DF

DG DI DJ DK DL DM DN

DO DP

DX DS DT DU

DV

DY DZ DAU E

EA EL F FA

38 1

CHAZARIN, J. and, PIRIOU, A. [l] I n t r o d u c t i o n l a T h e o r i e des i q u a t i o n s aux Derivees P a r t i e l l e s , G a u t h i e r - V i l l a r s , P a r i s , 1981 DYM, H. [l]and MCKEAN, H., F o u r i e r S e r i e s and I n t e g r a l s , Academic, 1972. - [2] and IACOB, A., Topics i n Op. Th., Birkhauser, 1984, pp. 141-240. - 131 and ALPAY, D., I n t e g . Eqs. Op. Th., 7 (1984), 589641 and 8 (1985), 145-180. - [4] and MCKEAN, H., Gaussian Processes, F u n c t i o n Theory, and t h e I n v e r s e S p e c t r a l Problem, Academic, 1976. - [5] and IACOB, A., T o e p l i t z Centen., B i r k h a u s e r , 1982, pp. 233-260 DODD, R., EILBECK, J., GIBBON, J., and MORRIS, H. [l]S o l i t o n s and Non1 i n e a r Waves, Academic, 1983 DEVANEY, R. [l]An I n t r o d u c t i o n t o Chaotic Dynamical Systems, Benjamin, 1986 DEIFT, P. and TRUBOWITZ, E. [l]Corn. Pure Appl. Math., 32 (1979), 121251 DOUGLAS, R. [l]Banach Algebra Techniques i n Operator Theory, Academic, 1972. - [2] C* Algebra Extensions and K-Homology, P r i n c e t o n , 1980 DIAZ, J. [l]A. W e i n s t e i n Selecta, Pitman, 1978 DITTRICH, W. and ROUTER, M. [l] S e l e c t e d Topics i n Gauge Theories, S p r i n g e r , 1986 DERRICK, G. [l]JMP, 5 (1964), 1252-1254 DOLLARD, J. [l]Rocky Mount. Jour. Math., 1 (1971), 5-88. - [2] and FRIEDMAN, C., Product I n t e g r a t i o n , Addison-Wesley, 1979. DEIMLING, K. [l]N o n l i n e a r F u n c t i o n a l A n a l y s i s , S p r i n g e r , 1985. - [2] ODE i n Banach Space, S p r i n g e r LNM 596, 1977. DIEUDONNE, J. [l]Foundations o f Modern A n a l y s i s , Academic, 1960 DOOB, J. [l]S t o c h a s t i c Processes, Wiley, 1953. DAPRATO, G. [l]A p p l i c a t i o n s C r o i s s a n t s e t Equations d ' E v o l u t i o n dans l e s Es aces de Banach, Academic, 1976. DIXMIER, J. E l ] Les Algebres d'Op4rateurs dans 1 'Espace H i l b e r t i e n , 1957 DEANS, S. [l]Theory of t h e Radon Transform and some o f i t s A p p l i c a t i o n s , Wiley, 1983 DALETSKIJ, Y. and KREIN, M. [l]S t a b i l i t y o f S o l u t i o n s o f D i f f e r e n t i a l Equations i n Banach Spaces, AMS T r a n s l a t i o n 43, 1974 DUNFORD, N . and SCHWARTZ, J. [l] L i n e a r Operators, 1-3, I n t e r s c i e n c e Wiley, 1958, 1963, 1971. DUVAUT, G. and LIONS, J. [l]Les InGquations en Mechanique e t en Physique, Dunod, 1972 DUPONT, J. [l]C u r v a t u r e and C h a r a c t e r i s t i c Classes, S p r i n g e r , 1978 DUBROVIN, B. and NOVIKOV, S . [l]and FOMIN, A., Modern Geometry, 1-2, S p r i n g e r , 1985. - [2] and MATVEEV, V., UMN 31 (1976), 55-136 DAUBECHIES, I.and KLAUDER, J. [l]JMP, 26 (1985), 2239-2256 EKELAND, I.and TEMAM, R. [l]Convex A n a l y s i s and V a r i a t i o n a l Problems, North-Holland, 1976 EARDLEY, D. and MONCRIEF, V . [l]CMP, 83 (1982), 171-191 and 193-212. EILENBERG, S. and STEENROD, N. [l]Foundations o f A l g e b r a i c Topology, P r i n c e t o n , 1952 FELSAGER, B. r l l Geometry, P a r t i c l e s , and F i e l d s , Odense Univ., 1981 FADEE , i. [ l j UMN, 14 ( i 9 5 9 ) , 57-119; SOV. ProbI Mat., 31 (1974), 9317 (1967), 323-350. - [3] and 180. - [2] Trudy Mosk. Mat. Ob3;., TAKHTAJAN, L., H a m i l t o n i a n Methods i n t h e Theory o f S o l i t o n s , Springer, 1987. - [4] and SLAVNOV, A., Gauge F i e l d s Benjamin, 1980. - [5] and KOREPIN, V., Physics Reports 42 (1978), 1-87. DEFIL PPO, S. [l]and LANDI, G., Quantum F i e l d Theory, E l s e v i e r , 1986, pp. 289-304. - [2] and MARMD, G., and VILASI, G., Quantum F i e l d Theory, E l s e v i e r , 1986, pp. 271-283.

...,

FB

382

ROBERT CARROLL

FREED, D, and UHLENBECK, K. [l]I n s t a n t o n s and 4 M a n i f o l d s , Springer, 1984 FE FELLER, W. [l]An I n t r o d u c t i o n t o P r o b a b i l i t y Theory 1-2, Wiley, 1950, 1966 FK FRIEDRICHS, K. [l]P e r t u r b a t i o n o f Spectra i n H i l b e r t Space, AMS, 1965 FL FLETT, T. [l] D i f f e r e n t i a l A n a l y s i s , Cambridge, 1980 FO FOIAS, C. [l]and TEMAM, R., Jour. Math. Pure Appl., 58 (1979), 339-368. - [2] and TEMAM, R. and SELL, G., I M A P r e p r i n t 234, 1986. - [3] and TEMAM, R. and SELL, G. and NICOLAENKO, B., IMA P r e p r i n t 279, 1986. - [4] and TEMAM, R. and MANLEY, O., Phys. F l u i d s , 30 (1987), 2007-2020. - [5] and TEMAM, R., D i r e c t . i n PDE, Academic, 1987, pp. 55-73. - [6] and PRODI, G., Rend. Sem. Mat. Padova, 39 (1967), 1-34. - [7] and SELL, G. and T I T I , E., Phase l o c k i n g and approx. o f i n e r t i a l m a n i f o l d s . . , t o appear. FP FARIS, W . [l] S e l f a d j o i n t o p e r a t o r s , S p r i n g e r , 1975 FR FRIEDMAN, A. [l]Generalized Functions and PDE, P r e n t i c e - H a l l , 1963. [2] PDE o f P a r a b o l i c Type, P r e n t i c e - H a l l , 1964. - [3] D i r e c t . i n PDE, Academic, 1987, pp. 75-88. FS FLANDERS, H. [l] D i f f e r e n t i a l Forms, Academic, 1963 FT FATTORINI, H. [l]The Cauchy Problem, Addison-Wesley, 1983.- [2] Second Order L i n e a r D i f f e r e n t i a l Equations i n Banach Space, North-Holland, 1985 FU FUCIK, S. [l]S o l v a b i l i t y o f N o n l i n e a r Equations and Boundary Value Problems, Reidel, 1980 FY FEYNMAN, R. and HIBBS, A. [l]Quantum Mechanics and Path I n t e g r a t i o n , McGraw-Hi 11 , 1965 G GELFAND, I. [l]and FOMIN, S., C a l c u l u s o f V a r i a t i o n s , P r e n t i c e - H a l l , 1963. - [2] and SILOV, G., Generalized Functions, Vol. 1-3, Moscow, 1958. GA GARDING, L. [l]Acta Math., 85 (1951), 1-62 GB GARABEDIAN, P. [l]PDE, Wiley, 1964 GC GOLO, V., MONASTYRSKY, M., and NOVIKOV, S. [l] CMP, 69 (1979), 237-246 GD GHIDAJLIA, J. [l]and TEMAM, R., Jour. Math. Pures Appl., 66 (19861, 273-319. - [2] F i n i t e Dimensional Behavior ..., t o appear GE GREENBERG, and HARPER, J. [l]A l g e b r a i c Topology, Benjamin, 1981 GF GLASSEY, R. [l]Math. Z e i t . , 177 (1981), 323-340 and 178 (1981), 233261. - [2] and STRAUSS, W., CMP, 65 (1979), 1-13; 67 (1979), 5167; 81 (1981), 171-187; 89 (1983), 465-482. GG GUNNING, R. and ROSSI, H. [l]A n a l y t i c Functions o f Several Complex Vari a b l e s , P r e n t i c e - H a l l , 1965 GH GINEBRE, J. and VELO, B. [l]CMP, 82 (1981), 1-28 G I GILBARG, D. and TRUDINGER, N. [l]E l l i p t i c PDE o f Second Order, S p r i n g e r , 1983 GJ GILL, T. and ZACHARY, W. [l]JDE, t o appear. - [2] Phys. Rev. L e t t . , t o appear. GK GUCKENHEIMER, J. and HOLMES, P. [l]N o n l i n e a r O s c i l l a t i o n s , Dynamical Systems, and B i f u r c a t i o n o f Vector F i e l d s , Springer, 1983 GL GLIMM, J. and JAFFE, A. [l]Quantum Physics, Second E d i t i o n , Springer, 1987 GM GOMES, A. [l]Equa@es D i f e r e n c i a i s e Semigrupos UFRJ, Textos 15, R i o de Janeiro, 1982 GN GRANAS, A. [I]Rozprzwy Mat., Warsaw, 1962 GO GOLDSTEIN, H. [l]C l a s s i c a l Mechanics, Addison-Wesley, 1980 GP GOPINATH, B. and SONDHI, M. [l]Jour. Acous. SOC. Amer., 49 (1971), 1867 -1873; Proc. IEEE, 59 (1971), 383-392 GIL GILBERT, R. [I] F u n c t i o n t h e o r e t i c methods i n PDE, Academic, 1969

FD

...,

.

...,

REFERENCES

383

GQ GODEMENT, R. [l]T h i o r i e des Faisceaux, Hermann, P a r i s , 1958 GR GRISVARD, P. [l]E l l i p t i c Problems i n Nonsmooth Domains, Pitman, 1985 GS GOLDSTEIN, J. [l]Semigroups o f L i n e a r Operators and A p p l i c a t i o n s , Oxf o r d , 1985 GT GROTHENDIECK, A. [l]Memoir 16, AMS, 1955 GU GUILLEMIN, V. and STERNBERG, S . [l]Symplectic Techniques i n Physics, Cambridge, 1984. - [2] Geometric Asymptotics, AMS, 1977, GW GLOWINSKI, R., LIONS, J., and TREMOLIERES, R. [l]Analyse Numerique des I n d q u a t i o n s V a r i a t i o n n e l l e s , Dunod, P a r i s , 1976 GV GERVER, M. [l]Geophys. Jour. Roy. A s t r . SOC., 21 (1970), 337-357 GX GREEN, M . , SCHWARTZ, J. and WITTEN, E. [l]S u p e r s t r i n g Theory, Cambridge Univ. Press, 1987 GY GILKEY, P. [l]I n v a r i a n t Theory, t h e Heat Equation, and t h e A t i y a h - S i n g e r Index Theorem, P u b l i s h - P e r i s h , 1984. - [2] and EGUCHI, T. and HANSON, A., Physics Reports 66, North-Holland, 1980, pp. 213-393 GZ GROETSCH, C. [l]The Theory o f Tikhonov R e g u l a r i z a t i o n , Pitman, 1984. [2] G e n e r a l i z e d I n v e r s e s o f L i n e a r Operators, Dekker, 1977 GER GERDZHIKOV, V., IVANOV, M., and KULISH, P. [l]Teor. Mat. F i z . , 44 ( 1 980) , 342-357 GUT GUTKIN, E. [l]Ann. I n s t . H. Poincar6, 3 (1986), 285-314; Advances Math., 6 (1985), 413-421; Annals Physics, 176 (1987), 22-48 H HORMANDER, L. [l]The A n a l y s i s o f L i n e a r PDO, 1-4, Springer, 1983-85. HA HUANG, K. [l]Quarks, Leptons, and Gauge F i e l d s , World Sci., Singapore, 1982 HB HOLMAN, H. [l]Vorlesung uber Faserbundel, Munster, 1965 HC HALE, J . [l]and MARTINEZ, P. (Eds.), Pitman Research Notes 132, 1985. - [2] and MAGALHAES, L. and OLIVA, W., An I n t r o d u c t i o n t o I n f i n i t e Dimensional Dynamical Systems - Geometric Theory, S p r i n g e r , 1984 HD HERMAN, R. [l]Geometry, Physics, and Systems, Dekker, 1973. - [2] Cart a n i a n Geometry, N o n l i n e a r Waves, and C o n t r o l Theory, A-By Math. S c i . Press, 1979-80. HE HESTENES, M. [l]C a l c u l u s o f V a r i a t i o n s and Optimal C o n t r o l Theory, K r i eger , 1980 HG HELGASON, S . [l]The Radon Transform, Birkhauser, 1980. - [2] Groups and Geometric A n a l y s i s , Academic, 1984. - [3] D i f f e r e n t i a l Geometry, L i e Groups, and Symmetric Spaces, Academic, 1978. HH HUGHSTON, L. and WARD, R. [l]Advances i n T w i s t o r Theory, Pitman, 1979 HI HIRSH, M. and SMALE, S . [l]D i f f e r e n t i a l Equations, Dynamical Systems, and L i n e a r Algebra, Academic, 1974 HK HOCKING, J. and YOUNG, G. [l] Topology, Addison-Wesley, 1961 HL HELLWIG, G. [l]Partielledifferentialgleichungen, Teubner, 1960. - [2] D i f f e r e n t i a l o p e r a t o r e n d e r Mathematische Physik, S p r i n g e r , 1964 HO HOCHSTADT, H. [l]D i f f e r e n t i a l Equations, Dover, 1975 HP HILLE, E. and PHILLIPS, R. [l]F u n c t i o n a l A n a l y s i s and Semigroups, AMS C o l l o g . Pub. 31, 1957 HT HURT, N. [I]Geometric Q u a n t i z a t i o n i n A c t i o n , R e i d e l , 1983 HU HUSEMOLLER, D. [l]F i b r e Bundles, McGraw-Hill, 1966 HV HORVATH, J. [l]T o p o l o g i c a l Vector Spaces, Addison-Wesley, 1967 HW HOWARD, M. [l]Geophysics, 48 (1983), 163-170 HY HARDY, G., LITTLEWOOD, J, and POLYA, G. [l]I n e q u a l i t i e s , Cambridge, 1934 I IOFFE, A. and TIKHOMIROV, V. [l]Theory o f Extremal Problems, North-Holland, 1979 I K IKEBE, T. [l]Arch. Rat. Mech. Anal., 5 (1960), 1-34 I T ITZYKSON, C. and ZUBER, J. [l]Quantum F i e l d Theory, McGraw-Hill, 1980 J JOHN, F. [l]PDE, S p r i n g e r , 1982

384

JA JB

JK JO K

KA

KB KD KE KF KH

KL KN KC KG

KP

ROBERT CARROLL

JAFFE, A. and TAUBES, C. [l]V o r t i c e s and Monopoles, Birkhauser, 1980 JORGENS, K. and WEIDEMAN, J . [l]S p e c t r a l P r o p e r t i e s o f L i n e a r Operat o r s , S p r i n g e r , 1973 JACKIW, R., TREIMAN, S., ZUMINO, B., and WITTEN, W. [l]C u r r e n t A l g e b r a and Anomal ies , P r i n c e t o n , 1 985 JONES, D. [l The Theory o f Generalized Functions, Cambridge, 1982 KNOWLES, G. 11 An I n t r o d u c t i o n t o A p p l i e d Optimal C o n t r o l , Academic, 1981 KATO, T. [l]Math. Z e i t . , 187 (1984), 471-480. - [ Z ] P e r t u r b a t i o n Theory f o r L i n e a r Operators, S p r i n g e r , 1976. - [3] Proc. Symp. Appl. Math., AMS, 1965, pp. 50-67. - [4] and KURODA, S., Rocky Mount. Jour. Math., 1 (1971), 127-171. - [5] and KURODA, S., Funct. Anal. and R e l a t e d F i e l d s , S p r i n g e r , 1970, pp. 99-131. - [6] Manuscripta Math,., 28 (1979), 89-99. - [7] and MASUDA, K., Ann. I n s t . H. Poincare, 3 (1986), 455-467. KINDERLEHRER, D. and STAMPACCHIA, G. [I]An I n t r o d u c t i o n t o V a r i a t i o n a l I n e q u a l i t i e s and t h e i r A p p l i c a t i o n s , Academic, '1980 KURODA, S. [l]An I n t r o d u c t i o n t o S c a t t e r i n g Theory, Aarhus, 1978 KELLEY, J. [l]General Topology, Van Nostrand, 1955 DEKERF, E. [l]C W I S y l l a b i , Amsterdam, 1985, pp. 71-89; 1980, pp. 1-58 KOBAYASHI, S. and NOMIZU, K. [l]Foundations o f D i f f e r e n t i a l Geometry, I n t e r s c i e n c e , 1963 KAILATH, T. [l]L e c t u r e s on Wiener and Kalman F i l t e r i n g , Springer, 1981 KNOPS, R. [l]Sym osium on Nonwellposed Problems, Springer, 1373 KLEINERMAN, S. [ l y D i r e c t i o n s PDE, Academic, 1987, pp. 113-143; Corn. Pure Appl. Math., 38 (1985), 631-641; L e c t . Appl. Math. AMS, 1986, pp. 293-326 KON, M. and RAPHAEL, L. [l]Some Negative Lp R e s u l t s f o r E i g e n f u n c t i o n Expansions t o appear. - [2] and YOUNG, J., On R e l a t i n g Genera l i z e d Expansions t o F o u r i e r I n t e g r a l s , t o appear. KAUP, D. [l]JMAA, 54 (1976), 849-864. - [2] and NEWELL, A., Advances Math., 37 (1976), 67-100. - [3] L e c t . Appl. Math., AMS, 1986, PP. [5] Jour. Math. 195-215. - [4] Physica 25D (1987), 361-368. Physics, 25 (1984), 277-281. - [6] Wave Phenomena, E l s e v i e r , 1984, pp. 163-174. - [7] Advances i n N o n l i n e a r Waves, Pitman, 1984, pp. 197-209. - [8] S I A M Jour. Appl. Math., 31 (1976), 121-133. - [ 9 1 [ l o ] and NEWELL, A., and NEUBERGER, H., JMP, 25 (1984), 282-284. L e t t . Nuovo Cimento, 20 (1977), 325-331; JMP, 19 (1978), 798-801. - [ll]and HANSEN, P., Physica 18D (1986), 77-84 and 25D (19871, 369-381. KREIN, S. [l]L i n e a r D i f f e r e n t i a l Equations i n Banach Space, MOSCOW, 1967 KRASNOSELSKIJ, M. [l]T o p o l o g i c a l Methods i n t h e Theory o f N o n l i n e a r I n t e g r a l Equations, MacMillan, 1964. - [2] and ZABREIKO, P., Geome t r i c a l Methods o f N o n l i n e a r Analysis, Springer, 1984 KANTOROVIE, L. and AKHILOV, G. [l]F u n c t i o n a l A n a l y s i s , Permagon, 1982 KUPERSHMIDT, B. [l]D i s c r e t e Lax Equations.. , , Aste'risque, 1985. - [2] Physica 27D (1987), 294-310. KOORNWINDER, T. [l]Ark. Mat., 1 3 (1975), 145-159. KAY, I. and MOSES, H. [l]I n v e r s e S c a t t e r i n g Papers, Math. S c i . Press, 1982 KULISH, P. and SKLYANIN, E. [l]S p r i n g e r LNP 151, 1981, pp..61-119 LANCZOS, C. [l]The V a r i a t i o n a l P r i n c i p l e s o f Mechanics, Univ. Toronto Press, 1949 LEVITAN, B. [l]I n v e r s e S t u r m - L i o u v i l l e Problems, MOSCOW, 1984. - [21 and SARGSYAN, I., I n t r o d u c t i o n t o S p e c t r a l Theory.. , MOSCOW,

2

...,

-

KR KS

KT KU KW KY

KZ L

LA

.

REFERENCES

LB LC LD LE LF LG LI

LJ LL

LM LN LO LP LS LV

LX

LY LZ M

MA

38 5

1970. - [3] The Theory o f G e n e r a l i z e d T r a n s l a t i o n Operators, Moscow, 1973. LIBERMANN, P. and MARLE, C. [l]S y m p l e c t i c Geometry and Mechanics, Reid e l , 1987. LAPIDUS, M. [l]S t u d i e s Appl. Math., 76 (1987), 93-132; Jour. F n l . Anal., 63 (1985), 261-275; I n t e g . Eqs. Op. Th., 8 (1985), 36-62. - [2] and JOHNSON, G., AMS Memoir 351, 1986. LADYZENSKAYA, 0. [l]Mathematical Q u e s t i o n s o f t h e Dynamics o f Viscous I n c o m p r e s s i b l e F l u i d s , MOSCOW, 1961. - [2] and URALTZEVA, N., L i n e a r and Q u a s i l i n e a r Equations o f E l l i p t i c Type, MOSCOW, 1964. LEE, E. and MARKUS, L . [l]Foundations o f Optimal C o n t r o l Theory, Wiley, 1968. LERNER, D, and SOMMERS, P. [l] Complex M a n i f o l d Techniques i n T h e o r e t i c a l Physics, Pitman, 1979 LANG, S. [l]D i f f e r e n t i a b l e M a n i f o l d s , Wily, 1962 LIONS, J. [l]Equations D i f f e r e n t i e l l e s O p e r a t i o n n e l l e s , SpTinger, 1961. - [2] and MAGENES, E., Problhmes aux Limite: Nonhomogene e t A p p l i c a t i o n s , Duno;, P a r i s , 1968-70. - [3] C o n t r o l e Optimal de Systemes Gouvernes p a r des Equations aux Deyivbes P a r t i $ l l e s , DunodG a u t h j e r - V i l l a r s , 1968. - [4] Quelques Methodes de R e s o l u t i o n des Problemes aux L i m i t e s N o n l i n e h i r e s , Dunod, 1969. - [ 5 1 P e r t u r b a t i o n S i n g u l l e r e des ProblPmes aux L i y i t e s e t en C o n t r o l e Optimal. - [6] C o n t r o l e des Systemes D i s t r i b u e s S i n g u l i e r s . - [7] L e c t u r e s 09 E l l i p t i c PDE, Tata I n s t . , Bpmbay, 1957. - [8] and LATTES, R., Methode de Q u a s i - r e v e r s i b i l i t e , Dunod, 1967. - [9] B u l l . SOC. Math. France, 84 (1956), 9-95. LIONS, P. [l]G e n e r a l i z e d S o l u t i o n s o f Hamilton-Jacobi Equations, Pitman, 1982; D i r e c t i o n s i n PDE, Academic, 1987, pp. 145-158. LANDAU, L. and LIFSHITZ, E. [l]C l a s s i c a l Theory o f F i e l d s , Permagon, 1975. - [2] Mechanics, Permagon, 1976. - [3] Quantum Mechanics, Permagon, 1977. - [4] Quantum Electrodynamics, Permagon, 1982. [5] S t a t i s t i c a l Physics, Permagon, 1980. LAMB, G. [l]Elements o f S o l i t o n Theory, Wiley, 1980. LUNEBERGER, D. [l]O p t i m i z a t i o n by Veqtor Space Methods, Wiley, 1969. LOEFFEL, J. [l]Ann. I n s t . H. Poincare, 8 (1968), 339-447. LOPES, J. [l]Gauge F i e l d Theory, Permagon, 1981 LOOMIS, L . and STERNBERG, S. [l]Advanced Calculus, Addison-Wesley, 1968 LAVRENTIEV, M. [l]Some I m p r o p e r l y Posed Problems o f Mathematical Phys i c s , S p r i n g e r , 1967. - [2] and VASILIEV, L. and ROMANOV, V., Mult i d i m e n s i o n a l I n v e r s e _ P r o b lems f o r D i f f e r e n t i a l Equations, S p r i n ger, 1970. - [3] and SISATSKIJ, S. and ROMANOV, V., Nonwellposed Problems i n Mathematical Physics and A n a l y s i s , Moscow, 1980 and AMS T r a n s l a t i o n , Vol. 64, 1986. LAX, P. [l]H y p e r b o l i c Systems o f Conservation Laws and Mathematical Theory o f Shock Waves, SIAM, 1973. - [2] and PHILLIPS, R., Scatt e r i n g Theory, Academic, 1967. - [3] and PHILLIPS, R., S c a t t e r i n g Theory f o r Automorphic F u n c t i o n s , P r i n c e t o n , 1976. - [4] and PHILLIPS, R., B u l l . AMS, 2 (1980), 261-295. - [5] Comn. Pure Appl. Math., 21 (19681, 467-490. LEVY, B. and YAGLE, A ; . [ l ] Acta Applicandae Math , 3 (1985), 255-289. [2] Jour. Acous. SOC. Amer., 76 (1984), 30 -308. LEE, T. [l]P a r t i c l e Physics and I n t r o d u c t i o n t o F i e l d Theory, Harwood, 1981. MCSHANE, E. [l]I n t e g r a t i o n , P r i n c e t o n , 1947. - 2 1 U n i f i e d I n t e g r a t i o n , Academic, 1983. MIRANDA, C. [l]PDE of E l l i p t i c Type, S p r i n g e r , 970

386

MB Mc

MD ME MF MG MI MJ MK ML MM MN MO MR

MS

ROBERT CARROLL

MARKETT, C. [l] Indag., 87A (1984), 299-313; I n t . S e r i e s Num. Math., Birkhauser, 1984, pp. 449-462; H a b i l i t a t i o n s c h r i f t , Aachen, 1985. MCLINDEN, L. [l]Thesis, Univ. Washington, 1971; N o n l i n . Anal., 6. (1982), 189-196. MANDL, F. and SHAW, G. [l]Quantum F i e l d Theory, Wiley, 1984. MENDEL, J. [l]Optimal Seismic Deconvol u t i o n , Academic, 1983 MACLANE, S. and BIRKHOFF, G. [l]Algebra, MacMillan, 1967 MALLET PARET, J. and SELL, G. [I] I M A P r e p r i n t 331, 1987 MIZOHATA, S. [l]B u l l . SOC. Math. France, 85 (1957), 15-50 MAJDA, A. [l]Compressible F l u i d Flow S p r i n g e r , 1984 MIKHLIN, S. [l]L i n e a r Equations o f Math. Physics, MOSCOW, 1964 MALGRANGE, B. [l]Ann. I n s t . F o u r i e r , 6 (1955/56), 271-355. MULDOWNEY, P. [l]A General Theory o f I n t e g r a t i o n i n F u n c t i o n Spaces, Pitman, 1987. MELLIN, A. [l]Journges Eqs. Der. Part., Saint-Jean de Monts, 1986. [2] I n t e r t w i n i n g Methods i n Mu1 t i d i m e n s i o n a l S c a t t e r i n g Theory, Univ. Lund, 1987 MORSE, P. and FESHBACH, H. [l]Methods o f T h e o r e t i c a l Physics, McGrawH i l l , 1953 MARtENKO, V. [l]S t u m - L i o u v i l l e o p e r a t o r s and t h e i r A p p l i c a t i o n s , Moscow, 1977. - [2] N o n l i n e a r Equations and Operator Algebras, Kiev, 1986 MOSES, H. [l]Stud. Appl. Math., 58 (1978), 187-207; S o l i t o n s i n A c t i o n , 1978, Academic, pp. 21-32. - [2] and DEFACIO, B., JMP, 21 (19801, 1716-1723. - [3] JMP, 16 (1975), 1044-1046; 20 (1979), 1151-1156 , JMP, 1 8 and 2047-2053; 21 (1980), 83-89. - [4] and KANAL, (1977), 2445-2447. - [5] JMP, 17 (1976), 73-75. - [6] and DEFACIO, B. and ABRAHAM, P., Jour. Phys., 16 (1’383), 303-316. MARTIN, R. [l]N o n l i n e a r Operators and D i f f e r e n t i a l Equations i n Banach Spaces, Wiley, 1976. MAURIN, K. [l]H i l b e r t Space Methods, Warsaw, 1959. - [2] General Eigenf u n c t i o n Expansions and U n i t a r y Representations o f T o p o l o g i c a l Groups , Warsaw, 1968. MUSKHELISHVILI, N. [l]S i n g u l a r I n t e g r a l Equations, MOSCOW, 1946. MURAT, F. [l]Ann. Sc. Norm. Sup. Pisa, 5 (1978), 489-507 MITCHELL, B. [l]Theory o f Categories, Academic, 1965 MAZYA, V. [l]Zur T h e o r i e Sobolew. Raume, Teubner, 1981 MOROZOV, V. [l]Methods f o r S o l v i n g I n c o r r e c t l y Posed Problems, S p r i n aer. 1984 MAHWIN; J: [l]T o p o l o g i c a l Degree Methods i n N o n l i n e a r Boundary Value Problems, CBMS 1140, AMS, 1979 MORAWETZ, C. [l] Comp. Maths. Appl. 7 (1981), 319-331 NOVIKOV, S.,MANAKOV, S., PITAEVSKIJ, L., and ZAKHAROV, V. [l]Theory o f Sol itons, P1enum, 1984 NASHED, M. [l]Advanced Seminar on Generalized Inverses, Academic, 1976 NACHBIN, L. [l L e c t u r e s on t h e Theory o f D i s t r i b u t i o n s , Recife, 1964. NIRENBERG, L. 13 Topics i n N o n l i n e a r F u n c t i o n a l Analysis, NYU, 1974 NARLIKAR, J. and PADMANABHAN, T. [l]G r a v i t y , Gauge Theories, and Quantum Cosmology, Reidel, 1986 NEWELL, A. [l]S o l i t o n s i n Mathematics and Physics, SIAM, 1985. - [2] S o l i t o n s , Springer, 1980, pp. 177-242. - [3] Rocky Mount. Jour. Math., 8 (1978), 25-52. - [4] and RATIU, T. and TABOR, M. and ZENG YANBO, Sol i t o n Mathematics, Univ. Montreal Press, 1986. NOVIKOV, R. and KHENKIN, G. [l]UMN 42 (1987), 93-152 NASH, C . [l]R e l a t i v i s t i c Quantum F i e l d s , Academic, 1978 NEWTON, P. [l]and SIROVICH, L., Q u a r t . Appl. Math., 1 (1986), 49-58

...,

.

MT MU MV MH MW MY MZ

MX MOR N

NA NC NI NK NL

NB NS NT

2

REFERENCES

NW

NZ

0 OA

P PA PB PC PE PG PK PL PD PO PP PR PT

PV PW PX PY PZ PAR

Q

QI R

RA

387

and 2 (1986), 367-374. - [2] and S I R O V I C H , L., Physica 21D (1986), 115-125. - [3] The p e r t u r b e d Non i n e a r Schrodinger Equation.. . , t o appear. - [4] and KELLER, J., S I A M Jour. Appl. Math. , 47 (1987), 959-964; Wave Motion, 10 (1988), t o appear. NEWTON, R. [l]S c a t t e r i n g Theory o f Waves and P a r t i c l e s , McGraw-Hill, N.Y., 1966. - [2] Conf. I n v e r s e S c a t t e r i n g , SIAM, 1983, pp. 1-74. - [3] JMP, 21 (1980), 493-505, 1698-1715, 2191-2200; 23 (1982), 594 - 604 - [4] S t a b i l i t y o f t h e G e n e r a l i z e d Marc'enko Method, t o appear. - [5] JMP, 23 (1982), 2257-2265. - [6] I n v e r s e Probs., 1 (1985) , 371 -380. NICKERSON, H., SPENCER, D., and STEENROD, N. [l]Advanced Calculus, Princeton, 1959 OLVER, P. [l]A p p l i c a t i o n s o f L i e Groups t o D i f f e r e n t i a l Equations, Springer, 1986 OHARU, S. and TAKAHASHI, T. [l]L e c t . Notes Num. Appl. Anal., 6 (1983), 125-142 POUNDSTONE, W. [l]The Recursive Universe, Contemporary Books, Chicago, 1985 PALAMODOV, V . [l]L i n e a r D i f f e r e n t i a l Operators w i t h Constant C o e f f i c i e n t s , MOSCOW, 1967 PARKER, T. [l]CMP, 85 (1982), 563-602 PASCALI, D. and SBURLAU, S. [l]N o n l i n e a r Maps o f MonotoRe Type, S i j t h o f f and Nordhoff, 1978 PETROWSKY, I. [l] L e c t u r e s on PDE, Wiley, 1959 PUGOVICKI, E. [l]Quantum Mechanics i n H i l b e r t Space, Academic, 1981 PARK, D. [l]I n t r o d u c t i o n t o t h e Quantum Theory, McGraw-Hill, 1964 PALAIS, R. [l]Annals o f Math. S t u d i e s , 57, 1965. - [2] Foundations o f Global N o n l i n e a r A n a l y s i s , Benjamin, 1968. POKORSKI, S. [l]Gauge F i e l d Theories, Cambridge, 1987 PONTRYAGIN, L., BOLTYANSKIJ, V., GAMKRELIDZE, R., and MIStENKO, E. [l] The Mathematical Theory o f Optimal Processes, Wiley, 1962 PAPOULIS, H. [l]The F o u r i e r I n t e g r a l and i t s A p p l i c a t i o n s , McGraw-Hill , 1962 PROTTER, M. and WEINBERGER, H. [l] Maximum P r i n c i p l e s i n D i f f e r e n t i a l Equations, P r e n t i c e - H a l l , 1967 PUTNAM, C . [ l ] Commutation P r o p e r t i e s o f Hi1 b e r t Space Operators.. . , S p r i n g e r , 1967 PAVEL, N. [l] D i f f e r e n t i a l Equations, Flow I n v a r i a n c e , . , ,Pitman, 1984 PAO, Y., SANTOSA, F., SYMES, W., and HOLLAND, C. [l]I n v e r s e Problems o f A c o u s t i c and E l a s t i c Waves, SIAM, 1984 POSCHEL, J. and TRUBOWITZ, E. [l] I n v e r s e S p e c t r a l Theory, Academic, 1987 PAYNE, L. [l]I m p r o p e r l y Posed Problems i n PDE, SIAM, 1975 PAZY, A. [l]Semigroups o f L i n e a r Operators S p r i n g e r , 1983. - [2] L e c t u r e s on A c c r e t i v e Operators and N o n l i n e a r D i f f e r e n t i a l Equat i o n s i n Banach SDaces. UFRJ. Rio de Janeiro. 1981 PERRY, P. [l]S c a t t e r i n g Theory b y - t h e Enss Method; Harwood, 1983. [2] Jour. F n l . Anal ., 75 (1987), 161-187 QUIGG, C. [l]Gauge Theories o f t h e Strong, Weak, and E l e c t r o m a g n e t i c I n t e r a c t ion, Benj a m i n , 1983 QILLAN, D. [l]Thesis, Harvard, 1964 REED, M. [l]and SIMON, B., Methods o f Modern Mathematical Physics, 1-4, Academic, 1975, 1978, 1979. - [2] A b s t r a c t N o n l i n e a r Wave Equat i o n s , Springer, 1976 RAPHAEL, L. [l]JMAA, 115 (1986), 93-104; Canad. Jour. Math., 38 (1986), 861-877; S I A M Jour. Math. Anal., 1 3 (1982), 676-689. - [2] and

.

...,

388

RB RC

RD RE RG RH RJ RK RF

RM RN

RO RI

RS

ROBERT CARROLL

KON, M. JDE, 50 (1983), 391-406 ROBINSON, E. 111 and SILVIA, M., Deconvolution o f Geophysical Time SerHoldeni e s , E l s e v i e r , 1979. - [2] M u l t i c h a n n e l Time S e r i e s Day, San Francisco, 1967. RICHTMYER, R. [l]P r i n c i p l e s o f Advanced Mathematical Physics, Springer, 1985 RUDIN, W . [l]Real and Complex A n a l y s i s , McGraw-Hill, 1966 RAIMOND, P. [l]F i e l d Theory. A Modern Primer, Benjamin, 1981 RODBERG, L. and THALER, R. [l]I n t r o d u c t i o n t o t h e Quantum Theory o f S c a t t e r i n g , Academic, 1967. DeRHAM, G. [I]V a r i e t 6 s D i f f g r e n t i a b l e s , Hermann, P a r i s , 1955 RAJARAMAN, R. [l S o l i t o n s and I n s t a n t o n s , North-Holland, 1982 ROCKAFELLAR, 'R. 1 1 Convex A n a l y s i s , Princeton, 1972. ROGERS, C. and SHADWICK. W. [l]Backlund Transformations and t h e i r App l i c a t i o n s , Academic, 1982 ROMANOV, V. [l]I n v e r s e Problems f o r D i f f e r e n t i a l Equations, N o v o s i b i r s k , 1973. - [2] I n v e r s e Problems o f Mathematical Physics, VNU, U t r e c h t , 1987 RAMM, A. [l]S c a t t e r i n g by Obstacles, Reidel, 1986. - [2] Theory and A p p l i c a t i o n s o f some New Classes o f I n t e g r a l Equations, Springer, 1980. - [3] I n v e r s e S c a t t e r i n g on t h e H a l f Line, t o appear. - [4] M u l t i d i m e n s i o n a l I n v e r s e Problems, t o appear. ROJANSKY, V . [l]I n t r o d u c t o r y Quantum Mechanics, P r e n t i c e - H a l l , 1946 RIVERS, R. [l]Path I n t e g r a l Methods i n Quantum F i e l d Theory, Cambridge, 1987 ROSE, J . and CHENEY, M. [l]and DEFACIO, B., Jour. Opt. SOC. Amer., 11 (1985), 1954-1957; JMP, 25 (1984), 2995-3000; JMP, 26 (1985), 2803-2813; Phys. Rev. L e t t . , 57 (1986), 782-786. - [2] JMP, 26 ( 1 985), 436-439; Phys. Rev. L e t t . , 59 ( 1 987), 954-956; General i z e d E i g e n f u n c t i o n Expansions t o appear. RODRIGUES, J. [l]O b s t a c l e Problems i n Mathematical Physics, North-Holl a n d , 1987 RABINOWITZ, P. [l]Minimax Methods . , CBMS R65, AMS, 1984 RODRIGUES, P. and DeLEON, M. [l] Generalized C l a s s i c a l Mechanics and F i e l d Theory, North-Holland, 1985 RYDER, L. [l]Quantum F i e l d Theory, Cambridge, 1985 RIESZ, F. and SZ.NAGY, B. [l]Lecons d ' A n a l y s e F o n c t i o n n e l l e , Budapest, 1953 SOMMERFELD, A. [l]Electrodynamics, Academic, 1964 SARWAR, A. [l]Proc. SEG-SIAM-SPE Conf., Houston, 1985, t o appear. - [2] and RUDMAN, A., Geophys. Jour. Roy. A s t r . SOC., 81 (1985), 551-562 SOBOLEV, S. [I]A p p l i c a t i o n s o f F u n c t i o n a l A n a l y s i s i n Mathematical Physics, AMS, 1963 SCHIFF, L. [l]Quantum Mechanics, 1955 SCADRON, M. [l] Advanced Quantum Theory, Springer, 1979 SHENK, N. and THOE, D. [l]Rocky Mount. Jour. Math., 1 (1971), 89-125 SHABAT, A. [l]Sel. Math. Sov., 4 (1985), 19-35 SIERSMA, J. [l] Thesis, Groningen, 1979 SCHECHTER, M. [l]Operator Methods i n Quantum Mechanics, North-Holland, 1981. - [2] Spectra o f P a r t i a l D i f f e r e n t i a l Operators, North-Holland, 1971. - [3] P r i n c i p l e s o f F u n c t i o n a l A n a l y s i s , Academic, 1971. SABATIER, P. [l] Problimes Inverses, CNRS, P a r i s , 1980. - [2] A p p l i e d I n v e r s e Problems, S p r i n g e r , 1978 SIMON, B. [l]B u l l . AMS, 7 (1982), 447-526. - [2] Quantum Mechanics f o r Hamiltonians D e f i n e d as Q u a d r a t i c Forms, Princeton, 1971. - [31

...,

!

...,

RT RW RX

RY RZ

S SA

Sl3

sc SD SE

SF SG SH

SI SJ

..

REFERENCES

SK SL SM

SN

so

SP

SQ

SR

ss ST

su sv sw

38 9

The P ( v ) Euclidean F i e l d Theory, P r i n c e t o n , 1974 SAKURAI, J. [I? Modern Quantum Mechanics, Benjamin, 1985 SCHWARTZ, L. [l]T h e b r i e des D i s t r i b u t i o n s , Hermann, P a r i s , 1966 SMOLLER, J. [l]Shock Waves and Reaction D i f f u s i o n Equations, S p r i n g e r , 1983 SPANIER, E. [l]A l g e b r a i c Topology, McGraw-Hill, 1966 SONDHI, M. and RESNICK, J. [l]Jour. Acous. SOC. Amer., 73 (1983), 9851002 SPERB, R. [l]Maximum P r i n c i p l e s and t h e i r A p p l i c a t i o n s , Academic, 1981 SANTOSA, F. [l]Thesis, Univ. I l l i n o i s , 1980. - [2] and SYMES, W., Cons t r u c t i o n o f t h e Newton Hessian.. , t o appear. - [3] Geophys. Jour. Roy. A s t r . SOC., 70 (1982), 229-243. - [4] and SCHWETLICK, H., Wave Motion, 4 (1982), 99-110. - [5] and SYMES, W., S I A M Jour. Sci. S t a t . Comput., 7 (1986), 1307-1330. - [6] and PAO, Y., Jour. Acous. SOC. Amer., 80 (1986), 1429-1437. - [7] and SACKS, P., Some Simple Methods f o r V e l o c i t y I n v e r s i o n , t o appear. SNIATYCKI, J. [I]Geometric Q u a n t i z a t i o n and Quantum Mechanics, S p r i n ger, 1980 SIMS, D. and WOODHOUSE, N. [l]L e c t u r e s on Geometric Q u a n t i z a t i o n , LNP #53, Springer, 1976 STRANG, G. [l]I n t r o d u c t i o n t o A p p l i e d Mathematics, Wellesley-Cambridge, 1986 SHULMAN, L. [l]Techniques and A p p l i c a t i o n s o f Path I n t e g r a t i o n , Wiley, 1981 SINGER, I.[l]CMP, 60 (1978), 7-12 SHOWALTER, R. [l]JMAA, 56 (1977), 123-135; A p p l i c a b l e Anal. 7 (1978), 297-308; S I A M Jour. Math. Anal., 3 (1972), 527-543. - [2] H i l b e r t [3] and BOSSE, M., HomogeniSpace Methods i n PDE, Pitman, 1977. z a t i o n o f t h e Layered Medium Equation, A p p l i c a b l e Anal., t o appear SCHUTZ, B. [l]Geometrical Methods o f Mathematical Physics, Cambridge, 1980 SYMES, W. [l]and SANTOSA, F., I n v e r s e Problems, SIAM, t o appear. - [2] JMAA, 94 (1983), 435-453. [3] S I A M Jour. Math. Anal., 17 (1986), 132-151 and 1400-1420. - [4] Math. Comp. Meth. Seismic Exp ...., SIAM, 1986, pp. 128-157. - [5] and SACKS, P., Comm. PDE, 1 0 (1985), 635-676 STAKGOLD, I. [l]Green's F u n c t i o n s and Boundary Value Problems, Wiley, 1979 SEGAL,-Il [l] Jour. F n l . Anal., 33 (1979), 175-194. SCHOEN, R . [l]Jour. D i f f . Geom., 20 (1984), 479-495 STRAUSS, W. [l]Jour. Fnl. Anal., 41 (1981), 110-133; 43 (1981), 281[3] and 293. - [2] and SHATAH, J., CMP, 100 (1985), 173-190. SHATAH, J. and GRILLAKIS, M., S t a b i l i t y Theory o f S o l i t a r y Waves . , t o appear. SOURIAU, J. [l] S t r u c t u r e des Systkmes Dynamiques, Dunod, 1970 SATTINGER, D. [l]Contemp. Math., AMS, 56 (1986), pp. 319-333. - [2] Stud. Appl. Math., 72 (1985), 65-86. - [3] Automorphisms and Backl u n d Transformations, t o appear. - [4] and ZURKOWSKI, V . , Physica 26D (1987), 225-250 and Proc. NATO Conf. on I n f i n i t e Dimen. Dynam. Syst., t o appear. - [5] N o n l i n e a r Probs., North-Holland, 1982, pp. 51-64. - [6] Dynamical S s t . 11, Academic, 1982, pp. 349-365. SYLVESTER, J. and UHLMANN, G. fl]Annals Math., 125 (1987), 153-169; . , t o appear I n v e r s e boundary v a l u e problems STERNBERG, S. [l]L e c t u r e s on D i f f e r e n t i a l Geometry, P r e n t i c e - H a l l , 1964 - [2] L e c t u r e s on PDE, Univ. Pennsylvania, 1967. SWEENEY, W . [l]Acta Math., 120 (1968), 223-277

.

~~

-

sx SY

sz SAG SCH STR

SOU SAT

SYL STE SWE

-

-

..

..

390

T TA TB TD TE

TC TU TH TI

TM TN TO TR TS

TT TW TY TZ

U V

VE VI VM VR W

WA WB

WC WD WR WE

ROBERT CARROLL

THIRRING, W. [l]A Course i n Mathematical Physics, 1-4, Springer, 1979 TANABE, H. [l]Equations o f E v o l u t i o n , Pitman, 1979 TAUBES, C. [I]CMP, 69 (1979), 179-193; 72 (1980), 277-292; 75 (1980), 207-227. - [2] Pitman Res. Notes Math., 122, 1985, pp. 257-271. [3] CMP, 91 (1983), 235-263; 97 (1985), 473-540. TONDEUR, P. [l] I n t r o d u c t i o n t o L i e Groups and Transformation Groups, Springer, 1965. - [2] F o l i a t i o n s on Riemannian Manifolds, t o appear. TEMAM, R. [l]Navier-Stokes Equations, North-Holland, 1984. - [2] Jour. Math. Pure Appl., 48 (1969), 159-172. - [3] Navier-Stokes Equat i o n s and N o n l i n e a r F u n c t i o n a l A n a l y s i s , CBMS, SIAM, 1983. - [4] Pitman Res. Notes. Math. #122, 1985, pp. 272-292. TSUTSUMI, Y. [l]N o n l i n . Anal., 11 (1987), 1143-1154; Funk. Ekvac., 30 (1987), 115-125. TOYNBEE, A . [l]A Study o f H i s t o r y , Abridged, Oxford, 1987 Halsted, N.Y., 1984 TOTH, G. [l] Harmonic and Minimal Maps TITCHMARSH, E. [l]Theory o f Functions, Oxford, 1932. - [2] I n t r o d u c t i o n t o t h e Theory o f F o u r i e r I n t e g r a l s , Oxford, 1937. - [3] Eigenfunct i o n Expansions A s s o c i a t e d w i t h Second Order D i f f e r e n t i a l Equat i o n s , Oxford, 1962. TRIMECHE, K. 113 Jour. Math. Pure Appl., 60 (1981), 51-98 TON, B U I AN [l]JDE, 25 (1977), 288-309 TROUTMAN, J. [l] V a r i a t i o n a l C a l c u l u s w i t h Elementary Convex Functions, Springer, 1983 TREVES, F. [l]B a s i c L i n e a r PDE, Academic, 1975 TERRAS, A. [l]Harmonic A n a l y s i s on Symmetric Spaces and A p p l i c a t i o n s , S p r i n g e r , 1975 TRAUTMAN, A. [l]D i f f e r e n t i a l Geometry f o r P h y s i c i s t s , B i b 1 o p o l i s , Naples, 1986 TSUTSUMI, M. [l]JDE, 42 (1981), 260-281; R I M S Kyoto Univ., 7 (1971), 329-344; S I A M Jour. Math. Anal., 15 (1984), 357-366. TARTAR, L. [l]Pitman Res. Notes Math. #39, 1984, pp. 136-2 2. TRUBOWITZ, E. [l]and POSCHEL, J . , I n v e r s e S p e c t r a l Theory, Academi c, 1987. - [2] and KAPPELER, T., Comm. Math. Helv., 61 ( 986), 442480. UHLENBECK, K. [l]CMP, 83 (1982), 11-29 and 31-42 VAINBERG, M. [ 1 ] - V a r i a t i o n a l Methods f o r t h e Study o f N o n l i n e a r Operat o r s , Holden-Day, 1964 VENKOV, A. [l]UMN, 34 (1979), 79-153 VILENKIN, N. [l]Special Functions and t h e Theory o f Group Representat i o n s , Moscow, 1965 VAISMAN, I. [l]Cohomology and D i f f e r e n t i a l Forms, Dekker, 1973 VARADARAJAN, V. [l] L i e Groups and L i e Algebras, P r e n t i c e - H a l l , 1974 WEINSTEIN, A. [l]Comm. Pure Appl. Math., 7 (1954), 105-116. - [2] B u l l . Acad. Roy. Belg., 37 (1951), 348-358. - [3] Proc. F i f t h Symp. Appl. Math., 1954, pp. 137-147. WANG, C. [l]Mathematical P r i n c i p l e s o f Mechanics and Electromagnetism, Plenum, 1979 WU, T. and OHMURA, T. [l] Quantum Theory o f S c a t t e r i n g , P r e n t i c e - H a l l , 1962 WILCOX, C. [l]Arch. Rat. Mech. Anal., (1966), 37- ; (19721, 280 WEIDEMAN, J. [l]L i n e a r Operators i n H i l b e r t Space, Teubner, 1976 WARNER, F. [l]Foundations o f D i f f e r e n t i a b l e M a n i f o l d s and L i e Groups,

...,

WIENER?Pi!ni!?iyTA?;ourier Cambridge, 1933.

-

I n t e g r a l and C e r t a i n o f i t s A p p l i c a t i o n s , [2] E x t r a p o l a t i o n , I n t e r p o l a t i o n , and Smooth-

REFERENCES

391

i n g o f S t a t i o n a r y Time Series, MIT Press, 1949. WEINSTEIN, A. [l]L e c t u r e s on S y m p l e c t i c M a n i f o l d s , CBMS, AMS, 1977 WONG, E. and HAJEK, 8. [l]S t o c h a s t i c Processes i n E n g i n e e r i n g Systems, Springer, 1984 WH WHITHAM, G. [l]L i n e a r and N o n l i n e a r Waves, Wiley, 1974 wo WOOOHOUSE, N. [l]Geometric Q u a n t i z a t i o n , Oxford, 1980 W I WEINSTEIN, M. [l]Comm. PDE, 11 (1986), 545-565; JDE, 69 (1987), 192203. - [2] S I A M Jour. Math. Anal., 16 (1985), 472-491. - [3] CMP, 87 (1983), 567-576; 106 (1986), 569-580 WL WAHLQUIST, H. and ESTABROOK, F. [l]JMP, 16 (1975), 1-7 WT WESTENHOLZ, C.VON [l]D i f f e r e n t i a l Forms i n Mathematical Physics, N o r t h Hol 1and, 1981 wu WOUK, A . [l]A Course o f A p p l i e d F u n c t i o n a l A n a l y s i s , Wiley, 1979 Y YOUNG, L. [l] L e c t u r e s on t h e C a l c u l u s o f V a r i a t i o n s and Optimal Cont r o l Theory, Saunders, 1969 YA YAU, S . [l]N o n l i n e a r A n a l y s i s , New Orleans, 1986 YE YAGLE, A. [l]D i f f e r e n t i a l and I n t e g . Meth. f o r M u l t i d i m e n . S c a t t e r i n g Probs., t o appear. - [2] D i f f . and I n t e g . Meth. f o r 3-D I n v e r s e S c a t t . Probs. w i t h a Non-local Pot., t o appear. - [3] and LEVY, B. Layer S t r i p . Solns. o f M u l t i d i m . I n v e r s e S c a t t . Probs., t o appear. - [4] Mu1 t i d i m . I n v e r s e S c a t t : An Orthogonal i z a t i o n Eorrnul , t o appear. - [5] A F a s t A l g o r . f o r L i n e a r E s t and Connect. b e t . t o appear. 3-D I n v e r s e S c a t t . and L i n e a r L e a s t Sq. Est YO YOSIDA, K. [l]F u n c t i o n a l A n a l y s i s , S p r i n g e r , 1965 ZA ZAANEN, A. [l]L i n e a r A n a l y s i s , Nordhoff, 1956 Z ZACHMANOGLOU, E. and THOE, D. _[ 1 1_ I n t r o d u c t i o n t o PDE, W i l l i a m s - W i l k i n , B a l t i m o r e , 1976 ZE ZEIDLER, E. [l]N o n l i n e a r F u n c t i o n a l A n a l y s i s Vol. 3, S p r i n g e r , 1985. [2] N o n l i n e a r F u n c t i o n a l A n a l y s i s , Vol. 1, S p r i n g e r , 1986. - 131 N o n l i n e a r F u n c t i o n a l A n a l y s i s , Vol. 2, eubner, 1977 [2] and SHAZK ZAKHAROV, V. [l]S o l i t o n s , S p r i n g e r , 1980, pp 243-285. BAT, P., Funkts. Anal. P r i l o ? . , 8 (1974 , 226-235 and 13 (1979), 166-174 ZY ZACHARY, W. [l]JMAA, 117 (19861, 449-495 WF WG

.

...., ....,

-

This Page Intentionally Left Blank

393

INDEX

A Abelian gauge, 296 Absolute c o n t i n u i t y , 7,272,337 Absorbing set, 84,315 Accretive, 236,272,343 Acoustic waves, 90 Action-angl e v a r i a b l e s , 199,200 Action functional , 55,78 Adjoint f u n c t i o n s , 206 Adjoint system, 34 Affine f u n c t i o n , 266 Affine minorant, 266 AKNS system, 192 Alaoglu's theorem, 271,279,323 d ' A1 embert s o l u t i o n , 10,13 Algebra o f o p e r a t o r s , 60 Almost everywhere, 236,337 A l t e r n a t i n g , 355 Angular momentum, 64 Annihilation o p e r a t o r , 60,287 Anticausal, 152,154 A n t i d e r i v a t i o n , 368 Antidual, 70 Approximate eigenvector, 103 Arzela-Ascol i theorem, 95,320,260 Asplund's theorem, 273 Asymptotic s t a t e s , 104 Attractor, 83 Autocorrelation, 47,155 B B-complete c o n t i n u i t y , 237 Backlund transformation, 191,216 Backward heat problem, 25 Baire space, 317 Banach-Steinhaus theorem, 239 ,31 7,320 Banach theorem homomorphisms, 324 Bang-bang control , 34 Barrel , 317 Barreled space, 317 B a r r i e r , 60 BBM equation, 76 Bianchi i d e n t i t y , 301,305 Bispinor, 297 Blowup, 228 Bogomolny formula, 79

duBois Reymond, 4 , 5 Boundary o p e r a t o r , 52 Bounded s e t , 316,318 Bounded map, 229 Bound s t a t e , 38,61,120 Bra. 63 Brachistochrone problem, 2 Brouwer fixed p o i n t theorem, 247, 253,259 Brownian motion, 291 Burger's equation, 76 C

C'-norm, 7 Canonical equations, 56 Canonical form PDE, 17,18 Canon i ca 1 t r a n s f orma t i on , 55 ,201 Casimir o p e r a t o r , 65 Category, 252,361 Catenary, 30 Cauchy d a t a , 14 Cauchy- Kowa 1evs ky theorem, 87 Cauchy n e t , 313 Cauchy Peano theorem, 74 Cauchy problem, 12,14,56,151 Cauchy Schwartz i n e q u a l i t y , 325 Causal, 141,146,152,154 C e l l , 52,366 Chain, 52,53,366 Chain r u l e , 354 C h a r a c t e r i s t i c s , 10,12,18,56 C h a r a c t e r i s t i c strip, 56 Charge, 293 Charts, 351 Chern c l a s s , 309 Chern number, 78 C h r i s t o f f e l symbol , 298 Closed form, 359 Closed map, 262 Closed graph theorem, 325 Closure, 313 C l u s t e r Point. 241 Coerci ve, 67,280,282 ,283 Color, 298 Comnutat ion re1 a t i o n s , 59,60,287 Compact homotopy, 264 Compactness , 233,319

394

ROBERT CARROLL

Compact o p e r a t o r , 233 Compensated compactness, 285 Complete c o n t i n u i t y , 229,233,237 Completely i n t e g r a b l e , 200,218 Completeness, 63, 204,220 Complete space, 313 Cone c o n d i t i o n , 345 Concave f u n c t i o n , 268 Conjugate f i e l d s , 296 Conj u g a t e f u n c t i o n , 29,265,267 Conjugate l i n e a r , 67,69 Conjugate p o i n t , 31 Conjugate v a r i a b l e s , 199 Connected, 256 Connected Green's f u n c t i o n , 294,295 Connection, 78,211,217,290,303 Conservation o f energy, 24,53 Conserved q u a n t i t y , 188 Continuous spectrum, 103,108 Continuous subspace, 105 C o n t r a c t i b l e , 253 C o n t r a c t i o n , 301 C o n t r a c t i o n map theorem, 230 C o n t r a v a r i a n t , 287 Convex d i s c e d envelope, 317,332 Convex f u n c t i o n a l , 243,265 Convex h u l l , 254,264 Convolution, 12 Cost o p t i m i z a t i o n , 27 Cotangent bundle, 51,63,355,365 Cotangent space, 64,353 Countable a t i n f i n i t y , 351 Coupling c o n s t a n t , 211 Covariant, 287,296 C o v a r i a n t d e r i v a t i v e , 78,218,296,298, 304 C r a n d a l l - L i g g e t t theorem, 277 C r e a t i o n o p e r a t o r , 60,287 C r i t i c a l p o i n t , 228,245 C r i t i c a l Sobolev exponent, 285,286 Crum transform, 191 Current, 80,292,296 Curvature, 78,217,290,304,212 Cycle, 366

.

D

D-monotone, 236,343 D a r b o u x - C h r i s t o f f e l formula, 203 Darboux t r a n s f o r m a t i o n , 191 Deconvolution, 45 Degenerate Cauchy problem, 1 8 Degree, 256,257 D e l t a f u n c t i o n , 12,42,149 Demi c o n t i n uous , 228,235,236 D e r i v a t i o n , 354 D e r r i c k ' s theorem, 80

D e t e r m i n i n g sequence, 332 Diatomic molecule, 64 D i f f e r e n t i a l form, 357 D i f f e r e n t i a l i d e a l , 216 D i f f e r e n t i a t ion, 22,330,334,348 D i r a c equation, 295 D i r e c t i o n a l d e r i v a t i v e , 29 Dirichlet functional, 5 D i r ic h l e t p r o b l em, 5,15,66,270 Disced s e t , 315 D i s c r e t e spectrum, 102 D i s c r e t i z a t i o n , 275 D i s p e r s i o n r e 1 a t ion, 209,214 D i s s i p a t i v e PDE, 84 D i s t r i b u t i o n , 329 Divergence form, 234 Domain, 263 Domain o f dependence, 13,47 Domain o f i n f l u e n c e , 1 3 Dominated convergence, 236,337 Dual i n v e r s e problem, 202 D u a l i t y , 316 D u a l i t y map, 273 Dual problem, 26,267 Dual f u n c t i o n s , 206 Dual space, 313 Dual v a r i a b l e , 26 D u f f i n g ' s equation, 75 Dynamical v a r i a b l e s , 59 E

.

€-approximate s o l u t i o n , 275 E i g e n f u n c t i o n , 11,39 E i g e n f u n c t i o n expansion, 12,39,146 Eigenstate, 59 Eigenval ue, 11 E f f e c t i v e domain, 265 E l 1 ip t i c equation, 15,17,234,240, 242,282 Embedding theorems, 346 Energy, 23 Energy-momentum tensor, 288,301 Epigraph, 265,284 Equicontinuous, 95,319 Equivalence c l a s s , 236 E q u i l i b r i u m , 27 E s s e n t i a l , 255,263 E s s e n t i a l l y s e l f a d j o i n t , 58,109 E s s e n t i a l spectrum, 103,108 Eul e r ' s equations, 4,49,74 Eul e r - Po is son-Darboux equation, 18, 22 E u l e r ' s t r i c k , 3,30 E v o l u t i o n equation, 12,197,248 Exact form, 359 E x c i s i o n , 253

-

395 Expectation, 57 Exponential map, 374 Extension by c o n t i n u i t y , 321 E x t e r i o r product, 355 E x t r a p o l a t i o n , 47 F F a c t o r i z a t i o n , 152,154 Fermat's p r i n c i p l e , 2 Feynman i n t e g r a l , 290 Feynman-Kac formula, 292 Feynman propagator, 295 F i e l d s t r e n g t h , 304 F i n e s t topology, 333 F i n i t e dimensional map, 258 F i n i t e l y continuous, 242 F i n i t e i n t e r s e c t i o n p r o p e r t y , 234 F i x e d p o i n t index, 264 F l a v o r , 298 Flow, 218 Forcing, 220 Forms extension, 104 Forward s c a t t e r i n g amplitude, 138 F o u r i e r method, 12 F o u r i e r t r a n s f o r m , 12,337 F o u r i e r t y p e o p e r a t o r , 123,153,181 F r a c t a l dimension, 84 Frechet d e r i v a t i v e , 174,198,209,229, 362 Frechet space, 313 Free boundary, 285 Fredholm o p e r a t o r , 216,234 F r i e d r i c h ' s extension, 82 Frobenius theorem, 216 F u n c t i o n a l l y i n v a r i a n t s e t , 83 Functor, 252 ,361 Fundamental system NBH, 312 Fundamental s o l u t i o n , 19 Fadeev-Popov ansatz, 300

G r - r e g u l a r i z a t i o n , 266,268 G a l e r k i n method, 240,283 Gateaux d e r i v a t i v e , 229 Gauge, 302 Gauge a l g e b r a , 306 Gauge f i x i n g , 299 Gauge i n v a r i a n c e , 81 Gauge p o t e n t i a l , 9,81,305 Gauge t r a n s f o r m a t i o n , 9,81,211,216, 292,296,306 Ghost f i e l d , 300 Gel fand-Levi t a n equation, 41,92,123, 129,151,152,160 General i z e d e i g e n f u n c t i o n s , 110,118, 148

G e n e r a l i z e d G a l e r k i n , 240,248 General i z e d i n v e r s e , 87 Generalized momentum, 49 General i z e d Radon t r a n s f o r m , 140 G e n e r a l i z e d s o l u t i o n , 249 G e n e r a l i z e d t r a n s l a t i o n , 152 Generating f u n c t i o n a l , 293,298 Germ, 353 Ghost f i e l d , 300 Ginzburg-Landau equations, 76-79,82 Global a t t r a c t o r , 85 Global a n a l y s i s , 77 Goldstone boson, 81 Gopi na th-Sondhi e q u a t i o n , 166 ,1 67 Goursat problem, 151,194,220 Graded a b e l i a n groups, 252 Gradient, 287 G r a d i e n t map, 232 Graph t o p o l o g y , 250 Green ' s o p e r a t o r , 122 Green's f u n c t i o n , 39,97,114,116,162, 163,290 Green ' s theorem, 5 ,66 G r i b o v a m b i g u i t y , 299 Gronwall lemma, 75 Grothendieck t o p o l o g i e s , 348 Ground s t a t e , 288 Gaussian i n t e g r a l , 295 H H-compact o p e r a t o r , 110 Haar measure, 299 Hademard example, 1 9 Hahn-Banach theorem, 238,313,316 Hami 1t o n i an , 29 ,34 H a m i l t o n i a n v e c t o r f i e l d , 52 Hamilton-Jacobi e q u a t i o n , 54,56,285 H a m i l t o n ' s equations, 51,198 H a m i l t o n ' s p r i n c i p l e , 8,49 Hamnerstein o p e r a t o r , 26 Hanging c a b l e , 26 Hankel k e r n e l , 162 Harmonic f u n c t i o n , 20 Harmonic o s c i l l a t o r , 59 H a u s d o r f f dimension, 83 H a u s d o r f f measure, 83 H a u s d o r f f space, 311,336 Heat equation, 13 Heat k e r n e l , 14 Heisenberg u n c e r t a i n t y , 58,59 H e i s e n b e r t p i c t u r e , 63 H e m i c o n t i n u i t y , 236 Heini-Bore1 p r o p e r t y , 234 Hermite polynomials, 60 H e r m i t i a n l i n e bundle, 308 H e r m i t i a n o p e r a t o r , 60

396

ROBERT CARROLL

Higgs f i e l d , 78,300 Higgs mechanism, 81 H i 1 b e r t i n t e g r a l , 31,55 H i l b e r t space, 325 H i l l e - Y o s i d a theorem, 342 Hodge * o p e r a t o r , 369 Hodge theory, 366 Holder continuous, 15,260 Homology, 252,253 Homotopy, 32,252 H o r i z o n t a l 1i f t , 304 Huyghens p r i n c i p l e , 23 Hyperbolic, 13,17 H y p e r a c c r e t i ve , 277 Hyperplane, 314 H y p o e l l i p t i c , 14 I Imaginary time, 292 Impedance, 41,148,156 Imp1 i c i t f u n c t i o n theorem, 230 Impulse response, 41,42,140,163 Index, 234 Index r a i s i n g , 78 I n d i c a t o r f u n c t i o n , 265 I n e r t i a l m a n i f o l d , 85 I n e s s e n t i a l , 255,263 I n f i n i t e s i m a l generator, 340 I n t e g r a l curve, 373 I n t e g r a l i n v a r i a n t , 53 I n t e g r a l m a n i f o l d , 21 7 I n t e r i o r product, 367 I n t e r p o l a t i o n , 47 I n t e r t w i n i n g , 105 ,147,151 ,1 73 I n v a r i a n c e o f domain, 242,255 I n v e r s e problem, 41 I n v e r s e f u n c t i o n theorem, 230 I n v e r s e s c a t t e r i n g t r a n s f o r m , 168,173, 179,180 I s o p e r i m e t r i c problem, 25 I s o s p e c t r a l , 183,217 J

Jacobi equation, 31 Jacobian, 53,292,355 Jacobi i d e n t i t y , 54,371 Jacobi necessary c o n d i t i o n , 31 Jacobi s u f f i c i e n t c o n d i t i o n , 32 J e t bundle, 302,216,364 J e t bundle exact sequence, 365 J o s t m a t r i x , 121,137 J o s t s o l u t i o n , 35,62 ,156,184 Jump c o n d i t i o n , 19

K Kadomtsev-Petviasvi 1 i e q u a t i o n , 76

K e p l e r ' s laws, 50 K1ein-Gordon equation, 76,79,286 Kontoroviz-Lebedev i n v e r s i o n ,154 , 178 KdV equation, 76,183,209 K r e i n equation, 161,162,167 K r e i n k e r n e l , 160 Krein-Levinson r e c u r s i o n s , 159 Kuhn-Tucker c o n d i t i o n , 28 Ket, 63 L Lagrange equations, 8,49 Lagrangian , 8,26,287,302,306 Lagrange's i d e n t i t y , 110,116,120 Lap1 ace-Be1 t r a m i o p e r a t o r , 150,370 Laplace o p e r a t o r , 15 Laplace t r a n s f o r m , 44 Layer s t r i p p i n g , 141,148,159 Lax equation, 186,189 Lax-Mi 1gram theorem, 67 Lax p a i r , 189 Least a c t i o n , 8,49 Lebesgue i n t e g r a l , 335 L e f t i n v a r i a n t v e c t o r f i e l d , 372 L e f t t r a n s l a t i o n , 371 Legendre cond it i on, 5,30 Legendre-Fenchel transform, 29,50, 56 Leray-Schauder theory, 256,259 Lewy's example, 87 L i e algebra, 371,372 L i e b r a c k e t , 59 L i e d e r i v a t i v e , 367 L i e groups, 370 L i e n a r d equations, 75 L i m i t c y c l e , 114 L i m i t i n g a b s o r p t i o n , 118 L i m i t i n g amplitude, 119 L i m i t p o i n t , 114 L i n e bundle, 79 L i n e a r Gateaux d i f f e r e n t i a l , 266 L i o n s theorem, 69 Lippman-Schwinger e q u a t i o n , 110,116, 120 L i p s c h i t z c o n d i t i o n , 74 L i p s c h i t z c o n t i n u i t y , 278 L i p s c h i t z m a n i f o l d , 85 Local c o o r d i n a t e s , 351 L o c a l i z a t i o n , 169,177 L o c a l l y bounded, 229 L o c a l l y connected, 256 L o c a l l y convex space, 312 L o c a l l y f i n i t e c o v e r i n g , 351 L o g a r i t h m i c c o n v e x i t y , 89 L o r e n t z gauge, 9,290

INDEX L o r e n t z group, 375 L o r e n t z m e t r i c , 80 Lower semicontinuous, 94,246,264 LSC r e g u l a r i z a t i o n , 268 Lagrange m u l t i p l i e r , 26,28,233,244

M m-accretive, 276 Mackey-Arens theorem, 322 Mackey's theorem, 323 Mackey topology, 322 Magnetic f l u x q u a n t i z a t i o n , 81 M a n i f o l d , 351 Marzenko equation, 137 ,143,153,155, 158.161 Maximai a c c r e t i v e , 277,343,345 Maximal D-monotone, 236,343 Maximal monotone, 279,344 Maximum p r i n c i p l e , 20,24 Maxwell ' s e q u a t i o n s , 8,289 Mean v a l u e theorem, 20 M e a s u r a b i l i t y , 69,337 M e t r i c space, 312,238 M i l d s o l u t i o n , 275 Minimal area, 6 Minimal coup1 ing, 300 Measure, 336 Minkowski space, 300,371 M i r a c l e , 139,143,144 M i u r a ' s transform, 191 M o d i f i e d G-L e q u a t i o n , 43,48,166 M o d i f i e d KdV e q u a t i o n , 191 Momentum e i g e n s t a t e , 63 Monodromy m a t r i x , 212 Monge cone, 56 Monotone o p e r a t o r , 238 Monotone sets, 272,279 Monte1 space, 94,334 Morphi sm, 361 M u l t i s o l i t o n , 186 Mazur's lemna, 254

N N a t u r a l t r a n s f o r m a t i o n , 252,361 Navier-Stokes equations, 72,82 Neighborhood (NBH) , 31 1 ,31 2 N e m y t s k i j o p e r a t o r , 261 Nets, 240,313 N o n l i n e a r Schrodinger e q u a t i o n , 76 Neumann problem, 66 Newton's method, 231 Newton's second law, 50 N o n d e s t r u c t i v e e v a l u a t i o n , 45 Nonexpansive map, 273 N o n l i n e a r programming, 27 N o n l i n e a r wave equations, 285

397 Norm, 311 N o r m a l i z i n g c o n s t a n t s , 62,185 Normal problem, 268 N u c l e a r space, 348 0

Observable, 58 Obs t a c l es , 284 Open map, 262,324 Open s e t , 311 Optimal c o n t r o l , 33 Ordinary d i f f e r e n t i a l equation, 1 O r i e n t e d frame, 52 Orthogonal ity, 11 ,11 7,122,123,196, 206 Overdetermined system, 222 P Paley-Wiener theorem, 42,121,151, 339 P a r a b o l i c , 13,17 Para.11e l ogram 1aw, 326 Para1 1e l t r a n s p o r t , 21 2 Parseval formula, 57,338 P a r t i a l d i f f e r e n t i a l equations, 1 P a r t i c l e f i e l d , 302 P a r t i t i o n o f u n i t y , 247,347,353 P a u l i m a t r i c e s , 65,296 Phase f l o w , 52 P h i l 1 ips-Miyadera theorem, 341 P i c a r d - L i n d e l o f f theorem, 74 Piecewise smooth, 7 P l a n c k ' s c o n s t a n t , 15 Plasma wave e q u a t i o n , 140,144 Poincarb-Cartan i n v a r i a n t , 54 PoincarC group, 292,371 P o i n c a r b ' s lemma, 366 P o i n t spectrum, 108 Poisson b r a c k e t , 53,198,209 Poisson i n t e g r a l formula, 1 6 Poisson-Jensen formula, 172,214 Poisson s o l u t i o n , 22 P o l a r f u n c t i o n , 265 P o l a r s e t , 316 P o n t r y a g i n maximal p r i n c i p l e , 34 Porous media, 285 Precompac t , 254 ,31 9 P r e d a t o r - p r e y model , 75 Prequan t i z a t i o n , 309 P r i m a l problem, 267 P r i n c i p a l bundle, 218,302 P r o b a b i l i t y d e n s i t y , 57 Product m a n i f o l d , 360 P r o g r e s s i n g wave expansion, 144 P r o j e c t i o n , 326 Pro1 o n g a t i o n , 21 7

ROBERT CARROLL

398 Propagator, 290 Proper map, 262, 265 Pseudoinverse, 87 Pseudomonotone, 280,282

Q Q u a d r a t i c form, 104 Quantiza t i o n , 308 Quantum electrodynamics , 296 Quark, 298 Q u a s i r e v e r s i b i l i t y , 89 R

r - m a t r i x , 211 R a d i a t i o n boundary cond t i o n , 45 Radon transform, 139 Real H i l b e r t space, 326 R e f l e c t i o n c o e f f i c i e n t , 154,170 R e f l e c t i o n l ess p o t e n t i a , 186 R e f l ec t iv i t y , 148,160,170 R e f l exive, 233,237,323 R e g u l a r i z a t i o n , 91,332 Resol vant, 39,118 Resolvant i d e n t i t y , 114 Resol v a n t k e r n e l , 39 Resolvant s e t , 102 R e t r a c t i o n , 253 Retrograde l i g h t cone, 23 Reynolds number, 84 deRham cohomology, 366 deRham complex, 366 R i c c a t i equation, 32,198,219 Riemann c u r v a t u r e t e n s o r , 298 Riemann f u n c t i ;n, 42 Riemannian geometry, 369 Riemann H i l b e r t problem, 121,137,159 Riemann Lebesgue lemna, 86,207 Riesz theorem, 326 Rigged H i l b e r t space, 12 R o t a t i o n , 64,324 R o t a t o r , 64 R e f l e c t i o n seismology, 45 S

s - a c c r e t i v e , 274 Saddle p o i n t , 269 Sard's theorem, 256 S c a l a r l y i n t e g r a b l e , 337 S c a l a r product, 325,327 S c a t t e r i n g by o b s t a c l e , 11 7 S c a t t e r i n g m a t r i x , 62,120 S c a t t e r i n g s o l u t i o n s , 61 SchrBdinger equation , 15,59,101,205, 21 1 Schrodinger p i c t u r e , 63 S e l f a d j o i n t r e a l i z a t i o n , 103

Schwartz space, 331 Semi convex, 243 Semigroup, 340 Semi i n n e r product, 273 Semi monotone, 242 Semi norm, 311 Semi r e f l e x i v e , 323 Separable, 234,237 Separation o f v a r i a b l e s , 10,13,16 Sequential compactness , 233,235 Sesquil i n e a r ( 1 -1/2) form, 67 Set o f a t t a i n a b i l i t y , 34 Shear waves, 41 Shur' s a1 g o r i thm, 159 Sideways Cauchy problem, 161,162 Sine-Gordon equation, 76,80 S i n g u l a r Cauchy problem, 1 8 S i n g u l a r spectrum, 1-08 Smoothing, 250,251 S n e l l ' s law, 2 Sobolev space, 66,94,345 Sokhotskij-Plemel j formulas, 214 S o l i d mean value, 21 S o l i t a r y wave, 190 S o l i t o n , 78,80 S p e c t r a l d e n s i t y , 42 S p e c t r a l f a m i l y , 102,114,118 S p e c t r a l measure, 39,115,139 S p e c t r a l p a i r i n g s , 46,151,159 S p e c t r a l r e s o l u t i o n , 114 S p e c t r a l theorem, 102 S p e c t r a l t h e o r y , 39 Spectrum, 102 Spherical f u n c t i o n , 150 Spherical mean, 21 Spin, 64 Spinor, 297 Spontaneous symmetry breaking, 81 Square e i g e n f u n c t i o n s , 177,196 Squeezing p r o p e r t y , 85 S t a b i l i z a t i o n , 89 S t a b i l i t y , 93 S t a b l e problem, 268 Steepest descent, 232 S t o c h a s t i c e s t i m a t i o n , 153 S t o k e ' s theorem, 53,366 S t r a t i f i e d medium, 40 S t r i c t l y convex, 273,279 S t r i c t i n d u c t i v e l i m i t , 330,332,336 S t r o n g l y complete, 106 S t r o n g l y e l l i p t i c , 242 Strong minimum, 7 Strong s o l u t i o n , 249 Strong topology, 316,318 Structure, v i , v i i i , l S u b d i f f e r e n t i a l , 266

INDEX Subgradient, 266 Subspace o f a b s o l u t e c o n t i n u i t y , 106 Summability, 91 S u p e r c o n d u c t i v i t y , 74,77 Support, 329 Support t a n g e n t plane, 28,34 S w i t c h i n g l o c u s , 35 Symplectic geometry, 51,53,368 Sympl e c t i c s t r u c t u r e , 52,204 S e q u e n t i a l convergence, 329 Schauder's f i x e d p o i n t theorem, 254 T Tangent bundle, 51,355,363 Tangent space f u n c t o r , 363 Tangent v e c t o r , 358 Tangent space, 354,362 Target, 32 T a y l o r ' s theorem, 230,362 Temporal gauge, 301 Tempered d i s t r i b u t i o n , 331 Tensor product, 347 T e s t f u n c t i o n , 5,68 Tikhonov r e g u l a r i z a t i o n , 2,87,90 Tikhonov theorem, 323 Time dependent e i g e n f u n c t i o n , 221 Time o p t i m a l c o n t r o l , 33 Time o r d e r e d p r o d u c t , 293 Tomography, 45 T o p o l o g i c a l v e c t o r space, 311 Trace, 66,82,346 Topology, 31 1 T r a n s i t i o n f u n c t i o n s , 302 T r a n s i t i o n m a t r i x , 212 Transmission c o e f f i c i e n t , 170 Transmission data, 45 Transmutation, 35,43,91,120,124,133, 150,155,165,173,184 Transmutation machinery, 46,148 Transonic gas dynamics, 18 Transparency, 61 T r a n s p o r t equations, 144 T r iangul a b l e , 253 T r a v e l time, 41,156 T r i a n g u l a r i t y , 152 T r i c o m i equation, 17 T r i v i a l i z a t i o n , 302 T r o t t e r ' s p r o d u c t formula, 290 Tunneling, 61 Turbulence, 286 U

Uniform c o n t i n u i t y , 229 Units, 9 U n i t a r y , 62 U n i v e r s a l a t t r a c t o r , 84

399

Upper semicontinuous, Uryson o p e r a t o r , 261

265

V van d e r P o l ' s e q u a t i o n , 75 V a r i a t i o n a l convergence, 285 V a r i a t i o n a l d e r i v a t i v e , 26 V a r i a t i o n a l i n e q u a l i t y , 267 V a r i a t i o n o f parameters, 35 V a r i e t y , 315 Vector bundle, 360 V e c t o r f i e l d , 359,371 V e c t o r valued d i s t r i b u t i o n , 347 Vector v a l u e d f u n c t i o n , 336 Very r e g u l a r , 345 V i b r a t i n g s t r i n g , 10 Vortex, 78 V o r t e x d i r e c t i o n , 54 V o r t e x tube, 54 Vol t e r r a - L o t k a model, 75 Vol t e r r a o p e r a t o r , 262 W Wave equation, 10,23 Wave f u n c t i o n , 15,57 Wave o p e r a t o r s , 104 Wave propagation, 40 Wave speeds, 189 Weak d u a l i t y , 26 Weakly LSC, 94,246 Weakly complete, 106 Weakly continuous, 240,321 Weak minimum, 7 Weak s e q u e n t i a l convergence, 94,235, 324 Weak s o l u t i o n , 249 Weak topology, 317,318,321 Weierstrass necessary c o n d i t i o n , 31 W e i l ' s i n t e g r a l i t y c o n d i t i o n , 309 W e i n s t e i n formula, 22 Well posed, 13,19,25 Weyl ' s lemma, 6 Wiener Hopf o p e r a t o r , 215 Wiener Hopf equation, 216 Wiener measure, 292 Winding number, 81 Wronskian, 37 Wick o r d e r i n g , 288

x-Y-z Yamabe problem, 77 Zakharov-Shabat system, 171,192

This Page Intentionally Left Blank