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 !