MATHEMATICAL PHYSICS
NORTH-HOLLAND MATHEMATICS STUDIES Notas de Matematica (121)
Editor: Leopoldo Nachbin Centro Bra...
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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 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
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
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"
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
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
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
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
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
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 )
7,
H ( T ) = (1/2)[G(r+T)
while f o r
-
T
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 = ?$I' 2 m - 2 dx = ( l / n ) Jlp Jldx = ( l / f i ) 2 ( A p ) 2 . It f o l l o w s t h a t ( b ) (Ax)(Ap) ~ h / 2
d e v i a t i o n i n s t a t i s t i c a l parlance. m
-/I
-
0 ) . The energy s t a t e a t l e v e l E corresponds t o HQ = E$ ( 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
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$a
(Sh
'L
s i n h and Ch
transparancy o f t h e b a r r i e r .
%
cosh).
That t h e p a r t i c l e may pass through t h e b a r r i e r
i n these circumstances i s r e f e r r e d t o as a t u n n e l i n g e f f e c t .
To c o n t i n u e
t h i s theme, suppose we have a p o t e n t i a l w e l l , U(x) as above b u t w i t h Uo < 0 and Uo
a, U(x) = 0 f o r 1x1 < a, and E < Uo. Then f o r 1x1 > a one has 2 -(Ti /2m)ILxx = (E-Uo)9 so 9 = Bexp(-Bx) f o r x > a and 9 = Cexp(Bx) f o r x < 2 above). I n t h e r e g i o n 1x1 < a one has -a where 6 = [(2m/h )(Uo-E)]' (= 9 = a e x p ( i k x ) + y e x p ( - i k x ) ( k as above).
It f o l l o w s upon matching s o l u t i o n s
a t x = ?a ( w i t h d e r i v a t i v e s ) t h a t t h e r e can o n l y be a s o l u t i o n when E = En where En i s a r o o t o f Tan[2a(En2m/h2)"]=2[En(Uo-En)]'/(2En-Uo)
w i t h 0 < En
Thus t h e r e i s a d i s c r e t e energy spectrum between 0 and Uo and t h e
< Uo.
r e s u l t i n g ,9,
a r e c a l l e d bound s t a t e s .
= AexpBn(atx)[exp(-iana
+_
The wave f u n c t i o n 9,
e x p ( i a n a ) ] f o r x .=-a,$,
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
(the f i r s t 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 [ 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
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
...
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 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 ) - 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)
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
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
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
9 ' ) .
We do n o t d i s c u s s
t h e method o f s o l u t i o n i n [Kyl] where one works from WU* = Uo and l a t e r asks for
u0
=
u-~.
RElllARK 4.13,
The n e x t s t e p i s t o determine W i n terms o f
5 (assume H has
o n l y a b s o l u t e l y c o n t i n u o u s spectrum); we deal h e r e o n l y w i t h t h e 1-D s i t u a t i o n f o l l o w i n g [Kyl].
To compare n o t a t i o n w i t h p a r t s o f [Kyl] we w r i t e e.g.
130
ROBERT CARROLL
$EUt(x)
(**a)
$Eut
= exp(ikx)
!bl i n (4.4).
-
A l s o as x
by
--
$EUt
(*6)
as I k (
-t
'L
= I
e x p ( i k x ) + b ( k ) e x p ( - i k x ) so
Another n o t a t i o n i n [Kyl] i s g i v e n
m.
!bk(x) = e x p ( i k x ) + / z ( x I K I x ' ) e x p ( i k x
(*u)
w i t h AL
f-(-k,x)
(in t h e sense
ky)2dk = (1/2n)[I,"
%
) d x ' so t h a t v i a say (**)
One t a k e s a l s o
K.
(*A*)
1
(x(H0,A ;E,a)
1
f o r k = JE and a = +1, so t h a t
e x p ( i a k x ) = $F(a,x)
=)I
-t
One w r i t e s b ( - k ) = b ( k ) and sometimes (**+) b ( k ) = g ( k )
b ( k ) = s12 = . lR
e x p ( - 2 i a k ) where g = 0 ( k - ' ) ILk(x)
so t h a t
( i / 2 k ) l z exp(ik1x-x' I)q(x')$FUt(x')dx'
= (1/2Jnk)
It I$E(a, 8 =))dE($L(a,
It $F(a,x)$F(a,y)dE = ( 1 / 4 n ) I ;I e x p ( i a k x ) e x p ( - i a J t e x p ( - i k x ) e x p ( i k y ) d k ] = (1/2n)
exp(ikx)exp(-iky)dk t
exp(ikx)exp(-iky)dk = 6(x-y)). t o t h i n k o f (Ho.Ao;E,a)
Note t h a t i t w i l l sometimes be convenient
as $ i ( a , x ) even though t h e x r e p r e s e n t a t i o n has n o t Note a l s o
been s p e c i f i e d ; i n case o f p o s s i b l e c o n f u s i o n we w r i t e $(a,-).
:I $ ~ ( a , x ) $ , ( a ' , x ) d x s i d e r s (xlH,A;E,a)+ (4.21)
= 6(E-E1)6(a,a') %
$ i ( a , x ) = $(a,x)
f
T
i c ) - l ( n o t e (4,21)
%
Now one con-
6(k-k')/2k).
(i/2k)Iz
1:
e
I
I dx
= +(i/Zk)exp(riklx-x'I)
where one uses ( x l y , ( E - H o ) l x ' ) Ho
(6(E-E')
$ i ( a , x ) determined v i a ( ( x l q l x ' ) = q ( x ) s ( x - x ' ) ) I(
x I I q 1x1' ) d x " $ i ( a ,x'I)
and y+(E-Ho)
i s e s s e n t i a l l y ( a & ) , up t o a m u l t i p l i e r ?
=
-
%
(E
-
$F(l,x)
=
t
( 1 / 2 4 1 ~ k ) $ ~ ( x ) ) . Thus i n p a r t i c u l a r f o r a = 1 qE % $- i n ( 0 6 ) and $; 'L $1 = $out f o r a = 1 (up t o a m u l t i p l i e r ) . Now r e f e r r i n g t o ( + a ) - ( + 6 ) i n ("0) i n 1-D t h e S(E-Ho) terms a r e g i v e n by 6(E-Ho) = 1 I $ F ( a " , - ) ) d a " ( $ F ( a " , - ) l and 9 f o r ex-
( t h e I d a " i s c r u c i a l l a t e r on f o r J I $ t ( a " , .))da"($;(a",-)l)
ample has a form ( $ ~ ( a , . ) l s l $ ~ , ( a ' , . ) ) ( o r b e t t e r ( H o , A o ; E , a ~ ~ ~ H o , A o ; E ' , a ' ~ = s(E-E') ( r e c a l l dE = 2kdk). I n o r d e r t o o b t a i n f u r t h e r i n s i g h t i n t o and a b e t t e r understan-
REInIARK 4-15.
d i n g o f t h e o p e r a t o r s W, M+, pi, e t c . used i n [ Y y l ] and developed above, we
w i l l s k e t c h t h e approach
of
[ F a l l and v a r i a t i o n s i n [C2,3,20]).
F i r s t con-
s i d e r t h e h a l f l i n e t h e o r y w i t h i z ' = Hz, z ( 0 ) = zo, z ( t ) = e x p ( - i t H ) z o , zo I e i g e n f u n c t i o n s o f t h e p o i n t spectrum o f H, z,
= Utzo, e t c .
one w r i t e s UWU" = I (as b e f o r e ) , U*UW = I ( W =-(U*U)-'),
Thus i n [ F a l l
W(k) = l/M(k)M(-k),
(UM-')(UM-l)*
e t c . ( s o w i t h M- % M(-k) % M* % M: one can w r i t e e.g. -1 -1 L e t us r e c a l l a l s o t h a t = I, W = M ( M )*, U = UM ,, etc.).
if B
E QUW
S ( k ) = M(-k)/M(k),
%
U then
One d i f f e r e n c e h e r e w i t h t h e ma-
( n o t a t i o n o f [C2,3]).
t e r i a l from [Kyl] i s t h a t t h e domains o f H and
Ho
a r e n o t so e x p l i c i t l y i n -
v o l v e d i n [ F a l l s i n c e t h e t h e o r y i s developed v i a t h e k r e p r e s e n t a t i o n ; t h i s i s c l a r i f i e d below.
2
-D 0
t
Thus t h e model h e r e i n v o l v e s e i g e n f u n c t i o n s 0 ( x , k ) o f
2
cp
= k 0 s a t i s f y i n g 0 (0,k)
tions f(x,k)
%
e x p ( i k x ) as x
e (x,k)
(4.26)
0(x,k)
(*+A)
+
= [F(-k)f(x,k)
0 and 0 ' ( 0 , k ) = 1 a l o n g w i t h J o s t s o l u -
Thus
a.
+
= [Sinkx/k]
f ( x , k ) = eikx Then
+
=
,a -
[Sink(x-t)/k]q(t)e (t,k)dt;
[Sink(t-x)/k]q(t)f(t,k)dt F(k)f(x,-k)]/2ik
and F ( k )
M(k) (note i n
%
N
Remark 2.6.11, f-) = f'f-
+
-
F
%
2ihM2 i s u n r e l a t e d ) i s g i v e n as c ( 0 , f ) = F ( k ) ( s i n c e W ( f ,
f f - ' = 2 i k ) so t h a t F ( k ) = f ( 0 , k ) .
$ (x,k)
= e(x,k)/M(k),
$+(x,k)k
2dk so t h a t TTG
T+g = 1: g(x)$+(x,k)dx = g and (**&) T+T:
p r o j e c t i o n o n t o Jcc (H = -D st(x,k)
=
+ $ (x,-k)
convenience.
One w r i t e s f u r t h e r ( * * a ) = G(k), and T g:
= Ik and TT :+
=
(2/~)/:
= I where I
G(k) %
2 + q h e r e and s i m i l a r formulas h oCl d f o r $ - (Cx , k )
=
L e t us assume no bound s t a t e s e x i s t f o r
= (x,k)/M(-k)). The boundary c o n d i t i o n s h e r e f o r H o r Ho a r e based on Sinkx/k;
t h u s f ( 0 ) = 0 and f ' ( 0 ) = 1. Note t h a t i n general, f o l l o w i n g [C2,3,13;Mrll, 2 2 -1 i f a t r a n s m u t a t i o n Bh: D + D -q = Q has k e r n e l B~ = 6 + Kh and Bh = Bh has k e r n e l yh = 6 + Lh then f o r B h f , one wants f ( 0 ) = 1 and f ' ( 0 ) = 0 b u t w i t h
134
ROBERT CARROLL
Bhg we r e q u i r e g ( 0 ) = 1 and g ' ( 0 ) = h.
T h i s i s an unsymnetrical transmuta-
t i o n and a r i s e s because o f t h e way i n which h i s i n s e r t e d i n t o t h e k e r n e l s Kh and Lh; 6h can be symnetrized t o map Cos x t h[Sinhx/h] + p!yh i f d e s i r e d . 2 From [ F a l l we n o t e t h a t s o l u t i o n s o f i!ht = HJ, ( H = -D t q ) w i t h J,(x,O) =
$0'
( x ) have t h e form (assume no d i s c r e t e spectrum) - i k 2 t k2dk
m
(4.27)
= (2/n)l
J,'(x,t)
*(k)?(x,k)e 0
f o r example where $ i ( k ) = :/
J,i(x)J,'(x,k)dx
= * ( k ) ( t h i n k h e r e o f *(k) as 2 2 *(k)[Sinkx/k]exp(-ik t ) k dk s a t i s f i e s 2 2 = ( 2 / n ) / F *(k)[Sinkx/k]k dk and i f *(k)k E L
Now (*++) x ( x , t ) = (2/71)/:
given).
Hox = -xxx w i t h x(x,O) one has (*+.) 0 = t+*'Iim I$'(x,t) ixt
=
/om
O(x,k)
%
(IM(k)l/k)Sin(kx
(4.28) Then
d(k) =
ixt
= H
:1
-
-
x(x,t)12dx.
argM(k)) as x
+
-
R e c a l l here from
( F = M).
(*+A)
Next, g i v e n J,o s e t 2
J,,(x)$'(x,k)dx;
x' w i t h xf(x,O)
= (2/71)J'"
X'
= (2/n)/:
0
*'(k)[Sink~/k]e-'~
*'(k)[Sinkx/k]k
2dk = x:(x)
tk2dk
= TT :$',
0
=
U:q0
(To i s t h e t r a n s f o r m based on Sinkx/k analogous t o ( * @ a ) and one t
U- = U S, w r i t e s U,- % TT :o i n x or U,- % TOT+ i n k ) . Here f o r m a l l y z+ = U:zo, zt = Sz-, e t c . and U, % l i m exp(itH)exp(-itH,) as t + 2 - . One has a theorem
-
i n [ F a l l ( r e s u l t i n g from
s t a t i n g t h a t i f J,, i s p e r p e n d i c u l a r t o t h e
(*+a))
e i g e n f u n c t i o n s o f t h e p o i n t spectrum o f H and i f i'Lt = HJ, w i t h $(x,O) = Go Thus J, = e x p ( - i t H ) $ O then (*.*I1 ; IJ,(x,t) - x ' ( x , t ) l 2dx -+ 0 as t (% z(t)) and say x t = T;[e~p(-ik~t)T,$~] = T:[exp(-ik 2 t)To(T;T,to3] (recall
+-.
-f
zt
%
l i m e x p ( i t H o ) $ = U*J,
w i t h exp(-itH,)J,,
T;[exp(-ik + O
= exp(-itHo)U:J,o
exp(-itH)bo
-
and e v i d e n t l y e x p ( - i t H ) $ o = T:[exp(-ik Hence x t = T;[exp(-ik
2 t)To$,]).
and one has ll$
-
xt1I2 = llexp(itHo)(J,-xt)l12
and exp(itH,)exp(-itH)
U:J,0112
-f
:U
t)T,$,] 2 t)To(U:$o)l
= llexp(itHo)
The boundary c o n d i 2 = T$[exp(-ik t)TtIL0] =
etc.
t i o n s a r e i m p l i c i t i n expressions l i k e exp(-itH)$, 2 "+ 2 t)$, k dk. I f one used d i f f e r e n t e i g e n f u n c t i o n s f o r (2/7r)/; $,(x,k)exp(-ik H based on s t r u c t u r a l l y d i f f e r e n t i n i t i a l values (say $(O,k) = 0 h e r e ) then e.g.
exp(-itHo)J,o
4
D(H).
T h i s however i s i n c i d e n t a l .
The p l a c e where i n i -
t i a l c a n d i t i o n s a r e used i m p l i c i t l y i n t h i s s k e t c h from [ F a l l i s s u l t s (*+.)
and (*.*)
which use (4.26),
(*+A),
in t h e r e -
etc.
RmARK 4.16.
Now c o n s i d e r t h e f u l l l i n e t h e o r y f o l l o w i n g [ F a l l . One works 2 L and JCo % JC x JC, p % $ = (pl,p2) (column v e c t o r ) , (v,;) = ;/ [(ply t (p2,?2)]dk ( k 2 % E, 2kdk % dE, and a f a c t o r o f 271 i s d i s p l a c e d here
w i t h JC
rl)
%
from t h e end o f Remark 4.14 s i n c e a symmetric F o u r i e r t r a n s f o r m w i l l be employed).
Set To:
J, .+ p : J C +
KO: p1 = (1/J2n)/;
$ e x p ( - i k x ) d x , p 2 = (1/J2n)
SCATTERING THEORY
lz
$exp(ikx)dx.
(l/JZn)/;
Then T:;
JCo
(exp(-ikx),exp(ikx))
+
I and T0T*0 = I ; n o t e T*q = 0 ( 1 / J 2 1 ~ ) [ r v,exp(ikx)dk ~ + J r v2exp
JC w i t h T;To
-+
+
lpdk =
=
Now u1 = !b2 and u2 = !bl i n [ F a l l so u;
(-ikx)dk].
'1.
=
(column v e c t o r ) i s d e f i n e d v i a components
+
( r e c a l l u1
+
u;(x,k)
=
;t(x,k)
u;(x,k)
=
5,
= T-f+
JC : J/ 0
-+
(l/JZn)/:p(x)
=
= T+T:
=
I.
(!b2,!b1)!bdx T h i s i s ach-
and one a l s o w r i t e s u - ( x , k ) = ;;(x,k = T - f + ( - k , x ) and 1 where we r e c a l l f + ( k , x ) = f + ( - k , x ) e t c . so
0-
= T-f+(-k,x)
-
and u;
= Vj2
Set t h e n
= T-f-(-k,x).
(*=A)
-
f
T(f-) = (-st 21
( -), ft
s e t (*=&) U+ = TT :o
Note from
- : 1-2 ) ( ~f+- ) -
0
+ JC
0
f- - u and T(f )+I-, , (
+
+
= U;U '-
c\,
while
-
2 -+
('1-) 92
= 9
=
T*T T*T and o + + o
?
=
-> (1/2n)J
JI
T ST* = T+T*. 0
I
0
+
R s ( k - q ) ( s i n c e RiT = -R,T-),
+ Rrs(k-q).
$4. integrals 1 J
(1/2n)
(A+*)
( 1 / 2 ~ ) J JI1 ( k , x N l (q,x)dx = (Tq/Ti)G(k+q) + and (1/2n)/
Note now t h a t (T /T-)a(k+q)
$2(k,x)$2(q,x)dx
= (T /T-)s(k+q)
q
k
: 6(k+q) so t h e e q u a t i o n (*==) can
6 ( k + q ) ) + Ss(k-q) where S appears q 5k = ( 0 v k w i t h (A*A) s(k+q) 0 The 6(k+q) terms seem t o g e t i n t h e way h e r e b u t i n f a c t t h e k-
be w r i t t e n as (*=A).
(1/2n)
('Z)($;,$i)q=
JJ'2J/1'1 '2'2'2) (here the + , ~ 1 + J / 2 ~ 2 I d k= ( 1 / 2 ~ ) ( , - ~1V$j 1 1 + @l$2p2 are i n x!). Now r e c a l l (4.25) - (*&=) i n Remark 4.14 so t h a t
J ('2)CJ1
Consequently
$1
92
= Ts(k-q),
w i t h pPf
Then (l/JZn)/ $ ( TT--ff+- - ) = ( l / J Z n ) / '('l-). JI2
i s determined v i a (*==) ('1)
J IL2(kyx)lL, (q,x)dx
in
and (4.23) (*=a)
. IL 0'
I and (*=+) S
=
(A)
and U- = T*To where T-: J c - + JC
( A ) = ( 1 / J 2 n ) / JI u;(x,JA)dx U*'U'
=
q =
-+
= Su- where S i s t h e m a t r i x w i t h T i n t h e d i a g o n a l s and e n t r i e s s.ii
f
in
and u;
$2 e t c . so t h i s corresponds t o (l/JZn)/:
= T f-(-k,x)
the (i,j) position.
?:JC
+ 9i(x)
We assume no d i s c r e t e s p e c t r a and t h e n TT :+
Then d e f i n e U+ = T*T
+ u
%
= Zhk$i(-l,x)
S i m i l a r l y T+: JC
2Jnk$i(1 , x ) ( c f . t h e end o f Remark 4.14).
+ + (q1, v 2 ) + ui(x,Jx)dx
135
-+
s c a l a r p r o d u c t comes from JE and i n v o l v e s h e r e and i n v o l v e s h e r e k
0 only
rr
so t h e r e i s no c o n t r i b u t i o n from s(k+q).
Consider S = T;15To, where an ac-
t i o n ( a ( k + q ) , e x p ( i k x ) ) = 0 s i n c e k,q 2 0 and N
(4.30)
S = ( 1 / 2 n ) l m (eiqy,e-iqy 0
T R exp(-iqx))dq )(RL Tr)(exp( i q x )
136
ROBERT CARROLL
Hence ( r e c a l l a,q 2 0 ) (A*@) ?exp(icxx) ilarly
+
x e x p ( - i a x ) = Rrexp(iay)
T = s l l = s 2 2 y Rr = sZ1 ,
Re = s 1 2 ) as
+ R / ( e x p ( - i a y ) ) and sim-
= Texp(icry)
It f o l l o w s t h a t ( r e c a l l
Texp(-iay).
-
i n (4.22)
( - 1 l$ll
(*Am)
)
(1/2n)
%
% RfS(0) % s12 and ( 1 Itill ) ( 1 / 2 n ) / exp(-iay)dexp(iax) s l l = sZ2 and s i m i l a r l y f o r t h e corresponding formulas based on
J exp(iay)dexp(iax)dy dy
T6(0)
%
‘L
t h e equations
(A*&).
F i n a l l y l e t us n e x t w r i t e o u t t h e formulas o f [ F a l l f o r t h e w e i g h t o p e r a t o r s . Thus one d e f i n e s (A1 (4.31)
UIJ’
=
(U.: JC-+ JC) and Vi: 1
/I$(x)fl
“u
A R y A2
,,” A1 ( x , y ) $ ( y ) d y ;
+
JC
-f
AL i n (**))
%
U2$ = 9 ( x )
(**+) Vl$
JC a r e g i v e n by 0
+
f,A2(x,Y)$(Y)dY
= q ; ql(k)
= (1/J2n)
( x y - k ) d x ; 1 ~ 2 ( k =)
J’(x)dx where fl = f+ and f2 = f - . t h i s c o n s i d e r e.g. (4.32)
VTToJ’= (1/2n)jom f;,fl (1/2iT)/z (UleikY)e-jkxdkdx etc.).
t a t i o n s U - s a t i s f y i n g HU = UH,
M- = U+M+,
W
e t c . w h i l e S = U;’U-,
where U,-
‘L
U = UM,,
etc.
and Nf = T;hiTo.
T:To
U = U-
Here Then
w i t h sZ1 = b/a = Rr s i n c e i n [ F a l l s12
w r i t i n g a = 1/T = c12 and b = cll our p r e v i o u s s,,,
Thus WU*U = I = UWU*,
S = M-/M,
1/MM-,
“u
U1$
Now we want t o deal w i t h transmu-
e t c . as b e f o r e .
we w r i t e f o l l o w i n g [ F a l l Ui = UN,:
=
etc.
+ a 0 a - O M i = (-b- l ) ; M = ( - ); M- = ( ’ b - ) 2 b 1 2 Oa
“+
One has N t = T;NiTo b = Rr/T,
with
4;
= (MY)* 1
sZ1 = Rry s12 = Ra = -b-/a, A
s t a t e s f o r convenience we have Wi
UiWiU?.
s2 1
Thus
and S = MY-lM;
a = 1/T, e t c . ) .
“+-I .’+*-I
= N;
lN1
i1 = ( s11521) and i2= (512 ^w2 %2h i n (*@A) o f Remark
i n [ F a l l and
+-1 -
= M2
-
’12
M2 ( n o t e here a g a i n
Assuming no bound
= (MfMf*)-l
= TW ; Tio
with I =
) ( r e c a l l o u r p r e v i o u s s12 i s
1 4.13
-
c f . a l s o 511 and [Fa3]).
SCATTERING THEORY
137
5. SCACCERINC CHE0Rg. I U (SPECCRAC MECH0D8 I N 3 4 ) .
We c o n t i n u e w i t h some
more r e c e n t developments, i n c l u d i n g a b r i e f t r e a t m e n t o f R. Newton's MarEenko method f r o m s e v e r a l p o i n t s o f view ( c f . [NwP;Rsl;Yel]).
Again a remark
f o r m a t i s used i n t h e same s p i r i t as before ( c f . a l s o [ N b l l ) . R e f e r r i n g back t o Remark 4.2 l e t us go now t o t h e M e q u a t i o n o f
REIRARK 5.1,
Newton ( c f . [ N w ~ ] ) . R e c a l l f i r s t f r o m d e f i n i t i o n s and a s y m p t o t i c p r o p e r t i e s # T # T t h a t J,' = Q8 J, ( n o t e 8#dT = b 8 = I, J/(-k,x) = J/(k,x)-, b = QSQ, e t c . ) . ( H = (;
Set y ( k , x ) = exp(-ikHx)J,(k,x)
8,
= exp(-ikHx)bexp(ikHx);
-9)
h e r e ) and t h e n y # = ;S,'Qy
thus
T
bx
(5.1)
=
1
Rrexp(2i kx) T
Rgexp( - 2 i k x ) One notes t h a t 5,
where
i s t h e m a t r i x which would a r i s e f r o m a p o t e n t i a l U(x+y).
Now i n o r d e r t o deal w i t h ( * 6 ) i n Remark 4.2 d i r e c t l y one w r i t e s i t i n t h e f o r m (*) f # Define
-
I = [u-I]Q[f-I]Q + Q[f-I]Q + 0 - 1 and t a k e s F o u r i e r t r a n s f o r m s .
(A) Z ( a ) =
(1/271)lI e x p ( i k x ) [ u ( k ) - I l d k Q ;
Then (*) becomes
[f(k)-Ildk.
( 0 )
F = 2:
*
F#Q
F(a) = (1/2n)/f
+
# QF Q
exp(-ikx)
+ ZQ. B u t f ( k ) - I i s
a n a l y t i c f o r Imk z 0 and vanishes as I k l + m t h e r e so F ( a ) = 0 f o r # Hence f o r a 2 0, F(a) = Z ( a ) Q + ( C * F ) ( a ) Q and we have Given F and Z d e f i n e d by
CHE0REM 5.2,
(A)
a
0 ( * a ) Zl(a,x)
-
=
-; (at2x) r
- 1;
1 ; ~ l ( a t ~ - 2 x ) H 2 ( ~ y x ) d ~where
ir(a+6
^R
m
(a) = r 9 . t
( 1 / 2 1 r ) / ~ Rrye(k)exp(ikoc)dk. (*b) -2DZ1(0,x)
Then a f t e r some m a n i p u l a t i o n one o b t a i n s a l s o We n o t e t h a t i f q ( x ) = 0 f o r x < 0
= q ( x ) = 2DxZ2(0,x).
t h i s reduces t o t h e G-L procedure ( c f . [ N w ~ ] ) . Again bound s t a t e s can be worked i n t o t h e t h e o r y b u t we o m i t t h i s . Now go t o 3-D so one considers -A9
RRnARK 5.4. equation
(*+I +(k,e,x)
where 9 (k,e,x) f u r t h e r v(k,e,x)
J/,(k,e,x)
e x p ( i ( ke,x)
=
0
=
(thus k
I k / i n say (3.2) and IL
- 9 0 (k,e,x)]
= -S-(k,e,;)
can see t h i s from ( 3 . 2 ) s h i f t e d as 1x1
+ m).
jii
and ST =
(*a)
8'
e,e')
= k(k,e',e), =
u n i t sphere S
= A(k,;,e)
-
where x = x/
XI
=
-Ixlexp(-iklxl) %
8 ' here (one
as t h e k e r n e l o f
( k / 2 a i ) A ( k ) ( t h i s i s t h e same as ( 3 . 6 ) ) .
= Zi(k,e,e')
and A(k,e,e')
QsQ where s( has k e r n e l i ( k , e , e ' ) ,
and Q f ( e ) = f ( - e ) .
8-'Q$.
-tm
For f i x e d k one t h i n k s o f A(k,e,e')
One has e v i d e n t l y A(-k,e,e')
9'
Set
or (*+) by n o t i n g t h a t t h e argument o f I x - y l g e t s
an o p e r a t o r A ( k ) and S ( k ) = I =
@-).
%
Note here from (3.5) t h a t - S (k,e',e)
and by (3.7) one expects (@-(x,k) = 9 ( k , e , x ) )
A(k,e',e)
(A*)
3 [ e x p ( i k l x - y l ) / ~ x - Y ~ I ~ ( Y ) ~ ( ~Y , ~ , Y ) ~
ke,x)) and (*=) A(k,e',e) = -(1/411) t h a t A(k,e,e) = - ( 1 / 4 1 ~ ) 1 q ( x ) y ( k , e , x ) d 3 x
(forward s c a t t e r i n g amplitude).
and
%
2 3 q9 = k 9 i n R w i t h L-S
= $(k,e,x)exp(-i(
I ~ o ( k , e ' , x ) q ( x ) I L ( k , e y ~ ) d 3 ~so
[@-(x,k)
- 1
t
A(k,-el,-e) so t h a t T T 8 has k e r n e l 8 ( k ,
=
It i s easy t o see t h a t
$*k
I
= .5$* =
One t h i n k s o f A and 8 a c t i n g on f u n c t i o n s o v e r t h e
2
as i n ( 3 . 6 ) and under reasonable hypotheses on q ( x ) ( c f . [ N w ~ ] ) one has f o r F E Lp(S2), p > 4 ( A A ) L; k211A(k)Fl12dk < cllFI12 Hence 2 2 P2 7 P' Jm II(5-1)FII dk 5 cllFII and s i n c e IIgll < cllgll i n Lp n L ( S ) i t f o l l o w s t h a t P p 2 P f o r s u i t a b l e F, if II (S-I)Fl12dk < cllFf2- so t h a t t h e F o u r i e r t r a n s f o r m o f P' ( 5 - 1 ) F w i l l make sense c l a s s i c a l l y .
-0.7
REIIIARK 5.5.
Now i n 3-D one cannot d e f i n e
v
by toundary c o n d i t i o n s ; i n s t e a d
t h e J o s t f u n c t i o n J i s determined f r o m a R-H problem f i r s t and then f i n e d v i a J. potheses
v
i s de-
Thus ( a g a i n assuming no bound s t a t e s and assuming s u i t a b l e hy-
on q ( x ) as i n [Nw2,3]) J i s determined as a s o l u t i o n o f Then J w i l l be a n a l y t i c i n C+ w i t h I I J ( k ) - Ill
QJQ$(R-H problem).
(Am) +
J#
=
0 as I k l
SCATTERING THEORY
139
m and 9 i s d e f i n e d as q ( k , x ) = J(k)$(k,x). Then Ip(-k,x) = Qp(k,x) and f o r f E Lp(S2), @(k,x) = 1 f ( e ) [ v ( k , e , x ) - $o(k,e,x)lde E Lk2 w i t h 11@11: 5
-+
t
cllfl12 Furthermore @ has a n a l y t i c c o n t i n u a t i o n s i n t o C and C-, e x p ( i k l x 1 ) P ' t @ i s bounded i n C , and e x p ( - i k l x l ) @ i s bounded i n C-. Hence @ E L 2 w i t h support i n [ - l x l , l x l l
and
Thus 9 i s e n t i r e i n k o f e x p o n e n t i a l t y p e 1x1 , b u t i t s e x i s t e n c e depends on s o l v i n g t h e R-H problem
[ N w ~ ] ) shows t h a t
(A&)
(Am)
f o r J.
2e-v[w(e.x
t
Some a n a l y s i s which we o m i t h e r e ( c f .
,e,x)
-
w(e.x-,e,x)]
c a l l e d t h e " m i r a c l e " ; t h e e q u a t i o n appears t o be i s a potential q(x)
(A&)
One can a l s o t h i n k o f
i s guaranteed ( i . e . t h e
e e
= q ( x ) and t h i s i s
dependent b u t when t h e r e dependence d i s a p p e a r s ) .
v - $ ~as a f u n c t i o n o f ke and produce a f o r m u l a (A+) 3 - J;y,~lxl h(x,y)$,(k,e,y)d y. A s p e c t r a l measure f o r t h e
Ip(k,e,x) = $,(k,e,x) 9 f u n c t i o n s can a l s o be o b t a i n e d as b e f o r e s t a r t i n g w i t h ( A = ) 6 ( x - y ) = ( 1 / 2 2nI3/ ?(k,e,x)$(k,e,y)k d dk. Assuming no bound s t a t e s one o b t a i n s 1 (q(k, = 6 ( x - y ) where (**) dp/dE = ( k / 1 6 n 3 ) [ J ( k ) J * ( k ) ] - '
x),dp(E)p(k,y)) 0).
Using
(A+)
ho (x,Y) =
I ($o( k, x ) ,d(P-PO No ( k ,Y)
3 where dpo/dE = ( k / l 6 n ) f o r E > 0. computes w v i a Radon t r a n s f o r m s uses
(A&)
RrmARK
(@A)
To o b t a i n t h e p o t e n t i a l f r o m h one f i r s t w(cr,B,x)
=
1 h ( x , y ) & ( a - e - y ) d 3y and t h e n
(we w i l l d i s c u s s Radon t r a n s f o r m a t i o n s i n more d e t a i l below). The procedure above has i n v o l v e d f i r s t d e t e r m i n i n g J v i a a R-H
5.6,
problem and t h e n u s i n g t h e G-L t y p e machinery.
It i s more d i r e c t t o use a
M method as f o l l o w s ( a g a i n assume no bound s t a t e s o c c u r ) . back t o
(E = k2 >
one o b t a i n s a l s o a G-L t y p e e q u a t i o n
(me)
JI# = $-'Q$ and s e t s y(k,e,x)
= $(k,e,x)exp(-i(
Thus one goes ke,x)).
I n the
= $(k,e,e')exp(ik(e-e',x)) same s p i r i t as b e f o r e one s e t s ( a & ) $,(k,e,e') # and t h e n Y = $,'QY. I t f o l l o w s a l s o t h a t bx a r i s e s f r o m a s h i f t e d potenn 2 t i a l q x ( z ) = q ( x + z ) . L e t 1 be t h e f u n c t i o n 1 on S and d e f i n e
(5.6)
G(a)
= i ( 2 n ) - 2 1 z kQAx(k)e kadk.
G(a,e,e')
= i(2a
-2
:I
kA
The corresponding M e q u a t i o n now i s ( a > 0 )
k,-e,e')e - i k [ a + x . ( e + e t ) l d k
140
ROBERT CARROLL
=
t G(a)? where y ( k , x )
q ( x ) t h e operators G
rf
t i c f o r Imk > 0, S;'Qy,
t
1; F ( a ) e x p ( i k a ) d a .
G#
G(a+B) and
%
e t c . ( n o t e y ( k , x ) = Y(k,e,x)
there i s a miracle equation
G ( - a - 8 ) a r e compact, y ( k , x )
%
2 11 dk
ei x )
(5.16)
Ts(xytyei)
=
[
2eS +
M(t-ei-x,es,ei)d
S
where M(t,es,ei)
= -(1/8n
-2ei-vGs(x,t=ei
x,ei)
2
I
)DtR(t,eS,ei)
and t h e m i r a c l e e q u a t i o n i s q ( x ) =
( c f . also (5.29)).
The c o n n e c t i o n w i t h t h e M e q u a t i o n
i n Remark 5.6 w i l l be e s t a b l i s h e d l a t e r ( c f . Remark 5.13).
REmARK 5.11.
The development i n Remark 5.10 i's somewhat c o m p l i c a t e d a t
times and t o shed some l i g h t on a l l t h i s we g i v e a n o t h e r d e r i v a t i o n o f t h e One begins w i t h t h e
M e q u a t i o n f o l l o w i n g [Rs1,2]. (5.17)
$'(k,e^,x)
so t h a t $+
'L
@-
= e ik'.x
-I
i n (3.2) and $ -
'L
L-S e q u a t i o n i n t h e form
[e'iklx-Y1/4nlx-y[]q(y)$'(k,$,y)dy Here t h e + s i g n
@+.
= (l/2n)lm f(k)exp(-ikt)dk t E v i d e n t l y (em) $-(k,s,x) = $ (-k,-$,x)
A
so t h a t f = e x p ( i k X I ) 6(t-lxl) . and $' k,G,x) = $' -k,e",x). Set now ( c f . -+
(5.18)
outgoing r a d i a t i o n m
condition since our Fourier transform i s F - ' f ( k ) (em))
+ [ e i k ' X ' / ~ x ~ ] A ( k y ~ s . ~ i ) + h(k,giYx); $+( k ,Gi x ) = e ik'i.x
si)
A
- ( 1 / 4 1 ~ ) [ e-ikes.x
=
q(x)$+(kyGi ,x)dx
U ) . T h i s i s t h e same as (em) except t h a t e x p ( - i k l x l ) i s c o n s i d e r e d (q (1/2n)exp(ikt)). outgoing there ( t h e Fourier transform being F-l I n the Q
p r e s e n t c o n t e x t (5.17) i n v o l v e s t h e same A as i n ( * w ) .
The i n v e r s i o n i s
( f o l l o w i n g (5.10)) (5.19)
f ( x ) = (1/16n3)11
[ 2$'(k,e",x)~
?'(k,g,y)f(y)dyd26k2dk
S which l e a d s t o a Radon i n v e r s i o n (5.11).
One denotes by u'(t,G,x)
verse F o u r i e r t r a n s f o r m of $' g i v e n by (5.17) (5.20)
u'= 6 ( t - G - x )
-
-
the i n -
thus ( r = I x - y I )
1 I [6(r~(t-~))/4n~~-y~]q~(y)u'(~,~,~)d~dy
144
(!b'
ROBERT CARROLL
i s extended t o n e g a t i v e k v i a (+.)
t
and one has u-(t,e",x)
= u (-t,-;,x)).
Now suppose a t l a r g e n e g a t i v e t a p r o b i n g wave uo = 6 ( t - e - x ) i s given, and t h a t q ( x ) has compact s u p p o r t . (PWE) [-A + q ( x ) ] u = -utt
The s o l u t i o n o f t h e plasma wave e q u a t i o n
f o r l a r g e n e g a t i v e t equals uo (so u i = 6 ' t h e r e )
and a f t e r s c a t t e r i n g one has u
^eS,')
-
= l i m Ixl[u(t,^ei.x)
1x1) so t h a t ( m A ) R($iyGsyt)
J exp(-ikt)A(k,e"s,;i)dk.
uo t us.
6(t-ei-x)] = -(1/4n)J
For t h e f a r f i e l d s e t (.*)
as x , t
-+
-with
T =
t-1x1
u+(t+e^s-y,6i,y)q(y)dy
T h i s a l l agrees w i t h
(0.)
-
(e"S
R(giY = x/
= (1/2n)
(6*) modulo t h e change
of s i g n a r i s i n g f r o m t h e use o f d i f f e r e n t F o u r i e r transforms.
Now t a k e a
p r o g r e s s i n g wave expansion ( Y = Heavyside f u n c t i o n and E ( s ) = s Y ( s ) ) (me) u(t,g,x) = 6(t-z.x) + F(t,e^,x) t B($,x)Y(t-$-x) t D($,x)E(t-g.x) ( F E C1 i s zero f o r t < G-x).
The " t r a n s p o r t " equations a r e determined by p u t t i n g (me)
i n t o t h e PWE and e q u a t i n g s i n g u l a r terms so t h a t one o b t a i n s
( t h i s fundamental i d e n t i t y i s a c t u a l l y t h e same as Newton's m i r a c l e ) .
Now f o l l o w i n g [Rsl] we g i v e two d e r i v a t i o n s o f t h e Newton M e q u a t i o n ( t i m e and frequency domain). (5.22)
$'(k,e",x)
F i r s t i n t h e frequency domain = JI-(k,G,x)
- (k/2ni)12A(k,;',;e)JI-(k,G',x)d~' S
+
To see t h i s n o t e t h a t t h e r e l a t i o n (-A-E)-' = (-Atq-E)-' ( c f . (5.15)). (-A-E)-lq(-A+q-E)-l f o r E = k 2 t i c can be w r i t t e n ( ~ 6 )(-Atq-k 2 t i c ) - ' = -[I-Go(+k+i~)q]-'G
0 (+k*iE)
where Go(k) i s t h e o p e r a t o r w i t h k e r n e l Go(k, I x -
y I ) = - e x p ( i k l x - y l ) / 4 n ~ x - y ~ ( c f . [Nwl;Rsl]). Apply t h i s t o q ( x ) e x p ( i k z - x ) and use (5.17) t o o b t a i n ( m + ) JI'(k,$,x) = exp(ike"-x) (-Atq-k 2 t i c ) - '
ii!
[q(x)exp(ik&?.x)] this). t
J, (k,^e,x)
the
(see here a l s o ( 6 4 ) i n 13 f o r an a l t e r n a t i v e f o r m u l a t i o n o f
S u b t r a c t i n g t h e + and - equations from ( m + ) g i v e s (H = - A t q ) - J/-(k,e^,x) = [ ( H - k 2 - i c ) - ' - (H-k2+ic)-'](qexp( i k g - x ) ) .
11;
L !l [ 1 ( i n a s u i t a b l e sense)
(ma)
But
i s ( - n i / k ) P k where Pk i s t h e s p e c t r a l pro-
j e c t i o n r e f e r r e d t o k i n s t e a d o f E (dE = 2kdk - see below and c f . (*+), ( * 6 ) , ( b m ) , e t c . i n 12.3). On t h e o t h e r hand by (5.19) - w r i t t e n as (1/8n 3 ) 1 ; Jsz - t h i s s p e c t r a l p r o j e c t i o n i s determined by t h e ISe and u s i n g t h e 9e i g e n f u n c t i o n expansion one o b t a i n s h
(5.23)
J,'(k,;,x)
-
J/-(k,$,x)
= ( - i k / 8 n 2 ) 1 2 J , - ( k y 2 ' , x ) j ?-k,G',y)qe ike -yr
S
(r
= dyd';').
Note h e r e i n (*+) o f 13 f o r example w i t h t
-2kdk one o b t a i n s ( E a i ) ( E ( A ) f , g )
= l i m JA(ARf,g)(-2kdk)
%
-k2 and d t =
so dE/dk
%
( l i m AR)
SCATTERING THEORY
(-2k/2ni). (5.24)
145
Now go t o ( 5 . 2 2 ) and w r i t e f i r s t
+
-
= $ (-k,-?,x)
$+(k,e",x)
(k/2ni)12 A ( k , 6 ' , ~ ) $ + ( - k , - ~ ' , x ) d 2 ~ ' S
+
M u l t i p l y by e x p ( - i k g - x ) and w r i t e B(k,G,x) = $ (k,C,x)exp(-ikG-x) t o g e t B(k,;,x) = B(-k,-G,x) - (k/2ni)Js A(k,s',;)exp[ik(t$'-6)-xIp(-k,-$',x)d 2.e '
-
from which ( s u b t r a c t 1 ) one has 6-1 = B--1 S e t t i n g (***) F(a,$,x)
G ' ,x)
= (l/Zn)!;
m
= ( 1 / 2 n ) L m exp[ik(a+(&;'
( k / Z n i ) J Ae
-
exp(-ikx)[B(k,s,x)
-
( k / 2 n i ) / Ae(B-1).
l ] d k and
M(a,e^,
(**A)
) - x ) ] i kA( k,$' ,$)dk one t a k e s i n v e r s e F o u r i e r
transforms i n the B equation t o obtain (5.25)
F(a,e^,x)
1;
+
(5.26)
A
A
2 e"' +
S
i2
A
4
M(a-6,e,e',x)F(-6,-g4
,x)d2g'dS
i s a n a l y t i c f o r Imk > 0 w i t h B-1 s u i t a b l y bounded
Now B(k,;,,x) F(a,G,x)
+ j2M(a,e,e',x)d
= F(-a,-e^,x)
Hence f o r
= 0.
F(a,$,x)
=
I
CY
>
SO
for a < 0
0
2 M(a,e,e' , x ) d e ' +
I" 1
S
0
A
A
M(a+G,e,e' ,x)F(6,-$'
,x)r
s
(I- = d2$'dS).
This leads t o
CHEBREIII 5.12,
One can d e r i v e t h e M e q u a t i o n (5.26) as i n d i c a t e d and t h e
miracle i s q ( x ) = -26-~F(O+,g,x).
REmARK 5-13,
- (*+)
For comparison purposes l o o k a t (5.6)
and
(**A)
-
(5.
26). Thus M(a,S,-;',x)/Zn = G(a,G',$) represents the kernel o f G ( G ( a ) : 2 2 2 2 L ( S ) + L ( S ) f o r example) and (5.26) i s a k e r n e l f o r m o f (.+).
REmARK 5-14. above.
One g i v e s a l s o i n [Rs1,2]
a t i m e domain f o r m o f t h e M e q u a t i o n
Thus t a k i n g i n v e r s e F o u r i e r t r a n s f o r m s i n (5.22) we g e t
(5.27)
u+(t,e^,x)
Now use u - ( t , s , x ) (5.28)
t = u (-t,-g,x)
usc(t,s,x)
-
= u-(t,;,x)
-
(1/2n)j S
I~u-(T,~',x)D,R(~,~',t-T)dTd'~'
w i t h u+(t,e*,x)
= u sc ( - t , - $ , x )
-
=
6 ( t - $ - x ) + uSC(t,g,x)
( 1 / 2 n ) 1 2 DtR(s,g',t-e S
I,u
m sc (T,-e',x)DtR(e^,~',ttr)dsd2he' S But by c a u s a l i t y u(t,e^,x) = 0 f o r t < 6 . x so f o r t > e - x
(5.29)
(1/2n)/
usc(t,e,x)
-
= - ( 1 / 2 ~ ) /Rt(&,e',t-e ~
( 1/ 2n ) I s *
lzl.*
S ( T ,- ^e ' ,X ) Rt ( e^,
x ) d 2, e'
' ,t + T
-
) d.r d2e"'
x ) d 2.e '
t o get
146
ROBERT CARROLL
T h i s i s t h e same as (5.26) ( c f . a l s o ( 5 . 1 6 ) ) w i t h +
-
h
-
&(a) = u
sc
= t
*
(t,e,x);
-
$ax and 4
A
M(a,e,e',x)
=
(1/2n)DaR(e",G
REmARK 5-15, Note i n pond t o
A
u (a+e-x,e,x)
F(a,e",x)
(5.30)
CL
U~'(T,-$',X).
6 = Tt $'.x.
5.26) we have F:&,-;',x) and we want t h i s t o c o r r e s sc But F(a,e^,x) - u (t,e,x) f o r a = t-e^-x so one t r i e s
Then t h e argument i s c o r r e c t f o r M
l i m i t i n (5.29)
R and we have -:'-x
%
as a
(instead o f $ l a x ; c f . [Rsl]).
The f o l l o w i n g from [ R s ~ ] serves t o u n i f y some o f t h e preceed2 i n g m a t e r i a l , One c o n s i d e r s i n 3-D ( * * a ) (A-Vw2-q+w ) $ = 0 under s u i t a b l e
R E M R K 5.16.
hypotheses on V and q.
For v a r i o u s V,q t h i s can be r e l a t e d t o v a r i o u s phy-
= exp(iw s i c a l problems. The L-S e q u a t i o n ( c f . ( 5 . 1 7 ) ) i s (*r) $'(w,;,x) $ - x ) t J G i ( q x - y ) [ q + w 2 V](y)$'(w,$,y)dy where G i ( o , z ) = -exp['+iw(z 11/41~1z I.
2
The Green's f u n c t i o n s s a t i s f y (**+) [A-w V-q+w2]G' i n (3.8) by (G here
-G i n ( 3 . 8 ) )
(**,)
G' = G'
0
= 5(x-y) and a r e g i v e n as
As i n
+ I Gi[q+w2V]G*dz. 0
(5.17)-(5.18) one has (e"' Q gSy e" Q S i ) (*A*) A(w,6',6) = - ( 1 / 4 ~ ) Ie x p [ - i w 2 ;'.y](w V+q)$+(w,$,y)dy. By r e s u l t s i n 53,5 one knows t h a t f o r t h e Schrod+ + A 2 i n g e r e q u a t i o n ( V = 0 ) (*AA) - [ 8 ~/ i w ] [ G (w,x,y) - G-(w,x,y)] = JS21L-(w,e,x)
-+
$-(q;,y)d6
( a l l formulas i n a d i s t r i b u t i o n sense).
Then w i t h some a d d i t i o n -
a l hypotheses an a n a l y s i s u s i n g F o u r i e r t r a n s f o r m and c a u s a l i t y y i e l d s
tHE0REm 5.17.
I f IT(x,Y)
I
5 Ix-yI/c,
(c-'
-2
= 1-V 2 c m ) then (*A@)
3 y ) = - ( 1 / 1 6 ~ )lf e x p ( i w ~ ) $'(qe^,x)S/'(w,~,y)de^dw ~2
(*A&)
,s
G+(O,x, 3 s ( x - y ) = - ( 1 / 1 6 ~)
lrnjS2 ( A - - q ) e ~ p [ i w ~ ( x , y ) ] $ - ( q $ , x ) ~ ~ ( w , $ , y ) d & ! (~h~e r e A-q can o p e r a t e i n x
- m
[I
o r y ) and f o r u' d e f i n e d v i a $'(w,z,x) = exp(iwt)u'(t,;,x)dt 2 (1 / 8 )~/f Js2 ( A-q)u '( t - T ( x,y) , , x ) u +(t,??,y)dGdt.
y) =
-
Thus f o r s u i t a b l e and one has
(*a*)
3
(*A+)
S(x-
-'
$'(w,z,x) = I d y $ (w,;,y)ip(y)exp(iwT(x,y)) 3 2. i ip(x) = - ( 1 / 1 6 ~ ) ( A - q ( x ) ) [ f &JJs2d e$ (w,$,x)ip'(w, y. EHZ0REIII 6-12. i s anticausal.
The M e q u a t i o n i s Upon c o n s t r u c t i o n B
= ( 1 / 2 ~ l ) i z@:(x)[A:(y)/c-]dA
-1
Q
=
and J ( t , x ) = 6 ( t - x ) + (1/21r)L:
RMlARK 6-13, t h e case A
Q
For x , t
20
5
one can w r i t e
has k e r n e l S ( t , x ) + J ( t , x ) w i t h ( s @,(x)dA G
= BH* = g(HAH*) where Y
-
i s causal and
'i
v i a K-L t h e o r y w i t h k e r n e l ;(x,y)
k-';
= HzH* where t h e r i g h t s i d e
G c / c - ) S ( t , x ) = ( 1 / 2 n ) L I sQ(X)@,(t) Q ( XQ) @GX ( t ) @ A G X(y)dh.
one has a l s o J ( t , x )
= 1 w i t h Qu = u "
6
Hence
= (1/2n)LI @f(t)@FA(x)dh.
In
qu one can w r i t e t h e G-L e q u a t i o n o f Remark
156
ROBERT CARROLL
B-l) while the
6.8 as A = TXW(S) = 33* (3
5s'.
=
5
Hk*
HTXw^(S)H*.
M e q u a t i o n o f Theorem 6.12 i s
The f a c t o r i z a t i o n p o i n t o f view i s u s e f u l f o r com-
p u t a t i o n a l purposes when one d i s c r e t i z e s t h e problems a p p r o p r i a t e l y .
We
w i l l g i v e i n 52.7 a n o t h e r d e r i v a t i o n o f t h e r e s u l t s i n Theorem 6.12 and generalizations thereof. We go now t o systems o f t h e form ( 6 . 1 ) .
L e t us f i r s t remark t h a t i f one i s
d e a l i n g w i t h a seismic problem pvtt = -Px ( P = p r e s s u r e ) , P = pv
X'
and w =
a c o u s t i c impedance, one obvt ( v e l o c i t y ) , t h e n f o r y = t r a v e l t i m e and A -1 t a i n s w = - A Pt and P = -Awt. Then s e t t i n g = A-%P and CP = A% ! with p = Y Y (*+4)/2 and q = (*-@)/2 one o b t a i n s (A+,) p + pt = -rq and qy - qt = - r p , Y where r = (1/2)D logA ( r e f l e c t i v i t y ) . Thus we a r e i n t h e c o n t e x t o f (6) Y - i k ) (A=) where p 'L WR and q 'L WL and t a k i n g F o u r i e r t r a n s f o r m s i n t (D Otl D f = i k H f - r A f , where H = - j = f 2, (!)Ay and A = ( o); n o t e a l s o
*
x f
-+
(b -p),
w (WR)
w i t h WR = ( V + I ) / 2 and WL = ( V - I ) / 2 from (6) ( - W = -r ).
This i s
t h e sake as (6.2) i n [Nw2] o r (11) i n [ H w l ] except f o r a f a c t o r o f 2 ( 2 r
V , 2f
2,
y, e t c . ) .
We r e f e r t o [H21;Nw2]
2r
f o r unsupported statements i n t h i s
s e c t i o n and l e t us n o t e t h a t t h e t h e o r y o f " w 2 ] i s a l s o sketched i n 52.4.
I n f a c t we r e p e a t here ( i n expanded form and s p e c i a l t o t h e h a l f l i n e prob& 1 lem) a few equations and r e s u l t s f r o m 552.4-2.5. Thus, s e t t i n g 1 ' = ( o ) , one d e f i n e s now J o s t o b j e c t s by ( n o t e r = 0 f o r x < 0 )
f,
(6.13)
= eikHxf'
= e ikHxAy,
fr(k,x) Here f,
e ikH(x-y)
t
r ( Y ) A $ (k,y)dy;
- jOX e kH('-'
) r ( y ) A f ( k, y ) dy
(resp. f r ) d e s c r i b e s a wave which i s r i g h t o r downgoing f o r x l a r g e
(resp. l e f t o r upgoing as x = 0 ) .
I n f a c t fl ( r e s p . fr) i s t h o u g h t o f as coming i n from t h e l e f t (resp. r i g h t ) ; f, c o n t a i n s no upward t r a v e l i n g waves s i n c e i t i s downgoing as x kHx)f
+
A?' as x
= ( l / T$) ( f1z )
Thus exp(-ikHx)f, and one w r i t e s ( a * ) f,(k,O)
-m
-+
( l i m as x - +
-m)
-f
a.
0, and f r2e x p ( i k x ) 1 as x -t ( - i k x ) f o r x 5 O), e x p ( - i k x ) f r -+
-m -f
-f
Now f o r k r e a l f 0
= TAfl(k,x)
-
0
1
( a c t u a l l y fr(k,x) 1
--.
= (l/T)(/r)
= 0, fr(k,O) 2
ir/? and e x p ( i k x ) f :
i l a r l y one has f,exp(-ikx) 1 + 1 and f,exp(ikx) 2 0 2 1/T and f,exp(ikx) R /T as x -+ 0
x -+ m and e x p ( - i = l i m exp(-ikHx)fl(k,x)
w i t h l i m exp(-ikHx)fr(k,x)
W r i t t e n o u t t h i s means i n p a r t i c u l a r fr(k,O) 1 -+
... 1 ' as
-f
-+
(x
= 0 and f r2( k , x ) -+
0 as x
l / ias x -+
.+ m ) .
= 1, f r1e x p ( - i k x )
my
-+
= exp m.
f,exp(-ikx) 1
Sim-+
(x) = f r,, r y e( - k , x ) and one has i n p a r t i c u l a r (QA) f r (-k,x) RIAfr(k,x). I n a s t a n d a r d way one f i n d s a l s o t h a t fr and fL
157
SYSTEMS
a r e a n a l y t i c f o r Imk > 0, c o n t i n u o u s up t o Imk = 0, w h i l e ( f o r Imk > 0) llfLll 5 cexp(-xImk), -+
1 as I k (
7'
-
x)fL(k,x)
[Hwl],
+ m,
IIfrll 5 cexp(xImk),
kz
and
i s analytic with
!t -+
0 as I k l
-+
2
-/I?' b e l o n g t o L (R).
and e x p ( i k x ) f r ( k , x )
-.
Further exp(-ik Now f o l l o w i n g
where t h e development i s somewhat d i f f e r e n t t o t h a t i n [ N W ~ ] , we t a k e and s e t ( @ @ ) G(k,x)
(@A)
l / i i s a n a l y t i c and nonzero w i t h l / i
-
= exp(ikx)fr(k,x)
Then u s i n g s t a n d a r d t e c h -
A?'.
niques and n o t a t i o n f r o m p h y s i c s w i t h G(k,x) = ( 1 / 2 n ) I I G ( k ' , x ) / [ k ' - k - i ~ ] d k ' from
'
e x p ( - i kx)A'i
=
-
0
-(A/2n)Lm Re(k' ) f r ( k ' , x ) [ l z e x p ( - i y ( k t k ' ) ) d y ] d k ' . m
0
fr(k,x)
(6.14)
-
e
W r i t e now
so t h a t ( 0 6 ) has t h e form
= (12/2n)jm Rl(k')fr(k',x)exp(-ik'y)dk'
A(y,x)
(e+)
-
( a f t e r some c a l c u l a t i o n ) i t f o l l o w s t h a t ( @ & ) f r ( k , x )
(@A)
-ikxi' = -1-t e-ikyA(y,x)dy Then
which we p r e f e r t o t h i n k o f as t h e d e f i n i t i o n o f A i n s t e a d o f ( W ) . Set X ( x ) = ( 1 / 2 n ) ~g L~( k ) e x p ( - i k x ) d k
CHE0REI 6.14.
w i t h A d e f i n e d by (6.14).
(so Z(x) = 0 f o r x < 0 )
Then one has an M t y p e e q u a t i o n ( - x 'y
Z(c+y) = 0 f o r 5 < - y ) A(y,x)
= Z(X+Y)Al'
X
A(S,x)Z(c+y)dc.
-Al-y
zx; The r e -
covery f o r m u l a i s r ( y ) = 2A1(y,y).
RRRAaK 6-15.
We n o t e t h a t i n t e g r a l equations as i n Theorem 6.14 w i t h k e r -
n e l s 2 ( ~ + y )(Hankel o p e r a t o r s ) a r e r e f e r r e d t o as M t y p e e q u a t i o n s .
One has
a l s o from (6.14) (which i s a k i n d o f t r a n s m u t a t i o n f o r m u l a ) and
DxA(y,
(Am)
x ) = HD A(y,x) + r(x)AA(y,x) = 0. Y We t u r n n e x t t o t h e s p e c t r a l r e p r e s e n t a t i o n s o f t h e o b j e c t s i n t h e system t h e o r y i n terms o f t h e us n o t e f i r s t from
+ 0 )
=
(@A)
In this direction l e t etc. described e a r l i e r . 1 2 t h a t one can determine Dfr(k,O+) = - r ( O ) and D f r ( k ,
q!"
2
- i k ( r e c a l l a l s o f', = 0 and fr = e x p ( - i k x ) f o r x 2 0 which g i v e s i n i -
fR
t i a l values f o r x
g i v e i n i t i a l values f o r 0). Similar calculations f o r x > 0 for w h i l e (even though d i s c o n t i n u i t i e s o c c u r f o r t h e d e r i v a t i v e s a t 1 + x = 0 ) t h e d e r i v a t i v e s from t h e r i g h t a t x = 0 can be o b t a i n e d as Dfe(k,O )
5
= ik(l/f)
-
r ( 0 ) E L / 8 and DfL(k,O+) 2
-
= -ik(EL/f)
scramble t h i n g s r e l a t i v e t o h and k as f o l l o w s .
Next l e t us un2 1 W r i t i n g Vr,L = fr,L + f r(O)/?.
r,e
?
A
and I = 1 we know t h a t and s a t i s f y second o r d e r e q u a t i o n s r,Lz ' r , ~ fr,L 2 2 ( i B = - A B, P I = -k21, QA = - A A, QV = -k V, e t c . ) . Hence we can i d e n t i f y h
h and k a t t h i s l e v e l and express Vr,[
ponding
v pxy Q , o : ' ~ ,e t c . ,
t i e s as x
-+
m
i n terms o f A, B, C, D o r t h e c o r r e s -
t a k i n g i n t o account t h e v a r i o u s a s y m p t o t i c p r o p e r -
and t h e values a t x = 0. Any r e f e r e n c e t o F o u r i e r t r a n s f o r m A
can be abandoned a t t h i s stage.
Thus e.g. Vr
A
( r e s p . Ir)w i l l be a l i n e a r
158
ROBERT CARROLL A
A
combination o f A and C (resp. B and D ) and one expects VA and 1; t o be d i s continuous a t t h e o r i g i n i n view o f t h e c o n s t r u c t i o n o f fr ( c f . a l s o Remark 6.17).
+
We can w r i t e t h e n f o r example
^v, =
(aa)
A
One has a l s o ( 6 * ) Vl
ihe!).
ixge,] Q
Z-'[v,
+
= Z-'[(l+&)v!/f
Q
-
Xe,];
Q
"Ir= Z% -vXP
iX(l-64)O:/?J
= Z-'[ap;
t
A
and I , i s r e a l l y n o t needed here. Now one can use e q u a t i o n (*) t o P P and compare v X y b = X B X l and a = -MI: i n terms o f v;, N
represent a =
t h e a s y m p t o t i c values as x
-f
m
0
CHE0REI 6.16.
The procedure j u s t i n d i c a t e d y i e l d s 1/T = c
FP/2, :r
-
= [cQ
F-/2]/[c-
Q
Q
+ F /2],
Q
g,
and
We n o t e i n passing f r o m
REmARK 6.17. 2
There r e s u l t s
w i t h those o b t a i n e d from (a*).
= [FQ/2
(@a)
-
t h a t e.g.
-
-
Q
+ F /2 Q
c i +
=
+ c$.
c$/[FQ/2
1 fr = A-D + i(B+C) w h i l e 1 2 D x f r = ikf!. rfr and
-
= A+D i(B-C). Hence from ( a ) ( r e c a l l from ( A m ) r 2 2 1 2 2 D f = - i k f r - rf,) (&A) D x f h = i h f h - rf,. Thus x r\, k i n (U) w h i l e D x f r x r 2 1 = - i X f r - rfr and A k again. Thus we see t h a t a l t h o u g h QU = X U g i v e s A C equations f o r t h e v e c t o r s ( B ) and ( D ) d i f f e r e n t from (+), c e r t a i n complex
f
Q
l i n e a r combinations o f t h e A,B,C,D
as i n
(@a)
s a t i s f y (+).
Now i n o r d e r t o o b t a i n a s p e c t r a l v e r s i o n o f A we t r a n s f o r m t h e fr problem (Am)
to
-i)fr
(:
=
(!)A
again as i n (+) and use t h e i d e n t i f i c a t i o n X
2,
k
i n d i c a t e d i n Remark 6.17 t o w r i t e f r o m (6.14)
V (Ir) r
(6.15)
=
e
-iAx
1 (-1)
x
- /-x
,-.,
Set A +A
A (A:
+ A2),-iXy
- A
dy
1 N
= A1 and A1-A2 = A2 and one o b t a i n s a f t e r some 2 = - [ k o ( x y y ) + kO(x,y)]/2 and = [kvp(xyy) + k p ( x , y ) l / 2 .
r2
v l
R e f e r r i n g t o [ C l o , l l ] 'we a n t i c i p a t e t h e development t o f o l l o w and s t a t e Given A d e f i n e d by (6.14) one has A1(y,x)
tHE0RBIl 6.18. (1/4)[kp +
kvp
where t h e mij
-
kQ
-
kvQ]
and A2(Ylx)
= -mll(xyY)
= -mzl(x,y)
r e f e r t o t h e M m a t r i x o f [Bal;C10,11,34],
+
k"P
=
+ k
+ ] Q Q K(x,y) = -A1(y,x),
= -(1/4)[kp
and t h e M t y p e e q u a t i o n i n Theorem 6.14 i s a v e r s i o n o f t h e M e q u a t i o n o f [Bal;C10]
( c f . below) upon i d e n t i f y i n g K(x,y) = -Al(x,y)
and Z ( x ) = R ( x ) .
Phooi: To pass f r o m t h e M e q u a t i o n i n Theorem 6.14 t o t h e M t y p e e q u a t i o n o f [Bal;C10,11,34] w r i t e Al(y,x) = -J-' A2(c,x) (S+y)dS and A2(y,x) = Y Adding these equations one has -K(x,y) = Z(x+y) Jx A1(5,x)Z(S+y)dS. Z(x+y) +'-I K(x,S)Z(S+y)dC which i s t h e M e q u a t i o n o f [Bal;C10] w i t h R = -Y Z ( c f . t h e development below). QED 1 x Consider (6.14) i n t h e form fr = REMARK 6.19, mZl(x,y)exp(-iky)dy with 2 fr = e x p ( - i k x ) + JX mll(xyy)exp(-iky)dy and r e c a l l Remark 6.15 where
-
-X
SYSTEMS fr(k,x)
= Ffr(t,x)
M2*(x,t),
(Dt
+
-ik).
M 2 1 ( x y - t ) = M12(x,t),
159
= R e c a l l a l s o f r o m [Bal;C10] t h a t Mll(xy-t) and M l l = 6 ( x - t ) t mll w i t h M12 = m 1 2 ( m d u 0 c f . below). The response ( i R = M * ( & ) t o i n -
l o some Heavyside f u n c t i o n s 0 i t i a l data ( & ) i n (6) i s t h e n ( m 2 2 m ~ 2 s ( x + t ) ) and F('R) =
ll (mil m2t 16 ( x - t ) ) e x p ( - i k t ) d t
= fr(k,x)
WL
=
( c f . [Bal;ClO]).
LLI
(!R)exp(ikt)dt
The i h p u t gf ( f )
i s n o t used i n [ B a l l s i n c e d i s t u r b a n c e s a r e propagated i n t o t h e medium f r o m t h e l e f t , w h i l e fr correspontis t o a wave coming from t h e r i g h t .
However
t h i s shows t h e background f o r t h e i d e n t i f i c a t i o n s i n Theorem 6.18 and we r e f e r t o [Bal;C10,11]
WL.
and remarks below f o r f u r t h e r machinery r e l a t i v e t o W R Y
For t h e Riemann-Hilbert problem and c o r r e s p o n d i n g M e q u a t i o n t e c h n i q u e
r e l a t i v e t o t h i s h a l f l i n e problem one s i m p l y adapts t h e procedure o f §4,5 ( c f . Theorem 5.2,
Remark 5.3, and [Nw2;C1lY34]).
We go now t o t h e model equations ( 6 . 1 ) which have served so w e l l i n s t u d y i n g i n v e r s e s c a t t e r i n g by l a y e r s t r i p p i n g methods e t c . ( c f . [BalY2;K12;Lyl ,2]).
A p a r t i c u l a r l y c l e a r and i l l u m i n a t i n g e x p o s i t i o n o f t h e c o n n e c t i o n s between l a y e r s t r i p p i n g , r e l a t e d Shur a l g o r i t h m s and Krein-Levinson r e c u r s i o n s , s c a t t e r i n g , e t c . appears i n [Ba2],
I n t h e e x p o s i t i o n below, based on [C10,11,
341, we use t h e t r a n s m u t a t i o n techniques o f [C2,3,13]
t o p r o v i d e spec r a 1
formulas f o r t h e fundamental q u a n t i t i e s appearing i n t h e u n i f i e d G-L, M y K, G-S, e t c . i n t e g r a l equations o f [Bal,2] ( c f . a l s o [Bpl;Gpl;LaZ;Sa1,2] for which some s p e c t r a l i n f o r m a t i o n i s a l s o g i v e n ) .
We a l s o g i v e s p e c t r a
for-
mulas f o r v a r i o u s t r a n s m u t a t i o n k e r n e l s and G-L t y p e equations appear ng i n a systems c o n t e x t as i n [C2,3,10,11,13,14]
and emphasize a g a i n t h a t one can
view G-L, M y e t c . t y p e equations e i t h e r from t h e p o i n t o f view o f p h y s i c s where t h e y a r i s e i n s o l v i n g i n v e r s e problems o r as i m p o r t a n t s t r u c t u r a l formulas
i n v o l v i n g the connection o f d i f f e r e n t i a l operators.
The r e s u l t s
i n v o l v i n g s p e c t r a l i n g r e d i e n t s h e r e seem t o p r o v i d e c o n s i d e r a b l e i n s i g h t i n t o t h e whole m a t t e r and c o u l d w e l l be u s e f u l f o r computations.
Our p o i n t o f
view i s t r a n s m u t a t i o n a l i n t h e sense t h a t one d e a l s w i t h c e r t a i n problems and s i t u a t i o n s i n terms o f l i n k i n g v a r i o u s u n d e r l y i n g o p e r a t o r s v i a s p e c t r a l p a i r i n g s o f g e n e r a l i z e d e i g e n f u n c t i o n s and e x p r e s s i n g t h e r e s u l t s i n terms o f fundamental o b j e c t s f o r t h e u n d e r l y i n g o p e r a t o r s .
L e t us n o t e f i r s t t h a t
a spectral form f o r transmutation kernels r e l a t i v e t o canonical operators
Q
-
W and Qo = JDx can be expressed a s f o l l o w s ( c f . [Arl;D2,3,5;Dtl; A o c C % ( B ) = X and Q = ( o ),, (,) La2;Mrl;Sfll). One s e t s R = AJa, Q, = SinXx w i t h Q,o = SinAx and qo = (-coshx) ( n o t a t i o n o f [C14]) and R ( r e l a t i v e t o We have ( c f . [C10,34] Q,) on (-m,m) i s even w i t h 2R = dw = dwp On [ O , m ) . = JDx
(i 0")
Q
Q
160
ROBERT CARROLL
f o r more d e t a i l ). Set @ ( A , f ) = 10" f(x)@(x,A)dx and z(A,f) = :/
tHE0REm 6.20, 2,
for
@*
x
-
real
notation with
a0
(g
Similarly
i: @ ( A , f ) G ( x , x ) d A = iI @(x,x) if @(x,X)@(A,f)dA w i t h c o r r e s -
Then f ( x ) = =
and *o where Ro = l / n .
as a r o l e model, B(Y,X)
@
@(x,A)f(x)dx
and f can be m a t r i x valued).
LI @(A,f)K(x,A)dA
and f ( x ) =
ponding formulas f o r
-
(A,B)
PI,
and ? ( x , f ) .
one d e f i n e s 3 ( A , f ) K(x,f)dx
&
i.e.
W r i t i n g now i n a s t a n d a r d
(l/n)[:
=
9(y,A$o(x,A)dA,
y(x,y)
=
r:
N
@,(x,A)R@(y,A)dA, F ( y , x ) = y*(x,y) = @(y,A)60(x,A)dA one o b t a i n s t r a n s m u t a t i o n o p e r a t o r s B,E: Qo + Q w i t h 3 = B-' ( B = k e r B , y = k e r 3 , ker
i) and
+ K(y,x),
-
i t f o l l o w s t h a t BDo =
d R , etc.
=
Writing ~ ( y , x ) = 6(x-y)
= -DsK(t,s)J
w i t h W ( t ) = JK(t,t)
There i s a G-L e q u a t i o n g ( y , c ) = ( ~ ( y , x ) , A ( x , c ) )
K ( t , t ) J and K(t,O) = 0. (
M0 =
@,
f o r example, one has Q(Dt)K(t,s)
w i t h A(x,c) and
=
Ic.
N
L:
@o(x,A)R@o(c,A)dX
SinAx,SinAc)R.
h a v i n g diagonal e n t r i e s
F u r t h e r B and
There a r e s i m i l a r formulas i n t h e
(
CosAx,CosXc ) R
have t h e s t a n d a r d t r i a n g u l a r i t y .
* t h e o r y which we o m i t here
( c f . [C10,14,
rz
L e t us n o t e t h a t f o r ( f , ~ =) ~ f(A)g(A)RdA one has ( A ( x , x ) , S i n x t ) R
341).
=(B(x,h),Cosxt)R
=
0 s i n c e B i s odd and A i s even i n A.
Thus, f o r example,
4. ,
B ( x , t ) has diagonal e n t r i e s ( A(x,A),Cosxt)R and ( B(x,A),SinAt)R. On t h e V o t h e r hand, one sees e a s i l y t h a t t h e s o l u t i o n q ( x , t ) = ( I ) t o (6.1) w i t h
+
6
+
i n i t i a l d a t a (o+) i s q ( x , t ) = (( A(x,x),Cosxt)R,( t o r , where q(0,t)
(
,
Cosxtdt.
A
)R means 2R on
t o (I)(x,O) V
kHE0REtl 6.21.
5 ) + :/
t
=
G(t) =
(it) (
A
= w ).
(2R = u
[O,m)
B(x,A),SinAt)R),
The impulse response
i s t h e n g i v e n as p( G ( t ) j l and one has
+
1 ,CosAt)R = ( 2 / n ) / r AmCosAtdA from which Am = 1; G ( t )
X)
+ A(x,
The G-L e q u a t i o n i n Theorem 6.20 takes t h e form 0 = K(x,c) K(x,n)A(q,c)dn
c
for
-
= 6(x-t)
Y ( t - x ) ] where Y
0 and Y ( t ) = 0 f o r t < 0 ) . Using
= 0 f o r t < x one o b t a i n s f r o m (6.16)
t h i s w i t h ( 6 6 ) and WR(x,t) = WL(x,t) f o r -x 2 t 5 x (6.17) WL(O,t+X) +
i-, t WR(O,t-T)mll(x,T)dT : j
WL(O,t+T)mll
+
l-xtkL(0,t+T)m21 (X,r)dT
= 0;
l,"WR(0,t-r)m2,(X,r)dT
=
(X,T)dT +
0
One assumes t h e p r o b i n g wave WR(O,t) t o have a l e a d i n g impulse and t h u s WR ( x , t ) = 6 ( t - x ) t w,(x,t)Y(t-x) takes t h e form
+
w i t h WL(x,t) = w,(x,t)Y(t-x).
Then (6.17)
162
ROBERT CARROLL w h i l e f o r -x 2 t 2 x there i s a propagating equation o f t h e form
2Dxm11(x,x)
Dxm = [JDt 0.
-
Wlm, m
(mll
m12), column v e c t o r , w i t h mll(O,O)
= m21(0,0) =
Now f o r d i f f e r e n t choices o f t h e p r o b i n g waves WR(O,t) and WL(O,t)
We c o n s i d e r ( A )
(6.18) one o b t a i n s v a r i o u s i n t e g r a l equations as f o l l o w s . wR(O,t)
= 0 and wL(O,t)
[Baly2;Ly1,2]
= R ( t ) Y ( t ) o r (B) w R ( O , t )
f o r philosophy e t c . ) .
0 = WL(O,t+X) t K(x,t) t 2
= G(x) = r - r '
where 2DxK(x,x)
= wL(O,t)
-
[C2,3;Fal;Chl]
= mll(x,t)
t
+ wL(0,t+r)] K ( X , T ) ~ T
[WR(O,t-T)
;'("')'I.
I f t h e p r o b i n g waves have t h e
form ( A ) we have a c l a s s i c a l M t y p e equation (WR(0,t-T) i s o f Hankel t y p e
= h ( t ) (see h e r e
We w r i t e now ( 6 ~ ) K(x,t)
+ m 1 2 ( x y t ) ( c f . (6+)). Then adding i n (6.18)
m21(x,t) and L ( x , t ) = mll(x,t) (6.19)
in
= 0 and t h e k e r n e l
we n o t e again how t h i s d i f f e r s from t h e M equations o f
and Theorem 6.12).
Taking (B) above as s c a t t e r i n g data and
i n t r o d u c i n g (+*) KS(x,t) = ( 1 / 2 ) [ K ( x , t )
+ K(x,-t)]
one o b t a i n s a G-L equa-
t i o n i n t h e form (6.20) (0
0, e t c . ) .
Take, f o r example, t h e e q u a t i o n Q(Dx)U
= 6(t)
This problem, considered as an upward Cauchy problem, was
s o l v e d i n f a c t above v i a IP i n t h e f o r m U ( x , t ) = ( CosAt,vA(x) Q ) Q (GQ =-;R on [O,.o) and r e c a l l A = aZ-' w i t h V = UZ-'). Relative t o Q with A = Z we know t h e standard t r a n s m u t a t i o n ( x , t ) = ( C o s A t , ~ ~ : ( x ) ) ~= y ( t , xQ) A Q-1 ( x ) = y(t,x)Z(x)
=
+ g ( t ) = G ( t ) ( r e a d o u t impulse response), and 6 ( x ) ( a d d i t i o n a l l y U = 0 f o r t < 0, Ut(x,O) = 0 ( = - Z Z x ) f o r
= Utty U(0,t)
Q
i s t r i a n g u l a r w i t h B ( x , t ) = 0 = U(x,t) f o r x
i s " c a u s a l " and
9
Q
>
t.
Thus U ( x , t )
( x , t ) w i l l i n f a c t p l a y t h e r o l e o f a causal Green's
SYSTEMS function.
Now l o o k a t t h e same problem sideways.
w i t h say Ux(O,t) U(x,t)
=
163
(
= 0.
= G be g i v e n
One t r i e s t o f i n d a sideways s o l u t i o n i n t h e f o r m
CosAt,F(A)vA(x)) Q
where dP = 2dA/n so t h a t U ( 0 , t )
PA
= G(t) =
(
CosAt,
Now i f t h i s i s t o r e p r e s e n t t h e s o l u t i o n
F ( A ) ) w i t h F ( A ) = FCG(t) = G ( A ) .
(Cos t.q!)x))u
L e t U(0,t)
it follows that W(A) =
A
c Q/;
(i.e.
= ;(A)
the spectral-factor
G i s e x a c t l y one w h i c h produces t h e c o r r e c t t r i a n g u l a r i t y f o r a causal s o l u tion).
We n o t e t h a t i f T ( r e s p . S ) r e f e r s t o g e n e r a l i z e d t r a n s l a t i o n r e l a -
tive to
o2
= P (resp.
Q)
t h e n TTf(T) t =
= 10" g(x)A,(x) ( Q g ( A k AQ( x ) , v $ c ) ) u general p r i n c i p l e s f r o m [C2,3]
U ( x , t ) =(Y Q ( t , S ) S t 6 ( S ) )
(6.22)
vA Q(x)dx).
P
= (yQ(tyS),6(X-S)/AQ)
f o r m a l l y s a t i s f i e s t h e Cauchy problem Q(D,)U Ut(xyO) = 0.
w i t h sxg(s) = 5 Thus i n keeping w i t h
( (~Cf)COS~T,COSht)
2
BQ(x,t)
= DtU w i t h U(x,O)
= 6 ( x ) and
On t h e o t h e r hand f o r t h e sideways Cauchy problem one c o n s i d -
e r s f o r m a l l y ( B ( x , t ) = ( q AQ( x ) , C o s A t ) = i?A(x,A),CosAt)P) t Q 1-I ( B Q ( X Y ~ ) . T T G ( ~ () = ) U(x,Y)).
(+A)
g(x,t)
The k e r n e l 8 ( x , t ) a c t s as an a n t i c a u s a l Green's f u n c t i o n s i n c e 8 ( x , t )
Q
f o r t > x (while
Q
(x,t)
= 0 f o r t < x).
t a t i o n and 8 (x,T)
Q
We n o t e t h a t
i s even i n
= (2/n)/z
G^(A)CosAtCosArdA
G i s even i n t h e c o s i n e r e p r e s e n -
t h a t (+A) g i v e s
TSO
CHEBREUI 6-23, The impulse response U ( x , t ) = (CosAt,qA(x))u Q f o r Vtt U,)
U(0,x) = 6 ( x ) , and Ut(O,x)
U(x,t)
= (I/Z)~~(X,-)
*
= 0 w i t h U(0,t)
G ( = ) = (1/2)1-'
sideways Cauchy problem w i t h V ( 0 , t ) causal s o l u t i o n U ( x , t ) tion B
Q
= 0
The c o m p o s i t i o n (+A) i s a k i n d o f
= FcG, T:G(T)
generalized convolution since f o r = (1/2)[G(tt~) t G ( l t - ~ l ) ] .
Q
=
= Z(Z-'
= G ( t ) can be expressed v i a
8 (x,T)G(t-r)dr x Q = G ( t ) and Ux(O,t)
as t h e s o l u t i o n o f a = 0.
Note t h a t a
a r i s e s as a c o n v o l u t i o n o f a noncausal Green's f u n c -
w i t h G.
Consider now i n an ad hoc manner i n (6.18) i n i t i a l c o n d i t i o n s (B) SO W,(O,t) V One has = 6 ( t ) + h ( t ) and WL(O,t) = h ( t ) and s e t h ( t ) = 0. = V To s o l v e t h e sideways Cauchy problme f o r t h i s v e c t o r so ( I ) ( o y t ) = (:). V ( T ) we c o n s i d e r f o r dv = (2/a)dA
(iIl)m
Note t h a t t h i s i s n o t t h e most g e n e r a l ' f o r m o f s o l u t i o n b u t i t s u f f i c e s h e r e ( c f . (6.26)).
Then one o b t a i n s from WR = M l l
and WL = M21
164
ROBERT CARROLL
mE0Rm
-
A),CosAt),
(C(x,A),Sinxt)
- (D(x,A),Cosxt)
CosAt)p
-
P
a r e o b t a i n e d as Mll(xy-t) 0 for
It1 >
P
+ (B(x,x),Sinht)
-
(B(x,A),Sinxt)u
( C ( x , A ) , S i r ~ x t ) ~ . M22 and M12
= M 2 2 ( x y t ) and M 2 1 ( x y - t ) = M12(x,t)
One r e f e r s now t o ( 6 4 ) t o w r i t e ( n o t e k ( x , t )
k'Q ( x , t )
= (A(x,x),
(Mll
= MEl
=
1x1.).
w h i l e c,,(x,t) t
w i t h 4M21(x,t)
P
+ (D(x,
= (A(x,h),CosAt)P
one has 4Mll(x,t)
For du = (2/lr)d
6.24.
and
+
;,,(x,t)
and k p ( x , t ) a r e even i n t
Q
k'Q ( x , t )
a r e odd i n t ) (4.)
4mll(x,t)
whereas ( 4 6 ) 4 1 n ~ ~ ( x , t )= kQ
-
kp
= k (x,t) v
-
Q v
kp t k
9'
+ kp(x,t) Hence
m21 s p e c t r a l l y v i a ( 4 0 ) and ( 4 6 ) w h i l e K, L, and KS a r e g i v e n by 2 K ( x , t ) = 2 ( m l l + m21) = k ( x , t ) + c Q ( x , t ) , One can express mll
mE0RRR 6.25, 2L(x,t)
+ m12)
= 2(mll
=
and
Q
k Q ( x , t ) + kYp(x,t),
+ K(x,-t)]
and K S ( x y t ) = [ K ( x , t )
(1/2) = (1/2)kQ(xyt). t L e t us n o t e here t h a t , s e t t i n g w ( x , t ) = J0 U ( x , ? ) d r f o r 4 as
REmARK 6.26,
we o b t a i n w ( x , t ) = ( ~ Q~ ( x ) , [ S i n A t / x ] w ) ~i t h wt(x,O)
i n Theorem 6.23, and w(0,t)
= Y(t)
+ 1; g(r)d.c.
T h i s i s t h e s o l u t i o n used i n [C2,3,7-9]
s t u d y i n g t h e one dimensional i n v e r s e problem f o r SH waves. N
tion A (x)
K i n [C3]).
, {(x)
Thus kQ(y,x) = Z%
= 2D k ( x , x )
(recall
9
-
6 = Z-'(y)tx(y,x)
= $("')'I).
in
I n that situaN
Z - l and one has ~ ( y , x ) = Z 6 ( y - x ) t Kx(y,x) w i t h K(x,x) = 1
%
( t=
(1/2)k
= 6+(x)
- Z4
and from KS =
One can a l s o compare G-L
Q x Q equations as f o l l o w s ( c f . Remark 6.29 f o r t h e MGL e q u a t i o n a s s o c i a t e d w i t h t h i s example).
From [C3] o r 51.6 t h e canonical G-L equation f o r k
t h e c o n t e x t k (y,x)
Q
= Z-4tx(y,x))
i s 0 = (6 (y,t),A(t,x))
Q
Q
(seen i n
f o r x < y where
From B =6'2 + we a l s o have 1 ; CCoshxCosAtdA = 6 ( x - t ) + A ( t , x ) . rl again f o r A = Z -4 a (a v Q~ ) A(x,A) , = Cosxx + :/ k (x,c)Cosxcdc. Thus kQ(y, Q F u r t h e r from U ( x , t ) = (a(x,x), ; kQ(y,t)A(t,x)dt = 0 (x < y). x ) + A(y,x) + 1
A(t,x)
=
Q ,
one has 6 + 2h = U ( 0 , t ) = V(0,t)
Cosxt),
V(0) =-6 + 2h w i t h I ( 0 ) = 6 ) .
= ;/
k o s x t d x ( r e c a l l Z(0) = 1 and
Hence, w r i t i n g do) = (2/n)dx t :dh
one ha
+
:1 CoSAtCoSAxGdA (1/2)1; [CoSx(t+x) + CosA(t-x)]GdA = h ( t + x ) h ( l t - x l ) since 2 h ( t ) = 1 ; SCosxtdA. T h i s y i e l d s k ( x , t ) + [ h ( x + t ) + h (
A(t,x)
=
X-
Q
tl 11 + $
[h(x+-c) + h( It-rl ) ] k Q ( X y ~ ) d= ~0 as i n (6.20). L e t us make a few o b s e r v a t i o n s here about t h e development i n
RElnARK 6.27.
[ B p l ] where G-L, M type, and G-S t y p e i n t e g r a l equations a r e d e r i v e d from t i m e domain p r i n c i p l e s .
+ hu(0,t)
= 0.
say Vx(O,t)
Thus one c o n s i d e r s utt = uxx
s ( x ) u w i t h -ux(O,t)
This corresponds t o a V problem, f o r example, w i t h s =
= -r(O)V(O,t)
- r ( O ) w i t h A(0,A)
-
= 1 so A
4
and
so - r ( O ) = - ( 1 / 2 ) Z 1 ( 0 ) = h ( r e c a l l a l s o A'(0,A) = 2 % lpQ f o r A " - 4 A = -A A ) . Such problems a r e x,h
SYSTEMS
165
t r e a t e d i n many places and we r e f e r , f o r example, t o [C2,3;Mrl]. I n [Bpl] one d e a l s w i t h a causal Green's f u n c t i o n o r impulse response G1(x,t) which i n f a c t equals i i Q ( x , t ) =
(vQ
such an i d e n t i f i c a t i o n ) .
(x),Cosxt)u
(we s e t i Q ( x , t ) =
o
i n s t e a d o f -K1 i n [ B p l ] ) .
The f u n c t i o n Kl(O,t)
o
for t
-
We have F = 1 + R
i n v o l v e s d r = dX/2n on (--,-).
a 1 t t l e calculation the and I
1 1
t (D,Gexp(iAt))
namely, 0 = V ( x , t )
Q
-
(1/2)[kp-ip]
1
+ k v ] / 2 ) one o b t a i n s (*m) f o r V ( x , t ) = 0 when Q + R ( x + t ) + f-xt R(t+r)K(x,T)dT.
= K(x,t)
166
ROBERT CARROLL
F i n a l l y we w i l l a l s o approach t h e G-S e q u a t i o n as i n [C34] t h r o u g h t h e deThus c o n s i d e r t h e system
velopment i n [Bpl] ( c f . a l s o [Gpl;SalY2;Sol]). and QU = Utt from (*). (6.1) a l o n g w i t h P I = Itt (-m,m)
We s e t diY = (1/2a)dA on
and c o n s i d e r f o r t h e sideways Cauchy problem a s o l u t i o n i n t h e form
(6.26)
[
=
(
Famexp(iAt)
)-
+ i(Ga,exp(iAt)
4
)-
!J
T h i s a l l o w s general
I % F % J and u U % 6 % K. Suppose U0 = 0. Then ( r e c a l l U = Z% and I now I ( 0 , t ) = lo i s g i v e n w i t h U(0,t) = Z-+I) U ( x , t ) = (1/2)A ( 6 - - 6' + k' ) * I, which can a l s o be developed d i r The i d e n t i f i c a t i o n s w i t h [ B p l ] a r e p
%
Q
Q
e c t l y from (6.26). Now f o l l o w i n g [ B p l ] c o n s i d e r D t I o = 26 + h ( 1 t l ) (= Ft), t h f o r t > 0 w i t h l o ( - t )= - t o ( t ) ,and Uo = 0 ( = G ) . I ( = F) = 1 + 1 For 0
t h e corresponding U ( x , t ) one shows i n [ B p l ] t h a t U ( x , t ) = -1 f o r and hence
(m*)
-1 = ( 1 / 2 ) A 4 i Q
* I,
f o r It1 < x.
t h i s i s t h e G-S e q u a t i o n ( c f . a l s o [Gpl;SalY2;Sol]). [Bpl] one c o n s t r u c t s J, K w i t h Kt = -Z% -=A(';To = - s g n ( t ) . It f o l l o w s t h a t K = (1/2)ZQ+ f3Q *
* yo] and *A i =
+x) +
Qb"
= (1/2)Z2BQ
EHEBREIII 6.28. (1/2)K
*
I;
for It[ < x
-Z%
Q
(=A)
It1
z0
I n the notation o f
-
k"
) and J ( 0 , t ) = - F0(t
= ( 1 / 2 ) Z +Q[ T 0 ( t - x )
* io+
K = i3[(1/2)tQ
as d e s i r e d and t h u s f r o m
+
6'
13.
E v i d e n t l y Kt
one has ( c f . [C34])
(m*)
The G-S e q u a t i o n has t h e form (It1 < x ) , 1 = (1/2)Kt = K
*
[6 + (1/2)h(
Itl)] =
x
0. Also ( c f . ( 4 . 3 ) ) R/T = ( 1 / 2 i k ) l I qf2(k,t)e-iktdt;
(7.1)
1/T = l - ( l / Z i k ) / :
qf2eiktdt
= 1
-
R2/T = ( 1 / 2 i k ) l I qfl(k,t)eiktdt;
(1/2ik)fm q(t)fl(k,t)e -iktdt m
which f o l l o w immediately from ( 4 . 1 ) . Next we c o n s i d e r a h a l f l i n e s i t u a t i o n and f o r completeness w i l l i n d i c a t e several p o i n t s o f view (as i n 86). =
Thus c o n s i d e r i n a system f o r m a t w = vt
v e l o c i t y and P = p r e s s u r e w i t h P = -pvx and p v
(PP)'
=
impedance and y = t r a v e l t i m e ( y ' =
be assumed) ( + ) w
Y
=
has ( c f . (.+)
=
-Px. One takes A = so thict (A(0) = 1 can
= -Awt; v = (Av ) / A ; wtt = ( A w ~ ) ~ / A Y tt Y Y S 6 a r e n o t used h e r e - A always r e f e r s t o an of
-A-'Pt;
( n o t e t h e symbols A,B,C,D impedance now).
t2
(P/!J)')
P
S e t t i n g p = (l/Z)[A-'P
+ A%]
i n S2.6) p + pt = -rAq; qy Y
-
and q = (1/2)[A-%'
- At]
qt = - r A p where rA= (1/2)Dy
one
170
ROBERT CARROLL
It w i l l be convenient i n r e l a t i n g t h i s t o t r a n s m i s s i o n l i n e s t o t a k e
logA.
Z = A - l as impedance ( w i t h r Z= -rAas r e f l e c t i v i t y ( t h i s c h o i c e Z = A - l i s made i n o r d e r t o connect n o t a t i o n h e r e t o t h a t o f 52.6 and [C10,11,341). Then we w r i t e V = UZ-'and I = I$ (U ?J w, I ?J P ) w i t h wR = ( 1 / 2 ) ( V + I ) and wL = ( l / Z ) ( V - I ) ( r i g h t and l e f t t r a v e l i n g waves). Thus wR % p and wL ?J -q and 0xwR t DtwR = -rZw L w i t h DxwL - Dt wL = -rZwR. A standard s i t u a t i o n now f o r t h e geophysical problem i s t o impose an impulse v ( 0 , t ) = - 6 ( t ) = P(0, Y = 6 ( y ) ) so t h a t an impulse response
t ) ( o r e q u i v a l e n t l y w(y,O) = vt(y,O)
w(0,t) = 6 ( t ) t 2 g ( t ) Y ( t ) i s o b t a i n e d ( 6 = 6+ here t ) = 6 t $Y and q ( 0 , t ) = - i Y . 6
+ GY and wL(O,t) = Y;
I n t h e wR,wL
- c f . §2.6).
Then p(0,
c o n t e x t t h i s becomes w R ( O , t )
F i n a l l y i n connecting t h i s n o t a t i o n t o 52.6 f o r example we w r i t e v ( 0 , t ) =
Y
t G
r
w i t h G' = 6
+ Y2:
= 6 t g = 6 t
Thus (.)
fl
= exp(ikHx)l' t
= G
G;.
Another approach t o t h e h a l f l i n e problem f o l l o w s [NwZ;Hwl] i n g (6.13).
=
which i s a standard s i t u a t i o n i n [ B a l l f o r example.
as i n 6 f o l l o w -
exp(ikH(x-y))r(y)Af,dy
and fr =
e x p ( i k H x ) A l ' - $ e x p ( i k H ( x - y ) ) r ( y ) A f r d y where r = rZ= ( 1 / 2 ) Z ' / Z as above, 1 0 1 1 0 1 ' = ( o ) , A = ( 1 o ) y H = ( o -1), fr,e = fr,[(k,x), and we t h i n k o f Z = 1 for x 2 0 .
It w i l l now be convenient t o assume r ( 0 ) = 0 ( i . e . A ' ( 0 ) = 0 = 0
0
Z ' ( 0 ) ) i n o r d e r t o have t h e t r a n s m i s s i o n and r e f l e c t i o n c o e f f i c i e n t s (T,R, and
k2 resp:)
f o r t h i s problem agree w i t h t h e corresponding f u l l l i n e co-
e f f i c i e n t s based on
= r2 - r ' = z5(Z-')"
- c f . below).
( = A-'(A')"
This
i s n o t a s e r i o u s r e s t r i c t i o n here and f o r c e r t a i n i d e n t i f i c a t i o n s i t i s n o t 3
needed a t a l l ( c f . 52.6 and [Nw2;Hwl]
f o r discussion).
r,
Note i n § 6 T, R[,
e t c . a r e used f o r t h e h a l f l i n e and t h e r e i s some d i s c u s s i o n f o l l o w i n g (6. 13) w i t h formulas i n Theorem 6.16.
For c e r t a i n i n v e s t i g a t i o n s however r ( 0 )
9 0 can be i m p o r t a n t and one can e a s i l y o b t a i n formulas r e l a t i n g R,T,R2 and 1 + e x p ( i k x ) , fl 2 + 0, i f ' + R,T,R2 f o r such s i t u a t i o n s . One has now (**) fe 0
0
0
fiexp(ikx), and if: + e x p ( - i k x ) a t m w h i l e a t -m, ?fL + e x p ( i k x ) and if:r+ 0 1 2 R2exp(-ikx) ( f r = 0 and fr = e x p ( - i k x ) f o r x 5 0 ) . F u r t h e r one has
To compare these w i t h (7.1) i s s i m p l y a m a t t e r o f r o u t i n e i n t e g r a t i o n by 2 p a r t s w i t h q = q = r - r ' and r ( 0 ) = 0; one uses a l s o ( c f . [ N w l ] and §2.6) 1 1 2 2 2 Dx{r,e = kHf r ,t - rAfr,L; Dxfr,g = ikfr,[ - rfr,e; and Dxfr,[ = -ikf,,[ -
rfr,e
.
I n t h i s d i r e c t i o n consider (from (4.3))
(*A)
cZ1 = 1 / T = 1
- (1/2ik)
POTENTIAL AND SPECTRUM 1
2
1 ; q e x p ( - i k t ) ( f e + 5 ) d t p r o v i d e d f+ = (I/~)z'/z,
t ) E1d t
=
q = r2-rl =
( - i k t ) d t = r ( 0 ) f e2 (k,O)
.
T?5(~-')11
-
1 ; r2 e x p ( - i k t ) ( $
fa1 + 52
171
Now 1 ; q e x p ( - i k t ) $1d t
+ r(O)f{(k,O) 1
f,)dt 2
+ 1 ; r2e x p ( - i k t ) ( $
-
Recall again r =
i s valid.
1 ; ( r2 - r ' ) e x p ( - i k
=
and s i m i l a r l y ; 1 qfeexp 2
5 1) d t
-
2ik1;
r e x p ( - i k t ) $ d2 t .
2
Consequently f r o m ( 7 2 ) (*@) 1/T = 1 +:1 r e x p ( - i k t ) $ ( k , t ) d t = 1/T.
ig =
calculations give
ir= Rr
RL and
0
t h a t t h e r e a r e d i f f e r e n t formulas r e l a t i n g RpT,Rr
RElRARK 7.1.
Similar
when r ( 0 ) = 0 ( c f . a l s o [Nw2] a 6 n o t e 0
0
when r ( 0 ) t 0).
and RL,T,Rr
We n o t e h e r e i n p a s s i n g t h a t t h e Zakharov-Shabat (Z-S) eigen-
-
f u n c t i o n s s a t i s f y ( c f . 19) ( * 6 ) Dxvl + isv = w 2 and Dxv2 iiy2 = wpl w i t h 1 1 9 = ('1 1 % ( 0 ) e x p ( - i s x ) as x --m ( s i m i l a r l y f o r J, = (JI, J , ~ ) , column v e c t o r , Ova JI % ( l ) e x p ( i s x ) as x -1. If we i d e n t i f y q and w w i t h -r = -rZ= rAand 1 2 2 1 w i t h k t h e n v1 % fr, v 2 % f 1 ry Q1 % yl f and J,2 % fL. -f
-f
We r e c a l l n e x t t h e h a l f l i n e f o r m a t o f 551.6,2.6,
f o r second G e n e r a l l y one makes assumptions A E C 1 , 0 < a 5
o r d e r e q u a t i o n s as i n (+).
A 5 B
0 and t h u s knowledge o f I c I f o r h r e a l leads t o c v i a t h e Poisson-Jensen formula f o r example.
A c t u a l l y i n 51.6 and [C8] a method i s
g i v e n t o r e c o v e r c - d i r e c t l y from readout impulse response a t a p o i n t y > 0
..
Q
( t r a n s m i s s i o n d a t a ) p r o v i d e d A ( y ) = Am f o r y 'y.
I m p l i c i t i n [C8] b u t n o t
developed i s t h e f o l l o w i n g r e s u l t which d i s p l a y s F/2c- as an i m p o r t a n t spect r a l q u a n t i t y ( c f . [C36,43]
Under t h e hypotheses i n d i c a t e d ( A
t?HE0REN 7.2. A
-+
from which much o f t h i s s e c t i o n i s e x t r a c t e d ) . E
C
1
,0
5A 5 6
x (m*) 0 = [Ep R PI f 1 (A,x)fr(A,
here).
(A+)
with Tf2
[Ep
/I
R f l = f;,
-
T h i s w i l l a l s o h o l d o f course w i t h P r e p l a c e d by Q and hence (u)
y)dA. T(x,y)
-
= ( 1 / 2 n ) / z [Ep
an obvious a b b r e v i a t i o n .
R ]f Q (A,x)fl(x,y)dx 4 9 1 This leads t o
= ( 1 / 2 1 ~ ) i I[Zp
-
EQ]f;f~dX
in
For E p = Bp o r -Sp one has an M e q u a t i o n ( y > x ) 0 = K(x,y) +
CHE0REm 7.6,
T(x,Y) + Jxm K(x,S)T(S,y)dS.
Next one wants a formula f o r K i n Theorem 7.6 analogous t o say (&*). from (em) (me) K(x,y)
=
(1/2n)l:
-
[f,(A,x) P
f lQ( ~ , x ) ] T f Q (A,y)dA
First
and we r e -
= T f2(A,y) Q - R f Q(A,y) Q 2 t o obtain (m&) c a l l (++) f o r y > x w i t h fl(-A,y) Q 4 1 f r ( A , x ) T f Q (h,y)dA = [RQ - E p ] f ( A , x ) f ~ ( A , y ) d A . It f o l l o w s t h a t f o r
LI
lI
4 2
P Q Ep]fl(X,x)fl(h,y)dA
Q Q f2dA Q -( s i n c e l: flT P 2n6(x-y) i n ( m e ) ) . S i m i l a r l y by symmetry i f 6 t L : fl + fy we have f o r y > x 4 P ( m m ) L(x,y) = (1/2n)j: [RP - ZQ]fl(h,x)fl(h,y)dA and o f course (&*) s t i l l
y > x
(m+)
holds ( i . e .
K(x,y)
(1/2a)l:
=
[RQ
= D K(x,x)
=
(7.10)
( q - b ) ( ~ ) = (l/n)Dxl:
X
-
K(x,y) = (1/21~)j: [RQ
-DxL(x,x)
For
ME0REI 7.7.
-
).
Rp]flQ(h,y)fp(A,x)d
Since ( l / Z ) ( i - C )
we g e t f o r m a l l y
Xp =
[HQ
B p o r -Sp,
-
zp]fl(h,x)fl P
Q (A,x)dh
K(x,y) i s g i v e n by
(m+)
when y > x and
(7.10) p r o v i d e s f o r m a l l y a recovery formula f o r t h e p o t e n t i a l s . Thus tip o r -Sp p l a y s a r o l e i n t h e h a l f l i n e t h e o r y p a r t i a l l y analogous t o
Rp i n t h e f u l l l i n e (compare (7.10) and (&*) e t c . ) . We w i l l l o o k a t some now and seek t h o s e w i t h t h e most n a t u r a l con-
h a l f l i n e spectral quantities
n e c t i o n s t o t h e p o t e n t i a l (Theorem 7.2 w i l l be a guide here a l o n g w i t h r e 2 It i s convenient t o assume A E C s u l t s i n [C2,3,10,11,13,34] and 52.6). and work w i t h
6=
-
D2
4 ( 4 = A-'(k')") again. L e t us r e c a l l ( A ' ( 0 ) # [SinA(y-x)/h]~(x)iA(x)dx Q and GhQ = c f l t c-f;
= 0)
(***) v.Q A ( y ) = Coshy
t
&!).
2 i A c - a t y = 0 (we drop s u b s c r i p t s Q e t c . when no con-
Now W(i,f,)
=
f u s i o n w i l l a r i s e ) and as y (cf.
(7.1))
(**A)
+
2ixc- = i h
m,W(4,fl)
- 1;
n,
exp(ihx)[iAi
qexp(iAx)Gh(x)dx.
-
(fl
@ ] which g i v e s
Also f r o m (A) w i t h
%
177
POTENTIAL AND SPECTRUM
W(i,f,)(O) = D f l ( 0 ) ( * * a ) 2ixc- = i x - 1 ; c$oshxfl(x,x)dx. Similar r e s u l t s ; 4[Sinhx/h] hold f o r F = f l ( x , O ) where W(S',f,)(O) = -F; thus (**O F = 1 + 1 fl(A,x)dx (from (A) a g a i n ) . Me will d e r i v e general formulas f o r 1 ; A$p[dx e t c . below b u t f o r now we i n d i c a t e just t h e l o c a l i z a t i o n s t e p . Thus take e. g. C = 2ixc- with W ( i , f l ) = C. Arguing a s in Remark 7.4 one o b t a i n s (**+) & ( v ) = ( f l / C ) $ i 2 vdx - (G/C)$ fl$vdx; d f l ( v ) = -(fl/C)Jx" + f l v d x + ( a / C )
-1; i AQ ( x ) f l ( x , x ) v ( x ) d x o r a C / a q = x -2 - i f l . Similarly one has from W(s',fl) = -F (*A*) d i ( v ) = - ( f l / F ) J o e vdx + (e'/F)$ e f l v d x ; d f l ( v ) = (f1/F)Jxme'flvdx - (i/F)JXm flvdx 2 a n d (*AA) d F ( v ) = ; 1 i ; ( x ) f , ( h , x ) v ( x ) d x or a F / a q = i f , . Now although c- and F a r e individuall y natural enough t h e l o c a l i z a t i o n s involving i f , or - i f l a r e s o r t of "mixed" and one expects a b e t t e r l o c a l i z a t i o n from R2 = F/2c-, ", o r S Thus 9' -2 and consequently -2ixdc % G f i we have & [c-dc - cdc ]/c or from (**=I, (*A*) a$ / a q = (1/2ixc-2)iG. Similarly from (*AA), dF- % i f ; a n d (*A&) asQ/ Q 2 .. a q = - ( 2 i x / F ) 0 e . These have a nice "square eigenfunction" form with half l i n e eigenfunctions ( c f . 552.9 and 2.10 f o r square eigenfunctions in s o l i t o n t h e o r y ) ; however asymptotic considerations via formulas 1 ike (AA) do not seem t o give an inverse. We will i n v e s t i g a t e t h e half l i n e square eigenfunct i o n s a t another time a n d r e f e r t o [Aol,Z;Cjl;Nll ,2;Nvl;Zkl ,2] f o r background. Finally consider F/2c- = R2 with dR2 % ( 1 / 2 ) [ c - i f l + F4f1/2ix]/c-2 = (fl/4ihc-')[F; + Eixc-s']. From (A*) we have then (*A+) a R 2 / a q = (1/4ih c-')f: a n d sumnarizing we s t a t e fxm flvdx. 2
Then a s in
(W)
(**,)
dC(v)
=
2,
&HE@REIR 7.8.
by (**.),
The l o c a l i z a t i o n s f o r c - , F , 3, S , a n d R2 = F/2c- a r e given (*A@), (*A&), and (*A+).
(*AA),
We see t h a t F/2c- = R2 l o c a l i z e s l i k e R2 in (+*) and in f a c t from (A&) e t c . ( * A m ) R2 = ( l + R 2 ) / ( 1 - R 2 ) = 1 + 2 R 2 / ( 1 - R 2 ) (with = 1 These expressions will a r i s e l a t e r in our half l i n e version of 2R2/(1+R2)).
REmARK 7.9.
Ril
a spectral transform ( o r IST).
We note a l s o from Fc + F-c- = 1 t h a t R 2 +
R -2 = & = 1 / 2 / c / 2 so t h a t some range r e s t r i c t i o n on t h e o p e r a t o r dR2 may
a r i s e as f o r R in [TzZ] ( c f . Remark 7 . 4 ) ; we will examine t h i s a t another time. In any event however, in c e r t a i n r e s p e c t s R 2 seems t o be a s i g n i f i cant s p e c t r a l quantity f o r t h e half l i n e . In developing t h e IST f o r t h e half l i n e below we will want t o extend t h e Fourier Sine o r Cosine transform. In t h i s d i r e c t i o n l e t us note t h a t (*a*) + x 2,.0 0 = f 2 f ; and xGe' = ( i / 4 ) ( f22 - f;').
R m R K 7.10,
Now finding an expression f o r AR2 (or 43 o r AS) i n terms of i n t e g r a l s of
Ai
178
ROBERT CARROLL
may n o t be p o s s i b l e " e x a c t l y " (which i s o f small concern s i n c e t h e IST i t s e l f i n v o l v e s b o t h R and S = 2ikR/T).
I n t h i s d i r e c t i o n l e t us w r i t e down
some p r e l i m i n a r y formulas c o n n e c t i n g p w i t h s p e c t r a l q u a n t i t i e s . One can Q P where W(0) = 2iAcp and s t a r t w i t h Wronskian formulas f o r say W = W(GA,fl) 2 W(m) = 2 i ~ c - Thus from (D i)i: = -A2$? and (D2 b ) f lP = -A 2 flP we have 9' P Q P W(i Q ,f P )I; = IF (fi-4)GAfldx so (*@A) 2iXAc- = -IF Af6:(x)fl(x)dx (Ac- 9 cp, A Ap.1 = 6-6, e t c . ) . S i m i l a r l y W1 = W(O,,fl) ' 9 P s a t i s f i e s W1(0) = - F p and- 'W1
-
-
-
-
(a)
= -F
Q
so (*@@)
mulas below.
AF =
:I
We w i l l expand upon these f o r -
A$:(x)fl(A,x)dx. P
Summarizing here g i v e s The formulas
CHE0REm 7-11,
r e l a t e A t t o Ac- and AF.
(*@A)-(*@@)
REIIIARK 7-12, L e t us say a few works now about t r a n s m u t a t i o n k e r n e l s
Gy
-
i!
-+
i n o r d e r t o produce some a d d i t i o n a l h a l f l i n e r e c o v e r y formulas analogous
Thus one has t r a n s m u t a t i o n k e r n e l s (*a&) i ( y , x ) = ( G QA ( y ) , v* P ( x ) ) ~ Q = (;:(x),GX(y)) where ( f , g ) Q I fgdo e t c . such t h a t i:i FJ, + and ;(x,y) -Q .P Q Q v- X 9 and +: v X + v X . It i s e a s i l y shown ( c f . [C2,3;Mrl] and 582.6, 1.6, 1.11 t o (7.10).
Q
f o r d e t a i l s ) t h a t b(y,x) = 6 ( x - y ) + i((y,x)
(*@+) 2D i ( x , x ) = X
x) = 0 for x
>
4-b
= -2Dxi(x,x).
y and ;(x,y)
and ;(x,y)
= 6(x-y)
+
i(x,y) with
F u r t h e r one has t h e t r i a n g u l a r i t y i ( y ,
= 0 f o r y > x.
I n order t o obtain a spectral Set
formula f o r t? d i r e c t l y one can s u b t r a c t o f f t h e 6 f u n c t i o n as f o l l o w s . N
~ ( y , x ) = T(x,y) and n o t e t h a t K(y,x) = 0 f o r x < y w i t h a 6 f u n c t i o n c o n t r i b u t i o n a l o n g t h e diagonal x = y.
Thus f o r x i y
(*OH)
t?(y,x)
= i(y,x)
-9 B(Y,x) = 10 v X ( y ) ~ ~ ( x ) [ d o-p doQ]. S i m i l a r l y one has a k e r n e l ( ( f , g ) 2 2 P Q 1 f g d v w i t h dvQ = 2x dh/nlF( ) (*&*) i ( y , x ) = ( i : ( ~ $ ~ ( x ) ) and ~ ;(x,y)
N
*P Q-Q (OA(x),OX(y)) w i t h
Q
f o r e and we w r i t e
$
g:
'P
Oh
+;:
= 6(x-y)
and
c:
is;.
-
? ( x , y ) = lo m O ' QA ( y ) i ! ( x ) [ d v p
[C10,11,34;Chl]
and 52.6) ( * b e )
4-6
-
T,
=
One has t r i a n g u l a r i t y as be-
+ t ( y , x ) w i t h ;(x,y)
c ( x , y ) = 0 f o r y > x and t?(y,x) = 0 f o r x i(y,x)
-
m
= 6(x-y)
Hence ( x ~
+
?(x,y) where n
(*&A) ) K(y,x) = y. dvQ]. F u r t h e r as i n (*@e) one has ( c f .
= 2Dxi(x,x).
y
We summarize t h i s i n theo-
rem 7.13 and w i l l w r i t e a r e f i n e d v e r s i o n l a t e r .
CHE0REm 7.13. (*&A ) - (*&@ )
.
One can w r i t e Ap i n terms o f Ado o r Adv v i a (*em)-(*@+) o r
L e t us now r e f i n e Theorem 7.13 by u s i n g a K-L v e r s i o n o f t h e t r a n s m u t a t i o n k e r n e l s due t o t h e a u t h o r ( c f . [C2,3,15] (7.11)
B'(Y,x) = (1/271)l;
and e.g.
52.6).
~:(Y)[~Y(A,X)/C~I~A;
B^(Y,XI= - ( i / n ) l I
!(Y )[fY(A,x)/FpldA
Thus
POTENTIAL AND SPECTRUM (cf.
(A*)
A f u l l l i n e version of
and [C2,3]).
B"
179 (g(y,x)
c o u l d a l s o be u s e f u l so we w r i t e ( r e c a y f r o m [C2,3] c h a r a c t e r i z e d by
(*&a) ;(Y,x)
E:
fl/c-
-,f l / c -
and i ( y , x )
%
= ( 1 / 2 1 ~ ) j zTp(c,/cQ)fp(X,,y)f~(X,x)dX
= ;(x,y))
;(x,y)
-8:
by
fl/F
v^
and is
and 52.6 t h a t +
fl/F)
and C(X,Y) = ( 1 / 2 ~ ) 1 1Tp
( F /F )fQ(A,y)fi(X,x)dX. Ifone r e t a i n s a i term i n (*U) ( f i r s t e q u a t i o n ) P Q 1 = (1/2 f o r example v i a K-L r e d u c t i o n i t would a r i s e i n t h e f o r m ( * b e ) ;(y,x) IT)/: i!(x)[fl Q (A,y)/c-]dh so t h a t combining w i t h i ( y , x ) would s t i l l i n v o l v e f u r t h e r reduction.
Q
I t s h o u l d n ' t r e a l l y m a t t e r whether we use ( * t b ) o r (*&+);
e v e n t u a l l y we want e x p r e s s i o n s f o r t h e k e r n e l s i n terms o f i n t e g r a l s o f "nat u r a l " spectral quantities against t i o n o f (**+) o r (7.12)
(*&a).
i l s
Thus u s i n g
and i l s w i t h a view toward a p p l i c a w i t h (7.11) and (*&+) we o b t a i n
(A*)
~ ( Y , x ) = (l/n)lI [?!(y)G!(x)R;
+
ihG~(y)e~(x)ld~;
S i m i l a r l y , as f o r (*&+)
(obvious n o t a t i o n ) .
We see t h a t t h e s p e c t r a l q u a n t i t y R2 a g a i n a r i s e s n a t -
u r a l l y and i n t e g r a l s o f t h e f o r m CHE0REm 7.14-
,I xeidx
= 0 since
i
and
i are
even.
Thus
i,
can Under t h e hypotheses i n d i c a t e d t h e k e r n e l s i, $, and 13) i n terms o f Re = F/2c-. I n p a r t i c u l a r formally l / R 2Q - l/R;, e t c . ) (*&.) (1/2)Ab = -(l/n)DxL: X 2
be w r i t t e n v i a (7.12)-(7
1/ 2 )A6
=
( 1/ n ) D x i I
1:
The v a n i s h ng o f i n t e g r a l s
REmARK 7.15.
formulas f o r t r a n s m u t a t i o n k e r n e l s .
i P, ( x )G Q( x )AR2dX. heGdx a l s o p r o v i d e s , i n t e r e s t i n g
Thus e.g.
i(y,x) =
(1/1~)1: G:(y)G!(x)
R2dX P and z ( y , x ) = ( 1 / 1 ~ ) j : 4x(y)GX(x)R2dA. Q P Q I f we now add t h e two e q u a t i o n s i n (*&.) t h e r e r e s u l t s f o r m a l l y (*+*) Af, = P-Q 2-P-Q 2 ( l / n ) D X I I [i,v,(~R~) - X eXeX~(l/R2)]dx. I n p a r t i c u l a r formally f o r P = D P w i t h 6 = 0 and R2 = 1 (7.14)
6=
(l/r)Dxf:
[CosXxC!(l-R2) Q
-
xSin~x~!(l/R~-l)]dh
and we n o t e t h a t 1-R2 = -2R2/(1-R2) w i t h 1/R2-l = -2R2/(1+R2) ( c f . (*Am)). T h i s w i l l r e p r e s e n t one h a l f o f o u r IST f o r t h e h a l f l i n e . For t h e o t h e r
180
ROBERT CARROLL
and w r i t e o u t fl = FG t 2iAc-e'; a l o n g w i t h t h i s we h a l f we go t o (*.A)-(**.) Q P one has ( w i t h a l i t t l e c a l c u l a t i o n ) W(0) = n o t e a l s o t h a t f o r W = W(Gx,f2)
- i x with
= 2ix[c-RP/T Q pp - c Q/T P1 w h i l e f o r W(s'hQ,fg) = W one has W(0) = + FQR ]/Tp. Consequently ( A ~ I= 4-6) (*+A) 2 i x [ ( 1 / 2 ) +
W(m)
-1 w i t h W(-)
= -[F-
P 1 - (F- + FQRP ) / T p = -1: A$!(x)f2(A,P (c-R P-c ) / T ] = -IoP A$:(x)f2(A,x)dx; Q Q P Q one o b t a i n s x ) d x and we r e c a l l f2 = 4 - i x i . Expanding (*+A) and (**A)-(*.*) f o r A; = cj-b, AC- = c i - cp, e t c . ( * W ) FpI: A$!$Fdx + 2 i x c p I r A K qA Phd x =
-2ixAc-;
-I:
+
A$%'dx 4 ?p
+
A$$:dx
FP$
2ixcpi:
ixIr A%p!dx -
i h 1 , A P % dx = 1
xx
A$F!dx
= AF
(from
and (*+&)
(*.A)-(*..))
-
= 2 i x [ ( 1 / 2 ) + (cQRP cQ)/TP]; -1; A$!$!dx + P P [F- + FQR ]/T Now some c a l c u l a t i o n s which we merciQm
.
f u l l y o m i t g i v e (*++) 10 A@$!dx = -iA[TPc- + 2c-c-RP - 2c-c 1; /:A@%,' P 1; A $ F F d x =Q -1 + PTQF /2 + cP- QF + c-F R ; dx = 1 + c-F R - c-TP - c F Q P Q CJ P; P Q PQ P Q 1 ,A $ y F d x = [TpF - Fp(FQ + FQRP)]/2ix. Some f u r t h e r c a l c u l a t i o n s w i l l Q Q-P P Q Q y i e l d ( c f . ( * A m ) , (A&), e t c . ) (*+a) 1 ; Ab[Gf! + e,lpx]dx = T R2/T + R p ( l R!R:)/TQT' + (R;Q - R ; ) / T ~ T ~ = [C-(RQ - i ) / c p + 2c-c ( 1 - R~-P ) ( RP~ t RQ ~ +) Q 2 Q P 2. P - Q + R2)]/(1 P + RF)(*=*)Jr Ab[$:Gf - h exO,]dx = ( i X / ( l + R 2P ) ) 2c,CQ(RiQ - R2)(1 P
b
.
-
[cQ(R!
l)/cp
here t h a t
-
(*mA)
2c c - ( 1 - R i P ) ( 1 + R PR Q ) + 2c-c ( 1 + RP2 ) ( 1 - RpR-Q)]. Note P Q 2 2P Q 2 2 R/T = c ( l - R i ) , 1/T = c (1 + R 2 ) , and R2/T = c (R2 - 1 ) .
Also some c a l c u l a t i o n shows t h a t e.g. ( * m * ) i s s u i t a b l y a n t i s y m n e t r i c . In P 2 p a r t i c u l a r t a k e now P = D again w i t h p = 0, Rp = R2 = 0, T = 1, e t c . and P (7.15)
lom $qxSinAx/A .Q
(7.16)
1"0
For
6
-f
+ i;Coshx]dx
~ [ C o s x x- ~hSinxxi):]dx ~
= c-(RQ
Q 2
1 ) + c (R-Q 4 2
-
1)
= i h [ c - ( R 9 - 1 ) + cQ(1-R;')] Q 2
D2 t h i s l a s t e q u a t i o n approaches
interesting.
-
:I
qCos2Axdx
2.
0 which i s n o t t o o
However (7.14) and (7.15) t o g e t h e r a r e o f i n t e r e s t and we have
EHE0REm 7-16,
The formulas (*+*) and (7.14) h o l d f o r m a l l y and (*++) has
been e s t a b l i s h e d l e a d i n g t o ( * W ) and (7.16).
The formulas (7.14) and (7.15)
p r o v i d e a form o f IST f o r m a l l y f o r t h e h a l f l i n e .
Phuud:
L e t us examine t h e l a s t statement.
e t c . and R$
small TQ
For
* 1 i s approximately r e a l w i t h c
Q
+
1/2.
%
1,
4:
Cosxx,
The f i r s t e q u a t i o n
Q (7.15) becomes a p p r o x i m a t e l y a S i n e t r a n s f o r m ( * m @ ) I; 6Sin2xxdx % A(R2-1) w h i l e (7.14) i s % ( l / n ) O x l I (1-R:)CosZxxdx = ( 2 / n ) / I x(R2-1)Sin2xxdx. Q
4
Treating A(R2-l)
as odd ( v i a (*me) f o r example) t h i s l a s t e q u a t i o n i s
(4/7r)1; X ( R 2 - l )SinZxxdx which means 4(5/2) i n v e r s e 2X(R2-1)
'L
%
(2/7)J;
%
2A(R2-1 )SinAgdx w i t h
:1 4(5/2)Sin2xgdg i n agreement w i t h
(*me).
QED
POTENTIAL AND SPECTRUM
181
It i s w o r t h n o t i n g what happens i f we b e g i n w i t h a h a l f
RflllARK 7.17.
b
problem i n v o l v i n g F o u r i e r t y p e o p e r a t o r s
-
= D2
f~ extended
with
ine
t o be an
-
even f u n c t i o n (and s u i t a b l y continuous, d e c r e a s i n g a t a, e t c . c f . 52 6). = A-'(A5)" = r + ri, I f i n f a c t t h i s a r i s e s from an impedance problem w i t h
rA = (1/2)A'/A,
and h = r ( 0 ) = ( 1 / 2 ) A ' ( O ) then i n general ( u n l e s s h = we 2 d e f i n e d as s o l u t i o n s o f I% = - A 4 w i t h F(0) = 1, z ' ( 0 ) = 0, -5 x 5, , \ v j I , - n "llU n \,lUL.C " - " ""L " r\u, -
Z,g, i ) . Since
distinguish " \ v j
v,
h so
=
UllU
we w r i t e ( c f .
(A*))
a l s o t h a t 2ih;-
-
N
IIUlll
~
T
N
A
0 rapidly a t
-+
(*.&)
= 2iXc-
m
+ hF).
=
1/2 y ) . These a r e o u r s t a n d a r d M equations. The i d e a a g a i n i s t h a t s p e c t r a l knowledge o f RR, R L y T ( o r some s u b s e t ) determines rRand rL and hence AR and AL ( r e c a l l we assume no bound s t a t e s f o r t h e moment).
Then as i n e1.6;
c f a l s o §§2.1-2.7) one can show
t h a t (*+) u ( x ) = -2D A ( x , x ) = 2D A ( x , x ) ( s o RR o r RL w i l l d e t e r m i n e u; r e x R X L c a l l t h a t t h e t dependence o f u and e v e r y t h i n g e l s e , except X -k2, has been momentarily suppressed).
I f t h e r e a r e now bound s t a t e s a t k = i k . s e t J
and (*. ) becomes
Thus t o determine AR (and hence AR) one needs RR(k) f o r k r e a l , t h e l o c a t i o n s k = i k . o f t h e bound s t a t e s , t h e i r number N, and t h e " n o r m a l i z i n g " c o n s t a n t s mRJ= y . c ( i k . ) = - i c l l ( i k . ) / i 1 2 ( i k j ) (c',2 = dc12/dk e v a l u a t e d a t j J11 J J k = i k . ) . The r e c o v e r y f o r m u l a (*+) has t h e same form (we r e f e r t o [Lml] J f o r t h e bound s t a t e c a l c u l a t i o n s - c f . a l s o [Chl ; F a l l ) . Now i n s e r t t h e t dependence i n u $xx(x,t) fl(k,x,t)
-
u(x,t)$(x,t) and f 2 ( k , x , t )
2
= -k $ ( x , t )
q and c o n s i d e r ( k independent o f t ) (*.) where ( 6 ) h o l d s .
and s i m i l a r l y (+)
-
(m)
t ) and AL(x,s,t).
F u r t h e r we must w r i t e c . . ( k , t )
c2,(k,t)f2(-k,x,t)
+ cZ2(k,t)f2(k,x,t)
1J
Then one has f u n c t i o n s
i n v o l v e s f u n c t i o n s AR(x,sY w i t h e.g.
fl(k,x,t)
=
and f i n a l l y ( w o r k i n g w i t h AL now)
ROBERT CARROLL
186
u ( x , t ) = 2DxAL(x,x,t) (8.8)
AL(x+Y,t)
(8.9)
AL(z,t)
where + A,-(x,y,t)
+
I,X
AL(S+y,t)AL(x,s,t)ds
= 0
(x
>
Y)
1;
= (1/21~)1: [ ~ ~ ~ / c ~ ~ ] ( k , t ) e - t~ ~ ' m d ki ( i k j , t ) e k j 2
(mjL = v j ( t ) c 2 2 ( i k j , t )
= - i ~ ~ ~ ( i k ~ , t ) / ; ~ ~t ) ( iwhere k 621(ik
t) % d ~ ~ ~ / d k j' j' e v a l u a t e d a t k = i k . i s a r e s i d u e term). The p o i n t here i s t h a t t h e t i m e J v a r i a t i o n o f A L ( Z , t ) i s v e r y simple. I n f a c t one can use t h e l i n k i n g o f u 2 and JI from t h e Lax f o r m u l a w i t h $xx - u$ = - k JI and ( o ) , namely JIt = -4JIxxx t 6 q X t 3ux$ i n t h e a s y m p t o t i c r e g i o n where u % 0 i s assumed so t h a t )Lt - 4 1 4 ~(we ~ ~o n l y need s c a t t e r i n g d a t a t o determine t h e t r a n s m i s s i o n and r e f l e c t i o n c o e f f i c i e n t s - and t h e n o r m a l i z a t i o n c o n s t a n t s as i n d i c a t e d a f t e r Q
2
(8.9)). Consider a s o l u t i o n o f yxx - uy = -k y and yt = -4yxxx o f t h e f o r m y = h(k,t)fl(k,x,t) p r o p o r t i o n a l t o fl as x + ( h i s needed s i n c e f % 1 e x p ( i k x ) cannot s a t i s f y ( 0 ) ) . E v i d e n t l y h must s a t i s f y ht = 4 i k 3 h so h ( k , t ) = h(k,0)exp(4ik3t).
Next c o n s i d e r fl
--with y
and w r i t e o u t yt = -4yxxx t o o b t a i n
2,
h(k,t)fl
%
~ ~ ~ e x p ( i tk xc 2) 2 e x p ( - i k x ) as x
+
.-.
L mJ. ( i k j , t ) hDtfl
+ htfl
+ Dt~22 =
=
LJ -8kft m.(ik.,O)e J J
3
+ D t ~ 2 1 = ( i k ) 3~ ~ ~ ( - 4 ) 3x 3 - 4 ( - i k ) c22 so D t ~ 2 1 = 0 and D t ~ 2 2 = - 8 i k c ~ ~ T) h.i s 3
= -4hD f i n v o l v e s 4 i k c21
g i v e s then (by r e l a t i o n s among t h e c .
i n d i c a t e d e a r l i e r ) cll(k,t)
= cll(k,
Ij mR. ( i k t ) = mR. ( i k O)exp(8k:t). O)exp(8ik 3t), c 1 2 ( k , t ) = c12(k,0), and 3 J j' J j' i l a r l y RL(k,t) = RL(k,O)exp(-8ik t ) and e.g. (8.11)
%(Z,t)
3
= ( 1 / 2 1 ~ ) 1 RL(k,O)e 1 -i(kzt8k t)
Sim-
I; m;(ikj,0)e-8kjt+kjz3
Thus summarizing we s t a t e ( f o r m a l l y and h e u r i s t i c a l l y ) L m.(ik.,O), and t h e p o l e l o c a t i o n s i k . f o r u(x,O) J J J have been determined, t h e subsequent temporal development o f t ( z , t ) i s d e t -
ME0REIR 8.1.
Once RL(k,O),
ermined by (8.11) and t h i s leads t o u ( x , t ) = 2DxAL(x,x,t).
RmARK 8.2.
Similarly
In o r d e r now t o a r r i v e a t an e x p l i c i t m u l t i s o l i t o n s i t u a t i o n
one wants t o deal w i t h r e f l e c t i o n l e s s p o t e n t i a l s ( i . e . RL(k) = 0 mark 8.3).
F i r s t however ( c f . [Chl;Lml;Kyl])
-
c f . Re-
we c o n s i d e r s i t u a t i o n s where
SOL ITON THEORY
187
I f t h e r e a r e no p o l e s o f RL (A*) RL(k) = c q (k-a,)/n; ( k - B . ) (m 5 n - 1 ) . J ( k ) f o r Imk > 0 t h e n i n ( 8 . 5 ) r L ( z ) = 0 f o r z < 0. I f i n a d d i t i o n T ( k ) has no poles f o r Imk > 0 (no bound s t a t e s ) t h e n (*&) a p p l i e s and AL(x,x) f o r x < 0 (so u ( x ) = 0 f o r x < 0 ) .
= 0
To f i n d u i t is e a s i e s t t h e n t o con-
R R ( k ) T ( - k ) + RL(-k)T(k) = 0 and 1 = + (RRI2 = s t r u c t RR(k) f r o m (") 2 l T I 2 + I R L I , e t c . ( c f . [Lml]) and t h e n use a s e p a r a b i l i t y t e c h n i q u e t o obt a i n AR. One has h e r e f o r T ( k ) h a v i n g n e i t h e r zeros n o r p o l e s i n Imk > 0
(
+ indicates
a s e m i c i r c u l a r c o n t o u r i n t h e upper h a l f p l a n e and R = RL o r
RR-note (1/2ni) $ [logT]dd(s-i)
= 0 and one t a k e s c o n j u g a t e s and adds t o
l o g T ( k ) = ( 1 / 2 n i ) P [ l o g T ( ~ ) ] / ( ~ - k )t o g e t ( 8 . 1 3 ) ) . More g e n e r a l l y when T ( k ) has e.g. f i r s t o r d e r zeros a i o r p o l e s B i n Imk > 0 j
(8.14)
T(k) =
n
[ ( k - a i ) ( k - B . ) / ( k - ~ i ) ( k - ~ . ) l e [(1/2ai J J
Thus suppose e.g.
( c f . [Lml]).
RL(k) = a B / ( k + i a ) ( k + i B ) ,
)J-.f% a,@>
& w ] d i 0 r e a l . Then
as above u ( x ) = 0 f o r x < 0 (we assume T has no zeros o r p o l e s f o r Imk > 0 ) . By (8.13),
u s i n g R = RL one o b t a i n s T ( k ) = k ( k + i y ) / ( k + i a ) ( k + i ~ ) where y2
a2+~' ( e x e r c i s e ) .
Then from (")
RR(k) = - a B ( k + i y ) / ( k + i a ) ( k + i B ) ( k - i y )
t h a t by (8.5) r R ( z ) = moexp(-yz) where mo = - 2 a ~ / ( y + a ) ( y + ~ )< 0. t o f i n d a s o l u t i o n now o f (*.)
=
SO
One t r i e s
i n t h e form AR(xyy) = p(x)exp(-yy) (separ-
a b l e ) and a l i t t l e c a l c u l a t i o n g i v e s ( e x e r c i s e ) moexp(-yx) + p ( x ) + p ( x ) (mo/2y)exp(-2yx) = 0 w i t h (Am) u ( x ) = -2DxAR(x,x) where exp@ = (2y/ImoI 1/2 .
2
2
2
= 2y Y(x)csch (yx++)
2 2
Another example o f t h i s t y p e i n v o l v e s RL(k) = - a /(a +k ) (a r e a l ) and i f one chooses t h e p o l e o f T ( k ) i n t h e upper h a l f p l a n e t o be a t i a = k a l s o t h e n working w i t h ?(k) = T ( k ) [ ( k - i a ) / ( k + i a ) ] and r e f e r r i n g t o (8.14) one o b t a i n s T ( k ) = k ( k + i a J Z ) / ( k 2 +a2 ). Then (") y i e l d s RR(k) and some c a l c u l a t i o n g i v e s AR(z) = r R ( z )
+
yocl,(ia)exp(-az)
( - a d z ) f o r z > 0.
( c f . ( 8 . 6 ) ) i n t h e f o r m A R ( z ) = 2aJ2exp
An assumption now o f AR(x,y) = f ( x ) e x p ( - a J 2 y )
2
leads t o
2
( e x e r c i s e ) (A&) u ( x ) = -4a Y(x)sech aJ2x which i s a t r u n c a t e d f o r m o f t h e 2 sech p o t e n t i a l o f ( * ) ( f o r t = 0; n o t e f ( x ) = -2yexp(-yx)/(l+exp(-2yx), Y
aJ2 i n AR above).
REMARK 8.3.
Going now f i n a l l y t o r e f l e c t i o n l e s s p o t e n t i a l s RL(k) = 0 (and
m u l t i s o l i t o n s o l u t i o n s ) one has r L ( z ) = 0 and, i n s e r t i n g now a t i m e dependence, we t a k e ( p o l e s a t k = i k . ) J
(A+)
AL(zyt) =
IN1 m Lj ( t ) e x p ( kJ. z )
where mL =
j
188
ROBERT CARROLL
3 m!(O)exp(-8k.t) as i n (8.10). One t r i e s f o r a s o l u t i o n o f say (8.8) i n t h e J J Set *(z) f o r m (suppressing t momentarily) ( A m ) AL(x,y) = 1 a . ( x ) e x p ( k . y ) . L J J = (m.exp(k.2)) and @(z) = ( e x p ( k . 2 ) ) (as column v e c t o r s ) so AL(x,y) = A T ( x ) J J J T @ ( y ) (A(x) ( a . ( x ) ) ) and (8.8) becomes (AL(x,y) = (x)@(y)) J
*
AT(x)[I +
(8.15)
1:
@ ( ~ ) * ~ ( s ) d s ] @ ( y+) q T ( x ) @ ( y ) = 0
where I = (( 6 . .)) and V(x) = I t
LE @ ( s ) * T ( s ) d s
i s an i n v e r t i b l e m a t r i x . I t T f o l l o w s t h a t AL(x,y) = -*T(x)V-l ( x ) @ ( y ) and AL(x,x) = -Tr[@(x)* ( x ) V - ’ ( x ) ] T 2 = -Dxlog detV ( s i n c e dV/dx = @(x)* ( x ) ) ( e x e r c i s e ) . Hence ( a * ) u ( x ) = -2Dx 2 l o g d e t V ( x ) and i n s e r t i n g now a t dependence one o b t a i n s u ( x , t ) = -2Dxlog T d e t V(x,t) where V(x,t) = I + @ ( s ) * ( s , t ) d s f o r * ( z , t ) = ( m h ( i k o)exp 3 J j’ (-8k.t)exp(k.z)). F u r t h e r i t i s e a s i l y checked t h a t t h e e n t r i e s i n V(x,t) J L J L 3 L w i l l be ( m . ( t ) = m . ( i k O ) e x p ( - 8 k . t ) ) (@A) V . . ( x , t ) = 6ij t [m.(t)exp(ki t J J j’ J !J J kj)x/(kitkj)]. I f one t a k e s a two s o l i t o n s i t u a t i o n ( N = 2 ) w i t h poles o f T ( k ) a t i k l and i k 2 ( k 2 > kl) then some r o u t i n e c a l c u l a t i o n ( c f . [Lml]) 3 3 y i e l d s , f o r y1 = klx - 4klt + 61, y2 = k2x - 4 k 2 t t 6 2 y and 6i = ( 1 / 2 ) l o g 1J
[(m, (0)/2ki )(K2-kl
11
)/(k2+kl
2 2 2 2 2 u ( x , t ) = -2(k2-kl )[(k2Csh y2+klCshyl
(8.16)
where Csh = csch, Cth = coth, and Tnh = tanh.
-
2 2 (y 1-A) -2klsech
u
)]
Now one can show ( e x e r c i s e
-
t h a t a t times l o n g b e f o r e o r l o n g a f t e r t h e 2 s o l i t o n s i n t e r a c t
c f . [Lml]) (00)
)/(k2Cthy2-klTnhyl
2k2sech 2 2 (y2+A); u
%
-2k 21 sech2 (y1+A) - 2k2sech 2 2 For kl = 1, k2 = 1.5,
(y2-A) r e s p e c t i v e l y , where A = 210g[(k2+kl)/(k2-k,)].
and 61
= 0 one has f o r example ( c f . [Lml])
REmARK 8.4, We want t o say j u s t a word here about conserved q u a n t i t i e s ( c f . [Cel;Dal;Kul;Lml;Nel;Ol] - more w i l l be s a i d l a t e r ) . To see how these a r i s e
- 6uuX + uxxx = 0 and i n t e g r a t e t o o b t a i n - uxx]dx = 0 (assuming s u i t a b l e b e h a v i o r a t f m ) .
t a k e f i r s t t h e KdV e q u a t i o n ut (06)
Hence
Dt{f
udx =
/f udx
/_fDx[3u2
S i m i l a r l y , m u l t i p l y i n g t h e KdV e q u a t i o n by u and 3 + (1/2)ux]dx 2 = 0 2 Dx[2u - uuxx i n t e g r a t e t o o b t a i n ( a + ) (1/2)Dti_1 u dx =
so
_/fu2 dx
= constant.
= constant.
[Z
Many more conserved q u a n t i t i e s can be d i s c o v e r e d i n
an ad hoc manner b u t t h e r e i s a l s o some meaning t o a l l t h i s and t h e m a t t e r
w i l l be discussed more s y s t e m a t i c a l l y below.
SOLITON THEORY
189
L e t us go n e x t t o t h e i m p o r t a n t paper [Lx5] and e x t r a c t a few b a s i c i d e a s . We work w i t h t h e KdV e q u a t i o n i n t h e form (om) ut
+
uux + uxxx = 0 and ob-
s e r v e t h a t i f v i s a n o t h e r s o l u t i o n t h e n w = u-v s a t i s f i e s ( 6 * ) wt + uwx +
wvx + wxxx = 0. M u l t i p l y by w and i n t e g r a t e by p a r t s o v e r (-a,-), assuming - 2 co w and w -+ 0 s u i t a b l y a t + m , e t c . , t o o b t a i n (6.) Dt(1/2)Lm w dx + la ( v x xx2 (1/2)ux)w dx = 0.
2 Set ( 1 / 2 ) { 1 w dx = E ( t ) and max 2vx-u,
D t E ( t ) 5 m E ( t ) from which by G r o n w a l l ' s lemma one has
= m t o obtain
(60)
E ( t ) 5 E(0)exprnt.
T h i s shows t h a t s o l u t i o n s a r e u n i q u e l y determined by t h e i r i n i t i a l values I n c i d e n t a l l y t h e e x i s t e n c e o f g l o b a l s o l u t i o n s o f (om) and r e l a t e d
u(x,O).
equations has been e s t a b l i s h e d i n v a r i o u s c o n t e x t s ( c f . [ B f l ;Ka6,7;Swl
;Te2;
Tul;Twl])
and we do n o t deal w i t h t h i s here. One notes a l s o t h a t , as b e f o r e 2 i n (*), u ( x , t ) = 3csech [ ( 1 / 2 ) J c ( x - c t ) ] i s a soliton solution o f (0.). L e t us w r i t e (om) i n t h e general form ( 6 6 ) ut = K(u) ( c f . a g a i n
REmARK 8.5, [Lx~]).
Now t h i n k o f a map u
+
Lu = L ( t ) ( L
Lu s e l f a d j o i n t i n some H i l b e r t space H.
'L
D2 + ( 1 / 6 ) u h e r e ) w i t h
event i f L(O)$(O,x) = h ( O ) $ ( O , x ) t h e n $ ( t , x ) = U(t)L(O)$(O,x)
=
Consider t h e requirement t h a t t h e
L ( t ) = Lu s h o u l d be u n i t a r i l y e q u i v a l e n t when u s a t i s f i e s ( 6 6 ) . L(t)U(t)$(O,x)
L
=
U(t)$(O,x)
= U(t)h(O)$(O,x)
I n that
s a t i s f i e s L$ =
= h(O)$(t,x)
(UU* =
u*u
=
I).Thus t h e eigenvalues h ( 0 ) would be i n t e g r a l s o f t h e e q u a t i o n ( 4 6 ) ( X ( t ) v a r i e s ) . NOW g i v e n (a+) U ( t ) - l L ( t ) U ( t )
= A ( O ) = h u ( t ) = c o n s t a n t as u ( t , x )
= L ( 0 ) independent o f t and w r i t i n g B = UtU*
we o b t a i n (6m) Lt = [B,L]
(which i s t h e same as ( 8 . 4 ) when Lt = -ut b u t h e r e Lt (6m) i s c a l l e d a Lax e q u a t i o n and (L,B)
by d i f f e r e n t i a t i n g ( 6 + ) ) . $t = Ut$(O,x)
'L
(1/6)ut).
Equation
a Lax p a i r ( n o t e (6m) i s o b t a i n e d
We n o t e a l s o t h a t w i t h $ = U$(O,x) as above (+*)
= U U*$
= B$ as i n t h e d i s c u s s i o n b e f o r e (8.3). I n the pret 2 s e n t s i t u a t i o n w i t h ( 0 . ) as t h e KdV e q u a t i o n one f i n d s t h a t L = D + ( 1 / 6 ) u 3 and B = 4[D + (1/8)uD + ( 1 / 8 ) u x ] = 4D3 + (1/2)uD + ( 1 / 2 ) u x y i e l d s [B,L] =
K(u) = ( 1 / 6 ) ( u x x x t uux) = ( 1 / 6 ) u t = Lt.
T h i s d i s c u s s i o n g i v e s a somewhat
cleaner version o f t h e h e u r i s t i c s e a r l i e r leading t o (8.4) etc.
We n o t e
a l s o t h a t f o r a f i x e d L w i t h p o t e n t i a l u ( o r u/6) t h e r e w i l l be an i n f i n i t e number o f odd o r d e r Bm such t h a t [B,,L]
= Km(u) i n v o l v e s o n l y u and i t s x
d e r i v a t i v e s ; consequently ut = Km(u) determines a h i g h e r o r d e r KdV t y p e equat i o n such t h a t t h e eigenvalues X(0) o f Lu w i t h p o t e n t i a l u a r e i n t e g r a l s .
REmARK 8.6, = 3csech
2
Again f o l l o w i n g [ L x ~ ] c o n s i d e r s o l i t a r y waves u ( x , t ) = s ( x - c t )
[ c ( x - c t ) / 2 ] which we n o t e w i l l s a t i s f y
(+A)
- c s x + s s x + s x x x = 0.
I t was d i s c o v e r e d by Gardner and Kruskal and d e r i v e d by Lax i n [ L x ~ ] t h a t
t h e wave speeds c o f such waves a r e i n t e g r a l s o f t h e m o t i o n s a t i s f y i n g c . ( u )
J
190
ROBERT CARROLL
= 4h.(u) where h i s an e i g e n v a l u e o f Lu. To see how t h i s goes t h e o r e t i c J j a l l y , f o l l o w i n g [ L x ~ ] , we c o n s i d e r ut = K(u) a g a i n and suppose ( 0 0 ) DEK(u+
E V ) I ~ == V(u)v ~ e x i s t s as a Frechet d e r i v a t i v e f o r example.
Differentiating
= K(u) i n E we o b t a i n then ( 0 6 ) vt = V(u)v where v = DEuEIE=O, uE b e i n g ‘jt a one parameter f a m i l y o f s o l u t i o n s w i t h say i n i t i a l d a t a uE(O,x) = u o ( x ) +
Ef(x).
L e t I ( u ) be an i n t e g r a l o f ut = K(u) and assume i t i s Frechet d i f -
f e r e n t i a b l e w i t h ( 0 0 ) D € I ( u + E v ) = ( G ( u ) , v ) where G
%
gradient.
I f uE i s a
one parameter f a m i l y o f s o l u t i o n s t h e n I ( u E ) i s independent o f t and hence
so i s ( G ( u ) , v ) where ut = K(u) and vt = V(u)v ( u = u ( t , x ) e t c . w i t h a c t i n g i n x,
,
)
symmetric, and D E I ( ~ E ) I E , o
= (G(u),v)
translation invariant.
)
(
,
)
f o r v = DEuE a t
Now assume s ( x - c t ) i s a s o l i t a r y wave s o l u t i o n w i t h ut
= 0). (
,
(
E
K(u) and
Then e v i d e n t l y ( G ( s ( x - c t ) ) , v ( x , t ) )
i s inde-
pendent o f t f o r v as i n d i c a t e d and s e t t i n g v ( x + c t , t ) = w ( x , t ) one o b t a i n s (G(s(x)),w(x,t))
independent o f t.
Consequently
s i n c e wt = cvx + vt = cwx + vt one has
,
(
)
,
G(s(x)),wt)
= 0 and
wt = [cD + V(s)]w ( t h e o p e r a t o r + V(s)]w) = 0 and thus ) i f c o n v e n i e n t ) ([-cD+V*(s)]G(s),w) = 0 (D* = -D). The v a l u e
V w i l l comnute w i t h t r a n s l a t i o n s ) . ((
(+a) (
(m*)
Hence (G(s),[cD
o f w a t any p a r t i c u l a r t i m e (e.g. t = 0 ) i s a r b i t r a r y and one concludes that
(W)
[-cD + V*(s)]G(s)
Next assume t h e equation ut = K(u) i s en-
= 0.
e r g y p r e s e r v i n g i n t h e sense t h a t ( u ( t ) , u ( t ) )
i s independent o f t when ut =
K(u) ( t h i s can be v e r i f i e d f o r KdV f o r example). 2(u,ut) E
(me)
V*(u)u
= 2(u,K(u))
Then ((
,
)
= (
,
)) 0 =
and p u t t i n g i n uE one o b t a i n s a f t e r d i f f e r e n t i a t i n g i n
(v,K(u)) + (u,V(u)v) = 0.
Thus s i n c e v i s a r b i t r a r y ( m b ) K(u) + 0 and f o r 0 = c s x + K ( s ) one has [cD - V*(s)]s = 0. Hence s be-
longs t o t h e n u l l space N o f cD
-
V*(s) so from
(aA)
G(s) =
KS
s i n c e dim N
This shows t h a t under t h e hypotheses i n d i c a t e d f o r I g i v e n w i t h grad I = G then a s o l i t a r y wave s ( x - c t ) s a t i s f i e s G(s) = K S where K = = 1 i s normal.
K(I,c) ( i . e . s i s an e i g e n f u n c t i o n o f G).
I f one a p p l i e s t h i s t o I ( u ) = - L = D 2 + 4 6 ) we
h ( u ) f o r Lw = hw ( i . e . w i s an e i g e n f u n c t i o n f o r X(u) obtain f i r s t
(i
3
D L = ;/6
=
v/6 when v =
DE~EIE,o)
L\;r + vw/6 = A ;
+iw.
Take s c a l a r products w i t h w and i n t e g r a t e t o g e t (wv,w)/6 = i(w,w) ( n o t e (Li,w) = (i,Lw) = A(i,w)). Normalize w so t h a t (w,w) = 1 and then i = ( 1 / 6 ) 2 2 It f o l l o w s t h a t (w ,v) = (G(u),w) which means t h a t gradh(u) = G(u) = w /6. 2 G(s) = w / 6 = K S so t h e e i g e n f u n c t i o n w o f L corresponding t o X(s) i s w = ^ C S ’ / ~ and we t a k e = 1. One then checks e x p l i c i t l y ( u s i n g -cs + s 2/ 2 + sxx =
0 etc. from
(‘A))
t h a t Lw =
f e r t o [ L x ~ ]f o r f u r t h e r discussion.
=
C S ~ / ~so/ t~h a t
c ( s ) = 4X(s).
We r e -
SOLITON THEORY
191
Consider ut + 6uux + uxxx = 0 w i t h M i u r a ' s t r a n s f o r m a t i o n ( m m ) 2 u = - v - v so t h a t (***) ut t 6uux + uxxx = -(Dx + 2 v ) ( v t - 6~ vX + vxXx). X 2 The e q u a t i o n vt - 6v vx + vxxx = 0 i s c a l l e d t h e m o d i f i e d KdV (mKdV) equa-
REmARK 8.7.
2
t i o n and we see t h a t e v e r y s o l u t i o n v o f mKdV i s mapped under (-) 2 s o l u t i o n o f KdV. I f we add t o ( m m ) t h e e q u a t i o n (**A) vt = 6v vx then (mm)
-
(**)
- vxxx
f o r m a Backlund t r a n s f o r m a t i o n (BT) between KdV and mKdV. = 0 and E(v,
Here one d e f i n e s a BT between d i f f e r e n t i a l equations D(u,x,t) = 0 as a s e t o f r e l a t i o n s i n v o l v i n g ( x , t , u ( x , t ) )
Y,T)
to a
and (Y,T,v(Y,T))
such
t h a t BT i s i n t e g r a b l e f o r v i f and o n l y i f D(u) = 0 and BT i s i n t e g r a b l e f o r
u i f and o n l y i f E ( v ) = 0 w h i l e g i v e n u ( r e s p . v ) such t h a t D(u) = 0 ( r e s p . E(v) = 0 ) BT d e f i n e s v (resp. u ) t o w i t h i n a f i n i t e s e t o f c o n s t a n t s and Such t r a n s f o r m a t i o n s w i l l be discussed l a t e r i n
E(v) = 0 (resp. D(u) = 0). more d e t a i l ( c f . §11).
E M I P L E 8.8.
Dx(u+v)/2 = a S i n [ ( u - v ) / 2 ]
and Dt(u-v)/2
= (l/a)Sin[(u+v)/2]
i s a BT t r a n s f o r m i n g t h e sine-Gordon e q u a t i o n qxt = Simp i n t o i t s e l f . The main p o i n t h e r e i s t h a t a s c a t t e r i n g problem as i n (9.1)
REmARK 8.9,
(9.2) ( i n 52.9 t o f o l l o w ) , namely, v vlt
= Avl
f o r A,B,C
+ Bv2 and v~~
= Cvl
t icvl
!x
= qv2,
v2x
-
-
i r v 2 = rvl w i t h
- Av2 l e a d s t o c o m p a t a b i l i t y e q u a t i o n s (9.3)
and an e v o l u t i o n e q u a t i o n D(u) = 0 f o r u = (q,r),given
nomial d i s p e r s i o n r e l a t i o n s f o r t h e l i n e a r i z e d problem.
say p o l y -
The e q u a t i o n s (9.1)
- (9.2) t h e n s e r v e as a BT between D(u)
= 0 and some E(v,&)
5 ) = 0 i s a p a i r o f PDE f o r v = (vl,v2)
n o t i n v o l v i n g u (see [ A o l ] f o r d i s -
cussion).
As an example o f how t h i s s i t u a t i o n l e a d s t o new i n f o r m a t i o n con-
s i d e r ( f r o m [Lml]) two S t u r m - L i o u v i l l e o p e r a t o r s (**.) ,y, and wxx =
= 0 where E(v,
(A +
J,(x,t)w
+ q(x,t)y w = A(x,t,x)y
= (1
i n which w and y a r e r e l a t e d v i a ( * 6 )
Given X~ = 0 we can be t a l k i n g about two s o l u t i o n s q , J , o f t h e KdV + y., equation. T h i s s i t u a t i o n i s a l s o r e l a t e d t o t h e Darboux t r a n s f o r m a t i o n d i s cussed i n [Lml] f o r example. c i e n t s t o o b t a i n Axx + (94)
qx
+
Now p u t ( * 6 ) A(q-J,)
and i n t e g r a t i n g one has A
-yx/F one
has
yxx = (y
2
i n t o (*a)
= 0 and 2Ax
- Ax
+
q
- q = T(t)
-
and equate c o e f f i J,
Eliminating
and l i n e a r i z i n g v i a A =
T ( t ) = 7 = c o n s t a n t we e q u a t i o n f o r x = ?. F u r t h e r
t q ) y " so s e t t i n g
i s a particular solution o f the y
= 0.
see t h a t
o f Darboux t r a n s f o r m a t i o n s ( o r t h e Crum t r a n s f o r m a t i o n ) we have 2 ( l o g ? ) " so t h e w e q u a t i o n becomes wxx = (A + q
-
and
-
T h i s i s a way o f i n -
t r o d u c i n g p o t e n t i a l changes which i s o f t e n p r o d u c t i v e .
z;
J, = q
E ( l o g 7 ) " ) w which i n v o l v e s
a p o t e n t i a l change f r o m t h e y e q u a t i o n o f -2(log;)". we i n t r o d u c e p o t e n t i a l f u n c t i o n s v i a q =
i n the s p i r i t
J,
I n t h e s p i r i t o f BT
z x so A = ( z - z ' ) / 2 i s
192
ROBERT CARROLL
a p a r t i c u l a r s o l u t i o n o f 2Ax
+v
-
J/ = 0 and A
2
-
Ax
- IP
h,
= X = -m/2 becomes
v = z X' (**+) p t p ' = m +(z - z ' ) / 2 . Now suppose 2 so t h a t (*.) zt - 3(Zx) + zxxx = 0 w i t h s u i t a b l e n o r m a l i z a t i o n (and s i m i l a r l y f o r z ' ) . Then d i f f e r e n t i a t i n g (**+) i n t and i n t e g r a t i n g i n x we o b t a i n (*A*) zt + z; = r ( z - z ' ) ( z t - z i ) d x ( a g a i n w i t h and s e t t i n g u = ztz', v = z - z ' one o b t a i n s some n o r m a l i z a t i o n ) . Using (*.) 2 zt + z; = 1 [ ( 3 / 2 ) ( v )xux - vvxxx]dx from which, a f t e r some c a l c u l a t i o n f o r p = ICI = zx and p ' =
Jlt
-
6ICICIX + ICIxxx
= 0
( n o t e h e r e from (*+)
z x x + z k X = uxx = vvx), 2 3/2)uX
The l a s t e q u a t i o n i n solution
-
2 2[P2 + PP' + ( P ' ) 2 1 vvXx + ( 1 / 2 ) v X = { - ( z - z ' ) ( z x x - ZAX)
8.18) p l u s (**+) i s a BT f o r t h e KdV e q u a t i o n .
-
Ifa
s known t h e n a n o t h e r s o l u t i o n z may be o b t a i n e d by
z' t o (*.)
s o l v i n g t h e BT which we r e w r i t e here as (8.19)
( A ) zt +
(B) p + p ' = zx
+ z;(
2
Note Dx(B) + (A) i m p l i e s Q ( z )
(**.) and Dx(A)
-
-
= 2[p 2 + pp' + ( p ' ) 2 ] = m t (1/2)(z
+ Q(z')
-
(Z-Z')(Z,~-Z'~~); 2 ' )
2
= 0 where Q ( z ) = zt
Dt(B) i m p l i e s ( z - z ' ) [ Q ( z )
-
Q ( z ' ) ] = 0.
i m p l i e s Q ( z ) = Q ( z ' ) = 0 so J/ and p s a t i s f y KdV.
-
2
3zx + zxxx i s Hence (A) + (B)
As an example o f how t o 2 2 and zt = 2p - zzxx
use t h e BT t h e o r y t a k e z ' = 0 t o f i n d z x = m + ( 1 / 2 ) z = 2mzx ( z x x = zp).
where m = -2k
2
.
T h i s leads t o
Then zx =
9. S0CI.t!0Q)NsV I A A W
q
SgXEW.
(*AA)
z = -2kTanh(kx-4k3t) f o r I z I < 2k 2 s o l i t o n f o r KdV.
i s t h e s t a n d a r d sech
We w i l l g i v e now some d i s c u s s i o n o f c e r t a i n
i m p o r t a n t f e a t u r e s o f s o l i t o n theory. i n [Aol;Cjl ;Ddl ;Fa3;N11,2;Nvl].
We f o l l o w standard source m a t e r i a l
We w i l l be p a r t l y h i s t o r i c a l i n t h e o r d e r
o f s e l e c t i n g m a t e r i a l and p a r t l y personal i n t h e e x p l i c i t choices; a l s o we
w i l l n o t be a b l e t o cover a l l o f t h e r e c e n t m a t e r i a l i n t h i s r a p i d l y moving f i e l d ( c f . however §2.11). L e t us b e g i n w i t h t h e AKNS approach ( A b l o w i t z , Kaup, Newel 1, Segur) which g e n e r a l i z e s a c o n t e x t o f Zakharov-Shabat (Z-S). F i r s t r e c a l l t h e Lax e q u a t i o n ( 6 . ) i n § 4 (Lt = [B,L]) a r i s i n g from "compatiFor L and B s u i t a b l y chosen
b i l i t y " o f qt = BJ/ and LJ/ = AJ/ when A t = 0.
(4.) i s t h e KdV e q u a t i o n f o r example b u t one must e.g. guess L and f i n d B i n o r d e r t o have t h e c o m p a t i b i l i t y be a meaningful e v o l u t i o n e q u a t i o n . We
i n d i c a t e t h e AKNS procedure now f o r t h e g e n e r a l i z e d Z-S system (9.1)
vlx
t i r v l = qv2; v2x
-
i s v 2 = rvl
193
AKNS SYSTEMS and c o n s i d e r an e v o l u t i o n o f t h e f o r m v
(9.2)
It
= Avl + Bv2; v~~ = Cvl + Dv2 = Cvl
-
Av2
One assumes A,B,C
( D = -A i n v o l v e s no l o s s i n g e n e r a l i t y ) .
a r e s c a l a r func-
t i o n s independent o f v and we n o t e t h a t r = -1 i n (9.1) y i e l d s v2xx + q ) v 2 = 0.
A
(9.3)
-
C o m p a t i b i l i t y o f (9.1) = qC-rB;
X
B +2ir,B = q -2Aq; X t
C -2icC = rt+2Ar X
and these equations can be s o l v e d f o r A,B,C t i o n equation) i s s a t i s f i e d . mention e.g.
(c2+
(9.2) r e q u i r e s ( e x e r c i s e )
i f another e q u a t i o n ( t h e e v o l u -
There a r e v a r i o u s approaches t o t h i s and we
t h e expansion o f A,B,C
i n t r u n c a t e d power s e r i e s i n
c.
Thus
2 2 EX:IURPI;E 9-1- L e t A = A. + 0. However one notes t h a t zeros o f a ( c ) a r e n o t n e c e s s a r i l y s i m p l e o r on t h e imaginary a x i s and t h e y may o c c u r on t h e r e a l l i n e (we w i l l t r y t o a v o i d this last possibility).
EXAMPLE 9.2. $1(x,c)
I f r = f q * one has e.g. $,(x,r,)
= W;(x,t),
$,(x,c)
= -vT(x,?),
'E(t),Ck
$2(x,s) = * $ ~ ( x , ~ ) ,a^( 0 from - 4 0
+
195
to
m+iO
+
passing over a l l zeros
o f a ( < ) and operate on (9.7) w i t h (1/2*)JC e x p ( i s y ) d s t o g e t f o r y > x
+ :\
0 = ^K(x,y) + (Y)F(x+y)
(9.8)
K(x,s)F(s+y)ds
where ( m ) F ( x ) = ( 1 / 2 s ) I C ( b / a ) ( s ) e x p ( i s x ) d s , S(x) = ( 1 / 2 s ) i c exp(isx)ds, A
and ( 1 / 2 s ) i c [ 9 ( x , s ) / a ( 5 ) ] e x p ( i s y ) d ~ = 0 f o r y
(c"
( 1 / 2 s ) / t (^b/$)(s)exp(-isx)ds has f o r y
>
(9.9)
K(x,y)
RrmARK
9.3.
(along w i t h (9.10)
>
x.
S i m i l a r l y f o r F(x) =
a contour passing below t h e zeros o f
a")
one
x
-
1 * (o)F(x+y)
-
fXm
t?(x,s);(s+y)ds
= 0
I f a ( s ) # 0 f o r 5 r e a l and a ( < ) has i s o l a t e d simple zeros
i ) then F ( x ) = (1/2n)lf
(b/a)(c)eiSxdc
i ( x ) = (1/2a)\-
--
- ilN 1 c.ei5jx; J
(6/$)(c)e-icXdc
+ i l N C.e 1
- i s .x J
J
$(2j)/$'(tj).
and C j = I n t h e event o f slower deJ A cay o f q,r (9.10) s t i l l holds b u t t h e n o r m a l i z i n g constants c and c . a r e J J found from 7 = and c = $./a! etc. Rigorous r e s u l t s o f existence and J j J J uniqueness a r e known f o r t h e G-L-M t y p e equations (9.8) - (9.9) i n many i m where c
J
= b(s.)/a'(sj)
Cjqj
portant situations.
REmARK
9.4,
Now go back t o (9.1)
-
(9.2) and one knows from s i t u a t i o n s as
-
i n Example 9.1 t h a t t h e r e w i l l be s o l u t i o n s t o t h e c o m p a t i b i l i t y equations
B(s) *
(9.3) w i t h A ( < ) * A _ ( < ) ,
0, and C ( s )
-+
0 as 1x1 *
(we mention here
o n l y the s i t u a t i o n r = Tq* w i t h a2 = 2 i again from Example 9.1, where A = 2 2 i s i iqq*, B =-2qs i q andC=f2q*c T iq; so, w i t h q -+ 0 and qx -+ 0 a t 2x' +, A- = lim A ( < ) = 2 i s ). Set then (withv,$,a,b, e t c . time dependent) w(**) v t = v e x p ( A - t ) , $ t = $exp(-A-t), v t = vexp(-A-t), and $ t = $exp(A-t)
-
and p u t t h i s i n (9.2) t o o b t a i n e.g. (9.11)
DF =
-A-A
1
-
1 (note v 2, (,)exp(-isx) cannot s a t i s f y (9.2) so v t i s introduced). From v = 4 a $ + b $ one obtains then from (9.11) as x * m (*A) atexp(-i5x) = 0 and bt e x p ( i s x ) = -2A ( s ) b e x p ( i s x ) from which (*a) b ( s , t ) = b(s,O)exp[-2A-(c)t] and a ( s , t ) = a(s,O) time).
( t h e l a s t equation showing t h a t t h e sk are f i x e d i n
S i m i l a r l y from c
J
= $./a!
J
J
one has c . ( t ) = c exp(-2A-(sj)t, J J,O
A
cj(t) =
196
ROBERT CARROLL
A
c 3. ( t ) = ~ j y 0 e x p ( 2 A - ( c j ) t ) , nb(c,t) ;(s,O)exp(ZA-(s)t) and a*(s,t) = :(s.O). It f o l l o w s as b e f o r e ( c f . 98) t h a t (*6) F ( x , t ) = ( 1 / 2 1 ~ [Z(b/a)(t,O)exp[icx 2 A - ( t ) t ] d t - N ~ ~ , ~ e x p [ i c ~ x - ' 2 A.)t] - ( i ( w i t h a s i m i l a r expression f o r ?) and J t h i s g i v e s us some connection w i t h t h e r e s u l t s o f § 8 about t h e t i m e v a r i a -
c1
t i o n o f s p e c t r a l data.
so K1(x,y)
PI,
Take now r = Tq* f o r i l l u s t r a t i o n and x
(i= +F*)
+F*(x+y)
and q = -2Kl(x,x)
+
becomes q ( x , t )
-
%
i n (9.9)
-(1/21~)
( t h e c . c o n t r i b u t i o n w i l l be small as x J 2 Then t h e problem tends t o a l i n e a r problem as x -+ m ( i q t = qxx + 2q q* 2 i q t - qxx w i t h s o l u t i o n s q = ( 1 / 2 1 ~ ) jaI ( k ) e x p i [ k x - ~ ( k ) t ] d k where w = -k
jm(b/a)-(e,O)exp[-2ic~-2A:(c)t]dc
-m -+
-+
m).
-
i s the dispersion r e l a t i o n
here A- = 2 i s
2
so
~ ( c =)
2iA-(-c/2)).
Next we broach t h e f a s c i n a t i n g s u b j e c t o f squared e i g e n f u n c t i o n s ( o r p r o i n ( 9 . 1 ) f o r v and m u l t i p l i e s t h e 2 equations b y q1 and q 2 r e s p e c t i v e l y t h e r e r e s u l t s (*+) (ql)x + 2icr: = 2 2 2 2 ( I ~ ~ =P w~2 ) + ~w l . Thus t h e squared e i g 29p1q2; ( P ~ -) 2~ i w 2 = e n f u n c t i o n s s a t i s f y a homogeneous f o r m o f (9.3) (*.) A x = qC - r B , B + A2 i 2 2icB = -2qA, and Cx - 2isC = 2Ar. S i m i l a r a n a l y s i s a p p l i e s t o rpl, q2, etc.
ducts o f e i g e n f u n c t i o n s ) .
I f one uses
and i n f a c t 3 s o l u t i o n s o f
(*B)
q
are
By v a r i a t i o n o f parameters one c o u l d t h e n s o l v e (9.3) (where qt and r t a r e t h e inhomogeneous terms) and i t i s n a t u r a l t o t a k e as boundary c o n d i t i o n s on t h e c o n d i t i o n s before, namely A
A,B,C
requirement a t x =
fm
-+
A - ( c ) w i t h B,C
This + q t 2 , and a3 = ('l?') qzq2 -19 (r)@l 0 = rql (9.2) as (D = ( ), Q =
2 Az - ( ~ )q( ~ ) ] @ ~ d x0 f o r @1 = qq2 f o r example). To see
( AC -A'), N = (O q)) r O t i n g ct = 0 g i l e s
p*)Nt
verse P-'
'l)
t PSXP-'
1
= (-q2
-
(AA)
-+
0 as 1x1
lz
c o n d i t i o n s (:*)
requires
m.
[(r ) -+
,
vx = i r D v t Nv; vt = Qv. = Qx t ic,[Q,D]
D i f f e r e n t i a t i 2 g and s e t -
+ [Q,N].
Set P =
('l f l ) q 2 Po2
with in-
and d e f i n e S such t h a t Q = PSP-' so t h a t Qx = PXSP-l
Vpzl -VL] PSP- PxP .
One f i n d s then from (")
-
(Am)
that
(Ad)
S =
S(-m)
t ' I- m P- NtPdx w i t h S ( - m ) = A - ( s ) ( b -:). T h i s l e a d s t o Q and hence A,B,C b u t here we o n l y look f o r c o n d i t i o n s needed f o r a s o l u t i o n t o e x i s t . Thus r e A h aexp( - i S x ) ) bexp(-iSx)) as C a l l Ip = a? + b$, $ = -$$ + bJ/ w i t h Lp % (bexp(irx) and (-aexp(iTx) T h i s leads immediately t o x -+
'
-.
a; (9.13)
2ab
Evaluating
2 8
b6
S(m) = (A&)
at
00
-(as
- bi)
and u s i n g (9.13) g i v e s
Q
AKNS SYSTEMS
-
(*+), and d e f i n i t i o n o f t h e s c a t t e r i n g data, one has a l 2 m 42 2 ) dx = [qp, + wl]dx = ab; -ab = lm (v1v2)xdj: = La ( W 2
However from (9.1), SO (A+)
197
rf
r,” (v 1 2 x , ~p
A
A
+ r$:)dx; and -a$+bb+l = L I [‘P,;, + v2GlIxdx = :2: [qp2Z2 + w l v l l d x . Def i n e ai as above ( a f t e r (A*)) and t h e n p u t t i n g (A+) i n (9.14) one a r r i v e s a t f o r i = 1,2,3
(A*)
( e x e r c se
q u i r e d f o r a s o l u t i o n of
09
ted =
A,B,C
at
* l t l ) and ( w 2
(9.15)
-
c f . [Aol]).
Thus
(A*)
i s the condition re-
9.3) t o e x i s t w i t h t h e boundary c o n d j t i o n s i n d i c a -
S i m i a r l y we c o u l d use q1 = (*?
q2 =
*5)’
?m.
(*35)’
and *3
show t h a t [(-;It
+
2A-(s)(;)Pidx
= 0
f o r i = 1,2,3. Next we use t h e o r t h o g o n a l i t y c o n d i t i o n s equations as f o l l o w s .
i:
From (*+)
v,v2
=
t o determine t h e e v o l u t i o n
(A*)
/,” [ q p 22 + w 21]dx
and p u t t i n g I - f =
f ( y ) d y w i t h I+f = Ix fm ( y ) d y we can w r i t e t h e r e m a i n i n g 2 e q u a t i o n s i n
(*+) as ( i = 1,2; (9.16)
( I f)g = If g )
t f = (1/2i)
Lei =
i’
[
-Dx+2q1-r - 2 r I - r D x2q1-q -2rI-q
I
I n t h e s p i r i t o f a n a l y t i c f u n c t i o n a l c a l c u l u s i f A - ( s ) i s a n a l y t i c one w r i t e s A-(s)al
Im
-m
from (9.16) and
becomes ( A m )
(A*)
It i s e a s i l y seen now t h a t
(9.17)
T 1: = ( 1 / 2 i )
[ Dx+2rI+q 2qI+q
(
u,aI-Bv)
[(-i)t*i
+ 2(i)A-ai1
/,”
dxu(x)a(x)/, B(y)v(y)dy T o f C is dyB(y)v(y)/ym d x a ( x ) u ( x ) = ( ~ I + a u , v ) and t h e a d j o i n t C
dx = 0. =
A-(C)ei
=
- 2 r I+r -Dx-2qI+r
=
I
T r 2A-(C )(q)l@idx = 0 SO T r t h a t a s u f f i c i e n t c o n d i t i o n f o r s o l v a b i l i t y i s (.A) ) + 2A-(C ) ( ) = 0. q t q One can show t h e c o n d i t i o n (*A) i s a l s o necessary under s u i t a b l e hypotheses. Then
(A*)
i n t h e f or m
(A=)
becomes ( a * )
[(-;It+
(r
-
(9.3) The n a t u r a l e v o l u t i o n equations assoc a t e d w i t h (9.1) say A _ ( < ) e n t i r e , i s and boundary c o n d i t i o n s A + A_, B,C 0 a t +my w i t h
WE0REIII 9.5.
-+
given v i a
R!ZJIARK 9.6.
(.A).
Consider (9.1) a g a i n w i t h s o l u t i o n ‘P =
one has v l e x p ( i s x )
a n a l y t i c and i t approaches 1 as
(Ti)where f o r rl
-+
-.
Ims > 0
The f u n c t i o n
198
ROBERT CARROLL
a ( < ) = W(v,lL) = l i m v l e x p ( i g x ) dependent o f t ( c f . (*A),
v,
= exp(-isxtv),
as x
etc.).
-
t o obtain a R i c c a t i equation f o r vx = u
v = v(x,t),
n
a l s o has these p r o p e r t i e s and i s i n -
+ m
Now e l i m i n a t e v 2 from (9.1) and s u b s t i t u t e (0.)
+ q ( p / q l x . Since v + o as 1c.1 + (Ims > 0 ) ( 0 6 ) u = ( 2 i c ) - ' c ) ~ i] s a permissable expansion ( e x e r c i s e ) and p u t t i n g t h i s n-1 -qr, u1 -wX, Pntl = q ( u n / q I x + uk-$!-k,.-l one can w r i t e now l o g a ( s ) = v(m) = _:I udx and hence (om )
(*+I -m
lo
p0 =
m
Lm pndx.
C , , / ( 2 i ~ ) ~ ~ + ' where )] Cn =
Since l o g a ( s ) i s indepen-
dent o f t f o r a l l c w i t h Imc > 0 t h e Cn a r e t i m e independent and these y i e l d an i n f i n i t e number o f c o n s e r v a t i o n laws. (9.18)
Co =
(-qr)dx; C3 =
I
C,
=
For example
1 (-qrx)dx;
C2 =
1 [-qrxx
2 + ( q r ) Idx;
2 2 [-qrxxx + 4q rrx + r q q x l d x
The d e r i v a t i o n does n o t use (9.2) so such i n t e g r a l s a r e c o n s t a n t s o f motion f o r any o f t h e e v o l u t i o n equations based on (9.1) o b t a i n e d v i a a r b i t r a r y We r e f e r t o [ A l o ] f o r s p e c i a l r e s u l t s r e l a t i v e t o KdV, NLS, e t c .
(9.2).
We go n e x t t o t h e q u e s t i o n o f H a m i l t o n i a n s t r u c t u r e ( c f . §§1.7, 1 .a, 3.8 f o r background and see [Aol ;N11,2;Nvl;Fa3] f o r s o l i t o n r e l a t e d t h e o r y ) . We r e c a l l f o r example i n a v i b r a t i n g s t r i n g problem one would c o n s i d e r k i n e t i c 2 2 energy T = (1/2) f mutdx and p o t e n t i a l energy V = (1/2) f cuxdx w i t h L = T-V t 2 and p = mut so T (1/2) f ( p /m)dx. The a c t i o n i n t e g r a l i s A = Itf;Ldxdt and t h e standard v a r i a t i o n a l argument m i n i m i z i n g A g i v e s 0 Dt(mut)]
d x d t which y i e l d s a wave t y p e equation f o r u.
o f [Aol] f o r example one w r i t e s f o r H(p,q,t)
f f [Ox(cux)
-
Now i n t h e n o t a t i o n
= f h(p(x,t,a),q(x,t,B)dx,
aH/
f (dH/6p)(ap /aa)dx so t h a t i f h i s a f u n c t i o n o f p and d e r i v a t i v e s pn
aa =
= D p:
t h e n (6*) (6H/6p) =
1;
(-1)'D;(ah/apn),
This involves various i n t e -
g r a t i o n s by p a r t s and s u i t a b l e " t e s t " f u n c t i o n s o f course so g e n e r a l l y we
w i l l j u s t t h i n k o f 6H/Gp f o r example as t h e Frechet d e r i v a t i v e ( c f . 53.2). Thus r e c a l l f o r i l l u s t r a t i o n D'vdx
=
(
6H/6pPv> = DEH(p+w,q,t)lE=O = f as r e q u i r e d i n (&*).
The Hamilton equations
a p / a t = 6H/6p and a p / a t = -6H/sq become then f o r H = (1/2) f ( p
4=
p/m and
EUAIRPLE 9.7. = 0 and irt m
f~ =
1 (ah/apn) 2/m + cqx)dx, 2
-Dx(cqx) (as r e q u i r e d ) .
Consider a2 = - 2 i i n (*) which becomes then i q t + qxx
-
rxx2+ 2qr2 = 0.
-ilm [qxpx + (qp) I d x .
Then
Set q = q ( x , t )
4 = 6H/6p
and
-
2 2q r
and p = r ( x , t ) w i t h H = = -6H/Gq.
Now Poisson brackets were d e f i n e d i n 51.7 b u t we do t h i s here from a some-
AKNS SYSTEMS
3.8).
what d i f f e r e n t p o i n t o f view ( c f . a l s o v a r i a b l e s (p,q)
-f
(P,Q)
and d e f i n e (A,B)
=
199
Thus one t h i n k s o f changing
/I [(GA/Gq)(SB/Sp)
-
(sA/sp)(sB/
sq)]dx.
The t r a n s f o r m a t i o n (p,q) + (P,Q) i s c a l l e d c a n o n i c a l i f ( Q ( x ) , Q ( y ) ) We t h i n k now o f = 0, and ( Q ( x ) , P ( y ) ) = s ( x - y ) ( c f . §1.7). T r equations (*A), namely ( r, + 2A ( t ) ( ) = 0, i n which i t t u r n s o u t t h a t q -q t ( q , r ) p l a y t h e r o l e o f c o n j u g a t e v a r i a b l e s (q,p), t h e map ( q , r ) -+ (Q,P) i s = 0, ( P ( x ) , P ( y ) )
c a n o n i c a l , P and Q a r e o f a c t i o n - a n g l e t y p e (H = H(P)) so aP/at = 0, and
aQ/at
= sH/sP = c o n s t a n t , and t h e i n f i n i t e s e t o f c o n s e r v a t i o n laws f o l l o w
from t h e a c t i o n - a n g l e v a r i a b l e r e p r e s e n t a t i o n ( P and Q t o be determined).
In o r d e r t o show a l l t h i s we remark f i r s t t h a t A - ( s ) i n
(*A)
can be w r i t t e n
i n terms o f t h e d i s p e r s i o n r e l a t i o n s f o r t h e l i n e a r problem. Indeed f o r x T D + m I+ -+ 0 and I: 'L ( 1 / 2 i ) ( 0 x -D:). Then as x m (*A) becomes ( 6 0 ) rt + -f
2A ( D x / 2 i ) r = 0 and -qt + 2A-(-Dx/2i)q t i o n s one o b t a i n s q = exp[i(kx-w
q
= 0.
S o l v i n g f o r p l a n e wave s o l u -
( k ) t ) ] and r = e x p ( i ( k x - w r ( k ) t ) ]
so t h a t
( v i a F o u r i e r t r a n s f o r m i n ( 6 . ) ) ( 6 6 ) A - ( s ) = .(1/2i)w ( - 2 5 ) = - ( 1 / 2 i ) w r ( 2 5 ) . q Now f o l l o w i n g [ A o l ] we show L e t A- be e n t i r e o f t h e f o r m A - ( s ) (1/2i)r: (-2c)'an w i t h an EHEBREIII 9.8. r e a l . Then t h e system (0.) i s H a m i l t o n i a n w i t h c o n j u g a t e v a r i a b l e s q and r and H(q,r) = aninCn(q,r) where Cn i s d e f i n e d by (am).
icz
Phaob:
From
(0)
= 1
r e w r i t t e n a l i t t l e one has ( b e ) q l ( x , s ) e x p ( i s x )
+ :/
q
~ ~ ( ~ , c ) e x p ( i s y ) dfrom y which (6m) 6v1(x,s)/6q(y) = e ( x - ~ k ~ ( ~ , s ) e x ~ [ i s ( ~ - x ) l where e i s t h e Heavyside f u n c t i o n . Now f o r any p i e c e w i s e d i f f e r e n t i a b l e A one can d e f i n e (**) ( 6 A / 6 q ) ( x ) = l i m ( & A ( x ) / s q ( y ) ) as y = q2(x,s)
sql(x,c)/sq(x)
and s i m i l a r l y 6 p 2 ( x , c ) / 6 r ( x )
6ql/&r = sq2/sq.
For t h e e i g e n f u n c t i o n s $:Jl,2
= 6$1,2/6q
/6r
( n o t e $1,2
t
x so i n p a r t i c u l a r
= vl(x,s)
while 0 =
one has i n t h e same manner 0
i n v o l v e s Ix
and use (**)).
Hence (*A) 192 s a ( s ) / s q ( x ) = ( 6 / 6 4 ) [ ~ 7 ~ -9 ~92$11 q2G2 and s a ( c ) / & r ( x ) = -q1JI1 where everyt h i n g i s d e f i n e d f o r Imc > 0. Now f o r r e a l 5 from (9.1), as i n (**), one
has
(**I
= 6J,
(
~
~ + 42i591$l ~ ) = ~~ ( I P ~+J v2J11); , ~ (92J/2)x -
+ v2J,l)x = 2qv7 2 J, 2 + 2w 1J,1 from which vlJ12 + v2G2 + r ~ ~ $ ~ ]( ndo yt e a = ql$2 a t m ) . I t f o l l o w s t h a t q2JI1);
and t h i s extends t o t h e upper h a l f plane.
Now l e t I s 1
r ( v JI
=
+
P
m,
~ =*
Imr
a~ -
> 0,
+
1 2m
24
[q
t o get
200
ROBERT CARROLL
From t h i s and ( W ) we o b t a n (9.21)
grad
qYr
loga =
A l s o we r e c a l l l o g a ( s ) =
1C
/(2ir,)"' from (am) w h i l e t h e e v o l u t i o n equa'T r o r ( r, + 2A-(C ) ( ) = 0 where A (5) (= ( 1 / 2 i ) ~ ( - 2 5 ) ) by as-q t q q sumption has t h e form A-(CT) = ( 1 / 2 i ) C (-2cT)nan. Hence grad C /(2i)"' q,r n = - ( 1 / 2 i ) ( C T ) n ( qr ) o r ( + 6 ) gradq,,.Cn = -(Zi)n(CT)n(i) and consequently (W) tion i s
(@A)
2 A - ( C T ) ( i ) = (l/i)l (-2)"an(C T ) n(q) r --
ili inangrad q,r C n '
T h e r e f o r e t h e ev-
o l u t i o n equations have t h e form rt = -6H/6q and qt = 6H/6r where H anCn and 6H/6r = gradrH w i t h sH/sq = grad
R?illARK 9.9.
=
il in
H.
QED q One says t h a t two f u n c t i o n s A,B o f v a r i a b l e s (p,q)
(here (r,q))
are i n involution i f (A,B) 0 ( c f . ( U ) ) . I n t h e p r e s e n t s i t u a t i o n one has f o r Cn as i n Theorem 9.8 ( r e c a l l t h e expression as i n t e g r a l s i n (9.18) etc.) 6r)
-
(+m)
0 = dCn/dt =
(6Cn/6r)(sH/6q)]dx
;:
[(sCn/Gq)qt
+ (GCn/Gr)rt]dx
=
lI [(6Cn/Gq)(6H/
I n p a r t i c u l a r t h i s would h o l d f o r H =
= (Cn,H).
Cm and by t h e Jacobi i d e n t i t y ( c f . 91.7) (Cn,Cm)
= 0.
L e t us r e c a l l a l s o
t h e n o t i o n o f complete i n t e g r a b i l i t y o f a dynamical system w i t h N degrees o f freedom; i t i s c o m p l e t e l y i n t e g r a b l e i f t h e r e e x i s t N independent constants o f motion i n i n v o l u t i o n .
I n t h a t case one can s o l v e t h e Hamilton
equations by quadrature ( c f . [ C j l ] ) .
For an i n f i n i t e dimensional system t h e
s i t u a t i o n i s more c o m p l i c a t e d s i n c e t h e number o f c o n s t a n t s r e q u i r e d i s n o t c l e a r i n general.
REmARK 9-10, L e t us w r i t e down t h e a c t i o n - a n g l e v a r i a b l e s as f o l l o w s ( c f . Assume a ( s ) and a ( 5 ) have o n l y s i m p l e zeros i n t h e i r r e s p e c t i v e [Aol]). h a l f planes o f a n a l y t i c i t y w i t h no zeros on t h e r e a l a x i s . Assume a l s o t h a t b ( 5 ) and b ( c ) can be extended o f f t h e r e a l a x i s .
Define
Given a ( < ) = 0 and cm = ( b / a ' ) ( s m ) (Imr; > 0, 1 ( m 5 N ) and 4
Am
?k = ( b / a ' ) ( t k ) (Imsk tk, and = -2il0g;~.
^Qk
etc. are also natural.
/ ( r , - $ , ) lt ( 1 / 2 n i ) ( a ( & ) = a(s)nl (~-i,,,)/(c-Q) ( m ~ )l o g a ( s ) = l1 -Jm w [ ( l o g ( a G ) / ( ~ - ~ ) ] d(Imr, ~ > 0 ) , Wronskian r e l a t i o n s , e t c . T h i s f o r m u l a (mA)
can a l s o be expanded t o determine Cn i n terms o f l o g [ a i ( S ) ] ,
HAMILTONIAN STRUCTURE
201
-
IO~[(C-F~)/(F-~,,,)I,
and l o g [ ( 6 - t k ) / ( c - t k ) ]
t i o n t h a t (p,q) * (P,Q)
We o m i t v e r i f i c a -
( c f . [Aol]).
i s c a n o n i c a l ( c f . [ A o l ] f o r d e t a i l s and Remark 10.7)
and w r i t e down t h e H a m i l t o n i a n as ( c f . 510 f o r more d e t a i l s )
(9.23)
H =
(2/n)lI
A - ( ~ ) [ l o g [ a ~ ( ~ +) I
11N l o g [ ( E - ~ k ) / ( E - ~ k ) l ] d E
c1N log[(F-$,)/(E-cm)l
+
$ A-(s)dc (-5
4iI
To show t h a t a P / a t = 0 and a Q / a t = SH/SP one observes t h a t D t l o g [ a i ( c ) ]
= 0, A.
= 0, and Dtlogc = Dtlogb(E) = -2A-(C), Dtcm = 0, Dtlogcm = -2A-(s,), Ak 2A-(2k) ( r e c a l l h e r e from Remark 9.4 b ( s , t ) = b(c,O)exp(-ZA-(c)t) and b = A
E v i d e n t l y then SH/SP (Z/n)A ( 5 ) = -(l/n)Dtlogb boexp(2A-(c)t)). i s i n d i c a t e d i n [Aol]. c a l c u l a t i o n o f 6H/6sm and SH/S;k
= Qt w h i l e
Q,
10. 5 0 L 1 & 0 N &HE@Rg(HAIRILt0NIAN $tWetuRE). ed some f a c t s and f e a t u r e s f o l l o w i n g “121
and e m b e l l i s h t h e procedure i n
Thus f i r s t w r i t e t h e system (9.1) as
v a r i o u s ways.
vX
(10.1)
We w i l l e x t r a c t f r o m § 9 as need-
PV; P = ( - irs
=
i s1
A
and s e t
@
cl).
= (‘I
Consider v a r i a t i o n s 6 V , 6@, e t c . i n t h e f o r m (*) (6V)x
p l a c i n g V by S@(-L)].
@
But
6-r
-i6< 6q
‘ 2 ‘ 2
= PSV + ( 6 P ) V ; S P = ( (@-16V)x = -@-’PSV + @
).
Now
[ P S Vi6s + SPV] =
@-’ Q,
=
( m a t r i x A e q u f t i o n ) one o b t a i n s
@-’
-@-bX@-’ = -@-’P and
hence
X1 6 P V ( v a r i a t i o n o f parameters). (A)
Re-
+ @-’
6@ = @[J-xL@-16PPdc = -1 and hence
=
iL
where t h e l a s t t e r m i s 6@(-L).
Then we assume q,r
0
(and 6q,Sr) a r e
Q,
0 for
x 5 - L and a t x = -L, 6q, = i L e x p ( i c L ) s s w i t h 6?, = i L e x p ( - i c L ) S c f r o m (9.4) exp( -icL ) 0 ( iLexp( igL)6 s 0 iL 0 so @-lS@(-L) = ( 0 -exp( i c L ) 0 iLexp(-icL)dc) = (0 - i L bexp(-iix)) and we d e f i n e 65. Now as x +m , @ + aexp(-i<x) (bexp( is x ) -aexp( ic x ) (10.3)
I(u,v) =
(” [-Squ2v2
-m
( s o I = if ( 6 qr ) ( -uJ 2v$ 2 ) d x ) . L e t us n o t e t h a t $ = from ( A ) i n 59 w h i l e a$ + b^b with a = ql$2 s e t x = L and l e t L
-f
a
b
- a? w i t h $
q2$l,
etc.
t o o b t a n ( a f t e r some c a l c u l a t i o n )
=
b
+
6
Now i n (10.2)
ROBERT CARROLL
202
(10.4)
6a = - I ( v , $ ) + i 6 c l z (a-v1$2
+ v 2 j l ) d x ; 6g
- -I($,$)
-
-~
~ $ ~ ) d6 bx ; I(v,$) + i s c l z (vlZ2
+ $2$1)dx; 6 2 = -I(;,?)
'651:
- i 6 c l l [ Idx
(where [ ] = ( $ t $ l $ 2+ $ 2 $ l ) ) . We note now t h a t f o r two s o l u t i o n s ( ' l ) u2 and ('l) o f (9.1) ( 0 ) i(ulw2 t u2w1) = Dx[-D u w t D u w 1. In p a r t i c u l a r w2 3 1 2 3 2 1 using the asymptotic values ( 9 . 4 ) one obtains e a s i l y from t h i s ( 6 ) Dclog a ( 3 ) = - i l z [(v $ + v 2 i b l ) / a - l l d x . Further i n (10.4) f o r example one sees 1m2 t h a t 6a/6c 'I, =iLm (v1$2 + v2!b1 - a)dx so t h a t (10.4) represents 6a i n terms of 64, 6r, and 6 5 a s a " t o t a l v a r i a t i o n " . The o t h e r equations i n (10.4) have the same meaning. Now (assuming simple zeros Ck and cm) f o r s c a t t e r i n g data one takes e.g. ( c f . (9.22)) (6) b/a and ^b/if o r 5 r e a l , (L,,,,c,,,), and ( t k y t k ) ( c a l l t h i s S,). Alternatively one'can take ( m ) (S-): i / a and b/a* f o r 5 r e a l , (c,,,,~,,,), and (;k98k) where B, = l / ( b a ' ) ( c m ) and B"k = 1 / ( 6 a " l ) ( f k ) . Using 2 (10.4) one shows e a s i l y t h a t when 6 3 = 0 (*) &($/a) = ( l / a ) I ( $ , $ ) ; 6(b/a^) = (1/$2)1($,$) and some c a l c u l a t i o n , which we omit ( c f . [N12]), y i e l d s (*A) n 4 4 65, = cmIm($,$) = c, [ m m ( -6r 6 q ) ($1 + $ ) ( s m ) d x ; 6c.k = AckIk($,$) where I k denotes evaluation a t ;k (here one is i n the context of S-1. I n p a r t i c u l a r 3 , remains an eigenvalue of a ( c ) s i n c e (vm = b,,$,, a t cm) s a [ r , ( t ) , t ] = -bmIm($,$) + Further in t h e S- context (*@) ar;16cm 0 by (10.4) and (*A) ( c m = bm/a;). 6(~,,,) = (l/aA) 2 [I,,!,(+,$) - ( a " / a l ) I ($,$)I; 6 ( i k ) = - (i;/ii) m ! '"6r Ik($,$)] where e.g. I,,!,($,$) = jm(-,q)D3('%3)dx. Thus a l l of t h e v a r i a t i o n s involve inner products of ( - 6 r ) w i t h squared eigenfunctions ( c f . §9 where 6q the notation i s s l i g h t l y d i f f e r e n t ) ) and one s e t s 2 *2 j2 lL2 E- = w = ; ; ~m = Xm = ; Slk; ;k> (10.5) 2 2 $2 D3 $: rn
(~/;i)~[~i($,$)
[$iJ * =IJ;]
[
A
42
;Irn;
I"']
42
where Gk and ;k involve $ i and D 4 . evaluated a t tk. The formulas based on 3 1 data S, a r e referred t o i n t h e context of a dual inverse problem and e.g. one writes ( * 6 ) I ( v , v ) = -lI (i:)(*,ST)dx t o e x h i b i t "duality". One f i n d s then as above (10.6)
6(b/a) = ( l / a2 )I(v,v); 6 ( 6 / $ )
Then f o r S, one i s dealing ( v i a
(*&)I
=
( 1 / i 2 ) I ( $ , $ ) ; 65, = (l/cmar;12)
w i t h square eigenfunctions
HAM1LTONI A N STRUCTURE
where J
AT
AT
G:
(10.8)
L = (1/2i) -Dx-2q1+r 2rI+r
and D
A 2
e v a l u a t e d a t ;k. t h i s " d u a l i t y " n o t a t i o n i s t h a t i f one d e f i n e s ( c f . (9.16) 11
qk and X k i n v o l v i n g
203
rdy [ -Dx-2qhm 2rfXm r d y
-'qr+q]
;L
T
Ip.
5 1
-2q/xm qdy
1
The reason f o r - (9.17))
-
DX+2rrxa qdy
= (1/2i)
2r1-r I-r -Dx+2q
Dx+2rI,q
( n o t e t h a t i f we w r i t e CT = (a b, t h e n L =
(-1-:!
w h i l e C = ("
1 t) w i t h LT
-'))
t h e n e.g. (L-c)(*,*) E d= 0 and (L-c)(xm,xk) = (fl,Zk)'while ( LT- 5 ) = ( -8 C( T "T T *T ( x i , i l ) = (Qk,Q ) ((LT-s)(* ,Q ) = 0 - we n o t e t h a t CGl = cG1 i n (9.10) i s T 9 T t h e same as L Q = c* f r o m (10.7) - (10.8))..
PElllARK 10.1.
The s e t s E+ o r E- a c t u a l l y f o r m a b a s i s i n s u i t a b l e spaces and
one can expand v e c t o r s such as ( sr) o r ( :)
-Q
i n terms o f them ( c f . [Kp1,2]).
F i r s t we n o t e t h e o r t h o g o n a l i t y r e l a t i o n s f o r square e i g e n f u n c t i o n s based m T Lm u-vdx. Thus suppose Lu = cu and L v = c ' v . Then a l i t t l e c a l -
on ( u , v ) =
c u l a t i o n g i v e s (*+) I_\u(c)v(r,')dx
= [1/2i(c-c')](u2v2
-
u
~
v so ~u s i n) g ~
t h e a s y m p t o t i c s (9.4)
These a r e sometimes c a l l e d D a r b o u x - C h r i s t o f f e l t y p e formulas and a s i m p l e T c o n t o u r i n t e g r a t i o n argument f o r example y i e l d s now (*.) ( Q ( s ) , * ( e l ) ) = 2 AT -aa 6 ( 5 - 5 ' ) ( 5 , ~ 'r e a l ) . S i m i l a r c a l c u l a t i o n s y i e l d (A*) ($(L),*( 5 ' ) ) = A 2 4T na S ( 5 - 5 ' ) ( a l s o (*(c),* ( 5 ' ) ) = 0, e t c . ) and d i f f e r e n t i a t i n g t h e s e r e l a T T 2 t i o n s one o b t a i n s (u) ( D Q ( c k ) , Q ( c . ) ) = ( Q ( c k ) , D Q ( 5 . ) ) = - ( i / 2 ) a i fikj, T 5 J 5 J ( Dc*(Ck),D,* ( c j ) ) = -(i/2)aia;skj, e t c . ( c f . "121). Some f u r t h e r c a l c u l a t i o n , u s i n g (*+I i n §9 f o r example, E, o r E- as a b a s i s )
(*m)-(A*)-(AA),
e t c . now y i e l d s ( g i v e n
~
~
ROBERT CARROLL
204
+
(10.10) where
r
+ ;k6;Gk];
62kGk
-
= 2 i c cm#
2ic
(-:)
2k*A k , and
we r e f e r t o [Kp1,2]
f o r proofs. Let C (resp.
connection l e t us n o t e a few formulas ( c f . [Kp1,2]). to
+
= - ( 1 / n ) l E [(b/a)*+(^b/;)*lds
In this
t ) be
a
i n t h e 5 p l a n e passing o v e r (resp. under) t h e zeros o f assume f o r s i m p l i c i t y t h a t q , r have compact support. Then
contour a (resp
m
*
(10.11 )
= ( l / n ) l c (b/a)*d
0 and and f l ( x , i ) = GT,(x,A) where 1 w r i t e T 1 (x,A.) = y . T 2 (x,A.) when a ( A . ) = 0 and residue(l/a(A))T-(x,A)lA=Aj 2 J J + J J A . ( j = 1 ,..., N) w i t h = c.T,(x,~.) (c = yj/a'(Aj)). Then g i v e n r, c J J j ji J 1 suitable properties one wants t o f i n d ,T, ,T, and ( l / a ) T w i t h t h e nece s s a r y a n a l y t i c p r o p e r t i e s . We w r i t e e.g. ( c f . (**)) (*&) T,1 :- ( 01) e x p ( - i A x / 1 2 0 0 r , ( ~ , y ) ( ~ ) e x p ( - i ~ y / 2 ) d y w i t h T, = (1 ) e x p ( i A x / 2 ) + IX rt(x,Y)(l w)exP 2) + ( i A y ) d y and p u t t h i s i n (0;) t o o b t a i n f o r y 2 x t h e f a m i l i a r (**) r,(x,y) t A(xty) t r,(x,s)A(sty)ds = 0 where A ( x ) = w(x)u- + E;(x)u+ w i t h w(x) =
0.
> 0 and 151 l a r g e ( c f .
[Aol]),
Thus as r e q u i r e d i n K(x,y),
w h i l e !blexp(-i
f o r y < x the in-
t e g r a l vanishes by c o n t o u r i n t e g r a t i o n i n t h e upper h a l f p l a n e so t o s i m p l y
i s formally correct o f course (and can be r i g o r o u s l y c o r r e c t f o r s u i t a b l e growth e t c . ) b u t i t l e a d s f o r example a t x = 0 which may encounter d i f f i c u l t i e s i n convert o (A&.) 2 gence ( u n l e s s s t a n d a r d t i m e v a r i a t i o n nb(c,t) 2, nbo(s)exp(4is t ) i s p r e s e n t i n a way which which may n o t happen). Thus i n o r d e r t o dea w i t h (A&.) c l e a r l y e x h i b i t s i t as a f o r m u l a i n F o u r i e r ype a n a l y s i s w i t h a d i s c o n t i n u i t y a t 0 ( i . e . q ( x ) = 0 f o r x < 0 ) one must l e a v e t h e e x p ( - i s y ) t e r m i n , i n i n s e r t y = x as i n
c o u l d be m i s l e a d i n g .
(A&+)
some way, and t a k e l i m i t s . small x, JI = =
^bp
+
b
- a,;
T h i s i s most eas l y accomplished by w r i t i n g f o r
so t h a t ( c f . [ A o l ] and 19) f o r s r e a l !bl
G2 'L ^bexp(-igx)
- v 2 exp(i5x) O(x 2 )l and
(A&+)
%
sy) 'L ^bexp(-2icx) as y more r i g o r o u s l y (A+A)
t
o ( x ) ( n o t e here f r o m
O(x) s i n c e
+
i=
%
ft
i2
).
(0)
%
&, -
i n 19 cPlexp(isx)
'L
a$, 1 t
Thus i n f a c t !bl(x,s)exp(-i
x and x -t 0 and t h e n i n p l a c e o f -(l/~) lz i ( c , t ) e x p ( - 2 i s x ) d s
(A&=)
we w r i t e
= x+o 1i m
f(t,x). The f a c t o r e x p ( - 2 i s x ) o f course a l s o h e l p s f o r convergence purposes. We n o t e t h a t t h i s k i n d o f procedure c o u l d a l s o a r i s e i n F o u r i e r a n a l y s i s i n f = 0 elsewhere, w i t h F f = ( l / i s ) [ e x p ( i c E ) - 1 1 so t h a t (A+.) 1 = lim (1/2r)l: F f e x p ( - i s x ) d c . F o r x f 0 conx-to t t o u r i n t e g r a t i o n g i v e s t h e c o r r e c t answers b u t f o r x = 0, f ( O ) must be ex1i m [ e x p ( i s ~ ) - l ] d s / i c b e i n g a t b e s t ambiguous). pressed as x-to f ( x ) ((1/2a)l:
d e a l i n g w i t h e.g.
f ( x ) = 1 for 0 < x
BIlf'(x,) J (h.-h.)ll - lIf(xo+Bhi) - f ( x o ) - ~ f ' ( x o ) h i l l - l l f ( x o + B h j ) - f ( x o ) - B f ' ( x )h.ll
t i v e l y compact.
1
> BE
J
- o(le1).
P i c k hn w i t h IIhnll = 1 and
Since
E
E
such t h a t llfl(xo)(hi
i s independent o f B t h i s i m p l i e s f(xo+Bhi)
O
has no
J
2 34
ROBERT CARROLL
convergent subsequence which c o n t r a d i c t s . L e t us r e c a l l h e r e t h a t a c l o s e d subset S o f a m e t r i c space i s compact i f every c o v e r i n g o f S by open s e t s
-
c f . a l s o D e f i n i t i o n A29 and n o t e h e r e S i s compact i f and o n l y i every f a m i l y o f c l o s e d s e t s i n S has a f i n i t e subcovering (Heini-Bore1 p r o p e r t y
w i t h the f i n i t e intersection property
-
.e. f i n i t e s u b f a m i l i e s have nonvoid
- has nonvoid i n t e r s e c t i o n )
intersection
Then i n f a c t S i s compact i f and
o n l y i f i t i s s e q u e n t i a l l y compact ( i . e . every i n f i n i t e sequence i n S has a convergent subsequence i n S ) .
To see t h s, f i r s t , g i v e n S compact, i f M =
I x n l ( n = 1,2, ...) has no convergent subsequence, cover S by b a l l s B(x,E) each o f which c o n t a i n s a t most one p o i n t o f M.
The r e s u l t i n g f i n i t e sub-
c o v e r i n g i m p l i e s M i s f i n i t e which c o n t r a d i c t s .
Conversely i f S i s sequen-
t i a l l y compact (and c l o s e d ) one notes f i r s t t h a t S i s separable ( i . e . t h e r e Indeed p i c k po a r b i t r a r y i n S w i t h D =
e x i s t s a c o u n t a b l e dense s e t pn).
S ; D i s f i n i t e s i n c e i f d(p ,q ) -+ m t h e r e e x i s t a convero n gent subsequence ^qn -+ q and d(po,q) = m which i s precluded. Choose now i n -
supd(p,p
0
), p
E
d u c t i v e l y pitl d(pn,q)
such t h a t mind(pn,pitl)
( 0 5 n 5 i t l ) where di = sup
di/2
f o r q E S and 0 5 n 5 i. E v i d e n t l y do 2 dl L... and i f dn
2E
> 0
f o r a l l n t h e n no subsequence o f t h e pn i s Caucby which c o n t r a d i c t s t h e convergence o f some subsequence. d(pn,p)
l / p - l / n ; in f a c t wP wrj i s compact f o r l / p wS-’(A) r - ( s - j ) / n < l / r . Then defining an o p e r a t o r 3: V -+ V ’ by ( B ( u ) , w ) = b ( u , w ) (as with A) one makes hypotheses on t h e B so t h a t 3: V V ’ will be com8 pact. For example i f one assumes estimates of the form (*&) then t h e analogue of (**I is (-1 I b ( u , v ) I 5 g ( h s - l ,p)llvlls;l ,E?‘ ^I, Let be V w i t h the ws-l topology and write b(u”,T) = (%,V) (where 3: V -+ V can a l s o be extenP ded by c o n t i n u i t y t o the c l o s u r e of v“ i n WS-l by (A+)). The argument of P h A Theorem 2.14 shows t h a t B i s demicontinuous from say V -+ V ’ and takes bounded sets i n t o bounded s e t s . Then l e t i : V +. V be the compact i n j e c t i o n . Since V i s r e f l e x i v e and separable we can v e r i f y the B complete c o n t i n u i t y of B by looking a t sequences un + u weakly ( B complete c o n t i n u i t y means complete c o n t i n u i t y on bounded s e t s - see below). Evidently B = i*
I (f-fn)(f,l,)l =
But t h e n f o r a l l n,
1 which c o n t r a d i c t s s i n c e t h e fn a r e dense on S1.
The s e t o f f i n i t e l i n e a r combinations
anf,l,
w i t h an r a t i o n a l complex i s
t h e n dense i n E ' which shows t h a t i t i s separable.
Next t o produce a m e t r i c
(weak) t o p o l o g y on t h e u n i t b a l l B1 o f a separable r e f l e x i v e Banach space E l e t e,!, be dense i n E ' and d e f i n e l(x-y,el;)l]. than o ( E , E ' ) . be i d e n t i c a l . 3.
(Am)
d(x,y) =
1 (1/2n)[1(x-y,e;l)l/(l
+
T h i s y i e l d s a m e t r i c t o p o l o g y on B, which i s e v i d e n t l y weaker But B1 i s o ( E , E ' )
IIIBNOCONE OPERACORk.
compact and hence t h e two t o p o l o g i e s must
The t h e o r y o f monotone o p e r a t o r s ( c f . D e f i n i t i o n 2.
15) was s y s t e m a t i c a l l y developed i n connection w i t h a p p l i c a t i o n s t o PDE i n t h e 1 9 6 0 ' s and 1970's.
We w i l l make no a t t e m p t t o g i v e h i s t o r i c a l documen-
t a t i o n n o r r e f e r e n c e s t o papers b u t r e f e r t o [ C l ;Bel;Brl;Dml this.
71)
We f o l l o w here [ C l ]
;Ze3] f o r a l l
(based i n p a r t on Browder's e a r l y work i n [Br2-
and t o [Dml] f o r o u r p r e s e n t a t i o n .
We g i v e m a i n l y t y p i c a l r e s u l t s and
ideas; no a t t e m p t i s made t o g i v e b e s t p o s s i b l e theorems.
The t h e o r y has
a t t a i n e d a v e r y s o p h i s t i c a t e d a b s t r a c t form and e a r l i e r f o r m u l a t i o n s seem more i n s t r u c t i v e and u s e f u l f o r t h i s book.
MONOTONE OPERATORS
REIIMRK 3.1.
239
L e t us n o t e t h a t m o n o t o n i c i t y o f A: V
V' (V' = antidual) i s
-+
2 0 ( w h i c h corresponds t o Re(Av,v) 2 0 when
expressed v i a Re(A(u)-A(w),u-w)
I t i s c o n v e n i e n t a t t i m e s t o t h i n k o f r e a l Banach spaces and
A i s linear).
o m i t t h e symbol Re; t h e n t o go back t o t h e complex s i t u a t i o n one s i m p l y i n s e r t s Re a t t h e a p p r o p r i a t e places.
We remark a l s o t h a t f o r A as i n Theorem
2.14 a s o l u t i o n u t o Au = f i s a weak t y p e " v a r i a t i o n a l " s o l u t i o n o f t h e problem based on ( - l ~ a l D U A a ( ~ , ~ , . . . y D S u ) = g where say (g,v),z = (f,v)V-Vl.
1
I n what f o l l o w s A: D(A) C V * V ' ,
D(A) i s dense, and V i s a r e f l e x i v e Banach
space ( = B space) as i n D e f i n i t i o n 2.15 (D i s a l s o 1 i n e a r u s u a l l y ) .
CHE0Rfill 3.2.
I f A i s maximal D-monotone and l o c a l y bounded t h e n i t i s
demicontinuous.
Phood:
L e t un E D, un
uo E D.
I f A(un) P A(uo) weakly t h e n as b e f o r e t h e r e i s an i n f i n i t e subsequence c o n v e r g i n g weakly t o some wo # A(uo) (by +
B u t i f u E D i s a r b i t r a r y , Re(A,(unk)
weak s e q u e n t i a l compactness). Unk
-
u) 1. 0 and as k
Hence f o r a l l u
E
-f
m,
+ Nuo).
-
unk
D (*) R d w o
-
uo
+
-
0 and ~ ( u , , ~ )
-
A(u),uo
-
~ ( u +) wo
-
A(u),
~ ( u weakly. )
u ) 2 0 which c o n t r a d i c t s s i n c e wo
QED
C H E 0 R B 3.3.
L e t A be D-monotone and hemicontinuous; t h e n A i s maximal D-
monotone.
Phood:
Suppose (*) h o l d s f o r uo E D and a l l u E D b u t wo
-
i s dense and l i n e a r t h e r e e x i s t s v E D such t h a t R e ( w o t l y and u o + t v E D f o r 0 5 t 5 1 say.
weakly i f tn > 0, tn + 0. ( p u t u = uo+tnv i n ( * ) ) .
A(uo),v)
Now w r i t e wo
-
Since D > 0 stric-
By h e m i c o n t i n u i t y A(uo+tnv) * A(uo)
Then by assumptions Re(A(uo+tnv)
A(uo+tnv) t o o b t a i n R d A ( u o + t n v )
-
A(uo+tnv) = w0
A(uo),v) 2 R H w o
-
-
-
wo,v) 2 0
A ( u o ) + A(uo)
A(uo),v)
t h e l e f t s i d e tends t o 0 we have a c o n t r a d i c t i o n .
C0R0CLARg 3.4.
# A(uo).
> 0.
-
Since
QED
I f A i s hemicontinuous, D-monotone, and l o c a l l y bounded t h e n
i t i s demicontinuous. C l e a r l y d e m i c o n t i n u i t y i m p l i e s h e m i c o n t i n u i t y ; f u r t h e r A demicontinuous i m p l i e s A i s l o c a l l y bounded s i n c e i f un
-+
u E D, un
E
D, t h e n { A ( u n ) } i s
weakly bounded ( b e i n g weakly Cauchy) and hence by Banach-Steinhaus t A ( u n ) l i s s t r o n g l y bounded.
CHE0RElll 3.5.
I f V i s f i n i t e dimensional one has
I f V i s f i n i t e dimensional and A ( d e n s e l y d e f i n e d as i n D e f i -
n i t i o n 2.15) i s D-monotone and hemicontinuous, t h e n A i s continuous.
Phood:
Using C o r o l l a r y 3.4 i t i s enough t o show t h a t A i s l o c a l l y bounded,
240
ROBERT CARROLL
s i n c e d e m i c o n t i n u i t y i s o b v i o u s l y e q u i v a l e n t t o c o n t i n u i t y f o r V f i n i t e dimWe s h a l l show t h a t i n f a c t A monotone and dimV
0 i s f i x e d h e r e ) , un
+.
u, and ailA(un)
Re(ai’A(un) -f
For w
+. m.
-1 0 5 an Re(A(un)
But a;’A(u+tw)
Without l o s s o f g e n e r a l i t y one
D, w i t h A(un) unbounded.
can assume IIA(un)II = an (3.1)
i s enough f o r
Suppose t h e c o n t r a r y ; t h e n t h e r e i s a u E D and a seq-
l o c a l boundedness. uence un
m
D i v i d i n g b y t and l e t t i n g n
D i s dense and ailA(un)
-
A(u+tw),un
-+
u
ailA(u+tw),un
m
tw)
one has l i m i n f Re(ailA(un),-w)
i s bounded t h i s h o l d s f o r any w E V.
i s bounded.
2 0.
Since
Replacing w by
-w (and by + i w i n complex spaces V ) one o b t a i n s l i m ( a ~ ’ A ( u n ) , w > =
0 for a l l
w E V which means ailA(un) 0 weakly and hence s t r o n g l y s i n c e dimV < But t h i s c o n t r a d i c t s s i n c e llailA(un)ll = 1. QED -f
-.
A v e r s i o n o f t h i s l a s t theorem was proved by Kato and we mention a l s o t h e background work o f M i n t y on monotone o p e r a t o r s and V i g i k on c o n c r e t e e l l i p t i c s i t u a t i o n s which c o n t r i b u t e d t o t h e development o f t h e t h e o r y . now some e x i s t e n c e r e s u l t s f o r n o n l i n e a r maps V
-+
We prove
V ’ d e f i n e d everywhere i n
V ( t h e s e correspond t o e l l i p t i c s i t u a t i o n s and a r e based on e a r l y work o f Browder i n t h i s a r e a ) . We g i v e f i r s t a theorem which i l l u s t r a t e s some asp e c t s o f a g e n e r a l i z e d G a l e r k i n method.
CHE0REIII 3.6.
F
-+
L e t F be a r e f l e x i v e B space, A: F -+ F ’ be hemicontinuous, B:
F ’ be B c o m p l e t e l y continuous, and f o r a l l u,v E F l e t ).(
A(v),u-v) + (3(u) m
as x
-f
m.
-
3 ( v ) , u - v ) 2 0 and
Then A maps F o n t o F ’
(q
(0)
Re(A(u)
-
Re(A(u),u) > q(IIuII)IIuII w i t h q ( x )
-+
may be n e g a t i v e f o r x s m a l l ) .
Phoo6: L e t w E F ’ be a r b i t r a r y and c o n s i d e r A1(u) = A(u) - w. I t i s easy t o see t h a t A, s a t i s f i e s t h e same c o n d i t i o n s as A (e.g. y , ( x ) = ‘p(x) - llwll) and hence i t s u f f i c e s t o show t h a t 0 E R ( A ) . L e t A be t h e d i r e c t e d s e t o f f i n i t e dimensional subspaces o f F o r d e r e d by i n c l u s i o n . For E E A l e t iE: E + F be t h e i n j e c t i o n . (To a v o i d n e t arguments one can assume V i s separa b l e r e f l e x i v e (hence V ’ separable) w i t h Vn C Vn+l a sequence o f f i n i t e dimin: Vn +. V, e t c . ensional supspaces, o r d e r e d by i n c l u s i o n , w i t h V = U V n’ Then argue as below w i t h inr e p l a c i n g iE; t h i s would correspond t o a t r a d i t i o n a l G a l e r k i n t y p e procedure. Net arguments a r e however e s s e n t i a l l y t h e same and i t seems w o r t h w h i l e t o develop some f a c i l i t y i n t h i s ( c f . [Kel;Bo4] and Appendix A).) Now i s weakly continuous (Appendix A ) and hence A E + B E
it
MONOTONE OPERATORS
E
= i$At3)iE:
241
E' i s hemicontinuous and monotone ( e x e r c i s e ) .
-+
Since E i s
f i n i t e dimensional Theorem 3.5 i m p l i e s t h a t A E t BEY and hence AEy i s continuous.
Also f o r u
E ( 6 ) Re(AEu,u) =
€
Re(
I F i E u y u ) = Re(Au,u) L v ( I I u l l ) I I u l .
Now we c i t e a lemna which i s proved i n 53.4 (needed t o f i n i s h t h e p r o o f ) .
LRRIRA 3.7, L e t A be a continuous map o f t h e f i n i t e dimensional B space E i n t o E ' such t h a t ( 0 ) h o l d s w i t h ~ ( x + ) m as x -+ Then A maps E o n t o E l .
-.
Given t h e t r u t h o f Lemna 3.7 we o b t a i n an element uE E E such t h a t AEuE = 0 and hence cp(UuIIE) 5 0 f o r each uE as E v a r i e s . l y i n E.
L e t now u
T h i s means IIuEll 5 M u n i f o r m -
F be a r b i t r a r y and c o n s i d e r a l l E
€
e. c o n t a i n i n g t h e 1 - D space Eo generated by u ) .
T h i s says t h a t t h e n e t AuE converges t o 0 weakly.
0.
€ A
c o n t a i n i n g u (i.
Then ( A u E Y u ) = (AEuE,u) = Moreover f o r such E Hence ( + ) Re
Re(AuE-Au,uE-u) = Re(AEuE,uE-u) t Re(Au,u-uE) = Re(Au,u-uE).
2 0.
[(3u-3uE,u-uE)+(Au,u-uE)]
Eo
E
Now g i v e n any Eo
E A,
w r i t e Fo = { U { u E l ;
E; E E A } ; c l e a r l y Fo C BM where BM i s t h e c l o s e d b a l l o f r a d i u s M cenNow t h e n e t uE ( a l l E) l i e s i n BM and by weak compactness has a
t e r e d a t 0.
weak c l u s t e r p o i n t uo cluster point).
E
BM (a s e t i s compact i f and o n l y i f each n e t has a
By s t a n d a r d f a c t s about n e t s (see [Kel;Dul]
and Appendix
A ) uo belongs t o t h e weak c l o s u r e o f each Fo ( c f . h e r e a l s o t h e p r o o f o f Theorem 3.22). Thus i n ( + ) l e t t i n g a subnet u E g + uo weakly we o b t a i n ( m ) > 0, s i n c e 3u o( + 3u s t r o n g l y by t h e compactness o f 3 E 0 and t h e f a c t t h a t I I u E ~ l l5 M. Moreover ( = ) h o l d s f o r any u E F s i n c e uo i s
Re(Au+3u-3uo,u-uo)
adherent t o each Fo.
Now a p p l y Theorem 3.3 t o conclude t h a t A t 3 i s maximal
F monotone and hence from (=), (A+3)uO = 3u0 o r Auo = 0.
QED
T h i s enables one i n p a r t i c u l a r t o f i n d u E V such t h a t a(u,v) all v
E
= (f,v)
for
V where f E V ' i s g i v e n a r b i t r a r i l y , whenever a ( - , - ) s a t i s f i e s (")
+ b(-,-) satisfies
Wi(A)
C V C as beP f o r e , and a,b a r e t h e v a r i a t i o n a l forms i n V a s s o c i a t e d w i t h t h e d i f f e r e n -
i n 53.2,
a(-,-)
(A*)
i n 53.2,
?(A)
t i a l o p e r a t o r s (**) and (A&) w i t h hypotheses o f t h e t y p e ( * b ) i n § 3 . 2 . Some g e n e r a l i z a t i o n s o f Theorem 3.6 i n v o l v i n g densely d e f i n e d o p e r a t o r s a r e i m p o r t a n t i n t h e t h e o r y o f n o n l i n e a r e v o l u t i o n equations and appear l a t e r ( c f . [Brly5;Cl]),
We mention f i r s t some v a r i a t i o n s and e x t e n s i o n s f o r ev-
erywhere d e f i n e d o p e r a t o r s . h y p o t h e s i s i n a f o r m a(u,v)
F i r s t i t i s possible t o r e f e r the monotonicity = (Au,v)
o r d e r d e r i v a t i v e s (see [Brly2;Dml;Cl]).
(**) a(u,v,w) u
E
ws-1
P
.
1I I_<s
t o t h e terms i n v o l v i n g o n l y t h e h i g h e s t To do t h i s one c o n s i d e r s t h e f o r m
cI
(Aa(x,u, ...,DS-l~,DS~),Da~)
where e.g.
v,w E W
S
and P T h i s i s w e l l d e f i n e d when ( * 6 ) o f 53.2 h o l d s w i t h C8 = DBu f o r =
242
ROBERT CARROLL
161 5 s-1 and 5
B
= D'v
for
161 = s; i t i s easy t o see then t h a t where g ( x ) = c ( l + x P - ' )
(*A)
f o r some c .
u,w)I I g ( l l u l l s-1 ,P )g(llvll~,p)llwllSYP (*A) i m p l i e s t h e r e i s an element S(v,u) E V ' such t h a t (**) (S(v,u),w)
la(v, Now =
S
where we assume momentarily t h a t C V C W w i t h u.v,w E V. Then P P t h e o p e r a t o r S(u,u) corresponds t o t h e A ( u ) t r e a t e d p r e v i o u s l y and as b e f o r e
a(v,u,w)
t h e map u
w i t h (BU,W) = b(u,w)
uous.
-f
S(v,u):
V
V ' w i l l be B c o m p l e t e l y c o n t i n -
-f
This leads t o t h e f o l l o w i n g simple abstract formulation f o r d i f f e r e n -
t i a l problems based on (**).
F ' i s semimonotone i f A ( u ) = S(u,u) where S: D E F I N Z C I ~ N3-8, A map A: F F X F F ' i s monotone i n t h e f i r s t argument and B c o m p l e t e l y continuous i n -+
-f
F ' i s f i n i t e l y continuous i f i t i s continuous f r o m f i n i t e dimensional subspaces E c F t o t h e weak t o p o l o g y o f F ' .
t h e second.
A map A : F
-f
EHE0REfll 3.9, L e t F be a r e f l e x i v e Banach space and A a f i n i t e l y continuous semimonotone o p e r a t o r F F ' ( A ( u ) = S(u,u)) w i t h u S(u,v) hemicontinuous, -f
s a t i s f y i n g Re(A(u),u) )'p(IIuII)IIuII
with ~ ( x )
-f
-
-f
as x
-.
-f
Then A maps F
onto F ' .
Pfiuua:
R(A) s i n c e S1(u,v) S(u,v).
=
S(u,v)
For E E A, AE =
holds a g a i n . M.
Again i t s u f f i c e s t o show 0 E
We f o l l o w t h e p r o o f o f Theorem 3.6.
Again AuE
-
y, y f i x e d , w i l l have t h e same p r o p e r t i e s as
itAiE:
E
-f
E ' i s continuous by assumptions and ( 6 )
By Lemma 3.7 we have elements uE E E w i t h AEuE = 0 and IIuEll 5 -f
0 weakly and t h e r e i s a uo E BM weakly adherent t o a l l Fo.
Given u E F and E E A, u E E, we have Re( S(u,uE)
-
S(uEYuE),u-uE) 1. 0 which
means, s i n c e S(uE,uE) = A(uE) w i t h (A(uE),u-uE) = ( A E ( u E ) , u - u E ) = 0, t h a t R e ( S ( u , u ~ ),u - u E ) > 0.
By B complete c o n t i n u i t y o f S i n t h e second argument
w i t h IIuEll 5 M we o b t a i n Re(S(u,uo),u-uo) a p p l y i n g Theorem 3 . 3 t o t h e map u
-+
2 0.
S(u,uo)
T h i s h o l d s f o r a l l u E F and
one has S(uo,uo) = A(uo) = 0.
We mention n e x t a n o t h e r t y p e o f theorem having a p p l i c a t i o n t o e l l i p t i c prob-
A t y p i c a l r e s u l t o f Browder i s c i t e d
lems which a r e n o t s t r o n g l y e l l i p t i c . g e n e r a l i z i n g a theorem o f Z a r a n t o n e l l o .
EHEBRETil 3-10, map.
L e t F be a r e f l e x i v e
B
space and T: F
Suppose ( * 6 ) I ( T u , u ) l ~ ' p ( I I u l l ) l l u l lw i t h ~ ( x )
-f
-
-
F ' a demicontinuous
-f
as x
-f
and suppose
f o r each N > 0 t h e r e i s a continuous ( s t r i c t l y ) i n c r e a s i n g f u n c t i o n $,(O)
= 0, such t h a t f o r IIuII, IIvII < N (*+) I(Tu-Tv,u-v)l
1. $N(IIu-vll)llu-vll.
Then T i s a 1-1 map o f F o n t o F ' w i t h a continuous i n v e r s e . The p r o o f i s s i m i l a r t o p r e v i o u s arguments b u t now one uses t h e Brouwer t h e orem on i n v a r i a n c e o f domain ( c f . §3.4) which i m p l i e s t h a t a b i c o n t i n u o u s
243
MONOTONE OPERATORS
1-1 map TE: E
E ' has open range (TE = i f T i E , dimE
0. Hence $(I-,) $
+IP(E,TI,X).
2)
h ( u o ) - h ( v ) where wo E F ' i s g i v e n i n advance.
-
If h
I f T = 0 t h e n h has a minimum a t uo.
3.3 Tuo = wo.
0 on a c l o s e d
convex s e t K, w i t h h = o f f K, t h e n f o r v € K, ( T u ~ - w ~ , v - u 2 ~ )0 which solves a problem i n monotone i n e q u a l i t i e s on convex s e t s ( c f . 553.5-3.7).
tmrmA 3.18,
Assume T: F i v e B space, h: F + (--,-]
+
F ' i s hemicontinuous and monotone, F i s a r e f l e x i s convex w i t h h # -, and w € F ' i s given. Then 0
uo s a t i s f i e s ( a * ) if and o n l y i f
Phaod:
0 ) . As t + 0, Tvt + Tuo weakly i n F ' by h e m i c o n t i n u i t y and hence ( T u ~ - w ~ , w -2u h(uo) ~~ - h(w) f o r any w E F which i s ( a * ) . Conv e r s e l y i f (.*) we have
(@A)
h o l d s t h e n s i n c e (Tv,v-uo) 2 ( T u ~ , v - u ~ ) by , monotonicity
QED
imnediately.
We s h a l l assume i n what f o l l o w s t h a t h i s l o w e r semicontinuous and t h i s i n a sense i s a weaker s i t u a t i o n than t h a t which can be expected t o p r e v a i l if h ( x ) = I ( x , x ) w i t h I o n l y semiconvex.
F i r s t we r e c a l l t h a t h i s s a i d t o be
weakly l o w e r semicontinuous (LSC) on a s e t B < c, x Wc(h) = t x ; h ( x ) -
w i t h t h e xa xa)
E
- I(x,,xo)
This means xa Theorem 3.13). xo) 5 c ' .
B I i s weakly c l o s e d i n B.
Wc(h) and B bounded.
Now l e t xa
+
xo weakly
By p r o p e r t y (B) o f D e f i n i t i o n 3.11
I(xa,
0 and hence i f c ' > c i s a r b i t r a r y I(xa,xo) 5 c ' e v e n t u a l l y .
+
E
F i f f o r each r e a l c t h e s e t
W
C'rXo
e v e n t u a l l y , which i s weakly c l o s e d (see t h e p r o o f o f
s i n c e xa xo weakly and t h u s h ( x o ) = I ( x o , c',x Since c ' > c i s a r b i t r a r y we have h ( x o ) 5 c and xo E Wc(h). Thus
CHE0REN 3.19. I(x,x)
E
C
Hence xo E W
-f
I f I(. ,- ) i s semiconvex on F X F, F Banach, t h e n x
-+
h(x) =
i s weakly LSC on bounded s e t s .
We n o t e t h a t a map x
+
f(x) : F
+
R i s weakly LSC means a l s o t h a t xa
-+
x
weakly i m p l i e s l i m i n f f ( x a ) 2 f ( x ) (one should compare t h i s w i t h t h e d e f i n i t i o n above).
&HE8)REm 3.20. +
Next we have L e t F be a f i n i t e dimensional B space, T a continuous map F
F ' , and h a LSC convex f u n c t i o n , h: F
t r a r y R, t h e r e e x i s t s uo E BR w i t h (.*)
-+
(--,m).
Given wo E F ' and a r b
t r u e f o r a l l v E BR.
P ~ C J C J ~We: can assume w i t h o u t l o s s o f g e n e r a l i t y t h a t wo = 0 s i n c e T ' x = Tx
247
MONOTONE OPERATORS
-wo s a t i s f i e s t h e same hypotheses. E BR t h e r e would be a v
BR such t h a t
E
BR f i x e d t h e s e t Sv o f u
I f t h e lemna were f a l s e t h e n f o r each u
E
(exercise) while i f h(v) =
-
(
(0.)
BR s a t i s f y i n g we s e t Sv =
(0.)
Tu,v-u)
< h(u)-h(v). For v E i s open i n BRy s i n c e h i s LSC
@. By
t h e compactness o f BR we
c o u l d t h e n cover i t by a f i n i t e number o f such s e t s S v j (1 5 j 5 p).
If
J,j
i s a c o r r e s p o n d i n g c o n t i n u o u s p a r t i t i o n o f u n i t y (see Appendix B) d e f i n e
1: J,.(u)vj. 1 J,j(uj = 1 ) (u)(Tu,vj-u) < 1 ;
J,(u) = with
Evidently then
- h(J,(u)).
-
J,j(u)[h(u)
c o n v e x i t y h(J,(u)) 5 < h(u)
J,
and i s continuous.
1;
maps BR i n t o BR ( s i n c e 0 ~ $ ~ ( 5u 1) Thus f o r u
h(vj)] = h(u)
!bj(u)h(v.)
E
- 1;
BR (Tu,J,(u)-u) E
1'1
J,
j However by
Gj(u)h(vj).
and hence f o r a l l u
J
=
BR ( 0 6 ) (Tu,J,(u)-u)
But by t h e Brouwer f i x e d p o i n t theorem (see 53.4)
J,
has
a f i x e d p o i n t u i n BR and p u t t i n g u i n ( 0 6 ) we g e t 0 < 0 which i s impossible.
T h e r e f o r e Lemna 3.20 i s t r u e .
QED
L e t F be a B space, h: F + (-m,m] a convex f u n c t i o n w i t h h ( 0 ) = CmrmA 3.21. 0, wo E F ' , and l e t R be g i v e n such t h a t f o r IIuII = R t h e map. T: F + F ' s a t s a t i s f i e s (**) f o r a l l v
0
0
Paood: Set v
If uo E BR s a t i s f i e s (.*)
+ h ( u ) > 0.
i s f i e s (Tu-wo,u) IIu II < R and u
= 0 in
t h i s means IIu II < R. 0
(.*)
E
2 h(uo); by t h e hypotneses
t o o b t a i n -(Tuo-wo,uo)
Let v
E
f o r v E BR t h e n
F.
F be a r b i t r a r y and c o n s i d e r vt = ( 1 - t ) u o + t v .
T h i s t r a c e s o u t t h e l i n e f r o m uo t o v and hence f o r t small l i e s i n BR. P u t t i n g vt f o r v i n (**) one has a f t e r c a n c e l i n g t > O,( Tuo-wo,v-uo) h(uo)
- h ( v ) which i s (.*), now v a l i d f o r any v.
CE0RElIl 3-22, a g i v e n wo
E
QED
L e t T be a monotone hemicontinuous map F
f l e x i v e B space, and l e t h: F
+
F ' t h e r e e x i s t s R such t h a t
Then t h e r e e x i s t s uo
E
+
F', with F a re-
be LSC w i t h h ( 0 ) = 0.
(-m,m]
BR such t h a t (.*)
(
Tu-wo,u)
+ h(u)
>
Suppose f o r 0 f o r IIuII = R.
holds.
The p r o o f f o l l o w s p a t t e r n s e s t a b l i s h e d above i n Theorem 3.6,
3.21,
2
Lennnas 3.20,
e t c . (see [ C l ] f o r d e t a i l s ) .
Assume t h e f i r s t h y p o t h e s i s o f Theorem 3.22 and assume t h a t C0R0ttARg 3.23. as IIuIl + m y {(Tu,u) + h(u)}/llull + a. Then a s o l u t i o n o f ( a * ) e x i s t s (any w0). F o r p r o o f n o t e t h a t f o r g i v e n wo
E
F', (Tu-wo,u) + h ( u ) ,(Tu,u)
IIw Illlull and t h i s i s > 0 f o r IIuII l a r g e enough. 0
Hence t h e second hypotheses
o f Theorem 3.22 h o l d s and hence a s o l u t i o n o f (.*) u s e f u l i n f o r m a t i o n about s o l u t i o n s o f theorem ( c f . [ C l ]
CHE0REIR 3.24.
(0*)
+ h(u) -
exists.
Some f u r t h e r
i s contained i n the f o l l o w i n g
f o r proof).
Under t h e hypotheses o f Theorem 3.22 t h e s e t o f s o l u t i o n s
248
ROBERT CARROLL
S(wo) o f (.+)
(w,)
i s a c l o s e d convex s e t i n F and i f T i s s t r i c t l y monotone S
i s a single point.
We go now t o some n o n l i n e a r e v o l u t i o n equations f o l l o w i n g [Br5] ( c f . a l s o [Cl])
and f o r r e l a t e d work see [Brl;Bdl;Ka3;Tal;Li4;Mtl;Pcl;Pzl Again we
§53.6-3.7).
kill
some i l l u s t r a t i v e cases.
(cf. also
n o t g i v e t h e b e s t known r e s u l t s b u t r a t h e r p i c k The monotone o p e r a t o r aspect o f t h e t h e o r y has
a t t a i n e d a h i g h l y s o p h i s t i c a t e d f o r m i n [ B r l ] f o r example and we w i l l n o t try t o cover t h i s .
We g i v e one theorem (Theorem 3.25) t o i l l u s t r a t e some
a p p l i c a b l e a b s t r a c t t e c h n i q u e based on Browder's g e n e r a l i z e d G a l e r k i n met h o d and t h e n s k e t c h an a p p l i c a t i o n t o a n o n l i n e a r e v o l u t i o n equation. r e s u l t s on n o n l i n e a r e v o l u t i o n equations w i l l be i n §§3.6-3.7;
More
t h e technique
and p h i l o s o p h y t h e r e w i l l be somewhat d i f f e r e n t .
CHE0RETll 3-25. L e t F be a r e f l e x i v e B space, L: F * F ' a c l o s e d densely def i n e d l i n e a r map w i t h domain D = D(L), G: F .+ F ' a hemicontinuous map d e f i n ed on a l l F c a r r y i n g bounded s e t s i n t o bounded s e t s and C: F map.
Define T = L
+
F' a compact
.+.
G w i t h domain D and assume t h a t T + C i s
D monotone;
suppose f u r t h e r t h a t L* i s t h e c l o s u r e o f i t s r e s t r i c t i o n t o D(L) n D(L*) as x + Then R(T) = F ' . and t h a t Re 0 i s a r b i t r a r y t h i s i m p l i e s (Cu,
E,
and
E
given
I( (T+C)v,uE-v) -
-
(T+C)v,uo-v)
2 -(2+M+llvll)
-
(T+C)v,uo-v)
0 for all
F u r t h e r s i n c e any l i n e a r L i s hemicontinuous ( c f . remarks a f t e r
Theorem B40) we know TtC i s hemicontinuous and hence by Theorem 3.3 T+C i s
D maximal monotone w i t h (T+C)uo = Cuo o r Tuo = 0. I n p a r t i c u l a r t h i s theorem a p p l i e s t o L,
1o f
i f L,
QED
= Lw, where Ls i s t h e c l o s u r e
a densely d e f i n e d l i n e a r L and Lw = L ' * where L ' = L*ID(L) n D(L*)
(since then
i' =
L;
We s h a l l a p p l y i t i n t h i s f o r m w i t h D(L) c D(L*)
= L;).
Thus l e t H
f o l l o w i n g [Br5] w i t h some m o d i f i c a t i o n s ( c f . a l s o [C1,39,40]). be a H i l b e r t space and E
C
H a dense l i n e a r subset c a r r y i n g t h e s t r u c t u r e o f
, ) be t h e H
a r e f l e x i v e B space w i t h continuous i n j e c t i o n i n t o H; l e t (
,
)E t h e E-E' d u a l i t y b r a c k e t ( c o n j u g a t e l i n e a r ) and one w r i t e s E C H C E ' where H ' i s i d e n t i f i e d w i t h H. L e t f be a map Ta X E s c a l a r p r o d u c t and
+
E ' (Ta = [r,r+a])
continuous ( 6 ) t g i v e n and u t
(
E
s a t i s f y i n g e.g.
* (f(t,u),v)E
E, I l f ( t , u ) l l E l
5 C [ I I UEI I ~ - +~ h ( t ) ] where h
0 belongs t o L u'(t)
(0.)
+
f(t,u):
E
E
+
E' i s
0 we p i c k a f i n i t e s e t
i n C such t h a t C C UB(zi,c/
..,zn
4 ) ( B ( x , r ) i s t h e b a l l o f r a d i u s r and c e n t e r x ) . L e t K be t h e convex h u l l of t h e zi; then t h e map (xi,zi) xizi: [O,1ln X {zl X X zn} -+ K w i t h -f
Xi
=
...
1
1 e x h i b i t s K as t h e continuous image o f a compact s e t and hence i n xi E C, A . as above, t h e r e i s
hixi,
Now i f x =
p a r t i c u l a r as precompact.
-1
-
1
111
-
< ~ / 4 and hence IIx Aizn(i)ll = A.(xi ~ ) t h a t Ilx a z ~ ( such ,.,i z n ( i ) 1I 1 z ~ ( ~ ) 5I I ~ / 4 . Now r(C) l i e s i n any NBH o f t h e s e t o f such p o i n t s x and
hence i n any NBH o f an ~ / 4NBH o f K.
K, j
=
1,
...,m,
r(C)
one has
-f
C uB(kj,E).
Let C
C
UB(xi,~/2),
x.11 f o r Ilx-xill f o r IIx-x.II 1
q
-
xi E C,
~ / 2 ;a . ( x ) =
1
1
E
Then l e t qi(x)
E.
-
i = 1,
... ,n
IIx-xill
f o r E/Z
= ai(x)/l
ai(x)
from
and d e f i n e (*) a i ( x )
5 IIx-xill 5 with q(x) =
i s continuous and maps C i n t o t h e convex h u l l ;(xi)
when Ilx-x.II 2
E
k. E
J
QED
which i s homeomorphic t o Bn and c o n t a i n e d i n C . r(xi)
UB(k ,E/Z), j
C i s continuous t h e n f has a f i x e d p o i n t .
Pmod:
dently
C
I f C i s a convex compact subset o f a B space E and
CHE6REGI 4.5 (3CHALIDER)f: C
In particular i f K
N
E;
=
IIx
-
ai(x) = 0
1 ni(x)xi.
Evi
o f t h e xi,
F u r t h e r s i n c e ni(x)
= 0
-
n(x)ll = I l l n q (x)(x-xi)ll 5 E . The map n o f o f 1 1 i i n t o i t s e l f has a f i x e d p o i n t y by t h e Brouwer theorem and by ( A )
n o f(y)
(A)
E,
IIx
= y f o l l o w s Ilf(y)
C with Ilf(yn)
-
ynll 5 l / n .
-
yl 5
E.
Thus we can o b t a i n a sequence yn
Then yn has a convergent subsequence yk
by compactness and by c o n t i n u i t y f ( y ) = l i m f ( y k ) = y.
C6ROCCARY 4.6.
-f
y
QED
L e t f be a continuous map o f a c l o s e d convex subset C o f a
B space E i n t o a compact subset Co C C . The p r o o f uses Lemma 4.4 t o say t h a t
Then f has a f i x e d p o i n t .
F(Co)i s
o f f t o < r ( C 0 ) has a f i x e d p o i n t by Theorem 4.5.
compact and t h e r e s t r i c t i o n There a r e many g e n e r a l i z a -
t i o n s o f t h e Schauder theorem b o t h t o l o c a l l y convex spaces, t o m u l t i v a l u e d maps, and t o c o n d i t i o n s on t h e i t e r a t e s o f a map; we c i t e e.g. 1 1 where f u r t h e r r e f e r e n c e s can be found.
D E F I N I Q I 6 N 4.7.
[Dul;ZeZ;Ks
Next ( c f . [Gnl] f o r homotopy)
L e t X and Y be t o p o l o g i c a l spaces.
A map f: X
+
Y is
2 55
TOPOLOGICAL METHODS
c a l l e d i n e s s e n t i a l i n Y i f i t i s homotopic t o a c o n s t a n t map c: X
-f
yo; i f
f i s n o t i n e s s e n t i a l t h e n i t w i l l be c a l l e d e s s e n t i a l . The case o f p r i n c i p a l i n t e r e s t i s when X = Sn i n which case a homotopy h(x, t ) c o n n e c t i n g h(0,O)
= f : Sn
-f
e x t e n s i o n f " o f f d e f i n e d on B"'. for x
E
ykgives
+
r i s e t o an
Indeed s i m p l y w r i t e f ( ( 1 - t ) x ) = h ( x , t )
Conversely i f f has an e x t e n s i o n
t E I = [0,1].
Sn,
Y and h ( - , 1 ) = c : Sn
t h e n f = c under t h e homotopy h ( x , t ) = T ( ( 1 - t ) x ) .
f":
8"'
+
Y
Thus when X = Sn one can
say t h a t f i s i n e s s e n t i a l i n Y i f and o n l y i f i t has an e x t e n s i o n F d e f i n e d on 8"'
Hence i f we t a k e Y t o be R+:'
w i t h values i n Y.
t h e n a map A: B"'
-f
t o Y must have a z e r o i n 8"' values i n Y ) .
(i.e.
R"'
- 0 )
which can be shown t o be e s s e n t i a l on Sn r e l a t i v e
R+:'
( i f n o t A i t s e l f would be an e x t e n s i o n w i t h
T h i s f a c t i s used f r e e l y f o r spheres o f r a d i u s R i n t h e f o l -
l o w i n g lemma (which i s needed f o r Lemma 3.3.7). L e t A be a continuous map o f a f i n i t e dimensional B space E i n t o
CEarmA 4.8,
E ' such t h a t Re(A(u),u) ~ I ~ ( I I u ~ I ) I I uw Ii It h q ( x )
Phood:
+ m
-f
m.
Then R ( A ) = E ' .
We can t h i n k o f E as a H i l b e r t space w i t h E = E ' and as i n t h e p r o o f
o f Theorem 3.3.6 i t s u f f i c e s t o show 0 E R(A). Re(Atu,u)
+ (l-t)llul12
= tRe(Au,u)
>
0.
{ x ; IIxII = R) and At f u r n i s h e s a homotopy i n where E+ = E
-
{O).
Choose R so l a r g e t h a t f o r
Then i f At = t A + ( 1 - t ) I , 0 < t < 1, one has f o r /lull
IIuII = R, Re(Au,u) > 0. = R,
as x
I n p a r t i c u l a r Atu # 0 on SR =
+ E
c o n n e c t i n g A w i t h I on SR
+
But on SR, I i s e s s e n t i a l i n E , o b v i o u s l y , s i n c e I i t -
s e l f has a z e r o on BR = I x ; IIxII < R1.
+ E .
t e n s i o n J o f I t o BR Extend t h i s t o 7 = 0 f o r x -f
( I f i n e s s e n t i a l t h e r e e x i s t s an ex-
Set F = I - J so F = 0 on SR and F ( x ) f x on BR. Rn+'-BR; F i s a compact map E + En+' w i t h
'+'
N
E
no f i x e d p o i n t s and t h i s i s e a s i l y shown t o be i m p o s s i b l e c o m p o s i t i o n o f homotopies A must now be e s s e n t i a l i n has a z e r o i n BR.
+ E
( e x e r c i s e ) . ) By
on SR and hence A
QED
We c i t e now t h e Brouwer theorem on i n v a r i a n c e o f domain which i s needed i n t h e p r o o f o f Theorem 3.3.10.
ZHE0REIIl 4.9.
L e t f: U
-f
Rn be continuous and l o c a l l y 1-1 w i t h U C Rn open.
Then f i s an open map.
Phaod: = B(xo,r)
It i s s u f f i c i e n t
t o show t h a t t o each xo E U t h e r e e x i s t s a b a l l B
such t h a t f ( B ) c o n t a i n s a b a l l w i t h c e n t e r f ( x o ) .
xo = 0 and f ( 0 )
= 0.
We can assume
Choose r such t h a t f i s 1-1 on B(O,r) and c o n s i d e r f o r
Evidently h i s continuous i n E [O,l],h(x,t) = f[x/(l+t)] - f[-tx/(l+t)]. ( x , t ) , h(x,O) = f ( x ) and h ( x , l ) = f ( x / 2 ) - f ( - x / 2 ) i s an odd f u n c t i o n . I f
t
256
ROBERT CARROLL
now h ( x , t ) = 0 f o r some ( x , t )
E
aE(0,r) X [0,1]
then x / l + t
-xt/l+t since
f i s 1-1 and t h u s x = 0 (a c o n t r a d i c t i o n s i n c e 0 4 aB). T h e r e f o r e d(f,B(O, r ) , y ) = d(h(x,l),B(O,r),O) # 0 f o r e v e r y y E B(0,s) (some s ) and t h i s i m Here d r e f e r s t o t h e t o p o l o g i c a l degree which i s
p l i e s B(0,s) C f ( B ( 0 , r ) ) .
QED
examined be1 ow.
DEFINXEI0N 4-10, The degree i s d e f i n e d as a f u n c t i o n d: { ( f , A J ) , open, f: y
-+
Rn continuous, y E R'/lf(aA)}
( 6 ) d(f,A,y)
E A
= d(f,Al,y)
open s e t s such t h a t y t
E
[0,1]
when h:
and y ( t ) $ h(aA,t)
#
f(iT/A,
X [0,1]
+
R s a t i s f y i n g ( A ) d(id.,A,y)
-f
+ d(f,A2,y)
A C Rn
whenever A , ,
A2
c
= 1,
A are d i s j o i n t
i s independent o f
U A2) ( C ) d ( h ( x , t ) , A , y ( t ) )
Rn i s continuous, y: [ O , l ]
-+
Rn i s continuous,
f o r t E [0,1].
T h i s d e f i n i t i o n i s o f course useless u n t i l we see what i t i s we want here. The i d e a o f degree goes back t o Brouwer and was extended by Leray-Schauder; r o u g h l y deg f ( f : A
-+
Rn as above) i s an i n t e g e r determined by f l a Awhich
when nonzero i m p l i e s t h a t f has a z e r o i n A .
The degree s h o u l d a l s o be a 1 H e u r i s t i c a l l y f o r say f E C
homotopy i n v a r i a n t (as i n D e f i n i t i o n 4.10).
we g e t an a l g e b r a i c c o u n t o f t h e number o f s o l u t i o n s o f f ( x ) = 0 i n A prov i d e d f ( x ) # 0 on
ai
(see below).
Then we say xo
€
A i s a regular point i f
d f ( x o ) = f ' ( x o ) i s n o n s i n g u l a r ; o t h e r w i s e xo i s c r i t i c a l .
On t h e o t h e r hand
yo i s c a l l e d a r e g u l a r v a l u e o f f i f f - ' ( y 0 )
c o n t a i n s no c r i t i c a l p o i n t s o f
f; o t h e r w i s e yo i s c a l l e d a c r i t i c a l value.
One knows by S a r d ' s theorem
Now
t h a t t h e s e t o f c r i t i c a l values o f f has measure zero ( c f . [Sml;Bel]). given f
E
1 C , f:
-t
R',
A open, bounded, and connected Kan open connected
s e t by d e f i n i t i o n cannot be w r i t t e n as t h e d i s j o i n t union o f 2 nonempty open sets
-
A i s l o c a l l y connected i f f o r e v e r y p o i n t xo E A and open U 3 xo
t h e r e e x i s t s a connected open V, xo
E
V c U
-
one knows a l s o t h a t any 2
-
p o i n t s o f a compact connected and l o c a l l y connected m e t r i c space S - A say can be j o i n e d by an a r c i n Sl],and
X;
f(x)
f ( x ) # yo on a z , t h e s e t f - ' ( y 0 ) = { x
yo} i s f i n i t e when yo i s a r e g u l a r value.
imp1 i c i t f u n c t i o n theorem ( c f . Theorems 3.2.7-3.2.8)
T h i s f o l l o w s from t h e which decrees t h a t f - l
( y o ) must be d i s c r e t e and thus cannot have a l i m i t p o i n t i n
i.
Now d e f i n e
t h e degree o f f a t yo b y
Thus d ( y ) i s an i n t e g e r ( p o s i t i v e , n e g a t i v e , o r z e r o ) . nonzero one knows f - ' ( y )
However i f i t i s
i s n o t empty. One can extend t h i s d e f i n i t i o n by 1 approximation t o f u n c t i o n s f E C a t p o i n t s where d f ( x ) i s s i n g u l a r and t o
TOPOLOGICAL METHODS functions f
E
Co.
However a n e a t e r way i s d i s p l a y e d i n [ S m l ]
which we i n d i c a t e here ( c f . a l s o [Br8;Ze2]). L e t yo E R n / f ( a i )
DEFINlCI0N 4-11.
257
f o r example
Thus
and P = q ( y ) d y ( d y = dyl
. . . 4 dyn
a Cm n - f o r m on Rn w i t h compact s u p p o r t K C R n / f ( a i ) such t h a t yo
l
The degree o f f a t yo can t h e n be d e f i n e d as d(f,A,yo)
P = 1.
be
E
K and
=
JAv o f .
Such d i f f e r e n t i a l forms P w i l l be c a l l e d admissable f o r (yo,f). To show t h a t t h i s i s w e l l d e f i n e d ( i . e .
independent o f P) one shows f i r s t
t h a t i f v = 9 d y i s a Cm n - f o r m on Rn w i t h compact s u p p o r t K such t h a t 1 v = 0 t h e n t h e r e e x i s t s an ( n - 1 ) - f o r m w w i t h v = dw and supp w C K ( e x e r c i s e -
see [Sml]). =
Then i f P and
v with
p-n
I,,
P
= 0 etc.
v
n a r e b o t h admisdable i n D e f i n i t i o n 4.11 one has - J n o f = J dw o f
Set v = dw so t h a t J P o f
J a i w o f = 0 s i n c e supp w C K i m p l i e s w = 0 on f ( a n ) . o f = J A n o f and d i s w e l l d e f i n e d . It i s now a b a s i c a l l y rou-
J d(w o f )
Hence
J =
t i n e m a t t e r t o show t h a t d(f,A,yo)
We w i l l s k e t c h some o f t h i s and r e f e r t o [ S m l ]
t e properties.
t a i l s ( c f . a l s o [Br8;Dml;Ze2]).
(think o f C
f o r f u l l de-
Thus
The degree o f D e f i n i t i o n 4.11 has t h e f o l l o w i n g p r o p e r t i e s
tBE0RRn 4-12. 1
i n D e f i n i t i o n 4.11 has a l l t h e a p p r o p r i a -
f u n c t i o n s f o r 1 - 9 ) : ( 1 ) I f Iyl-yoI
i s small d(f,A,yl)
yo) ( 2 ) I f yo i s a r e g u l a r v a l u e f o r f t h e n d.(f,A,yo)
and i n p a r t i c u l a r d(f,A,yo)
= 0 i f yo
ft i s a continuous 1-parameter f a m i l y
t f i x e d and yo
'#
ft(A)
f o r t E [0,1]
= d(f,A,
= d(yo) (from (4.1))
f ( i ) ( 3 ) (Homotopy i n v a r i a n c e ) If 1 X [O,l] Rn which i s C on A f o r +
t h e n d(ftyA,yo)
i s independent o f t
( 4 ) I f f = g on aiz and yo B f(ai;) = g(ai;) then d(f,A,yo) = d(g,A,yo) ( 5 ) I f Ai C A , A . n A = @ and yo $ f ( i / u A i ) t h e n d(f,Aiyyo) = 0 f o r a l l but a fin1 j i t e number o f i and d(f,A,yo) = 1 d(f,Ai,yo) (6) (Excision) I f Q C i i s c l o s e d and yo 4 f ( Q )t h e n d(f,A,yo) = d(f,A/Q,yo) ( 7 ) L e t A and be bounded 1 - n open s e t s o f dimension n and m r e s p e c t i v e l y w i t h f E C (A,R ) and f E Rm). I f yo E R n / f ( a x ) and yo E Rm/?(ai) t h e n d ( f X A X 7, (yo,F0)) =
C1(f,
7,
- 4 -
d(f,A,yo)d(f,A,yo) for x
E
0) i f 0 Rn,
ai ( i . e . 9 g(ai)
( 8 ) I f f ( x ) and g ( x ) never p o i n t i n o p p o s i t e d i r e c t i o n s f(x)
+ hg(x)
4 0 f o r X 2 0, x E a x ) t h e n d(f,A,O) = d(g,A, g E C(V,Rn), U,V bounded open s e t s i n
( 9 ) L e t f E C(U,V),
and V . be t h e s e t o f open connected subsets of
J
a r e d i s j o i n t compact subsets o f V.
V / f ( a n ) , whose c l o s u r e s
Then i f zo E Rn/(g o f)(aG),d(g
o f,U,
zo) = 1 d(f,U,Vj)d(g,Vj,zo) w i t h a f i n i t e sum ( h e r e d(f,U,v) i s c o n s t a n t f o r v E Vj). One shows t h a t t h e p r o p e r t i e s 1 - 9 a r e v a l i d f o r c o n t i n u o u s funct i o n s by u n i f o r m a p p r o x i m a t i o n o f a continuous f by C1 f u n c t i o n s fn. Then one has (10) I f f and g a r e continuous, yo f$ f ( a x ) ,
and f = g on
ai
then
2 58
ROBERT CARROLL
= d(g,A,yo)
d(f,A,yo)
(11) Ifv
C(a?,Rn) and yo $
E
v(aK)
pends o n l y on t h e homotopy c l a s s o f IP ( h e r e d(f,A,yo) continuous e x t e n s i o n f o f tf
v
-
to A
t h e n d(v,A,yo)
de-
i s t h e same f o r any
indeed i f g i s any o t h e r e x t e n s i o n ft =
+ ( 1 - y ) g can be used w i t h (3)). (1) i s t r i v i a l ( e x e r c i s e ) and
P R U O ~ : See here [ S m l ] f o r m i s s i n g d e t a i l s . f o r (2) i f f-’(y0)
w i t h Ni a NBH o f xi on which f i s a homeomorphism,
= {xi}
Ni n N . = 0, and N = nf(Ni), J
c o n s i d e r f o r suppu c N ( p a d m i s s a b l e f o r y$N)
( c f . ( 4 . 1 ) and Appendix C).
For ( 3 ) l e t Y = { f t ( x ) ,
$ Y and Y i s compact.
Then yo
n Y =
E
ax,
0 5 t 5 11.
L e t u be admissable f o r yo and f w i t h suppp
= IA u o f t i s continuous i n t and hence c o n s t a n t
Then d(ft,Ayyo)
0.
x
For ( 4 ) one a p p l i e s ( 3 ) t o t h e f a m i l y t f + ( 1 - t ) g .
s i n c e d i s an i n t e g e r .
( 5 ) i s e s s e n t i a l l y a r o u t i n e c a l c u l a t i o n ( e x e r c i s e ) and ( 6 ) f o l l o w s from ( 5 ) We l e a v e ( 7 ) as an e x e r c i s e w h i l e ( 8 ) f o l l o w s f r o m c o n s i d e r a -
immediately.
t i o n o f t h e homotopy t f ( x ) + ( I - t ) g ( x ) . t i o n (exercise). we t a k e fn defined.
-f
F i n a l l y (9) i s a r o u t i n e calcula-
f, + f E Co(/l,Rn)), For t h e approximations
f u n i f o r m l y on/1.‘
E
t h e a p p r o x i m a t i n g sequence one c o n s i d e r s a homotopy tfn + ( 1 - t ) g E
C’,
g,
-f
C1(A,Rn)), is
To see t h i s i s independent o f
= l i m d(fnyA,yo).
Then d(f,A,yo)
f,
For n l a r g e yo q! f n ( a i ) and d(fn,A,yo)
f; t h e c o n c l u s i o n i s v i r t u a l l y immediate ( e x e r c i s e ) .
9
where g, The v e r i f i -
c a t i o n o f ( 1 ) - ( 9 ) now f o r continuous maps i s easy ( e x e r c i s e ) and p r o p e r t i e s ( 1 0 ) - ( 1 1 ) a r e a l s o immediate.
QED
I n o r d e r t o extend degree t h e o r y t o reasonable c l a s s o f maps.
CEIIBIIA 4.13.
If
dimensional spaces one has t o f i n d a
We r e f e r back t o Theorem 4.5 now and check
K c B i s a c l o s e d bounded subset o f a
B i s compact then T i s a u n i f o r m l i m i t ( i . e . maps TE (dimR(TE)
0 it
can be covered by open s e t s Ni
( 1 5 i 5 j(&)) w i t h c e n t e r s xi and we t a k e a p a r t i t i o n o f u n i t y suppGi c Niy 847).
and
Set TE(x)
-+
i n norm) o f f i n i t e dimensional
1 d ~ ~ ( x= )1 f o r x E T(K) ( c f . Lemma = 1 gi(T(x))xi and e v i d e n t l y T E ( x )
-f
54.4,
$iy
$i
2 0,
Theorems 4.5 and
T ( x ) u n i f o r m l y i n x.
By Theorem 4.5 we know t h a t a compact map o f a c l o s e d convex bounded s e t D B y B Banach, i n t o i t s e l f has a f i x e d p o i n t . Now l e t U C B be bounded and
C
TOPOLOGICAL METHODS open, B Banach, and T = I
-
K:
u’ + B
259
where K i s compact.
F i r s t n o t e t h a t T(aG) i s c l o s e d s i n c e i f T(xn) = xn K(xn)
+
k (subsequence) and xn
K(x) = k and y = x-K(x). set
E
< d/2.
Let
-+
k+y = x
4
F o r yo
E
aq
L e t yo E B/T(afi).
- K(xn)
by c l o s u r e
-
-f
y we have
b u t by c o n t i n u i t y = d > 0 and we
T ( a i ) then dist(yo,T(a@)
KE be a f i n i t e dimensional E-approximation o f K as i n Lem-
ma 4.13 w i t h range i n a f i n i t e dimensional space VE c o n t a i n i n g yo.
-
KE e v i d e n t l y TE(x)
fined.
One s e t s d(T,U,yo)
= I
# yo f o r x
E
au and
hence d(TE,VE n E,yo)
usapproximation
= d(TE,VE n U,yo) and a l i t t l e c a l c u l a t i o n ,
i n g homotopy again, shows t h a t t h i s does n o t depend on t h e ( c f . [Bel;Sml]
F o r TE i s de-
E
A l l o f t h e p r o p e r t i e s i n Theorem 4.12 ex-
and Theorem 4.25).
t e n d now t o t h e B space s i t u a t i o n f o r T = I - K and v i a ( 1 0 ) - ( 1 1 ) one need onl y deal w i t h homotopy c l a s s e s o f maps T:
d(T,U,y,)
-
( c f . h e r e Theorem 4.25).
yo) = 1 f o r yo E U ( c f .
REmARK 4.14,
(4.1));
ac
+
R/IyoI i n o r d e r t o d e t e r m i n e
L e t us a l s o n o t e e x p l i c i t l y t h a t d(I,U,
a l s o d I-K,U,yo)
= d( I-K-yo,U,O).
L e t us s k e t c h a p r o o f o f t h e Brouwer f i x e d p o i n t theorem (Then+l We must show t h e r e i s no r e t r a c t i o n r: B
orem 4.3) u s i n g degree ideas. .+
Sn.
0
Suppose t h e r e were and n o t e t h a
o f degree d(r,BntlO) i s impossible.
RElllRRK 4.15,
= d(I,Bntl,O)
r(dBntl)
so t h a t by p r o p e r t y 4 C Sn which
The s t u d y o f o p e r a t o r s T = I - K as above i s c a l l e d Leray-Schau-
d e r t h e o r y and has many a p p l i c a t i o n s . veloped so f a r i n 112-4. rem 2.6),
4
T h i s i m p l i e s 0 E r(Bntl)
= 1
L e t us summarize t h e main p o i n t s de-
Thus F and G d e r i v a t i v e s , c o n t r a c t i o n maps (Theo-
i n v e r s e f u n c t i o n theorem (Theorem 2.7) which used a Newton t y p e
a p p r o x i m a t i o n technique, descent (Theorem 2.9),
imp1 i c i t f u n c t i o n theorem (Theorem 2.8),
monotone o p e r a t o r t h e o r y i n 53.3,
steepest
homotopy arguments
and i n v a r i a n c e o f domain from §3.4 were used i n p r o v i n g r e s u l t s i n §3.3, as w e l l as t h e Brouwer f i x e d p o i n t theorem, and a p p l i c a t i o n s were i n d i c a t e d i n §3.3 t o d i f f e r e n t i a l problems.
I n §3.4 we proved t h e Brouwer and Schauder
f i x e d p o i n t theorems a l r e a d y a l o n g w i t h i n v a r i a n c e o f domain, and have developed some homotopy and degree ideas.
Further applications a r e i n order.
EMRIPLE 4.16, Consider Au t f(x,u,Du) = 0, x E A , and u = 0 on an w i t h say For A t h e r e a l i z a t i o n o f A c o r r e s p o n d i n g t o 0 O i r i c h l e t boundary f E Cm. c o n d i t i o n s A - l w i l l make sense as a compact o p e r a t o r i n s u i t a b l e spaces and o u r e q u a t i o n becomes u
- KF(u)
= 0 ( F ( u ) = f(x,u,Du)).
F o r s u i t a b l e spaces
one t h i n k s e.g. o f t h e c l a s s i c a l Schauder e s t i m a t e s ( c f . [Nil;ZeE;B2;Gil; S m l ] ) and spaces o f t h e form C”@(A) o r 2 say UuU < c w h i l e u E C2”(A) n W (A) 1,B P
W 1( A ) ; t h e n a p r i o r i e s t i m a t e s g i v e p w i t h a compact i n j e c t i o n o f t h i s
260
ROBERT CARROLL
space i n t o C l Y B f o r example (C2ya denotes Holder continuous second d e r i v a More s i m p l y ( f o l l o w i n g [Sml])
t i v e s etc.).
i n t h e p r e s e n t s i t u a t i o n use l u
It m -< 1, IIKulll 5 c where II Ill i s a C 1 norm (I1 II c o u l d a l s o be used - c f . 1,B For reasonable f, n o t growing t o o r a p i d l y , t h e a p r i o r i e s t i m [Gil;Ze2]). a t e s w i l l i n v o l v e ( 0 ) IIuII < c ( f o r u = KFu - I I U I I ~5, ~c can a l s o be used) 1 1 and one c o n s i d e r s a b a l l U = {u; IIulll 5 l t c l i n t h e B space C ( A ) ( w i t h u = 0 on 3;). L e t Tu = u-KFu and by ( a ) Tu # 0 f o r Ilulll = l t c ( i . e . u E aG). Thus d(T,U,O) i s d e f i n e d and one c o n s i d e r s Tt(u) = u - tKF(u) (0 5 t 5 1).
By t h e homotopy p r o p e r t y o f degree d(T,U,O)
= d(I,U,O)
= 1.
Hence Tu = 0
1 has a s o l u t i o n i n U i n C ( A ) ( c f . a l s o Theorem 4.25) and a l i t t l e f u r t h e r work w i l l e s t a b l i s h smoothness. Note t h a t t h e a p r i o r i e s t i m a t e i s essent i a l here f o r e x i s t e n c e .
EW\WI;E 4-17,
As an example o f t h e use o f f i x e d p o i n t theorems d i r e c t l y
consider f i r s t y ' = f(x,y)
w i t h y ( 0 ) = yo o r y ( x ) = yo
T x ( y ) where f ( s , y ) E F, y E F ( F Banach).
J , f(s,y(s))ds
t
=
I n t h e p r o o f o f Theorem 2.11 we
had a s i m i l a r s i t u a t i o n ( w i t h f ( s , y ( s ) ) r e p l a c e d by f ( y ( s ) ) E H = H i l b e r t 1 and f E C ) and remarked t h a t f o r x small Tx was o b v i o u s l y a c o n t r a c t i o n so l o c a l l y t h e r e e x i s t s a unique y ( x ) w i t h y ( x ) = T x ( y ) .
T h i s remark i s based
on working w i t h T x ( y ) i n Co(H) (Co(B) would i n v o l v e t h e same arguments f o r 1 B bounded) and n o t i n g t h a t IITx(y) yoIIH 5 :f l l f ( y ( s ) ) l l d s . When f E C one
-
-
has by D e f i n i t i o n 2.3 and Theorem 2.4 I f ( y ( s ) )
f(yo)l 5 [llf'(yo)l
-
IIy(s)-yoll f o r s small and hence f o r x small enough IITx(y) f(s,y)
i s say compact on [O,a]
X F
-+
F and e.g.
llf(s,y)ll
yoll
t
1.
€1 Now i f
5 c ( l + / l y / / ) then we
can use degree t h e o r y f o r example t o o b t a i n a s o l u t i o n o f y ( x ) = T x ( y ) on (no s h r i n k i n g o f t h e i n t e r v a l i s needed
[O,a]
-
t h e bound
t
theorem would g i v e a unique s o l u t i o n on a s m a l l e r i n t e r v a l ) . Tx(y) f o r y
E
Co(F) w i t h Y ( S ) E B = B ( y o y b ) = {y; Ily(S)-yon
c o n t r a c t i o n map F i r s t the set
5 b l and 0 5
S
i x l i e s i n a t r a n s l a t e by yo o f t h e c l o s e d convex h u l l o f t h e r e l a t i v e l y
X B
compact image C o f f: [O,x]
F times x (B = B(0,b)
-+
-
n o t e h e r e :f
f(s,
y ( s ) ) d s i s a l i m i t o f Riemann sums ~ ~ f ( s ~ , y ( s ~ ) [ ( s ~ - s ~where - ~ ) / si-l x]
?(c)
si
5
Referring t o Thus T x ( y ) C which i s compact by Lemma 4.4. 5 si). t h e Arzela-Ascol i theorem (Appendix A , Theorem A33) we see by e q u i c o n t i n u i t y t h a t t h e image under T,[T(y)](x) i s r e l a t i v e l y compact ( 0 5 x ( y ) (y = ATy i n Co(F); 0 c1 t
CJ:
= Tx(y), o f Co(B) = Mb i n Co(F)
5 a).
x 5 1)
(sup norm)
F u r t h e r g i v e n a s o l u t i o n o f y ( x ) = ATx one has Ily(x)ll (IIyoII
+ 1: c ( l + l l y ( s ) l l ) d s 5
l y ( s ) I d s from which Ily(x)ll 5 c l e x p ( a c ) 5 c2 by Lemna 1.10.4.
r > c2 so (1- T)y # 0 f o r any y ( x ) w i t h Ily(x)ll = r anywhere on [O,a]
Choose (i.e.
TOPOLOGICAL METHODS
sup y ( x )
= r).
Hence d(1-T,Br(0),O)
a s o l u t i o n ( c f . Theorem 4.25).
261
= 1 and ( I - T ) y = 0 has
= d(I,Br(0),O)
L e t us n o t e here t h a t i f one had worked w i t h
t h e Schauder f i x e d p o i n t theorem i t would have been necessary i n general t o find a set A
C
Co(F) such t h a t T(A) C A.
s h r i n k i n g t h e i n t e r v a l i n general.
T h i s c o u l d be o b t a i n e d o n l y by
One n o t e s h e r e (and i n Theorem 2.11)
t h a t a l o c a l s o l u t i o n f o r 0 5 x 5 a (a < a ) which remains bounded as x
-+
a
w i l l n o r m a l l y have a l i m i t y ( a ) which can then be used as an i n i t i a l v a l u e f o r a c o n t i n u a t i o n y ( x ) f o r a 5 x 5 B ( B < a ) . For t h i s one needs o n l y t h a t 1 e.g. Ilf(s,y)ll ~ ~ p ( s ) $ ( I I y l lw) i t h $ ( c ) < m f o r 5 < m and say v E Lloc ( c f . [ C 13).
I n Theorem 2.11 we were a b l e t o c o n t i n u e t h e s o l u t i o n f o r a l l t be-
cause o f t h e hypotheses on f which l e t t o bounds on x ( t ) e t c . One can a l s o use f i x e d p o i n t ideas f o r a s e m i l i n e a r h y p e r b o l -
EXAntPCE 4.18.
f o r example ( u can a l s o be i n c l u d e d by Y Thus f o r t h e c h a r a c t e r i s t i c i n i t i a l v a l u e 1 1 problem u = f, u(x,O) = v ( x ) E C on [O,a], u(0,y) = $(y) E C on [O,b] XY ( ~ ( 0 )= $(O), f continuous, w i t h If(x,y,u,O)l 5 M ( l + l u l ) and If(x,y,u,v) i c problem such as u
XY
= f(x,y,u,uX)
expanding t h e equations below).
-
5
f(x,y,u,v)I
(4.3)
C I V - V I one
writes
U(X,Y) = P ( X ) + $ ( Y ) V(X,Y) = 9 ' ( x ) +
- ~ ( 0 +)
I,";1
f(5,~,U(c,~),v(S,~)d5d~;
I Y f(x,~,u(x,~),v(x,~)d~ 0
Then s o l v e t h e second e q u a t i o n v = Tu by c o n t r a c t i o n , o b t a i n a p r i o r i e s t i m a t e s v i a t h e Gronwall lemma (Lemma 1.10.4),
and use Schauder's f i x e d p o i n t
theorem f o r t h e f i r s t e q u a t i o n w i t h v = Tu ( c o n t i n u a t i o n s as i n Remark 4.17 may a l s o be needed b u t we o m i t t h i s ) . There a r e many d i f f e r e n t t y p e s o f i n t e g r a l o p e r a t o r s t o which
EXAIIIPCE 4.19.
v a r i o u s n o n l i n e a r techniques can be a p p l i e d ( c f . [Ksl,2;Mtl ;Ze2]). g. an i n t e g r a l o p e r a t o r ( K u ) ( t ) = J A K(t,s,u(s)ds son o p e r a t o r .
Thus e.
i s r e f e r r e d t o as a Ury-
Here one t h i n k s o f a f u n c t i o n K ( t , - , - ) :
A X Cn
-f
Cn f o r exam-
p l e ( t a k e n = 1 here and t i s a parameter); f o r f u n c t i o n s u ( - ) : A * Cn one Cn ( K i s c a l l e d a N e m y t s k i j o r s u b s t i t u -
writes (Ku)(t,-)
= K(t,-,u(.)):
t i o n operator).
Various hypotheses a r e p l a c e d on K so t h a t e.g. measurable
functions u
-+
A
-+
measurable f u n c t i o n s K u ( t , - ) .
(t)= JA G(t,s)f(s,u(s))ds
An o p e r a t o r o f t h e form (Ku)
i s c a l l e d a Hammerstein o p e r a t o r and t h e s e a r i s e
f r e q u e n t l y i n d i f f e r e n t i a l e q u a t i o n s ( c f . Example 7.16). a r i s e s from say u '
+
Au = f ( t , u )
w i t h u ( 0 ) = uo E E.
Another o p e r a t o r
Thus ( c f . Appendix B)
i f -A generates a s t r o n g l y c o n t i n u o u s semigroup i n a B space E, G ( t ) =
262
ROBERT CARROLL t h e n ( 6 ) u ( t ) = G(t)uo
exp(-At),
equation o f Volterra type.
G(t-s)f(s,u(s))ds
f
which i s an i n t e g r a l
There a r e many theorems f o r a l l o f these t y p e s
o f o p e r a t o r s and we w i l l n o t even a t t e m p t t o g i v e references. The f o l l o w i n g g e n e r a l i z a t i o n o f a r e s u l t o f Dolph-Minty i s w o r t h c i t i n g ; t h e p r o o f i s i n s t r u c t i v e b u t we o m i t i t h e r e i n r e f e r r i n g t o [ C l ]
(it involves
monotone o p e r a t o r techniques as i n 53.3). L e t H be a H i l b e r t space,
tHE0REIII 4.20.
K E L(H) be monotone, and F: H 2 0 f o r IlxII
be hemicontinuous, bounded, and monotone, w i t h Re(F(x),x)
+
H
M.
>
Then t h e e q u a t i o n y t KFy = 0 has a s o l u t i o n .
REmARK 4-21, L e t us say a few works about p r o p e r maps ( c f . [Bel;Ze2;Dml]). As n o t e d i n [Ze?] t h e r e i s a sense i n which uniqueness i m p l i e s e x i s t e n c e ( g o i n g back t o Schauder).
Thus i f
A i s an n X n m a t r i x and Ax
most one s o l u t i o n t h e n r a n k A = n and detA # 0 so t h a t x = A-’y tion.
= y has a t
i s a solu-
One expects t h i s s i t u a t i o n t o p r e v a i l a l s o f o r l i n e a r compact p e r t u r -
b a t i o n s o f t h e i d e n t i t y which w i l l be Fredholm o p e r a t o r s o f i n d e x 0 ( c f . G e n e r a l l y one wants t o know when a (conRemark 2.13 and see e.g. [Sh3]). t i n u o u s ) o p e r a t o r f: E -+ F (E,F Banach) has open range so t h a t f ( x o ) = yo i m p l i e s f ( x ) = y has a s o l u t i o n f o r y in some NBH o f yo.
Recall here t h e
open mapping theorem f o r l i n e a r o p e r a t o r s (Theorem A45) which says t h a t f
i s open i f i t i s o n t o F ( t h e n f ( U ) i s open f o r U open).
Similarly for fin-
i t e dimensional spaces we have i n v a r i a n c e o f domain (Theorem 4.9) so t h a t f ( U ) i s open f o r U open when f i s l o c a l l y 1-1.
A r e l a t e d question i s o f
course whether x depends c o n t i n u o u s l y on y = f ( x ) ( s t a b i l i t y ) .
One c o u l d
d i s c o u r s e a t l e n g t h i n t h i s area o f ideas b u t we w i l l t r y t o e x t r a c t a few key ideas and theorems.
Thus f i r s t one d e f i n e s a continuous map f:
1 (E,F Banach) t o be p r o p e r i f f- ( K ) i s compact f o r K compact. f o r a p r o p e r map t h e s e t o f s o l u t i o n s S = { x E Y
E; f ( x )
E
+
F
In particular
= y l i s compact.
Next (see [ B e l l f o r m i s s i n g d e t a i l s )
CHE0REll 4.22.
The f o l l o w i n g a r e e q u i v a l e n t . ( 1 ) f i s proper ( 2 ) f i s a c l o -
sed map ( i . e .
f ( C ) i s c l o s e d f o r C c l o s e d ) and S i s compact f o r y f i x e d Y ( 3 ) I f E and F a r e f i n i t e dimensional then f has t h e p r o p e r t y I l f ( x ) l l + m when llxll + m. Phuod:
-
F o r ( 1 ) i m p l i e s ( 2 ) we need o n l y show f i s closed.
and l e t y,
= f ( x n ) + y , xn E C.
compact and hence a subsequence x, f ( x ) = y.
L e t C be c l o s e d
Since Y = {yn} i s compact, u = f - ’ ( Y ) E u converges t o x E C.
For ( 2 ) i m p l i e s (1) one t a k e s a compact subset
is
By c o n t i n u i t y
K
C F with f-l(K)
TOPOLOGICAL METHODS
263
= D and covers D w i t h c l o s e d s e t s D, h a v i n g t h e f i n i t e i n t e r s e c t i o n p r o p e r t y ( c f . Remark 3.2.13). A l i t t l e argument ( c f . [ B e l l ) shows t h a t "0, = @ which i m p l i e s D i s compact.
( 3 ) i m p l i e s ( 2 ) and c o n v e r s e l y f o r E,F f i n i t e
dimensional i s l e f t as an e x e r c i s e .
QED
Under hypotheses o f t h e f o r m made i n §§3.2-3.2 f o r o p e r a t o r s Au = 1 S la1 5s (-l)lulDaAa(x,u, 0 u), A: Ws -+ W-' one can show t h a t A i s p r o p e r ( c f . P P Now f o r g l o b a l r e s u l t s one r e c a l l s f i r s t f r o m Theorem 2.7 t h a t i f [Bell). g ' i s 1-1 o n t o t h e n g: E F i s a l o c a l homeomorphism. The e x t e n s i o n as a
...,
-+
g l o b a l theorem i s ( c f . [ B e l l f o r p r o o f ) L e t f: E
CHE0RElil 4.23.
-+
F be continuous.
Then f i s a homeomorphism i f and
o n l y f i s a l o c a l homeomorphism and i s p r o p e r . We conclude t h i s s e c t i o n w i t h a few more a p p l i c a t i o n s o f degree t h e o r y , properness, e t c . t o a r r i v e a t an i n v a r i a n c e o f domain theorem f o r B spaces; t h e p r e s e n t a t i o n f o l l o w s [ B e l l and we p r o v i d e some f u r t h e r d e t a i l s e s t a b l i s h i n g t h e degree p r o p e r t i e s o f maps IiK i n B spaces which were o m i t t e d a f t e r Theorem 4.12.
One can deal w i t h continuous maps f:
E
-f
F f o r which a
I-K where K i s compact - b u t we w i l l F - c f . [Bel;Dml;Ze2] ( v a r i o u s o p e r a t o r s a r e admissable, E n o t d w e l l on t h i s h e r e ) . R e c a l l f r o m D e f i n i t i o n 4.5 t h a t f: Sn Y i s indegree f u n c t i o n i s d e f i n e d as i s t h e case f o r f
=
-f
-f
e s s e n t i a l i f i t i s homotopic t o a c o n s t a n t and i n t h i s event f extends t o a map 8"' 8"' B"'.
-+
-f
Y; t h i s p r o p e r t y c h a r a c t e r i z e s i n e s s e n t i a l i t y .
Rn+'
i s e s s e n t i a l as a map Sn
-f
Rn+'/{OI
Further i f f:
t h e n f must have a 0 i n
E v i d e n t l y e s s e n t i a l i t y i s a homotopy i n v a r i a n t and one extends t h e s e
ideas t o B spaces by w o r k i n g w i t h compact p e r t u r b a t i o n s o f a f i x e d c o n t i n uous map f (e.g. f = I ) . Thus t a k e f = I f o r s i m p l i c i t y ( w i t h E = F ) , S C be t h e s e t o f E c l o s e d ( E Banach), and C a component o f E/S. L e t CI(S,E) -+ E compact, and CY(S,E) c CI(S,E) t h e subset where g # 0 on S . The map g E CY(S,E) i s i n e s s e n t i a l w i t h r e s p e c t t o C i f g has an
maps g = I+K, K: E extension
9".
C y ( C u S,E)
-
otherwise g i s essential.
Thus t o show t h a t a
g i v e n g E C I ( C U S,E) has a z e r o i n C one need o n l y show t h a t g i s essent i a l w i t h r e s p e c t t o C. One shows i n a s t a n d a r d manner ( c f . [ B e l l ) t h a t e s s e n t i a l i t y (and i n e s s e n t i a l i t y ) o f maps g i n CY(S,E) what i s c a l l e d compact homotopy where h ( x , t ) : g(x,t) = I t h(x,t), gl.
gl(x)
=
I + h(x,l),
i s i n v a r i a n t under
S X [O,l]
and g o ( x ) = I
-+
E i s compact,
+ h(x,O) l i n k s go and
As one expects now (see [ B e l l f o r p r o o f )
CHE0REfil 4.24,
L e t D be a convex bounded domain i n
E
(domain = open connec-
264
ROBERT CARROLL
ted set).
Then f , g
E
CY(ai,E)
a r e compactly homotopic i f and o n l y i f d ( f ,
Further i f f E CY(ai,E) then f i s essential r e l a t i v e t o D D,O) = d(g,D,O). i f and o n l y i f d(f,D,O) # 0. I n p a r t i c u l a r i f d(f,D,O) 4 0 t h e n f ( x ) = 0 has a s o l u t i o n i n D.
REmARK 4.25.
There a r e many deep and b e a u t i f u l r e s u l t s i n n o n l i n e a r a n a l y -
s i s which can be developed u s i n g degree t h e o r y ( c f . [Bel;Br8;Dml
;Sml;Ze2]).
We have o n l y t r i e d t o g i v e t h e f l a v o r o f (some) degree arguments here. us c i t e however a few f u r t h e r r e s u l t s .
I f D C E i s a bounded domain and f = I+K,
EHE0REflI 4.26,
l o c a l homeomorphism w i t h d(f,D,p)
Let
First K compact, i s a
= 21 t h e n f ( x ) = p has e x a c t l y one s o l u -
t i o n i n D.
f ( x ) = p l i s d i s c r e t e and f i n i t e . and by ( 5 ) i n Theorem 4 . 1 2 , ~ t l =
1
6
= t x E E; P Cover S w i t h d i s j o i n t open s e t s 0 C D P P d(f,OX,p). One then shows v i a homotopy
P J L V V ~ : f w i l l be p r o p e r on a bounded
( e x e r c i s e ) so t h e s e t S
t h a t f o r any p a t h p ( t ) i n D w i t h s u i t a b l e 0 Op(t),f(p(t)))
i s constant ( c f . [ B e l l
Hence d(f,Ox,p)
i s c o n s t a n t and t h e r e can o n l y be one x ( n o t e x~Ox,OxnOy=O). Thus under t h e hypotheses o f Theorem 4.26 i f f ( D ) n f ( a 6 ) = @
REmARK 4-27. then f : D
-
3 p ( t ) the function d(f, P(t) homotopy and connectedness a r e used).
+
f ( D ) i s 1-1.
To see t h i s n o t e f ( D ) C E / f ( a i ) so d(f,D,p')
d e f i n e d f o r a l l p ' E f ( D ) and i s c o n s t a n t ( = 21).
is
I n view o f t h i s one asks
whether a 1-1 map f o f a bounded open s e t D o n t o f ( D ) i s a homeomorphism ( i n v a r i a n c e o f domain).
Such a theorem ( i . e .
showing f i s open) i s t r u e f o r
f = I+K f o r example and we r e f e r t o [ B e l l f o r d e t a i l s .
f i x e d p o i n t index i(T,G)
= d(1-T,G,O)
T a r e compact w i t h ( I - T ) x d(1-S,G,O)
+ 0 and ( I - S ) x
f e r e n t ( c f . Theorem 4.25). to < 1.
# 0 on
= t(1-T)x + (1-t)(I-S)x
t h i s cannot be a compact homotopy on
IIx-x"llfor
x
> 0 and a l l (x,
y ) , (:,$) i n G(A) = { ( x , y ) ; y E Ax}. We mention a l s o t h a t a map A: O ( A ) C E' E - 9 2 i s monotone if ( x - y , x ' - y ' ) 2 0 f o r a l l x,y E D(A), x ' E Ax, y ' E Ay.
NONLINEAR SEMI GROUPS
273
L e t us n o t e t h a t when A i s a c c r e t i v e t h e e q u a t i o n x+hy = z f o r ( x , y ) (i.e.
(l+AA)x 3
z ) has a t most one s o l u t i o n .
Indeed i f xi+Ayi
=
E
G(A)
z t h e n IIxl
-
-x2 + h(yl y2)11 2- IIx 1 - x 21I i m p l i e s x1 = x2 and hence y1 = y2. Thus i f z E R(l+r\A) t h e r e e x i s t s a u n i q u e (x,y) E G(A) w i t h x t h y = z and hence ( 1 +
i s s i n g l e valued. I n f a c t (1 + A A ) - ' i s nonexpansive s i n c e i f x1 + hyl = z1 and x2 + hy2 = z t h e n Ilx - x + x ( y y )ll = IIz -z II > IIx - x II = l l ( 1 t hA)-'
1 (l+hA)- z211.
-
xA)-lzl
2 1 2 1-2 1 2 - 1 2 The argument can be r e v e r s e d so t h a t i f (l+AA)-'
nonexpansive t h e n A i s a c c r e t i v e .
is
Now one d e f i n e s a k i n d o f s c a l a r p r o d u c t
( c a l l e d s e m i - i n n e r p r o d u c t ) on E so t h a t one can handle t h e i d e a o f a c c r e t i v e i n a manner s i m i l a r t o t h e s i t u a t i o n i n H i l b e r t space. first
(A)
z2;Dml]) (0)
[x,y],
= [IIx+Ayll
that A 1i m
[x,y]
=
-
(A>
IIxll]/A
0).
Thus d e f i n e
One checks ( e x e r c i s e
-
c f . [P
i s nondecreasing and t h i s l e a d s t o t h e d e f i n i t i o n inf [x,ylA = [x,yIA. It i s r o u t i n e t o prove ( c f . Cpz2-j)
+
[x,y-JA
( 1 ) [ , 3: E X E + R i s USC ( 2 ) [crx,By] = IBI[x,y] ( 3 ) PR0P05fCf0N 6.2. [x,y] 1. 0 i f and o n l y i f Ilx+Xyll 2 IIxII f o r a l l A > 0 ( 4 ) [x,y+z] 5 [x,y] + [x,zl
(5) [x.ax+yI
= a l x l + [x,yl
[O,y] = llyll ( 8 ) [x,y] - [x,z] t i v e i f and o n l y i f [x-;,y-;]
RrmARK
(6) -[x,-yI
5 lly-zll.
5
[x,YI (7) ~ [ x I IIyll ,Y and I ~ + eE i s a c c r e -
It f o l l o w s t h a t A: E
2 0 f o r every (x,y),
(i,;) i n
G(A).
For a ( r e a l ) H i l b e r t space E = H one has [ x , y l A = ( l / A ) [ l l x + 2 nxn ] / ( i i X + ~ y n + tixi) and [X,y] = ( ~ , ~ ) / I I ~ I I For . E = LP(A), 1 < p 1i m m y one can e v a l u a t e i n a s t r a i g h t f o r w a r d way ( l / A ) [ ( I A lu+hvlpds)l'p (JA l u I p d s ) l / P 1 - [u,v] t o o b t a i n [u,v] = I I u l l " ~ I A v ( s ) ) u I p - ' s g n u d s ( e x e r iyn2
6.3,
-
-
The s e m i - i n n e r p r o d u c t a s s o c i a t e d w i t h
cise).
[XJ]
i s ( y , x ) + = /lx/l[x,y]
(cf. [Dmll). Now a map J: E {x'
J(x)
E
3
E';
2
t'
i s c a l l e d a d u a l i t y map ( r e l a t i v e t o a gauge 9 ) when
(x',x)
= IIx'IIIIxII;
IIx'II = 9 ( l I x l l ) l .
Here 9 s h o u l d be a con-
t i n u o u s and i n c r e a s i n g f u n c t i o n w i t h 9 ( 0 ) = 0 and 9 ( r ) [Brl;Pcl;Dml;Bdl]). F(X)
= {XI E
(x',x)
The most common
E'; ( x ' , x )
9
+
m
as r *
m
(cf.
i n v o l v e s 9 ( r ) = r and we w r i t e t h e n
2
= 11x11~ = 11x811I .
I n [pz21 one uses J(X) = t x ' E E ' ;
= IIxII when IIx'II = 1 1 b u t t h i s i s n o t perhaps b e s t c a l l e d a d u a l i t y
map; i t i s more a p p r o p r i a t e t o n o t e s i m p l y (as i n [ P z ~ ] ) t h a t F ( x ) = IIxII J ( x ) where F
2,
9 ( r ) = r i s a d u a l i t y map.
Then i f x ' E J ( x ) s e t y ' = IIxIIx'
so ( y ' , x ) = IIxII 2 and Ily'II = IIxII. Now when E ' i s s t r i c t l y convex, one sees immediately t h a t F and hence J i s s i n g l e valued (see Remark 6.16 f o r p r o o f a space E ' i s s t r i c t l y convex i f t h e u n i t sphere does n o t c o n t a i n l i n e segments
-
i . e . IIxII = llyll = 1 and x = y i m p l i e s llAx+(l-A)yll < 1 f o r A
E
(0,l))
and we r e c a l l a r e s u l t o f Asplund which says t h a t any r e f l e x i v e B space E
-
274
ROBERT CARROLL
can be p r o v i d e d w i t h an e q u i v a l e n t norm under which E i s s t r i c t l y convex w h i l e E ' i s a l s o s t r i c t l y convex under t h e new dual norm.
EHE0REN 6.4.
[x,y]
Let x '
Ptuud:
< IIxtAyll
-
= max ( x ' , y )
for x'
E
J(x).
J ( x ) and A > 0 so ( x ' , x + A y ) = IIxIl
E
IIxII whence ( x ' , y ) ( [ x , y ] .
E J ( x ) such t h a t ( x ' , y )
= [x,y].
Ife.g. E
+
= allxlI
115'11 5 1 then 5 '
E
E ' w i t h t h e same norm.
B[x,y]
Define a l i n e a r functional 5 '
( t h u s ( 5 ' , x ) = IIxII and(C',y)
= [x,~]).
V ' and by Hahn-Banach w i l l have an e x t e n s i o n x ' To see 115'11 5 1 n o t e f i r s t [x,By]
B by P r o p o s i t i o n 6.2 ( e x e r c i s e ) .
E'
2 B[x,y]
Hence (E',crx+By) 5 allxII + [x,By]
By] 5 IIax+Byll ( a g a i n u s i n g Prop. 6.2). By11 o r II€,'II 5 1 and x ' E
Thus A ( x ' , y )
A(x',y).
F o r t h i s l e t x,y be l i n e a r l y independent
and V C E t h e subspace generated by I x , y l . on V by (c',ax+By)
+
One needs t h e r e f o r e o n l y t o f i n d x '
It f o l l o w s t h a t
f o r any = [x,ax+
5 IIax+
(C,',ax+By)
e x t e n d i n g 5 ' w i t h IIx'II 5 1 e x i s t s s a t i s f y i n g
9 ED
( x ' , x ) = IlxII and ( x ' , y ) = [x,y].
REMRK 6.5, From Theorem 6.4 i t f o l l o w s immediately t h a t [x,y] '> 0 if and o n l y if t h e r e e x i s t s x ' E J ( x ) w i t h ( x ' , x ) 2 0, and moreover t h e f o l l o w i n g ( 1 ) A i s a c c r e t i v e ( 2 ) (l+AA)-'
c o n d i t i o n s a r e now seen t o be e q u i v a l e n t . i s nonexpansive f o r A >
(x*,y^)
For every (x,y),
o
o
(3) [x-;,y-jr]
f o r (x,y),
i n G(A) t h e r e e x i s t s x '
(;,;I
i n G(A) ( 4 )
J ( x - i ) such t h a t
E
(
x',y-;)
A c t u a l l y ( 4 ) w i l l h o l d f o r a l l x ' E J ( x - $ ) as w i l l be seen i n t h e
> 0.
course o f subsequent developments.
-
cise
One a l s o shows i n a r o u t i n e way ( e x e r -
c f . [ P z ~ ] ) t h a t i f A i s a c c r e t i v e then
fi
i s accretive
an a c c r e t i v e A i s c l o s e d i f and o n l y i f R(1tA) i s closed.
(A
G(A)) and
%
We remark t h a t
c a l c u l a t i o n s w i t h m u l t i v a l u e d o p e r a t o r s a r e modeled on standard o p e r a t o r c a l c u l a t i o n s ; one s i m p l y works w i t h p a i r s (x,y) E G ( A ) ( i . e . y
E
A ( x ) ) . NOW
d e f i n e A t o be s - a c c r e t i v e i f ( x ' , y - y ) 5 0 f o r a l l x ' E J ( x - ? ) and a l l ( x , y ) and
(x*,;)
tive.
E v i d e n t l y when J i s s i n g l e valued s - a c c r e t i v e
E G(A).
f
accre-
A number o f s i m p l e r e s u l t s i n v o l v i n g s - a c c r e t i v e o p e r a t o r s f o l l o w .
EHE0REN 6.6.
( 1 ) I f A i s a c c r e t i v e and
t i v e . ( 2 ) I f S: E
-f
E is
B
i s s-accretive then A t B i s accre-
nonexpansive then 1-S i s s - a c c r e t i v e ( 3 ) I f S ( v )
i s nonexpansive d e f i n e D(A) = I x E then A i s s - a c c r e t i v e ( 4 ) IfA: E
-f
E; E
l i m [x-S(p)x]/p
= Ax e x i s t s ( P
t
* 0 )I;
i s a c c r e t i v e and continuous ( c o n t i n u -
ous i m p l i e s s i n g l e v a l u e d ) then A i s s - a c c r e t i v e .
Pkood:
We sketch t h e p r o o f o f a few p o i n t s and l e a v e t h e r e s t as an e x e r -
cise (cf. [ P z ~ ] ) .
~XII - ( x ' , s x - s i )
For ( 2 ) l e t S be nonexpansive and x ' E J(x-;);
2 o since ( x',sx-sx') < I I S ~ - S ~ ^ I 0.
y ) E G(A) which c o n t r a -
1' 1
d i c t s ( r e c a l l a l s o F i s s i n g l e valued i n t h e p r e s e n t s i t u a t i o n ) .
On t h e
o t h e r hand t o show A maximal monotone i m p l i e s E ' = R(A+XF) l e t us check f i r s t t h a t F i s c o e r c i v e and e.g. hemicontinuous (hence demicontinuous by Theorems 3.2 and 3.3 s i n c e F i s o b v i o u s l y bounded v i a IIF(x)ll = IIxll). Thus 2 ( c f . [ T a l l ) i f f,g E F ( u ) t h e n ( f , u ) = ( g , u ) = IIul12 = IIfl12 = IIgll w h i l e f o r
0
IIuII so ll(1-
h)f+xgll = IIuII = Ilfll = Ilgll and t h i s i m p l i e s f = g when E ' i s s t r i c t l y convex. T h i s proves F i s s i n g l e v a l u e d f o r E ' s t r i c t l y convex.
x
X)u+xv] and as verges w h i l e IIf,II
.+
A.
Ilf,
1I 0
Il(1-h )u+x vII. 0
= F[(1-
t h e r i g h t s i d e o f ( f A , ( l - h ) u + X v ) = ll(l-x)u+Xvl12 con-
= l l ( l - h ) u + ~ v l l i s bounded.
n e t f s .+ f weakly and consequently ( f , ( l - A IIfn
Next p u t f,
0
By t h e A l a o g l u theorem a sub-
0 )u+Xov) =
I l ( l - ~ o ) u + ~ o v l 1so 2 that
B u t by d e f i n i t i o n s IIfll 5 IIf
II so IIfll = IIf 1I 10 xo
280
ROBERT CARROLL
and hence f E F[(1-Xo)utxov] y i e l d s f,
-+
must equal f,
s i n c e IIF(x)H = IIxl,F
0
-
f weakly ( e x e r c i s e
.
A l i t t l e f u r t h e r argument
c f . arguments i n S3.3).
i s bounded and coercive.
We n o t e a l s o t h a t
Now one can p r o v e a r e s u l t
s a y i n g t h a t ifA i s maximal monotone and B i s e.g. monotone, s i n g l e valued, hemicontinuous, bounded, and c o e r c i v e t h e n t h e r e e x i s t s xo such t h a t (Am) (
u - x o y B x o t v ) 1. 0 f o r a l l (u,v)
trary y
0
E
G(A); a p p l i e d t o Bx = AF(x)
0
etc. f o r r e l a t e d situations).
§3.3, e.g.
The r e s u l t
ment below and we f i r s t n o t e t h a t F: E +
(Fx,x-y),
x weakly and l i m sup for all y
E
(
+
(Am)
+
+
3.3.6,
w i l l f o l l o w f r o m some argu-
5 0 i m p l i e s l i m i n f ( Fxa,xa-y) 2
Fxa,xa-x)
E ( E = D(F) here).
x weakly ( o r xu
Theorems 3.3,3,
E ' i s pseudomonotone which means This n o t i o n w i l l a r i s e again i n To see t h a t F i s pseudomono-
t h e s t u d y o f v a r i a t i o n a l i n e q u a l i t i e s i n §3.7. t o n e l e t xn
yo f o r a r b i -
E Axo o r yo E R ( A t x F ) which w i l l prove t h a t A max-
we g e t y -,F(x0)
i m 1 monotone i m p l i e s R(A+xF) = E ' (cf.
t h a t xu
-
x as a n e t
- e i t h e r argument i s t h e same);
2 ( Fx,xn-x) so l i m i n f ( Fxn,xn-x) 2 0 and hence 2 0. For y E E = D(F) a r b i t r a r y , a g a i n by m o n o t o n i c i t y (*A) l i m i n f ( Fxn,xn-y) 2 l i m ( Fy,xn-y) = ( Fy, x-y). L e t w = (1-A)x+Ay and p u t t h i s i n (.A) i n place o f y t o obtain l i m by m o n o t o n i c i t y lim
(a*)
inf
(
(
(
Fxn,xn-x)
Fxn,xn-x)
= 0 when l i m sup ( Fxn,xn-x)
(Fw,x-y)
f r o m which l e t t i n g A
i n f (Fx.,x.-y) 1
1
Hence u s i n g (@*) l i m i n f
L ( Fw,h(x-y)).
Fxi,xi-x+X(x-y))
+
0, l i m i n f
[( Fxiyxi-x)
= lim inf
t
(
(
(
Fxiyx-y)
2
2 ( Fx,x-y). Hence l i m 2 ( Fx,x-y) as r e q u i r e d t o
Fxi,x-y)
Fxiyx-y)]
show F pseudomonotone. Now t o prove
(Am)
one may f i r s t assume w i t h o u t loss o f g e n e r a l i t y t h a t 0
E
D ( A ) and we can assume B i s pseudomonotone, c o e r c i v e , bounded, and demicon tinuous.
F o l l o w i n g [Dml;Tal;Bdl]
and work w i t h u
E
8,(0)
f o r example assume f i r s t dimE
0 f o r a l l
such t h a t ( A ( u ) - f , v - u )
6
.u
(f,-l) E V ' (exercise
- note
with
i(T)
Then A i s pseudomonotone and (**) i s equiva-,.d
lent t o finding
v
ry
,u
E
where f =
N
= ( u , ~ )E K involves a '>v(u)).
This i s a l N
most e q u i v a l e n t t o t h e problem o f Remark 7.4 b u t n o t q u i t e s i n c e K i s n o t bounded and t h e r e i s n o t c o e r c i v i t y o f t h e form m a n i p u l a t i o n u s i n g t h e p a r t i c u l a r form o f
REmARK 7.4,
(*A).
7 suffices
A l i t t l e technical ( c f . [Li4]).
Recall now from 53.5 t h e i d e a o f s u b g r a d i e n t e t c . so t h a t f o r
a proper convex f u n c t i o n ~ ( u on ) V, alp(u) C V ' i s t h e s e t o f x such t h a t ) ~ ( v -) ~ ( u '>(x,v-u) i s now e q u i v a l e n t t o A(u)
-
f + av(u).
(i.e. (*m)
V
alp:
-t
2").
The problem (**) o f Remark 7.3
F i n d u E V such t h a t - ( A ( u )
- f ) E alp(u) o r
0 E
I n p a r t i c u l a r one sees how m u l t i v a l u e d o p e r a t o r s a r i s e
n a t u r a l l y i n t h e study o f v a r i a t i o n a l i n e q u a l i t i e s .
REmARK 7.5,
L e t us l o o k a t some c o n c r e t e examples from [Bw2;Bd2;Dvl;Kbl].
Thus f i r s t c o n s i d e r " o b s t a c l e " problems i n 2 dimensions f o r example. Thus 2 d e s c r i b e d say by y = $ ( x ) , continuous, and 0 5 x
g i v e n an o b s t a c l e A c R
< L, one wants t o connect x = 0 and x = L by an e l a s t i c s t r i n g passing be-
l o w A which cannot p e n e n t r a t e A.
Thus u ( x ) z $ ( x ) and a t y p i c a l example i s
Thus u ( x ) i $ ( x ) , u ( 0 ) = u ( L ) = 0, u" > 0, and u ( x ) < $ ( x ) i m p l i e s u " = 0. Note t h a t t h e c o n d i t i o n (u-$)u" = 0 d e s c r i b e s t h i s l a s t s i t u a t i o n . L e t K = 1 L 2 Iv,v(O) = v ( L ) = 0, v ~ $ i n1 say Ho w i t h energy E ( v ) = (1/2)J0 v ' dx and o f course one must r e q u i r e $ ( x )
0 a t 0 and L e t c .
We c o u l d ask now f o r u E
K such t h a t E(u) < E(v) f o r a l l v E K o r e q u i v a l e n t l y one asks f o r u E K Lsuch t h a t (A*) I. u ' ( x ) [ u ' ( x ) - v ' ( x ) ] d x 5 O f o r a l l v E K. The e q u i v a l e n c e
i s an easy e x e r c i s e f o l l o w i n g p r e v i o u s p r a c t i c e . A h i g h e r dimensional v e r 3 v i a A(u) = -Au t hu f o r exam1 p l e and a(u,v) = A vu-vvdx + f A Auvdx. L e t yo be t h e t r a c e map H ( A ) -t I?' s i o n o f t h i s problem can be phrased i n say R
( r ) and
take $
E
H'(A)
w i t h yo$ '> 0 and K = I v E HA, v i $ i n A } .
By prev-
i o u s r e s u l t s we know t h e r e e x i s t s a unique s o l u t i o n o f t h e problem (u) Find
VARIATIONAL INEQUALITIES
u
K such t h a t a(u,u-v)
E
5 JA f ( u - v ) d x f o r a l l v E K, f o r f
Some a n a l y s i s as above g i v e s t h e n (-Au+Au-f).(u-$) -Au
t
Au
= f when u
01 ( c f . [ B w ~ ] Y f o r d e t a i l s ) . One s h o u l d n o t e t h a t v a r i a t i o n a l i n e q u a l i t i e s a r e an e f f i c a c i o u s way t o deal w i t h many f r e e boundary problems ( c f . [Li4;Bw2;Kbl]).
REmARK 7.7,
We had o r i g i n a l l y i n t e n d e d t o s k e t c h i n t h i s s e c t i o n some t e c h -
niques i n v o l v i n g p e n a l t y and r e g u l a r i z a t i o n methods from [ L i 4 ] f o r example as we1 1 as t h e existence-uniqueness p r o o f s f o r t h e Navier-Stokes theorems c i t e d i n 51.10 ( f o l l o w i n g [ L i 4 ]
-
c f . a l s o [Fol-6;Te1,3,4]).
f o r c e s us t o c u r t a i l t h i s e x p o s i t i o n .
Lack o f space
S i m i l a r l y we t h i n k i t would be appro-
p r i a t e t o t r e a t a b s t r a c t Hamilton-Jacobi equations ( f o l l o w i n g [Ljl;Bd3]), v a r i a t i o n a l convergence ( d ' a p r e s [ A c l ] ) ,
compensated compactness ( c f . [Mhl]),
homogenization, e t c . t o c i t e b u t a few areas o f c u r r e n t i n t e r e s t and a g a i n must work w i t h i n page l i m i t a t i o n s .
Another t o p i c would be t h e c o n s i d e r a t i o n
o f e l l i p t i c problems near t h e c r i t i c a l Sobolev exponent (where compactness f a i l s ) and f o r t h i s we r e f e r t o [ B x l ] .
A number o f i n t e r e s t i n g r e s u l t s on
n o n l i n e a r wave equations can be found i n [ G f l ;Str1-3;Fr3;Bfl-3;Ghl
Ka6,7; Te2;Swl; Tcl ;Swl-3;Wil-3]
;Oal ;Kcl;
f o r example, a1 ong w i t h n o n l i n e a r e v o l u t i o n
286
ROBERT CARROLL
r e s u l t s f o r o t h e r equations, b u t we w i l l n o t have space t o even s k e t c h any o f t h i s , a l t h o u g h i t has s i g n i f i c a n t impact on t h e study o f quantum f i e l d theory, s o l i t o n theory, e t c .
We a l s o o m i t any d i s c u s s i o n o f t h e t r a n s i t i o n
t o t u r b u l e n c e and chaos i n d e a l i n g w i t h n o n l i n e a r e v o l u t i o n equations ( c f . [ B i r l ; N t l - 3 1 f o r example). 6. QUANenm FtELD eHE0R&J. We w i l l s k e t c h here some o f t h e machinery which
a r i s e s i n quantum f i e l d t h e o r y and t h e s t u d y o f gauge f i e l d s .
No a t t e m p t i s
made t o d e s c r i b e t h e p h y s i c s b u t i t i s p o s s i b l e t o g e t an i d e a o f wh,at i s going on m a t h e m a t i c a l l y by s i m p l y w r i t i n g o u t t h e equations, v a r i a t i o n a l principles, etc.
F o r r e f e r e n c e s we mention [ L l ;Nkl ;Gul ;Hal ;J a l ;G11 ; C i 1 ;L p l ;
J k l ;H t l ;Bql ;Sxl ;F1 ;Fa3; F y l ;I t 1 ;Sul ;Re1 ; T t l ;Q1;Cgl ;A1 1 ;By1 ; D j l ;Fa4;Fdl ;Gxl ;
A number o f t h e mathematical problems i n v o l v e e.g. a n a l y s i s
B l e l ;Mdl ; R j l ] .
o f c r i t i c a l Sobolev i n d i c e s where standard compactness r e s u l t s f a i l ( c f . [Tbl,2;Ul]),
o r t h e s t u d y o f t o p o l o g i c a l - g e o m e t r i c questions, e t c . which
can be e a s i l y understood a t l e a s t , even i f s o l u t i o n s t o t h e problems a r e d i f f i c u l t ( i f available).
We f e e l t h a t enough background d i s c u s s i o n o f v a r -
i a t i o n a l ideas, Lagrange and Hamilton equations,
e t c . has been g i v e n a l r e a d y
so t h a t we can s i m p l y w r i t e down Lagrangians, a c t i o n i n t e g r a l s , e t c . , i n f l a t o r curved spaces, w i t h o u t a l o t o f formal j u s t i f i c a t i o n .
Geometric
ideas such as connections, c u r v a t u r e , e t c . and d i f f e r e n t i a l forms a r e a l s o s i m p l y used as needed; t h e d e f i n i t i o n s a r e e i t h e r g i v e n i n t h e t e x t o r can be found i n Appendix C.
Although t h e p h y s i c i s t s approach t o mathematics may
seem c a v a l i e r a t times (and o c c a s i o n a l l y l e a d s t o e r r o r ) s t i l l t h e exposure t o t h i s k i n d o f c r e a t i v e h e u r i s t i c mathematical t h i n k i n g i s w e l l w o r t h an e r r o r o r two; one can w o r r y about r i g o r once t h e r e i s some substance w o r t h w o r r y i n g about. L e t us r e c a l l from § § 1 . 7 - 1 . 8 t h e c l a s s i c a l f c r m a l i s m ( s e t h e r e I H , F I = -(H, F) =
1 (aH/api)(aF/aqi) -
IH,pkI and Dtqk = IH,qk},
(aH/aqi)(aF/api)
f o r t h e Poisson b r a c k e t ) Dtpk =
a l o n g w i t h t h e o p e r a t o r form h d A / d t = i [ H , A ( t ) ] ,
A(0) = A (we w i l l t a k e h = 1 and/or c = 1 a t t i m e s h e r e ) . procedure r e q u i r e s a l s o [p,q]
= -i (Q I p , q l = 1 ) and
The q u a n t i z a t i o n
iJ't = HJ/ f o r s t a t e s J,
i D w h i l e pt = i[H,p] and qt = i[H,q]). ( r e c a l l as o p e r a t o r s p Q, - i D and q q P We r e c a l l a l s o Example 1.8.3 where t h e q u a n t i z a t i o n procedure f o r a harmonic Q,
o s c i l l a t o r leads n a t u r a l l y t o c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s . a f i e l d t h e o r y one r e p l a c e s t h e sum
1 by
Now i n
an i n t e g r a l , j u s t as i n t h e case o f
continuous e l a s t i c s,ystems i n t h e c a l c u l u s o f v a r i a t i o n s , o n l y now t h e r e i s an a d d i t i o n a l q u a n t i z a t i o n i n v o l v e d .
Thus e.g.
f o r a f r e e Klein-Gordon (K-
287
QUANTUM FIELD THEORY
G)
f i e l d one has
2 2 (Dt-d+m )IP= 0 c o r r e s p o n d i n g t o a Lagrangian L0 w i t h den-
s i t y to ( c f . §1.10) (*)
C,(v,av)
( h e r e x'
D ~ D ~DU,
%
= (ct,x),
x
2
= (ct,-x),
= (i3j =~ (a/ax,), )
(a/a(ct),-a/ax)
2
-
= ( 1 / 2 ) [ 1 D'qDUlp
Lo(lp,a9)
gii
(1,-1,-1,-1),
'L
D~
2 2 m IP 3 w i t h Lo =
d3x
'=
u = DUD o r
(a/a(ct),a/ax)
%
f
ru
1
(ap)
=
1
(a/ax'), = ( l / c I D t - A, g gij = 6ij ( h e r e gA' = gA,), = gpvxVy etc. cf. [Lll]). R e c a l l a l s o t h a t one r e f e r s t o v e c t o r s vJ as c o n t r a v a r k b i jJ, i a n t and v j = gjkv as c o v a r i a n t , l e n g t h s a r e 1 [ g . .x x ] ds f o r a c u r v e
-
'
1
1 g'"(af/ax ) eIJ I J d f s ( a f / a ( c t ) e o i z1 ( c f . 51.10 and Appendix C - 9:, TxM TEM, g = 1 g..dx Id x j , i j i 1J gx(v)w = gx(v,w), 1 v Dx 1 g . . v dx .(gi = gji) o r 9:, vi 1 gijvJ = viy g;:' w = c widxi 1 wJDx =l J w P, w 1 = giJwj, and i n p a r t i c u l a r 1 D v f dx" 1 (1 g'"Dvf)DP as i n d i c a t e d ) . One sometimes w r i t e s 6q = 1 D!Jqdx' and
x ( s ) (sum on repeated i n d i c e s ) , D f =
3 (af/axi)ei
%
-f
*
1
-+
-+
*
-+
-+
( c f . 51.10) i s t h e g r a d i e n t o p e r a t o r on f u n c t i o n s t o 1-forms
we see t h a t D
U
( c f . §1.10) i s t h e g r a d i e n t from f u n c t i o n s t o t a n g e o t v e c t o r s . Note 3 2 p,p' = p: - C1 pi, and n o t a t i o n a l l y we can w r i t e e.g. D' =
w h i l e 0'
1
a l s o t h a t e.g.
1 gU"Dv also
9
= (l/c2)D2 - A. I n p h y s i c s one w r i t e s ' t It w i l l be c o n v e n i e n t t o t a k e c 1 w i t h xo
= gxDP so t h a t D DY = D'D
,i
= Diq
and q'
= t and x = ( t , x )
i 'I = D q.
=
( t h e r e s h o u l d be no c o n f u s i o n here i n d i s t i n g u i s h i n g 3-D
o r 4-0 x ) f r o m now on; a l s o t a k e h = 1 and t h e a c t i o n f u n c t i o n a l w i l l be (A)
4
S = 1 L d t = J t d x.
t i o n s o f t h e f i e l d as
A now s t a n d a r d v a r i a t i o n a l argument g i v e s t h e equa-
( 0 )
( a / a x ' ) [ a ~ / a ( D ~ p ) ] = aL/aq and one d e f i n e s a con-
Then TI ( t o q ) by ( & ) n ( x , t ) = a L ( q , a q ) / a + where i 'L Dtq. H = 1 H(n,q)d3x where H = nq - c i s a H a m i l t c n i a n d e n s i t y . I f t h e r e
j u g a t e momentum (0)
a r e a number o f f i e l d s q , one sums o r i n t e g r a t e s o v e r a and na = a C ( q , a q , ) / a(aoqff).
A s t a n d a r d n o t a t i o n a l s o expresses
at/a
(aoqa)as s L / s ( a O q a ) ( c f
T h i s i s formal so f a r and one must t a k e i n t o account e.g.,
83.2).
orderings, l o c a l properties, etc.
operator
The q u a n t i z a t i o n s t e p i s e a s i l y expressed
f o r m a l l y v i a (m) [q(x,t),lp(x',t)l = 0; [ T ( x , t ) , n ( x ' , t ) l = 0; [ n ( x , t ) , q ( x ' , t ) ] = - i 6 3 ( x - x ' ) where fi3 i s a D i r a c measure i n R 3 and t h e dynamical equat i o n s a r e f o r m a l l y (**) i,(x,t)
= i[H,qff]
= na and ;,(x,t)
It
= i[H,n,].
seems t o be e a s i e r and more i n s t r u c t i v e f o r m a l l y (and u s e f u l as i n 53.2) t o t h i n k o f Poisson b r a c k e t s however and w r i t e (e.g.
aL/aGff = ac/a(aoq,), ~ ~ ) ( a c / a ~ , ) l I, , 1 i C , 1, e t c . ) t,
= 6~/6+,
-+
{n,(x.t).n,(y,t)} aH/aT ; CL
{F,P;j =
=
(*A)
1 [(aF/ana)(aP;/aq,)
{n,(xyt),q,(y,t)l
= { ~ ~ ( ~ , t ) ~ q , ( Y , t =) l 0; aoq,(xlt)
a 0n, ( x , t ) = {H,T,} = -aH/aqa. TI = a c o / a i = i and Ho = ml,
(*) we have
-
3
L = J Ld x, H =
3
1 n,ia - (aF/a
= 6 (x-Y);
= {HYq,l = For t h e K-G f i e l d w i t h to g i v e n by 2 2 2 co = ( 1 / 2 ) [ n + I v 3 v I 2 + m IP 1. I n = n,(X¶t)
2 p a r t i c u l a r aHo/an = n and aHo/aq = m 9 ; one has a L0/ a q = -aHo/aq = Dtn =
288
2 -m q
ROBERT CARROLL
+ 1,3 Diq2
and Dtq = -aHo/an =
so D2q = -m 2v + h p .
IT
t
I n a d d i t i o n t o t h e K-G Lagrangian to o f (*) l e t us mention ex-
REmARK 8.1.
p l i c i t l y a l s o t h e q 4 s e l f i n t e r a c t i n g t h e o r y based on t = to - (A/4!)v
t
t h e Sine-Gordon (S-G) Lagrangian ( c f . 552.9-2.11)
A)[Cos(JAq/m) R m R K 8.2.
-
and
+ (m4/
1 1 ( t h i s l a t t e r 1; l e a d s t o a good f i e l d t h e o r y i n 2-0, n o t 4 ) .
L e t us i n d i c a t e a s t a n d a r d q u a n t i z a t i o n o f t h e K-G f i e l d v i a
F o u r i e r t h e o r y ( c f . a l s o 51.8). form (nq +
= ( 1 / 2 ) 1 D’qD’q
4
m2v
=
Thus w r i t e a H e r m i t i a n s c a l a r f i e l d i n t h e 3
0 ) v ( x , t ) = / d k [ a ( k ) f k ( x ) + a*(k)f;:x)]
e x p ( - i k . x ) / [ ( 2 1 ~ ) ~ 2 ~ , ] ’ , wk = (k2+m2)’, t c a l l e d a i n physics, ko w k Y and x Q
and k - x = w k t %
-
where f k ( x ) =
( k , x ) (a* i s u s u a l l y
- t h e n o t a t i o n i s standard, as
(t,x)
a r e o t h e r s , and we w i l l t r y t o be c l e a r and c o n s i s t e n t i n o u r use o f sym-
- however we w i l l o c c a s i o n a l l y use d i f f e r e n t n o t a t i o n s ) . One w r i t e s a(t)L$b(t) = abt - a b and i t f o l l o w s t h a t ( e x e r c i s e - c f . [ B q l ; I t l ] ) (*@) a ( k ) = i/ d3xf;(x,t)a Lq ( x , t ) ( n o t e / f;(x,t)ia:fk(x,t)d 3x -- 63 ( k - k ) and bols
3O
-
/ f k ( x , t ) i g f (x,t)d x = 0 a ( k ) i s t i m e independent). Then e.g. [a(k),a* ( k ) ] = / / ogk d xd 3y[fi(xYt)a”dp(xYt)Yfk(YYt)a:~(YYt)] = i/ d 3 X f { ( X ~ t ) c f k ( X , t ) = 63(k-k) and s i m i l a r l y [ a ( k ) , a ( k ) ]
= [a*(k),a*(k)]
a r e o p e r a t o r s and t h e f k a r e f u n c t i o n m u l t i p l i e r s
= 0 ( t h e a ( k ) and v ( x , t )
-
one uses h e r e ( m ) s i n c e 2 2 a 0v = IT). I n t h i s t e r m i n o l o g y we have a l s o ( H = ( 1 / 2 ) ( n 2 + 103v12 + m v ) ) 3 3 I t i s i n s t r u c t i v e here t o d i s (*6) H = / Hd x = ( 1 / 2 ) / d kwk[a*a + aa*]. C r e t i Z e / d 3k
AVky 63 (k-;)
Q
(1/2)wk(aiak + aka;),
m
1 Hky Wk
= (Ikl2+m2)’,
= &k$, e t c . ( c f .
[ai,a;]
Hk
=
Example 1.8.3
One t h i n k s o f energy e i g e n f u n c t i o n s as pro-
The ground s t a t e i s
1 wk/2 (which w i l l be fuss - c f . [ B q l ; I t l ; G l l ] ) .
energy E =
without a l o t o f
H =
v k ( n k ) , Hkvk = wk(nk++)vky and v k = (l/nk!)’(a*)nkqk
%
( 0 ) (nk = O y l y 2,...). i t has
Gk;/AVk,
ak = JAVka(k),
f o r a p a r a l l e l development). d u c t s ‘IPk where v k
Q
vo
‘IPk(0) w i t h a k p k ( 0 ) = 0 and
s i m p l y removed f r o m t h e t h e o r y We do n o t g i v e h e r e a l o n g d i s -
cussion b u t simply r e c o r d some formulas i n d i c a t i n g what i s g o i n g on.
Thus
f i r s t ( f o r general background) v i a c o n s i d e r a t i o n s o f symmetry and conservat i o n laws ( c f . [ B q l ; G l l ; I t l ] ) one has an energy-momentum v e c t o r (H,P) where 3 3 P Q - / 1 ~ 0 ~ x9 d ( 1 / 2 ) / d k k(a*a+aa*) Q ( 1 / 2 ) k(aCak + aka;) here w i t h
1
Q
P kv k (,kn ) = k ( n k f 4 ) q k ( n k ) e t c . a n d , s e t t i n g N~ = a;akywith =
1k
Nk ( P = (P’),
1 wk/2 one 1 wk/2 = 1 wka;ak
o f energy
H
-
k = (k’)).
Nkqk = nkqk and P’
Now t o remove t h e vacuum e x p e c t a t i o n v a l u e
s t i p u l a t e s a z e r o energy ground s t a t e w i t h = (1/2):aLak
+ aka;:
(*+I H’ =
where : : denotes t h e so c a l l e d
Wick o r d e r i n g where a n n i h i l a t i o n o p e r a t o r s ak appear t o t h e r i g h t o f c r e a t i o n o p e r a t o r s a;
(see [ G l l ]
f o r a good d i s c u s s i o n o f t h i s
- g e n e r a l l y if
QUANTUM FIELD THEORY
Q = ( A t A*)/42 one has :Qn: = 2-”‘1
289
(;)A*jAn-J).
Note h e r e t h a t wkagakvk
= Wkakgk and observe t h a t c l a s s i c a l l y , where t h e commutators a r e a l l zero,
t h e r e i s no z e r o p o i n t energy.
H
c o n t i n u o u s case v i a
=
This n o t a t i o n i s then t r a n s f e r r e d t o t h e
J d3 k w k a * ( k l a l k ) e t c .
Somewhat more g e n e r a l l y i f
one wants t o deal w i t h p a r t i c l e s and a n t i p a r t i c l e s a f i e l d i s used (charged s c a l a r f i e l d
3 d X : S * I I + vv*vv +
w*v:
- v1
where n =
v 2 as i n
and
v*
appropriate quantization.
’ (F”)
O
=
( r e c a l l i n g E = -vv
o
(F’”
- At, B
=
v
X E = -B
- aAV/axu).
= aA!’/axv
= -1 ( i
-
E Ox By EX -Bz 0‘
v E
t’
row, w = column)
0, v X B = Et,
=
Dp(DwAv) = 0 and D‘F’”
Note f o r gij
2 l ) ,one w r i t e s Fuv
(FL) =
(8.2)
(11 =
E
p a r t i c u l a r (*.) aFuv/axv = 0 o r nA’ =
(cf. [Bql;Itl;Gll]).
E E ’ - E Ox By -BZ -Ex -B Ox By -E: B i -Bx Ox
i
F’”
=
We r e c a l l f i r s t t h e c l a s s i c a l f r e e Maxwell equa-
t i o n s ( c f . Example 1.2.11 and §1.10) and w r i t e
and gii
H
We go n e x t t o t h e e l e c t r o m a g n e t i c f i e l d and w i l l i n d i c a t e t h e
REmRRK 8.3.
(8.1)
= (vl+iv2)/J2
say) w i t h e.g.
(*A)
= (v1-iv2)/J2
v
= gij
etc.)
In
+ DV
t D’F”’
= 0 (i= j ) , goo = 1,
and FPv = -gVaFg w i t h
= gu,Ft
-iy
We w i l l phrase a l l t h i s below i n terms o f d i f f e r e n t i a l forms, c o n n e c t i o n s , c u r v a t u r e , e t c . b u t i t w i l l be i n s t r u c t i v e t o have v a r i o u s p o i n t s o f view. 2 Now t h e c l a s s i c a l Lagrangian 1; = -(1/4)FpvFpv = ( 1 / 2 ) ( E 2 - B ) ( E = I E I , etc.) leads t o tion.
ac/aio
IT =’
= 0 and t h i s i s n o t s a t i s f a c t o r y f o r q u a n t i z a -
=
v
’ = -A’
( n o t e (A’)
%
(Ao,-A)
Du
while
t h e n a s u i t a b l e Lagrangian d e n s i t y i s aL/ai
’w i t h x 0 ( n o( tt,ex ) )
I f one uses a L o r e n t z gauge as i n Example 1.2.11 so D A’
t h i s means d i v A + v t = 0 s i n c e A’
(A*)
and (A’)
’ -(l/Z)(D =
a
= a/ax’
c
=
‘L
(Ao,A)).
=
=
A )(DVAp) w i t h
V ’
IT’ =
We r e f e r h e r e t o Re-
mark 9.3 f o r f u r t h e r d i s c u s s i o n ) . L e t us now p u t t h e Maxwell t h e o r y i n t h e language o f d i f f e r e n -
RfiltARK 8.4.
t i a l forms ( c f . Appendix C ) . (again c = 1 )
(AA)
Thus f o l l o w i n g [Gul;Wtl;Fsl;Cul]
one w r i t e s
B dz A dx - Bzdx A dy + Exdt A dx + Y Then t h e equations V X E = -Bt and 0.B = 0 a r e e q u i -
F = -Bxdy A dz
-
E d t A dy + E,dt A dz. Y v a l e n t t o dF = 0. The equations v X B
Et and v.E = 0 ( i n t h e absence o f
c u r r e n t s and charges) can be w r i t t e n as 6 F = 0 ( o r 6 F = -u0J when v X B =
Et +
poJ).
We r e c a l l h e r e t h a t 6 = *d* where
(Am)
*(dx A dy) = -dz A d t ,
290
ROBERT CARROLL
*(dy A dz)
-dx A d t , *(dx A dz) = -dy A d t , *(dx A d t ) = dy A dz, *(dy A d t ) = -dx A dz, and *(dz A d t ) = dx A dy. The L o r e n t z gauge w i t h d i v A t qt : :
Since dF = 0 we w r i t e ( a t l e a s t l o c a l l y ) F = dA
= 0 l o o k s now as f o l l o w s .
( A = a 1-form) and any A t d f = A ' g i v e s t h e same F. dS which i s
IY
( n o t e d i v S = -65, 5 =
Lorentz gauge i n v o l v e s 6A' = 0 o r 6 A have SdA' = -AA' =
nA' =
= 0 and coupled w i t h SF = 0 we
Note a l s o t h a t f o r A ' =
0.
1 AidxHone
has S A ' =
We w i l l see l a t e r i n c o n n e c t i o n w i t h g e o m e t r i c quan-
as d e s i r e d .
-D,,[A')'
1 viei, - Af
R e c a l l now -A = Sd t i 5 = 1 vidx , e t c . ) . Then t h e
t i z a t i o n how one deals w i t h A as a c o n n e c t i o n and F as a c u r v a t u r e i n , s u i t a b l e f i b r e bundles ( c f . a l s o [Gul;G11;Blel;Cul;Tr1;Sxl]).
REII~ARK 8.5, As a n o t h e r i n t r o d u c t o r y p o i n t o f view we want now t o develop a l i t t l e t h e use o f a c t i o n i n t e g r a l s and t h e Feynman p a t h i n t e g r a l . T h i s p o i n t o f view has become fundamental i n many i n v e s t i g a t i o n s and t h e r e a r e many a p p l i c a t i o n s ( c f . [Bql ;Nkl ;J a l ;B11 ;F1 ;Fyl ;Sul ; A l l ;L1 ;Re1 ;Mml ;Lcl ,2]). We w i l l g i v e f i r s t a p a r t i a l l y h e u r i s t i c d e r i v a t i o n f o l l o w i n g [Fyl;Sul;Rel]. 2 2 2 Thus s t a r t w i t h ih!bt = W w i t h H = TtV = p /2m + V -h Dx/2m t V(x). We
-
t h i n k o f a propagator o r Green's f u n c t i o n G s a t i s f y i n g as an o p e r a t o r (H ihDt)G(t,to)
o r i n c o o r d i n a t e f o r m (Hx
= -itiS(t-to)I
-ihS(x-y)6(t-to).
-
ihDt)G(x,t,y,tO)
One w r i t e s i n s t a n d a r d n o t a t i o n G(x,t,y,to)
G(t,to) = o(t-to)exp[-iH(t-to)/h]
= (xlG(t,to)l
F o r m a l l y o f course
y ) and t h e s t a t e s a r e t o e v o l v e v i a # ( t ) = G(t,to)!b(to). (46)
=
where 0 i s a Heavyside f u n c t i o n . Now
it/n
f o r m a l l y and h e u r i s t i c a l l y ( f o l l o w i n g [ S u l ] ) we w r i t e X =
.
and s e t G(x,
t,y) = (xlexp[-h(T+V)/N]. .exp[-h(T+V)/N]ly) w i t h exp[-X(T+V)/N = exp(-XT/N) 2 2 2 2 exp(-XV/N) + O ( A /N ). One assumes t h a t t h e O ( X /N ) t e r m i s w e l l behaved
when a p p l i e d t o s t a t e s , etc.,
and f o r reasonable p o t e n t i a l s t h i s can be j u s N Next one wants t o r e p l a c e t h e t e r m (A+) (exp[-h(TtV)/N]) by [exp
tified.
(-XT/N)exp(-XV/N)]
N
and i n t h i s d i r e c t i o n one notes t h a t [l + (x+yn)/nIn
e x p ( x ) as l o n g as yn (A+)
-f
one can w r i t e e.g.
(-XT/N)exp(-XV/N)
-
0 when n
-f
-
(exercise
- c f . [Sul]).
-+
To work w i t h
[ e ~ p ( - X T / N ) e x p ( - h V / N ) ] ~ - [exp(-X(TtV)/NIN = [exp
e~p(-x(T+V)/N)][exp(-x(T+V)/N)]~-~ t
... t
[exp(-XT/N)
exp(-XV/N)IN-' [exp(-xT/N)exp(-XV/N) - exp[-X(T+V)/N]] and t h e N [ ] terms 2 2 a r e O(X /N ) . With some r e f i n e m e n t t h i s l i n e o f reasoning can be developed t o prove t h e T r o t t e r p r o d u c t f o r m u l a ( c f . Appendix B f o r semigroups and see
- c f . a l s o 53.6).
[Sul;Dwl;Pz]
f o r proof
tHE@REIn 8.6
L e t A and B be l i n e a r o p e r a t o r s i n a B space E such t h a t A, B,
exp(At), Qt and A+B a r e generators o f c o n t r a c t i o n semigroups Pt t - lim t / n t / n nxx. exp[(A+B)t]. Then f o r x E E,R x - n- ( P Q ) Q
and Rt
Q
Q
exp(Bt),
QUANTUM FIELD THEORY From t h i s one w r i t e s t h e n G(x,t,y)
= lim
291
( X I [exp(-xT/N)exp(-xV/N)]
N I y ) and
= I between terms ( j = l,...,N-1) one o b t a i n s ( c f . j §§2.3-2.4) ( A m ) G(x,t,y) = l i m 1 dxl...dXN-lf-l ( xjtl lexp(-xT/N)exp(-xV/N) I x . ) (xo = y and x = xN). Now V i s a m u l t i p l i c a t i o n o p e r a t o r so exp(-xV/N)I J x . ) = I x . ) exp(-xV/N) and t o t r e a t t h e o p e r a t o r exp(-xT/N) one i n s e r t s momJ J entum s t a t e s 1 = 1 d p l p ) ( p I w i t h ( P I E ) (2rh)-'exp(-ipg/h) so t h a t ( n l e x p ( - x T / N ) I c ) = J dp( n l e x p ( - x T / N ) l p ) ( P I S ) = (1/2nh)/: dpexp(-xp 2/2mN)exp[ip(n-
inserting 1 dx.Ix.)(x
J
J
The i n t e g r a l s can be e v a l u a t e d v i a /: exp(-ay 2+by)dy = J(n/a)exp(b 2/ 2a) ( e x e r c i s e ) and hence ( nlexp(-xT/N) 15) = (mN/2aAh 2 )4exp[-mN(n-E) 2/2Ah2].
c)/h].
P u t t i n g a l l t h i s together then y i e l d s (8.3)
G(x,t,y)
m(x.tl-x.)
,+
= 1i m
Letting ( E i = exp[-XV(xj)/N]). t h e p a t h i n t e g r a l f o r G as
(Aj"
= xy,-xj).
2 ) N/2nN-1e-[
dxl...dxN-l(mN/2sxh
3
Now t h e [
E
2hh2
2
N
lEJ N
= t / N = t i x / i N and r e o r g a n i z i n g one o b t a i n s
t e r m i s an a p p r o x i m a t i o n t o t h e Riemann i n -
t e g r a l 10 d ~ [ ( 1 / 2 ) m X ~- V ( x ) ] d r o v e r a p a t h
X(T)
t
j o i n i n g y = x 0 and x = x N One t h i n k s o f sumning
and o f course t h i s r e p r e s e n t s an a c t i o n S = Jo LdT.
o v e r a l l p o s s i b l e ( c o n t i n u o u s ) broken l i n e paths c o n n e c t i n g y and x; hence by a p p r o x i m a t i o n one i n t e r p e r t s t h e i n t e g r a l (@*) G(x,t,y) (-))/h]
= c
1 exp[iS(x
as a sum o v e r a l l p o s s i b l e c o n t i n u o u s paths x j o i n i n g y and x ( h e r e
c = ( r n / 2 a i f i ~ ) ~+/ ~m as N rigorous theory).
-f
m
which was one o f t h e problems i n d e v e l o p i n g a
One extends t h e s e c o n s i d e r a t i o n s t o 3-D and t o a Lagran-
- V(x) i n a more o r l e s s s t r a i g h t f o r w a r d way, e x c e p t t h a t t h e v e c t o r c o n t r i b u t i o n ( i e / n c ) 1(xjtl - x . ) A ( x ) must i n v o l v e
g i a n L = ( 1 / 2 ) m l i I 2 + (e/c);-A
e v a l u a t i o n o f A(x) a t x = ( 1 / 2 ) ( x . + x ) o r e.g. ( 1 / 2 ) f A ( x j ) + A(xjtl)] must J j+l be used. T h i s f e a t u r e i s standard i n t h e t h e o r y o f Brownian m o t i o n and t h e I t o i n t e g r a l and w i l l n o t be discussed h e r e ( c f . [Sul;Wgl]). [Sul ;Fyl ; G l l ;Nkl ;Bql ; I t 1 ;Mml ;Lcl,2]
We r e f e r t o
f o r t h e p h i l o s o p h y o f t h e Feynman i n t e -
g r a l i n c o n n e c t i o n w i t h d i f f r a c t i o n , quantum mechanical paths, p r o b a b i l i t y , e t c . and remark h e r e o n l y t h a t s t a t i o n a r y phase arguments l e a d one t o conc l u d e t h a t t h e main c o n t r i b u t i o n t o t h e i n t e g r a l occurs when 6 s 6x = 0 and t h i s leads one t o e v a l u a t i o n o v e r t h e c l a s s i c a l p a t h determined by t h e Lagrange equations.
The i n t e g r a l (@*) i s o f t e n w r i t t e n as
1 exp[iS(x)/h]D(x)
where t h e magic D ( x ) i s used s y m b o l i c a l l y as some s o r t o f
" t h i n g " i n p a t h space.
(@A)
G x,t,y)
=
The r e a d e r s h o u l d n o t e t h a t we have f a r t o o much
2 92
ROBERT CARROLL
r e s p e c t f o r t h e p a t h i n t e g r a l and i t s consequences t o be mocking about D ( x ) b u t i t s use and i n t e r p e r t a t i o n a r e s t i l l perhaps i n an e x p l o r a t o r y stage; f o r a good t r e a t m e n t o f t h e f u n c t i o n a l i n t e g r a t i o n p o i n t o f v i e w see e.g. The mathematical p h i l o s o p h y has i n v o l v e d i n s e r t i n g convergence
[ G l l ;Mml].
f a c t o r s , w o r k i n g t h r o u g h imaginary t i m e
and u s i n g a n a l y t i c c o n t i n u a t i o n ,
e t c . and t h e p a t h i n t e g r a l may be one o f t h e most i m p o r t a n t and p r o d u c t i v e n o n r i g o r o u s o b j e c t s y e t conceived i n mathematical physics; however c o n s i d e r ,2]). a b l e r i g o r can be p r o v i d e d ( c f . [Gll;Mml;Lc1,2;All 2 -2 one w r i t e s T -t - i r i n (.*) w i t h h = 1, j , + - x , and t formally
(0.)
t V(x))ds].
In particular if -f
- i t t h e n one has
n dx(s)exp[-J; (mi2/2 G(x,-it,y) = ( x l e x p ( - t H ) l y ) = J W(Y 9x9 t 1 t 2 Here t h e t e r m exp[-Jo m i ds/2] w i l l serve as a convergence f a c -
t o r and w i t h some r e o r g a n i z a t i o n , u s i n g t h e T r o t t e r p r o d u c t formula again, one o b t a i n s t h e Feynman-Kac formula ( 0 6 ) ( x l e x p ( - t H ) l y ) = J exp[-Jot V(x)ds] dWt
YX
where dWt
YX
denotes c o n d i t i o n a l Wiener measure i n t h e p a t h space W(y,x,
t ) (see h e r e [ G l l ]
REmARK 8-7.
f o r details).
L e t us now see how t h i s approach l o o k s i n f i e l d t h e o r y ( c f . Thus t a k e S(tl,t2,[v])
[Fa4;Rel;Nkl;Itl;NxlI).
=
c2
d4xI;(~,av) where [v]
c o u l d be any c o l l e c t i o n o f l o c a l f i e l d s and we f o l l o w [Rel] i n o r d e r t o g i v e 4 a s k e t c h o f what i s g o i n g on. F i r s t as i n ( 0 ) e t c . (@+) 6 s = 1 d x[aL/av -
+ ID dop(ac/a(apv))sv where Is i s a s u i t a b l e s u r f a c e i n t e -
ap[ar/a(apv)]]6v gral.
Given
[ar/a(apv)]
6q =
0 on u one o b t a i n s t h e E u l e r equations
(a=)
aL/av
-
a
’
One s h o u l d remark h e r e t h a t v a r i o u s c o n s e r v a t i o n
= 6 S / S q = 0.
laws a r i s e from t h e i n v a r i a n c e o f L under L o r e n t z t r a n s f o r m a t i o n s + t r a n s l a t i o n s ( t h e Poincar;
group); a l t e r n a t i v e l y one s t i p u l a t e s t h a t t h e a c t i o n be
i n v a r i a n t under such t r a n s f o r m a t i o n s o f c o o r d i n a t e s and f i e l d s .
Thus c o o r -
denotes a L o r e n t z t r a n s f o r m a t i o n ) ’ 2 6x’a f up t o o r d e r (Sx) w h i l e f u n c t i o n s change a c c o r d i n g t o ’ ( h e r e s,f denotes t h e f u n c t i o n a l change a t x - i . e . t h e form o f f i s p o s s i b d i n a t e changes a r e x’
-f
a’
+ Atx”
(A’
6f =
l y c o o r d i n a t e o r frame dependent).
4 4 Then 6(d x ) = d xJ6x’ P
’
’
t
( J a c o b i a n ) and
st = 6 t + 6x’a 1: = 6xpa 1; + ( a t / a L p ) ~ ~ +9 (at/a(aup))60apv w h i l e s a q = 0 O p4 [ ~ ~ , a ~+ ] av 6 q = a 6 p ( e x e r c i s e ) . We can w r i t e t h e n ( 6 * ) 6s = J d x[SC P O 4 P O 4 + aPsxpc] = 1 d x[capsxp t 6xpa c + a [ ( a ~ / a ( a ~ q ) ) ~= ~ 1~ d~ ]xau[c6xp ] + ( a t / a ( a q ) ) 6 0 v ] (where (-) has been used). T h i s can a l s o be w r i t t e n as 6s = J d4 x p ,[[tg; - ( a ~ / a ( a g ) ) a ~ ~ ] +s x( ~a c / a ( a p ~ ] ( g i = 6;). Ifone
’
’
imagines now some g l o b a l (x-independent) t r a n s f o r m a t i o n fixp = (SxP/Swa)Swa and 6q = ( b / 6 w a ) 6 w a t h e n SS = - J d 4 xa j’6wa where -j: = [tg: - ( a C / a ( a p v ) ) p a apq]6xP/6wa t (aC/a(a v ) ) ( s q / s w a ) i s c a l l e d a c u r r e n t d e n s i t y ; thus when SS
’
QUANTUM FIELD THEORY
293
a j' = 0.
T h i s i s a k i n d o f E. Noether u a theorem f o r c l a s s i c a l f i e l d t h e o r y r e l a t i n g a c o n s e r v a t i o n l a w t o i n v a r i a n c e t o f t h e a c t i o n . I n p a r t i c u l a r ( c f . Remark 1.10.8) c o n s i d e r ((A) . . 0 = L * dxo = 0 f o r a l l 5wa one concludes t h a t
.
t l
Zm d3xa j' = It2 dxoao[z d 3 x j i t /tt'dxo/z d3xaijd The l a s t t e r m w i l l vanP a I i s h i f boundary c o n d i t i o n s a r e s u i i a b l e and hence t h e charges ( 6 0 ) Q"z ( T ) = 3 .o -Jm m d XJ,(T,X) a r e t i m e independent ( i . e . f r o m ((A) Qa(T1) = Q a ( T 2 ) ) . Hence -m
6 s = 0 i m p l i e s e x i s t e n c e o f conserved charges.
It w i l l be h e l p f u l t o i n t r o d u c e a few a d d i t i o n a l ideas.
RfiRARK 8.8.
we c o n s i d e r t h e p a t h i n t e g r a l i n phase space f o l l o w i n g [Sul;Fa4]
First
since i n
p a r t i c u l a r t h i s l e a d s t o many i n t e r e s t i n g r e s u l t s d i r e c t l y and i s f r e q u e n t l y used i n t h e Russian l i t e r a t u r e .
As n o t e d i n [ S u l ] i t i s p o s s i b l e t o make
e r r o r s b u t we w i l l n o t d i s c u s s t h e p h i l o s o p h y o f paths e t c . here. Thus as N b e f o r e one w r i t e s down G(x,t,y) = l i m <XI [exp(-AT/N)exp(-AV/N)] l y > and i n s e r t s now a l t e r n a t i v e l y r e s o l u t i o n s o f t h e i d e n t i t y u s i n g c o o r d i n a t e and = lim
momentum e i g e n s t a t e s t o o b t a i n ( ( 6 ) G(x,t,y)
( p2€/2m) I pN) ( pNI exp(VE) I xN-l) -it/hN, xo = y, and xN = x. (
pi(exp(VE)lxi-l)
. . .( x1 I exp( p 2 ~ / 2 m )I pl) Now
(
1 [$-'dp (
xi/exp(p2€/2m)lpi)=
= exp(V(xi-l)~)(pilxi-l)
dx.]dp& x l e x p P ' p1 I exp(Vc) ly) where E = e x p ( p : ~ / 2 m ) ( x ~ / p ~ and )
w h i l e ( x l p ) = (21~fi)-'exp(ipx/li)
It f o l l o w s t h a t gives a 6 function normalization J d x ( p l d ( x 1 p ' ) = 6(p-p'). im N-'dpid~i]dpN(2rrh)-Nexp[( i / f i ) J o t ( p i H)d.r] where ( b e ) G(x,t,y) % Jhno 2 1; ( p i - H1d.r % ( t / N ) C o [ ~ ~ ( x ~ - x ~ - ~ ) / ( t pj/2m / N ) - V(xj_,)l.
-
RRRARK 8.9, S(t,T,x)
(S =
t u d e yT + xt.
= (xt1yT) = J D(x)exp(i/h) L e t us w r i t e (@A) as (6.) G(x,t,y,T) t Cd-r) where ( x t l y T ) i s w r i t t e n t o denote t r a n s i t i o n a m p l i -
/T
One expects o f course by c o n s t r u c t i o n t h a t (+*) G(x,t,y,T)
jm G(x,t,z,~)G(z,r,y,T)dz -m
and $ ( x , t )
press G v i a Remark 8.8 as G(x,t,y,T)
=
Z l
G(x,t,y.T)$(y,T)dy.
=
We a l s o ext
= J [dpdq/2~]exp[(i/h)/T
(pG-H)d.r]
(sometimes h e r e D ( x ) i s w r i t t e n as ~ [ d q ] w i t h an u n s p e c i f i e d n o r m a l i z i n g constant
K
and sometimes s i m p l y as [dq]).
I n f i e l d t h e o r y one i s concerned
w i t h c a l c u l a t i n g Green's f u n c t i o n s d e f i n e d by G(x l,...,xn)
= (OITv(xl)
q ( x n ) l 0 ) f r o m which S m a t r i x elements can be o b t a i n e d ( c f . 53.9).
...
Here T
denotes a t i m e o r d e r e d p r o d u c t where t h e f a c t o r s a r e o r d e r e d so t h a t l a t e r times s t a n d t o t h e l e f t o f e a r l i e r times.
We w i l l n o t d w e l l h e r e on t h e
p h i l o s o p h y o f p e r t u r b a t i o n t h e o r y , t i m e o r d e r e d products, Feynman diagrams, Feynman propagators, e t c . ( c f . [It l ;G11; F y l ;Nsl ;Bql ;Cgl ;L1; Fa4;Ael ;Bog1 b u t we w i l l e x h i b i t some i n t e r e s t i n g formulas and p o i n t s o f view.
1)
L e t us
f o l l o w [Cgl] here t o s k e t c h t h e c o n s t r u c t i o n o f a g e n e r a t i n g f u n c t i o n W(J) f o r such Green's f u n c t i o n s ( c f . a l s o [ A e l ] ) .
Thus r e c a l l f i r s t ( c f . Remark
2 94
ROBERT CARROLL
1.8.6) t h e Heisenberg dynamical v a r i a b l e s w i t h t i m e independent s t a t e vect o r s l a ) and t h e Schrodinger t i m e dependent s t a t e v e c t o r s l a , t ) ' = e x p ( - i t H ) l a ) (TI = 1 ) . Consider e.g. G(tl,t2) = (OITQH ( t , ) Q H ( t 2 ) 1 0 ) = I dqdq'(01 q ' , t ' ) ( q ' , t ' ( T Q H (t,)Q H( t 2 ) 1 q , t 1 0 ) where ( O l q , t ) = v o ( q , t )
i s t h e ground s t a t e w a v e f u n c t i o n (and QH
%
= vo(q)exp(-iEot)
Heisenberg r e p r e s e n t a t i o n o f Q
2,
S
q - i . e . Q I q ) = 914)). Now f o r t ' > tl > t2 > t we can w r i t e (.A) (q',t'l T QH ( t l ) Q H ( t 2 ) l q , t ) = ( q'lexp(-iH(t'-tl))QSexp(-iH(tl-t2))Q Se x p ( - i H ( t 2 - t ) ) l q 2 ) = J (q'Iexp(-iH(t'-tl))(q,)(ql
I Q s e x p ( - i H ( t l - t 2 ) ) l q 2 ) ( q 2 ( q Se x p ( - i H ( t 2 -
(QH = e x p ( i H t ) Q S e x p ( - i h t ) o r A ( t ) = U*AU i n Remark 1.8.4). Then u s i n g t h e p a t h i n t e g r a l we o b t a i n (one t h i n k s o f tl and t2 as break p o i n t s t))1q)dqldq2
i n the paths) [pt
- H(t,q)]]
(
q',t'lTQH(tl)QH(t2)lq,t)
=
I [ d p d q / 2 ~ ] q ( t ~)q(t2)eXp[iI;'d.r
and e v i d e n t l y t h e same r e s u l t h o l d s f o r t2 > tl.
G(tl,t2) = J [dpds/2.k0(s',t)v~(q,t)q(tl )q(t2)exp[it'd.r[p4 where we have added a " c u r r e n t " Jq t o 1; ( J = 0 o u t o f [T,T']); one can change H
-+
H-Jq.
One can remove t h e
which we r e f e r t o [Cgl;Ael]
Hence ( + a )
- H
+ Jqll equivalently
vo,v,* terms by an argument f o r
and t h e r e r e s u l t s f o r m a l l y (qi = q ( t i ) )
The c o n s t r u c t i o n extends t o G ( t l,...,tn) = ( O I T q ( t l ) ...q( t n ) I O ) and t h e s e c a l l a l l be generated f o r m a l l y v i a
w i t h G ( t l,...,tn) = (-i)'6'W(J)/6J(tl)
...6J(tn)/J,0
( n o t e one takes func-
t i o n a l d e r i v a t i v e s and t h e n l a b e l s tl, t2, e t c . ) .
Here W(J)
'L
(010) ( f o r H
-Jq) and W(0) = 1.
The formula l o o k s somewhat more " r i g o r o u s " i f one goes T,,) 'L in G(-iTl ,..., -kn) i n which case t o imaginary t i m e and w r i t e s S ( T ,..., ~
S ( T1
,. . . ,T,
= lim
.
GnWijE ( J ) / 6 J ( T ) . . .6J ( T~ ) I J= where ( c f [Cgl ;Ae 13 ) (A6 ) GE( J ) 1 [ d q ] e ~ p [ J ~ ~ ' d s ( - m ; ~ / 2- V(q) + J ( s ) q ) ] ( f o r 1: = m42/2 - V where con-
vergence can be e n v i s i o n e d more e a s i l y ) . normalization f a c t o r
4
= (q',t'lq,t)
Here one o m i t s t h e J independent
f r o m (8.6) s i n c e i n f i e l d t h e o r y one i s
o f t e n i n t e r e s t e d i n "connected" Green's f u n c t i o n s where (1/WE(J))6'WE(J)/ 6J(
x = (it,x)). We f i r s t c o n t i n u e t h e d i s c u s s i o n o f 58 a l i t t l e
i n c e r t a i n d i r e c t i o n s i n o r d e r t o have a v a i l a b l e a few more ideas and cons t r u c t i o n s from c l a s s i c a l quantum f i e l d t h e o r y .
GAUGE FIELDS
RmARK 9.1.
295
L e t us f i r s t extend t h e g e n e r a t i n g f u n c t i o n o f Reamrk 8.9 t o a Thus we have f i e l d s v and 4 / d x(na v H + Jv)]
f i e l d s i t u a t i o n f o l l o w i n g [Cgl;Acl;Fa4;Itl;Nsl]. 'TI
say and one d e f i n e s (*) Z(J) = W(J) = 1 [dpdn]exp[i
= 1 [&]exp[i
/d4x(t t &)I.
replacing x
(it,x)
J(x)
u
= (t,x)
(A)
ZE(J) = i E ( J ) = I
We d e f i n e t h e n ( G = G ( ~ l y . . . y x n ) y
(x)]).
0
-
This i s r e l a t e d t o a Euclidean version
I
?lJ
=
[dv]exp(ld4x"[L(v(~)) +
N
r
y
WE = WE(J))
Based on Feynman t h e o r y t h i s i s c a l l e d a connected Green's f u n c t i o n and t h e f o r example. For t h e s t a n d a r d 2 2 i l l u s t r a t i v e example t ( v ) = t o ( v ) + tl(v), cO(v) = ( 1 / 2 ) a A d v - (1/2)m v , and t l ( v ) = -X/4!v 4 one o b t a i n s ( 0 ) ZE(J) = 1 [ d v ] e ~ p [ - / d ~ Z [ ( a ~ v ) ~+/ 2 p h i l o s o p h y i s discussed i n [Cgl ; I t l ; B o g l ; R y l ]
2
+ m2v 2 /2
-
Jv]]. I f we drop t h e E s u b s c r i p t f o r t h e Euclidean s i t u a t i o n and w r i t e Zo(J) = / [dv]expJ d4 x"(Co+Jv) ( n o t e to - ( 1 / 2 ) ( a v ) 2 ( 1 / 2 ) ( v 3 v ) 2 - m2v 2 /2 i n 2 c o o r d i n a t e s ) then f o r m a l l y ( b ) Z ( J ) = e x p [ J0d 4 y (1/2)(V3v)
Q ,
( a l i t t l e s t r a n g e perhaps b u t f o r m a l l y an obvious n o t a t i o n ) . 2 2 2 Replace now - ( a o v ) 2 - (v3v) i n ( 0 ) by v ( a T + v3)v ( d i v e r g e n c e theorem f o r 2 4 - 4s u i t a b l e v , A3 % v 3 ) and w r i t e ( + ) Z,(J) = J [dv]exp[-(1/2)id xd yK(2,y) 4t 1 d zJy] where K(;,?) = ~~(;-y)(-a' - v32 + m2 ) . T h i s can be t r e a t e d as t1(6/6J)1Zo(J)
an
m
dimensional Gaussian i n t e g r a l , namely as a l i m i t i n g f o r m o f (.)
...r h N e x p [ - ( 1 / 2 ) l
viKijvj
1
/ dvl
Jkvk] ( l / d e t K ) e x p [ ( l / Z ) I Ji(K-l)ijJj] as N 4 and one w r i t e s t h e n (**) Zo(J) = e x p [ ( l / 2 ) i d x" 4 Working t h r o u g h momd 4 ~ J ( ~ ) A ( ~ , ~ ) Jwhere ( ~ ) ] / d4yK(2,y")A(7,?) = 6 (x-z). +
Q
( c f . [Fyl;Itl;Cgl;Nsl])
-t
C
I
U
entum space i t can be shown t h a t ( e x e r c i s e - c f . [ F y l ; B q l ; I t l f o r v a r i o u s 4 2 w = / [ d k e x p ( i k . ( X " _ ? ) ) / ( 2 ~ ) ~ ( r+~ m ) ] ( k = ( i k o ,
d e r i v a t i o n s ) (*A) A(;,?)
z0
N
= T = i t and ko = i k o we o b t a i n t h e Feynman propagator (*@) Letting = i A F ( x - y ) = i/ ( d4 k l ( 2 n ) 4 ) e x p [ - i k . ( x - y ) ] / ( k 2 m2 + i E ) ( c f . a l s o
k)). );A -,:(
-
2 t h e i c t e r m corresponds t o a damping f a c t o r i c v / 2 i n t h e Lagran2 g i a n and one can t a k e +k i n t h e e x p o n e n t i a l ) . Note h e r e x - p % t E - x - p , k 2 2 Q k ki, e t c . ( t h e r e l a t i o n s between 3-D and 4-D terms a r e o b v i o u s ) . 552.3-2.4;
0
-1
RRilARK 9.2. equation.
For completeness we s h o u l d say something h e r e about t h e D i r a c Thus ( c f . [ B q l ; I t l ] )
( * b ) i$t = (-ia.v
=
aiak
+ akaio=
t i o n now i s Y
{:
apylJ.
c l a s s i c a l l y one l o o k s a t a wave e q u a t i o n
+ Bm)$ = H9 where 9 i s a 4 - v e c t o r wave f u n c t i o n ,
{ai,ak}
2
0 f o r i # k, Iai,B} = 0, and a: = B = I. A s t a n d a r d n o t a i J v = B, Y = Bai ( i = 1,2,3), {-rl ,Y I = 2guv, *'Y = y0ylJyo, and
Then ( * b ) becomes (*+) (iv'alJ
-
m)$ : ( i y
- m)$
The K-G e q u a t i o n i s o b t a i n e d by m u l t i p l y i n g by ( i y + m).
= 0 (c = h = 1).
One w r i t e s a l s o
2 96
ROBERT CARROLL
0
(9.2)
y
where t h e
5
=
(note
53
=
I 0 i -I); y
i
1;
= ( O-5 i
0
i a r e the Pauli spin matrices i i s o f t e n as
5
o r -ti).
T~
ai
= (5i
ai
5
1
(A
0 -i
0 1
= (l
o),
o), and
a2 = (i
We r e f e r t o [ I t l ; B q l ]
f o r the philos-
ophy o f t h e D i r a c e q u a t i o n and w i l l o n l y mention a few f e a t u r e s here.
Y (*+) and
-
G(i$
ia.$y’
(*m):
-
P
+
I$
= 0,
m ) $ l ( x l ) = 0).
-+
11 t o
such t h a t
?(A)
$ ‘ ( X I ) = % ( x ) t h e n t h e equations
5 = $*ro, a r e
covariant (i.e.
i n particular
An a p p r o p r i a t e Lagrangian here i s
o b t a i n (*+) and ( * m ) .
a t p i and ii = a t l a i t so H
1: =
(A*)
f t d4x independently w i t h r e s p e c t
m)$ and one v a r i e s t h e a c t i o n S =
t o $ and =
and i f $
= yo?*yo,
(iy”(a/ax’’)
4 matrix
= f1 then there e x i s t s a 4 x
First
= A t xu ,
one notes t h a t if e.g. A i s a L o r e n t z t r a n s f o r m a t i o n x ’ = Ax, x ” A:
“1,
The a p p r o p r i a t e c o n j u g a t e f i e l d s a r e
+ m]$ ( j = 1,2,3). j Now t o i n t r o d u c e gauge f i e l d s we’ proceed f i r s t f r o m a p h y s i c s
RrmARK 9.3.
f d3x$[-iyja
=
p o i n t o f view as i n [Cgl;Fa4;Itl;Rel;Ql;Mdl]. 11 t h e e l e c t r o m a g n e t i c s i t u a t i o n where
v
=
We r e c a l l from Example 1.2.
vo - x t
+ vx a r i s e
and A = A.
( ( I P ~ , A ~o)r (q,A) a r e c a l l e d gauge p o t e n t i a l s and g i v e t h e same E and B); we r e c a l l a l s o t h e L o r e n t z gauge where d i v A + q t = 0 ( c = 1 h e r e ) . w r i t e here A’ axlax
v
%
= (q,A)
(xt,-V3x)).
-
w i t h gauge t r a n s f o r m a t i o n s A’ + A p
L e t us
apx (where a’”x
Now c o n s i d e r a c l a s s i c a l wave f u n c t i o n
9 satisfying a 9 i s then
Schrodinger e q u a t i o n ( s i m i l a r c o n s i d e r a t i o n s a p p l y t o f i e l d s and Products o f t h e form $*A$
called 9).
9
+
e
e x p ( i e ) $ and one asks about
volve derivatives o f
\t. and
=
a r e unchanged under a phase r o t a t i o n e ( x ) here.
these t r a n s f o r m v i a
Operators A g e n e r a l l y i n -
all$
+
a
$ ’ = exp(ie(x))[aP$
!J
+ i(a,e)$]. I f now one r e p l a c e s a by a gauge c o v a r i a n t d e r i v a t i v e D = a !J ” + ieA ( e b e i n g a charge) and i f A t r a n s f o r m s v i a A A ’ = A’ - ( l / e ) a p e !J 1-1 P U then DU$ exp(ie)Dp$ and 9*Dp$ i s i n v a r i a n t . Thus t h e e l e c t r o m a g n e t i c -+
-+
c o u p l i n g i s i n t i m a t e l y connected t o l o c a l phase i n v a r i a n c e .
Note here t h e
Lagrangian 1: o f ( * b ) i n Remark 9.2 ( a p p l i e d t o f i e l d s ) i s to =:(if = ?(iy’aP
J’A’
-
m)q and t h i s i s now r e p l a c e d by
where J’ = wy’q
(”)
1; =
?(iypD
’
-
i s a (conserved) e l e c t r o m a g n e t i c c u r r e n t .
g i a n 1: i s i n v a r i a n t under A
-+
”lJ
A
- a e
and
c+-+
exp(ie)p.
grangian f o r quantum electrodynamics (QED) i s t h e n 4)FPvFpv ( c f . Remark 8.3).
(Aa)
-
m)q
m ) q = to -
The Lagran-
A “complete” La-
t = to
-
J’A
’-
(l/
One can a l s o imagine o t h e r gauge i n v a r i a n t La-
grangians b u t t h i s one seems d i s t i n g u i s h e d by l e a d i n g t o a r e n o r m a l i z a b l e t h e o r y (a s u b j e c t we w i l l m e r c i f u l l y o m i t ) . t h e o r y based on U ( l ) (= phase r o t a t i o n s ) .
QED i s c a l l e d an A b e l i a n gauge Now c o n s i d e r gauge groups more
GAUGE FIELDS
I n p a r t i c u l a r c o n s i d e r S U ( 2 ) which was
c o m p l i c a t e d t h a n phase r o t a t i o n s .
used i n t h e o r i g i n a l Yang-Mills t h e o r y . t i o n s JI
+
e x p ( i ~ . a / 2 ) J I where
T =
2 97
i n e r e f e r s here t o isospin rota-
u = ( T ~ , T ~ , Ta ~ r e) t h e P a u l i m a t r i c e s o f
T-cx = 1 T . C Y . and JI ( i n t h e sim1 1' JI p l e s t v e r s i o n ) w i l l have a f o r m such as ( ' ) where JIi a r e complex f u n c t i o n s
Remark 9.2,
i s a v e c t o r (a1,a2,a3)
CY
with
$2
( 2 component s p i n o r s ) .
For t h e 4-D t h e o r y one o f t e n d e a l s w i t h composite
s p i n o r s JI = (')
( b i - s p i n o r s ) where
e q u a t i o n (iy'la,
-
X
a r e 2 component s p i n o r s and a D i r a c
m)JI = 0 becomes (A&) i v t = mp -iu.v+;
= (01,02,u3)).
(U
v,x
Now S U ( 2 )
o f 2 x 2 m a t r i c e s A = (( a . .)) 'J w r i t e A = a + i u - a where all 4 2 = a - i a 3 w i t h a + a2 = 1 ( I 4 a gauge t r a n s f o r m a t i o n JI + JI'
ixt
-mx
=
-iu-v
v
3
r o t a t i o n s i n R 3 ( c f . Appendix C) and c o n s i s t s
%
w i t h detA = 1 and A*A = AA* = I. One can = a4+ia 3; a12 = ial+a2,
and t h e u
aZ1 = ial-a2,
form a b a s i s o f SU(2)).
and aZ2 We c o n s i d e r
= GJI on 2-component s p i n o r s w i t h G = e x p ( i T . a
( x ) / 2 ) so t h a t aUJI G(a J I ) + (aPG)JI. Set D = a + i g B = 13 + i g B where 1 bllP - b 3!J BP = (1/2)T-bl, = (1/2)~':' = ( 1 / 2 ) ( ( bij)) where P P %b12 = bP-1bp, b21 3 P 1 2 = bP+ibP, and bZ2 = -bP ( p = 0,1,2,3). = G(DPJI) One wants then DPJI * D;JIl
.Y
-f
(ap
which means
+ (a,G)JI + igB;(GJI) LJ T h i s r e q u i r e s igB;(GJI) = igG(BUJI)
+ i g B ' ) I L ' = G(a,JI)
G(aPJI) + igG(BPJI).
" a r b i t r a r y " JI, and hence g)(aPG)G-' -(l/g)a out
1-1
(A+)
(A+)
B;
= G[BP
( c f . a l s o Remark 9.4).
= G(aP + i g B
-
!J
(a,,G)JI,
+ (i/g)G-'(aPG)]G-'
)JI
=
t o hold f o r
= GBPG-l
+
(i/
Note f o r G = e x p ( i e ) one o b t a i n s B ' !J - BP = e). It i s i n s t r u c t i v e t o work
e which i s t h e c o n d i t i o n i n QED ( g on an i n f i n i t e s i m a l l e v e l
t h i s t o [Ql;Itl;Cgl].
(%
G = 1 + (i/2)r.a)
and we r e f e r f o r
W i t h o u t d e l v i n g i n t o t h e p h i l o s o p h y t o o much a t t h i s
t i m e we s i m p l y remark now t h a t an a p p r o p r i a t e Yang-Mills Lagrangian f o r t h e i n t e r a c t i o n o f a D i r a c b i s p i n o r w i t h a Y-M f i e l d w i l l be (JI
L
(Am)
(cjkm
-
= $(iyPDP
m)JI
-
(1/2)TrFUVFuV where "F:
= avb!
-
(') X
aPb:
as above)
+ gsjkmbibt
i s t h e L e v i C i v i t a symbol which i s 1 f o r even p e r m u t a t i o n s o f t h e i n -
d i c e s , -1 f o r odd permutations, and 0 o t h e r w i s e ) . ngth tensor i s F
= (l/ig)[Dv,DP]
= avBP
-
Note h e r e t h e f i e l d s t r e -
aPBu + ig[Bv,BP]
and observe
t h a t yPDP(:)
= y'yg)
REmARK 9.4.
It i s probably worth w r i t i n g o u t the s i t u a t i o n f o r scalar
(D
Q
DP) makes sense.
fields
v
v
= e x p ( i T - a ( x ) ) v where T = ( T . ) (row v e c t o r ) .
v'
= (pi)
(column v e c t o r , i = 1,2,3)
w i t h Y-M gauge t r a n s f o r m a t i o n s
Here T. generates i s o J s p i n r o t a t i o n s about t h e j a x i s and s a t i s f i e s t h e S U ( 2 ) a l g e b r a r u l e s [T j' Tk] = i E j k m P ( n o t e h e r e [u./2,uk/2] = isjkmum/2 where u = r j ) . Now DP = J j I a t i g T - b and a s u i t a b l e Lagrangian can be based on i: = ( l / 2 ) ( D ! J v ) - ( D P v )
-
+
P
J
!J
(1/4)TrFPvFP"
-
V ( v - v ) ( g i v e n a p o t e n t i a l V , e.g. V ( v - v )
= (1/2)mp-p
+
2 98
ROBERT CARROLL
I n t h i s connection one notes a l s o t h a t f o r quarks qf =
(A/~!)(v.v)~).
-
( q a f ) (column v e c t o r 1,2,3,
a = 0,1,2,3
= m2 = m3 = m, f = f l a v o r = l y . . . y n y a = c o l o r =
ml
and each qaf i s a D i r a c b i s p i n o r ) one has an SU(3) t h e o r y
w i t h gauge t r a n s f o r m a t i o n s (**) q ' ( x ) = Gq = l o r (A = (Ak) a row v e c t o r , k = 1,...,8, matrices
-
see e.g.
[Lpl;Cgl]).
exp[(l/2)giA.A(x)]q(x)
A k = generators
i n co-
o f SU(3) a r e 3 x 3
The c o l o r gauge f i e l d based on (a*) i s
ii
Eu
= gA (A / 2 ) = (g/2)A . A and t h e t r a n s f o r m r u l e i s = G[gP + i G - l ( a u G ) ] uk k u G-' as b e f o r e ( n o t e a l s o from GG-' = I one has (auG)G-l t Ga ( G - l ) = 0 so u
-
B; = GKuG-l
iGa (G-l) VAd
again DP =
t iB
(au
1.1
)
G[EP - i a u ] G - l ) . : (a I t ig ) and f u u
u
The c o v a r i a n t d e r i v a t i v e ,is
=
= q(iyPDu
- m)q
-
(1/4)TrF'"FPV
(summed o v e r f l a v o r s ) ( c f . below f o r t h e f o r m a t i o n o f such Lagrangians).
REmARK 9.5,
L e t us p o i n t o u t here t h a t i n l o o k i n g a t t h e above formulas f o r
c o v a r i a n t d e r i v a t i v e s and Lagrangians e t c . one sees immediate i n t e r p e r t a t i o n i n terms o f c l a s s i c a l d i f f e r e n t i a l geometry.
Covariant d e r i v a t i v e s a r e o f
-
course based on t h e i d e a o f connections ( c f . 53.10) so t h a t e.g. D
lJ
=
aP
t
igP where
trie components o f : i
(g/2)A .A o r
=
u
u
iiP
= gT-bu o r Bu = (g/2).r-bP i d e n t i f y
A:r
w i t h C h r i s t o f f e l symbols
0-4 w h i l e u,w r u n over v a r i o u s s e t s
-
terms l i k e
(here A i s a Dirac index
we a r e t h u s t h i n k i n g o f geometry i n
t h e i n t e r n a l charge space o r c o l o r space, o r whereever).
The Riemann c u r -
v a t u r e t e n s o r i s d e f i n e d v i a RV = a r" - a r" t Y A r" - r Ar" and i n pa8 a PR R ua u 8 Xu ua A6 t h e case o f = (g/2).r.bu f o r example t h i s w i l l correspond t o ( c f . [Cgl]) w
-
aaiBB
u a iia -
t o T r F a B FI R
.
REmARK 9.6,
[iga,iii,]
= ( i g / 2 ) ~ %and ~ Tr(.r.Fa,)(T-FaB)
i s proportional
T h i s w i l l be e x p l i c a t e d i n 510 i n a c o o r d i n a t e f r e e manner. The n e x t s t e p i s t o q u a n t i z e t h e Y-M f i e l d s and one must r e -
gard some o f t h e development h e r e as h e u r i s t i c ( b u t h o p e f u l l y s t i m u l a t i n g ) . We saw how t h e p a t h i n t e g r a l f o r m u l a t i o n i n Remark 8.9 l e a d s t o t h e connect e d Green's f u n c t i o n s G(X"l Remark 9.1). =
1 [&d7i]exp[i/
N
v i a the generating functional ( c f . also
To be more e x p l i c i t l e t us r e c a l l (*) i n t h e f o r m Z ( J ) = i ( J ) 4 4 d x(doq H ( n , q ) t Jp)] = 1 [Clp]exp[i/ d x(C t Jp)] w i t h
-
N
WE(J)
,.. . ,xn)
G(;l,...lgi)
g i v e n by (A) and
via
(0).
We a l s o r e c a l l from Remark 8.
3 t h a t i n t h e e l e c t r o m a g n e t i c f i e l d one uses f i e l d v a r i a b l e s A
ac/ai\
Fi
(=
-AP
f o r the f o f
(A*)
i n § 8 ) ; one s e t s a l s o
IT
P
=
P.
with
ar/aAP.
'TI
=
For QED
i n t h e U(1) gauge f o r m u l a t i o n as i n Remark 9.3 one has an 1; as i n (A@), name l y 1:
=7(ivPau -
m)q
-
(1/4)FuvFu" (Fuv = auAu
-
apAv).
For gauge t h e o r y
however i t w i l l be necessary now t o impose a gauge f i x i n g c o n d i t i o n and work w i t h s t a t e s IL where e.g. a P A IIL) = 0 ( L o r e n t z gauge f o r example); t h i s amP
ounts t o removing redundant degrees o f freedom and w i l l be r e f l e c t e d i n t h e
GAUGE FIELDS
299
p a t h i n t e g r a l f o r m u l a t i o n ( c f . [Cql ;Fa4;Rel ; D , i l ] ) .
There a r e g l o b a l o b s t r u -
-
c t i o n s t o g l o b a l gauge f i x i n g ( G r i b o v a m b i g u i t y
c f . [Svl;Jkl])
and t h i s i s
b r i e f l y discussed l a t e r . Note f o r t h e QED gauge w i t h 9 = 0 one has e.g. no = o and v 3 e n = o (c = - ( I / ~ ) ( ~ F ~ ~ F O + ' F..F i j SO 71P = ai:/a(aoAu) = F ). FPv = - ( 1 / 4 ) 1 Fv;
Thus t a k e now SU(2) Y-M f i e l d s w i t h i: = -1(J1 / 4 ) T r F (a = 1,2,3
Zo(J)
(9.3)
(a'Aav-avAau)
[dAUle
= =
- a Aa
a Aa
say) where Fa''=
P4v x(i:otJ$
iJ
t
Id4xA:(gPVa2-auav)A:;
gEabci'Ac P
" Y1;
d4xc0 =
I
Z(J) =
V
-
p P V
and w r i t e 1 /d4x[a
[dAp]e
Aa-a A a )
u v
v u
i/ d4x(LtJPAP)
Now one would l i k e t o d u p l i c a t e t h e procedure o f ( = ) i n Remark 9.1 i n v o l v i n g detK, A, e t c . b u t t h i s w o n ' t work because e.g. have an i n v e r s e ( n o t e e.g. t i o n operator
-
t o o many f i e l d s ;
c f . [Cgl]).
K
PV
KPV = g P v a 2
Kv i s p r o p o r t i o n a l t o K
PA
-
a a
so Kuvuii::
does n o t a projec-
The gauge freedom has i n v o l v e d i n t e g r a t i n g o v e r
i n p a r t i c u l a r o r b i t s which a r e gauge e q u i v a l e n t and have
f u n c t i o n a l volume must now be f a c t o r e d o u t .
Some examples a r e g i v e n i n [Cg
13 t o m o t i v a t e t h e procedure b u t we s i m p l y i n d i c a t e h e r e t h e general method. Thus e.g.
t h e a c t i o n i s i n v a r i a n t under A
A i - . r / 2 = U[Au..r/2
t
U
+
A' where U = e x p ( - i ~ . e / 2 ) w i t h P
( c f . G i n Remark 9.3,
(l/ig)U-laPU]U-l
G
n,
exp(i7.42)).
The a c t i o n i s c o n s t a n t on t h e o r b i t o f t h e gauge group SU(2) formed by a l l
Ae and one wants t o r e s t r i c t t h e p a t h i n t e g r a l t o a " h y p e r s u r f a c e " which
P I f f a ( A p ) = 0, a = 1,2,3, i s such a hyperi n t e r s e c t s each o r b i t o n l y once. 8 s u r f a c e then f a ( A P ) = 0 s h o u l d have a u n i q u e s o l u t i o n e f o r A g i v e n and P
Since gauge f i x i n g i s g e n e r a l l y o n l y pos-
t h i s i s a gauge f i x i n g c o n d i t i o n .
s i b l e l o c a l l y t h i s procedure i s o n l y h e u r i s t i c ; however i t i s p r o d u c t i v e i n terms o f p e r t u r b a t i o n t h e o r y (see [ J k l ] f o r a good d i s c u s s i o n o f gauge f i x ing).
The d i s c u s s i o n i n [ S v l ] shows t h a t i f t h e c o n d i t i o n s a t m amount t o 4 U t m l f o r example then t o p o l o g i c a l o b s t r u c t i o n s e x i s t .
s t u d y i n g M = S4 = R
The a p p r o p r i a t e i n t e g r a l i n SU(2) i s w r i t t e n [de] = D e f i n e now (**) A[';],
where (Mf)ab
= 6f,/6eb
=
so t h a t Af[A,]
1 [de]6[fa(A;)]
( n o t e detMf # 0 s i n c e fa(A:)
s o l u t i o n and 6 h e r e r e f e r s t o a 6 f u n c t i o n
-
-
= de'
w i l l provide e x p l i c i t f o r -
l e f t i n v a r i a n t Haar measure f o r which [Vil;Hg2] mulas).
3 n1 dea ( d ( 8 0 ' )
e.g.
= detMf
i s t o have a unique formally J des(g(e)) =
J (ae/ag)dgs(g) = ( a e / a g ) l g = O ) .
One checks e a s i l y t h a t A(A';), i s gauge i n 4 variant (exercise cf. [ C g l ] ) and (am) J [dAP]exp(i/ d x C ) = J [de][dAP] J [de][dAP]A (A ) 6 ( f a ( A P ) ) e x p ( i / d 4 xC) ( n o t e Af(AP)6(fa(A:))exp(i/ d4xL)
-
f C
4 Af(AP) and e x p ( i / d x C ) a r e i n v a r i a n t under Au i s independent o f 8 w i t h / [de] t h e
m
-f
AP).
The i n t e g r a n d i n
(0.)
volume one wishes t o f a c t o r out; one
300
ROBERT CARROLL
a r r i v e s a t t h e Fadeev-Popov ansatz i n v o l v i n g t h e c o n s i d e r a t i o n o f ( 0 6 ) Zf
( J ) = I[dAu]detMfs(fa(Au))exp[iI
d4x(C
+ J,A')].
Thus [dAA]
a term detMf6(fa(Au)) i n o r d e r t o e l i m i n a t e t h e
i s m o d i f i e d by
orbital integral.
Further The f u r -
d e t a i l s and d i s c u s s i o n can be found i n [Cgl ;Ga4;Djl;Nsl;Ryl;Pdl].
t h e r a n a l y s i s o f detMf i n terms o f Feynman diagrams e t c . l e a d s t o t h e Fadeev-Popov ghost f i e l d s and i n t h i s r e s p e c t f o r Mf = 1 + L one w r i t e s detMf = exp(Tr[logMf]) = exp[TrL + ( 1 / 2 ) T r L 2 + + ( l / n ) T r L n + ...I = e x p [ I d4 x
...
4
4 Laa(X,X) + ( 1 / 2 ) l d xd YLab(XYY)Lba
RFmARK 9.7.
...I
I n Remark 1.10.7 and 1 10.8 f o l l o w i n g [ T b l ; J a l ]
remarks on Y-M-H f o l l o w i n g [Sagl;Tb2;
L e t us make a few a d d i t i o n a
;Gfl].
Ja1;Eal;Ghl
we discussed
f e l d s i n connection w i t h t h e G-L equa-
some f e a t u r e s o f Yang-Mills-Higgs tions.
+
Y,X)
I n p a r t i c u l a r we g r a d u a l l y i n j e c t more and more machinery
i n t o t h e t h e o r y i n p r e p a r a t i o n f o r t h e general f o r m u l a t i o n i n 53-10. Thus 4 t h e idea now i s t o work o v e r Minkowski space M = R w i t h rl i j = -gij = (-1,
l,l,l)here and t o f i r s t l o o k f o r l o c a l e x i s t e n c e o f s o l u t i o n s o f t h e e v o l u t i o n problem w i t h Cauchy t y p e d a t a s p e c i f i e d a t say t = 0 ( c = 1 - t h e temp o r a l gauge A'
= 0 i s used i n [Eal;Ghl]
f o r example a l t h o u g h f o r c e r t a i n
l o c a l e s t i m a t e s a gauge t r a n s f o r m a t i o n i s u s e f u l ) . group w i t h L i e a l g e b r a v
g
+
5 and
g l ( p , R ) ) so t h a t [ea,eb]
L e t G be a compact L i e
l e t 8 be a r e a l m a t r i x r e p r e s e n t a t i o n o f
fabcec
=
( t h e ea a r e g e n e r a t o r s ) .
(0:
The Y-M po-
valued l - f o r m over M o f t h e form A = A a e dx' = Audx' w i t h u a e )dx' A d x V = F dx' A dx" where F = a A - a A + PV a P" 'V 1!" V l J The n o t a t i o n h e r e and i n §3.10 i s e q u i v a l e n t t o p r e v i o u s n o t a t i o n
tential i s a
c u r v a t u r e F = (Fa
'
[A ,Av].
-
b u t we w i l l n o t always a t t e m p t t o a d j u s t
s i g n s f o r example s i n c e d i f f e r e n t
authors use d i f f e r e n t conventions; anyone who i s s e r i o u s l y concerned can On a f l a t t = to s p a c e l i k e h y p e r s u r f a c e i and F = F . . d x d x j ( i , j = 1,2,3). The
e a s i l y produce a u n i f i e d n o t a t i o n . i n M one w r i t e s A = Ayeadxi = Aidxi
'J
Higgs f i e l d i s a v e c t o r valued f u n c t i o n 9 on t o r space
E
%
M
w i t h values i n t h e r e a l vec-
Rp which serves as a r e p r e s e n t a t i o n space f o r t h e m a t r i c e s Ba.
The c o v a r i a n t d e r i v a t i v e i s D
q = a,p+
u
Auq
( t h e use o f t h i s c o v a r i a n t d e r i -
v a t i v e i s o f t e n r e f e r r e d t o as minimal c o u p l i n g o f A t o 9 ) and i f U i s a smooth G v a l u e d f u n c t i o n on M i t generates gauge t r a n s f o r m a t i o n s o f (A,@
' f e r e n c e a r i s e s from t h e U - l and
(Dug)' =
Tr[-(1/4)FpVFu"]
U(Duv).
-
-' +
' i factors there).
v i a (a+) 9 ' = Up, A ' = UApU
It f o l l o w s t h a t
-
the sign d i f -
(0.)
F L Y = UFpv
For a gauge i n v a r i a n t Lagrangian one has ( 6 * ) 1; =
(1/2)(Dp9).(D'v)
v a r i a n t polynomial ( i . e .
Ua Uml ( c f . Remarks 9.3-9.4
-
P(9) where P(9) i s assumed t o be an i n -
P(Uq) = P ( 9 ) ) o f degree 5 4 ; T r denotes here (nega-
GAUGE FIELDS
301
a t i v e ) t r a c e o v e r t h e m a t r i c e s B a where e a i s chosen t o be r e a l a n t i s y m m e t r i c and fabci s c o m p l e t e l y a n t i s y m m e t r i c ((DUq)-(D'q)
w i t h Treaeb = 6ab,
= (Du
V ) ~ ( D ' ~ f) o~r some i n v a r i a n t " c o n t r a c t i o n " i n E = Rp). The c o r r e s p o n d i n g 3 H a m i l t o n i a n i s (6.) H = I d x [ n - n / 2 + Tr(EiEi/2 + F . . F . . / 4 ) + (Diq)-(Div)/2 1J 1J + P ( v ) + Tr(AoC)] where Ei = E:ea = atAi - aiAo + [A 0' A i] = Foi, n = a tq + A q = Doq, and C = Caoa = -a . E . + [E.,A.] - (n-eaq)ea. The i n i t i a l v a l u e 0 J J J J c o n s t r a i n t i s C = 0 and (Ei,n) a r e momenta c o n j u g a t e t o (Ai,q). Most o f t h e terms i n H s h o u l d make sense by now ( c f . a l s o 510); we remark ( c f . [ E a l ] ) t h a t t h e energy momentum t e n s o r i s (6.) (D'v)*(D"q)
-
(1/2)n'"(Dav).(Daq)
-
T'"
n'"P(q)
o f t h e equations o f m o t i o c ( 6 6 ) VvF'"
-
= Tr[FPaF:
= -((D'q)
+
(1/4)n'"FaBFaB]
= 0 as a consequence
where aVT'" -eaq)ea
and (D,(D'P))~
=
aP/avk ( h e r e vyFaB = a F + [ A ,F I ) . Note a l s o t h e B i a n c h i i d e n t i t y ( 6 + ) Y aB y aB + v B Fy a = 0. Now one w r i t e s o u t t h e Hamilton e q u a t i o n s i n t h e vyFaB VaFBy +
temporal gauge An = 0 as (9.4)
Dt
Ei
-aj[Ai,:j]-[A.,F.
=
9 n
J
1
.]-((Dilp)-@a~)Oa 1J
For t e c h n i c a l reasons one now s p l i t s E = Eidx i v i a Ei = EiT + EiL ( t r a n s v e r s e = divergence f r e e and l o n g i t u d i n a l = c u r l f r e e ) where
aiEi
T
= 0 and
E
ijk
L c L j Ek = 0 and t h e n EL i s r e p l a c e d by a smoother o b j e c t EC such t h a t Ei = Ei
when t h e c o n s t r a i n t aiE: 11 f o r details).
=
aiEi
= [E.A.]
-
J .J
(n.eaq)e
a
i s s a t i s f i e d (see [Ea
The e v o l u t i o n equations ( 9 . 4 ) can t h e n be w r i t t e n as ( 6 = )
d u / d t = Au + J(u) (where u = (Ai Ei q n ) as a column v e c t o r - h e r e Au i s t h e T The o p e r a t o r A same as t h e f i r s t t e r m i n (9.4) w i t h Ei r e p l a c e d by Ei). 2 generates a 1-parameter group on (HS+l x H s ) where H, i s t h e a p p r o p r i a t e Sobolev space and f o r s 2 1 one can e s t a b l i s h l o c a l e x i s t e n c e and uniqueness o f s o l u t i o n s t o t h e i n t e g r a l e q u a t i o n (+*) u ( t ) = e x p [ A ( t - t ) ] u ( t o ) + 0
ItQ d s e x p [ A ( t - s ) ] J ( u ( s ) ) .
F u r t h e r t e c h n i c a l argument l e a d s t o g l o b a l s o l u -
t i o n s and we r e f e r t o [Eal;Ghl;Sagl;Gfl]
f o r theorems i n t h i s s p i r i t .
p r a c t i c e one i s o f t e n i n t e r e s t e d i n t h e s t a t i c Y-M-H s i t u a t i o n on R ample where (AM) i s independent o f time, A. [1qI2
-
1 1 ( c f . [Jal;Tb2]
= 0, and e.g.
P(v)
%
3
In f o r ex-
(x/8)
and 51.10.
10. GAUGE F I E L D S (WCHEmACICS) AND GE0mECRIC qllANCIZACI0N. We go now t o a more mathematical f o r m u l a t i o n o f t h e gauge i n v a r i a n c e ideas u s i n g t h e l a n guage o f f i b r e bundles, connections, c u r v a t u r e , e t c . ( c f . [Gy2;Trl ;Sxl ; J a l ; Tbl-3;Eal;Fdl;Ul
;Boul;Ghl ;Pbl ; B l e l ;Gy2;Thl ;Burl ;Gul ;Dhl ; D j l ;Ael ; K h l ] ) .
All
o f t h e necessary g e o m e t r i c a l d e f i n i t i o n s and ideas a r e presented i n t h e t e x t
302
ROBERT CARROLL
There w i l l be no space t o g i v e i n f o r m a t i o n about i m p o r t -
o r i n Appendix C.
a n t a p p l i c a t i o n s o f index t h e o r y and cohomology f o r example t o mathematical ;C1; J k l ;Gul ; H t l ;Wol ; S r l ; S S l ]
physics ( c f . [Gyl,2;Bsl
RElllARK 10.1.
f o r this).
The general f o r m a t f o r gauge f i e l d t h e o r y i n v o l v e s a p r i n c i -
4
p a l f i b r e bundle 7 : P -+ M w i t h s t r u c t u r e group G ( c f . Remark 10.2 - M = R w i t h Minkowski m e t r i c f o r example and G = SU(2) f o r Y-M t h e o r y i s t y p i c a l c f . [ B l e l ;Burl;Gul;Gy2;Ttl;Dhl;Djl;Aell). feomorphism y: P G.
+
-
A gauge t r a n s f o r m a t i o n i s a d i f -
P such t h a t y ( u g ) = y ( u ) g and noy =
TI
f o r u E P and g E
The s e t o f gauge t r a n s f o r m a t i o n s w i t h t h e composition l a w o f maps' f o r m
t h e gauge group Y. t i a b l e map
T:
P
+
Generally a " p a r t i c l e " f i e l d i s a s u f f i c i e n t l y d i f f e r e n F such t h a t T(pg) = g-'T(p);
here F i s a d i f f e r e n t i a b l e
m a n i f o l d on which G a c t s f r o m t h e l e f t and i n p r a c t i c e F i s o f t e n Rp.
Q be t h e s e t o f such f i e l d s and one assumes i n p h y s i c s t h a t
Let
for y
T I TOY
E
w i l l be c a l l e d t h e space o f p h y s i c a l c o n f i g u r a t i o n s . Now k k one d e f i n e s a Lagrangian on Q v i a a map L: J (P,F) + R where J i s t h e kk k Y.
Hence Q/Y =
@
j e t bundle ( c f . Appendix C ) and i t i s r e q u i r e d t h a t L ( j ( f ) ) = L ( j ( f ) ) f o r P P9 k p E P and g E G. This i m p l i e s L ( j ( f ) ) i s c o n s t a n t a l o n g f i b r e s so i t can P k k be considered a f u n c t i o n on M and we w i l l want a l s o L ( j ( f ) ) = L ( j ( f o y ) ) 1 P P f o r y E Y. In p r a c t i c e one w i l l d e f i n e L on J (P,F) and i n t h i s s i t u a t i o n 1 a coordinate free d e f i n i t i o n i s possible v i a j ( f ) = (p,f(p),df(p)) so f o r P n ( p ) = x, L ( j p1 ( f ) ) ( x ) = L o ( f ) ( x ) where Lo: @ + R ( e x e r c i s e - c f . [ B l e l ; B u r 11 f o r details).
REmARK 10.2.
The q u e s t i o n o f gauge i n v a r i a n c e i s considered i n Rem.10.8.
L e t us d e f i n e a p r i n c i p a l f i b r e bundle P ( o v e r M w i t h L i e
group G ) as f o l l o w s . late a triple
F i r s t i n o r d e r t o connect n o t a t i o n t o [ B l e l ] we s t i p u -
TI^
= (P,M,G) w i t h n: P M t h e p r o j e c t i o n and G a L i e group For each g E G t h e r e i s t o be a diffeomorphism R * P + P g' (Rg(p) = pg) such t h a t p(g1g2) = (pg1)g2 and pe = p. One r e q u i r e s t h e map P X G P t o be e.g. C" and i f pg = p then g = e ( f r e e a c t i o n ) . The (C") -1 map TI: P + M i s o n t o and n - ' ( n ( p ) ) = Ipg; g E G ; 7 (x)) i s c a l l e d t h e f i b r e -+
( c f . Appendix C ) .
+
over x and TI-'(x) i s d i f f e o m o r p h i c t o G ( b u t t h e r e i s no n a t u r a l group s t r u c t u r e on T I - ' ( x ) ) .
F i n a l l y f o r each x E M t h e r e e x i s t s an open U 3 x and a
diffeomorphism TU: .-'(U)
-+
su(pg) = s u ( p ) g (su: n-'(U) ( o r a c h o i c e o f gauge).
U X G o f t h e form TU(p) = (n(p),s,(p)) -f
G).
where
T h i s TU i s c a l l e d a l o c a l t r i v i a l i z a t i o n
T h i s d e f i n i t i o n can be souped up i n terms o f t h e
c o n s t r u c t i o n s i n Appendix C b u t we o m i t d e t a i l s ( c f . [Cl;Tdl;Khl;Ttl]). us a l s o f o r m a l l y mention t h e t r a n s i t i o n f u n c t i o n s guv: U n V t r i v i a l i z a t i o n s Tu and TV around x
E
My x
E
U
-t
Let
G for. 2 l o c a l
n V, d e f i n e d v i a guv(x)
=
303
GAUGE FIELD THEORY sU(p)sv(p)-'.
These have t h e p r o p e r t i e s ( 1 ) guu(y) = e ( 2 ) gvu(y) = g;i(y)
( 3 ) guv(y)gvw(y)gwu(y) = e . Local s e c t i o n s o f IT^ a r e Cm maps U : M + P such TI o o = 1 l o c a l l y . E v i d e n t l y t h e r e i s a n a t u r a l correspondence l o c a l
that
s e c t i o n s and l o c a l t r i v i a l i z a t i o n s . One can d e f i n e connections i n v a r i o u s ways and we f o l l o w [ B l
REilARK 10.3.
e l ] i n g i v i n g several equivalent d e f i n i t i o n s ( t h e
5
i s demonstrated i n [Bl
e l ] and some r e l a t i o n s w i l l become c l e a r h e r e v i a t h e d i s c u s s i o n and use o f t h e concepts). t a i n e d i n e.g.
M o t i v a t i o n a l d i s c u s s i o n and g e o m e t r i c a l i n t u i t i o n can be ob[Khl ;Blel;Gg2;Ttl;Bsl;Ae1;Spil;Dzl].
f i b r e bundle nG: P
-f
M
Thus g i v e n a p r i n c i p a l
(dimM = n ) ( 1 ) A c o n n e c t i o n a s s i g n s t o e v e r y p
E
P a
subspace H C T (P) ( h o r i z o n t a l subspace) such t h a t T ( P ) = H 8 V ( d i r e c t P P P P P One assumes H depends smoothly on p sum) where V = { x E Tp(P); aX, = 01. P P i n t h e sense t h a t t h e r e w i l l be l o c a l "frames" o f n v e c t o r f i e l d s spanning H ( n o t e 8 does n o t i n v o l v e o r t h o g o n a l i t y n e c e s s a r i l y ) ( 2 ) A c o n n e c t i o n i s P a q v a l u e d 1 - f o r m w d e f i n e d on P ( 0; t h e n
G.
s i n c e z / a E G we have y 5 p ( z / a t x o ) - L ( z / a ) ; axo)-L(z);
-s, s
ay 5 ap(z/a+xo)-aL(z/a)
and L(z+axo) = a y + L ( z ) (p(z+axo).
> 0; t h e n z / a = -z/s
+ L(z/s) 5
and -p(z/s-xo)
E
y;
f o r x i n some NBH V o f 0 i n F.
choose a FSN which i s symmetric ( i . e .
x
-p(x-sx0) + L ( z ) 5
Since p i s now c o n t i n u o u s a t 0, i t
ys; and y a t L ( z ) = L(z+axo) 5 p ( z + a x o ) .
follows that p(x) 5
= p(z+
On t h e o t h e r hand, l e t a =
E
But we can always
V i m p l i e s -x
E
Then
V ) i n a LCS.
suppose t h e e x t e n s i o n o f L t o have been c a r r i e d o u t t o a l l F w i t h L ( x ) 2 p ( x ) on F. Therefore,
IL(x) I
5 E f o l l o w s a l s o now - L ( x ) = L ( - x ) c p ( - x ) 5 E. f o r x r5 V and L i s continuous. Also t h i s shows t h a t
From L ( x ) ( p ( x ) IL(x)
(p(x)
C0R0ttARg A.11.
I
5
E
since -L(x) = L(-x) 5 p(-x) = p(x).
QED
For any continuous seminorm p on a LCS F t h e r e i s a u E F'
( F ' = dual o f F ) w i t h l u ( z )
I
(p(z)
and u ( x o ) = p ( x o ) , xo a r b i t r a r y , can be
p r e s c r i b e d i n advance.
Phaod:
{axel)
D e f i n e L on { x o l ( = subspace spanned by xo =
.p(x0) and extend L.
REmARK A.12.
by L(ax,)
=
QED
Thus i n p a r t i c u l a r , i n a LCS F, where continuous seminorms a l -
ways e x i s t t h e r e a r e a u t o m a t i c a l l y n o n t r i v i a l elements o f F ' , and t h i s i s a good reason t o work i n LCS. We say now t h a t a s e t B i s d i s c e d i f
AX E
B whenever x E B and 1x1 5 1.
t h e r , i f F i s a TVS o v e r R, then B i s convex i f x,y E B f o r 0 < A < 1.
E
B implies
AX t
Fur-
(1-x)y
I f F i s a TVS o v e r C, l e t Fo be t h e same space c o n s i d e r -
ed as a TVS o v e r R ( m u l t i p l i c a t i o n b y i, then, i s t r e a t e d as an automorphism o f Fo and n o t as a d i a l a t i o n ) .
Then B
C
F i s convex i f i t i s convex i n Fo.
I n a complex TVS E a complex hyperplane determined b y f ( x ) = a t i g i s t h e i n t e r s e c t i o n o f two r e a l hyperplanes Re f ( x ) = a and Re f ( i x ) = - B ( n o t e t h a t
I m F ( x ) = -Re f ( i x ) ) .
Conversely, i f
Ho i s a r e a l homogeneous hyperplane de-
termined b y g ( x ) = 0, g a r e a l l i n e a r f o r m on E, t h e n Ho n iH, homogeneous hyperplane determined by g ( x ) correspond t o continuous forms ( f - l ( O )
-
ig(ix)
0.
i s a complex
Closed hyperplanes
i s c l o s e d when f i s c o n t i n u o u s ) .
We
s t a t e n e x t a geometrical v e r s i o n o f t h e Hahn-Banach theorem w i t h o u t p r o o f (see [ B o ~ ]
- a p r o o f f o r normed spaces i s sketched below i n Theorem A.16).
31 5
APPENDIX A
We remark here a l s o t h a t a TVS F i s o f t e n s a i d t o be l o c a l l y convex i f N ( x ) in F
0
has a FSN c o n s i s t i n g o f convex s e t s .
Then i t can be shown ( c f . [ B o ~ ] In
f o r d e t a i l s ) t h a t seminorms can be found which determine t h e t o p o l o g y .
t h i s d i r e c t i o n we c o n s i d e r s e t s B which a r e convex, d i s c e d ( o r symmetric as above), a b s o r b i n g ( i . e . point.
x E
pB
for
1p1
2 p o ) , and have 0 as an i n t e r i o r
Then d e f i n e t h e gauge o r Minkowski f u n c t i o n a l o f B by
i n f p f o r x E pB and 0 < p . a seminorm ( e x e r c i s e ) .
= i x E F; p ( x )
It follows t h a t
Note p ( x ) i n
( 0 )
( 0 )
p(x) =
5 1 1 and p i s
i s d e f i n e d f o r B o n l y convex and
The p r o p e r t y o f b e i n g d i s c e d t r a n s l a t e s i n -
a b s o r b i n g w i t h 0 E i n t e r i o r B.
t o p(ax) = I a I p ( x ) b u t i s n o t needed t o d e f i n e p ( x ) f o r B. I f E i s a TVS o v e r R and A i s a convex open nonempty s e t i n E
CHE0RElll A.13,
with M a linear variety (i.e.
t r a n s l a t e o f a l i n e a r subspace) n o t i n t e r s e c -
t i n g A, t h e n t h e r e i s a c l o s e d hyperplane H c o n t a i n i n g
M and n o t i n t e r s e c -
t i n g A. I n p a r t i c u l a r ( c f . [ B o ~ ] ) , if A i s convex open and nonempty, and B i s convex nonempty w i t h A n B = Since 0
4
C,
t h e n C = A-B i s convex open and nonempty ( e x e r c i s e ) .
by Theorem A.13 t h e r e i s a l i n e a r c o n t i n u o u s f o r m f # 0 on E
such t h a t f ( x ) =
@,
>
0 i n C.
Set a
For x E A, y E B one has then f ( x ) > f ( y ) .
i n f f ( x ) f o r x E A and n o t e t h a t a i s f i n i t e w i t h f ( x ) ? a f o r x E A and
f ( x ) 5 a f o r x E B.
Thus A and B l i e i n t h e h a l f - s p a c e s f ( x )
< a determined by t h e c l o s e d hyperplane f ( x ) = a.
now A # @ be c l o s e d and convex and x W o f 0 such t h a t A+W n x+W =
$
A.
l y (exercise).
Then t h e r e i s a convex open NBH
(exercise).
open, by t h e above t h e r e i s a hyperplane
2 a and f ( x )
As an a p p l i c a t i o n , l e t
Since A+W and x+W a r e convex and
H s e p a r a t i n g them - i n f a c t s t r i c t -
We conclude Every c l o s e d convex s e t A
CHE0REm A.14-
C
E i s the intersection o f the
closed half-spaces containing it.
A v a r i a t i o n on t h i s which i s o f t e n u s e f u l i s g ven by L e t G be a c l o s e d l i n e a r subspace o f a Banach space E and xo E
LEilllllA A.15,
E an element n o t i n G.
Then t h e r e i s a uo E E
w i t h ( u o , x o ) = 1 and
(
uo,z)
= 0 f o r a l l z E G.
Pmag: L e t d
= i n f IIxo-zII f o r z
E G.
C l e a r l y d > 0, s i n c e o t h e r w i s e xo
would be a l i m i t of elements o f G and hence would belong t o G.
z
E G,
Ilx
0
-211
2 d.
xo and G by t h e r u l e L(axo + z ) = a. s i n c e - z / a E G.
Thus f o r a l l
D e f i n e a l i n e a r form L on t h e l i n e a r space spanned by Then IIax0+zII = lallxo+z/al1 2
Thus / L ( a x o + z ) l = IuI 2 IIaxo+zll/d.
laid,
Then by Theorem A . l l
31 6
ROBERT CARROLL
we can extend L t o a continuous l i n e a r f u n c t i o n uo w i t h Thus
(
uo,xo) = 1 and
(
I( uo,x)I
cllxfl/d.
Also IIuoll 2 l / d .
uo,z) = 0 f o r z E G.
QED
L e t E be a r e a l normed space and A a convex open (nonempty)
CHE0Rm A.16,
M be a l i n e a r v a r i e t y n o t i n t e r s e c t i n g A. Then t h e r e e x i s t s a c l o s e d hyperplane H 3 M and n o t i n t e r s e c t i n g A ( i . e . t h e r e e x i s t s e ' E E ' and c E R such t h a t ( e y e ' ) c f o r e E M and ( e y e ' ) < c f o r e E A ) . set.
Let
Let 0
P4006:
E
A w i t h o u t l o s s o f g e n e r a l i t y and l e t G be t h e subspace o f E
generated by M ( n o t e A i s a convex open (NBH) o f 0 and hence absorbing). Then M i s a hyperplane i n G, 0
f
M, so t h e r e e x i s t s a l i n e a r f u n c t i o n a l f
on G such t h a t M = { x ; f ( x ) = 11. t i o n a l ) o f A d e f i n e d by f ( a x ) = a 2 p(ax) f o r x
(0)
E
Thus I f ( x ) l 5 p ( x ) f o r x
E
u on E w i t h l u ( x ) l 2 p ( x ) .
L e t p be t h e gauge ( o r Minkowski f u n c -
so t h a t f ( x ) = 1 5 p ( x ) f o r x
M and
E
M.
Evidently
a > 0 w h i l e f o r a < 0, f ( a x ) 5 0 5 p ( a x ) .
G and one extends f t o a continuous l i n e a r f o r m
Let H = I x
6
E; u ( x ) = 11 be t h e corresponding
c l o s e d hyperplane i n E which does n o t i n t e r s e c t A ( u ( x ) < 1 f o r x E A).QED
DEFZNZEI0N A.17,
A s e t B i n a TVS i s bounded i f i t i s absorbed by any NBH
o f 0 o r e q u i v a l e n t l y by any V i n a FSN o f 0. for
B absorbed by V means B
C
XV
I A ~ 2 x0.
DEFZNZCION A . M .
F and G a r e s a i d t o be i n d u a l i t y i f t h e r e i s a b i l i n e a r
form ( , ) on F X G w i t h t h e p r o p e r t i e s (A) For any x # 0 i n F t h e r e i s a y E G such t h a t ( x , y ) # 0 ( 6 ) For any y # 0 i n G t h e r e i s an x 6 F such t h a t (
x,y)
#
0.
If F i s a LCS and F ' = G i t s dual, t h e n t h e n a t u r a l a c t i o n o f F ' on F g i v e s a bracket
(
,
)
as d e s c r i b e d e a r l i e r .
t o assert t h a t i f x
E
Here one uses t h e Hahn-Banach theorem
G, x # 0, t h e r e i s a seminorm p w i t h p ( x ) f 0 ( a l l
# 0 (see C o r o l l a r y
spaces a r e H a u s d o r f f ) and hence an x ' E F ' w i t h (x,x') A.ll).
We s h a l l assume a l l spaces a r e LCS from now on.
DEFINZEIBN
A.19.
L e t F ' be dual t o F, F a LCS; t h e n t h e s t r o n g topology on
F ' i s t h e t o p o l o g y o f u n i f o r m convergence on bounded s e t s o f F.
S E F ' converges t o S E F ' means (S,,b)
-f
(S,b) u n i f o r m l y f o r b
Thus a n e t E
B with B
a bounded ' s e t i n F. I t i s e a s i l y checked ( e x e r c i s e ) t h a t t h e c l o s e d convex d i s c e d envelope of a
bounded s e t i s bounded and hence a FSN o f 0 i n F ' f o r t h e s t r o n g t o p o l o g y i s formed o f t h e p o l a r s o f c l o s e d bounded d i s c e d s e t s B of a d i s c e d B C F i s d e f i n e d by Bo = { x '
E
F';
C
F.
The p o l a r Eo
I ( x ' , x ) l 5 1 f o r x E B}.
C
F' In
31 7
APPENDIX A
general, f o r a r b i t r a r y B C F, one d e f i n e s Bo = I x ' E F ' : R e ( x ' , x ) 5 1 f o r x E
BI; t h e n o t i o n s a r e e q u i v a l e n t f o r d i s c e d B.
3
fro.
5 implies
Bo
= { f i n i t e sums 1 h . b J J' I f F i s a normed space, t h e n t h e s t r o n g t o p o l o g y i n F'
r B i s t h e convex d i s c e d envelope o f B; r B
b . E By
J
Note t h a t B C
1 Ihjl
5 11.
i s t h e s t a n d a r d norm t o p o l o g y determined, f o r example by ( 6 ) IIx'II = sup 1(x, x ' ) I (sup f o r # x l l = 1).
One s h o u l d check t h a t i f F i s a Banach space, t h e n so i s F' w i t h t h e norm ( 6 ) ( e x e r c i s e ) .
DEFINICL0N A.20.
The weak t o p o l o g y on F ' , denoted by u(F',F),
s e s t ( = weakest) t o p o l o g y on F ' making a l l t h e l i n e a r maps x '
C continuous ( x E F ) .
i s t h e coar-+
(x,x'
F'
):
+
Thus i n s t a n d a r d n o t a t i o n F C CF = IIF C and o(F',F)
i s t h e induced t o p o l o g y o f t h e p r o d u c t w i t h a FSN c o n s i s t i n g o f f i n i t e i n tersections o f sets I x '
E
F';
l(x',x)l
5
E}.
I t i s e a s i l y seen ( e x e r c i s e ) t h a t t h e weak o r s t r o n g t o p o l o g i e s on F " make
F ' i n t o a H a u s d o r f f space.
A b a r r e l i n a TVS i s a closed, convex, disced, a b s o r b i n g
DEFZNZCI0N A.21,
A TVS i s tonne16 o r b a r r e l e d i f e v e r y b a r r e l i s a NBH
s e t (a French j o k e ) .
R e c a l l t h a t a s e t V i s a b s o r b i n g i f i t absorbs any p o i n t x.
o f 0.
A B a i r e space i s a t o p o l o g i c a l space such t h a t e v e r y coun-
DEFINZCI0N $1.22,
t a b l e union o f c l o s e d s e t s w i t h o u t i n t e r i o r p o i n t s i t s e l f has no i n t e r i o r point.
CHEBREIII A.23. P400d:
A complete m e t r i z a b l e space i s B a i r e .
L e t E be a complete m e t r i z a b l e space w i t h d i s t a n c e f u n c t i o n d ( x , y ) .
Suppose E = UE,
En closed, w i t h no En c o n t a i n i n g a nonvoid open s e t .
El # E and CE1 = complement o f El open b a l l B1 = B
P ~ , E ~=)
does n o t c o n t a i n B(p1fi1/2),
I x E E; d(x,pl)
5 ~~1 w i t h
1/2.
j
(la1
5 j ) f o r some 9 j E D C One can assume IDalp.( 5 l / j f o r la( 5 j ( i . e . v j +. 0
= 1 (by l i n e a r i t y ) and then
i n Dt when we l e t j r u n ) a l t h o u g h
(
T,'p
J
.)
J
+
0.
T h i s c o n t r a d i c t s and hence
we can s t a t e
A l i n e a r map T: C i
tHE0REm 8.2.
+.
C i s a d i s t r i b u t i o n (T E D'(R)) i f and
o n l y i f T i s a continuous l i n e a r map DK
C f o r every K C R compact.
By Theorem 8.2 i n o r d e r t o t e s t whether o r n o t a s p e c i f i c ob-
REmARK 3.3, j e c t (e.g.
+
a d e l t d o b j e c t d e f i n e d by ( 6 , ~ )= ~ ( 0 ) )i s a d i s t r i b u t i o n one E
needs o n l y check i t s a c t i o n on convergent sequences o f t e s t f u n c t i o n s
j However l e t us mention t h a t t h e r e i s
I n p r a c t i c e t h i s i s a l l we need. DK. a t o p o l o g y on D = C i , c a l l e d a s t r i c t i n d u c t i v e l i m i t topology, which i s c h a r a c t e r i z e d by t h e p r o p e r t y t h a t a l i n e a r map T: D t i n u o u s i f and o n l y i f T: D K m i n i n g " sequence K
C
n
Kn+l
+
+.
F, F a LCS, i s con-
F i s continuous f o r each Kn i n any " d e t e r -
o f compact s e t s which exhaust R ( i . e .
R =
UK,).
T h i s a l l o w s one t o s p e c i f y d i s t r i b u t i o n s T E D'(n) as continuous l i n e a r maps T: D
+
D has t h e s t r i c t i n d u c t i v e l i m i t t o p o l o g y (and accounts
C when
f o r t h e d u a l i t y n o t a t i o n D-D'. Let R = R
EXAAIRPCE 3.4. clearly
( 6,q
j
)
=
'p
.(O)
J
1 +
'p + 0 i n D be a g e n e r i c sequence. Then j K 0 so t h e 6 o b j e c t i s a d i s t r i b u t i o n . For any f E
and l e t
E v i d e n t l y ( f , q ) + 0 so f = /f f ( x ) ' p ( x ) d x f o r 'p E .C: L1oc j determines a d i s t r i b u t i o n . I n p a r t i c u l a r one d e f i n e s t h e Heavyside funcdefine
(f,'p)
t i o n Y by Y(x) = 0 f o r x < 0 and Y(x) = 1 f o r x > 0. Now t h e main reason f o r c o n s t r u c t i n g a t h e o r y o f d i s t r i b u t i o n s was t o be a b l e t o d i f f e r e n t i a t e enough o b j e c t s so t h a t a t h e o r y o f l i n e a r p a r t i a l d i f f e r e n t i a l equations was p o s s i b l e .
Thus D i s c o n s t r u c t e d v i a a t o p o l o g y
based on d i f f e r e n t i a t i o n and by d u a l i t y we w i l l be a b l e t o d i f f e r e n t i a t e objects i n Dl. -(T,DkP):
D
More p r e c i s e l y l e t T +
D
+
C.
E
D' and c o n s i d e r t h e map M:
C l e a r l y M i s l i n e a r and Dk: DK
-f
'p .+
Dkv
+
DK I s continuous;
hence ( g i v e n t h a t t h e t o p o l o g y o f DK i s i n f a c t t h e t o p o l o g y induced by D) by Remark 8.3,
Dk: D + D i s continuous.
Since T: D
+
C i s c o n t i n u o u s by de-
f i n i t i o n , M i s continuous and hence determines an element i n D ' ( R ) -(T,Dk'p)).
=
T h i s leads t o
DEFZNltI0N 8-5- Given T T,Dkq ).
q ) = -(
(M(lp)
E
D' one d e f i n e s DkT by t h e formula
(IP E D) (DkT,
APPENDIX B
Given T = f
E)tAmPI;E 3.6.
E
1 C (0)we see t h a t D e f i n i t i o n 8 . 5 reduces t o t h e
standard f o r m u l a o f i n t e g r a t i o n by p a r t s . one has DY = 6 s i n c e (DY,qP= -(Y,v') DEFINZtI0N 8.7-
331
A p p l i e d t o T = Y o f Example 8.4
= -/rv'(x)dx = q(0) = ( 6 , ~ ) .
L e t E denote C"(n) w i t h t h e t o p o l o g y o f u n i f o r m convergence T h i s w i l l be a m e t r i z a b l e
on compact s e t s o f f u n c t i o n s and a l l d e r i v a t i v e s .
space ( t h e t o p o l o g y i s d e f i n e d by a c o u n t a b l e number o f seminorms) and conI f K C Kn+l w i t h R = UKn i s n a d e t e r m i n i n g sequence o f compact s e t s t h e n a sequence v k +. 0 i n E means
vergence can always be r e f e r r e d t o sequences. t h a t f o r any p and n f i x e d , supIDaqkI
-f
0 for x
Kn and la1 5 p.
E
The dual
space E ' ( = t h e space o f continuous l i n e a r maps E * C) i s i n f a c t t h e space o f d i s t r i b u t i o n s T w i t h compact s u p p o r t (we o m i t t h e p r o o f o f t h i s b u t i t i s routine
-
see t h e references c i t e d e a r l i e r ) .
Here one says t h a t T = 0 i n an
open s e t A C n i f ( T , q ) = 0 f o r a l l q E C i ( A ) .
The complement i n R o f t h e
u n i o n o f a l l such A where T = 0 i s c a l l e d supp T. DEFINt&I0N 8.8.
For R = Rn now l e t
5 denote t h e space o f Cm f u n c t i o n s IP n
(x E R ) f o r every a =
and B = ( B 1' B,). Such f u n c t i o n s a r e c a l l e d r a p i d l y decreasing and one says v k 0 2 m a i n 5 i f f o r any m and p f i x e d , sup I ( l + l x l ) D v k l 0 ( x E Rn) f o r la1 5 p.
such t h a t sup ( x B D a v ( x ) (
n
F i s an e n t i r e f u n c t i o n o f e x p o n e n t i a l t y p e T w i t h f E LL o n 2 F ( t ) e x p ( i t z ) d t where F E L (-T,T). t h e r e a l l i n e i f and o n l y i f f ( z ) = 1 F ( z ) i s an e n t i r e f u n c t i o n o f e x p o n e n t i a l t y p e T w i t h f E L tJHE0REIR 3.32. T F ( t ) e x p ( i t z ) d t where F(T) = F(-T) = 0 f o r z r e a l i f and o n l y i f f ( z ) =
CHE@REIII 8.31,
and t h e f u n c t i o n o b t a i n e d by e x t e n d i n g F t o be 0 o u t s i d e o f [-T,T]
has an
a b s o l u t e l y convergent F o u r i e r s e r i e s on any i n t e r v a l [-T-E,T+E]. Theorem 8.31 can be proved by d i r e c t e x t e n s i o n f r o m C; formula and a l i t t l e t h o u g h t
-
u s i n g t h e Parseval L e t us i n c l u d e h e r e
Theorem B.32 i s harder.
some i n f o r m a t i o n about l i n e a r semigroups which w i l l be u s e f u l a t v a r i o u s times ( c f .
[Cl;Dul;Ftl;Bzl;Hpl;Ka2;Tal;Yol]
A basic
for further details).
m o t i v a t i o n i s t o s o l v e t h e Cauchy problem f o r t h e simple case ( * a ) u ' + Au = 0, U ( T ) = u
0
where A i s a c o n s t a n t unbounded o p e r a t o r i n
A family S(t)
DEfZNltI@N 3-33,
a
6anach space.
L ( F ) ( L ( F ) = bounded l i n e a r o p e r a t o r s on
E
2t
t h e Banach space F) w i l l be c a l l e d a s t r o n g l y continuous semigroup ( 0
-) i f S(t+T) = S ( t ) S ( T ) , S(0) = I, and t
-+
0, d S ( t ) y / d t =
- A S ( t ) y = - S ( t ) A y when y E D(A) ( h e r e d / d t denotes o r d i n a r y v e c t o r valued d i f f e r e n t i a t i o n as a l i m i t o f d i f f e r e n c e q u o t i e n t s ) .
x
Phoud: y,
If y, = 10 S ( t ) y d t , then e v i d e n t l y x-'yx
= (l/t)/k
-
F.
a r e dense i n
W r i t i n g -At
-
[S(S+t)y
= [S(t)
-
-+
y as
x
-f
I ] / t we have as t
S(S)yIdE; = (l/t)I$+t S(S)ydS
-
0, and hence such +
0, (*&) -kty,
( l / t ) f b S(S)ydS
-,S(h)y Fur-
Since l i m Atyh e x i s t s , y, E D(A), which i s consequently dense.
y.
t h e r , f o r t > 0, A > 0, and y S ( t ) y ] = -S(t)A,y
-f
D(A) c o n s i d e r -A,S(t)y
€
x
- S ( t ) A y as
-+
-AS(t)y = d+S(t)ydt = -S(t)Ay.
Similarly, x-l[S(t)y
Ay,
-f
-f
d - S ( t ) y / d t = - S ( t ) A y as h
=
y, and AY,,
w.
-+
-AI:
S(S)ydS.
( * r ) we
E v i d e n t l y , AS(c)yn = S(c)Ayn
-f
w as t
-f
will
obtain
(**I
S(t)y
-
y =
Then t o show A i s c l o s e d l e t yn E D ( A ) , yn
(llS(~,)Il 5 c t h e r e by Banach-Steinhaus), = ( l / t ) $ S(S)wdg
S(t-A)y] = -S(t-A)
which means t h a t d S ( t ) y / d t e x i s t s as
Also, i n t e g r a t i n g and u s i n g
-1: AS(S)ydS
-
0, by s t r o n g c o n t i n u i t y ( s i n c e S ( t - A )
remain bounded by Banach-Steinhaus), described.
-
= h-l[S(t+x)y
0, which i m p l i e s S ( t ) y E D(A) w i t h
0.
-+
+
S(5)w u n i f o r m l y on [ O , t ]
and hence from (**) Aty = l i m Atyn
Thus y E D(A) w i t h Ay = w.
QED
APPENDIX B
341
Thus i f -A i s t h e i n f i n i t e s i m a l g e n e r a t o r o f a s t r o n g l y c o n t i n u o u s semigroup S ( t ) , i t f o l l o w s t h a t f o r any uo
E
D(A) t h e Cauchy problem (*a) has a s o l u -
t i o n u = S(t)uo, which i s i n f a c t s t r o n g l y d i f f e r e n t i a b l e f o r t > 0 w i t h u ' = -S(t)Auo continuous t h e r e .
tions u
E
This s o l u t i o n i s unique ( i n t h e class o f func1 E C ( F ) f o r t > 0 w i t h y ' t Ay = 0 and
C 1 (F) f o r t > 0 ) s i n c e i f y
c
y ( 0 ) = uo t h e n c o n s i d e r f o r 0
0, ( * m ) A a i x ( A ) y = - ( l / a ) i T e x p ( - h t ) [ S ( t + a ) y - S ( t ) y l d t = -(l/cx) exp( - A t ) S ( t ) y d t t (1 / a ) e x p ( h a ) / : exp( - A t ) S ( t ) y d t . Hence [exp( xa )-l]I," E
kh(A)
A
6
a "A
(A)y
-f
-XR",(A)y
t y as u
-f
0, and hence $(A)y E D ( A ) w i t h Aih(A)y =
F u r t h e r , f o r y E D(A) b o t h e x p ( - A t ) S ( t ) y and A e x p ( - A t ) S ( t ) y
-XR,(A)y t y. a r e c o n t i n u o u s and i n t e g r a b l e ; hence s i n c e A i s closed, i t can be c a r r i e d under t h e (Riemann) i n t e g r a l s i g n i n Theorem 8.35 t o o b t a i n ARVh(A)y = K,(A)Ay
(use here t h e d e f i n i t i o n o f t h e Riemann i n t e g r a l as a l i m i t o f sums)
Thus we have shown (A+XI)gh(A)y = y and f o r y E D ( A ) ,
[Afih(A)
t h I ] y = y which determines Gh[A)
as ( A + X I ) - l .
Rv,(A)(A+XI)y
=
QED
We c i t e n e x t t h e P h i l 1 ips-Miyadera e x t e n s i o n o f t h e H i l l e - Y o s i d a theorem.
342
ROBERT CARROLL
It i s o f f r e q u e n t use i n app i c a t i o n s o f semigroup t h e o r y t o s o l v i n g t h e
b u t w i l l n o t be needed i n f u l l s t r e n g t h here; we s k e t c h
Cauchy problem (*.)
i n s t e a d a p r o o f o f t h e c o r o l a r y which i s used more o f t e n .
A necessary and s u f f i c i e n t c o n d i t i o n f o r a c l o s e d l i n e a r
CHEBREIII 3.36.
o p e r a t o r -A w i t h dense domain t o generate a s t r o n g l y continuous semigroup S ( t ) i s that there exist
m
> 0 and w such t h a t Il(X1
C@RBCCARY B.37
+ A)-'II
5 m/(A-w)n
for
I n t h i s event IlS(t)ll 5 mexp(wt).
> w and i n t e g e r s n.
a l l real
A necessary and s u f f i c i e n t c o n d i t i o n f o r a
(HIftE-YBdIDA).
c l o s e d densely defined l i n e a r o p e r a t o r -A t o generate a s t r o n g l y continuous
5 l / X f o r X > 0.
semigroup o f c o n t r a c t i o n o p e r a t o r s i s t h a t 1I (xI+A)-lll
Ptooh:
F i r s t n o t e t h a t I I ( I + c Y A ) - ~ I5 I 1 for
f o r t 1. 0. E
F ) as t
-f
0 (exercise).
[Sm(t-s)Sn(s)xJds t h a t Sn(t)x
2 0 and s e t S n ( t ) = ( l + t A / n ) - n
Then IISn(t)ll 5 1, so t h a t t h e S n ( t ) a r e u n i f o r m l y bounded. f o r t > 0 and S n ( t ) x
thermore one has S;l(t) = -A(I+tA/n)-"'
(x
CY
-
Consider now [ S n ( t )
= l i m J:-'[-S,'(t-s)Sn(s)x
S,(t)x
[I+(~/n)A]-~-'xds.
= l i m I:-'[(s/n)
2
-
-
+
Fur-
Sn(0)x = x
Sm(t)]x = l i m j i E ( d / d s )
+ Sm(t-s)S;l(s)x]ds. It follows (t-~)/m]A~[I+((t-s)/m)A]-~-'
Then i f x E D(A ) we have an easy e s t i m a t e s i n c e i n p a r -
t i c u l a r t h e r e s o l v a n t o f A commutes w i t h A; one has t h e r e l a t i o n S n ( t ) x -m-1 [I t (s/n)A]-"' A'xds. S,(t)x = Jot [ ( s / n ) - ( t - s ) / m l [ I + ( ( t - s ) / m ) A l f o l l o w s t h a t IISn(t)x - S,(t)xll 2 11A2xllJ~[ ( s / n ) + (t-s)/m]ds = ( t 2/ 2 )
It
2 ( l / n + l/m)llA xII.
Hence t h e S n ( t ) x form a Cauchy sequence, and l i m S n ( t ) x 2 However, D(A ) i s dense i n 2 o u r Banach space F ( e x e r c i s e ) ; n o t e f o r example t h a t D ( A ) = (A+hI)-lD(A)
e x i s t s u n i f o r m l y i n t i n any f i n i t e i n t e r v a l .
f o r h > O a n d (A+XI)-l has range D ( A ) which i s dense.
By v i r t u e o f t h e u n i -
form boundedness o f t h e S n ( t ) i t f o l l o w s t h a t S ( t ) x = l i m S n ( t ) x = l i m ( I + t A / n ) - n x e x i s t s f o r any x E F ( c f . Theorem A.31).
I t i s easy now t o
check t h a t S ( t ) has t h e d e s i r e d p r o p e r t i e s f o r a s t r o n g l y c o n t i n u o u s semigroup and we l e a v e t h i s as an e x e r c i s e . As an a p p l i c a t i o n c o n s i d e r A = -A
+
BED
c d e f i n e d i n H = L L ( Q ) by D i r i c h l e t o r
Neumann c o n d i t i o n s ( c f . §1.9). Then i n e i t h e r case Re((A+h)u,u) = (ctReA) 2 I l u l l ~ t J l v u l dx and Theorem 1.9.2 i m p l i e s t h a t A+h i s 1-1 o n t o H f o r Rex 2 + c > 0. From Re((A+h)u,u) 2 (c+Reh)Null we o b t a i n furthermore II(A+X)ull 2 >
(c+Rex)llull and hence II(A+h)-lI1
< l/(c+Reh).
Consequently f o r c = 0 Coro-
l l a r y 6.37 i m p l i e s t h a t -A generates a s t r o n g l y continuous semigroup o f cont r a c t i o n o p e r a t o r s , s o l v i n g s t r o n g l y t h e r e f o r e t h e Cauchy problem f o r u ' Au = 0 w i t h D i r i c h l e t o r Neumann boundary c o n d i t i o n s .
APPENDIX 6
343
To r e l a t e t h e semigroup ideas t o monotone o p e r a t o r s as i n
3.3 we d e f i n e a
l i n e a r monotone o p e r a t o r A t o be maximal l i n e a r , o r s i m p l y maximal, i f i t i s n o t t h e p r o p e r r e s t r i c t i o n o f a n o t h e r monotone l i n e a r o p e r a t o r .
We empha-
s i z e t h a t t h i s i s n o t t h e same as s a y i n g t h a t A i s maximal D ( A ) monotone i n t h e sense o f D e f i n i t i o n 3.2.15.
R e c a l l a l s o t h a t monotone o p e r a t o r s i n
H i l b e r t spaces a r e c a l l e d a c c r e t i v e ( c f . 13.3).
ZHE0REIII 3-38, - A i s t h e i n f i n i t e s i m a l g e n e r a t o r o f a s t r o n g l y c o n t i n u o u s semigroup o f c o n t r a c t i o n o p e r a t o r s i n t h e H i l b e r t space H i f and o n l y i f A i s a ( c l o s e d ) maximal a c c r e t i v e l i n e a r o p e r a t o r w i t h dense domain. For p r o o f we n o t e f i r s t t h a t i f S ( t ) i s a c o n t r a c t i o n semigroup w i t h generat o r -A t h e n f o r y
E
D ( A ) and A
-f
0,
(A*)
2
0 L ( l / ~ ) { i l S ( ~ ) y -l l ~IIyll 1 =
- (YJY). Hence A
([S(A)Y-Y]/A,S(A)Y) + ( Y , [ S ( ~ ) Y - Y I / A ) + - ( A Y , Y )
t i v e and D(A) i s dense by Theorem 6.34 w i t h A closed.
i s accre-
For t h e r e m a i n i n g de-
t a i l s we r e f e r t o 13.3 and remarks below.
RrmARK 8-39, A maximal a c c r e t i v e l i n e a r o p e r a t o r i n a H i l b e r t space need n o t be closed, b u t a densely d e f i n e d maximal a c c r e t i v e o p e r a t o r A i s closed.
w, and u E D ( A ) t h e n r d e f i n e d on D(K) = D(A);A E C } by T(V+AU) = Av + A W i s an a c c r e t i v e e x t e n s i o n o f A,
Indeed, i f un E D(A), un .(v+Au;v
E
+
u, Aun
-f
which c o n t r a d i c t s ; however, i f u E D(A) t h e n by Theorem 3.3.3 Au = w s i n c e any l i n e a r A i s a u t o m a t i c a l l y hemicontinuous (see t h e p r o o f o f C o r o l l a r y 6. 41 below f o r r e l e v a n t d e t a i l s and see Remark 3.6.16
f o r a d i s c u s s i o n o f max-
imal m o n o t o n i c i t y i n t h e c o n t e x t o f m u l t i v a l u e d maps).
S i m i l a r l y a closed
maximal a c c r e t i v e o p e r a t o r i s densely d e f i n e d ( e x e r c i s e ) . We p r o v e now a v a r i a t i o n on Theorem 3.3.6
due t o Browder which can be used
i n c i d e n t a l l y t o complete t h e p r o o f o f Theorem 6.38.
CH€@REm 8.40.
F be a r e f l e x i v e Banach space and A: F
Let
hemicontinuous o p e r a t o r (D = D(A) -2
s i o n A t o any
3
D
f o r u E D with q(x)
Pmvd:
(E n o t + m
C
-f
F ' a D-monotone
F ) which has no proper monotone extenIfRe ( A ( u ) , u ) ~ q ( l l u I )IIull I
necessarily linear).
as x
-f
m,
then R(A) = F ' .
By Theorem 3.3.3 A i s maximal D-monotone.
R e f e r r i n g f o r comparison
t o t h e p r o o f o f Theorem 3.3.6 l e t w E F ' be a r b i t r a r y and A be t h e f a m i l y o f f i n i t e dimensional subspaces o f D o r d e r e d by i n c l u s i o n . F i s t h e i n j e c t i o n and we d e f i n e again A E = i f A i E .
F o r E E A, iE: E +
Then as b e f o r e , AE i s E-
monotone and hemicontinuous and hence continuous ( b y Thecrem 3.3.5) Re(AEu,u) ~ q ( l l u l ~ ) U u fl lo r
E'
U E
E ( c f . §3.3).
and we p i c k uE E E such t h a t AEuE = i f w .
with
By L e m a 3.3.7 AE maps E o n t o Then q(lI uil
)IluEII 5 Re
(
AEuE,uL)
344
ROBERT CARROLL
) = Re (w,u ) < IIwllIIuEll. Consequently, v(IIuEll) 5 IlwII and hence E E E Ilu II < M f o r some M independent o f E. L e t BM = { u E F;IIull 5 M I and then by E weak compactness ( c f . Theorem A.43) t h e r e e x i s t s uo E BM such t h a t f o r each
= Re ( i*w,u
Eo E A, each f i n i t e s e t vl,
I( u,-uE,vj
w i t h E 3 Eo and
...vm i n F ' , and each ) I 5 f o r 1 5 j 5 m. E
Eo E A c o n t a i n i n g v and l e t E E A w i t h Eo
E
> 0, one can f i n d E E A
Now f o r any v E D p i c k some
E; t h e n (AA) Re (Av-w,v-uE) = Re (Av-AuE,v-u ) > 0 s i n c e AEuE = i * w means AiEuE AuE = w. But choosing E E v1 = Av-w above t h e r e e x i s t s E 3 Eo w i t h l( uo-uE,Av-w)( 5 E . Hence Re ( A v -w,v-uo) -w,v-u
O
from (")
-E
)
and, s i n c e
C
i s a r b i t r a r y , we conclude t h a t Re ( A v -
E
Then i f uo E D one deduces t h a t Auo = w s i n c e A i s maximal D-
5 0.
But uo must be i n D, s i n c e if n o t we c o u l d d e f i n e a monotone op-
monotone.
N
N
erator
2
A w i t h D(K) = D(A) u uo, Au
0
QED
c o n t r a d i c t s t h e hypotheses. Now suppose A: V
+
This
w, and Au = Au f o r u E D ( A ) .
=
V ' i s a l i n e a r monotone o p e r a t o r w i t h dense ( l i n e a r ) do-
main D = D(A) i n a r e f l e x i v e Banach space V.
It i s a u t o m a t i c a l l y hemicon-
tinuous, s i n c e from u E 0, w E V, u+tnw E D (which i m p l i e s w E D),
it fol-
lows t h a t A(u+tnw) = Au + t n A w Au, s t r o n g l y i n f a c t ( c f . D e f i n i t i o n 3.2. 2 1 5 ) . I f Re ( A u , u ) 2 cIIuII f o r example then f o r any w E V ' Theorem B.40 -f
y i e l d s uo E V such t h a t Re (w-Av,uo-v)
5 0 for a l l v
be maximal D(A)-monotone by Theorem 3.3.3
Hence i f uo
t o n e o p e r a t o r s a r e D maximal monotone). But again u monotone.
0
further A w i l l
E D(A);
( t h u s densely d e f i n e d l i n e a r mono€
D ( A ) t h e n Auo = w.
must be i n D(A) i f we assume f o r example t h a t A i s maximal N
Indeed, i f uo $ D(A),
d e f i n e A 3 A on D(X) = Iu+auo;u
E
D(A);a E
N
Cl by A(u+auo) (
,
= Au
+
aw.
Then s e t t i n g v = -u/a E D(A) and r e c a l l i n g t h a t N
i s c o n j u g a t e l i n e a r d u a l i t y , t h e r e r e s u l t s Re (A(u+auo),u+auo)
)
Re (wo-Av,uo-v)
C0R0LCARM 3.41.
5 0 f o r a l l u E D(A).
=
2
T h i s c o n t r a d i c t s and we have proved
L e t V be a r e f l e x i v e Banach space and A: V
-f
d e f i n e d maximal monotone l i n e a r o p e r a t o r s a t i s f y i n g Re ( A u , u )
R(A)
= la1
V ' a densely
1. cIIuII 2 . Then
V'.
I n p a r t i c u l a r i f A i s maximal a c c r e t i v e i n a H i l b e r t space H and h > 0 t h e n 2 Moreover, r e f e r r i n g back t o t h e p r o o f
A+h s a t i s f i e s Re((A+h)u,u) 2 XIIuII o f Theorem 8.40,
.
l e t A = A + X w i t h AEuE =
itw,
uo as b e f o r e ; t h e n (")
r e w r i t t e n as Re((A+h)v-w,v-uE) = Re((A+h)v-(A+h)uE,v-uE) v
E
D(A) ( s i n c e as b e f o r e AEuE =
o f course
it
i t w means AiEuE = AuE =
can be
XIIv-u E112 f o r a l l (A+h)uE = w
-
here
i s taken w i t h r e s p e c t t o t h e s c a l a r product i n H ) . Then by 2 1Iv-u 112 (upon s e l e c t i n g a subnet
weak l o w e r s e m i c o n t i n u i t y l i m i n f IIv-uEll
0
uEa converging weakly t o u o ) and, as before, Re((A+h)v-w,v-uEa)
can be made
APPENDIX a r b i t r a r i l y c l o s e t o Re( (A+A)v-w,v-uo). -w,v-u
> Allv-u 112 > 0. 0)0 -
B
345
Hence one concludes t h a t Re( (A+h)v
Now a r g u i n g as before, if uo E D(A) = D(A+A) t h e n
(A+A)uo = w, s i n c e A+A i s maximal O(A)-monotone.
4
On t h e o t h e r hand, i f uo N
cy
D(A),
c o n s t r u c t (A+h)-
A + h as above ( b u t now (A+A)uo = w, e t c . ) and N
XIIv-u It2 w i l l i m p l y t h a t A i s an
observe t h a t t h e n Re((A+x)(v-uo),v-uO)
0
a c c r e t i v e l i n e a r e x t e n s i o n o f A which i s excluded.
Hence we have proved
t h a t i f A i s maximal a c c r e t i v e .(and densely d e f i n e d ) t h e n R(A+A) = H f o r any A > 0; moreover,
2 hllul12 we a l s o o b t a i n II(A+A)-lII
from Re((A+h)u,u)
Then, u s i n g C o r o l l a r y B.37 and Remark B.39,
5 l/A.
i t f o l l o w s t h a t -A generates a
s t r o n g l y continuous semigroup o f c o n t r a c t i o n s , which proves one h a l f o f Theorem B.38. To complete t h e p r o o f o f Theorem B.38 i t i s o n l y necessary t o show t h a t t h e c l o s e d densely d e f i n e d a c c r e t i v e o p e r a t o r A o f
(A*)
i s maximal a c c r e t i v e .
Now we know by C o r o l l a r y 8.37 t h a t A+A i s 1-1 o n t o H f o r A > 0.
Then assume
N
A i s a maximal a c c r e t i v e e x t e n s i o n o f A (whose e x i s t e n c e f o l l o w s f r o m a v e r s i o n o f Z o r n ' s lemma f o r example
- exercise - c f . [Kel]).
From what we have
j u s t proved x + h i s 1-1 o n t o H f o r a l l A > 0 ( C o r o l l a r y 8.41). C
x, one has (Z+A)(A+A)-'x
= x so A+:
maps D ( A ) 1-1 o n t o H.
But s i n c e A Hence D(A) =
N
D ( A ) and A i s i n f a c t maximal a c c r e t i v e .
Theorem 8.38 i s t h u s c o m p l e t e l y
proved, a l b e i t somewhat c i r c u i t o u s l y , and we have i n c i d e n t a l l y proved
A densely d e f i n e d l i n e a r a c c r e t i v e o p e r a t o r i n a H i l b e r t space
LEarmA B.42.
H i s maximal a c c r e t i v e i f and o n l y i f R(A+hI) = H f o r a l l A > 0.
L e t us mention here a few f a c t s about Sobolev spaces f o r r e f e r e n c e a t v a r i ous p l a c e s i n t h e book (see [Adl;Mgl]
f o r further details).
bp(R) i s t h e space o f ( e q u i v a l e n c e c l a s s e s o f ) f u n c t i o n s P u E L P ( n ) such t h a t D"u E L P ( n ) f o r la1 2 m. The norm i s d e f i n e d by IIuI1 m,P = Jn I D " ~ l ~ d x 1 (sum ~ ' ~ o v e r la1 ( m ) . Wm(n) i s a r e f l e x i v e Banach space P i n Wm f o r p # 1,- ( e x e r c i s e ) . Wm(n)o i s d e f i n e d t o be t h e c l o s u r e o f P P' Now we s h a l l c a l l a bounded r e g i o n R C Rn v e r y r e g u l a r i f i t s boundary r i s
DEFZNZCZ0N %.43.
[I
Cy
a Cm compact (n-1)-dimensional m a n i f o l d w i t h R l y i n g on one s i d e o f
r.
We
say t h a t R s a t i s f i e s t h e cone c o n d i t i o n i f t h e r e i s a f i x e d cone K such t h a t a t any p o i n t p E i n a.
r
one can p l a c e t h e v e r t e x a t p and have K-p l i e e n t i r e l y
E v i d e n t l y a v e r y r e g u l a r r e g i o n s a t i s f i e s t h e cone c o n d i t i o n .
CHE0REIII 3-44.
L e t n C Rn be a bounded open s e t s a t i s f y i n g t h e cone c o n d i -
t i o n and Rs t h e i n t e r s e c t i o n o f R w i t h any t r a n s l a t e o f RS ( s 5 n; m and s integers).
Then @(R) P
C
L q ( R S ) a l g e b r a i c a l l y and t o p o l o g i c a l l y ( i . e .
346
ROBERT CARROLL
continuous i n j e c t i o n ) f o r n > mp, n-mp < s, q 5 sp/(n-mp).
If n i s (e.g.)
holds f o r n-m 5 s.
mp, n-mp < n-1, q 5 (n-l)p/(n-mp)
For p = 1 t h i s
v e r y r e g u l a r , then $(n) C Lq(T) f o r n > ( t h e sense i n which "values" on ns o r
r
a r e determined i s i n d i c a t e d below). Thus i n p a r t i c u l a r one has Wm(n) C Lnp/(n-mp)(n) and W"(n) C L(n-l)P/(n-mp) P1 ( r ) w h i l e f o r n = 3 we have W, (n) C L 6 (n) and W,1(n) C Lp4 ( r ) . We s h a l l c a l l such theorems Sobolev t y p e embedding theorems.
One should a l s o r e c a l l t h e
elementary f a c t s ( e x e r c i s e ) t h a t f o r bounded n, LP(n) C Lq(n) f o r p 2 q, a l g e b r a i c a l l y and t o p o l o g i c a l l y , and f u r t h e r i f f E Lp, g E Lq, t h e n f g E Ls f o r 1/s = l / p
LP norm and p,q,s compact f o r l / p
5 IIfll IIgll
(see [ D u l l ) ; here II II denotes t h e P . P q is 2 1. We r e c a l l a l s o t h a t t h e embedding WS(n) + W:(n) P ( s - j ) / n < l / r ( c f . [Adl;Myl]).
l / q w i t h Ilfgll,
t
-
To d e s c r i b e t h e boundary values (and values on l o w e r dimensional s l i c e s ) i n -
r
d i c a t e d i n Theorem 8.44 we s h a l l d i s c u s s t h e t r a c e you on E Wm(s?). D e f i n e D(fi) t o be t h e r e s t r i c t i o n o f D(Rn) t o
P r e g u l a r D ( 6 ) i s dense i n Wm(n) ( c f . [ l i 2 ] ) . P where n i s t h e u n i t i n t e r i o r normal and u E
When n i s v e r y
D e f i n e y . u = a j u / a n J on J
r,
D(6). Then y . can be t h o u g h t o f J 1, m > 0, and extends by c o n t i n -
as a map i n t o Wm-J-l'p(r) f o r example, p > P u i t y t o a l l Wm(n). I n f a c t one has ( c f . [ L i z ] ) P QXE0REfll 3-45, For m > 0 an i n t e g e r , p > 1, t h e map y : u ul: $(a)
o f a function u
c.
Iyou,ylu,
-f
...,
nWm-J-l/p(r) i s continuous and onto. Thus Ily .uII i n ' h - 1 /PI P J t h e boundary space i n d i c a t e d i s bounded by c1IuII i n Wm S i m i l a r l y given g j P' E Wm-j-l'p(r) t h e r e e x i s t s J E Wm(n) such t h a t y . u = g . and IIuII i n Wi(n) 5 P P J J c~ Ilg.11 (norm g i n w m - j - l / P , sum f o r o 5 j 5 [m-l/p]). J j p For p = 2 one w r i t e s Hm(n) = W!(n) f o r example and we c i t e a n o t h e r embedding -+
theorem o f Sobolev t y p e which i s f r e q u e n t l y u s e f u l
tHE0REB B.46,
I f R i s v e r y r e g u l a r (bounded) t h e n Hm(R) C C
c a l l y and t o p o l o g i c a l l y f o r 2k wn():
c
k (6) a l g e b r a i -
2m-n ( k and n a r e i n t e g e r s ) ; s i m i l a r l y
i f j < Zm-n/p.
c
We w i l l have occasion t o use f u n c t i o n s 9 = 1 i n a NBH o f a g i v e n compact K with
J, E
and K
C:
f o r example.
To c o n s t r u c t such f u n c t i o n s l e t
t h e compact s e t o f a l l p o i n t s a t d i s t a n c e -