Differential Equations (North Holland Mathematics Studies,92)

DIFFERENTIAL EQUATIONS

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NORTH-HOLLAND MATHEMATICS STUDIES

DIF'F'ERENTIAL EQUAnONS Proceedings of the Conference held at The University of Alabama in Birmingham, Birmingham, Alabama, U.S.A. 21-26 March, 1983 Edited by

Ian W. KNOWLES and Roger T. LEWIS The University ofAlabama in Birmingham Birmingham Alabama U.S.A.

IS84 NORTH-HOLLAND - AMSTERDAM. NEW YORK. OXFORD

92

Elsevier Science Publishers B . V . 1984 AN rights reserved. N o part of this publication may be reproduced. stored in a retrieval system, @

~

or transmitted, in any form o r by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

I S B N : 0 444 84875 5

Publishers: ELSEVIER SCIENCE PUBLISHERS B . V . P.O. Box 1991 1000 BZ Amsterdam The Netherlands Sole distributors for the U . S .A . and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY. INC 52 Vanderbilt Avenue NewY0rk.N.Y. 10017 U.S.A.

Library of Congmi Cataloging in Publication Data

Main entry under title: Differential equations. (North-Holland mathematics studies ; 9 2 ) Lectures given at the International Conference on Differential Equations, held at the University of Alabama in Bi&ngham during March 21-26, 1983”--Pref. Bibliography: p. Includes index. 1. Differential equations--Congresses. I. Knowles, Ian W. 11. Lewis, Roger T. 111. International Conference on Differential Equations (1983 : University of Alabama) IV. Series.

~ 3 7 0 D53 . 1984 ISBN 0-444-86a75-5

515 -3 5

84-4060

PRINTED IN THE NETHERLANDS

V

PREFACE

T h i s volume forms a permanent r e c o r d o f l e c t u r e s q i v e n a t t h e I n t e r n a t i o n a l Conference on D i f f e r e n t i a l Equations h e l d a t t h e U n i v e r s i t y o f Alabama i n Birmingham d u r i n g March 21-26,

1983.

The c o n f e r e n c e was supported by about 150 mathematicians from t h e f o l l o w i n g countries:

Belgium, Canada, China, Eqypt, Enqland, Federal R e p u b l i c o f Germany,

France, I n d i a , I r e l a n d , I s r a e l , Japan, Mexico, N i g e r i a , Norway, R e p u b l i c o f South

U. S. S. R. and t h e U. S . A . I t s main t o p r o v i d e a forum f o r t h e d i s c u s s i o n o f r e c e n t work i n t h e t h e o r y o f purpose was

A f r i c a , Scotland, Sweden, S w i t z e r l a n d , t h e

o r d i n a r y and p a r t i a l d i f f e r e n t i a l e q u a t i o n s , b o t h l i n e a r and n o n - l i n e a r , under t h e g e n e r a l heading o f t h e e q u a t i o n s o f mathematical p h y s i c s . hour l e c t u r e s were g i v e n by

s.

unified

I n v i t e d one-

Agmon, F. V. A t k i n s o n , H. B r e z i s , R. C a r r o l l ,

M. C r a n d a l l , I. Ekeland, V . Enss, J . F r o h l i c h ,

T. Kato,

P. Lax,

E. L i e b ,

C . Morawetz, P. R a b i n o w i t z , M. Reed, P. Sarnak, M. Schechter, B. Simon, J . Smoller,

R. Temam, and K. Yajima.

The remainder o f t h e proqramme c o n s i s t e d o f i n v i t e d one-

ha1 f hour 1ec t u r e s . On b e h a l f o f t h e p a r t i c i p a n t s , t h e c o n f e r e n c e d i r e c t o r s acknowledge, w i t h g r a t i t u d e , t h e qenerous f i n a n c i a l s u p p o r t p r o v i d e d b y t h e N a t i o n a l Science Foundat i o n , under g r a n t number MCS-8214420, and b y t h e School o f N a t u r a l Sciences and Mathematics, U n i v e r s i t y o f Alabama i n Birmingham.

We a r e e s p e c i a l l y g r a t e f u l t o

P r o f e s s o r P e t e r O ' N e i l , Dean o f t h e School, and P r o f e s s o r L o u i s Dale, Chairman o f t h e Department o f Mathematics, f o r t h e i r s u p p o r t and encouragement.

We acknow-

l e d g e a l s o t h e v a l u a b l e s u p p o r t p r o v i d e d b y t h e f a c u l t y and s t a f f o f t h e Department o f Mathematics; here, we a r e p a r t i c u l a r l y g r a t e f u l t o P r o f e s s o r Fred Martens f o r h i s c o n s i d e r a b l e c o n t r i b u t i o n i n d i r e c t i n a t h e l o c a l arrangements, and t o Mrs. E i l e e n Schauer f o r u n d e r t a k i n g t h e enormous t a s k o f t y p i n g much o f t h e c o n f e r e n c e m a t e r i a l , i n c l u d i n g many o f t h e papers a p p e a r i n g i n t h i s volume. F i n a l l y , i t i s a p l e a s u r e once a g a i n t o acknowledge t h e f r i e n d l y and p a t i e n t a s s i s t a n c e o f Drs. A r j e n Sevenster, E d i t o r o f t h e Mathematics S t u d i e s S e r i e s o f N o r t h - H o l l a n d , d u r i n q t h e p r e p a r a t i o n o f t h e s e proceedinqs. I a n W . Knowles Roqer T. Lewis Conference D i r e c t o r s

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vii

CONTENTS

Preface L e c t u r e s Not Appearing i n t h e Proceedings Address L i s t o f C o n t r i b u t o r s

V

xiii xv

U n i f o r m D i s s i p a t i v e S o l u t i o n s f o r a T h i r d Order Non-Linear D i f f e r e n t i a1 E q u a t i o n A.U. Afuwape

1

On P o s i t i v e S o l u t i o n s o f E l l i p t i c Equations w i t h P e r i o d i c C o e f f i c i e n t s i n R n , S p e c t r a l R e s u l t s and Extensions t o E l l i p t i c Operators on Riemannian M a n i f o l d s S. Agmon

7

C l a s s i f i c a t i o n o f I n i t i a l Data f o r t h e Porous Medium E q u a t i o n i n R N N.D. A l i k a k o s and R. Rostamian

19

S t a b i l i t y o f Quantum Mechanical Shape Resonances v i a R i c c a t i ' s Equation M. Ashbaugh and C. Sundberg

25

Remarks on t h e I n v e r s e Square P o t e n t i a l i n Quantum Mechanics P. Baras and J.A. G o l d s t e i n

31

J u l i a Sets and Autonomous D i f f e r e n t i a l Equations M.F. B a r n s l e y and A.N. H a r r i n g t o n

37

A L i m i t i n g A b s o r p t i o n P r i n c i p l e f o r a Sum o f Tensor Products M. B e n - A r t z i and A. D e v i n a t z

43

An A l g e b r a i c G e n e r a l i z a t i o n o f S t o c h a s t i c I n t e g r a t i o n M. Berger and A. Sloan

49

V a r i a t i o n a l Problems i n v o l v i n g Lack o f Compactness and R e l l i c h ' s Conjecture H. B r e z i s

53

A F a c t o r i z a t i o n Method f o r Symmetric D i f f e r e n t i a l Operators and i t s A p p l i c a t i o n s t o D i r i c h l e t I n e q u a l i t i e s and t o t h e D i r i c h l e t Index R . C . Brown

61

S o l u t i o n s w i t h Asymptotic C o n d i t i o n s o f a N o n l i n e a r Boundary Value Problem N.P. CLc

71

A b s o l u t e Continuous Spectrum o f One-Dimensional Schrodinger Operators R. Carmona

77

viii

Contents

Some Topics i n Transmutation R. C a r r o l l

87

An Equation Modeling t h e E l e c t r i c B a l l a s t R e s i s t o r N. Chafee

105

Nonexistence o f P o s i t i v e S o l u t i o n s f o r S i n g u l a r Hyperbolic D i f f e r e n t i a l Inequalities C.Y. Chan

111

Almost P e r i o d i c i t y o f Bounded S o l u t i o n s t o Nonlinear A b s t r a c t Equations C. Corduneanu and J.A. G o l d s t e i n

115

A P r i o r i Estimates i n Nonlinear Eigenvalue Problems f o r E l l i p t i c Systems C. Cosner

123

Developments i n t h e Theory o f Nonlinear F i r s t - O r d e r P a r t i a l D i f f e r e n t ia1 Equations M.G. Crandall and P.E. Souganidis

131

R e l a t i v i s t i c Molecules w i t h Coulomb I n t e r a c t i o n I . Daubechies and E.H. L i e b

143

Non-Linear Delay D i f f e r e n t i a l Equations and F u n c t i o n Algebras L.D. Drager and W. Layton

149

Transformations o f D i f f e r e n t i a l Equations, t h e Levinson Asymptotic Theorem and D e f i c i e n c y I n d i c e s M.S.P. Eastham

155

A Morse Theorem f o r Hamiltonian Systems I.Ekeland

165

S c a t t e r i n g and Spectral Theory f o r Three P a r t i c l e Systems V . Enss

173

A L e f t D e f i n i t e Two-Parameter Eigenvalue Problem M. Faierman

205

A Semigroup Approach t o Burgers' System W.E. F i t z g i b b o n

213

On t h e Eigenvalues o f Non-Definite E l l i p t i c Operators J . F l e c k i n g e r and A.B. M i n g a r e l l i

219

Existence o f Generators and D i f f e r e n t i a b i l i t y o f E v o l u t i o n s M.A. Freedman

229

On t h e Asymptotic Behavior o f t h e P o s i t i v e S o l u t i o n s o f a D i f f e r e n t i a l Equation w i t h a Discontinuous Nonlinear Term J.R. Graef, P.W. Spikes, and M.K. Grammatikopoulos

237

Energy Estimates f o r Symmetric Hyperbolic I n t e g r o - D i f f e r e n t i a l Equations R. Grimmer and M. Zeman

241

Resolvent and Heat Kernels f o r Operators o f Schrodinger Type w i t h A p p l i c a t i o n s t o Spectral Theory 0. Gurarie

249

Contents

ix

V o r t i c i t y , I n c o m p r e s s i b i l i t y , and Boundary Conditions i n t h e Numerical S o l u t i o n of t h e Navier-Stokes Equations K. Gustafson and K. H a l a s i

257

Asymptotic Completeness f o r Few Body Schrodinger Operators G.A. Hagedorn and P.A. Perry

265

Asymptotics o f t h e Titchmarsh-Weyl m-Function, a Bessel-Approximative Case S.G. Halvorsen

271

L a t t i c e M u l t i s c a l e S i n g u l a r P e r t u r b a t i o n Theory C.R. Handy

279

S o l u t i o n s f o r Model Boltzmann Equations proposed by Z i f f D.P. Hardin and J.V. Herod

285

Asymptotic Behavior o f S o l u t i o n s o f Disconjugate D i f f e r e n t i a l Equations D. H i n t o n

293

Boundary Conditions f o r D i f f e r e n t i a l Systems i n I n t e r m e d i a t e L i m i t Situations A.M. K r a l l , J.K. Shaw, and D.B. H i n t o n

301

On t h e Spectrum o f a Hamiltonian System w i t h Two S i n g u l a r Endpoints J.K. Shaw and D.B. Hinton

307

A Product Formula f o r C e r t a i n Q u a d r a t i c Form P e r t u r b a t i o n s R.J. Hughes

313

On t h e Existence o f Resonant States K. I n g B l f s s o n

32 1

Asymptotic Behavior o f t h e S c a t t e r i n g Amplitude a t High Energies H. I s o z a k i and H. Kitada

327

F u l l - and Half-Range Theory o f an I n d e f i n i t e S t u r m - L i o u v i l l e Problem H.G. Kaper

335

Remarks on Holomorphic F a m i l i e s o f Schrodinger and D i r a c Operators T. Kato

341

Necessary and S u f f i c i e n t Conditions f o r S o l v a b i l i t y o f Non-Solvable 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 S. K i r o

353

R e g u l a r i t y P r o p e r t i e s o f Schrodinger Operators on Domains of Rn M.A. Kon

359

R e l a t i v e Symmetries o f D i f f e r e n t i a l Equations B.A. Kupershmidt

367

Necessary and S u f f i c i e n t Conditions f o r O s c i l l a t i o n s o f Higher Order Del ay D i f f eren t ia 1 Equations G. Ladas, Y.G. Sficas, and I . P . S t a v r o u l a k i s

373

Boundary Behavior o f S o l u t i o n s o f Degenerate E l l i p t i c Equations and Generation o f Semigroups M. L a n g l a i s

381

X

Contents

The Zero D i s p e r s i o n L i m i t o f t h e KdV Equation P.D. Lax

387

Recent M u l t i p l i c i t y R e s u l t s f o r N o n l i n e a r Boundary Value Problems A.C. Lazer and P.J. McKenna

391

Nonstopping I t e r a t i o n f o r O r d i n a r y D i f f e r e n t i a l O p e r a t o r E q u a t i o n S.J. Lee

397

Some Vector F i e l d Equations E.H. L i e b

403

Weight D i s t r i b u t i o n s and Moments f o r a C e r t a i n Class o f Orthogonal Polynomials L.L. L i t t l e j o h n

413

On t h e L i m i t - P o i n t C l a s s i f i c a t i o n o f a Class o f N o n - S e l f - A d j o i n t O r d i n a r y D i f f e r e n t i a l Operators J.-L. L i u

42 1

A Moving Boundary Problem d e s c r i b i n g Oxygen Consumption i n S o i l R.C. McCann and P.K. McConnaughey

427

Converse BVP f o r A s s o c i a t e d E l l i p t i c and P a r a b o l i c F r a c t i o n a l P a r t i a l D i f f e r e n t i a 1 Operators P.A. McCoy

431

Bounds f o r C o n s t r u c t e d S o l u t i o n s o f Second and F o u r t h Order I n v e r s e Eigenvalue Problems J.R. McLaughlin

437

D i f f e r e n c e E q u a t i o n Models o f D i f f e r e n t i a l Equations h a v i n g Zero Local Truncation Errors R.E. Mickens

445

A Remark on Continuum E i g e n f u n c t i o n s o f N-Body Schrodinger Operators P.A. P e r r y

451

A C u r i o u s S i n g u l a r P e r t u r b a t i o n Problem P.H. Rabinowitz

455

A S i m p l i f i e d C h a r a c t e r i z a t i o n o f t h e Boundary C o n d i t i o n s which determine J - S e l f a d j o i n t E x t e n s i o n s o f J-Symmetric ( D i f f e r e n t i a l ) Operators D. Race

465

E s s e n t i a l S e l f - A d j o i n t n e s s f o r Powers o f Schrodinger Operators T.T. Read

47 1

Geometry and D i s c r e t e V e l o c i t y Approximations t o t h e Boltzmann E q u a t i o n M.C. Reed

477

Domains i n H y p e r b o l i c Space and L i m i t Sets o f K l e i n i a n Groups P. Sarnak

485

S e l f a d j o i n t Operators, S p e c t r a l and S c a t t e r i n g Theory, V a r i a t i o n a l Techniques, Non-Linear Phenomena, L i n e a r and Non-Linear P a r t i a l D i f f e r e n t i a l Equations, and R e l a t e d T o p i c s M. Schechter

501

Contents

xi

On Quasiuniqueness V . Schuchman

511

m-Functions and t h e A b s o l u t e l y Continuous Spectrum o f One Dimensional Almost P e r i o d i c Schrodinger Operators B. Simon

519

The Complete S o l u t i o n Space f o r a System o f R e a c t i o n - D i f f u s i o n Equations J.A. S m o l l e r

521

V a r i a t i o n a l Problems w i t h S i n g u l a r S o l u t i o n s R. Temam

537

B i f u r c a t i o n o f Subharmonic S o l u t i o n s : A Generic Approach A . Vanderbauwhede

545

A Class o f i s o p e r i m e t r i c V a r i a t i o n a l Problems on C e r t a i n Or1 icz-Sobolev Spaces P.A. V u i l l e r m o t

553

The R e l a t i o n o f S o l u t i o n s o f D i f f e r e n t ODES i s a Commutation R e l a t i o n B.F. W h i t i n g

561

P o i n t w i s e I n i t i a l - V a l u e Problems f o r F u n c t i o n a l D i f f e r e n t i a l Equations

J . Wiener

571

Asymptotics and S p e c t r a l Theory f o r High Order O r d i n a r y D i f f e r e n t i a l Equations w i t h Power C o e f f i c i e n t s A.D. Wood

581

Large Time Behaviors o f T i m e - P e r i o d i c Quantum Systems K. Yajima

589

Eigenvalues o f t h e L a p l a c i a n : An E x t e n s i o n t o H i g h e r Dimensions (11) E.M.E. Zayed

599

A u t h o r Index

607

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xiii

LECTURES NOT APPEARING I N THE PROCEEDINGS

Y. A k y i l d i z V.

Alexiades

S y m p l e c t i c geometry o f s h a l l o w w a t e r waves On b i n a r y a1 l o y s o l i d i f i c a t i o n

F. V. A t k i n s o n

Asymptotics o f the g e n e r a l i z e d Embden-Fowler e q u a t i o n

A. Azzam

S i n g u l a r i t i e s n e a r c o r n e r s o f s o l u t i o n s o f mixed boundary v a l u e problems f o r e l l i p t i c e q u a t i o n s i n s e c t i o n a l l y smooth domains

C . Bandle

Free boundaries i n n o n l i n e a r plasma problems

Edward Be1bruno

A f a m i l y o f p e r i o d i c o r b i t s i n t h e three-dimensional r e s t r i c t e d three-body problem

K. J . Brown

Steady s t a t e s o l u t i o n s o f systems o f r e a c t i o n d i f f u s i o n equations modelling c o - e x i s t i n g populations

Saber Elayd

N e g a t i v e a t t r a c t i o n and s t a b i l i t y i n semiflows

M. Hossam E -Din and Chen-Han Sung

Asymptotic b e h a v i o r on s o l u t i o n s o f a p e r t u r b e d 1i n e a r system o f o r d i n a r y d i f f e r e n t i a l equations

L . S. Frank

On a c l a s s o f s e m i l i n e a r e l l i p t i c s i n g u l a r perturbations

J. Frohlich

Unbounded, symmetric semigroups and v i r t u a l r e p r e s e n t a t i o n s o f symmetric spaces

I. M. G a l i and H. A. E l - S a i f y C o n t r o l o f systems governed by i n f i n i t e order equation o f hyperbolic type B a s i l i s Gidas

Free boundary problems f o r degenerate n o n l i n e a r e l l i p t i c d i f f e r e n t i a l e q u a t i o n s and a p p l i c a t i o n s t o physics

Louis G r i m

A n a l y t i c solutions o f d i f f e r e n c e equations

John R . Haddock

Converqence p r o p e r t i e s o f a semigroup on t h e memory space C y

Evans M. H a r r e l l , I 1

S p e c t r a l t h e o r y as a t o o l f o r n o n l i n e a r POEnonexistence o f s o l u t i o n s

I r a Herbst

Absence o f p o s i t i v e e i g e n v a l u e s f o r t h e Schr-odinger equation

x iv

Lectures Not Appearing in the Proceedings

R. Jensen

Domain of dependence f o r v i s c o s i t y solutions of f i r s t order p a r t i a l d i f f e r e n t i a l equations

Robert M. Kauffman

Limit-point c r i t e r i a f o r a c l a s s of p a r t i a l d i f f e r e n t i a l operators with l a r g e positive higher order coefficients

Ronald A. Knight

Fundamental dynamic r e l a t i o n a l rninimality

Tibor Krisztin

On the convergence of solutions of functional d i f f e r e n t i a l equations with i n f i n i t e delay

R . Mennicken

Eigenvalue problems non-linear in the parameter

Pedro Morales

Topological properties of the s e t of solutions f o r some Cauchy problems in l o c a l l y convex spaces

C. Morawetz

Some new phenomena f o r a nonlinear wave equation

J . W . Neuberger

Steepest descent f o r systems of nonlinear p a r t i a l di fferen t ia1 equations

Bill Patula

Riccati type transformations f o r second order l i n e a r difference equations

R . Vittal Rao and K . B. Athreya F i n i t e section integral operators spectrum and Kac-Akhiezer formula

-

V . Sree Hari Rao

Existence of solutions f o r a system of second order periodic boundary value problems

Louise A. Raphael

C r i t e r i a f o r equisummability under a n a l y t i c m u l t i p l i e r s

Eric Schechter

Gihman’s convergence c r i t e r i o n and compact perturbations of m-dissipative operators

R . Schianchi and E. Mascolo

Recent existence r e s u l t s f o r non convex v a r i a t i o n a l problems

Penny Smith

Regularity f o r sinqular non-linear e l l i p t i c systems i n weighted Sobolev spaces

Clasine van Winter

Non-selfadjoint Schrodinger operators generating groups

W. Walter

A simple proof of t h e Cauchy-Kowal ewski theorem

James R. Ward, J r .

On the s o l v a b i l i t y of some weakly nonlinear boundary

Val ue problems R. Weder

Spectral a n a l y s i s s c a t t e r i n q theory and eigenfunctions expansions f o r strongly propagative systems

xv

ADDRESS LIST OF CONTRIBUTORS

Anthony U. Afuwape

Department o f Mathematics, U n i v e r s i t y o f I f e I l e - I f e , NIGERIA

Shrnuel Agrnon

I n s t i t u t e o f Mathematics, Hebrew U n i v e r s i t y o f Jerusalem, Jerusalem, ISRAEL

Nicholas

D. A l i k a k o s

Department o f Mathematics, Purdue U n i v e r s i t y , West L a f a y e t t e , I n d i a n a 47907

Mark Ashbaugh

Department o f Mathematics, U n i v e r s i t y o f M i s s o u r i , Columbia, M i s s o u r i 65211

P i e r r e Baras

L a b o r a t o i r e IMAG, Tour des Mathgrnatics, Analyse Numerique, BP 68, 38402 S t . M a r t i n d'Heres Cedex, FRANCE

Michael F. B a r n s l e y

School o f Mathematics, Georgia I n s t i t u t e o f Techn o l o g y , A t l a n t a , Georgia 30332

Matani a B e n - A r t z i

Department o f Mathematics, Technion ITT, H a i f a 32000, ISRAEL

Marc B e r g e r

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

Haim B r e z i s

U n i v e r s i t e P. e t M. C u r i e , 4, p l . J u s s i e u , 75230 P a r i s Cedex 05 FRANCE

R i c h a r d C. Brown

Department o f Mathematics, U n i v e r s i t y o f Alabama, U n i v e r s i t y , Alabama 35486

Nguyen Phuong Cac

Department o f Mathematics, The U n i v e r s i t y o f Iowa, Iowa C i t y , Iowa 52242

Ren6 Carmona

Department o f Mathematics, U n i v e r s i t y o f C a l i f o r n i a a t I r v i n e , I r v i n e , C a l i f o r n i a 92717

Robert C a r r o l l

Department o f Mathematics, U n i v e r s i t y o f I l l i n o i s , Urbana, I l l i n o i s 61801

N a t h a n i e l Chafee

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

C. Y.

Department o f Mathematics and S t a t i s t i c s , U n i v e r s i t y o f Southwestern L o u i s i a n a , L a f a y e t t e , L o u i s i a n a

Chan

Address Lisf of Contributors

xvi

C. Corduneanu

Department of Mathematics, University o f Texas a t Arlington, Arlington, Texas 76019

Chris Cosner

Department of Mathematics and Computer Science, University o f Miami, Coral Gables, Florida 33124

Michael G. Crandall

Department of Mathematics and Mathematics Research Center, University of Wisconsin, Madison, Wisconsin

Ingrid Daubechies

TENA, Fab. WE, V U B , Pleinlaan 2, B-1050 Brussels, BELGIUM

Allen Devinatz

Department of Mathematics, Northwestern University, Evanston, I l l i n o i s 60201

Lance D. Drager

School of Mathematics, Georgia I n s t i t u t e o f Technology, Atlanta, Georgia 30332

M. S. P. Eastham

Department o f Mathematics, Chelsea College, (University o f London), London SWlO OUA, U. K.

Ivar Ekeland

CEREMADE, Universite Paris-9 Dauphine, 75775 Paris Cedex 16, FRANCE

Volker Enss

I n s t i t u t e f u r Mathematik I , Freie U n i v e r s i t a t , D-100 Berlin 33 GERMANY

Melvin Faierman

Department of Mathematics, University of the Witwatersrand, Johannesburg, SOUTH AFRICA

W. E. Fitzgibbon

Department of Mathematics, University of Houston, Houston, Texas 77004

J . Fleckinger

Department Math. Universite Paul S a b a t i e r , 118, Route de Narbonne, 31062 Toulouse CEDEX FRANCE

Michael A. Freedman

Department of Mathematics, Vanderbil t University Nashville, Tennessee 37235

Jerome A. Goldstein

Department of Mathematics and Quantum Theory Group, Tulane University, New Orleans, Louisiana 70118

John R. Graef

Department o f Mathematics and S t a t i s t i c s , Mississippi S t a t e University , Mississippi S t a t e , Mississippi

Myron K . Grammatikopoulos

Department o f Mathematics, University of Ioannina, Ioannina, Greece

Ronald Grimmer

Department of Mathematics, Southern I l l i n o i s University, Carbondale, I11 i n o i s 62901

David Gurarie

Mathematics Department, Case Western Reserve University, Cleveland, Ohio 44106

Kar

Gustafson

Ge o ge A. Hagedorn

Department of Mathematics, University o f Colorado, Boulder, Colorado Department o f Mathematics, Virginia Polytechnic I n s t i t u t e and S t a t e University, Blacksburg, Virginia 24061

Address L i s t of Contributors

xvii

Kadosa H a l a s i

Department o f Mathematics, U n i v e r s ty o f Colorado, Boulder, Colorado

S. G. H a l v o r s e n

Department o f Mathematics, U n i v e r s t y o f Trondheim, 7000 Trondheim, NORWAY

C a r l o s R. Handy

Department o f Physics, A t l a n t a U n i v e r s i t y , A t l a n t a , Georgia 30314

D. P. H a r d i n

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

Andrew N. H a r r i ngton

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

J . V. Herod

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

Don H i n t o n

Mathematics Department, U n i v e r s i t y o f Tennessee, Knoxvi 1l e , Tennessee 37996

Rhonda J. Hughes

Department o f Mathematics, Bryn Mawr College, Bryn Mawr, Pennsylvania 19010

K e t i l l Ingolfsson

Department o f Mathematics, U n i v e r s i t y o f Alabama, U n i v e r s i t y , Alabama 35486

Hiroshi Isozaki

Department o f Mathematics, Kyoto U n i v e r s i t y , Sakyo, Kyoto, JAPAN

Hans G. Kaper

Mathematics and Computer Science D i v i s i o n , Argonne N a t i o n a l L a b o r a t o r y , Argonne, I 1 1 i n o i s 60439

T o s i o Kato

Department o f Mathematics, U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , C a l i f o r n i a 94720

Shmuel K i r o

Department o f Mahtematics, Rice U n i v e r s i t y , Houston, Texas

H i t o s h i K i tada

Department o f Mathematics, U n i v e r s i t y o f Tokyo, Meguro , Tokyo, JAPAN

Mark A. Kon

Department o f Mathematics, Boston U n i v e r s i t y , Boston, Massachusetts, 02215

A l l a n M. K r a l l

Mathematics Department, Pennsylvania S t a t e U n i v e r s i t y , U n i v e r s i t y Park, Pennsylvania 16802

B. A . Kupershmidt

U n i v e r s i t y o f Tennessee Space I n s t i t u t e , T u l lahoma, Tennessee

G. Ladas

Department o f Mathematics, U n i v e r s i t y o f Rhode I s l a n d K i n g s t o n , Rhode I s l a n d 02881

M i c h e l Langl a i s

Department o f Mathematics, Purdue U n i v e r s i t y , West L a f a y e t t e , I n d i a n a

P e t e r D. Lax

Courant I n s t i t u t e o f Mathematical Sciences, Mercer S t r e e t , New York, New York 10012

Address List of Contributors

xviii

W i l l am Layton

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

A. C

Lazer

Department o f Mathematics, U n i v e r s i t y o f Miami, C o r a l Gables, F l o r i d a

Sung J. Lee

Department o f Mathematics, U n i v e r s i t y o f South F l o r i d a , Tampa, F l o r i d a 33620

Elliott

H. L i e b

Departments o f Mathematics and P h y s i c s , P r i n c e t o n U n i v e r s i t y , P r i n c e t o n , New J e r s e y 08544

Lance L. L i t t l e j o h n

Department o f Mathematics, Computer Science and Systems Design, U n i v e r s i t y o f Texas a t San Antonio, San Antonio, Texas 78255

Jing-lin Liu

Department o f Mathematics, U n i v e r s i t y o f I n n e r Mongolia, Huhehot, Inner-Mongol i a , PEOPLE'S REPUBLIC OF CHINA

Roger C. McCann

Department o f Mathematics and S t a t i s t i c s , M i s s i s s i p p i S t a t e U n i v e r s i t y , M i s s i s s i p p i State, M i s s i s s i p p i

Paul K. McConnaughey

Department o f Agronomy, M i s s i s s i p p i S t a t e U n i v e r s i t y , M i s s i s s i p p i State, M i s s i s s i p p i

P e t e r A. McCoy

Mathematics Department, U n i t e d S t a t e s Naval Academy, Annapol i s , Mary1 and 21402

P. J. McKenna

Department of Mathematics, U n i v e r s i t y o f F l o r i d a , Gainesville, Florida

Joyce R. McLaughlin

Department o f Mathematical Sciences, Rensselaer P o l y t e c h n i c I n s t i t u t e , Troy, New York, 12181

Ronald E. Mickens

Department o f Physics, A t l a n t a U n i v e r s i t y , A t l a n t a , Georgia 30314

A. B. M i n g a r e l l i

Department o f Mathematics, U n i v e r s i t y of Ottawa, Ottawa, CANADA, K1N 984

P e t e r A. P e r r y

Department o f Mathematics, C a l i f o r n i a I n s t i t u t e o f Technology, Pasadena, C a l i f o r n i a 91125

Paul H. Rabinowitz

Department of Mathematics, U n i v e r s i t y o f WisconsinMadison, Madison, Wisconsin

David Race

Mathematics Department, U n i v e r s i t y o f t h e Witwatersrand Jan Smuts Avenue, Johannesburg, SOUTH AFRICA

Thomas T. Read

Department o f Mathematics, Western Washington U n i v e r s i t y , B e l l ingham, l l a s h i n p t o n 98225

Michael C. Reed

Department o f Mathematics, Duke U n i v e r s i t y , Durham, N o r t h C a r o l i n a 27706

Rouben Rostamian

Department o f Mathematics, Pennsylvania S t a t e U n i v e r s i t y , U n i v e r s i t y Park, Pennsylvania 16802

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xix

P. Sarnak

Courant I n s t i t u t e , New York U n i v e r s i t y , Mercer S t r e e t , New York, New York 10012

M a r t i n Schechter

Courant I n s t i t u t e o f Mathematical Sciences, New York U n i v e r s i t y , New York, New York 10012

V l a d i m i r Schuchman

Department o f Mathematics, Texas Tech U n i v e r s i t y , Lubbock, Texas 79409

Y. G. S f i c a s

Department o f Mathematics, U n i v e r s i t y o f Ioannina, Ioannina, GREECE

J. K. Shaw

Department o f Mathematics, V i r g i n i a Tech, Blacksburg, V i r g i n i a 24061

B a r r y Simon

Departments o f Mathematics and Physics, C a l i f o r n i a I n s t i t u t e o f Technology, Pasadena, C a l i f o r n i a 91125

A l a n Sloan

School o f Mathematics, Georgia I n s t i t u t e o f Technology, A t l a n t a , Georgia 30332

J o e l A. S m o l l e r

Department o f Mathematics, U n i v e r s i t y o f Michigan, Ann Arbor, M i c h i g a n

P a n a g i o t i s E. Souganidis

D i v i s i o n o f A p p l i e d Mathematics, Brown U n i v e r s i t y , Providence, Rhode I s l a n d

Paul W . Spikes

Department o f Mathematics and S t a t i s t i c s , M i s s i s s i p p i S t a t e U n i v e r s i t y , M i s s i s s i p p i State, M i s s i s s i p p i

I . P. S t a v r o u l a k i s

Department o f Mathematics, U n i v e r s i t y o f Rhode I s l a n d , Kingston, Rhode I s l a n d 02881

C a r l Sundberg

Department o f Mathematics, U n i v e r s i t y o f Tennessee, K n o x v i l l e , Tennessee 37996-1300

R. Temam

L a b o r a t o r i e D'Analyse Numerique, U n i v e r s i t e de P a r i s Sud, B a t i m e n t 425, 91405 Orsay, Cedex FRANCE

A. Vanderbauwhede

I n s t i t u u t voor T h e o r e t i s c h e Mechanica, R i j k s u n i v e r s i t e i t Gent, K r i j g s l a a n 281, B-9000

Gent, BELGIUM P i e r r e A. V u i l l e r m o t

Department o f Mathematics, U n i v e r s i t y o f Texas, A r l i n g t o n , Texas 76019

Bernard F. I J h i t i n g

Groupe d ' A s t r o p h y s i q u e , O b s e r v a t o i r e de P a r i s , 92 Meudon, P a r i s FRANCE

Joseph Wiener

Department o f Mathematics, Pan American U n i v e r s i t y , Edinburg, Texas

A l a s t a i r D. Wood

School o f Mathematical Sciences, N a t i o n a l I n s t i t u t e f o r H i g h e r Education, D u b l i n 9, IRELAND

K e n j i Yajima

Department o f Pure and A p p l i e d Sciences, U n i v e r s i t y o f Tokyo, 3-8-1, Komaba, Meguroku, Tokyo 153 JAPAN

xx

Address List of Contributors

Elsayed M. E. Zayed

Mathematics Department, Faculty o f Science, Zagazig University, Zagazig, EGYPT

Marvin Zeman

Department o f Mathematics, Southern Illinois University, Carbondale, Illinois 62901

DIFFERENTIAL EQUATIONS LW. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

1

UNIFORM DISSIPATIVE SOLUTIONS F O R A T H I R D O R D E R N O N - L I N E A R DIFFERENTIAL EQUATION A n t h o n y Uyi Afuwape Department o f Mathematics U n i v e r s i t y of I f e Ile-1fe.NIGERIA.

The f r e q u e n c y - d o m a i n t e c h n i q u e , a r i s i n g f r o m p r o b l e m s o f c o n t r o l , h a s b e e n v e r y useful i n d i s c u s s i n g t h e qualitative properties of solutions f o r non-linear d i f f e r e n t i a l equations. Earlier r e s u l t s are g e n e r a l i zed i n t h i s p a p e r t o e q u a t i o n s o f t h e form:

xr11+ fix") + g(x’l + hlxl = p(t,x,x’,x") where t h e f u n c t i o n s f , g , h and p a r e c o n t i n u ous i n t h e i r r e s p e c t i v e a r g u m e n t s . Uniform d i s s i p a t i v i t y is discussed for t h i s equation, using the frequency-domain t e c h n i q u e . 1.INTRODUCTION I n a n e a r l i e r work, B a r b s l a t and Halanay [ 4 1 o b t a i n e d a g e n e r a l r e s u l t , w h i c h was m o t i v a t e d by t h e w o r k s o f Popov V . M . , Kalman R . E . and Yacubovich V . A . T h i s r e s u l t i s g e n e r a l l y r e f e r r e d t o as t h e ) turned out t o frequency-domain t e c h n i q u e . The t e c h n i q u e ( c f [4-6] b e v e r y u s e f u l i n d i s c u s s i n g p r o b l e m s o f oscillations,dissipativity, u n i f o r m d i s s i p a t i v i t y a n d many o t h e r q u a l i t a t i v e p r o p e r t i e s o f solutions f o r non-linear d i f f e r e n t i a l equations. The a u t h o r i n [1,2] h a s used t h i s technique t o g e n e r a l i z e e a r l i e r results in [ 4 , 9 t o e q u a t i o n s o f t h e t h i r d - o r d e r w i t h two nonl i n e a r f u n c t i o n s . I n t h e p r e s e n t p a p e r , t h e frequency-domain t e c h n i que is u s e d t o g e n e r a l i z e t h e s e r e s u l t s f u r t h e r t o a t h i r d - o r d e r d i f f e r e n t i a l e q u a t i o n w i t h t h r e e n o n - l i n e a r terms. P r e c i s e l y , we c o n s i d e r e q u a t i o n s o f t h e form:

xPr1+ flx") + g l x ' ) + hlx)

=

p(t,x,x’,x"J

(1.1)

where f , g , h and p a r e c o n t i n u o u s i n t h e i r r e s p e c t i v e a r g u m e n t s . A s r e q u i r e d by t h e f r e q u e n c y - d o m a i n t e c h n i q u e , w e s h a l l a s s u m e t h a t t h e r e e x i s t p o s i t i v e c o n s t a n t s a , b , c w i t h ab > c a n d n o n - n e g a t i v e c o n s t a n t s p1 , l.~? , p 3 s u c h t h a t

hold

for

a

Q

f(z)/z b G glz)/z

Q

c

< hfz)/z

Q


l z I1 > A o l

a

< f’(zl

Q

;

a + p, b G g ' ( z ) G b + p*

;

c

< h"z)

Q

a + l . ~ ;~ c + p,

The c o n d i t i o n s a > 0 , b > o , ab > c > o Hurwitztr c o n d i t i o n s f o r t h e t h i r d - o r d e r 5"’

c + p3

X, l a r g e e n o u g h .

+ m'r + bx’ + cx

represent the l i n e a r equation =

0

which a r e s u f f i c i e n t f o r uniform d i s s i p a t i v e e q u a t i o n s .

It

Rout h

-

A.U. Afuwape

2

2.MAIN RESULTS The f o l l o w i n g t h e o r e m h o l d s f o r e q u a t i o n Theorem ( 2 . 1 ) : S u p p o s e t h a t i n ( 1 . 1 1, f(0) Po > O some c o n s t a n t I p ( t , X I , x 2 , 2 , ) I Q p, f o r a l l t , x ,x2, x 3 . Suppose f u r t h e r ties (2.11, junctions f I g and h

l i m (1/z2){&zf(a)da

-

(1.1).

=

= 910)

0

= h(01

a nd f o r

(2.1)

t h a t i n addition toinequalisatisfy

%z;Zflz)

1

0

2

Iz l-

Then, t h e s o l u t i o n s of e q u a t i o n ( 1 . 1 ) are u n i f o r m l y d i s s i p a t i v e .

p

Remark ( 2 . 2 ) . C o n d i t i o n (2.1) i m p o s e d on t h e f u n c t i o n c a n b e weakened t o I p1t,x1,x2,xJl Po f p, ( l X , l f I x21 f I x 3 II for > 0 1 j =o,r) a n d 4 > 0 s u f f i c i e n t l y s m a l l . Remark ( 2 . 3 ) . and

4

If

0, then equation

( 1 . 1 ) be c om e s

xrrrf f g1x’) + h ( x ) = p(t,x,x’,x”) Theorem ( 2 . 1 ) r e d u c e s t o a n e a r l i e r r e s u l t i n

If

Remark ( 2 . 4 ) .

,

$ .!

(2.4)

I11

If

Y

:0

,

equation

12.5)

(1.1) reduces t o

f f x ” ) f g f x ’ ) f- c x = p f t , x , x ’ , x ” ) I n t h i s case, t h e f o l l o w i n g i n d e p e n d e n t r e s u l t h o l d s : 5 ) ) )f

Theorem ( 2 . 6 ) . S u p p o s e t h a t i n e q u a t i o n ( 2 . 6 1 , c > 0 p o s i t i v e numbers a b a n d n o n - n e g a t i v e n u m b e r s ?.+ a b > c a n d f o r I z I 2 X, , X, l a r g e e n o u g h , a G f(z)/z

< a + p, ; a G

b C gfzl/z with

f(0)

= 0

g(0).

b

f

p2 ; b

Q

all

t, xi , z 2 , x 3

f’lzl g‘(zl

a

f

b +

(2.61

.

and t h e r e e x i s t , p2 s u c h t h a t

p,

p2

(2.7)

S u p p o s e f u r t h e r t h a t f o r some c o n s t a n t Po > 0

I p f t , x , ,z2’ X 3 )I G for

.

t h e n e q u a t i o n ( 1 . 1 ) be c om e s

x”‘ f f f x ” ) f bx’ f h ( x l = p ( t , z , x ’ , x ” ) [21. a n d Theorem ( 2 . 1 ) r e d u c e s t o a r e s u l t i n

Remark ( 2 . 5 ) .

12.31

12.81

P,

and

g f d d a - % zg(z) 1 > 0 I z b t h e s o l u t i s n s o f e q u a t i o n (2.6) a r e u n i f o r m l y d i s s i p a t i v e . -l fi 1m/ z 2 ) f c

Then

Theorem ( 2 . 6 ) g e n e r a l i z e s e a r l i e r r e s u l t s o f B a r b g l a t Remark ( 2 . 7 ) . [ 4 ] a n d o f H a l a n a y i n [51 and H al anay i n

.

3

Dissipative Solutions for a Non-Linear Equation

3.PROOF OF THEOREM ( 2 . 1 ) . h Let ffx") a ax" +F(Yh g & f ) = b x ' + $ ( x l J and h l x ) ex + h ( x l Then, equation ( 1 . 1 ) can be written in the vector form: X’ = AX - B Q ( u ) i- P(t,X), U = C*X (3.11 where 0 0 0 0

0

A = -a

-c -b

0

0

Hith

Thus a function

=

C*X

=

C(iw) =

X

and C*(iwI

o ( 0 )

- A)-’B

= is equivalent to

C(iw) where A = ( c - a w ' ) + iwIb - w 2 ) . , we have that matrix A in (3.1) is stable. ' The frequency-domain inequality (cf. 141 is ~ ( w )E D, + Re { Dldiag( cl.1 iL,,D 2 G(i9 1 + J + 02[ Re ID3 I + diag( l J j ) G(iw)>] > 0 (3.2) for all w 'R , where D . (j 1,2,3) are diagonal matrices with D > 0, D , D > 0 dnd pj are bounds for the elements o f vectar @ ( o f , sasisfying Choos T. >

J

Ti

(w

where

A. L! Afuwape

4

) ( c

r

=

{ ( T~

and

s

(

W{

-

el +

( c

W ~ T ,

W2e3

)

(c

-

aw2 )

+

au2 )

-

w2(

-

I

ad

d))

T 3 ) ( b el-W2ej ( b

-

( T ~

+

W~T,)

(b

)I

-

u2 )I.

F o r (3.3) to be true for all & g , Sylvester s criteria for positive definiteness of matrices can be applied. That is, we must show that for all 6 fl, (3.3) is valid if:

>

11

o

’:y2 )

(3.4)

u + iv det

-

u

iv

(3.5)

0

>

and > 0 (3.6). det n(w1 As in [ 1 ] , inequality (3.4) is valid if ( e l / T~ and w * e , + w ‘ ( % - b e , ) --c -1 > p1 ( c - a w ~ +) W’ ~ ( b - w 2 ) ’ On evaluating inequality (3.51, we have Till 7122 (u2 + v 2 ) / 4 1 A I 4 > 0

-

(

> (a/b)

.

This will be true, for all w and -1 -1

T ~ / T ~ ) (+ Lc ~ ~)

Finallv. detv ( w )

)

IR , if p2

a-’<

>

will be positive for

+

(40 all

1 -

<

( e L / T 2 )

(b/c)

(3.7).

W G W if

{ u(pr + qs) + v(ps 4 1 ~ 1 ~

-

qr)} > 0 (3.8).

Considering the degrees of p , q , r! s , u, v and ( ~ 1 as~ polynomials in w , we observe that for inequality (3.8) to be true, it suffices to have 1 ‘2’3 -(-+

4C2

1)

(3.9)

‘3’2

a situation obtained in the limiting case with O 3 0. Equivalent to this is to have 0J-I + c- ) bounded below by a positive linear function in p 2 and p 3 . The supplementary conditions (2.2) follow since 2,3). Hence the conclusion of Theorem (2.1) follows generalized Theorem of Yacubovich ( [4] 1 .

0 . 2 0 (j J

from the

4.PROOF OF THEOREM ( 2 . 6 1 A In order to prove Theorem Let f(xff)=aff + f(x )and g(3:f)=b3.r+g(zf) (2.6), we first write equation (2.6) in vector form X = AX - B @ ( a ) + P(t,X), u C*X (4.1)

.

1,

5

Dissipative Solutions for a Non-Linear Equation

where

A = ( :

y)

0 1 --c

; B = [ y 0

C0 l ) ; c = [ O 1 l

0

0

0

-a

-b

For this system, the transfer function given by --a - i-cw 1 G(iw) -

(

A

C(iw) -a

C*(iwI

- i-cw --c

-e

-

A)-’B

is

1

where as before, A = (c - aw2J + iw(b -w2). The frequency-domain condition (3.2) € o r this becomes

/

iRz

dl

\

-cT2

d2 R,

(e,d

=

[ (a2

- hie,+ C ~ , I W +~

+ fb2e, and

L,

=

-

W{T,W"

+

-cTL )1w2

[ ( a 2 - b ) ~ ,i-c(e,+

ra-c(T2

I

-

+ c(h, ae, - T ,

8, - b e , m + - cr,) 1 ~ ] w ~+

+ [ f b 2 - a d T , + bclT, - el - c e,/bi] 1 Clearly, if we choose 6, > 0, we have for all w G E , d, > 0 and if ( 8 , / T,) > ( a / b ) we have d2 > 0 with (I/uJ > 1. and e 2 we have for all Equally, for these choices of wt?R det n ( w ) d,d2 - (R: + &: )/(4 ( 4 " ) > 0 (4.3). The supplementary conditions (2.8) follow by the choice of e . J (j = 1 , Z ) . The conclusion of Theorem (2.6) thus follows from the generalized Theorem of Yacubovich. REFERENCES [I] Afuwape, A.U., Frequency-Domain Criteria for Dissipativity of some Third-Order Differential Equations, Analele Sti. Univ, ’AL. I. CUZA’, xxiv (1978) 271-275. 1 2 1 Afuwape, A.U., An Application of the Frequency-Domain Criteria For Dissipativity of a Certain Third-Order Non-linear Differential Equation, Analysis 1 (1981) 211-216. I 3 1 Barbzlat, I.and Halanay, A., Nouvelles Applications de la Methodes Frequentielle dans la Theorie des Oscillations, Rev. Roum. Sci. Techn. Elect. et Energ., '16, 4(1971) 689-702. [ 4 I Barbzlat, I.and Halanay, A., Conditions de Comprtement

6

A. U.Afirwape

’presque lineaire’ dans la Theorie des Oscillations, Rev. Roum. Sci. Techn. Elect. et Energ., 29, 2(1974) 321-341. [5] Halanay, A., Frequency-Domain Criteria For Dissipativity, in: Weiss, L(ed.) Ordinary Differential Equations, MRL-MRC Conference (1972) 413-416. 161 Popov, V.M., Dischotomy and Stability by Frequency-Domain Methods, Pro. I.E.E.E., 62, 5 (1974) 547-562. [ T I Reissig, R. Sansone, G. and Conti, R.,Nonlinear Differential Equations of Higher Order (Noordhoff Int. Publ. 1974). [81 Yacubovich, V.A., The matrix method in the theory of the stability of non-linear control systems, But. Rem. Control, 25 (1964) 905-9 16. [g] Yacubovich, V.A., Frequency-Domain conditions for absolute stability and dissipativity of control systems withone differentiable non-linearity, Soviet Maths., 6 (1965) 98-101.

DIFFERENTIAL EQUATIONS

LW.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

7

ON POSITIVE SOLUTIONS OF ELLIPTIC EQUATIONS WITH PERIODIC COEFFICIENTS IN En,SPECTRAL RESULTS AND EXTENSIONS TO ELLIPTIC OPERATORS ON RIEMANNIAN MANIFOLDS Shmuel Agmon Institute of Mathematics Hebrew University of Jerusalem Jerusalem, Israel

1. Introduction.

It is well known that there are close connections between spectral properties of second order elliptic operators and properties of positive solutions of elliptic equations. Here are some examples (see also Allegretto [1,2,31, Piepenbrink 19,101, Moss-Piepenbrink 181 and Simon [ll]). Theorem 1.1. Let P =

-

A

+

q(x)

be a Schrodinger operator defined in a

domain R c En where q E Lioc(R) with r > n/2, q real. A necessary and sufficient condition for P to have a non-negative spectrum in the sense that (W,(P) =

for all cp

E Ci(R),

f(lWI2 + q1VJ12)dx 2 n

0

is that the equation Pu = 0

admits a positive solution in

n . Theorem 1.2.

Let P be as above and suppose that the form

bounded from below on of

P

in L 2 ( a ) .

C;(Q).

Let H

u

ess

(HI

is

Write 1 = inf u

where

(Pcp,cp)

be the Dirichlet self-adjoint realization

ess

(HI

denotes the essential spectrum of H.

Then

(i) For every A < C the equation (P-A)u = 0 admits a positive solution in some neighborhood of infinity in R (i.e., in Q \ K for some compact K C n). (ii) For every X > C the equation any neighborhood of infinity in n.

(P-A)u = 0

admits no positive solution in

Positive solutions are also useful as majorants for solutions of elliptic equations which do n o t grow too rapidly at infinity. For instance, we have the following result. Theorem 1.3. Under the same conditions as in Theorem 1.2 let $(XIbe an eigenfunction of H with eigenvalue X 3 C and let u(x) be a positive Then solution of cP-A)u = 0 in some neighborhood of infinity in R 5 Cu(x) in a neighborhood of infinity for some constant C. [$(x)l

.

In this lecture we shall discuss properties of positive solutions and related spectral results for elliptic operators with periodic coefficients on lRn and also discuss extensions of the results to a larger class of elliptic operators on certain non-compact Riemannian manifolds. FOK the sake of presentation we shall at first limit ourselves to the periodic case. The starting point here is the observation (proved in §2) that whenever the elliptic

8

S. Agmon

equation Pu = 0 admits a positive solution on Bn then it also admits a positive exponential type solution. The family of positive solutions of Pu = 0 which are "exponentials" has a major role in our discussion. Various properties of this family are presented in 52. Some results in spectral theory of elliptic operators with periodic coefficients which are based on knowledge of the family of positive exponential solutions are discussed briefly in 03. Finally in § 4 we describe extensions of the results to a more general class of elliptic operators defined on certain non-compact Riemannian manifolds. 2.

Positive exponential type solutions We consider a second order elliptic operator P

.

lRn

We assume that P n

ai

a/xi, x

=

n

=

(xl,

acting on functions u

on

is in a divergence form:

...,x,).

n

We assume that the coefficients of

P are real

measurable functions and that n

a

ij

a positive constant. We assume further that

E Lm(lRn) , bi E L~oc(l17n) , bi E L~occlRn) with p

with p1 > n/2. xl,

ni" , Y

and 5 in

for all x

...,xn

> n

and

Finally we assume that the coefficients of P

c E Lyl&lRn)

are periodic in

with period 1.

By a solution of the differential equation Pu = 0

in some open set

in Rn

we shall understand as usual a function u

(2.2) in

which verifies (2.2) in the weak (once integrated) sense. It follows

Hioc(a) from the results of Stampacchia 1121 that all solutions of (2.2) are Holder continuous functions. Furthermore, Harnack’s inequality holds for the class of non-negative solutions of (2.2) in n. We shall be interested in the structure of the family of positive solutions of (2.2) in Rn. The first basic result is the following Theorem 2.1. Suppose the equation Pu = 0 has a positive solution in Rn . Then the equation also has a positive solution of the form,

where

i s a periodic function

rp

in Rn

and 5 is some vector in En.

We give two proofs of the theorem. First Proof: Let

En

\\,

S

be the set of all solutions of the equation Pu = 0 in

and consider S as a locally convex linear space with a sequence of semi= max \u(x> I , u E S . Let K be norms ( \ , j = 1,2,. given by 11 <j Rn Ix Ithe set of positive solutions of Pu = I1 in normalized at the origin by u ( 0 ) = 1. Clearly K is a non-empty convex subset of S. Furthermore, using Harnack’s inequality and a-priori Holder norm estimates (see [12]) it is easy to see that K is a compact set in S.

-

. .,

.\\

Positive Solutions of Elliptic Equations ei, i

...,n, denote

=

1,

Rn and denote by

T~

Let

u(x) -+ u(x+ei).

Fi : u Since P

t h e u n i t v e c t o r i n t h e d i r e c t i o n of

xi i n

t h e corresponding t r a n s l a t i o n o p e r a t o r a c t i n g on

Rn , -ri:

f u n c t i o n s on

9

c o m u t e s with

+ T

Introduce t h e n maps:

uEK.

u/u(ei),

i

i t follows t h a t

T.

(i = 1,. ..,n) i s a continuous

F.

map which t a k e s K i n t o i t s e l f . Hence, applying t h e Schauder-Tychonoff f i x e d p o i n t theorem [ 1 3 ] t o F1 we conclude t h a t F1 has a f i x e d p o i n t i n K which implies t h a t there e x i s t s a positive solution

X1.

some p o s i t i v e number subset of

N e x t denote by

c o n s i s t i n g of a l l

K

commutes with

in K

u

TIU1 =

Xlul

for

t h e non-empty closed convex

K1

such t h a t

F2 maps

i t follows t h a t

T~

such t h a t

u1

X u. 1

T u =

1

K1 i n t o i t s e l f .

Since T2 Invoking again

t h e Schauder-Tychonoff f i x e d p o i n t theorem i t follows t h a t t h e r e e x i s t s a Continuing i n t h i s p o s i t i v e s o l u t i o n u2 i n K1 such t h a t T ~ =U A 2 ~u 2 . manner we conclude t h a t t h e r e e x i s t s a p o s i t i v e s o l u t i o n u of ( 2 . 2 ) i n Rn s a t i s f y i n g T . U = X iu f o r i = l,....,n, where t h e Xi are c e r t a i n p o s i t i v e constants.

The p o s i t i v e s o l u t i o n

u

thus obtained i s of t h e form ( 2 . 3 )

described i n t h e theorem. To s e e t h i s s e t ~ ( x )= u(x)e-Q'x> where 5 =(logX1, log An) and n o t e t h a t Ti'p = ( D f o r i = 1, n. This completes

...,

...,

t h e proof. This proof i s due t o Yehuda Pinczover ( p r i v a t e communication).

Second proof:

Let S and K be defined as i n t h e proof above. Then as was noted K i s a compact s e t i n t h e l o c a l l y convex l i n e a r space S . From t h e Krein-Milman theorem i t follows t h a t t h e set of extremal p o i n t s of K i s not-empty. Let u be any extremal p o i n t of K. We s h a l l show t h a t u must be of t h e form ( 2 . 3 ) . To t h i s end introduce as i n t h e proof above t h e t r a n s l a t e d f u n c t i o n s

...,

T u = u(x+e 1, i = 1, n. Then T u and u a r e p o s i t i v e s o l u t i o n s of i i i Pv = 0 i n R". From Harnack's i n e q u a l i t y applied t o u (see ( 2 . 4 ) ) i t

follows t h a t t h e r e e x i s t s a number i = 1,

...,n. v

Then vi

i

wi

w

i

belong t o u = a v

i i

with

> 0

such t h a t

u - ET u i

> 0

in

Rn f o r

Define

= .siu/u(ei),

and

E

ai

follows t h a t

vi

-

K

and

i i

=

u(0) - Eu(e.)

=

u(0)

=

1.

i = 1,

so t h a t

Since

must be a m u l t i p l e of Xi,

Eu(ei)).

+ B W

Bi

positive constants

-

ETiu)/(u(0)

+ Bi

ai = Eu(ei),

numbers with

= (u

...,n.

u

ai

and

Bi

are positive

i s an extremal p o i n t of K i t

u, that is: T u = X u for certain i i This implies t h a t u i s of t h e form

( 2 . 3 ) and completes t h e proof. Note t h a t t h e second proof of Theorem 2 . 1 y i e l d s a d d i t i o n a l information on t h e s t r u c t u r e of p o s i t i v e s o l u t i o n s of Pu = 0 v i a t h e Krein-Milman theorem. This a d d i t i o n a l information i s needed f o r t h e proof of t h e r e p r e s e n t a t i o n theorem f o r p o s i t i v e s o l u t i o n s which we d e s c r i b e l a t e r on.

10

S. Agmon

With some abuse of terminology we s h a l l r e f e r t o a s o l u t i o n u of t h e form ( 2 . 3 ) as an exponential s o l u t i o n of Pu = 0. The v e c t o r 5 i n t h e r e p r e s e n t a t i o n ( 2 . 3 ) w i l l be c a l l e d t h e exponent of U . The following two theorems d e s c r i b e p r o p e r t i e s of t h e manifold of exponents of a l l p o s i t i v e exponential s o l u t i o n s of (2.2). Theorem 2 . 2 . Under t h e c o n d i t i o n s of Theorem 2 . 1 any two p o s i t i v e exponential s o l u t i o n s of (2.2) with t h e same exponent 5 a r e l i n e a r l y dependent. Denote by

r t h e s e t of a l l 6 E IRn such t h a t 5 i s t h e exponent of some p o s i t i v e Then e i t h e r exponential s o l u t i o n of C2.2) i n IR"

.

(i)

r

c o n s i s t s of a s i n g l e p o i n t

(ii)

r r

=

or

i s a compact n-1 dimensional a n a l y t i c manifold embedded i n IRn such t h a t aK where K i s a s t r i c t l y convex n-dimensional s e t i n a n .

*

Theorem 2.3. Comider the o p e r a t o r P-X and P -1 w h e r e P * i s t h e formal a d j o i n t of P and A E R . There e x i s t s a r e a l number A = A@) such t h a t both equations: (P-X)u = 0

and

(P*-X)u = 0

possess p o s i t i v e s o l u t i o n s i n

possess no p o s i t i v e s o l u t i o n s i n

g E En

Ti) t h e s e t of

besp.

exponential s o l u t i o n i n (resp. K*)

an

( P - x ) ~=

be t h e convex body i n lRn

X

Theorem 2 . 2 ) .

Then

K

x

K;

=

-

o

(resp.

such t h a t

aK

.

c i n t KX f o r X < 1-1 5 A u o KA = { < 1 (a s i n g l e p o i n t ) and

(ii)

For any X 5 A denote by 'a 5 i s t h e exponent of some p o s i t i v e (P*-x)~= 0 ) . =

rh(resp.

Let

* * 3KX =Ti)

K~ (by

i s a continuous set valued f u n c t i o n of 1 such t h a t

(i) K (iii)

and

i f A > A.

such t h a t

I R ~ of

if A 5 A

Rn

K-_

=

IR"

KX.

. *

It follows from t h e l a s t theorem t h a t i f P = P then K, i s s m e t r i c It is c o n v e n i h t o introduce w i t h r e s p e c t t o t h e o r i g i n and t h a t KA = CO}. a t t h i s p o i n t t h e following n o t a t i o n : Definition. 0 E aK,.

We denote by

Ao(P)

t h e unique number X 5

A(P)

such t h a t

A(P)

if

n

It i s c l e a r t h a t self-adjoint. Note t h a t equation

A,(P)

=

no@*)

and t h a t

We s h a l l see l a t e r chat

A,@)

Ao(P)

=

P

i s formally

has a s p e c t r a l i n t e r p r e t a t i o n .

A (P) can a l s o be defined a s t h e unique r e a l 1 such t h a t t h e 0 (P-X)u = 0 has a p o s i t i v e p e r i o d i c s o l u t i o n i n R n .

We s h a l l not g i v e h e r e proofs of Theorem 2.2 and Theorem 2 . 3 . Instead we s h a l l e s t a b l i s h t h e following r e l a t e d theorem which i s used i n t h e proof of Theorem 2.2. Theorem 2 . 4 .

A s s u m e t h a t t h e c o n d i t i o n s of Theorem 2.2 hold and t h a t F i s

defined as i n Theorem 2 . 2 . Then r i s a compact s e t i n R n . L e t K(T) denote t h e convex h u l l of r and l e t TI be a p o i n t i n K(7) which i s n o t a n extremal point.

then t h e r e does not e x i s t a s o l u t i o n

change sign) such t h a t

uCx)e-<"'x'

u

2 0 of

Pu = 0 i n

i s a bounded f u n c t i o n i n

Rn

.

Rn (u may

It follows from t h e theorem t h a t a l l p o i n t s of T are extremal p o i n t s of and t h u s i n p a r t i c u l a r i t follows t h a t no t h r e e p o i n t s of r are on t h e same l i n e . This r e s u l t i s used i n t h e proof of Theorem 2.2 t o e s t a b l i s h t h e s t r i c t convexity of K(T).

K(T)

Proof of Theorem 2.4.

The proof t h a t

r

i s a closed s e t is easy and w i l l be

11

Positive Solutions o f Elliptic Equations o m i t t e d (it u s e s t h e p o i n t w i s e and L2 e l l i p t i c estimates f o r s o l u t i o n s of To show t h a t r i s a bounded s e t we invoke Harnack's i n e q u a l i t y which (2.2)). i m p l i e s ( s i n c e t h e c o e f f i c i e n t s of P are p e r i o d i c ) t h a t f o r any R > 0 t h e r e e x i s t s a c o n s t a n t C = C(R) such t h a t i f u i s any p o s i t i v e s o l u t i o n of Pu = 0

i n R", t h e n UCX) 5 C u b )

(2.4)

ai"

f o r any x , y i n such t h a t Ix-y( 5 R . It f o l l o w s from ( 2 . 4 ) t h a t t h e r e e x i s t p o s i t i v e c o n s t a n t s A and a such t h a t i f u i s a p o s i t i v e s o l u t i o n of

(2.2)

Rn

in

,

Applying (2.5) that 151 5 CY

then

t o p o s i t i v e e x p o n e n t i a l s o l u t i o n s of (2.2) f o r gE r Hence r i s bounded.

.

i n R"

Next l e t I7 be a p o i n t i n K ( T ) which i s n o t a n extremal p o i n t . show t h a t t h e r e d o e s n o t exist a s o l u t i o n u(x)e- i s bounded i n

Rn.

u f 0

of (2.2)

i n Rn

w e conclude W e wish t o

such t h a t

To prove t h i s we may assume w i t h no l o s s of

g e n e r a l i t y t h a t rl = 0. Indeed, i f u i s a n y s o l u t i o n of Pu = 0 i n R" t h e n - i s a s o l u t i o n of t h e e q u a t i o n P v = 0 i n Rn where Po is an v = ue 0 e l l i p t i c o p e r a t o r w i t h p e r i o d i c c o e f f i c i e n t s similar t o P. Thus, r e p l a c i n g P by Po and t h e s e t r by ro = r - rl (which i s t h e set of exponents of p o s i t i v e e x p o n e n t i a l s o l u t i o n s of t o t h e s p e c i a l case where

P u = 0 1 , w e r e d u c e t h e r e s u l t t o b e proved 0

i s the origin.

17

Hence by l e t t i n g rl = 0 we assume t h a t O E K ( T ) and t h a t t h e o r i g i n i s n o t a n e x t r e m a l p o i n t of K ( T ) . Invoking C a r a t h e o r d o r y ' s theorem i t t h u s f o l l o w s i t h a t thrire e x i s t p o i n t s 5 E r , i = I, m w i t h 2 5 m 5 n+l and numbers t . > 0 such t h a t

...,

m C

t i 5 i = 0.

(2.6)

i=l Let

Si,

ui(x) be a p o s i t i v e e x p o n e n t i a l s o l u t i o n of (2.2) in m m, and set u (x) = C ui(x). We have:

,...,

i = 1

0

Rn w i t h exponent

i=l (2.7)

where

and

c

i s some p o s i t i v e c o n s t a n t . L e t L b e t h e l i n e a r subspace i n Rn 1 m 5 ,...,c , and l e t n d e n o t e t h e o r t h o g o n a l p r o j e c t i o n on L . It

spanned by

i s c l e a r t h a t p(x) i s a continuous homogeneous f u n c t i o n of degree 1 on Rn such t h a t p (x) = p (nx) Furthermore, w e c l a i m t h a t p 1. 0 and t h a t i f IIx # 0. Indeed, i f t h i s were n o t the c a s e t h e r e would e x i s t a p(x) > 0 0 0 0 p o i n t x E L , x # 0, such t h a t p(x ) 5 0 which would imply ( i n view of 0 ( 2 . 8 ) ) t h a t < s i , x > 5 0 f o r i = l,.. . . , m . A t t h e s a m e t i m e t h e r e l a t i o n (2.6)

.

S. Agmon

12 implies that

m i C ti<< ,xo> = 0. Thus, since ti> 0, it would follow that i=1

= 0 for i = I,...,m. This however contradicts that fact that the p(x) > 0 if EX # 0. span L, proving that

-

From (2.7) and the properties of +

and that uo(x)

as

ITIxl

+

it follows that uo(x) 2 c > 0 in Rn With these preliminary considerations

p

.

m

ci

we turn to our main task and show that the equation Pu = 0 cannot possess a

Hi".

non-zero bounded solution in

Thus assume that the equation admits a

solution u E L"(lRn), u $ 0 and assume also (with no loss of generality) that u(x 0) > 0 at some point x 0 We shall show that this leads to a contradiction. To this end introduce the family of functions u6 defined for 6 > 0 by u (x) = uo(x) - 6u(x). Denote by tio the supremum of all numbers 6 2 0 such

.

6

>

that u

0 in

R n . Then clearly 0

< 6o <

Consider the function u

m.

6

60

which is a non-negative (continuous) solution of (2.2) in Rn. It verifies: (i)

(x): x E R n }

inf{u &O

u6 (x) +

(ii)

0

((i)

-

=

as 111x1+

0, and m

.

follows from the maximality of

and (ii) follows from the corresponding

6o

property for uo, since u is bounded.)

i !0

Since u

it follows further from

6O

u

Harnack’s inequality that

>

0

in Rn

,...,

{yj61 o, j = 1,2

of a sequence of points

&)

u

+

o

so

that (i) implies the existence Iyj I +

with

as j + -

6O

-,

such that

.

(2.9)

Also it follows from (ii) that there exists a number R > 0 such that

(x) 2 1 if

u

/TIXI

(2.10)

R

6O

Pick now a unit-vector

($1,

j

z such that Tlz

=

z and define a sequence of points

...

=

lnxj I 1. R.

1,2, , by xJ = yJ 5 Rz where the sign is chosen so that Thus in view of (2.10) we have:

On the other hand, since inequality (2.4) that u

Ixj-yj 1 5 R it follows from (2.9) and Harnack’s Cxj) + 0 as j + m. This yields a contradiction

&O

which shows that the equation (2.2) does not possess a non-zero bounded solution in lRn

.

The proof is now complete.

W e conclude this section with the following

Representation Theorem. Under the assumption that the equation Pu a positive solution in R" , let

=

0 admits

Positive Solutions of Elliptic Equations

13

be t h e family of p o s i t i v e exponential s o l u t i o n s of Pu = 0 normalized by u (0) = 1 (K as in Theorem 2.2). I f u i s any p o s i t i v e s o l u t i o n of Pu = 0 E,

then t h e r e e x i s t s a unique p o s i t i v e measure m ub) =

I

on

aK

such t h a t

u (xIdm(5).

aK

The proof of t h e theorem follows by combining t h e previous r e s u l t s with t h e Krein-Milan theorem ( s e e second proof of Theorem 2 . 1 ) .

We note t h a t the r e p r e s e n t a t i o n theorem i s known t o hold when P has constant c o e f f i c i e n t s . The r e s u l t was proved i n t h e more general context of symmetric spaces by Karpelevic [ 6 ] . Another more elementary proof of t h e r e p r e s e n t a t i o n theorem f o r p o s i t i v e s o l u t i o n s of e l l i p t i c equations with constant c o e f f i c i e n t s was given by C a f f a r e l l i and Littman IS].

3.

On t h e spectrum and on t h e r a t e of decay of eigenfunctions of perturbed e l l i p t i c o p e r a t o r s with p e r i o d i c c o e f f i c i e n t s

The r e s u l t s given i n Theorem 2 . 2 and Theorem 2 . 3 which d e s c r i b e the s t r u c t u r e of t h e manifold of exponents of t h e p o s i t i v e exponential type s o l u t i o n s of t h e equation (P-X)u = 0 i n W n admit a p p l i c a t i o n s t o s p e c t r a l problems. Before we d i s c u s s such a p p l i c a t i o n s we s h a l l make some g e n e r a l remarks. Suppose t h a t A i s a second order e l l i p t i c operator i n an unbounded A s s u m e t h a t A has t h e same form as P except t h a t t h e c o e f f i c i e n t s domain R of A need not be p e r i o d i c . It can be shown t h a t i f t h e equation Au = 0 admits a p o s i t i v e s o l u t i o n i n R then t h e r e a l s o e x i s t s a p o s i t i v e s o l u t i o n uo i n a

.

neighborhood of i n f i n i t y
$2 such t h a t

uo

has minimal growth a t i n f i n i t y i n

v i s any p o s i t i v e s o l u t i o n a t a neighborhood of i n f i n i t y i n i n some neighborhood of i n f i n i t y f o r some constant C . More

p r e c i s e l y , i t can be shown t h a t when t h e r e e x i s t s a p o s i t i v e s o l u t i o n of in

R , and

xo

R

i s any p o i n t i n

,

Au = 0

then t h e r e e x i s t s a p o s i t i v e s o l u t i o n i n

0

R {x 1 which has minimal growth a t i n f i n i t y . Such a s o l u t i o n is e i t h e r a p o s i t i v e Green's f u n c t i o n a s s o c i a t e d with t h e D i r i c h l e t problem i n R o r i n t h e r a r e s i t u a t i o n when t h e s i n g u l a r i t y a t xo i s removable t h e s o l u t i o n i s a "ground s t a t e " . These remarks make i t c l e a r t h a t when one s t u d i e s p o s i t i v e s o l u t i o n s of an e l l i p t i c equation in an unbounded domain, i t i s very u s e f u l t o determine t h e r a t e of decay of p o s i t i v e s o l u t i o n s a t i n f i n i t y which possess minimal growth a t i n f i n i t y .

n We now r e t u r n t o t h e e l l i p t i c operator P with p e r i o d i c c o e f f i c i e n t s i n P discussed i n t h e previous s e c t i o n . We consider a perturbed operator: P = P

+ V(x)

where

V

i s a r e a l function i n

.

L;oc(En)

with

r > n/2

such

t h a t V -+ 0 as 1x1 + m The following r e s u l t d e s c r i b e s t h e asymptotic behavior of p o s i t i v e s o l u t i o n s of minimal growth a t a neighborhood of i n f i n i t y in

lRn

of t h e equation

Theorem 3.1.

F o r any

@X)u A
=

0.

there e x i s t s a p o s i t i v e solution

uo

of

(p-1)~ = 0 defined i n some neighborhood of i n f i n i t y i n Rn having minimal growth a t i n f i n i t y . The following asymptotic r e l a t i o n holds:

where

14

KX

of

S.Agrnon

is the compact convex set associated with the positive exponential solutions ( p - 1 ) ~ = 0 (see Theorem 2.3).

-

Note that px(x) is a continuous homogeneous function of degree 1 and that directions as 1x1 + only if p,(x) > 0 u (x) decays exponentially in 0 in En { O } . This will be the case iff X < Ao(P) (see definition of A. in 52). Theorem 3.1 yields an asymptotic formula for the behavior at infinity of Green’s functions of P - 1 It also gives asymptotic formulas for ground states associated with the operator p. One can also use the theorem to determine the bottom of the essential spectrum of realizations of the operator

.

?

in Lp(IRn).

Theorem 3.2.

We state some of these results in Let H

DCH)

=

{u:

UE

HU

Then Ao(P) E uess(H).

P in L2(lRn)

be the realization of

=

defined by

L ~ ( R ~ )n H:~~(R~), P ~ LE2(Rn ) I

PU

u E D(H).

for

Moreover

AO(P) = inf Re u

ess (H).

Let $(x) such that

be an eigenfunction of Rev < Ao(P). Then [$(x)I

where

px

eigenfunction

4.

5 c exp(-

is defined by (3.1), C

eigenvalue p with $(x)

H with a (possibly complex) eigenvalue

Rep
(x)) in E? Rev a constant. Finally suppose that H p

has an

Then H has a ground state, i.e., a positive

with eigenvalue X

=

inf Reo(H).

The following holds:

Extensions to Riemannian covering spaces The results which were described in the previous sections for elliptic

operators with periodic coefficients in lRn can be generalized to a wider class of elliptic operators on certain non-compact Riemannian manifolds. We now discuss briefly these extensions. and let P be a Let M be a compact Riemannian manifold of class ’C second order elliptic operator defined on M. We shall assume that in local coordinates the operator P is of the form i2.1) with coefficients satisfying the same local conditions as before. Let M be a covering space of M and assume that M is nonzcompact. We shall lift the elliptic operator and the Riemannian meiric to M and we shall continue to denote by P the elliptic operator on M. We shall be interested as before in-properties of positive solutions of the differential equation Pu = 0 in M and in related spectral properties of P. I n the special situation which we have already discussed,

15

Positive Solutions of Elliptic Equations namely when M = Tn

and

&

= Rn,

it was shown that whenever the equation

Pu = 0 admits a positive solution in Rn it also admits a positive exponential type solution and that the class of positive exponential solutions has a basic role in applications to spectral theory. In order to generalize these results from the n-dimensional torus to other compact Riemannian manifolds we must first define the analogue of exponential type functions on the covering space M. Let G te the group of covering transformations of (G is an infinite group since M is non-compact.) Given a real valued function u in M one says the u is G-multiplicative if for any g E G: for all x

u(gx> = y,u(x)

f4

in

'a is a constant. When yg = 1 for all g in G one says that u is a G-invariant funstion. Similarly one defines the notion of a G-invariant differential on M. It is clear that a G-invariant function (differential) on M can be considered as a function (differential) on M. where

Let now u be a positive G-multiplicative function of class C1 on M. Clearly dlogu is a G-invariant differential on M and thus can be considered as a closed differential on M. Consider the real vector space of all closed differentials obtained in this way modulo the real vector space of exact differentials. Invoking De Rham’s theorem it follows that this quotient space has a finite dimension m, 0. Hence, if m 1.1, there exist m positive G-multiplicative functions u on -fi of class C1 such that every positive G-multiplicative function u on M of class C1 has a unique representation of the form u(x) = w(x)exp(Sllog where

hl(x)

.

+. .+ Em log hm(x))

~(x) is a G-invariant function on M

and

,E,

...,em

(4.1) are real numbers.

Returning to the question of generalization of the results described in this lecture to elliptic operators on M, we now mention that all results admit such extensions in the case where G is an infinite abelian group. In the general case the role of the positive "exponential" solutions ( 2 . 3 ) is taken over by solutions of the form ( 4 . 1 ) , i.e., by positive G-multiplicative solutbrs of Pu = 0 in M. In particular when M is of parabolic type (that is when is abelian) the results described in the previous the fundamental group a,(M) sections can be extended to the (lifted) operator P in 2 where M is any non-compact covering space of M. A s an example we state the following Theorem 4.1. Under the assumption that G is abelian, suppose that the equation Pu = 0 admits a positive solution in- M. Then the equation also admits a G-multiplicative positive solution in M. Similarly when G is abelian one can establish a representation theorem for positive solutions of Pu = 0 in M which generalizes the representation theorem given at the end of section 2. In conclusion we shall show that if G is-not abelian then Theorem 4.1 need not be true. To this end we assume now that M is the universal cover of the We shall also take the compact Riemannian manifold M so that G = nl(M). operator P to be -A where A is the Laplace-Beltrami operator on M. We denote : by A the bottom of the spectrum of -A A = inf I -

M

2 IVV\

/ I _ 1v12 1. o

M where the infimum is taken over all smooth functions v M. We have the following

with compact support in

16

S. Agmon

Theorem 4 . 2 . Suppose t h a t A > 0 and l e t 0 < X 5 A _ . Then t h e equation ( A + X ) u = 0 does not admit a p o s i t i v e s o l u t i o n i n M which i s G-multiplicative.

+

It i s w e l l known Cseg I n t r o d u c t i o n ) t h a t the equation ( A A)u = 0 admits Thus Theorem 4 . 2 shows t h a t Theorem 4 . 1 a p o s i t i v e s o l u r i o n on M i f X 5 A i s f a l s e i f A > 0. This will be f o r i n s t a n c e t h e case i f M h a s a negative s e c t i o n a l curvature (see McKean 171). Recently Brooks [41 has shown t h a t A = 0 i f and o n l y i f wl(M) i s a n amenable group. (A group G i s amenable i f and only i f t h e r e i s a f i n i t e l y a d d i t i v e l e f t i n v a r i a n t measure on G.) Thus i t follows from Theorem 4 . 2 t h a t Theorem 4 . 1 f a i l s t o hold whenever nl(M) i s not amenable.

.

This i s t h e case, f o r i n s t a n c e , when group on two generators.

~i

CM) 1

c o n t a i n s a subgroup which i s a f r e e

(A + A)u = 0 admits We s h a l l show t h a t

[Sketch) Suppose t h a t t h e equation Proof of Theorem 4 . 2 . a p o s i t i v e s o l u t i o n uo i n M which i s G-multiplicative. t h i s leads t o a contradiction. o p e r a t o r s on

uo = eh

Set

i h +h A& = e Ae-

.

-

h

Since

and consider t h e two e l l i p t i c

M:

i s a G-additive f u n c t i o n i t follows t h a t

and

A+

a r e G-invariant

A-

e l l i p t i c o p e r a t o r s on 8. Thus we can consider A+ and A- a s being defined on 2 M. Since A i s formally s e l f - a d j o i n t i n L CM,dm) where dm i s the measure induced by t h e Riemannian m e t r i c i t follows r e a d i l y t h a t A- i s t h e formal a d j o i n t of

in

A+

2

L (H,dm). Now, i t i s n o t d i f f l l c u l t t o prove i n a g e n e r a l

s i t u a t i o n t h a t i f A i s a n e l l i p t i c operator i n M such t h a t t h e equation Au = 0 admits a p o s i t i v e s o l u t i o n i n M then t h e a d j o i n t equation A*u = 0 a l s o admits a p o s i t i v e s o l u t i o n i n M. We u s e t h i s r e s u l t i n our case. By X )u = 0 has t h e s o l u t i o n u E 1. Hence t h e c o n s t r u c t i o n the equation (A+

+

(A-

a d j o i n t equation denote by

+

A)u = 0

has a p o s i t i v e

C

2

solution in

Multiplying by a constant we s h a l l assume t h a t

I).

M

which we

min I)= 1. Next

M l i f t J,

to

and consider i t a s a G-invariant

s o l u t i o n of

(A-

+

A)$

=

0

i n 8 . Using ( 4 . 2 ) we thus f i n d t h a t (4.3) By a simple computation we o b t a i n from ( 4 . 3 ) t h a t U

U

(3) = 2 2

(A-x)

uO

J,

Since

(A+X)uo

=

0

in

I v(-$ l 2

U

2 1 on Recalling t h a t I) uo(l point

- 9-l) 0

(4.5).

G.

in

(4.4)

i t follows from ( 4 . 4 ) by s u b t r a c t i o n t h a t

I?,

d(u,

o

-

fi

0 T) 5-

while

i s a non-negative

C2

X(u,

+

uO 9

-)

< 0

in

0

M.

J,[x ) = 1 a t some p o i n t function i n

fi which

(4.5) xo, i t i s c l e a r

a t t a i n s i t s minimum a t a

.

x This i s however impossible i n view of t h e d i f f e r e n t i a l i n e q u a l i t y We have a r r i v e d a t a c o n t r a d i c t i o n which proves t h e theorem.

References

[l]

W. A l l e g r e t t o , On the equivalence of two t y p e s of o s c i l l a t i o n f o r e l l i p t i c

Positive Solutions of Elliptic Equations operators, Pac. J. Math. 55(1974),

319-328.

[2]

W. Allegretto, Spectral estimates and oscillation of singular differential operators, Proc. Amer. Math. SOC. 73(1979), 51.

[3]

W. Allegretto, Positive solutions and spectral properties of second order 15-25. elliptic operators, Pac. J. Math. 92(1981),

[4] R. Brooks, Amenability and the spectrum of the Laplacian, B u l l . h e r . Math. SOC. (New Series) 6(1982), 87-89. [5]

L. A. Caffarelli and W. Littman, Representation formulas for solutions to AU-U = o in R" , to appear.

[6] F. I. Karpelevic, The geometry of geodesics and the eigenfunctions of the Beltrami-Laplace operator on symmetric spaces. Trudy Moskow. Mat. Obsc. 14 (1965), 48-185 = Trans. Moscow Math. SOC. 1965, 51-199, h e r . Math. SOC., Providence, R.I., 1967. [7]

A on a manifold of negative curvature, J. Diff. Geometry 4 (1970), 359-376.

H . P. McKean, An upper bound to the spectrum of

[8] W. Moss and J. Piepenbrink, Positive solutions of elliptic equations, Pac J. Math. 75 (1978), 219-226.

[9] J. Piepenbrink, Nonoscillatory elliptic equations, J. Diff. Eqn. 15 (1974 541-550. [lo] J. Piepenbrink, A conjecture of Glazman, J. Diff. eqn. 24 (1977), 173-177 [ll] B. Simon, Schrodinger semigroups, B u l l . Amer. Math. S O C . (New Series) 7 (1982), 447-526. [12] G. Stampacchia, Le probleme de Dirichlet pour les equations elliptlques du second ordre a coefficients discontinus, Ann. Inst. Fourier 16 (1965), 189-258. [13] A. Tychonoff, Ein Fixpunktsatz, Math. Ann. 111 (19351, 767-776.

17

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

19

CLASSIFICATION OF I N I T I A L DATA FOR THE POROUS MEDIUM EQUATION I N RN N i c h o l a s D. A l i k a k o s Department of Mathematics Purdue U n i v e r s i t y West L a f a y e t t e , I N 47907 U.S.A.

1.

Rouben Rostamian Department o f Mathematics Pennsylvania State University U n i v e r s i t y P a r k , PA 16802

U.S.A.

INTRODUCTION

In t h i s paper w e a r e concerned w i t h t h e Cauchy problem f o r t h e porous medium e q u a t i o n w i t h n o n i n t e g r a b l e non-negative i n i t i a l d a t a :

The i n t e r p l a y between t h e o r d e r s of magnitude of

uo(x)

at large x

and

t h a t of u ( x , t ) a t l a r g e t was s t u d i e d i n [l]. Here w e d i s c u s s t h e meaning and i m p l i c a t i o n s of some of t h e r e s u l t s t h e r e and add some e x t e n s i o n s and observations,

A u s e f u l measure of t h e "order o f magnitude" of

uo(x)

for large

is the

x

s-average which w e d e f i n e by

s 1 0

f o r any radius quantity

r

f o r which t h e l i m i t e x i s t s .

centered a t

c ERN

and

IB

1

Here

Br(c)

is i t s measure.

i s t h e b a l l in RN of

With

s = N

the

ag(N), i f d e f i n e d , is in some s e n s e q u i t e c l o s e t o t h e i n t u i t i v e

c o n c e p t of t h e a v e r a g e , however see S e c t i o n 3 f o r a n o t h e r p o i n t of view. The a p p a r e n t p r i v i l a g e d r o l e of t h e o r i g i n i n t h i s d e f i n i t i o n is s u p e r f i c i a l , as one can show u s i n g t h e n o n - n e g a t i v i t y of uo t h a t i f t h e l i m i t above e x i s t s t h e n t h e N replacement o f Br(0) by Br(c) w i l l produce t h e same r e s u l t f o r each c € R

.

A somewhat d i f f e r e n t s - a v e r a g e , and one more c l o s e l y a s s o c i a t e d w i t h (PM) i s t h e q u a n t i t y a D ( s ) d e f i n e d a s f o l l o w s . For any s > 0 l e t

fj - 1

A (x)

=

ilB1l

,

N x € R

-

{O}

20

N.U. Afikakos and R. Rostamian

and p a s s i n g

t o z e r o set

s

Ao(x) = 6(x) = D i r a c ' s d e l t a f u n c t i o n .

The s-average

i s d e f i n e d as t h e c o n s t a n t ( i f it e x i s t s ) f o r which t h e f o l l o w i n g

a,(s)

convergence

(S

-

rN-S u o ( r x )

AVD) :

+

aD(s)As(x)

D'(lRN).

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

and

a,(s)

= aB(s),

as

r

If

+

e x i s t s , t h e n so does

a,(s)

but t h e converse i s n o t t r u e , [ l ] .

For t h e f u t u r e r e f e r e n c e , l e t ' s i n t r o d u c e h e r e one f i n a l d e f i n i t i o n . t h a t t h e a v e r a g e of uo i s uniformly e q u a l t o a i f

(UAV):

(x)dx = a

lim

uniformly i n

We s a y

N

c C R ,

r-

Almost p e r i o d i c f u n c t i o n s i n The q u a n t i t i e s

have t h i s p r o p e r t y , see Fink [ 4 , page 3 2 1 .

R1

aB(s),

aD(s),

Thus i f a n a v e r a g e o f

under (PM).

and uo

a,

whichever e x i s t , are i n v a r i a n t

e x i s t s t h e n t h e c o r r e s p o n d i n g a v e r a g e of

u ( - , t ) a l s o e x i s t s and remains c o n s t a n t f o r a l l t 3 0 ( s e e [l]). O f c o u r s e t h i s i s t u r e provided t h a t t h e s o l u t i o n of (PM) i s d e f i n e d f o r a l l t i m e . A r e s u l t of B e n i l a n , C r a n d a l l and P i e r r e [ 3 ] combined w i t h a theorem o f Aronscn and C a f a r e l l i [Z] shows t h a t a n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r g l o b a l - i n - t i m e e x i s t e n c e of s o l u t i o n s of (PM) i s

2

a (B m-1

+

N)

= 0.

Thus throughout t h i s paper w e w i l l

assume t h a t

2

0 5 s < - + N m- 1

(*) :

t o e n s u r e t h e e x i s t e n c e o f s o l u t i o n s . We w i l l d i s c u s s t h e i m p l i c a t i o n s of ( s -AVD) i n r e l a t i o n t o s o l u t i o n s o f (PM) i n S e c t i o n 2 , and t h e "meaning" of aD(s)

2.

and i t s r e l a t i o n t o t h e more i n t u i t i v e q u a n t i t y

THE IMPLICATIONS OF

(S

i n Section 3 .

-AVD)

Recall t h e d e f i n i t i o n of q,(x,t;E)

aB(s)

hs(x)

i n S e c t i o n 1.

For any

E 2 0

d e n o t e t h e s o l u t i o n of (PM) w i t h t h e i n i t i a l c o n d i t i o n

let

cAs(x).

This

s o l u t i o n e x i s t s f o r a l l t > 0 provided t h a t s s a t i s f i e s (*). The f o l l o w i n g theorem from 111 s a y s t h a t i n some s e n s e any s o l u t i o n of (PM) e v e n t u a l l y l o o k s l i k e some * qs'

Theorem 2.1:

Suppose t h a t (*) h o l d s .

Then f o r any c o n s t a n t

c > 0

w e have

The Porous Medium Equation in RN

i f and only i f

aD(s> f a r

a =

2

+

uo

N - s (m-l)(N-s)

The asymptotic states

e x i s t s and equals

’

’ q,(x,t;C)

identical f a r a l l i n i t i a l data

uo

=

2

+

8.

21

Here w e have l e t

1 (m-l)(N-s)

a r e u n i v e r s a l , i n t h e sense t h a t they a r e

with a common s-average

aD(s).

The c a s e

s = 0 already occurs i n Kamin [ 6 ] and Friedman and Kamin [ 5 ] . The case s = N is p a r t i c u l a r l y i n t e r e s t i n g s i n c e q N ( x , t ; d ) z 8 . We s t a t e t h i s a s a c o r o l l a r y : Corollary 2 . 2 :

For any constant

i f and only i f

aD(N)

for

Thus t h e e x i s t e n c e of

uo

c

0

we have

e x i s t s and equals

aD(N)

8.

i s necessary and s u f f i c i e n t f o r t h e uniform

convergence of u ( x , t ) t o a constant on b a l l s expanding a t t h e r a t e of r a t e fi i s c r u c i a l a s i t can be seen i n t h e following example. N = L

The s o l u t i o n of (PM) corresponding t o

uo(x) =

(2.2) :

i

1

if

x > O

0

if

x
converges uniformly t o a constant CT<

1/2, but not f a r

for this

uo

and t h e i n i t i a l condition

on b a l l s expanding a t t h e r a t e

Am > 1 / 2

The

See [l] f o r a proof.

t

U

,

a (N) does n o t e x i s t D can be d i r e c t l y v e r i f i e d , t h u s confirming t h e statement of t h e

c o r o l l a r y above. euqals

A.

1/2, not

IS=

1/2.

That

a (N) does e x i s t and B a s might be expected from t h e l i m i t i n g s t a t e of t h e

Note t h a t on t h e o t h e r hand s u r p r i s i n g l y

Am

s o l u t i o n . We w i l l have more t o say on t h i s i n t h e next s e c t i o n . The source of on t h e l e f t and r i g h t semi-axes. d i f f i c u l t i e s h e r e i s t h e s e v e r e unbalance of uo Radial symmetry produces a b e t t e r behavior: Theorem 2 . 3 :

Suppose t h a t

u0

i s r a d i a l , (*) h o l d s , and t h a t

lim t a ( u ( o , t ) - qs(O,t;t)l

=

o

.

t-

Then

aD(s)

far

uo

e x i s t s and equals

8.

Combined with Theorem 2 . 1 t h i s says t h a t f o r r a d i a l s o l u t i o n s t h e a t t h e o r i g i n implies t h e uniform convergence convergence of s o l u t i o n s t o qs over b a l l s expanding a t t h e r a t e t p . The proof of t h i s theorem i s long and t e c h n i c a l s o we o m i t i t here.

N. D. Alikekos and R. Rostamian

22

The next theorem considers t h e o t h e r extreme, where t h e convergence i s N uniform on e n t i r e R : Theorem 2 . 4 : (2.3):

uo

lim t+-

s a t i s f i e s (UAV)

i f and only i f

sup l u ( x , t ) - a1 =

o

N

x €R

Proof:

The s u f f i c i e n c y of (UAV) w a s proved i n [l]. To show t h e n e c e s s i t y l e t ' s note f i r s t t h a t (2.3) i n p a r t i c u l a r i m p l i e s (2.1) (with s = N) which i n t u r n implies t h e e x i s t e n c e of aD(N), t h a t is (2.4):

lim u (rx) = a 0

in

0.c~~)

r+=

What is more, due t o t h e uniformity of convergence i n (2.3) with respect t o t r a n s l a t i o n s i n space,(2.4) a l s o holds uniformly with r e s p e c t t o t r a n s l a t i o n s , that is

uniformly in

3.

c.

T h i s , however, i s equivalent t o (UAV) by Theorem (3.2)

THE MEANING OF

(S

-AVD)

We have already mentioned t h a t a B ( s ) = a,(s),

implies ( s -AVB)

( s -AVD)

t h e converse being f a l s e i n general.

and

In t h i s s e c t i o n we examine

some e x t r a hypotheses which make t h e converse t r u e . But before we proceed l e t ' s s t r e s s t h a t t h e condition ( s - A V ) above i s not adequate f o r d e s c r i b i n g a concept

B

of average i n any meaningful way i n general. Indeed one can give examples of functions uo f o r which a B ( s ) e x i s t , but when t h e b a l l s a r e replaced with cubes centered a t t h e o r i g i n i n ( s - A V ) , t h e l i m i t s e i t h e r f a i l t o e x i s t o r i f B they do they a r e o t h e r than a,(s). This of course i s somewhat obvious i f uo allowed t o t a k e both p o s i t i v e and negative s i g n s , ( w i t h t h e condition s i t u a t i o n i s more d e l i c a t e ) .

I t can be shown on t h e o t h e r hand t h a t i f

r a d i a l then t h e two l i m i t i n g procedures produce t h e same r e s u l t . more: Theorem 3.1:

If

uo

is r a d i a l then

( s -AV

Suppose t h a t

uo(x) = f ( 1x1)

and t h a t

B

c2 > c1 z 0.

Then f o r any

r z 0

uo

is

) and ( s - A V ) a r e equivalent and

n

( s -AVB)

holds.

is the

In f a c t we have

; g ( s ) = a,(s).

Proof:

uo 2 0

Fix

we have by a change o f v a r i a b l e s :

The Porous Medium Equation in RN

Jcc2

ITN-s

f(ra)aN-ldo

Thus a p p l y i n g ( s -AVB)

Now t a k e a in

cp

in

we obtain

N

CO(R )

and f o r s i m p l i c i t y assume t h a t it i s s u p p o r t e d

For a n a r b i t r a r y p o s i t i v e i n t e g e r

B1(0).

23

L

let

uk = k/L, k = 1,2, ...,L ,

and compute

Applying t h e mean v a l u e theorem r e p l a c e t h e i n n e r i n t e g r a l by f o r some :a r

-+

-

Taking

Isl(,)cp(o*,e)de k

E ( U ~ - ~ , U ~t h) e, n u s i n g t h e l i m i t i n g r e s u l t o b t a i n e d above l e t

to arrive a t

L

--t

=

t h e n w e o b t a i n from t h i s Riemann sum t h e f o l l o w i n g i n t e g r a l :

as r e q u i r e d . Another s p e c i a l case o f i n t e r e s t i s when ( s -AV B ) holds u n i f o r m l y w i t h r e s p e c t t o t r a n s l a t i o n s of c o o r d i n a t e s , t h a t i s

aB ( s ) = rlim

1 ~ -SIN ~ 'Br(c) 1

u (x)dx,

0

It can b e shown t h a t t h i s can happen o n l y i f

uniformly i n

ag(s)

=

0

c 6R

or

N

.

s = N.

The second

N.D. Alikakos and R. Rostamian

24

a l t e r n a t i v e i s t h e same as (UAV) of S e c t i o n 1. i m p l i e s t h e uniform v e r s i o n of ( s -AVD) w i t h Theorem 3 . 2 :

The n e x t theorem shows t h a t t h i s s = N:

(UAV) h o l d s i f and o n l y i f

+

u,(r(x-c)

uniformly i n

N c C R

c)

+

a

in

V'CR~)

.

The proof i s l o n g and t e c h n i c a l so w e omit it h e r e . by the uniform convergence here we mean t h a t f o r any

uniformly i n

c C R

N

c in

L e t ' s mention t h a t m N Co(R ) :

.

REFERENCES : N.D.

Alikakos, and R . Rostamian, On t h e u n i f o r m i z a t i o n o f s o l u t i o n s of t h e

porous medium e q u a t i o n i n

RN,

t o appear.

D . G . Aronson, and L.A. C a f f a r e l l i , The i n i t i a l t r a c e of a s o l u t i o n of t h e porous medium e q u a t i o n , m a n u s c r i p t . Ph. Bgnilan, M.G.

equation i n

RN

C r a n d a l l and M. P i e r r e , S o l u t i o n s o f t h e porous medium under o p t i m a l c o n d i t i o n s on i n i t i a l v a l u e s , m a n u s c r i p t .

A.M. Fink, A l m o s t P e r i o d i c D i f f e r e n t i a l E q u a t i o n s , L e c t u r e Notes i n Math., N o . 377, Springer-Verlag, 1 9 7 4 .

A. Friedman and S . Kamin, The a s y m p t o t i c b e h a v i o r of g a s i n an n-dimensional porous medium, T r a n s . h e r . Math. S O C . 262 (1980) 551-563. S . Kamin, S i m i l a r s o l u t i o n s and t h e a s y m p t o t i c s of f i l t r a t i o n e q u a t i o n , Arch. B a t l . Mech. Anal. (1976) 171-183.

DIFFERENTIAL EQUATIONS

I.W.Knowles and R.T.Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

25

STABILITY OF QUANTUM MECHANICAL SHAPE RESONANCES VIA RICCATI’S EQUATION Mark AshbaughX Department of Mathematics University of Missouri Columbia, MO 65211

Carl Sundberg Department of Mathematics University of Tennessee Knoxville, TN 37996-1300

The problem of locating resonance energies of Schrgdinger’s equation with a barrier potential is considered in the limit where the barrier is sent to infinity., The sphericallysymmetric three-dimensional case is treated by the methods of ordinary differential equations using an outgoing wave boundary condition to define resonance energies.

A detailed

analysis of the associated Riccati equation, S(x) = 1

-

(Wn(x)

- E)S(Xl2,

plays a major role in our approach; here W

(XI

is the barri-

er potential which is assumed to approach infinity as n for almost all x in the barrier region. n +

m

-+m

We show that as

the resonance energies associated with the above prob-

lem approach the eigenvalues of a related "unperturbed" problem.

We consider the quantum mechanics problem, -$'I

i (U(X) i

Wn(X))$ = E$

on 0 2

X

<

(1)

m,

where supp U c [0,1] and the Wn form a sequence of barrier potentials with the m for properties W integrable, W 2 0, supp Wn c [l,a] (1 < a < m ) , and Wn(x) n n a.e. x E [l,a] as n +a. This problem results upon separating variables in a -f

three-dimensional spherically symmetric problem and taking the angular momentum to be 0 , i.e.

we restrict ourselves to finding spherically symmetric solutions.

From a physical standpoint, the potential barrier W having certain energies E.

serves to trap particles

For energies near these values a trapped particle

will take a long tine to tunnel out through the barrier and escape to infinity. Corresponding to these energies, we define certain complex values of E to be resonance energies.

The somewhat unintuitive imaginary part is a measure of the

lifetime of the resonance; it is inversely proportional to the characteristic decay time of an associated physical state.

In the present paper we address the

problem of locating the resonance energies in the limit as n

+m,

i.e.

as the

potential barrier is sent to infinity. A similar problem with AW(x) replacing Wn(x) as the barrier potential in eq. (1)

M. Ashbaugh and C Sundberg

26

above was treated by Ashbaugh and Harrell [l] in the limit as A

They were

+m.

able to obtain more detailed results than we do here owing to the more specific nature of their problem.

An

extension of their basic stability result to the

general n-dimensional case was effected by Simon 181. For further details of the work presented here see 121. In the present context we define resonance energies as follows: Definition. E

E

C is a resonance ener y if equation (1) has a non-trivial solu-

-.e

-+Ex

tion $ satisfying $(O) = 0 and $ ( x )

as x

+

-.

(Our

convention cn the

square root function is that it is positive on the positive real axis and has a cut along the negative real axis). Remark. [5,7]

The condition IJJ(x)-eifi

boundary condition.

is called an outgoing wave or Gamow-Siegert

Since our potential vanishes for x > a this is equi-

valent to the condition $ ' (a) = i&

$(a) so that our problem may be viewed as a

(non-self-adjoint) Sturm-Liouville problem on the interval [Oral with eigenvalue parameter appearing in the boundary conditions. For use in our subsequent discussion we define two functions, $ (x,E) and qrn(xIE), 0

as solutions to the differential equation (1) satisfying the initial conditions

$0 (0,E) = 0, $’(O,E) = 1 and $Jrn(a,E) = 1, $:(a,E) = ifi. These are essentially 0 the regular and Jost solutions, respectively, from the theory of potential scattIn terms of $,

ering [3,6].

and Qm, the condition for aresonanceenergy is is the Wronskian of $0 and

simply W{$Jor$rn))x=l= 0, where W{$Jo,$,)

$rn

and we have

used the fact that it is constant for two solutions to equation (1) in choosing to evaluate it at the convenient point x = 1.

Actually, to isolate $

0

and qrn we

prefer to find resonances using F (E) = 0 where JIO(lrE)

$rn(lrErn)

$A(lrE)

- Jbb(llEI")

F (E) !

W{$or’$rn~lx=l =

$;(1rE)$;(lrErn)

*

(In the course of our investigation it transpires that the extra factors introduced in the denominator here do not vanish atresonance energies if n is sufficiently large.) In the following section we will show that $m (1,Ern) 0 as n - + m $L(l,E,n) uniformly for E in certain compact subsets of C. -+

Armed with this information we

now conclude the argument showing stability of resonances. Fn(E)

+

First we note that

$O(lrE)/$~(lrE) E F(E) which has the values e such that $O(lre) = 0 as

its roots.

These will be called the"unperturbedeigenva1ues" and are in fact the 2

eigenvalues of the operator h = -d / d x conditions.

Since

9,

2

+

U(x) on [0,1] with Dirichlet boundary

is analytic in E and

$m

is analytic in E except at E = 0

we can use Hurwitz’s Theorem from complex variable theory 141 to conclude :

27

Stability of Resonances via Riccati's Equation Theorem 1. (stability of Resonances).

For any unpertured eigenvahe e > 0 and for

n sufficiently large there is a unique nearby resonance energy E of equation (1) with En

e as n

-+

-+ oJ.

THE RICCATI EQUATION -+ 0 as n

To show that $m(lrE)/q:(l,E) equation which it satisfies. from this equation.

we will have recourse to the Riccati

m

-

(Wn(x)

-

. E)y 2 with

We must now extract the behavior of $m(l,E)/$~(lrE)

where y = $-/$:.

y(a) = l/i&

-f

This equation is y’ = 1

In the remainder of this paper we content ourselves with

only outlining the essential steps of our argument; for detailed proofs see 121. As a preliminary step we consider the simpler equation S’(X) = 1 on [0,1] where A(x) t 0 and integrable. E

- A(x)S(x) 2

We prescribe S(0)

and examine the behavior of S(1).

[O,n/2]

(2) =

if3 ce c

5 Or

This problem should be viewed as

coming from the above problem concerning y by a reflection and a rescaling of the x-axis (so that a becomes 0 and 1 remains 1) and by dropping E. S(x) = f(x)

+

Letting

ig (x) where f (x) and gfx) are real-valued we have f’ (x) = 1

- A(x) [f(x)2 -

2

g(x) I

g’ (X) = -ZA(x)f (x)g(x). From this system of equations it is easy to show the following facts: (1) f > 0 on (0,ll; (2) g’ 5 0 on [0,1] and either g E 0 or g > 0;

most one point to the left of which f

(3) f = g a t

g

and to the right of which f > g; (4) f < 1

+

c on [0,11.

We now insert A (x) in place of A(x) in equation ( 2 ) where A (x) 2 0 and An(x) - + m n for a.e. x E [0,1] as n + m . For the family of differential equations so obtained iB we define S (x) as that solution which obeys S (0) = ce Then we have: n n

.

Theorem 2.

Sn(x) exists on [0,1] and Sn(x)

compact subset of [ O , - ) , e eventually arg Sn(x) Remark.

E

E

[O,IT/21,

[0,11/4)

-f

0 as n

+CO

uniformly for c in any

and x in compact subsets of (0,lI. Moreover,

for any x > 0.

This shows that by taking n sufficiently large fn will surely pass above

gn and, further, that the crossing point can be made as near as we like to x = 0. Here we have set S (x) = fn(x) + ign(xl. n

To deal with resonances we shall need to extend the above result to values of greater than T / 2 .

We take f3

E

[n/2,

5'T/8]

as a convenient choice.

The extended

result will follow from the above once we have shown that for sufficiently large n

M. Askbaugh and C Sundberg

28

fn has a zero z arbitrarily near x = 0 and that gn does not grow too rapidly while f is negztive; in particular, we show gn ( zn ) 5 c m These results

.

yield; Set B

Let S (x) and An (x) be as above.

Theorem 3 .

P

= {zlO 5

arg

z 5

x Then s (x) exists on [0,1] and S ( x ) 3 0 uniformly in S (0) E B n n n PI any p > 0 and x1 > 0. Also for any x1 > 0 there exists N such that 0 5 arg S (x) < 1T/4 for all n 2 N and all x

E

E

5n/8,lz/ p}.

[x,,ll

for

[xl,ll.

To finish off the argument showing that +m(l,E,n)/Jbb(l,EIn)

*0

as n

+ m

we put E

back into our equation and use a comparison technique. Passing back to second order linear equations for a moment we define u and v by u; = A (x)u and n n n n v" = (An(x) E)vn with initial conditions u ( 0 ) = 1 = v ( 0 ) and n -1 -iB - I v (x) = w (x)u (x) and fin((x) = wA(x)/wn(x) we uA(0) = c e - ~ ~ ( 0 )Setting . n n n

-

find that

From this equation and our previous results about Sn(x) that i2 (x) is bounded independently of n for large n.

+ w’fx)/wn(x)

= u;(x)/Unix) v’ix)/vn(x) n

it follows that v~(x)/vn(x) vn(l)/vA(l)

*o

as n

+a,

= l/SnfX)

*

0.

In particular,

or more precisely we have:

Theorem 4. With vn(x) as defined above, vn(l)/vA(l)

E

+ ,.(nif

for x > 0 or that vn(x)/vi(x)

+ m

all E in any compact subset of

iz

= u,(x)/u’ (x) one can show n Then from the relation

CIO

C

+

0 uniformly as n

for

and for all initial conditions in

< arg z 5 57~/8, p1

5

1.

5

~ ~ 1 .

When translated back into the notation of the resonance problem we began with, this shows that i/i,(l,E,n)/i/i~(l,E,n) E

E

{z

E

cI-r/4 5 arg z <

71,

R1

5

0 as n

+

IzI

5

1 2

R

+ -

uniformly over

where 0 < R1 5 R2 <

m.

This, in turn,

is precisely what is needed in order to prove our stability of resonances result (Theorem 1) using Hurwitz’s Theorem. REFERENCES Ashbaugh, M.S., and E.M. Harrell, Perturbation theory for shape resonances, and large barrier potentials, Commun. Math. Phys. 151-170 (1982). 121 Ashbaugh, M.S., and Carl Sundberg, A n improved stability result for resonances to appear in Trans. Amer. Math. SOC. 131 Chadan, K., and P.C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer-Verlag, New York, 1977. 141 Conway, J., Functions of One Complex Variable, Springer-Verlag, New York, 1973. [51 Gamow, G., Constitution of Atomic Nuclei and Radioactivity, Oxford University Press, oxfcrd, 1931. I61 Reed, M., and B. Simon, Methods of Modern Mathematical Physics, Vol. 111: Scattering Theory, Academic Press, New York, 1979.

[l]

z,

Stability of Resonances via Riccati’s Equation

173

181

Siegert, A.J.F., On the derivation of the dispersion formula for nuclear reactions, Phys. Rev. 56, 750-752 (1939). Simon, B., Exterior complex scaling and molecular resonances in the BornOppenheimer approximation (to appear).

29

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

31

REMARKS ON THE INVERSE SQUARE POTENTIAL I N QUANTUM MECHANICS PIERRE BARAS

and

JEROME A. GOLDSTEIN*

Department of Mathematics and Quantum Theory Group Tulane U n i v e r s i t y New O r l e a n s , LA 70118 U.S.A.

L a b o r a t o i r e IMAG Tour d e s M a t h i h a t i c s Analyse Numerique, BP 68 38402 S t . Martin d’H5res Cedex FRANCE

let = -A - c / l x (

H

be d e f i n e d on

CT(WN\ { O ] ) .

2

T h i s (symmetric) Schr‘ddinger o p e r a t o r (on

2 N L (R ))

h a s been much s t u d i e d because i t h a s t h e c u r i o u s p r o p e r t y o f b e i n g well-behaved o r p o o r l y behaved depending on t h e v a l u e o f t h e c o n s t a n t c*(N) = [ ( N - 2)/2]‘ = N(N

C,(N)

C*(N) = C,(N)

Note t h a t

adjointness of ‘MEOREM 1.

HC

(i)

+ 1.

-

4)/4

for

N = 1,2,

for

N

2

1

Let

c.

...,

.

Concerning t h e semiboundedness and e s s e n t i a l s e l f -

we have t h e f o l l o w i n g r e s u l t .

is nonnegative if

Hc

c

5

Hc is not semibounded if

C*(N).

c > C*(N). (ii)

Hc

is essentiazly self-adjoint if and onZy if

c I C,(N).

This theorem c a n be found i n Kalf-Schmincke-Walter-Wkt [4, p.1871. a l s o Fefferman [ 3 ] and Reed-Simon [ 5 , pp.161,186,325].

See

Both [4] and [S] c o n t a i n

f u r t h e r i n f o r m a t i o n and r e f e r e n c e s . I n t h e a f f i r m a t i v e p a r t o f t h e s t a t e m e n t of Theorem l ( i ) i n [ 4 ] , t h e assertion is t h a t i s nonnegative.

c 2 C*(N).

i s bounded below i f

H

For t h e p r o o f suppose

But t h i s i m p l i e s t h a t

f o r some

Hc 2 -a1

a > 0.

Let

6 > 0

and l e t (U6f)(X) for

f

c

2 N L (W )

and

x

E

IR

N

.

Then

=

U6

-N/2

6

f(6x)

is u n i t a r y on

L2(WN)

and a s i m p l e

c a l c u l a t i o n shows t h a t

U6HcUi’ so that

*

Hc

i s u n i t a r i l y equivalent t o

P a r t i a l l y s u p p o r t e d by an NSF g r a n t .

= 6 -2 Hc

6-

,

times i t s e l f .

But

H

2 -a1

Hc

P. Baras and J. A. Goldstein

32 2

implies 6- H

2

-6-2aI , and since

~ )O(H~) inf ~ ( U ~ H ~ U= -inf

(u

spectrum), we may let 6 + m to conclude H 2 0. can have no eigenvalues. A similar scaling argument shows that Hc 2 N Namely, suppose Hc@ = E+ with @ 6 L (W ) . For @,(x) = @(Ax) with A > 0 2 and x E WN we get Hc$A = A E$*. The orthogonality relation between eigen=

vectors corresponding to different eigenvalues gives ( f o r

as A

1. Thus $ E 0. With respect to semiboundedness C*(N)

X

f

1)

+

value of c.

is clearly the cutoff point for the

If Kc denotes a (semibounded if possible) self-adjoint extension

of Hc , then the initial value problem for the parabolic equation au/at 2 N is well-posed in L (R ) for c

5

+ K ~ = U

C*(N).

o

It is not well-posed if c > C*(N) ,

b u t it is not clear to which extent it is solvable o r not.

can be replaced by (Let us add parenthetically that the domain C:(RN\{O)) 2 N 2 5. This is because E Lloc(WN) if N 2 5.) Recently Baras and Goldstein [I] proved a result which, among other things,

XI-^

C i ( R N ) if

provides an alternate proof and strong sharpening of the second assertion of Theorem l(i). Of concern is the heat (rather than the Schrodinger) equation au/at

+

u + f(x,t) 1x1 T > 0 , c > 0 , and 0

= AU +

(1)

1 where S f E L (RN X ( 0 , T ) ) are given. A solution in the sense of distributions is sought whose initial value is a (nonnegative) finite Radon measure LI in the sense that

for

(x,t)

for each

E

RN x ( 0 , T )

@ E C;(W

N

)

.

We approximate (1) by

where

THEOREM 2 [l]. provided

(i)

If c

5

C*(N)

, then (l), (2) has a nonnegative solution u

33

The Inverse Square Potential in Quantum Mechanics

where cx i s the smallest root of

(N-Z-a)a=c. The solution i s given by solution o f (3,) constant

,

(2)

.

(4)

u(x,t) = lim un(x,t) where u i s the nonnegative n* For each E E (0,T) and R > 0 there i s a p o s i t i v e

C such t h a t u(x,t)

2

(u,f) #

i f 1x1 < R , t E (E,T) , and nonnegative solution, then

ClxJ-"

(51

ConuerseZy, i f

(0,O).

(11, (2) has a

f o r each R > 0 and each E E (0,t) where a is as above. (ii) If c > C*(N) , and i f a t Zeast one of p , f i s not zero, then (I), (2) has no nonnegative solution.

for each

t > 0 and each x

Moreover, f o r

E

lim u (x,t) = n-rm N

u

as i n (i) above,

m

w .

Note that a is given by ~1

= (N - 2)/2

- [((N

-

2)/2)'

- 4cI1"

Thus IxI-" E Lloc(R Let 1-I be absolutely 2 N ) i f c 5 C*(N). 1 N continuous, s o that u(dx) = u0 (x)dx with 0 2 uo E L ( R ) . If also 2 N uo E L (R 1 (and, say, f E 0), then ( 6 ) holds automatically by the Schwarz inequality. Thus the necessary condition (6) is consistent with the L2 theory. We shall give a heuristic argument to make the statement of Theorem 2 seem plausible. F o r a detailed proof see [l]. Let cx s (N - 2 ) / 2 . Then f o r k > 0 , $(x)

klxl-"

satisfies

where c is given by (4). By Maximum Principle and comparison arguments, we expect to have good existence results f o r c as in (4) corresponding to a < (N - 2 ) / 2 , i.e., f o r c s C * ( N ) . Now let c > C*(N). Rewrite

P. Baras and J.A. Goldstein

34

au/at

=

AU

+

C 2 u 1x1

as au/at

=

nu

-u

+ C*"Z'

+ f(x,t)

1x1

where

The n e c e s s a r y c o n d i t i o n s (5) and (6) imply

since

a = ( N - 2)/2

(which c o r r e s p o n d s t o t h e v a l u e

C*(N)

i n applying p a r t

T h i s c o n t r a d i c t i o n shows a s o l u t i o n cannot e x i s t .

( i ) o f t h e theorem).

A d i r e c t approach t o p a r t ( i i ) may b e based on a s t u d y of t h e k e r n e l

u(dy).

)1

, as b e f o r e , so t h a t uo(y)dy

can b e r e p l a c e d

An a l t e r n a t e r e p r e s e n t a t i o n i s g i v e n by t h e Feynman-Kac formula

N

lR ) is t h e s p a c e o f c o n t i n u o u s p a t h s i n IRN and Px Wiener measure s t a r t i n g a t x . T h i s c l e a r l y i m p l i e s t h a t u n ( x , t ) i s where

S = C([O,m);

nondecreasing

t > 0

l i m G (x,y,t) n-rm

x,y

An

Let

> 0

,

= m

IRN .

E

=

i n f o(H:)

be t h e ground s t a t e energy o f

c o r r e s p o n d i n g wave f u n c t i o n , i . e . $,(x)

for all

x ; Qn

@n

2

t

N

L (1R ) ,

II@nl(

An

+

-m

as

n

+a.

H: = 1

and l e t

, H:

9,

@n =

L

be t h e

Xn qn

and

e x i s t s and i s u n i q u e l y determined by t h e s e c o n d i t i o n s

by a s u i t a b l e v e r s i o n o f t h e Perron-Frobenius theorem. show

is

.

I t i s enough t o e s t a b l i s h t h a t , f o r

for all

Gn

,

can b e a measure

uo

Note t h a t by

= -A + V,(x)

H:

d e f i n e d by:

If

c > C *( N )

one can

The Inverse Square Potential in Quantum Mechanics

35

2

Since Vn(x) = n Vl(nx) , a scaling argument similar to those given above imp1ies =

nN’2

@l(nx)

,

xn

n

=

2

x1 .

Consequently -tX

exp(-tHn)

2

e

>

-tAn Gn(x,y;t) = LexP(-tHpxl(Y) 2 e -tn2A = n e b,(nx)@,(ny) But

A1 <

(since A

0

=

n2A

1

-+

and

N

R

-+

m

Pe-"IZI

a = (-A )''* 1

-

E

where 0 <

+

lyl)}

E

<

(-A

1

Thus

Gn(x,y;t) as n

t

by [2; pp.73-741; here

B = 8, > 0.

.

and

-m)

@,(z)

holds on

@n(x)@nCY)

for all

2

(t,x,y)

BnN expi-tn2 A, E

(0,m)

x

- an(lxl

+

m

1~~ x 1 ~ ~ .

Our original interest in these matters resulted from suggestions o f

H. Brezis and J.-L. Lions. We also thank Brezis for his continued interest and encouragement. Finally we thank C. Fefferman, H. Kalf, and E. Lieb for stimulating and helpful discussions.

REFERENCES [l]

P. Baras and J. A . Goldstein, In preparation.

[2]

R. Carmona and B. Simon, Pointwise bounds on eigenfunctions and wave packets in N-body quantum systems, Corn. Math. Phys. 80 (1981), 59-98.

[3]

C. L. Fefferman, The Uncertainty Principle, Lecture Notes for the h e r . Math. SOC. Colloquium Lectures, Denver, Colorado, January, 1983.

[4]

H. Kalf, U.-W. Schmincke, J. Walter, and R. W%t, On the spectral theory of Schradinger and Dirac operators with stronEly singular potentials, in Spectral-Theory and Differential Equations-(ed. b; W. N-. Everitt) Lecture Notes in Math. No. 448, Springer, Berlin (1975), 182-226.

;

[5]

M. Reed and B. Simon, Methods of Modern Mathematical Physics 11: Fourier Analysis, Self-Adjointness, Academic, New York, 1975.

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DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis(Editors) 0Elsevier Science Publishers B.V. (North-HoUand), 1984

37

JULIA SETS AND AUTONOMOUS DIFFERENTIAL EQUATIONS Michael F. B a r n s l e y and Andrew N. H a r r i n g t o n School o f Mathematics Georgia I n s t i t u t e o f Technology A t l a n t a , Georgia 30332

We r e v i e w t h e t h e o r y o f moments o f balanced measures on J u l i a s e t s , and i l l u s t r a t e i t s c o n n e c t i o n w i t h t h e s t u d y o f c e r t a i n autonomous o r d i n a r y d i f f e r e n Our emphasis here i s on how, s t a r t i n g f r o m a r a t i o n a l mapping

t i a l equations.

F ( z ) o f t h e complex p l a n e i n t o i t s e l f , under t h e a p p r o p r i a t e c o n d i t i o n s one can c o n s t r u c t an a s s o c i a t e d s e l f - a d j o i n t o p e r a t o r whose spectrum i s a f r a c t a l (see The procedure may i l l u s t r a t e how

[ l ] ) r e l a t e d t o F ( r ) , namely i t s J u l i a set.

s i m p l i f i e d H a m i l t o n i a n s f o r c o m p l i c a t e d p h y s i c a l systems, w i t h f r a c t a l s t r u c t u r e , m i g h t be developed; we have i n inind snowflakes, t h e r i n g s o f S a t u r n and c o a s t l i n e s , f o r example.

To i l l u s t r a t e how a f r a c t a l can be a s s o c i a t e d w i t h an i t e r a t e d r a t i o n a l map, we c o n s i d e r Newton's method a p p l i e d t o f i n d t h e z e r o s i n t h e complex p l a n e (1: o f 3 2 The a s s o c i a t e d r a t i o n a l map i s F ( z ) = z - f ( z ) / f ' ( z ) = 2 z / ( 3 z - 1 ) , f ( z ) = z3-z. 0 which has -1, 0, and +1 as a t t r a c t i v e f i x e d p o i n t s . Using t h e n o t a t i o n F ( z ) =

z and Fn+'(z) = F ( F n ( z ) ) , we say t h a t a p o i n t z i s a t t r a c t e d t o a Lim F " ( z ) n-

=

6

ic if

a.

I n t h e f o l l o w i n g s k e t c h we show a p a r t o f Ic which i n c l u d e s t h e a t t r a c t i v e f i x e d

0 and +1

p o i n t s ; t h e d i f f e r e n t shadings show which p o i n t s a r e a t t r a c t e d t o -1, respectively.

The boundary o f t h e s e t of p o i n t s a t t r a c t e d t o -1 i s t h e same as

t h a t f o r t h e s e t o f p o i n t s a t t r a c t e d t o 0; i t i s a f r a c t a l because i t i s not s i m p l i f i e d by m a g n i f i c a t i o n ,

as i l l u s t r a t e d i n t h e second s k e t c h w i t h a blowup

o f one o f t h e components i n t h e f i r s t sketch; t h i s boundary i s t h e J u l i a s e t f o r

F(z). Now l e t F ( z ) be an a r b i t r a r y r a t i o n a l f u n c t i o n o f degree N > 1, t a k i n g t h e A

extended complex p l a n e Ic =

(1: u Em}

o f F i s a s e t o f p o i n t s {z1,z2

into itself.

,...,z k )

For k

such t h a t F(zl)

6

{1,2,3,

= z2,

...1

F(z2)

=

a k-cycle

z 3 ,...,

and zi # z . f o r i # j . I f z1 # t h e k - c y c l e i s s a i d t o be a t t r a c J t i v e , i n d i f f e r e n t o r r e p u l s i v e a c c o r d i n g as l F ( k ) ( z l ) l < 1, =1 o r >1 respecF ( z k ) = z,,

tively.

If z1

=

l e t G(z) = l / F ( l / z ) and use G ( k ) ( 0 ) i n p l a c e o f F ( k ) ( z l )

t h e p r e v i o u s sentence.

The J u l i a set J o f F ( z ) can be d e f i n e d as t h e s e t of

p o i n t s z such t h a t z belongs t o a r e p u l s i v e k - c y c l e of F ( z ) f o r some k

E

in

M. F. Barnsley and A. N. Harrington

38

9

points a t t r a c t e d

/ to 0

@ ;;i;;s

attracted

unshaded points a t t r a c t e d t o +1

{1,2,3,...1, [2,3,4]. J i s completely i n v a r i a n t under F, namely F-lJ = J ; i t contains no i s o l a t e d points; and usually i t has no i n t e r i o r and has a complicated f r a c t a l s t r u c t u r e -- t h i s i s the case f o r example i f there a r e t h r e e d i s t i n c t points, each o f which belongs t o a n a t t r a c t i v e k-cycle of F f o r some k E {1,2,3, I. The J u l i a s e t f o r F(z) = z 2 - A , X E R , i s important t o the theory 2 o f t h e i t e r a t i o n s o f F(x) = x -1, X E IR , a s studied by Feigenbaum and o t h e r s ,

...

see [51. The J u l i a s e t J and i t s i n t e r p l a y with F can be characterized by t h e f o l lowing:

THEOREM [6,7]. Let F ( z ) be a r a t i o n a l function of degree N > 1. There e x i s t s a unique p r o b a b i l i t y measure p such t h a t p(E) = p(FY1E)/N whenever E i s a Bore1 J 1 subset o f $, where {F: ( z ) l ~ = ,i s a complete assignment of inverse branches of F. J

We c a l l p t h e balanced measure f o r F, [ 8 ] . F i s an ergodic transformation of J w i t h respect t o p. In order t h a t we may be able t o construct a s e l f a d j o i n t operator whose spectral density i s j ~ , we consider the moments o f u.

39

Julia Sets and Autonomous Differential Equations THEOREM [9].

L e t F ( z ) be a r a t i o n a l f u n c t i o n o f degree N > 1.

p o l e a t c, where a,B,y

and 6

L e t c belong t o

L e t $ ( z ) = (az+B)/(yz+6) have

an a t t r a c t i v e o r i n d i f f e r e n t k-cycle o f F(z).

A l l o f t h e moments

E (c.

f o r n E ~0,1,2y...}, f (@(z))'ddu(z), J e x i s t and can be c a l c u l a t e d r e c u r s i v e l y i n terms o f t h e c o e f f i c i e n t s i n F(z). Mn =

-

.

2 Then c = i s an a t t r a c t i v e 1-cycle f o r F(z) and L e t F(z) = (z-2) we can choose $ ( z ) = z. The theorem asserts we can c a l c u l a t e t h e moments EXAMPLE.

Mn =

I

We f i n d Mo = 1, M1 = 2, M3 = 6, M4 = 20, t h i s example t h a t J = [0,4]

for n

zndp(z)

J

E

{OY1,2,...1.

...; i n f a c t i t i s w e l l

known

[lo]

for

JX.(4-x)) f o r 0 < x < 4.

and dp(x) = l / ( n

The n e x t step i s t o consider t h e orthogonal monic polynomials { P , ( Z ) } ~ = ~ where Pn(z) i s o f degree n, has u n i t leading c o e f f i c i e n t , and

f

Pn(z)

dv(z) = 0

J f o r n # m, where the bar means the complex conjugate i s taken. When

F(z) i s a polynomial o f degree g r e a t e r than one, i t i s always p o s s i b l e

t o c a l c u l a t e r e c u r s i v e l y i n terms o f t h e c o e f f i c i e n t s i n F ( z ) an i n f i n i t e subsequence o f t h e associated orthogonal polynomials, [8, 17, 121.

F(z) i s any r a t i o n a l t r a n s f o r m a t i o n f o r which m belongs t o an a t t r a c t i v e o r i n d i f f e r e n t k-cycle and f o r which J c IR u Cm}, one can use t h e moments When

M,

=

f

xndp(x)

J for n

E

{Oyl,Z,

...I

t o c a l c u l a t e a l l o f t h e orthogonal polynomials associated w i t h

F, according t o

1

zn

These polynomials u n i q u e l y determine r e a l numbers an and bn, n where a.

= 0 and an > 0 f o r n

E

I0,l ,2,..

.I,

# 0, such t h a t

Pn+l ( 2 ) = (z-bn)Pn(z)

2 - anPn-l (z).

I n t u r n , t h e an's and bn's i n the three-term recurrence formula f i x a h a l f i n f i n i t e Jacobi m a t r i x

40

M.F Barnsley and A. N. Harrington

-

THEOREM [9]. which

{-I.

L e t F ( z ) be a r a t i o n a l f u n c t i o n , o f degree g r e a t e r t h a n one, f o r

belongs t o an a t t r a c t i v e o r i n d i f f e r e n t k - c y c l e and f o r which J

c

iR

u

The a s s o c i a t e d J a c o b i m a t r i x d i s s e l f - a d j o i n t w i t h spectrum J and s p e c t r a l

d e n s i t y p, t h e balanced measure f o r F(z). EXAMPLE.

L e t F ( z ) = z 2 -A

with

X > 2.

The c o n d i t i o n s o f t h e l a t t e r theorem a r e

met, J b e i n g a Cantor s e t i n R . In [131, i t i s shown t h a t bn = 0 f o r a l l n , 2 2 2 2 w h i l e a2m+, = h - a I n [14], i t i s and a2m+2 = X/ a2m+l f o r rn E {0,7,2, }. 2m sholvn t h a t t h e sequence {am} i s almost p e r i o d i c .

...

F i n a l l y , we d e s c r i b e an example which p r o v i d e s a c o n n e c t i o n between t h e t h e o r y of moments o f balanced measures on J u l i a s e t s and c e r t a i n autonomous o r d i n a r y d i f f e r e n t i a l equations.

Consider t h e e q u a t i o n

on E w i t h z(1) = P(eia)

and a / 2 ~any s u i t a b l e i r r a t i o n a l number. Here P(z) 2 3 denotes t h e W e i e r s t r a s s P - f u n c t i o n which obeys ( P ' ( z ) ) = 4P(z) -4P(z). The branch o f t h e square r o o t i s chosen t o ensure t h a t t h e m o t i o n proceeds " f o r w a r d " along the t r a j e c t o r y .

I n 191 i t i s shown t h a t t h e w - l i m i t s e t o f t h e t r a j e c t o r y

h

i s t h e whole o f t, and t h a t t h e c o r r e s p o n d i n g i n v a r i a n t measure i s e x a c t l y t h e balanced measure f o r t h e r a t i o n a l f u n c t i o n , f i r s t s t u d i e d by L a t t 2 s [13],

2

2

F ( z ) = ( z +1) /(4z

2

(z -l)),

whose J u l i a s e t i s t h e whole o f

-

be shown t o be d p ( z ) = ( c o n s t a n t ) d x d y / < l z /

n

E

{0,1,2,

...1 ,

2 I z -1

1).

?. T h i s

measure can

The moments J@(z)'dU(z),

do n o t a l l e x i s t , regardless o f t h e choice f o r t h e pole c o f

442).

Herman [15] has r e c e n t l y p r o v e d t h a t t h e r e e x i s t r a t i o n a l t r a n s f o r m a t i o n s A

which possess an i n d i f f e r e n t f i x e d p o i n t c =

m,

and have J = &. For t h e s e

examples a l l of t h e moments

J zndv(z) , J

n

t

{0,1,2

,...I,

do e x i s t and can be c a l c u l a t e d r e c u r s i v e l y . REFERENCES

[l]

Mandlebrot, B., 1982)

.

The F r a c t a l Geometry of N a t u r e

(W.

H. Freeman, San F r a n c i s c o

Julia Sets and Autoizornous Differential Equations

[Z]

Fatou, M. P., Sur l e s equations f o n c t i o n e l l e s , Bulletin de SOC. Math. de France 47 (1919), 161-271; ibid. 48, 33-94; i b i d . 48, 208-314.

[3]

J u l i a , G., Memoire sur L ' i t e r a t i o n des fonctions r a t i o n e l l e s , J r n l . de Math. Pures e t Appl. 4 (1918), 47-245.

41

141 Brolin, H., Invariant sets u n d e r i t e r a t i o n of r a t i o n a l functions, A r k i v f l r Matematik 6 (1965), 103-144. [5]

Collet, P. and Eckmann, J., I t e r a t e d Maps on t h e Interval a s Dynamical Systems (Birkhauser, Boston 1980).

[6]

Demko, S., In preparation.

[7]

Freire, A., Lopes A. and Ma;, R., An i n v a r i a n t measure f o r rational maps, Preprint, I.M.P.A., Rio d e daneiro (1982).

[8]

Barnsley, M., Geronimo, J. S. and Harrington, A. N., Orthogonal polynomials 7 (19821, associated with invariant measures on J u l i a s e t s , Bulletin A.M.S. 381 -384.

[9]

Barnsley, M. and Harrington, A. N., Moments o f balanced measures on J u l i a sets, P r e p r i n t , School of Mathematics, Georgia Tech., (Dec. 1982).

[lo]

Ulam, S. and von Neumann, J . , On combination of s t o c h a s t i c and deterministic processes, Bulletin A.M.S. 53 (1947), 1120.

[ll]

B e s s i s , D. and Moussa, P., Orthogonality properties o f i t e r a t e d polynomial mappings, Commun. Math. Phys. 88 (1983), 503-529.

[12]

Pitcher, T. S. and Kinney J. R., Some connections between ergodic theory and i t e r a t i o n o f polynomials, Arkiv f g r Matematik 8 (1968), 25-32.

[13]

L a t t k , S., Sur L ' i t e r a t i o n s des s u b s t i t u t i o n s r a t i o n e l l e s e t Les fonctions de Poincare, Note aux C. R. Acad. Sc. P a r i s 166 (1918), 26-28.

[14]

B e l l i s s a r d , J . , Bessis, D. and Moussa, P., Chaotic s t a t e s of almost periodic Schradinger operators, Phys. Rev. L e t t e r s 49 (1982), 701-704.

[15]

Herman, M. R., Examples de f r a c t i o n s r a t i o n e l l e s ayant une o r b i t e dense sur l a sphere de Riemann, Preprint, Centre de Mathematiques de L'Ecole Polytechnique, France, (March 1983).

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DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

43

A L I M I T I N G ABSORPTION PRINCIPLE FOR A SUM OF TENSOR PRODUCTS

Matania Ben-Artzi Department of Mathematics Technion I T T Haifa 32000, I s r a e l Allen Devinatzl Department of Mathematics Northwestern University Evanston, I L 60201, USA

A l i m i t i n g absorption p r i n c i p l e i s obtained f o r a sum of t e n s o r products of t h e form

H = HI 8 I2

+ I, 69

H2

under v a r i o u s hypotheses on t h e s e l f - a d j o i n t o p e r a t o r s H, and H 2 .

§ 1.

INTRODUCTION

In t h i s paper we s h a l l consider t h e problem of o b t a i n i n g a l i m i t i n g a b s o r p t i o n p r i n c i p l e f o r s e l f - a d j o i n t o p e r a t o r s of t h e form H = HI 69 I 2 + I, @ H 2 , (1.1) where H1 and Hp a r e s e l f - a d j o i n t o p e r a t o r s on H i l b e r t spaces Hiand &, and 11 and I 2 a r e t h e i d e n t i t y o p e r a t o r s on t h e s e spaces r e s p e c t i v e l y . A s i s w e l l known, o p e r a t o r s of t h e form (1.1) a r i s e q u i t e n a t u r a l l y i n c l a s s i c a l physics, i n quantum mechanical s e t t i n g s , a s well a s elsewhere. The most usual s i t u a t i o n s occur when HI and Hp a r e d i f f e r e n t i a l o p e r a t o r s a c t i n g i n L2 spaces. For example, t h e SchrEdinger o p e r a t o r s

a r e n a t u r a l examples. Limiting a b s o r p t i o n p r i n c i p l e s f o r t h e s e o p e r a t o r s are immediate consequences of t h e a b s t r a c t r e s u l t s we s h a l l p r e s e n t . Thus we supply a general framework i n t o which t o f i t known r e s u l t s (see e.g. [ 1 ] , [ 3 ] , [ 4 ] )a s w e l l as a general framework f o r o b t a i n i n g new r e s u l t s . The methods of proof of the theorems i n t h i s paper a r e elementary i n n a t u r e i n t h e sense t h a t w e use only well known c l a s s i c a l r e s u l t s and methods. These include t h e o p e r a t i o n a l c a l c u l u s i n H i l b e r t space, p r o p e r t i e s of t h e Poisson k e r n e l and t h e most elementary and c l a s s i c a l p r o p e r t i e s of H i l b e r t transform theory. We s h a l l o b t a i n a l i m i t i n g a b s o r p t i o n p r i n c i p l e f o r H under varying hypotheses on In t h e f i r s t s e c t i o n w e s h a l l g i v e t h e simplest of such r e s u l t s . In H I and H p . t h e n e x t s e c t i o n s w e s h a l l g i v e v a r i a t i o n s on t h e theme of t h e f i r s t s e c t i o n as w e l l as a f u r t h e r r e s u l t w i t h hypotheses of a somewhat d i f f e r e n t c h a r a c t e r .

44 52.

M. Ben-Artzi and A. Devinatz A LIMITING ABSORPTION PRINCIPLE

L e t us i n t r o d u c e some terminology which w e s h a l l u s e i n t h i s and i n succeeding s e c t i o n s . We s h a l l t a k e H1 and H 2 t o be s e l f - a d j o i n t o p e r a t o r s on t h e H i l b e r t I m z f 0, j = 1,2, Their r e s o l v e n t s a r e R j ( z ) = ( H . - .z)-', and H,. spaces and t h e r e s o l v e n t o f t h e o p e r a t o r (1.1) i s denoted $y R ( z ) . I n t h i s and i n t h e next s e c t i o n we s h a l l s t a t e our hypotheses i n terms of t h e d i f f e r e n c e of resolv e n t s , r a t h e r t h a n i n t e r m s of r e s o l v e n t s , s i n c e o u r hypotheses w i l l u s u a l l y b e e a s i e r to v e r i f y f o r t h e former t h a n f o r t h e l a t t e r . Toward t h i s end we set

We s h a l l a l s o suppose t h a t x l , y, a r e H i l b e r t s p a c e s w i t h

where t h e embeddings a r e d e n s e and continuous. The c o l l e c t i o n of bounded l i n e a r t r a n s f o r m a t i o n s from x t o y* w i l l be d e s i g n a t e d by B(x.,y?) and t h e i r norms by j J J J " 'I' B(xj ,yi). There a r e v a r i o u s t e n s o r product H i l b e r t s p a c e s which c a n be formed from t h e above and t h e s p a c e s . For example t h e norms on X I 8 H, w i l l be d e s i g n a t e d by 11 - 1 1 x 1 8 K2 norms i n t h e space of bounded l i n e a r t r a n s f o r m a t i o n s from x i 8 x 2 t o yp 8 H 2 by We t h i n k t h e s e i l l u s t r a t i o n s w i l l make c l e a r t h e n o t a t i o n s 11 -11 B(xl ~ ~ H,)., ~ f we s h a l l u s e i n s i m i l a r s i t u a t i o n s . ~

The assumptions we s h a l l make a r e as f o l l o w s : For every A f R

t h e r e exists an o p e r a t o r A1(A)

f B(xl.y:)

so t h a t

(2.3) There e x i s t s an M > 0 and an a , 0 < a < 1, s o t h a t f o r a l l A1,A2 w i t h I X g - All 5 1,

f 1R

F i n a l l y we s h a l l suppose t h a t (2.5a) and f o r every compact I

W,

(2.5b)

A s examples of o p e r a t o r s which s a t i s f y t h e p r e v i o u s hypotheses w e n o t e t h e following:

(2.6)

H1 = -A

(2.7)

H

1

=

acting i n L2(iRn),

d2 -7 -F dx

ex

n t 3,

a c t i n g i n L2(IR1),

x1 = y1 = L2"(Xn),

s > 1.

x1 = y1 = L z Y s ( R 1 ) , s > 114.

The r e l e v a n t p r o p e r t i e s f o r t h e example (2.6) a r e e a s i l y o b t a i n e d u s i n g t r a c e e s t i m a t e s i n Sobolev s p a c e s o r by t r a n s f o r m i n g t o p o l a r c o o r d i n a t e s and u s i n g w e l l known p r o p e r t i e s of Bessel f u n c t i o n s . They have a l s o been o b t a i n e d by s l i g h t l y d i f f e r e n t techniques [ 2 ] . For t h e example (2.7) t h e r e l e v a n t p r o p e r t i e s a r e e a s i l y shown because of t h e w e l l known p r o p e r t i e s of t h e Airy f u n c t i o n . A c t u a l l y t h e s e p r o p e r t i e s were obtained f o r t h e r e s o l v e n t i n [ 3 ] . However, s i n c e we r e q u i r e t h e s e

45

A Limiting Absorption Principle p r o p e r t i e s o n l y f o r t h e d i f f e r e n c e of r e s o l v e n t s , t h e proof is somewhat more direct. I t s e e m s h i h l y l i k e l y t h a t a wide v a r i e t y of o r d i n a r y d i f f e r e n t i a l o p e r a t o r s a c t i n g i n L ( R ) w i l l s a t i s f y t h e above hypotheses. For example, from t h e work i n [ l ] , i t would appear t h a t f o r 0 < 8 5 2 t h e o p e r a t o r

9

H~ = -

d2

-

e sgn x l x l '

x1

i n ~ 2 R( ) ,

= y1 =

L 2 , s - 8/4

,

s . 5 ,

But t h i s may r e q u i r e a f a i r amount of

w i l l a l s o s a t i s f y t h e g i v e n hypotheses. computation.

We now s t a t e t h e theorem of t h i s s e c t i o n . For s i m p l i c i t y of n o t a t i o n we s h a l l d e s i g n a t e t h e norm i n B(xl 0 f$, Y: 0 H2) by 111 * 111

-

THEOREM 2.1. Under t h e hypotheses (2.3) t o ( 2 . 5 ) , f o r every h ER, t h e r e e x i s t R'(A) f B(x, 0 H,, y: 8 SO t h a t

%)

l i m s u p 111 R ( A + i c ) - R'(X)III ESO ACE

(2.8)

Moreover, (2.9)

SUP

Ill R'(A)

111

<

A€R

and for every h1,x2 f IR with 11,

-

All

5

= 0.

-

1,

D e t a i l s o f t h e proof of t h i s r e s u l t a s w e l l a s t h e p r o o f s o f o t h e r r e s u l t s o f t h i s paper w i l l appear elsewhere.

§3.

VARIATIONS ON THE THEME

For ease of a p p l i c a t i o n t o c o n c r e t e s i t u a t i o n s i t would be w e l l t o have a r e s u l t when one of t h e o p e r a t o r s i n t h e decomposition (1.1) is t h e two dimensional Laplacean, o r a n o t h e r t y p e o f o p e r a t o r f o r which t h e hypotheses of s e c t i o n 1 are n o t s a t i s f i e d . With t h i s i n mind w e s h a l l a l t e r s l i g h t l y t h e hypotheses ( 2 . 3 ) t o (2.5) of t h e l a s t s e c t i o n . However, t h e s e a l t e r a t i o n s w i l l b e enough t o r e q u i r e v a r y i n g hypotheses on t h e o p e r a t o r H2. We s h a l l make two d i f f e r e n t s e t s of assumptions on H1 and H2. The f i r s t s e t of assumptions on H1 and H2 are as f o l l o w s . For t h e o p e r a t o r H1 we make t h e f o l l o w i n g assumptions:

so t h a t

For e v e r y h C R\{O) t h e r e e x i s t s a n o p e r a t o r A I ( X ) f B(xl,y;)

There e x i s t an M > 0 and a n a , 0 < a < 1, so t h a t i f e i t h e r b o t h X 1 and X 2 a r e p o s i t i v e , o r i f b o t h h l and h 2 are n e g a t i v e , and j X 2 - A 1 5 1,

1

and f o r e v e r y compact s e t I cIR\{O)

M. Ben-Artzi and A. Devinatz

46 (3.3b)

SUP11 Al(Ln)llB(xl,y;)

<

-*

XEI

n20 F i n a l l y , w e s h a l l suppose t h a t

E1({O})

(3.4)

= 0.

For t h e o p e r a t o r H, w e make t h e f o l l o w i n g assumption: There i s a n open set U cIR so t h a t f o r every K

cc

U,

(3.5)

n> 0 THEOREM 3.1.

operators

Under the conditions (3.1) to (3.5), for every X C U there exist x2,y; B y;) s o that

P ( Ac) B ( X ~ B

R(X t i E )

(3.6)

-+

Ri(h)

in the norm topology of g ( x l B x2,yf Q y t ) , the convergence being uniform in every satisfy a uniform local H6lder condition of order a K c c U. Furthermore, R'(A) in K. REMARKS. A p r o t o t y p e example of an o p e r a t o r which s a t i s f i e s t h e hypotheses (3.1) t o (3.4) i s HI = -A a c t i n g i n L 2 ( a'), w i t h x1 = y1 = LzYs( l R 2 ) * s > 1. Thus we may apply t h i s theorem t o t h e o p e r a t o r

(3.7)

I

w i t h x2 = y2 = L 2 s s - B / 4 ( R ) , s > f . Indeed w e may r e p l a c e e sgn x1 x l I B by c o n s i d e r a b l y more g e n e r a l V(x1) s i n c e t h e c o n d i t i o n (3.5) can b e e a s i l y e s t a b l i s h e d once we have good asymptotic e s t i m a t e s f o r s o l u t i o n s t o t h e o r d i n a r y d i f f e r e n t i a l e q u a t i o n -d2/dx2 V(x) - A = 0 a t t m . Thus i f we a r e w i l l i n g t o a c c e p t a s t r o n g e r weight f u n c t i o n t h a t one could p o s s i b l y g e t w i t h a more p r e c i s e a n a l y s i s of t h e o r d i n a r y d i f f e r e n t i a l o p e r a t o r s involved, w e may apply Theorem 3.1 w i t h ease i n a wide v a r i e t y of c o n c r e t e s i t u a t i o n s .

+

A s a second s e t o f assumptions on H1 and H2 w e t a k e t h e following:

For t h e o p e r a t o r HI we make t h e f o l l o w i n g assumptions: For every A '2 R\{O} t h e r e e x i s t s a n o p e r a t o r A 1 ( X ) '2 B(xl,yT)

so t h a t

There exists a n a, 0 < a < 1, and f o r every 6 > 0, t h e r e exists an M6 > 0 so t h a t i f e i t h e r X1,X2 Z 6 o r A ~ , h 25 -6 and I X 2 - A l 1 2 1, t h e n (3.9) For every 6 > 0 (3.10a) and f o r every compact s e t I c _ I R \ C O l , (3.10b) n20

For t h e o p e r a t o r H2 w e make t h e f o l l o w i n g assumption:

A Liniititig Absorption Principle

There i s an open set U 533 so t h a t f o r every K t h e r e a r e o p e r a t o r s R*(A) f B(x,,y$) s o t h a t

47

cc

U and f o r every A f K,

2

THEOREM 3.2. Under the hypotheses (3.8) t o (3.11), operators R?(h) i n B(xl 63 x2,y: 4 y t ) s o that

(3.12)

R(A2 iE)

+

Rf(h),

f o r every h f U there e x i s t

EJ. 0 ,

i n the norm topoZogy of B(x1 8 x2,yT 8 y;), the convergence being uniform i n every K cc U. Further, i f R:(X) s a t i s f y a Z o c d uniform HttZder condition i n K, so do R?(A). REMARKS. A p r o t o t y p e example o f an o p e r a t o r which s a t i s f i e s c o n d i t i o n s ( 3 . 8 ) t o (3.10) i s H1 = -A i n L 2 ( R n ) , n 2 1, and X I = y1 = L2"(Bn), s > t . It i s e a s i l y shown t h a t a wide v a r i e t y of o r d i n a r y d i f f e r e n t i a l o p e r a t o r s s a t i s f y t h e c o n d i t i o n i n p a r t i c u l a r , t h e o r d i n a r y d i f f e r e n t i a l p a r t of t h e p u t on H 2 , namely (3.11): operator ( 3 . 7 ) . Thus Theorem 3.2 g i v e s a somewhat s h a r p e r r e s u l t i n t h e s e cases. F u r t h e r r e s u l t s a r e a v a i l a b l e when N1 ( a n d / o r H2) h a s more t h a n one s i n g u l a r p o i n t where l i m i t i n g a b s o r p t i o n d o e s n o t h o l d . The r e s u l t s , of c o u r s e , depend on what hypotheses are made concerning H1 and H2. 54.

A FURTHER RESULT

I n t h i s s e c t i o n we s h a l l make hypotheses of a d i f f e r e n t n a t u r e t h a n were made i n t h e previous section. I n some i n s t a n c e s t h e c r u c i a l h y p o t h e s i s below a p p e a r s t o b e e a s i e r t o v e r i f y t h a n t h e p r e v i o u s ones. For s i m p l i c i t y we s h a l l g i v e a set of hypotheses w i t h o u t g l o b a l u n i f o r m i t y c o n s i d e r a t i o n s , such a s uniform boundedness of t h e l i m i t i n g r e s o l v e n t s on unbounded sets, o r w i t h o u t c o n s i d e r a t i o n s of unbounded s e t s of e i g e n v a l u e s f o r e i t h e r o p e r a t o r HI o r H2. The assumptions we make on HI and H2 a r e a s f o l l o w s :

(4.1)

€I1 and H2 a r e bounded below.

(4.2)

A l i m i t i n g a b s o r p t i o n p r i n c i p l e i s v a l i d f o r H1 and H2 uniformly on comp a c t s e t s i n R+ = (O,-) i n t h e t o p o l o g i e s of B(x1,yT) and B(x2,yZ) resp e c t i v e l y . More p r e c i s e l y t h e r e e x i s t R*(A) C B(xj,y;) so t h a t f o r every I K c c R f and j = 1 , 2

(4.3)

For every C$ F C o ( IR),

m

4(H1)xI 5 x l -

+

Under the previous asswnptions, f o r every A f B , there are operTHEOREM 4 . 1 . ators R + _ ( x )c B ( X , 4 x2,yq 4 y*,) s o t h a t for every K ==R+,

REMARKS. The c o n d i t i o n (4.3) i s e a s i l y e s t a b l i s h e d f o r H = -A i n L 2 ( B n ) , n 2 1, and X I = y1 = L 2 , s ( R n ) , s > 3. Another example which c a n e a s i l y be shown t o f i t t h e p r e v i o u s assumptions i s g i v e n by H1 = Hg = -A + 1/ 1x1 i n L2 ( a n ) , n 2 2 and xj = y . = L 2 , S ( R " ) , s > 3. A l s o t h e hypotheses on Hi can b e shown t o hold f o r a I . wlde v a r l e t y of o r d i n a r y d i f f e r e n t i a l o p e r a t o r s of t h e form -d2/dx2 V(x) i n L 2 ( R ) , where x1 and y1 are t a k e n a s s u i t a b l e weighted spaces.

+

48

M. Ben-Artzi and A. Devinatz

REFERENCES [ I ] Ben-Artzi, M., U n i t a r y e q u i v a l e n c e and s c a t t e r i n g t h e o r y f o r S t a r k - l i k e Hamiltonians, ( t o appear).

[Z] G i n i b r e , J. and Moulin,

M., H i l b e r t s p a c e a p p r o a c h t o quantum mechanical three-body problem, Ann. I n s t . H e n r i P o i n c a r 6 , Sec. A, XXI (1974) 97-145.

[3] H e r b s t , I . W., (1977) 55-70.

U n i t a r y e q u i v a l e n c e of S t a r k H a m i l t o n i a n s . Math. Z.,

155

[ 4 ] Y a j i m a , K . , S p e c t r a l and s c a t t e r i n g t h e o r y f o r S c h r o d i n g e r o p e r a t o r s w i t h S t a r k e f f e c t , J. Fac. S c i . , Univ. Tokyo, Sec. l A , 2 6 (1979) 377-389.

IResearch p a r t i a l l y s u p p o r t e d by NSF G r a n t MCS-8200898.

DIFFERENTIAL EQUATlONS I.W.Knowles and R.T.Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

49

AN ALGEBRAIC GENERALIZATION OF STOCHASTIC INTEGRATION Marc Berger and Alan Sloan School of Mathematics Georgia I n s t i t u t e o f Technology Atlanta, Georgia 30332 U.S.A.

The authors use a generalization of s t o c h a s t i c i n t e g r a t i o n t o analyze real constant c o e f f i c i e n t e l l i p t i c operators of order n which generate strongly continuous semigroups on L2 (IR k ) . Let p n ( x , t ) be the fundamental solution t o u t = Au where

A

=

(-1)

(n/2)-1

c -an n

j = l ax

j

f o r n even. For real polynomials ql, ...,qk on Rm l e t q = (ql,...,qk) and define i n t e g r a l operators F ( t ) , f o r each n ,q t > 0 on functions on lRk by ( Fn ,q ( t) f ) Ix) =

.f

f(x+q ( 2 ))pn ( z , t )dz.

IRm

The authors determine when etQ = lim F ( t / j ) j strongly on 2 k as j a, f o r some q on some R". L (R -f

In (Berger and Sloan generalization of stochast application of t h i s genera c o e f f i c i e n t 1 i near p a r t i a l

1982) and March (1983)) the authors presented a Here, the authors will discuss an i z a t i o n t o the c l a s s i f i c a t i o n of real constant d i f f e r e n t i a l operators, Q , o f order n which generate strongly continuous semigroups on L 2 (IR k ) . T h e s e t of a l l such 4 ' s i s denoted by I ( n , k ) . The simplest example in I ( n , l ) , f o r n even, i s

c integration.

For t h i s Q , t h e solution t o t h e i n i t i a l value problem, u(x,O) corresponding evolution equation ut = Qu may be written as m

u(x,t)

=

I f(x+yhnb,t)dy -03

where

and p n ( y , t ) = t-"npn(yt-'/n).

=

f ( x ) for the

M.Berger and A. Sloan

50

The authors have investigated the following question: I n what sense and t o what extent i s the kernel pn s u f f i c i e n t f o r t h e description of solutions t o t h e i n i t i a l value problem formore general evolution equations? To be more s p e c i f i c , f o r z = (z ,zm) i n W m , l e t p n ( z , t ) = p , ( z l , t ) ... p n ( z m , t ) . For real polym nomials q l , . . . , q k on IR of degree no l a r g e r than n and with no constant term, k l e t q ( z ) = ( q l ( z ) ,...,qk( 2 ) ) . Define an integral operator on functions f on W

,,...

by

1 f(x+q(z))p,(z,t)dz. Rm For which Q i n I ( n , k ) i s t h e r e a q on some Rm such t h a t (Fn,,(t)f)(x)

Let n be even.

=

strongly on L 2 (IRk ) ? Such Q's a r e called pure. Before describing an answer t o t h e question posed above some examples will be presented which i l l u s t r a t e the nature of purity. In one dimension, k = 1 , or f o r second order operators, n = 2 , a l l generators in I ( n , k ) a r e pure. These values, k = 1 o r n = 2 , a r e the only ones f o r which every Q in I ( n , k ) i s pure so t h a t the c l a s s i f i c a t i o n of e l l i p t i c generators i n t o pure and not pure i s a multidimensional , higher order theory. I n p a r t i c u l a r , a4 + A a4 a4 -(2 2+$ ax ax ay i s i n I ( 4 , Z ) f o r -2 5 A b u t i s pure i f and only i f 0 5 A 5 6. While

a2

-(-7 ax

a 2 t 7) a2 2 t a - a3 axayaz

t

ay2 az i s pure f o r a l l real a i t turns o u t t h a t

a4 a4 +a4+ a -(-T ax ay4 t az

a3

+

i s pure only when a = 0.

Q.

Whether o r not a Q i n I ( n , k ) i s pure i s l a r g e l y determined by the symbol o f That i s , l e t Q =

...,a k ) ,

where a = ( a l y aa # 0 f o r some

10.

ai = 0,1,2,

c

O< I a 1 Ln

...,

aaaa

la1 = a1 +

... '

= n and

a'

= (

a,)a1 ax

... (--I a

axk The symbol f o r such a Q i s the polynomial on R k ,

Q ( Y )=

C aaya O
ak

.

"k3

aa real constants with

An A Ige bruic Cen eraliza t io w o f Stochastic Integration

51

where

f o r y = (y ,,... , y k ) . C1,. . ,Cm such t h a t

.

Q has a Hermite symbol i f f o r some m there a r e n x k matrices m

where t h e Hn's a r e t h e generalized Hermite polynomials defined by

H n ( x l,...,x

1 an . atn

) = ;;~-exp[ n

1 2

n

E i=l

t=O

H2(x,y) = y + 7 x while H 4(w,x,y,z) = z + wy + The s e t of a l l generators in I ( n , k ) with Hermite symbols i s a real cone, H(n,k). Although membership i n H ( n , k ) i s a geometric property, i t i s equivalent t o an a n a l y t i c property: Q in I ( n , k ) has a Hermite symbol i f and only i f i t i s formally pure in t h e sense t h a t , f o r some q , Notice t h a t

$(x2 + w 2y )

+

$ w4.

The following theorems a r e proved in (Berger and Sloan (April 1983)). Theorem 1 : Let Q in I ( n , k ) have no lower order terms. The following a r e equivalent: ( a ) Q i s pure; ( b ) Q has a Hermite symbol ; ( c ) The symbol of Q i s E~ times t h e sum of nth powers of l i n e a r forms; ( d ) e x p ( t Q ) = F n , , ( t ) f o r some q. Theorem 2 : Let Q have no lower order terms and l e t Q be pure of order n. There i s a number c such t h a t i f B i s any 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 operator with real constant c o e f f i c i e n t s of order i e s s than n such t h a t cQ + B has a Hermite symbol , then Q + B i s pure. The examples discussed e a r l i e r follow d i r e c t l y from these theorems. This discussion will conclude with a concise d e s c r i p t i o n of the generalized s t o c h a s t i c i n t e g r a t i o n and the technical problem of showing an operator to be pure. The main elements of t h e generalization are: ( 1 ) a complex vector space, V, whose elements a r e c a l l e d t h e generalized c h a r a c t e r i s t i c s ; ( 2 ) an operator D which extends ordinary d i f f e r e n t i a t i o n t o formal functions of generalized c h a r a c t e r i s t i c s i n a manner c o n s i s t e n t with "composition" and a "chain r u l e " ; and ( 3 ) a "generalized expectation" operator L , which i s a projection from formal functions of c h a r a c t e r i s t i c s t o complex functions on [ O p ) and intertwines D and d L. ordinary d i f f e r e n t i a t i o n : LD = dt Associated w i t h each constant c o e f f i c i e n t operator Q in I ( n , k ) i s a flow

52

M. Berger and A. SIoan

w: Rk

+

V such t h a t D ( g ( w ( x ) ) ) = ( Q g ) ( w ( x ) ) and w(x)(O) = x f o r a l l p o l y n o m i a l s

g and a l l x i n R

k

.

I f u s a t i s f i e s ut = Qu, u(x,O) Du(w(x),s-t)

= Qu

-

= f ( x ) and

if w i s a f l o w f o r Q t h e n

tu = 0 f o r each f i x e d s > 0.

i s independent o f t f o r each f i x e d s

0.

Consequently ( L u ( w ( x ) , s - t ) ) ( t )

Equating values o f L a t t = 0

and

t = s l e a d s t o t h e b a s i c r e p r e s e n t a t i o n o f t h e t h e o r y : u(x,s) = L f ( w ( x ) ) ( s ) . each n even, thehe a r e b a s i c c h a r a c t e r i s t i c s 01,02,... flow o f Q =

E~

such t h a t x

t

Bi

For

i s the

a and such t h a t t h e Oils a r e independent r e l a t i v e t o L i n t h e axn

sense t h a t L ( Bml ~ f o r a l l j = 2,3,

...

(z)J

... e mj J. )

and m. = 0,1,2 1

m

...

= ~ ( 6 , ~ )

*... .

m. L ( ~ ~ J )

The problem o f f i n d i n g p o l y n o m i a l s q

converges t o e x p ( t Q ) i s f o r m a l l y t h e same as f i n d i n g a p o l y so t h a t F n,q J nomial q so t h a t L g ( w ( x ) ) = L g ( x t q ( 8 l,...,em)) f o r a l l p o l y n o m i a l s g, when w i s a f l o w f o r Q. REFERENCES 1.

Berger, M. and Sloan, A., A method o f g e n e r a l i z e d c h a r a c t e r i s t i c s , Memoirs o f t h e AMS, 266 ( J u l y , 1982).

2.

Berger, M. and Sloan, A., C h a r a c t e r i s t i c methods f o r m u l t i d i m e n s i o n a l e v o l u t i o n equations, Georgia Tech p r e p r i n t (March, 1983).

3.

Berger, M. and Sloan, A., P r o d u c t f o r m u l a s f o r semigroups w i t h e l l i p t i c generators, Georgia Tech p r e p r i n t ( A p r i l , 1983).

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

53

VARIATIONAL PROBLEMS INVOLVING LACK OF COMPACTNESS AND RELLICH'S CONJECTURE Haim B r e z i s U n i v e r s i t e P. e t M. C u r i e 4, p l . J u s s i e u 75230 P a r i s Cedex 05 France

1.

INTRODUCTION

L e t r C R3 be a Jordan curve; more p r e c i s e l y we assume t h a t c l o s e d b a l l of r a d i u s R. Consider t h e f o l l o w i n g problem:

r c

BR

-

a

Problem 1 Given a c o n s t a n t

H > 0

f i n d a s u r f a c e o f mean c u r v a t u r e

H

spanning

r.

Example: Suppose r i s a c i r c l e o f r a d i u s R. I f H < 1/R we have ( a t l e a s t ) 2 s o l u t i o n s o f Problem 1, a " s m a l l " s p h e r i c a l bubble and a " l a r g e " s p h e r i c a l bubble. I f H = 1 / R t h e r e i s one s o l u t i o n o f Problem 1, namely a h a l f - s p h e r e . I f H > 1/R t h e r e i s no s o l u t i o n o f Problem l . ( T h i s i n t u i t i v e f a c t i s e s t a b l i s h e d by Heinz i n [ 5 ] ) . A c o n j e c t u r e a t t r i b u t e d t o R e l l i c h a s s e r t s t h a t f o r any c u r v e T t h e r e e x i s t a t l e a s t 2 s u r f a c e s o f mean c u r v a t u r e H spanning r when H i s s m a l l enough. I n a j o i n t work w i t h Coron [2] we have o b t a i n e d t h e f o l l o w i n g r e s u l t : THEOREM 1.

If

H

<

1/R,

t h e r e e x i s t a t l e a s t 2 s o l u t i o n s o f Problem 1.

I n 1969 S. H i l d e b r a n d t [6] had e s t a b l i s h e d ( u n d e r t h e same assumption) t h a t t h e r e i s a t l e a s t one s o l u t i o n o f Problem 1. S t e f f e n [8] had shown t h a t t h e r e e x i s t s a sequence Hn -t 0 such t h a t f o r H = Hn, Problem 1 has a t l e a s t 2 s o l u t i o n s . More r e c e n t l y , and i n d e p e n d e n t l y o f o u r work, Struwe [ l o ] and S t e f f e n [9] have o b t a i n e d t h e same c o n c l u s i o n as i n Theorem 1 p r o v i d i n g H i s s m a l l enough, b u t w i t h o u t g i v i n g an e x p l i c i t bound f o r H. The p r o o f o f Theorem 1 i s c l o s e l y r e l a t e d t o an e a r l i e r work o f N i r e n b e r g and m y s e l f [4] d e a l i n g w i t h : Problem 2 Find a function

u

satisfying

i

+

in

R

U ' O

in

R

u = o

in

an,

- A U = Up

AU

where 0 C Rn i s a smooth bounded domain, p = ( n + 2 ) / ( n - 2 ) and A i s a c o n s t a n t . The exponent p = (n + 2 ) / ( n - 2 ) corresponds t o a l i m i t i n g case f o r t h e Sobolev embedding ( a s i m i l a r d i f f i c u l t y o c c u r s i n t h e Yamabe problem). Problem 2 possesses There ( i n [4]) we p r o v e t h a t , under some assumptions on A, a s o l u t i o n . I t i s e a s i e r t o understand how we s o l v e Problem 2 ( a t l e a s t when

H. Brezis

54

n 2 4) and I s h a l l e x p l a i n t h i s f i r s t . d e s c r i b e i t s analogy w i t h Problem 2.

Then I s h a l l c o n s i d e r Problem 1 and

2. DISCUSSION OF PROBLEM 2 One of t h e main r e s u l t s i n [4] i s t h e f o l l o w i n g :

THEOREM 2. Assume n 5 4. of Problem 2. Moreover i f E (OJ,). Here

X1

Then f o r e v e r y h E ( 0 , ~ ~ t) h e r e e x i s t s a s o l u t i o n R i s starshaped no s o l u t i o n o f Problem 2 e x i s t s when

denotes t h e f i r s t e i g e n v a l u e o f

- A w i t h zero D i r i c h l e t c o n d i t i o n .

Remark 1 . S u r p r i s i n g l y when n = 3, Problem 2 i s much more d e l i c a t e and t h e conclusion i s quite.different. I n f a c t , we have a complete s o l u t i o n o n l y when R is a b a l l ; t h e case o f a general domain i s n o t w e l l u n d e r s t o o d - f o r f u r t h e r det a i l s see [4]. For t h e p r o o f o f Theorem 2 we use an argument i n s p i r e d by t h e work of Th. Aubin [ l ] d e a l i n g w i t h Yamabe‘s c o n j e c t u r e . Sobolev’s i n e q u a l i t y , and more p r e c i s e l y t h e b e s t c o n s t a n t i n S o b o l e v ’ s i n e q u a l i t y p l a y s an i m p o r t a n t r o l e . Sobolev’s i n e q u a l i t y asserts t h a t 1I Vull

p + 1 = 2n/(n

(note t h a t

-

W u E

all UII

P+l 2 ) ) f o r some

a

>

0.

HA(n) The b e s t c o n s t a n t i s g i v e n by

n

S =

inf

{IIVUII~}.

uEHZ)(0) 1I UII p+, =1

I t i s known t h a t :

i s independent o f

depends o n l y on

n.

(a)

S

(b)

The infimum d e f i n i n g

(c)

S i s a c h i e v e d when R = IR ; t h e extremai functions are ~ ( x )= ( 1 + ~ x ~ z ) - ( n - 2 ) ~ *o, r a f t e r s c a l i n g , u c ( x ) = ( E + 1x12 -(n-2)/2 E

>

R;

S

i s n e v e r achieved when

S

n

i s bounded.

0.

Set (1)

S,

= inf

2 2 i I 1 0 u l l ~- X I I U ~ I ~ ~ ,

x

0.

u€HA( n) 1I ull pfl =1

The major s t e p i n t h e p r o o f o f Theorem 2 i s t h e f o l l o w i n g lemma: LEMMA 1 .

Assume

n

4.

Then

s,

i

s

wx>o.

The proof o f Lemma 1 r u n s as follows.

Set II VUII

Q(u)

=

2

- ail UII

2 II UII p+l

2

55

Vuriut io rial Pro b lr nis

and UJX) where

Ci(fi)

@ E

(assuming

0

d x ) ( c + 1x1

=

2 -(n-2)/2

1

i s any f i x e d c u t - o f f f u n c t i o n such t h a t As

E 0).

0

E +

@

1

near

x = 0

one o b t a i n s t h e f o l l o w i n g expansion

where K depends o n l y on n; Lemma 1 f o l l o w s immediately.

f o r f u r t h e r d e t a i l s see [4].

The c o n c l u s i o n o f

The n e x t Lemma was suggested t o us by E. L i e b ; a somewhat r e l a t e d argument appears i n [7]. LEMMA 2.

Proof.

If

Let

S,

then t h e i n f i m u m d e f i n e d by ( 1 ) i s achieved.

S,

<

(uj)

be a m i n i m i z i n g sequence f o r ( 1 ) t h a t i s

(2)

J ivuJ12

(3)

1

IUjlP+1

1 luj12 =

x

-

s,

o(1)

f

1.

=

A f t e r e x t r a c t i n g a subsequence we may assume t h a t uJ

-u

1

weakly i n

Ho(0),

uJ

-+

u

strongly i n

uJ

-+

u

a.e.

vq

LQ,

<

p + 1,

u weakly i n N o t i c e t h a t , so f a r , we can o n l y a s s e r t t h a t uJ 5 1 . The m a j o r d i f f i c u l t y c o n s i s t s i n p r o v i n g t h a t II uII A

lull^+^

VJ

= UJ

-

x J

lu12 =

LP+l P+1

= 1.

and t h u s We s e t

u.

From ( 2 ) and ( 3 ) we deduce t h a t (4)

1

(5)

1 lUlP+l

J

f

lVVjl2

-

+ J IVjlP+1

1

=

f

s,

+ o(1)

o(1).

( I n o r d e r t o pass f r o m ( 3 ) t o ( 5 ) i t i s convenient t o use a Lemma f r o m [ 3 ] ) . f o l l o w s from ( 5 ) t h a t

1

(6)

< II1,1I2

-

+

p+l

llV.1l2 J

p+l

+

o(1).

Combining ( 4 ) and ( 6 ) we f i n d

J Since

1

-

IVUI2 +

ju2

>

-

j

lVVjI2

S,II~II~

- h j U 2 5 S,(llUll2

(W u

E

H1~ ) ,

P+l

we conclude t h a t

J

IVVjlZ <

ShllVJlI2 P+l

+ o(1)

P+l

+ IlV.1l2 ) J P+l

+

o(1).

It

56

H. Brezis

F i n a l l y , using Sobolev's i n e q u a l i t y

w i t h t h e b e s t c o n s t a n t , we o b t a i n

1 Consequently VJ + 0 s t r o n g l y i n Ho and t h e r e f o r e UJ + u s t r o n g l y i n LP+l. Thus from ( 2 ) and ( 3 ) we deduce t h a t u i s a m i n i m i z e r f o r ( 1 ) . The f a c t t h a t e v e r y m i n i m i z i n g sequence converges s t r o n g l y was f i r s t p o i n t e d o u t by F. Browder a t some e a r l i e r stage. (The o r i g i n a l proof was somewhat more c o m p l i c a t e d ) . P r o o f o f Theorem 2 concluded. L e t u denote a m i n i m i z e r f o r ( 1 ) . We may always assume t h a t u 2 0, o t h e r w i s e r e p l a c e u b y I u I . The E u l e r e q u a t i o n f o r ( 1 ) says t h a t

f o r some Lagrange m u l t i p l i e r

D E 11

=

R,

1

and

J v u I2 -

1 u2 .

Since A < x1 i t f o l l o w s t h a t u > 0. The s t r o n g maximum p r i n c i p l e i m p l i e s t h a t u > 0 i n a. S t r e t c h i n g t h e Lagrange m u l t i p l i e r o u t of ( 7 ) we o b t a i n a s o l u t i o n o f Problem 2. The nonexistence c l a i m i n Theorem 2 when A 1. hd 1s o b v i o u s - j u s t m u l t i p l y the equation by @ an e i g e n f u n c t i o n of - A c o r r e s p n d i n g t o nonexistence when A 5 0 aXd R i s starsbaped f o l l o w s f r o m Pohozaev's

ihtii;:

where v i s t h e outward normal t o R i s starshaped a b o u t t h e o r i g i n .

as2,

so t h a t

(x.v)

>

0

a.e.

on

an when

Remark 2. We emphasize a g a i n t h e main s t r a t e g y i n t h e p r o o f o f Theorem 2. B a s i c a l l y , we have t o deal w i t h a v a r i a t i o n a l problem which l a c k s compactness (because t h e embedding

HA

C

LP+l

i s n o t compact).

However, we n o t i c e t h a t below a c r i t i -

c a l value some f o r m o f compactness i s r e s t o r e d back i n t o t h e problem. Lower o r d e r terms h e l p t o show t h a t we a r e below such a c r i t i c a l value. We s h a l l enc o u n t e r t h e same phenomenon i n t h e c o n t e x t o f Problem 1.

3.

DISCUSSION OF PROBLEM 1 .

Problem 1 i s reduced t o an e l l i p t i c system. L e t E IR2; x2 + y2 < 1 ) . F i n d u: n + R 3 s a t i s f y i n g

n = {(x,y)

u(aR)

=

r and u i s nondecreasing on an.

I f u s a t i s f i e s ( 8 ) , t h e s u r f a c e u(R) p r o v i d e s a s o l u t i o n o f Problem 1. Our main r e s u l t (see [ Z ] ) a s s e r t s t h a t i f r c BR and H < 1/R t h e r e e x i s t s a t least 2 d i s t i n c t solutions o f (8).

I s h a l l n o t t r y t o e x p l a i n how we s o l v e Problem 1-the argument i s r a t h e r t e c h n i c a l . I n s t e a d , I s h a l l d i s c u s s t h e same e q u a t i o n , b u t w i t h D i r i c h l e t boundary c o n d i t i o w i n p l a c e o f t h e P l a t e a u boundary c o n d i t i o n which occurs i n ( 8 ) . The s o l u t i o n o f t h e D i r i c h l e t problem i s t h e n used as a t o o l i n o r d e r t o s o l v e (8). More p r e c i s e l y we c o n s i d e r t h e problem

57

Variational Problems

i,

(9)

where ing:

y: a R

+

Au = 2H ux A u Y u = Y

R3

in

n

on

an

i s a g i v e n (smooth) f u n c t i o n .

Gur main r e s u l t i s t h e f o l l o w -

THEOREM 3. Assume I I Y I I , 5 R, y i s n o t i d e n t i c a l l y a c o n s t a n t and then t h e r e e x i s t a t l e a s t 2 s o l u t i o n s o f ( 9 ) .

H < 1/R,

Remark 3. S u r p r i s i n g l y i f y : C i s a c o n s t a n t t h e n Problem ( 9 ) has o n l y one s o l u t i o n , namely u :c (see Wente [ 1 2 ] ) . We s k e t c h t h e p r o o f o f Theorem 3. The e x i s t e n c e of a f i r s t s o l u t i o n u o f He o b t a i n s i t t h r o u g h t h e f o l l o w i n g m i n i m i z a t i o n ( 9 ) i s due t o H i l d e b r a n d t [63. p r o b 1em :

(10)

where R ' i s any c o n s t a n t such t h a t R < R ' < 1/H. I t i s n o t d i f f i c u l t t o see t h a t t h e minimum i n ( 1 0 ) i s achieved by some u such t h a t II~II, 5 R and t h u s s a t i s f i e s ( 9 ) . Using t h e f a c t t h a t t h e second-variation i s 0 we see t h a t

i

1V@I2

4H

+

Iu

1.0

@x

v

u

1 Ho.

4

By a more c o m p l i c a t e d argument one can even prove t h a t

1 1001~

(11)

+ 4H

1y

.

@ x A $y 2 6

Next, we l o o k f o r a second s o l u t i o n

/ 10$1~

V @

E

1 Ho,

f o r some

6 > 0.

u o f ( 9 ) o f t h e form u = g - v .

Hence

(12)

v

1

satisfies: Lv

-Av

+ 2H(llx A

V

Y

+

V x I\

u ) = ~Hv, A

Y

v = o

Vy

in on

an.

I n a d d i t i o n we r e q u i r e o f c o u r s e v 0. Problem (12) has a v a r i a t i o n a l s t r u c t u r e . Namely, 1 i s a s e l f - a d j o i n t o p e r a t o r and i s t h e F r e c h e t d e r i v a t i v e o f t h e func1 t i o n a l T( Lv,v) where (Lv,v) The n o n l i n e a r t e r m

f

=

vx

I lvvl 2 + 4H v Y

Q ( v ) where

/

g

.

v x A vy.

i s t h e Frechet d e r i v a t i v e o f t h e volume f u n c t i o n a l

1 v.

vX A v Y' I n view o f t h e approach used i n S e c t i o n 2 i t i s n a t u r a l t o c o n s i d e r : Q(v) =

(13)

J

= Inf

t(Lv,v)l

.

*H;(Rl Q(v)=l I f t h e infimum i n (13) i s achieved, we o b t a i n , a f t e r s t r e t c h i n g o u t t h e

H. Brezis

58

Lagrange m u l t i p l i e r , a nonzero s o l u t i o n o f ( 1 2 ) . The main d i f f i c u l t y w i t h t h i s p r o g r m comes f r o m t h e f a c t t h a t Q ( v ) i s not c o n t i n u o u s under weak convergence i n Ho. To s t a r t w i t h , i t i s n o t even c l e a r t h a t Q ( v ) i s w e l l d e f i n e d f o r

B

1 v E Ho. However, Q ( v ) can be d e f i n e d b y c o n t i n u i t y f o r h e l p o f t h e i s o p e r i m e t r i c i n e q u a l i t y (see [ l l ] ) :

:C 1

1Q($)12'3

(14)

W

1v;l2

$ E

C;(n;R

3

v

E

1 Ho

w i t h the

1.

Here, i n e q u a l i t y ( 1 4 ) p l a y s t h e r o l e of S o b o l e v ' s i n e q u a l i t y i n S e c t i o n 2. In f a c t , t h e r e a r e s t r i k i n g s i m i l a r i t i e s between ( 1 4 ) and S o b o l e v ' s i n e q u a l i t y : (a)

The b e s t c o n s t a n t f o r ( 1 4 ) , t h a t i s ,

i s n o t achieved. (b)

S remains t h e same if 9 i s r e p l a c e d by any domain i n n e v e r achieved f o r a bounded domain.

(c)

The b e s t c o n s t a n t function i s

i s achieved i n a l l o f

S

R2

lR2 , b u t

it i s

and t h e e x t e r n a l

o r after stretching, U,(X,Y)

1

= c2

+ x2 +

y

I n o r d e r t o prove t h a t t h e infimum i n (13) i s achieved we proceed i n 2 s t e p s :

Step 1. We show t h a t J Hence,there i s some p o i n t ratio

S . Here we use t h e f a c t t h a t (xo,y,) E n where vy (xo,yo)

y

1

# 0.

C

and t h u s y $ Y. We e s t i m a t e t h e

i s any f i x e d c u t - o f f - xo. y - y o ) where 4 E C,"(Q) near (xo,yo). As E -+ 0 we have t h e expansion 2 c > 0, R(vE) = S + O ( E ) - c c ,

f o r V(X,Y) = @(x,Y) U,(x f u n c t i o n such t h a t 0 1

which i m p l i e s t h a t

J

Step 2.

be a m i n i m i z i n g sequence f o r ( 1 3 ) , t h a t i s

Let

(vJ)

<

S.

(15)

(Lvj,vj) = J +

(16)

Q(vJ) = 1

Using ( 1 1 ) we see t h a t

Set

o(1)

.

(vj)

vj

-t

WJ

= VJ - v

v

i s bounded i n weakly i n

.

HA

.

HA

and t h e r e f o r e we may assume t h a t

59

Vuria tiorzal Problem I t i s n o t d i f f c u l t t o see

Q(vJ) = Q and t h u s , by ( 6 ) we have 1 5 lQ(v)

(17)

On t h e o t h e r h a n d , ( l 5 ) l e a d s t o (18)

(Lv,v) +

1

IVWJ

=

J

t

o(1).

1

I n a l l t h e c e computations i t i s m p o r t a n t t o n o t i c e f o r examp e t h a t t h e mapping GI-+ J y . 4 x A oY i s c o n t i n u o u s under weak convergence i n Ho. Combining

( 1 7 ) and ( 1 8 ) we see t h a t

(LV,~)+

J

$

5 J I Q ( V ) I ~ +/ ~ J 1 v w J 1 ~

~ o W J / ~

1 and t h e r e f o r e w j -t 0 s t r o n g l y i n Ho. and hence v i s a m i n i m i z e r f o r ( 1 3 ) .

In particular

vJ

+

v

strongly i n

HA

REFERENCES Aubin, Th., Equations d i f f g r e n t i e l l e s non l i n i a i r e s e t probleme de Yamabe concernant l a courbure s c a l a i r e , J . Math. Pures e t Appl. 55 (1976) 269-296. B r e z i s , H. and Coron, J . M., M u l t i p l e s o l u t i o n s o f H-systems and R e l l i c h ’ s c o n j e c t u r e , Comm. Pure Appl. Math. ( t o appear). B r e z i s , H. and L i e b , E . , A r e l a t i o n between p o i n t w i s e convergence o f f u n c t i o n s and convergence o f f u n c t i o n a l s , Proc. Amer. Math. SOC. ( t o appear) B r e z i s , H. and N i r e n b e r g , L., P o s i t i v e s o l u t i o n s o f n o n l i n e a r e l l i p t i c e q u a t i o n s i n v o l v i n g c r i t i c a l Sobolev exponents, Comm. Pure Appl Math. ( t o appear).

.

Heinz, E . , On t h e nonexistence o f a s u r f a c e o f c o n s t a n t mean c u r v a t u r e w i t h f i n i t e a r e a and p r e s c r i b e d r e c t i f i a b l e boundary, A r c h i v e Rat. Mech. Anal. 35 (1969) 249-252. H i l d e b r a n d t , S., On t h e P l a t e a u problem f o r surfaces o f c o n s t a n t mean c u r v a t u r e , Comm. Pure A p p l . Math. 23 (1970) 97-114. L i e b , E., Sharp c o n s t a n t s i n t h e Hardy-Littlewood-Sobolev and r e l a t e d i n e q u a l i t i e s , Annals o f Math. ( t o appear). Flachen k o n s t a n t e r m i t t l e r e r Krummung m i t vorgegebenerm S t e f f e n ,,.K., Volumen oder F l a c h e n i n h a l t , Arch. Rat. Mech. A n a l . 49 (1972) 99-128. S t e f f e n , K., On t h e nonuniqueness o f s u r f a c e s w i t h p r e s c r i b e d mean c u r v a t u r e spanning a g i v e n c o n t o u r ( t o appear).

[lo]

Struwe, M . , Nonuniqueness i n t h e P l a t e a u problem f o r s u r f a c e s o f c o n s t a n t mean c u r v a t u r e ( t o appear).

[ll]Wente, H., An e x i s t e n c e theorem f o r s u r f a c e s o f c o n s t a n t mean c u r v a t u r e , J. Math. Anal. A p p l i c . 26 (1969) 318-344. [12] Wente, H . , The d i f f e r e n t i a l e q u a t i o n Ax = 2H(x, A x v ) w i t h v a n i s h i n g boundary values, Proc. Amer. Math. SOC. 50(1975) 131-137.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS

I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland),1984

61

A FACTORIZATION METHOD FOR SYMMETRIC DIFFERENTIAL OPERATORS AND ITS APPLICATIONS TO D I R I C H L E T INEQUALITIES AND TO THE DIRICHLET INDEX

R i c h a r d C . Brown Department o f M a t h e m a t i c s U n i v e r s i t y o f Alabama U n i v e r s i t y , Alabama, 35486 U.S.A.

We w r i t e c e r t a i n s e l f - a d j o i n t e x t e n s i o n s o f minimal s y m m e t r i c o r d i n a r y d i f f e r e n t i a l o p e r a t o r s whose nonl e a d i n g c o e f f i c i e n t s may h a v e s u i t a b l y bounded n e g a t i v e p a r t s a s a Von Neumann c o m p o s i t i o n l " 1 where 7 i s a closed densely defined operator n a t u r a l l y i n d u c e d by t h e q u a d r a t i c form. This device is then u s e d t o p r o v e D i r i c h l e t i n e q u a l i t i e s and t o e x t e n d t h e Kauffman t h e o r y o f t h e D i r i c h l e t I n d e x t o t h e negative coefficient case. 1.

NOTATION AND INTRODUCTORY REMARKS

p o , p l , . . . , p n , w a r e r e a l m e a s u r a b l e f u n c t i o n s on t h e i n t e r v a l I : = [ a , b ) , - m < a < b 5 m s u c h t h a t w,po > 0 a . e . , and -1 po , p l , . . . , p n , w a r e l e b e s g u e i n t e g r a b l e on compact s u b i n t e r v a l s o f 1 2 I ( L L o c ( I ) ) . L ( w ; I ) i s t h e w e i g h t e d H i l b e r t s p a c e o f complex s q u i r e i n t e g r a b l e f u n c t i o n s on I w i t h i n n e r p r o d u c t [ f , g ] : = w f g and W When w = 1 we w r i t e L 2 ( I ) . For t h i s c a s e and norm l/f11:: = 1;wlf 1 '

/,b

.

f o r o t h e r H i l b e r t s p a c e s t h e n o t a t i o n II.(I a n d w i l l denote norms and i n n e r p r o d u c t s ( t h e c o n t e x t i n d i c a t i n g which s p a c e s a r e meant). F i n a l l y , ACRoc s i g n i f i e s t h e s p a c e o f l o c a l l y a b s o l u t e l y [ - , a ]

c o n t i n u o u s f u n c t i o n s on I . L e t M[y] b e t h e s y m m e t r i c d i f f e r e n t i a l e x p r e s s i o n w-1y[2n1 where Y [ 2 n 1 i s t h e q u a s i - d e r i v a t i v e o f o r d e r 2n w i t h r e f e r e n c e t o t h e coefficient s e t {pi}(cf.[13,§15.2]). The symbols T and T o r e p r e s e n t 2 t h e maximal and minimal o p e r a t o r s i n L ( w ; I ) d e t e r m i n e d by M , and TA i s t h e " p r e m i n i m a l " r e s t r i c t i o n o f To t o compact s u p p o r t f u n c t i o n s . The above a s s u m p t i o n s a n d d e f i n i t i o n s y i e l d t h e s o c a l l e d "minimal 1 c o n d i t i o n " s e t t i n g , t h e end p o i n t a b e i n g r e g u l a r and b s i n g u l a r . The r e a d e r i s d i r e c t e d t o [13,ch.V] o r t o [ l o ] f o r f u r t h e r b a s i c i n f o r m a t i o n on t h e o p e r a t o r t h e o r y a s s o c i a t e d w i t h s y m m e t r i c differential expressions. 1. The r e s u l t s o f t h i s p a p e r e x t e n d t o t h e c a s e s where b o t h e n d p o i n t s a r e s i n g u l a r o r both a r e regular. For r e a s o n s o f economy w e omit s p e c i f i c t r e a t m e n t s of t h e s e c a s e s .

R.C. Brown

62

2.

PRELIMINARY R E S U L T S ON THE CLOSURE OF D I R I C H L E T FORMS

. . . 'Pn

Let p o ,

be a s a b o v e . D e f i n e a s e s q u i l i n e a r form t[u,v]: -

t

2 i n L ( w ; I ) by

(n-i)-(n-i) yi = o jbPiU a

=

V

and t h e a s s o c i a t e d q u a d r a t i c form

L.

L e t V d e n o t e t h e domain of t o converge a b s o l u t e l y ,

D:

= { y & 1,2

Since t h e i n t e g r a l s a r e understood

( w ; I ) : y n- 1€ A 4 0 c , j b ; p i / ( y ( n - i ) , 2 <

mf.

a We assume t h r o u g h o u t t h a t &[y]

A2

- i s c l o s e d ( i n t h e s e n s e o f Kato [ 8 , p . 3 1 5 ] ) . t

I f &[y]

E

2

//y//, f o r some

S u f f i c i e n t Conditions

+

Pi = P i

-

Lemma 1. Proof.

-

Pi

-

Then

L e t :p --

E

+

E.

for A 2 . For i = L = L+ - t- where

EW

> 0.

- y ~ ~ y ~ we~ cwa n, a r r a n g e A 1 by c o n s i d e r i n g

2 dllyIIw where 6 = y

+

has t h e following p r o p e r t i e s :

A1

2 . 1 Remark. h[y]

2

t

f o r some --

E

> 0.

1,

...,n,

Then

+

-

define p.,p so that i i

t+ i s c l o s e d .

Lemma 1 was d e m o n s t r a t e d i n [ 4 ] i f po a n d w a r e bounded

For t h e l o c a l l y away from z e r o on compact s u b i n t e r v a l s o f I . g e n e r a l c a s e w i t h w = 1, s e e [S,Lemma 11. The c a s e w # 1 i s s i m i l a r .

--

t+b e c l o s e d and w - l locally integrable. Suppose Lemma 2 . f o r each 6 > 0 , i = l,...,n, there is decomposition I into a c o u n t a b l e s e t o f nonoverlapping i n t e r v a l s {JiLI each of f i n i t e such t h a t l e n g t h ( L ( J i l l ) : = l e n g t h o f J i L ) --

and

a

of

63

A Factorization Method and Dirichlet Inequalities

Then t is closed and bounded below. Proof. It is shown in [S] that (i inequality (2.1)

lPIlY

(n-i) 2

I 5

K(E) j w

I

for all

E

> 0, whence

and (ii) guarantee the sum

I

t- is

r+ bounded

t is closed and boundea below

with t+ bound zero. [8,Theorem 1.33,p.320].

Hence

2.2 Remarks. If i > 1 (2.1) holds (see [S]) if there exists a partition such that (i) and (ii) are merely bounded. Lemma 2 also implies some other sufficient conditions known to imply (2.1), in particular those of Kwong and Zettl [ll,Theorem 5 1 . However, conditions which seem independent of Lemma 2 follow from the theory of weighted Gabushin product inequalities (see [121), since these inequalities also imply sum inequalities like (2.1). 2.3 Examples.

case.

1.

The com act s u rt+aveP:ompact

2.

Monomial coefficients. Let po = t", pn = tY. Then any pi may be O(t B ) where 6 5 minIy + Z(n i), in y + * na I . The proof (which is comp1icated)may be found in [ 5 ] .

3.

To see Exponentials. If po = eat = w, p i can be O(eclt). this consider the partition IJi,I = {[L,L +1]I in Lemma 2.

4.

Point Evaluation Functionals. By approximating point evalution functionals by a Dirac sequence of compact support functions, Lemma 2 may be applied to derive the inequality

ort

Lemma 2 is easily seen to apply support.

-

R.C Brown

64

3.

THE OPERATORS I ,

zb, ro,

r i V y 7;

AND THE F A C T O R I Z A T I O N OF

I+,

SOME SELF-ADJOINT EXTENSIONS OF To

0 i s a subspace of L2 (w;I).

From A1 and A 2 2 a l s o i n d u c e s a H i l b e r t s p a c e 14 on D w i t h i n n e r p r o d u c t t [ u , v j . L e t I : L 2 ( w ; I ) -+ fl be t h e i d e n t i t y i n j e c t i o n , 1; t h e r e s t r i c t i o n o f 1 t o t h e s e t U A o f f u n c t i o n s o f compact s u p p o r t i n ( a , b ) , a n d Z 0 t h e r e s t r i c t i o n o f 1 t o t h e domain Do:

=

(i) {y E D : y ( a ) = 0,

D(y,F)(b-)

= 0,

v

i = 0,

...,n

z E D(T)

n D)

-

1;

where n-1 D(Y,3(S) =

(3.11 (N.R.,

[ i j [Zn-i-11 ci = o Y(S) T(s)

t h e s u p e r s c r i p t s i n ( 3 . 1 ) denote q u a s i - d e r i v a t i v e s . )

3.1 Remark.

I f p O c L1Q o c ( I ) , Cauchy's i n e q u a l i t y a n d t h e l o c a l

i n t e g r a b i l i t y of t h e o t h e r c o e f f i c i e n t s t e l l us t h f t UA, D o , U 3 C z s o t h a t t h e o p e r a t o r s a r e a l l d e n s e l y d e f i n e d i n L'(w;I). For a p r o o f o f t h e d e n s i t y o f t h e domains i n t h e g e n e r a l c a s e c o n s u l t [4],Lemma I ] . S t i l l a n o t h e r p r o o f w i l l emerge from P r o p o s i t i o n 1 below ( c f . Remark 3 . 2 ) .

-

Lemma 3. Proof.

I

normally solvable.

By A 1 and A 2 .

Next w e d e f i n e I + : H L 2 ( w ; I ) by 7 + ( z ) : = M [ z ] on U + : = D ( T ) fl 0. ( I n o t h e r words I + i s t h e r e s t r i c t i o n o f T t o t h e D i r i c h l e t i n t e g r a b l e f u n c t i o n s i n i t s domain endowed w i t h t h e H i l b e r t s p a c e s t r u c t u r e -+

o f H.) Let ’:Z b e t h e r e s t r i c t i o n o f 1' r e s t r i c t i o n of I+ t o

t o D(T;),

and 1; t h e

A Factorization Method and Dirichlet Inequalities

Proof.

65

T h i s is j u s t t h e D i r i c h l e t formula w y M [ z ] = D(y,T) ( b - )

a i=o

-

D(y,T) ( a + ) .

a

I:*

P r o p o s i t i o n 1.

=

Sketch of t h e Proof.

I+;

I+*

=

it,=

By Lemma 4

lo;

lt,* C I+,

1: C I + , e t c .

Hence i t

s u f f i c e s t o prove t h e r e v e r s e i n c l u s i o n s . Let (a,B) E G(1:); thus i n p a r t i c u l a r a E H. Let l : , 1' s i g n i f y f o r a moment t h e r e s t r i c t i o n s o f t h e s e o p e r a t o r s t o a compact s u b i n t e r v a l A . Thus we a r e 2 ) € G ( l + ) for in the regular case. S i n c e 1' i s o n t o L ( w ; A ) , ( a ' , B IA some c l ' E V + . By t h e F r e d h o l n a l t e r n a t i v e a n d t h e t e c h n i q u e o f [4,Lemma 11 ( s e e a l s o ibid.,Lemma

5),

R(lo)+

=

C o n s e q u e n t l y , a E D+ a n d ( a , B ) l A 6 G ( 1 ' ) . la l + C lA*. T h a t 1; = I + and l * = 1; f o l l o w I0

I *0

1 3 1; =>

'

I*C

I + and Lemma 4 .

3.2 Remark.

=

N(7').

Since A i s a r b i t r a r y from t h e i m p l i c a t i o n s

S i m i l a r a r g u m e n t s w i l l show

f i r s t t h a t l : ' * = 1 and t h e n t h a t lA* = I

0

Proposition 1 implies t h a t 7 ,

densely defined o p e r a t o r s .

N(IA*)

.

IO’

I + and I:

are closed

Lemma 1, P r o p o s i t i o n 1, and t h e c l o s e d

r a n g e t h e o r e m [ 8 , T h e o r e m 5 . 1 3 , ~ . 2341 i m p l y t h a t t h e r a n g e s of l o , I + , and 1; a r e c l o s e d . Moreover s i n c e I and I a r e 1 - 1, 7' and 0

1: a r e o n t o .

Lemma 5 .

7+lo

and

1il

are s e l f - a d j o i n t

domains a r e c o r e s r e s p e c t i v e l y

and D

T

=

of

1,

of

extensions To. 1, F u r t h e r D

and

Their T = Do

D.

P r o o f . A t h e o r e m o f Von Neumann [8,Theorem 3 . 2 4 , ~ . 2 7 5 1 combined w l t h t h e Second R e p r e s e n t a t i o n Theorem [8,Theorem 2 . 2 3 , ~ . 3311. 3 . 3 Remark.

I+lo, l + I are 1 - 1 positive operators. 0

Therefore t h e i r

r a n g e s a r e c l o s e d and o n t o . 3.4 Remark.

I n c o n c r e t e t e r m s Lemma 5 s a y s t h a t t h e r e s t r i c t i o n s o f ( n - i ) I 2 < a, s a t i s f y i n g e i t h e r t h e boundary b n T s u c h t h a t Ja i $ o pi Iy conditions

R.C. Brown

66 (1) y ( i ) ( a )

= 0,

i =

D(y,y)(b-) = 0 ,

o ,...,n -

YZ E

1,

D(T) fl D ,

or ( 2 ) yLi1(a

= 0,

Dfu,7l b - ) are self-adjoint

4.

=

i = n, 0,

VLI

...,2n - 1 , E

D

.

APPLICATIONS TO DIRICHLET INEQUALITIES

Theorem 1. Let E be bounded below and A 2 h o l d f o r a o s i t i v e T I , i = 1 , . .&n d e f i n i t e t r a n s l a t i o n of t. Let 0 < Bi all y E U

.

where u > - m restriction T

andi s-t he least of 7'1

for

--

element in t h e spectrum of t h e

d e f i n e d by t h e boundary c o n d i t i o n s

..

i = 1,. , n , D[y,f] (b-) = 0 , s i n B . f pji-11 a Y 1 E q u a l i t y h o l d s f o r $ & D i f a n d o n l y if i s an v y E 21. f u n c t i o n of T B c o r r e s p o n d i n g 5 u 4 . O t h e r w i s e t h e r e i s sequence o f f u n c t i o n s <$I,> & D s u c h t h a t -

==

Proof.

:-?-I

WLOG assume A 1 h o l d s .

Case 1.

A l l t h e Bi

+

a

=

5.

(4.1) and

t h e p r o p e r t i e s o f T ( i n c l u d i n g t h e f a c t t h a t u > - m ) a r e immediate B B c o n s e q u e n c e s of t h e F i r s t R e p r e s e n t a t i o n Theorem ( 4 . 2 ) f o l l o w s from [8,Theorems 2 . 1 o r 2 . 6 , ~ . 322-3233 and Lemma 5 . standard theory. 2. The form on t h e l e f t s i d e o f ( 4 . 1 ) i s c l o s e d and bounded below by A 2 , t h e i n e q u a l i t y o f 2 . 3 , example 4 , s e c t i o n 2 , and 18,Theorem 1 . 3 3 , ~ . 3201. L e t S be t h e s e l f - a d j o i n t o p e r a t o r guaranteed by t h e r e p r e s e n t a t i o n theorem. C a l c u l a t i o n shows t h a t t [ y , z ] = [T y , z J f o r a l l z E D. By [S,Theorem 2 . 1 , B T i s symmetric; a n d i f T B i s s e l f - a d j o i n t , iii., p . 3 2 2 1 , T B C S.

Case

B

2. I f n o t , we t r a n s l a t e t h e form and p r o v e t h e t h e o r e m i n a p o s i t i v e d e f i n i t e s e t t i n g . A s p e c t r a l mapping argument i s t h e n u s e d t o recover the "untranslated" inequality. For d e t a i l s see the argument i n [4,Theorem 2 1 .

A Factorization Method and Dirichlet Inequalities

61

T = S. Let S’ be the n dimensional extension of 7 A l with no B boundary conditions imposed at a. S’ is closed and S ' * is the n dimensional restriction of 7'1 having all zero boundary conditions 0 at a. So dimension (D(S’)/D(S’*)) = 2n;and T is an n dimensional B Therefore T is self-adjoint. The symmetric extension of S ' * . remainder of the argument follows case 1. 4.1 Remarks. 1. We note that T B is the Friedrichs extension of To representing the form in (4.1); in particular 1'1 are the Friedrichs and 1’7 0 extensions representing the forms 1 Ioyll2 and ~[y;). For n = 1, the boundary conditions at a of Ta are the most 2. general possible for self-adjoint extensions. F o r n > 1 such boundary conditions are described in terms of a hermitian matrix Q [ 7 , p . 2921. In this setting the cotangent terms in (4.1) become (n-1) t -(n-1) 1 (a). [ Y , ,Y 1 (a)Q [Y ,Y

--

---

3. Since minimal conditions, more general boundary conditions, no Dirichlet index hypothesis, and larger pi are assumed, Theorem 1 is more general than the inequalities established in [l].

Inequalities -for quasi-differential operators. The technique developed above works for quite general differential expressions provided the quadratic form is closed and bounded below. As aE illustration, we derive the simplest forms of Dirichlet inequalities for the ShinZettl quasi-differential expression M[y]:

=

[-sly’ +

We assume that a-’ 1

(ao

+

+

(i bo

alc

, ao, b o ,

2

IY

+

'

a,c)y]

+

(i bo

- alc)y’

.

1 c E LLoc(I) and are nonnegative.

As before T, T0 signify the maximal and minimal operators determined by M. (See [6] or [141 for accounts of the theory of symmetric quasi-differential operators.) We define the sesquilinear form t[u,vl: -

=

I,"

alu’v’

+

(ao

+ (ibo - alc)u

v'

+

alc2 )uv

- (ibo + alc)urv.

68

R.C Brown

Thus

+ 2 bo T E I ( Y

-

Lemma 6.

Let

holds where $ :

6 > 0 , e suppose

al

=

7’) -

bt

+

(alc)’.-

t

the

2 alcRc(yy’).

inequality

i s c l o s e d and bounded below.

P r o o f . The c o n c l u s i o n f o l l o w s from [8,Theorem 1 . 3 3 , ~ . 32C] and some routine integral estimates.

D[y,y] ( s ) : = y ( s ) [-a,:’ Again WLOG assume

+

( i bo

+

a,c)y] ( s ) .

t o be p o s i t i v e d e f i n i t e .

f,

I:,

Io, T

+

, To,+

+: fo a r e

d e f i n e d a s b e f o r e , b u t w i t h r e f e r e n c e t o t h e new c h o i c e s o f t, D[Y,F] ( s ) , and M . The same a r g u m e n t s m u t a t i s m u t a n d i s e s t a b l i s h t h e r e s u l t s o f s e c t i o n s 3 a n d 4 i n t h i s s e t t i n g . We a r r i v e a t t h e inequalities

uo b e i n g t h e i n f . o f t h e s p e c t r u m o f I;T

5.

(T+To)

i f y E i?(Uo).

THE D I R I C H L E T INDEX

I n t h i s s e c t i o n we s k e t c h how t h e D i r i c h l e t i n d e x t h e o r y o f R.M. Kauffman c a n be e x t e n d e d t o t h e n e g a t i v e c o e f f i c i e n t c a s e . We However t h e r e a d e r w i l l n o t f i n d i t d i f f i omit n e a r l y a l l p r o o f s . c u l t t o r e c o n s t r u c t them from [ 2 ] o r 131 s i n c e o u r l o g i c f o l l o w s t h e same p a t t e r n a s t h e e a r l i e r p o s i t i v e c o e f f i c i e n t t r e a t m e n t . D e f i n i t i o n 1.

z

€ D(T)

i s D i r i c h l e t (D) i f and o n l y i f z E H.

Definition 2. The D i r i c h l e t Index (DI) of M _ i s -t h e d i m e n s i o n o f t h e s ~ o fn t h e D s o l u t i o n s of T.

A Factorization Method and Dirichlet Inequalities

The

D e f i n i t i o n 3. (D ( T I / D ( T o ) 1 * Lemma 7 .

D e f i c i e n c y I n d e x (DFI)

D I = codimension R ( I )

1.

d i m e n s i o n ( D ( 1 ' 1 ) /D( I Z 0 )

=

0

of

M

69

1 / 2 dimension

d i m e n s i o n (Q/Q ) = 0

I t i s e a s y t o e s t a b l i s h t h a t 2n DI 2 n . I n g e n e r a l DFI - D I i s t h e d i m e n s i o n o f t h e s p a n o f t h e non-D s o l u t i o n s o f T . Thus DI = DFI <=> T i s D. Following t h e terminology f o r o r d i n a r y d i f f e r e n t i a l o p e r a t o r s , w e c a l l I " l i m i t p o i n t " when DI = n . Hence I i s l i m i t p o i n t <=> I 0

i s a n n d i m e n s i o n a l r e s t r i c t i o n o f I ; i . e . , D ( y , F ) ( b - ) = 0 , v y E: Q and z E D(T) r ) Q. I t c a n a l s o b e shown t h a t t h e i n d e x i s minimal <=> D(T1) i s a c o r e o f t h e r e s t r i c t i o n o f Q s u c h t h a t 0

y ( i ) ( a ) = 0, i =

o ,...,n -

1.

The r e l e v a n t c o n j e c t u r e i s t h a t t h e i n d e x i s a l m o s t a l w a y s n, a t l e a s t when w = 1.

...,PA

a r e r e a l m e a s u r a b l e f u n c t i o n s on I s u c h t h a t 1 P 0 + P I , > 0, ( P o + P ; ) - l , P 1 + P ; , . . * , p , + p ; c L ~ o c ( I ) ' Define Suppose pA,p;,

t

E'

M, ?, 6 , ?, ?',

Define

< E).

{pi + p i 1 and Theorem 3.

Suppose

6

t1 Q.

t h e r e e x i s t s

/I

S i n c e (17 ( a i +

Then T and 7

(ai- a)i2 L

GA,

a , a. E:

+

2

/I I( a i - a111 2

L € 1 1 1 ~ ~ - a(Iw .

-E

1

2

h a v e t h e same DI.

1

E

and ? ( a i ) 2

/I a i - a /lW

+

fi

?(a) in

.

If Now

.

We c o n c l u d e from ( 5 . 1 ) t h a t

I ( a ) i n H , and s o a € Do. and 7 ( a . ) + I ( a ) i n H , t h e n I(ai)

1.

I t i s s u f f i c i e n t t o c h e c k t h a t Do = ,D . o.

a E

15.1)

(so that L1[y]

bounded.

Clearly

co,

+

r1 .

Proof.

=

t’

))yll, f o r ?0 w i t h r e s p e c t t o t h e c o e f f i c i e n t s e t e t c . , r e l a t i v e t o t h e form +

Assume t h a t A l , A 2 h o l d f o r

O n t h e o t h e r hand i f a .

1

-f

a in Do

s a t i s f y t h e h y p o t h e s e s o f [8,Theorems 1 . 3 1 , 1 . 3 3 , e . g . , t, I' 3. p. 319,320,l.

R. C Brown

70

REFERENCES

.

On the minimiBradley J.S., Hinton, D.B., and Kauffman R )I toy. SOC. zation -f singular quadratic functionals, ’roc Edinburg, 87A (1981) 193-208. Brown, R . C . , An approach to the Dirichlet index for operators satisfying minimal conditions, in: Everitt, W.N. and Sleeman, B.D. (eds.), Ordinary and Partial Differential Equations (Lecture Notes in Mathematics 964; Springer-Verlag, Berlin, Heidelberg and New Y o r k , 1982) 110-117. Brown, R . C . , The Dirichlet index under minimal conditions, submitted Brown, R . C . , A von Neumann factorization of some selfadjoint extensions of positive symmetric differential operators and its applications to inequalities, to appear in the Proceedings of the 1982 Dundee Symposium on Differential Equations. Brown, R . C . and Hinton, D.B., Sufficient conditions f o r weighted inequalities of sum and product form, in preparation. Everitt, W.N. and Zettl, A., Generalized symmetric ordinary differential expressions 1: The general theory, Nieuw Archief voor Wiskunde (3) 27 ( 1979) 363-397. Hinton, D.B., On the eigenfunction expansions of singular ordinary differential equations, J. Differential Equations 24 (1977) 282-308. Kato, T., Perturbation Theory f o r Linear Operators, (SpringerVerlag, New York, 1966). Kauffman, R.M., The number of Dirichlet solutions to a class of linear ordinary differential equations, J. Differential Equations 31 (1979) 117-129. [lo] Kauffman, R.M., Read, T.T., and Zettl, A., The Deficiency Index Problem for Powers of Ordinary Differential Expressions (Lecture Notes in Mathematics 621; Springer-Verlag, Berlin, Heidelberg and New Y o r k , 1977). [ll] Kwong, M.K. and Zettl, A., Weighted norm inequalities of sum form involving derivatives, P r o c . Roy. SOC. Edinburg, 88A (1981) 1 2 1 - 1 3 4 . [12] Kwong, M.K. and Zettl, A., Norm inequalities of product form in weighted Lp spaces, Proc. Roy. SOC. Edinburg, 89A (1981)293-307. [13] Naimark, M.A., Linear Differential Operators, Part 11, (Ungar, New York, 1968). [14] Zettl, A., Formally self-adjoint quasi-differential operators, Rocky Mountain J. of Math. 5 (1975) 453-474.

.

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

71

SOLUTIONS WITH ASYMPTOTIC CONDITIONS OF A NONLINEAR BOUNDARY VALUE PROBLEM Nguygn Phuong Ca'c Department o f Ma thema t i c s The U n i v e r s i t y o f Iowa Iowa City, Iowa 52242 U.S.A.

INTRODUCTION L e t D be a bounded domain i n RN ( N 2 2 ) and l e t n = R N - E where E i s t h e e a r e conc l o s u r e o f D. We assume t h a t t h e boundary an o f n i s smooth. W cerned w i t h t h e boundary v a l u e problem ( a b b r e v i a t e d t o BVP h e r e a f t e r ) Au = p ( x , u , v u ) + f

in

n, u

= 0

on

an

(11

where A i s a n o n l i n e a r e l l i p t i c d i f f e r e n t i a l o p e r a t o r i n d i v e r g e n c e f o r m o f L e r a y - L i o n s t y p e , vu = g r a d u , f i s a d i s t r i b u t i o n on n. We s h a l l assume t h a t 1 the function p(x,t,q) d e f i n e d on n x R x R N has L -growth i n x and unres t r i c t e d growth i n t h e second v a r i a b l e t. I t w i l l be proved t h a t t h e BVP ( 1 ) i s s o l v a b l e i f i t has an upper s o l u t i o n Ji and a l o w e r s o l u t i o n rp w i t h rp i Ji and

-

c p , ~E ~ ~ ( nn ~) q ( n )

for some

E (I,-).

S i m i l a r problems a r e c o n s i d e r e d , among o t h e r s , by P. Hess i n [3],[4] and t h e a u t h o r i n [1],[2]. In [3] t h e growths o f t h e f u n c t i o n p a r e d i f f e r e n t from t h o s e assumed h e r e and i n [4] t h e domain i s bounded. Our r e s u l t seems t o p r o v i d e t h e answer t o a q u e s t i o n r a i s e d i n a remark a t t h e end o f [ 4 ] as t o whether i t s r e s u l t f o r bounded domains c o u l d be extended t o unbounded ones. In [l]t h e c o n d i t i o n s imposed on t h e upper and l o w e r s o l u t i o n s JI and cp a r e weaker t h a n t h o s e assumed i n t h i s paper: namely, i t i s assumed i n [I]t h a t JI and cp have o n l y l o c a l p r o p e r t i e s . Then we have t o r e s t r i c t o u r s e l v e s t o l i n e a r o p e r a t o r s A and t h e s o l u t i o n o b t a i n e d i s a l s o l o c a l i n n a t u r e . Furthermore, i t seems t o US t h a t t h e method o f [l]cannot be adapted t o n o n l i n e a r o p e r a t o r s . While [2] concerns m a i n l y w i t h t h e s o l v a b i l i t y o f t h e BVP (1) i n w e i g h t e d Sobolev's spaces u s i n g t h e r e s u l t o f [l] ( t h u s t h e e l l i p t i c o p e r a t o r s c o n s i d e r e d i n [2] a r e l i n e a r ) , b y t a k i n g t h e w e i g h t s equal t o 1, we have a l r e a d y o b t a i n e d i n [ Z ] ( c f . i t s Theorem 2 ) f o r l i n e a r o p e r a t o r s a r e s u l t s i m i l a r t o o u r Theorem I below. The method o f upper and l o w e r s o l u t i o n s i s c o n c e p t u a l l y s i m p l e and p a r t i c u l a r l y u s e f u l i n p r o v i n g t h e e x i s t e n c e o f a s o l u t i o n f o r n o n c o e r c i v e and p o s s i b l y s t r o n g l y n o n l i n e a r BVPs. However, i n r e a l i t y i t i s s e v e r e l y l i m i t e d by t h e d i f f i c u l t y encountered i n c o n s t r u c t i n g an upper s o l u t i o n Ji and a l o w e r s o l u t i o n cp w i t h cp i 4. T h i s d i f f i c u l t y i s g e n u i n e l y n o n t r i v i a l i f t h e domain i s unbounded, t h e o p e r a t o r i s n o n l i n e a r and we want, as i n o u r Theorem I below, t h e upper and l o w e r s o l u t i o n s t o be s i m u l t a n e o u s l y bounded and t o b e l o n g t o some space u

W''q(n) n L q ( n ) ( 1 c q , T < m) because c o n s t a n t s cannot t h e n serve as upper and 1oc l o w e r s o l u t i o n s . T h e r e f o r e , f o r i l l u s t r a t i v e purposes, we s h a l l g i v e an example f o r w h i c h o u r Theorem I a p p l i e s . We s h a l l e x p l i c i t l y c o n s t r u c t upper (and l o w e r ) s o l u t i o n s by " g l u i n g " t o g e t h e r upper (and l o w e r ) s o l u t i o n s on subsets of n. THE RESULT

Let

N.P. G i c

72

0. =

AU = - O i [ A i ( ~ , ~ , v ~ ) ] ,

1

-,a axi

w i t h t h e c o n v e n t i o n t h a t i f t h e i n d e x i i s r e p e a t e d t h e n summation o v e r t h a t i n d e x f r o m 1 t o N i s i m p l i e d . Throughout t h e paper we assume: (Hl)

For each

i = 1 ,.

.. ,N,

f o r each

(t,q)

E R x RN the function

A i : n x R x R N 3 R i s o f Caratheodory's t y p e , i.e.,

f o r a l m o s t a l l (a.a.) uous. tion

x E

(q* = &). k o ( x )

E Lq*(n)

f o r a.a.

ko(x)+co([tlq-l+

v (t,q)

n,

x E

S-

Ai (x,t,q)qi x E

n, b'

2

i s measurable and

( t , q ) -+ Ai(x,t,q)

0

a.a.

Iqlq-'),

-,

x E

c

0

i s contin2

0

and a f u n c -

n such t h a t

i = l , .,N, .-

E R x R ~ ;

(H3) f o r a.a.

+ Ai(x,t,q) q, 1 < q <

Furthermore t h e r e e x i s t c o n s t a n t s ko(-)

IAi(x,t,q)l

(H4)

x

n the function

2

vlql 9

( t , q ) E R x R N w i t h some c o n s t a n t

v > 0.

N + R i s o f C a r a t h e o d o r y ' s t y p e and t h e r e e x i s t s a c o n s t a n t E , 0 < E s q, a f u n c t i o n k l ( . ) E L1(n), x E R and a c o n t i n u o u s f u n c t i o n c ( - ) : [0,-)-+ [O,m) kl(x) 2 0 a.a.

The f u n c t i o n

p(x,t,q)

:R x R x R

such t h a t Ip(x,t,q) f o r a.a.

If

f E

E n

x

MiA;q*(n)

v (t,q)

for all where) i n

c(P)[kl(x)

E R x R~

then a f u n c t i o n

l o c a l sense o f t h e BVP ( 1 ) i f rp

JnAi

I

(x,rp,vcp)Oi

5

cp

0

v dx

with

E W;i:(Q) on

an,

+

it1

Irllq-El 5 p.

i s c a l l e d a lower s o l u t i o n i n the 1 E Lloc(n) and

p(x,rp,vcp)

. ~ ~ ( x , c p , v c p )v d x + (f,v)

v E WAyq(n) fl Lm(n) o f compact s u p p o r t and v 2 0 a.e. n, where (.,.) denotes t h e p a i r i n g between WA"(n)

(= almost e v e r y and i t s dual

w-1 * q * ( R ) . An upper s o l u t i o n i n t h e l o c a l sense i s d e f i n e d by r e v e r s i n g t h e i n e q u a l i t y s i g n i n t h e above d e f i n i t i o n . THEOREM I .

Let

f E W-lYq*(n).

and a l o w e r s o l u t i o n

cp

(T

cp i 0 s

= min(q,:))

1 u E Wo'q(R)

with

Suppose t h a t t h e BVP ( 1 ) has an upper s o l u t i o n

Lm(R) n Lq(cp) 0. Then t h e BVP ( 1 ) has s o l u t i o n w E WAyq(R) fl Lm(R) fl Lq"(R)

i n t h e l o c a l sense, b o t h b e l o n g i n g

4

on

a.e. i n t h e sense t h a t f o r e v e r y

jl

A Nonlinear Boundary Value Problem

the function

&

p(x,u,vu)w

73

i n t e g r a b l e and we have:

JL

A.(x,u,vu)D~ w dx =

n

p(x,u,vu)

w d x + (f,w).

PROOF. We s k e t c h t h e proof. I t i s hoped t h a t a d e t a i l e d p r o o f w i l l appear somewhere e l s e i n a more g e n e r a l c o n t e x t . For a number r > 0 l e t

6, We f i x

no > 0

R

= {x E

such t h a t

N

nr

: 1x1 < r ] ,

ac B

n n.

= B~

n > no

For each i n t e g e r

c o n s i d e r t h e BVP

“O’

Au = p ( x , u , v u ) + f By t h e r e s u l t of [ 4 ]

nn:

F o r each

i t has a s o l u t i o n

n Lm(nn)

v E Wb’q(fln)

Ai (x,u,,vun)Di We deduce t h a t

in

llunll ,,q

u = 0

fin,

u E W:’q(62n)

ann.

on with

cp s un i li,

a.e.

in

we have

v dx =

i

wo CnJ

p(x,un.vun) v d x + ( f , v ) nn h e r e and i n t h e sequel Ki ( i = l , 2 , . . - )

K1,

de-

notes a p o s i t i v e c o n s t a n t independent o f n, n o t n e c e s s a r i l y always t h e same. We e x t e n d un t o t h e whole domain R b y d e f i n i n g u n ( x ) = 0 when x E f l - n n . We can e x t r a c t f r o m

{u,]

We s h a l l show t h a t

a subsequence, s t i l l denoted by [un]

converges weakly i n

[u,]

converges a.e.

u

and

rn > n

Vx E

0

a function

n, c,(x)

l v c m ( . ) \ i s bounded on

m > no

WAyq(0) t o

h2

=

1

for

such t h a t

u,

u.

For t h a t purpose, we con1

& ( - ) E C (n) w i t h t h e f o l l o w i n g x E

nm, crn(x)

=

by a c o n s t a n t independent o f

o m.

for

x

4

F o r any i n t e g e r

we can show t h a t lim p(x,un.vu,)&,(un-u)dx n-- R

Taking

to

i s a s o l u t i o n o f t h e BVP ( 1 ) .

s t r u c t f o r each i n t e g e r properties: ~ , ( x ) E [O,1]

on

tun],

v = &(un-u)

= 0

l i m (f.c,(un-u)) = 0. n-= i n ( 2 ) we t h e n o b t a i n lim

n--

n

A . ( ~ , u ~ , v ~ , ) ~ ~ D ~ ( u ~ - u )= d0. x 1

It f o l l o w s from t h i s equation t h a t lim [Ai(x,un,vun) - Ai(~,un,vu)]Di(un-u)dx = 0. nR We deduce f r o m t h i s ( c f . e.g. [5], P r o o f o f Lemma 2.2, page 184) t h a t we can exsuch t h a t [vu,] cont r a c t f r o m r u n ] a subsequence, s t i l l denoted by {u,],

verges a.e.

to

vu

on

nm.

Since t h i s i s t r u e f o r any i n t e g e r

d i a g o n a l process, we see t h a t we can e x t r a c t from n o t e d by

{u,],

i = 1 , ... , N ,

such t h a t

{vu,]

converges a.e.

rn > no,

using a

r u n ] a subsequence, s t i l l det o v u on n. Then f o r each

74

N.P. Cbc {Ai(x,un.vun)~

A . (x,u,vu)

converges weakly t o

w E Lq"(0)

We n e x t show t h a t f o r e v e r y lim

nu-

n

in

1

n WA'q(n)

r l Lm(n)

Lq*(n)

we have

w dx.

p(x,un,vun) w dx = .fp(x,u,vu)

(3)

I n f a c t , l e t E’ > 0 be a r b i t r a r i l y g i v e n . B e a r i n g i n mind t h a t see t h a t an i n t e g e r m > no can be chosen such t h a t

Furthermore we can f i n d

6

> 0 such t h a t i f

Eo

By E g o r o f f ' s theorem, t h e r e e x i s t s a subset

nm-Eo,

t h a t on

cp(x,un,vun)]

E cRm,

of

cp

mes E < 6

nm

converges u n i f o r m l y t o

with

5

un

i f

we

then

mes E < 6

0

p(x,u,vu).

such

Then i t i s n o t

d i f f i c u l t t o see t h a t ( 3 ) i s t r u e . Finally, i f i n (2) w i t h

n a m+l

m*we s h a l l see t h a t theorem.

u

we t a k e

v =

5mw and t h e n l e t n and t h e n

i s a s o l u t i o n o f t h e BVP ( 1 ) i n t h e sense o f t h e

AN EXAMPLE

As an a p p l i c a t i o n o f Theorem I we prove THEOREM 11. E R;

d : R+

Suppose t h a t

vt

Ip(x,t,ql)[

where

2 s q < -, 0 <

k(x)lxla BVP

+ 0 as

E

is d i f f e r e n t i a b l e ,

k(x)

uniformly

P

for a.a.

0

with a >

-Di[lDi~1q-2Di~]+d(u)

n

let

p

k(-) E

Lm(n),

Then the s t r o n g l y n o n l i n e a r in

n,

(4) (5)

I, i t s u f f i c e s t o c o n s t r u c t an 0 5 $ a.e. on n;

tp i

a(t)

2

At V t > 0

and

i s s u f f i c i e n t l y l a r g e t h e n we have

-Di[lDif,lq-2Di$lI+ 1x1 > p .

N.

and

F o r t h a t purpose we f i r s t f i x a number

$ , ( X I = M ~ ~ l x l - Since ~ .

can be shown t h a t i f

for

P,

an

on

of t h e BVP ( 4 ) , ( 5 ) .

rp,$ E LOD(n) L'(I1) M > 1 such t h a t

x # 0

5

A >0

u E W:¶~(Q).

1 PROOF ( s k e t c h ) . Since k ( . ) E L ( n ) , by Theorem upper s o l u t i o n $ and a l o w e r s o l u t i o n cp w i t h

For

d ( 0 ) = 0, d ' ( t ) z

x E 0

= p(x,u,vu)

u = 0

has 5 s o l u t i o n

Vltl

k(x)+c~p)1rllq-'

s q-1,

1x1 +

R

d(Jil)

2

p(x,$l,v$l)

We s e t

$ ( x ) = $l(x)

if 1x1 2 p ,

$(x) = M

if 1x1

p.

0 <

E i

q-1

it

15

A Nonlinear Boundary Value Problem

4 > 0 on n, $ ( - ) E Lm(n) n Lq(,). To prove t h a t JI i s an upper with s o lu t i o n of t h e BVP ( 4 ) , ( 5 ) i t s u f f i c e s t o show t h a t f o r each v E C;(n) v 2 0 we have

Then c l e a r l y

4’D i t l q - * D i t D i v d x + S d ( J I ) R

v dx

2

n p(x,$,v$)vdx.

This i s done by d i r e c t computation bearing i n mind t h a t the f i r s t i n t e g r a l on the l e f t hand s i d e can be wr i t t en as t h e sum of an inte gra l on R-C2 and a boundary i n t e g r a l on

aB . P

P

A negative lower s o l u t i o n

rp

can be constructed s i m i l a r l y .

REFERENCES Ca'c, Nguy& P., Nonlinear e l l i p t i c boundary value problems f o r unbounded domains, J . D i f f e r e n t i a l Equations 45(1982) 191-198. Ca'c, NguyGn P . , On some q u as i l i n ear e l l i p t i c boundary value problems with conditions a t i n f i n i t y . To appear i n J . D iffe re ntia l Equations. Hess, P . , Nonlinear e l l i p t i c problems i n unbounded domains. Inte rna tiona l Summer School on Nonlinear Operators, B erlin, Sept. 1975, i n "Abhandlung der Akademie der Wissenschafter der DDR." Hess, P . , A second o r d er nonlinear e l l i p t i c boundary value problem, i n "Nonlinear Analysis: A C o l l ect i o n of Papers i n Honor of Erich H. Rothe," Academic Pr e s s , New York, 1978. Lions, J.L., Quelques me'thodes de r 6 s o l u t ion des problemes aux l i m i t e s non l i n g a i r e s , Dunod, Gau t h i er - Vi l l ar s , P a r i s , 1969.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

77

ABSOLUTE CONTINUOUS SPECTRUM OF ONE-DIMENSIONAL SCHRUDINGER OPERATORS Ren6 Carmona D e p a r t m e n t of M a t h e m a t i c s U n i v e r s i t y of C a l i f o r n i a a t I r v i n e I r v i n e , C a 92717

We g i v e s u f f i c i e n t c o n d i t i o n s on t h e a s y m p t o t i c behavior of t h e p o t e n t i a l f u n c t i o n q ( t ) f o r t g o i n g t o o n e of t h e i n f i n i t y f o r t h e s p e c t r u m of

--d2 + q ( t ) t o be p u r e l y a b s o l u t e l y c o n t i n u o u s i n dt some s u b s e t s o f IR .

I.

INTRODUCTION I n what follows,

q(t)

w i l l a l w a y s be a l o c a l l y s q u a r e i n -

t e g r a b l e r e a l v a l u e d f u n c t i o n which s a t i s f i e s :

f o r some

a > 0 , and w e i n v e s t i g a t e t h e s p e c t r a l p r o p e r t i e s of t h e

unique s e l f - a d j o i n t

If

q(t)

extension, say

H , of t h e o p e r a t o r d e f i n e d f o r

i s random, w e know i n some cases a n d w e b e l i e v e i n

g e n e r a l , t h a t i f i t i s random enough t h e s p e c t r u m w i l l b e a l m o s t s u r e l y p u r e p o i n t ( S e e [7] and [ 2 ]

f o r example).

On t h e o t h e r h a n d ,

w e know a l s o t h a t some c o n t r i b u t i o n s t o t h e p o t e n t i a l c a n a n i h i l a t e t h e t e n d a n c y t o l o c a l i z e and c r e a t e p u r e p o i n t s p e c t r u m d u e t o t h e randomness.

F o r e x a m p l e ( s e e [ S ] and m a i n l y [l])

t u r n i n g on a

c o n s t a n t e l e c t r i c f i e l d g i v e s i m m e d i a t e l y a p u r e l y a b s o l u t e l y cont i n u o u s s p e c t r u m e q u a l t o t h e w h o l e r e a l l i n e , i n d e p e n d e n t l y of t h e

R. Cannon0

I8

magnitude of the disorder parameters.

This is screeming for a

better understanding of what can force an absolutely continuous spectrum - Another motivation can be borrowed from the study of models like the following.

Lf.

n ,where ql(t)

E(t

w ) = F(Xt(w))

is a continuous periodic function and with

{Xt]telR

mot on on the unit circ e from

C

S1

H(w) = -

of

S1

.

[-e, fc]

onto

Let:

and

a stationary process ofBrownian

F

a smooth Morse function

Almost surely in

ci E

n the spectrum

d2

7+

q(t,w) is the set U[ai - e,b. + e ] where dt i is the spectrum of the (non random) operator

Cl = U[ai,bi] i H1 = d22 + ql(t) , C dt pure point in C\Cl

C1

is purely absolutely continuous in

~

.

and

A complete study of this example (and some

other ones) can be found in [ S )

.

Let us try to argue i t , at least at a heuristic leve1,by considering the asymptotics of the solutions of the eigenvalue equation : ( t , ) + q(t)y(t)

-y"

If

h e C\C,

, A

Schradinger operator

=

hy(t)

.

(E . V .E)

is in a gap of the spectrum of the periodic H1

so Floquet’s theory tells us that one of

the solutions of (E.V.E.)

decays exponentially as

t

-+

all the linearly independent ones explode exponentially. other hand, since

w

t

-+

m

while

On the

h e C , the Ljapunov exponent of ( E . V . E )

strictlypositivefor dichotomy.

-

is

and we may expect the same exponential

Moreover, since the noise process

ie(t);t)O]

is

random enough we may expect to be able to patch these random exponentially decaying solutions to the exponentially decaying

Spectrum of Schrijdinger Operators solution near A

4

.

T h i s c a n b e d o n e as i n [ Z ]

f o r a d e n s e s e t of

a n d t h i s t a k e s care of t h e claim o n t h e p u r e p o i n t s p e c -

C\Cl

trum.

-m

79

For

A

E

Xl

t h e random s o l u t i o n s on

are s t i l l f o r m i n g

IR,

t h e same e x p o n e n t i a l d i c h o t o m y w h i l e now, o n

a l l the

(-=,O]

s o l u t i o n s are bounded and ( a g a i n by F l o q u e t t h e o r y ) bounded u n i formly i n t h e i n i t i a l c o n d i t i o n a t t h e o r i g i n and c o m p a c t s u b s e t s of t h e i n t e r i o r o f

Cl

.

restricted t o

A

W e will see i n t h e f o l -

l o w i n g s e c t i o n t h a t t h i s is e n o u g h t o i n s u r e t h e p u r e a b s o l u t e Z1

c o n t i n u i t y o f t h e s p e c t r u m on

for

2

t

(E.V.E)

0

, i n d e p e n d e n t l y of w h a t

q(t)

is

and c o n s e q u e n t l y t h e b e h a v i o r of the s o l u t i o n s of

for

t

2

.

0

To b e c o m p l e t e w e m u s t s a y t h a t t h e a b s o l u t e c o n t i n u i t y of t h e s p e c t r u m of S c h r a d i n g e r o p e r a t o r s h a v i n g d i f f e r e n t a s y m p t o t i c s when

t

+ --m

and

t

-B

+-

h a s a l r e a d y b e i n g i n v e s t i g a t e d i n [5]

b u t i t seems t h a t o u r r e s u l t s a r e b e t t e r e v e n t h o u g h o u r m e t h o d i s

less g e n e r a l s i n c e r e s t r i c t e d t o t h e o n e d i m e n s i o n a l case.

11.

THE M A I N RESULT: I n o r d e r t o i n v e s t i g a t e t h e s p e c t r a l p r o p e r t i e s of

u s e t h e O.D.E.

(ordinary d i f f e r e n t i a l equation)

plained i n [4]

f o r example.

that

--

< a < 0 < b < +

w

Then w e c o n s i d e r t h e o p e r a t o r

HLY” [a,bl

LY

by :

0 =

a

and

and

p

in

b

such

[O,rr)

d e f i n e d on i t s domain:

Be” = i f E L 2 ( [ a , b ) , d t ) ; f is C 1 , f‘ a,b l u t e l y continuous, -f“+ qfsL2([a,b],dt)

f(a)cosCu+ f ‘ ( a ) s i n

we

a p p r o a c h a s ex-

F i r s t we p i c k reals and t w o a n g l e s

H

is a b s o and

f(b)cosp+ f‘(b)sinp=

01

.

R. Carmona

80 whenever

f

..

ho < h l <

.

D

This operator has a compact resolvent.

Let

be the eigenvalues of

and let $ o , Q l , a * Ca,b3 be corresponding eigenfunctions which constitute a complete orthonormal system of

.

L2([a,b],dt)

eigenfunctions also depend on

Note that the eigenvalues and the

a,b,a and

B

.

Now we set:

and

where :

and

-y"

y,(t,h)

(resp.

y2(t,h))

(t) + (q(t) -h)y(t) = 0

y;(O,h)

(resp.

= 0

is the solution of

which satisfies

y2(0,h) = 0

yi(0,h) = 1 )

yl(O,h) = 1 ,

. If

f cL2([a,b],dt),

we set:

so defined is a unitary transformation of

Then

U

onto

L’@E,

measure :

L2([a,b],-dt)

M ~ , ~(dh)) , ~ where , ~ Ma,b,a,B is the positive matrix

Specfrum of Schrodinger Operators

i n t o t h e o p e r a t o r of m u l t i p l i c a t i o n

which t a k e t h e o p e r a t o r

Ca,bI by t h e independent v a r i a b l e A .

a -+ q(t)

-03

and

b

-t

+a

.

W e would l i k e t o t a k e t h e l i m i t

Our a s s u m p t i o n s on t h e p o t e n t i a l f u n c t i o n

i s i n t h e s o c a l l e d Weyl l i m i t

H

imply t h a t t h e o p e r a t o r

p o i n t case a t p l u s and m i n u s i n f i n i t y . c o n v e r g e n c e of t h e m e a s u r e s

t o measures

is p o s i t i v e .

E,c

81

and

'a,b,w,B

T h i s i n s u r e s t h e vague

' 'a,b,@,B

and

'a , h ,ct , B

such t h a t t h e m a t r i x measure

By l i m i t i n g a r g u m e n t s o n e c h e c k s t h a t t h e o p e r a t o r

U

defined by:

is a u n i t a r y t r a n s f o r m a t i o n from which t a k e s t h e o p e r a t o r

H

t h e independent v a r i a b l e

h

L2@1,dt)

onto

L2@2,M(dA))

i n t o t h e o p e r a t o r of m u l t i p l i c a t i o n by

.

So:

" t h e s p e c t r a l c h a r a c t e r i s t i c s of t h e s e l f - a d j o i n t

operator

H

are c o n t a i n e d i n t h e measure a ( d h ) = S(dh) + ((dA)"

in t h e sense t h a t , i f

E

d e n o t e s t h e r e s o l u t i o n of t h e i d e n t i t y o f

H , w e have f o r a l l B o r e l s u b s e t s

u(A) = 0

of

E ,

2 Vf o L ( l R , d t ) < E ( A ) f , f > = 0

An i m p o r t a n t f a c t i s t h a t of t h e boundary c o n d i t i o n s

A

(i

Q

and

.

d o e s n o t d e p e n d on t h e c h o i c e

p

.

Hence t h e p r o p e r t i e s

Of

R. Cannona

82

u

w h i c h a r e o b t a i n e d by c o n s i d e r i n g i t as t h e l i m i t o f t h e m e a s u r e s :

s h o u l d be o b t a i n e d i n t h e same w a y by t a k i n g t h e l i m i t of t h e

m e as u r e s 0

a,b,B

( d h ) = LJnu (dh)du 0 a,b,a,B

.

( - ) can b e t h o u g h t o f as an a v e r a g e , o r a,b,B ( . ) when t h e b o u n d a r y expected v a l u e , o f t h e measures u a , b , a ,B c o n d i t i o n (Y is c o n s i d e r e d t o b e random and u n i f o r m l y d i s t r i b u t e d .

The m e a s u r e

(5

I t t u r n s o u t t h a t t h e measure

cs

a,b,B

is a b s o l u t e l y c o n t i n u o u s a n d

t h e formula g i v i n g its d e n s i t y i s t h e s t a r t i n g p o i n t o f our investigations.

In f a c t :

{ U h ( t , s ) ; t , s eR]

where

denotes t h e propagator of t h e d i f f e r e n t i a l

equation :

with

The a b o v e r e s u l t is c o m p l e t e l y p r o v e d i n [ 3 ] compute t h e d e n s i t y f o r

B

condition

at

for

t

2

0

.

So, i n o r d e r t o

w e f i r s t s o l v e (E.V.E) w i t h t h e b o u n d a r y

b , o r e q u i v a l e n t l y w e s o l v e (ODE)

with i n i t i a l condition

at the origin.

h

.

[zy:

!]

at

b

on

, and w e normalize

[O,b]

the solution

Note t h a t t h i s i n v o l v e s o n l y t h e v a l u e s of

q(t)

T h e n , w i t h t h i s n o r m a l i z e d v e c t o r as i n i t i a l c o n d i t i o n

Spectrum of Schrodinger Operators

a t t h e o r i g i n w e s o l v e (ODE) i n

83

and t h e d e s i r e d r e s u l t is

Ca,O]

t h e i n v e r s e o f t h e s q u a r e o f t h e norm o f t h e s o l u t i o n a t t h a t t h i s i n v o l v e s only t h e v a l u e s of t h e fundamental formula for

t

2

t 5 0 .

and

0

(FF)

q(t)

t

for

5

0

i n d e p e n d e n t l y of t h e u n i t v e c t o r

8

(and i n t h e l i m i t s a,b,B dependent of t h e v a l u e s of

,b

a

.

-t-m

q(t)

Note

So t h a t

s e p a r a t e s t h e c o n t r i b u t i o n s of

Moreover, if w e c a n c o n t r o l

Q

.

a

q(t)

IIUh(a,O)BiI

, w e c a n g e t estimates on

-

-t

t

for

, of

2

0

.

u)

which a r e i n -

This motivates the

following:

Definition: The d i f f e r e n t i a l e q u a t i o n ( 0 . D . E ) stable in

near

A ClR

--o)

i s s a i d t o be u n i f o r m l y

if

Examples : i)

q

L

0

and

A

closed i n t e r v a l contained i n

(0,m)

.

By

a s i m p l e p e r t u r b a t i o n argument w e h a v e : ii)

q(t)

integrable near

contained i n

(0,m)

.

-m

and

A

closed i n t e r v a l

Moreover, a s r e c a l l e d i n t h e i n -

troduction, Floquet's theory gives: iii)

q(t)

p e r i o d i c and

A

closed i n t e r v a l contained i n t h e

spectrum. We a r e now r e a d y t o s t a t e and p r o v e our main r e s u l t :

Theorem :

Let A.

A.

be a n open s e t o f u n i f o r m s t a b i l i t y n e a r

i s c o n t a i n e d i n t h e s p e c t r u m of

-m

. Then

H , t h i s spectrum is p u r e l y

R. Carmona

a4

absolutely continuous i n

and t h e d e n s i t y o f

A.

a b o v e a n d b e l o w away from z e r o on

0

i s bounded

.A

Proof: The e x i s t e n c e of bounded s o l u t i o n s of ( E . V . E ) w i t h bounded derivatives for t h e spectrum.

h o ho

Now, l e t

e a s i l y implies t h a t

is c o n t a i n e d i n

A.

b e open.

A c A.

by t h e p r o p e r t i e s o f v a g u e convergence of measures

5

l i m inf

I n

a+ -a d

o Qa , b ,a ,B (A)dff

b++m by F a t o u ’ s lemma

b e c a u s e of t h e v a l u e of t h e d e n s i t y given by ( F . F ) , because IIA911-2

a

5

2 x 2

1 and

0

IIA-1112

= llA1I2

if

A

is

matrix with determinant a u n i t v e c t o r , and because

of o u r assumption of uniform stability in This proves t h a t

a

A.

.

is a b s o l u t e l y continuous i n

A.

and t h a t

85

Spectrum of Schrodiizger Operators

To p r o v e t h e l o w e r bound w e go t h r o u g h a s i m i l a r a r g u m e n t w i t h

A

closed t h i s t i m e .

According t o t h e examples given above t h i s theorem i m p l i e s that i f

is i n t e g r a b l e n e a r

q(t)

contains

t h e n t h e s p e c t r u m of

--o)

H

( 0 , ~ ) and i t i s p u r e l y a b s o l u t e l y c o n t i n u o u s t h e r e ,

m a t t e r what

q(t)

is p e r i o d i c and

t

looks l i k e f o r q(t) = ql(t)

for

2

0

.

Similarly i f

no

ql(t)

t <_ 0 , t h e n t h e s p e c t r u m of

H

2

c o n t a i n s t h e s p e c t r u m of

-+

a n d i t is p u r e l y a b s o l u t e l y ql(t) dt q(t) for t l 0 . c o n t i n u o u s t h e r e , i r r e s p e c t i v e l y o f t h e v a l u e s of

-

Obviously t h e above theorem is p e r f e c t l y t a y l o r e d t o treat t h e random p o t e n t i a l s d i s c u s s e d i n t h e i n t r o d u c t i o n .

W e would l i k e t o m e n t i o n an a p p l i c a t i o n t h a t h a s b e e n p o i n t e d o u t t o u s by B a r r y S i m o n .

I n t h e i r work on q u a s i - p e r i o d i c p o t e n -

t i a l s , D i n a b u r g and S i n a i d e v o t e d h a l f of [ 6 ] t o show t h a t some s u b s e t s of

were s e t s o f u n i f o r m s t a b i l i t y and t h e s e c o n d h a l f

IR

t o c o n c l u d e t h e e x i s t e n c e of a n a b s o l u t e l y c o n t i n u o u s component o f t h e spectrum. t h e i r work.

Our r e s u l t e n l i g h t e n a n d s h o r t e n t h e s e c o n d h a l f o f I t is even improved i n t h e s e n s e t h a t i t r e q u i r e s t h e

Dinaburg-Sinai's assumptions near one of t h e i n f i n i t i e s only. F i n a l l y w e would l i k e t o e m p h a s i z e t h a t o u r method c a n be u s e d t o t a c k l e cases f o r which t h e u n i f o r m s t a b i l i t y a s s u m p t i o n i s not s a t i s f i e d .

F o r e x a m p l e , i f w e set

q(t)

= - Flt/U = anything

then one can e a s i l y check t h a t

+ v(t) for

for

t 3 0

t < 0

R. Carrnona

86

for each

A

6

iR , but nevertheless, by making random not only the

boundary condition

cy

, but also the length of the interval

the method gives that the spectrum is IR absolutely continuous.

[a,b]

and that it is purely

(The details are given in [33)

This last

result improves on [ll.

References [l]

F. Bentosela et al., Schr6dinger operator with electric field and random or deterministic potential. Comm. Math. Phys. 88 (1983) 387-397.

c2]

R. Carmona, Exponential localization in one-dimensional disordered systems. Duke Math. J. 49 (1982) 191-213.

[S]

R. Carmona, One-dimensional Schradinger Operators with Random or deterministic potentials: Analysis 51 (1983) 229-258.

[4]

New spectral types.

J. Funct.

E . A. Coddington and Levinson, "Theory of ordinary differential equations", McGraw Hill, New York 2 9 5 5 .

[5] E. B. Davies and B . Simon, Scattering theory for systems with different spatial asymptotics on the left and right. Comm. Math. Phys . (1978) 277-301. [6]

E. I. Dinaburg and Ya. G . Sinai, The one-dimensional Schradinger equation with a quasiperiodic potential. Funct. Anal. Appl. 9 (1976) 279-289.

[7]

I. Ja. Gol’Sheid, S. A. MolEanov and L. A . Pastur, A pure point spectrum of the stochastic one dimensional Schrsdinger equation. Funct. Anal. Appl. 11 (1977) 1-10.

[83

I. Herbst and J , Howland, The stark ladder and other one dimensional external field problems. Comm. Math. Phys. 80 (1981) 23-

DIFFERENTIAL EQUATIONS

I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science PublishersB.V.(North-Holland), I984

87

SOME TOPICS I N TRANSMUTATION

Robert C a r r o l l University of I l l i n o i s Urbana, I l l i n o i s

A few of t h e p r i n c i p a l themes from transmutation theory a s developed over t h e p a s t f o u r years o r so i n [ 5-351 a r e f o r m a l l y i n d i c a t e d i n t h i s survey and some of the more r e c e n t r e s u l t s (mainly work c a r r i e d o u t s i n c e t h e appearance of t h e book [ 191) a r e sketched i n more d e t a i l .

1. INTRODUCTION. form

&I

Qou

=

+

~ ( X ) U or

W e consider second order d i f f e r e n t i a l o p e r a t o r s of the

-

Qu

2

= x

Q u

+

-Q(x)u

(AQ r e a l ) and w e

where Q u = (AQu’)’/AQ A

concentrate here on s i t u a t i o n s where e i t h e r (A) Q i s modeled on a r a d i a l Laplace-

B e l t r a m i o p e r a t o r ( 0 5 x < -) i n a noncompact Riemannian symmetric space (e.g. A =

with

Q

q^

a (AQ

(x) with C

Q

t

Cm) where l i m A’/A

Q Q

= 2p

as x

Q

< A <

m,

and A

*

Q

+

A (-) r a p i d l y with p

Q

Q

while G(x)

involves A

s u i t a b l e ( c f . [ 39;44;45;52;70;74]) o r (B) -

3 m

=

Q

E C

321) o r ( C ) Q i s a Fourier type operator P u = u" - q(x)u on (--,-) even, p o s i t i v e , and continuous with qexp2Hx E L ( O , - ) w

than necessary - c f . [ 4 8 ] ) o r (D) Q involves A

-

where q i s s u i t a b l e - thus s e t t i n g x2{k2

-

<(x)Ip = X

2

ip

(cf. [ 3;38]).

ip

=

Q

- ;(x)

on [ 0 , m ) ,

0 <

say continuous ( c f . [ 28-

0 and

1

1

2

= pQ

where q i s r e a l ,

( t h e hypotheses a r e s t r o n g e r

-

2 2 x 2 mainly and Q(x> = x { k -

= xu an equation

-

Qu

2

2

= A u becomes x 9"

We r e c a l l t h a t an operator B: P

+

t

Q i s said to

be a transmutation i f i t i n t e r t w i n e s P and Q i n t h e form BP = QB a c t i n g on s u i t a b l e objects.

This i s

not

g e n e r a l l y an equation in L2 f o r example; P, Q, and B w i l l

u s u a l l y a c t on v a r i o u s f u n c t i o n s or d i s t r i b u t i o n s i n t h e course of an investigat i o n and B Co).

=

B

-1

will e x i s t ( i n the sense of V o l t e r r a o p e r a t o r s f o r example i n say

In t h i s review we w i l l discuss various B mainly i n terms of t h e i r k e r n e l s

represented a s s p e c t r a l i n t e g r a l s .

The transmutation machine runs v i a various pro-

p e r t i e s of c e r t a i n eigenfunctions and transmutation k e r n e l s and such p r o p e r t i e s (known t o hold f o r l a r g e c l a s s e s of s u i t a b l e p o t e n t i a l s ) w i l l be b u i l t i n t o the

R. Carroll

88

h y p o t h e s e s and theorems ( r a t h e r t h a n h y p o t h e s e s on

4 o r :).

One p r o v e s ( i n v a r i -

ous ways) t r i a n g u l a r i t y theorems f o r t h e t r a n s m u t a t i o n k e r n e l s which l e a d t o "cano n i c a l " c o n n e c t i o n f o r m u l a s of Riemann-Liouville Kober (E-K)

(R-L)

,

t y p e between s p e c i a l f u n c t i o n s ( c f . 1 1 9 1 ) .

Weyl (W), and E r d g l y i The s t u d y of s p h e r i c a l

t r a n s f o r m s i s connected t o t h e t r a n s m u t a t i o n theme and one o b t a i n s i n p a r t i c u l a r some "canonical" [ 22;23;25]).

e x t e n s i o n s of t h e Kontoroviz-Lebedev

(K-L)

i n v e r s i o n t h e o r y (cf

.

G e n e r a l Gelfand-Levitan (G-L) and Marzenko (M) e q u a t i o n s a r e d e r i v e d

by c o n n e c t i n g v a r i o u s t r a n s m u t a t i o n s i n terms of t h e i r k e r n e l s (cf. [ 1 9 1 and see [ 26;27] f o r some new M t y p e f o r m u l a s ) . Bergman-Gilbert

W e a l s o g i v e a c h a r a c t e r i z a t i o n of t h e

(B-G) i n t e g r a l o p e r a t o r i n terms of a s p e c t r a l r e p r e s e n t a t i o n of

t h e k e r n e l ( c f . [ 20;211).

L e t u s remark t h a t t r a n s m u t a t i o n methods (sometimes d i s -

g u i s e d ) have p l a y e d an i m p o r t a n t r o l e i n t h e s t u d y of d i f f e r e n t i a l e q u a t i o n s and t h e i n t e r a c t i o n w i t h p h y s i c s i s h i g h l y n o n t r i v i a l ( s e e e.g. [ 3;4;19;28-33;37;38;

40-43;51;56-58;62-69;72;73I).

I n p a r t i c u l a r t h e r e have been p r o d u c t i v e a p p l i c a -

t i o n s t o c e r t a i n i n v e r s e problems i n g e o p h y s i c s ( c f . [ 1 9 ; 2 8 - 3 3 ; 3 7 ] ) a b o u t which w e w i l l r e p o r t i n 16.

L e t u s a l s o mention h e r e some work of B. Whiting d e s c r i b e d

a t t h i s c o n f e r e n c e where some t r a n s m u t a t i o n a s p e c t s of Darboux t r a n s f o r m a t i o n s a r e

indicated. 2. PARSEVAL FORMULAS.

The Russian s c h o o l h a s used t r a n s m u t a t i o n methods

f o r y e a r s i n t h e s t u d y of d i f f e r e n t i a l e q u a t i o n s ( c f . [ 1 9 ; 4 5 ; 5 7 ; 5 9 ] ) a n d Marrenko i n t r o d u c e d a method of h a n d l i n g t h e s p e c t r a l t h e o r y and e i g e n f u n c t i o n expansions f o r n o n s e l f a d j o i n t Sturm-Liouville problems i n v o l v i n g D

2

- q(x) v i a a generalized

s p e c t r a l f u n c t i o n ( d i s t r i b u t i o n ) which we f o l l o w h e r e ( c f . [ 571).

Some a s p e c t s of h

t h i s approach w e r e extended by Gasymov [ 4 5 ] t o s i n g u l a r o p e r a t o r s Q u = u" -& m ((2m+l)/x)u' - q^(x)u

OK

e

=

m-% a n i n t e g e r .

I n [ 7;81 ( c f . a l s o [ 15;191) we i n d i -

c a t e d how t h e b a s i c Marrenko p r o c e d u r e could be a d a p t e d i n a c o n t e x t of g e n e r a l 1

transmutation theory t o Q f o r general m m

2 -4 and

t e r n a t i v e d e r i v a t i o n of Gasymov's r e s u l t s .

t h i s l e d i n p a r t i c u l a r t o an a l -

T h i s was f u r t h e r extended t o o p e r a t o r s

A

Q a s i n 51 modeled on r a d i a l L a p l a c e o p e r a t o r s i n [ 1 7 ; 1 9 ] ( c f . a l s o [ 2 ; 3 9 ; 7 0 ; 7 4 ] ) .

Thus t a k e

6 with

h y p o t h e s e s (A) and w e w i l l a l l o w

4E

1

Lloc

t o be complex v a l u e d i n

Some Topics in Trarismutatiori

89

For s i m p l i c i t y w e e x c l u d e s t r o n g s i n g u l a r i t i e s

d e r i v i n g t h e P a r s e v a l formula.

Q

B/x2 n e a r x = 0 i n o r d e r t o d e a l w i t h t r a n s f o r m s based on s p h e r i c a l f u n c t i o n s PA T h e b a s i c e i g e n f u n c t i o n e q u a t i o n h e r e i s (*)

( c f . [ 1 9 ; 3 9 ] f o r more g e n e r a l i t y ) .

&I

=

-A 2 u and

ably a t

m

For

0.

$

vanishing suit-

-

A W(VJ:,@!~)

and ( f o r

-t

where W(f,g) = f ' g

Q

.,"

n Q ( x ) d x where A

Q

real)

A

+

c (A)*:

Q

Define Q f ( X )

fg'.

-

=

Q A

= 1/27r(cp(A)l

2

(

c A!@)A-(

=

aQ (x) +A

%

=

so t h a t 2 i h c (A)

Q

Q

?(A)

= (

Q ( x ) ) = Jmf(x) f(x),nA 0

+ p Q2u

A

= A q Q and f o r Qu = P u = (A u ' ) ' / A

generally has t h e formP?(x) = f(x) =

with R

=

one d e f i n e s a l s o J o s t s o l u t i o n s @Q+ A ( x )s a t i s f y i n g (*) w i t h

e*ixx a s x

Q

= ;)

Q s a t i s f i e s a l s o ~ y ( 0 )= 1 w i t h DxqA(0)

p:

t h e i n v e r s i o n of P (Pf

Q Q " Q f(A),qx(x))v where dv(A)

=

Ro(A)dA

is

( s e e e.g. [ 1 9 ; 3 9 ; 4 4 ; 5 2 ; 7 4 ] ) . We assume h e r e t h a t R

known a l t h o u g h t h e r e are t e c h n i q u e s f o r " d i s c o v e r i n g " i t through a t r a n s m u t a t i o n Bp:

D2

+

$

i f i t i s n o t known ( c f . [ 7 ; 8 ; 1 9 ] ) . The o b j e c t now i s t o produce an in-

v e r s i o n f o r Q ( f o l l o w i n g a P a r s e v a l formula) v i a a g e n e r a l i z e d s p e c t r a l f u n c t i o n

- "

RQ, a c t i n g on a c e r t a i n space o f e n t i r e f u n c t i o n s ; t h u s f o r s u i t a b l e f , g w e w i l l

produce a formula (?) ( RQ,Q?Qg)A A " (Aki,A',") Qgy Q Q

(t )

that for

= (A-'f,A-'g),

Q

9"

Q f" (X)9p,(y)) Q

and

Q

= (

A4i,A4g).

4;

= &(x-y)/A (x) one h a s

Q

Q

Here one writes a l s o f = A

Q

= Qf = ( f ( x ) , q Q A(x)).

@z

f",

g

=

We n o t e f o r m a l l y from

= q Q ( y ) and ( A4i,A4<)

Q

A

Q

= i(y) =

Q

RQ,

i s t h e e i g e n f u n c t i o n expansion formula.

x

A

L e t now B: P

h

Q be t h e t r a n s m u t a t i o n c h a r a c t e r i z e d by W PA = p:;

-f

we em-

p h a s i z e t h a t B can b e determined by s o l v i n g a c e r t a i n Cauchy problem f o r example w i t h o u t any u s e of s p e c t r a l i n f o r m a t i o n ( c f . [ 1 8 ; 1 9 ; 5 3 ; 5 5 ; 7 0 ] ) . One can e x p r e s s t h e B and 8 = E-'

a c t i o n through k e r n e l f o r m u l a s Bf ( y ) =

(

~ ( y , x ) , f ( x ) ) and Bg(x) =

( y ( x , y ) , g ( y ) ) where B and y are t r i a n g u l a r i n t h e s e n s e t h a t B(y,x) = 0 f o r x > y and y ( x , y ) = 0 f o r y > x w h i l e y ( x , y ) = &(x-y) K(y,x) ( n o t e AQ = Ap).

= A (x)L(x,y)A-'(y)

Q

P

*

one d e f i n e s l ( x ) = l ( x , O ) ( c f . [ 4 5 ; 7 0 1 ) .

Q, 9,

and

Q

L ( x , y ) and B(y,x) = 6(x-y)

I n t h i s c o n n e c t i o n o u r h y p o t h e s e s on

r e q u i r i n g t h a t A (x)$(x,y) P A

+

act (P(Af) =

Pf

=

f*)

+

4 a r e e x p r e s s e d by

be continuous f o r 0

5 y 5 x and

One d e f i n e s c e r t a i n s p a c e s i n which P ,

and by Paley-Wiener

type r e s u l t s (cf. [ 3 9 ; 4 4 ; 4 5 ; A

"C

5 2 ; 7 4 ] ) one i s l e d t o a s p a c e Ep of even e n t i r e f u n c t i o n s f of e x p o n e n t i a l t y p e (determined by (;(A)\

5 c e x p ( o ( I m h ( ) f o r suppf

C [

&A

0,oJ) w i t h Rf:

E L

2

.

We t a k e

R. Carroll

90

W t o be t h e s p a c e of even e n t i r e f u n c t i o n s F of e x p o n e n t i a l t y p e ( a s above) such t h a t f m (F(X)(Rodh <

a.

0

= P W and IK = PW = A P K

Set K

W.

t r a n s l a t i o n Sy a s s o c i a t e d w i t h X

and g i v e n

it follows t h a t f g E

t h e form ( c f . [ 7;8;16;18;19;49;53;541)

= ( R Q , p l f Y ( hQ ) p X ( ~Q) I P X ( y ) ) X .I n o r d e r t o e s t a b l i s h

f i r s t ( c f . [ 6;7;18;191)

nA

:,;

P- 1) and r e c a l l t h e i d e a of a g e n e r a l i z e d

(I€' =

6 in

gE

c

Note t h a t W

(t)

f o r s u i t a b l e f , g , PB*f = ?.f

w e need v a r i o u s i n g r e d i e n t s ;

m

A (x)dx = Jo SYZ(x)?(x)A (x)dx =

Q

Q

(

E

One h a s t h e n ( S:i(x),g(x))

f (x) ,S:g(x)

)

In particular

and @ * g = p g .

B* p r e s e r v e s compact s u p p o r t so t h a t f o r s u i t a b l e f , Qf = PB*f f o r s u i t a b l e f , g of compact s u p p o r t .

ic P and !?&g =

sn 1. 0, (6;

perimentally"

f , g w i t h gk c o n t i n u o u s , gk above t h a t

of S:

(

=

gk

AQEk,

= (

o

x

(t).

(

Q

= R>o

(t),

One sets "ex-

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

*

(A:'f,Ai'g)

( i . e . SYSn = x Q

E

W'

( s i n c e Qfgg

E W)

V

(

= ( Rx(X)Ro,QfQg), ,

and one wants t o d e t e r m i n e a d i s t r i b u t i o n

( s o t h a t Tn

Q

Rn,pA (x) ) v ; B6:(y)

(L(x,y)G;(y)dy).

Thus w e

%

-f

(

RQ,Qf!?g)).

(@:)

= ( R ,:

F i r s t note

RQ,gfQg), = (A-'f,A-'g)

Q

P

(y) ) v = ( R:,IPx

(y) ) v

CfQg), i t f o l l o w s t h a t ( P R v , h ) =

Then f o r m a l l y , o m i t t i n g many d e t a i l s , one has

Under t h e h y p o t h e s e s i n d i c a t e d one h a s a P a r s e v a l formula

THEORM 2.1. (

1.

o p e r a t i n g on f ( y ) g ( x ) , w i l l l e a d t o

Consequently f o r h E K, P h = H E W (H

+ ( h(x),

=

where

0

f(y),Ek(y))

,Qf (X)Qg(X))v

= ( R:(X)

Q

6n,h)

6 (x)dx

E’

On t h e o t h e r hand t h e r i g h t s i d e of ( f ) , o p e r a t i n g on s u i t a b l e

from ($) w i t h y = 0 t h a t 6 n ( x ) =

(

I

,p y(x)IPA Q ( y ) )".

R:(A)

RQ E W’ t o which Rn converges weakly i n W’

=PRn.

m n

g , i t f o l l o w s from t h e b a s i c c o n s t r u c t i o n

+

f(y),( SY6n(x),gk(x))) * x Q

f (y)g(x), leads t o a t e r m T t h i n k of R,

sY6"(x) x Q

Hence t h e l e f t s i d e of

6(x-y)/AQ(y)). one s i d e of

(t)

= sn/aq)

W

.

6n = 0 n e a r 0 and f o r x > l / n , and

0'

E

lmSy?(x)p"(x>

Now l e t s n ( x ) b e an approximation t o t h e d e l t a f u n c t i o n 6 ( x ) i n

6" E C"

S:~(X)

(t)

f o r s u i t a b l e f , g of compact s u p p o r t where RQ = R E W’ can

Q

+

be w r i t t e n f o r m a l l y a s Ro

RZP. A

A

3 . BASIC FORMULAS I N THE Q THEORY. W e w i l l e x p r e s s t h e Q s p e c t r a l p a i r ing (

(

RQ,FG)X by

F,G)\, =

(

(

F,G)u.

Now l e t

?

and

6 be

of t y p e ( A ) o r (B) f o r example ( w i t h

4 P R ,FG)) and t h e fundamental t r a n s m u t a t i o n B: P

n -f

Q with

P

BpA = p y

can b e

o b t a i n e d a s Bf(y) = p ( 0 , y ) where IP i s t h e unique s o l u t i o n of t h e Cauchy problem n

P ( D )p = Q(D

Y

) q , p(x,O)

=

f ( x ) (extended t o

(-m,m)

as an even f u n c t i o n ) , and

Some Topics in Tvatisinutation

ip

18;19;53;551).

(x,O) = 0 ( c f .

Y

91

On t h e o t h e r hand i n [ 4 9 1 t h e t h e o r y of g e n e r a l A

A

U i z e d t r a n s l a t i o n s r e l a t e d t o a d i f f e r e n t i a l o p e r a t o r Q i s b a s e d on s o l v i n g Q(D ) X

=

$(D

Y

2

)U f o r y

0,

-m

< x <

w i t h U(x,O) = Af(x) and U (x,O) = C f ( x ) where A and Y

m,

I n [ 1 6 ; 1 9 ] w e extended t h i s

C are s u i t a b l e l i n e a r o p e r a t o r s w i t h (Af)’(O) = C f ( 0 ) .

*

approach t o s i n g u l a r Q t y p e o p e r a t o r s ( w i t h h y p o t h e s e s (A)) and d i s c u s s e d transmuA

t a t i o n s B: P Af(x);

ip

n

A

+

(x,O) = 0

Y

A

Q determined by Bf(y) = ip(0,y) where (A) P(D )p = Q(D ) i p ; ip(x,O) = X Y (ip

Y

(x,O)

=

0 because of t h e s i n g u l a r i t y i n

6).

F i r s t one c a n

show t h a t i f A and P commute ( a c t i n g on s u i t a b l e f ) and (A) h a s a unique s o l u t i o n A

*

t h e n B f ( y ) = p ( 0 , y ) d e t e r m i n e s a t r a n s m u t a t i o n B: P A

Now l e t B: P

(B

-+

A P Q b e determined by B p h =

+

Q.

-

N

= B-’)

where ( $ ( x ) u ( x ) , B v ( x ) )

=

-1

One h a s ( B = B

( A (y)v(y),Bu(y)).

Q

P Q P Q B(Y,x) = ( a X ( x ) , i p X ( ~ ) ) vkerB , = B(y,x) = ( a x ( x ) , i p h ( y ) ) , , N

-

B#

a s above and d e f i n e B a s

ip;

N

) kerB =

P kerB = Y ( X , Y ) = ( p h ( x ) ,

” P Q and k e r 8 = v ( x , y ) = (ipx(x),CIX(y))y. Now B a r i s e s from a problem (A) w i t h

a:(y)),,

A = 1 where ip(x,y) =

associated with

^P

(

B ( y , 5 ) , T x f ( 5 ) ) (Tx i s t h e s t a n d a r d g e n e r a l i z e d t r a n s l a t i o n

5

5

d e t e r m i n e d by TXf(E;) = ( P f ( X ) , i p

5

P X

P (x)p ( 5 ) ) ) . A

One of t h e main

V

u

r e s u l t s o f [ 161 w a s t o c h a r a c t e r i z e B i n a s i m i l a r way ( c f . a l s o 1 1 9 1 ) . THEOREM 3.1.

For g e n e r a l

gf (y)

t e r i z e d as T ( 0 , y ) = = A(x,S)

P

P^

and

6 w i t h hypotheses

=

w

=

can b e charac-

5

P

= ( n , ( C ) , i p , ( ~ ) ) ~s o t h a t A i s t h e o p e r a t o r a r i s i n g i n t h e g e n e r a l i z e d G-L c

&:

%

where y ( x , y ) = ( F ( y , C ) ,TXf ( 5 ) ) s a t i s f i e s (A) w i t h kerA

N

e q u a t i o n BA = B o r ( B ( y , t ) , A ( t , x ) ) = B(y,x). =

(A) o r (B)

Thus.

Q

Another f e a t u r e of t h e f o r m u l a s

Q

( y ( x , y ) , V A ( y ) ) = ~ Y ( X , - and ) ipA(y) = WJi’ =

from Paley-Wiener functions

ip

(

P

LpA

P B(y,x),iph(x)) = p B ( y , - ) arises

t y p e theorems developed i n [ 3 9 ; 4 4 ; 5 2 ; 7 4 ] s i n c e t h e s p h e r i c a l

Q (y) f o r example w i l l be e n t i r e i n X of e x p o n e n t i a l t y p e y under t y p i c a l A

hypotheses ( A ) .

It r e s u l t s t h a t ( c f . [ 1 9 ; 3 5 ] ) B(y,-)

E

E’ w i t h suppB(y,-)

c

L0,yl

N

and y ( x , - ) E E’

w i t h s u p p y ( x , - ) C [ O,x].

S i n c e B(y,x) = Ap(x)A-’(y)y(x,y)

Q

this

N

a l s o s a y s t h a t suppB(y,-) C [ y,-). rr

Consequently t h e B and B i n t e g r a t i o n a c t i o n i s of R-L

( r e s p . W) t y p e and

by i n t e r p o l a t i o n one c o n s t r u c t s i n t e r e s t i n g combinations of E-K o p e r a t o r s ( c f .

[14;19]).

One n o t e s t h a t t r i a n g u l a r i t y above i s a l s o i n t i m a t e l y r e l a t e d t o t h e

c o n n e c t i o n of B f o r example t o a Cauchy problem f o r a h y p e r b o l i c e q u a t i o n .

We

92

R. Carroll

<

w i l l assume from now on t h a t

i s r e a l and t h a t t h e r e s u l t i n g measure r e p r e s e n t i n g

RQ i s a b s o l u t e l y c o n t i n u o u s i n which c a s e RQ

for

5

Id

2Qdh

dw =

A

where w ( A )

Q

= 1/2 71

1 'Q I 2

Under t y p i c a l h y p o t h e s e s (A) f o r example one h a s IP Q (x) e n t i r e i n X w i t h h

on 1 0 , m ) . IIPh(x)1 Q

%

> 0 and x

e(iX-p)x

2

K(x)exp(lImX\x), x

2

Q

\?I(.W e

and l / c (-1) a r e a n a l y t i c

Q

c > 0 (cf. [ 19;39;44;52;741).

(P = p ) f o r s a y

nomials i n

0 , a n d QQX ( x ) , Xc ( - A ) ,

x E [ c,")

w h i l e Xc (-A)

Q

Q

F u r t h e r when I m h > 0 , and c-'(-h)

are bounded by poly-

Q

w i l l assume such p r o p e r t i e s now and refer t o t h e s e ( r e a l and

absolutely continuous)

6 as "regular".

L e t us w r i t e W ( A )

= I c /c

P

For r e g u l a r

n

s i o n f o r Q a r i s i n g from

and

(t)

$, %f

=

*:

where

*:

=

aQ/c (-1). A

1'

Q

P = W(A)qlQ i n t h i s s i t u a t i o n . r e g u l a r and r e c a l l from [ 191 t h a t -Box THEOREM 3.2.

aQ(x) x %

Q

with

P^

A

and Q

Further.

.

Also t h e i n v e r -

can b e w r i t t e n a s (*) f ( x ) = (1/2TT)L I Q f (h)*:(x)dX.

The f i r s t a s s e r t i o n w a s proved i n [ 1 9 ; 3 5 ] 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 arguments and we mention h e r e a new proof i n 1 2 7 1 .

Thus f i r s t w e c a n r e w r i t e t h e i n v e r s i o n

formula a r i s i n g from ( ? ) i n 5 2 as i n d i c a t e d - namely (*) f ( x ) = {cQ(h)9:(x)

+

YQdx = ~ ( X - U ) !J

Q (y)dX.

cQ(-h)@QX(x)]dh = ( 1 / 2 1 ~ )iIf(h)Y:(x)dh.

Now m u l t i p l y by 0

=

Then f o r m a l l y (1/2rr)

Jmn! 0

P Q P m P and F ( y , x ) = ( a h ( x ) , q X ( y ) ) w= Q { n h ( x ) I ( y ) s o g ( y , x ) = ( 1 / 2 1 ~ ) i P , ( x )

*P (x) !J

- P and i n t e g r a t e t o o b t a i n B{* ( x ) ] (y) =

m P P Q { ( 1 / 2 1 ~ ) i ~ X ( x ) * u ( x ) d x ~ d=h Y ( y ) .

and : Y

4 j I f ( h ) E (A) Q

!J

5::w i t h

u

iI *:(y)

P W e remark t h a t formulas of t h e t y p e q y = W h

a s s o c i a t e d t r i a n g u l a r i t y , which can b e d e r i v e d a b s t r a c t l y a s

above ( i n v a r i o u s ways), f u r n i s h a u n i f i e d and c a n o n i c a l p r o c e d u r e f o r d e r i v i n g many i n t e r e s t i n g c o n n e c t i o n formulas between s p e c i a l f u n c t i o n s ( c f . [ 1 ; 1 9 ; 5 2 I ) .

4 . FOURIER TYPE ANALYSIS AND THE MAREENKO EQUATION.

The G-L and M equa-

t i o n s arise i n quantum s c a t t e r i n g t h e o r y € o r example as p a r t of t h e machinery t o s o l v e i n v e r s e problems.

V a r i o u s a d a p t i o n s of t h e t e c h n i q u e t o o t h e r i n v e r s e prob-

l e m s a s w e l l a s o t h e r approaches t o a p p l i e d i n v e r s e problems have l e d t o a number of g e n e r a l i z a t i o n s and v e r s i o n s of G-L and M e q u a t i o n s ( s e e e . g . [ 4 ; 9 ; 1 0 ; 1 1 ; 1 6 ; 2 8 -

32;38;51;58;63]).

The p o s s i b i l i t y of r e l a t i n g t h e G-L and M e q u a t i o n s v i a t r a n s -

m u t a t i o n f i r s t came t o o u r a t t e n t i o n v i a [ 4 2 ] and w e gave a p a r t i a l g e n e r a l i z a t i o n of t h i s t e c h n i q u e i n 1 9 ; 1 0 ; 1 9 ] r e l a t i v e t o

P^

= D2 and r e g u l a r

6.

I n I 2 6 ; 2 7 1 we

A

were a b l e t o e x t e n d t h i s c o n n e c t i o n t o a l a r g e r c l a s s of P .

One works w i t h

93

Some Topics in Transmutation

* F o u r i e r type P as i n 91 ( c f . 1 4 8 1 ) f o r which t h e c l a s s i c a l F o u r i e r theory (involv+ihx) ing CosXx, Sinhx, and ecan be extended.

n

We develop and e x p l o i t f o r such P

t h e analogues of the c l a s s i c a l transforms and e s t a b l i s h some new technique (of independent i n t e r e s t ) using i d e a s of generalized t r a n s l a t i o n and convolution.

Once

t h i s machinery i s i n p l a c e t h e G-L and M connection follows e x a c t l y t h e l i n e s s e t f o r t h i n [ 10; 1 9 1. 4

A

Thus l e t P be of F o u r i e r type on

2

4

For (*) Pu = -A

be r e g u l a r .

O,m),

t i o n s and J o s t s o l u t i o n s p:

u ( c f . [ 4 8 ; 7 1 ] ) i n a d d i t i o n t o s p h e r i c a l func-

P

and

P a s above we d e f i n e x A ( x ) a s t h e s o l u t i o n of

-

P P (*) s a t i s f y i n g XA(0) = 0 and DxXA(0) = -1.

+ M1(-A)e

M1(A)r-iAX

i X x and X,(x) P

and l e t Q , considered only on

(-m,m)

For r e a l A a s x

+ M(-A)eiAx

M(A)e-iXx

%

P one has p X ( x ) n,

-t m

where M and M

1

a r e analy-

t i c f o r I d > -H (except f o r a simple pole a t A = 0) and n e i t h e r vanishes f o r ImA > -6.

The f u n c t i o n p ( h ) = 1 / 4 i h M M

+

=

M-M1j/2MM1

P

= 2ih{MPA - M

X (M-

P

i s a n a l y t i c f o r ImA > -6 and one d e f i n e s A(X) = M(-A)

=

M(A)

for X real).

P

P

W e can write

P

X 1 and one d e f i n e s z A ( x ) = 2ih{WX(x) + M1XA(x)l; t h e s e a r e defined

*P (-x)

P

= z,(x)

A

P zX play

(also note t h a t

*P-A (x)

P

= pzA(x)

P + AQX(x)).

The funcA

a n a t u r a l r o l e i n the f u l l l i n e s c a t t e r i n g theory f o r P ( c f .

[ 38;43;511) and i n f a c t a s x s i c a l reflection coefficient.

+ m,

$;(x)

%

e-iXx - A(h)eiAx, s o t h a t -A i s a clas= c (-A)

One can show a l s o d i r e c t l y t h a t M ( A )

P

1

P F(A) = 2iAM(A) where F i s a c l a s s i c a l J o s t f u n c t i o n a r i s i n g f r o m x X =

{m--

and F-@+I/

P The s p e c t r a l measure f o r the eigenfunction expansion r e l a t i v e t o p X ( r e s p .

2iA. P x,)

P

1 A

f o r a l l x with P t i o n s Q A and

for real

1

i s du = d A / 2 n j c p )

P where 'PA = @:/cp(-A) (2AAM/M;)qA

P

2

2 ( r e s p . do = dA/2nlM/ ) .

(dv = GPdh) and

( t h e decomposition of

P

P We remark a l s o t h a t 2 R e q = 2nGppA

P $ = 2ih{(M/M1)pAP - XA) with

I m Ag = - 2 X X pA -

i n t o r e a l and imaginary p a r t s was s t a t e d in-

c o r r e c t l y i n some i l l u s t r a t i v e m a t e r i a l i n 1 1 9 1 , pp. 325-329 and i n 1311, pp, 5457 - t h e a p p r o p r i a t e modifications a r e obvious). We assume

5 is

N

r e g u l a r and B: p f

-t

p y with B = (B

can be represented i n terms of s p e c t r a l k e r n e l s a s i n 5 3 . f

i

A

A

t i o n B: P

+

Q : @:

A

t h a t B and e . g .

-f

with s p e c t r a l k e r n e l ;(y,x)

=

-1 i / * P )

. LpX

-f

ICp/CQI

2 4

PA

We d e f i n e a transmuta-

P Q (n,(x)/~~(-X),@,(y))~/2n

(note

B transmute on d i f f e r e n t o b j e c t s i n general - n o t e a l s o a typogra-

R. Carroll

94

A

p h i c a l e r r o r i n [ 2 6 1 i n t h e formula f o r E(y,x)). Note a l s o t h a t O(y,x) = 0 f o r y > h

x under normal h y p o t h e s e s on Q. -

"

A

B v i a B.

2

Now t h e i d e a of t h e M e q u a t i o n i s t o r e l a t e B and

o u r development of t h i s m a t e r i a l i n [ 191 w a s based on [ 421

When P = D

Here w e can e s t a b l i s h analogous machinery f l 2 F i r s t following [48;71] f o r f o r P and the r e s u l t r e d u c e s t o [ 1 0 ; 1 9 ] when P = D

u s i n g t h e F o u r i e r t r a n s f o r m on

(-m,m).

.

6

iz

suitable f , @(f) =

P f ( x ) Q A ( x ) d x can be i n v e r t e d i n t h e form f ( x ) = (1/2a) i z @ ( f )

P p(A)Z: (x)dA ( n o t e o u r c o n f i r m a t i o n of t h i s s o r t of i n v e r s i o n i n [ 191, Chap. 2 , ReA mark 1 0 . 1 2 and a t t h e end of 1 1 2 1 w a s b a d l y p h r a s e d - a c o r r e c t v e r s i o n a p p e a r s i n One d e f i n e s a k i n d of g e n e r a l i z e d t r a n s l a t i o n C Y f ( x )

[ 271). P

X A (x)p(A)dX and a g e n e r a l i z e d c o n v o l u t i o n (g P

= ( 1 / 2 s ) lz@(f)@(g)BA(x)p(X)dhs o t h a t @ ( f

-P s i b l e t o express t h e r e s u l t N = P ': A

*

*

f ) ( x ) = (f

*

P

( 1 / 2 s ) Lz@(f)@ ( y )

=

A

g ) ( x ) = (g(y),E:f(x)) It i s t h e n pos-

g) = @(f)@(g)e t c . N

i n t h e form ( r e c a l l B(y,x) and t ( y , x ) v a n i s h

f o r y > x and we t a k e both k e r n e l s t o be 0 f o r a l l x < 0)

.

N

THEOREM 4.1

A

-

{B(y,-) (1/271)

.

H}(x) where @ ( H ) = M / c ( - A ) 1 Q

~ z @ ( H ) ~ ~( t() y P (1) P ~d i .

B(y,x) =

4

and w e can w r i t e t h e n B = B x w h e r e kerJC=

U N

One t a k e s t h e g e n e r a l i z e d G-L e q u a t i o n of [ l 0 ; 1 9 ] now i n t h e form B = BW A

Y

W

B;Kw where kerW

=

P P 42C2 W(x,y) = ( q ( x ) p , ( y ) , v / w U

=

' I

-

x

u

(u = vp, dw = GdX) and t h e i d e a Y-

t h e n i s t o s e l e c t a n o p e r a t o r JC such t h a t B X i s s u i t a b l y t r i a n g u l a r and (Jcwc) i s "nice".

-

A f t e r some c a l c u l a t i o n w e o b t a i n XWg =

Jm ( c /M-)(l

+

1)

is

-m

Q

[ 10;19;42

I

(p/p-))@i(t){@y(x) z

K(t,x) ,g(x)) with K(t,x) = ( 1 / 4 ~ ) N

For Jc a f i r s t c h o i c e ( f o l l o w i n g

ZP(x)}pdA. u

5

3(h

+

(

= (z(x,s) , h ( s ) ) w i t h X(x,s) = ( 1 / 2 1 ~ )

P P _/I<. / c - ) @ (s)ZA(x)pdX. 1 . Q X

P = 0 f o r x > y and With t h i s c h o i c e L ( y , x ) = ker(B%J = ( 1 / 2 1 ~ )L ~ ( l / c - ) @ Q ( y ) Q x ( x ) d A Q A

one h a s .

THEOREM 4 . 2 . With h y p o t h e s e s a s i n d i c a t e d above one h a s a g e n e r a l i z e d M

e q u a t i o n f o r k e r n e l s , namely, f o r x > y , 0 = S ( t , x ) = -(1/28)

D

2

$

P

@-x

=

P

+

P A@A).

r n h

Y

P P L ~ ~ QIm(MM1)@A(t)@A(x)dA PS and

@K(t)@!A(x)dA ( r e c a l l

the only

J

B(y,t)[S(t,x)

J(t,x) = (1/4n)

9

form one can w r i t e J ( t , x )

=

where

L~(~+(Q/P-))(Q/P-)

.

This is an i n t e g r a l equation f o r

i n p u t i n t o t h e k e r n e l s S and J comes from s

t h i s r e d u c e s t o [ 10;19 1.

+ J(t,x)}dt

= c /c-

Q Q

in S.

and For

^P

In o r d e r t o p u t t h e i n t e g r a l e q u a t i o n i n t o a s i m p l e r 6(x-t) - T(t,x) t o o b t a i n ( 6 ) i ( y , x )

=

J

m

Y

A

B(y,t)iT(t,x) , .+

- S(t,x)ldt.

=

A s l i g h t f u r t h e r s i m p l i f i c a t i o n can be a c h i e v e d by t a k i n g K w i t h

Some Topics iii Transmutation

-

95

P P 2 kernel x(x,s) = ( 1 / 2 ~ 1 )/ I ( M , / C ~ ) @(s)zA ~ (x)p dX and then we can w r i t e t h e i n t e g r a l A

equation

( r ) for B

i n t h e same form with s l i g h t l y d i f f e r e n t S and T. u

5. BASIC ?THEORY.

We t u r n now t o operators Q and t h e theory has a num-

ber of d i f f e r e n t f e a t u r e s ( c f . [ 3 ; 2 2 ; 2 3 ; 3 8 ; 4 0 - 4 3 ; 4 6 ; 5 1 ; 5 6 - 6 3 1 ) .

There i s f i r s t a

considerable l i t e r a t u r e on i n t e g r a l o p e r a t o r s which t r a n s f o m a n a l y t i c f u n c t i o n s or harmonic f u n c t i o n s i n t o s o l u t i o n s of e l l i p t i c equations ( i n p a r t i c u l a r t h e B-G operator does t h i s ) and i n a d d i t i o n c e r t a i n d i r e c t and i n v e r s e problems i n s c a t t e r N

ing theory have been i n v e s t i g a t e d using Q t y p e operators.

Let u s w r i t e A u

2 n- 1ur)r/rn-' F f r )u = 0, A u = ( r n

as depends only

or s p h e r i c a l v a r i a b l e s .

n 2 fonn Pu = P u = r urr N

+

+ Asu, n

AS, n

=

OSu/r2 where n

+ on angle

We express t h e r a d i a l p a r t of An m u l t i p l i e d by r 2 i n t h e @ n u ( s e t a l s o Qu = Q u = Pu

(n-l)rur

+ r -3F ( r 2 )u).

Gilbert's

+

method of ascent then shows t h a t s o l u t i o n s ( r e g u l a r around t h e o r i g i n ) of Anu =

0 for n

(xl,

22

...,xn)

+

r(Grr

s h a l l r e f e r t o t h i s a s t h e B-G

+ ( p"-3z(r,p),h(p,.))

tion.

+

FG) = 0; G(O,T) = 0 ; G(r,O) = -

i n t e g r a l operator.

r

I

0

2 F(p )pdp.

Now change v a r i a b l e s and w r i t e with Y t h e Heavyside func-

where K(r,p) = K(r,p)Y(r-p)

The d i f f e r e n t i a l equation s a t i s f i e d by G l e a d s t o a d i f f e r e n t i a l equation

This i s a l l embodied i n t h e d i s t r i b u t i o n context 2

-r2F(r )6(r-p)

(?$e

=

2 (p 8 ) ' ' - (n-l)(pg)').

t o r r e p r e s e n t s a transmutation u a-1

(pg(r,p),p

-4J0r F ( p 2 )pdp/rn-' a t P = r. -* n-3vKI = where one has (9, - P p ) { p

=

One determines then t h a t t h e B-G V

%(h) =

) =

(

B(r,p),h(p,')).

M{pg(r,p)I(o) (Mellin transform p T =

-a-n+2)

equation 68 = a(okn-2)9 with QQ(r)r-'

0(o+n-2)oT).

-

.

+

+

opera-

" a

Then w r i t i n g B f p ) - v

choosing $ Q ( r ) (= ILQ(r) with

=

-*v

a) we w i l l have Q B = P B and P

t o be t h e s o l u t i o n of t h e eigenfunction

1 as r

+

0 we have (note :pT

=

~(~i-n-2)

"

THEOREM 5 . 1 . The extended B-G k e r n e l B(r,p) i s c h a r a c t e r i z e d by v

t h e s p e c t r a l formula (with i n t e g r a l s c-im + c+im)% B(r,p) = ( l / Z v i ) I p =

We

V

f o r K when p < r with a jump d i s c o n t i n u i t y K ( r , r ) =

=

where x =

and t h e Bergman k e r n e l G (which does not depend on n) s a t i s f i e s t h e

equation 2(1--r)GrT - Gr

u = h

+ J01 an-1 G ( r , l - c 2 )h(xa 2 ) d a

can be w r i t t e n a s u(x) = h(x)

Fu

(1/2iri) i ~ ~ - ~ p ' $ : ( r ) dand ~ N

g{p'}

=

$z(r).

-

.

-0-1 Q $o(r)do

The s p e c t r a l p a i r i n g i n Theorem 5 . 1 ,

corresponding t o t h e P spectrum, i s handled expeditiously v i a t h e Mellin transform and we w i l l s e e below how t o handle t h e Q spectrum more generally.

For n = 3 ,

=

96

R. Carroll

6 = x 2D 2 + 2xD + k 2x2

0,

and t h e n $:(x)

0+5

F u r t h e r i n t h e n o t a t i o n above one knows t h a t ( c f . [ 4 7 1 ) K ( r , p ) = -Sk

and z = k x .

(~/r)'~~{kr(l-(p/r))'}/(l-(p/r))~

-4

and i n p a s s i n g w e n o t e t h a t M{pK(r,p)j = -k-'

( k r ) where s d e n o t e s t h e s t a n d a r d Lommel f u n c t i o n .

Sa+3 / 2 , u+

(kr)

where v =

= 2a~k-4.(a+3/2)z-SJv(z)

Y

N

+

P = Pn = r2D2

2 Recall t h a t r A

(n-l)rD.

= P"

n

+ Q:

- -+

U

e x p l i c i t an i m p o r t a n t p r o p e r t y of t r a n s m u t a t i o n s B: P

L e t u s make

Q where Q = P

+

where

n:

2 2 r F ( r ) and

d o e s n o t depend on

2 n 2 It f o l l o w s t h a t f o r s u i t a b l e F , B{p A u} = B{Pnu +n>} = Q Bu +a> = r tan n 2 2 2 2 2 F ( r )}Bu and t h u s B{p An} = r {An F}B ( i . e . B: p A n + r2An + r F ) . Thus one

r.

+

+

wants m u l t i p l i c a t i o n by r L i n o r d e r t o e x t e n d t h e t r a n s m u t a t i o n t h e o r y from r a d i a l operators t o certain operators i n several variables. We w i l l t r e a t now t h e c a s e A

X

2 2 xu s o (x u ' ) ' = ~ ( x u ) " = x4" and Qu = X u b e c a n e s

2

(one w r i t e s a l s o A 2 = a ( u + l ) = v2

@

We d e n o t e by

Ip

(v,k,x) t h e "regular"

N

t i o n f(v,-k)

-

0

%. s o a

%

f(v,-k,x)

e

%

-

(ip

Q,

xv+'

eikx as x

(W(f,g) = f g '

+ x 2 { k2

2 xip"

(0)

r e a l , and set c

- q(x)b

= a n g u l a r momentum and

(0)

Y

= W(f(v,-k,x),ip(v,k,x))

and s a y fm x 2 ( T \ d x <

-

s o l u t i o n of

f ( v , t k , x ) t h e "Jost" s o l u t i o n s (e.g.

<

as a model and t h i s w i l l e x h i b i t t h e

Thus t a k e T(x) = x2{k2 - y ( x ) } ,

guidelines f o r a general theory. @ =

= x L.

-

f'g).

as x

+

v

Q,

=

L+%).

0) and by

w i t h t h e J o s t func-

+ m)

Assume e.g.

fomx \ < ( d x

a s i n [ 31 b u t a g a i n we do n o t emphasize h y p o t h e s e s on

a

( c f . [ 3 8 ; 5 6 ; 6 2 ] ) ; we want mainly v a r i o u s a n a l y t i c i t y p r o p e r t i e s of t h e eigenfunct i o n s and J o s t f u n c t i o n p l u s a s y m p t o t i c p r o p e r t i e s of t h e form

as

1.

-+

-,

Rev > 0 , where ( c o r r e s p o n d i n g t o

J v ( k x ) and f o = f o ( v , - k , x ) I-(v+l)k-v+%v-' t h a t f(v,-k,x)

7=

0),

@

IpO

and f

%

fo

( v , k , x ) = 2vr(v+l)k-v$

L 1 = (krkx) 'exp(&in(w+5))Hv(kx)

(i" = expkina).

ip

w i t h fo(v,-k)

= 2"(2/8)'

W e t h i n k of k a s f i x e d h e r e and one knows t h e n

i s e n t i r e i n v w h i l e @ ( v , k , x ) and f ( v , - k )

are a n a l y t i c f o r Rev > 0

( t h e range of a n a l y t i c i t y c a n b e e n l a r g e d w i t h s u i t a b l e h y p o t h e s e s on

q).

In order

t o d e a l w i t h s p e c t r a l q u e s t i o n s w i t h o u t i n t r o d u c i n g an u n n e c e s s a r y parameter deN

s c r i b i n g s e l f a d j o i n t r e a l i z a t i o n s of Q ( c f . 156 I) we f o l l o w f o r m a l l y t h e p r o c e d u r e of [ 31.

and l e t Z d e n o t e t h e s e t of z e r o s v . ( i f 3 m 2 i n Rev > 0 w i t h M ( v . , k ) = 1 g (v - k , r ) d r . With such s i m p l e J 0 j'

Thus s e t g(v,-k,x)

any) of f(v,-k)

= f(v,-k,x)/x

2

z e r o s v . one s e t s d p ( v ) = I s ( v - v . ) / M J J

2

(v

k ) f o r v E Z and f o r v E [O,i-) dp(v) =

j’

91

Some Topics in Transmutation 2 2iv dv/Tf (v,-k)f (-v,-k). = (

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

,., an operator Q,

From 13 1 one has the formal completeness relation 6 ( r - s ) THEOREM 5.2. Let gl, P I , etc. refer to

and we show then.

P

based on yl. Define B(r,s)

= (

g(w,-k,r),gl(w,-k,s)) N

-4

(

g(v,-k,r),gl(u,-k,s))

P

1 with B f ( s )

= (

6(r,s),f(r))

and Bf(r)

-

P

and y(r,s)

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

suitable f. The r and s brackets refer to distribution pairings on [O,-) has triangularity B(r,s) = 0 for s > =

Im f(s)g(u,-k,s)ds +

r

r

Then B: Q,

-f

T

and B(r,s)

"B1): 6

+

(G

-Q,

and B{gl(v,-k,-)l(r)

and correspondingly 8

=

?(u)

(:(v),g(v,-k,r))

P

are transmutations with B{g(v,-k;)j(r) Set

B

-* = B

t 1f and b f

=

6,Bf

g(v,-k,r).

,-* = B ; then GBf

=

.

N

u

gl(v,-k,r)

=

for

and one

Set tf(v)

= 0 for r > s .

-1A = G f(r) = f(r)

that formally G?(r)

so

and 8

0

N

=

=

Bf(r)

(so

(with

B

=

.

B(r,s),f(s))) -1 B 1.

= (

=

Although the technique follows [ 31 there are significant variations and the results in Theorem 5.2 represent a considerable extension and refinement of [ 31.

Note also that

based on

4. m

=

x u"

+

= h(r, - )

tions of (An

)

represents Jm 0

in contrast to certain pairings in [ 31

. * 2 Thus one considers Qu = x u"

+

(n-1)xu’

+ ;I

,

It is appropriate at this point to mention an exterior transmuta-

tion of [ 4 0 ; 4 1 ] . 2

(

+

(n-1)su’

+ x2 {k2 - - q(x)lu

2 2 x k u which arise in working with formulas u(r,*)

~~-~K(r,s)h(s, -)ds linking solutions h of (An

+ F)u

with 2rn-2K(r,r)

= 0 for F =

=

transmutation P

the map Bef(r)

N

3

../-

s ) ) 1 (where Q, P

K(r,s)

Jr sq(s)ds and setting z(r,s)

THEOREM 5.3. For suitable U

2 k - q(r).

Q and for n

=

3 , 6(s-r)

=

satisfies =

f(r)

+ i(r,s)

+ k2)h $K

B(r,s)

=

P and p1 is the "free" measure arising when

1 has no zeros for Rew > 0 and d p (v) = -(v/rk)Sinvsdu).

0 with solu-

N

=

+ (g(r,s),f(s)) Q

Be{hj(r,*)

(

.

P K for s > r

one can show

K(r,s)Y(s-r)

-

=

=

and ?u

determines a g(u,-k,r),gl(u,-k, =

0 so that f(w,-k)

-

In the "free" case q

= 0

the inversion theory for b is the K-L theory which can be treated in various ways

-

(cf. 1501).

-q

The version which we obtain below in a general manner specializes for m 1 j m = 0 to G(v) = J G(s)Hv(ks)ds and rG(r) = 4 ,imwG(v)Jv(kr)dv. In order to arrive H

0

at a general form of this suppose f(v,-k) has no zeros for Rev > 0 From properties of f (tv,-k,x) and q(*v,k,x)

c(v)du. =

-(l/Zv){f(-v,-k)q(v,k,x)

rf(r)

=

(f(v),f(u,-k,r)>

P

- f(v,-k)lp(-v,k,x)) im * = % L,f(v)f(v,-k,r);(v)dv

so

that dp(v)

=

(note in particular f (v,-k,x)

and f(v,-k,x) is even in v) one has and from this (*) rf(r) =

R. Carroll

98 i m

A

= - ( i / n ) /imVf(v)@(u,k,r)dv

where @ ( v , k , r ) = I p ( v , k , r ) / f ( v , - k )

t h e f r e e c a s e Q 0 = (1rr/2k)'i-('-')

Jv(kr)).

=

?)

THEOREM 5.4.

w e have

$(v)du t h e i n v e r s i o n (*) h o l d s .

If

and in a d d i t i o n B t * l ( v , k , - ) > ( r )

= ( B(r,s),*l(v,k,s))

=

Given

5 and 5,

have continuous s p e c t r a t h e n B is c h a r a c t e r i z e d by B{g ( v , - k , - ) j ( r ) 1 -k,r)

in

Using t h e formal r e l a t i o n (*) - ( i u / n )

IOm ~ ( ~ , k , s ) g ( v , - k , s ) d s / s= 6(V-V) a r i s i n g from a b s o l u t e l y continuous & ( v )

-

( c f . Theorem 3.2

both

.

(6/sl)g(u,

=

(* = @ / r ) .

*(v,k,r)

One can c o n s t r u c t a f o r m a l proof of t h e l a s t r e l a t i o n f o l l o w i n g [ 1 9 1 (using analyt i c c o n t i n u a t i o n ) b u t a s i m p l e r f o r m a l v e r i f i c a t i o n can b e obtained ( c f . 53) by looking a t $(v,s)

(

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

= gl(v,-k,s),

=

a s an e x t e n s i o n of G t o B , so t h a t

gl(v,-k,s)

using the inversion

c"),

Further f o l l m i n g [ 191 t h e equations g(v,-k,r) t) =

(

B(u,t),g(v,-k,u))

and t h e n applying (*) f o r

and g

Q1

.,

1’

and gl(v,-k, * g i v e r i s e t o a G-L e q u a t i o n i n t h e form @ ( r , t ) = ( B ( r , s ) , = (

A ( s , t ) ) where A ( s , t ) = ( gl(v,-k,s),gl(v,-k,t))p’ v

-

t o r above where K arises from B: P

-

Y

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

In connection w i t h t h e B-G opera-

Q w i t h P = x2D2

+

+

M

2xD and Q = P

+ k 2x 2

one

c a n modify t h e c o n s t r u c t i o n l e a d i n g t o Theorem 5.3 and c o n s t r u c t a n i n t e r i o r t r a n s v

, . . I

Q + P i n v e r t i n g B i n t h e form Biu = h = u

mutation Bi:

L i ( x , < ) = 4kx+(x-<)-1?11{k<4(x-<)1?}

-

( c f . [ 40;56;64]

+7 L (x,<)u(L)dE o f

Iv(z) = Jv(iz)i-v).

where Such ker-

n e l s have i n t e r e s t i n g f e a t u r e s i n v o l v i n g g e n e r a t i n g f u n c t i o n s ( c f . 1641). The c o n s t r u c t i o n s of [ 2 2 ; 2 3 ] were phrased t o depend only on c e r t a i n asymp t o t i c behavior i n v of t h e J o s t s o l u t i o n s and t h e r e g u l a r s o l u t i o n ( c f . a l s o [ 601). We show i n [ 251 t h a t t h e p r o c e d u r e s l e a d i n g t o s p e c t r a l measures e t c . in 1 3 ; 2 2 ; 2 3 ]

are a l s o v a l i d when t h e asymptotic behavior i s modeled on t h e Whittaker f u n c t i o n s = a / x (not included i n [ 3;231).

which a r i s e from a Coulomb p o t e n t i a l

We o b t a i n

a v e r s i o n of t h e g e n e r a l i z e d K-L i n v e r s i o n above and t h i s g i v e s a t r a n s f o r m theory

In t h i s d i r e c t i o n w r i t e

f o r c e r t a i n Whittaker f u n c t i o n s which a p p e a r s t o be new. t h e Whittaker e q u a t i o n in t h e form (+) u 2

x2{k2 - (a/x)}u = h u where z = Zikx, a / x and v = tions M

e++ a s

(2)

above

and W

-

cf. [62]).

zz

K =

+ I-% + -a/2ik,

A'

2 2 2 ( K / z ) -I-(k-v ) / z j u = 0; x u" = v

2

= u

(thus

=

For s o l u t i o n s of (9) we have Whittaker func-

Set

K

=

1 as x

+

0) and J o s t s o l u t i o n (e

(2).

-k, and u'

+

i n s o n = a / 2 k and one h a s t h e n a s t h e r e g u l a r

K,V

K,'J

4-1ve(k,x)

s o l u t i o n (x

+

-ikx f L ( k , x )

+

1 as x

+ m)

Same Topics in Transmuintion VLe(k,x) = ( 2 i k ) x - b ikx).

(2ikx) = (-2ik)

in,v

W e set f i ( k , x ) = Win,v(2ikx)

+ Fa = F L

t h e z e r o s of

,., = W(ff,vs)

99

-e- h - i n , v ( - 2 i k x )

and f L ( k , x ) = W

-in,v (-2

and n o t e t h a t ( z = 2ikx and - 2 i k x = zexp(-in))

= F(v,n) = (2k)

-v+'*

v-k i ?'(2v+l)/K'(vf?Ttin)

occur f o r

v + ' d i n = -m (Rev < 0 ) w h i l e fL = .IF(v,n).P(-v,k,x) - F ( - v , n ) p ( v , k , x ) ) / 2 v . f i n e a Whittaker t r a n s f o r m and i t s i n v e r s e by w f ( v ) = i ( v ) = J

0

i m

I

with r f ( r ) =

f(v)fL(k,r)p(v)dv.

0

above w e w r i t e * ( v , k , r )

=

We de-

m

f(s)fL(k,s)ds/s

In o r d e r t o u s e t h e g e n e r a l i z e d K-L t e c h n i q u e j m

V(v,k,r)/F(v,n)

The asymptotic behavior i s s l i g h t l y d i f f e r e n t h e r e b u t t h e methods of

*(v,k,r)dv.

THEOREM 5.5.

[ 3;231 apply completelv and one o b t a i n s

The completeness r e l a t i o n

f o r t h e J o s t s o l u t i o n s i n t h e Coulomb c a s e can b e w r i t t e n 6(r-s) = J

i m

{fL(k,r)/r>

0

{f,(k,s)/sJdp;

r (vq5-h). J

m

0

r\

and o b t a i n r f ( T ) = -(i/a) _Iimvf(v)

2

dp = -2v d u / n f i (v,n)F(-v,n)

=

2iuSinZnvr(v+'~in)dv/nCosn(u-in)

The Whittaker i n v e r s i o n i n K-L f om i s ( f o r f s u i t a b l e ) wf

im f(S)W-~~ ( - ,2~i k s ) d s / s w i t h r f ( r ) = (1/2nk) &,vf

4

A

= f (U) =

(V)

( ~ ) M _ ~ ~ , ~ ( - 2 i(v+%+in)dv/ kr)r N

r (2vfl).

One can a l s o c h a r a c t e r i z e t h e fundamental t r a n s m u t a t i o n s B and B of

c

t h e Q t h e o r y i n terms of Cauchy problems and c o n s t r u c t e x p l i c i t formulas f o r generalized translations.

For example, assuming g ( v , - k , l )

i s r e a l i z a b l e i n examples. U ( r , s )

+

0 on t h e spectrum, which

= ( ;(v)/g(v,-k,l),g(v,-k,r)g(v,-k,s)

)p =

Tif(s)

A

(f(w) = G f ( v ) ) makes s e n s e f o r m a l l y and f o r s u i t a b l e d

Y

t r a n s l a t i o n f o r Q determined by QrU = ( r ( s , n ) , f ( q ) ) where

r(s,n)

2 0 and

U, U ( 1 , s )

= f ( s ) , and DrU(l,s) = C f ( s ) =

(VC =

= ( g(v,-k,s)g(v,-k,n>,Dg(v,-k,l)/g(v,-k,l))p

The "Cauchy" problem i n d i c a t e d i s t o b e c o n s i d e r e d i n two

and Cf(1) = f ' ( 1 ) ) . regions n

<

represents a generalized

q

5 0 r e s p e c t i v e l y by s e t t i n g n

l o g r and 5 = l o g s , from which

=

<

5

-1 -1 p M If(U)l

=

s t a n d a r d uniqueness r e s u l t s can be t r a n s p o r t e d ; t h e d a t a i s given on

In t h e same s p i r i t s e t ;(o) (1127ri)

/p

-u-lA

N

Q,.

Let

v

J m f(p)p'dp 0

=

Mipf(p)} ( f ( p ) =

=

x

n-1

I

.

4

e

m.

is

-

L e t Q and Q,

f ( ~ ) r - ~ - ~ s - ~ - ~ d o / 2 nTHEOREM i. 5.6.

a s above and l e t A and C b e l i n e a r o p e r a t o r s commuting w i t h

be t h e unique s o l u t i o n of

Trv

= clV

(6'

n,

Gl),"(1,s)

=

A f ( s ) , and Dr N

~ ( 1 , s )= C f ( s ) .

<

*

2 f ( o ) d o ) and then a g e n e r a l i z e d t r a n s l a t i o n f o r Pu = ( r u ' ) '

g i v e n f o r m a l l y by U ( r , s ) = be based on A

=

-a

Then B f ( r ) = 9 ( r , l ) determines a t r a n s m u t a t i o n B: Q,

r

V

* Q.

In

p a r t i c u l a r f o r n = 3 l e t U ( t , s ) have t h e form above w i t h ( p , g , e ) r e p l a c e d by (p', 1

R. Carroll

100

N

gl,bl)

Set 7 ( r , s ) =

etc.

(

6(r,t),Ul(t,s))

N

and V ( r , s ) =

(

6(r,t),Ul(t,s))

so t h a t

N

q ( r , l ) = Bf(r) a n d F ( r , l ) = %f(r).

= z(s)

For s u i t a b l e f we o b t a i n e.g. IP(1,s)

=

h.

C f(a),x(s,cr)),

= E f ( s ) = ( f(u),E(s,cr)),

Dr$(l,s)

where f o r m a l l y A(s,o) = ( a ( v , k )

gl(v,-k,s),gl(w,-k.u))

P 1 and r ( s , o )

g(v,-k,l)/gl(v,-k,l);

y(v,k) = Dg(v,-k,l)/g,(v,-k,l)).

corresponding formulas.

.

6 . INVERSE PROBLEMS.

P 1 (a(v,k)

= ( v(v,k)g,(w,-k,s),g,(W,-k,of)

=

Similarly f o r 7 there a r e

L e t u s mention b r i e f l y some r e s u l t s i n [ 3 2 ; 3 7 1 .

One works h e r e w i t h a one dimensional i n v e r s e problem p ( x ) v t t = (11 (x)vxlx w i t h i n i t i a l impulse d i s t u r b a n c e a t t h e o r i g i n and readout a t x = 0 o r x = x

l a r g e enough

(assume p and 11 a r e k n m c o n s t a n t s f o r l a r g e x).

7 (P/u)'dS 0

(with A = (pp)'

=

impedance normalized t o A.

Q(Dy)v, v = 0 f o r t < 0 , vt(y,O)

= 6(y)

In t r a v e l t i m e y =

= A(0) = 1) one h a s v t t

=

( A v ) /A = Y Y

and t h e readout i s

(or v (0,t) = -6(t)), Y

With r e a d o u t v ( 0 , t ) (impulse response)

e i t h e r v ( 0 , t ) = G(t) o r v ( y o , t ) = H(t).

one can determine A(y) by G-L methods f o r example ( c f . 119;28;29;32;61;67-69;72;

731) and i n [ 32 1 i t was shown t h a t 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 t h e s p e c t r a l measure

t o reflection data)

-

$cGr(y+x)

.

= 1 / 2 ~ r l c =~ /(2/n) ~ JmG(t)SinAtdt/A and ( f o r G = 1

( d e n s i t y ) :(A)

Gr(y-x))

=

0

THEOREM 6.1.

+ Gr -

The G-L e q u a t i o n has t h e form K(y,x)

-

4 1Y K(y,s){G:(~+x)

Gr r e f e r s

+

G k ( l s - x J ) ) d s f o r x < y (and A-'(y)

=

0

1 - K(y,y)).

In p a r t i c u l a r i f

s t a b i l i t y estimate l l A ~ ( ~ , - ) l l ",Y s i d e r e d "transmission" =

Hl(t)

= G*

-

Y

II

In s t a n d a r d n o t a t i o n (7;

(2Am/7r)lHll

'.

Fff where f f ( t )

. -+

m)

",2Y

.

small then t h e r e is a 1 L (2Y)

-F M I I E ' I I

Y

1

L (2Y)

1,

I n 1 3 7 1 we con-

d a t a H(t) = v ( y o , t ) f o r yo s u f f i c i e n t l y l a r g e and s e t H ' ( t ) b

when y

Gr w i t h 11 E ' I I

< 2N {IIEII

-

(H = 0 f o r t < yo, A Z H = Y(t-yo)

THEOREM 6.2.

A2eixy

E

one h a s l / c (-A)

4

=

+ h(t), c Q (A)@':

L A'H'

= 6(t-yo)

+ cQ

= 2 ~ e x p ( - i A y o1 ) (~A )

T h i s can b e e x p l o i t e d i n s e v e r a l ways.

+ hl(t)).

Then.

w i t h GQ normalized a s

A

(2,

=

FH1)

and ;(A)

F i r s t note that

=

IH"1 l 2 =

H (t+T)H (T)dT i s an a u t o c o r r e l a t i o n t y p e f u n c t i o n . Consequent1 yo l y w i t h s u i t a b l e m a n i p u l a t i o n and i n t e r p e r t a t i o n G ' ( t ) = A,fi'ft). This leads t o = Jm

s t a b i l i t y e s t i m a t e s based d i r e c t l y on h and hl on [ y o , - ) .

Another i n t e r e s t i n g

formula a r i s e s by some F o u r i e r t y p e manipulation which shows a f i n i t e i n t e r v a l deY0+t pendence of G' and H' v i a G' (t) = J H (T)K(t-T)dT where however K ( t - r ) = yo

Some Topics in Transmutation 1,

101

m

(A:/2n)

imexp(-iX (t+yo--t)>*,9 (0)dX

contains an unknown factor @:(O).

Another re-

sult in [ 371 allows one to construct an extended G-L equation (which looks promising for numerical computation) in the form H ( T ) - K(y where G(yo,~) = $tG(T+yo)

,T) =

G(yo,’)

+ QK(y,,-)

* G’

+ G(T-Y~)~. We remark that although connections between

autocorrelation ideas and spectral measures are known via the theory of Krein strings (see e.g. the book "Gaussian processes, function theory, and the inverse spectral problem" by H. Dym and H. McKead, the technique above based on the transmutation machine provides a new link directly connected with a geophysical problem, This appears promising in connecting various approaches in geophysics involving time series analysis and the inverse problem.

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3.

G . Burdet, M. Giffon, and

4.

R.

5.

R. Carroll, Applicable Anal., 8 (19791, 253-263

6.

R.

7.

R. Carroll, Osaka Jour. Math., 1 9 (19821, 815-831

8.

R.

9.

R. Carroll, Applicable Anal., 1 2 (1981), 153-157

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35. R. C a r r o l l and J. G i l b e r t , Math. Annalen, 258 (1981), 39-54 3 6 . R. C a r r o l l , A survey of t r a n s m u t a t i o n t h e o r y , t o a p p e a r

37. R. C a r r o l l and F. Santosa, Some t r a n s m u t a t i o n methods i n geophysics. Proc. Conf. I n v e r s e S c a t t e r i n g , Univ. Tulsa, 1983, t o a p p e a r 38. K. Chadan and P . S a b a t i e r , I n v e r s e problems i n quantum s c a t t e r i n g t h e o r y , S p r i n g e r , N.Y., 1977 39. H. Chebli, J o u r . Math. P u r e s Appl.,

58 (1979), 1-19

40. D. Colton and R. Kress, Quart. J o u r . Appl. Math., (1979), 53-62

31 (19781, 9-17 and 32

41. D, Colton and W. Wendland, Proc. Royal SOC. Edinburgh, 75A (1975/76), 97-107

42. L. Fadeev, Uspekhi Mat. Nauk, 14 (1959), 57-119 4 3 . L. Fadeev, Sov. Prob. Mat., 31 (1974), 93-180

Some Topics in Transmutation 44. M. Flensted-Jensen,

Ark. Mat.,

103

19 (1972), 143-162

45. M. Gasymov, Trudy L e t . sk. Spekt. T e o r i i Oper., I z d . E l m , Baku, 1975, pp. 20-45 46. R. G i l b e r t , Function t h e o r e t i c methods i n p a r t i a l d i f f e r e n t i a l e q u a t i o n s , Academic P r e s s , N.Y., 1969 47. R. G i l b e r t , SIAM J o u r . Math. Anal., 48. V. Hutson, Jour. Math. Mech.,

1 (1970), 96-114

1 4 (1965), 807-819

49. V. Hutson and J. Pym, Proc. London Math. SOC., 24 (1972), 548-576 50. D. Jones, J o u r . I n s t . Math. Appl.,

26 (1980), 133-141

51. I. Kay and H. Moses, I n v e r s e s c a t t e r i n g p a p e r s , 1955-1963, Math. S c i . P r e s s , 1982 52. T. Koornwinder, Ark. Mat.,

1 3 (1975), 145-159

53. B. L e v i t a n , The t h e o r y of g e n e r a l i z e d t r a n s l a t i o n o p e r a t o r s , I z d . Nauka, Moscow, 1973 54. B. L e v i t a n , Uspekhi Mat. Nauk, 6 (1951), 102-143 55. J . Lions, B u l l . SOC. Math. France, 84 (1956), 9-95 56. J. L o e f f e l , Ann. I n s t . H. Poincar:,

8 (1968), 339-447

57. V. Marrenko, Sturm-Liouville o p e r a t o r s and t h e i r a p p l i c a t i o n s , I z d . Nauk. Dumka, Kiev, 1977 58. H. Moses, Jour. Math. P h y s i c s , 2 1 (1980), 83-89 59. M. Naimark, L i n e a r d i f f e r e n t i a l o p e r a t o r s , I z d . Nauka, Moscow, 1969 60. D. Naylor and P. Chang, SIAM J o u r . Math. Anal.,

13 (1982), 1053-1071

61. R. Newton, Geophys. J o u r . Royal A s t r . SOC., 65 (1981), 191-215 62. R. Newton, S c a t t e r i n g t h e o r y of waves and p a r t i c l e s , McGraw-Hill, N.Y.,

1966

63. R. Newton, J o u r . Math. P h y s i c s , 21 (1980), 493-505 and 1698-1715 64. P. S a b a t i e r , J o u r . Math. P h y s i c s , 8 (19671, 1957-1972 65. P. S a b a t i e r , Comptes Rendus Acad. S c i . P a r i s , 278 (1974), 545-547 and 603-606 66. P . S a b a t i e r , J o u r . Math. P h y s i c s , 1 2 (1971), 1303-1326 67. F. Santosa, T h e s i s , Univ. I l l i n o i s , 1980 68. F. Santosa, Geophys. J o u r . Royal A s t r . S O C . , 70 (1982),

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DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

105

AN EQUATION MODELING THE ELECTRIC BALLAST RESISTOR

Nathaniel Chafee School of Mathematics Georgia I n s t i t u t e of Technology Atlnata, Georgia 30332 U.S.A.

W e imagine a s t r a i g h t segment of very t h i n wire lying along the u n i t i n t e r v a l of an x-axis. This wire i s surrounded by a gas whose temperature has a c e r t a i n fixed value a > 0, and heat flows from t h e wire i n t o t h e g a s o r viceversa. A t the same time, an e l e c t r i c c u r r e n t I i s passing through t h e wire from one end t o the other, and, because of the e f f e c t s of e l e c t r i c a l r e s i s t a n c e , t h i s c u r r e n t I generates heat within the wire. We a r e i n t e r e s t e d i n the r e s u l t i n g d i s t r i b u t i o n of temperature u along t h a t wire. W e s h a l l regard u a s a function u ( x , t ) of the position x , 0 2 x 5 1 , along t h e wire and of the time t w i t h 0 ~t < +a. I n a l l t h a t follows we s h a l l work in a regime where u > 0. The device we have just described -- wire, c u r r e n t , and ambient gas -- i s sometimes called an e l e c t r i c b a l l a s t r e s i s t o r , and over the years i t has been t h e object of several s c i e n t i f i c i n v e s t i g a t i o n s , both experimental and t h e o r e t i c a l . Indeed, two e a r l y i n v e s t i g a t i o n s a r e reported i n [4] and 121, and several Much o f the t h e o r e t i c a l work on [5], and [3]. recent studies appear i n [1,6], t h e b a l l a s t r e s i s t o r concerns the nature of behavior of temperature d i s t r i b u t i o n s u ( x , t ) which might appear in t h e given wire. This present paper i s i n t h a t same vein. S p e c i f i c a l l y , we a r e going t o report here two new r e s u l t s concerning the boundedness of temperature d i s t r i b u t i o n s u ( x , t ) a s the time t approaches +m. Our f i r s t task i s t o formulate s u i t a b l e mathematical r e l a t i o n s governing u. To begin, we shall r e q u i r e t h a t u s a t i s f y the following equations:

Here, r ( u ) denotes the r e s i s t i v i t y of our given wire, t h a t i s , i t s e l e c t r i c a l r e s i s t a n c e per u n i t length. g ( u ) denotes the time-rate per u n i t length of wire a t w h i c h heat flows from the wire i n t o the gas. As t h e notation i n d i c a t e s , we a r e assuming t h a t both r ( u ) and g ( u ) a r e functions o f u and of u alone. Later, we shall impose precise conditions on these functions r ( u ) and g ( u ) .

N. Chafee

106

I n Eq. ( l c ) t h e q u a n t i t y + ( x ) s i g n i f i e s an i n i t i a l d i s t r i b u t i o n o f temperat u r e along t h e given w i r e , and as ( l c ) i n d i c a t e s we r e q u i r e t h a t $ ( x ) be p o s i t i v e a t a l l x i n [0,1]. Eqs. ( 1 ) a r e i n c o m p l e t e i n one v i t a l r e s p e c t : t h e manner i n which I i s r e l a t e d t o u, x, and t.

we have n o t y e t s p e c i f i e d We s h a l l now r e n d e r such a

r e l a t i o n s h i p i n t h e form o f the f o l l o w i n g hypothesis: c o n s t a n t V,

There e x i s t s a p o s i t i v e

independent o f x, t, o r u, such t h a t 1

V

=

I i

(21

r(u(x,t})dx.

0 W i t h r e g a r d t o o u r b a l l a s t r e s i s t o r , V i s t o be i n t e r p r e t e d as t h e v o l t a g e o r d i f f e r e n c e i n e l e c t r i c p o t e n t i a l g o i n g f r o m one end x = 0 o f t h e w i r e t o t h e o t h e r end x = 1. Eq. ( 2 ) i s a s t a t e m e n t o f Ohm's Law, t o g e t h e r w i t h an assumpt i o n t h a t I has no x-dependence. The h y p o t h e s i s we have j u s t i n t r o d u c e d in c o n n e c t i o n w i t h ( 2 ) embodies a r e q u i r e m e n t t h a t t h e v o l t a g e V be h e l d f i x e d w i t h r e s p e c t t o t h e t i m e t and t h e temperature d i s t r i b u t i o n u. A t t h i s j u n c t u r e , we f i n d i t c o n v e n i e n t t o i n t r o d u c e a f u n c t i o n a l p[u] setting

by

1 p [u] =

i r(u(x,t))dx.

(3)

0

C l e a r l y , ~ [ u ]r e p r e s e n t s t h e t o t a l e l e c t r i c a l r e s i s t a n c e o f f e r e d by t h e g i v e n w i r e i n t h e presence o f t h e t e m p e r a t u r e d i s t r i b u t i o n u ( x , t ) .

On t h e b a s i s o f ( Z ) , ( 3 ) , and o u r h y p o t h e s i s o f c o n s t a n t v o l t a g e V, we can now r e w r i t e ( 1 ) as

F o r l a c k o f a b e t t e r term, we c a l l ( 4 a ) an i n t e g r o - d i f f e r e n t i a l e q u a t i o n o f p a r a b o l i c type.

Eq. ( 4 a ) , t o g e t h e r w i t h (4b,c), i s o u r mathematical model f o r t h e b a l l a s t r e s s t o r , and t h e r e m a i n i n g p o r t i o n o f t h i s paper i s an i n v e s t i g a t i o n o f Eqs. ( 4

.

H e n c e f o r t h , d u r i n g t h a t i n v e s t i g a t i o n , we s h a l l suppose t h a t

t h e f u n c t i o n s g and r i n Eq. (4a) s a t i s f y t h e f o l l o w i n g hypotheses.

fH1)

2 and r a r e C -smooth f u n c t i o n s mapping t h e i n t e r v a l [O,+m)

into

(-m,*t.

(H2) r i s p o s i t i v e on [O,+m)

and i t s d e r i v a t i v e r ' i s non-negative on

[O,+m).

(H3) The d e r i v a t i v e g ' i s p o s i t i v e on n e g a t i v e on

[O,+m).

[O,+m)

and t h e d e r i v a t i v e g" i s non-

At1

107

Equation Mudcliiig ihc Electric Ballast Resistor Indeed, a i s

0 such t h a t g ( a ) = 0.

(H4) There e x i s t s a u n i q u e number a

t o be regarded as t h e t e m p e r a t u r e o f t h e ambient gas mentioned s e v e r a l paragraphs above. We s h a l l l e t X denote t h e space o f a l l c o n t i n u o u s f u n c t i o n s @: [0,1]

+

and on X we s h a l l impose t h e usual Co-supremum norm, which we h e r e

(-m,+m),

denote by

[I 11.

Under

11 11

o f c o u r s e X becomes a Banach space. We s h a l l a l s o 1 impose an a u x i l i a r y norm on X, namely, t h e L - i n t e g r a l norm, and we h e r e d e n o t e sisting o f a l l @

+

.

11 11,

t h a t norm by

F i n a l l y , we s h a l l l e t Xo denote t h e h a l f - s p a c e i n X con-

X such t h a t @ ( X I > 0 on 0

t

5

x

5 1.

+

By a s o l u t i o n o f

Now w i t h r e f e r e n c e t o ( 4 c ) , l e t @ be any element i n Xo. Eqs. ( 4 ) we s h a l l mean a r e a l valued f u n c t i o n u = u ( x , t ) t h e f o r m [0,1]

x

[O,s),

with 0 < s

p o z i t i v e everywhere on [ O , l ]

x

(+m,

such t h a t :

d e f i n e d on a domain of

( i ) u i s c o n t i n u o u s and

[O,s); (ii) t h e p a r t i a l d e r i v a t i v e s ux, uXX, u t

x (0,s); ( i i i ) u s a t i s f i e s Eq. (4a) e x i s t and a r e c o n t i n u o u s everywhere on [ O , l ] x (0,s); ( i v ) u s a t i s f i e s Eqs. ( 4 b ) and ( 4 c ) on t h e i n t e r v a l s everywhere on [ O , l ]

5 x 5 1 respectively.

0 < t < s and 0

U s i n g arguments o f t h e s o r t a p p e a r i n g i n [7], element $

E

+ Xo,

one can p r o v e t h a t , f o r any

Eqs. ( 4 ) have a u n i q u e s o l u t i o n u = u ( x , t )

d e f i n e d on a domain o f t h e f o r m [0,1]

x

= u(x,t;@)

[ O , s ( @ ) ) w i t h 0 < s($) <

+m.

well Now we can

s t a t e o u r f i r s t main r e s u l t . Theorem 1.

L e t V i n Eq. ( 4 a ) be any p o s i t i v e c o n s t a n t .

number v > 0, t h e r e e x i s t s a c o n s t a n t M1 > 0 such t h a t , i f @ %then ]lu(*,t;@)]/l

<

M~ f o r a l l t

t

X L and if 11$11,

[o,s(@)).

By h y p o t h e s i s , t h e two numbers r ( 0 ) and g ' ( a ) a r e each p o s i t i v e .

Proof.

Hence, g i v e n v > 0 we can d e f i n e a number

M, Now choose any f u n c t i o n $ u(x,t)

Then, f o r any E

= u(x,t;$)

= v + a +

E

M 1 > 0 by s e t t i n g

v 2r ( o ) - ' g f ( a ) - ' .

(5)

+

Xo and c o n s i d e r t h e c o r r e s p o n d i n g s o l u t i o n u =

o f Eqs. ( 4 ) .

We s u b s t i t u t e t h i s s o l u t i o n i n t o Eq. ( 4 a ) and t h e n i n t e g r a t e b o t h s i d e s

o f t h a t e q u a t i o n w i t h r e s p e c t t o x f r o m x = 0 t o x = 1.

Taking i n t o account

Eqs. ( 4 b ) and ( 3 ) we a r r i v e a t t h e r e l a t i o n I

2 d ' - J udx 5 - 1 g ( u ) d x + V p [ u ] - l dt 0 0 From Hypotheses (H?), ( H 3 ) , r(0).

(0 < t <

(H4), i t f o l l o w s t h a t g ( u )

S($)).

2 g'(a)(u-a)

and P[U]

Hence,

7

d I dt 0

1 udx

5

-g'(a)

J udx 0

f

2 g'(a)a + V r(O)-l

(0 < t < s($))

2

<

.h! Chafee

I08 and, t h e r e f o r e , 1

J udx 5 e x p [ - g ' ( a ) t ] 0

1 J @dx + g ' ( a ) - l { g ' ( a ) a 0

(0 < t <

t V2r(0)-'}

(6)

S(@)).

The r i g h t hand member o f (6) i s less t h a n M, i n ( 5 ) f o r a l l t

E

Eo,S(@)).

On t h e o t h e r hand, we know t h a t u i s p o s i t i v e everywhere on i t s domain. t h e l e f t hand member of ( 6 ) i s equal t o ~ ~ u ( ~ , t ; @ ) f[o~r l a l l t y i e l d s the inequality llu(*,t;@)ll

< Ml

E

Hence,

[O,s($)).

This

f o r a l l t c [ O , s ( @ ) ) , and t h u s our

1

p r o o f i s complete.

Theorem 1 g i v e s us a bound on an L 1 - i n t e g r a l norm of s o l u t i o n s u o f Eqs. ( 4 ) . We s h o u l d l i k e t o develop a c o r r e s p o n d i n g bound on a C"-supremum norm o f such s o l u t i o n s u. We reason as f o l l o w s . L e t us c o n s i d e r t h e c l a s s i c a l h e a t e q u a t i o n ut = uxx on t h e u n i t i n t e r v a l o f t h e x - a x i s w i t h t h e boundary c o n d i t i o n s ux = 0 a t x = 0,l

and w i t h t h e i n i t i a l

c o n d i t i o n u = @(x) a t t = 0, where @ i s any g i v e n f u n c t i o n i n X.

This i n i t i a l -

boundary v a l u e problem generates a s t r o n g l y c o n t i n u o u s semigroup { T ( t ) } o f bounded l i n e a r o p e r a t o r s t a k i n g

X

into itself.

t i o n f o r t h e h e a t e q u a t i o n [8, pp. 30-32,

Using t h e w e l l known source s o l u -

122-1311, i t i s easy t o c o n s t r u c t t h e

f o l l o w i n g e x p l i c i t r e p r e s e n t a t i o n f o r C T ( t ) 1:

(I$

€

x;

0 < t <

0 < x < 1).

fm,

From t h i s r e p r e s e n t a t i o n of { T ( t ) } we can draw two c o n c l u s i o n s p e r t i n e n t t o

our p r e s e n t l i n e o f reasoning. IIT(t)$li Second, i f $ x

E

[0,1]

$

E

Xo.

E

5

F i r s t , t h e r e i s a c o n s t a n t K > 0 such t h a t

Kt-'"

ll@lll

(4

X and if $ ( x ) < 0 for all x

and a l l t

t

E

E

[O,l],

(7)

then [ T ( t ) @ ] ( x ) < 0 for all = u(x,t;$)

of E

5.

(4) w i t h

S u b s t i t u t i n g t h i s s o l u t i o n i n t o ( 4 a ) , we r e w r i t e ( 4 a ) as

-

u t = uxx + { g ( o ) Next, we choose any number tl formu 1a

0 c t < 1).

[O.+-).

Now, l e t us c o n s i d e r any s o l u t i o n u = u ( x , t ) t

X;

E

g ( u ) } + { v 2 [ u ~ - ~ r ( u -) g ( 0 ) I . [O,s(@))

and t h e n n o t e t h e v a r i a t on o f c o n s t a n t s

109

An Equation Modeling the Electric Ballast Resistor

( 0 5 tl 5 t <

S(@)).

-

By v i r t u e of Hypotheses (H3) and (H4) we have Cg(0)

t.

Therefore, T(t-T){g(O) u(.,t)

-

g ( u ( * , ~ ) ) l 5 0 f o r tl

5

50

g(U(.,T)} T

5

t.

t

5 T(t-tl)u(*,tl)

+ f

f o r tl

5T

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

-

T(t-T){V2p[U(*,T)]-2r(U(b,T))

g(0)Id.r (8 1

t1

5 tl 5 t <

(0

S(@)).

With t h e a i d o f t h i s i n e q u a l i t y , we can p r o v e t h e f o l l o w i n g theorem. Theorem 2.

L e t V i n Eq.

( 4 a ) be any p o s i t i v e c o n s t a n t .

number v > 0, t h e r e e x i s t s a c o n s t a n t M > 0 such t h a t , if$

v, t h e n l l u ( * , t ; @ ) / l < M f o r a l l t

E

:X

and if 11 $

11

<

[O,s(@)).

We choose o u r number v > 0.

Proof.

Then, f o r any E

Using arguments o f t h e sort employed

by Pazy i n [ 7 ] one can p r o v e t h a t t h e r e i s a number h, 0 < h < 1, such t h a t , i f @

t

E

Xo and if I/ $

I/

<

~ for all t v, t h e n s ( $ ) > h and ~ ~ u ( - , t ; @ <) ~2v

E

[O,h].

L e t V be as i n t h e hypotheses o f o u r p r e s e n t theorem and l e t K be as i n ( 7 ) Also, w i t h

above.

L,

as i n t h e p r e c e d i n g paragraph, l e t

M, be as i n Theorem 1.

Finally, l e t

t

We choose any element @

E

responding s o l u t i o n u = u(x,t) the r e l a t i o n \lr(u(-,t)f\\,

Xo such t h a t

11 @ I/

= u(x,t;@) o f Eqs.

= p[u(*,t)],

Next, we choose any number t

E

< v and we c o n s i d e r t h e c o r -

(4).

valid for a l l t

S t r a i g h t a w a y , we n o t e E

[O,s($f).

[ h , s ( @ ) ) and t h e n we s e t tl = t - h .

Using

< M. ( 7 ) , ( 8 ) , ( 9 ) , as w e l l as Theorem 1, one can show t h a t l l u ( * , t ; $ ) l l Thus, we have ~ ~ u ( ~ , t ; <@ M) ~f o~r e v e r y t E [ h , s ( $ ) ) . But, we a l r e a d y know It f o l and by ( 9 ) we have 2v < M. t h a t l l u ( - , t ; @ ) l l < 2v f o r e v e r y t E [O,h], lows t h a t \ l u ( * , t ; $ ) l / c M f o r a l l t E [O,s(@)), and o u r p r o o f i s complete. Theorem 2 e s t a b l i s h e s t h e boundedness o f s o l u t i o n s f o r Eqs. ( 4 ) w i t h r e s p e c t t o an a p p r o p r i a t e C"-supremum norm, and t h i s i s t h e t a s k we s e t o u r s e l v e s earlier.

TO conclude t h i s paper, we want t o p o i n t o u t a s t r a i g h t f o r w a r d b u t

i m p o r t a n t c o r o l l a r y o f Theorem 2. C o r o l l a r y 2.1.

L e t V i n Eq. ( 4 a ) be any p o s i t i v e c o n s t a n t .

Then, f o r any

N. Chafee

110

+

element @ E Xo, we have s ( @ ) = +=; t h a t i s , u ( x , t ; @ ) i s well defined a t every point ( x , t ) lying i n [0,1] x [o,+-).

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D. Bedeaux, P. Mazur, and R. A. Pasmanter, The b a l l a s t r e s i s t o r ; an e l e c t r o thermal i n s t a b i l i t y i n a conducting wire I ; the nature of t h e s t a t i o n a r y s t a t e s , Physica 86 A (1977), 355-382.

2.

H. Busch, Eber d i e erw'drming von drahten i n verdunnten gasen durch den e l e k t r i s s e n storm, Ann. Physik 64 (19211, 401-450.

3.

N . Chafee. The e l e c t r i c b a l l a s t r e s i s t o r : homoseneous and nonhomogeneous equi l i b r i a , Nonlinear Differential Equations : Invariance, Stabi 1 i ti, and Bifurcation, ed. by P. de Motoni and L. Salvadori, Academic Press, New York, 1981, pp. 97-127.

4.

R. P. Gifford and N . C. Page, Experiments with B a l l a s t s of a Nernst Lamp, B.S. Thesis, MIT, Cambridge, Mass., 1902.

5.

R. Landauer, The b a l l a s t r e s i s t o r , Phys. Rev. A

6.

R. A. Pasmanter, D. Bedeaux, and P. Mazur, The b a l l a s t r e s i s t o r ; an e l e c t r o thermal i n s t a b i l i t y in a conducting wire 11; f l u c t u a t i o n s around homogeneous s t a t i o n a r y s t a t e s , Physica 90 A (1978), 151-163.

7.

A. Pazy, A c l a s s of semi-linear equations of evolution, I s r a e l J . Math. 20 (1975), 23-36.

8.

D. V. Widder, The Heat Equation, Academic Press, New York, 1975.

(3)

15 (1977),

2117-2119.

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

111

NONEXISTENCE OF POSITIVE SOLUTIONS FOR SINGULAR HYPERBOLIC DIFFERENTIAL INEQUALITIES C.Y.

Chan

Department of Mathematics and S t a t i s t i c s U n i v e r s i t y of Southwestern L o u i s i a n a Lafayette, Louisiana U.S.A.

A Sturmian theorem l e a d i n g t o n o n e x i s t e n c e o f p o s i t i v e s o l u t i o n s i s e s t a b l i s h e d f o r s i n g u l a r h y p e r b o l i c d i f f e r e n t i a l i n e q u a l i t i e s under mixed boundary c o n d i t i o n s . F o r i l l u s t r a t i o n , an example i s g i v e n . The r e s u l t i s t h e n extended t o a coupled system o f q u a s i l i n e a r s i n g u l a r h y p e r b o l i c d i f f e r e n t i a l i n e q u a l i t i e s s u b j e c t t o mixed coupled boundary c o n d i t i o n s . 1.

INTRODUCTION

R e c e n t l y , Chan C21, and Chan and Young 161 s t u d i e d e x i s t e n c e and uniqueness of Cauchy problems f o r s e m i l i n e a r two-dimensional Euler-Poisson-Darboux e q u a t i o n s w i t h a r e a l parameter k > 0. I n p a r t i c u l a r , t h e l a t t e r paper d e a l t w i t h nonn e g a t i v e s o l u t i o n s . The main purpose here i s t o e s t a b l i s h comparison r e s u l t s f o r n o n e x i s t e n c e o f p o s i t i v e s o l u t i o n s f o r i n i t i a l mixed boundary v a l u e problems i n v o l v i n g n-dimensional s i n g u l a r h y p e r b o l i c d i f f e r e n t i a l i n e q u a l i t i e s f o r k 2 0. R e s u l t s o f a s i m i l a r n a t u r e f o r t h e p a r a b o l i c case were g i v e n by Chan and Young [51, and more r e c e n t l y f o r time-dependent m a t r i x and v e c t o r d i f f e r e n t i a l systems by Chan C11. R e l a t e d r e s u l t s f o r l e s s general h y p e r b o l i c problems were g i v e n by K r e i t h [71, T r a v i s C81, and Chan and Young C3; 41. I n s e c t i o n 2 , we s t u d y n o n s e l f a d j o i n t h y p e r b o l i c d i f f e r e n t i a l i n e q u a l i t i e s ; t h e r e s u l t o b v i o u s l y h o l d s f o r t h e s e l f a d j o i n t case. An example i s c o n s t r u c t e d t o i l l u s t r a t e t h e r e s u l t . I n s e c t i o n 3, we e x t e n d t h e r e s u l t t o a coupled system o f quasilinear nonselfadjoint hyperbolic d i f f e r e n t i a l inequalities.

2.

NONEXISTENCE

L e t D be a bounded n-dimensional domain i n t h e r e a l n-dimensional E u c l i d e a n space, ..., un) t h e outward u n i t normal t o aD. A l s o , l e t x = (xl, x2, x3, ..., x n ) , R = D x (0, -), S = aD x D- i t s c l o s u r e , aD i t s boundary, and ( v ~ ~, 2 ~, 3 ,

k Lu : utt + --u t t

-

DiCa..(x, 1J

t ) D . u l + 2bi(x, J

t)Diu + c ( x , t ) u ,

$t - DiCA..(x,

t ) D . v l + 2Bi(x, t)Div + C(x, t ) v , J where Di denotes d i f f e r e n t i a t i o n w i t h r e s p e c t t o xi, and t h e r e p e a t e d n d i c e s a r e t o be summed f r o m one t o n. The m a t r i c e s ( a i j ) and ( A i j ) a r e symmetri and p o s i t i v e d e f i n i t e w i t h elements b e l o n g i n g t o Mv : vtt +

1J

c'(G)

n c(R-).

The mixed boundary c o n d i t i o n s f o r u and v a r e g i v e n r e s p e c t i v e l y by a..v.D.u + p(x, t ) u 1J 1 J

=

0

on S,

(2.1)

C.Y. Chan

I12

A..v.D.v + P(x, t ) v = 0 on S , 1J 1 J where - m < p < +m w i t h p ( y , t o ) = +m d e n o t i n g u ( y , t o ) = 0. aD i s p i e c e w i s e smooth, u and v a r e i n

C2(R)

n

(2.2) I t i s assumed t h a t

cl(n-),

and f o r ( x , t ) i n R, b i , B i , c, C , p and P a r e c o n t i n u o u s . Since ( A i j ) i s p o s i t i v e d e f i n i t e , t h e r e e x i s t s a c o n t i n u o u s f u n c t i o n g 1 ( c f . 111) such t h a t

where A = ( A ' j ) , B i s t h e column v e c t o r ( B i ) , and BT denotes t h e t r a n s p o s e o f B. S i m i l a r l y , i$( a j . - A - j ) i s p o s i t i v e d e f i n i t e , we can choose a c o n t i n u o u s f u n c t i o n g 2 such { h a t + o r a l l r e a l (n + 1 ) - t u p l e s ( y l , y2, y3, . . , Yn + the quadratic form

.

2

-

i s positive definite. (2.4) A . . ) Y . Y . + 2yn + l(bi - Bi)yi + g2yn + 1J 1 J I f f o r i = 1, 2 , 3, n , B i a r e i d e n t i c a l l y zero, t h e n g 1 can be chosen t o be i d e n t i c a l l y zero; s i m i l a r l y , g2 can be chosen t o be i d e n t i c a l l y z e r o if bi 5 Bi. (a..

1J

...,

By d i r e c t computation, we can d e r i v e t h e f o l l o w i n g i d e n t i t y :

+(uMv - vLu) =

-

- v a 1J ..D.u)1 t 1 V 1 J J J v 2 C A ~ ~ D ~ ( + ) D+~ z(+)B~D~(:) ($ + gl(+)

-

[(a..

+

(C

+ -w) k - D.C!(UA..D.V

$wt

-

1J

where w = uvt

-

c

A . .)(Diu)(D.u) 1J J +

91

+

+ 2u(bi

-

2

1

Bi)Diu

2

+ g2u 1

2 g2)u ,

(2.5)

vut.

The f o l l o w i n g r e s u l t a s s e r t s t h a t v must v a n i s h somewhere

on R-.

THEOREM 1. Assume ( a 1J . . - A i . ) i s p o s i t i v e d e f i n i t e . L e t g 1 and g 2 s a t i s f y ( 2 . 3 ) Also, l e t u s a t i g f y Lu 5 0 i n a, t h e boundary c o n d i t i o n (2.1) such and ( 2 . 4 ) . t h a t u ( x , 0 ) = 0, and u > 0 i n n. I f

JD ( C - c

+ g1 + g 2 ) u

JaD( p - P)u2 dS

20

2

dx 5 0

f o r t > 0,

f o r t > 0 when p #

(2.6) +m,

(2.7)

<

t h e n every f u n c t i o n v s a t i s f y i n g Mv 0 i n n, and t h e boundary c o n d i t i o n ( 2 . 2 ) cannot be p o s i t i v e t h r o u g h o u t n-, p r o v i d e d ut(x, 0) = 0 when k = 0. P r o o f . Suppose v > 0 t h r o u g h o u t R-. t > 0, we have

I n t e g r a t i n g t h e i d e n t i t y (2.5) o v e r 0 with

u v - l ( u A . . ~ . D . v - v a . . v . D . u ) dS (2.8) 1J 1 J 1.I 1 J by u s i n g (2.3), (2.4), ( 2 . 6 ) and t h e d i v e r g e n c e theorem. L e t us c o n s i d e r t h e JD t-k uv -1( t kw ) dx ~

113

Nonexistence of' Positive Solutions

boundary i n t e g r a l on t h e r i g h t hand s i d e o f ( 2 . 8 ) . I f p = +m a t a p o i n t on S, t h e n u = 0 t h e r e ; t h i s i m p l i e s t h a t t h e i n t e g r a n d i s z e r o t h e r e . Ifp # +-, t h e n f r o m t h e boundary c o n d i t i o n s ( 2 . 1 ) and ( 2 . 2 ) , t h e i n t e g r a n d i s ( p - P)u2, and hence t h e boundary i n t e g r a l i s n o n n e g a t i v e by ( 2 . 7 ) . Thus,

fD t- k uv -1 ( t kw ) dx ~ 20

for t

0.

k F o r each t > 0, t h e r e e x i s t s a subdomain Dt of D such t h a t ( t w ) 2~ 0. S i n c e tkw i s z e r o a t t = 0, we have t k w 0 i n Dt f o r t > 0. I t f o l l o w s from 2 w = -v ( u / v )

t

that

This c o n t r a d i c t s t h e hypothesis Because u/v = 0 a t t = 0, we have u/v 5 0 i n Dt. t h a t u > 0 i n R and t h e assumption t h a t v > 0 on X . Thus, v cannot be p o s i t i v e t h r o u g h o u t n-. From t h e p r o o f o f t h e theorem, we n o t e t h a t i f p = +m on S, t h e n t h e boundary c o n d i t i o n ( 2 . 2 ) on v, and t h e h y p o t h e s i s (2.7) can be o m i t t e d ; furthermore, if f o r i = 1, 2, 3, ..., n , b i - B i a r e i d e n t i c a l l y zero, t h e n t h e theorem i s v a l i d even when a i j E A j j . L e t us g i v e an example t o i l l u s t r a t e t h e above theorem and remarks. Example 1.

Let

Lu

=

Utt

Mv z v tt

- uxx - (1 + 2 t - Z ) u , 0 < x <

TI,

0

<

t,

-

TI,

0

<

t.

vxx

-

2(1 + t-2)v, 0 < x <

The f u n c t i o n u = t 2 s i n x s a t i s f i e s t h e hypotheses o f Theorem 1 w i t h p :+- and k = 0. Since (2.6) i s s a t i s f i e d , i t f o l l o w s t h a t e v e r y f u n c t i o n v s a t i s f y i n g MV 2 0 , 0 < x <

TI,

0 < t,

cannot be p o s i t i v e t h r o u g h o u t t h e r e g i o n g i v e n by 0 5 x 5

TI,

0 5 t.

For example,

2 t . v = t e sinx satisfying

t Mv = 4 t e s i n x > 0, 0 < x

<

TI,

0 < t,

i s n o t p o s i t i v e throughout t h e region. 3.

A COUPLED QUASILINEAR SYSTEM

Here, we extend t h e r e s u l t o f t h e p r e c e d i n g s e c t i o n t o a coupled system o f q u a s i l i n e a r s i n g u l a r hyperbolic d i f f e r e n t i a l i n e q u a l i t i e s subject t o mixed coupled boundary c o n d i t i o n s . Below, we g i v e a t y p i c a l r e s u l t . Let

+ +kJ

-

D.[a. . ( x , t, u ) D . u l + 2bi(x, 1 1J J + c ( x , t, u ) u + r ( x , t, v ) v ,

L*u : u

tt

t

t

t , u)Diu

114

C. Y. Chan

+ C(X, t, V ) V - R(x, t,

U)U,

where t h e m a t r i c e s ( a i j ) and ( A i j ) a r e symmetric and p o s i t i v e d e f i n i t e w i t h n, b i , B i , c, C, continuously d i f f e r e n t i a b l e elements, and f o r i = 1, 2, 3, r and R a r e continuous. Using the same technique i n t h e proof o f Theorem 1, we have the f o l l o w i n g r e s u l t .

...,

THEOREM 2. L e t u and v s a t i s f y the i n e q u a l i t i e s L*u 5 0 and M*v in n, and the boundary c o n d i t i o n s aijviDju

+ p ( x , t ) u + q(x, t ) v < 0

0 f o r (x, t )

on S,

A..u.D.v - Q(x, t)u + P(x, t ) v 0 on S 1J 1 J r e s p e c t i v e l y , where p and P a r e continuous, and q and Q a r e continuous and nonnegative, such t h a t u(x, 0) = 0, and u > 0 i n a. If ( a i j - A i j ) i s p o s i t i v e d e f i n i t e , r and R are nonnegative, c o n d i t i o n (2.7) holds, and t h e r e e x i s t g 1 and 92 such t h a t ( 2 . 3 ) , (2.4) and (2.6) hold, then v cannot be p o s i t i v e throughout R-, provided ut(x, 0 ) = 0 when k = 0. REFERENCES S i n g u l a r and unbounded m a t r i x s o l u t i o n s f o r b o t h time-dependent m a t r i x and v e c t o r d i f f e r e n t i a l systems, J . Math. Anal. Appl. 87 (1982) 147157. A l t e r n a t i n g bounds converging t o the s o l u t i o n o f a semilinear Chan, C.Y., Euler-Poisson-Darboux equation, i n : Lakshmikantham, V . (ed.), Proceedings o f t h e Vth I n t e r n a t i o n a l Conference on “Trends i n Theory and P r a c t i c e o f Nonlinear D i f f e r e n t i a l Equations” h e l d a t t h e U n i v e r s i t y o f Texas a t A r l i n g t o n , June 14-18, 1982. Comparison theorems f o r a coupled system of Chan, C.Y. and Young, E.C., s i n g u l a r h y p e r b o l i c d i f f e r e n t i a l i n e q u a l i t i e s . I . Time-independent uncoupling c o e f f i c i e n t s , A t t i Accad. Naz. L i n c e i Rend. C1. S c i . F i s . Mat. Natur. 66 (1979) 250-254. Chan, C.Y. and Young, E.C., Comparison theorems f o r a coupled system of s i n g u l a r h y p e r b o l i c d i f f e r e n t i a l i n e q u a l i t i e s . 11. Time-dependent c o e f f i c i e n t s w i t h mixed coupled boundary c o n d i t i o n s , A t t i Accad. Naz. L i n c e i Rend. C1. Sci. F i s . Mat. Natur. 66 (1979) 361-364. Chan, C.Y. and Young, E.C., Sturmian theorems f o r degenerate p a r a b o l i c i n e q u a l i t i e s , J . Math. Anal. Appl. 74 (1980) 560-567. Chan, C.Y. and Young, E.C., Maximal and minimal nonnegative s u l u t i o n s of a semilinear Euler-Poisson-Darboux equation, J . Math. Anal. Appl. 90 (1982) 209-212. K r e i t h , K., Sturmian theorems f o r h y p e r b o l i c equations, Proc. Amer. Math. SOC. 22 (1969) 277-281. T r a v i s , C.C., Comparison and o s c i l l a t i o n theorems f o r h y p e r b o l i c equations, U t i l i t a s Math. 6 (1974) 139-151.

c11 Chan, C.Y., 121

L31

C41

[51 161 C71 [81

DIFFERENTIAL EQUATIONS

I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

ALMOST PERIODICITY OF BOUNDED SOLUTIONS TO NONLINEAR ABSTRACT EQUATIONS

J. A. G o l d s t e i n

C. Corduneanu

Department o f Mathematics The U n i v e r s i t y o f Texas a t A r l i n g t o n A r l i n g t o n , Texas 76019

*

Department o f Mathematics Tulane U n i v e r s i t y New Orleans, L o u i s i a n a 70118

The paper i s devoted t o t h e problem o f Bohr-Neugebauer f o r t h e e q u a t i o n s ( 1 ) and (2), i n a H i l b e r t space H, i . e . , p r o v i d i n g c o n d i t i o n s under which a bounded s o l u t i o n i s necess a r i l y a l m o s t p e r i o d i c . The p r o o f s o f t h e main r e s u l t s a r e based on t h e s o l u t i o n o f some d i f f e r e n t i a l i n e q u a l i t i e s , v a l i d on t h e e n t i r e r e a l a x i s . INTRODUCTION L e t H b e a k ! i l b e r t space o v e r t h e r e a l s , and c o n s i d e r t h e a b s t r a c t d i f f e r e n t i a l equations i n H (1 1

ut

Au + F ( t , u ) ,

utt = Au + F ( t , u ) ,

(2) where A below.

=

and

F

t E R, t

R,

s t a n d f o r some o p e r a t o r s whose p r o p e r t i e s w i l l be s p e c i f i e d

It i s known t h a t ( 2 ) can be a l s o w r i t t e n i n t h e form ( 1 ) , b u t i n t h e p r o d u c t space H x H. The reason we pay s p e c i a l a t t e n t i o n t o t h e e q u a t i o n ( 2 ) comes from t h e f a c t t h a t i t s r e d u c t i o n t o t h e f o r m ( 1 ) changes t h e b a s i c p r o p e r t i e s of t h e i n v o l v e d o p e r a t o r s , making i t i m p o s s i b l e t o d e r i v e r e s u l t s f o r ( Z ) , u s i n g t h o s e obtained f o r ( 1 ) .

The b a s i c problem we a r e g o i n g t o d i s c u s s i n t h i s paper i s whether boundedness o f a s o l u t i o n t o ( 1 ) o r ( Z ) , on t h e whole a x i s , i m l i e s i t s a l m o s t p e r i o d i c i t (Bohr-Bochner a.p.), under t h e assumption t h a h u ) i s a l m o s t p e r i o d i c Y i n t ( n o t n e c e s s a r i l y i n t h e Bohr-Bochner sense) L e t us p o i n t o u t t h a t t h e problem f o r m u l a t e d above has been t h o r o u g h l y i n v e s t i g a t e d d u r i n g t h e l a s t s i x decades. Indeed, a f t e r a f i r s t s o l u t i o n was g i v e n by Bohr and Neugebauer i n 1926 ( s e e [5], [ll],[14]) f o r t h e f i n i t e dimensional l i n e a r case, t h e problem a t t r a c t e d t h e i n t e r e s t o f many r e s e a r c h e r s . Both f i n i t e dimensional and i n f i n i t e - d i m e n s i o n a l cases have been i n v e s t i g a t e d , t h e concent r a t i o n on t h e l a t e r becoming e v i d e n t d u r i n g t h e l a s t two decades (see [l],[14]). The i n i t i a l problem f o r m u l a t e d by Bohr and Neugebauer has been t h u s g e n e r a l i z e d t o p a r t i a l d i f f e r e n t i a l e q u a t i o n s , and a good deal o f r e s u l t s has accumulated. Besides ['I J and [1$], t h e f o l l o w i n g r e f e r e n c e s c o n t a i n n o t a b l e c o n t r i b u t i o n i n t h i s r e g a r d : [ 2 ] , [4], [ l o ] , [12], [13], [15]. F u r t h e r r e f e r e n c e s can be found i n t h e books o r papers quoted above. Our aim i s t o o b t a i n c o n d i t i o n s under which t h e Bohr-Xeugebauer problem has a p o s i t i v e s o l u t i o n , i n case o f e q u a t i o n s ( 1 ) and ( 2 ) . The b a s i c t o o l s a r e two In this q u a l i t a t i v e r e s u l t s on d i f f e r e n t i a l i n e q u a l i t i e s , g i v e n i n [ 6 ] and [7]. approach, t h e case o f monotone o p e r a t o r s ( i n a s t r o n g sense) i s covered. T h i s

*

P a r t i a l l y supported by an NSF g r a n t .

C Cordurieanu and J. A. Goldsteiri

116

case received p a r t i c u l a r a t t e n t i o n during the l a s t years (see [4]. [12], [13], [14], [15]). B u t operators which do n o t belong t o t h i s c l a s s can a l s o be d e a l t with. Several applications t o p a r t i a l d i f f e r e n t i a ? equations w i l l be emphasized. In t h i s paper, t h e e x i s t e n c e o f aboundedsolution (on R ) t o ( 1 ) or ( 2 ) i s assumed, and only those properties for A and F will be required which guarantee the almost p e r i o d i c i t y of the bounded solution. In a b s t r a c t framework, t h i s problem has been a l s o investigated in [3], [16]. MAIN RESULTS Let us s t a t e now t h e conditions concerning operators A and F , occurring in equations ( 1 ) and ( 2 ) . These properties will be assumed f o r both equations ( 1 ) and ( 2 ) .

I.

A: D(A)

(31 where for r 11.

$:

s a t i s f i e s a monotonicity type condition of t h e form

H

->

[O,-)

->[O,m)

and

$(r)

>

0

0.

>

F: R

x

H

H ->

i s such t h a t (F(t,x)-F(t,Y), x-Y) 2 - $(IIX-YII), V t t R , x,yEH,

(4)

with

(Ax-AY, X - Y ) 1. ~ ( ( I x - Y I]), v x , Y E D ( A ) , i s continuous, and such t h a t $ ( O ) = 0 ,

JI:

[0,m)

->

[O,m)

continuous, and

$ ( r ) 2 0.

111. The map t -> F ( t , - ) , from R i n t o H, i s (Bohr-Bochner) almost periodic f o r every fixed x E H, uniformly on every bounded s e t of H.

IV.

The functions

$(r)

and

q(r) a r e such t h a t

$ ( r ) = o ( r ) as

(5)

r

->

O+,

E >

0,

while the g r e a t e s t root o f the equation (7)

say

r

= r(c),

$ ( r ) = q ( r ) + Er, v e r i f i e s t h e condition

(8)

lim

r(E)

=

0 as

E

->

0,

We can now s t a t e t h e following r e s u l t in regard t o the equation ( 1 ) : CCJYLLLWL equm5un IT), ~ ~ d ~.~nump&Ltioa e h 1 - JV. 16 U : R -> H LA boLu2ion 0 6 ( 7 I , bounded o n R, then u & (Bohh-Bochlzeh) &nont pehiodie. 16 F in pehiadic i n t w d h pehiod T > 0 , t h a t u 0 d o p e h i odic, with .the name pehiod.

TheoEiem I

1-

Proof.

Let

c E

R

be a fixed number, and denote

V ( t ) = I l U ( t + T ) - U ( t ) l I , V t E R. (9) From ( l ) , and the s i m i l a r equation we obtain replacing t a f t e r t h e i r subtraction and s c a l a r multiplication by v ,

(10)

by

t

+ T,

one derives

Almost Ptriodicity for Nonlinear Equations

i f one takes i n t o account conditons I number E > 0 i n (10) i s defined by

-

117

I11 f o r t h e operators A

and

F.

The

E = Sup{llF(t+T,u(t))-F(t,u(t))/l; t E RIB (11 1 and is f i n i t e because of our assumption I11 on F, and because u i s bounded on R. Of course, E can be made a r b i t r a r i l y small, provided one chooses T among the E-almost-periods of F.

The i n e q u a l i t y (10) f o r Namely, (10) implies

v

can be d e a l t with by using the lemma given i n [7].

v ( t ) = IIu(t+T)-u(t)ll 5 r ( E ) , v t E R, (12) from which one obtains the almost p e r i o d i c i t y of the solution u. Indeed, taking i n t o account t h e condition (8) f o r r ( E ) , a s well a s formula (11) t h a t defines E, t h e a s s e r t i o n follows immediately. The periodic case i s obvious, on behalf of the above considerations. Assume t h a t instead of I and 11, one supposes:

Corollary 1 . 1'.

A: D ( A )

(3' 1 where 11'. (4'

i s such t h a t

(Ax-AY,x-Y) 5 - $ ( I \ X - Y \ ~v) ,X, Y s a t i s f i e s the conditions described i n I .

@

F: R

1

with

H

-->

x

H

->

H

E

D(A),

i s such t h a t

$

( F ( t , x ) - F ( t , Y ) , x - ~ ) 5 $(IIX-Y\I)~ V t E R, V x, y E H , s a t i s f y i n g t h e same conditions as in 11.

Then, under conditions I ' , 1 1 ' , 111, and IV, t h e v a l i d i t y o f the conclusion of Theorem 1 i s assured. Indeed, a l l reduces t o previous case by the change of variable

t

=

- s, s

E

R.

-

a1

Corollary 2. A kery special case of Theorem 1 can be obtained assuming A monotone f o r some CY > 0 , and F ( t , x ) f F ( t ) .

Then any bounded (on R ) solution Sf u t = Au + F ( t ) is (Bohr-Bochner) almost periodic: indeed, l e t $ ( r ) = a r and + ( r ) 3 0 in Theorem 1. An application o f t h i s corollary t o a system of two equations i n in the sequel.

H

will be given

Let us now s t a t e a r e s u l t of almost p e r i o d i c i t y f o r the equation ( 2 ) .

Conn.ideh e q u a t t u n ( Z ) , undm answnpfiunh 7 - IU. 14 u: R -> H (6-ttLongI doLLLtion od "21, bounded an R, t h e n u .LA iBohh-&achneti) &ant peniadic. Id F A p o i o d i c i n t , w i t h p d a d T 0, t h w u 0 & a peniodic, w i t h t h e name p o i a d .

Tkeafim 2

Proof. Let us take T E R , and define v ( t ) m o u n d e d solution of ( 2 ) . One obtains f o r

by formula ( 9 ) , where u ( t ) i s v ( t ) the following inequality: EV,

where

E > 0 shown above.

i s defined by ( 1 1 ) .

a

v t

E

R,

I t i s a f i n t e number f o r the same reasons

The general case of an inequality of the form

1 3 ) , valid on the whole real a x i s ,

118

C Corduneanu and J.A. Goldstein

has been discussed i n [6]. I f one a p p l i e s t h e lemma i n [ 6 ] then an e s t i m a t e o f t h e form ( 1 2 ) f o r I I u ( t + T ) - u ( t ) [ l i s o b t a i n e d , which l e a d s immediately t o t h e conc l u s i o n o f t h e Theorem 2. L e t us p o i n t o u t t h a t c o n d i t i o n s ( 5 ) and ( 6 ) guarantee the existence o f t h e function r = r ( E ) f o r a l l p o s i t i v e E s u f f i c i e n t l y small. The p e r i o d i c case a l s o f o l l o w s f r o m t h e above scheme s i n c e ( 1 3 ) i s v a l i d f o r e v e r y E > O ( f r o m ( l l ) , one would o b t a i n E = 0; w i t h E > 0 one s t r e n g t h e n s t h e inequality). C o r o l l a r 3. I f A -7-5 and F t , x F(t), a1rnos p e r iod ic . 5

i s such t h a t A - a1 t h e n any bounded (on

i s a monotone o p e r a t o r f o r some C( R) s o l u t i o n o f t h e e q u a t i o n (2) i s

>

0,

t

Indeed, choosing @ ( r ) = obviously v e r i f i e d .

and

ar2

$ ( r ) E 0, t h e c o n d i t i o n s o f Theorem 2 a r e

COMMENTS The c o n d i t i o n s I - I V , i n b o t h Theorems 1 and 2 i m p l y t h e uniqueness o f t h e bounded s o l u t i o n on R. L e t us prove t h i s a s s e r t i o n i n t h e case ( 2 ) . Assuming u1 and u2 a r e two does n o t have t o be bounded s o l u t i o n s (on R ) o f e q u a t i o n ( 2 ) , i n which F ( t , x ) n e c e s s a r i l y almost p e r i o d i c , t h e n v = JIu1-u211 s a t i s f i e s t h e f o l l o w i n g d i f f e r ential inequality

_ - d2

v2 + ( v ) - $ ( v ) , V t E R. dt2 I f E > 0 i s an a r b i t r a r y number, then ( 1 4 ) i m p l i e s (13). Consequently, II u1 ( t ) - u 2 ( t ) II < r ( E ) , V t E R, which proves t h e uniqueness.

(14)

On a h a l f - a x i s , [4], [9], [12].

b o t h e q u a t i o n s ( 1 ) and ( 2 ) have i n f i n i t e l y many bounded s o l u t i o n s The a s y m p t o t i c b e h a v i o r has been i n v e s t i g a t e d i n [4] and [12].

Another comment we would l i k e t o make i s concerned w i t h t h e use o f weaker assumpt i o n s o f almost p e r i o d i c i t y on F which w i l l s t i l l guarantee t h e v a l i d i t y o f Theorems 1 and 2. These assumptions, i n which Bohr-Bochner almost p e r i o d i c i t y i s r e p l a c e d by Stepanov a l m o s t p e r i o d i c i t y , appear i n [41, [121. [141, [151. In o r d e r t o o b t a i n r e s u l t s s i m i l a r t o those o f Theorems 1 and 2 u s i n g t h e Stepanov almost p e r i o d i c i t y we would need adequate g e n e r a l i z a t i o n s o f t h e lemmas i n [ 6 ] A g e n e r a l i z a t i o n a l o n g t h e s e l i n e s appears i n [12] f o r f i r s t o r d e r and [7]. i n e q u a l i t i e s , and t h e e s t i m a t e s a r e o b t a i n e d o n l y on a h a l f - a x i s . W i t h o u t a i m i n g a t t h e g r e a t g e n e r a l i t y , we s h a l l i n d i c a t e h e r e an example o f a n o n l i n e a r second o r d e r i n e q u a l i t y f o r which i t i s p o s s i b l e t o o b t a i n an e s t i m a t e 0, t E R , be bounded such t h a t i n v o l v i n g t h e Stepanov norm. Namely, l e t v ( t )

2

__ >

(15) where

a > 0, f: R ->

R

CYV - I f ( t ) l & , t E R, dt2 i s l o c a l l y i n t e g r a b l e , and

.t+l

Then, t h e r e e x i s t s a c o n s t a n t

K = K(a)

>

0,

independent o f

f,

such t h a t

(17) The p r o o f can be demonstrated a l o n g t h e same l i n e s as i n [ 8 ] where t h e l i n e a r

119

Almost Periodicifj' for Nordincar Equations c o u n t e r p a r t has been discussed. By means o f such e s t i m a t e s one can e a s i l y p r o v e t h e a l m o s t p e r i o d i c i t y o f t h e in bounded s o l u t i o n s assuming o n l y Stepanov t y p e almost p e r i o d i c i t y f o r F ( t , x ) cases ( 1 ) o r ( 2 ) .

APPLICATIONS L e t us c o n s i d e r t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n s (see [8]) ut = A u f f ( t , x , u ) , t 6 R, x E R C Rn, (18) (19) where

+ Au

utt

= f(t,x,u),

t E R, x E R C R n , S = an.

i s a bounded domain w i t h smooth boundary

R

We s h a l l assume t h a t f : R x SL x R -> R (20) 2 i s c o n t i n u o u s t o g e t h e r w i h i t s d e r i v a t i v e fu, and i s L -almost p e r i o d i c ( i . e . , as a map f r o m R i n t o L (a), u n i f o r m l y w i t h r e s p e c t t o u E R ) . Moreover, we assume t h a t

8 .

(21 1 where

fu 5 u < il,(t,x,u) X1

>

E

R

x

5x

R,

i s t h e s m a l l e s t e i g e n v a l u e o f t h e problem

0

AV

(22)

+ xv

= 0

in

R, v I S = 0.

Under t h e above mentioned assumptions, any C(2)-sol u t i o n u ( t ,x) o f t h e e q u a t i o n ( 1 8 ) , v a n i s h i n g on t h e boundary o f Q , i n w h i c h t h e r e e x i s t s M > 0 w i t h

L

u (t,x)dx

(23) 2

i s L -almost periodic. $ ( r ) = XlrZ

and

$(r) =

5M,

V t E

R,

Indeed t h i s f o l l o w s f r o m Theorem 1 b y l e t t i n g 2

.

pr

A c o m p l e t e l y s i m i l a r r e s u l t h o l d s i n t h e case o f t h e e l l i p t i c e q u a t i o n ( 1 9 ) . d e t a i l s see [ 8 ] . H,

We s h a l l now c o n s i d e r a system o f two e q u a t i o n s i n

(24) where u > 0 conditions:

U'

f

AU

BV + g(t,v)

f

-

= 0, V '

i s a c o n s t a n t , and t h e o p e r a t o r s

namely

BU + vv = 0,

A,B,g

s a t i s f y the following

b)

A i s a l i n e a r o p e r a t o r on H , w i t h A - a1 monotone f o r some B i s a n i n v e r t i b l e s e l f - a d j o i n t o p e r a t o r on H.

c)

g: R

a)

x

H

->

H

a > 0.

s a t i s f i e s a uniform L i p s c h i t z condition, w i t h constant

(25) and i s almost p e r i o d i c i n

IIg(t,x)-g(t,Y)ll t,

For

Imllwl

m > 0

I

u n i f o r m l y w i t h r e s p e c t t o t h e second argument.

Moreover, t h e f o l l o w i n g c o n d i t i o n w i l l be needed: d)

The c o n s t a n t s

(26 1

a , ~ , and

m are subject t o m2 < 4av.

L e t us p o i n t o u t t h a t t h e case v = 0 i n ( 2 4 ) l e a d s t o t h e second o r d e r e q u a t i o n 2 w" f A w ' f B w f g(t,Bw) = 0, (27) which can be viewed as d e s c r i b i n g n o n l i n e a r o s c i l l a t i o n s i n presence o f a f r i c t i o n

C Corduneanu and J.A. Goidstein

120

force with (nonlinear) damping.

The system ( 2 4 ) can be rewritten as a s i n g l e equation in t h e product space H w i t h usual s c a l a r product generated by t h e product o f H . Indeed, i f

U

=

col(u,v),

and A ( t =

[;

:I’

=

B1 + B2

B2 =

f

B3

f

G(t,-),

x

H

where

[-;’j! =I: 4’ B3

and G ( t , U ) = c o l ( g ( t , v ) , O ) , then 24) becomes (28) U’ + A(t)U = 0.

Summing up ( 2 9 ) (33)

-

(32) one obtains ( A ( t ) U - A ( t ) U ,U-i)

IJ

where =

Because o f ( 2 6 ) B

rnin{a

m - --, 28

( ( U - i 112,

2

v

- 2

>

0.

can be chosen i n ( 3 2 ) t o s a t i s f y

The inequality ( 3 3 ) gives the monotonicity of A(t) which allows f o r a d i r e c t application of Theorem 1. Therefore, under conditions a ) , b ) , c ) , and d ) , the system (24) s a t i s f i e s t h e Eohr-Neugebauer property: each bounded (on A ) solution i s almost periodic. REFERENCES [l]

Amerio, L. and Prouse, G . : Almost Periodic Functions and Functional Equat i o n s (Van Nostrand-Reinhold, New York, 1978).

[2]

B i r o l i , M. S u r l e s s o l u t i o n s born6es o u presque-p6riodiques des 6quations multivoques sur un espace de Hilbert, Ricerche Mat. 21 (1972) 17-47.

[31

Boles, B. and Tsend, L . , A generalized Bohr-Neugebauer theorem, Differential Equations 8 (1972) 1031-1035.

On the weak asymptotic almcst p e r i o d i c i t y of bounded solutions of u" E Au + f , f o r monotone A , J . Math. Analysis Appl. 37 (1980) 309-31 7. Corduneanu, C . Almost Periodic Functions, (John Wiley and Sons, New York, 1968).

c41 Bruck, R. E . , [5]

Almost Periodicity f o r Nonlinear Equations

121

[6]

Corduneanu, C. Bounded and almost p e r i o d i c s o l u t i o n s o f c e r t a i n n o n l i n e a r e l l i p t i c e q u a t i o n s , TGhoku Math. J o u r n a l 32 (1980) 265-278.

[7]

Corduneanu, C. Bounded and a l m o s t p e r i o d i c s o l u t i o n s o f c e r t a i n n o n l i n e a r p a r a b o l i c e q u a t i o n s , L i b e r t a s Mathematica I1 (1982) 131-139.

[8]

Corduneanu, C. Almost p e r i o d i c s o l u t i o n s t o n o n l i n e a r e l l i p t i c and p a r a b o l i c e q u a t i o n s , N o n l i n e a r A n a l y s i s , TMA 7 (1983) 357-363.

191 Corduneanu, C. and G o l d s t e i n , J . A. t o nonlinear abstract equations, ( i n preparation).

[lo]

P e r i o d i c and almost o e r i o d i c s o l u t i o n s

Dafermos, M. Almost p e r i o d i c processes and a l m o s t p e r i o d i c s o l u t i o n s o f e v o l u t i o n e q u a t i o n s , A. R. Bednarek and L. Cesari (eds), Dynamical Systems, (Academic Press, I n c . , New York, 1977).

[ll] F i n k , A. M. Almost P e r i o d i c D i f f e r e n t i a l Equations, L e c t u r e Notes i n Mathem a t i c s , No. 377, ( S p r i n g e r - V e r l a g , B e r l i n , 1974).

[12]

Haraux, A . N o n l i n e a r E v o l u t i o n Equations, L e c t u r e Notes i n Mathematics, No. 841 , ( S p r i n g e r - V e r l ag , B e r l i n , 1981 )

[13]

Haraux, A. G e n e r a l i z e d a l m o s t p e r i o d i c s o l u t i o n s and e r g o d i c p r o p e r t i e s o f quasi-autonomous d i s s i p a t i v e systems, J. D i f f . Equations 48 (1983) 269-279.

[14]

L e v i t a n , B. M. and Zhikov, V . V. Almost P e r i o d i c F u n c t i o n s and D i f f e r e n t i a l Equations, (Cambridge Univ. Press, Cambridge, 1982).

[15]

Pankov, A. A. Bounded and a l m o s t p e r i o d i c s o l u t i o n s o f e v o l u t i o n a r y v a r i a t i o n a l i n e q u a l i t i e s , Math. USSR S b o r n i k 36 (1980)519-533.

[16]

Zaidman, S. Remarks on d i f f e r e n t i a l e q u a t i o n s w i t h Bohr-Neugebauer p r o p e r t y , J . Math. Anal. Appl. 38 (1972) 167-173.

.

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DIFFERENTIAL EQUATIONS I.W. Knowles and R.T.Lewis (Editors) QElsevier Science Publishers B.V. (North-Holland), 1984

123

A PRIOR1 ESTIMATES IN NONLINEAR EIGENVALUE PROBLEMS FOR ELLIPTIC SYSTEMS

Chris Cosner Department of Mathematics and Computer Science University of Miami Coral Gables, FL. 33124

The systems considered have the form A; = ):(?A in R , -+ u = 0 on an, where u : R + 7Rm, f ; !Rm + m m , R C 7Rn is a bounded domain, A is a matrix of second order elliptic operators, and X is a real parameter. For simplicity the results are stated for a single equation, but the range of validity for systems is discussed. The first type of a prior1 estimates give lower bounds for supl;l in terms of X and IRI when f is superlinear, upper bounds for supl
A

2

2

A

A great deal of study over many years has been given to the linear eigenvalue problem (1)

nu + hu u = o n

=

0 in n in aR

where R S l R is a smooth, bounded domain. One common question is that of tinding bounds for those values of A which permit a nontrivial solution of (1). To obtain upper bounds for the first eigenvalue is simple, at least in theory, because of its characterization as the minimum of the Rayleigh quotient. Lower bounds are more difficult: but many results are known. The best known is the Faber-Krahn inequality [ 4 ] , [ 8 1 , [ 9 ] , which states that the first eigenvalue for (1) on R is at least as large as that for the n-ball with n-volume equal to that of R. Other bounds have been obtained by various investigators; see [ 7 ] , 1121 , 1131, [17]. Of course it is not possible to bound the eigenfunctions for (l), because any multiple of an eigenfunction is also an eigenfunction. In the case of the nonlinear analog to (11, that is, (2)

Au + Xf(u) = 0 u = o

in R on ao,

124

C. Cosner

where A i s a r e a l p a r a m e t e r , t h e s i t u a t i o n i s much d i f f e r e n t . f ( u ) = up, p > 1, L e v i n s o n [lo] t i o n s f o r a l l X E (0,~).On t h e t h e r e may b e r e l a t i o n s between solution. The p r e s e n t work i s

When

showed t h a t ( 2 ) h a s p o s i t i v e s o l u o t h e r h a n d , f o r s o m e c h o i c e s of f , X and t h e s i z e of t h e a s s o c i a t e d concerned w i t h f i n d i n g such r e l a t i o n s .

F o r s i m p l i c i t y , t h e r e s u l t s of t h e n e x t s e c t i o n w i l l be s t a t e d f o r ( 2 ) w i t h n 2 3 ; however, most r e m a i n t r u e w i t h only minor m o d i f i c a t i o n s f o r t h e c a s e o f v e c t o r f u n c t i o n s u = ( u' , . . . , u r n ) s y s t e m s of t h e form (3)

q[ f

B=

( a q y ( x , u ) ux. R) x . i,]=1 1 J

+

+

n

+

bq'(x,u)u:

satisfying

caB(x,u)uR]

i

i=l

XfCl(x,u) = 0

i n R, a = l , . . . , m u = 0 on 2 8 , p r o v i d e d t h a t f o r ri = ( q q ) E ' I R ~ ~< , = either

€!Elm,

X E R and u € T R m ,

o r t h e n-volume of R i s s u f f i c i e n t l y s m a l l and

f o r some c o > O .

I n g e n e r a l , s t r o n g l y c o u p l e d s y s t e m s s u c h a s ( 3 ) do

n o t s a t i s f y a maximum p r i n c i p l e , so methods b a s e d on t h e maximum p r i n c i p l e d o n o t a p p l y . The d e t a i l s of t h e a n a l y s i s f o r t h e g e n e r a l c a s e ( 3 ) are g i v e n i n [21. Some i n e q u a l i t i e s a r e needed f o r t h e a n a l y s i s . t h e f i r s t e i g e n v a l u e f o r (1) on R; t h e n i f u

where p 0 i s t h e f i r s t e i g e n v a l u e f o r - A

E

First,

let u,(R)

be

W k J 2 (Sl) I

on t h e u n i t b a l l i n 'IR",

wn

t h e n-volume of t h e u n i t b a l l , and IRI i s t h e n-volume of R. T h i s combines t h e Faber-Krahn i n e q u a l i t y w i t h R a y l e i g h ' s c h a r a c t e r i z a t i o n Another i m p o r t a n t t o o l i s t h e S o b o l e v i n e q u a l t i t y : f o r of pl(R). p* = 2 n / ( n - 2 ) ,

125

A Priori Estimates in Nonlinear Eigenvalue Problems

b e t h e u s u a l norm on t h e Sobolev s p a c e W k r p ( Q ) , s t a n d a r d Lp i n t e r p o l a t i o n i n e q u a l i t i e s a p p l i e d t o ( 6 ) and ( 7 ) y i e l d

f o r u ~ W ~ ' ~ ( f i1 ) <, p < p * , where I < ( n , p , O ) 0 f o r p =p*.

+

0 as ( R /

+

0 except

The b a s i c r e s u l t i s as f o l l o w s : Lemma 1: Suppose t h a t u i s a n o n t r i v i a l smooth s o l u t i o n t o ( 2 ) f o r some g i v e n A , where f ( u ) i s smooth a n d (9)

/f(u)I < folul

for s o m e y

E

2 I

(0, (n+Z)/(n-Z)

7.

Then -

Proof:

M u l t i p l y i n g by u i n ( 2 ) and i n t e g r a t i n g by p a r t s y i e l d s

it follows t h a t

By ( 8 ) and ( 9 ) ,

(Note t h a t 1 < q + l 5 p*, s i n c e 0 < q f o l l o w s immediately. The i n e q u a l i t y of q .

5

(n+2)/(n-2) .)

(10) h a s v a r i o u s i m p l i c a t i o n s , d e p e n d i n g on t h e v a l u e

Theorem 1 ( L i n e a r g r o w t h ) : solution, then P1(Q)/fo

Proof:

When q = 1, t h e t e r m

K(n,2,Q) = l/ul(fi)

If

q = 1 i n ( 9 ) and ( 3 ) h a s a n o n t r i v i a l

5 A.

(12)

11 u I I o,q-lq+l

i n ( 1 0 ) becomes 1, and

by ( 6 ) .

Theorem 2 ( S u p e r l i n e a r r o w t h ) : I f 1 < q 5 ( n + 2 / ( n - 2 ) ( 3 ) h a s t h e n o n t r i v i a l z o l u t i o n u f o r some A # O , t h e n

(13)

sup

R

I'I

I n e q u a l i t y (10)

(t) (k) 1/ ( q - 1 )

1/ ( q + l )

(

i n (91,

1 foK(n, q+l,R)

and

C Cosner

126

Also,

I n e q u a l i t i e s (14) and

(15) y i e l d ( 1 3 ) .

Theorem 3 ( S u b l i n e a r g r o w t h ) : I f 0 < q < 1 i n ( 9 ) a n d ( 3 ) h a s t h e n o n t r i v i a l solution II f o r s o m e T # otfien ,

f o r some c o n s t a n t M ( n , q , R . f o ) . Proof:

H e r e M w i l l denote a generic constant.

From (lo),

The u s u a l e l l i p t i c a p r i o r i e s t i m a t e s t h e n i m p l y t h a t s i n c e u s a t i s f i e s (3),

L e t po = 1 + l / q ;

t h e n a p p l y i n g t h e S o b o l e v embedding t h e o r e m t o ( 1 9 )

yields

S e t t i n g pk+l = n p k / ( n - 2 p k )

>

[n/(n-Z)]p

and i t e r a t i n g t h e arguments

k i n i n e q u a l i t i e s ( 1 8 )- ( 2 0 ) y i e l d s (16) v i a t h e u s u a l " b o o t s t r a p "

process, s i n c e w2'p(n) Remarks:

c

CO(Q)

for p sufficiently large.

Theorems 1 a n d 2 e x t e n d t o s y s t e m s s u c h a s ( 3 ) with only

minor c h a n g e s .

Theorem 3 a l s o e x t e n d s i f t h e L o e f f i c i e n t s a?$, b y a ,

A Priori Estimates in Nonlinear Eigenvalue Problems

127

and c a b do n o t depend on u. Some r e s u l t s s i m i l a r t o Theorem 2 have been o b t a i n e d by S c h a e f e r and S p e r b 1141 f o r Au + Xup = 0 ; however , t h e i r r e s u l t s are based on t h e maximum p r i n c i p l e and do n o t e x t e n d t o s y s t e m s s u c h a s ( 3 ) . An o p t i m a l v a l u e f o r t h e c o n s t a n t C ( n ) i n (7) h a s been o b t a i n e d by T a l e n t ? 1151; u s i n g t h a t bound it i s p o s s i b l e t o give a n e x p l i c i t e x p r e s s i o n f o r K(n,p,R) when 2 5 p 5 2 n / ( n - 2 ) . These methods c a n a l s o b e a p p l i e d t o c e r t a i n h i g h e r o r d e r e q u a t i o n s and s y s t e m s . V a r i o u s e x t e n s i o n s and g e n e r a l i z a t i o n s of t h e r e s u l t s p r e s e n t e d h e r e , a l o n g w i t h p r e c i s e e x p r e s s i o n s f o r s o m e of t h e c o n s t a n t s , and a d d i t i o n a l t e c h n i c a l d i s c u s s i o n and r e f e r e n c e s , a r e g i v e n i n L2]. The above e s t i m a t e s d o n o t g i v e upper bounds f o r s u p l u l i n t h e super-

R

l i n e a r case, so t h e y a r e n o t s u i t a b l e u s e i n e x i s t e n c e t h e o r y . There a r e f e w if a n y r e f e r e n c e s i n t h e l i t e r a t u r e t o s u c h bounds f o r s y s t e m s , b u t f o r some s y s t e m s less g e n e r a l t h a n ( 3 ) i t i s p o s s i b l e t o g e n e r a l i z e c e r t a i n r e s u l t s f o r s i n g l e e q u a t i o n s . Suppose t h e system has t h e f o r m (21)

~

"

u =

+u f "~( u ) o on a n

= 0, i n R a = I,

...,m

ff

where t h e o p e r a t o r s L are s e c o n d o r d e r and u n i f o r m l y e l l i p t i c , s u c h t h a t La and La* s a t i s f y t h e maximum p r i n c i p l e , and t h a t f " 2 0 f o r a l l a p r o v i d e d ua > O for a l l a. Then one t y p e of estimate s u i t a b l e f o r u s e i n o b t a i n i n g t h e e x i s t e n c e of n o n n e g a t i v e n o n t r i v i a l s o l u t i o n s t o (21) i n t h e s u p e r l i n e a r c a s e i s a n a p r i o r i bound from above on t and s u p l u l f o r s o l u t i o n s to (22)

~~u~

u

= 0

i f a ( u ) iti"

on a R ,

a

=

u '0

o

i n R, a = l , . ..,m i n R, a = l , . . . , m

where (I' i s t h e f i r s t e i g e n f u n c t i o n o f La*, s u i t a b l y n o r m a l i z e d . Such bounds f o r a s i n g l e e u a t i o n have been o b t a i n e d by B r e z i s and T u r n e r [l], P . L . Lions 1119, and Gidas and Spruck L6], among o t h e r s . ( I n t h e case of s y s t e m s i t i s h i g h l y d e s i r a b l e t o a v o i d t h e req u i r e m e n t o f a v a r i a t i o n a l s t r u c t u r e o r e x i s t e n c e of a p r i m i t i v e f o r f , b e c a u s e such a s t r u c t u r e r e q u i r e s afa/aua = afB/aua f o r a # B , which i s a s e v e r e r e s t r i c t i o n on t h e n o n l i n e a r i t y . Thus it i s worthw h i l e t o look f o r e x i s t e n c e r e s u l t s which a v o i d t h a t r e s t r i c t i o n . ) The r e s u l t s o f [l] r e q u i r e f (u) t o have growth o f o r d e r \ u [ ' a t i n f i n i t y w i t h 1 < q < ( n + l ) / ( n - 1 ) ; t h o s e of 1113 a l l o w 1 < q < n / ( n - 2 ) b u t r e q u i r e t h a t L = A and t h a t R be convex. The bounds of L6] o n l y r e q u i r e t h a t 1 < q < ( n + 2 ) / ( n - 2 ) . The r e s u l t s of Ll] g e n e r a l i z e t o s y s t e m s such a s ( 2 2 ) ; so do t h o s e o f [ll] i f w e r e q u i r e t h a t L " = A , t h a t R be convex, and t h a t f { u ) be quasimonotone, t h a t i s , afcl/au B 0 f o r a # 0, The e s t i m a t e s of [6] do n o t a p p e a r t o e x t e n d s o e a s i l y t o s y s t e m s . The g e n e r a l i z a t i o n t o s y s t e m s of t h e r e s u l t s of [I] and

128

C. Cosner

[ll] is discussed in \3] and applied to existence theory. Theproofs are somewhat technical and are similar to those for the case of a single equation. One key ingredient in extending the results of 1111 to systems is the generalization to that case b Troy 1161 of the symmetry results of Gidas, Ni, and Nirenberq L53 for a single equation.

REFERENCES : 111

H. Brezis and R. Turner, On a class of superlinear elliptic equations, Comm. P.D.E. 2(6) (1977) 601-614.

I21

C. Cosner, Estimates for eigenfunctions and eigenvalues of nonlinear elliptic problems, to appear, A.M.S. Transactions.

t3 1

C. Cosner, Positive solutions for superlinear elliptic systems without variational structure, preprint.

L4 1

C. Faber, Beweis, dass unter allen homogenenen Membranen van gleicher Flache und gleicher Spannung die Kreisformige tiefsten Grundton gibt, Sitzungsber. Bayer. Akad. der Wiss. Math-Phys., Munich (1923) 169-172.

L5

1

L6 1

B. Gidas, W.-M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243. B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. in P.D.E. 6 ( 8 ) (1981) 883-901

.

171

J. Hersch, Physical interpretation and strengthening of M.H. Protter’s method for vibrating nonhomogeneous membranes; its analog for Schrodinger’s equation, Pacific J. of Math. 11 (1962) 971-980

.

18 1

E. Krahn, Uber eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann. 94 (1925) 97-100.

191

E. Krahn, Uber Minimaleigenschaften der Kugel in drei und Mehr Dimensionen, Acta Comm. Univ. Tartu (Dorpat) A9 (1926) 1-44.

LlO]

N. Levinson, Positive eigenfunctions for Au+Af(u) = 0, Arch. Rational Mech. Anal. 11 (1962) 258-272.

1111 P.-L.

Lions, A priori estimates and existence results for some semilinear elliptic problems, M.R.C. Technical Summary Report #1975, Madison, Wisconsin (1979).

1121 L. Payne and H. Weinberger, Lower bounds for vibration frequencies of elastically supported membranes and plates, J. SOC. Indust. Appl. Math. 5(4) (1957) 171-182.

A Priori Estimates in Nonlinear Eigenvalue Problems

129

1131 M. Protter, Lower bounds for the first eigenvalue of elliptic equations, Ann. of Math. 71(3) (1960) 4 2 3 - 4 4 4 . Ll4] P. Schaefer and R. Sperb, Maximum principles and bounds in some

inhomogeneous elliptic boundary value problems, SIAM J. Math. Anal. 8 ( 5 ) (1977) 871-878. Ll5] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura (1976) 353-372. Appl. l l O ( 4 )

Ll6] W. Troy, Symmetry properties in systems of semilinear elliptic equations, J. Differential Equations 4 2 (1981) 4 0 0 - 4 1 3 .

Ll7] H. Weinberger, Upper and lower bounds for eigenvalues by finite deifference methods, Comm. Pure and Appl. Math. 9(3) (1956) 613-623.

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DIFFERENTIALEQUATIONS I.W. Knowles and R.T. Lewis(Editors) 0 Elsevier Science PublishersB.V. (North-Holland),1984

I3 1

DEVELOPMENTS IN THE THEORY OF NONLINEAR FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS

Michael G. Crandall Department of Mathematics and Mathematics Research Center University of Wisconsin-Madison Madison, Wisconsin

Panagiotis E. Souganidis Division of Applied Mathematics Brown University Providence, m o d e Island

INTRODUCTION There has been a substantial development of the theory of scalar, nonlinear, first-order partial differential equations in the last two years. It is our goal to provide a brief tour of this work and a guide to the recent literature. In view of the rate the literature is growing, this report will be somewhat out of date even on its publication, but we hope it will be useful. In the discussion below, Sa is an open subset of R", T > 0, pT = Qx(O,T), "x" denotes points of Rn, u is a real-valued function of x or (t,x), u 1 stands for the spatial gradient of u and H, $, $ are given Du = (ux 1 xn functions of the indicated arguments. We discuss the initial-boundary value problem

,....,

( IBVP)

u + H(x,t,u,Du) = 0 for 0 < t < T, x e 0, t utx,t) = +(x,t) for o < t < T, x e an, u(x,O) = $ ( X Ifor x e Q,

together with its stationary form H(x,u,Du) = 0 for x U(X)

=

e a,

$(x) for x e 352.

A very simple case of (IBVP) arises when 52 = R and H is independent of (x,t,u). The problem then has the form u + H(u ) = 0 for x e R, t > 0, t X u(x,O)

=

$(x) for

x

e R.

The classical nethod of characteristics reveals that any smooth solution u of on RX"O,TI satisfies ux(y + tH’($’(y)),t) = $’(y) for O 0 as y varies, this information is contradictory. Thus ( 1 ) cannot have smooth solutions defined for all time except in extremely special circumstances. Simple examples also show that solutions of ( 1 ) with slightly less regularity than continuous first derivatives are not unique. For example, if H(p) = -p2 and 6 = 0, then u E 0 and v = max(t - I x l , O ) are distinct compactly supported (for bounded t) and piecewise linear solutions of ( 1 ) which satisfy the equation except on the lines 1x1 = t and x = 0, where v is n o t differentiable. The above remarks recall the classical observations that ( 1 ) does not have global smooth solutions in general and that the most natural weakenings of the

M. G. Crundull and P.E. Sougunidis

132

c l a s s i c a l n o t i o n of s o l u t i o n l e a d t o nonuniqueness. However, i n view of t h e way t h e s e problems a r i s e i n a p p l i c a t i o n s - i n p a r t i c u l a r , i n t h e c a l c u l u s of v a r i a t i o n s , c o n t r o l t h e o r y and d i f f e r e n t i a l games one e x p e c t s a n o t i o n of s o l u t i o n of (IBVPf and ( B V P ) f o r which t h e r e i s b o t h e x i s t e n c e and u n i q u e n e s s

-

.

The f i r s t d e m o n s t r a t i o n of u n i q u e n e s s f o r a n o t i o n o f g e n e r a l i z e d s o l u t i o n of ( I B V P ) o r (BVP) a d e q u a t e t o c o v e r a p p l i c a t i o n s t o , e . g . , d i f f e r e n t i a l games, w a s g i v e n i n M. G. C r a n d a l l and P. L. Lions [71. This n o t i o n of s o l u t i o n i s e x p l a i n e d

i n S e c t i o n 1 where u n i q u e n e s s r e s u l t s a r e a l s o d i s c u s s e d . The t o p i c of e x i s t e n c e i s t a k e n up i n s e c t i o n 2, w h i l e S e c t i o n 3 i s c o n c e r n e d w i t h t h e i n t e r p l a y between t h e s e t o p i c s , c o n t r o l t h e o r y and t h e t h e o r y of d i f f e r e n t i a l games. Approximation and r e p r e s e n t a t i o n of s o l u t i o n s a r e d i s c u s s e d i n S e c t i o n 4. As o u r g o a l is a b r i e f o u t l i n e of r e c e n t developments, t h e v e r y s u b s t a n t i a l l i t e r a t u r e which p r e d a t e s t h e work d i s c u s s e d h e r e i n w i l l n o t b e r e f e r r e d to. The book t281 of P. L. Lions w i l l p r o v i d e t h e i n t e r e s t e d r e a d e r w i t h an a p p r o p r i a t e h i s t o r i c a l view and r e f e r e n c e s t o t h e o l d e r l i t e r a t u r e .

There is a t h e o r y of second o r d e r e q u a t i o n s and t h e i r r e l a t i o n s t o s t o c h a s t i c c o n t r o l and games which c o r r e s p o n d s t o t h e f i r s t o r d e r t h e o r y reviewed i n t h i s paper. W e have n o t d i s c u s s e d t h i s t h e o r y h e r e i n and r e f e r t h e i n t e r e s t e d r e a d e r t o t h e p a p e r s [291, [301 of P. L. L i o n s i n t h i s r e g a r d . The t o p i c of q u a s i v a r i a t i o n a l i n e q u a l i t i e s and Hamilton-Jacobi e q u a t i o n s , which i s n o t o t h e r w i s e mentioned h e r e i n , is t a k e n up i n [ l l .

SECTION 1.

NOTIONS OF SOLUTION AND UNIQUENESS

I t w i l l be c o n v e n i e n t t o c o n s i d e r a g e n e r a l e q u a t i o n of t h e form

F ( y , u , D u ) = 0 i n 0,

(1.1) where 0

C fl i s open, y

= (yl,y2,.

.. , y m ) ,

Du = ( u Y1'

...,uYm

),

and F:OXRX#

+ R.

O f c o u r s e , t h i s g e n e r a l form i n c o r p o r a t e s the e q u a t i o n s i n b o t h (IBVP) and (BVP). For u e C ( 0 ) and z e 0 p u t

where a * b i s t h e s c a l a r p r o d u c t of a , b e #. E. g., i f 0 = ( - 1 , l ) and The r e l a t i o n p e D+u(z) can b e u ( y ) = l y l , t h e n D+u(O) = p and D-u(O) = [-1.11. w r i t t e n u ( y ) < u ( z ) + p - ( y - z ) + o(y - z ) , w i t h t h e u s u a l meaning of o ( y - z ) , and C l e a r l y u i s d i f f e r e n t i a b l e a t z e 0 e x a c t l y when a s i m i l a r remark a p p l i e s t o D-. b o t h D+u(z) and D-u(z) are b o t h nonempty and t h e n

D+u(z) = D-ulz)

=

{

Du(z)

}

where Du(z) d e n o t e s t h e u s u a l ( F r s c h e t ) d e r i v a t i v e of u c o n t i n u o u s f u n c t i o n s u which a r e nowhere d i f f e r e n t i a b l e , f u n c t i o n s u such t h a t a t l e a s t one of D+u(z) and D-u(z) z e 0. One way t o d e f i n e a v i s c o s i t y s o l u t i o n of ( 1 . 1 ) D e f i n i t i o n 1. (1.3)

Let u

e

C(0).

at Z. Since t h e r e are t h e r e are continuous i s empty a t e v e r y p o i n t is:

Then u i s a v i s c o s i t y s o l u t i o n of F

~ ( y , u ( y ) , p )6

o

for all y

e o

and p

e

~+u(y).

< 0 in

0 if

Nonlitreur Fmt-Order Equations Similarly, u is a v i s c o s i t y s o l u t i o n of 0 (1.4)

0 C ~ ( y , u ( y ) , p )f o r a l l y

<

F in

e o

o

133

if

and p e D - u ( ~ ) .

F i n a l l y , u i s a v i s c o s i t y s o l u t i o n o f F = 0 in 0 i f it is a v i s c o s i t y s o l u t i o n of b o t h F C 0 and 0 6 F. W e W i l l also c a l l a v i s c o s i t y s o l u t i o n of F C 0 ( r e s p e c t i v e l y , 0 < F ) a v i s c o s i t y s u b s o l u t i o n ( s u p e r s o l u t i o n ) of F = 0. The o b v i o u s r e l a t i o n s between D, D+, D- g u a r a n t e e t h a t t h e n o t i o n of a v i s c o s i t y s o l u t i o n is c o n s i s t e n t w i t h t h a t of a c l a s s i c a l s o l u t i o n . For example, c o n t i n u o u s l y d i f f e r e n t i a b l e s o l u t i o n s of F = 0 i n t h e c l a s s i c a l s e n s e a r e v i s c o s i t y s o l u t i o n s and a v i s c o s i t y s o l u t i o n u of F = 0 s a t i s f i e s F ( y , u ( y ) , O U ( y ) ) = 0 a t each p o i n t y e 0 a t which u is differentiable.

The t e r m " v i s c o s i t y s o l u t i o n " a r i s e s b e c a u s e a s t a n d a r d method t o approach t h e e x i s t e n c e of s o l u t i o n s o f , e . g . , (IBVP) i s t o solve ( i f p o s s i b l e ) u

(1.5)

E t

+ H(x,t,uE,DuE)

-

E A U ~= 0 ,

where A d e n o t e s t h e L a p l a c i a n i n R", s u b j e c t t o i n i t i a l and boundary c o n d i t i o n s and t h e n t o s h o w t h a t u h a s a l i m i t p o i n t i n C f O ) as E f O w i t h t h e desired properties. This i s t h e m&hod of " v a n i s h i n g v i s c o s i t y " . H e r e a r e two o t h e r ways t o d e f i n e a v i s c o s i t y s o l u t i o n .

Definition 2 .

A

v i s c o s i t y s o l u t i o n of F = 0 i s a u e C ( 0 ) such that i f

0

$ e C ( 0 ) i s n o n n e g a t i v e and k e R, t h e n ( i )and ( i i )below h o l d = (i) I$ 4 ( u - k ) h a s a p s i t i v e maximum i n 0, t h e n

a t s o m e p o i n t y a t which t h e maximum i s a c h i e v e d . ( i i ) I f $ ( u - k) h a s a n e g a t i v e minimum i n 0, t h e n ( 1 . 6 ) h o l d s w i t h t h e o p p o s i t e i n e q u a l i t y a t some p o i n t y a t which t h e minimum i s a c h i e v e d . D e f i n i t i o n 3.

A v i s c o s i t y s o l u t i o n of F = 0

is a u

e

C ( 0 ) such t h a t f o r every

$ e C'(01, F(y,u(yf,D$(y)) C 0 (respectively, 0 < F f y , ~ ( y ) , D @ f y > > h> olds a t each r e l a t i v e maximum ( r e s p e c t i v e l y , minimum) p i n t y e 0 of u - 0. The e q u i v a l e n c e o f t h e s e t h r e e d e f i n i t i o n s is proved i n 171 ( i n which Definition 3 i s related t o ideas D e f i n i t i o n 2 was t a k e n a s t h e b a s i c p r o p e r t y ) . used by L. C. Evans i n 1101. D e f i n i t i o n 2 is o b v i o u s l y c o n v e n i e n t f o r l o c a l i z a t i o n and f o r t h i s r e a s o n u n i q u e n e s s was f i r s t o b t a i n e d u s i n g it. S u b s e q u e n t l y , i n [61, a v a r i e t y of t h i n g s w e r e proved u s i n g D e f i n i t i o n s 1 and 3 d i r e c t l y . W e recommend 16) f o r an i n t r o d u c t i o n t o properties of v i s c o s i t y s o l u t i o n s and f o r e a s y access t o p r o o f s of model cases of r e s u l t s l i k e t h o s e g i v e n below. L e t u s f o r m u l a t e c o n d i t i o n s on H under which one can p r o v e u n i q u e n e s s BR w i l l d e n o t e t h e R-ball i n d. The f i r s t t w o c o n d i t i o n s are t h e b a s i c results. c o n t i n u i t y r e q u i r e m e n t and a m o n o t o n i c i t y c o n d i t i o n c o n c e r n i n g t h e dependence of H on U. (For t h o s e who may wonder, it i s t h e m o n o t o n i c i t y c o n d i t i o n t h a t rules Out t h e case of a c o n s e r v a t i o n l a w i n what f o l l o w s ) .

-

H:S?x[O,T]XKd

+

R i s uniformly continuous

(HI 1

on Rx[O,T]x[-R,R]xBR f o r e a c h R > 0.

M. G. Crundull und

134

P.E. Sougunidis

F o r each R > 0 t h e r e i s a c o n s t a n t y R > 0 s u c h t h a t

-

H(x,t.r,p)

f o r x e Q, p

e R"

H(x,t,s,p)

and -R

<

s

> YRfr -

s)

< R.

The n e x t t w o h y p o t h e s e s c o n c e r n the b e h a v i o r of t h e uniform c o n t i n u i t y i n x as p becomes large. They are e x p r e s s e d i n terms o f t h e f u n c t i o n

A ( R , ~ , E ) = sup{ I H ( x , t , r , p ) - H ( y , t , r , p ) I : O < t < T , l r l < R , Ix-yI I p l < p , l x - y l < E } . Here

I I

d e n o t e s b o t h t h e norm i n $ and i n R.

The weaker o f t h e t w o h y p o t h e s e s

is

l i m A(R,E,E)

= 0 for R

> 0.

EtO

S i n c e A i s n o n d e c r e a s i n g i n i t s arguments a m o r e s e v e r e r e s t r i c t i o n is l i m A(R,p,E)

(H4 )

= 0 f o r R,

p

> 0.

EC 0

-

The c o n d i t i o n s (HI) (H4) are m e a n i n g f u l f o r t h e problem (BVP) as w e l l when H is i n t e r p r e t e d a s a f u n c t i o n of t which happens t o be i n d e p e n d e n t o f t . The u n i q u e n e s s r e s u l t o f 171 for ( B V P ) is:

Theorem "71). Let u, v be bounded c o n t i n u o u s f u n c t i o n s on Q which a r e v i s c o s i t y s o l u t i o n s of H = 0 i n Q. L e t $ b e c o n t i n u o u s on aSl and u ( x ) - $ ( z ) , and v ( x ) $(z) t e n d t o z e r o as x + z e a Q u n i f o r m l y i n Z . Set R = max(nuii

,lVn Lrn(Q)

) Lrn(Q)

and l e t [ H l ) and ( H 2 ) h o l d w i t h YR > 0 i n (H2). Then: ( i ) If (H4) h o l d s , t h e n u = v. ( i i )I f u and v are u n i f o r m l y c o n t i n u o u s and (H3) h o l d s , t h e n u = ( i i i ) I f u and v are L i p s c h i t z c o n t i n u o u s , t h e n u = V .

V.

T h i s r e s u l t i n f a c t f o l l o w s from m o r e g e n e r a l estimates comparing v i s c o s i t y sub-and-super s o l u t i o n s o f d i f f e r e n t problems. W e w i l l not formulate t h e s e r e s u l t s here. Observe t h a t a s t h e h y p o t h e s e s on u and v a r e s t r e n g t h e n e d , less i s r e q u i r e d of H. The c o r r e s p o n d i n g r e s u l t f o r (IBVP) is q u i t e s i m i l a r . The s t a t e m e n t f o r t h i s case arises upon r e p l a c i n g S l by QX[O,T], 352 by t h e u n i o n of anx[O,Tl and Qx{O}, and $ ( X I b y $ ( x , t ) on aQx[O,TI and by u o ( x ) when t = 0. Moreover, t h e r e q u i r e m e n t yR > 0 i n (H2) i s dropped. (The l i n e a r f u n c t i o n y R ( r 6 ) i n ( H 2 ) i s r e p l a c e d by a n o n l i n e a r f u n c t i o n i n t h e r e s u l t f o r (BVP) i n [ 7 ] . ) The n e c e s s i t y of c o n d i t i o n s l i k e (H3) o r (H4) i s shown v i a examples i n [71.

-

B e f o r e t h e r e s u l t s mentioned above, t h e main u n i q u e n e s s r e s u l t s which w e r e e s t a b l i s h e d i n a g e n e r a l i t y f o r which t h e r e w a s a c o r r e s p o n d i n g g l o b a l e x i s t e n c e t h e o r y concerned t h e case of convex H a m i l t o n i a n s . These r e s u l t s concern s o l u t i o n s o f t h e e q u a t i o n s i n an a l m o s t everywhere s e n s e which also s a t i s f y a t y p e of " s e m i concavity" condition. Concerning such r e s u l t s w e r e f e r to t h e book of P. L. L i o n s t281. The v i s c o s i t y n o t i o n i s u s e d i n C281, b u t t h e main emphasis i n t h i s book i s t h e i m p o r t a n t s p e c i a l case of v i s c o s i t y s o l u t i o n s which a r e L i p s c h i t z c o n t i n u o u s ( a n d hence s a t i s f y t h e e q u a t i o n a l m o s t e v e r y w h e r e ) . O t h e r u n i q u e n e s s r e s u l t s c o n c e r n domains of dependence ( e . g . , [ 7 1 ) , unbounded f u n c t i o n s , ( H . I s h i i [ 2 1 1 ) and H a m i l t o n i a n s which a r e n o t n e c e s s a r i l y c o n t i n u o u s i n t ( H . I s h i i [ 2 0 1 ) . With r e s p e c t t o domains o f dependence, o b s e r v e t h a t i f w e

135

Nonlin m r First-Order Equations

r e g a r d (IBVP) as a special case o f (BVP) by t h i n k i n g of t as as " s p a c e v a r i a b l e " , G e n e r a l r e s u l t s c o n c e r n i n g which p a r t t h e n w e have n o t p r e s c r i b e d d a t a a t t = T. o f t h e boundary of C2 i s i m p o r t a n t f o r u n i q u e n e s s i n (BVP) a r e t h e subject o f work of R. J e n s e n ( [ 2 2 1 ) i n p r o g r e s s a t t h e t i m e o f t h i s symposium.

SECTION 2. EXISTENCE The e x i s t e n c e t h e o r y f o r v i s c o s i t y s o l u t i o n s of (IBVP) and (BVP) is much m o r e a c o n t i n u a t i o n of t h e e x i s t e n c e t h e o r y which p r e d a t e s t h e n o t i o n o f v i s c o s i t y s o l u t i o n s t h a n t h e c o r r e s p o n d i n g u n i q u e n e s s t h e o r y (which is quite d i s t i n c t from what e x i s t e d b e f o r e ) i s a c o n t i n u a t i o n o f more c l a s s i c a l r e s u l t s . Roughly s p e a k i n g , known methods a d a p t t o p r o v i n g t h e e x i s t e n c e of v i s c o s i t y s o l u t i o n s and t h e f l e x i b i l i t y o f t h e n o t i o n allows one t o t a k e l i m i t s f r e e l y and o b t a i n new r e s u l t s . T h e r e are also new arguments which arose p a r t l y i n t r y i n g t o g o t t h e e x i s t e n c e t h e o r y i n harmony with t h e g e n e r a l i t y of t h e u n i q u e n e s s t h e o r y . W e are g o i n g t o d e s c r i b e , i n m o r e or less c h r o n o l o g i c a l order, r e s u l t s o b t a i n e d s i n c e t h e i n t r o d u c t i o n o f v i s c o s i t y s o l u t i o n s and a s k t h e r e a d e r t o b e aware t h a t t h i s d o e s n o t g i v e an accurate h i s t o r i c a l view. W e a g a i n r e f e r t o [281 f o r a m o r e b a l a n c e d view o f t h e earlier t h e o r y . The s o r t of d r a m a t i c e x i s t e n c e and u n i q u e n e s s theorems which are now p o s s i b l e may b e i l l u s t r a t e d b y t h e model problems u + H(Du) = v i n

(2.1)

R",

and

(2.2) I t w a s proven i n [71 t h a t if n is c o n t i n u o u s from If t o R and v and uo are bounded and u n i f o r m l y c o n t i n u o u s , t h e n ( 2 . 1 ) and ( 2 . 2 ) h a v e v i s c o s i t y s o l u t i o n s t o which t h e u n i q u e n e s s theorem a p p l i e s . The o n l y r e g u l a r i t y r e q u i r e d i s c o n t i n u i t y of H and u n i f o r m c o n t i n u i t y o f v and u o , and t h e n t h e r e i s a u n i q u e g l o b a l s o l u t i o n .

P. L. L i o n s , i n f271 and f281, c o n s i d e r s problems of the forms (IBVPf and Two t y p e s of a s s u m p t i o n s on H a r e i m p o r t a n t i n h i s work. One i s a c o n t i n u i t y a s s u m p t i o n of L i p s c h i t z t y p e which i s u s e d t o e s t a b l i s h u n i f o r m L i p s c h i t z estimates on s o l u t i o n s o f (IBVP) and (BVP) ( u s i n g r e s u l t s o f [261). A s i m p l e special c a s e o f t h i s h y p o t h e s i s r e a d s

(BVP).

For R

>

0 t h e r e i s a c o n s t a n t CR s u c h t h a t

for t e [O,TI,

X,

y

The o t h e r assumption r e a d s

(H6)

H(t,x,r,p)

+ m

as IpI +

e h,

p

e f

and Irl

<

R.

m,

u n i f o r m l y f o r x e 0 , O
e R.

The c o n c l u s i o n - i n e i t h e r case - i s t h e e x i s t e n c e of bounded, L i p s c h i t z c o n t i n u o u s v i s c o s i t y s o l u t i o n s p r o v i d e d t h e boundary c o n d i t i o n s are s u i t a b l e ( s e e below). The p r o o f s i n [281 a r e b a s e d upon t h e v a n i s h i n g v i s c o s i t y method, w h i l e i n [27] a method of p e n i l i z a t i o n of domain ( i n t r o d u c e d i n [ 2 9 1 ) is employed.

M. G. Crandall and P.E. Souganidis

136

The p a p e r 1331 d e a l s w i t h (IBVP) and t h e “resolvent-type‘lstationary

u + AH(x,u,Du)

=

v for 1

>

problem

0

i n R”x[O,Tl and F? r e s p e c t i v e l y . The r e s u l t i s t h e e x i s t e n c e of bounded and u n i f o r m l y c o n t i n u o u s v i s c o s i t y s o l u t i o n s under h y p o t h e s e s (H4) or ( H 5 ) , where i n t h e l a t t e r case t h e dependence of H on ( t , r , p ) i s less r e s t r i c t i v e t h a t t h a t u s e d I n b o t h c a s e s t h e v a n i s h i n g v i s c o s i t y method is u s e d and t h e proof i n 1271, [281. p r o c e e d s v i a e s t i m a t i n g t h e modulus of c o n t i n u i t y of s o l u t i o n s o f t h e a p p r o x i m a t e problems w i t h r e s p e c t t o x. However, t h e best theorem c o n c e r n i n g e x i s t e n c e of bounded u n i f o r m l y c o n t i n u o u s v i s c o s i t y s o l u t i o n s ( w i t h o u t ( H 6 ) ) w a s o b t a i n e d by G. B a r l e s [ l ] . B a r l e s was a b l e t o show e x i s t e n c e under t h e a s s u m p t i o n ( H 3 ) , which i s t h e weakest assumption under which k n i q u e n e s s o f bounded u n i f o r m l y c o n t i n u o u s s o l u t i o n s h a s b e e n proved. The n o t a t i o n B U C ( K ) w i l l mean t h e bounded and u n i f o r m l y c o n t i n u o u s f u n c t i o n s on K. B a r l e s p r o v i d e d t h e e x i s t e n c e a s s e r t i o n o f t h e n e x t theorem which i s s t a t e d f o r t h e g e n e r a l c a s e i n which Sl i s n o t #, b u t it i s v a l i d e q u a l l y w e l l when Sl i s # ( t h e r e simply b e i n g no c o n d i t i o n s on an i n t h i s e v e n t ) .

1 .

B U C ( n ) b e , r e s p e c t i v e l y , a sub- and a s u p e r s o l u t i o n of H = 0 i n Q . Assume, = ii on an. Then t h e r e is a u n i q u e v i s c o s i t y s o l u t i o n u €? B l J C t d ) moreover, t h a t of H = 0 i n 0 such t h a t u = 2 = ii on 352. ( b ) Assume t h a t H:GX[O,T]xRXR” + R s a t i s f i e s ( H I ) , ( H Z ) , and ( H 3 ) . L e t 2, u e BUC(~X[O,T])b e , r e s p e c t i v e l y , a sub- and a s u p e r s o l u t i o n of u + H ( x , t , u , D u ) = 0 i n fix[O,Tl s u c h t h a t g ( x , t ) = E ( x , t ) = $ ( x , t ) f o r ( x , t ) i n aQxf0,Tl a n d Then t h e r e i s a u n i q u e v i s c o s i t y s o l u t i o n u ( x , O ) =-ii(x,O) = $ ( x ) in Sa. u e BUC(nx[O,T]) o f ( I B V P ) .

B a r l e s a t t a c k e d t h e e x i s t e n c e problem by r e d u c i n g t h e q u e s t i o n t o one of t h e form v + fi(y,v,Dv) = 0 i n F? f o r s o m e m and t h e n a p p r o x i m a t i n g t h e Hamiltonian 7 i n v o l v e d by smooth ones s a t i s f y i n g an (H51-type assumption. The r e s u l t s of [271 t o g e t h e r w i t h t h e u n i q u e n e s s estimates o f [71 a r e used t o complete t h e p r o o f . Of c o u r s e , g i v e n t h e above theorem, i n t e r e s t s h i f t s t o t h e q u e s t i o n of when s u b - and s u p e r s o l u t i o n s w i t h t h e d e s i r e d p r o p e r t i e s can be found. There a r e many E s p e c i a l l y when Sl i s n o t R”, it i s o n l y i n t h e c a s e of open q u e s t i o n s h e r e . convex Hamiltonians t h a t t h i s i s w e l l u n d e r s t o o d , and one can f i n d n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s i n [281. R e c e n t l y , L. C. Evans and H. I s h i i [131 gave a d i f f e r e n t proof f o r t h e e x i s t e n c e of a v i s c o s i t y s o l u t i o n of u + H(x,Du) = 0 i n Sl, u = $ on an, where H s a t i s f i e s ( H 3 ) . Their t e c h n i q u e i n v o l v e s a v e r s i o n of t h e l o c a l i z a t i o n method of 1291 *

SECTION 3.

CALCULUS OF VARIATIONS, OPTIMAL CONTROL THEORY AND DIFFERENTIAL GAMES.

Hamilton-Jacobi e q u a t i o n s a r e c l a s s i c a l l y d e r i v e d i n t h e c l o s e l y r e l a t e d a r e a s of t h e c a l c u l u s o f v a r i a t i o n s , o p t i m a l c o n t r o l t h e o r y (where t h e y are c a l l e d Bellman e q u a t i o n s ) and d i f f e r e n t i a l games ( I s a a c s e q u a t i o n s ) . A s w a s o b s e r v e d by Bellman, a v e r y good way t o e x h i b i t t h e r e l a t i o n s h i p between t h e e q u a t i o n s and t h e problems i n which t h e y a r i s e i s v i a t h e dynamic programming p r i n c i p l e (see, for example, Fleming and R i s h e l [ 1 7 1 ) . T h i s r e m a i n s t r u e i n t h e c o n t e x t o f v i s c o s i t y s o l u t i o n s a s w a s p o i n t e d o u t by P . L. L i o n s . I n t h i s s e c t i o n w e w i l l expand a b i t on t h e s e r e m a r k s and g i v e s e v e r a l r e f e r e n c e s .

137

No n [ineiw First- Order Equations

We begin by reviewing the relationship between the notion of a viscosity solution and control theory in the simplest possible case: We consider a "finite horizon" control problem without boundary conditions and formulate two typical theorems (omitting all the technical assumptions - the references contain precise statements). After this we quickly outline the situation as regards differential games. Consider a system whose state at time t if started at x is y (t) and whose evolution is governed by the following Cauchy problem for an ordinary differential equation:

(3.1)

where f:RNxV + 2,V is a given compact set in some @ (for example) and v(t), to V. Under natural called the control, is a measurable function from LO,-) assumptions on f, (3.1 ) has a unique solution for all x e # denoted by yx(t). now define a pay-off J(x,t;v(-)) for each given x,t and control v(-):

J(x,t;v(.))

(3.2)

=

uo(y (t)) +

We

t h(yx(s),v(s))ds,

1 0

Here h(x,v) and uo are qiven continuous functions which satisfy certain hypotheses. The problem is to choose the control function v(*) so as to minimize J. An important quantity in this situation i s the value (or the optimal cost): u(x,t)

inf J(x,t;v(.)). V(.) Two of the main goals of control theory are to characterize u and to design optimal (or nearly optimal) controls. One of the most important tools for the study of these problems is the dynamic programming principle. In the case under consideration here it says: (3.3)

=

For all 0

Theorem (dynamic programming).

<

s

< t

S

u(x,t) = inf V(’)

{

1

h(yx(o),v(a))do + u(yx(s),t

-

s)}.

0

Using this principle P.L. Lions proved [281: Theorem.

The optimal cost u given by ( 3 . 3 ) is the viscosity solution of u

t

+ sup {f(x,v)*Du VeV

-

h(x,v)} = 0 in RnX(O,m),

u(x,a) = u (x) in R? 0

Observe that the Hamiltonian here, that is ~ ( x , p )= s~p(f(x,v).p

-

h(x,v): v

e v),

is the supremum of affine functions and is therefore convex. For the proofs and generalizations of these results and references to the classical literature, see P. L. Lions [281, [241, and L. C. Evans [121. Other recent references which are relevant for the interaction between viscosity solutions and contol theory are: P. L. Lions and M. Nisi0 [31], I. Capuzzo Dolcetta and L. C. Evans 141, L. C. Evans and H. Ishii [13], I. Capuzzo Dolcetta [31, I. Capuzzo Dolcetta and H. Ishii [51. Differential games resemble contol theory in involving a system of differential equations and a pay-off function. However, now there are two players

M. G. Crundull and P.E. Sougunidis

138

a n d t w o c o n t r o l sets. The f i r s t p l a y e r a t t e m p t s t o make t h e pay-off l a r g e by c h o o s i n g h i s c o n t r o l , w h i l e t h e s e c o n d p l a y e r s t r i v e s t o k e e p t h e pay-off s m a l l by judicious choice of h i s co n tro l . The n o t i o n of v a l u e becomes much m o r e s u b t l e b e c a u s e , a s t h e words s u g g e s t , w e a r e d e a l i n g w i t h " s u p - i n f " o r " i n f - s u p " o p e r a t i o n s (as opposed t o t h e s i m p l e "sup" of t h e c o n t r o l case) depending on which p l a y e r "goes f i r s t " and t h i s must be i n t e r p r e t e d i n some i n f i n i t e s i m a l way. However, t h e v a l u e may be d e f i n e d i n v a r i o u s ways which t u r n o u t t o be e q u i v a l e n t - a n d a v e r y good way t o see t h e e q u i v a l e n c e is t o use the r e l a t i o n w i t h HamiltonJ a c o b i e q u a t i o n s and t h e u n i q u e n e s s t h e o r y . The f o r m a l r e l a t i o n s h i p s between d i f f e r e n t i a l games and Hamilton-Jacobi e q u a t i o n s were o b s e r v e d by Isaacs [191 i n t h e e a r l y 1950's: h e showed t h a t i f t h e v a l u e s of v a r i o u s d i f f e r e n t i a l games w e r e regular enough, t h e n t h e y s o l v e c e r t a i n f i r s t - o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n s ( c a l l e d Isaacs e q u a t i o n s ) w i t h "max-min" or "min-max" t y p e n o n l i n e a r i t i e s ( i n t h e s e n s e t h a t t h e Hamiltonian f o r t h e c o n t r o l problem i s a "max" n o n l i n e a r i t y ) . The d i f f i c u l t y t h a t t h e s e e q u a t i o n s seldom have s o l u t i o n s r e g u l a r enough t o j u s t i f y t h e f o r m a l arguments w a s worked a r o u n d i n v a r i o u s ways by, among o t h e r s , W. Fleming [151, 1161, A. Friedman [181, E l l i o t and K a l t o n [91 (see als o t h e r e f e r e n c e s i n t h e s e w o r k s ) . A main i d e a h e r e w a s t o approximate t h e d i f f e r e n t i a l game by a s t o c h a s t i c d i f f e r e n t i a l game whose v a l u e would t h e n s a t i s f y a v a n i s h i n g - v i s c o s i t y t y p e o f a p p r o x i m a t i o n of t h e f i r s t - o r d e r e q u a t i o n . It p r o v e d p o s s i b l e t o t a k e a unique l i m i t as t h e n o i s e went t o 0 a n d t o i d e n t i f y t h i s l i m i t w i t h t h e value. Again, t h e u n i q u e e x i s t e n c e of t h i s l i m i t - which w a s r e l a t i v e l y d i f f i c u l t t o show - is made t r a n s p a r e n t by u n i q u e n e s s r e s u l t s l i k e t h o s e of S e c t i o n 1. For d i r e c t p r o o f s of t h e f a c t s t h a t ( v a r i o u s n o t i o n s o f ) t h e v a l u e o f d i f f e r e n t i a l games s a t i s f y t h e I s a a c ' s e q u a t i o n i n t h e v i s c o s i t y s e n s e as w e l l as s o m e a p p l i c a t i o n s w e r e f e r t o P. S o u g a n i d i s [321, E. Barron, L. C. Evans and R . J e n s e n [ Z ] , L. C. Evans and H. I s h i i [131, L. C. Evans and P. S o u g a n i d i s [141, H. I s h i i 1201, and P. L. L i o n s [241.

SECTION 4. REPRESENTATION AND APPROXIMATION OF SOLUTIONS I t w i l l be a d v a n t a g e o u s t o s k e t c h t h e t o p i c s of t h i s s e c t i o n i n t h e c o n t e x t of t h e model problem

u

t

+

o f o r t > 0,

H ( D ~ )=

u(x,o)

=

$(XI, for x

e

x

e

R ~ ,

R ~ ,

where H : d + R i s c o n t i n u o u s . The proof of t h e u n i q u e n e s s r e s u l t s of S e c t i o n 1 and t h e e x i s t e n c e r e s u l t s of S e c t i o n 1 and 2 p r o v i d e a unique bounded v i s c o s i t y s o l u t i o n u of ( 4 . 1 ) which i s u n i f o r m l y c o n t i n u o u s on # x [O,T] f o r each J, e BUC(Rn) and T > 0. Moreover, t h e s e l f - m a p S ( t ) of BUC(Rn) g i v e n by S ( t ) $ ( x ) = u ( x , t ) where u is t h e s o l u t i o n of ( 4 . 1 ) is a semigroup ( i . e . , S ( t ) S ( s ) = S ( t + s ) ) which p r e s e r v e s t h e n a t u r a l o r d e r of BUC(Rn) and s a t i s f i e s !S(t)J,

-

S(t)qll

<

ll$

-

1111

for 0

<

t and $, 11 e BUC($1).

H e r e II ll d e n o t e s t h e e s s e n t i a l supremum norm. W e r e f e r t o t h i s norm e s t i m a t e by s a y i n g s i m p l y t h a t S ( t ) i s "nonexpansive". W e f i r s t f o r m u l a t e a n a b s t r a c t theorem

a b o u t a p p r o x i m a t i n g S ( t ) and t h e n g i v e some s p e c i a l c a s e s of it. For p > 0 , l e t F(p):BUC(Rn) + B U C ( R " ) s a t i s f y : (Fl)

F ( p ) i s and o r d e r - p r e s e r v i n g , map of B U C ( # ) .

(F2)

For each c$ e B U C ( R n ) ,

t r a n s l a t i o n i n v a r i a n t and nonexpansive s e l f -

p + F(p)c$

is c o n t i n u o u s from LO,-)

i n t o BUC(Rn).

139

(F3)

There i s a c o n s t a n t C s u c h t h a t

for p derivatives

4 e B U C ( R " ) whose f i r s t and second p a r t i a l are c o n t i n u o u s and bounded on rc".

>

0 and t h o s e

2 I n ( F 3 ) , IID $11 d e n o t e s t h e l a r g e s t supremum norm of any second d e r i v a t i v e o f 6, e t c . . W e w i l l approximate S by t h e f o l l o w i n g o b j e c t b u i l t from F: (4.2)

s (t)+ = F(t

x

Theorem ( [ 3 2 ] ) .

-

n i ) F ( i ) " + i f nX 6 t

l e t F s a t i s f y ( F l ) , (FZ), ( F 3 ) . J,

l i m IIS(t)J, - S ( t ) $ l l A+ 0

(4.3)

< (n+l)X, n

x

e

= 0,1,2,.

,

and

BUC(R") and T

= 0 uniformly f o r 0 C t

<

x

> 0.

> 0. Then

T.

I f also H and J, a r e L i p s c h i t z c o n t i n u o u s , t h e n t h e r e i s a c o n s t a n t K such t h a t

iis(t)$- sx(t)+n< KJX

(4.4)

for

o <

t

<

T.

W e remark t h a t t h e v a l i d i t y of ( 4 . 3 ) o n l y r e q u i r e s t h a t ( F ( p ) $ - $ ) / p + 2 -H(D$) u n i f o r m l y when $ i s bounded i n C The s t r o n g e s t i m a t e ( 4 . 4 ) r e q u i r e s t h e stronger criteria i n (F3).

.

L e t u s g i v e some examples. Assume t h a t Y and Z are compact s u b s e t s of some Min-max r e p r e s e n t a t i o n s : L e t H be g i v e n by h:YxZ + R, f:YXZ + F? a r e bounded.

fl

and

If we set F ( p ) $ J ( x )= s u p i n f (-ph(y,z) + $ ( x - p f ( y , z ) ) f , y e y zez t h e n ( F l ) - ( F 3 ) h o l d , a s a s t r a i g h t f o r w a r d a n a l y s i s shows. The theorem, i n t h i s c a s e , r e p r e s e n t s S ( t ) J , as t h e l i m i t of i t e r a t e d "max-min" o p e r a t i o n s and p r o v i d e s a n error estimate. A d i f f e r e n c e approximation:

Here w e t a k e N = 1 t o s i m p l i f y t h e w r i t i n g . The e q u a t i o n becomes u + H ( u x ) = 0. Assume t h a t H i s L i p s c h i t z c o n t i n u o u s and c o n s i d e r t h e d i f f e r e n c e t approximation

of t h i s e q u a t i o n , i n which 0 i s a c o n s t a n t t o be c h o s e n , Ax and A t a r e s t e p s i z e s T h i s approximation is c o n s i s t e n t with t h e and U? c o r r e s p o n d s t o u ( jAx,nAt). + U?-l - 2U?), which e q u a t i3 o n and i n v o l v e s t h e v i s c o s i t y t e r m -(QAx)(U? 3 "Fa c o r r e s p o n d i n g t o t h i s c o r r e s p o n d s r o u g h l y t o -B(Ax)uXx. L e t a = At/Ax.'+:n a p r o x i m a t i o n is

T h i s F h a s t h e d e s i r e d p r o p e r t i e s (F1) - ( F 3 ) if 2 a B < 1 and L C 0, where L is t h e The theorem i m p l i e s t h a t i f 2aB C 1 and L < 0. $ i s Lipschitz constant for H.

M. G. Crandall arid P.E Souganidis

140 0

Lipschitz continuous and U J and all j satisfy

$(jAx), then the

=

-

tun 1

for a suitable

C.

S(nAt)+(jAx)l

Un

J

given by 14.5) for n = 1,2,...

< CJ(At1

(We leave it to the reader to trace this relation.)

A product formula: Let the Hamiltonian H in ( 4 . 1 ) be given as a sum: H(p) = H1(p) + H2(p) where each of H, H,, and H2 is Lipschitz continuous. Let S(t), Sl(t), and S2(t) be the semigroups associated with the problem ( 4 . 1 ) for the Hamiltonians H, H1, and H2 respectively. Put F(P)

=

S1(p)S2(p).

It is again possible to verify (F1 - (F3) , although it is not so easy as in the previous two cases. As a consequence, one has Ils(t)* for n = 1,2,.

.. and bounded

0

-

(Sl(t/n)S2(t/n))nqll < c/Jn

< t.

With these examples to set the scene, we outline the current situation. Approximation of viscosity solutions by a max-min representation was first established by L. C . Evans 1111. The abstract theorem is a very special case of results of Souganidis 1321, wherein one finds more general applications to max-min representations. These are closely related to Fleming’s notion of the value of a differential game (see 1321). The Elliot - Kalton notion yields a different representation theorem ([131, 1141 ) , as does the Friedman notion (1321, 121 1. A different max-min representation (where the index sets are finite) is given by Lenhart [231. The first error estimates like ( 4 . 4 ) were given in 181 in the case of numerical approximations. A general class of difference approximations of (4.1) are discussed in 1321 and error estimates are made. Moreover, error estimates are made in the same spirit for the approximation of (4.1) by (1.5). These reproduce in a much simpler way estimates inherent in the older literature. The proofs of the abstract results of f321 are related to those of 181. The application of the general convergence results in I32I to numerical approximations extend the results of [81 to more general Hamiltonians, etc.. Implicit schemes are also considered. Product formulae related to the example above are a l s o found in [321. Other works of interest include [ 5 1 , in which a stationary problem with a convex Hamiltonian is approximated by a "min" method and the convexity permits a sharper error estimate.

REFERENCES Barles, G., Thsse de Doctorat de 3 ;me Cycle, Universit6 de Paris IX Dauphine, 1982 - 1983. [2] Barron, N. E., L. C. Evans and R. Jensen, Viscosity solutions of Isaacs’ equations and differential games with Lipschitz controls, J. Diff. !?q., to appear. [3] Capuzzo Dolcetta, I., On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming, Appl. Math. and Op., to appear.

[l]

No ti1i)z car First-Order Equations

141

[41 Cappuzzo Dolcetta, I. and L. C. Evans, Optimal switching for ordinary differential equations, SIAM J. of Control and Optimization, to appear. [51 Cappuzzo Dolcetta, I. and H. Ishii, Approximate solutions of the Bellman equation of deterministic control theory, Appl. Math. and Op., to appear. [ 6 ] Crandall, M. G . , L. C , Evans and P . L. Lions, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. SOC., to appear. [71 Crandall, M. G. and P. L. Lions, Viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. SOC. 277 (1983), 1-42. [81 Crandall, M. G. and P. L. Lions, Two approximations of solutions of Hamilton-Jacobi equations, Math. Comp., to appear. 191 Elliot, R. J. and N. J. Kalton, The existence of value in differential games, Memoirs of Amer. Math. Society 126 (1972). [lo] Evans, L. C., On solving certain nonlinear partial differential equations by accretive operator methods, Israel J. Math. 86 (19801, 225247. [ll] Evans, L. C., Some max-min methods for the Hamilton-Jacobi equation, Indiana U. Math. J., to appear. [121 Evans, L. C., Nonlinear systems in optimal control theory and related topics, in J. M. Ball (ed.), Systems of Nonlinear Partial Differential Equations, (NATO AS1 Series C No. 1 1 1 , Reidel, Boston, 1983) 1131 Evans, L. C. and H. Ishii, Nonlinear first order PDE on bounded domains, to appear. [14] Evans, L. C. and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Mathematics Research Center TSR 2492, University of WisconsinMadison ( 1983 ). 1151 Fleming, W. H., The convergence problem for differential games, J. Math. Analysis and Applications, 3 (1961), 102-116. [161 Fleming, W. H., The convergence problem for differential games 11, Advances in Game Theory, Ann. Math. Studies 52, Princeton U. [171 Fleming, W. H., and R . Rishel, Deterministic and Stochastic Optimal Control, (Springer Verlag, New York, 1975 1. (181 Friedman, A . , Differential Games, (CBMS publications 18, AMS, Providence, 1974). I191 Isaacs, R., Differential Games (Wiley, New York, 1965). [ZO] Ishii, H., Viscosity solutions of Hamilton-Jacobi equations with discontinuous Hamiltonian and differential games, in preparation. [ 2 1 ] Ishii, H., Uniqueness of unbounded solutions of Hamilton-Hacobi equations. to appear. 1221 Jensen, R., in preparation. [23] Lenhart, S. M., Semilinear approximation technique for max-min type HamiltonJacobi equations over finite max-min sets index sets, Non. Anal., Theor. Meth. Appl., to appear. [241 Lions, P. L., Isaacs equations, to appear in the Encyclopedia of Systems and Control, Pergamon, Oxford. 1251 Lions, P. L., Sur les Bquations de Monge-Ampsre, I and 11, Manuscripta Math. 41 (19831, 1-43 and Arch. Rat. Mech. Anal., to appear. 1261 Lions, P. L., Rgsolution d’gquations quasilingaires, Arch. Rat. Mech. Anal. 74 (1980), 335-353. [271 Lions, P. L., Existence results for first-order Hamilton-Jacobi equations, Ricerche Mat. Napoli (to appear). [28] Lions, P. L., Generalized Solutions of Hamilton-Jacobi Equations, (Research Notes in Mathematics 69, Pitman, Boston, 1982).

M.G. Crandatl und P.E.

142

Sougunidis

[29] Lions, P. L.,

[30]

[31]

[32]

[331

Optimal Control of diffusion processes and Hamilton-JacobiBellman equations, Parts 1-3, (to appear in Corn. P. D. E. and Nonlineas Partial Oifferential Equations and Their Applications, College de France Seminar, Vol. V, Pitman. London). Lions, P. L., Hamilton-Jacobi-Bellman equations and the optimal control of stochastic systems, Proceedings of the International Congress of Mathematicians, Warsaw, 1983. Lions, P. L. and M. Nisio, A uniqueness result for the semigroup approach associated with the Hamilton-Jacobi-Bellman equations, Proc. Japan Acad. 58 (1982). 273-276. Souqanidis, P. E., Approximation schemes for viscosity solutions of Hamilton-Jacobi equations, Thesis and Mathematics Research Center TSR 2511, University of Wisconsin-Madison, 1982. (Revised and submitted in two parts.) Souganidis, P. E., Existence of viscosity solutions of Hamilton-Jacobi equations, J. Diff. Equa., to appear.

Sponsored in part by the United States Army under Contract NO. DAAG29-80-C-0041 and in part by the National Science Foundation under Grant No. MCS-8002946.

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

143

RELATIVISTIC MOLECULES WITH COULOMB INTERACTION Ingrid Daubechiesl, E l l i o t H. Liebl Departments of M a t h a t i c s and Physics Princeton University Princeton, New Jersey 08544 U. S.A.

=/=-

As an approxination t o a r e l a t i v i s t i c one-electron

m l e c u l e , we study the operator H

Z e ZK j = 1 X-R j

2

'

with Z . 2 0 for a l l j . H is bounded below i f f 32 e 2 Z j G f o r a l l j Under t h i s condition, w e show t h a t

.

1) t h e system i s stable when t h e nuclear repulsion is taken i n t o account, i.e. Eo + E

inf spec H.

0

2 ) the ground state energy E is an increasing function 0 of the internuclear distances j k

IR-R

I.

1. INTRODUCTION

The problem of " s t a b i l i t y of mTter" consists i n proving t h a t a system of charged p a r t i c l e s (electrons and nuclei), interacting electromgnetically, does not collapse. In t h e f r m o r k of n o n r e l a t i v i s t i c Schrijdinger quantum mechanics, with Coulomb interactions between the p a r t i c l e s , a f i r s t p m f of t h i s w a s given by F. &son and A. L e n 4 [ 11 A shorter proof , leading t o a much b e t t e r bound on t h e binding energy per electron, was l a t e r given by E. Lieb and W. Thirring [ 2 1 . The strategy these p m f s followed was first t o consider t h e nuclei fixed (i.e. with i n f i n i t e mss) ; t h e general case (nuclei with f i n i t e mss) then follows easily. , t h e problem then consists in With t h e K nuclei a t fixed positions R proving t h a t 1) t h e Hamiltonian descri&g * ' . N e ectmns and K nuclei (including a l s o t h e repulsion term between the electrons, and between the nuclei) is bounded below by a constant independent of t h e ; 2 ) t h e energy per p a r t i c l e , i.e. t h e ground state energy of t h e system of N e ectrons and K nuclei, divided by (N + K), is bounded below by some constant.

.

%

7

If electrons w e r e bosons, it i s known [ 3 ] t h a t mtter would not be s t a b l e : statement 1) above s t i l l holds f o r bosons, but 2 ) does not.

A firstapproxirration t o a r e l a t i v i s t i c approach of t h e problem of s t a b i l i t y of m t t e r is given by a study of the Hamiltonian

where a = e2/hc

B

(137)-l, and where w e use u n i t s such t h a t h = c = 1.

I. Daubechies and E.H. Lieb

I44

It turns out t h a t , unlike t h e Schriidinger case, even statement 1) above (i.e. t h e existence of an g-independent lower bound for t h e ground state of t o the antisymmetric subspace of L2 ( IR3N)) i s not This i s t h e problem w e want t o discuss here.

For t h e one-electron-one-nucleus-case, N K = 1, t h e operator H i 1 ( Z ) was already studied independently by k d e r [ 41 and Herbst [ 51 W e rechl some of t h e i r r e s u l t s in S2. The case where K i s a r b i t r a r y and N 1 is studied i n 5 3 , where w e show t h a t an E-independent lower bound indeed e x i s t s . W e have no results f o r t h e general case (both N and K a r b i t r a r y ) , but we present a few c o m n t s on t h i s case in 54. For each of our theorems we s h a l l give a sketch of t h e proof, with t h e min ideas ; t h e f u l l d e t a i l s s h a l l be published elsewhere [ 6 1 .

.

2 . ‘IRE ONE ELECTRON, ONE NUCLEUS CASE

For N = K

1, t h e operator HlY1

i s given by

H1,1(Z) = (-A t m 2 ) l j 2

-

Z

~Ix)-~

.

There e x i s t s a c r i t i c a l value Zc for t h e nuclear charge Z i n t h i s operator [ 4 , 5 ] For 2 Q Zc t h e operator H 1 , 1 ( Z ) , considered as a quadratic form on Q( IpI 1 D ( ( P [ ~ /=~ I)f E L2 ( R3) ; ld3k l f ( k ) ) 2 ) k ) < - 1 , is bounded below by zero, and can therefore be extended unambiguously (Fbiedrichs extension) t o a positive s e l f a d j o i n t operator. For Z > Zc t h e quadratic form i s unbounded below, and it i s not c l e a r which s e l f a d j o i n t extension should be chosen. The c r u c i a l lemma establishing t h e existence of t h i s c r i t i c a l charge Zc i s t h e following estimate, where IpI denotes t h e operator LennTk31

This e s t k t e can be traced back t o &to 171. The constant n / 2 is t h e sharp constant i n t h i s inequality as w a s shown by Herbst [ 51, and in a d i f f e r e n t way by Kovalenko e t a l . [ 81

.

Using t h i s l e m one e a s i l y sees t h a t t h e numerical value of t h e c r i t i c a l charge is Zc 2/(na) z 87.2. The existence of a critical nuclear charge i s typical f o r r e l a t i v i s t i c d e l s involving point nuclei ; f o r t h e Dirac equation t h e critical charge is cr-l z 137, for t h e Klein-Gordon equation ( 2 a ) - ’ Z 68.5.

For s t r i c t l y s u b c r i t i c a l nuclear charge (Z < Zc) more i n f o m t i o n concerning t h e spectrum of H 1 , 1 is hown 4,Sl : t h e e s s e n t i a l spectrum of H1 1 is then [my:) , and i s purely absolutely continuous ; t h e d i s c r e t e spectrum of ’H1,l i s contamed i n the i n t e r v a l 0,m) and consists of i n f i n i t e l y m y eigenvalues accumulating a t m and separated from 0 by a gap which increases with Zc - Z :

3. THE

ONE-ELECTRON MOLECULE:

I n t h i s section we s h a l l discuss t h e case N = 1, K a r b i t r a r y . A s an h e d i a t e consequence of lm 1, H ~ , K , considered as a quadratic form on Q ( l p l ) , i s unbounded below whenever any Z is s u p e r c r i t i c a l (Z > Z ) . W e shall therefore Itcis r a t h e r straightforward always restrict ourselves t o the case zk G z,, a l l t o show t h a t in t h i s case H ~ , Kcan be defined as a semibounded quadratic form on Q( \ p i ) , and hence extended t o a self-adjoint operator by means of t h e Friedrichs extension. It i s not obvious, however, t h a t an E-independent bound f o r exists.

E.

145

Relativistic Molecules with Coulomb Interaction Let us define HY,K t o be t h e 1-electron hamiltonian without nuclear repulsion :

H;,~ I,

K

I:K zk a Ix -$I -1 k=l It i s a simple consequence of t h e unboundedness below

(-A

+ m2 ) 1 / 2 -

C $ > Zc. k=l of H 1 , 1 ( Z ) f o r Z > Zc t h a t t h e ground state energy for t h e internuclear distances shrink t o zero :

Suppose t h a t

Zk
,

K

C

k.1

Z

k

HiyK w i l l tend t o -

m

if

> Z

C

Under $$e same conditions however, t h e nuclear repulsion term U(Z,R) = C Zk Z1 a IRk - R1l-l tends t o + m. The existence of an E->Zependent lower k,l=l k < l bound for H1 is therefore only possible i f t h e nuclear repulsion is strong Hq K. No e t h a t f o r A enough t o ov&ome t h e "collapsing tendency" present t e n d k g t o z e r o both U(z, AR) and t h e leading terni of inf u [ H I K (2,AR)] behave like a constant multiplied by X - l . A proof t h a t t h e nuclea?? repulsTon is stronger than t h e collapsing tendency k~ Hq,K requires therefore a r a t h e r d e l i c a t e estirrate of these constants. In order t o achieve t h i s , we s h a l l need t h e following refinement of l e m 1.

K(Z,E)

15

By scaling one e a s i l y sees t h a t it is s u f f i c i e n t t o pmve (1) f o r R l.breover, it i s s u f f i c i e n t t o pmve (1) for $ positive, spherically symmetric and decreasing i n 1x1 : since 1pl-l i s equivalent to a convolution by (2rr2) XI-^, one can use the generalized rearrangement i n e q u a l i t i e s proved in [ 9 ] t o show t h a t both t h e l e f t hand s i d e s of (1) and t h e second term of t h e r i g h t hand s i d e increase if $ is replaced by P, t h e symmetric rearrangement of $. For $(r),defined f o r 0 G r < 1, w e consider f ( r ) = $(r)i f r G 1, f ( r ) = F3$ (r-l) if r > 1. Because of l e m 1, w e have

0

0

(1) follows then f r o m min uq 0,ll

(u-'

l+u I n -1 1-u = 2

This e s t k m t e can then be used t o prove t h e following proposition, s t a t i n g t h a t t h e nuclear repulsion i s indeed stronger than t h e collapsing tendency, in t h e case Zk Zc ( i . e . = 2 / a ) , a l l k.

Props it ion

146

where

1. Daubechies and E. H. Lieb

Vkfx) = Ix -

51-l -

3

cK

1=1 1% 1fk

-1 51 .

-

It is c l e a r l y s u f f i c i e n t t o prove ( 3 ) with vk replaced by (vk)+(x) = mx (0, vk(x)). For each k, (Vk)+ has support i n t h e b a l l B(Rk,tk) where ti1=3 Z 1Rk-RlI-l. l#k In particular, for each pair (k, 1) with k f 1, tk + tl Q ( 2 / 3 ) IRk - R1I ; t h i s implies t h a t t h e balls %, B1 are d i s j o i n t , with a mhinnun distance superior t o I % - R1 I /3 between them. 2 f i R O V f T , On each ball Bk, (vk)+ G I X Xk 4 , where Xk i S t h e c h a r a c t e r i s t i c function of t h e set Hence

(Vk)+ G

Z

k

Wk2

C

k

(z W,)2 k

-’

%.

where w e have used t h e fact t h a t t h e balls

%

are d i s j o i n t in t h e last equality. Therefore ( 3 ) will be established i f 1pl - (2111) equivalently, i f ( 2 / * ) (Z W have k k

(E Wk)2

]pi-’ (Z W 1 < 1 on L2( 1 1

3

0 on Q ( f p I), or

1. For f in L 2 ( R 3 )

,w e

kfl

For t h e f i r s t terms we can apply l e m 2 , and write

Ix/-~

For t h second set of terms i n (4), w e rewrite \ p i - ’ as convolution with (2r2)-’ ; t h e f a c t t h a t t h e distance between t h e balls %, B1 i s l a r g e r than IRk - R 1 / 3 leads t o an upper und on these terms, which turns out t o be outbalanced by t h e t e r n in tipin t h e upper bounds ( 5 ) . Hence k

1

k

which pmves ( 3 ) Since IpI Q (-A + m2 ) 1/2, and since (12/a) < 2,2 a = ( 2 / 1 r ) ~ a - l , it is clear t h a t (2) proves H1 (Z,R) 2 0 provided t h a t each zk = 2, or 0. A simple concavity argument (see’f61T Tnables us t o extend t h i s r e s u l t t o t h e general case 0 G zk G a l l k. W e have thus proved t h e following theorem :

zc,

Theorem

For any {Zk ; k = 1 kfll:

. ..K, 0

Zk

Z,}

a d f o r m y {Rk ; k

1

. -. K, Rk

# R1 i f

147

Relativistic Molecules with Coulomb lnteruction with b = (12/~)(2: CX)-’%

6.88

x

.

(Z,R) is bounded below by zero, Since b 1, this theorem proves that H1 independently of E. W e h o w even mre : bmce b is s t r i c t l y smller than 1, t h e nuclear repulsion not only prevents collapse, but even pushes t h e nuclei apart i f they COE t o c l o s e t o each other. A m t from t h e above result, w e also have moved mnotonicitv of the m u n d state energy of Hq (Z,R) in the-nuclear coord&ates g. Explicitly, we sh&ed that i f V i , j : \ R i - fit1 Z T R ~ - Ril , then inf G (Hi,K (Z,&))g inf 0 (HlYK U (z,g')) This result w a s hown f o r t h e schrijdinger case 10,111 ; the proofs of [10,111 , using path integral techniques, e s s e n t i a l l y carry over t o t h i s case (mdulo some technical changes ; see [ 61 ) . 4. N ELECTRONS, K NUCLEI

A s w e already m t i o n e d in t h e introduction, w e have no general result f o r %,K yet. It is nevertheless possible t o g e t some insight i n t o t h e problemusing e define t h e d i l a t i o n operator scaling arguments. W ( U ( a ) +)(XI

...

a3N'2

+ (a x1 ..

,

a%)

.

W e have then (at least forrrally)

z,~)

($

U(a)* % (m ; U ( a ) = a $,K ; 2, a R) Therefore t h e existence of an s-independent lmer bound for HN,K ( m ; 2, &) implies t h a t Hu K ( 0 ; Z,R) ( i . e . w i t h k i n e t i c energy (-A)1/2 instead of (-A t m2l1i2) 16 boundez Eelow by 0 , independently of R. On t h e other hand, since (p2 t m2I1/*, a lower bound f o r % (0 ; Z,R) is obviously a l s o a lower bound for HN %(rn ; 2,s). It is therefor; necessary and s u f f i c i e n t t o prove %,k (OY ZYRT 'O'

(PI

the Fermi statistics. For general N, K, it i s obvious t h a t Fermi statistics and t h e electrun repulsion w i l l have t o play a role. The strategy of 121 , which used a density functional lower bound f o r t h e k i n e t i c energy of N fermions, and then applied results of Thorns - Fermi theory, would not work however : t h e lower bound on t h e k i n e t i c energy f o r the operator Ip I has t h e form C . l l $(x) I 8 / 3 , and t h e corresponding Thorns - Fermi functional is not bounded below. FOGTNOTES "Work p a r t i a l l y supported by U. S. National Science Foundation grant PHY-8116101-AO1. "Research fellow a t t h e I n t e m i v e r s i t a i r I n s t i t u u t vcor Kernwetenschappen, Belgium. Present address : Theoretische Natuurkunde, Fakulteit Wetenschappen, Vrije Universiteit Brussel, Pleinlaan 2 , B - 1050 BruSsels, Belgium. REEERENCES

[11 Dyson F . , L e n d A . , S t a b i l i t y of mtter I, J. Math. Phys. 8 (1967) 423-434. Len& A . , @son F., S t a b i l i t y of mtter 11, J. Math. Phys. 9 (1968) 698-711.

1 Daubechies and E H Lieb Lieb E., Thirring W., A bound f o r t h e k i n e t i c energy of fermions which proves t h e s t a b i l i t y of mtter, E hys. Rev. Lett. 35 (1975) 687-689 ; b a t a : Phys. Rev. Lett. 35 (1975) 1116. See also : Lieb E., The s t a b i l i t y of m t t e r , Rev.

Mod. Phys. 48 (19761 553-569. Dyson F., Ground s t a t e energy of a f i n i t e system of charged particles, J. Math. Phys. 8 967) 1538-1545. Lieb E., The N5j3 l a w f o r bosons, phys. Lett. A 70 (1979) 71-73. Weder R . , Spectral analysis of pseudodifferential operators, J. Funct. Anal. 20 (1975) 319-337. 2 2 112 - Ze2 /r, Corn. &th. Herbst I., Spectral theory of t h e operator (p im ) Phys. 53 (1977) 285-295. Daubechies I., Lieb E . , One electron r e l a t i v i s t i c m l e c u l e s with coulomb interaction, t o be published in Corn. Math. E’hys

.

G t o T., Perturbation theory for l i n e a r operators (Springer, Berlin and New York, 1966 ; 2nd. ed. 1976). Kovalenk V., Perelmuter M., Semenov Ya., Schr6dinger operators with L1l2 R ) p o t e n t i a l s , J. Math. Phys. 22 (1981) 1033-1044.

w

P

Brascamp H., Lieb E . , L u t t h g e r H . , A general rearrangement inequality for multiple integrals, J. F’unct. Anal. 17 (1974) 227-237. Lieb E., S h n B., Monotonicity of t h e electronic contribution t o t h e BornOppenhekr energy, J. Phys. B 11 (1978) L 537-542. Lieb E., Monotonicity of t h e m l e c u l a r electronic energy in t h e nuclear coordinates, J. Phys. B 15 ( 1 9 8 2 ) L 63-65.

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

1 49

NON-LINEAR DELAY DIFFERENTIAL EQUATIONS AND FUNCTION ALGEBRAS Lance D. Drager and W i l l i a m Layton* School o f Mathematics Georgia I n s t i t u t e of Technology A t l a n t a , Georgia 30332 U.S.A.

We w i l l d e s c r i b e some r e s u l t s c o n c e r n i n g q u a l i t a t i v e p r o p e r t i e s o f bounded s o l u t i o n s o f t h e non-linear ( s c a l a r ) delay d i f f e r e n t i a l equation x ' ( t ) + g ( x ( t ) , X(t-T)) = f ( t ) (11 where T i s a f i x e d c o n s t a n t ( n o t n e c e s s a r i l y p o s i t i v e ) and f i s a g i v e n f o r c i n g f u n c t i o n , which we w i l l always assume i s bounded and continuous.

I n p r e v i o u s work, we have s t u d i e d a non-resonance c o n d i t i o n which assures t h a t ( 1 ) has a unique ( g l o b a l l y ) bounded s o l u t i o n . i t i s a l s o shown t h a t if f i s p e r i o d i c [resp.

T h i s i s proven i n [5],

where

almost p e r i o d i c ] t h i s bounded s o l u -

t i o n i s p e r i o d i c [almost p e r i o d i c ] (see a l s o [7],

[2-41).

Our goal here i s t o e x t e n d t h e s e r e s u l t s and p l a c e them i n a g e n e r a l framework by c o n s i d e r i n g c e r t a i n subalgebras o f t h e a l g e b r a o f bounded c o n t i n u o u s funct i o n s on t h e l i n e . To do t h i s , we w i l l need t o o u t l i n e t h e p r o o f g i v e n i n [5] o f t h e e x i s t e n c e and uniqueness o f t h e bounded s o l u t i o n o f ( 1 ) . We f i r s t s t a t e o u r non-resonance c o n d i t i o n and d i s c u s s some p r e l i m i n a r i e s . 2 2 L e t R 5 R be t h e r e g i o n R = { ( a , b ) E R I la1 I l b l l ( s e e F i g . 1 ) .

Our

non-resonance c o n d i t i o n is: CONDITION (NR):

g: R 2

+

R s a t i s f i e s c o n d i t i o n (NR) i f g i s C 1 and t h e

image o f t h e g r a d i e n t o f g i s bounded and bounded away f r o m R (see F i g . 1 ) .

=

0, so o u r r e s u l t s w i l l Note t h a t c o n d i t i o n ( N R ) may be s a t i s f i e d i f g Y a p p l y t o ODES o f t h e form x ' ( t ) + g ( x ( t ) ) = f ( t ) , where c o n d i t i o n (NR) reduces t o

t h e r e q u i r e m e n t t h a t g ' i s bounded and bounded away f r o m zero.

0 denotes t h e Banach a l g e b r a o f bounded c o n t i n u o u s f u n c t i o n s R R with k t h e supremum norm I / - 1 ) . BC denotes t h e space o f f u n c t i o n s w i t h k d e r i v a t i v e s BC

i n BC

0

.

-f

If A

BCk i s a subspace, Ak 5 BCk denotes t h e space o f f u n c t i o n s w i t h

k d e r i v a t i v e s i n A. x,y

E

We use t h e n o t a t i o n x,(t)

BCo, we denote by g(x,y)

the f u n c t i o n t

= x(t-T)

t+

f o r t r a n s l a t e s , and i f

g(x(t),y(t)).

Purely t o avoid

t h e t e c h n i c a l i t i e s o f c o u n t i n g d e r i v a t i v e s , we assume g i n ( 1 ) i s Cm. We need t o r e c a l l some f a c t s about t h e elementary ODE

*Work o f t h e second a u t h o r p a r t i a l l y supported by NSF Grant MCS-820202s.

L. D. Drager and W. Lavton

150

Lax(t) = x ' ( t )

+

a E R, a # 0 (2) 0 BC , ( 2 ) has a unique bounded

ax(t) = f ( t )

By elementary techniques, we see t h a t i f f

E

In fact

s o l u t i o n , which we denote by La'f.

From ( 3 ) , i t i s easy t o see t h a t La1 i s a bounded l i n e a r o p e r a t o r 6C0 -1 k k+l w i t h o p e r a t o r norm s l / l a l . O f course, La BC c BC .

THEOREM ( 1 ) : solution x BCk

+

BCk+',

E

BC

0

The p r o o f , and

We now o u t l i n e t h e p r o o f o f t h e f o l l o w i n g theorem f r o m [5]. condition ( N R ) , are i n t h e s p i r i t of

+

[a].

I f g s a t i s f i e s c o n d i t i o n (NR) and f c BC0 , ( 1 ) has a unique

BCo. which we denote by x = S ( f ) .

S i n a (non-linear) b i j e c t i o n

w h i c h we c a l l t h e s o l u t i o n o p e r a t o r o f ( 1 ) .

To prove t h e theorem, we r e w r i t e ( 1 ) as L ~ X=

+ ax = ax - g(x,x )

X I

where a # 0 w i l l be chosen below.

-

f

T h i s i s equivalent t o

x = L~-1 ( a x - g ( x , x , ) ) where we d e f i n e N a ( x ) = ax

+

+ Lilf

= F ~ ( X )+

-1

L~-1 f

(4)

(4) i s a fixed point

g ( x , x 7 ) and Fa = La N.,

problem i n 8C0, which we w i l l s o l v e w i t h t h e C o n t r a c t i o n Mapping Lemma.

To begin,

an a p p l i c a t i o n o f t h e Mean Value Theorem y i e l d s

IINa(x)

-

Na(y)lI

5

(5)

Ka IlX-Yll

I

K~ = s u p i l a - g x ( u , v ) 1 + l g ( u , v ) ~ Y

(u,v)

R* I

and so lIFa(x)

-

Fa(y)Il

5 aa

Ilx-yll

,

aa =

Ka/lal.

The t r i c k i s t o use C o n d i t i o n ( N R ) t o choose a so t h a t an we p u t t h e image o f t h e g r a d i e n t of g i n s i d e a square

<

1.

I n t h e n o t a t i o n of F i g . 1, an elementary e s t i m a t e g i v e s l a - h l + I k l l a - s l ) f o r a l l a and a l l (h,k) e max( l a - r l , l a - s l ) .

To do t h i s ,

D, as i l l u s t r a t e d i n F i g . 1.

D (see [6] for d e t a i l s ) .

5

max(la-rl,

Thus Ka

5

I f we choose (a.0) t o be t h e c e n t e r o f 0, a = (r+s)/2 we have

Ka ( r - s ) / 2 < ( r + s ) / 2 = a, so f o r t h i s c h o i c e of a, aa c 1. I f a a < 1 , we say a i s c o m p a t i b l e w i t h g. The theorem now f o l l o w s from t h e C o n t r a c t i o n Mapping Lemma k and a b o o t s t r a p arguiiient. I n [ 5 ] i t i s shown t h a t S i s a d i f f e o m o r p h i s m f r o m BC t o BCk+'. We now observe t h e f o l l o w i n g lemma, which f o l l o w s f r o m t h e p r o o f o f t h e

15 1

Non-Lineur Delay Difjercwtiul Equutions and Function Algebras

Fig. 1 C o n t r a c t i o n Mapping Lemma. Assume g s a t i s f i e s c o n d i t i o n ( N R ) .

LEMMA ( 2 ) :

If A

c

BCo i s a c l o s e d sub-

space s a t i s f y i n g t h e c o n d i t i o n s (Al)

x

(A2)

LilA 5

E

A =>

x

A,

t

A.

fir some a c o m p a t i b l e w i t h g.

g(A,A) 5 A, i . e . ,

(A3) t h e n S(A) 5 A,

i.e.,

x,y

if f

6

c

A => g(x,y)

6

A.

A, t h e bounded s o l u t i o n o f ( 1 ) i s i n A.

The problem now i s t o f i n d ways t o v e r i f y t h e s e c o n d i t i o n s . s i d e r (A3), which i s handled by c o n s i d e r i n g subalgebras o f BC

0

.

We f i r s t conI f x,y

c

0

BC ,

we can a p p l y t h e Stone-Weirstrass Theorem t o approximate g u n i f o r m l y by p o l y n o m i a l s on t h e image o f ( x , y ) . LEMMA ( 3 ) : algebra.

L e t g : R2

-f

T h i s g i v e s us t h e f o l l o w i n g s t a n d a r d lemma.

R be c o n t i n u o u s and l e t A 5 BCo be a c l o s e d sub-

I f A contains the constant functions ( i . e . ,

1 k A b u t g(0,O)

= 0 we s t i l l have g(A,A)

1

E

A), g(A,A)

5 A.

5 A.

To s t a t e our main theorem, we make t h e f o l l o w i n g d e f i n i t i o n s . Let g s a t i s f y c o n d i t i o n (NR) and l e t A 5 BCo be a c l o s e d DEFINITIONS:

If

152

L.D. Drager and W.Layton

I f A s a t i s f i e s (Al) and ( A 2 ) , we say A i s a p a r t i a l l y (9-T)compatible subalgebra. I f , in addition, we have e i t h e r 1 t A o r g(0,O) = 0 we say A i s a (g,T)-compatible subalgebra. W e have now proven most of t h e following theorem. THEOREM ( 4 ) (Compatible Subalgebras Theorem): I f g s a t i s f i e s condition ( N R ) and A i s a (g,T)-compatible subalgebra, then S(A) 5 A. I n f a c t S(Ak) 5 Ak+’. W e p o i n t o u t what happens when A i s only p a r t i a l l y (g,T)-compatible. As a preliminary observation, note t h a t i f G: R lR i s defined by G(n) = g(n,q), subalgebra.

-f

condition (NR) implies 9’ i s bounded a n d bounded away from zero, so i s a diffeomorphism from I? t o R . COROLLARY ( 5 ) : Assume g s a t i s f i e s dondition (NR) and A i s a p a r t i a l l y (g,?)-compatible subalgebra. I f f c A and c i s a constant, S(f+c) E <+A, where 5 = ;-‘(c). In f a c t S(c+Ak ) 5 S+Ak+l. In p a r t i c u l a r S(Ak ) 5 SO+Ak+l,where E0 = -1 9 (0). To prove t h e c o r o l l a r y , assume 1 k A. Define s ( q , c ) = g(q+c, <+c)-c. g a l s o s a t i s f i e s condition ( N R ) and G(0,O) = 0. T h u s , i f f E A k , t h e bounded solution of + g(x,xT) = f i s in Ak+’. B u t then (;+t)’ + g(x+<,(x+C),) = ftc, k so S ( f + c ) = <+S c <+A . We remark t h a t t h i s corollary i s a simple case o f a more general theorem, which we will discuss b r i e f l y in the closing remarks. Our theorem w i l l n o t be useful until we have a way of checking condition ( A Z ) . As a f i r s t approach note t h a t ( 3 ) says, i n t u i t i v e l y speaking, t h a t Li’f

-

<

m

y

-

i s a weighted sum of t r a n s l a t e s of f and t h a t only forward [resp. backward] t r a n s l a t i o n s appear i f a > 0 [a < 01. I t i s not, however, s u f f i c i e n t f o r A t o be a closed subspace i n v a r i a n t under t h e appropriate t r a n s l a t i o n semigroup in order t o have 5 A. The d i f f i c u l t y i s t h a t t h e t r a n s l a t i o n group i s not well 0 behaved on BC . J u s t applying t h e d e f i n i t i o n s , we see t h a t t h e map IR + BCo: s I+ f s ( f s ( t ) = f ( t - s ) ) i s (norm) continuous exactly when f i s uniformly continuous. As matters stand, we a r e n o t able t o completely characterize t h e L,’ 0 invariant subspaces of BC (note t h i s i s purely a question about the simple ODE

Li’A

( 2 ) ) . I n the case where A c o n s i s t s of uniformly continuous functions, we can use ( 3 ) t o write L a 1f , f e A , as a BC 0 -valued Riemann i n t e g r a l , which proves the

following standard lemma.

LEMMA ( 6 ) : Let A 5 BCo be a closed subspace consisting of uniformly continuous functions. I f A i s i n v a r i a n t under a l l forward [resp. backward] t r a n s l a t i o n s , L;’A 5 A f o r a l l a > o [a < 01. We now give some examples of the applications o f these r e s u l t s . We assume g s a t i s f i e s condition ( N R ) and, unless otherwise s t a t e d , (A2) i s v e r i f i e d by applying Lemma ( 6 ) . When Lemma 6 does not apply, we give an ad hoc argument. EXAMPLE 1: CONSTANT FUNCTIONS: The subalgebra C of constant functions i s (g,T)-compatible f o r any g and

T,

so S ( C ) c C. Of course, t h i s can be seen

153

Non-Linear Delay Differential Equations and Function Algebras

d i r e c t l y from ( l ) , and we see t h a t the bounded solution of x ’ + g(x,x,)

=

c is

;-1 ( C ) . EXAMPLE 2: PERIODIC FUNCTIONS: Let Pw 5 BC 0 be the w-periodic functions. Pw i s t r a n s l a t i o n i n v a r i a n t , and we s e e t h a t Pw i s (g,T)-compatible f o r any g , T and S(Po) 5 Pu. EXAMPLE 3: ALMOST PERIODIC FUNCTIONS, MODULE CONTAINMENT, QUASI-PERIODIC 0 FUNCTIONS: Let AP 5 BC be t h e (uniformly) almost periodic functions (see [ l ] f o r a l l the f a c t s we need about almost periodic functions). AP i s a closed transl a t i o n i n v a r i a n t subalgebra o f BCo, and AP i s (g,r)-compatible f o r a l l g and T , so S(APk) c APk+’. I f f E AP, modff) denotes the module ( i . e . , t h e a d d i t i v e subgroup of R ) generated by t h e Fourier exponents of f . I f mod(f) i s f i n i t e l y generated, f i s called quasi-periodic. I f A 2 R i s a module, l e t AP,, = {f E APlmod(f) 5 A}. I t can be shown t h a t AP,, i s a closed t r a n s l a t i o n invariant 0 subalgebra of BC , and AP,, i s (g,r)-compatible f o r a l l g and T , so S(APA) 5 AP,,. Applying t h i s r e s u l t with h = mod(f), we see mod(S(f)) 5 mod(f). In f a c t , we have mod(S(f)) = mod(f). To see t h i s , l e t x = S ( f ) and A = mod(x). We know x ’ t AP, so modjx’) 2 mod(x) (see Thm. 1.17 of [I]). Thus f = x ’ + g(x,xT) t APA, so mod(f) 2 A = mod(x) 5 mod(f) and we conclude mod(S(f)) = rnod(f). I n p a r t i c u l a r , i f f i s quasi-periodic, S ( f ) i s quasi-periodic with the same fundamental frequencies. Of course, t h e periodic case i s j u s t t h e case where mod(f) has one generator. EXAMPLE 4: STEADY STATES AT +m: Let L+ be the subspace BC 0 consisting of functions f f o r which f ( - ) = lim f ( t ) e x i s t s . L+ i s a closed t r a n s l a t i o n 0 tinvariant subalgebra of BC . In this case, we cannot apply Lemma ( 6 ) , b u t i f we 1 + + apply the Dominated Convergence Theorem in ( 3 ) , we see t h a t L a L 5 L f o r a l l + a # 0. L+ i s (g,T)-compatible f o r a l l g and T , and S ( L ) 5 A+, i . e . , i f the forcing f approaches a steady s t a t e f(m) as t

+

-, t h e bounded solution x

= S(f)

of ( 1 ) approaches a steady s t a t e x(-). W e can use Corollary ( 5 ) t o determine + + + . + Lo = If E L If(-) = 01. Let L o i s p a r t i a l l y (g,T)-compatible. If f t L , x(~). + f = f(m) + fo. where f o E Lo. By Corollary ( 5 ) , x = S ( f ) = S(f(m) + f ) = + xo, 0 where xo E Li and 5 = ;-’(f(m)). Thus x(-) = 5 . O f course, s i m i l a r considerations apply a t -m. EXAMPLE 5: CAUSAL SUBSPACES: Up t o t h i s p o i n t , we have not needed any r e s t r i c t i o n s on g and T beyond condition ( N R ) . In t h i s example we need f u r t h e r r e s t r i c t i o n s . For s i m p l i c i t y , assume g(0,O) = 0. Let K = {f E BC 0 I f ( t ) = 0 f o r t 0 3 . We can i n t e r p r e t x = S ( f ) , where f E K , as the response of a system governed by ( l ) , t o which no external forcing i s applied u n t i l time t = 0. I t i s natural t o ask i f t h e system shows any response (x # 0 ) before time t = 0, i . e . , i s S ( K ) 5 K? We note t h a t K i s i n v a r i a n t only under forward t r a n s l a t i o n s . From ( 3 ) we see L i ’ K 5 K i f a > 0 , b u t not i f a < 0. T h u s , K will be ( g , T ) -

L. D.Drager and W. Layton

154

compatible i f we require T 2 0 (which i s n a t u r a l , b u t has n o t previously been necessary) and t h a t the image of the gradient of g l i e s t o t h e right of R. In t h i s case, we say g i s forward non-resonant. T h u s , forward non-resonance can be interpreted as a c a u s a l i t y condition and backwards non-resonance an " a n t i c a u s a l i t y " condition. W e More d e t a i l s on t h e s e r e s u l t s a n d f u r t h e r examples may be found in [6]. a l s o prove there an extension of these r e s u l t s , which we c a l l t h e Perturbation by Ideals Theorem. As a n example, we consider t h e following question. Example 5 makes i t natural t o ask i f the operator S i s causal i f T 2 0 and g i s forward non-resonant. To say S i s causal means t h a t i f f l ( t ) = f 2 ( t ) f o r t I to, S ( f l ) ( t ) = S ( f 2 ) ( t ) f o r t 5 to. The reasoning o f Example 5 shows t h i s i s t r u e f o r f = 0. S i s indeed causal, and the crucial observation in p r o v i n g t h i s i s t h a t 1 0 K i s an w i n B C . We a l s o note t h a t Corollary ( 5 ) i s a special case of the Perturbation by Ideals Theorem. applications, see [6].

For t h e statement of t h i s theorem and f u r t h e r

REFERENCES

1

121

Corduneanu, C . , Almost Periodic Functions, Wiley Interscience, New York, 1961. Drager, L. 0. and Layton, W., Parameter dependent solutions t o functional d i f f e r e n t i a l equations subject t o almost periodic forcing.

C31 -,

On the smoothness of t h e solution operator f o r non-resonant non1 inear functional di f f e r e n t i a1 equations.

r41

-,

On nonlinear difference approximations t o nonlinear functional d i f f e r e n t i a l equations.

r 5 1 -,

Periodic, almost periodic and bounded s o l u t i o n s t o functional d i f f e r e n t i a l equations.

" Q u a l i t a t i v e properties of bounded s o l u t i o n s of delay d i f f e r e n t i a l [61 -, equations and algebras and i d e a l s of bounded continuous functions," preliminary report.

c 71

Layton, W . , Existence of almost periodic s o l u t i o n s t o delay d i f f e r e n t i a l equations with Lipschitz n o n l i n e a r i t i e s , t o appear.

181

Mawhin, J . 3 Contractive mappings and periodically perturbed conservative systems, Seminaire de Mathematiques AppliquGes e t Mgcanique, Rapport 80. Louvain-la Neuve; Presses Universitaires, 1974.

DIFFERENTIAL EQUATIONS LW. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

TRANSFORMATIONS OF DIFFERENTIAL EQUATIONS, THE LEVINSON ASYMPTOTIC THEOREM AND DEFICIENCY INDICES

M.S.P.

Eastham

C h e l s e a C o l l e g e ( U n i v e r s i t y of London) London SWlO OUA, U.K.

The l e c t u r e

s u r v e y s r e c e n t r e s u l t s , a n d g i v e s new o n e s ,

c o n c e r n i n g t h e a s y m p t o t i c form of s o l u t i o n s of h i g h e r order equations.

A s y s t e m a t i c method f o r t r a n s f o r m i n g

t h e e q u a t i o n s t o t h e L e v i n s o n form Y '

= ( A + R)Y is

d i s c u s s e d a n d a p p l i e d t o o b t a i n a n a l o g u e s of t h e Liouville-Green asymptotic formulae.

Further appli-

c a t i o n s c o n c e r n t h e e v a l u a t i o n of d e f i c i e n c y i n d i c e s f o r even a n d o d d - o r d e r d i f f e r e n t i a l e q u a t i o n s a n d , f o r r i g h t - d e f i n i t e problems, t h e D i r i c h l e t i n d i c e s .

1,

THE GENERAL METHOD

W e consider the first-order

d i f f e r e n t i a l system

Y’

(1.1)

= AY

on a n i n t e r v a l [ X I = ) , where A i s a n nxn m a t r i x a n d Y i s an n-component vector.

I n d e t e r m i n i n g t h e a s y m p t o t i c n a t u r e of n l i n e a r l y

i n d e p e n d e n t s o l u t i o n s Y, t h e aim i s t o t r a n s f o r m ( 1 . 1 )

u'

= (A

f

i n t o t h e form (1.2)

R)U

and a p p l y t h e b a s i c a s y m p t o t i c theorem of L e v i n s o n [ 2 , p . 6 ] . A is a diagonal matrix,

A = dg

(A .), 3

u s u a l dichotomy c o n d i t i o n on t h e A

U as x

+-

j

( iXX

Aj(t)dt)

is the j-th coordinate vector.

form of t h e c o r r e s p o n d i n g s o l u t i o n s of t r a n s f o r m i n g back from U t o Y .

Here

Under t h e

( 1 . 2 ) h a s s o l u t on s

(x) = {ej + o ( l ) } e x p

j

=, where e

1'

and R i s L ( X , = )

(1.1)

(1.3)

The a s y m p t o t i c

i s t h e n o b t a i n e d on

M. S.P. Eastham

156

we assume that A has n simple eigen-

A s a first step towards ( 1 . 2 ) ,

values p . and we express A as I A = TA1 T

-1

,

(1.4)

where h l = dg ( b . ) and the columns of T are the eigenvectors v . of 7 7 A. The substitution Y = TZ

in (1.1 ) then gives

z'

= (Al

R1)Z,

+

where -1 R 1 = -T

T’

.

We have next to consider various possibilities concerning the relative sizes of h l and R 1 as x + -, as follows.

(i) R, E L(X,m). In this case (1.6) already has the form ( 1 . 2 ) and no further transformation is called for. This case represents a relatively simple situation for ( l . l ) , and a typical example is -6 - ) , where 6 > 0 and A. is constant with n A ( x ) = A. + O(x ) (x simple eigenvalues. Here we have, correspondingly, T (x)= T o + O ( x - ~ ) -f

and then R, = -T-’T’ = O(x

-6-1

).

(ii) R1 f L ( X , m ) and R 1 = o ( A l ) . entry in R 1 is o ( p . ) for each j. 7 transformation of the form 2 =

(I

By R1 = o ( A 1 ) we mean that each

In this case we apply a second

f

(1.8)

S)U

with S(x) + 0 as x -+ m . A systematic construction of S in certain m is situations where A 1 -+ A o , a constant limiting matrix, as x An alternative method which is given by Harris and Lutz 1 1 3 , 1 2 1 . also applicable in cases where A l has large diagonal entries is to repeat the procedure which led from ( 1 . 1 ) to ( 1 . 6 ) but starting now Thus we write with (1.6) -f

.

2 = TAU,

(1.9)

Asymptotic Theory of'Differentia1 Equations

where T1 d i a g o n a l i z e s A

1

5'

+

Then T1 h a s t h e form I + S i f

A t y p i c a l example of t h i s Case ( i i ) i s where t h e e n t r i e s

R1 = o ( A 1 ) .

i n A a r e ( c o n s t . ) x a f o r v a r i o u s v a l u e s of a. where R1

=

157

-T

-1

-1

T’ = O ( x

W e obtain situations

and t h e s e c o n d t r a n s f o r m a t i o n ( 1 . 8 ) l e a d s

)

t o a s y s t e m of t h e r e q u i r e d f o r m ( 1 . 2 ) .

o(R.,). H e r e a g a i n w e c a n a p p l y t h e p r o c e d u r e of d i a g o n a l i z i n g t w i c e , u s i n g ( 1 . 5 ) and ( 1 . 9 ) . Now, how(iii)R1

ever, T

and A l

L(X,m)

=

w i l l n o t n e c e s s a r i l y h a v e t h e form I+S.

1

T h i s case i s l e s s

w e l l u n d e r s t o o d t h a n C a s e ( i i ) . I t may b e o b j e c t e d t h a t t h e f i r s t transformation (1.5)

i s i n a p p r o p r i a t e because it f o c u s s e s a t t e n t i o n which i s i n f a c t n o t dominant i n t h e s e c o n d

on a d i a g o n a l m a t r i x A l transformation.

T h i s i s no known t e c h n i q u e a v a i l a b l e f o r d e c i d i n g

i n a d v a n c e , on t h e b a s i s of m a t r i x i n ( 1 . 6 ) w i l l be.

( l . l ) , what t h e d o m i n a t i n g d i a g o n a l

Thus i t i s n o t c l e a r what i s t h e b e s t

c h o i c e f o r t h e i n i t i a l t r a n s f o r m a t i o n of ( i v ) R1 L L(X,-) a n d n e i t h e r A , t u r e of t h e t e r m s i n A., (1.6).

(1.1).

= o ( R ) nor

1

R1

=

o(Al).

Here a mix-

and R1 form t h e d o m i n a n t p a r t o f h l + R 1 i n

The s a m e r e m a r k s a s i n ( i i i ) a p p l y , b u t w e a l s o r e f e r t o

C a s s e l l [l]

f o r a d i f f e r e n t s y s t e m a t i c p r o c e d u r e f o r matrices A

which have a p e r i o d i c component. I n 552-3 w e c o n s i d e r C a s e s ( i i ) - ( i v ) f o r s y s t e m s (1.1) from h i g h e r - o r d e r d i f f e r e n t i a l e q u a t i o n s .

I n 554-5

which a r i s e we Five appli-

c a t i o n s t o t h e e v a l u a t i o n of t h e d e f i c i e n c y and D i r c h l e t i n d i c e s i n t h e s p e c t r a l t h e o r y of d i f f e r e n t i a l o p e r a t o r s . 2.

ASYMPTOTIC FORMULAE OF LIOUVILLE-GREEN TYPE

W e begin with the f a m i l i a r equation

where p and q are nowhere z e r o i n ( X , m ) .

The u s u a l method of e s t a b -

l i s h i n g t h e a s y m p t o t i c form of s o l u t i o n s i n v o l v e s t h e t r a n s f o r m a t i o n

Amonst t h e c o n d i t i o n s imposed a r e t h a t p a n d q ' a r e r e a l - v a l u e d and that t

+

m

as x +

m.

T h i s l a s t c o n d i t i o n i s a-0

<

2 when p ( x ) = x a

M. S.P. Easfham

158

and q ( x ) (2.2)

=

con st.)^

8

.

I n t h e o p p o s i t e case when t f

is n o t a p p l i c a b l e a n d ( 2 . 1 )

t h e u s e of a t r a n s f o r m a t i o n [ 2 ,

-

as x

+

m ,

is investigated d i r e c t l y without

13.61.

The method of d i a g o n a l i z i n g t w i c e , which w a s o u t l i n e d i n 51, p r o v i d e s a u n i f i e d a p p r o a c h t o t h e a s y m p t o t i c t h e o r y of

(2.1).

This

method of d e a l i n g w i t h ( 2 . 1 ) h a s n o t been d e v e l o p e d i n t h e l i t e r a t u r e u n t i l now a n d , e v e n f o r ( 2 . 1 1 , which s e e m t o b e new.

it l e a d s t o a s y m p t o t i c f o r m u l a e

F u l l d e t a i l s and d i s c u s s i o n a p p e a r i n [ 7 1 b u t

o n e p o i n t t h a t c a n b e mentioned i s t h a t t h e r e i s no r e s t r i c t i o n t o real-valued c o e f f i c i e n t s .

W e w r i t e ( 2 . 1 ) i n t h e form ( 1 . 1 ) w i t h

Then, w i t h T c h o s e n s o t h a t t h e f i r s t e n t r y i n e a c h column i s u n i t y , (1.6) i s Z'

=

(A,

-

(2.3)

DA2D)Z,

where

When w e a p p l y t h e s e c o n d d i a g o n a l i z a t i o n p r o c e d u r e t o ( 2 . 3 ) , a s indicated i n § I

( i i ) - ( i v ), t h e s e t h r e e c a s e s a r i s e a c c o r d i n g a s

where c i s a n o n - z e r o c o n s t a n t .

F u r t h e r c o n d i t i o n s i n v o l v i n g second

d e r i v a t i v e s h a v e a l s o t o b e imposed, and w e r e f e r t o [ 7 ] f o r t h e s e details.

S i m p l i f y i n g t h e main r e s u l t s i n [ 7 ] ,

w e m e n t i o n t h e asymp-

t o t i c r e s u l t s which f o l l o w f r o m t h e t h r e e cases i n ( 2 . 4 ) . The f i r s t

As.ymptotic Theory of Differential Equations

159

case leads t o t h e usual Liouville-Green solutions

The s e c o n d leads t o s o l u t i o n s

and t h e t h i r d c a s e g i v e s s o l u t i o n s

where k = (1-16 c

- 2 ) 1/ 2 (C

# f4); when p ( x )

= x a and q ( x ) = c o n s t . ) x ' ,

t h e t h r e e cases a r e a-B

< 2,

a-B

> 2,

a-B

= 2

W e can a l s o consider higher-order

( p y ( " ) ) (k)

-

(an Euler equation).

e x t e n s i o n s of

( 2 . ) such a s

qy = 0 ,

a n e q u a t i o n which w a s i n t r o d u c e d by H i n t o n [ 1 4 ] , o r more g e n e r a l l y (2.6) The r e s u l t c o r r e s p o n d i n g t o (2.51, f o r example, i s t h a t t h e r e a r e solutions

where Q = W(P1

...

Pn-l)P'n

,

n

w .

7

= 1.

T h e r e a r e c o n d i t i o n s t o b e imposed on q a n d t h e pr a n d , when ar and q ( x ) = ( c o n s t . ) x ' , t h e c o n d i t i o n s become pr(x) = x

...

a + + a - B < n. W e r e f e r t o 1 n-1 a s y m p t o t i c t h e o r y of ( 2 . 6 ) .

[a]

f o r f u l l e r d e t a i l s of t h e

M. S.P. Eastham

160

3. HIGHER-ORDER EQUATIONS OF SELF-ADJOINT TYPE We consider, as an example, the odd-order equation

is nowhere zero on ( X , m ) . If q2r and iq where D=d/dx and q 2n+l 2r+l are real-valued, the left-hand side of (3.1) defines a symmetric differential expression, but we do not confine ourselves to this situation at this stage. Everitt and Zettl [lo] have shown that (3.1) can be written in the quasi-derivative form

Y

= AY,

where the first component of Y is y and A involves the q, but no The characteristic equation of A is derivatives of the 9,. 2n+1

c

qr flr = 0 .

r=O One set of conditions under which this equation can be solved (asymptotically) for the eigenvalues p . is 7

(a) qr are nowhere zero in (XI-) (b) qr-l/qr = o(qr/qr+l)

(1

5 r 5 2n)

(3.2)

Then we have

and p . = O ( W . ) . When the qr behave like powers of x as x -+ m , in 7 ]+I the sense that (x) = 0 {x- qr(x) } (s=1 2,), the transformed system ( 1 . 6 ) has the form

qis)

Z

=

{A, + O(x- )}Z

if T is chosen suitably. If, further, -1 x = O ( l . l , ) = 0tqo/q1), then x-l = o(Al) and we have Case (ii) of f l

.

The second diagonal-

Asymptotic Theory of Differential Equations

161

i z a t i o n , i n v o l v i n g a t r a n s f o r m a t i o n ( 1 . 8 ) , t h e n l e a d s t o a system of t h e form ( 1 . 2 ) .

Transforming back, w e o b t a i n t h e asymptotic n a t u r e

of s o l u t i o n s y . o f 3

( 3 . 1 ) i n t h e form (3.3)

Thus w e h a v e t r u e a s y m p t o t i c f o r m u l a e f o r

I t i s n o t c l e a r w h e t h e r t h e r e i s any s p e c i a l r e a s o n why a s y m p t o t i c

d o n o t a r i s e d i r e c t l y f r o m t h e method.

formulae f o r

F u l l d e t a i l s f o r ( 3 . 1 ) a r e g i v e n i n [ 3 ] , a n d t h e d e t a i l s f o r evenorder equations are i n [4,9,11].

A l s o i n [ 6 ] and [ l l ] i s a d i s c u s s -

§ I occur.

i o n o f s i t u a t i o n s where C a s e s ( i i i ) a n d ( i v ) of

4 . DEFICIENCY INDICES Here w e c o n s i d e r t h e case where ( 3 . 1 ) i s a s s o c i a t e d w i t h a symmetric o p e r a t o r i n t h e H i l b e r t space L 2 ( X I m ) .

Thus i q 2 r + 1 a n d q 2 r a r e r e a l -

v a l u e d , e x c e p t t h a t q o i s r e p l a c e d by qo+A,where t h e new q o i s r e a l v a l u e d and

i s t h e non-real spectral parameter.

i n d i c e s N + a n d N-

The d e f i c i e n c y

a r e t h e number o f l i n e a r l y i n d e p e n d e n t s o l u t i o n s of

(3.1) t h a t are L2(X,-)

for i m A

>

0 and i m A

< 0.

One d e f i c i e n c y

i n d e x l i e s i n t h e r a n g e ( n , 2 n + l ) and t h e o t h e r i n t h e r a n g e (n+1,2n+l).

Also,

N + = 2 n + l if a n d o n l y i f N-

= 2n+l.

The s i z e of t h e s o l u t i o n s y . i n ( 3 . 3 ) i s l a r g e l y d e t e r m i n e d by 7

A calculation gives

r e w 3. where

-

( i m A ) R I’ .

M.S.P. Eastham

162 W e have

R1 = -R2 =

4

,

i q -1 l

(4.1)

By by ( 3 . 2 ) , and R 2 r + l and R 2 r + 2 h a v e o p p o s i t e s i g n s ( 0 5 r 5 n - 1 ) . (4.1) a n d ( 4 . 2 ) , t h e r e i s a j, ( 0 L j 0 -< 2 n + l ) , j, # 1 ) s u c h t h a t

diverges f o r 1

5

j

5

j,

and c o n v e r g e s f o r j

0

+ 1 < j

5

Then w e

2n+l.

have (a)

L e t j, b e e v e n , j ,

(b)

L e t j, b e o d d , j ,

N-

These r e s u l t s g i v e v a l u e s of N + a n d N-

(0

= 2M

= 2M-1

= N+

-

5

(2

M

5

5

M

n)

5

.

Then

n+l).

1 = 2n+l-M

Then

(R2M-1 < 0 ) .

g e n e r a l c r i t e r i a on t h e q . f o r a l l p o s s i b l e 3 Note that, in t o o c c u r , s u b j e c t t o IN+-N-/(l.

(b), M#1 s i n c e j O # l .

Hence, i f one d e f i c i e n c y i n d e x i s 2 n + l , w e

must b e i n ( a ) and t h e n t h e o t h e r i s a l s o 2 n + l .

W e mention i n

p a r t i c u l a r t h e r e s u l t t h a t N + = N-

is L(X,-)

b e i n g t h e c a s e M=O of

(a).

= 2n+l i f

g-’ 1

,

this

W e r e f e r t o [ 3 ] f o r t h e p r o o f s of t h e s e

r e s u l t s , a n d t o [ 4 , 9 , 1 1 ] f o r t h e e v e n - o r d e r case. 5.DIRICHLET INDICES By i n t r o d u c i n g a s p e c t r a l p a r a m e t e r A i n t o o t h e r c o e f f i c i e n t s q . i n 1 ( 3 . 1 ) , w e can c o n s i d e r t h e d i f f e r e n t i a l e q u a t i o n

where

T~

and

T~

a r e symmetric d i f f e r e n t i a l e x p r e s s i o n s .

p o s i t i v e d e f i n i t e Dirichlet

i n t e g r a l D ( y ) , t h e Dirichlet

If

T~

has a

indices M+

Asymptotic Theory o f Difjerential Equations

163

and M- a r e d e f i n e d i n t h e same way a s N + and N- b u t now w i t h D(y)<m i n s t e a d of y

E

L2 ( X , m ) .

W e c o n c e n t r a t e h e r e on a p r o p e r t y of t h e

Dirichlet i n d i c e s which w a s n o t e d by K a r l s s o n [15] and w h e r e , i n c o n t r a s t t o N+ and N-, o n e of M+ and M-

c a n t a k e i t s maximum v a l u e

but not the other. Let

h a v e odd o r d e r 2n+l and l e a d i n g c o e f f i c i e n t i q ( x ) w i t h q ( x ) > O , w h i l e T~ h a s even o r d e r 2n and l e a d i n g c o e f f i c i e n t ( - 1 ) n p ( x ) w i t h T,

p(x)>O.

Then K a r l s s o n showed t h a t

(a)

L e t p / q be L ( X , m ) .

(b)

L e t p / q b e n o t L(X,-). M-

< 2n+l.

Then M+=2n+ i f a n d o n l y i f M-=2n+l.

If n is add a n d M + = 2 n + l , t h e n I f n i s even a n d M- = 2 n + l , t h e n M+ < 2 n + l .

The r e s u l t ( b ) raises t h e q u e s t i o n of e x a c t l y what v a l u e s a r e p o s s i b l e f o r o n e D i r i c h l e t i n d e x when t h e o t h e r D i r i c h l e t i n d e x h a s t h e maximum v a l u e 2 n + l .

The a s y m p t o t i c t h e o r y o f 13 h a s b e e n u s e d t o

show t h a t t h e v a l u e s 2n a n d 2n-1

c a n be r e a l i s e d b u t i s n o t known

whether any lower v a l u e s can b e r e a l i s e d .

W e r e f e r t o [5] f o r the

detailed analysis.

REFERENCES C a s s e l l , J.S., Q u a r t . J . Math.

(Oxford) 33 ( 1 9 8 2 ) 281-296.

Eastham, M.S.P., Theory of o r d i n a r y d i f f e r e n t i a l e q u a t i o n s (Van N o s t r a n d R e i n h o l d , 1970) Eastham, M.S.P.,

Proc. Roy. SOC. Edinburgh 90A (1981) 263-279.

Eastham, M.S.P.,

J. London Math.

Eastham, M . S . P . ,

Proc. Roy.

Eastham, M.S.P.,

P r o c . Roy. SOC. London 383A (1982) 465-476.

SOC. 26 (1982) 113-116.

S O C . Edinburgh 9 1 A ( 1 9 8 2 ) 347-360.

Eastham, M.S.P., Proc. Dundee 1982 Symposium on D i f f e r e n t i a l E q u a t i o n s , S p r i n g e r L e c t u r e Notes i n M a t h e m a t i c s , t o a p p e a r . Eastham, N.S.P.,

J . London Math. S O C . , t o a p p e a r .

Eastham, M.S.P. and G r u d n i e w i c z , C.G.M., 2 4 (1981) 255-271. [ l o ] E v e r i t t , W.N. 363-397.

and Z e t t l ,

1111 G r u d n i e l i i c z , C . G . M . ,

A.,

J . London Math. SOC.

Nieuw Arch. Wisk. 2 7 (1979)

Proc. Roy.

SOC. Edinburgh 87A (1980)53-64.

164

M.S.P. Eastham

[ I 2 1 H a r r i s , W.A.

and Lutz, D.A.,

J. Math. Anal. Appl. 4 8 ( 1 9 7 4 ) l - 1 6 .

[ I 3 1 H a r r i s , W.A. 571-586.

and L u t z , D . A . ,

J . Math. Anal. Appl.

57 (1977)

E q u a t i o n s 4 (1968) 590-596.

[I41 Hinton, D.B.,

J. D i f f .

[15] Karlsson, B.,

J . London M a t h .

SOC. 9 ( 1 9 7 4 ) 1 3 1 - 1 4 1 .

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) OElsevier Science Publishers B.V. (North-Holland), 1984

165

A MORSE THEORY FOR HAMILTONIAN SYSTEMS Ivar Ekeland CEREMADE, Universit6 Paris-9 Dauphine 75775 P a r i s Cedex 16 France

I.

INTRODUCTION

Let us begin by r e c a l l i n g some well-known f a c t s about Hamiltonian systems. We a r e given a function H : R~~ + R , c a l l e d t h e Hamiltonian. The i n t e g e r n i s c a l l e d t h e number of degrees of freedom. Let J E L(RZn) be given by the

ern

We are i n t e r e s t e d in the d i f f e r e n t i a l equation:

x

(2)

I t has

JH'(x).

as a f i r s t integral: H(x(t)) = H(x(0)) ,

(3) T

H

=

all

t.

Assume we have a solution x of equation ( 2 ) which i s 0. Consider the l i n e a r i z e d equation around ST:

(4)

=

JH"(x(t))y.

The c o e f f i c i e n t s a r e Consider the resolvent

T-periodic in R

R = JH"(x(t))R,

(5)

T-periodic, f o r some

t.

of equation ( 4 ) : R(0) = I Z n .

The c h a r a c t e r i s t i c m u l t i p l i e r s of known t h a t :

x a r e the eigenvalues of

R(T).

It is

(a)

1 i s a c h a r a c t e r i s t i c m u l t i p l i e r , because the equation ( 2 ) i s autonomous. The associated eigenvector i s i ( 0 ) = i ( T ) .

(b)

1 i s again a c h a r a c t e r i s t i c m u l t i p l i e r , because t h e equation ( 2 ) has a f i r s t i n t e g r a l . So 1 i s a double eigenvalue of R ( T ) . The correspondi n g eigenspace, however, i s usually one-dimensional, generated by i ( 0 ) . For specjal systems, l i n e a r systems with constant c o e f f i c i e n t s , f o r instance, x = JAx, w i t h A* = A , i t can be two-dimensional.

(c)

i f i i s a c h a r a c t e r i s t i c m u l t i p l i e r , so i s real.

(d)

i f 1 i s a c h a r a c t e r i s t i c m u l t i p l i e r , so i s symplectic: R(T)*JR(T) = J .

-

A,

A

-1

because

R(T)

is

, because R(T) i s

I66

I. Ekeland

I t f o l l o w s from t h i s t h a t c h a r a c t e r i s t i c m u l t i p l i e r s come i n p a i r s (A,A-l) o r (A,A) i f A i s r e a l o r has modulus 1, o r i n quadruples

(A,T,X-~, A - ~ ) otherwise. L e t A # 1, 1x1 f 1 be a simple c h a r a c t e r i s t i c m u l t i p l i e r . The theory o f Krein ( [ 6 1, [ 7 1 , [ll];see also [ 9 1) assigns t o i t a d e f i n i t e sign, i n such a way t h a t i f A i s p o s i t i v e i n t h e sense o f Krein, then A i s negative. More generally, i f A # 1 i s an m-fold m u l t i p l i e r on t h e u n i t c i r c l e , we introduce the r o o t subspace

V

C

C2n:

V = Ker (R(T) -

(6)

hI)m.

I t i s m-dimensional and the r e s t r i c t i o n t o V o f t h e h e r m i t i a n form i s non-degenerate. L e t q be i t s index, i.e., the number o f n e g a t i v e squares i n a d i a g o n a l i z a t i o n . We then say t h a t A has Krein t y p e (m-9.9) ( q times n e g a t i v e and (m - q ) times p o s i t i v e ) .

-i(Js,c)

We now agree on a couple o f d e f i n i t i o n s : D e f i n i t i o n 1.

A T-periodic solution

(7)

dim Ker(1

-

x

of

x

= JH'(x)

i s non-deqenerate if

R ( T ) ) = 1.

0

D e f i n i t i o n 2. A T - p e r i o d i c s o l u t i o n x o f = JH'(x) i s non-resonant i f 1 has m u l t i p l i c i t y 2 as a c h a r a c t e r i s t i c m u l t i p l i e r , i f t h e remaining m u l t i p l i e r s on t h e u n i t c i r c l e can be w r i t t e n : A. =

(8) with

0

<

C(

J < 1/2,

j -

(9)

(1 ,a,

e

2imj

-

and

A.

J

= e

-2imj

,

l r J i m

and i f the numbers: ..**

'am)

are l i n e a r l y independent over t h e r a t i o n a l s .

0

I f x i s h y p e r b o l i c , i.e., i f i t has no c h a r a c t e r i s t i c m u l t i p l i e r on the u n i t c i r c l e except 1, counted t w i c e , i t i s non-resonant.

We now have t o say what we mean by a non-resonant f a m i l y o f p e r i o d i c solutions. be a f i n i t e L e t x ,".,xN Assume w i t h period& T1 ,-:-,T m u l t i p l i e r o f multipli!;ty two. o f the u n i t c i r c l e , 0 5 mk 5 n -

f a m i l y o f p e r i o d i c s o l u t i o n s o f equation (2), each x k , 1 < k 5 N, has 1 as a c h a r a c t e r i s t i c I t has m i oyher m u l t i p l i e r s on the upper h a l f 1 . C a l l them

(10) integer I n the f i f t h s e c t i o n o f t h i s work, we w i l l associate w i t h x c o e f f i c i e n t s nkj, 1 5 j 2 1 + m k and d e f i n e a p o s i t i v e r e a l number Vk by:

where we s e t

ako =

0

and

ak(l+mk)

-_ - I

We d e f i n e the normalized exponents of

xk

t o be the

(1 + mk) numbers:

167

A Morse Theorem for Hamiltonian Systems

W k j -- 1 ' k j '

(12)

l z j i l + m k .

We note t h a t the system o f r e a l numbers t l , W k j I l ~ k ~ N 1, ( j ( m k + l }

(1 3)

has a t l e a s t corank N over t h e r a t i o n a l s , because i t s a t i s f i e s N independent l i n e a r r e l a t i o n s w i t h i n t e g e r c o e f f i c i e n t s , which we g e t by d i v i d i n g equation (11) by Vk: l+m k

'

(14)

k,

1 =

J=2

2 n k j (Wkj -

Wk(

j - 1 )) +

2nk.wkl

*

be a f i n i t e f a m i l y o f p e r i o d i c s o l u t i o n s o f - D e f i n i t i o n 3. L e t x , . - - , x = JH'(x1. non-besonant i f 1 i s a double c h a r a c t e r i s t i c m u l t i p l i e r o f We c a l l every xk, and i f t h e system (13) has corank N over t h e r a t i o n a l s . 0

It

tion

Note t h a t , i f the f a m i l y xk, 1 5 k 5 N, i s non-resonant, then every soluxk is non-resonant i n t h e sense o f d e f i n i t i o n 2. Note a l s o t h a t h y p e r b o l i c t r a j e c t o r i e s never resonate.

bolic,

mk = 0,

and

xk

contributes only

uk, =

;(nk1)-'

If

xk

i s hyper-

E Q.

In o t h e r words, a resonant f a m i l y must c o n t a i n a resonant subfamily c o n s i s t i n g o n l y o f non-hyperbolic s o l u t i o n s . We can now s t a t e o u r main r e s u l t :

THEOREM.

Let

h E

IR

be given.

S = {x E R

(15)

2n

I H(x)

Assume the l e v e l s e t : = h}

i s a non-empty compact C" submanifold o f IRZn, which i s (2n - 1 ) dimensional and has s t r i c t l y p o s i t i v e Gaussian curvature everywhere. Consider t h e d i f f e r e n t i a l equation: = JH'(x).

(16)

If

n 23,

t h e f o l l o w i n g a l t e r n a t i v e holds:

(a)

e i t h e r t h i s equation has an i n f i n i t e number o f p e r i o d i c s o l u t i o n s l y i n g on S

(b)

o r t h e ( f i n i t e l y many) p e r i o d i c s o l u t i o n s which l i e on resonant f a m i l y .

S

form a 0

Using Smale's version o f Thom's T r a n s v e r s a l i t y theorem, we g e t a consequence which we s t a t e l o o s e l y as f o l l o w s :

(i:,

THEOREM. F o r almost a l l the sense o f Baire, and the C" topology) compact C" submanifolds S o f IR , w i t h dimension (2n - 1 ) and s t r i c t l y p o s i t i v e Gaussian curvature, t h e f o l i a t i o n : (17)

i

E

JN(X)

where N ( x ) i s the normal subspace t o t r a j e c t o r i e s on S.

S

at

x,

has i n f i n t e l y many closed 0

I. Ekeland

168

11.

FIXED ENERGY PROBLEM

We a r e g i v e n a Cm s u b m a n i f o l d S ciRZn, which i s compact, dimensional, and has everywhere p o s i t i v e Gaussian c u r v a t u r e . L e t t h e normal subspace t o S a t x E S.

(2n - 1) N ( x ) denote

Consider t h e f o l i a t i o n : (18)

x E S.

E JN(x),

To understand t h i s e q u a t i o n , t a k e any f u n c t i o n as a l e v e l s e t :

S = tx

(19)

I H(x)

=

H: IRZn

+

lR

which has

hl.

Then N(x) i s generated by H ' f x ) , p r o v i d e d t h i s v e c t o r i s non-zero, t h e curves s a t i s f y i n g (18) a r e j u s t t h e s o l u t i o n s o f :

x

(20)

= JH'(x),

S

and

H ( x ) = h,

up t o r e p a r a m e t r i z a t i o n w i t h r e s p e c t t o t i m e correspond t o p e r i o d i c s o l u t i o n s o f ( 3 ) .

t.

Closed curves s a t i s f y i n g ( 1 )

We a r e f r e e t o choose H, p r o v i d e d H ' ( x ) does n o t vanish and S i s a l e v e l s e t . We choose H t o be p o s i t i v e l y homogeneous o f degree 3/2 (any degree a w i t h 1 < a < 2 w i l l do):

(211

H(Ax) =

(22)

s

= {x

A.~'~ H(x)

I H(x)

=

tj h >

0

1).

From now on i t i s understood t h a t H t o g e t h e r , t h e y determine i t c o m p l e t e l y . 111.

w i l l s a t i s f y c o n d i t i o n s (21) and ( 2 2 ) ;

FIXED-PERIOD PROBLEM

The l e v e l s e t s Sh = { x 1 H ( x ) = . h } a r e a l l h o m o t h e t i c o f S1 = S. I n t h i homothety, T - p e r i o d i c s o l u t i o n s o f x = J H ' ( x ) l y i n g on S correspond t o Th1lS and a l l these s o l u t i o n s have t h e p e r i o d i c s o l u t i o n s of k = J H ' ( x ) l y i n g on Sh, same c h a r a c t e r i s t i c m u l t i p l i e r s . Now f i x

T and c o n s i d e r t h e problem:

(23)

= JH'(x),

x ( 0 ) = x(T).

Any s o l u t i o n o f t h e f i x e d - p e r i o d problem (23) can be b r o u g h t back t o t h e energy l e v e l 1 b y homothety, and y i e l d s a s o l u t i o n o f t h e f i x e d - e n e r g y problem (20). However, i t i s n o t t r u e t h a t two d i f f e r e n t s o l u t i o n s of ( 2 3 ) w i l l y i e l d two d i f f e r e n t s o l u t i o n s o f (201, as t h e f o l l o w i n g lemma shows:

B. Problem

(23) has i n f i n i t e l y many s o l u t i o n s .

P r o o f . It i s known t h a t p r o b l e m ( 2 0 ) has a t l e a s t one s o l u t i o n ( W e i n s t e i n [TO], C l a r k e [3]). L e t T be i t s p e r i o d . L e t k be an i n t e g e r g r e a t e than 1 . Then t h e h o m o t h e t i c o f t h e s o l u t i o n we found on t h e energy l e v e l h = k - 5 w i l l be Tk-1 p e r i o d i c , and hence T - p e r i o d i c . We b u i l d i n t h i s way an i n f i n i t e sequence o f s o l u t i o n s of problem ( 2 3 ) .

A Morse Tlieorein .for Hamiltonian Systems

IV.

169

FUNCTIONAL ANALYSIS

F. Clarke a n d the author ([3] and [4]) have proved t h a t problem (23) i s equivalent to the following problem:

where T

Li

{u

=

E

L3

11

u d t = 01

0

& nu

(25)

\

T

=

and

u,

,/ n u d t

=

0

0

G(y) = Sup {xy

- H(x) 1

x EIRzn1

.

To be precise, we have the following:

x

THEOREM (131, 141) I f x solves ( 2 3 ) , t h e n u = solves ( 2 4 ) . u solves ( 2 4 ) , then t h e r e i s some solution x of (23) such t h a t

Conversely, i f

i

= u.

The function G i s t h e Fenchel conjugate ( o r Legendre transform) o f I t i s convex and p o s i t i v e l y homogeneous of degree 3: G(Ay)

(26)

The function

= A3

F on

G(y)

Li

fl

A >

0

H.

0.

has marvellous p r o p e r t i e s :

(a)

i t is

(b)

i t i s twice

(c)

every c r i t i c a l point has f i n i t e index and f i n i t e n u l l i t y . By this we mean t h a t t h e Hessian (F"(x)y,y) can be diagonalized; t h e number of squares with negative (resp. vanishing) c o e f f i c i e n t s i s the index (resp. t h e nu11i t y )

C'

Gateaux-differentiable

the n u l l i t y of every non-zero c r i t i c a l point i s a t l e a s t 1 and a t most 2n. 1 ( e ) F i s S -invariant. S e t t i n g S1 = lR/ Z and u , ( t ) = u ( t + e T ) , we have : (d)

(27)

F(u,)

=

F(u)

fl 0

E

S

1,

O f course ( e ) explains why t h e n u l l i t y has t o b e a t l e a s t 1: a c r i t i c a l point, so i s ue, f o r a l l e E S 1 , and ( U ~ I ~ E 1 SI

if

u # 0 is

(28)

i s an imbedded c i r c l e consisting only of c r i t i c a l points. All i t s points correspond t o the same periodic solution of = J H ' ( x ) u p t o phase s h i f t s x + x,. We now adapt t o t h i s infinite-dimensional case B o t t ' s Morse theory f o r functions with non-degenerate c r i t i c a l manifolds ( [ l ] , [Z]; see a l s o [ 81). with the following r e s u l t :

170

1. Ekeland

A c r i t i c a l c i r c l e i s nondegenerate i f t h e n u l l i t y o f a l l i t s p o i n t s

Definition. i s 1.

THEOREM. I f a l l t h e c r i t i c a l c i r c l e s o f F a r e nondegenerate, t h e n t h e r e i s a t l e a s t one c r i t i c a l c i r c l e o f each even i n d e x i E 214. 0 I t f o l l o w s i m m e d i a t e l y t h a t F has i n f i n i t e l y many c r i t i c a l c i r c l e s , and hence t h a t p r o b l e m ( 2 3 ) has i n f i n i t e l y many s o l u t i o n s . However, as we p o i n t e d o u t i n s e c t i o n 111, t h i s does n o t i m p l y t h a t problem (20) has i n f i n i t e l y many solut i o n s . More work i s r e q u i r e d .

V.

ITERATING CLOSED TRAJECTORIES

be a s o l u t i o n of problem ( Z O ) , t h a t i s , a T - p e r i o d i c s o l u t i o n o f l y i n g on S. L e t xk be t h e T p e r i o d i c s o l u t i o n which i s komot h e t i c t o x on t h e l e v e l s e t H ( x ) = k-511-We say t h a t xk i s t h e k t itera t e o f x, and we have x1 = x.

k

Let

x

= JH’(x)

xk

a l l a r e c r i t i c a l p o i n t s o f F, and b e l o n g t o d i f f e r e n t Now t h e uk = c r i t i c a l c i r c l e s . We can s o r t them o u t b y t h e i r i n d e x , thanks t o t h e f o l l o w i n g formula. THEOREM.

ikbe t h e i n d e x o f

Let

ik=

(29)

p

where that:

kz= I j ( p )

uk.

We have: k E N

for all

i s an i n t e g e r - v a l u e d , non-negative,

j

(a)

t h e p o i n t s where

f u n c t i o n on t h e u n i t c i r c l e such

j i s discontinuous a r e c h a r a c t e r i s t i c m u l t i p l i e r s o f

X

(b)

i f x # 1 i s a c h a r a c t e r i s t i c m u l t i p l i e r on t h e u n i t c i r c l e , o f K r e i n t y p e (P,q), t h e n :

l i m [j(heio) 0-+0

(30) (c)

x

if

-

j(xe-”)l

= q

-

p

i s non-degenerate:

Note t h a t , if x i s a non-degenerate p e r i o d i c s o l u t i o n , c o n d i t i o n s ( a ) t o e now d e f i n e t h e numbers n . which (d) c o m p l e t e l y determine t h e f u n c t i o n j . W we used i n t h e i n t r o d u c t i o n : n i s t h e c o n s t a n t v a l u e taken by j, kJbetween two c h a r a c t e r i s t i c m u l t i p l i e r s k J o f xk on t h e u n i t c i r c l e , namely 2irra 2ina k j and k(j-1) The sum in f o r m u l a ( 2 9 ) is t a k e n o v e r a l l give corollaries: LEMMA.

(32)

If

x

kth r o o t s

of u n i t y .

L e t us

i s h y p e r b o l i c , we have: ik= il + ( k - l ) ( i l t n ) .

Proof. There a r e no m u l t i p l i e r s on t h e u n i t c i r c l e , e x c e p t 1. So j i s c o n s t a n t o n e u n i t c i r c l e d e p r i v e d o f t h e p o i n t 1 , and i t s v a l u e i s g i v e n by ( 3 1 ) . W r i t i n g t h i s i n t o ( 2 9 ) , we g e t t h e announced r e s u l t .

171

A Morse Theorem j o r Humiltoniun Systems

LEMMA.

If

x

i s non-degenerate:

(33)

0

Proof. V.

An easy exercise on t h e Riemann i n t e g r a l .

PROOF OF MAIN THEOREM

The proof now goes by showing t h a t f i n i t e l y many s o l u t i o n s o f problem ( cannot supply enough i t e r a t e s t o account f o r a l l even indices.

)

Denote by x t h e s o l u t i o n s of problem (20). Then a l l solutions of problem (23) m u s t be i t e r s t e s of one of the x " Since F has c r i t i c a l c i r c l e s of every even index, the i t e r a t i o n formulas: (34) must account f o r a l l even numbers. I n o t h e r words, every even number must be written in the form (34) f o r some p and k . To simplify matters, assume the the i t e r a t i o n formulas become :

i

(35)

Pk

=

i

Now, suppose rn + n m

=

Pl

xp, 1 5 p 5 N

a r e a l l hyperbolic.

Then

+ (k - l ) ( i p l + n ) . is a prime number.

Then we w i l l not be able t o w r i t e

k ( i p l + n) - n

+ n = m + n , that. i s i = m. Pl PI Since there i s an i n f i n i t e number of prime numbers, there must be an i n f i n i t e hence an i n f i n i t e number of solutions x numer o f i P' Pl ' A f u l l proof is given in [5].

unless

i

BIBLIOGRAPHY

Bott, R . , 248-261.

Non-degenerate c r i t i c a l manifolds, Annals of Math. 60 (1954)

Bott, R . , Morse t h e o r y , o l d and new, Bull. AMS (New S e r i e s ) 7 (1982) 287-446. Clarke, F . , Periodic s o l u t i o n s t o Hamiltonian inclusions, J . Diff. Eq. 40 (1981) 1-6. Clarke, F . a n d Ekeland, I . , Hamiltonian t r a j e c t o r i e s having prescribed minimal period, Comm. Pure App. Math 23 (1980) 103-116. Ekeland, I . , Une t h e o r i e de Morse pour l e s systemes hamiltoniens, submitted t o Annalesde I'IHP: Analyse non l i n e a i r e (new s e r i e s ) . Gelfand, I . and Lidski, V., On the s t r u c t u r e of the regions of s t a b i l i t y of 1 inear canonical systems of d i f f e r e n t i a l equations with periodic coefficients, Upsekli Math. Naouk 10 (1955) 3-40 (AMS Translation 8, 1958, 143-131).

172

I. Ekeland

[7]

K r e i n , M., G e n e r a l i s a t i o n o f c e r t a i n i n v e s t i g a t i o n s of A. M. Liapounov on l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h p e r i o d i c c o e f f i c i e n t s , Doklady Akad. Naouk SSSR 73 (1950) 445-448.

[8]

Meyer, W., K r i t i s c h e M a n n i g l a l t i g k e i t e n i n H i l b e r t m a n n g l a l t i g k e i t e n , Math. Ann. 170 (1967) 45-66.

[9]

Moser, J., New a s p e c t s i n t h e t h e o r y o f s t a b i l i t y o f H a m i l t o n i a n systems, Cornm. Pure App. Math, 11 (1958) 81-114.

[lo] Weinstein, A.,

P e r i o d i c o r b i t s f o r convex H a m i l t o n i a n systems, Annals o f Math, 108 (1978) 507-518.

E l l ] Yakubowitz and S t a r z h i n s k i i , L i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h p e r i o d i c c o e f f i c i e n t s , H a l s t e d Press.

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

173

SCATTERING AND SPECTRAL THEORY FOR THREE PARTICLE SYSTEMS

Volker Enss I n s t i t u t f u r Mathematik I Freie Universitat D-1000 Berlin 33 Germany

We give a time-dependent, geometrical proof of asymptotic completeness for the quantum mechanical t h r e e body problem. The concepts and methods a r e presented i n d e t a i l , i n p a r t i c u l a r t h e physical i n t u i t i o n c l o s e l y r e l a t e d t o the s t e p s i n the mathematical proof. To be e a s i l y a c c e s s i b l e and t o reduce the purely technical e f f o r t we r e s t r i c t ourselves t o simple p a i r p o t e n t i a l s of s h o r t range. I

Introduction, t h e Model Properties o f Two-Body Systems I11 Local Decay IV Two-Cluster Configurations Phase Space Localization and Asymptotic Observables V VI Wave Operators VII Completeness and Spectral Consequences VIII Concluding Remarks and References

I1

I . Introduction, the Model

The aim of s c a t t e r i n g theory i s t o study t h e long time behavior f o r s o l u t i o n s of evolution equations. In p a r t i c u l a r one i s i n t e r e s t e d in approximations of t h e (complicated) time evolution by simpler ones which a r e e a s i e r t o control. The approximation should be good f o r large times. Asymptotic completeness means t h a t the time evolution f o r any nc~z.tXe~in5soLu.xXan asymptotically i s well approximated by simpler ones. For a l i n e a r evolution equation with i n i t i a l values i n a Hilbert space one f i r s t s i n g l e s out t h e s t a t i o n a r y or periodic s o l u t i o n s . For them t h e asymptotic time evolution i s the same as f o r f i n i t e times and i t i s simple in many cases. The best one can hope f o r i s t h a t a l l i n i t i a l values orthogonal t o those giving periodic s o l u t i o n s belong t o s c a t t e r i n g s o l u t i o n s w h i c h "move towards l n f i n i t y " f o r l a r g e times. Let us consider the Schrodinger equation which describes the dynamics o f a system of f i n i t e l y many quantum mechanical p a r t i c l e s . d i Y ( t ) = H "(t) ,

Y(0)

= Y

.

Here Y ( . ) i s a mapping from the real l i n e i n t o some Hilbert space SC ( t h e s t a t e space of the system), H i s a (generally unbounded) s e l f - a d j o i n t operator on Jf . The functional calculus permits t o construct exponentials of s e l f - a d j o i n t operat o r s , i n p a r t i c u l a r the time evolution operator exp(-iHt). The unique solution of the i n i t i a l value problem i s

174

V. Ems

I f the i n i t a l s t a t e Y i s an eigenvector of H : HY=EY , then t h e time evolution i s t r i v i a l Y(t) = exp(-iEt) I , the t r a j e c t o r 6 u ( t ) s t a y s within t h e one-dimensional subspace of Jc . The closed l i n e a r span P p(H) Jc of eigenvectors i s c a l l e d the subspace of bound s t a t e s . I t s orthogonal complement i s ch r a c t e r i z e d by t h e spectral theorem as t h e continuous spectral subspace Pcont(H) .K . To show asymptotic completeness c o n s i s t s of the following two t a s k s ( a ) find (one or s e v e r a l ) simple comparison time evolutions, ( b ) f o r l a r g e times T find a decomposition of the s t a t e Y ( T ) such t h a t f o r l a t e r times the comparison dynamics a r e good approximations of the t r u e dynamics on the corresponding p a r t s of the s t a t e . Experience from physics suggests appropriate simple time evolutions. I f the i n t e r a c t i o n between p a r t i c l e s becomes weak towards i n f i n i t y , then i t should b e possible t o neglect the forces between f a r separated p a r t i c l e s . I n the special case o f "simple s c a t t e r i n g systems" there i s a s i n g l e comparison time evolution, t h e " f r e e " time evolution and the s p l i t t i n g of the s t a t e i s not necessary. Prototypes of such systems a r e the internal motion of a two body system ( p o t e n t i a l s c a t t e r i n g ) i f the forces decay s u i t a b l y towards i n f i n i t y ; or the s c a t t e r i n g of c l a s s i c a l waves by an obs t a c l e or by localized inhomogeneities of the medium. The simple f r e e dynamics neglects the forces o r t h e o b s t a c l e . This s i t u a t i o n i s well understood. More complicated i s the mu1 ti-channel s c a t t e r i n g theory with d i f f e r e n t asymptotic time evolutions. The t h r e e body system which we t r e a t here, i s a model f o r t h i s case. I f a11 p a r t i c l e s asymptotically f l y away from each other,then the f r e e time evolution ( a l l forces neglected) i s a good approximation. Another p o s s i b i l i t y f o r t h e asymptotic behavior of a s c a t t e r i n g s t a t e i s , t h a t a s i n g l e p a r t i c l e moves f r e e l y away from a bounded p a i r , the internal motion of the p a i r being t r i v i a l . This i s a complete l i s t of asymptotic time evolutions f o r three p a r t i c l e quantum scattering states. For the proof we e x p l o i t our d e t a i l e d knowledge about two body systems. They a r i s e as subsystems of the f u l l system. Moreover, a bounded p a i r and a t h i r d part i c l e behave l i k e a two body system as long as they s t a y separated. I f three part i c l e s a r e f a r separated from each other and a l l t h e i r r e l a t i v e v e l o c i t i e s point outwards (such t h a t a l l distances increase) then in t h e f u t u r e a l l forces can be neglected. This i s shown using our knowledge of the f r e e time evolution. So f a r we had to t r e a t only simpler time evolutions. I t remains t o show t h a t any t h r e e body s c a t t e r i n g s t a t e which does n o t have a bounded p a i r asymptotically, event u a l l y will have the kinematic p r o p e r t i e s , mentioned above, i . e . asymptotic alignment of the r e l a t i v position and momentum operators. This f a c t has t o be shown f o r the f u l l y i n t e r a c t i n g time evolution, b u t l't turns o u t t o be e s s e n t i a l l y independent of the p a r t i c u l a r forces. I t can be shown using t h e asymptotic time evolution of some s u i t a b l e observables which can be controlled because they a r e not s e n s i t i v e t o i n t e r a c t i o n s . They represent "modest questions" which can be answered even f o r the complicated f u l l time evolution. The complex problem of the asymptotic behavior of the f u l l time evolution could be simplified and solved by studying easy special aspects of the f u l l time evolution only and by exploit i n g the d e t a i l e d knowledge about t h e simpler approximations, For the simple time evolutions the spectral properties of t h e i r generators a r e e x p l i c i t l y known. The asymptotic e q u a l i t y of the t r u e time evolution with the simple ones on the corresponding subspaces allows t o construct a unitary equivalence between H and a known operator. E.g. one can conclude the absence of singular continuous spectrum. Here we consider the following model. There are three p o s s i b i l i t i e s , l a b e l l e d by a , t o group the three p a r t i c l e s i n t o a pair and a t h i r d p a r t i c l e . The f r e e motion of the center of mass i s separated o f f . We use Jacobi coordinates t o describe the internal degrees of freedom. xa denotes the r e l a t i v e coordinate of the p a r t i c l e s in t h e p a i r , ua t h e i r reduced mass and

175

Scattering TIieorl' .for Tlircc, Particlcs

p,

=

-

i a/ax"

i s t h e c a n o n i c a l l y c o n j u g a t e momentum o p e r a t o r . The r e l a t i v e v e l o c i t y between t h e p a r t i c l e s i n t h e p a i r i s t h e n pa/ua . The c o r r e s p o n d i n g q u a n t i t i e s f o r t h e t h i r d p a r t i c l e r e l a t i v e t o t h e c e n t e r o f mass o f t h e p a i r a r e denoted as f o l l o w s : ya f o r t h e p o s i t i o n , ua t h e reduced mass, q, t h e c o n j u g a t e momentum o p e r a t o r and qa/va t h e v e l o c i t y . The p a r t i c l e s move i n u-dimensional space. The s t a t e space f o r t e i n t e r n a l m o t i o n o f t h e t h r e e body system i s t h e H i l b e r t space 3c = L ( R 2 U , dvxa duya). The f r e e H a m i l t o n i a n Ho d e s c r i b e s t h e k i n e t i c energy

9 .

The f o r m (1.4) i s independent o f t h e chosen p a i r i n g a . The i n t e r n a l k i n e t i c energy o f t h e p a i r i s denoted b y ha, , we use t h e same l e t t e r f o r o p e r a t o r s on SC and on t h e " f a c t o r space" l i k e L2( Rv , duxa). S i m i l a r l y , k g i s t h e k i n e t i c energy o f t h e t h i r d p a r t i c l e r e l a t i v e t o t h e c e n t e r o f mass o f t h e p a i r . A l l o p e r a t o r s a r e s e l f - a d j o i n t on s u i t a b l e domains. A wide c l a s s o f p o t e n t i a l s can be t r e a t e d by t h i s method as w i l l be d i s c u s s e d i n S e c t i o n VIII. Only t o s i m p l i f y t h e p r e s e n t a t i o n and t o emphasise t h e main i d e a s we r e s t r i c t o u r s e l v e s here t o a s p e c i a l case. The i n t e r a c t i o n s a r e d e s c r i b e d by p a i r p o t e n t i a l s V,(xa) which a r e m u l t i p l i c a t i o n o p e r a t o r s i n x-space. They s a t i s fy the short-range c o n d i t i o n IV,(X")l

>C(1 +

H = H t V = H 0

0

I XUl

)++El,

+ Z V

E >

0

(1.5)

(1.6)

a

i s t h e H a m i l t o n i a n which generates t h e f u l l y i n t e r a c t i n g t i m e e v o l u t i o n . I t i s s e l f - a d j o i n t on t h e same domain as Ho. To d e s c r i b e t h e approximate t i m e e v o l u ti ons we use t h e t h r e e channel Hami 1t o n i ans H a = H O + V a = h a + k a0

'

(1.7)

ha qenerates t h e i n t e r a c t i n q i n t e r n a l m o t i o n of t h e p a i r . w h i l e t h e t h i r d D a r t i l e moves f r e e l y , i t s i n t e r a c t i o n s w i t h the p a r t i c l e s i n the p a i r a r e o m i t t e d . The diagram s h o u l d h e l p t o remember t h e n o t a t i o n and s e r v e as a r e f e r e n c e .

We denote by Ppp(H) and PCont(H) t h e o r t h o g o n a l p r o j e c t i o n s on t h e p o i n t s p e c t r a l and c o n t i n u o u s s p e c t r a l subspaces, r e s p e c t i v e l y , f o r a s e l f a d j o l n t o p e r a t o r H. F(.) denotes a s p e c t r a l p r o j e c t i o n f o r a s e l f a d j o i n t o p e r a t o r and a p a r t o f t h e spectrum as g i v e n i n t h e parentheses, e.g. F(S < H < E), 6, E E lR, o r F ( I xal 5 R ) . The main r e s u l t i s s t a t e d and shown i n S e c t i o n V I I . The " a u x i l i a r y " r e s u l t s i n S e c t i o n s I1 - V I a r e rounded o f f t o g i v e a more complete p i c t u r e o f t h r e e body s c a t t e r i n g t h e o r y . References a r e g i v e n i n S e c t i o n V I I I . 11. P r o p e r t i e s o f Two-body Systems I n t h e three-body system t h e r e a r e two ways i n which two-body systems a r i s e

V. Enss

176

n a t u r a l l y , ( i ) as subsystems o f two p a r t i c l e s o u t o f t h e t h r e e , ( i i ) as a system o f one s i n g l e p a r t i c l e and a bounded p a i r . The r e s u l t s o f t h i s s e c t i o n w i l l be a p p l i e d l a t e r t o b o t h cases. We denote h e r e t h e c o o r d i n a t e s by x E RV , t h e conj u g a t e momentum o p e r a t o r by P = -i vX , and t h e H a m i l t o n i a n s a r e

(2.1) We assume t h a t t h e p o t e n t i a l 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

V = V(x) w i t h I V ( x ) l < C (1 + I x I ) - ( ~ + ' ) , E > O .

(2.2)

I t i s w e l l known t h a t o e S S ( h ) = 10, m ) . h may have f i n i t e l y o r i n f i n i t e l y many e i g e n v a l u e s which can accumulate o n l y a t zero. Neg t i v e e i g e n v a l u e s a r e a t most a ) c & s c ( h ) f o r a l l a > 0. P o s i t i v e f i n i t e l y degenerate, i . e . oPP(h) r- ( - m , e i g e n v a l u e s a r e absent, b u t 0 may be an e i g e n v a l u e . S i n c e zero-energy e i g e n f u n c t i o n s may have v e r y slow decay we w i l l g i v e p r o o f s below w h i c h a r e independ e n t o f any p a r t i c u l a r decay r a t e o f e i g e n f u n c t i o n s . The degeneracy o f t h e e i g e n v a l u e 0 may be a r b i t r a r y . A l s o zero-energy resonances a r e i r r e l e v a n t f o r o u r proof.

-

Our f i r s t o b j e c t i v e w i l l be t o s t u d y some p r o p e r t i e s o f t h e f r e e H a m i l t o n i a n h, and p r o p a g a t i o n p r o p e r t i e s o f t h e f r e e t i m e e v o l u t i o n exp(- i h, t ) . The s t a r t i n g p o i n t i s an easy t e c h n i c a l lemma which c o n t r o l s t h e p o s s i b l e d i s l o c a l i z a t i o n due t o t h e a p p l i c a t i o n o f f u n c t i o n s o f t h e momentum o p e r a t o r . F u n c t i o n s o f s e l f a d j o i n t o p e r a t o r s a r e d e f i n e d by t h e f u n c t i o n a l c a l c u l u s and F ( . ) denotes m u l t i p l i c a t i o n w i t h the c h a r a c t e r i s t i c f u n c t i o n o f the i n d i c a t e d region. Lemma 2.1. LeX M, M'

c Rw be r n e a u ~ ~ a b & 4&5

w d h r := d i s t (M,

M I ) ,

$ E C"(W").

=GK@iZ&N€l4 IIF(x

M ' ) $(P) F(x

E

< min

{sup 1 $ 1 ; c

E

.

M)II

$1

r-NJ dVq I [ ( - ,)(N+v)'2

(q)l ] ;

The elementary p r o o f can be found i n [ 4 , Sect. 21. I t uses t h a t $ ( P ) a c t s i n x-space as c o n v o l u t i o n w i t h $ , t h e i n v e r s e F o u r i e r t r a n s f o r m . The l a t t e r i s c o n t r o l l e d by p a r t i a l i n t e g r a t i o n . To be u s e f u l we have t o s p e c i f y t h e f u n c t i o n s 4 d $ E C z ( R " ) o t $ ( p ) = ( p2/ 2 m - z ) - ~[ i . e . $ ( P ) = (ho - ~ ) - ~ 1 C o r o l l a r y 2.2. L doh k E N , .then 40% ab? N E M

I I F ( E~

MI)

$(PI

F(X

E

M)II < ~ ~ ( ($1 1 + r)-N

Proof. S u f f i c i e n t l y high d e r i v a t i v e s o f

$

.

are integrable. U

C o r o l l a r y 2.3. IIF(x <

M’) exp(- i ho

c o n s t (1

+

t ) (ho

1 tl)2tV . r-'

-

z)-(~")

F ( x E M)II

.

2 Proof. 1 ( - A )(2+w)/2 l e x p ( - i t q /2m) [ (q2/2m)-z1 -('+V))I 9 < c o n s t ( I + i q t i y + v (1 t q2)-(2+v)

.

which i s

integrable i n R v .

177

Scattering Theor?,f o r Three Particles

This c o r o l l a r y has a simple physical i n t e r p r e t a t i o n . To t r a v e l a distance r within a time t the p a r t i l e must have a minimal speed v = r/l tl , or a minimal energy proportional t o r 2 / t . Due t o the power of the resolvent t h i s component of the s t a t e i s bounded by (t2/r2)vi2 . ( 2 . 5 ) i s a crude estimate, which i s good enough for l a t e r a p p l i c a t i o n s . I t i s useful only f o r s h o r t times. For long times we need a sharper estimate. Consider a c l a s s i c a l f r e e p a r t i c l e localized somewhere in a s e t M a t time zero, and with possible v e l o c i t i e s i n a s e t V . Then a t time t t h e p a r t i c l e will c e r t a i n l y not be outside t h e s e t {x, + v t I x E M , v E V ) . A standard estimate of geometrical s c a t t e r i n g theory s t a t e s t h a t t8e f r e e quantum time evolution has the same properties in a very good approximation.

5

Cr

E ( R ' ) have s u p p g C { p E R"I I p - m voI < m v } R V , v > 0 . L e t M, M ' bc meanwrable nubne,i2 0 6 Rv , and LeA M t vo t ) - v I tl > 0 . Then doh ah? n E N

Proposition 2 . 4 . L e L g doh name vo E r := d i s t ( M ' ,

I I F ( E~ M I ) e-i ho

g(p)

F(X E

M)II 5

cn

(1

+ I ti

t t-)-n

.

(2.6)

.

The cavLltutclou2 Cn depend only an t h e nhape 0 6 g 16 a 6amZq 0 6 6uvzcLiavu g depe n h b t n O O t h & j (e.9. in t h e topology 06 .S(R")) on a paharn&m vmqing i n a compact s d , t h e n ,the c o n ~ t a m 2Cn can be chosen unLt(0tztdy 6vh t k i h ~m.2.q.

For the proof one can use a G a l i l e i transform t i o n and show t h a t the inverse Fourier transform $ ( u ) of @ ( p ) = exp(- i . t p h /2m) g ( p t m v o ) decays rapidly in I u I and I tl f o r I u I L v l t l by the s t a t i o n a r y phase estimate. Note t h a t t h e support condition on g says t h a t there i s an a > 0 such t h a t supp g c i p E Rw 1 I p - m voI < m ( v - a ) ] . The additional separation al tl takes care of t h e rapidly decaying "quantum t a i l s " . [ 2,4]. be any localized operator such t h a t - ( 1tE) IIF(Ixl > R) A II const ( 1 + R ) 9

Let A

and l e t

+

E

C:(R")

or +(P) = (ho

-

z ) - ~ .Then by Corollary 2.2

IIF(lxl > R ) g ( P ) All

R/2) All+IIF(I X I >R) @ ( P ) F ( I XI
const ( 1 + R )

(2.8)

-( 1 t E )

T h u s application of $ ( P ) does not d i s t u r b the (power-) decay r a t e . We will use t h i s f a c t below repeatedly. Next we show t h a t the d i f f e r e n c e between functions of the k i n e t i c energy and o f the f u l l energy decays a t l e a s t a s f a s t as the POt e n t i a1 . Lemma 2.5. L e A h , h o naLL56y ( 2 . 1 ) and (2.2), and LeL g ( h o ) = ( h o - z ) - k , z E P (ho) n c h ) , t h e f i e ~ o L w e n At & ; u ~ LleL g E C E ( R ) $ ( p ) = g ( p /2m) t C ; ( R " ) I . Then

k

E

N,

5

[*

IIF(I XI >R) [ g ( h )

-

g ( h o ) ] I1 5 const(1tR)

-( 1+E)

(2.9)

Proof. For the resolvents by (2.2) and ( 2 . 8 ) llF(1 XI >R) [ ( h - z ) - l <

const( 1+R) - (

1tE)

-

(ho-z)- 1] 1I (II

F ( l x l >R) (h0-z)-'

VII .ll(h-z)-lll

(2.10)

C: Ems

178

F o r h i g h e r powers o f t h e r e s o l v e n t s t h e same f o l l o w s f r o m t h e expansion N- 1 - (N-k-1 ) (2.11) (h-z)-N - ( h o - z ) - N = c (ho-z)-k [ ( h - z ) - l - (ho-z)-'] (h-z) k=o

.

For g

C;(R)

E

set

CE(R ) . W i t h

$ ( w ) := ( U - Z ) " + ~ g(w) E

(2.12)

i d t i ( t ) eihOt

$(ho) = (2n)-'"

and t h e analogous e x p r e s s i o n f o r $ ( h ) we o b t a i n l l F ( l x l >R) I g ( h )

-

g(ho)lll

< IIF(I XI >R) [ ( h - z ) -(v*2)

- (ho-z)-("+2)] 1I .II$(h)ll

+

{$(h)

JIF(IXI>R) (ho-z) -("+*)

-

$(ho)lll.

(2.13)

The f i r s t summand has t h e d e s i r e d decay, we e s t i m a t e t h e second, u s i n g t h e Duhamel f o r m u l a by

It i ds IIF(I XI >R) (ho-Z) - ( ~ + 2 ) ,isho

x f

(iv) ei (t-S)hll

(2.14)

0

The r a p i d decay of ensures r a p i d decay i n R o f t h e i n t e g r a l i n t h e f i r s t summand f o r any 6>0. The second t e r m i n (2.14) i s bounded by const i

l l F ( l x l >R) (ho-z) - ( w + 2 ) eish~ F ( I XI
SUP

I SI
+

II ( ho-z)-(U+2)ll

e l l

F( I XI >R/2) Vll 1 .

(2.15)

The d e s i r e d decay o f t h e l a s t t e r m f o l l o w s f r o m t h e decay assumption on V (2.2). To t h e f i r s t l i n e o f ( 2 . 1 5 ) we a p p l y C o r o l l a r y 2.3 and o b t a i n t h e bound c o n s t ( 1+R6)'+'

R-'

5 c o n s t R - ( 1+E) Y

if

6

i s chosen O < s < ( l - E ) / ( v t 2 ) . 0

We o b t a i n e a s i l y t h a t smooth f u n c t i o n s o f t h e f u l l energy do n o t s p o i l l o c a l i z a t i o n t o o much. C o r o l l a r y 2.6.

Le*

g

be a

iiz

Lemma 2 . 5 ,

llF(I xi > r t R ) g ( h ) F ( I XI< R ) i l < c o n s t ( l t r ) - (

the^ 1+E)

(2.16)

P r o o f . IIF(I X I >r+R) g ( h ) F ( I XI
c IiF(I XI > r t R ) [ g ( h ) t

-

g ( h o ) l 1I

!IF([ XI >r+R) g ( h o ) F(I XI
We a p p l y Lemma 2.5 and C o r o l l a r y 2.2,

(2.17) respectively 0

179

Scattering Tlwori f o r Three Particles

We conclude t h i s s e c t i o n by showing t h a t "low-energy p a r t i c l e s t r a v e l s l o w l y " . I n c o n t r a s t t o P r o p o s i t o n 2.4 we w i l l t r e a t h e r e t h e i n t e r a c t i n g t i m e e v o l u t i o n . The e s t i m a t e o f t h e "quantum t a i l s " i n t h e " c l a s s i c a l l y f o r b i d d e n r e g i o n " i s s h a r p enough t o be u s e f u l i n l a t e r a p p l i c a t i o n s ( i n t e g r a b l e decay i n t i m e ) . Theorem 2.7. a>O, R>1,

LeL

8E

C E ( R ) w L t h supp

llF(I XI >(1+2a)(RtvI tl ) ) e

< const(R+vl tl )

-iht

6c

(-

2

m,

m v / Z ) , v>O. the^ g o t any

o

g ( h ) F(1 XI
- ( 1+E)

(2.18)

P r o o f . There i s a f u n c t i o n g E Cm R ) as i n Lemma 2.5 w i t h g ( o ) = 1 f o r supp 8 and supp g c ( - m , m oh v / 2 ) . Then g ( h ) . g ( h ) = g ( h ) . W i t h o u t l o s s o f gen e r a l it y we may assume II q( h)llLlI g( h)ll =l.

w E

By Lemma 2.5 IIF(I XI > ( 1 + 2 a ) ( R t v l t l ) ) [ g ( h )

-

(2.19)

g ( h o ) ] IIzconst(R+vI t l ) - ( 1 t E )

Therefore i t i s s u f f i c i e n t t o study 1I F( I XI > ( 1+2a)( R+vl t l ) ) g ( h o ) e-jht

< II F ( I XI > ( 1+2a) ( R t v l t l

+

IIF(I XI > ( l + Z a ) ( R t v l t l

d ( h ) F ( I X I
1

g ( h o ) { e- j h t

-

1

-ihot

o

g(ho) e

e-ihot}

$(h) F(lxl
(2.20)

g ( h ) F ( I XI
The l a s t t e r m i n (2.20) can be s p l i t as IlF(I XI > ( l t Z a ) ( R + v l t l ) ) g ( h o ) e - j h o t F ( I XI < R t a ( R t v l t l ))I1 t

IIF(I X I >R+a(R+vt)) t ( h ) F ( I XI
<

const(R+vl t l ) - ( 1+E)

(2.21)

Here we have used C o r o l l a r y 2.6 and P r o p o s i t i o n 2.4 w i t h vo=O, r=a(R+vl t l ) . The d i f f e r e n c e o f t h e e x p o n e n t i a l s i n ( 2 . 2 0 ) i s expanded w i t h t h e Duhamel formula t o g i v e t h e bound t .f ds A F ( l x l ~ ( l + 2 a ) ( R + v lt l ) ) g ( h o ) e-iho(t-s) V e-ihs g ( h ) F(I XI
t

< f dsllF(I XI >(1+2a)(R+vl t l ) ) g ( h o ) e-iho(t-s)

(2.22a)

F ( I X I < a ( R t v l t l ))I1 .IIVII

0

+

t IIF(I X I >a(R+vl tl ) ) V l i . J ds IIF(I X I > ( l t Z a ) ( R + v l s1 ) ) e-ihs 0

t t J

ds llF(I X I >(l+Ea)(R+vl t l ) ) g ( h o ) e-ihO(t-s)

8 ( h ) F ( I XI
F ( I x 1 < ( l t Z a ) ( R + v l s1 ))I1

x

0 x

IIF(I X I >a(R+vl tl ) ) Vll

.

(2.22c)

The i n t e g r a n d in (2.22a) decays r a p i d l y i n ( R + v I t l ) u n i f o r m l y i n O i sld tl a g a i n by P r o p o s i t i o n 2.4. T h i s i m p l i e s f a s t decay o f t h e i n t e g r a l . S i m i l a r l y t h e i n t e grand i n t h e f i r s t l i n e o f (2.22~)decays r a p i d l y i n I t - s l . T h e r e f o r e t h e i n t e g r a l i s bounded n i f rmly i n R and t and t h e decay IIF(I XI >a(Rtvl tl ) ) Vll 5 < c ( l t a ( R t v 1 tl ) ) - r l + E y remains.

V. Enss

180

With the shorthands y(s)

= SF(1 XI =

>(1+2a)(R+vlsI ) ) e

(R+vl S I ) - ( I + € )z ( s )

-ihs o g ( h ) F ( I XI
(2.23)

9

we can sum u p o u r e s t i m a t e s a s y ( t ) 5 const(R+vl tl )

{I

t

t i ds y ( s ) l ,

(2.24)

0

(2.25) By Gronwall's i n e q u a l i t y [ 11 ] m

s u p z ( t ) z c ' e x p t C ' i d s ( R + ~ )I- (~' +I € ) I < M t 0

<

-

v ~21.

Thus (2.26)

y ( t ) 5 M(R+vI tl ) - ( 1+E) which proves ( 2 . 1 8 ) .

Remark. The s p l i t t i n g ( 2 . 2 2 ) and the e s t i m a t e ( 2 . 2 2 ~ )can be used t o s i m p l i f y m r a b l y p a r t s o f the o r i g i n a l proof i n [ 4 I . 111. Local Decay

The s t a r t i n g p o i n t o f geometrical methods i n s c a t t e r i n g theory i s t h e following o b s e r v a t i o n . Any s t a t e orthogonal t o a l l bound s t a t e s w i l l e v e n t u a l l y decay i n t o two subsystems which a r e f a r s e p a r a t e d from each o t h e r . One can prove t h i s by compactness arguments. A useful e s t i m a t e i s 1eZ H be a n d d - a d j o i n t opehatoh on borne H a b e h t npace Jc , denoten ,the phojeotion t o c o n t i n u o w npecakal bubopace, and L e i

be any compact opuuztoh un JC. For

C

C ( t ) : = exp(iHt) C exp(-iHt)

I

1. d t C ( t ) PCont(H)Il=o

lim :1 1TI-t-

0

This r e s u l t i s a key e s t i m a t e i n e r g o d i c theory. Usually i t i s s t a t e d n s t r o n g form, b u t t h e uniform e s t i m a t e (3.1) can be shown along t h e same l i n e s ( s e e e . g . Lemma 4 . 2 i n [ 2 1 .) C o r o l l a r y 3.2. L e A H and f o r a l l I TI LTE. 1 :'Tdt

sup TER

IIC

C

be

emiHt pCont(H)

an

above. Foh any

42

a T(E

such t h a t

~ll<~llyII.

( a d on

Proof. By t h e Schwarz i n e q u a l i t y tT1 T dt IIc ,-iHt pcont (H) e -iHT u l l . 1 l 2 2 0

thetie

T

T ( E ) depenh on t h e compact opexa-toh C

~

E

H I , buA Lt

independent

v6

I

Scattering Theor!) f u r Three Particles

1 T -iHt pcont -iH-r (H) e -< T- J0 d t IIC e =

(pcont(,)

e- i H T Y ,

yrll

181

2

T J d t (C*C)(t)

PCont(H)

e-iHT Y )

0 <

-

INT l T 1 d t (C*C)(t) Pcent(H)II~II~ll 2,

(3.3)

0

Since C*C i s compact, the f i r s t f a c t o r i s smaller than Lemma 3.1.

E'

f o r ITILT(E) by 0

These a b s t r a c t estimates a r e linked t o propagation properties of quantum s c a t t e ring s t a t e s by L v c d compactnenh (and by more s u b t l e equations l a t e r in Section V ) . For R < m denote by F(llxll
F(llxll
(3.4)

Here F(H < E ) denotes the obvious spectral projection f o r Ho and s i m i l a r l y f o r other se?f-adjoint operators below. A simple perturbation argument shows t h a t f o r almost any i n t e r a c t i n g quantum Hamiltonian H=Ho+V local compactness holds, i . e . f o r any R , E < = F(lIxII > p i f f o r any pairing a t h e conditions I xal

R imply I xol >p f o r a l l o t h e r pairings o # a . I n l a t e r applications depending on the admissible e r r o r E , P has t o be chosen large and R much l a r g e r . We show f i r s t t h a t f o r a s c a t t e r i n g s t a t e i n the time average e i t h e r a l l p a r t i c l e s a r e f a r separated from each o t h e r or two p a r t i c l e s may be close and the t h i r d p a r t i c l e i s f a r from t h e pair. Lemma 3.3. Fur, any o
[n F(I xal

Y E

>p) t

c1

l l F ( H > E ) YII +

PCont(H)

f(E;p,R;T)

Jc

c F(I xal R ) } e-iHt

YII

a

il~ll,

avid

lim I TI

f(E;p,R;T)=O v E,p<
+-

Proof. The f o u r projections i n the curly bracket of (3.6) a r e pairwise orthogonal. T h e t e g r a n d i s bounded by

c IIF(I x'I E)'ill. (3.8) By local compactness ( 3 . 5 ) a n d by Corollary 3.2 the time average of the f i r s t sum tends t o zero f o r any E,p, and R .

0

182

V.

Elm

The o p e r a t o r F(lxal
a,p,E,

T)Tdt IlF(I xal

sup TER

und

Y E X

Pcont(h")

'0)

ebiHat

yll

T

+

< IIF(ha>E) Yll

f(E,p;T)

IIiyll,

(3.9)

whetle CLee

l i m f(E,p;T)=O ITl+m

P r o o f . S i n c e ha and k: Therefore SF(I x'l <


(3.10)

E,p

commute t h e t i m e e v o l u t i o n can be w r i t t e n as a p r o d u c t .

Pcont(ha)

e-ihat

e-ikzt

IIF(I x a l < p ) F(ha<E) Pcont(hu)

F(h"<E) yll

e - i h a t ;I

(3.11)

-

Here t h e f i r s t norm i s i n t h e t h r e e body H i l b e r t space L2(iRZv ,d"xad"ya), t h e l a t t e r i s i n t h e two body subspace L2(!Ru ,duxa) w i t h

? %\$(x")12

id"ya I ~ ( x " , y " ) 1 2 ; l i ? I I = I ~ ~ I I .

=

while (3.12)

On t h i s subspace F ( I xal
F O ~at.t a , p , ~ , ~E

d t IIF(I xal


Pcont(ha)

e-iHt

P ~ ' " ~ ( H ) J Cand ,

€>o

YII

T O

(3.13)

P r o o f . From t h e r e l a t i v e boundedness o f t h e H a m i l t o n i a n s i t f o l l o w s t h a t f o r g i and E t h e r e i s an E ' = E ' ( E , € ) such t h a t f o r a l l a

VCi-E

(3.14)

IIF(h">E') F ( H < E ) l l < c , l l F ( k ~ > E ' ) F(H<E)II<E. We keep t h i s E ' f i x e d i n t h e sequel. By Lemma 3.4 t h e r e i s a TI p,E, and E such t h a t f o r a l l CY 1 T+T1 SUP T E

!R

j

d t IIF(1 xCil < p ) Pcont(ha) F ( h " < E ' ) ,-iHat

~II<EII@II.

(3.15)

T

Keep T i f i x e d i n t h e sequel and determine R t h e f o l l o w i n g e x p r e s s i o n i s bounded by 3 ~ : Sup

depending on

IIF(I xal


l a r g e enough such t h a t f o r a l l

Pcont(ha) F(h"<E') F ( I y a l > R ) F ( k E < E ' )

x

I t l (TI x

Ie

-iH"t

- ,iHtlll

(3.16)

C(

Scatteririg Thrnrj~for Tliree Purtic1r.s

183

U s i n g t h e Ouhamel f o r m u l a t h i s i s m a j o r i z e d by

c

sup

T1'I SIT'I

I I F ( I x a l < p ) PCont(ha)

F ( h u < E ' ) e- i h a s

@ #u x

F ( l y a l >R) e - i k 3

F ( k I < E ' ) Vg("o)ll.

(3.17)

On t h e s m a l l H i l b e r t space t h e compact o p e r a t o r v a l u e d f u n c t i o n

F ( I xaI


F(ha<E') e-ih"s

Pcont(h")

i s norm c o n t i n u o u s i n

(3.18)

s . T h e r e f o r e t h e r e i s a p ' such t h a t

(3.19)

Next determine

c

R'

such t h a t f o r a l l

I I F ( I x ~cI p ' ) F ( l y a l

a

>R’) V D ( x 6 ) I l < ~ / T 1 .

(3.20)

B#a

The same argument as f o r (3.18) sup I s1 T

I I F ( l y a I >R)

schows t h a t t h e r e i s an

R with

e-lkgs F(ka<E') 0 F ( I y a I
1 (3.21)

Thus R can be chosen i n such a way t h a t (3.16) i s bounded by 3 e . It depends , E l ) on p , E , and E o n l y . The p h y s i c a l c o n t e n t o f t h i s e s t i ( t h r o u g h R',p',T mate i s v e r y s i m b l e . Consider a c o n f i g u r a t i o n where a p a i r o f p a r t i c l e s has sep a r a t i o n l e s s than p and t h e t h i r d p a r t i c l e i s f a r away f r o m t h e p a i r . Energy bounds r e s t r i c t t h e maximal speeds. Then u n i f o r m l y i n a g i v e n t i m e i n t e r v a l [ - T ,T1l t h e s e p a r a t i o n o f t h e t h i r d p a r t l ' c l e f r o m each p a r t i c l e i n t h e p a i r w i l l rembin l a r g e ( i. e . l x g l > R ' - p ' f o r @ # a ) .The i n t e r a c t i o n between f a r separated p a r t i c l e s i s weak, t h e r e f o r e t h e channel t i m e e v o l u t i o n generated b y Ha is a good approximation o f the f u l l time evolution. Summing up o u r e s t i m a t e s we have shown

U n i f o r m l y f o r 1 TI

l Tr d t . . . 52

-

TO Choose

(3.22)

6 ~ l l ~ .l l

<

Tll SUP T

T2(€)2 T1l

l JT d t IIF(I xal 0

< Ell'II

.

t h i s implies

1 T+T1 - f dt

...

<12 EIIYII

.

(3.23)

T

such t h a t f o r 1 TI 2T2( E) '0)

Pcont(ha)

F ( l y a l < R ) F(H<E) e-iHt

YII

(3.24)

V. Enss

184

This i s possible by Corol ary 3.2 because F ( I Ix' < p ) Pcont(ha) F I yal < R ) F ( H < E ) i s compact. T ~ ( E )depends only on p,E, and (3.13) i s s a t i s f i e d .

E.

With T ( E ) = T ~ ( E / ~the ~ ) estimate 0

As a Corollary of the proof we obtain

Lemma 3.6. Fm g lim SF(I X'I

E

>p)

C i ( R ) , any g(H) F ( I x'l

a d

a

r


(3.25)

P+

Proof. For any R i g ( H ) F(I x'l < r ) F(ly"l < R )

i s compact and the r e s u l t follows f o r t h a t p a r t . On the o t h e r hand the previous proof shows

-

ll[ g(H)

lim R+

-

< r ) F(Iy"1 zR)II=O.

g(H")] F(I xI'

(3.26)

A d i r e c t integral representation with respect t o qa reduces the problem with g ( H a ) t o the two-body s i t u a t i o n t r e a t e d i n Section 11.

0

IV. Two-Cluster Configurations

I n the l a s t section we have shown t h a t t w o p a r t i c l e s which l i e in the continuous spectral subspace of thelr subsystem Hamiltonian, have t o separate in t h e time average. I n this section we will study bounded p a i r s as subsystems of t h r e e part i c l e scattering states. The s e t of thresholds T =

U

of the three body system

T

a P P ( h a ) U 10)

C

,0] .

( - m

IS

(4.1)

a

We know t h a t zero i s t the only possible accumulation polnt of threshold values and t h a t T i s bounded below. Therefore

u

[ T-6,Tt6]

TET

i s a union o f f i n i t e l y many d i s j o i n t i n t e r v a l s whose t o t a l length tends t o zero as 640. By A=A(a,E) we denote the following bounded subsets of R A =

{W

E o(H)l

I w - T I >6 V

T

E

7 , o<E}.

(4.2)

Then the s e t of vectors ZI = { Y E P C o n t ( H ) J c l

F(HEA)

Y = '?

, 0<6<E<-}

(4.3)

i s a dense subset of PCont(H) X . The lower cutoff 6>0 i s crucial f o r our approach. I t avoids t h a t f a r separated subsystems might a l l have r e l a t i v e veloc i t i e s near zero. The upper cutoff E i s convenient b u t n o t e s s e n t i a l . Now choose a p a r t i c u l a r s e t A, i - e . f i x the values of 0<6<E<m . Suppose t h a t the p a i r a is in a bound s t a t e with energy EY E opp(h"). Then t h e conditions

185

Scattering Theory for Three Particles

Ha = ET

+ ki

EY + (2va)-'(q,)

=

2

E

A

(4.4)

determine s p h e r i c a l s h e l l s i n momentum space w h i c h f o r a l l i and a have a common minimal d i s t a n c e f r o m t h e o r i g i n . Thus t h e r e e x i s t s a f i n i t e c o l l e c t i o n o f v e l o c i t i e s v E Rv w i t h v := mi,nl vj1/5 > O such t h a t j J { q a E R" I vj-q,/ual
y

1

i s a f i n i t e open c o v e r o f t h e s e t s g i v e n i n (4.4) f o r a l l E$ E oPp(h") and f o r a l l a . Moreover t h e r e i s a &kLte decomposition o f t h e i d e n t i t y on t h i s s p h e r i cal shell:

4 g j 2(q,/v,)=l

ifq, s a t i s f i e s (4.41, any E; , any a. J We denote by gJ(qa/ua) a l s o 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 s i n q,-momentum space. F o r s t a t e s i n t h e range o f gj(q,/ua) the v e l o c i t i e s o f the t h i r d p a r t i c l e r e l a t i v e t o t h e c e n t e r o f mass o f t h e p a i r a r e c o n c e n t r a t e d n e a r v . w i t h a maximal spread o f vcl v j l /5. C e r t a i n l y t h e whole c o n s t r u c t i o n depends on $he p a r t i c u l a r c h o i c e o f t h e c u t o f f s 6 and E . The goal o f t h i s s e c t i o n i s t o show t h e f o l l o w i n g . Consider a f i x e d s t a t e Y E Pcont(H)3C w i t h energy s u p p o r t i n some s e t A . T h a t p a r t o f t h e s t a t e , where a p a i r i s i n a bound s t a t e and t h e t h i r d p a r t i c l e i s f a r away f r o m t h e p a i r f o r some p o s i t i v e t i m e T , w i l l have a s i m p l e r t i m e e v o l u t i o n i n t h e f u t u r e . F o r times l a t e r t h a n T we can n e g l e c t t h e e f f e c t o f t h e p o t e n t i a l s which c o u p l e t h e t h i r d p a r t i c l e w i t h those I n t h e p a i r . The e r r o r i s s m a l l f o r l a r g e separat i o n o f t h e t h i r d p a r t i c l e u n i f o r m l y i n TLO. Moreover these c o n f i g u r a t i o n s become a s y m p t o t i c a l l y s t a b l e , T h i s w i l l be p r o v e d u s i n g p r o p a g a t i o n p r o p e r t i e s . The m o t i o n o f t h e t h i r d p a r t i c l e r e l a t i v e t o t h e c e n t e r o f mass o f t h e p a i r i s cont r o l l e d as i n t h e two-body case. I n a d d i t i o n we need bounds on t h e d i s t a n c e b e t ween t h e p a r t i c l e s i n t h e p a i r which f o l l o w f r o m t h e r e s u l t s o f S e c t i o n 2. We b e g i n w i t h a f i n i t e decompositon o f t h e phase space f o r t h e t h i r d p a r t i c l e . F o r v ' and g. as d e f i n e d i n ( 4 . 5 ) , (4.6) we w i l l use 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 s w i t heh.-! c h a 4 a c t e r i s t i c functions o f the sets M.={ya J E R"I

Iy'IR

, y"-vj~O}.

(4.7)

I n analogy w i t h t h e two-body s i t u a t i o n we denote

R

:=

gj(q,/",)

F(Ya

Mj) gj(q,/V,),

(4.8)

Both o p e r a t o r s a r e p o s i t i v e , t h e sums a r e f i n i t e . We suppress t h e dependence on a i n t h e n o t a t i o n . The "remainder" KR =

j"

gj(q,/"a)

F ( l Y " l R ) 4j(9,/Va)

(4.10)

2 i s compact i n L (R' ,duy,). The l a b e l " o u t " i n d i c a t e s t h a t under t h e f r e e t i m e e v o l u t i o n t h e t h i r d p a r t i c l e moves away f r o m t h e c e n t e r o f mass of t h e p a i r i n t h e f u t u r e and s i m i l a r l y f o r " i n " i n t h e p a s t .

V. Enss

186

P r o p o s i t i o n 4.1. Fan a n y N E

1 e-ikst a ) IIF(lyal < -ZR+2vt)

kst

b ) 1I P p e-i

N , t L 0 , v>O a g i v e n above,

P ~ U t llcCN(ltttR)-N,

(4.11)

F ( l yal < $ t 2 v t ) l l C N ( l t t t R ) - N .

(4.12)

P r o o f . I t i s s u f f i c i e n t t o t r e a t any o f t h e f i n i t e l y many summands

1 -ikgt IIF(IyaI < 9 t 2 v t ) e g.(q /v J

dist(M.+v.t,O)

J

a

=(R2t ( v . t ) ' ) l / '

J

J

a

1

(4.13)

F(ya E Mj)ll.

1. &(Rt

v. t )

R t3vt

J

(4.14)

h o l d s with vsl v . I / 5 as g i v e n above. Now P r o p o s i t i o n 2.5 w i t h r = R ( l J Z ( 4 . 1 1 ) . F o r (4.12) a n a l o g o u s l y .

-

1 / 2 ) shows 0

R e c a l l t h a t Ha=h"a+k: , and denote by Pq t h p r o j e c t i o n o n t o an e i g e n v e c t o r o f ha Pauf r e l a t e s t o t h e same p a i r i n g cx w i t h e i g e n v a l u e E j E &P(ha)c(- m , O ] . Operators on t h e f a c t o r spaces and t h e i r n a t u r a l e x t e n s i o n t o t h e f u l l t h r e e body H i l b e r t space a r e denoted by t h e same l e t t e r s .

.

(4.15) (4.16) l i m E(R)=O. R+ m P r o o f . There i s a

Then

Py =g(ha) P;

8E

Ct(R ) w i t h 8 ( w ) = l i f

w E

opp(ha), supp

8

.

c ( - m,pav2/2).

(4.17)

Next choose p ( 2 ) l a r g e enough such t h a t f o r g i v e n E ' > O , P L ~ ( E ' ) IIF(I XaI

'0)

PYII < E 1 / 4 .

(4.18)

To prove (4.15) i t i s t h e n s u f f i c i e n t t o show t h a t (4.19) F o l l o w i n g Cook's method t h i s i s bounded by t h e i n t e g r a l o f t h e norm o f t h e d e r i vative

7 dt

I1 (H-Ha)

< 1

7 d t IlVB(x3) e-ihat

0

-

emH i at

8(ha) F(I xaI


Piutll

8 ( h a ) F ( l x " l < p ) e - i k a0 t PR outll

B#a 0

Each i n t e g r a n d i s bounded b y l l V B ( x B ) F ( I xal < Z3 ( p t v t ) ) F ( l y a l

>

p1t 2 v t ) l l

t

.

(4.20)

187

Scutteririg Theory for T h e e Purticlrs +IIV I1 .IlF(I xal > Z3 ( p + v t ) ) emihat

e

t(h")

+IIV II.IIF(lyal < -ZR+2vt) 1 e- i k o0 t PR outll

e

F ( I xal q ) I I

.

(4.21)

By Theorem 2.7 t h e second summand i s bounded b y const(p+vt)

-(l+E)

Choose p ~ p ( ~ 'l a) r g e enough such t h a t m

c OZU

IIV II- c o n s t l3

(4.22)

I dt (~+vt)-(~+')<~'/6. 0

S i m i l a r l y choosing R>R(E') t h e c o n t r i b u t i o n f r o m t h e t h i r d t e r m i n ( 4 . 2 1 ) i s s m a l l e r t h a n ~ ' / 6by P r o p o s i t i o n 4 . 1 . F o r o#a one has I x @ I ~ y -aI lx U l . W i t h t h e decay assumption ( 1 . 5 ) on t h e p o t e n t i a l t h e f i r s t summand i n (4.21) i s bounded by I1VB(x3) F ( I x'l,

T1( R + v t - 3 p ) ) l l c o n s t ( l + R - 3 p + v t )

- ( 1tE)

(4.23)

F i n a l l y choose R r R ( € ' ) l a r g e enough t o make t h e i n t e g r a l o f (4.23) s m a l l e r t h a n ~ ' / 1 2 . Then ( 4 . 1 9 ) h o l d s and (4.15) i s proved. The p r o o f o f ( 4 . 1 6 ) i s analogous. 0

We have shown t h a t f o r t h e outgoing t w o - c l u s t e r p a r t t h e t h i r d p a r t i c l e t r a v e i s l i k e a f r e e p a r t i c l e . I t remains t o show t h a t a g i v e n s t a t e \y does n o t have incoming p a r t s w i t h l a r g e R i n t h e f u t u r e . T h i s e s t i m a t e cannot be u n i f o r m i n Y as t h e p r e v i o u s one was.

(4.24)

P r o o f . By P r o p o s i t i o n 4.2

(4.25) Using P r o p o s i t i o n 4 . 1 we see t h a t

l i m s u p IIP;

R+

-

pR in

e-iHaT

yII

720

(4.26) 0

(4.27)

V. Ems

188

(4.29) (4.30) p 4 o j e c t . i a ~ n . The bum oueh i 0 undetcn.tood a.4 oucx a cvmp.kte p a i n w h e o h t h a g v n d eigenvectotL.J, i . e . c Pq = P P P ( h a ) .

dedine o h t h o g o n d

06

be,t

i Remark. The d i s t i n c t i o n of t h e f u t u r e and past is necessary, the operators will b e e r e n t i f rearrangement c o l l i s i o n s are possible. The ranges of these proj e c t i o n s will be discussed in connection with the wave operators in Section VI. Proof. Since the approximating sequence of operators i s uniformly bounded and commutes with the time evolution i t i s s u f f i c i e n t t o show convergence on the s e t U as defined i n ( 4 . 3 ) . For the p a r t i c u l a r set A corresponding t o a given Y E U, and f o r given LY , construct the operators P a u t , KR as in (4.8) - ( 4 . 1 0 ) . Their sum i s a function of the momentum o the ' and third p a r t i c l e only, i t i s the i d e n t i t y on the range of PqF(Ha E A ) by ( 4 . 6 ) , a n d f o r € 4there i s a small neighborhood A ' of A such t h a t l l I P T t + PRi n + KR - li 1 F ( H ~E A ' ) I I < E . (4.31)

pcu"?H)

P4n

PP

Choose $ Moreover

E

C i ( l R ) such t h a t $ ( w ) = l f o r

P; [ + ( H I =

-

i(21~)-'"

o E

A

and s u p p

F(H 6 A) t I d t $ ( t )J d s e +iH"(t-s) Pq(fi-Ha) R 0

I$ c A ' .

Then $ ( H ) Y = Y . (4.32)

$(Ha)]

F(H

E

A ) e+iHs

i s compact as a norm-convergent integral o f a norm continuous compact operator valued function, since P q V e F ( H E A ) i s compact f o r ~ # a. Also PT KR i s a compact operator on Jc f o r any R < - . Since the time average of compact oper a t o r s tends t o zero on t h e continuous spectral subspace by Lemma 3.1 we obtain (4.33) f o r Y E U , any R< - . By Lemma 4 . 3 the contribution from the term with P a n i s bounded by E f o r a l l R > R ( E ) . Thus we can conclude: Let Rnf- be any increasing sequence of c u t o f f - r a d i i , then t h e r e i s a sequence T,,.C~ such t h a t (4.34)

T h i s i s an expression o f t h e i n t u i t i v e l y c l e a r f a c t t h a t f o r a s c a t t e r i n g s t a t e with a bounded p a i r the t h i r d p a r t i c l e has eventually t o be f a r separa ed from the p a i r and be moving away from i t (be outgoing). The subsets Pq ’:NP are "absorbing" f o r the p a r t Pq e-iHt Y of the t r a j e c t o r y . The f u t u r e time evol u t i o n beyond T~ i s well approximated by exp(-iHat) which leaves Pq unchanged. This allows t o show t h a t a l s o the range o f Py i s absorbing. Hence exp(iHt) P y exp(-iHt) i s monotone and the l i m i t will e x i s t ,

2, 2,

sup e

t o

iHt ,-iHt

p;

,-iHTn

Rn

2, "u

189

Scattering Tlzeorv j o r Three Particles

2 sup e

iHt e-iHat p;

tzo

,-iH.rn

Y

Rn

(4.35) Here 5 denotes an a r b i t r a r i l y small e r r o r f o r n l a r g e enoy h and we have used (4.34) and Proposition 4 . 2 . For two p r o j e c t o r s l i k e Pq and elityPq e-iHt t h e property I1 ( P2P1 - P1)d < E imp1 i e s (O,(P2 -P 1 ) '3)~(@,(P1P2P1- P1) Q ) ~ - ~ ~ P."(P2P1-P1) 1@~~ Therefore the sequence

IIPf e-iHt YII

(4.36)

@I!>-€.

i s asymptotically increasing and

(4.37) e x i s t s f o r a l l Y E D . This implies strong convergence because.of the projections on the subspace Pcont(H)Jc. Note t h a t s-lim P; Ppp(H) e-1Ht = 0 uniformly n-t m in t E B . T h u s Ran P"m) C Ran PCont(H). W i t h the r o l e s of P i n and interchanged the same r e s u l t s i o l ow f o r t - t - m . The orthogonality f o r i # j i s obvious. For d i f f e r e n t pairings P i P j i s compact and

PiUt

(4.38) For large 4.4.

T

t h i s i s small by Lemma 3.1. This f i n i s h e s the proof of Proposition 0

V. Phase Space Localization and Asymptotic Observables

W e will derive i n the next s e c t i o n t h a t s t a t e s in t h e ranges of P(FQ-) as defined in (4.30) will have simple asymptotic time evolutions. To t r e a t the orthogonal complement in Pcont(H)JC we will show again t h a t c e r t a i n subsets of the s t a t e space a r e "absorbing". The subsets a r e r e l a t e d t o t h e c l a s s i c a l phase space of the three-body system, i . e . we obtain simultaneous information about r e l a t i v e positions and momenta ( v e l o c i t i e s ) of the p a r t i c l e s . The method i s very d i f f e r e n t from t h a t of the previous section where we could use o u r d e t a i l e d knowledge about two c l u s t e r systems. Here we ask instead very modest questions which a r e expressed through the asymptotic behaviour of some observables. They a r e chosen such t h a t they can be controlled even under the f u l l y l n t e r a c t i n g time evolution.

an g i u m in S e c t i o n 7 and

Theorem 5.1. Foh H Y E P"rn'LW IK obey lim n-t

II[f(pa

lim n+ w

Il[f(vcr

Xa

n

Q-

$n

P(&-m)

)

-

Then - iH T ~

Y = 0.

f(Pa)l e

60h

Yll =o,

- f ( q a ) l e-iHTn lvl1=0,

P(+m

any f

E

) dehined i n ( 4 . 3 0 ) Le,t C;(R") and any a

(5.1) (5.1')

V. Ens3

190 604

f E(:C

what

T,,

b'2.gUelzCU

W 2' )

and any

a

m e nlLitabee sequenceo tcndiiig t o 5 T n may be chobefl ifldependent 0 4 f .

m

604

P(+

m),

4 e o p e c t i v d y . The

Two c l a s s i c a l f r e e p a r t i c l e s s t a r t i n g f r o m one p o i n t a t t i m e zero w i t h r e l a t i v e momentum pa and reduced mass ua w i l l s a t i s f y paxa/~,,=pa a t t i m e T~ where xa i s t h e i r s e p a r a t i o n . One s h o u l d t h i n k o f t h e f ' s as smoothed c u t o f f s for s m a l l r e gions. That p a r t o f t h e s t a t e which has g i v e n momenta pa w i l l be l o c a l i z e d asymptot i c a l l y where one i n t u i t i v e l y expects i t t o be. The simultaneous i n f o r m a t i o n about t h e p o s i t i o n and t h e momentum does n o t c o n t r a d i c t t h e u n c e r t a i n t y p r i n c i p l e . Due t o t h e smoothness o f f t h e u n c e r t a i n t y o f t h e p o s i t i o n grows l i n e a r l y i n I T n l w h i l e t h e u n c e r t a i n t y o f t h e momenta i s f i x e d . Thus t h e p r o d u c t w i l l e v e n t u a l l y be l a r g e r t h a n P l a n c k ' s c o n s t a n t . Moreover we see f r o m ( x - x o ) / ( t - t o ) x/t + 0 as I t l + - t h a t t h e c h o i c e o f t h e o r i g i n o f t i m e does n o t m a t t e r a s y m p t o t i c a l l y and t h a t any f i n i t e i n i t i a l s e p a r a t i o n w i l l l e a d t o t h e same r e s u l t . Thus any t r a j e c t o r y which a s y m p t o t i c a l l y approaches a s t r a i g h t l i n e , r u n a l o n g w i t h c o n s t a n t speed, s h o u l d l e a d t o (5.1), t h e d e t a i l s o f t h e i n t e r a c t i o n as e.g. a f i n i t e t i m e d e l a y need n o t be c o n t r o l l e d t o o b t a i n t h e theorem. C e r t a i n l y t h e same i n t u i t i o n a p p l i e s t o t h e m o t i o n o f t h e t h i r d p a r t i c l e r e l a t i v e t o a p a i r . In p a r t i c u l a r we l e a r n f r o m Theorem 5 . 1 t h a t a s y m p t o t i c a l l y non-zero momenta a r e p a r a l l e l t o t h e p o s i t i o n v e c t o r s i n t h e f u t u r e and a n t i p a r a l l e l i n t h e p a s t , i . e . o u t g o i n g o r i n coming, r e s p e c t i v e l y , i n an even sharper sense t h a n used above.

-

We i n t r o d u c e t h e f o l l o w i n g s e l f - a d j o i n t f u n c t i o n s o f t h e p o s i t i o n and momentum operators :

x2

Ixa?

=

+

,

walya I 2

(5.3)

Both o p e r a t o r s a r e independent o f t h e p a i r i n g a s i m i l a r l y as Ho which i s g i v e n i n ( 1 . 4 ) . C e r t a i n y i t i s s u f f i c i e n t t o v e r i f y Theorem 5 . 1 on a dense subset o f , we r e s t r i c t i t l a t e r t o a l l o w a p p l i c a t i o n o f a l l un[I- P ( k pcont(Hj bounded o p e r a t o r s X2,D, and H. L e t

U4.

-

=

F(H<E) [ 1 -

u

P(k

m)]

PCont(H)

Jc

.

(5.5)

E<m

These a r e dense i n t h e g l v e n s e t . T y p i c a l l y a l s o t h e s e t s

D+ n U ( l

dense, where U(I XI ) i s t h e domain o f t h e o p e r a t o r I XI =(X2)172. n e c e s s i t y t o prove i t by a f u r t h e r a p p r o x i m a t i o n . Y A =(I +

,x2)-1

Y

.

XI ) w i l l be

B u t we a v o i d t h e (5.6)

The f o l l o w i n g domain p r o p e r t i e s a r e m a i n l y t e c h n i c a l , t h e y a r e easy t o show. F o r l a t e r use we c o l l e c t them w i t h o u t p r o o f . Lemma 5.2. L e R a)

lim A+ 0

? E

U* arid

Y

x

a$ de6ined in ( 5 . 6 ) .

x x ? ]=O, I I I Y - Y I I + I I H ~ o ) ( ~ - )ll

(5.7)

Scattering Theory f o r Three Particles

C)

l i m s u p {IIP(+ -)Y t++-

d)

e-iHt

e)

e-iHtH~A

A

A II+IIP(+ -) H Y I I } < E v O < A ~ A ( E ) ,

v t

E

W , A>O

.

’Y E U ( I XI ) n D ( H ( o ) ) E

191

U(lXI ) v t E

(5.9)

R , x>O;

(5.10) (5.11)

We w i l l reduce Theorem 5.1 t o t h e f o l l o w i n g P r o p o s i t i o n 5.3. LeA Y E Qk and y A an dt.dined i n ( 5 . 5 ) and ( 5 . 6 ) . F a t any E>O thehe A a A ( E ) ouch Rhu-t .doh UUO
l i m s u p I (yA,eiHT

-

:{

-

ZHI e

-~HT A y

)ICE

,

(5.12)

T++m

b)

A

l i m s u p I ( Y ,e

iHT

1 x2 {27

H} e

-iHT

A Y )I <E

,

(5.13)

.r++ -m

c)

l T l i m s u p 17 J ds (YA,eiHs

-

{Ho

HI e

-iHs

A

Y )I<€

.

(5.14)

0

T++-

Remark 5.4. The shorthand

O(T) = e

iHr

,,

e-iHT

(5.15) D (and s i m i l a r l y f o r other s e l f a d j o i n t o p e r a t o r s ) i n t h e Heisenberg p i c t u r e . We u r g e t h e r e a d e r t o conv i n c e h i m s e l f o f t h e s i m p l e meaning o f t h e P r o p o s i t i o n u s i n g paper and pen. F o r t h e two body s i t u a t i o n (= one p a r t i c l e moving i n a p o t e n t i a l ) t h e s t r a i g h t l i n e i n t h e space-timediagram d e s c r i b e s a f r e e p a r t i c l e while the dotted l i n e depicts the m o t i o n under s t r o n g i n t e r a c t i o n s (elastic reflection a the rigin). The o b s e r v a b l e (m/2)x (T)/T does n o t "see" whether an i n t e r a c t i o n t o o k p l a c e o r n o t , and a s y m p t o t i c a l l y equals H&H.

denotes t h e t i m e e v o l u t i o n o f t h e o b s e r v a b l e

r I I

/

/ I I

/

I

t

/ I I

\

\

Proof

$

8

S i m i l a r l y f o r t h e o t h e r observables. The t h r e e body case i s now a s t r a i g h t f o r w a r d g e n e r a l i z a t i o n . F o r sketches i t i s c o n v e n i e n t t o use t h e c e n t e r o f mass o f a p a i r as t h e o r i g i n i n space. M a t h e m a t i c a l l y t h i s observat i o n i s expressed by t h e f a c t , t h a t a l l terms i n v o l v i n g t h e p o t e n t i a l a s y m p t o t i c a l l y d i s a p p e a r by compactness arguments.

o f Theorem 5.1. By t h e f u n c t i o n a l c a l c u l u s

f ( pi,xa/~,)

=

IT)-'/^

Jd"u ?( u ) exp( iu.paxa/Tn),

(5.16)

Y. Enss

192

and analogously f o r the o t h e r operators. Due t o t h e rapid decay of f ( u ) the norm (5.1) i s dominated by a constant times (5.17)

By the canonical c o m u t a t i o n r e l a t i o n s t h e difference of t h e exponentials i s exp(iu.p,)

{expl iu,(~ax,/Tn-p,-uu,/2T~)]

(5.18)

- 11 1 .

We w r i t e t h i s difference a s an integral 1 i i ds exp[ isu. ( ~ a ~ a / ~ n - p a - ~ u a /u.(u,xa/Tn-p,-uuu/2Tn) 2~n)~

.

(5.19)

0

(5.20) Up t o a constant t h i s i s a l s o a bound f o r the norm in ( 5 . 1 ) f o r any Y. I t i s s u f f i c i e n t t o v e r i f y Theorem 5.1 f o r E Q+ - . For given E approximate Y by Y A and find A ~ ( E ) such t h a t f o r O < ~ < A ~ ( E )

2 sup1 fl IIY-YAll

<E

.

(5.21)

Uniformly f o r I T n l >T(E) the contribution from the f i r s t summand i n (5.20) i s bounded by E . T h u s i t remains t o f i n d sequences T n (which a r e independent o f f ! ) such t h a t (5.22)

f o r s u i t a b l y chosen for

1 11 ( v a 2va

n

-

x

E

( O , X ~ ( E ) ) . Analogously f o r ( 5 . 1 ’ ) we have t o f i n d a bound

9,) e-iHTn i u ~ i l * .

(5.23)

The introduction of the cutoff Y + Y ~ guarantees t h a t these norms a r e f i n i t e by (5.10). Summing the contributions from (5.22) and ( 5 . 2 3 ) we obtain

+(Y’,

{ H O ( ~ , , -) H I yA).

(5.24)

The f i r s t two terms obey the desired estimate f o r any sequence T n + k m by Propos i t i o n 5.3 a ) and b ) . For t h e t h i r d summand, which i s continuous, we know from (5.14) t h a t i t s time average is small. Therefore j t i s small f o r some sequences T ~ ++ - . The proof of (5.2) i s analogous. 0 Proof of Proposition 5.3. The formal calculation d ,iHT e-iHr -- e i H ~i [ H , D l e - i H ~

a;

(5.25)

193

Scuttering Tlreorj, for Three Particles

i s j u s t i f i e d i n a s c a l a r p r o d u c t between v e c t o r s Y h b y Lemma 5.2. U s i n g t h e canon i c a l commutation r e l a t i o n s we g e t

i[H,D]

=

i[Ho.D]

i[V,DI

t

= 2Ho t

i Z[ Va,D] a

.

(5.26)

F o r t h e commutator w i t h V, we use t h e r e p r e s e n t a t i o n ( 5 . 4 ) o f . Then t h e o n l y c o n t r i b u t i o n comes f r o m d = 1{ p axa t x".p a ' 2 a i z[Va,D]

} =

-

x".p

= z{i(Vaxa-p,

-

iv/2.

p;xaVa)

t

D

w i t h t h e same (5.27)

v.Va3 ,

(5.28)

a

a

i[H,D]

= 2 H + Cii(Vaxa-pa a = 2H+

-

+

p;XNVa)

(w-2)Va}

.

Z Ka

(5.29)

a

By t h e fundamental theorem o f c a l c u l u s

1 (vX, D ( T )

yx)

= i ( ~x . D a~ ) +l ; Tf d t ( x~ , i eH t i.[ H , D ]

e- i H t Y h )

0

1 (u', D u x )

= -

t (Y',

~Hu') t Z a

-rI

T

f d t (Y', 0

Ka(t) Y ' ) .

(5.30)

F o r any a d t h e f i r s t summand i s f i n i t e and i t vanishes as I T I + - . The second term i s t h e d e s i r e d l i m i t , and i t remains t o show t h a t t h e t h i r d t e r m i s s m a l l . I t s s t r u c t u r e i s v e r y s u m i l a r t o (5.14):

1’ x c 7 J d t (Y , VJt)

Yx ).

(5.31)

c l o Let g

E

C;(IR)

be such t h a t

I ( Y A Y Ka(t) ya) <

211[ 1 -g(H)]

-f

0

-

as

x-

-

(H-z)

0

g(H) Y = Y . Then

( y h 9 g(H) K a ( t ) g(H) y X ) l

(Y'-Y)II e l l (H-z)-'

uniformly i n

t

Ka(H-z)- 1II .I1 (H-z) d I I (5.32)

,

Since g(H) K, g ( H ) a r e bounded we may r e p l a c e ,yA by Y making an a r b i t r a r i l y s m a l l e r r o r . Moreover observe t h a t I xa Va(xa)l 5 const (1+1 xaJ ) - E , t h u s

lim

(5.33)

llg(H) Ka g(H) F(I xal > p ) l I = O

P'-

by Lemma 3 . 6 . Consequently i t i s s u f f i c i e n t t o e s t i m a t e

1

d t ( u , eiHt

g(H) Ka g(H) F ( I x a l < p ) e-iHt

Y)

5

v. Enss

194 IIg(H) K,

<

g(H)II

l T J'

dtll F ( I x " l < p ) e -iHt

yII

.

(5.34)

0

The l a s t norm can be bounded by IIF(I xal < p ) Pcont(ha) e-iHt

+

'

1 P ~ \ I+

A F ( Ixa\

j>N

yll

z

IIPY e-iHt

jiN

J

(5.35)

y~t.

Now we use f i n a l l y t h e s p e c t r a l asymptions about

Y . By P r o p o s i t i o n 3.5 t h e t i m e average o f t h e f i r s t term vanishes as I T I + - . Choosing N l a r g e enough t h e second term i s s m a l l e r than any E . The l i m i t as t + m o f t h e f i n i t e l y many remain i n g terms vanishes because Pg(+ -)Y=O ( c . f . P r o p o s i t i o n 4 . 4 ) . Thus a l s o t h e f u t u r e t i m e average vanishes a s d m p t o t i c a l l y . S i m i l a r l y f o r t h e p a s t i f P(- -)\y=O.

0

The e s t i m a t e o f ( 5 . 3 4 ) w i l l be used a g a i n . Therefore we s i n g l e i t o u t a s a separ a t e statement.

Let

Lemma 5.5.

lim T ++ m T

1

Y E

d t CllF(I x'l 0

..)I PCont(H).

Ran[ ]I - P(+ cp)

Then

dub. any

p<

-

e-iHtyll =o.

(5.36)

CL

C o r o l l a r y 5.6. L e t P E Ran[ ]I - P(+ -)I Pcont ( H ) . The& t h m c L M ~dequenct~ 1e.g. t h e dame nequence, 60k uJhich Theonem 5.1 h a been pkoved) duch t h a t du.t =fly g C;(R 1 f

a) b)

-

lim n+

C)

-

lim

n-f Id

II[ g(H) - g(Ho)[ e-iHTn

Ill g(h')

SUPP

-

g(h:)]

(5.37)

ull=O,

e-iHTn

(5.38)

~ll=O,

g c ( - -,O), iheM (5.39)

g(H) \Y = 0. P r o o f . By Lemma 2.5 lim P

'

-

II[ g(ha)

g(h:)]

F(I xCLl>p)ll=O.

m

S i m i l a r l y one shows

-

l i m I][ g(H) P*

-

g(Ho)l

n F(I

xQI > p ) l l = O .

a

With Lemma 5.5 t h i s i m p l i e s (5.38) and (5.37), g(Ho)=O and (5.37).

r e s p e c t i v e l y . (5.39) f o l l o w s f r o m 0

V I . Wave Operators For a t h r e e body system one d e f i n e s t h e f o l l o w i n g f o u r wave o p e r a t o r s i Ht e- i H o t QO = s - l i m e t-

m

Scattering Theory f o r Three Particles

aa

= s-lim

ei H t e - i H ' t

195

pPp(ha);

t+

and s i m i l a r l y f o r t h e p a s t . To s i m p l i f y n o t a t i o n we have o m i t t e d t h e s i g n s den o t i n g t h e t i m e d i r e c t i o n . I n t h i s s e c t i o n we show e x i s t e n c e o f t h e l i m i t s and b e g i n t h e a n a l y s i s o f t h e i r ranges. T h i s w i l l be c o n t i n u e d i n t h e n e x t s e c t i o n where we show completeness. I t i s c o n v e n i e n t t o use t h e f o l l o w i n g c r i t e r i o n f o r existence. Lemma 6.1. a) QO ~ X A 4L 6~ Tn+

b)

ha.

a t o t a t b e * 04 u e o t o a

Y

and

hume. btqueMCe

m

na exdin;tn 4 4 lim

sup I l ( e- i H t

n-t

t2o

-

i and a n ~ . t 06 ueotuc,

any

t h m e 0 u 4equence

-

~

~

which h tutu!. i n

Y

P?

Wm ~AL

+

,-iH"t

e-iHaTn

yII

= 0.

(6.4)

We o m i t t h e s i m p l e p r o o f which i s a r e w r i t i n g o f t h e Cauchy convergence c r i t e r i o n ( s e e Lemma 8.1 i n [ Z ] ) . P r o p o s i t i o n 6.2.

The wave opehatoh no exhin;tn-

P r o o f . The wave f u n c t i o n f o r a g i v e n s t a t e Y depends on t h e p a i r i n g a , i . e . on t h e o s e n c o o r d i n a t e s pa,q,(or xa,y"). We choose a t o t a l s e t o f v e c t o r s h a v i n g s t r i c t l y non-zero r e l a t i v e momenta f o r a l l p a i r s a s i m u l t a n e o u s l y . n p a r t i c u l a r we assume t h a t t h e r e a r e v e l o c i t i e s O#va E l R v (depending on t h e s t a t e Y ) such t h a t t h e momentum space wave f u n c t i o n s $a(pa,qa) E C;(RZv) (wh ch depend on a ) f o r a s t a t e P s a t i s f y t h e f o l l o w i n g s u p p o r t p r o p e r t y f o r a l l a SUPP

$,(Pa4,)

c tP,

6

RU I

I

P,-LIaVB1

<

l

a

3 !Jal v 1

}.

I t i s c l e a r t h a t t h e s e t i s t o t a l i n JC. The f r e e H a m i l t o n i a n has p u r e l y cont i n u o u s spectrum and t h e same h o l d s f o r t h e subsystem H a m i l t o n i a n s . T h e r e f o r e P ( k m ) = 0 and Theorem 5.1 a p p l i e s t o t h e f r e e t i m e e v o l u t i o n on t h e whole H i l b e r t space. I f

fB ( PB( Pa 4,) '9@(Pa 99,)

1

= fa

( P, 4,),

16-51

t h e n f"(p,,q,) a r e d i f f e r e n t a- ependent e x p r e s s i o n s f o r t h e same m u l t i p l i c a t i o n o p e r a t o r i n momentum space. vBxg and w4ya a r e expressed by t h e same l i n e a r t r a n s f o r m a t i o n s i n terms o f pax, and way" as p and qB a r e expressed i n terms o f p and q., . T h e r e f o r e fff( vaxa/7,u ye/,) a r e t 8 e d i f f e r e n t o o r d i n a t e system depen%ent e x p r e s s i o n s of t h e .sme o p e r a t o r . We choose f a E CG( R 5 v ) , O l f a l such t h a t f o r all 0

Then [ 1 1im n-. m

V. E m s

196

Cook's e s t i m a t e y i e l d s a bound on t h e d i f f e r e n c e o f t h e f r e e and i n t e r a c t i n g t i m e e v o l u t i o n s on v e c t o r s o E D ( H o ) :

Thus

it i s sufficient to m

l i m i d t IIVa e-iHOt n-t- o I t i s p o s s i b l e t o choose t h e same CI f o r V, and fa s i n c e as o p e r a t o r s t h e fa(*;) a r e independent o f a . We e s t i m a t e t h e i n t e g r a n d i n ( 6 . 7 ) by

IlVa(xa) F ( I xal > 1 I vaI ( t t ~ ~ ) ) ti l t

IIV II - I I F ( Ixal <

1 I val ( t + T n ) ) e- i H o t

a f

(Pa.Sa)

a f

(11,

xa T ,v n

a )II

.

a Tn

The i n t e g r a l o f t h e f i r s t summand i s f i n i t e and i t vanishes as T ~ + - by t h e decay t h e second t e r m i s bounded by assumption (1.5) on t h e p o t e n t i a l . Up t o IIV,II

2 where t h e orm i s on L (W") and qa and yCc/~,,a r e c o n s i d e r e d as parameters. Since fa E CG(!R") t h e parameters may be r e s t r i c t e d t o a compact s e t . Then P r o p o s i t i o n 2.4 w i t h r=I vaI Tn/3 shows r a p i d decay o f (6.8) i n ( t t T ) . T h i s shows ( 6 . 7 ) and thus completes t h e p r o o f o f P r o p o s i t i o n 6.2. 0 Remark. I f we were i n t e r e s t e d o n l y in t h e e x i s t e n c e p r o o f we c o u l d have g i v e n a m u c h o r t e r argument. An easy e x t e n s i o n o f t h e e s t i m a t e s o f S e c t i o n I 1 a l l o w s t o show f o r t h e g i v e n t o t a l s e t o f v e c t o r s Y I I F ( IX

~
1 I vCl ( t t T ) ) e

for any a and N E N . From t h i s one e a s i l y concludes m l i m 1 d t c IIVa e - i H o t HOT Yll = o. T+mo

a

By Cook's e s t i m a t e t h i s i m p l i e s ( 6 . 3 ) . B u t f o r t h e completeness p r o o f we w i l l need t h e f u n c t i o n s f a and t h e e s t i m a t e (6.7) anyway, so we have used i t h e r e as w e l l t o save space. A s i m i l a r remark a p p l i e s t o t h e channel-case c o n s i d e r e d n e x t .

P r o p o s i t i o n 6.3. The

d w t m wave upmdom

Qa

ex0.t.

P r o o f . We use t h e n o t a t i o n , t h e energy r e s t r i c t i o n s o f t h e t o t a l s e t , and t h e r e s u l . o f S e c t i o n I V . F i r s t f i n d f o r g i v e n E>O by P r o p o s i t i o n 4.2 an R ( E ) such that sup II (eSiHt

t>n

-

e-iHat)

P?i PoUtll<~/3 R v

Lemma 4.3 guarantees t h a t f o r some R>R(E)

R>R(E).

(6.9)

Scattering Theor.),for Three Particles

197

(6.10)

I f Y = Py C gS(q,/ua) on Y i s

Y

t h e r e m a i n i n g t e r m i n t h e decomposition o f t h e i n d e n t i t y

C = P !i K R '

(6.11)

( c . f . (4.10)) which i s compact on t h e f u l l t h r e e body H i l b e r t space f o r any Since Ha has p u r e l y c o n t i n u o u s spectrum t h e t i m e average o f IIC e-iHaT yll

R.

(6.12)

decays by C o r o l l a r y 3.2. I n p a r t i c u l a r we can f i n d f o r t h e a T = T ( E ) such t h a t (6.12) i s bounded by ~ / 6 . F o r t h i s T

R

determined above (6.13)

By Lemma 6 . 1 t h i s i s s u f f i c i e n t f o r t h e e x i s t e n c e .

0

The ranges o f t h e wave o p e r a t o r s c o n s i s t o f t h o s e v e c t o r s which a s y m p t o t i c a l l y have a s i m p l e r t i m e e v o l u t i o n . Lemma 6.4. a)

0

Ran n

= {Y

I t h e r e i s a sequence

l i m sup n+m t L 0 b)

Ran

II(e

-

-iHt

= ( Y I t h e r e i s a sequence

,im

sup

n+m

t20

-

lim n-

II[

Tn+m

,-iH0t)

T ~ + -

I l ( e - i H t - ,-iH"t)

II - ppp(h")] e-iHTn

such t h a t

,-~HT,

such t h a t

,-iHTn

Yll =

Remark. Here (and i m Lemma 6.1)

yll = 0,

E

Ran no i f f t h e r e i s a

0 = l i m IIY t+m

-

e

i H t ,-iH0t

0 E Jc

and

0. I .

t h e l i m f o r some n- m both sets o f conditions are equivalent.

Proof. a) Y

(6.14)

Yll = 01.

(6.15) (6.16)

can be r e p l a c e d by l i m

-rn+m

w i t h y=no@

,

T+ m

.

ID 1I

(6.17) S t r o n g convergence i n t h e l a t t e r e x p r e s s i o n i s c h a r a c t e r i z e d by (6.14), Lemna 6 . 1 a) w i t h t h e r o l e s o f H and H, i n t e r c h a n g e d . b)

C o n d i t i o n (6.15) i s f o u n d a n a l o g o u s l y , i n a d d i t i o n which i s guaranteed by ( 6 . 1 6 ) .

ID

see

must l i e i n Ppp(ha) 0

V. Ems

198

We conclude t b i s s e c t i o n r e l a t i n g t h e ranges o f t h e c l u s t e r wave o p e r a t o r s t o t h e p r o j e c t i o n s P (+ -) i n t r o d u c e d i n S e c t i o n I V .

The ohthogun& p t v j ~ X . o P"(+ ~ me ,the hange ptrojectionn 06 na .

- P

-) dehined .in PhapobiLLon 4.4

C e r t a i n l y a l l r e s u l t s o f t h i s s e c t i o n h o l d a n a l o g o u s l y i n t h e remote

Remark 6.6. past (t+ -

m).

Proof. IfPa(+ -) Y = 0 t h e n P i s n o t i n t h e range o f na by p r o p e r t y (6.16). On t h e e r hand t h e v e c t o r s ( u s i n g n o t a t i o n ( 4 . 1 ) - ( 4 . 3 ) ) {Y E

Ran P;(+

w ) l F(HEA) Y = I, O < s < E < - }

(6.18)

f o r m a t o t a l s e t i n t h e range o f P ( + - ) . M i m i c k i n g t h e e x i s t e n c e p r o o f o f na we use t h e r e s u l t s o f S e c t i o n I V t o show t h a t t h e r e i s a sequence T n + = such t h a t

lim n+m Thus

sup ti0

Ran P*(+

I,(e-iHt

m)

-

e-iHatl

,-iHTn

Yll = 0.

(6.19)

c Ran a", and t h e p r o o f i s complete.

0

V I I . Completeness and S p e c t r a l Consequences I n t h i s s e c t i o n we show o u r main r e s u l t , systems. T r a d i t i o n a l l y t h i s i s s t a t e d as Ran no @ Ran na = PCont(H)Jc

a s y m p t o t i c completeness f o r t h r e e body

.

a

An e q u i v a l e n t , p h y s i c a l l y more i n t u i t i v e s t a t e m e n t i s t h e f o l l o w i n g . Any s c a t t e r i n g s t a t e y ( i . e. an s t a t e o r t h o g o n a l t o a l l bound s t a t e s : Y E (PPP(H)J€)~ = Pcoxt(H)X ) a s y m p t o t i c a l l y has a s i m p l e t i m e e v o l u t i o n : A t a l a t e time T there i s a s p l i t t i n g

(7.3) 17.4) E(T)

-f

0

as

T

-f

m

.

(7.5)

The e q u i v a l e n c e o f t h e s e p r o p e r t i e s w i t h (7.1) i s e v i d e n t f r o m Lemma 6.4 and t h e remark f o l l o w i n g i t . C l e a r l y Ho generates a s i m p l e t i m e e v o l u t i o n , b u t a l s o Ha PPP(ha) i s s i m p l e as a d i r e c t sum o f f r e e subsystem H a m i l t o n i a n s

We have shown a l r e a d y i n P r o p o s i t i o n 6.5 t h a t t h e ranges o f Pa(+ -) c i d e . I t remains t o show t h a t Ran no = Ran [

P - P(+ -)I PCont(H).

and R" c o i n -

(7.7)

199

Scattering Tlieorj, f o r Three Particles

Theorem 7 . 1 . L e t H be t h e SchhGdingeh ope.ha.toh doh a t h e e body bybtekti M deXined i n S e c t i o n I . Then t h e -time e w o P d o n 0 aympto-ticdCey c o m p l e h , i.e. ( 7 . 1 ) hoP& o h e q u i w d e d y (7.2) - ( 7 . 5 ) . S i m d d y doh t h e p a n t . Proof. By Proposition 6 . 5 and t h e orthogonality of the ranges of the wave operat o r s one has Ran no

C

Rani11

-

P(+

(7.8)

m)]PCont(H).

W e w i l l prove the opposig inclusion by showing t h a t f o r a t o t a l s e t o f vectors Y E Ran[ 11 - P(+ -)I Pcon ( H ) t h e property (6.14) holds. W e choose

D

-

u

=

Ran F(G
-

P ( t =)I PCont(H).

(7.9)

o<S<E<

By Corollary 5.6 c ) D i s t o t a l i n t h e given s e t . Now pick a sequence Tn'" such t h a t Theorem 5 . 1 and Corollary 5.6 hold. For any 6’<6 and E ' > E we conclude from (5.37) (7.10) The elimination of zero k i n e t i c energy Ho does not y e t exclude the p o s s i b i l i t y o f zero r e l a t i v e momentum f o r a p a i r . I f , however, t h e r e l a t i v e momenta f o r one p a i r a r e very small, then the r e l a t i v e momenta must be s i g n i f i c a n t l y higher f o r the o t h e r two p a i r s . W e make a s p l i t t i n g accordingly. We can pick a s u f f i c i e n t l y small velocity v>D such t h a t I gether imply f o r a l l

.

I q a l > 5 WaV

ga E C i ( R ) , Ozgazl, g a ( w ) = 1 i f w q -a Then f o r a#B:ga(hE) g'(h8) F(Ho>6) = 0. Moreover there i s a dinite s e t o f v e l o c i t i e s vg

There a r e

33

pal5pav

and H 0 s t o -

(7.11) 2 v / 4 and ga(u)=O f o r WLU,V 2/ 2 .

Rv .I v:l~5v,and

functions

C t ( W U ) , z[3g1*51,

E

210

g J. ( q a

=

o

z [ ?$(q ) I 2 j

J

if =

lqa

1 if

-

v

V?I>

a J -

6Ho<E

v

a ' '

and

hi 5

(7.12) pav 2 / 2 .

B

This decomposition i s e s s e n t i a l l y the same as ( 4 . 6 ) . On the range of F ( 6 < H o < E ) we then have (7.13) The g"(h;) ( a n d a l s o t h e i r squares) take care of t h a t p a r t of the 2u-dimensional momentum space, where t h e r e l a t i v e momenta of a p a i r might be close t o zero. On the remainder in Ran F(6
V. Enss

200

The sum o f f; over k may be evaluated f o r any u s i n c e the operators a r e independent of a by ( 6 . 5 ) . With r e l a t i o n (7.10) we can conclude t h a t lim

-

p

n+ m

For the vectors Y E U ( s e e ( 7 . 9 ) ) Theorem 5.1 and Corollary 5.6 a r e applicable and we obtain as an approximation o f the l a t t e r term i n (7.16)

z

j

fg(Pa,qa) f g ( u a

Xa

va

yn

1 e n

-iHTn

(7.17a)

'v+

The estimate of the f u t u r e time evolution on (7.17a) was given above in ( 6 . 6 ) and ( 6 . 7 ) . There we have shown lim

sup

z

n+-

t20

j

U (e

= 0.

(7.18)

F o r ( 7 . 1 7 b ) observe t h a t s t a t e s i n the range of

a r e outgoing with Rn=4V~nin the sense o f Section 4, Propositions4.la) and 4 . l b ) hold analogously. In p a r t i c u l a r the estimate o f (4.19) can be repeated almost l i t e r a l l y w i t h pn=VTn and one obtains f o r a l l a

Let now

then

T ~ > T ~ ,

-iHt

sup 1I ( e t2o

-

,-iH0t

) e

-iHTm

Y 1I

- ,-iHot) e -iHo(Tm-Tn) ... e - 1 ' H T n Y + 2 s u p I I (-iHt ~ t2o + z ( e- i H a t - ,-iH0t) ,-iH"(Tm-'n) . .. , - i H ~ n

vr,i,

(7.21)

a

... i n d i c a t e s the s p l i t t i n g as given i n (7.17). By choosing T n = T n ( E ) l a r g e enough the e r r o r i s smaller than E uuLi@~n?Rg i n TmLTn and a l s o the f i r s t summand is smaller than E uniformly i n T k T n . An addition by Proposition 4.4 and P ( + ~ ) Y = O we may require t h a t f o r T ~ T ~

where

20 1

Scattering Theory f o r Three Particles

IIPPP(ha) ga(h")

xa 2

(T)

g"($

m

)e

-iHTm

Yll <

E

.

(7.22)

T h e r e f o r e t h e p r o o f i s complete i f we can show t h a t f o r any ( l a r g e ) a T~ such t h a t

sn

there i s

(7.23) The norm i s bounded by

which i s a p u r e two body e x p r e s s i o n . F o r t h i s a s y m p t o t i c completeness i s w e l l known, i . e . f o r any Y lim T

-ba

rn

sup l l ( e - i h " t t20

-

,-ih:t)

,-iha(Tm-?n)

pcontt(ha) yI, =

o.

(7.25)

2 . 2 F o r g i v e n Tn t h e o p e r a t o r g"(ha)g"((u / 2 ) ( x * / T ~ ) ) i s compact on L (R’ ) . Thus t h e p r o p e r t y o f (7.25) h o l d s f o r (7.247 as w e l l and (7.23) i s shown.

0

Remark. We would l i k e t o s t r e s s t h a t t h e p r o o f i s s i m p l e i n essence a l t h o u g h i t m o u s t o w r i t e dowa t h e decomposi t i p s e x p l i c i t l y . Theorem 5 . 1 assures t h a t a s y m p t o t i c a l l y pa % uax /Tn and qa % uay /Tn. F o r s t a t e s with t h e s e c h a r a c t e r i s t i c s i n phase space t h e f u t u r e t i m e e v o l u t i o n has been c o n t r o l l e d i n S e c t i o n s I V and V I . The n o n - c o m m u t a t i v i t y o f p o s i t i o n and momentum as w e l l as t h e d i f f e r e n t s u i t a b l e c o o r d i n a t e systems depending on t h e p a i r i n g f o r c e upon us c a r e f u l s p l i t t i n g s and a cumbersome n o t a t i o n , b u t n o t h i n g i s d i f f i c u l t . W i t h t h e obvious changes of t h e s i g n s t h e c o r r e s p o n d i n g r e s u l t s h o l d i n t h e p a s t . We may use t h i s deep r e s u l t f r o m s c a t t e r i n g t h e o r y t o o b t a i n d e t a i l e d i n f o r m a t i o n about t h e c o n t i n u o u s spectrum o f t h e H a m i l t o n i a n H. C o r o l l a r y 7.2. H.PCont(H)

i5 unitatLiey eqlLivaRevLt t o t h e dihect A w n

06

Ho and 06 (7.26)

fB t~ (ES + k i ) a i

w h a t E: m e t h e eiiyenvduen 0 6 t h e 5ubnybtem Hamietvrtiau ha, t h e y me tepeated in t h e n m accotding t o ,thi.t mLLetipLLcity. Ho -in ( u p La can~tav~&,]a negative LapLucian opcrrp4 on L 2 ( R 2 v ) , tu,kiee t h e k; m e negative muRtipLes a6 LapLucian opetaivhn o n L ( Ru ) . I n patiZcURm H h a no n i n g u h t cvntinuow bpechum. P r o o f . I t i s w e l l known t h a t no i s a u n i t a r y o p e r a t o r f r o m X 3nt; i t s range, a n d a t t h e channel wave o p e r a t o r s a"PP a r e u n i t a r y f r o m P T P L ( R ) o n t o t h e i r ranges. The i n t e r t w i n i n g r e l a t i o n s

no* H

R'

(7.27)

= Ho

P s Q"" Hn" PS = H" P y = (ES

+

k):

Py

(7.28)

i m p l y t h a t H r Ran no i s u n i t a r i l y e q u i v a l e n t t o H (on aC) and H r Ran Q"PT $ r e u u n i t a r i l y e q u i v a l e n t t o Ha on PYX which i s e q u i v a ? e n t t o ( E T + k): on L (R ) f o r t h e m o t i o n o f t h e t h i r d p a r t i c l e . A s y m p t o t i c completeness i n t h e form (7.1)

202

V. Enss

i m p l i e s t h a t t h e s e components add up t o H-PCont(H). The absence o f s i n g u l a r cont i n u o u s spectrum f o l l o w s from t h e a b s o l u t e c o n t i n u i t y o f L a p l a c i a n o p e r a t o r s .

V I I I . C o n c l u d i n g Remarks and References

As a g e n e r a l r e f e r e n c e f o r s p e c t r a l and s c a t t e r i n g t h e o r y t h e r e a d e r may c o n s u l t [ 1 7 ] . We have r e s t r i c t e d o u r p r e s e n t a t i o n h e r e t o t h e case o f s i m p l e bounded p a i r p o t e n t i a l s o f s h o r t range i n o r d e r t o s t r e s s t h e main i d e a s and methods o f t h e g e o m e t r i c a l , t i m e dependent approach and t o a v o i d t e c h n i c a l i t i e s as f a r as possible. Nevertheless t h i s class o f p o t e n t i a l s includes the p o s s i b i l i t y o f i n f i n i t e l y many s c a t t e r i n g channels, o f z e r o energy bound s t a t e s and resonances f o r subsystems, where t i m e independent methods f a c e d i f f i c u l t i e s . Moreover t h e d i mension v of t h e space i s n o t r e s t r i c t e d t o ~ 1 b3u t a r b i t r a r y , and c o u p l i n g c o n s t a n t s can be i n s e r t e d w i t h o u t r e s t r i c t i o n s . The e x t e n s i o n t o r e l a t i v e l y oper a t o r bounded p e r t u r b a t i o n s i s s t r a i g h t f o r w a r d , w h i l e t h e i n c l u s i o n o f form bounded and h i g h l y s i n g u l a r p o s i t i v e p e r t u r b a t i o n s and o f n o n - l o c a l p o t e n t i a l s i s p o s s i b l e a t t h e expense o f some t e c h n i c a l i t i e s . Long-range f o r c e s l i k e t h e p h y s i c a l l y i m p o r t a n t Coulomb f o r c e r e q u i r e m o d i f i e d f r e e t i m e e v o l u t i o n s f o r t h e a p p r o x i m a t i o n o f t h e i n t e r a c t i n g one. A c o u n t e r p a r t t o t h e p r e s e n t r e p o r t w i t h an elementary e x p o s i t i o n o f t h e Coulomb case w i l l be g i v e n i n [6]. S i n c e t h e fundamental work o f Faddeev [ 81 and G i n i b r e - M o u l i n [ 91 t h e t i m e - i n d e pendent approach was dominant,which c e n t e r s around t h e s t u d y o f t h e r e s o l v e n t o f H n e a r t h e r e a l s a x i s and Kato-smoothness p l a y e d a m a j o r r o l e . ( F o r r e f e rences see [ 17, V o l . 1111 , [ 181 , o r [ 71 ) The methods o f p r o o f impose r o u g h l y t h e f o l l o w i n g c o n d i t i o n s on t h e systems: ( i ) t h e dimension o f space must be ~ 2 3 ; ( i i ) t h e p a i r p o t e n t i a l s have t o decay l i k e I XI - ( Z t c ) , E > O towards i n f i n i t y ; ( i i i ) t h e r e a r e no zero-energy bound s t a t e s o r resonances i n any two-body subsystem. The f i n i t e n e s s o f t h e n e g a t i v e s p e c t r a o f subsystem H a m i l t o n i a n s , i . e . a f i n i t e number o f s c a t t e r i n g channels, i s i m p l i e d by (ii). Recently these l i m i t a t i o n s were overcome i n two d i r e c t i o n s . Loss and S i g a l 1121 c o u l d a v o i d t h e i m p l i c i t c o n d i t i o n ( i i i ) assuming f a s t e r decay i n ( i i ) ; and M e r k u r i e v [ 131 handles t h e a d d i t i o n o f Coulomb i n t e r a c t i o n s i n t h r e e dimensions as w e l l . Hard and v e r y d e t a i l e d e s t i m a t e s o f t h e s i n g u l a r two-body r e s o l v e n t s o r Coulomb r e s o l v e n t s , r e s p e c t i v e l y , a r e needed. Under t h e assumptions ( i ) - ( i i i ) Hagedorn and P e r r y [lo] use m a i n l y time-depend e n t methods. T h e i r p r o o f seems t o e x t e n d much e a s i e r t o h i g h e r p a r t i c l e numbers t h e n t h e r e s o l v e n t methods do. R e c e n t l y Mourre [ 151 a p p l i e h i method o f c o n j u g a t e o p e r a t o r s t o t r e a t p a i r p o t e n t i a l s decaying l i k e I XI towards i n f i n i t y w i t h o u t c o n d i t t o n ( i ) and ( i i i ) . Regarding a s y m p t o t i c completeness o u r g e o m e t r i c a l , time-dependent method covers a l l t h e s e r e s u l t s and i n g e n e r a l i t i s t e c h n i c a l l y l e s s demanding. I t p e r m i t s t o t r e a t more g e n e r a l long-range p o t e n t i a l s and t h e two body subsystems may have p o s i t i v e energy e i g e n v a l u e s (we t h e n need i n f o r m a t i o n about t h e decay o f these e i g e n f u n c t i o n s , w h i c h i s known). Moreover we a d m i t i n many cases s t r o n g e r l o c a l s i n g u l a r i t i e s and we can h a n d l e e a s i l y nonl o c a l p o t e n t i a l s , which d e s c r i b e v e l o c i t y dependent f o r c e s . The advantage o f t h e r e s o l v e n t e q u a t i o n method, however, i s t h a t i t y i e l d s a d d i t i o n a l r e s u l t s , e.g., t h e Faddeev e q u a t i o n s a r e a u s e f u l t o o l f o r c a l c u l a t i o n s .

-htef

V a r i o u s r e s u l t s about absence of s i n g u l a r continuous spectrum were c o n t a i n e d i n t h e papers on a s y m p t o t i c completeness. The most thorough s t u d y o f t h i s q u e s t i o n was g i v e n by Mourre [ 141 and extended i n [ 161 T h e i r assumptions a r e more r e s t r i c t i v e w i t h r e g a r d t o l o c a l s i n g u l a r i t i e s , b u t t h e y can a d m i t more general smooth l o n g - r a n g e f o r c e s t h a n we do.

.

Our method was f i r s t developed f o r p o t e n t i a l s c a t t e r i n g [ 1 , 2 and r e f e r e n c e s

203

Scattering Theor?,fur Three Particles

g i v e n t h e r e ] and t h e n extended t o three-body systems [3]. The f u l l y worked o u t p r o o f i n t h e g e n e r a l case w i l l be g i v e n i n [ 71. The a u x i l i a r y r e s u l t s g i v e n i n S e c t i o n I 1 have been shown under weak assumptions i n [4], and those o f S e c t i o n V i n [5] f o r t h e two-body case ( t h e g e n e r a l i z a t i o n t o t h e t h r e e body system f o r a l a r g e c l a s s o f p o t e n t i a l s i n s t r a i g h t f o r w a r d ) . The e x t e n s i o n t o h i g h e r p a r t i c l e numbers i s p r e s e n t l y under i n v e s t i g a t i o n . The e x i s t e n c e o f channel wave o p e r a t o r s w i t h o u t i m p l i c i t c o n d i t i o n s on t h e subsystems seems t o be new i n dimensions U = 1 and 2.

References

1. V . Enss: A s y m p t o t i c completeness f o r quantum mechanical p o t e n t i a l s c a t t e r i n g I , Commun. Math. Phys. 61, 285-291(1978). 2. --: Geometric methods i n s p e c t r a l and s c a t t e r i n g t h e o r y o f S c h r o d i n g e r operat o r s , i n : Rigorous Atomic and M o l e c u l a r Physics, G. V e l o and A.S. Wightman eds, Plenum, New York 1981 . (Proceedings Erl’ce 1980).

3. --: Completeness of t h r e e body quantum s c a t t e r T n g , i n : Dynamics, Algebras, Processes, P. Blanchard ed., L e c t u r e Notes i n Mathemati’==, p r e s s . (Proceedings B i e l e f e I d 1981).

S p r i n g e r , in

4. --: P r o p a g a t i o n p r o p e r t i e s o f quantum s c a t t e r i n g s t a t e s , J. Func. Anal. 52, 219-251 (1983). 5 . --: A s y m p t o t i c o b s e r v a b l e s on s c a t t e r i n g s t a t e s , Commun. Math. Phys.

E,

245-268(1983). 6. - - : Three body Coulomb s c a t t e r i n g t h e o r y , i n : Proceedings 11. 1nt.Conference on O p e r a t o r Algebras, I d e a l s , and t h e i r A p p l i c a t i o n s i n T h e o r e t i c a l Physics, L e i p z i g 1983, Teubner, L e i p z i g , t o appear.

7. --: Three body quantum s c a t t e r i n g t h e o r y , i n p r e p a r a t i o n . 8 . L.D. Faddeev: Mathematical Aspects o f t h e Three Body Problem i n t h e Quantum S c a t t e r l n g Theory, I s r a e l Program f o r S c i e n t i f i c T r a n s l a t i o n s , Jerusalem,

1965. 9. J. G i n i b r e and M. M o u l i n : H i l b e r t space approach t o t h e quantum mechanical three-body problem, Ann. I n s t . H. Poincari! A 21, 97-145(1974). ~

10. G.A. Hagedorn and P.A.

P e r r y : A s y m p t o t i c completeness f o r c e r t a i n t h r e e 36, 213-232(1983). body S c h r o d i n g e r o p e r a t o r s , Commun. Pure Appl Math. -

.

11. P. Hartmann: O r d i n a r y D i f f e r e n t i a l Equations, W i l e y , New York 1964; p. 24. 12. M. Loss and I . M . S i g a l : The t h r e e body problem w i t h t h r e s h o l d s i n g u l a r i t i e s , p r e p r i n t , ETH Z u r i c h , 1982. 13. S . P . M e r k u r i e v : Three-body Coulomb s c a t t e r i n g , i n : New Developments i n Mathem a t i c a l Physics, H. M i t t e r and L. P i t t n e r eds., A c t a Phys. A u s t r i a c a , Suppl. ’23, - 6 5 - T T o ( ) , (Proceedings Schladming 1981); and r e f e r e n c e s g i v e n t h e r e .

14. E. Mourre: Absence o f s i n g u l a r c o n t i n u o u s spectrum f o r certal’n s e l f - a d j o i n t o p e r a t o r s , Commun. Math. Phys. 78,391-408( 1981).

204

V. Enss

15. E. Mourre: Operateur conjugues e t p r o p r i e t e s de p r o p a g a t i o n 11, remarques s u r l a compl’etude asymptotique des systemes i t r o i s c o r p s , p r e p r i n t CNRS-Marseille, 1982. 16. P. Perry, I . M . S i g a l , B. Simon: S p e c t r a l a n a l y s i s o f N-body S c h r o d i n g e r operat o r s , Ann. Math. 114, 519-567(1981). 17. M. Reed and B. Simon: Methods o f Modern Mathematical P h y s i c s , V o l . I - I V , demic Press, New York, 19/2-19/9.

Aca-

18. J . G i n i b r e : S o e c t r a l and s c a t t e r i n q t h e o r y of t h e S c h r o d l n s e r e a u a t i o n f o r three-body systems, i n : The S c h r o d i n g e r G u a t i o n , W. T h i r r i n g and P. Urban eds., A c t a Phys. Austr., Suppl. 95-138(1977) (Proceedings Vienna 1976).

17,

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

205

A LEFT DEFINITE TWO-PARAMETER EIGENVALUE PROBLEM

Melvin Faierrnan Department of Mathematics University of t h e Witwatersrand Johannesburg South Africa

I n t h i s work we discuss t h e spectral p r o p e r t i e s , and i n p a r t i c u f a r the eigenfunction expansion problem, f o r a l e f t d e f i n i t e two-parameter system of ordinary d i f f e r e n t i a l equations o f the second order. INTRODUCTION W e c o n s i d e r t h e s i m u l t a n e o u s two p a r a m e t e r s y s t e m s

are r e a l - v a l u e d smooth f u n c t i o n s w i t h pi > 0 . We where p i , qi, A i , and Bi a l s o suppose t h a t : ( i ) A = A1B2 - A2B1 assumes b o t h p o s i t i v e and n e g a t i v e ( t h e p r o d u c t of t h e i n t e r v a l s 0 xr Q 1 , r = 1 , 2 ) , and v a l u e s i n I, (ii) t h e r e e x i s t r e a l numbers T l , T, s u c h t h a t T I A r ( x r ) + T2Br(xr) > 0 i n 0 Q xr 1 f o r r = 1, 2 . On a c c o u n t o f ( i i ) t h e r e i s no l o s s o f g e n e r a l i t y xr 1 f o r r = 1, 2 i n assuming h e n c e f o r t h t h a t A, > 0 i n 0

.

f o r (il, x 2 ) , w e c a l l h* an e i g e n v a l u e o f t h e s y s t e m (1-4) i f f o r ( 2 r - 1) h a s a n o n - t r i v i a l s o l u t i o n , s a y yr!xr, X*) , s a t i s f y i n g ( 2 r ) The p r o d u c t y1 ( X I , A * ) y2 ( x p , A * ) 1 s c a l l e d an e i g e n f u n c t i o n o f f o r r = 1, 2 (1-4) c o r r e s p o n d i n g t o A * . Then u n d e r t h e g i v e n a s s u m p t i o n s t h e o b j e c t o f t h i s paper w i l l accordingly be t o d i s c u s s t h e s p e c t r a l p r o p e r t i e s of t h e system (1-4) , and t h e n t o i n v e s t i g a t e t h e a s s o c i a t e d e i g e n f u n c t i o n e x p a n s i o n problem.

Writing

A

=

A* ,

.

I n o r d e r t o a r r i v e a t t h e r e q u i r e d r e s u l t s , it i s n e c e s s a r y t o i n v e s t i g a t e t h e s p e c t r a l p r o p e r t i e s o f a c e r t a i n e l l i p t i c boundary v a l u e problem a s s o c i a t e d w i t h The r e a s o n f o r t h i s w i l l b e made c l e a r i n t h e s e q u e l . t h e s y s t e m (1-4) denote t h e A c c o r d i n g l y , w r i t i n g x f o r ( X I , x2) and l e t t i n g R and t h e i n t e r i o r and b o u n d a r y , r e s p e c t i v e l y , of I2 , l e t us f i x o u r a t t e n t i o n upon t h e boundary v a l u e problem

.

LU

where

L

r

-

XA(x)u = 0

for

x E

R ,

u(x) = 0

for

x E

r,

(5)

denotes the e l l i p t i c o p erato r

D~ = a / a x r ,

a1 =

P1A2

,

a2 = p2A1

, and

q = qiA2 + qzA1

.

With

(

,

)

and

M. Faierman

206

11 11 denoting t h e i n n e r product and norm, r e s p e c t i v e l y , i n a l s o consider t h e D i r i c h l e t form a s s o c i a t e d with ( 5 ) , 2 B(v,u) = (Drv, arDru) + (v,qu) f o r v, u 6 V = r=l

1

,2? = L2

(R) ,

l e t us

;)(a) ,

where we r e f e r t o Agmon (1965) f o r terminology. W e note from Faierman (1978) t h a t V i s a closed subspace of H I (R) and t h a t an element u i n H I belongs t o V i f and only i f i t s t r a c e on is zero. I f y denotes t h e lower bound of B , then it is usual i n t h e i n v e s t i g a t i o n of t h e s p e c t r a l p r o p e r t i e s of t h e system (1-4) t o consider t h e c a s e s y 2 0 , y < 0 s e p a r a t e l y ; and s i n c e t h e f i r s t of t h e s e c a s e s has already been t h e s u b j e c t of some i n v e s t i g a t i o n by H i l b e r t (1953) and Faierman ( 1 9 8 1 ) , we s h a l l henceforth r e s t r i c t our i n v e s t i g a t i o n t o t h e case

(a)

r

I t w i l l always be supposed i n t h e sequel t h a t

ASSUMPTION 1.

y <

0

F i n a l l y , we denote by @ ( x r , A) t h e s o l u t i o n of ( 2 r - 1) s a t i s f y i n g and l e t $ * ( x , h ) = @ i ( x i , ) @,(O,A) = 0 , p r ( 0 ) @ i ( O , x )= 1 , r = 1, 2 , PRELIMINARY RESULTS

The o b j e c t of t h i s s e c t i o n i s t o c o l l e c t some r e s u l t s which we r e q u i r e i n t h e sequel. Accordingly, w e f i r s t note t h a t the form B(v,u) i s symmetric, Moreover, it i s a l s o bounded from below, c l o s e d , and densely defined i n 7 . c l e a r t h a t B(v,u) i s coercive over V and IB(v,u)I cIIvII IIuII1 for some constant c and every v, u i n V , while we may argue a h ' k Agm6n (1965) t o show t h a t B(v,u) i s i n d e f i n i t e on V . I f A denotes t h e s e l f a d j o i n t operator a s s o c i a t e d with B(v,u) and AF denotes t h e Fredholm domain of A , then THEOREM 1.

I t i s t h e case t h a t

0 E

AF

.

Proof. This r e s u l t follows from a standard t e s t f o r a s c e r t a i n i n g t h a t A = 0 i s not i n t h e e s s e n t i a l spectrum of A and frcin t h e f a c t t h a t B i s coercive over V . ASSUMPTION 2 . 0 E P(A)

.

To s i m p l i f y our c a l c u l a t i o n s we s h a l l henceforth suppose t h a t

We next consider t h e boundary value problem LU = f

for

x E R , u(x)

for

= 0

x E

r

(6)

for f E X . Then t o d e a l with (6) l e t u s f i x our a t t e n t i o n upon t h e generalized boundary value problem: given f E X I f i n d a u E V such t h a t B(v,u) =

(7)

(v,f)

f o r every v E V . From assumption 2 it i s c l e a r t h a t (7) has a unique s o l u t i o n u , namely u = Tf , where T = A-' Moreover, from Faierman (1978) we a l s o know t h a t : loc ( i ) u E H2 , u E H 2 ( G ) f o r every open semi-disc G with edge on and whose closure does not contain a n y corner p o i n t s of r and u i s both a s t r o n g and weak s o l u t i o n of ( 6 ) ; ( i i ) u is continuous i n any compact subset of I2 which excludes t h e corner p o i n t s of and u = 0 a t each p o i n t of r which i s not a c o m e r p o i n t ; (iii) i f f E H z ( G ) f o r every open d i s c G i n R o r f o r every open semi-disc of t h e kind described above, then u i s of c l a s s C2 i n any compact subset of 1 2 which excludes t h e corner p o i n t s of

.

r

(a)

r

r

.

I f u denotes t h e s o l u t i o n of ( 7 ) , then we have seen that u = Tf . Arguments s i m i l a r t o those used i n Agmon (1965) show t h a t T i s a bounded l i n e a r transformation of B i n t o V and a s a mapping o f B i n t o i t s e l f , T i s

207

A Left Definite Two-Parameter Eigenvalue Problem compact and s e l f a d j o i n t .

1

From Faierman (1978) we a l s o know t h a t

D ( A ) = R(T) = {u u E V , u 6 H F c ( s 2 ) , Lu E .u) , and hence it follows t h a t Moreover, if Q denotes t h e bounded s e l f a d j o i n t Au = Lu f o r u E D ( A ) o p e r a t o r i n 2 defined by (Qf) (x) = A ( x ) f (x) and K = TQ , then it follows t h a t K i s a compact o p e r a t o r i n X which i s s t r o n g l y symmetrisable with r e s p e c t t o Q and which i s a l s o symmetrisable with r e s p e c t t o t h e bounded s e lfadj oi nt operator S = Q K . I t i s c l e a r t h a t N ( Q ) = N ( K ) , while we may

.

a l s o show t h a t t h e i n n e r product in a

.

(S.,.)

i s e i t h e r i n d e f i n i t e o r non-negative

W e now show t h e connection between t h e ei-genvalue problem (1-4) and t h e boundary Accordingly, i f At = (A’, , $2) i s an eigenvalue of t h e value problem ( 5 ) . , then it follows from t h e above r e s u l t s t h a t system (1-41 and $ = $ * ( x , A') $ E D(A) and A$ = h% Q$ , and hence h%K$ = @ # 0 Thus t h e second components of t h e eigenvalues of (1-4) a r e c h a r a c t e r i s t i c values of K , while t h e corresponding eigenfunctions a r e c h a r a c t e r i s t i c f u n c t i o n s and a l s o form a W e s h a l l see l a t e r t h a t t h e converse (with l i n e a r l y independent set i n X ' . some r e s t r i c t i o n ) i s a l s o t r u e . Hence t h e study of t h e s p e c t r a l p r o p e r t i e s of the system (1-4) i s reduced t o t h e study of t h e s p e c t r a l p r o p e r t i e s o f K , and accordingly we now i n v e s t i g a t e K i n g r e a t e r d e t a i l .

.

considered only a s a vector space, i s equipped with t h e inner product , then it becomes an i n d e f i n i t e i n n e r product space and K i s symmetric with r e s p e c t t o t h i s i n n e r product. Accordingly, i n what follows we u s e t h e terminology of i n d e f i n i t e i n n e r product spaces and w e r e f e r t o Bognar (1974) f o r t h e appropriate d e f i n i t i o n s . Moreover, i n o r d e r t o d i s t i n g u i s h and [ , 1 , between o r t h o g o n a l i t y w i t h r e s p e c t t o t h e i n n e r products ( , ) we say t h a t the v e c t o r s f , g E 8 are Q-orthogonal i f [f,g] = 0 ; o r t h o g o n a l i t y alone w i l l always mean o r t h o g o n a l i t y w i t h r e s p e c t t o the i n n e r ( , ) . Also i f 1 i s an eigenvalue of K , then we denote by 9 product h t h e p r i n c i p a l subspace corresponding t o h Since K i s compact, we know t h a t gh is f i n i t e dimensional i f A # 0 When [

, I

X ,

= (Q. ,.)

.

.

THEOREM 2 .

K

has an i n f i n i t e number of d i s t i n c t non-zero r e a l eigenvalues.

Proof. This follows immediately from t h e foregoing r e s u l t s and Faierman (19791, where it i s shown t h a t t h e system (1-4) possesses r e a l eigenvalues which form a ( r e a l 2-space) having no f i n i t e p o i n t s of denumerably i n f i n i t e subset o f E z accumulation (we c a l l an eigenvalue of ( 1 - 4 ) r e a l i f both i t s components a r e real). From arguments s i m i l a r t o t h o s e used i n Bognar (1974) w e a l s o have

11,

I f h and p a r e eigenvalues of K such t h a t h # then 9h i s Q-orthogonal t o ? I Hence t h e p r i n c i p a l subspace corresponding t o a non-real eigenvalue is'neutral.

THEOREM 3 .

.

. ,

THEOREM 4. If i s a non-zero eigenvalue of K , then so i s h Moreover: ( i ) dim N ( ( K - XI) r, = dim N ( ( K h I ) r ) f o r every i n t e g e r r 0 (ii) 9 and 9 - form a dual p a i r i n t h e i n d e f i n i t e i n n e r product space X , (iii) i f A A i s

-

x real,

then

i s not degenerate.

THEOREM 5.

The eigenspaces corresponding t o t h e d i s t i n c t eigenvalues of a r e d e f i n i t e with r e s p e c t t o t h e i n n e r product [ , ] , with t h e p o s s i b l e exception of a t most a f i n i t e number of them.

I f t h e theorem i s f a l s e , then we may use t h e f a c t t h a t Proof. coercive over V t o a r r i v e a t a c o n t r a d i c t i o n . A s a consequence of t h e s e r e s u l t s we now have

B(v,u)

K

is

M. Fuiermun

208

The d i s t i n c t e i g e n v a l u e s o f K are r e a l and semi-simple, t h e p o s s i b l e e x c e p t i o n o f a t m o s t a f i n i t e number of them.

COROLLARY 1.

with

COROLLARY 2 . K h a s a t most a f i n i t e number of n o n - r e a l e i g e n v a l u e s . These e i g e n v a l u e s come i n complex c o n j u g a t e p a i r s , w i t h e a c h such e i g e n v a l u e h a v i n g the same a l g e b r a i c and g e o m e t r i c m u l t i p l i c i t y as i t s complex c o n j u g a t e .

F i n a l l y , l e t 9 b e t h e sum of a l l t h e s u b s p a c e s f o r which i s n o t real and f o r which h i s r e a l , # 0 , and t h e i n n e r p r o h c t [ , ] is n o t d e f i n i t e Then from s i m p l e arguments i n v o l v i n g t h e above r e s u l t s w e may in N(K - hI) deduce t h a t

.

.

Suppose t h a t 9 # 0 THEOREM 6. s u b s p a c e s 9 and 81, where P and i n v a r i a n t under K

.

*1

Then m i s t h e d i r e c t sum o f c l o s e d a r e Q-orthogonal and a r e b o t h

MAIN RESULTS From t h e r e s u l t s of t h e p r e v i o u s s e c t i o n and from s i m p l e arguments i n v o l v i n g the system (1-4) it r e a d i l y f o l l o w s t h a t THEOREM 7 . The t o t a l i t y o f t h e e i g e n v a l u e s o f t h e s y s t e m (1-4) forms a c o u n t a b l y i n f i n i t e subset o f C2 (complex 2-space) h a v i n g no f i n i t e p o i n t s of accumulation. Moreover, t h e s e e i g e n v a l u e s are a l l r e a l , w i t h t h e p o s s i b l e e x c e p t i o n of a t m o s t a f i n i t e number of them, and i f t h e r e a r e n o n - r e a l e i g e n v a l u e s , t h e n t h e y must o c c u r i n complex c o n j u g a t e p a i r s , i.e., the Finally, c o r r e s p o n d i n g components are t h e complex c o n j u g a t e s o f e a c h o t h e r . and h# are any two d i s t i n c t e i g e n v a l u e s o f t h e system (1-4) and i f At , $ ( x ) = $*Cx, 1) , t h e n = 0 @ ( x ) = $ * ( x , it)

.

[$,TI

.

THEOREM 8. Suppose t h a t V # 0 and V-l i s an e i g e n v a l u e o f K Then t h e r e are p r e c i s e l y a f i n i t e number of d i s t i n c t e i g e n v a l u e s o f t h e s y s t e m ( 1 - 4 ) , , whose second components are V . Moreover, i f V i s r e a l , t h e n s a y A?[;:

{$*(x, h:)}: is a basis of N ( V K - I ) eigenvalue of t h e system ( 1 - 4 ) , t h e n

.

V

# 0

The c a s e where t h e e i g e n v a l u e c o m p l i c a t e d and w i l l n o t b e d i s c u s s e d h e r e .

REMARK 1.

C o n v e r s e l y , i f h = (u,V) i s an and V-l i s an e i g e n v a l u e of K V-’

of

K

.

i s n o t real i s somewhat

Proof. I f KU = V-’u # 0 , t h e n u = VTQu , and hence u i s t h e s o l u t i o n o f In light of the t h e g e n e r a l i z e d boundary v a l u e problem ( 7 ) w i t h f = UQu p r o p e r t i e s of s o l u t i o n s of 17) d e s c r i b e d above, t h e a s s e r t i o n s o f t h e theorem f o l l o w i m d i a t e l y from t h e f o r e g o i n g r e s u l t s and arguments s i m i l a r t o t h o s e used by H i l b e r t ( 1 9 5 3 ) .

.

L e t us now p a s s t o t h e problem o f expanding an a r b i t r a r y f u n c t i o n d e f i n e d on i n terms of t h e e i g e n f u n c t i o n s o f t h e s y s t e m ( 1 - 4 ) . To simplify our c a l c u l a t i o n s w e assume t h a t

ASSUMPTION 3.

I n t h e s e q u e l it w i l l be supposed t h a t

12

P = 0 .

THEOREM 9. The d i s t i n c t non-zero e i g e n v a l u e s o f K are a l l real, semi-simple, o f f i n i t e m u l t i p l i c i t y , denumerably i n f i n i t e i n number, and e a c h c o r r e s p o n d i n g [ , ] Moreover, i f eigenspace i s d e f i n i t e w i t h r e s p e c t t o t h e i n n e r product w e d e n o t e t h e non-zero e i g e n v a l u e s o f K by {uj’ where e a c h e i g e n v a l u e i s 11-1. a r e arranged i n i n c r e a s i n g r e p e a t e d a c c o r d i n g t o i t s m u l t i p l i c i t y and t h e o r d e r o f magnitude, t h e n 0 < I F 1 ] G I F 2 / G as j + m . a n i IUjl * m F i n a l l y , t o t h e sequence o f e i g e n v a l u e s {u?)? t h e r e c o r r e s p o n d s t h e sequence of e i g e n f u n c t i o n s {uj}T which s a t i s f y t h e r e l a t i o n s [ u j , uk] = f 6jk ,

}TI

...

I

.

A Left Definite Two-Parameter Eigenvalue Problem

209

Proof. The a s s e r t i o n s of t h e theorem follow immediately from t h e p r o p e r t i e s of K c i t e d i n t h e previous s e c t i o n and from assumption 3 . From theorems 8 and 9 we see t h a t t h e d i s t i n c t eigenvalues of t h e system (1-4) ( U . , p.) and 3 1

{xf}?,

may be arranged i n t o a sequence of t h e form where :1 = Uj is realHence, i n l i g h t of theorem 7 we may assume t h a t

W e henceforth suppose t h a t t h e u j i n theorem 9 a r e p r e c i s e l y ASSUMPTION 4 . t h e eigenfunctions of t h e system ( 1 - 4 ) defined by u j (x) = ( x ) / l [$j, + j l f o r j > 1 , where $ j ( x ) = $*(x, A T )

I "'

.

From t h e p o i n t of view of r e s o l v i n g t h e eigenfunction expansion problem, we s h a l l now consider a p a r t i c u l a r decomposition of .Y. Accordingly, l e t {E(X)) ( E ( h + O ) = E ( X ) ) denote t h e s p e c t r a l family of Q and l e t P+ = I - E ( 0 ) , P o = E ( 0 ) - E ( - 0 ) , P- = E ( - 0 ) , ( E + = P'X, 2 ’ ’ = Po*. Then it i s c l e a r t h a t X = Z+ i9' i2’- , with t h e s e l a t t e r subspaces being both pairwise orthogonal and Q-orthogonal, t h a t Q i s reduced by t h e above i s precisely decomposition of 2 , and t h a t t h e i s o t r o p i c p a r t of P I = P + + P - , K 1 = P I K Yl Then L e t fll = Y+ i 9 - , Y o = N ( Q ) = N(K) it i s c l e a r t h a t t h e i n n e r product [ , 1 i s i n d e f i n i t e i n T 1 and t h a t K1 i s a compact o p e r a t o r i n t h e H i l b e r t space Vl which i s symmetric w i t h r e s p e c t t o [ , I From theorem 9 it i s a l s o easy t o see t h a t

I

.

.

.

The d i s t i n c t non-zero eigenvalues of K1 a r e a l l r e a l , semisimple, of f i n i t e m u l t i p l i c i t y , denumerably i n f i n i t e i n number, and each corresponding eigenspace i s d e f i n i t e with r e s p e c t t o t h e i n n e r product [ , 1 Indeed, these eigenvalues, when arranged i n t o a sequence a s i n theorem 9 , a r e p r e c i s e l y t h e {ujl}T and t o t h i s sequence of eigenvalues t h e r e corresponds t h e

THEOREM 1 0 .

.

sequence of eigenfunctions

rvj}T,

'where

I t i s not d i f f i c u l t t o v e r i f y t h a t both

v

= P

u.

1 1 .

9 ’ and 2 - a r e not i n t r i n s i c a l l y Hence i f we complete ( i . e . , complete with r e s p e c t t o t h e norm [x,x] ) l e t Ati denote t h e i r i n t r i n s i c completions and d e f i n e t h e i n n e r product [ , I on ’ A , and hence on T2 = A +x - a M ,i n t h e usual manner, then ,f12 i s a Krein space with r e s p e c t t o t h i s i n n e r product and may be w r i t t e n a s a d i r e c t sum P2 = .M+ (i) , where (i)denotes t h e d i r e c t sum of subspaces of f12 which [ , ] I f J denotes t h e a r e orthogonal with r e s p e c t t o t h e i n n e r product fundamental symmetry corresponding t o t h i s decomposition of dyi , then it i s c l e a r t h a t J I X l = P f - P - , and n o t i n g t h a t flz i s a H i l b e r t space with r e s p e c t [ , IJ = [J.,.] ( w i t h norm 11 llJ = [ , 1":) , we a l s o t o t h e i n n e r product see t h a t F1 i s dense i n ’’’A.

I "' .

I

.

.

K 1 may be extended t o a l l of X 2 a s a compact o p e r a t o r and t h i s [ , ] Moreover, i f extension i s symmetric with r e s p e c t t o t h e i n n e r product we denote t h i s extension by Kz , then U ( K z ) = U(K1) , while i f E U(K2) - { O } , then i s semi-simple and N ( K 2 - X I ) = N ( K 1 - 11) THEOREM 11.

.

.

Proof. Observing t h a t JK1 i s a compact o p e r a t o r i n xT1 which i s symmetric [ , IJ and t h a t ~ ~ x ~ I~I Q I IJ IIxk2 ' , it with r e s p e c t t o t h e i n n e r product follows from Lax (1954) t h a t J K 1 , and hence K , , can be extended t o a l l of z2 as a compact o p e r a t o r . The remaining a s s e r t i o n s follow from t h e arguments of Lax i f we appeal t o theorem 1 0 and Bognar (1974). In l i g h t of theorems 1 0 and 11, we may assume without loss of g e n e r a l i t y t h a t t h e r e a r e a denumerably i n f i n i t e number of d i s t i n c t non-zero eigenvalues of Kz

M. Faierman

210

whose c o r r e s p o n d i n g e i g e n s p a c e s are p o s i t i v e d e f i n i t e w i t h r e s p e c t t o t h e i n n e r [ , 1 L e t 9, d e n o t e t h e c l o s u r e of t h e s p a n o f a l l s u c h p o s i t i v e product d e f i n i t e eigenspaces.

.

THEOREM 1 2 . K, h a s a n i n v a r i a n t maximal p o s i t i v e s u b s p a c e 9,3 Y o and a n i n v a r i a n t maximal n e g a t i v e s u b s p a c e 9, such t h a t 9, and Y, a r e t h e o r t h o g o n a l companions of e a c h o t h e r .

Proof. T h i s f o l l o w s e a s i l y from Bognar (1974) i f w e make u s e of t h e f a c t t h a t &W, = Y o , where & = c l o s u r e and U d e n o t e s t h e Cayley t r a n s f o r m o f K 2 .

a,

Next l e t @ d e n o t e t h e d i r e c t sum o f s u b s p a c e s of re spect t o t h e inn er product [ , 1J .

which a r e o r t h o g o n a l w i t h

.

THEOREM 13. L e t dl d e n o t e t h e i s o t r o p i c p a r t o f Y 1 Then A i s i n v a r i a n t u n d e r K, , U ( K , 14 = 101 i f ..4/ # 0 , and A i s a l s o t h e i s o t r o p i c p a r t of 9, Moreover, A', = a d @ C L ( 9 , + 9,) and i f h # 0 i s i n U ( K , ) and N = N ( K , - h I ) i s p o s i t i v e d e f i n i t e ( r e s p . n e g a t i v e d e f i n i t e ) , t h e n N c Y , ( r e s p . 5Ep)

.

.

1

I

Ye

.

Proof. W e have Y1 = 4 = le ($) J# and X, Ye Y Hence it i s now e a s y t o see t h a t t h e o r t h o g o n a l companion of 9 ,Y , i s j u s t J9’ , t h a t B (G) ’9 i s dense i n K, , and t h a t 'Y = J./(@ Y 2 The remaining a s s e r t i o n s now f o l l o w e a s i l y from t h e d e f i n i t i o n s of t h e terms i n v o l v e d and arguments similar t o t h o s e o f Langer ( 1 9 6 2 ) .

.

THEOREM 14.

If

g E 2,

+ 9,,

[

~

- ~1 Lg I [~g , v . ] [ v v 1

1

1'1

then

n

n

IV

j’

- 1

~ , g

f9,v.l[V.,V.IV.I

1

1

1

1

1

+

o

as

n

+. m

Proof.

T h i s f o l l o w s from Zaanen (1953) o r Reid (1951) i f we n o t e t h a t i n PiJ , where Pi is the o r t h o p r o j e c t o r mapping #2 o n t o Y'i

K2

. 91,

i s f u l l y symmetrisable w i t h r e s p e c t t o

.

Next l e t {xn}y d e n o t e an a r b i t r a r y e n u m e r a t i o n o f t h e e i g e n v a l u e s of t h e s y s t e m ( 1 - 4 ) , p u t $,(x) = $*(x, An) , and l e t $,(x) = $ n ( x ) / [$n, $nl for n > 1. Note t h a t t h e s e q u e n c e {$n} i s b u t a r e a r r a n g e m e n t of t h e sequence L e t u s a l s o s a y t h a t a series o f {un} of theorem 9 (see a s s u m p t i o n 4 ) . f u n c t i o n s a l l d e f i n e d on a g i v e n s e t X c o n v e r g e s r e g u l a r l y on X i f t h e s e r i e s of a b s o l u t e v a l u e s of t h e s e f u n c t i o n s c o n v e r g e s u n i f o r m l y on X . Finally, let = {x x E I, , A(x) = 01 .

I '"

I

I

THEOREM 1 5 . Suppose t h a t S 0 i n A' and l e t f ( x ) b e a f u n c t i o n of c l a s s C 4 i n I, which v a n i s h e s i n a r e l a t i v e l y open s u b s e t o f I p c o n t a i n i n g X 1

.

Let f(x) v a n i s h on

and i t s p a r t i a l d e r i v a t i v e s up t o and i n c l u d i n g t h e second o r d e r a l l Then

r

.

m

f(x)

=

1 tf.

$nl [ $ n r $ n l $ n ( x )

+

h(x)

,

n=l where t h e s e r i e s c o n v e r g e s r e g u l a r l y , and hence u n i f o r m l y , on a c o n t i n u o u s f u n c t i o n on I2 s a t i s f y i n g A ( x ) h ( x ) = 0

.

I,

and

is

h(x)

Proof. Since [Kf, f ] = ( S f , f ) > 0 f o r f E K , [K,f, f ] > 0 f o r f E a, , and hence it f o l l o w s from theorem 1 3 t h a t K+/ = 0 and K,X, c d ( Y 1 + Y 2 ) I t now a p p e a r s p o s s i b l e t o use theorem 14 t o a r r i v e a t t h e r e q u i r e d e i q e n f u n c t i o n e x p a n s i o n : however t h i s w i l l n o t b e done h e r e s i n c e a d i r e c t p r o o f may b e o b t a i n e d by u s i n g t h e r e s u l t s o f Faierman ( 1 9 8 1 ) . Indeed, w e f i r s t n o t e t h a t

.

21 1

A Left Definite Two-Parameter Eigenvalue Problem

Also i n theorem 9 w e now h a v e [u,, uk] = sgn p, 6,k, ( S u j , u,) = 1pj[-' 6jk. w e n o t e from t h e above r e f e r e n c e t h a t : ( i ) t h e sequence o f non-zero e i g e n v a l u e s is precisely and t h e c o r r e s p o n d i n g sequence o f o f S1" T S"' e i g e n f u n c t i o n s i s C V, ' I 2s'" uj}y , which forms an orthonormal sequence i n X I m

(i i )

1

)pn1-3 uA(x) n= 1

{uy>y

I 1

<

c

for

x E I,

,

and

(iii) f

=

KZg

f o r some

g E X

The a s s e r t i o n o f t h e theorem now f o l l o w s e a s i l y from t h e s e r e s u l t s and t h o s e of Zaanen ( 1 9 5 3 ) .

We hope in l a t e r p a p e r s t o use t h e above decomposition of X , and REMARK 2 . i n p a r t i c u l a r theorems 13-14, t o r e s o l v e t h e e i g e n f u n c t i o n e x p a n s i o n problem f o r t h e s y s t e m (1-4) f o r t h o s e c a s e s n o t c o n s i d e r e d i n theorem 15.

REFERENCES Agmon, S . , L e c t u r e s on E l l i p t i c Boundary Value Problems ( V a n N o s t r a n d , New York, 1 9 6 5 ) . Bognar, J . , I n d e f i n i t e I n n e r P r o d u c t S p a c e s ( S p r i n g e r , New York, 1 9 7 4 ) . Faierman , M. , E i g e n f u n c t i o n e x p a n s i o n s a s s o c i a t e d w i t h a two-parameter s y s t e m of d i f f e r e n t i a l e q u a t i o n s , P r o c . Roy. SOC. Edinburgh 8 1 A (1978) 79-93. Faierman, M . , An o s c i l l a t i o n theorem f o r a two-parameter s y s t e m of d i f f e r e n t i a l e q u a t i o n s , Q u a e s t i o n e s Math. 3 (19791 313-321. Faierman, M . , An e i g e n f u n c t i o n e x p a n s i o n a s s o c i a t e d w i t h a two-parameter system o f d i f f e r e n t i a l e q u a t i o n s . I , P r o c . Roy. SOC. Edinburgh 89A (1981) 143-155. H i l b e r t , D . , GrundzGge e i n e r A l l g e m e i n e r T h e o r i e d e r L i n e a r e n I n t e g r a l g l e i c h u n q e n ( C h e l s e a , N e w York, 1 9 5 3 ) . L a n g e r , H . , Zur s p e k t r a l t h e o r i e ; I - s e l b s t a d j u n g i e r t e r o p e r a t o r e n , Math. Ann. 146 (1962) 60-85. Lax, P.D., Symmetrizdble l i n e a r t r a n s f o r m a t i o n s , Comm. P u r e A p p l . Math. 7 (1954) 633-647. R e i d , W.T., Syrrnnetrizable c o m p l e t e l y c o n t i n u o u s l i n e a r t r a n s f o r m a t i o n s i n H i l b e r t s p a c e , Duke Math. J. 1 8 (1951) 41-56. Zaanen, A.C., L i n e a r A n a l y s i s ( N o r t h H o l l a n d , Amsterdam, 1 9 5 3 ) .

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DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

213

A SEMIGROUP APPROACH TO BURGERS’ SYSTEM W.E.

Fitzgibbon

Department o f Mathematics U n i v e r s i t y o f Houston Houston, Texas 77004 U.S.A.

1.

Introduction

We s h a l l be concerned w i t h Burgers system which couples one o r d i n a r y and one p a r t i a l d i f f e r e n t i a l e q u a t i o n and i s i n t e n d e d t o p o r t r a y f l o w i n a channel. The e q u a t i o n s have t h e form:

1.1

*(t) dt

= P(t)

v(0,t)

U(0)

-

= v(1,t) =

uo

vU(t)

-

L’(v(x,t))*dx

= 0

v(x,O)

= v (x) 0

The dependent v a r i a b l e s U and v a r e i n t r o d u c e d t o r e p r e s e n t v e l o c i t i e s . The f i r s t one w i l l be t h e analogue o f p r i m a r y m o t i o n o f l i q u i d f l o w i n g i n t h e channel; i n t h e model i t i s a f u n c t i o n o f t i m e o n l y . The o t h e r v a r i a b l e r e p r e s e n t s second a r y motion; when i t d i f f e r s f r o m z e r o we say t h a t t u r b u l e n c e i s p r e s e n t i n t h e system even i n t h e case o f v being independent o f t; t h e independent v a r i a b l e x r e p r e s e n t s t h e w i d t h o f t h e channel. The q u a n t i t y P ( t ) i s t h e analogue o f e x t e r i o r f o r c e a c t i n g on t h e system. Viscous e f f e c t s a r e r e p r e s e n t e d by t h e terms i n v o l v i n g v . L o o s e l y i n t r e p r e t e d , t h e f i r s t e q u a t i o n r e p r e s e n t s a balance between t h e a c c e l e r a t i o n o f t h e p r i n c i p a l f l o w , t h e a p p l i e d f o r c e , v i s c o u s damping and damping due t o t u r b u l e n c e which i s r e p r e s e n t e d by t h e i n t e g r a l term. The second e q u a t i o n i s a balance between t h e a c c e l e r a t i o n o f t h e p r i n c i p a l f l o w , v i s c o u s d i f f u s i o n and c o n v e c t i v e c o u p l i n g w i t h t h e p r i n c i p a l f l o w . The system was proposed by J . Burgers 111 and i s i n t e n d e d t o be a s i m p l i f i e d model which i n c o r p o r a t e s some o f t h e f e a t u r e s o f t h e more c o m p l i c a t e d Navier-Stokes system. Horgan and Olmstead E l 2 1 have o b t a i n e d a uniqueness r e s u l t and examined s t a b i l i t y o f l a m i n a r f l o w . Olmstead and Davis [151 have addressed q u e s t i o n s o f I n a r e c e n t s e r i e s of s t a b i l i t y and b i f u r c a t i o n f o r a modulated Burgers system. papers D l o t k o uses G a l e r k i n methods t o o b t a i n weak s o l u t i o n s and t h e n p r o v i d e s c o n d i t i o n s s u f f i c i e n t t o guarantee t h a t t h e weak s o l u t i o n s a r e s t r o n g s o l u t i o n s . Our approach i s d i f f e r e n t . We use t h e t h e o r y o f a n a l y t i c semigroups t o o b t a i n g l o b a l s o l u t i o n s and a v a r i a t i o n o f parameters r e p r e s e n t a t i o n o f s o l u t i o n s . E x p l o i t a t i o n o f t h e s e m i l i n e a r s t r u c t u r e o f t h e problem a l l o w s us t o p r o v i d e an e x p o n e n t i a l r e p r e s e n t a t i o n o f s o l u t i o n s and t o f u r t h e r d i s c u s s t h e qua1 i t a t i v e s t r u c t u r e o f t h e problem. 2.

A b s t r a c t Equations

I t i s o u r i n t e n t i o n t o view Burgers system as an o r d i n a r y d i f f e r e n t i a l e q u a t i o n i n a Banach space. Toward t h i s end we o v e r v i e w a p o r t i o n o f t h e t h e o r y of a b s t r a c t e q u a t i o n s and p r o v i d e a r e p r e s e n t a t i o n theorem f o r t h e i r s o l u t i o n s .

214

W.E. Fitzgibbon

We consider equations of the form

p ' ( t ) = AHt) + J ( t ) P ( t ) p(0) = yo E x

(2.1)

Where Io( ) maps the interval 10,Tl t o a Banach space X, A i s the infinitesimal generator of an a n a l y t i c semigroup, and { J ( t ) 1 t E C0,tll i s a family of nonl i n e a r unbounded mappings of X t o i t s e l f . By a strong solution we mean a funct i o n p( ) : [O,TI + X having the following properties ( i ) I.( ) : COYTI X i s continuous on COYTI and p(0) = p , ( i i ) ) i s absolutely continuous on compact subsets of [O,Tl and ( i i i ) ' p ( ) i s d i f f e r e n t i a b l e f o r a.e. t E COYTI. Whenever strong s o l u t i o n s e x i s t they have v a r i a t i o n of parameters representation. -f

~

P(t) = T(t)po

(2.2) where If

0

I

{T(t) E

p(A)

t z 01

+

J

T(t

-

s)J(s)Y(s)ds

i s the semigroup generated by

we can define f r a c t i o n a l powers of

A.

For

A.

0

<

y

1 we l e t

<

where r ( ) i s the gamma function. Because the A-y a r e bounded one t o one l i n e a r operators on X we have AY = The f r a c t i o n a l powers of A have the desired algebraic properties and they themselves can be shown t o be i n f i n i t e simal generators o f a n a l y t i c semigroups. We d e f i n e i n t e r p o l a t i o n spaces, X

Y

=

{x

X

I

x

E

11x1 I

with norm

O(Ay)}

and continuously embedded in

X.

=

Y

I lAYxl 1 .

The spaces

X

Y

a r e dense

We have the following local existence r e s u l t which i s a special case of the Sobelevskii Theorem f o r a b s t r a c t equations of parabolic type 1173. Theorem 2.3. Let A be t h e infinitesimal generator of an a n a l y t i c semigroup on a Banach space X w i t h 0 E p ( A ) . Assume there e x i s t s a y E (0,l) so t h a t f o r each t E [O,T] J ( t ) i s an everywhere defined operator from xy t o X such the following hold: L( ): R

( i ) There e x i s t s a continuous

I

IJ(t)lP- J(t)PlI 5 L(SUP{l IYI Iy.I f o r t E 10,TI , 9 ,ri, E Xy M (

( i i ) There e x i s t s a continuous

I IJ(t)P for t,

T

J ( ~ ) l b Il It E C0,Tl ,

i f Ip E X there e x i s t s a which'satixfi es

P(t) Moreover p( )

T(t)lPO +

T*

,

+

such t h a t

R

1uJ1 lyn IY ): R

R

-f

AHt)

+

ly

9

such t h a t

- ? I M ( I PI I,) $ 0 Xy ~

5

and a unique continuous y( ) : CO,T*I

T

kT(t-

+

X

s)J(s)y(s)ds

i s continuously d i f f e r e n t i a b l e on

p'(t) =

YI

(O,T*I

and s a t i s f i e s

J(t)v(t)

T h e material discussed t h u s f a r i n t h i s section i s well known and nice treatments of the subjects a r e found i n 191, 1101, 1111, C161, C171. We define a family o f nonlinear operators formulae

{F(t)

I

t

E

10,TIl on

D(Ay)

by the

215

A Semigroup Approucfi to Burger's System

2.4

W + J(t)P

F(t)y =

w i t h P ( F ( t ) ) = { y D(Ay) 1 (A + J ( t ) ) y c D(Ay)} The f a m i l y o f o p e r a t o r s { F ( t ) 1 t c [O,Tl} i s t h e generator o f a nonlinear evolut i o n system. We have t h e f o l l o w i n g theorem which p r o v i d e s an e x p o n e n t i a l f o r m u l a for strong solutions t o (2.1). Theorem 2 . 5 . L e t A and { J ( t ) 1 t E [O,TI} s a t i s f y t h e c o n d i t i o n s o f Theorem 2 . 3 and l e t { F ( t ) t E IO,Tl} be d e f i n e d v i a ( 2 . 4 ) . I f po E D(Ay) and Ip( ) i s a s t r o n g s o l u t i o n o f ( 2 . 1 ) on [ O , T I , t h e n n

I

= o

t

uniformly f o r

COYTI.

I n d i c a t i o n o f p r o o f . T h i s r e s u l t i s a n e x t e n s i o n o f a r e s u l t o f Webb El71 t o t h e case o f l o c a l l y L i p s c h i t z t i m e dependent n o n l i n e a r terms. The e x i s t e n c e o f t h e l i m i t b a s i c a l l y i s a s t a t e m e n t t h a t t h e backward d i f f e r e n c e scheme f o r ( 2 . 1 ) converges and we can a p p l y t h e e x i s t e n c e and r e g u l a r i t y t h e o r y f o r ( 2 . 1 ) t o i n s u r e t h e d e r i v e d convergence. The argument i s r e a s o n a b l y s t r a i g h t f o r w a r d b u t complicated i n t h e technical d e t a i l .

3.

Burgers System

We now a p p l y t h e a b s t r a c t t h e o r y o f t h e p r e c e e d i n g s e c t ' o n t o t h e Burgers system. We b e g i n by s p e c i f y i n g a d e n s e l y d e f i n e d o p e r a t o r on L C 0 , l I f o r v > 0, we s e t

h

(Au)(x)

= vull(x)

with D ( A ) = cu

E

for

2

I

L (o,i)

I t i s v e r y w e l l known t h a t L ~ ( o , I ) . We now l e t

A

u

x E

E

[0,11

H ~ ( o , n~ HI 2 ( 0 ~ 1 ) )

generates an a n a l y t i c semigroup

{T(t)

1

t

2

01

on

A

and d e f i n e

= R 8 h

L2(0,1)

A

X

X

-+

by

;)

i =(-;

3.1 with

2

A:

D(A)

= R

e

D(A).

I t i s n o t d i t f i c u l t t o show t h a aemigroup { T ( t ) I t 2 O} on w i l l have norm

A

6

I / C ~ Y V I I l y = VYl"l Henceforth we s h a l l use y = 1/2 15 P ( ) : [O,T] { (t)

I

t

E

+

CO,Tl}

R

A i s t h e i n f i n i esimal g e n e r a t o r o f an a n a l y t i c and t h a t 0 p(A). The i n t e r p o l a t i o n spaces +

IIvlly

i s c o n t i n u o u s l y d i f f e r e n t i a b l e we d e f i n e by

We have t h e f o l l o w i n g g l o b a l e x i s t e n c e theorem Theorem 3.3. Let and r e s p e c t i v e l y ; suppose t h a t

t) I t (t) 1 t

[O,Tl} t

0}

be d e f i n e d v i a ( 3 . 1 ) and ( 3 . 2 ) i s t h e a n a l y t i c semigroup on 2

216

W.E. Fitzgibbon

A.

generated by If p f I U , vo( 11 E 2112 t h e n f o r any [O,TI -+ X s a t i s f y i n g o n l y one c o n t i n u o u s f u n f t i o n

(3.4) Moreover

(3.5)

v:

P ( t ) = ? ( t ) P o +Jt ? ( t - s)i(s)p(s)ds i s c o n t i n u o u s l y d i f f e r e n t i a b l e on C0,TI

p( )

T

>

0

there exists

and s a t i s f i e s

p ’ ( t ) = ;iY”(t) f h y q t )

I n d i c a t i o n o f p r o o f The S a b o l e v s k i i Theorem p r o v i d e s a l o c a l r e s u l t . I f one can e s t a b l i s h t h e boundedness o f t h e e x p r e s s i o n 3 ( t ) p ( t ) one can use t h e methods of c l a s s i c a l o r d i n a r y i f f e r e n t i a l e q u a t i o n s t o e x t e n d o u r s o l u t i o n t o C0,Tl. i s e s t a b l i s h e d by showing t h a t I ( v x ( - , t ) I I i s The d e s i r e d bound on ! ( t ) v ( t ) bounded. M u l t i p l i c a t i o n o f t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n i n (1.1) by v and m u l t i p l i c a t i o n by v and i n t e g r a t i o n on C 0 , l l produces t h e energy e q u a l i t y

1/2 d / d t ( l l v l 1 2 ) + v ( l l v X 1 l 2 ) = U / \ V / / ~ T h i s shows t h a t 1 ( v ( - , t ) I and an bounded and r e l a t e s t h e boundedness The boundedness o f t h e t i m e d e r i v a of I I v x ( * , t ) I I t o t h a t o f I I v t i v e i s a consequence o f a u n i q u h e s s estimate. More s p e c i f i c a l l y we show t h a t i f (Ui, vi) i = 1 o r 2 a r e s o l u t i o n s t o ( 1 , l ) w i t h i n i t i a l c o n d i t i o n s (Ui(0), v i ( x ) ) and I ( t ) = {lU,(t) - U 2 ( t ) 1 2 + ( l v l ( * , t ) - v,(.,t)l(l then there

I

exists a

K(t)

so t h a t

dI(t)dt

2

I(t)

(!:;

1.

satisfies the differential inequality

K(t)I(t)

t h e desired r e s u l t follows. We immediately o b t a i n t h e f o l l o w i n g r e p r e s e n t a t i o n P r o p o s i i i o n 3.6.

Assume t h a t t h e c o n d i t i o n s o f Theorem 3.3.

il/* i n the

t h a t { F ( t ) I t 2 O} i s d e f i n e d o n and y( ) i s t h e s o l u t i o n t o (3.4) then

a r e s a t i s f i e d and

manner o f (2.4).

If

Po E &2

I n 171 t h e a u t h o r has u t i l i z e d t h e Hopf b i f u r c a t i o n Theory o f n o n l i n e a r semigroups o f Marsden C131 t o d i s c u s s t h e q u a l i t a t i v e b e h a v i o r o f a r e l a t e d autonomous model. Forthcoming work s h a l l e x p l o i t t h e s e m i l i n e a r s t r u c t u r e o f (1,l) t o examine t h e b e h a v i o r o f s o l u t i o n s t o modulated Burgers systems.

A Semigroup Approuck to Burger's System

211

REFERENCES

Ill

C.M. Burgers, " A mathematical model i l l u s t r a t i n g t h e t h e o r y o f t u r b u l e n c e , " Advances i n A p p l i e d Mechanics (R. von M i l e s and T. von KFirm'an, ed.) Academic Press, New York (1948), 171-199.

a,

r 2 1 J.D. Cole, "On a q u a s i l i n e a r p a r a b o l i c e q u a t i o n o c c u r i n g i n aerodynamics," Q u a r t , Appl. Math. 9 (19511, 225-236. L31

T. D o l t k o , "On t h e one dimensional Burgers e q u a t i o n e x i s t e n c e , uniqueness, s t a b i l i t y " , Zeszyty Nauk. Univ. J a g i e l l o n Prace Mat 23 ( i n p r e s s ) .

141

, "On c l a s s i c a l s o l u t i o n o f t h e one dimensional Burgers equation," Zeszyty Nauk Univ. J a g i e l l o n Prace Mat. 3 ( i n p r e s s ) .

I51

, "Some remarks c o n c e r n i n g t h e u n s t a b i l i t y o f t h e on dimensional Burgers equation," Ann. Polon, Math ( t o appear).

C 61

"The two d i m e n s i o n a l B u r g e r ' s t u r b u l e n c e model Univ.-2 1 ( i 9 8 1 ) , 809-823.

171

W. F i t z g i b b o n , " A two dimensional model f o r turbulence,"

C 81

, " R e p r e s e n t a t i o n and a s y m p t o t i c b e h a v i o r o f s t r o n g l y damped e v o l u t i o n equations," N o n l i n e a r Phenomena i n Mathematical Sciences (V. Lakshmikantham, ed.), Academic Press, NewTork, 1982, 389-396.

C 91

A. Friedman, P a r t i a l D i f f e r e n t i a l Equations Hol t, R h i n e h a r t and Winston, New York, 1 9 6 7

,"J

Math. Kyoto

( t o appear).

C 101 J . G o l d s t e i n , "Semigroups o f o p e r a t o r s and a b s t r a c t Cauchy problems,"

lec-

t u r e notes, T u l a n e U n i v e r s i t y , 1970.

C 111 D. Henry Geometric Theory o f S e m i l i n e a r P a r a b o l i c Equations, L e c t u r e Notes i n Mathematics, v o l 840 S p r i n g e r - V e r l a g , B e r l i n , 1981.

C 121 I131

C 141

C.O. Horgan and W.E. Olmstead, " S t a b i l i t y and uniqueness f o r a t u r b u r l e n c e model o f Burgers," Q u a r t Appl Math, (1978), 121-127.

J. Marsden, "The Hopf b i f u r c a t i o n f o r n o n l i n e a r semigroups", B u l l .

Math. SOC. , 79 (1973) , 537-541.

w.

J . Marsden and M. McCracken, The B i f u r c a t i o n and I t s A p p l i c a t i o n s , A p p l i e d Mathematical S c i e n c e s x , S p r i n g e r - V e r l a g , B e r l i n , 1976.

C 151 W.E.

Olmstead and S.H. Davis. " S t a b i l i t y and b i f u r c a t i o n i n a Modulated Burgers system," A p p l i e d Mathematics T e c h n i c a l No. 7925, Northwestern Univ e r s i ty, 1980.

C 161

A. Pazy, "Semigroups o f 1 i n e a r o p e r a t o r s and a p p l i c a t i o n s t o p a r t i a l d i f f e r e n t i a l equations," L e c t u r e Notes No. lJ, U n i v e r s i t y o f Maryland, C o l l e g e P a r t , MD. 1974.

i 171

P.E. SobolevskJ;, "On e q u a t i o n s o f p a r a b o l i c t y p e i n a Banach space," Trud Moscor. -Mat Orsc 10 (1961), 297-350. T r a n s l . Amer. Math SOC. 240 ( 1 9 7 8 e 129-143.

1181 G.F. Webb, " E x p o n e n t i a l r e p r e s e n t a t i o n o f s o l u t i o n s t o an a b s t r a c t semiJ . Math. 70 (1977), 269-279. l i n e a r d i f f e r e n t i a l equation,"

w.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

219

ON THE EIGENVALUES OF NON-DEFINITE ELLIPTIC OPERATORS

J. F l e c k i n g e r k and A.B.

Mingarelli

Department of Mathematics U n i v e r s i t y of O t t a w a Ottawa, Canada, K1N 9B4

I.

INTRODUCTION L e t R be a bounded open s e t i n IRn

w i t h a smooth boundary

r

.

Let A be a

l i n e a r d i f f e r e n t i a l o p e r a t o r of o r d e r 2m d e f i n e d on R which i s uniformly s t r o n g l y e l l i p t i c and f o r m a l l y s e l f - a d j o i n t .

We d e n o t e by A t h e p o s i t i v e r e a l i z a -

(1)

t i o n , s e l f - a d j o i n t and unbounded i n LL(Q) of a boundary v a l u e problem a s s o c i a t e d 2 w i t h A on R. W e suppose t h a t t h e imbedding of t h e domain D(A) i n t o L (R) i s compact w i t h dense r a n g e and t h a t t h e r e e x i s t s y > 0 s u c h t h a t f o r e v e r y U E D ( A ) , 2 (Au,u) 2 v l l ~ l l

.

Consider now c , g : R + I R

e a c h o f which i s c o n t i n u o u s on.

5

and g a t t a i n s b o t h

p o s i t i v e and n e g a t i v e v a l u e s : - m S g ( x ) 5 M and meas {x: g ( x ) = O } = 0 . W e s t u d y h e r e t h e e i g e n v a l u e s of t h e boundary v a l u e problem Lu: (') where B

1

( A + c ) u = Xgu

{BuI,

on

R ,

U E

D(A),

= 0

d e n o t e s t h e u s u a l homogeneous D i r i c h l e t o r Neumann boundary c o n d i t i o n s

Because of o u r hypotheses t h i s problem i s , i n g e n e r a l , non-definite i n t h e s e n s e t h a t L i s a n o t n e c e s s a r i l y p o s i t i v e o p e r a t o r and g a f u n c t i o n which may change its sign.

A t y p i c a l problem i s , f o r example,

W e o b t a i n h e r e some r e s u l t s on t h e e i g e n f u n c t i o n s and e i g e n v a l u e s (which w e

together c a l l eigenpairs, f o r brevity),

and on t h e e x i s t e n c e ( o r non-existence)

of p r i n c i p a l e i g e n v a l u e s ( i . e . ,

an e i g e n v a l u e whose a s s o c i a t e d ( r e a l ) eigenfunc-

t i o n d o e s n ' t change i t s s i g n i n

R).

The problem (P) h a s i t s o r i g i n s i n a p a p e r of R . G . D .

Richardson [ 8 ] i n t h e

(2)

J. Fleckinger and A. B. Mingarelli

220

one-dimensional case, and was studied more recently by Mingarelli [ 6 , 7 ] .

In the multi-dimensional case, when c is positive , results on eigenvalues and eigenfunctions can be found in Manes-Micheletti [5] and results on principal

...1 .

eigenvalues in [1,2,3,4,10, 11.

EXAMPLES IN THE ONE-DIMENSIONAL CASE. In his paper [ 8 ] Richardson mentions (without an example) that in such pro-

blems non-real eigenvalues may occur. We now list some examples of what may happens.

Consider the problem -y"+c(x)y y(a)

y(b)

2

-917 /16 on R

Let c(x)

Ex.1,

=

Xg(x)y

= =

0

=

(a,b),

.

( 0 , 2 ) and g(x)

=

R

on

=

1,

XE

(O,l), g(x) =-1, x t [1,2)

This problem has precisely one pair of non-real eigenvalues situated at about

.

+4.3628i Ex.2,

2

:1 - (9n /4),

Let c(x)

in ex.1 above,

R

Let R

Then h

=

=

In this case we note that (gy,y) 2

1 - (9n /4),

(0,4), c(x)

X E (CJ,1);

c(x)

[1.2) and g(x)

XE

Then X = l is an eigenvalue and y(x)

= (0,2).

associated eigenfunction. Ex.3,

2 (0,l); c ( x ) 5 -1- (917 / 4 ) ,

X E

2

=

f R

as

sin(3?rx/2) is an glyI2

=

0.

2 3 -1- (9T /4), X E (1,4)-

1 is an eigenvalue and yet its associated eigenfunction satisfies

f gy2
THE EICENVALUES. By hypothesis L is bounded below and has a pure point spectrum; let u s denote

by pi, $i the eigenpairs of (Q)

(Wi !JiuJi =

Bailr

= 0

,

with -m

and p . + m 3

as

j+m

.

<

1-I

1

i

u2

l-Ij

2

...

on ODE - the proofs being similar.

PROPOSITION 1 Two eigenfunctions

Proof.

... c

We have the following results which generalize to PDE.

Mingarelli’s results [6,7]

eigenvalues h l ,

s

a1, a2

associated respectively with two distinct

X 2 are linearly independent.

Suppose that 9,

=

a$2 ; then a ( h -A ) g $ = O and this gives a contradiction. 1 2 2

221

On the Eigenvalues of Non-Definite Elliptic Operators COROLLARY 1. There exists an at most countable set of distinct eigenvalues. PROPOSITION 2

Proof.

Two eigenfunctions @ 1, Q2 associated respectively with two eigen-

A2 , h l f x 2 , are such that (g@1 @ 2)

values h 1,

By hypothesis we have L@i

implies h1(gQ1,a2)

= (LQ1,412)

=

= (L@ @ ) =

1 2

Xig41i, i=1,2. Thus X1

= x2($l,g@2).

0.

The self-adjointness of L now f

1,

implies (g@ 1

c$

The eigenfunction $ associated with a non-real eigenvalue

COROLLARY 2.

2 ) = 0.

A is such

that (g41341)= 0 = (L$,@). We note that when A is a non-real eigenvalue with associated eigenfunction

4 , then

x, 5 is also an eigenpair and so

real eigenfunctions must arise from real

eigenvalues (since g is assumed not to vanish an open sets, with non-empty interior, having positive Lebesgue measure). In the sequel, when h is a real eigenvalue we will suppose that the associa-

ted eigenfunctions are real; this is always possible since the real and imaginary parts of a complex eigenfunction are themselves real eigenfunctions. We are interested in the positive eigenvalues. same for the negative eigenvalues as Lu THEOREM 1. Assume hypotheses (1)-(3) distinct positive eigenvalues h l , A 2 , tisfy (g@i,@i) Proof.

5

0, i = 1 , 2

,...,M.

Indeed the results are the

hgu can be written as L u = (-A)(-g)u.

=

. There exists an at most finite number of

...,hM

whose associated eigenfunctions

We may assume that 0 is not an eigenvalue of L.

If there exists

ui

ai sa=

0,

then consider the related problem

(P )

I

(A+(c+ag))u

IEulY = 0

-+

(c+Bg)

=

L+Bg.

(A+B)gu

5 Vgu

,

is chosen small enough so that

where B.0 A

=

V 4

is not an eigenvalue of

If theorem 1 holds for (p ) then a fortiori it holds for (P).

Consider now the infinite sequence of the eigenvalues of L; - m < p 1 < p 2 < uN
... where N

... 5

is such that p N < O c ~ N + land $. satisfies ( Q ) .

(1s i < M ) the eigenpairs of (P) for which (g@i,@i) S 0 , (recall that @ . is real when A. is real). First we deduce by an induction argument and proposition 1 that 411,...,QM are linearly independent. Since M > N we can choose a function j, in the subspace spanned by 41l,...,$M which If possible let M > N and denote by Ai, @.

J. Fleckinger and A.B. Mingarelli

222

..

is L2-orthogonal to each Jii, 1 5 i S N : Then Ji f 0, (@,Jii) = 0, i = 1,. ,N, and M 9 = C ciQi It is readily verified that (g$,$) S 0. Thus (LJI,Ji)
.

UN+l< (L$,@)/(Ji,$) THEOREM 2.

S O , a contradiction. Hence M S N and this proves the theorem.

There exists an at most finite number of distinct non-real eigenvalues,

under the hypotheses (1) - ( 3 ) . Proof. IV.

This is similar to the previous proof.

THE PRINCIPAL EIGENVALUES. We are now concerned with the existence of a principal eigenvalue L o , i.e.,

an eigenvalue of (P) whose associated eigenfunction does not change sign. This problem which is obtained by linearization of non-linear PDE is of cur-

...

rent interest in many physical problems [1,2,3,4,5,9,10, ] .

Some of the results

which follow are extensions of earlier results (with similar proofs) of [1,2]. For the rest of this paper we will suppose that hypotheses (1)- ( 3 ) are satisfied and, in addition, that:

L is an operator of order 2 with real-valued coefficients belonging to ce(,),

(0 <

(4)

e < 1).

We deduce from this that the smallest eigenvalue of (Q), namely pl, is simple and that its associated eigenfunction @,

A.

does not change its sign.

Furthermore

is the only eigenfunction of (Q) which does not change its sign. THE GENERAL CASE;

Proof.

E

D(L)

If h

5 L - 1 g

0

UNRESTRICTED AS TO ITS SIGN.

A 0 is a principal eigenvalue of (P) then Q

LEMMA 1. If f o r all u

u1

0

=

( u ) :((L-Aog)u,u)>O,

D(A).

is a principal eigenvalue of (P) then 0 is an eigenvalue of

whose associated eigenfunction doesn’t change sign.

smallest eigenvalue of T

and since D(Th )

=

0

from which the conclusion follows immediately

D(A) we have

Hence 0 is the

On the Eigenvalues o f Non-Definite Ellippbic Operators

A*

We d e f i n e

by

A*

=

u

.

1 g l u I 2 > 01 R THEOREM 3

l y xo5 A*

A*

If

where D+ =

CUE

D(A) : ( g u , u ) 5

Note t h a t when p1 > 0 , A" > 0 ,

A*

If

>

-m

and i f t h e r e e x i s t s a p r i n c i p a l e i g e n v a l u e A.

then necessari-

. t h e n t h e r e d o e s n o t e x i s t any p r i n c i p a l e i g e n v a l u e .

= -m

By d e f i n i t i o n of A",

Proof.

{ ik::z; }

inf E D+

223

i f ho > A"

,

then there e x i s t s v

E

Df

such t h a t

.

T h e r e f o r e Qx (v) < 0 f o r some V E D(A) a n d h e n c e by lemma 1, ( L v , v ) / ( g v , v ) c Xo A. c a n n o t b e p r i n c i p a l . T h i s p r o v0 e s t h e f i r s t s t a t e m e n t . The p r o o f of t h e o t h e r i s similar.

COROLLARY 3 .

I f t h e r e exists u

E

D+ s u c h t h a t ( L u , ~ )5 0 t h e n t h e r e e x i s t s no po-

s i t i v e p r i n c i p a l eigenvalue.

COROLLARY 4 .

i, qi,

Let

If there exists

l S i S N ,

?A

JIn’

1 5n S

b e t h e e i g e n p a i r s of (Q) w i t h p. < 0 , (p. 5 0 ) .

N , s u c h t h a t (gQn,Un)

2

0 , ( (gqn,qn) > 0 ) , t h e n t h e r e

e x i s t s no p o s i t i v e p r i n c i p a l e i g e n v a l u e .

EXAMPLE.

L e t L = - A w i t h Neumann boundary c o n d i t i o n s .

If

1 g>O, n

then t h e r e is

no p o s i t i v e p r i n c i p a l e i g e n v a l u e , ( T h i s f o l l o w s from C o r o l l a r y 3 ) . THEOREM 4

I f A" >

--m

and i f Q

eigenvalue. Proof.

(u) t 0, f o r a l l u

Denote by s t h e smallest e i g e n v a l u e o f T

But by d e f i n i t i o n o f A",

11 vn 11

A*

A*

D(A), t h e n h* i s a p r i n c i p a l

= L-h*g

+

X*,

i.e.,

E D + such t h a t n Hence Qhx(vn) + 0 as n - t m

THE CASE p

1

A*

is p r i n c i p a l .

20.

The f o l l o w i n g r e s u l t s c a n now b e p r o v e n as i n [ 1 , 2 ] .

If

.

So s = 0 i s t h e smallest e i g e n v a l u e o f TA* ; a n a s s o c i a t e d e i g e n -

f u n c t i o n d o e s n ' t change s i g n and s o

LEMMA 2 .

: Then

t h e r e i s an i n f i n i t e sequence of v

= 1 and (Lvn,vn)/(gvn,vn)

sSQ,,~(V,)+O.

B.

E

vl>

0 t h e n Q,,(u)>-O,

for a l l UED(A).

J. Fleckinger and A. B. Mingarelli

224 LEMMA 3 .

> 0 and

0< A

<

A" then A can’t be an eigenvalue

of ( P ) .

If p l > O then there exists exactly two principal eigenvalues, one po-

THEOREM 5. sitive

ul

If

A+0 and one negative If u l = O and

THEOREM 6 .

0

> pl/suplgl

.

9, is an associated eigenfunction then:

q1 t D-

If

+

, and ho

1-

=

{u

E

D(A) : (gu,u) < 0) then

A*

>

0

is the unique positive principal eigenvalue.

+

If $ E D 1

then 0 is the only non-negative principal eigenvalue.

When p l < O the problem is more intricate C.

THE CASE pl

0.

Suppose that there exists a real number k such

We first give an example. that c ( x ) 5 kg(x).

Then L u = Agu becomes Au

theorem 5, the problem Au

+ > 0 and

0

0

A-0 u'

=

(5-

0

E ’D

0-

0

< 0.

+ k where

(A

- k)gu

and now, according to

ogu admits twv and only two principal eigenvalues

Hence we obtain here two principal eigenvalues A+= 0

A' can be of any sign. 0

respectively).

THEOREM 7.

=

=

++k

(5

0

and

(The two corresponding eigenfunctions

We have a more general result.

Let kt W be such that c(x)-kg(x)

>-?.(A)

where s(A)

is the smallest

eigenvalue of A: s(A)

=

u

inf D(A)

{(Au,u)/llu\l2}

.

> 0

E

Then there exists precisely two principal eigenvalues of ( P ) . Proof.

(A+c)u

Agu is equivalent to (A+c-kg)u

=

=

(A- k)gu and A + c - k g being

positive by hypothesis, we can apply theorem 5.

D.

ESTIMATE ON A

0

IN THE CASE U1< 0

.

We notice that Lu=Agu can be written as (L+k)u sen large enough so that L + k is positive. Consider now the following eigenvalue problem: (L+k)u

=

= 0

Xmu ,

=

A(g+k/h)u

where k is cho-

225

On the Eigenvalues of Non-Definite Elliptic Operators

.

where m i s c o n t i n u o u s on

I f t h e r e e x i s t s a n x s u c h t h a t m ( x ) > 0 t h e n (by

theorem 5) t h e r e e x i s t s a p o s i t i v e p r i n c i p a l e i g e n v a l u e h (m) = 0

inf ( ( ( ~ ~ ~ ~ ; u ) } u E D+ m It i s s t r a i g h t f o r w a r d t o show t h a t h (m) i s non-

+=

{ u E D ( A ) : (mu,u) > O } . m i n c r e a s i n g i n m. C o n s i d e r now t h e problem

where D

for p>O.

0

S i n c e t h e r e e x i s t s x f o r which g ( x ) > O , t h e p r e v i o u s r e s u l t s a l s o h o l d

for m = g+k/p

,

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

v a l u e , d e n o t e d by h ( p ) , s u c h t h a t

The p o s i t i v e p r i n c i p a l e i g e n v a l u e of ( P ) , when i t e x i s t s , i s g i v e n b y t h e f i x e d p o i n t s of X ( p ) , i . e . ,

X(p)=p

.

a).

When p + + = ,

A(p)

+

A)(.

A 0 (m) w e deduce from But h ( a + k / p ) fa(p) = 0 ( t h e l a t t e r d o e s n ' t depend on

Using t h e m o n o t o n i c i t y of

h y p o t h e s i s ( 2 ) t h a t Xo(M+k/p) < h ( p ) 5 h (-m+k/p) 0 ( p , + k ) / ( a + k / p ) and f A ( u ) + ( p , + k ) / k as p + 0.

.

which i s t h e p o s i t i v e p r i n c i p a l e i g e n v a l u e of

( L + k ) u = Xgu and h ( p ) + 0 a s !.I

+

0.

We have t h e f o l l o w i n g bounding c u r v e s f o r

X(I1) :

7WJ

h

,/' Atp)=p

30 (-m + k/p)

.

- W M + k/p)

__

-

-P

J. Fleckinger and A. B. Mingarelli

226

It is now clear from the diagram that the following estimate holds: PROPOSITION 3. value A

0

In the case u l < O , if there exists a positive principal eigen-

+

of (P) whose associated eigenfunction is in D A

0

2 -p

/m = 1~ /inf g

1

1

then,

.

ACKNOWLEDGEMENTS. The first author is g:ateful

for financial support from the National Research

Council of Canada and the Minist’ere des Affaires Extgrieures, France. We should also wish to thank the Department of Mathematics of the University of Ottawa for making possible her visit there during Sept. 1982 and March-April 1983. The research of the second author is funded by grant U0167 from the Natural Sciences and Engineering Research Council of Canada. REFERENCES [l]

K.J. Brown and C.C. Lin, On the existence o f p o s i t i v e eigenfunctions f o r an

eigenvalue problem with i n d e f i n i t e weight-function, J. Math. Anal. Appl., 75, (1980), 112-120. [2] W.H. Fleming, A selection-migration model i n population genetics, J. Math. Biol.,

[3]

2,

(1975), 219-233.

P. Hess, On t h e principal eigenvalue of a second order linear e l l i p t i c pro-

blem with an i n d e f i n i t e weight-function, Math. Z.

m, (1982),

237-239.

[41 P. Hess and T. Kato, On some linear and non-linear eigenvalue problems with an i n d e f i n i t e weight-function. Corn. Partial Differential Equations, 5, (1980), 999-1030. [5]

A. Manes and A.-M. Micheletti, Un'estensione della teoria variazionale

classica degli autovalori per opemtori e Z 2 i t t i c i del secondo ordine. U.M.I.,

1. (1973),

Boll

285-301.

[6] A . B . Mingarelli, I n d e f i n i t e Stwm-Liouville problems, in Lecture Notes in Mathematics 964, Springer-Verlag, 519-528.

On the Eigenvalues of Non-Definite Elliptic Operators

[7]

227

A.B. Mingarelli, On the existence of non-simple real eigenvalues f o r general

Sturm-Liouville problems, To appear in Proc. Amer. Math. Soc. [8]

R.G.D.

Richardson, Contributions t o the study of oscillation properties of

the solutions of Zinear d i f f e r e n t i a l equations o f the second order., h e r . J. Math., [9]

40, ( 1 9 1 8 ) ,

283-316.

J.C. Saut and B. Scheurer, Remarks on a non-linear equation arising in popu-

Zation genetics, Corn. Partial Differential Equations, 2, ( 1 9 7 8 ) , 907-931.

[lo]

N. Senn and P. Hess, On positive solutions of a linear e l l i p t i c eigenvalue

probZem w i t h Newnann boundary conditions, Math. Ann., 258, ( 1 9 8 2 ) , 459-470.

*

Permanent Address ( J F ) : Department of Mathematics Universitc Paul Sabatier 118 rte de Narbonne 31062 Toulouse, Ccdex France.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

229

EXISTENCE OF GENERATORS AND DIFFERENTIABILITY OF EVOLUTIONS M i c h a e l A. Freedman D e p a r t m e n t of M a t h e m a t i c s Vanderbilt University N a s h v i l l e , T e n n e s s e e 37235 U.S.A. Let U ( t , s ) be a continuous c o n t r a c t i v e e v o l u t i o n a c t i n g i n R Under t h e sup-norm a n d

1 1 . I I 1-norm

s a r i l y differentiable a.e. Euclidean, U ( t , s ) §

1.

N

.

i s necesN Under o t h e r t o p o l o g i e s on R , e . g .

topologies, U ( t , s )

may b e nowhere d i f f e r e n t i a b l e .

In t h i s p a p e r we d i s c u s s e x t e n s i o n s of t h e H i l l e - Y o s i d a t h e o r e m w h i c h i n c l u d e

(nonautonomous) e v o l u t i o n s . convince t h e reader that

The a u t h o r h o p e s , i n t h e c o u r s e of t h i s p a p e r , t o

sum a n d

product i n t e g r a l s p l a y a fundamental r o l e i n t h e

t h e o r y of g e n e r a t o r s and e v o l u t i o n s . The H i l l e - Y o s i d a t h e o r e m i t s e l f s e r v e s t o c h a r a c t e r i z e s e m i g r o u p o p e r a t o r s T(t).

S p e c i f i c a l l y , suppose t h a t T ( t ) , t

o p e r a t o r s on a n a r b i t r a r y Banach s p a c e (X, (i) (ii) (iii) (iv)

2

0 , i s a f a m i l y o f bounded l i n e a r

I I I I)

Semigroup P r o p e r t y : T ( t ) T ( s ) = T ( t + s ) ,

w h i c h s a t i s f i e s the

V t,s

0,

I n i t i a l Property: T(0) = I d e n t i t y Operator I,

I / T ( t ) x - x l I + 0 as 1 I T ( t ) x / I 5 11x1 I, V t -> 0 , V x E

Strong Continuity Property:V x Contractive Property:

f

X,

t

+

+,

0

X.

Then, e v e r y s u c h s t r o n g l y c o n t i n u o u s c o n t r a c t i v e s e m i g r o u p T ( t ) i s s t r o n g l y d i f f e r e n t i a b l e ; t h a t i s , f o r a l l x i n a d e n s e set

p:

e x i s t s f o r a l l t > 0, while a t t = 0 , t h e one-sided

exists.

X, t h e s t r o n g l i m i t

limit

+

T h i s l a s t l i m i t y i e l d s a d e n s e l y d e f i n e d o p e r a t o r A = dT ( O ) / d t known a 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 T , f o r A g e n e r a t e s T a s t h e s t r o n g l i m i t of r e s o l vents:

A s well, A i s r e l a t e d t o T t h r o u g h t h e a b s t r a c t d i f f e r e n t i a l e q u a t i o n

M. A. Freedman

230

Finally, the infinitesimal generator A satisfies the so-called dissipative property

I I [I-AAlxl I 2

11x1 1 ,

A >

0,

x

E

P.

Now, a statement of the Hille-Yosida theorem is [l]: A necessary and sufficient condition for a densely defined operator A to be

the infinitesimal generator of a unique strongly continuous contractive semigroup,

in the sense of (l), is that A be dissipative.

5

The differential equation (2) is autonomous since the operator A does not

2.

In the study of the nonautonomous generalization of (2), we are

change with t.

lead to a two parameter family of bounded linear operators called an evolution U(t,s),

t

2 s,

which satisfies the

(I) Evolution Property: U(t,s)U(s,r) (11) (111)

Initial Property: U(t,t)

=

U(t,r),

V t

2 s 1. r,

I, V t,

=

Strong Continuity Property:

for all x

E

X, the map (t,s)

(IV) Contractive Property:

H

U(t,s)x

I IU(t,s)xl I 5

is continuous,

11x1 I,

x

E

X, v t 2

Evolutions are generally not as well behaved as semigroups.

s.

For example, a

strongly continuous contractive evolution U may be nowhere differentiable, as seen by U(t,s) = exp(i(g(t)-g(s)),

where g is real-valued, continuous and nowhere

However, the following result [2, p . 5 7 2 1 does hold in

differentiable and X = C.

a real finite dimensional Banach space. Proposition 1. Let X

N R with sup-norm

=

tinuous contractive evolution U(t,s)

N I I {xi}I=l I loo=lsyp ]xi[. <
(i.e. U satisfies TI)-(IV))

Then every conon X is differen-

tiable a.e. in both the t and s variables. Central to the proof of this proposition is Proposition 2. Let {A(t,s)}t,s A(s,s)

(b)

(t,s)

(c)

I IA(t,s)l I

Then for each t

=

A(t,s)

is a continuous map,

< 1,

m -

E

be a family of operators on X satisfying

I, V s ,

(a)

H

-

v t,s.

R , there exists an open neighborhood Nt of t such that for all

5 > 5 ENt 0 < det A < Aii(S,<),

i

=

1,2

,...,N

23 1

Existence and Differentiability Here A . , ( < , 3 ) 11

th denotes t h e i d i a g o n a l e l e m e n t of t h e NxN m a t r i x A(€,,3)

and t h e

norm i n ( c ) d e n o t e s t h e o p e r a t o r norm of A ( t , s ) i n d u c e d by t h e sup-norm on X. A s shown i n [ 2 , pp.567,568],

P r o p o s i t i o n 2 i m p l i e s P r o p o s i t i o n 3 below.

F u r t h e r m o r e , a s t h e r e a d e r can check f o r h i m s e l f , t h i s i m p l i c a t i o n r e m a i n s v a l i d N

i n d e p e n d e n t of w h a t e v e r norm i s used on R

.

P r o p o s i t i o n 3. The e v o l u t i o n U of P r o p o s i t i o n 1 i s of bounded v a r i a t i o n i n t h e

sense t h a t f o r e a c h b > a t h e r e e x i s t s a number y s u c h t h a t f o r e v e r y p a r t i t i o n

a

= t

.? I

Im 5 y.

< t -II 01- n 1=1 With P r o p o s i t i o n 3 i n h a n d , t h e Mac Nerney

t h e o r y of m u l t i p l i c a t i v e a n d

a d d i t i v e t y p e f u n c t i o n s of bounded v a r i a t i o n [ 5 ] may b e a p p l i e d t o show t h e exi s t e n c e of t h e

=

integral

[TI

=l~~rnn~~i-~i-l~.

< T < < sn = t and V i s c a l l e d t h e gene0 1r a t o r f o r U s i n c e i t c a n b e shown t h a t U h a s t h e p r o d u c t i n t e g r a l r e p r e s e n t a t i o n

where s = T

i n t e r m s of V:

C o r r e s p o n d i n g t o p r o p e r t i e s (1)-(IV),

(1’)

Additive Property:

(11')

I n i t i a l Property:

(111')

Proposition 4.

< t

2

V t

r,

s

X, t h e map ( t . s ) t+ V ( t , s ) x i s c o n t i n u o u s ,

< 1-

I I [ I - V ( t , s ) ] x l Im2

11x1

Im

x

E

X,

s.

V t

imply P r o p o s i t i o n 3 , so do p r o p e r t i e s ( I ' ) - ( I V ' )

Every c o n t i n u o u s d i s s i p a t i v e g e n e r a t o r V on (R

bounded v a r i a t i o n . 0-

= V(t,r)

V ( t , t ) = 0 tr t ,

Dissipative Property:

J u s t as p r o p e r t i e s ( I ) - ( I V )

a = t

+ V(s,r)

Strong Continuity Property: for all x

(IV')

V(t,s)

V satisfies the

N

, I / * 1 I-)

imply

i s of

T h a t i s , f o r e a c h b > a t h e r e exists y > 0 s u c h t h a t i f n < t n = b i s a p a r t i t i o n of [ a , b l t h e n G lV(ti,ti-l)l i=1

1

lco 5 Y.

Now, n o t e v e r y c o n t i n u o u s i n t e r v a l f u n c t i o n F ( t , s ) of bounded v a r i a t i o n i s d i f f e r e n t i a b l e a.e.

C o n s i d e r F ( t , s ) = I t - s l g ( t s ) where g i s c o n t i n u o u s a n d no-

where d i f f e r e n t i a b l e .

T h e r e f o r e , P r o p o s i t i o n 3 d o e s n o t i m m e d i a t e l y imply d i f f e r -

e n t i a b i l i t y of U ( t , s ) . t h e d i f f e r e n c e H(t)-H(s)

However, s i n c e V ( t , s ) i s a d d i t i v e a n d h e n c e e x p r e s s i b l e a s f o r some m a t r i x - v a l u e d

i n d e e d imply t h a t V ( t , s ) i s d i f f e r e n t i a b l e a.e. t h e product i n t e g r a l formula ( 4 ) .

point function, Proposition 4 does P r o p o s i t i o n 1 now f o l l o w s from

I n f a c t , a s shown i n [ 2 ] ,

U is related t o its

generator V through t h e a b s t r a c t p a r t i a l d i f f e r e n t i a l equations

M. A. Freedman

232

5 3.

Does P r o p o s i t i o n 1 h o l d f o r e v e r y r e a l f i n i t e d i m e n s i o n a l Banach s p a c e X?

The answer i s no.

Take X e q u a l t o E u c l i d e a n

RN

w i t h N > 1.

Next l e t g b e nowhere

d i f f e r e n t i a b l e and c o n t i n u o u s a n d l e t P b e any N x N n o n z e r o skew-adjoint

t

that P

,

t h e t r a n s p o s e of P , e q u a l s -P.

matrix so

Then t h e c o n t i n u o u s e v o l u t i o n U ( t , s ) =

exp [ (g ( t ) -g ( s ) ) P ] s a t i s f i e s u(t,s)u(t,s)x

= I,

And U h a s no p a r t i a l

which makes e a c h o p e r a t o r U ( t , s ) u n i t a r y , hence c o n t r a c t i v e . d e r i v a t i v e s , f o r , a s t h e r e a d e r may v e r i f y ,

where R ( 6 )

-t

0 as 6

+

0.

N Does P r o p o s i t i o n 1 h o l d f o r any o t h e r norm on R ? The answer N N Then, sup-norm i s r e p l a c e d by t h e sum-norm {xili=ll = igl/xil. i s an NxM matrix, t h e operator-norm of A i n d u c e d by t h e sum-norm i s N IAl iLllAijl [ 4 ] . Armed w i t h t h i s f o r m u l a , t h e r e a d e r i s §

4.

I1

I

I

=p&

e x e r c i s e , t o modify t h e p r o o f of Lemma 1 of [ Z ,

pp.566,567]

i s y e s when t h e if A = [A. . I 1J g i v e n by

l e f t , as an

so as t o prove t h a t

II*II

-norm s e t t i n g . Now, a p p l y 1 i n g t h e s e n t e n c e p r e c e d i n g t h e s t a t e m e n t of P r o p o s i t i o n 3, w e deduce t h a t Propo-

P r o p o s i t i o n 2 above r e m a i n s t r u e i n t h e p r e s e n t s i t i o n 3 a l s o r e m a i n s t r u e i n t h i s new s e t t i n g .

Hence f o r e v e r y c o n t i n u o u s , sum-

norm c o n t r a c t i v e e v o l u t i o n U, t h e r e e x i s t s a g e n e r a t o r V g i v e n by e q u a t i o n ( 3 ) a n d which s a t i s f i e s e q u a t i o n ( 4 ) . P r o p o s i t i o n 5. sum-norm Proof.

A s w e l l , w e have

P r o p o s i t i o n 4 h o l d s when t h e sup-norm

I 1* I 11 L e t V . .(c,<) 13

I I * [ Ic0

i s r e p l a c e d by t h e

d e n o t e t h e e l e m e n t i n t h e ith row and j t h column of t h e NxN

S i n c e V i s a d d i t i v e , e a c h V . . i s t o o . Hence, t h e r e e x i s t c o n t i n 1J uous f u n c t i o n s a such t h a t V . . ( 5 . 5 ) = a . - a 1. J. ( < ) f o r a l l 6 > 5 E [ a , b I . ij 11 13 With compactness of [ a , b ] a n d t h e e q u i v a l e n c e of a l l norms on RN i n mind, i t w i l l m a t r i x V(c,<).

.(c)

s u f f i c e t o show t h a t f o r e a c h t

E

[ a , b ] , t h e r e i s a n open neighborhood N t ,

t a i n i n g t , on which e a c h f u n c t i o n a

I n the p r e s e n t expressed as

l ( . I l 1-norm

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

con-

ij s e t t i n g , t h e d i s s i p a t i v e p r o p e r t y of V may b e

233

Existence and Differentiability

for a l l vectors {x.}E

I

Given t

E

R

N

and a l l

[a,b], let

E

t

5

>

where

6

ij

=

0 i f i # j and

> 0 be such t h a t f o r a l l

5> 5

6

= 1. ii i n N t = (t-&t,t*t),

We i n i t i a l l y show t h a t e a c h d i a g o n a l f u n c t i o n ci ( 5 ) i s of bounded v a r i a t i o n on PP N F i r s t d e f i n e x ( I ) , x ( ~ ) ,x C 3 )and x ( ~ ) a s t h o s e v e c t o r s i n R having compoNt.

nents

,

S u c c e s s i v e l y a p p l y i n g t h e s e v e c t o r s t o i n e q u a l i t y (5) y i e l d s , f o r a l l

N

5

> 5 E N

N

Adding i n e q u a l i t i e s (6) t o ( 7 ) and (8) t o (9) g i v e s , r e s p e c t i v e l y ,

( 5 ) + h ( 5 ) and 2a (5) + h ( 5 ) are d e c r e a s i n g on N t ' PP PP T h i s i m p l i e s t h a t t h e d i f f e r e n c e f u n c t i o n a ( 5 ) i s of bounded v a r i a t i o n on N t ' PP Therefore the functions

ci

t:

M.A. Freedman

234

Next c o n s i d e r t h e f u n c t i o n a for q P9

# p.

Applying t h e v e c t o r x ( ~ )w i t h

components

t o i n e q u a l i t y ( 5 ) , we o b t a i n f o r a l l

5

> 6

E

Nt:

T h a t is, t h e f u n c t i o n

I s d e c r e a s i n g o n N,,

w h i l e from (8) w e see t h a t t h e f u n c t i o n

N

N

i s too.

Hence t h e d i f f e r e n c e of t h e s e two f u n c t i o n s ,

N

C ( 3 a -a i=l

ip

iq

N ) + 2 C a

j=1 p j

-4a

+2a

(11)

’

P4

Now e a c h of i n e q u a l i t i e s ( 6 ) , (7) and (8) s t a t e s

i s of bounded v a r i a t i o n on N t .

t h a t a c e r t a i n f u n c t i o n , which w e d e n o t e as f l ,

f 2 and f 3 , r e s p e c t i v e l y , i s de-

Hence t h e f u n c t i o n s

c r e a s i n g on Nt.

N

f l - f

PP

=

l a i=l i p

and

(fl - f 2 ) / 2

-

(f3 - fl)

+

N

2a

PP

=

1a .

j=1 PJ

N

are of bounded v a r i a t i o n on N t .

L i k e w i s e , so i s

1

i= 1

i s of bounded v a r i a t i o n on N t . we see t h a t a P4

Uiq.

T h e r e f o r e , from (11)

235

Existence and Differentiability E a r l i e r we r e f e r r e d t o t h e p o s s i b i l i t y of g e n e r a l i z i n g t h e H i l l e - Y o s i d a theorem.

Here we o f f e r t h e f o l l o w i n g p r o p o s i t i o n , w h i c h h o l d s f o r X

b o t h t h e sup- and sum-norm t o p o l o g i e s .

=

RN u n d e r

The p r o o f , which i s l e f t t o t h e r e a d e r ,

r e l i e s on t h e p r e v i o u s p r o p o s i t i o n s . Proposition 6.

A n e c e s s a r y and s u f f i c i e n t c o n d i t i o n t h a t a n a d d i t i v e opera-

t o r - v a l u e d i n t e r v a l f u n c t i o n V ( t , s ) b e t h e g e n e r a t o r of a u n i q u e c o n t i n o u s cont r a c t i v e e v o l u t i o n U on X, i n t h e s e n s e of

( 4 ) , i s t h a t V be continuous and

dissipative.

5 5.

F i n a l l y , we a s k i f t h e p r e v i o u s p r o p o s i t i o n s which h o l d on

a l s o t r u e when N =

a.

RN

for N <

m,

are

The f o l l o w i n g example shows t h a t a t l e a s t w i t h r e s p e c t t o

P r o p o s i t i o n 4 , t h e answer i s no. Example.

D e f i n e t h e s e q u e n c e of p i e c e w i s e l i n e a r d e c r e a s i n g f u n c t i o n s

S e t X = Rm and f o r e a c h x

E

X, l e t V ( t , s ) b e t h e o p e r a t o r on X which maps x t o t h e

vector

W e c l a i m t h a t t h e f a m i l y of o p e r a t o r s V = { V ( t , s ) }

t h r o u g h (IV')

s a t i s f i e s p r o p e r t i e s (1') s
t i e s (I’),

( 1 1 ' ) and (IV’) f o l l o w i m m e d i a t e l y from t h e d e f i n i t i o n o f w a n d V. m,n E > 0, l e t n b e so l a r g e t h a t < E, a n d l e t d 0 b e small enough t h a t 0 5 t - .A 5 d i m p l i e s I w ~ , ~ ( .-~ wm ) n ( ~ ~5 ) ]E / 2 f o r a l l

l/Ko

With r e s p e c t t o (HI’), g i v e n

1 5 m 5 n 5 no. Then g i v e n T~ < 6 , w e have f o r a l l x E X ,

5

ao, whenever T

20

satisfies

10

-

0

0

I

i- I T

-

To[

T h e r e f o r e , beyond b e i n g s t r o n g l y c o n t i n u o u s , w e see t h a t V i s norm c o n t i n u o u s . I n o r d e r t o show t h a t V i s n o t of bounded v a r i a t i o n , c o n s i d e r t h e s e q u e n c e

M. A. Freedman

236 of p a r t i t i o n s of [ O , l ] : { - r

n n }n=l where T = i / n , i = O , l , . . . , n . i

p o s i t i v e i n t e g e r n and x = (l,l,

Hence i t is

not

t r u e t h a t f o r each x

f o r every p a r t i t i o n

1

i=O

Then, f o r e v e r y

...) ,

E

of [ 0 , 1 ] .

X there exists a constant K

such t h a t

T h a t i s , V i s n o t of ( s t r o n g ) bounded

v a r i a t i o n on [ 0 , 1 ] . T h i s example and t h e examples of s e c t i o n s 2 and 3 b r i n g o u t t h e n e e d f o r a t h e o r y of e v o l u t i o n s more g e n e r a l t h a n t h e bounded v a r i a t i o n b a s e d t h e o r y developed by Mac Nerney i n [ 5 ] . We r e f e r t h e r e a d e r , i n t h i s r e g a r d , t o [ 3 ] .

REFERENCES

[I] B u t z e r , P. L. a n d B e r e n s , H . , (Springer-Verlag,

Semi-groups of o p e r a t o r s and a p p r o x i m a t i o n

N e w York, 1967) 34,79.

[ Z ] Freedman, M. A . , A f a i t h f u l H i l l e - Y o s i d a theorem f o r f i n i t e d i m e n s i o n a l e v o l u -

t i o n s , T r a n s . Amer. Math. SOC. 265(1981) 563-573.

131 Freedman, M. A . , O p e r a t o r s of p - v a r i a t i o n a n d t h e e v o l u t i o n r e p r e s e n t a t i o n problem, Trans. Amer. Math. SOC., t o a p p e a r .

[ 4 ] L a n c a s t e r , P . , Theory of m a t r i c e s (Academic P r e s s , N e w York, 1969) 212. [ 5 ] Mac Nerney, J . S., (1963), 148-173.

I n t e g r a l e q u a t i o n s and semigroups, I l l i n o i s 3. Math. 7

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

231

ON THE ASYMPTOTIC BEHAVIOR OF THE POSITIVE SOLUTIONS OF A DIFFERENTIAL EQUATION WITH A DISCONTINUOUS NONLINEAR TERM John R . Graef and Paul W. Spikes Department o f Mathematics and S t a t i s t i c s Mississippi State University Mississippi State, M i s s i s s i p p i U.S.A. Myron K. Grammati kopoulos Department o f Mathematics U n i v e r s i t y o f Ioannina I o a n n i na, Greece The e x i s t e n c e and a s y m p t o t i c b e h a v i o r o f p o s i t i v e c o n t i n u a b l e s o l u t i o n s o f a higher order f u n c t i o n a l d i f f e r e n t i a l equation w i t h a d i s c o n t i n u o u s n o n l i n e a r t e r m a r e discussed. A s u f f i c i e n t c o n d i t i o n f o r t h e e q u a t i o n t o have no p o s i t i v e continuable s o l u t i o n s i s a l s o given. INTRODUCTION While much has been w r i t t e n about t h e a s y m p t o t i c b e h a v i o r o f s o l u t i o n s o f EmdenFowler x" + q ( t ) f ( x ) = 0 and Thomas-Fermi x" - q ( t ) f ( x ) = 0 t y p e e q u a t i o n s where q ( t ) > 0, x f ( x ) > 0 f o r x # 0, and f i s i n c r e a s i n g on (--,-), r e l a t i v e l y l i t t l e i s known about e q u a t i o n s o f these t y p e s when f i s p o s i t i v e and decreasing on ( 0 , ~ ) . F o r example, t h i s i s t h e case f o r t h e e q u a t i o n XI' + q ( t ) x - A = 0

(*1

where x i s a p o s i t i v e c o n s t a n t . We c i t e as r e c e n t c o n t r i b u t i o n s t o t h e s t u d y o f e q u a t i o n s o f t y p e ( * ) t h e paper o f T a l i a f e r r o [5] f o r second o r d e r e q u a t i o n s and t h e paper o f N a i t o [ 4 ] f o r h i g h e r o r d e r e q u a t i o n s . THE EXISTENCE AND BEHAVIOR OF POSITIVE SOLUTIONS

We a r e concerned w i t h t h e e x i s t e n c e and a s y m p t o t i c b e h $ v i o r o f t h e p o s i t i v e continuable solutions o f the nonlinear functional d i f f e r e n t i a l equation [r(t)x(n-v)(t)l(v) + f(t,x(g(t))) where 1 1 v < n - 1, r , g : [to,m)+R g ( t ) + m as t - f m , [to,-) x ( 0 , m ) .

=

o

a r e continuous, r ( t ) > 0,

(El

im

[l/r(s)]ds

=

m,

and f ( t , y ) i s p o s i t i v e and n o n i n c r e a s i n g w i t h r e s p e c t t o y on N o t i c e t h a t g i s a d e v i a t i n g argument, t h a t i s , i t may be

r e t a r d e d ( g ( t ) y t ) . advanced ( g ( t )

t ) , o r otherwise.

For convenience t h e f o l l o w i n g n o t a t i o n w i l l be used i n t h e remainder o f t h i s to and a l l t T we l e t paper. For any T

238

J.R. Graef e t al.

and [( t - ~ ) " - ~ s v-- l' / r ( s ) ] d s / ( n - v - 1 ) ! .

I t i s not d i f f i c u l t t o see t h a t t h e assumptions imposed on f , g , and r i n ( E ) imply t h a t any positive continuable solution x ( t ) of ( E ) belongs t o one of the two classes 0n- 1 ( t ) = z ( " - ' ) ( t ) + A > 0 as t + (1)

or w n-1

( t ) = z ( u - ' ) ( t ) + O as

t + m .

(11)

A t t h i s point we should point o u t t h a t i t i s possible t h a t equation ( E ) may have noncontinuable s o l u t i o n s . I n f a c t , Theorem 6 below gives s u f f i c i e n t conditions f o r (E) t o have no p o s i t i v e continuable s o l u t i o n s .

Next we w i l l s t a t e two lemmas which will be used i n proving some of our r e s u l t s . Lemma 1 . ([2,3; Lemma 11). Let u be a p o s i t i v e (n-u)-times continuously d i f f e r e n t i a b l e function on the interval [a,m) and l e t p be a p o s i t i v e continuous function on [a,m) such t h a t f[l/u(t)Idt = and the function w Moreover, l e t

uu

E

n-u)

-

i s u-times continuously d i f f e r e n t i a b l e on [a,-).

1

LI(~), i f 0 5 k

n-v-1

I f w n ( t ) E w ( " ) ( t ) i s of constant sign and n o t i d e n t i c a l l y zero f o r a l l l a r g e t , then there e x i s t t u ? a a n d a n i n t e g e r P , 0 5 P 5 n, with n + P even f o r on nonnegative o r n + L odd f o r on nonpositive, and such t h a t f o r every t tu L

>

0 implies w k ( t ) > 0 ( k = O,l,

. . . ,L - 1 )

and

P 5n

-

1 implies ( - 1 ) L+kw k ( t )

>

0 ( k = L , L + 1 , ..., n - 1 ) .

([2,3; Lemma 2 3 ) . If the functions u , U , w a n d wk a r e as i n Lemma 1 and f o r some k = 0 , 1 , ...,n - 2 w k ( t ) + c as t + m , then t ~ ~ + ~ ( t ) as + Ot-fm.

Lemma 2.

Our f i r s t theorem gives a g r o w t h r e s u l t f o r solutions o f ( E ) . Theorem 3. I f x ( t ) i s a positive continuable solution of ( E ) , then there e x i s t s a constant B 2 0 such t h a t x ( t ) / J ( t O , t )+ B as t + m .

239

Asymptotic BchaL'ior o f Positive Solutions

Proof. If x ( t ) > 0 i s a continuable solution o f ( E ) of type ( I ) , then successive integrations show t h a t w k ( t ) + m as t + - f o r k = O , l , ...,n - 2 . By L'HGpital's rule lim [ x ( t ) / J ( t o , t ) ] = lim z ( ' - ' ) ( t ) / ( v - l ) ! t + m

= B >

0.

t - f m

On the other hand, i f x ( t ) i s of type ( I I ) , then Lemmas 1 and 2 ensure t h a t t h e r e e x i s t s an i n t e g e r N with 0 5 N 5 n - 1 such t h a t U N ( t ) + E l 0 as t + - and

w k ( t ) + - as t + = f o r k Remark.

<

N.

L ' H o p i t a l ' s rule shows t h a t x ( t ) / J ( t o , t ) + O as t + - .

Naito [4] showed t h a t equation ( E ) has a p o s i t i v e solution s a t i s f y i n g x(t)/J(t,,t)+A

>

0 as

t + m

(11

<

(2)

i f a n d only i f r f ( s ,cJ ( to.g ( s 1) I d s

m

f o r some c > 0. We did n o t require ( 2 ) and were able t o show t h a t any solution of type ( I ) s a t i s f i e s ( 1 ) and any solution of type (11) s a t i s f i e s (1) with A 0. T a l i a f e r r o [5] proved a second order version of N a i t o ' s r e s u l t . Theorem 4.

I f there e x i s t s c

>

0 such t h a t

r f ( s , c J ( t O , g ( s ) ) ) d s=

(3)

m,

then every positive continuable solution of ( E ) i s of type ( I ) . Proof. If x ( t ) were o f type (11) then x ( g ( t ) ) 5 c J ( t O , g ( t ) ) f o r t 2 T f o r some T An i n t e g r a t i o n of ( E ) then y i e l d s a contradiction of ( 3 ) . to. Corollary 5. I f ( 3 ) holds f o r some c > 0 , then every p o s i t i v e continuable solut i o n x ( t ) of ( E ) s a t i s f i e s x ( t ) / J ( t O , t ) + B c as t + - . While Theorem 4 gives s u f f i c i e n t conditions f o r a l l positive continuable solutions o f ( E ) t o be cf type ( I ) , the following theorem gives conditions u n d e r which equation ( E ) has no p o s i t i v e continuable s o l u t i o n s . Theorem 6.

If

f o r every c

>

0 , then ( E ) has no positive continuable s o l u t i o n s .

For many functions f i f ( 4 ) holds f o r some c > 0 then i t holds f o r a l l c > 0. example where t h i s i s not the case i s the equation

( t x ' ) ' + k 2 e - l / t k/ t k e x

An

0, t > 0 where k > 0 i s a constant. Here ( 4 ) holds f o r any c > 0 such t h a t c + k 5 1 a n d f a i l s t o hold i f c + k > 1. Hence by Theorem 4 every p o s i t i v e continuable soluk t i o n i s of type ( I ) . Here x ( t ) = t n t - l / t i s such a solution. =

The equation

x" + t - 2 ( t 1 ' 2 + L n t

1/2

) x -1 ( t 1/2 )

=

0, t > 1 ,

s a t i s f i e s the hypotheses of Theorem 3. Note t h a t x ( t ) = t + Ln t i s a solution of type ( I ) ( u l ( t ) = x ' ( t ) = l + l / t ) which s a t i s f i e s x ( t ) / J ( l , t ) = ( t + L n t ) / ( t - l ) + l as t + = . Neither ( 3 ) nor ( 4 ) holds f o r t h i s equation.

240

J. R. Graef e t at.

DISCUSSION

F o r purposes o f comparison we now c o n s i d e r t h e e q u a t i o n ~ r ( t ) x ( ~ - ~ ) ( t ) -l ( f~( t), x ( g ( t ) ) ) = where r, g, f, and v a r e as b e f o r e .

(El)

o

In t h i s case a p o s i t i v e c o n t i n u a b l e s o l u t i o n

x ( t ) o f ( E ) w i l l be o f one o f t h e two t y p e s w n-1 ( t ) > 0

(111)

(t) < 0

(IV)

or w n-1

f o r sufficiently large t

to. i n g r e s u l t s f o r equation (E): i)

If x ( t )

>

o

I n [l]t h e a u t h o r s were a b l e t o prove t h e f o l l o w -

i s a solution o f

( i )o f

t y p e (111)

and

f o r every c ii) iii)

>

0, t h e n x ( t ) / J ( t O , t ) + A

If x(t)/J(tO,t)+A

>

0 as

t + m ,

> 0 as

t+m;

t h e n ( 5 ) h o l d s f o r some c

0;

If

f o r every c > 0, t h e n x ( t ) / J ( t O , t ) + m

as t + - .

While t h e r e i s n o t x a d i r e c t correspondence between s o l u t i o n s o f (E) o f t y p e ( I ) and s o l u t i ~ n so f (E) o f t y p e ( 1 1 1 ) , t h e r e i s an i n t e r e s t i n g s i m i l a r i t y between Theorem 3 (and N a i t o ' s r e s u l t ) and p a r t s i ) and i i ) above. One o f t h e b a s i c d i f f e r e n c e s i n t h e two e q u a t i o n s can be seen by comparing Theorem 6 and iii). REFERENCES

1.

Graef, J. R., Grammatikopoulos, M. K., and Spikes, P. W., On t h e p o s i t i v e s o l u t i o n s o f a h i g h e r o r d e r f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n w i t h a discont i n u i t y , I n t e r n a t . J. Math. Math. S c i . 5 (1982), 263-273.

2.

Gramrnatikopoulos, M. K., On t h e e f f e c t o f d e v i a t i n g arguments on t h e b e h a v i o r o f bounded s o l u t i o n s o f n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s , Ukrain, Mat. Z . 30 (1978), 462-473 (Russian).

3.

Grammatikopoulos, M. K . , A s y m p t o t i c and o s c i l l a t i o n c r i t e r i a f o r n o n l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h d e v i a t i n g arguments, t o appear.

4.

N a i t o . M.. E x i s t e n c e and asvmDtotic b e h a v i o r o f o o s i t i v e s o l u t i o n s o f d i f f e r e n t i a l i n e q u a l i t i e s w i t h d & i a t i n g argument, F u n k c i a l . Ekvac. 22 (1979), 127-1 42.

5.

T a l i a f e r r o , S . , On t h e p o s i t i v e s o l u t i o n s o f y" + + ( t ) y - ' Anal. 2 (1978), 437-446.

= 0, N o n l i n e a r

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), I984

241

ENERGY ESTIMATES FOR SYMMETRIC HYPERBOLIC INTEGRO-DIFFERENTIAL EQUATIONS Ronald Grimmer* and M a r v i n Zeman Department o f Mathematics Southern I l l i n o i s U n i v e r s i t y Carbondale, I l l i n o i s 62901

1.

INTRODUCTION

We c o n s i d e r t h e Cauchy problem a s s o c i a t e d w i t h t h e l i n e a r i n t e g r o - d i f f e r e n t i a l equation au Ao(x,t)at(x,t)

= P(x,t,ax)u(x,t)

(I.E.)

+

.bt B ( x , t ; r

,ax)u(x,T ) & + f ( x , t )

u(x,O) = g ( x ) , x € R n , 0 5 t 5 T,

where t h e p a r t i a l d i f f e r e n t i a l e q u a t i o n au AO(x, t )%(x , t ) = p ( x ,t ,ax)u(x, t 1 i s symmetric h y p e r b o l i c . We s h a l l a l s o assume t h a t B(x,t,T,ax) t and T , F and

can be f a c t o r e d as B = F P + K where f o r f i x e d

K a r e bounded o p e r a t o r s on Hs(Rn) and F i s s t r o n g l y d i f f e r e n t i a b l e

as a f u n c t i o n o f

T.

One way i n which symmetric h y p e r b o l i c d i f f e r e n t i a l e q u a t i o n s have been s t u d i e d i s by t h e use of a - p r i o r i L2 e s t i m a t e s ( s o - c a l l e d energy e s t i m a t e s ) . l e a d t o t h e e x i s t e n c e o f u n i q u e weak s o l u t i o n s .

These e s t i m a t e s

Moreover t h e e s t i m a t e s p r o v i d e

r e g u l a r i t y r e s u l t s as w e l l as show t h a t t h e s o l u t i o n s propagate w i t h f i n i t e speed. The aim o f t h i s paper i s t o p r o v e energy e s t i m a t e s f o r

(I.E.) which a r e analogous

t o t h e e s t i m a t e s found f o r symmetric h y p e r b o l i c d i f f e r e n t i a l e q u a t i o n s . We w i l l p r e s e n t two examples a s s o c i a t e d w i t h continuum mechanics f o r m a t e r i a l s w i t h memory which s a t i s f y t h e c o n d i t i o n s we impose on ( I . E . ) .

The f i r s t example

d e a l s w i t h t h e g e n e r a l i z e d l i n e a r t h e o r y o f h e a t c o n d u c t i o n p o s t u l a t e d by G u r t i n and P i p k i n [ 4 ] .

Our second example d e a l s w i t h t h e e l e c t r o m a g n e t i c t h e o r y f o r

inhomogeneous a n i s o t r o p i c s t a b l e media w i t h memory.

We c o n s i d e r t h e c o n s t i t u t i v e

r e l a t i o n s proposed by V o l t e r r a [ 7 ] . 2.

PRELIMINARIES

We s h a l l w r i t e x = ( x n-tuple

*

ci =

,,..., x n )

o r 5 = (El

,..., En)

f o r a c o o r d i n a t e i n Rn.

( al,... ,an) o f n o n n e g a t i v e i n t e g e r s x" =

X ~ I

2x;

...

F o r an

x"n. n

Work by t h i s a u t h o r p a r t i a l l y supported by t h e N a t i o n a l Science Foundation under Grant No. MCS-8201322.

242

R. Grimmer and M. Zeman

Hs(Rn) i s t h e H i l b e r t space w i t h norm IluI/, d e f i n e d by

where u i s t h e F o u r i e r t r a n s f o r m o f u .

( U , V ) ~i s i t s i n n e r p r o d u c t .

By Q we s h a l l always mean an open s e t i n Rn.

We n o t e f i n a l l y t h a t C w i l l denote a

c o n s t a n t which may v a r y from l i n e t o l i n e w h i l e B ( X ) w i l l denote t h e Banach Algebra o f bounded o p e r a t o r s on t h e space X . I n o r d e r t o s t u d y ( I . E . ) we s h a l l impose a number o f c o n d i t i o n s .

( I ) The a s s o c i a t e d p a r t i a l d i f f e r e n t i a l e q u a t i o n

au

A0 ( X ,t )=( x ,t 1 = P ( X ,t ,a x 1u ( X ,t n au = C Ak(X,t)-(X,t) + A(x,t)u(X,t) k= 1 axk i s symmetric h y p e r b o l i c . By t h i s we mean t h a t Ao, Ak and A a r e N X A

A

where Ak,

k

Rnx[O,T]

= 0,

...,n,

a r e symmetric w h i l e A.

Friedrichs [2]).

(c.f.

N matrices

i s positive definite i n

F u r t h e r , we r e q u i r e t h a t A k ( x , t ) ,

k=O

,...,n,

la1 5 s , so t h a t Ak y i e l d bounded o p e r a t o r s on k = 1 , ...,n, a r e i n C([O,T],B(HS(Rn)))

have bounded d e r i v a t i v e a;Ak,

I n a d d i t i o n we ask t h a t Ak,

HS(Rn). w h i l e A.

i s i n C'([O,T),B(HS(Rn))).

( 1 1 ) The f u n c t i o n B(x,t,T,aX)

can be w r i t t e n as

= F(x,t,r,ax)P(x,t,ax)

B(x,t,r,ax)

05

+ K(X,t,r,aX),

T

5 t,

X

€Rn,

where F and K a r e z e r o o r d e r p s e u d o - d i f f e r e n t i a l o p e r a t o r s and t h e s e g i v e r i s e t o bounded o p e r a t o r s on Hs(IRn)). F u r t h e r F i s C 1 as a f u n c t i o n of T and f o r f i x e d t, 0

3.

5

t

5 T, F, K, FT E C([O,t],B(Hs(Rn)))

as a f u n c t i o n o f

T.

THE B A S I C A - P R I O R 1 LL ESTIMATES

Before we p r e s e n t t h e L

2 e s t i m a t e s , we f i r s t t r a n s f o r m t h e e q u a t i o n

Ao$

= Au

n

a . c Ak ax

where A =

k=l

Since

is C

1 Ao$

1

in

*

oat

t ,b (FA + K')ud?+ f

= Au and

= Au

+ iu +

= Au

.

1Aog

= FAO and c o n s i d e r t h e system

$ F(x,t,T,aX);$h

+

,bt K ' u d? + f ( x , t )

we can i n t e g r a t e by p a r t s and g e t , a f t e r l e t t i n g v(x,O)

T , A

Ao$$

We l e t A

k

A + Au +

= Au + Au

-

+ F ( x ,t,t , a x ) v

-&

tFT(x,t

A*=Au.

oat

We n e x t c o n s i d e r t h e change o f v a r i a b l e s

so t h a t u = u ' and v = u ' + v ' .

,?

t

,ax)v(-r)dr +& K ' u d? + f ( x ,t)

= 0,

243

Energy Estimata .for Intcgro-Differential Equations

Then (1 ) becomes :

i

-

-&t

au' = A u ' + ;\u' + F u ' + F v ' A -

oat

A&'=

A % -

oat

n

, this

+,rt f O

T

- &t -FTv'd-r +,(, t K'u'd-r + f

3

oat

u'dT +,rt

O

FT v'd-r - &t K'u'd-c -

f.

system becomes

0 AJUt O

and

oat

-

- Fu' - F v '

[*o where

A

=A?!.-

oat

-

= -Au'

, .

A?!'-!

oat

-FTu'd.r

=

+ i U +,giUdT + F

:]U

a r e o p e r a t o r s o f o r d e r 0.

define

LU = A U

O t

-

AU

[t

Let A =

., - AU

-

:]

and A.

=

(Ao O

'1

and

A.

' .bt l3Ud-r.

We w i l l p r o v e t h e f o l l o w i n g e s t i m a t e f o r L : THEOREM 1 :

F o r e v e r y r e a l s, t h e r e i s a c o n s t a n t C ( s ) independent o f U ( b u t

depending on T) such t h a t (2)

llu(-J)lls

5 c(s)E(lU(-,O)llo

+

0 5 t 11

,$llLU(*,r)jlsdT)l,

f o r U E C' ([O,Tl,C;(n)). Proof: A

0

W e ' l l prove t h e r e s u l t i n a s e r i e s o f steps.

= I and s = 0.

We w i l l f i r s t assume t h a t

We w i l l t h e n p r o v e t h e e s t i m a t e f o r more general Ao, s t i l l under

t h e assumption t h a t s = 0. W e w i l l t h e n use t h e e s t i m a t e t o p r o v e t h e r e s u l t i n t h e g e n e r a l case. Step 1 : d 2 d ~ l l U ( t ) l I O= ( # t ) , U ( t ) ) O Using t h e f a c t t h a t Ak i s symmetric and k = 1,

...,n,

we have

[ ,a-:Vk U

d (U(t)&(t)),.

+

lo

=

,-k::[ - V

lo,

and l e t t i n g A k =

[? :]

R. Grimmer and M. Zeman

244

By Gronwal 1 ’ s inequal it y ,

liu(t)/lo f c

~ ~ u ( o ) l l+o

,t

/lLu(T)~lodTl.

Step 2: We now p r o v e e s t i m a t e ( Z ) , a g a i n f o r t h e case s = 0, f o r LU = A

(3) Since A

*

oat

- AU - AU

is p o s i t i v e - d e f i n i t e , d e f i n e Ail’‘

0 a positive

s p e c t r a o f A. V = Ao1/2U.

=

,[t B^ U ~ T . 0

-

1 6 X-+ 1 / 2 (XI 2 m ‘r

-AO)-’dA,

where l i s

p a t h o f i n t e g r a t i o n such t h a t Re X > 0 and i t c o n t a i n s a l l p o i n t i n i t s interior.

We n o t e t h a t Ail’‘

i s also positive-definite.

Then e q u a t i o n ( 3 ) becomes:

h

where A ‘ i n c l u d e s a l l o f t h e “bounded“ terms.

Hence

By G r o n w a l l ’ s i n e q u a l i t y , we t h e n have

Next, we o b t a i n a s i m i l a r e s t i m a t e f o r L*, t h e f o r m a l a d j o i n t o f L .

Taking

Let

Energj, Estimates for Iritegro-DifferentiulEquations

{U,V}

=

.(,T

245

(U(T),V(T))Od? as t h e L2 s c a l a r p r o d u c t on R n x [O,T],

L*

i s d e f i n e d by

the r e l a t i o n ILU,V}

i f U(x,O) = V(x,T) = 0.

{U,L*Vl

=

au - AU * au t c A av - A*V * cA - ,I' t B ( t , r ) U ( T ) d T , L*V = -A k = l kaXk 0 o a t k = l kaxk T * B * ( T , t ) V ( T ) d T , where A* i s t h e a d j o i n t o f t h e m a t r i x A .

au Since LU = A -

oat

.k

I t i s e v i d e n t t h a t L* i s s i m i l a r i n f o r m t o L and an argument s i m i l a r t o t h e p r o o f

o f Theorem 1 y i e l d s : COROLLARY 1:

F o r e v e r y r e a l s , t h e r e i s a c o n s t a n t C ( s ) independent o f V such t h a t Ilv(t)lls

f o r V E C1([O,Tl,

4.

5 c(s)

{ Ilv(T)(ls +

,kT ~

~ ~ * v ( ~ 0) 5~ t~ 5s T,d ~ ~ ,

C;(R)).

APPLICATIONS OF THE ENERGY ESTIMATES

The use o f energy e s t i m a t e s i s s t a n d a r d i n p r o v i n g well-posedness o f t h e Cauchy problem f o r p a r t i a l d i f f e r e n t i a l e q u a t i o n s .

(See L. Nirenberg, [6], f o r i n s t a n c e . )

One can prove, i n a s i m i l a r manner, t h e we1 1 -posedness o f ( I .E. ) : THEOREM 2: Consider ( I . E . ) . I f g E Hs(Rn) and f E Co([O,T],Hs(Rn)) are given, C 1 ([O,Tl,Hs-,(Rn)). t h e n t h e r e e x i s t s a u n i q u e s o l u t i o n u ( x , t ) E C 0 ([O,T],Hs(Rn))

n

I n a d d i t i o n , t h e energy e s t i m a t e s may be used t o show t h a t t h e s o l u t i o n o f ( I . E . ) has a f i n i t e domain o f dependence.

The domain o f dependence p r o p e r t y i s e q u i v a l e n t

t o t h e p r o p e r t y t h a t a s o l u t i o n propagates w i t h f i n i t e speed. Let

T =

X,,(x,t,E),

j=1,

...,N

n

be t h e r o o t s o f d e t ( T A 0 ( x , t ) -

c

E k A k ( x , t ) ) = 0.

k= 1

We d e f i n e Amax =

max

lXJ(x,t,c)l

( x , t Ed%O,TI 151=1 15j5 N THEOREM 3 : satisfying t

L e t D be t h e i n t e r i o r o f a backward cone i ( x , t ) : I x - x o l = X m a x ( t o - t ) } ,

2

0.

Suppose u i s d e f i n e d i n D s a t i s f y i n g

A au = P ( x , t , a x ) u ( x , t )

oat

i f u(x,t) = 0 f o r x E

then u ( x , t ) Proof:

131.

,(,t B u ( x , s ) d s ,

D

n

It = 01, t

0 in

and

5 0,

D.

The p r o o f i s s i m i l a r t o t h a t used t o p r o v e t h i s r e s u l t f o r p a r t i a l d i f f e r -

e n t i a l equations. Remark:

t

See

s.

Mizohata [5].

Theorem 3 i s a l s o proved by t h e use o f semi-groups by Grimmer and Zeman

R. Grimmer and M. Zeman

246 5.

EXAMPLES

We w i l l p r e s e n t two examples w h i c h can b e p u t i n t o a framework i n which t h e y s a t i s f y t h e c o n d i t i o n s o f Theorems 2 and 3. The f i r s t example d e a l s w i t h t h e g e n e r a l i z e d h e a t e q u a t i o n f o r m u l a t e d by G u r t i n and P i p k i n

[4].

Example 1 :

The g e n e r a l i z e d l i n e a r h e a t e q u a t i o n f o r m u l a t e d by G u r t i n and P i p k i n

is: B(0)Ut(x.t)

+

CUtt

(5)

f

,j_t_B'(t-~)U~(x,~)d~

2 t 2 = a ( o ) v u(x,t) + J m a ' ( t - r ) v U(X,T) + r ' ( x , t ) ,

where c > 0 i s t h e i n s t a n t a n e o u s s p e c i f i c h e a t , u ( x , t ) difference, r ( x , t )

i s t h e temperature

i s t h e h e a t s u p p l y and O ( s ) and a ( s ) a r e t h e energy r e l a t i o n

f u n c t i o n and t h e h e a t f l u x r e l a t i o n f u n c t i o n . It i s assumed t h a t B ( s ) and a f s ) 1 a r e o f c l a s s C and C2, r e s p e c t i v e l y , and t h a t a ( 0 ) > 0. We t r a n s f o r m t h e second o r d e r e q u a t i o n ( 5 ) i n t o a f i r s t o r d e r system. t h e change o f v a r i a b l e s ut

=

vo, u

x1 as e q u a t i o n s , t h e c o m p a t i b i l i t y c o n d i t i o n s

= v3 i n

= v2, u

u

= vl,

x2

We make ( 5 ) and add,

x3

- ( v ~ ) ~ , =] 0, i = 1,2,3.

a ( 0 ) [(viIt Assuming t h e h i s t o r y o f u ( x . t )

i s known, e q u a t i o n ( 5 ) i s then t r a n s f o r m e d i n t o

t h e system where

A0 =

('

0

0

O0

0 o a (00 ) a (00 ) 0 0 0 0 a(0)

1,

0

A1

0 0

0 0 0 0

0 a(0) 0

A 2 = ( a (00 ) 00 0 0

0 0

o0 ) ,

0

0

h

A and K a r e bounded m a t r i c e s , f i s a v e c t o r f u n c t i o n and U = (vo,v1,v2,v3) e a s i l y seen t h a t Ai, B = FA, where F = -

i

=

0,1,2,3,

a r e symmetric and A.

.

Remark:

.

i s positive-definite.

For t h i s example Xmax

t h e p r o p a g a t i o n speed o f t h e s o l u t i o n i s p +

T

It i s

Also

=e .

Hence

.

The f a c t t h a t t h e s o l u t i o n u has f i n i t e p r o p a g a t i o n speed was f i r s t proved

by F i n n and Wheeler [l]under t h e a d d i t i o n a l c o n d i t i o n B ( 0 )

1. 0.

E-nevgj, Estimates f o r It1 tegro-DifyerentialEquations

241

The second example deals with electromagnetic theory f o r an inhomogeneous anisotropic stab1 e di a1 e c t r i c . Example 2: The basic laws underlying the phenomena of electromagnetism a r e given by Maxwell's equations: f

where 6 i s the magnetic induction, H i s the magnetic f i e l d , 0 i s the e l e c t r i c induction, E i s the e l e c t r i c f i e l d , J i s the e l e c t r i c current and G i s an outside force representing charge. In a s t a b l e medium, Ohm's law a p p l i e s : J = oE, where a i s t h e r e s i s t i v i t y .

(7)

We consider t h e c o n s t i t u t i v e r e l a t i o n s which were f i r s t proposed by Volterra [ 7 ] : t 0 = E ( x , t ) E ( x , t ) + ,[,$(x,t,T)E(x,-r)d.r (8) B

(9)

=

p(x,t)H(x,t)

t

+ ,[,+(x,t,T)H(x,T)dT,

where C ( x , t ) , u ( x , t ) , $ ( x , t , - r ) , I ) ( X , t , T ) a r e tensors. e l e c t r i c and magnetic r e l a t i o n t e n s o r s , respectively. a r e p o s i t i v e - d e f i n i t e and t h a t 4 and IJJ a r e C1 in T .

@ and I) represent t h e We require t h a t E and p

Equations ( 6 ) - ( 9 ) lead t o t h e following system f o r U = ( E l ,E2,E3,H1 ,H2,H3)' :

where A.

1

=

[:

;I

'

0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0-1 0 0 0

0 0 0 0

, A =

A and K a r e bounded matrices.

[;: ;A) 0 0 0

0 0-1 0 0 0-1 0 0 0 0 0 0 0 0

1

0

0

0

0

0

[;::w) 0 0 0 0 0 0

=

0

1

0

0

0

0

- 1 0 0 0 0 0 0

In t h i s example B = FA + K, where F

0

0

0

0

0

0.

Remark: Vol t e r r a considered the q u a s i - s t a t i c problem when he proposed the c o n s t i t u t i v e r e l a t i o n s ( 8 ) and ( 9 ) . REFERENCES

[ l ] J.M. Finn and L . T . Wheeler, Wave propagation aspects o f the generalized theory of heat conduction, ZAMP, 23 (1972) 972-940.

[ Z ] K.O. Friedrichs, Symmetric hyperbolic l i n e a r d i f f e r e n t i a l equations, Comm. Pure and Appl. Math., 7 (1954) 345-392.

'

248

R. Grimmer und M. Zernun

[3] R.C. Grimmer and M. Zeman, Wave propagation for linear integrodifferential equations in Banach space, J. Diff. Equations, to appear. [4] M.E. Gurtin and A.C. Pipkin, A generalized theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968) 113-126. [5] S. Mizohata, The Theory of Partial Differential Equations, Cambridge

University Press, London, 1973. [6]

L. Nirenberg, Pseudo-differential operators, Global Analysis, Proc. Symp. Pure Math 16, Am. Math. SOC. Providence, 1970, 149-167.

[7] V. Volterra, Sur les gquations int6gro-differentielles et leurs applications, Acta. Math. 35 (1912) 295-355.

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

249

RESOLVENT AND HEAT KERNELS FOR OPERATORS OF SCHRODINGER TYPE WITH APPLICATIONS TO SPECTRAL THEORY David

Gurarie

Mathematics Department Case Western Reserve University Cleveland, Ohio 44106. Dedicated to Yu. I. Ljubich on his 50-th birthday. INTRODUCTION In this paper we shall discuss properties of kernels 'it; = ( 5 - A1-l (resolvent) and U = e-tA (semigroup) and related problems of spectral theory for elriptic operators A . on En and certain of their singular perturbations A = A + B (order B 5 order A ) , modeled - A + V(x). Typical proglems that after Schrodinger operators: arise here are (I) construction of kernels ( 5

-

A) -1

and

e-tA

(11) estimates of kernels (111) asymptotic expansions of kernels in terms of parameters 5 and t. In solving problem I we first construct and study free (unperturbed) resolvent Ro = ( 5 - AO)-l using psedodifferential calculus. Then we 5 relate R = ( 5 - A)-1 to Ro via the perturbation series 5 5

The convergence of ( 1 ) as well as other properties of R depend on the operator BRo = B(5 - A 0 ) - ' . We give conditions for LP-boundedness or compactness of BRo ( in other words "A -boundedness" or "Ao0 compactness" of B 1 and derive explicite estimate of the operator 0

norm 1/BR 11 in terms of 5 E a: (Theorem l).This Theorem generalizes earlier known results [ R S ] ,[Schl , [We]. We also prove that in some cases the resolvent R_(x,y) and other related kernels admit radial convolution-type bouhd

Function H h a s local singularity of Green’s function and decay at 00 which depends on regularity properties of the leading symbol of A, a(x,5), at 5 = 0. Precisely,

D. Gurarie

250

Lr , hence LP-operator norm of R is dominated Er
En

1

As a consequence of Theorems 1 and 2 we obtain a variety of results including LP-closedness of A, bounds on its LP-spectrum, essential selfadjointness, accretivity, summability, semigroup generation, etc. In the case of A -compact perturbation B we get a "fine" structure of LP-spectrum o f o A. Namely, (a) IJ (A) = IJ e ( A ) uIJd(A) consists of . essential and discrete spectra: (b) Ue(A) is identical f o r A and A (spectral stability) and equal to the range of the leading symbol of A . (cf. [Sch], [We]); (c) IJd(A) consists of a discrete set of m eigenvalues {A . I . of finite multiplicity, accumulating to IJ e (AO I 1=1 (cf.[Ag21 , [Kul ) . We also show that whole spectrum (essential for all LP.

+ discrete) is identical

Finally in section 4 we apply our calculus to the problem of asymptotic of spectral function N(A) = #{eigenvalues (A3 for selfadjoint operators of Schrodinger type: A = A + V(x) with potential V(x)+m 0 The classical "volume counting" conjecture of H. Weyl asserts that for large X

.

N(A)

2

Val{ (x,E)I "symbol A"5A - 1

=

i(X-V(X)

-n/m

P(X)

dx

(4)

where ~ ( x =) vol{c: a(x,c) 5 11. This problem was extensively studied in different contexts and settings ([Ag 11, [Me], [Ro], [RSI ,[Sj], [Fll to mention only a few). In particular a recent paper [Fll by J. Fleckinger proved [4] for Schrgdinger-type opgrators on unbounded domains of Rn. We reproduce this result on Q by a different method. The method of [Fl] was further elaboration of classical "minimax technics" (cf. [RS]) along with known "local asymptotics". Our method is based on study of heat-kernel (semigroup) U = exp(-tA), which in its turn is obtained from RS by Cauchy integrition

First we show that the operator U is of trace-class and derive "small time" asymptotic of tr U t= "Laplace transform of N ( A ) " . Then by Karamata’s Tauberian Theorem twe go back to asymptotic of N ( A ) for large A. Most Of results of sections 1-3 appeared in our papers [GK],[Gu 1-31. Results announced in section 4 will appear elsewhere. 1. CLASSES OF OPERATORS.

We consider elliptic operators A = A + B on Illn, whose leading part A 0 belongs to one of the following clgsses:

Resolvent and Heat Kernels

25 1

(Al) constant-coefficient pseudodifferential operators ($DO’S) with m symbol a(6) E S 1 , o l or uniformly elliptic differential operators A. = 1 aa(x) Da. The latter means that the leading symbol a ( x , ~ )= IallTL 1 aa(x)ca satisfies: cl1c1m 2 a(x,c) -5 c"lclm uniformly in x E R". For the sake of argument coefficients a (x) are assumed to be smooth and bounded with sufficient numbep of their derivatives, But in fact by Theorem 1 below one can proceed to limit in L -norm, i.e. allow all continuous coefficients a a (A2)"conformaldilations" of type 1 operators, A , = p(x)Ao . The dilating factor p is subject to so called "finite propagation speed condition" ( [Gu 23 )

.

I

P(") (XI I

c pl-lal/m for all partial derivatives p ( " ) , la1 5 m. Condition some limitatioas on the rate of growth of p(x) at not exceed 1x1 .

m

(6) ( 6 ) imposes

, which can

=I

Perturbations B to be considered, B ba (x)D", have "singular" r a coefficients ba LlOc. We require the number d

=

d

a

=

r

+ l a ] (a

"fractional order" of the a-th term)be 5 m = order of A

(7) 0 In particular, top-order coefficients {b 1 must be locally bounded (Lyoc). One also needs some limibk;; on the --behavior of b - s . In the first case (Al) we require a r : b Lm (Bl) baEL '+ Lm, i.e. b = b’ + b",where bkE Lr, E a a In the second case certain growth of ba at 00 can be allowed, depending on p. Precisely, r (B2) baE Lw, + LG -the sum of weighted LP-spaces with weights w’= P

-da/m and w

= p-

Ia

By weighted LP-norm we mean ( 1Ifw Ipdx)? ' W

Conditions (Bl) and (B2) will be shown to imply Ao-boundedness of-B. For A 0 -compactness one needs a stronger assumpt on. Namely, the L m space in all definitions above must be, replaced by a subspace L of all functions vanishing at m , {fE L :suplf (x) -to, as r+-} in ?he m IxI>r L; : wfE L ~ Iin the second case. first case, and by Lo,w = Examples of such operators are 1) Schrodinger operators : - A + V(x) , V E Lr+Lw, r =<1. 2 2) A = p(x) (-A)m +"lower order terms", or p(x) (1-A) '+. . (fractional S ) 3) Laplace-Beltrami operators on "asymptotically Euclidean" manifolds.

.

Let us notice that both classes of operators (Al)-(Bl) and (A2)-(B2) are closed under conjugation, in particular, include all symmetric operators with leading symbol a(x,c).

252 2.

D. Gurarie

MAIN THEOREMS.

Now we can state the following Theorem 1: Under t h e a b o v e a s s u m p t i o n s ( A l ) - ( B l ) o r ( A 2 ) - ( B 2 1 t h e o p e r a t o r B (<-A~)-’ i s L P - b o u n d e d f o r "L" - c o n d i t i o n " and LP-compact Ip=l m u s t b e e x c l u d e d i n t h e s e c o n d c a s e ) , and a l l 5 i n a c o m p l e m e n t o f a p a r a b o Zic r e g i o n & b o u t p o s i t i v e r e a l a x i s i n E. M o r e o v e r , t h e o p e r a t o r

for

- c o n d i t i o n " i n a l l L P - s p a c e s , lLp<min{ra}

norm

Here d = max d =max(n/ra+lal) and c o n s t a n t C=C,l a d e p e n d i n g o n p and t h e l e a d i n g s y m b o l a(x,c).

(lba/l w i t h Co

Different cases of Theorem 1 were proved in [GK] and [Gu 1 - 3 1 . We shall outline some basic ideas. 1) Free resolvent

R!

=

-1

is constructed via $D-calculus. 1 Namely, we consider a $DO K = $(x,D) with symbol $(x,c) = 5-a (XIS1 in the class S-m ( [ H o ] ) , whose kernel 110 K(x,z) =

(c-AO)

]$(x,c)eiS’’

dl; z = x-y

(9)

For constant-coefficient A. a convolution kernel K = K(x-y) is itself a resolvent. For variable-coefficient A. R factors into the product 03 5 0 -1 R = K ( I - L) = K 1 Lk (10) 5 1 DB (-a)a’((l). Using symbol class where L = L has symbol cp= 1O = x 5-a 5 B! 5 estimates for $ and I$ and radial bounds ( 3 ) for kernels K and L , one can show that (10) converges absolutely in the complement of the parabolic region determined by the right hand side of ( 8 1 , ie C ={c, = re : r-’ 21 (see [GK] [Gu] for details). 1 sine/2 I n+l 2 ) Operator M=b (x)DaRo :boundedness and compactness. Operator M is a product of multiplication with function b(x)sLr and a $DO K =$ (x,D) c1 a . S ~ ,(we ~ can ignore the second factor (I-L)-’ in R ) . (x,s) E -9, Depending on whether L= m -la1 is positive or zero K is either Calderon-Zygmund kernel (CZ) or admits radial convolutign-type bound H

S

( 3 ) with s = n-R. In particular, H s s

Lq (lzq
-

25 3

Resolvent and Heat Kernels

C o n s t a n t C i n b o t h cases depends on a f i n i t e number o f symbol c l a s s This e s t i m a t e a l s o e x p l a i n s n e c e s s i t y of c o n d i t i o n ( 7 ) seminorms

.

r on L -class

o f b a , and t h e l i m i t a t i o n on L P - s c a l e i n Theorem 1.

Indeed, i n o r d e r f o r c a u s e d by

bH t o b e LP-bounded t h e " s h i f t " i n LP-scale

1 < r =

-

= m-lal. n n lest t h e convolution t e r m H

b must b e "compensated" by

H,

On t h e o t h e r hand w e c a n n o t t a k e p > r

i.e.

1

w i l l "push" LP-class o u t of t h e i n t e r v a l [ 0 , 1 ] . 0

Compactness of k e r n e l s b and

BR f o l l o w s from e s t i m a t e s H (see [ S c h l , [ G u 3 1 ) .

3 ) Conformal d i l a t i o n s A.

(11) by " c u t t i n g o f f "

= p ( x ) A O . The argument

One c a n e x p r e s s k e r n e l s K and K

2

is t r i c k i e r h e r e .

i n terms o f s o c a l l e d p - d i l a t i o n s A

h

a

h

<-a(:,<)

o f " u n i f o r m 1 y e l l i p t i c " k e r n e l s K and K =+ x,D) = (see [Gu2] ) -n/m -l/n a L4 K(x,z) = p ( x ) K(x, p ( x ) 2 ) . S i n c e p (x)+m t h e r e i s no l o n g e r a s i n g l e c o n v o l u t i o n - t y p e bound f o r K . I n s t e a d , m a x i m a l - f u n c t i o n t e c h n i c s c a n be u s e d ( [ G u 2 1 ) . A

S i m i l a r argument a p p l i e d t o e a c h t e r m of series (10) a n d (1) y i e l d s r a d i a l bounds of r e s o l v e n t .

, [Gu 1 1 )

For c o n s t a n t - c o e ~ f i c i e n t - u n i ~ o r m l ey l l i p t i c -1 A. and I B l l - t y p e p e r t u r b a t i o n B t h e r e s o l v e n t k e r n e l R = ( < - A ) 5 i s e s t i m a t e d a s follovs Theorem 2 ( [GK]

:

H (r1lml x-yl ) ; 5 = re i e (12) r -l+n/m w i t h r a d i a l f u n c t i o n H = Hs ( 3 ) , w h e r e s = n-m, y = y o l s i n e / m l . IR5 (x,Y)

I 5

C

*Y

Notice t h a t q u a l i t a t i v e l y e s t i m a t e ( l 2 ) is t h e b e s t possible. f o r t h e L a p l a c e o p e r a t o r on R3 k e r n e l R = ( < + A ) - ' 5 1.Cy7-e i f i l x - y ! i.e. y=lsine/21.

Indeed

i s computed

e x p l i c i t e l y : R5 =

3. COROLLARIES OF THE M A I N THEOREM. W e s h a l l l i s t some c o r o l l a r i e s o f Theorems 1 and 2.

C1: O p e r a t o r s A s p e c t r u m of A

i8

and A = A. + B a r e L P - c l o s e d 1 'p c o n t a i n e d i n R.

D (B)r>D ( A o ) = D ( A ) P P P bolev space with weight p ( [ G u 1-23). C2:

LP-domains:

=

LPfm P

-

< -m i n ( r } . The Lp-

t h e m-th w e i g h t e d So-

C3: A c c r e t i v i t y : f o r any s m a l l E > O t h e r e e x i s t s C s u c h t h a t llBfllp < E I I A g f l / p + CIlfllp C4:

I

f

ED(A0).

E s s e n t i a l s e l f a d j o i n t n e s s ( c f . [Co]

A = A

0

+

B

,[Du], [Ka] , [RS]) :

i f operator

of Theorem 1 i s f o r m a l l y s y m m e t r i c t h e n A i s e s s e n t i a l l y

D. Gurarie

254 m

s e l f a d j o i n t on C o . C5: R e s o l v e n t s u m m a b i l i t y ( [GX], [Gu 1-21) :

R f (x)

-+

5

f (x) i n Lp-norm

and a.e. as 5+muniformly i n any s e c t o r { l a r g 5 / 28> 01. I n t h e p r o o f o f C5 r a d i a l bound o f Theorem 2 i s e s s e n t i a l . From t h e l a s t c o r o l l a r y by Cauchy i n t e g r a t i o n w e g e t a v a r i e t y of o t h e r " a n a l y t i c m u l t i p l i e r s " I$ ( A ) and "summation f a m i l i e s " { QJA)} Among them i s s e m i g r o u p ( h e a t ) k e r n e l Ut

= e

-tA -

-

1 5

dy ;

/,e-t5(<-A)-1

I$ = I$t = e-tc, ( 1 3 )

One c a n show Theorem 3 ( [GK] , [Gu 1 - 2 1 ) : S e m i g r o u p U = e-tA i s h o l o m o r p h i c i n t h e

t

r i g h t h a l f pZane Re t > O ,

a d m i t s r a d i a Z bound ( 1 2 ) and i s s t r o n g l y

Utf(x) + f ( x )

c o n t i n u o u s at t = O ,

i n LP-norm a n d a . e .

a s t +O.

So f a r w e Qave u s e d o n l y t h e f i r s t s t a t e m e n t o f Theorem 1, boundedn e s s o f BR The s e c o n d s t a t e m e n t ( c o m p a c t n e s s ) h a s a l s o i m p o r t a n t i m p l i c a t i o n i n s p e c t r a l t h e o r y . Namely,

.

Theorem 4 ( c f . [Schl , [We] , [Gu 3 1 ) : U n d e r L m - h y p o t h e s e o f T h e o r e m I i t 0

foolows

a ) t h e d i f f e r e n c e of

two r e s o l v e n t s R

5

-

Ro = R B R

0

5

5 5 i s compact; b ) t h e L p s p e c t r u m of A , a ( A ) , d i f f e r s from o ( A o ) b y a m d i s c r e t e s e t of e i g e n v a l u e s o f f i n i t e m u Z t i p Z i c i t y I A j I l accumulating t o

u(Ao).

The s t r u c t u r e o f c o n t i n u o u s s p e c t r u m i s w e l l known f o r c o n s t a n t - c o e f f i c i e n t o p e r a t o r s (see, f o r i n s t a n c e , [ S c h l ) : (5 ( A o ) = r a n g e of a ( E ) . By Theorem 4 it f o l l o w s t h a t oe(A) is s t a b l e und& s m a l l perturbat i o n s B. One a l s o h a s a n o t h e r form o f s t a b i l i t y Theorem 5 ( [ G u 3 1 ) : T h e whoZe L p s p e c t r u m of A ( c o n t i n u o u s

+

discrete)

i s i d e n t i c a Z f o r a l l 1 5 p 5 min{rc,}. 4 . ASYMPTOTIC OF SPECTRAL FUNCTION.

I n t h i s s e c t i o n w e s h a l l a p p l y r e s u l t s of s e c t i o n s 1-3 t o s t u d y s p e c t r a l f u n c t i o n of a s e l f a d j o i n t S c r o d i n g e r - t y p e o p e r a t o r A = A + V ( x ) W e assume t h e l e a d i n g p a r t A. t o b e of AZ-type, A. = p (x)Ao Oand p o t e n t i a l V ( x ) t o go t o m as x *a

.

L e t us introduce t h e following notations:

a ) V C ( x ) = min{c; V(x)} b ) V c ( x ) = VC ( x )

-

-

c; and

t h e cut-off Ac = A.

+

V a t t h e l e v e l c >O.

of Vc.

c ) ~ ( x =) v o l { < : a ( x , c ) c 11 = p -n/m ( x ) f i ( x ) ; $ ( x ) c o r r e s p o n d i n g t o t h e u n i f o r m l y e l l i p t i c symbol 2 ( x , < ) . / V c ( x ) dx b e i n g f i n i t e f o r a l l c > O , t h e oper a t o r A h a s d i s c r e t e s p e c t r u m , e q u i v a l e n t l y t h e r e s o l v e n t R i s comp a c t . I n d e e d , t h e f a m i l y o f L 2 - o p e r a t o r s AT+ A a s c+ +m 5. On t h e Of c o u r s e , o t h e r hand by Theorem 4 ( b ) : u ( Ac) = { h . 3 [ c , +m) 7 Notice t h a t i n t e g r a l s

u

.

255

Resolvent and Heat Kernels N(X;A) 5 - N(X;Ac)

i s f i n i t e f o r any h > O f h e n c e o ( A ) = od(A)

1 1 i s p u r e l y d i s c r e t e , and A .+ +-. 3 A s t r o n g e r a s s u m p t i o n on V ( x ) i s r e q u i r e d i n o r d e r t o h a v e t r a c e c l a s s k e r n e l s R and Ut. Namely ( c f . [Fl]) , 5 h ( s ) = I v ( x ) < s P -n/m dx = O ( s y ) , y>O, o r e q u i v a l e n t l y h(s) <

c

h ( s / 2 ) f o r some C > 0 .

(14)

W e have t h e f o l l o w i n g

Theorem 6 : For l a r g e

A

JI

I

( A ) 2. a ( x ,5 ) +v ( x ) < A d t d x = [ A - V ( X ) I " ' ~ ~ ( x ) dx, A s y m p t o t i c e q u i v a l e n c e i n ( 1 5 1 means t h a t N ( A ) / J J

...

(15) = 1

+

o(1). = e- t A

L e t u s o u t l i n e t h e p r o o f . F i r s t w e c o n s i d e r t h e h e a t k e r n e l Ut and d e r i v e a " s m a l l - t i m e " a s y m p t o t i c o f t r Ut (t > O )

tr

ut - r ( g

I

+1) t -n/m

m

.-tV(x)

P(X)

(16)

dx-

Observe t h a t i n t e g r a l ( 1 6 ) i s f i n i t e f o r a l l t > O by o u r a s s u m p t i o n ( 1 4 ) . To prove (16) w e c a n n o t a p p l y o u r c a l c u l u s d i r e c t l y t o A = A. + V , a s p e r t u r b a t i o n V i s n o t Ao-bounded.So w e " a p p r o x i m a t e " A by c u t - o f f ' s

But

Ut

and l e t c go t o

Ac = Ao+VC

i s no l o n g e r of t r a c e - c l a s s ,

spectrum [c;+m).

since

m.

For

Ac w e c a n w r i t e

Ac has a c o n t i n u o u s

W e s u b t r u c t t h e "continuous p a r t "

e

-t(Ao+c)

Of

and w r i t e t h e d i f f e r e n c e u s i n g p e r t u r b a t i o n series (1) a s f o l l o w s

ut

W e n e e d t o f i n d a s y m p t o t i c o f e a c h summond i n ( 1 7 ) . A s i n Theorem 1 0 t h e r e s o l v e n t R c a n b e r e p l a c e d by a YDO K = K = $ ( X , D ) , $ =

1

5

.

Then t h e k - t h t e r m o f ( 1 7 ) becomes t h e p r o d u c t KVK...VK. <-a (x,5 ) The f o l l o w i n g Lemma i s c r u c i a l f o r o u r argument Lemma 1: L e t K = J,(x,D) w i t h J, =

n e l M(x,y) = M = K V K . . . V

-1-1

2~ii

5-a

and V E C E

.

Consider t h e ker-

ik- t i m e s ) . T h e n

K k

I

e-t5 M 5 ( x , y ) dr, = (- t ) V ( X ) ~ e-t a ( x f 5 ) d c x ( l + 0(tlim) k!

The p r o o f of Lemma 1 i s b a s e d o n s y m b o l i c c a l c u l u s ( a s y m p t o t i c expan. K k , where K . = @ . ( x , D ) , and @ . = s t o n of p r o d u c t s of +DO’S K K1 3 3 7 ) T a y l o r e x p a n s i o n of V ( x ) and e s t i m a t e s o f k e r n e l s s i m i l a r t o D!J, Theorems 1 and 2 .

..

A f t e r Lemma 1 a s y m p t o t i c ( 1 6 ) e a s i l y f o l l o w s . I n d e e d s u b s t i t u t i n g ( 1 8 ) i n t o (17) a n d summing it up g i v e s

D.Gurarie

256

Letting c go to m and evaluating trace of both sides of (19) we arrive at the formula tr e-tA = lim tr

. ..

=

J/

e-t(V(x)+a(xlc))dgdx x ( +l o(1))

C

which easily reduces to (16) by integrating the 6-variable. Let us a l s o observe that (19) implies polynomial growth of N(X;A ), consequently of N (A ;A). Thus conditions of Karamata’s TauberianC Theorem (see [Sj]) are satisfied and we can proceed from asymptotic of heat kernel (16) to asymptotic of spectral function (15), q.e.d. REFERENCES. [Ag 11 Agmon,S. Lectures on elliptic boundary value problems (Van Nostand, 1965). [Ag 21 Agmon,S. Spectral properties of Schrodinger operators and scattering theory, Ann-Scuola Norm.Sup. Pisa,ll,2 (1975) 151-218. [Co] Cordes H 0. A global parametrix for pseudodifferential operators over En with applications, Preprint. [Du] Dung,N.X. Essential selfadjointness and selfadjointness for even order elliptic operators, Preprint. [GK] Gurarie,D. and Kon,M. Radial bounds for perturbations of elliptic operators, J.Funct.Ana1. (to appear). [Gu 13 Gurarie,D. Kernels of elliptic operators: bounds and s m a b i lity, J.Diff.Eq.(to appear). [Gu 21 Gurarie,D. Finite propagation speed, maximal function and kernels of strictly elliptic operators, Preprint,l983.

[Gu 31 Gurarie,D. On LP-spectrum of elliptic operators, Preprint,l983. [Ho] Hormander,L. Pseudodifferential operators, Comm.Pure Appl.Math. 18 (1965) 501-513. [Kal Kato,T. Remarks on the selfadjointness problem and related problems for differential operators, in Knowles,T.W. and Lewis, R.I. (eds.) Spectral theory of differential operators(North Holland, Amsterdam,1981). [Ku] Kuroda,S. Scattering theory for differential operators, I,II, J.Math.Soc.Japan 25 (1973) 75-104; 222-234. [Me] Melrose,R. Weyl’s conjecture for manifolds with concave boundary Proc.Symp.Pure Math.26 (1980) 257-274. [Ro] Rosenblum,G.V. Distribution of the discrete spectrum of singular differential operator, Sov.Math.(Izv. VUZ) 20 (1976) 63-71. [Schl Schechter,M. Spectra of partial differential operators (NorthHolland, 1971). [ S j l Sjostrand,J. On the eigenvalues of a class of hypoelliptic operators IV, Ann.Inst.Fourier , XXX-2 (1980) 109-169. [We] Weder,R. The unified approach to spectral analysis, I, Comm. Math.Phys.60 (1980) 291-299; 11, Proc.AMS 75,l (1979) 81-84. [Fl] Fleckinger, J. Distribution of the eigenvalues of operators of Schrodinger type, in Knowles, I.W. and Lewis, R.I.(eds) Spectral theory of differential operators (North Holland,Amsterdam,l981).

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

251

VORTICITY, INCOMPRESSIBILITY, AND BOUNDARY CONDITIONS I N THE NUMERICAL SOLUTION OF THE NAVIER-STOKES EQUATIONS* Karl Gustafson Kadosa H a l a s i U n i v e r s i t y of Colorado B o u l d e r , C o l o r a d o USA

We examine t h e e f f e c t s of i n c o m p r e s s i b i l i t y and b o u n d a r y c o n d i t i o n s on v o r t i c i t y d e v e l o p m e n t f o r t h e N a v i e r - S t o k e s equations. These a r e i l l u s t r a t e d by n u m e r i c a l r e s u l t s f o r driven c a v i t y flows.

1.

INTRODUCTION

I t is p e r h a p s c o n s t r u c t i v e , e s p e c i a l l y w i t h i n t h e c o n t e x t of t h i s I n t e r n a t i o n a l C o n f e r e n c e on D i f f e r e n t i a l E q u a t i o n s w i t h i t s b r o a d s c o p e y e t l e a n i n g t o some p r e p o n d e r a n c e of t r e a t m e n t of d i f f e r e n t i a l e q u a t i o n s from quantum m e c h a n i c s , t o c o n t r a s t t h e l a t t e r t o t h o s e from f l u i d m e c h a n i c s :

r’ BAS I CS

QUANTUM MECHANICS

FLUID MECHANICS

Equations Properties

Schrodinger Equation Linear Whole s p a c e domain S p e c t r a l Theory F o u r i e r Transform

Navier-Stokes Equation Nonlinear C o n f i n e d s p a c e domain B i f u r c a t i o n Theory Numerical A n a l y s i s

Mathematics Methods

I

Table 1 Contrasted Contexts

Although Table 1 o v e r s i m p l i f i e s th e s i t u a t i o n , t he bottom l i n e is: numerical methods n e c e s s a r i l y come t o t h e f o r e when o n e is d e a l i n g w i t h p r o b l e m s from f l u i d dynamics. T h e r e is a v a s t l i t e r a t u r e on such t h e o r y and r e s u l t s and we s h a l l n o t a t t e m p t t o s u r v e y i t . Our o b j e c t i v e h e r e i s t o g i v e some p r e l i m i n a r y r e s u l t s o f a s t u d y o f i n c o m p r e s s i b l e f l u i d f l o w i n bounded t w o d i m e n s i o n a l r e g i o n s , i n c l u d i n g ( a ) v o r t i c i t y development, ( b ) n e c e s s i t y , or l a c k of i t , of a d h e r i n g c l o s e l y t o t h e incompressibility condition, ( c ) r o l e of t h e p r e s s u r e g r a d i e n t i n t h e e v o l u t i o n of t h e f l o w , ( d ) c r i t e r i a f o r d e t e r m i n i n g “ s t e a d y s t a t e ” as l i m i t o f t i m e - e v o l v e d systems.

I n p a r t i c u l a r we g i v e n u m e r i c a l r e s u l t s f o r a d r i v e n c a v i t y flow. T h e s e i n c l u d e n o r m a l i z e d v e l o c i t i e s , p r e s s u r e g r a d i e n t s , k i n e t i c e n e r g y c o n t o u r s and g r a d i e n t s . F u r t h e r r e s u l t s and a n e x t e n s i v e r e l e v a n t b i b l i o g r a p h y w i l l a p p e a r i n a p a p e r i n preparation [l]

.

K. Custafson and K. Halasi

258 2.

NUMERICAL (NAVIER-STOKES) EQUATIONS

The N a v i e r - S t o k e s e q u a t i o n s i n p r i m i t i v e v a r i a b l e s ( v e l o c i t y , p r e s s u r e ) f o r v i s c o u s i n c o m p r e s s i b l e f l o w i n a domain 61 a r e v t = ( l / R e ) A v - (v*V)v

(1) (2)

V'V

-

,

Vp

,

= 0

where v i s t h e v e l o c i t y f i e l d , Re i s t h e R e y n l d s number, d i s c r e t i z e d for numerical a n a l y s i s according t o

p

the pressure, a r e

V * ++ D , V H G , A + + L ,

where t h e r i g h t hand s i d e c o n n o t e s t h e d i s c r e t i z e d form. We m e n t i o n t h a t f o r i r r e g u l a r regions Q there are i n t e r s t i n g questions concerning c o r r e c t f a c t o r i z a t i o n s L = DG n e a r t h e boundary , b u t we w i l l n o t go i n t o t h o s e here; see [1,2,3,4] As c o n c e r n s t h e p r e s e n t d i s c u s s i o n , we u s e d c e n t e r e d d i f f e r e n c e s on a MAC mesh ( i . e . , p r e s s u r e s c a l c u l a t e d a t c e l l c e n t e r s , v e l o c i t i e s a t c e l l mid-sides). The d i s c r e t i z e d q u a t i o n s become

.

V"+l

(1')

+ At[F(Vn) - G ( P n + l ) ]

= V"

D(Vn+l) = 0

(2') where F(Vn) = The n u m e r i c a l i n t e g r a t i o n of (1")

(Vn*G)Vn + ( l / R e ) L ( V n )

-

(1') (2')

.

was a c c o m p l i s h e d by f i r s t p r e d i c t i n g

= Vn + AtF(Vn)

Vn+l

a t the (PI)

-

n + l s t t i m e s t e p and t h e n c o r r e c t i n g Vn+1 = ;n+l - AtG(pn+I)

i n terms of t h e p r e s s u r e g r a d i e n t

(3)

G(PnC1)

,

where

Pn+l

is found from

D G ( P ~ + =~ )D F ( V ~ )

was made t o v a n i s h at t h e The l a t t e r e q u a t i o n r e s u l t s from ( 1 ' ) b e c a u s e DV" n t h time s t e p . T h i s then guaranteed t h a t th e equat i on ( 2 ' ) a l s o holds a t the n + l s t time s t e p . [ 4 , 1 ] ) shows t h e i m p o r t a n c e f o r i n c o m p r e s s i b l e flow o f r e q u i r i n g t h a t t h e flow v e l o c i t y f i e l d have v a n i s h i n g normal components at t h e boundary 861 T h a t amounts t o a p p l y i n g t h e n * o p e r a t o r t o t h e i n t e r i o r momentum e q u a t i o n (1) and c o n t i n u i n g i t t o t h e b o u n d a r y , from which An a n a l y s i s ( s e e

.

(4)

- n*G(Pn+l)

= n*\(Vn*G)Vn]

.

F i c t i t i o u s boundary v e l o c i t i e s a r e a s s i g n e d , i n t e r m s of i n t e r i o r v e l o c i t i e s a d j a c e n t t o t h e b o u n d a r y , s u c h t h a t one m a i n t a i n s d i v e r g e n c e - f r e e n e s s i n b o t h t h e i n t e r i o r and e x t e r i o r c e l l s a l o n g t h e b o u n d a r y .

3.

NUMERICAL TESTS AND EXPERIMENTS

The p r e s s u r e e q u a t i o n ( 3 ) must b e s o l v e d a t each t i m e s t e p . Hence one n e e d s f a s t P o i s s o n s o l v e r s . A number of t h e s e a r e a v a i l a b l e , e . g . , t h e FACRCR) s o l v e r s , t h e C a p a c i t a n c e M a t r i x methods s u c h a s t h o s e of P r o s k u r o s k i - W i d l u n d , and o t h e r s . For i r r e g u l a r domains one e n c o u n t e r s some i n t e r e s t i n g d i f f i c u l t i e s i n r e f o r m u l a t i n g t h e d i s c r e t e o p e r a t o r s a t t h e boundary i n a t t e m p t i n g c o n s i s t e n c y between i n c o m p r e s s i b i l i t y and t h o s e s o l v e r s . However, f o r r e c t a n g u l a r c a v i t y f l o w

Navirr- Sto kes Equations

259

s u c h p r o b l e m s do n o t a r i s e . We employed t h e BLKTRI ( c o m p l e t e c y c l i c r e d u c t i o n , b l o c k t r i d i a g o n a l ) s o l v e r from t h e NCAR FISHPAK s o f t w a r e . F i g u r e s A1 t h r o u g h A4 show t h e r e s u l t s o f t h e d r i v e n c a v i t y f l o w a t t i m e s t e p 500 The s i d e w a l l and bottom i n i t i a l v e l o c i t i e s were z e r o , t h e t o p l i d s l i d e s , the fluid within the cavity, u n i f o r m u l y t o t h e l e f t w i t h v e l o c i t y (-1,O) i n i t i a l l y a t r e s t , q u i c k l y d e v e l o p s a p r i m a r y v o r t e x and ( s e e t h e d i s c u s s i o n below) a l s o v e r y s h o r t l y t h e r e a f t e r d e v e l o p s t h e s e c o n d a r y v o r t i c e s shown. The q u a n t i t i e s h a v e been n o r m a l i z e d t o e m p h a s i z e t h e v o r t i c i t y p a t t e r n s r a t h e r t h a n a m p l i t u d e s . The Reynolds number was Re = 400 , a 32x32 g r i d was u s e d , and t i m e s t e p A t = 0 . 0 4 a p p e a r e d s t a b l e . In F i g u r e s Al-A4, i n c o m p r e s s i b i l i t y was m a i n t a i n e d , and a t t i m e s t e p 500 one h a s a c h i e v e d

.

-

I I V ~ vn+lii,

-< 0.8

x 10-4

.

F i g u r e s B1 t h r o u g h B4 d e p i c t t h e same s i t u a t i o n b u t w i t h i n c o m p r e s s i b i l i t y d i s r u p t e d t h r o u g h t h e mechanism o f a p e r t u r b e d p r e s s u r e e q u a t i o n (3' )

DG(Pn+')

= DF(Vn) +

ET

where E r = O.Ol(x + y ) on t h e mesh, 0 < x < 1 0 < y < 1 . Tde e f f e c t a t t i m e s t e p 500 was, where V,P d e n o t e Too: P5d0 %om-the A-flow and where d e n o t e s t h e c o r r e s p o n d i n g q u a n t i t y from t h e p e r t u r b e d B-flow :

*

(i) (ii) (iii) (iv) (v)

D(V*) V - V* K.E. P - P Slower

O(O.1)

,

globally;

, measured n e a r t h e geometric c e n t e r ; K.E.* = 8 3 . 5 2 - 8 1 . 9 9 = 1.5 , of t h e e n t i r e flow; = 0 ( . 1 ) - O(.l)XIO1l , roughly g l o b a l l y ; convergence. = <0.002,0.0132>

F o r f u r t h e r c o m p a r i s o n s when t h e boundary c o n d i t i o n i s a l s o p e r t u r b e d , s e e

4.

[l]

.

SOME CONCLUSIONS AND DISCUSSION

1 . The g e o m e t r y of t h e f l o w d e v e l o p s e a r l y . I n f i r s t r u n n i n g t h e p r o b l e m we d i d n o t s e e t h i s , f o r t h e v e l o c i t y a m p l i t u d e s a r e v e r y s m a l l away from t h e p r i n c i p a l a c t i o n . The u s e of n o r m a l i z e d q u a n t i t i e s showed s t r i k i n g l y how q u i c k l y t h e v o r t i c i t y p a t t e r n e s t a b l i s h e s i t s e l f . T h i s s h o u l d p e r h a p s h a v e been e x p e c t e d i n a d v a n c e on t h e o r e t i c a l g r o u n d s : v o r t i c i t y d i f f u s e s a p p r o x i m a t e l y a c c o r d i n g t o t h e h e a t e q u a t i o n and t h u s r a t h e r i n s t a n t a n e o u s l y i s d i s p e r s e d from t h e i n i t i a l s t a t e t o a l l p a r t s o f t h e domain even though of s m a l l a m p l i t u d e . 2 . A s i d e from g e n e r a t i n g l a r g e s p u r i o u s p r e s s u r e s , f a i l u r e of e x a c t i n c o m p r e s s i b i l i t y need n o t p l a y a m a j o r r o l e i n t h e e v o l v i n g b a s i c f l o w v i s c o s i t y geometry

.

3 . The p r e s s u r e e q u a t i o n c a n be i g n o r e d a t many i n t e r v e n i n g t i m e s t e p s ( t h i s h a s been o b s e r v e d by o t h e r s t u d i e s ) . 4 . The n o t i o n of " s t e a d y s t a t e " i s p r o b a b l y mesh d e p e n d e n t . We p l a n f u r t h e r e x p e r i m e n t s w i t h f i n e r mesh t o l o c a t e t e r t i a r y and h i g h e r e d d i e s i f p o s s i b l e . On p h y s i c a l g r o u n d s : v o r t i c i t y k e e p s e n t e r i n g t h e s y s t e m and one wonders i f eddy development e v e r s t o p s . 5 . A c e r t a i n n o n l i n e a r dependence between n u m e r i c a l i n s t a b i l i t y and Reynolds number was o b s e r v e d . V o r t i c i t y i s r e a l l y a t h r e e - d i m e n s i o n a l phenomena and f a s t P o i s s o n s o l v e r s f o r t h a t c a s e a r e needed t o p u r s u e t h e s e s t u d i e s .

6.

09Z

Navier-Stokes Equations

F i g u r e A3 K i n e t i c E n e r g y Contours

Figure A4 Normalized K i n e t i c Energy G r a d i e n t

26 1

262

K. Gustafson and K. Haiasi

Figure B1 Nurmalized V e l o c i t y F i e l d

(perturbed)

Figure B2 Normalized Pressure Gradient (perturbed)

Nauier--Sro kes Equations

F i g u r e B3 K i n e t i c Energy C o n t o u r s ( p e r t u r b e d )

F i g u r e B4 N o r m a l i z e d K i n e t i c Energy G r a d i e n t ( p e r t u r b e d )

263

K. Gustafsoii und K. Halasi

2 64 REFERENCES 1. 2.

K. Gustafson and K. Halasi, Aspects of the Driven Cavity Problem, to appear. K. Gustafson, Hybrid fast Poisson solvers for fluid dynamics, Proc. lqth IMACS

E Montreal , (1982), Vol. 1, 233-235.

3.

K. Gustafson, Hybrid fast Poisson solvers for fluid dynamics, transonic flow, solenoidal projection, and the Navier-Stokes equations for general domains, IMACS Transactions on Scientific Computation 1 , North Holland (1983), i 0 pp.

4.

K. Gustafson and K. Halasi, On the divergence-free (i.e., mass conservation, solenoidal) condition in comuutational fluid dvnamics: how imortant is it?. Proc. Third International Conference on Numerical Methods in Laminar and Turbulent Flow, Seattle, August, 1983, to appear.

*Partially supported by NSF MCS 80-12220 and by an NCAR grant of computing resources.

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) @ Elsevier Science Publishers B.V. (North-Holland), 1984

265

ASYMPTOTIC COMPLETENESS FOR FEW BODY SCHRODINGER OPERATORS George A. Hagedorn*

P e t e r A. Perry**

Department o f Mathematics V i r g i n i a Polytechnic I n s t i t u t e and S t a t e U n i v e r s i t y B1 acksburg, V i r g i n i a 24061 U.S.A.

Department o f Mathematics C a l i f o r n i a I n s t i t u t e o f Technology Pasadena, C a l i f o r n i a U.S.A.

91 125

We d i s c u s s a method f o r p r o v i n g a s y m p t o t i c completeness f o r many body quantum mechanics. The main i d e a depends on,an a n a l y s i s of t h e t i m e r a t e s o f decay o f t h e wave f u n c t i o n i n c e r t a i n r e g i o n s o f c o n f i g u r a t i o n space. These r a t e s o f decay depend on t h e s c a t t e r i n g channel t o w h i c h t h e wave f u n c t i o n belongs. The method has been a p p l i e d t o p r o v e asympt o t i c completeness f o r g e n e r i c t h r e e and f o u r body systems w i t h s h o r t range two body p o t e n t i a l s . I n t h i s n o t e we d e s c r i b e a method f o r s t u d y i n g s c a t t e r i n g t h e o r y f o r c e r t a i n N p a r t i c l e Schrodinger o p e r a t o r s . T h i s method has been used t o prove a s y m p t o t i c completeness f o r g e n e r i c p o t e n t i a l s when N 5 4 and t h e space dimension i s n t 3. The d e t a i l e d p r o o f s may be found i n [6,7]. The S c h r o d i n g e r o p e r a t o r f o r N p a r t i c l e s moving i n n dimensions i s

2 Nn on L ( R )

.

Here mi

i s t h e mass o f t h e ith p a r t i c l e ; ai i s t h e n dimensional

L a p l a c i a n i n t h e ith v a r i a b l e ; and t h e p o t e n t i a l energy between p a r t i c l e s i and j 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 V . .(xi - x j ) By removing t h e t r i v i a l c e n t e r o f mass m o t i o n from H ( s e e [16])

1J

.

we o b t a i n

The s c a t t e r i n g t h e o r y f o r H i n v o l v e s t h e a s y m p t o t i c b e h a v i o r o f t h e s o l u t i o n s t o the Schrodinger equation

as

If t h e p o t e n t i a l s V . .(xi - x . ) decrease s u f f i c i e n t l y r a p i d l y as 1J J Ixi - x j l + = , t h e n from p h y s i c a l c u n s i d e r a t i o n s one expects a l l s o l u t i o n s t o ( 1 ) t + ? m .

t o be s u p e r p o s i t i o n s of s o l u t i o n s which have t h e f o l l o w i n g a s y m p t o t i c s f o r e i t h e r t+-or t + - - : -itE 2 (N-l)n $ , where H $ = EJ, and $ E L (R 1. Bound s t a t e s o l u t i o n s , i . e . , e 1. 2.

S c a t t e r i n g s o l u t i o n s , those which a s y m p t o t i c a l l y d e s c r i b e k bound c l u s t e r s o f p a r t i c l e s , w i t h t h e c l u s t e r s a s y m p t o t i c a l l y moving f r e e l y r e l a ,N. t i v e t o one a n o t h e r . Here k = 2,3,

...

The a s y m p t o t i c completeness problem i s t o p r o v e t h a t t h e r e a r e no o t h e r t y p e s of solutions.

266

G. A. Hagedorn and

P.A. Perry

To s t a t e t h i s problem p r e c i s e l y we need some n o t a t i o n . A cZuster decomposition

.

is a p a r t i t i o n o f t h e N p a r t i c l e s i n t o k d i s j o i n t clusters Ci

D = {C.}k

1 1=1

d e f i n e HD = Ho + V D ,

where V D i s t h e sum o f a l l V . . w i t h i and j i n t h e same

c l u s t e r . The H i l b e r t space H = L (R(N-l) O H k @ H(D) so t h a t H = H1 O f f 2 @

.-.

H

D

hl@l

=

@[email protected]@1

1J

')

may be decomposed as

0 1

+ 1 O h 2 0 1 0 . - - @ 1

+

We

...

@1

+ 1 0 1 [email protected].@hk@l + l 0 1 @ 1 0 . - * @ 1 OKD The d u s t e r HumiZtoniun hi corresponds t o t h e energy o f t h e p a r t i c l e s i n c l u s t e r

Ci a l o n e ( i n t h e i r c e n t e r o f mass frame), and KD corresponds t o t h e k i n e t i c energy o f t h e k c e n t e r s o f mass o f t h e c l u s t e r s i n D r e l a t i v e t o one another ( i n t h e c e n t e r o f mass frame f o r a l l N p a r t i c l e s ) . F o r each i = 1,2, ,k we choose e i g e n f u n c t i o n s n . . € H i o f h . so t h a t I n . ) is an orthonormal b a s i s o f 1J 1 i .J t h e subspace o f Hi spanned by t h e e i g e n f u n c t i o n s o f hi. A channeZ CL i s a c l u s t e r

...

decomposition D ( a ) t o g e t h e r w i t h an e i g e n f u n c t i o n ni E { Q . . I f o r each hi. We 1 ,J k d e f i n e E, = 1 E i , where h . n . = Eiqi , and l e t Pa : H - H denote t h e p r o j e c t i o n 1 1 i =1 o n t o a l l v e c t o r s o f t h e f o r m n1 @ n 2 0 0 nk 0 J, , where $ E H(D(a)) is a r b i trary.

L e t Ta = 1 @ 1 0

-..@ 1 @ (KD(,)

-..

+ E a ) so t h a t Ho(,)

Pa = T a p a .

I n n 2 3 dimensions, i f t h e p o t e n t i a l s V . . b e l o n g t o L n ' 2 + E ( R n ) n Ln" 1J

-'(Rn)

t h e n one can prove t h e e x i s t e n c e o f t h e channe2 wuzw opsrutozs

n+ = s t r o n g - l i m e i t H i i t T a t+Tm

01

(see [16 , 181). Any s t a t e $ E H which belongs t o th.e range of no corresponds - 1 tT, t o a + E H ( D ( c x ) ) so t h a t e-itH$ is a s y m p t o t i c t o e (nl@q2@--.@nk@$). -itH Thus, a s t + - , e J, e v o l v e s a s y m p t o t i c a l l y as k f r e e l y moving c l u s t e r s o f p a r t i c l e s w i t h each c l u s t e r i n a bound s t a t e . The c o l l e c t i o n o f a l l channels d e s c r i b e s a l l t h e d i f f e r e n t p o s s i b l e ways i n which t h e p a r t i c l e s can move t o i n f i n i t y as f r e e l y moving bound c l u s t e r s . The ranges o f t h e channel wave ope r a t o r s a r e o r t h o g o n a l ( s e e [16]), so p h y s i c a l i n t u i t i o n suggests t h a t

H = Hbound 0 0 Run R,

a = Hbound 0 0 Run 0;

I f t h i s is t r u e , t h e n t h e s c a t t e r i n g f o r H i s c a l l e d asymptoticuZZy conpZetc.

Roughly speaking, t h e methods we d e s c r i b e here y i e l d a s y m p t o t i c completeness f o r "generic" p o t e n t i a l s V . .

1J

E

L"'

''(lRn)

n L"'

- '(Rn)

when n

2

3 and t h e

number of p a r t i c l e s is N 5 4. F o r d e t a i l e d statements o f theorems see [ 6 , 71. For o t h e r r e s u l t s on m u l t i p a r t i c l e a s y m p t o t i c completeness see [ Z , 3, 4, 5 , 8 , 9, 10, 11,12, 13,17, 19, 201. L e t us now d e s c r i b e o u r p r o o f o f a s y m p t o t i c completeness.

For s i m p l i c i t y

,

267

Asvmptotic Completencss f o r Few Body Schrodinger Operators

N = 3. L e t Pbound denote t h e o r t h o g o n a l p r o j e c A l o c a l Kato smoothness e s t i m a t e o f P e r r y , S i g a l and Simon

r e s t r i c t a t t e n t i o n t o t h e case t i o n o n t o Hbound.

[15] shows t h a t f o r a l l J, i n a dense subset of ( 1

- Pbound) H , one

has

Here p ( x ) i s any bounded f u n c t i o n which decays more r a p i d l y t h a n as 1x1 + a . The e s t i m a t e ( 2 ) i n t u i t i v e l y means t h a t i f JI i s n o t bound, t h e n i t l e a v e s any bounded r e g i o n o f c o n f i g u r a t i o n space a t a c e r t a i n r a t e . From t h i s r e s u l t one can prove t h e e x i s t e n c e o f t h e s t r o n g l i m i t s itH -itH P + = s-lim e (l- 'bound) ti+-

p,

if a i s a two c l u s t e r channel whose channel e i g e n f u n c t i o n qa decays exponen( F o r " g e n e r i c " p o t e n t i a l s which decay l i k e I x I - * - ' a t i n f i n i t y , t h i s tially. decay can be proved [1, 14, 161.) One proves t h e e x i s t e n c e o f P& by u s i n g t h e ( 1 - Pbound) as t h e i n t e g r a l fundamental theorem o f c a l c u l u s t o w r i t e eltH :,-ItH o f i t s d e r i v a t i v e , and t h e n a p p l y i n g t h e P e r r y , Siga , Simon e s t i m a t e (see [6,7 Once one has t h e e x i s t e n c e o f Pd i t i s t h e n n o t h a r d t o show t h a t P i = (a:)* and hence, t h a t P: i s t h e o r t h o g o n a l p r o j e c t i o n o n t o Ran a,'. We c a l l t h i s r e -

,

s u l t two c l u s t e r completeness. S i n c e we a r e s t u d y i n s t h e case N = 3, a l l channels except one a r e two c l u s t e r channels. Asymptotrc completeness f o l l o w s f r o m two c l u s t e r completeness i f t h e ranges of t h e t h r e e c l u s t e r wave o p e r a t o r s a: c o n s i s t of a l l v e c t o r s which a r e o r t h o g o n a l t o Hbound and t h e ranges o f P i , channels.

T h i s subspace i s ( 1

- 0 P:)(1

-

where a ranges o v e r two c l u s t e r Pbound) H = Q'H

t h e channel wave o p e r a t o r s a r e m u t u a l l y o r t h o g o n a l .

,

s i n c e t h e ranges of

To p r o v e Run R i = Q'H

we

f i r s t prove a l o c a l Kato smoothness e s t i m a t e which shows t h a t f o r J, i n a dense

+

subset o f Q H , we have f o r a l l p a i r s i j ,

0

and Q- H . Here p . . i s any bounded f u n c t i o n o f 1J x . which decays f a s t e r t h a n I x . .I-'-' as I x . . I + - , The e s t i m a t e ( 3 ) i s ij 1J 1J proved by a m u l t i p l e c o l l i s i o n expansion and an a p p l i c a t i o n o f t h e P e r r y , S i g a l , Simon e s t i m a t e (see [6,7]). The e s t i m a t e ( 3 ) i n t u i t i v e l y says t h a t i f jl i s o r t h o g o n a l t o t h e bound s t a t e s and a l l two c l u s t e r channels, t h e n i n t h e s t a t e $ , t h e d i s t a n c e s between p a i r s of p a r t i c l e s d i v e r g e s a t a c e r t a i n r a t e . Once t h i s e s t i m a t e i s proved i t i s easy t o prove (see [6,7]) t h a t t h e s t r o n g l i m i t itH0 - i t H s-lim e e Q' (4) t + tm e x i s t s and equals t h e a d j o i n t o f 0;. Since t h e s t r o n g l i m i t ( 4 ) i s a p a r t i a l

A s i m i l a r r e s u l t holds f o r

-m

i s o m e t r y w i t h i n i t i a l subspace c o n s t r u c t i o n , Ran ’Q f o l l o w s immediately.

= (

Q’,

H0

~

i t f o l l o w s t h a t Ran,; -L

0 ~ m

(two c i l s t e r )

~ an a:)~

,

= RanQ'.

Since, by

a s~y m p t o t i c completeness

G. A. Hagedorn and P.A. Perrj,

268

The N = 4 case i s s i m i l a r , although i t i s technically more complicated. the d e t a i l s o f this case s e e [7].

For

REFERENCES Combes, J . - M . and..Thomas, L . , Asymptotic Behavior of Eigenfunctions f o r Multiparticle Schrodinger Operators, Comm. Math. Phys. 34 (1973) 251 - 270. Enss, V.,..Completeness o f Three Body Quantum S c a t t e r i n g , P r e p r i n t , RuhrUniversitat, Bochum. (1982) Faddeev, L.D., Mathematical Aspects of t h e Three Body Problem i n Quantum Scattering Theory, I s r a e l Program f o r S c i e n t i f i c Translations, Jerusalem, 1965. Ginibre, J . and Moulin, M . , Hilbert Space Approach t o the Quantum Mechanical Three Body Problem, Ann. Inst. H . Poincari Sect. A. 21 (1974) 97-145. Hagedorn, G.A., Asxmptotic Completeness f o r Classes of Two, Three, and Four P a r t i c l e Schrodinger Operators, Trans. Amer. Math. SOC. 258 (1980) 1 - 75. Hagedorn,..G.A. and Perry, P . , Asymptotic Completeness f o r Certain ThreeBody Schrodinger Operators, Comm. Pure Appl. Math. 2 (1983) 213 - 232. Hagedorn,..G.A. and Perry, P . , Asymptotic Completeness f o r Certain Four Body Schrodinger Operators ( i n preparation). Howland, J.S., Abstract Stationary Theory o f Multichannel S c a t t e r i n g , J . Functional Anal. 22 (1976) 250 - 282. I o r i o , R. and O'Carroll, M., Asymptotic Completeness f o r M u l t i p a r t i c l e Schrodinger Operators with Weak P o t e n t i a l s , Comm. Math. Phys. 7(1972) 137 - 145. Lavine, R . , Commutators and Scattering Theory, 11. Repulsive I n t e r a c t i o n s , Comm. Math. Phys. 20 (1971) 301 -323. Loss, M . and S i g a l , I.M., The Three Body Problem w i t h Threshold Singularit i e s , preprint E . T . H . , Zurich, 1982. Mourre, E . , Application de l a mithode de Lavine au Problime a Trois Corps, Ann. Inst. H . Poincar6 Sect. A. 3 (1977) 219-262. Mourre, E. Operateurs Conjug& e t Prop!&& de Propagation, 11. Remarques sur l a ComplEtude Asymptotique des Systemes a Trois Corps, p r e p r i n t CNRSLuminy 1982. O'Connor, A.J., Exponential Decay of Bound S t a t e Wave Functions, Comm. Math. Phys. 32 (1973) 319 - 340. Perry, P . , S i g a l , I.M., and Simon, B., Spectral Analysis of N-Body Schrodinger Operators, A n n . Math. 114 (1981) 519 - 567. Reed, M. and Simon, B . , Methods of Modern MathematicaZ Physics, III. Scattering Theory, New York, London: Academic Press, 1979. S i g a l , I.M., Mathematical Foundations o f Q u a n t u m S c a t t e r i n g Theory f o r Multiparticle Systems, Mem. Amer. Math. SOC. No. 209, 16 (1978). Simon, B., Scattering Theory and Quadratic Forms: On a Theorem of Schecht e r , Comm. Math. Phys. 55 (1977) 259 - 274. Thomas, L . , Asymptotic Completeness in Two- and Three-Particle Quantum Mechanical S c a t t e r i n g , A n n . Phys. 90 (1975) 127-165. Yajima, K . , An Abstract Stationary Approach t o Three Body S c a t t e r i n g , J. Fac. Sci. Univ. Tokyo Sect. 1A 25 (1978) 109-132.

Asymptotic Completenrss f o r Few Body Sclirijdinger Operators

*

T h i s m a t e r i a l is p a r t l y based on work s u p p o r t e d by the National Science Foundation under g r a n t MCS-8301277. ** B a n t r e l l Fellow i n Mathematical P h y s i c s .

269

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS

I.W.Knowles and R.T. Lewis (Editors) 271

0 Elsevier Science Publishers B.V. (North-Holland), 1984

ASYMPTOTICS OF THE TITCHMARSH-WEYL m-FUNCTION A BESSEL-APPROXIMATIVE CASE Halvorsen

S.G.

Department o f Mathematics U n i v e r s i t y o f Trondheim 7000 Trondheim Norway

The a s y m p t o t i c f o r m o f t h e Titchmarsh-Weyl f u n c t i o n m ( h ) f o r t h e weighted S t u r m - L i o u v i l l e e q u a t i o n y " + (Aw - q ) y = 0 on [O,-) is s t u d i e d , where w and q a r e r e a l and l o c a l l y X

integrable, w

2

I

0, and

Iw(t)

- tald t

= o(xatl)

,

X+O

,

a > - 1 . The r e s u l t s give$ extend e a r l i e r ones o f E v e r i t t and Halvorsen (1978) f o r t h e case -a = 0 . We c o n s i d e r t h e f o l l o w i n g o r d i n a r y d i f f e r e n t i a l e q u a t i o n , I

where w ( x ) 2 0 and q ( x ) a r e r e a l - v a l u e d f u n c t i o n s on t h e i n t e r v a l ( 0 , - ) , A a comp l e x parameter. w and q a r e supposed t o be l o c a l l y i n t e g r a b l e on [a,-) . L e t l i n e a r l y independent s o l u t i o n s e(x,A) , $(x,A) o f ( 1 ) be determined by i n i t i a l conditions e(o,h) =

0

(2) $(O,X)

= -1

, e’(0,h)

=

, $'(O,A)

= 0

1

e ' ( x , A ) and $ ' ( x , A ) w i l l be l o c a l l y a b s o l u t e l y c o n t i n u o u s on [ 0 , - ) . According t o t h e general t h e o r y , t h e r e e x i s t s f o r e v e r y n o n - r e a l A a t l e a s t one s o l u t i o n y =

e+m$

of ( 1 ) such t h a t

,/ -

2

wlyl

dx

<

-.

I n one c l a s s o f cases, t h e s o - c a l l e d

0

l i m i t p o i n t case ( a t t h e s i n g u l a r e n d p o i n t , h e r e -) t h i s s o l u t i o n i s unique, and t h e c o r r e s p o n d i n g m - c o e f f i c i e n t i s g i v e n by

More g e n e r a l l y , eq. ( 3 ) r e p r e s e n t s a c i r c l e i n t h e complex m-plane ( p r o v i d e d t h e i n t e g r a l on t h e r . h . s i d e e x i s t s f o r a l l v a l u e s o f m), W e y l ' s l i m i t c i r c l e , w i t h a l i m i t p o i n t as a degenerate case. The Titchmarsh-Weyl m - f u n c t i o n s a r e a n a l y t i c f u n c t i o n s o f f t h e r e a l A-axis s a t i s f y i n g ( 3 ) . General r e f e r e n c e s a r e Weyl [ l o ] , T i t c h m a r s h [8], H i l l e [ 7 3 , E v e r i tt and Bennewitz [ 5 ] . The a s y m p t o t i c b e h a v i o u r o f m ( h ) has been s t u d i e d by H i l l e [7], E v e r i t t [4], E v e r i t t and Halvorsen [ 6 ] , A t k i n s o n [ l ] and v e r y r e c e n t l y H a r r i s [ll], [12].

212

S. G. Halvorsen

Applications of the asymptotics of the m-function a r e made t o spectral theory by Atkinson [2], t o i n t e g r a l i n e q u a l i t i e s involving d e r i v a t i v e s by Evans and The main aim of t h i s paper i s to prove t h e following E v e r i t t [3]. THEOREM. Let m ( A ) be any Titchmarsh-Weyl function corresponding t o an eq. ( 1 ) and i n i t i a l conditions ( 2 ) and suppose t h a t -(a+l)

X

I w ( t ) - t"l d t + O as x + O f o r some 0 Then the following asymptotic formula holds f o r m(h) (4)

X

(5)

m(x) =

J

r(u) r(l -v)

"1-2 v e 1' n u x-I) [l + o ( l ) ] , u

as X + m in any one of the s e c t o r s 0 defining 0 5 a r g a U < 2nu when 0 < v

<

<

=

CY

-1

.

1 a+2

~

5 argh 5 n - 6 or 1 , 0 s arga < 2n.

6

>

I T + &s

argh

5

2n-6,

T h i s follows r a t h e r closely the l i n e s of proof of t h e corresponding theorem given by E v e r i t t and Halvorsen [6] f o r t h e case a = 0 , t h e main d i f f e r ence being the order of technical d i f f i c u l t y , as Bessel functions appear instead of sines a n d cosines. We f i r s t observe t h e following integral equations f o r e , 4 , e a s i l y obtainable from eq. ( l ) ,

I'YIooJ'.

X

(6)

e(x)

=

e0(x)

+

f

0

[ e o ( x l $o ( t ) - e o ( t ) $o

s a t i s f y i n g corresponding i n i t i a l conditions ( 2 ) . find

We next d e f i n e G ( x , t ) : e,,(x)

$,(t) - e O ( t ) $,(x)

( x ) l C x ( w - t")-ql e ( t ) d t

Using Watson [9], p . 97, 43 we

=

where we have written

Again, from Watson [9], p . 206 and p . 168 respectively

273

Asymptotics of the Titchmarsh- WeJ3l m-Function

with m

1 21'(v+~) 0

iQ

e U uv

- 112 22

/du ,

- 71 . Using now (12) and ( 1 3 ) and s i m p l i f y i n g , we g e t t h e f o l l o w i n g i m p o r t a n t f o r m u l a f o r t h e k e r n e l G ( x , t ) , v a l i d when l a r g z l

(14)

G(x,t)

, Re(v)

<

-

=

(xtra14 J5;

>

[(PxPt + Q x Q t ) s i n ( X - T ) + ( P t Q x - P x Q t ) c o s ( X - T ) ] ,

where X, T a r e defined above ( l l ) , and where we have w r i t t e n Px, Pt P(T,v) e t c . , w i t h v =

-

~

a + 2 .

Next, we need g l o b a l bounds f o r Px, P t

f o r s m a l l and l a r g e v a l u e s o f 1x1 and T I s i m u l t a n e o u s l y . J v ( z ) a t 0 and t h e f o l l o w i n g ( g l o b a l

f o r P(X,v), etc., v a l i d

From t h e b e h a v i o u r o f

Hence f r o m ( 9 ) , ( 1 0 )

w r i t i n g here and i n t h e sequel

u+2 1 -

V .

We now need t o e s t i m a t e G ( x , t ) .

From f o r m u l a s f o r P and Q i n [9] p. 205, o r f r o m t h e a s y m p t o t i c expansions on p. 199, combined w i t h t h e b e h a v i o u r o f J v ( z ) as z + O , f r o m which P , Q can be e s t i m a t e d , e x p r e s s i n g P and Q i n terms o f Jv

,J-",

we have

hence t h e bound f o r t h e t e r m PxPt w i l l dominate t h e ones f o r Q,Qt,

IGI

(20)

5

K l x ( - ' ( l + ( X I )v

- 112

(1

PtQx and PxQt.

,

T h i s y i e l d s t h e f o l l o w i n g bound on G = G(x,t)

+

IT^)'-''^

exp(IImX1 - IIrnTI)

We s h a l l need t h e f o l l o w i n g L e m a . L e t u,v, y and z be r e a l f u n c t i o n s , v L 0, uv, y v , v z l o c a l l y i n t e g r a b l e on some i n t e r v a l I , and suppose t h a t f o r [xo , x ] c I

(21 1

z(x)

5

y(x) + u(x) I x v ( t ) z ( t ) d t XO

.

214

S. G. Halvorsen

Then

(22)

z(x)

5

y(x) + u(x)

J

X

X

(jt u v d s ) d t

y ( t ) v ( t ) exp

f o r [xo

,x]

c

I

.

XO

This i s a Gronwall-type i n e q u a l i t y , i t s proof i s straightforward. Using t h e lemma, we g e t t h e f o l l o w i n g p r e l i m i n a r y bounds f o r e , $ , s t a r t i n g w i t h ( 6 ) , ( 7 ) r e s p . and u s i n g ( 1 6 ) , ( 1 7 ) , ( Z O ) , o m i t t i n g some d e t a i l s o f c a l c u l a tion.

Using (23) and ( 2 4 ) t o e s t i m a t e anew t h e i n t e g r a l s i n ( 6 ) , ( 7 ) we g e t

(26)

e(x)

=

(27)

w

=

eo(x) + 0

v - l/i

(IA(-V(l

+

v

+

0 ((1 + 1x1)

1x1)

- 1/2

EO(X)CE1( X I

- 11)

EO(X)[E1 (X)

- 11)

] ~ + as 0 h - t m ) such t h a t We now r e s t r i c t x t o an i n t e r v a l [0, ~ ( h ) (where

1111

(1 + 1T1)2v-1 d t = o(lxlv) 0 assumed t o h o l d u n i f o r m l y as h + - i n t h e x - s e c t o r s g i v e n i n t h e main theorem. Jp

(28)

It

t

i s e a s i l y seen t h a t ( 2 8 ) i s i n f a c t a c o n d i t i o n on w alone; no c o n d i t i o n on q beyond l o c a l i n t e g r a b i l i t y i s i m p l i e d . I n a d d i t i o n , we impose t h e f o l l o w i n g c o n d i t i o n on p ( h )

(29)

\ h ~ ~ p + m as

~

This i s equivalent t o saying t h a t

+

.

m

1x1 + m

f o r x = p , X being defined i n (11).

When ( 2 8 ) i s s a t i s f i e d , we see t h a t E 1 ( X ) + l , such t h a t i e ( x ) l , I $ ( x ) I w i l l

l e g / , l$ol i n (16), e ,0 . Writing

have t h e same upper bounds as t h e ones found f o r We now need t o e s t i m a t e t h e w e i g h t e d norms o f

/

P

2 wl8l

0

d e t a i 1s

dt =

/

li

0

2

tale( dt +

/

!J

0

( w - t*)

lel

2

r e s p . (17).

d t = I1 + I2 we f i n d , a g a i n o m i t t i n g

Asyinptotics of the Titchmarsh- Weyl rn-Function

=

We n o t e t h a t I 2 z 0 i f w

-1

t h a t E, ( x )

.

t'

Using now ( 2 6 ) and assuming (28) t o h o l d , such

, we f i n d f o r I,

= o(1)

equation (8) f o r e o .

d i r e c t l y from t h e d i f f e r e n t i a l

Omitting (the) c a l c u l a t i o n s

t h a t uses a s y m p t o t i c formulas f o r Bessel f u n c t i o n s ([9], r e n c e formula zJv' ( z ) = Z J ~ - ~ ( Z- ) v J v ( z ) ([9],

u

(31)

JuwIel2dt 0

I

'L

t'lel'dt

0

where we have w r i t t e n

(32)

I'

215

T

for Im(6)

1-!

wl$I2dt-

taI$I2dt

0

-

0

p. 199) and t h e r e c u r -

p. 45) we g e t

[r

.

L i k e w i s e , we g e t

[r

when T + m . We now use t h e l a s t two f o r m u l a s t o f i n d an upper bound c o n c e r n i n g m ( A ) , p r o v i n g t o be s h a r p ( v . Theorem)

We n o t e t h a t ( 2 9 ) g i v e s t h a t Tu1'2v*a

lV I m ( h ) I

(33) proof.

5

1

From ( 3 ) we have

Using ( 3 1 ) , ( 3 2 ) we o b t a i n

f r o m which ( 3 3 ) f o l l o w s . Now, c o n s i d e r t h e f o l l o w i n g i d e n t i t y

mo = ~-$L+)

where

" 1 - 2 ~ e i n v A- v .

From ( 2 6 ) , ( 2 7 ) , w r i t i n g ' $ =

JI = JI0

(35) We have

$

0

+

( 0 ) = -1,

O((1

@d(O)

+

+

+ m i 1 e , $ o = $o + mo-1 ,e

1x11v - 1/2 =

EO(X)[E1(X)

- 11)

-1 , and moreover $,(x)

mo

s a t i s f i e s eq. (18) and

m

/ X ' I $ ~2~dx 0

< =

.

From Watson [9], p. 73 and p. 198 we f i n d

S. G. Halvorsen

276 3 (x) = ( - i sin 0

(36)

fur Im(x) Hi2)(X)

>

.

0.

For Im(h)

<

) ( v ~ x1'2 ~ ) ~H ( ' )

(XI

0 , i n (36) change s i g n and change HL1)(X) i n t o

From t h e b e h a v i o u r o f H ( l ) ( X ) f o r s m a l l and l a r g e X we g e t I q o ( x ) l c K(l +

(37)

r(1tw

nw)

lXl)w-1'2

-'

Eo ( X I

Using ( 3 ) and (28) we f i n d

Again, from ( 3 ) we f i n d

L a s t l y , we f i n d from ( 3 5 ) and ( 2 8 ) , w r i t i n g w = xa t ( w - x a )

a +2

where X

v

= a t 2 !J

and f r o m ( 2 5 )

where we have w r i t t e n , w i t h

M

=

1/2w

~ W T P

u

!J

(42)

c = u- ( a + l )

1

[w-xaldx, 6 =

0 From (34) and ( 3 5 ) f o l l o w s ,

using ( 3 2 ) , (38)

= o(e-M)

m

(43)

[qldx, Y =

u M- m i n ( l , 2 w )

0

+

-

(41)

, 2 ~ ) ~ O(M-min(l ) , 2 ~ ) ~ ~ )

0(,,,2-min(l

Choosing t h e f i r s t two terms on t h e r i g h t hand s i d e o f ( 4 3 ) t o be o f equal magn i t u d e , t h e Theorem i s proved. The f a c t t h a t t h i s c h o i c e i s p o s s i b l e , b e i n g a d e t e r m i n a t i o n of )-1 as a f u n c t i o n o f T , may be seen a n a l o g o u s l y t o t h e t r e a t m e n t 1 of t h e case C( = 0 ( V = 7 ) g i v e n i n [6]. L e t w - xa = O ( x 5

ExampZe.

1%-

11

d e t a i 1 s.

=

0

T

-2v(

B-a)

(Log

( @ > a )as

X+O

T) 2 v ( ~ + ' ) - m i n ( '

, q + o

.

We g e t

9 2 ~ ) )f r o m ( 4 3 ) o m i t t i n g computational

277

Asymptofics oj t h e Titchniarsli- Weyl in-Function

Remark: A f t e r c o m p l e t i o n o f t h e p r e s e n t work, t h e a u t h o r was made aware o f t h e paper [13] o f I.S. Kac, t h e r e s u l t s o f which b e a r a r e l a t i o n s h i p t o those presented h e r e . REFERENCES

F.V. A t k i n s o n , On t h e l o c a t i o n o f t h e Weyl c i r c l e s . A 8 8 (1981), 3 4 5 - 3 5 6 .

Proc. R. SOC. Edinb.

F.V. A t k i n s o n , On t h e a s y m p t o t i c b e h a v i o u r o f t h e Titchmarsh Weyl m - c o e f f i c i e n t and t h e s p e c t r a l f u n c t i o n f o r s c a l a r second-order d i f f e r e n t i a l expressions, L e c t u r e Notes i n Mathematics, v o l . 964 (1982), 1 27.

-

Evans, W.N. E v e r i t t , A r e t u r n t o t h e Handy-Littlewood i n t e g r a l i n e q u a l i t y . Proc. R. SOC. Lond. A 3 8 0 (1982), 447 - 4 8 6 .

W.D.

W.N. E v e r i t t , On a p r o p e r t y o f t h e m - c o e f f i c i e n t o f a second-order l i n e a r d i f f e r e n t i a l e q u a t i o n , J . Lond. Math. SOC. ( 2 ) , 4 (1972), 443 - 457. W.N. E v e r i t t and C. Bennewitz, Some remarks on t h e Titchmarsh-Weyl mcoefficient. I n ' T r i b u t e t o 8ke P l e j e l ' , Univ. o f Uppsala (1980), 49 - 108.

W.N. E v e r i t t , S.G. Halvorsen, On t h e a s y m p t o t i c form o f t h e Titchma shWeyl m - c o e f f i c i e n t . A p p l i c a b l e A n a l y s i s 8 ( 1 9 7 8 ) , 153 - 169.

E. H i l l e , L e c t u r e s on o r d i n a r y d i f f e r e n t i a l e q u a t i o n s . London, 1969. E.C. Titchmarsh, E i g e n f u n c t i o n expansions I . Sec. ed., 1962.

Addison-Wes Oxford Univ

Press,

G.N. Watson, A T r e a t i s e on t h e Theory of Bessel F u n c t i o n s , Sec. ed., b r i d g e U.P., 1962.

Cam-

H. Weyl, Uber gewohnliche D i f f e r e n t j . a l g l e i c h u n g e n m i t S i n g u l a r i t a t e n und d i e zugehorigen Entwicklungen w i l l k u r l i c h e r Funftionen, Math. Ann. 68 (1910) 220 - 269. 8.3. H a r r i s , On t h e Titchmarsh-Weyl m - f u n c t i o n . i n Proc. R. SOC. Edinb.

P r e p r i n t , 1983.

To appear

B.J. H a r r i s , The a s y m p t o t i c form o f t h e Titchmarsh-Weyl m - f u n c t i o n . p r i n t , 1983.

Pre-

I.S. Kac, Power-Asymptotic Estimates f o r S p e c t r a l F u n c t i o n s o f G e n e r a l i z e d Boundary Value Problems o f Second Order, Sov. Math. Dokl. 13 ( 2 ) (1972), 453 - 457.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

219

LATTICE MULTISCALE SINGULAR PERTURBATION THEORY C a r l o s R. Handy Department o f P h y s i c s Atlanta University A t l a n t a , Georgia 3031 4

U.S.A.

S i n g u l a r p e r t u r b a t i o n t h e o r y i s a F o u r i e r s p a c e problem i n which extens i v e q u a n t i t i e s , s u c h a s t h e power moments, d e f i n e a r e p r e s e n t a t i o n on which t o b u i l d a n a n a l y t i c a l framework. F o r some l i n e a r systems, t h e moments a r e d i r e c t l y o b t a i n a b l e . For n o n l i n e a r ODEs t h i s i s i m p o s s i b l e . However, by a p p e a l i n g t o t h e h i g h t e m p e r a t u r e l a t t i c e expansion p h i l o sophy o f Bender e t a 1 ( a l t h o u g h n o t n e c e s s a r i l y t o t h e i r c o n t i n u a t i o n methods) t h e moments f o r some n o n l i n e a r ODEs may be g e n e r a t e d . Through s u c c e s s i v e a p p l i c a t i o n o f Fourier-Resonance r e c o n s t r u c t i o n (and i t s g e n e r a l i z a t i o n t h r o u g h Polya-Pade r e p r e s e n t a t i o n s ) I g l o b a l approximate s o l u t i o n s can be o b t a i n e d from knowledge o f t h e moments. INTRODUCTION For t h e i l l u s t r a t i v e p u r p o s e s o f t h i s p r e s e n t a t i o n , w e s h a l l c o n s i d e r t h e s i n g u l a r p e r t u r b a t i o n a n a l y s i s (A2-expansions) f o r t h e systems

-

A2Fxx

g x2F

-t

=

e F

(Schrodinger equation)

(1.1)

For t h e l a t t e r system we a r e i n t e r e s t e d i n t h e s o l u t i o n s a t i s f y i n g Fkink(0)

=

0

,

Fkink(+m) =?r m_ Jg

(for

=’A

g-')

(1.3)

Because o f t h e n a t u r e o f (1.2) and ( 1 . 3 ) we s e e t h a t t h e g e n e r a l s o l u t i o n , f o r a l l A, s a t i s f i e s a s c a l i n g r e l a t i o n

Consequently, one might e x p e c t t h a t a A-expansion i s i m p o s s i b l e . However. by r e g a r d i n g t h e s o l u t i o n as a d i s t r i b u t i o n , a c t i n g on a s u i t a b l e s p a c e of t e s t funct i o n s , T T ( x ) , i n d e x e d by 'T',One can d e f i n e a new r e p r e s e n t a t i o n s p a c e i n which A- e x p a n s i o n s a r e d e f i n a b l e

.

Fsub(x;A) 5 m_ {H(x) Jg 00

I(A;{u(p)}) =

A C

-

H(-x)} ( HE H e a v i s i d e f u n c t i o n )

. . A p T F ) ~,

(1.6) p=o P! where t h e T $ ) s may b e r e g a r d e d as known f u n c t i o n s o f ' ~ l , c o r r e s p o n d i n gt o t h e T a y l o r expansion c o e f f i c i e n t s o f t h e t e s t f u n c t i o n s . Thus, (1.6) i s a f o r m a l l y s e n s i b l e A-expansion a f f o r d e d by t h e new r e p r e s e n t a t i o n s p a c e , i n v o l v i n g t h e unknown Hamburger moments, { u ( p ) } Provided t h e s e a r e known ( i . e . a s u f f i c i e n t number up t o some g i v e n 0 r d e r . P ) and a r e c o n s t r u c t i o n p r o c e d u r e h a s been d e f i n e d , t h e n one may claim t o have d e f i n e d a A-expansion a n a l y s i s f o r t h e c o n f i g u r a t i o n

.

280

CR.Handy

space problem.

.

F o r t h e l i n e a r ODE problem, (1.1 ), t h e r e i s no s p a t i a l s c a l i n Nontheless, a s i m p l e F o u r i e r t r a n s f o r m a t i o n will r e s u l t i n - A2Fxk + A2k2Fft?k), s u g g e s t i n g t h a t a momentum space T a y l o r expansion i s t h e a p p r o p r i a t e method o f a n a l y s i s f o r t h e A-expansion problem. O f c o u r s e , t h e power moments are t h e c o e f f i c i e n t s of such a n expansion. More s p e c i f i c a l l y , t h e d i r e c t a p p l i c a t i o n o f t h e i n t e g r a l I d x XP t o both s i d e s o f (1.1) w i l l y i e l d u(pt2) =

.

( e / g ) u ( p ) + (A2/g)p(p-l h ( p - 2 )

(I.7 )

F o r symmetric s o l u t i o n s , u(p=odd) = 0, and s e t t i n g u ( 0 ) = 1 g e n e r a t e s r e c u r s i v e l y a l l t h e moments. Note t h a t t h e moments a r e t r i v i a l l y a n a l y t i c i n X ! Thus, t h e moments p l a y a k e y r o l e i n any A-expansion a n a l y s i s (i.e. s i n g u l a r p e r t u r b a t i o n t h e o r y ) . They a r e a s s o c i a t e d w i t h r e p r e s e n t a t i o n s p a c e s i n which such expansions a r e d e f i n a b l e . I n t h e n e x t s e c t i o n we i l l u s t r a t e one p r o c e d u r e t h a t may be used f o r o b t a i n i n g t h e moments f o r a given s e l f - i n t e r a c t i n g n o n l i n e a r s y s tem. LATTICE MULTISCALE SINGULAR PERTURBATION EXPANSIONS FOR THE MOMENTS L e t (1.2) r e p r e s e n t a t y p i c a l s e l f - i n t e r a c t i n g n o n l i n e a r system. The s o l u t i o n c o r r e s p o n d i n g t o (1.3) c o r r e s p o n d s t o a boundary l a y e r problem. We s h a l l s e e t h a t one can d e f i n e on t h e l a t t i c e a ' g l o b a l ' a n a l y t i c expansion which i s c a p a b l e of a d d r e s s i n g t h e g l o b a l b e h a v i o r o f t h e c o n f i g u r a t i o n s p a c e problem, r e f . 1 , a s w e l l as such l o c a l p r o p e r t i e s of t h e s o l u t i o n a s t h e d e r i v a t i v e s a t t h e boundary l a y e r l o c a t i o n . 2 3 Thus t h e l a t t i c e r e p r e s e n t a t i o n , and s u b s e q u e n t h i g h temperat u r e l a t t i c e expansion, s i m u l t a n e o u s l y i n c o r p o r a t e t h e ' i n n e r ' and ' o u t e r ' behav i o r o f t h e system w i t h o u t a p p e a l i n g t o a s y m p t o t i c matching p r e s c r i p t i o n s . I n a d d i t i o n , even though w e a r e working w i t h d i f f e r e n c e e q u a t i o n s f o r a s i n g u l a r p e r t u r b a t i o n problem, t h e s t r u c t u r e o f t h e t h e o r y i s such t h a t mesh s i z e c o n s i d e r a t i o n s a r e m a n i f e s t l y a b s e n t , as will become c l e a r i n t h e d i s c u s s i o n s t h a t f o l l o w , p a r t i c u l a r l y as t h e y r e l a t e t o c o n t i n u a t i o n i n ' a ' , t h e l a t t i c e s p a c i n g , i n t o t h e continuum l i m i t , a+O. Consider t h e d i f f e r e n c e e q u a t i o n v e r s i o n o f ( 1 . 2 ) .

A(')Fa(L)

= F a ( L t l ) t Fa(L-l)

-

2Fa(L) ,

Upon s u b s t i t u t i n g i n t o (2.1 ) and i d e n t i f y i n g t h e c o r r e s p o n d i n g o r d e r s we o b t a i n

3

(ctll d e n o t e s a l l c u b i c p r o d u c t s w i t h 2

2

w . < w, and

w t w t w =w) 1 2 3

V(F) =-:F4+3(m / g ) F i s t h e e f f e c t i v e p o t e n t i a l . Note t h a t among a l l t h e p o s s i b l e c h o i c e s f o r t h e z e r o t h order l a t t i c e f i e l d ( i n c l u d i n g p e r i o d i c s t e p f u n c t i o n s ) (2.3) was s e l e c t e d because of i t s o b v i o u s f a i t h f u l n e s s t o t h e a s y m p t o t i c b e h a v i o r of t h e d e s i r e d s o l u t i o n . I n a d d i t i o n , i t s image s p a c e v a l u e s correspond t o t h e

Singular Perturbation Theory

28 1

c r i t i c a l p o i n t s o f t h e e f f e c t i v e p o t e n t i a l , V. It i s a l s o noteworthy t h a t , b e c a u s e o f s c a l i n g , FO(L) = F ~ ~ ~ ~ ( L / o + ) . The l a t t i c e power moments, f o r a s u i t a b l y s u b t r a c t e d l a t t i c e f i e l d :

2.5) W

=

c

al+P

b2/a')W c ( p ; w )

(2.6)

w=l W

c(p;w) =

z

L=-W

LP F ( ~ ) ( L ), f o r

( c ( ~ ; o )= o

w 2 I

)

(2.7)

The C(p;w)s a r e n u m e r i c a l l y c a l c u l a b l e because o f t h e i m p o r t a n t p r o p e r t y of bounded l a t t i c e s u p p o r t : F ( w ) ( ~ =)

o ,

for

.

I

I L I >L ~ , w ,

(2.8)

It i s s u s p e c t e d t h a t b e s i d e s t h e a b i l i t y t o c a l c u l a t e moments n u m e r i c a l l y , (2.8) a l s o r e f l e c t s t h e capacity o f t h e type of l a t t i c e representation being generated t o s i m u l t a n e o u s l y a d d r e s s b o t h t h e g l o b a l and l o c a l p r o p e r t i e s o f t h e s o l u t i o n , as d e s c r i b e d a t t h e o u t s e t o f t h i s s e c t i o n . Only a f i n i t e number o f C ( p ; w ) s can be g e n e r a t e d . Thus we a r e l e f t t o contemplate how t o c o n v e r t t h e f i n i t e expansion W u a ( p ) = a l t P C (A2/a2)WC(p;w) t a1tPO((X2/az)W) (2.9) W=l i n t o a n e x p r e s s i o n r e l e v a n t for c a l c u l a t i n g

l i m ua(p) a+ 0

=

u

continuum'p)

.

The answer t o t h e above i s a f f o r d e d by r e g a r d i n g ( 2 . 6 ) and ( 2 . 9 ) as expansions 'valid' for 'large' a Thus we must c o n t i n u e t h e s e e x p a n s i o n s from such a regime i n t o t h e continuum regime, a=O. One s u c h g e n e r a l c l a s s of methods for implementing t h i s i s t h a t o f Pade a n a l y ~ i s .Through ~ t h e u s e o f l i n e a r a l g e b r a one may s o l v e f o r t h e f o l l o w i n g numerator and denominator Pade polynomial c o e f f i c i e n t s , i n terms o f t h e C(p;w)s:

.

d=O Where c1 i h 2 / a 2 ; W = TSB ( s o l v a b i l i t y ) ; T - B - & ( l t p ) = 0 ( c o n d i t i o n for o b t a i n i n g a f o r m a l l y f i n i t e e x p r e s s i o n ) ; and D(0) S 1 , c o n v e n t i o n a l n o r m a l i z a t i o n . Note, by a n t i s y m m e t r i c b e h a v i o r o f F ( L ) , u (p=even)=O, hence o n l y ua(p=odd)fO, 1 tp=even. P r o c e e d i n g a c c o r d i n g l y , w e gave a

From r e f . 1 w e have t h e t a b u l a t e d r e s u l t s : P 1

3 5 7 9

Pade o f L a t t i c e Moments -1.6~

-5.63 -61.66 -1,298 -44,721,

Actual -I .6L

-5.68 -59.16 -1.260

45,360

CR. Handy

282

It should b e s t r e s s e d t h a t no d i f f i c u l t a n a l y s i s h a s been done. So f a r we have used r e c u r s i v e a l g e b r a i c r e l a t i o n s and l i n e a r a l b e g r a i n g e n e r a t i n g t h e moments. RECONSTRUCTION Although t h e function-moments problem s u f f e r s from n o n ~ n i q u e n e s st~h a t i s n o t t o imply t h a t f o r a given d i f f e r e n t i a l system we c a n n o t i n f e r , f r o m g e n e r a l arguments, how t o r e c o n s t r u c t i t s s o l u t i o n from a g i v e n , f i n i t e s e t o f moments. O f p a r t i c u l a r i n t e r e s t t o U S i s t h e assumption t h a t t h e s o l u t i o n i n q u e s t i o n h a s a d i s c r e t e Lap l a c e r e p r e s e n t a t i o n ( o f t h e ' s p l i n e ' t y p e t o be given below). T h i s w i l l a l l o w u s t o u s e a second Pade a n a l y s i s w i t h r e s p e c t s t o a t r u n c a t e d F o u r i e r space T a y l o r expansion. T h i s type o f Fourier-Resonance r e c o n s t r u c t i o n h a s been a n a l y z e d by Chisholm: and u t i l i z e d i n a n t e n n a d e s i g n by Tesche! The g e n e r a l i z a t i o n o f t h i s approach t o problems m a n i f e s t i n g m u l t i - s i n g u l a r r e g i o n b e h a v i o r ( i . e . m u l t i boundar y l a y e r phenomena; m u l t i - t u r n i n g p o i n t problems) m o t i v a t e s t h e c o n s i d e r a t i o n , a n d d e f i n i t i o n , of 'Polya-Padel r e p r e s e n t a t i o n s ? L e t u s assume t h a t a s u f f i c i e n t number o f Hamburger moments are known (i.e. u ( p ) s ) . Thus t h e momentum s p a c e T a y l o r expansion for t h e F o u r i e r t r a n s f o r m o f t h e s o l u t i o n (or a properly subtracted solution) i s P P ? ( k ) = ( 2 ~ r ) - ' C L f E d (-ik)' t O( ( - i k ) ) (3.1) p=o P! C l e a r l y , we cannot perform a n i n v e r s e F o u r i e r t r a n f o r m a t i o n on t h e above. However, by g e n e r a t i n g s u i t a b l e Pade r e p r e s e n t a t i o n s , w e can e f f e c t u a t e such a n i n v e r s e F o u r i e r t r a n s f o r m a t i o n t h r o u g h t h e s u c c e s s i v e i d e n t i f i c a t i o n o f s i m p l e p o l e s and r e s i d u e s f o r t h e Pade e x p r e s s i o n . rn I

x

n=O N(n) ( - i k ) n d D(d) ( - i k )

F(k) =

(3.2)

O((-ik)')

(3.3)

c1

=

C

v

V

(assuming no m u l t i p l e p o l e s )

1 t iBvk

Leading t o :

(3.4) (3.5)

C(')

correspond t o summations o v e r B

V

s such t h a t

>

Re(@" ) <

0

.

I n r e f e r e n c e 1 i t i s shown how t h e above c o r r e c t l y g e n e r a t e s t h e d i s c r e t e Laplace r e p r e s e n t a t i o n f o r t h e t r u e s o l u t i o n Fkink(x) = (m//g) Tanh(mx//2)

.

Singular Perturbation Theory

283

POLYA PADE REPRESENTATIONS AND MULTI-SINGULAR REGIME SCLUTIONS

For a m u l t i - s i n g u l a r regime problem, such as t h e l i n e a r S c h r o d i n g e r e q u a t i o n two t u r n i n g p o i n t problem, ( I . l ) , one n e e d s t o c o n s i d e r a g e n e r a l i z a t i o n o f t h e p r e v i o u s s e c t i o n ’ s d i s c u s s i o n on Fourier-Resonance r e c o n s t r u c t i o n . The p r o c e d u r e d i s c u s s e d t h e r e i s s u i t a b l e f o r one s i n g u l a r regime s o l u t i o n s (i.e. one boundary l a y e r t y p e problems). Motivated by t h e Pade a n a l y s i s of Polya f r e q u e n c y s e r i e s 4 we d e s c r i b e t h e f o l l o w i n g t y p e o f s t r u c t u r e a s o f a Polya-Pade t y p e : (4.1 ) a F(k) = C (-ik)’ & = X exp(ikAT) Pade(T;B)=C k] T V WT P! T

ltiR6

‘

The mot.ivation f o r a t t e m p t i n g t o d e f i n e such a r e p r e s e n t a t i o n i s a p p a r e n t from t h e one s i n g u l a r regime r e c o n s t r u c t i o n c a s e o f (3.5). Thus, t h e i n v e r s e F o u r i e r t r a n s form o f (4.1 ) would c o r r e s p o n d t o v a r i o u s s p l i n e ’ - l i k e e x p o n e n t i a l a p p r o x i m a t i o n s to t h e s o l u t i o n , which i s made up o f m u l t i s i n g u l a r regimes l o c a t e d a t t h e A s. For t h e p u r p o s e s o f o u r two t u r n i n g p o i n t problem, ( 1 . 1 ) . i t i s expected t h a x t h e l a t t e r correspond t o t h e a c t u a l t u r n i n g p o i n t s themselves

‘

Indeed, from (1.7). upon s e t t i n g A+ 0. t h e r e s u l t i n g z e r o t h o r d e r s o l u t i o n i s f o r m a l l y s a t i s f i e d by a symmetric sum o f d e l t a f u n c t i o n s , f o r t h e symmetric solutions:

a e f o r e c o n t i n u i n g . i t should be n o t e d t h a t t h e T/B o r d e r chosen f o r t h e Pade i n ( 4 . 1 ) w i l l g u a r a n t e e t h a t t h e approximant have c o n t i n u o u s d e r i v a t i v e s a c r o s s t h e t u r n i n g p o i n t s up t o o r d e r N d e t e r m i n e d by:

T-B = -2 -N dnF -(At0 dxn

e n s u r e s t h a t f o r n=O,

t

dnF ) = -(AtO-) dxn

.

...,N

, (4.4)

Thus, f o r a system demanding c o n t i n u o u s d e r i v a t i v e s up t o second o r d e r , ( I . l ) , follows t h a t T-B= - 4 , i s r e q u i r e d . a t l e a s t .

it

For t h e c a s e ( 1 . 1 ) we have been a b l e t o d e f i n e t h e r e p r e s e n t a t i o n (4.1) by c o n s i dering t h e formal r e l a t i o n

(4.5)

The c o n t o u r o f i n t e g r a t i o n a t t h e extreme R.H.S. encloses a l l singularities. we Pade t h e c o r r e s p o n d i n g i n t e g r a n d , t h e r e r e s u l t s :

If

(4.6)

CR. Handy

284

Thus, t a k i n g t h e c o n t o u r i n t e g r a l w i t h r e s p e c t s t o t h e B-poles. a formal expansion F ( k ) = (2n)-'

C j

there results

A.exp(-ikB.) J J

(4.7)

Once a g a i n , t h e implementation o f an i n v e r s e F o u r i e r t r a n s f o r m on ( 4 . 7 ) w i l l r e s u l t i n a d i s c r e t e d e l t a f u n c t i o n r e p r e s e n t a t i o n , which i s u n d e s i r a b l e . A t t h e formal l e v e l , because t h e B s may be complex, t h e r e may r e s u l t d e l t a f u n c t i o n s w i t h complex arguments. T h i s s i t u a t i o n i s r e m i n i s c e n t o f t h e F o u r i e r s p a c e m u l t i p l e s c a l e a n a l y s i s o f J e f f r e y and Kawahara'.

It i s n u m e r i c a l l y observed t h a t a s A+ 0 , s u b s e t s o f t h e B . s c l u s t e r around each of t h e t u r n i n g p o i n t s . L e t { B . } l and { B . I 2 d e n o t e t h e J s u b s e t s which c l u s t e r a b o u t t h e r e s p e c t i v e t u r n i n g p o i A t s a l r 2 . For e a c h of t h e s e s u b s e t s , we may consider t h e following a n a l y s i s

(4.8) exp(-ikAT)

{I A e x p ( - i k ( B . -AT 1)) jT

=

exp(-ikAT) Pade(T;B;-ik)

JT

Upon g e n e r a t i n g t h e {a,B} for e a c h o f t h e t u r n i n g p o i n t s , t h e t o t a l Polya Pade r e p r e s e n t a t i o n , (4.1 ).becomes t h e o b j e c t o f a n i n v e r s e F o u r i e r t r a n s f o r m a t i o n Good r e s u l t s a r e o b t a i n e d ' f o r (1.1) i n t h e neighborhood of each of t h e t u r n i n g p o i n t s , as d e t e r m i n e d by t h e e x t e n t o f v a r i a t i o n o f t h e r e s p e c t i v e s u b s e t s IB.11

.

J

n

1 .C.R.Handy, " S i n g u l a r P e r t u r b a t i o n S t r o n g Coupling F i e l d Theory and The Moments Problem", Phys.Rev. D I I , Vol. 24, No. 2 ( 8 1 ) . 2.C.Bender. F.Cooper, G.S. G u r a l n i k , E.Mjolsness, H.A.Rose, and D.H.Sharlp, Adv. Appl. Math.l,22 (1 980). 3.C.Bender. Los Alamos S c i e n c e Vo1.2, 76, (1981). &.Baker,G.A.,Jr., E s s e n t i a l s o f Pade Approximants (Academic, New York, 1975) S.Shohat, J . A . , and Tamarkin. J . D . , The Problem o f Moments, American Mathematical S o c i e t y , Providence, Rhode I s l a n d , (1963). 6.Chisholm.J.S.R.. and Common. A.K., The Pade Approximant i n T h e o r e t i c a l P h y s i c s , E d i t e d by G. A. Baker and J . L . Gammel,(Academic P r e s s , New York,1970). 7.Tesche.F.M. ,"On t h e A n a l y s i s o f S c a t t e r i n g and Antenna Problems Using S i n g u l a r i t y Expansion Technique", IEEE T r a n s a c t i o n s on Antennas and P r o p a g a t i o n , Vol. AP-21, No. 1 , Jan. 1973. B.Jeffrey, A., Kawahara, T . , "A Note On t h e M u l t i p l e S c a l e F o u r i e r Transform," Nonlinear A n a l y s i s , Theory, Methods.& A p p l i c a t i o n s , Voi. 5 , No. 1 2 , pp. 1331-1340 (1981). 9. Handy, C.R., IIApplications of Polya-Pade a n a l y s i s To ODES", i n p r e p a r a t i o n .

DIFFERENTIAL EQUATIONS

I.W.Knowles and R.T. Lewis (Editors)

285

@Elsevier Science Publishers B.V. (North-Holland), 1984

SOLUTIONS FOR MODEL BOLTZMANN EQUATIONS PROPOSED BY ZIFF 0. P. H a r d i n and 3. V. Herod

School o f Mathematics Georgia I n s t i t u t e o f Technology A t l a n t a , Georgia 30332 USA

I n r e c e n t i s s u e s o f P h y s i c a l Review L e t t e r s and P h y s i c a l Review, Robert M. Z i f f p r e s e n t e d a f a m i l y o f k e r n e l s P(y,z;x) where by t h e k i n e t i c e q u a t i o n m

aF ,(x,t)

m

m

= /dyF(y,t)

0

/dZF(Z,t)P(y,z;x) 0

- F(x,t)

m

I d y F ( y , t ) I dzP(x,y;Z) 0 0

forms a v a r i e t y o f Boltzmann-type models. W e establish that i f F o ( x ) i s a c o n t i n u o u s bounded f u n c t i o n on [O,m) w i t h m

m

f Fo(x)dx

= 1

and

0

/xFo(x)dx < 0

m

and i f P i s a k e r n e l i n t h e f a m i l y proposed by Z i f f , t h e n t h e r e i s a s o l u t i o n i n t h e space BC[O,m) f o r t h e a s s o c i a t e d model w i t h F(x,O) = F o ( x ) . The s o l u t i o n e x i s t s f o r a l l t 2 0.

Two Bol tzmann models which a r e f r e q u e n t l y s t u d i e d a r e t h o s e f o r m u l a t e d by Bobylev, Krook, and Wu [2,4]

Both these models assume t h e

and by T j o n and Wu [5].

system i s s p a t i a l l y u n i f o r m and t h a t t h e s c a t t e r i n g i s i s o t r o p i c .

Among s e v e r a l

n i c e f e a t u r e s o f t h e s e models, one i s t h a t an e x a c t , n o n - e q u i l i b r i u m s o l u t i o n can be g i v e n f o r each. The models d e s c r i b e t h e e v o l u t i o n i n t i m e o f an energy d i s t r i b u t i o n f subj e c t t o t h e Boltzmann Energy E q u a t i o n : m

(BEE)

af = at

m

m

Jdyf(t,y) /dzf(t,z)P(y,z;x) 0 0

-f(t,x)

m

I d y f ( t , y ) /dzP(y,x;z) 0 0

O f course, t h e model and i t s r e s u l t i n g p r o p e r t i e s change by u s i n g d i f f e r e n t s c a t t e r i n g k e r n e l s P.

The k e r n e l P r e p r e s e n t s t h e c o l l i s i o n p r o b a b i l i t y f o r

p a r t i c l e s h a v i n g energy y and z t o produce a p a r t i c l e h a v i n g energy x .

For

p h y s i c a l reasons, one expects t h e s c a t t e r i n g k e r n e l s t o have c e r t a i n p r o p e r t i e s o f symmetry: P(y,z;x)

I n [7,8]

= P(z,y;x),

P(y,z;x)

= P(y,z;y+z-x),

P(y,z;x)

=

0 i f x > y + z.

Z i f f i d e n t i f i e d a f a m i l y Pm, m > 0 o f p r o b a b i l i t y d i s t r i b u t i o n s .

D.P. Hurdin afld J. V. Herod

28 6

Each has t h e p r e v i s o u l y l i s t e d symmetry p r o p e r t i e s .

And, P1 i s t h e d i s t r i b u t i o n

f o r the Tjon-Wu model; P 3 l 2 i s the d i s t r i b u t i o n f o r t h e Bobylev-Krook-Wu

In

161,

model.

T u r c h e t t i addresses t h e two fundamental questions f o r d i s t r i b u t i o n s

i n t h e f a m i l y o f Z i f f models: (i)

Find an a p p r o p r i a t e f u n c t i o n space such t h a t i f f ( 0 , x ) evolves s a t i s f y i n g (BEE) then f ( t , x )

(ii)

i s i n t h e space and

2

i s i n t h e space f o r a l l t

0.

E s t a b l i s h t h e asymptotic p r o p e r t i e s f o r f ( t , x ) as t increases. H i s answer f o r q u e s t i o n ( i ) i s L 1 [ O , - ) and t h e methods used appeal t o an

e x i s t e n c e r e s u l t from f u n c t i o n a l a n a l y s i s ; see [l] f o r the same r e s u l t s and techniques a p p l i e d t o the Tjon-Wu model.

I n t h i s paper, we answer question ( i ) i n t h e space

B[O,m)

o f bounded func-

t i o n s on [0,m)

and having norm I f / = sup I f ( x ) l . Furthermore, t h e s o l u t i o n i s o<x given as an e x p l i c i t i n f i n i t e s e r i e s where t h e t i m e dependence and energy depen-

Thus, i n s i g h t i s p r o v i d e d f o r t h e n a t u r e o f t h e t i m e

dence a r e separated. evol u t i on.

The a n a l y s i s o f t h e paper i s made by demonstrating i n e q u a l i t i e s t o establish

t h e n a t u r e o f t h e convergence f o r t h e s e r i e s s o l u t i o n .

But, t h e essence

o f t h e paper i s t h a t t h e method provides a r o b u s t technique t o s o l v e more general equations having t h e form of (BEE). Section 2.

THE CLASS OF ZIFF MODELS

The c l a s s o f models proposed by Z i f f a r i s e s from v a r y i n g choices f o r t h e p r o b a b i l i t y d i s t r i b u t i o n t o be used i n t h e equation (BEE). Pm(y,z;x),

if 0

<x

rn

21

i s d e f i n e d as f o l l o w s :

if x

2

The d i s t r i b u t i o n

y+z, then Pm(y,z;x) = 0; and,

< y+z, then

m

p (y,z;x)

= (y+z)2m-3 qm [ F 1

(yzlm-’

A where qm(A) = (m-1)

f

[ u ( l - ~ ) ] ~ - ~ d u0,

-

1 m a x ( l u - z l , ly+z-2x Y+Z Y+Z

5 A 5 1, m # 1, and q 1 (A)

I 11

= 1

if 0 < X < 1,

0 w i t h q (0) = 0, q l ( l ) = 2. 1

1 As Z i f f p o i n t s out, P (y,z;x)

= l/y+z i s the p r o b a b i l i t y d i s t r i b u t i o n

I t i s , then, o f i n t e r e s t t o consider l i m qm(X). m4 1 1 If 0 < A < 1 then l i m qm(A) = q ( A ) . mJ.1

associated w i t h t h e Tjon-Wu model. Lemma 1.

We now i n d i c a t e t h e i n e q u a l i t i e s and i d e n t i t i e s f o r t h e Z i f f d i s t r i b u t i o n which we w i l l use. Lemma 2.

If 1

im

then t h e r e i s a p o s i t i v e number Bm

then

I n f a c t , B~ =

i. f m < 2 and =2 i f m

2

2.

54

such t h a t i f 0

5X 5I

287

Boltzinann Equations

Proof. m

Suppose 0 Thus

2.

5X5

< 22-m i f m < Then ( l - A ) m - 2 -

1/2.

qm(A)

5

Cm

-

I f rn > 1 t h e n Pm(y,z;x)

Proof.

Suppose z < y < x.

< 1 if

h

(m-1) I u m - 2 d u = Cm Am-’ 0

where C = 22-m i. f 1 < m < 2 and Cm = 1 i f t h e n qmm (A) 5 qm(l) = 2qm($) 5 2 cm. A ~ - ’ . Lemma 3.

2 and (1-A)”-’

m 2 2.

On t h e o t h e r hand, i f

1 2 5A

<

1

< 2m-1Bm/(y+z).

I n t h i s case ly+z-Zxl = 2x-y-z > l y - z l so t h a t

Thus,

I n t h i s case, y - z > y - x > y+z-Zx.

Suppose z < x < y. ~ ( Y - x )L Z X

-

(Y+z).

1 [l 2

maY xl(Z -I Y+Z

Thus , Pm(y,z;x) F i n a l l y , suppose x < z < y. )

z

-

y

l

Also, y - z = x-z + X-y +

Thus,

(

3

5

’

ly+z-2xl ) I = - .Z y+z Y+L

6 [TIm-’ B

< Zm-lBm/(y+z).

Then ( y + z - Z x l > l y - z ( so t h a t

x[ 1 a- m

Y+Z

’

X

lY+z-2xl y+x ) I = -Y+Z .

Thus, pm (y,z;x)

(yfz ” x(y+z),m-l yz <

These t h r e e cases, t o g e t h e r w i t h t h e symmetry

2m-1Bm/(y+z).

- Pm(y,z;x)

= Pm(z,y;x)

-

establishes

t h e lemma. m

Lemma 4.

If m

2

1, t h e n / d x Pm(y,z;x) = 1. 0 The p r o o f o f t h i s lemma i s i n [6].

S e c t i o n 3.

I N I T I A L VALUE ZIFF MODELS

I n t h i s s e c t i o n , we show t h a t i f f o i s a bounded f u n c t i o n on [O,-) J f o ( x ) d x = 1 t h e n t h e r e i s a s o l u t i o n f o r (BEE) w i t h f ( 0 , x ) 0 t h i s s o l u t i o n i s g i v e n by

where

W,(x)

and

= f o ( x ) and

= fo(x)

and

m

Wn+l(x) = l Cn JdyW ( Y ) /dzWn+l-p(Z)pm(~,z;x) p=l 0

p

0

The s e r i e s g i v e n above has been used i n [3] t o p r o v i d e s o l u t i o n s i n normed

288

D.P. Hardin and J, V. Herod

l i n e a r spaces E f o r o r d i n a r y d i f f e r e n t i a l e q u a t i o n s which have t h e f o r m Y ' + Y = A(Y,Y)

Y ( 0 ) = E. Here, A i s a q u a d r a t i c f u n c t i o n f r o m E x E t o E. More s p e c i f i c a l l y , l e t X be a s e t and { E , I - I }

be a Banach space o f f u n c t i o n s

f r o m X t o t h e r e a l numbers R .

L e t t h e space € have t h e usual o r d e r i n g , so t h a t f > g i n {E, > , \ - I } p r o v i d e d f ( x ) L g ( x ) f o r each x i n X. And, t h e norm i s t o be

c o m p a t i b l e w i t h t h e o r d e r i n g i n t h e sense t h a t i f f ) g > 0 i n { E l >-, ( - / ) ,

If1 2

then

lgl. We suppose t h a t A i s a mapping f r o m E r E t o E which has t h e f o l l o w i n g

quadratic character: (i)A(f,g)

L

0 if f

1. 0

( i i ) A(f,g) = A(g,f), ( i i i ) A(af+g,h) = aA(f,h)

and g

+ A(g,h)

0,

f o r cx i n R, and

( i v ) t h e r e i s a convex s e t C c o n t a i n e d i n

E and c o n t a i n i n g t h e c o n s t a n t z e r o

f u n c t i o n such t h a t ( a ) i f f and g a r e i n C t h e n A ( f , g ) function A(f,f)

t

C,

and ( b ) t h e

f o r f i n E i s c o n t i n u o u s on C.

We now d e f i n e t h e b a s i c s e r i e s . Definition:

L e t f be i n E and I W In be a sequence w i t h values i n E g i v e n by P P=l n W1 = f and Wn+l -- 1- C A(b n -

n

L e t S n ( t ) = e-t Theorem 1 [3].

c (l-e-t)PW_ , T . . pfI p=o Suppose T > 0, f i s i n t h e convex s e t C, f

2

0 in

{E,L,[-[),

and l i m Sn(T) e x i s t s i n E. Then, t h e r e i s a continuous f u n c t i o n Y: [O,T) nsuch t h a t i f 0 5 t < T t h e n Y(t) = l i m Sn(t) nand

+

E

m

Y ( t ) = e-tf

+

i d s eS-tA(Y(s),Y(s)). 0

I n o r d e r t o a p p l y t h i s r e s u l t , we r e w r i t e (BEE).

F i r s t , n o t e t h i s standard

consequence o f t h e i n h e r e n t symmetry o f Pm which was i n d i c a t e d i n S e c t i o n 1 . Lemma 5 [l]. I f f i s a bounded s o l u t i o n o f (BEE) t h e n m

m

if(0,x)dx = if(t,x)dx 0 0 and

f o r a l l t > 0,

/mxf(O,x)dx = /mxf(t,x)dx f o r a l l t > 0. 0 0 T h i s lemma, t o g e t h e r w i t h Lemma 4, enables u s t o r e w r i t e (BEE) as

af (t,x) at

m

m

+ f ( t , x ) = / d y f ( t , y ) idzf(t,z)Pm(y,z;x) 0

f(0,x)

0 = fo(X).

Boltzmurin Equations

289

m

Here, i t i s assumed t h a t i f ( 0 , x ) d x = 1.

To w r i t e (BEE) i n t h e i n t e g r a t e d form,

0

identify

m

m

We seek s o l u t i o n s f o r

( IBEE) i n a f u n c t i o n space E. with

E

I n what f o l l o w s ,

llfll =

sup O<Xf u n c t i o n f such t h a t [O,m),

1.

If(x)

i s t h e space o f bounded f u n c t i o n s on

We t a k e C t o be t h e convex s e t of nonnegative UI

/f(x)dx 0 Lemma 6.

5

1.

I f f and g a r e i n C then m

u3

iA(f,g)(x)dx 0 ( i i ) for x > 0 ,

0

m

= if(x)dx.

0

llfll

5 A ( f , g ) ( x ) 5 4B,[

ig(x)dx 0

and

- l l g l l ll/*.

m

Proof.

Statement ( i ) f o l l o w s s i n c e /dxP(y,z;x) = 1. To e s t a b l i s h statement ( i i ) , 0 r e c a l l t h a t P(y,z;x) = 0 i f y+z < x and do a change o f v a r i a b l e i n t h e i n t e g r a t i o n

t o conclude t h a t m

m

i dy i dzf ( y ) g ( z ) P ( y , z ;x) 0

0 m

u

= J d u I d v f(v)g(U-v)P(v,u-v;x)

x

o m

u

< i d u i d v f(v)g(u-V)2Bm/u x o

The l a s t i n e q u a l i t y i s o b t a i n e d by r e a l i z i n g t h a t m

x

Lemma 7.

% o? d v f ( v ) g ( u - v )

0

m

and

du 7 i

0

0 and i f ( x ) d x

Suppose f

(4Bm)p/2 llfll

C

5 1

1 W (x)dx o

p

=

1.

dvf(v)g(u-v) +

1

0

i” d u i d v f ( v ) g ( u - v ) 0

=

1.

F o r p = 2,3,4

...

0

and x > 0, 0

2

W (x)’ P

D.P.Hardin and J. V. Herod

290

Proof.

m

That i W ( x ) d x = 1 f o l l o w s by i n d u c t i o n f r o m Lemma 6.

o

p

5 4Bm I l f

0 5 W,(x)

11 .

There i s a number T

Lemma 8.

(1-e t P

c p=o

and t h i s converges i f 1

-

rn

>

Then

wp+l

IBEE).

converges i n E and d e f i n e s a s o l u t i o n f o r Suppose t > 0 and

f o r 2 5 p < n.

0 such t h a t i f 0 5 t 5 T t h e n

>

e-t

Proof.

Ilf 11

Suppose W ( x ) (4Bm)p’2 P

Also,

n.

l/(%)

e

<

-t

.

That t h i s provides a l o c a l s o l u t i o n

f o r (IBEE) f o l l o w s f r o m Theorem 1. Lemma 9.

There i s a f u n c t i o n Y :

[O,m)

-t

E which i s a s o l u t i o n f o r (IBEE) f o r

a l l t. Proof.

We e s t a b l i s h t h a t i f T > 0 then any s o l u t i o n on [O,T)

ilv(t)II

2

e

< e

-t

llfll

+

J

0

-t

+

e

s-t

/” es-t

remains bounded.

I I A ( V ( s ) , v ( s ) ) l I ds

4Bm I l V ( s ) l l ds

0

Thus, by Gronwal 1 Is i n e q u a l i t y , IlV(t)\l 5

llfll

exp([4Bm-llt)

Hence, by s t a n d a r d arguments t h e s o l u t i o n may be c o n t i n u e d on [ O , - ) . m

Theorem 2.

I f f i s bounded on [ 0 , m ) ,

f ( x ) 2 0 f o r a l l x, and i f ( x ) d x = 1 then 0

converges i n E t o t h e o n l y f u n c t i o n Y: [O.m) -t E such t h a t Y s a t i f i e s t Y ( t ) = e - t f + J et-’A(Y(x),Y(s))ds.

0

Proof.

We know t h a t (IBEE) has a g l o b a l s o l u t i o n Y : [ O , - )

L e t uo(t,x) = e - t f ( x )

+

E f r o m Lemma 9.

and, f o r n 2 1 t

un( t , x ) = e - t f ( x ) + i et-sA(un-,

0

( s , ) ,un-, ( s ,-) ) ( x ) d s

Boitztnann Equations m

iun(t,.)ln=O

I n d u c t i v e l y , u n ( t , x ) 5 ~ , , + ~ ( t , x ) 5 Y(t,x). (see [3]).

29 1 converges i n Ll t o Y ( t , . )

Hence, l i m un(t,x) n-

By [3], t h e convergence o f I u n ( t , x ) l

converges t o Y ( t , x ) .

= Y(t,x).

t o Y(t,x)

implies

F o r each n and t,

i s a c o n t i n u o u s f u n c t i o n and v ( t , - )

-

e-tf(.)

i s a continuous function.

Hence,

B u t a l s o , as i n

t h e convergence i s u n i f o r m f o r x i n compact subsets o f [ O , - : . Lemma 6, A(f,g)(x)

x Hence, W

P+1

28

du

m

( x ) 5 2Bm i f x

0

o

1. 1 . .Thus, f o r t

converges u n i f o r m l y f o r x i n

[O,m).

m

I f ( x ) d x J g(x)dx.

5 2Bm I T I d v f ( v ) g ( u - v ) z $ >

0

0

T h i s completes t h e p r o o f .

REFERENCES:

M. F. Barnsley, J . V . Herod, V . V . J o r y , and G. B. Passty, The Tjon-Wu e q u a t i o n i n Banach space s e t t i n g s , J. F u n c t i o n a l Anal. 43 (1981), 32-51. A. V. Bobylev, Exact s o l u t i o n s o f t h e Boltzmann e q u a t i o n s , S o v i e t Phys. Dokl. 20 (1976), 822-824.

J . V. Herod, S e r i e s s o l u t i o n s f o r n o n l i n e a r Boltzmann e q u a t i o n s , J o u r n a l o f N o n l i n e a r A n a l y s i s : Theory, Methods, and A p p l i c a t i o n s , accepted f o r p u b l i c a tion. M. Krook and T. T. Wu, E x a c t s o l u t i o n s o f t h e Boltzmann e q u a t i o n , Phys. F l u i d s 20 (1977), 1589-1595.

J . A. T j o n and T. T. Wu, Numerical aspects o f t h e approach t o a M a x w e l l i a n d i s t r i b u t i o n , Phys. Rev. A 19 (1979), 883-888. G. T u r c h e t t i , On t h e s t r u c t u r e and a p p r o x i m a t i o n schemes o f t h e g e n e r a l i z e d Tjon-Wu e q u a t i o n , p r e p r i n t . R. M. Z i f f , Model Boltzmann e q u a t i o n s , Phys. Rev. A 23 (1981), 916-923. R. M. Z i f f , Model Boltzmann e q u a t i o n s 11. Rev. A 24 (1981), 509-513.

The M a x w e l l i a n Molecule, Phys.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS LW. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

293

ASYMPTOTIC BEHAVIOR OF SOLUTIONS

OF DISCONJUGATE

DIFFERENTIAL EQUATIONS Don H i n t o n Mat hema ti cs Department U n i v e r s i t y o f Tennessee K n o x v i l l e , TN 37996 U.S.A.

Asymptotic s o l u t i o n s a r e d e r i v e d f o r a c l a s s o f d i f f e r e n t i a l e q u a t i o n s which a r e d i s c o n j u g a t e . The method o f p r o o f r e q u i r e s a v a r i a t i o n of t h e c l a s s i c a l L e v i n s o n theorem. A p p l i c a t i o n s t o spectral theory a r e given.

1.

INTRODUCTION We c o n s i d e r here t h e d i f f e r e n t i a l e q u a t i o n

(1.1)

LY

LnY + Pn-1 L,-lY

=

where t h e o p e r a t o r s

L 0Y

Lk

+

...

+

Po LoY = 0

are defined by

= Y ; L p = rk(Lk-l

,

y)'

k = 1 ,..., n

-

1; Lny = (Ln-ly)'

The c o e f f i c i e n t f u n c t i o n s s a t i s f y

(1.2)

... '

pn-, , rl ,. . . , rn-1 a r e Lebesgue measurable, complex-valued a 5 x < b w i t h each o f p o , ..., Pn-1' l / r l ,. . . , l/rn-l Lebesgue i n t e g r a b l e on e v e r y compact s u b i n t e r v a l

PO'

f u n c t i o n s on i n t e r v a l of

[a, b)

The e q u a t i o n

2

.

Ly = 0

has t b e f o l l o w i n g m a t r i x f o r m u l a t i o n .

Define

6 and

by

(1.3)

ci

=

L 1-1 . Y , i = 1 ,..., n l/ri

, j = i + l

,

i

i = n, j = 1,

=

1,

...,

..., n , n ,

, otherwise. Then ( 1 . 1 ) i s e q u i v a l e n t t o (1.4)

C'

= d 6.

Standard e x i s t e n c e and uniqueness theorems a p p l y t o t h e system ( 1 . 4 ) .

Suppose

R

D. Hinton

294

i s a diagonal m a t r i x where t h e i t h d i a g o n a l e n t r y

(1.5)

i s a complex-valued,

Qi

s a t i s f i e s for

Qi

i = 1,

...,

n,

l o c a l l y a b s o l u t e l y c o n t i n u o u s f u n c t i o n which

never vanishes. D e f i n e now t h e t r a n s f o r m a t i o n

n

(1.6)

so t h a t

I-

=

R 5,

s a t i s f i e s the d i f f e r e n t i a l equation

I-’ =

(1.7)

[a’ a-1

+ R 4 n-']r-

We w i l l t r e a t t h e term R 4 f2-I as a p e r t u r b a t i o n term. For t h e theorem o f s e c t i o n 2 we d e f i n e a nonvanishing, complex-valued f u n c t i o n f on [a, b ) t o be e s s e n t i a l l y d e c r e a s i n g (ED) i f f o r some M > 0 , (1.8)

[f(x)/f(s)/

and we d e f i n e

f

M’

a 5 s 5 x < b

for

and

t o be e s s e n t i a l l y i n c r e a s i n q ( E I ) i f f o r some

Note t h a t if f i s (ED), t h e n l / f i s ( E I ) ; i f i f l i m sup I f ( x ) l = m as x -t b (which i n case l / f i s ( E I ) i f l i m s u p l f ( x ) l < m as x -t b

.

2.

l i m f(x) = 0 , x-tb

M > 0 ,

f i s ( E I ) , then l / f l i m I f ( x ) l = m as x

i s (ED) b ) and

+

AN ASYMPTOTIC THEOREM FOR SYSTEMS Our a s y m p t o t i c theorem f o r systems i s proved f o r a n o n l i n e a r v e r s i o n o f ( 1 . 7 ) ,

i.e., Q’ =

(2.1) where (2.2)

R

[a’ f2-l

+

i s as b e f o r e .

BIT-+

f(x,

I-)

F u r t h e r , assume

B, f a r e complex-valued, Lebesgue measurable f u n c t i o n s which a r e i n t e g r a b l e on compact s e t s , and I [ f ( x , II,)

- f ( x 3 I-1 )[I

f o r some measurable

We want t o t a k e advantage of s ume (2.3)

Sa",(x)=

:1

B ( s ) ds

y

B

satisfies

f

I r ( x ) \lo* - o,"

with

:j

y ( x ) dx <

f ( x , 0) = 0

and

-.

b e i n g c o n d i t i o n a l l y i n t e g r a b l e and t h e r e f o r e as-

exists.

Our t e c h n i q u e of p r o o f w i l l be t o a p p l y an i n t e g r a t i o n by p a r t s method used by T. Hallam [8, 91 t o a standard f i x e d p o i n t argument. By t h e v a r i a t i o n o f cons t a n t s f o r m u l a we may w r i t e ( 2 . 1 ) as

29 5

Asymptotic Behuvior of Disconjugate Equations

where

.

co = n ( a ) - ’ q ( a )

8; =

Since

, we

-B

make use o f t h e m a t r i x d i f f e r e n -

t i a t i o n formula (ABC)’ = A ’ B C + AB’C + A B C ’ i n (2.4) t o o b t a i n

1

X

[I +Bo(x)]n(x) = n(x){cl +

(2.5) where

c1 = n ( a ) - ’ G =

(2.6)

1

Q-’[Gv

+

a

-ill

[I +PBo(a)]il(a) S2-l

8, +pB,

Theorem 1 . Assume ( 1 . 5 ) , n Suppose

.

(i)

n i F c

+ 3,

0 C2-l (2.2)

for

n

such t h a t

(2.7)

as

x * b ,

___ where

ek

+

ek

i s t h e v e c t o r with

m. L e t

P1

kth

.

B

and ( 2 . 3 ) .

Then (2.1) has a s o l u t i o n n(x)lnk(x)

k

i = 1,

1

component

...,

n

Let

P2 = I

-

P,

.

be a f i x e d i n t e g e r f r o m

,

and o t h e r components zero.

be t h e d i a g o n a l m a t r i x d e f i n e d by

(ED) and zero o t h e r w i s e .

n1l

and

(EI)

(ED)

( I +B,)f(-,

(Pl)ii

= 1

if

niCi’kl

is

We c o n s i d e r t h e f o l l o w i n g s i n g u l a r

i n t e g r a l e q u a t i o n f o r m u l a t i o n o f (2.5),

1

X

(2.8)

[ I + f l o ( x ) l n ( x ) = S2(x)Iek +

Q-

P1[Gn + ( I

a

+8,)f ( - , n )

A c a l c u l a t i o n shows t h a t a s o l u t i o n o f (2.8) i s a s o l u t i o n of (2.1). @ ( x ) = n ( x ) / Q k ( x ) . Then (2.8) becomes (2.9)

[ I + ~ , ( X ) ] @ ( X ) = ek + Q ( x ) Qk(x)-’

Set

-

A s t a n d a r d argument now a p p l i e s which we o n l y s k e t c h . L e t W be t h e Banach space o f c o n t i n u o u s complex-valued, bounded v e c t o r f u n c t i o n s on [a, b ) w i t h sup norm. For @ E W , d e f i n e ( T @ ) ( x ) = -9,(x) @ ( x ) + r i g h t - h a n d s i d e of (2.9). The assumptions mdde ensure

T i s defined.

Let

M

be a common bound as i n (1.8) o r

D.Hinton

296 (1.9) f o r fb

.

Ri/Rk

It i s s u f f i c i e n t t h a t f o r a l l rb

IlGll

, and

< 1,/4M

.

Y < 1/5M

Ja ~ \ ( T @ ) ( x-) ( T $ ) ( x ) l l +

+

Thus

T

is a

x

+

b

.

A calculation yields that

\Fo(X)[@(x)

:j

M{\jG(@ -

:1

M{IIG(@ - ili)II + ( 5 / 4 )

$)\I

(5/4)

+

-

ili(x)I\I

vll+ - + \ \ I

n ( x ) Rk(x)-’

Since

q(x)

P1

0

+

as

Rk(x) @ ( x )

=

x

+

, the

b

we c o u l d w r i t e

50(x) =

G ( s ) ds

:j

which i m p l i e s t h a t

q’ =

[a

+ B + C]q

Hypothesis @(x)

-

ek

+

0

D. L u t z [ l o ] .

The

For example, i n

and i . n t e g r a t e by p a r t s as b e f o r e .

I n our a p p l i c a t i o n s below we o n l y use f ( x , q ) = C ( x ) n so t h a t (2.1) becomes (2.10)

1+3 .

p r o o f i s complete.

T h i s theorem i s s i m i l a r t o Theorem 3.1 of W. H a r r i s and method p e r m i t s i t e r a t i o n on a c o n d i t i o n a l l y i n t e g r a b l e term.

(2.5),

$11 .

~110- $\\I 5 3/4 ]I@ -

c o n t r a c t i o n mapping and has a u n i q u e f i x e d p o i n t

(i) gives t h a t as

2

x , Ilf?o(x)\l < 1 / 4 ,

f(x, q )

as a l i n e a r term, i . e . ,

.

Thus we have two small terms; C i s small i n t h e sense o f a b s o l u t e i n t e g r a b i l i t y ; B i s small i n t h e sense of c o n d i t i o n a l i n t e g r a b i l i t y . Theorem 1 i s p a r t i c u l a r l y easy t o a p p l y t o a c l a s s o f d i s c o n j u g a t e e q u a t i o n s . R e c a l l an e q u a t i o n (1 .l)i s c a l l e d d i s c o n j u g a t e on an i n t e r v a l I i f no n o n t r i v i a l s o l u t i o n has more t h a n n - 1 zeros, c o u n t i n g m u l t i p l i c i t i e s . General a s y m p t o t i c b e h a v i o r 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 e q u a t i o n s i s much more d i f f i c u l t t o o b t a i n i n case some s o l u t i o n s a r e o s c i l l a t o r y . The d i f f i c u i t y i s i n o b t a i n i n g a systems f o r m u l a t i o n t o which t h e systems theorems w i l l a p p l y . Several changes o f v a r i a b l e s a r e f r e q u e n t l y necessary. A sampling o f r e c e n t work on t h e s e more d i f f i c u l t problems may be found i n R . I . Becker [l], A. Oevimatz [3], M.S.P. Eastham [4, 51, M.P.S. Eastham and C.G.M. Grudniewicz [6], R. G i l b e r t [7], W. H a r r i s and D. L u t z [10,11], 0. H i n t o n [12], R.B. P a r i s and A.D. Wood [15], T.T. Read [16], and P . Walker [17, 181. 3.

OF THEOREM 1 TO ( 1 . 1 ) . I n (1.7) we may w r i t e R d Q - l = B + C where

APPLICATION

(3.1)

Pj-1

/n.

n

J

,

i = n, j = 1 ,

9.. = 1J

, otherwise.

0

I

0

, otherwise.

...,

n

297

Asymptotic Behavior of Disconjugute Equations

Assume now i n a d d i t i o n t o ( 1 . 2 ) and ( 2 . 3 ) , we can choose that

(i)

Ri/Rk

(iii)

:1

IlGll

is <

where

m

y i e l d s a fundamental m a t r i x

R(x) Y(x) R(x)-’

(3.2)

C o r o l l a r y 1. a > a .

some

+

I

as

m

, and

as

x

i s d i s c o n j u g a t e on [a,,

b)

for

R(x)-’

+

+

b ;

o f (1.4) such t h a t

Y

x

<

I

o f (2.10) s a t i s f y i n g ;(x)

t h u s t h e r e i s a fundamental m a t r i x

1, llCll

Then a p p l i c a t i o n o f Theorem 1

i s g i v e n by ( 2 . 6 ) .

G

k , (ii)

i and

( E I ) o r (ED) f o r each

R s a t i s f y i n g ( 1 . 5 ) so

.

b

+

(3.2) h o l d s , t h e n (1.1)

7 -

-P r o o f .

I f R“),

Y(p)

denote t h e m a t r i c e s o b t a i n e d f r o m

t a k i n g t h e elements which belong t o b o t h t h e f i r s t then (3.2) implies t h a t

p

rows and

R and Y by p

columns,

Hence by Z. N e h a r i ’ s e x t e n s i o n o f a theorem o f Polya [14], (1.1) i s e v e n t u a l l y d i scon j ugate. For t h e n e x t c o r o l l a r y suppose t h e measurable f u n c t i o n w s a t i s f i e s w(x) > 0

(3.3)

C o r o l l a r y 2.

+

[a, b )

on

with

Suppose (3.3)

i n t e g r a b l e on compact s e t s .

w

holds,

=

t h e weighted H i l b e r t space c ( a , b ) ator o f

,

w-l L

T

, and

n = 2m

L+ where L i s t h e formal a d j o i n t o f L h o l d s w i t h po r e p l a c e d by E~ - xw . Then i f

L

.

L

g i v e n by (1.1) s a t i s f i e s

Suppose f o r each X > 0 , (3.2) T i s a s e l f - a d j o i n t operator i n

which i s an e x t e n s i o n o f t h e minimal oper-

has a spectrum which i s bounded below and d i s c r e t e .

Proof. Since (3.2) h o l d s we have t h a t L - Xw i s e v e n t u a l l y d i s c o n j u g a t e f o r each A > 0 . Thus t h e r e a r e e v e n t u a l l y no n o n t r i v i a l s o l u t i o n s o f Ly = Xwy which have a p a i r o f z e r o s o f m u l t i p l i c i t y m . T h i s i s e q u i v a l e n t t o t h e spect r u m o f T b e i n g d i s c r e t e and bounded below ( c f . [ 1 3 ] ) . 4.

EXAMPLES Suppose i n (1.1 ) t h a t b = and f o r some a 1Iri(x)ILKx ,i=l,.. n. - , l , O < a < x < m .

Example 1 .

Let

E

and t a k e

> 0

.,

,I.-l-E

Qi+lcdx

1

Rn = 1

hence

, and f o r

IQi(x)/ri(x)

i = n - 1 ,..., 1

Q i + l ( ~ ) I 5 1/K

a.

satisfies

a

liCll

<

.

m

Suppose f u r t h e r t h a t

I, m

(4.1)

Qi(x) =

pi/Ri+l

exists,

i

=

0 ,..., n - 1

m

and f i n a l l y t h a t a

[lG[l

<

~0

which i s e q u i v a l e n t t o

K > 0

and

, take $ ( x ) so t h a t

ai

,

=

C

o f (3.1)

D.Hinton

298

r’ l(Qi-l Qi

14.2)

-

a

Qn-l Pi-l)/Ril

<

m

,i

1 ,..., n

=

-

1

-1

8, :0) . Hence a l l t h e hypotheses o f Theorem 1 h o l d and t h e r e i s a fundamental m a t r i x of ( 1 . 1 ) s a t i s f y i n g ( 3 . 2 ) . As a s p e c i a l case cons id e r (Note t h a t

-0 Q

U

. . . (x 2 ( x u1 y ’ ) ’ ) ’

( x n-l

(4.3)

+ poy = 0

...)I

.

Then t h e c o n d i t i o n s (4.1) and ( 4 . 2 ) reduce t o

I

co

(4.4)

Qo(x) =

po(t) t

-A

A

d t exists,

= c1

X

... +

B - (ul +

(1 +

E)(n

-

l),

-

6

a

.

) + n < 0

n-1

s i n xy , y > 0 , c o n d i t i o n s ( 4 . 4 ) h o l d i f

p o ( x ) = x’

For

...

(al +

+

-

) + n

c1

n -1

y < 0

.

Exam l e 2. I n t h i s example we show how t h e theorem o f G h i z z e t t i ( c f . [2, 2 h g e n e r a l i z e d . Suppose i n ( 1 . 1 ) t h a t a l l ri :1 on [ l , -J) . I n

p. 9

5

(1.4) s e t

=

exJ D n ( x ) r

where

J..

1

=

1J

if

j = i

+ 1 and 0 o t h e r w i s e , and

i s the diagonal m a t r i x D,(X)

= [l, x

Then f o l l o w i n g [2,

-1 , x -2

,...,

x

1 -nl

p. 911, B

on

Hence t h e r e i s a fundamental m a t r i x

Y

+ e-J D’;

= ( - 0 1n D-1 n

J

exJ D,(X)

= D,(X)

(4.5)

~~(X1~ - l ( x )o n ( x )

if

... + a n -1 -

+

p o ( x ) = xB , c o n d i t i o n s ( 4 . 4 ) h o l d i f

For

Dn

1

G =

01;

e

s a t i s f y i n g as

3o - 8O D n D-n

’0,

1,

-+

+

eJ)r

.

of ( 1 . 1 ) ( n o t e t h a t x

-f

m

R

=

D,

and

,

eJ ,

2,

-’

emJ Dn

B Dn eJ

satisfies

[ \lGll

<

where

m

z 0 ( x ) = e-J The a s y m p t o t i c s on Y i< k

and

i = 0,

...,

for

0

n

-

1 ,

D;, B D~ eJ

i n (4.5) are: i

k

.

. Yik(x)

xi-k

-+

l / ( k - i ) !as

S u f f i c i e n t conditions f o r

1;

\\GI[ <

x

+

for

are f o r

Asj,mptotic Bcliavior of' Disconjugate Equations

1

299

m

P.(x) = 1

Thus f o r and

[

Pi(t)

exists,

tn-l-i d t

x-' I P i ( x ) (

,

dx <

X

+ po(x)y

y"

= 0

x - l IPo(x)I dx <

m

, these , e.g..

conditions are

Po(x) =

, CI

p o ( x ) = ( s i n x")/x

L

t po(t) d t

> 1

exists

, s a t i s f i e s these

conditions. Example 3. An e i g e n v a l u e problem a r i s i n g d e s c r i b i n g t h e i n t e r n a l f l o w i n a gas c e n t r i f u g e i s L(y)

(4.6)

=

,0 5

( e x ( e x y " ) " ) " + Xy = 0

x <

,

m

where c e r t a i n boundary c o n d i t i o n s a r e imposed. T h i s problem i s d e s c r i b e d i n H. Wood and J . Morton [ l Y ] . T h i s i s an e q u a t i o n of t y p e (1.1) w i t h rl = r3 = r5 E 1 and r ( x ) = r 4 ( x ) = e x . Theorem 1 a p p l i e s (on [l,a)) w i t h 2

= x -1 -E

R5

x -2-2E R = e x 4 x -3-3E R3=e x

,

R6 = 1

,

I n f a c t t h e o n l y non-zero e n t r y o f

2x

,

R2=e

-4-4E 3

2X x-5-5E

,

R1 = e

[

is

G

x

0,’

-X/Rl

which i s i n x ( 1 ,

m).

Using ( 3 . 2 ) , t h e a s y m p t o t i c b e h a v i o r o f a b a s i s o f s o l u t i o n s o f ( 4 . 6 ) may be deduced. From t h e s e c a l c u l a t i o n s i t f o l l o w s t h a t (4.6) has f o u r l i n e a r l y dependent X2(0, m) s o l u t i o n s f o r e v e r y A . Thus t h e d e f i c i e n c y i n d e x o f t h e minimal operator associated w i t h (4.6) i s four. Example 4.

For t h e 2nd o r d e r e q u a t i o n w i t h r e a l c o e f f i c i e n t s .

~ ( y =) w - ' [ ( r y ' ) '

R2 = 1 , R1 ( x ) = r ( x ) / x

we may choose (i)

R1

(EI)

is

], lfii

or

1

Q,lfil

there are solutions (ry;)(x)

+

1

(ED),

(ii)

,1 < x <

1 +E

.

,

For Theorem 1 t o apply, we need

J

Qo(x) =

m

(q

- Xw)

fii' dx

e x i s t s , and

X

I-

(iii)

+ q y 1 = ~y

.

<

m.

y,,

When t h e s e c o n d i t i o n s h o l d , i t f o l l o w s f r o m ( 3 . 2 ) t h a t y2

of

Ly = Xy

such t h a t as

x

+ m

, y,(x)

+

1

and

F u r t h e r , we have t h a t

From these we have t h e f o l l o w i n g c o r o l l a r y . C o r o l l a r y 3. circle at

I f ( i ) - ( i i i ) above h o l d f o r some r e a l

i f and o n l y i f

!:

w(j:

r - 1 ) 2 dx <

.

A

, then

L

is limit-

300

D.Hinton REFERENCES

1.

R . I . Becker, "Asymptotic expansions o f second o r d e r l i n e a r d i f f e r e n t i a l equat i o n s h a v i n g c o n d i t i o n a l l y i n t e g r a b l e c o e f f i c i e n t s , " J. London Math. SOC. ( 2 ) 20( 1979), 472-484.

2.

W.A. Coppel, S t a b i l i t y and a s y m p t o t i c b e h a v i o r o f d i f f e r e n t i a l e q u a t i o n s (Heath, Boston, 1965).

3.

A. Devinatz, "The d e f i c i e n c y i n d e x o f a c e r t a i n c l a s s o f o r d i n a r y s e l f - a d j o i n t d i f f e r e n t i a l o p e r a t o r s , " Adv. i n Math. 8(1972), 434-473.

4.

M.S.P. Eastham, "Asymptotic t h e o r y and d e f i c i e n c y i n d i c e s f o r d i f f e r e n t i a l equations o f odd o r d e r , " Proc. Royal SOC. E d i n . 90A(1981), 263-279.

5.

, "Asymptotic f o r m u l a e o f L i o u v i l l e - G r e e n t y p e f o r h i g h e r - o r d e r d i f f e r e n t i a l equations," J . London Math. SOC., t o appear.

6.

M.S.P. Eastham and C.G.M. G r e i d n i e w i c z , " A s y m p t o t i c t h e o r y and d e f i c i e n c y i n d i c e s f o r h i g h e r - o r d e r s e l f - a d j o i n t d i f f e r e n t i a l equations," J. London Math. SOC. ( 2 ) ( 1 9 8 1 ) , 255-271.

7.

R.C. G i l b e r t , "The d e f i c i e n c y i n d e x o f a symmetric o r d i n a r y d i f f e r e n t i a l opera t o r w i t h complex c o e f f i c i e n t s , " J . D i f f . Equations 25(1977), 425-459.

8.

T. Hallam, " A s y m p t o t i c i n t e g r a t i o n o f second o r d e r d i f f e r e n t i a l e q u a t i o n s w i t h i n t e g r a b l e c o e f f i c i e n t s , " S I A M J. Appl. Math. 19(1970), 430-439.

9.

, "Asymptotic i n t e g r a t i o n o f a nonhomogeneous d i f f e r e n t i a l e q u a t i o n w i t h i n t e g r a b l e c o e f f i c i e n t s , " Czechoslovak Math. J . 21 ( 9 6 ) 1971, 661-671.

10.

W.A. H a r r i s and D.A. Lutz, "Asymptotic i n t e g r a t i o n o f a d i a b a t i c o s c i l l a t o r s , " J . Math. A n a l . Appl. 51 (1975), 76-93.

11.

"A u n i f i e d theory of asymptotic integration," 5 7 0 , 5 7 1 -586.

12.

D. H i n t o n , "Asymptotic b e h a v i o r o f s o l u t i o n s o f D i f f . Equations 4(1968), 590-596.

13.

0. H i n t o n and R. Lewis, " S i n g u l a r d i f f e r e n t i a l o p e r a t o r s w i t h s p e c t r a d i s c r e t e and bounded below," Proc. Royal. SOC. Edin. 84A(1979, 117-134).

14. 15.

Z. Nehari, " D i s c o n j u g a t e l i n e a r d i f f e r e n t i a l o p e r a t o r s , " Trans. AMS 129(1967), 500- 51 6. 2 R.B. P a r i s and A.D. Wood, "On t h e L n a t u r e o f s o l u t i o n s o f n t h o r d e r symm e t r i c d i f f e r e n t i a l e q u a t i o n s and McLeod's c o n j e c t u r e , " Proc. Royal. SOC. Edin. 90A(1981), 209-236.

16.

T.T. Read, " H i g h e r - o r d e r d i f f e r e n t i a l e q u a t i o n s w i t h small s o l u t i o n s , " London Math. SOC. ( 2 ) 19(1979), 107-122.

17.

P.W. Walker, "Asyrnptotica o f t h e s o l u t i o n s t o J. D i f f . Equations 9 ( 1 9 7 1 ) , 108-132.

18.

J. Math. Anal. Appl.

+ qy = 0," J . (ry(m))(k) -

( ( r y " ) ' - p y ' ) ' + qy

=

J. oy,"

, " A s y m p t o t i c s f o r a c l a s s o f f o u r t h - o r d e r d i f f e r e n t i a l equations," J . D i f f . Equations 11 (1972), 321-334.

19.

H.G. Wood and J.B. Morton, "Onsager's pancake a p p r o x i m a t i o n f o r t h e f l u i d dynamics o f a gas c e n t r i f u g e , " J . F l u i d Mech. 101(1980), 1-31.

DIFFERENTIALEQUATIONS I.W. Knowles and R.T.Lewis (Editors) 0 Elsevier Science PublishersB.V. (North-Holland),1984

30 1

BOUNDARY CONDITIONS FOR DIFFERENTIAL SYSTEMS IN INTERMEDIATE LIMIT SITUATIONS

J. K. Shaw

Allan M. Krall Mathematics Department Pennsylvania State University University Park, PA 16802

D. B. Hinton

Mathematics Department Mathematics Department Virginia Polytech Institute University of Tennessee and State University Knoxville, TN 37916 Blacksburg, VA 24061

1. INTRODUCTION. One of the major stumbling blocks in the development of n

the singular Sturm-Liouville theory f o r

dimensional vector systems is the

appropriate definition and proof of existence of boundary conditions. While a number of authors [ I ] , [2], [ll], [3-101 have examined various special cases and have on occasion used limits of Wronskians, to date a general development of boundary conditions has been missing. ment.

The present article presents such a develop-

It was motivated by [2] which considered the 4th order, scalar, limit-4

situation, which generalized to include the n-th order, scalar, limit-n case, as well as the n 2.

dimentional limit-n, first order case.

THE SYSTEM CF EQUATIONS. We consider the S : Jy'

where

A(t)

and

B(t)

(hA(t)

=

are complex

able over the interval

(a,b),

constant, skew hermitian

(J*

n

+ A(t)f

B(t))y

n, hermitian matrices, locally integr-

where =

+

dimensional vector system

n

A(t)

J

is nonnegative definite and

satisfying J2

-J),

=

is

-I.

We assume that Atkinson’s definiteness condition [l, p. 2531 holds, namely d

*

Jc y Aydt > 0, a

where

*

c

denotes the conjugate transpose, when

Associated with

S

c

y

d < b, satisfies S

with

f

=

0.

is its transpose system

T : -z*'J Formal

i

premultiplication of

z"(XA(T)

=

*

by

S

z

+

B(t))

+

g*A(t).

and postmultiplication of

T by

y , and

subtracting yields Lagrange’s formula (z*Jy)'

DEFINITION. We denote by

2 L,(a,b)

llyll In

L2(a,b) A

z*Af

-

g*Ay.

those elements y(t) b *

=

=

(fa y

in Rn

satisfying

AY dt)li2 <

we can define a (maximal)

L

in order to develop Green’s formula.

A.M. Krall et aL

302 DEFINITION.

We d e n o t e by

D,

those elements

y

2 LA(a,b) with t h e following

in

properties.

1. y

i s a b s o l u t e l y c o n t i n u o u s on compact s u b s e t s of

(a,b).

Hence

y’

e x i s t s a .e .

2.

For e a c h

t h e r e i s a n element

y

f

For a l l

y

in

D

such t h a t

+ A(t)f

Jy’ = B ( t ) y DEFINITION.

2 L (a,b)

in

we d e f i n e t h e o p e r a t o r

L

by s e t t i n g

Ly = f

whenever

+ Af

J y ‘ = By I n o t h e r words Ly

f

i f and o n l y i f

Lz = g

i f and o n l y i f

=

Jy‘

-

By

Jz

-

Bz = Ag.

2

Since t h e inner product i n

LA(a,b)



=

Af,

=

i s g i v e n by

Jb z*Aydt,

G r e e n ’ s formula can b e s t a t e d : (GREEN’S FORMULA).

THEOREM.

y

Let

and

D , Ly = f , Lz = g.

be i n

z

- =

(Z

*

Jy)

Then

b a

PROOF =

=

b

*

Ja z

Ji b

Afdt -

Jab

z*[Jy’-Byldt

*

= J a [ z Jy’+z

*

* g Aydt

-

Ja[Jz’-Bz]*ydt b

*’Jyldt

b

= ( z JY),

While t h e l i m i t s on t h e r i g h t o f Green’s formula c e r t a i n l y e x i s t , i n d i v i d u a l

terms which make u p t h e sum a t of a r e s t r i c t i o n of (z*Jy);

=

3.

0.

L

a

and

b

do n o t i n g e n e r a l .

For s e l f - a d j o i n t n e s s

t o o c c u r , boundary c o n d i t i o n s must be imposed so t h a t

I t i s n o t obvious from t h i s approach t h a t t h i s i s a t a l l p o s s i b l e .

BOUNDARY CONDITIONS.

I t i s w e l l known t h a t € o r each complex

homogeneous system HS : Jy‘

=

(XA(t)

+ B(t))y

A,

the

303

Boundury Conditions for Differential Systems

2 h a s a number of s o l u t i o n s which a r e i n LA(a,c) as w e l l as a number of s o l u t i o n s 2 which are i n LA(c,b) where c i s any f i x e d number a < c < b . W e examine t h e interval

(c,b)

2 L (c,b) A

f i r s t and show how e a c h s o l u t i o n i n

g e n e r a t e s a boundary

c o n d i t i o n i n t h e s e n s e of Dunford and Schwarz. Let

U

b e a fundamental m a t r i x f o r

JU’ Let

-1

V = JU

,

*

so that

the adjoint equation.

(Note t h a t

but with

ho

r e p l a c e d by

y

in

D,

For

-

when

X

=

Xo.

So

(hoA(t) -t B ( t ) ) U

=

and

V JU = I

-v*’J

U,

HS

= v*(~,,A+B),

V

s a t i s f i e s t h e same homogeneous e q u a t i o n a s

Sy

=

hO’

we s e t

*

V Jy.

Then (Sy)’

=

= =

+ V*Jy’ V*(hoA+B)y + V*(By+Af) * V A(f-Xoy). V*’Jy

Let

where

v

THEOREM.

* j

- .,..., =J

.).

= (v.

If

v.

The elements

nJ

2 LA(c,b),

is i n

v. J

satisfy

HS

with

h =

%0 ‘

then B.y

=

l i m S.y x+b

exists. Proof. If

y

( S j y ) ‘ = v* A(f-hoy) j

is in

then

D,

f

and

y

S.y(b’) 3 has a l i m i t a s THEOREM.

Let

b‘

+

2

are in

LA(c,b). =

b‘

Sc

Thus

*

+

v.A(f-X0y)dx J

S.Y(C)

I

b.

S4y(b’)

have a l i m i t as

b’

+

b

for a l l

y

in D .

Then

A.M. Krall et al.

304 2

is in

v,

Proof.

L,(c,b).

Note t h e e q u i v a l e n c e of t h e problems Jy’ - By

=

Af

and J y ’ - By -

X0 y

= Ag,

where f That i s , i f w e a r e given kOy

may b e a r b i t r a r y .

domain i n c l u d e s a l l of Define

T

.,

g, Thus

A

h y

=

0

w e may f i n d

Sb c

y

j

b

i s f i n i t e , o r by s e t t i n g

when

b

is infinite.

By a s s u m p t i o n

4

converges f o r each

J

b-c

sc

T

.

“J

=

By t h e Hahn-Banach theorem

2

s e n t a t i o n theorem

d e f i n e s a l i n e a r f u n c t i o n a l whose

* v

j

A 4 dx

4

exists for all

in

2 LA(c,b)

and

to

T.4

-.

f -

T h i s shows t h a t

by s e t t i n g

“J

when

*

f.

and t h e n

v* A(f-hoy)dx

b-(-)

n

g.

L (c,b). A

T .(4) = nJ

as

+

is i n

v

Sb

C

T. J

v . A$ dx J

i s bounded.

Thus by t h e R i e s z r e p r e -

L (c,b).

j THEOREM.

B;y = l i m S 4 y i s a boundary c o n d i t i o n f o r

is in

v.

y

in

D

i f and o n l y i f

Lf;(c,b).

Thus w e h a v e showed t h a t e a c h column

2

L (c,b)

A

v&~, is

...v

not i n

U = -JV

*-1

2

...,

LA(c,b),

.

A t the point

4.

v . o f t h e m a t r i x V which i s i n J I f t h e s e are o r d e r e d v l , v , with m t h e a p p r o p r i a t e fundamental m a t r i x U t o b e g i n w i t h

g e n e r a t e s a boundary c o n d i t i o n .

x = a,

GREEN’S FORMULA,

s i m i l a r c o m p u t a t i o n s can b e made w i t h l i k e r e s u l t s .

WITH BOUNDARY C O N D I T I O N S .

Green’s f o r m u l a , i n t r o d u c e d

e a r l i e r , c a n b e m o d i f i e d t o i n c o r p o r a t e t h e r e s u l t s of t h e p r e v i o u s s e c t i o n .

term

( z * J ~ ) c~a n b e r e p l a c e d a s f o l l o w s .

If

x

is near

b

Sy = V*Jy

is

The

Bouridarjl Conditions f o r Differential Sjssterns equivalent t o z*

=

y = -JV

(Sz)*V-l J .

*-1

(Sy)

and

(Sz)*

=

*

(V J z )

>k

305

is equivalent t o

Hence (z*Jy) (b) = l i m ( S Z ) * ( V - ~ J V * - ~(Sy) ) x-tb

Similarly near by

6, ?,

x

=

a,

i f w e d e n o t e t h e a p p r o p r i a t e fundamental m a t r i c e s

etc., ( z * ~ y (a) )

=

lim x+a

(~~)*(V-'.JV*-')

(~y)

Consequently, THEOREM (GREEN’S FORMULA, WITH BOUNDARY CONDITIONS).

Ly

=

f, Lz

=

g.

Let

y

and

z

be i n

D,

Then < L y , z > - =

l i m ( s ~ ) * ( v - ~ J v ~( -s y ~ )) x-tb

- lim

(Sz)

* (V--I JV *_I) (Sy)

x-ta The l i m i t s above must e x i s t , b u t i n d i v i d u a l t e r m s i n t h e m a t r i x p r o d u c t s 2 Only i n t h e l i m i t - n c a s e , when a l l s o l u t i o n s of HS are i n LA(a,b)

may n o t .

w i l l a l l the individual l i m i t s exist.

S i n c e we wish t o e v e n t u a l l y conclude t h a t

t h e s e l i m i t s c a n be made t o b e z e r o , a n o t h e r approach i s r e q u i r e d .

306

A.M. Krall et al. REFERENCES

[l]

Atkinson, F. V., DISCRETE AND CONTINUOUS BOUNDARY VALUE PROBLEMS (Academic Press, New York, 1 9 6 4 ) .

[2]

Fulton, C . T. and Krall, A. M., "Self-adjoint 4-th order boundary value problems in the limit4 case," Symp. Ord. Diff. Eq. Ops. 1982, Lect. Notes Math. Spring Verlag, W. N. Everitt, R. T. Lewis Editors, 1983.

[3]

Hinton, D. B. and Shaw, J. K., "On the spectrum of a singular Hamiltonian system," Quaestions. Math, 5 (1982) 29-81.

[41

, "On boundary value Problems for Hamiltonian systems with two singular points." submitted.

[51

, "Well posed boundary problems for Hamiltonian systems of limit point or limit circle type," Ord. and Part. Diff. E q . Proc., Dundee 1982, Lect. Notes Math. 964, 614-631.

[61

, "Titchmarsh’s -dependent boundary conditions for Hamiltonian systems," Ord. and Part. Diff. Eq. Proc., Dundee 1982, Lect. Notes Math. 964- 298-317.

~ 7 1

, "On Titchmarsh-Weyl M( )-Functions for linear Hamiltonian systems," J. Diff. Eq., 40 (1981) 316-342.

[81

, "Titchmarsh-Weyl theory for Hamiltonian systems," Spec. Th. of Diff. Op., North-Holland, I. W. Knowles and R. T. Lewis, Eds., (1981) 219-231.

[gl

, "Parametrization of the MQ.) Hamiltonian system of limit-circle tape," submitted.

[I01

, "Hamiltonian systems of limit-point or limitcircle type with both endpoints singular," submitted.

function for a

[ll] Kimura, T. and Takahasi, M., " S u r l e s operateurs differentials ordinaires linears formellement autoadjoint I," Funkcial. Ekvac. 7 (1965) 35-90.

DIFFERENTIAL EQUATIONS 1.W. Knowles and R.T. Lewis (Editors) QElsevier Science Publishers B.V. (North-Holland), 1984

307

ON THE SPECTRUM OF A HAMILTONIAN SYSTEM WITH TWO SINGULAR ENDPOINTS J.K.

D.B.

Shaw

Department o f Mathematics V i r g i n i a Tech Blacksburg, VA 24061 U.S.A.

Hinton

Department o f Mathematics U n i v e r s i t y o f Tennessee K n o x v i l l e , TN 3 7 9 9 6 - 1 3 0 0

U.S.A.

I n t h i s paper we connect t h e p o l e s t r u c t u r e o f t h e TitchmarshWeyl m - c o e f f i c i e n t s w i t h t h e spectrum o f a H a m i l t o n i a n system w h i c h i s s i n g u l a r a t each end o f an i n t e r v a l . C h a r a c t e r i z a t i o n s a r e g i v e n f o r t h e r e s o l v e n t s e t , p o i n t spectrum, c o n t i n uous spectrum and p o i n t - c o n t i n u o u s spectrum. We a l l o w t h e system t o b e o f e i t h e r l i m i t p o i n t o r l i m i t c i r c l e t y p e a t each end. INTRODUCTION We c o n s i d e r a ( 2 n ) t h o r d e r l i n e a r H a m i l t o n i a n system ([1],[5])

,

J$I = [ h A ( t ) + B ( t ) ] ; + A ( t ) ? ( t )

- m

5

a

i

t < b

5

m

,

(1.1)

and t h e a s s o c i a t e d homogeneous e q u a t i o n

5;'

= [hA(t)+B(t)]$,

a < t < b,

(1.2)

where A ( t ) and B ( t ) a r e 2nx2n m a t r i c e s , h i s a complex parameter,

?

Lo -I1

w i t h A nonnegative d e f i n i t e (A r x r

=

$ ( t ) and

J = , where I = In i s t h e nxn i d e n I We t a k e A ( t ) and B ( t ) t o be l o c a l l y i n t e g r a b l e , complex H e r m i t i a n m a t r i c e s ,

= ? ( t ) a r e 2n*1 v e c t o r f u n c t i o n s and

tity.

$

and i n v e r t i b l e ( 1

"definiteness"

2

r s 2n).

5

hypothesis

:],

0) and o f t h e f o r m A =l : [

where A,(x)

is

F i n a l l y , suppose t h a t ( 1 . 2 ) s a t i s f i e s A t k i n s o n ' s

I," PA$

>

0

,a

5

c < d

5

b , f o r every n o n t r i v i a l solu-

t i o n y' o f ( 1 . 2 ) . T h i s i s an assumption on b o t h t h e m a t r i c e s A ( x ) and B ( x ) . D e f i where n i t e n e s s f a i l s f o r t h e system yl' = b ( t ) y , , - y i = ( X p ( t ) + b l , ( t ) ) y , + b y , , = ('l),y , ( t ) E O , y 2 ( t ) = y 2 ( 0 ) e - b t , Y ; p(t) y t ( t ) d t However, ( 2 n f t h o r d e r l i n e a r o r d i n a r y d i f has v a n i s h i n g norm 1 f e r e n t i a l e q u a t i o n s ([5] , [ 1 5 ] ) , as w e l l as D i r a c systems ( [ 1 2 ] ) , a r e imbeddable as " d e f in i t e " sys terns.

b i s a constant, f o r t h e n o n t r i v i a l s o l u t i o n

.

We p a r t i t i o n 2 n r l v e c t o r s i n t o f i r s t and second n x l components by w r i t i n g o r sometimes we w i l l use s u b s c r i p t s y

=

(Ek)

$

when i t seems c l e a r e r t o do so.

=

( Yx ) ,

We say t h a t a v e c t o r f u n c t i o n ? ( t ) i s o f " i n t e g r a b l e square towards b" i f

1 ; ?*A?

<

m ,

for a

<

d

<

b , and " i n t e g r a b l e square towards a " i s d e f i n e d

These c o n d i t i o n s a r e w r i t t e n ? E L i ( d , b ) b r e s p e c t i v e l y ; ? ~ L a ( a , b ) w i l l s i g n i f y ia ?*A? < m

similarly.

and

.

?E

La(a,c)

(a

<

c < b)

308

J. A'. Shaw mid D.4. Hinton

Given a complex number A , l e t Na(h) and N b ( h ) denote the numbers of l i n e a r l y independent s o l u t i o ns of ( 1 . 2 ) which belong t o L 2 ( a , c ) A a < c < b . Kogan a n d Rofe-beketov ( [ l l ] ) have shown s t a n t i n t h e upper and lower half A-planes. Furthe r, r e a l A", then N a ( A ) = 2n f o r a l l A , a n d s i m i l a r l y f o r

and L 5 ( c , b )

, re spe c tive ly,

t h a t these indices a r e coni f N ( A , ) = 2 n f o r some t h e aendpoint b .

W e term a n endpoint having 2n l i n e a r l y independent s o l u t i o n s of ( 1 . 2 ) of c l a s s a " l i m i t c i r c l e " endpoint. I f t h er e a r e only n l i n e a r l y independent L$ solu-

L5

t i o n s t h e r e , we c a l l the endpoint " l i m i t p o i n t" . The principal hypothesis in t h i s paper i s t h a t each erzdpoint of la,b) i s e i t h e r l i m i t p o i n t or l i m i t circle. Our purpose here w i l l be t o o u t l i n e a theory which r e l a t e s the spe c tra of boundary problems associated with ( 1 . l ) t o the Titchmarsh-Weyl rn-coefficients of each endpoint. Full d e t a i l s may be found in t he forthcoming a r t i c l e [SJ. We c a rr y forward much of t h e a n a l y s i s of [6], and we continue t o r e l y upon and f u r t h e r extend the b as i c paper [ 2 ] of Chaudhuri and E v e r i t t .

AN OPERATOR EQUATION We wish t o think of ( 1 . 1 ) as a Hi l b er t space operator equation T : D(T)+H, and t o study i t s spectrum i n t h i s l i g h t . Introducing the notation Er = , where r n r i s t h e s i z e of t h e A , ( x ) block of A ( x ) , we note t h a t ( 1 . 1 ) can always be w r i t t e n

[ir :]

[ :](

JT' - )B;

or equivalently ( T - A E r ) P -+

g = E,.?

+ Eif

= hEr$ =

$,

(2.1)

where T;

denotes the l e f t s i d e of ( 2 . 1 ) a n d where

i s an a r b i t r a r y element o f t h e space ErLa (a ,b)

=

tEr? 1

$E

La(a,b)}.

Thus E r L i ( a , b ) c o n s i s t s of members of LZA(a,b)with l a s t 2n-r components equaling

0; by i n v e r t i b i l i t y of A , ( x ) , E r L a ( a , b ) i s a Hilbe rt space under the inner proTo d ef i n e the domain D(T), we begin w i t h the requirement duct (?,$) = J:?*AC. (i)

["

]

0 12n-r

(J;'

- )B;

=

0 for

;E

D(T) , under which ( 2 . 1 ) and ( 1 . 1 ) a r e seen t o

be equivalent. Under t h e canonical imbedding of a ( 2 n ) t h order l i n e a r ordinary d i f f e r e n t i a l equation as a Hamiltonian system, condition ( i ) i s the i d e n t i t y O = O ; f o r Dirac systems ( i ) i s vacuous becquse r = 2n = 2 . Further requirements f o r E ; D ( T ) a r e t h e standard onss ( i i ) y i s l o c a l l y a bsolute ly continuous on ( a , b ) , :](JT' -B;) i E r L a ( a , b ) (compare [3], p.244) . Fi( i i i ) Y E La(a,b) , (iv)

[:+

n a l l y , we need t o impose boundary conditions a t t h e endpoints t o determine the members of D ( T ) . F i r s t l y , a t a l i m i t point end, Y Z O bozinJarg ~ 0 7 ~ J i t i 0 1 will t 11s needed. A t a l i m i t c i r c l e endpoint, we wi l l use the boundary conditions which were developed by t h e authors i n [ 7 ] a n d [S]. Supposing t h a t , sa y, x = b i s l i m i t c i r c l e , we choose a real i. = a n d a fundamental matrix Y b ( t ) of ( 1 . 2 ) , corresponding t o h = Xb, which a l s o s a t i s f i e s Y b * ( t ) J Y b ( t ) = J , a i t < b . i s n o t d i f f i c u l t t o show ([S]) t h a t t h e l i m i t exists for

It

Jiy,(r;(t);(t))l

y

E

L a ( a , b ) , where the s u b s cr i p t denotes t h e f i r s t n x l component; we abbreviate I f x = a i s l i m i t c i r c l e , we may analogously form the l i m i t

this to (?t;),(b).

(r:y')2(a)

, t h i s time using the second nxl component and a fundamental matrix Y a ( t ) (Ya* JYa J ) corresponding t o a r eal i = 4a . The boundary conditions ( Y 2 $ ) l ( b ) = 0 and = 0 , t o be imposed only a t a l i m i t c i r c l e endpoint, lead t o well-posed boundary problems f o r ( 1 . 1 ) with I m ( x ) # 0 ,f E L i ( a , b ) , a n d

(r:$)2

Spectrum

for

T E Li(a,b)

.

' f L%(a,b) ~

If

and Im(A) # 0 ,

by ( 1 . 1 ) w i t h c o n d i t i o n s ( r & $ ) l ( b )

.

Li(a,b)

E ;

309

Hatiiiltotiiati Svstrms

Indeed, t h e f o l l o w i n g r e s u l t h o l d s .

Theorem A ( [ 7 ] ) . for

of

=

t h e n t h e boundary problem g i v e n

$ and (Y:$)*(a)

i s uniquely solvable

=

I t i s understood t h a t a t a l i m i t p o i n t e n d p o i n t no boundary

c o n d i t i o n i s imposed. The r e s u l t o f Theorem A g i v e s t h e f i n a l r e q u i r e m e n t f o r

(r:;)2(a)

is l i m i t c i r c l e , (Y2$ll(b)

= 0 if x = a

Y E

D(T)

TT

=

[:*

-

( v ) , we l e t T : D ( T ) + E r L i ( a , b )

:](JT'-B$)

ti(x,t,x)

l[G?llA

$

(v)

W i t h D(T) now

be t h e o p e r a t o r d e f i n e d by

; t h u s (2.1) becomes (T-AEr);

rhe u n i q u e s o l u t i o n

namely

= 0 i f x = b i s l i m i t circle,

no boundary c o n d i t i o n t o be imposed a t a l i m i t p o i n t e n d p o i n t . defined- by ( i )

,

=

5.

i n Theorem A i s g i v e n i n terms o f a Green's f u n c t i o n

and c o r r e s p o n d i n g r e s o l v e n t o p e r a t o r $ ( x ) = (GA?)(x) which s a t i s f i e s 5

II.II A denotes t h e norm coming from t h e i n n e r p r o Thus each n o n - r e a l A l i e s i n t h e resoZvent s e t p ( T ) of T,

l I m ( ~ ) [ ll?lIA, -~ where

.

duct o f ErLi(a,b)

meaning t h a t T - x E :~D(T) + E r L i ( a , b )

i s one-to-one and o n t o w i t h bounded i n v e r s e .

The spectrwn 5 ( T ) i s d e f i n e d as t h e complement o f p ( T ) i n t h e complex p l a n e ; u ( T ) t h e r e f o r e c o n s i s t s o f r e a l numbers o n l y . The p o i n t spectrum P(T) c o n s i s t s o f i s o l a t e d p o i n t s ( e i g e n v a l u e s ) o f u ( T ) ; t h e e s s e n t i a l spectrum i s E(T) = u ( T ) - P ( T ) ; t h e p o i n t - c o n t i n u o u s spectrum PC(T) c o n s i s t s o f ( n o n - i s o l a t e d ) e i g e n v a l u e s i n E(T) ( t h o s e X f o r which (1.2) has a n o n t r i v i a l L i ( a , b ) s o l u t i o n ) ; t h e c o n t i n u o u s spectrum i s C(T) = E ( T ) - PC(T) . I t w i l l be c o n v e n i e n t t o g i v e t h e f o r m o f t h e Green's f u n c t i o n used i n c o n n e c t i o n w i t h Theorem A, and t h i s r e q u i r e s i n t r o d u c t i o n o f t h e Titchmarsh-Weyl r n - c o e f f i c i e n t s ma(h) and mb(A). F i x a base p o i n t x = e, a < e < b , and l e t Z(t,A) be a

fundamental m a t r i x s o l u t i o n o f ( 1 . 2 ) which s a t i s f i e s Z(e,A) = Ipn f o r a l l complex

A.

Break Z(t,X)

i n t o nxn b l o c k s by w r i t i n g Z =

[i

$1 .

p o i n t , l e t us d e f i n e ma(A) = -

A t a l i m i t p o i n t end-

=-di+%<

l i m < ' ( t , X ) e(t,A) o r mb(A) l(t,h)e(t,h), t-ta I m ( x ) # 0 , i f e i t h e r i s a p p r o p r i a t e . These a r e nxn m a t r i x a n a l y t i c f u n c t i o n s which belong t o t h e s o - c a l l e d P i c k - N e v a n l i n n a c l a s s e s o f t h e h a l f - p l a n e s I m ( i ) > 0 , I m ( h ) < 0 ( [ 5 ] , [6], 1[7]). I n case an e n d p o i n t i s l i m i t c i r c l e , we d e f i n e i n s t e a d ma(x) =

-

(ri& ( r i z ) 2 ( a )

o r mb (A)

=

-

1 (r$$)-l (rda)l(b)

,

i f e i t h e r i s a p p r o p r i a t e . These d e f i n i t i o n s may be shown t o agree w i t h t h e c l a s s i c a l l i m i t c i r c l e d e f i n i t i o n s i n t h e s c a l a r case ([14], [B]); t h e y " p a r a m e t e r i z e " t h e m - c o e f f i c i e n t s , a t a l i m i t c i r c l e e n d p o i n t , i n t h e sense o f [B].

F o l l o w i n g T i t c h m a r s h ' s n o t a t i o n ([14, p.251) we l e t Z a ( t , A ) = $(t,A) +$(t%A)ma(X), + + $Jb(t,X) = $ ( t , x ) +$(t,h)ma(A), f o r Im(h) f 0. I n a l l cases we have $a E L i ( a , e ) , + $,~L$(e,b) ([7]). The Green's f u n c t i o n f o r t h e boundary problem of Theorem A i s g i v e n by ( [ 7 ] ) G(x,t,A) = Gb(x,X)(ma(h) - m b ( h ) ) - l $,*(t,x), x > t, and G(x,t,X) = $,(x,X)(ma(A) -mb(h))-l$b*(t,i) , x < t , I m ( x ) # 0 (compare [ l o ] ) . The r e s o l v e n t o p e r a t o r i s t h e i n t e g r a l o p e r a t o r G(.,A,-), where G(x,A,?) =

:J

G(x,t,A)A(t)?(t)

dt

.

J.K. Shuw und D.B. Hinton

310 THE CLASSIFICATION THEOREM

R e c a l l i n g t h a t t h e base p o i n t x = e i s f i x e d , l e t us d e f i n e f o r Im(h) # 0 F ( x ) = (1/2)[G(e,e+,h) + G(e,e-,A)]; F ( h ) i s t h e " c h a r a c t e r i s t i c f u n c t i o n " o f A t k i n s o n ([1, p.2681). One c a l c u l a t e s ([9],[7]) that

CZassifiication Theorem ( [ 9 ] ) . T holds

The f o l l o w i n g c l a s s i f i c a t i o n o f t h e spectrum o f

.

( i ) The p o i n t h o ~ p ( T )i f and o n l y if F i s a n a l y t i c a t X o ; t h e r e s o l v e n t o p e r a t o r a t such p o i n t s i s g i v e n by G(-,XO,.)

.

(ii)

The p o i n t h E P ( T ) i f and o n l y i f F has a s i m p l e p o l e a t h o ; i f 0 (h-ho) + then F(X) = l - l ( h - X O ) - l + +

.--

lo

Z(t.ho) (iii)

E

The p o i n t

A

Li(a.b), ~

g i v i n g t h e eigenfunctions.

C(T) E i f and o n l y if F i s n o t a n a l y t i c a t h o and

vF(ho+iv) = 0 . +?I The p o i n t hO E PC(T) i f and o n l y i f

li

V

(iv)

,,-l i %-v F ( x o +

i v ) = S # 0 and

-f

F(h) - i ( h

- A0)-l

S i s n o t a n a l y t i c a t ho; t h e eigenfunctions

a r e g i v e n by Z(t,ho) S

.

b u t a few r e D e t a i l s of t h e p r o o f s may be found i n [9] (compare a l s o [6],[2]) marks a r e i n o r d e r c o n c e r n i n g them. The s u f f i c i e n c y o f (i) i s a consequence o f t h e i d e n t i t y ([9])

(-1 ( t , x ) * , x < t , Making h = h + i v + h o i n (+I 0 (2.3), t h e l e f t s i d e approaches F ' ( h O ) ; t h u s t h e r i g h t s i d e i s f i n i t e a t A 0' meaning t h a t t h e Green's f u n c t i o n G(x,t,xo) and r e s o l v e n t o p e r a t o r G ( - , h O , - ) have Now t h e Green's f u n c t i o n may be w r i t t e n G(x,t,h)

G(x,t,x)

=

- Z(x,x)

a sense t h e r e .

Z

= -Z(x,h)Z

x > t , Im(x) f 0 .

(t,x)*,

I t i s n o t d i f f i c u l t t o see t h a t G ( * , h O , - )then a c t s as a bounded

i n v e r s e f o r T - aOEr,

so t h a t

A

~

p(T). E

I n t h e o t h e r d i r e c t i o n f o r ( i ) , a stand-

a r d f o r m u l a o f [2] ( c a r r i e d o v e r t o t h e p r e s e n t general s e t t i n g ) i n v o l v i n g t h e r e s o l v e n t o p e r a t o r G(.,h,-) a f f o r d s t h e a n a l y t i c c o n t i n u a t i o n o f F t o h O € p ( T )

.

The s u f f i c i e n c y o f ( i i ) i s a l s o a consequence o f ( 2 . 3 ) , w i t h h = h o + i v and h o a s i m p l e p o l e o f F. M u l t i p l y i n g ( 2 . 3 ) by v 2 and t h e n l e t t i n g v + O r e s u l t s i n b t h e i d e n t i t y ja (Z I - l ) * A ( Z I - l ) = . A f t e r arguing t h a t Z satisfies a p p r o p r i a t e boundary c o n d i t i o n s a t a l i m i t c i r c l e e n d p o i n t , one sees t h a t i t produces t h e e i g e n f u n c t i o n s c o r r e s p o n d i n g t o h o . The n e c e s s i t y o f ( i i ) may be proved u s i n g an extended T i t c h m a r s h - K o d a i r a f o r m u l a as i n [S]. Regarding ( i v ) , f o r t h e s u f f i c i e n c y one needs o n l y t o roduce e i g e n f u n c t i o n s . r e p l a c e d by S. The T h i s i s accomplished j u s t as i n p a r t ( i i ) above, w i t h o t h e r h a l f o f ( i v ) , and p a r t ( i i i ) , a r e e s t a b l i s h e d much t h e same as t h e corresponding r e s u l t s i n [6].

f-,

Ezmple.

no i n d i v i d u a l p o l e o f ma

I f i n ( 2 . 2 ) ma and mb a r e s c a l a r f u n c t i o n s ,

o r mb w i l l be a p o l e o f F, u n l e s s i t i s s i m u l t a n e o u s l y a p o l e o f b o t h , due t o c a n c e l l a t i o n . T h i s c a n c e l l a t i o n of p o l e s does n o t always o c c u r i n cases beyond t h e s c a l a r one, as examples show ( [ 9 ] ) . I f A D i s a p o l e o f F , b u t ma and mb a r e a n a l y t i c t h e r e , l e t ( m b ( h ) - m a ( A ) ) - l = ( x - A ~ ) - ~ u ~ ~ + u ~ + u , ( x, - x ~ ) + ~ ~ ~ a a b b rna(A) = u o + u l (1- X o ) + a * . , mb(A) = a. + u l ( A - h o ) + ; t h e U ‘ S a r e a l l nxn o-lcob “ - 1 “oa = 0; m a t r i c e s . One may compute t h a t = b b a b] ;

[

.] [ . a -

0 “-1

“0

0 0 0-1

“-100

-t

-f

t h u s $ a ( t , A o ) a - l = $ b ( t , A o ) u - ~, g i v i n g t h e e i g e n f u n c t i o n s .

00 “-1QO

To i l l u s t r a t e ,

-

, c o n s i d e r t h e “square w e l l “ p o t e n t i a l problem -y” + V ( x ) y = hy , - m < x < where V(x) = -B, 0 < x < x o , and V ( x ) = 0 o t h e r w i s e . S i n c e b o t h ends a r e l i m i t P o i n t , m+,(A) = - l i m [e(x,A) / $(x,A)] . The s o l u t i o n s e and Q may be X

++m

computed e x a c t l y , and t h e y g i v e m-_(h) = - i

and m _ ( A )

[i~ 1 ~ 2 c o s ( x o ( ~ + E 3 ) 1+~ (2h)+ B ) 1 / 2 ~ i n ( x u ( A + B ) 1 / 2 ) x

x

[ c o s ( x o ( ~ + ~ ) ’ / 2 )- i A 1 / 2 ( h + ~ ) - 1 / * s i.n ( x o ( A + B ) 1 / 2 ) ]

Then t h e c o n t i n u o u s spectrum i s C(T) = f o r which m-,

=

m,

; i.e.,

[O,m),

=

-’ .

and t h e p o l e s o f F a r e t h o s e h f o r

r e a l v a l u e s h f o r which 2 c o t ( x o ( x + B ) 1 / 2 )

( 2 h + B ) / ( ( A + B ) ’ j 2 1 ~ 1 ” ~ ) (see [13, p. 1101). p o i n t s i n t h e i n t e r v a l (-B,O), say X I , A,,

=

There a r e f i n i t e l y many such iN. Since n = 1, t h e term ue1

.--,

-f

may be dropped from t h e e i g e n f u n c t i o n e q u a t i o n For i n s t a n c e , when x < 0, 8(x,A) g i v i n g m-_(l) = - i hl”. cos(hi/2x) - isin(hi/2x) .

=

= ~ J ~ worked u - ~ o u t above. = ~ - ~ ’ ~ s i n ( ~ ’, / ~ x )

C O S ( A ’ ’ ~ X ) and $(x,h)

Thus a t a p o l e hk, $-,(x,hk)

I f we l e t

A i l 2 = i Bk

=

e(x,xk) +,6-m)$(x,Ak) =

, we g e t qJ-,(x,Ak)

exp [ - i ( i p k x ) ] = exp(Bkx) , w h i c h i s square i n t e g r a b l e toward

-m

=

s i n c e B~ > 0

.

REFERENCES

[l] F.V. A t k i n s o n , D i s c r e t e and Continuous Boundary Problems (Academic Press, New York, 1964). [Z]

J. Chaudhuri and W.N. E v e r i t t , On t h e spectrum o f o r d i n a r y second o r d e r d i f f e r e n t i a l o p e r a t o r s , Proc. Roy. SOC. Edin., 6 8 A (1967 - 6 8 ) , 9 5 - 119.

[3]

E.A. Coddington and N. Levinson, Theory o f O r d i n a r y D i f f e r e n t i a l Equations (McGraw-Hill, New York, 1955).

[4]

M.S.P.

[5]

D.B. H i n t o n and J.K. Shaw, On Titchrnarsh-Weyl M ( A ) - f u n c t i o n s f o r H a m i l t o n i a n systems, J. O i f f . Eqs., 4 0 ( 3 ) (1981), 3 1 6 - 3 4 2 .

[6]

D.B. H i n t o n and J.K. Shaw, On t h e spectrum o f a s i n g u l a r H a m i l t o n i a n system, Quaestiones Mathematicae, 5 (1982), 2 9 - 8 1 .

[7]

D.B. H i n t o n and J.K. Shaw, H a m i l t o n i a n systems o f l i m i t p o i n t o r l i m i t c i r c l e t y p e w i t h b o t h e n d p o i n t s s i n g u l a r , t o appear i n J . D i f f . Eqs.

[8]

D.B. H i n t o n and J.K. Shaw, P a r a m e t e r i z a t i o n o f t h e m(h) f u n c t i o n f o r a H a m i l t o n i a n system o f l i m i t c i r c l e t y p e , Pros. Roy. SOC. Edin.,

Eastham and H. K a l f , S c h r o d i n g e r - t y p e o p e r a t o r s w i t h c o n t i n u o u s s p e c t r a ( P i ttman , London, 1982).

312

J. K. Shaw and

D.B. Hiiiton

93A(1983), 349-360. [9]

D . B . Hinton a n d J.K. Shaw, On the spectrum of a s i n g u l a r Hamiltonian system, 11, submitted t o Quaestiones Mathematicae.

[lo]

K. Kodaira, The eigenvalue problem f o r ordinary d i f f e r e n t i a l equations of the second order and Heisenberg's theory o f S-matrices, Amer. J . M a t h . , 71 (1949), 921 - 945.

[ll]

V.I. Kcgan and F.S. Rofe-Beketov, On square-integrable solutions of symmetric systems of d i f f e r e n t i a l equations of a r b i t r a r y o r d e r , Proc. Roy. SOC. Edin., 74A(1974), 5 - 4 0 .

[12]

B.M. Levitan and I.S. Sargsjan, Introduction t o Spectral Theory : Selfa d j o i n t Ordinary Differential Operators (A.M.S. Translations of Math. Monographs, v . 39, Providence, R . I . , 1975). J.L. Powell and B . Crasemann, Q u a n t u m Mechanics (Addison-Wesley, Reading, 1961).

[13] [14]

[15]

E.C. Titchmarsh, Eigenfunction Expansions Associated with Second Order Differential Equations, P a r t I ( 2 n d e d . , Clarendon Press, Oxford, 1962). P.W. Walker, A vector-matrix formulation f o r formally symmetric ordinary d i f f e r e n t i a l equations with applications t o solutions of integrable square, J . London Math. SOC. ( Z ) , 9 (1974), 151 - 159.

DIFFERENTIAL EQUATlONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

3 13

A PRODUCT FORMULA FOR CERTAIN QUADRATIC FORM PERTURBATIONS Rhonda J . Hughes (1) Bryn Mawr C o l l e g e B r y n Mawr, P A 19010

I n s t i t u t e f o r Advanced Study P r i n c e t o n , N.J. 08540

We i n t r o d u c e t h e n o t i o n o f a r e g u l a r i z a t i o n o f a q u a d r a t i c form, and use t h i s t o o b t a i n a version o f t h e T r o t t e r product formula ,-t(A+B) = lim (,-t/n Ae-t/n B1n t h a t is n+m v a l i d i n case one o f t h e summands f a i l s t o correspond t o an o p e r a t o r on t h e ambient Hi l b e r t space. I n a number o f r e c e n t papers, v a r i o u s e x t e n s i o n s o f T r o t t e r ’ s c e l e b r a t e d p r o d u c t f o r m u l a have been o b t a i n e d . For example, i n [7] Kato shows t h a t t o r an a r b i t r a r y p a i r o f nonnegative s e l f - a d j o i n t o p e r a t o r s A and B a c t i n g i n a H i l b e r t space H, e -t(AiB)n

= s - l i m (e-t/n

B)n, where

n

i s the orthogonal p r o j e c t i o n

n-tm

o f H o n t o D(A’/’) n D(B 1/ 2 ) , and AGB i s t h e f o r m sum o f A and B. More r e c e n t l y , Lapidus [9] o b t a i n s a p r o d u c t f o r m u l a f o r u n i t a r y groups by u s i n g “ i m a g i n a r y r e s o l v e n t s “ i n t h e a p p r o x i m a t i o n ; he proves t h a t f o r a l l x e D(A1/*) r l D(B”2), w i t h A and B as above, e -it(A’B)x t E R (cf.

[9,

=

l i m ([I+i(t/n)A]- [I+i(t/n)B]-l)nx, n+m

for all

References] f o r s e v e r a l r e l a t e d r e s u l t s by Lapidus, and a l s o [ 8 ] ) .

I n t h s oaDer. we d e s c r i b e a v e r s i o n of t h e T r o t t e r Droduct f o r m u l a t h a t i s a p p l i c a b l e i n ’ t h e e v e n t t h a t one of t h e summands i s d e f i n e d merely as a s e s q u i l i n e a r form, and need n o t correspond t o an o p e r a t o r on t h e ambient H i l b e r t space. Our approach, which we d e s c r i b e below, i n v o l v e s t h e i n t r o d u c t i o n o f a n o t i o n t h a t i s analogous t o t h e r e g u l a r i z a t i o n o f a d i s t r i b u t i o n b y C m - f u n c t i o n s w i t h compact s u p p o r t , and serves t o r e p l a c e e-tB, by a more t r a c t a b l e e x p r e s s i o n .

which i n o u r case has no meaning,

L e t A be an m - s e c t o r i a l o p e r a t o r w i t h v e r t e x z e r o ( c f . [6, V.3.101) a c t i n g 1 L e t H be t h e comi n a H i l b e r t space H whose i n n e r p r o d u c t we denote b y <,>. p l e t i o n o f D(A) under t h e i n n e r p r o d u c t < $,$ >1 = Re <(A+I)g,$ >, and

T h i s r e s e a r c h was s u p p o r t e d i n p a r t by a g r a n t f r o m t h e N a t i o n a l Science Foundation, and was completed w h i l e t h e a u t h o r was a Member o f t h e I n s t i t u t e f o r Advanced Study.

R. J. Hughes

314

1 H C H C H - l b e t h e s c a l e o f spaces a s s o c i a t e d with A, where t!-l

i s t h e d u a l space

w i l l denote t h e c o n t i n u o u s e x t e n s i o n o f A t o an element o f L(H 1 ,H -1 ) 1 -1 ( c f . [2] and [ll]). Furthermore, we assume t h a t i E L(H ,H ) i s a c c r e t i v e ; i.e. Re < &,g > 0 f o r a l l g e H1 . We have t h e f o l l o w i n g key d e f i n i t i o n ( 1 l.Ila,b

o f HI;

denotes t h e norm o f an o p e r a t o r i n L(Ha,Hb)):

6

Definition A r e g u l a r i z a t i o n of ( w i t h r e s p e c t t o A) i s a s t r o n g l y continuous of bounded a c c r e t i v e o p e r a t o r s on H such t h a t f a m i l y i l l h ->

(i)B[01 = 0,

(ii)s - l i m + 1 B [ h l = B i n L(H1 ,H -1 ), and h-+O

ll,o

(iii) I l B [ h ] l

= O(ha) as h

I lB[h]*l ll,o

for

-+

0

+

f o r some c o n s t a n t

a >

( f o r a possibly d i f f e r e n t constant n

21 , and s i m i l a r l y a >

1 7).

^

Nowy l e t AGB denote t h e r e s t r i c t i o n o f A+B e L(H 1 ,H -1 ) t o a n o p e r a t o r i n H w i t h D(A4B) = {g e HII(A+B)g e H I . Then A+B i s m - a c c r e t i v e , and so - ( A i B ) 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 c o n t r a c t i o n semigroup i n H ( c f . [ l , P r o p o s i t i o n 31). We then have t h e f o l l o w i n g analogue o f T r o t t e r ' s formula: Theorem 1.

L e t {B[h]Ih e

->

b e any r e g u l a r i z a t i o n o f

- t ( A i B ) = s-lim

(,-t/n

Ae-B[t/n]

n-+-

i.

Then f o r a l l t

0,

n

1

i n L(H). The method of p r o o f i s e s s e n t i a l l y t h a t used by F a r i s ( [ l , Theorem 11) and We s h a l l c o n t e n t o u r s e l v e s h e r e with a b r i e f o u t l i n e , i n d i c a t i n g o n l y t h e main p o i n t s of d e p a r t u r e from F a r i s ' p r o o f , and r e f e r t h e r e a d e r t o [5] we f i n d t h a t i t s u f f i c e s t o f o r f u r t h e r d e t a i l s . Assuming f a m i l i a r i t y w i t h [l], Proof.

m

n [lo].

prove t h a t f o r 4 e H 1 , l i m h + 0'

[ I F1

h h h (P Q -R

= 0, where Ph = e -B[hI

Qh = ,-hA

and Rh = e-h(A4B) ( t h e Ph ’s a r e c o n t r a c t i o n s , s i n c e B[h] i s a c c r e t i v e f o r each h _> 0). Since

Fl ( hP

h h 1 Q -R ) g =

h dQ -l)g

+

(Ph-l)Qh+

-

k (Rh-l)+,

and t h e f i r s t and l a s t terms on t h e r i g h t - h a n d s i d e converge t o respectively, f o r

k (Ph-l)Qh+

-+

+

E

- i g i n H-',

as h

+

0'.

i n f i n i t e s e r i e s converging i n L(H).

(1)

F1 (P h - 1 ) g

=

-

-A$ and (A+i)+,

H1, t h e main d i f f i c u l t y l i e s i n p r o v i n g t h a t S i n c e B[h] E L(H), Ph i s d e f i n e d b y an 1 Thus f o r e H

+

1 B[h]+ + -zr; 1 ( B [ h ] ) * @ + h i g h e r - o r d e r terms.

3 15

Certain Quadratic Form Perturbations

Because o f p r o p e r t y ( i i ) o f t h e d e f i n i t i o n o f a r e g u l a r i z a t i o n , t h e f i r s t t e r m i n ( 1 ) converges t o -B$

i n H-l;

t h e r e m a i n i n g terms converge t o z e r o as a

consequence o f ( i i i ) , t h e f a c t t h a t I l B [ h ] * I I c o n t i n u i t y of I B [ h ] l

(use t h e e s t i m a t e

130 = llB[h]llo,-l, 2 I lB[hIl

1 IB[h] 2 I 11,-1

ll,ol

and s t r o n g lB[h]l lo,-l).

Moreover, f o r k t l , t h e ( k + l)St t e r m i n ( 1 ) i s dominated i n H-'-norm by Mk-lh 2 ~ - 1 const-o!, where h i s s u f f i c i e n t l y s m a l l and M i s a c o n s t a n t independent o f h. Thus we may t a k e t h e l i m i t term-by-term, r e s u l t i s proved.

as h + O

+ , and

the desired

As an i l l u s t r a t i o n o f Theorem 1, we o f f e r t h e f o l l o w i n g example, l e a v i n g t o t h e r e a d e r t h e v e r i f i c a t i o n t h a t t h e proposed {B[h]) i s indeed a r e g u l a r i z a t i o n . I n s t e a d , we g i v e a d i r e c t p r o o f o f t h e p r o d u c t f o r m u l a i n t h i s s p e c i a l case. 2 1 L e t H = L (R 1, A = d e l t a f u n c t i o n ) , where Rec =O i f k ? 1 . .

L

L e t b denote t h e q u a d r a t i c f o r m d e f i n e d by >,

transform o f B). v e r i f i ed : H

m? 2, and B = ~ 6 ' ~( t h) e k - t h d e r i v a t i v e o f t h e C, k E Z+, 0 : kc[?], ( i ) Rec :O i f k = 0, and ( i i )

^

b($,$)= < B * $ , $

Lemma.

C E

w i t h D ( b ) = { $ C H I < lBl*(&\,\il> < m l ( h e r e 6 i s t h e F o u r i e r 1 If H denotes t h e f o r m domain o f A , t h e f o l l o w i n g i s e a s i l y

1

CD(b ) , b i s a c c r e t i v e on

I b($,$)l 5

C(

H1,

I $ ( Il I l $ l 11,

and for all

$,$E

H1,

1 -1 Thus t h e r e e x i s t s b~ L(H ,H ) such t h a t b($,$)

=<

and some c o n s t a n t C. b$,$

>;

we d e f i n e A $ B

t o be t h e m - a c c r e t i v e o p e r a t o r c o r r e s p o n d i n g t o t h e c l o s e d - f o r m sum< A$,$> + 1 Now, l e t B [ t ] denote t h e f o r m d e f i n e d on H1 as f o l l o w s : f o r $ . $ E H , <&.$I >.

=I L

~

B[t]($,$)

isA $, eisA$ > d s ;


0

these forms were c o n s i d e r e d e x t e n s i v e l y i n [ 3 ] and [4]. well-defined, < B *(eisA$)*

f o r i f ~ E H ' , t h e n eisA$EH1, ,(eisA$).

>

Note t h a t B [ t ] i s

and s i n c e
The c r u c i a l f a c t , proved i n [4,Theorem

2.11,

i s that for k

i n t h e g i v e n range,

B [ t ] extends t o a bounded l i n e a r o p e r a t o r ( w h i c h we a l s o c a l l B [ t ] ) fact, B[t]

>=

, t h e integrand i n ( 2 ) i s a continuous f u n c t i o n o f s . on H.

In

i s an i n t e g r a l o p e r a t o r w i t h k e r n e l

Clearly B[t]

i s a c c r e t i v e , and s o e-B[t3

i s a c o n t r a c t i o n on H f o r a l l t ? 0.

R.J. Hughes

316 Moreover, now prove

Theorem 2.

IlB[t]lI

~ L i p ( 6 )f o r s u i t a b l e 6 > 0 ( c f . [4,

P r o p o s i t i o n 2.21).

We

For a l l t z 0,

where A = above.

and B = ~ 6 ‘ ~i n ) L2(R’),

w i t h m,

2 and c and k as r e s t r i c t e d

A i n H-’, Once again, t h e p r o o f reduces t o showing t h a t h1( P h - l ) Qh$ + b$

Proof.

w i t h Ph, Qh as d e f i n e d i n Theorem 1. s e r i e s converging i n L ( H ) .

B[h]EL(H),

so

Ph i s s i m p l y an i n f i n i t e

Thus i f @ , $ c H1 ( i n which case Q h$ E H1 ) .

+,<(B[h1])

2Qh $ , $ > + h i g h e r o r d e r terms.

N o t i c e t h a t t h e nth t e r m i n t h e o p e r a t o r s e r i e s i s bounded by lh1n66-1 , s i n c e n 1, t h e remainder i s bounded by I l B [ h ] l I E L i p ( 6 ) . Hence as soon as n ? 6k-1 , p r o v i d e d O < h c y . Thus we may t a k e t h e l i m i t , as h - 0 + , term-byk=n

+I:[

%

t e r m i n (3), and f o r n

+ 1, t h e s e terms converge t o z e r o i n L(H), and hence

i n L ( H , ~H - ~ 1. We now c o n s i d e r t h e f i r s t - o r d e r t e r m i n ( 3 ) . The i n t e g r a n d i s a c o n t i n u o u s + h 1 f u n c t i o n o f s, and Q $ + $ ( i n H ) as h i 0 . F o r m a l l y , t h e l i m i t appears t o be &,I)>

.

To check t h a t we a c t u a l l y have convergence i n H-’,

we r e v e r t t o t h e

Fourier transform representation o f the operators i n question. 1

$,I)EH

9

h

h

We have f o r

Certuin Quudratie Form Perturbations

317

h

by t h e Cauchy-Schwarz i n e q u a l i t y . The e x p r e s s i o n i n b r a c k e t s converges t o zero as h+O+ by t h e dominated convergence theorem, and t h e f a c t t h a t b o t h $ and i 1 Therefore E L ( R ) i f $ E H1. IZ

'i

To show t h a t t h e h i g h e r - o r d e r terms converge t o zero, we c o n s i d e r ( f o r j , 2)

318

R.J. Hughes

But

By an argument s i m i l a r t o t h e p r o o f o f [4, Theorem 2 . 3 ( i i ) ] ,

i t i s easy t o see

that

where C i s a c o n s t a n t t h a t i s bounded f o r h i n a bounded i n t e r v a l .

Since

1 = 0 p o i n t w i s e , i t f o l l o w s f r o m F u b i n i ' s Theorem, dominated I i m (Kl(h,z,xl) hO 'f 1 1 convergence, and t h e f a c t t h a t ( 1 + l y I k ) i E L ( R ) , t h a t t h e i n t e g r a l i n ( 4 ) converges t o zero, as h - 0

lim,

1

t

.

We have t h e r e f o r e shown t h a t

I I B [ h I j $ l I-1

=

0, f o r j 2 2,

hO and o u r theorem i s proved

A d d i t i o n a l a p p l i c a t i o n s may be found i n [ 5 ] .

Certain Quudratic Form Perturbations REFERENCES 1.

W.G.

2.

W.G. F a r i s , S e l f - A d j o i n t Operators, L e c t u r e Notes i n Mathematics, 433,

F a r i s , The p r o d u c t f o r m u l a f o r semigroups d e f i n e d by F r i e d r i c h s e x t e n s i o n s , P a c i f i c J. Math. 22 (1967), 47-70. S p r i n g e r - V e r l a g , B e r l i n , 1975.

3.

R.J. Hughes and I . E .

Segal, S i n g u l a r p e r t u r b a t i o n s i n t h e i n t e r a c t i o n r e p r e s e n t a t i o n , J. F u n c t i o n a l A n a l y s i s 38 (1980), 71-98.

4.

R.J. Hughes, S i n g u l a r p e r t u r b a t i o n s i n t h e i n t e r a c t i o n r e p r e s e n t a t i o n , I I , J . F u n c t i o n a l A n a l y s i s , 49 (1982), 293-314.

5.

R.J.

6.

T . Kato, P e r t u r b a t i o n Theory f o r L i n e a r Operators, S p r i n g e r - V e r l a g , New York, 1966.

7.

T. Kato, T r o t t e r ' s p r o d u c t f o r m u l a f o r an a r b i t r a r y p a i r o f s e l f - a d j o i n t c o n t r a c t i o n semigroups, T o p i c s i n F u n c t i o n a l A n a l y s i s , Adv. Math. Suppl. S t u d i e s , I . Gohberg and M. Kac, eds., Acad. Press, New York 3 ( 1978), 185-1 95.

8.

T. Koyama and T . I c h i n o s e , On t h e T r o t t e r p r o d u c t formula, Proc. Japan

Hughes, A v e r s i o n o f t h e T r o t t e r p r o d u c t f o r m u l a f o r q u a d r a t i c f o r m p e r t u r b a t i o n s , t o appear.

Acad. Ser. A Math. S c i . 57 (1981), no. 2, 95-100. 9.

M. Lapidus, The problem o f t h e T r o t t e r - L i e f o r m u l a f o r u n i t a r y groups o f

o p e r a t o r s , B u l l . S c i . Math.,

t o appear.

10.

E. Nelson, Feynman I n t e g r a l and t h e Schrodinger e q u a t i o n , J . Math. Phys. 5 (1964), 332-343.

11.

E . Nelson, Time-ordered o p e r a t o r p r o d u c t s o f sharp-time q u a d r a t i c forms, J . F u n c t i o n a l A n a l y s i s 11 (1972), 211-219.

3 19

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

32 1

ON THE EXISTENCE OF RESONANT STATES Ketill Ingglfsson Department of Mathematics The University of Alabama University, Alabama 3 5 4 8 6 U.S.A.

These remarks ought to contribute to a simple but consistent form expressing quantummechanical resonant states. They consider a well known feature, which in this context will be referred to as the "paradox of resonance". It has been used in many ways in order to show the inadequacy of quantumdynamical forms. The procedure developed here exploits some properties of the semigroups appropriate for the decay. INTRODUCTION It is well known that difficulties arise if one attempts to explain quantumphysical resonances as solutions of the general Schroedinger equation for appropriate initial states. This has been demonstrated O R numerous occasions with the most varied demands for mathematical sophistication. B.Simon has, in his own words, "caricatured" resonances (1) in order to stress the importance of a more elaborate formal language. It seems, however, that the old formulation can still be given some consideration. It is, for exemple, wortwhile to know what exactly is contradictory in the old dynamics and which assumptions on its form make a consistent redefinition of resonant states possible. Consider a dynamical structure of "pure states" in reference to John von Neumann (2). A particular evolution is then expressed by an oneparameter family of vectors,)l)t;t20 ] , in the unit sphere of the separable infinite-dimensional complex Hilbert space unique up to isometrical isomorphism, generally denoted by 2 . It is of some convenience to consider the vector family as the range of a dynamical mapping , which takes the nonnegative reals into the prescribed unit sphere. By the rules of causality the evolution of states implies that the ranges of two mappings either intersect as one of the ranges or that they are disjoint. When H is the selfadjoint operator of energy this can be accomplished by stating the dynamical law as follows: 1) If $) s a ) ( H ) is given as the image under a m a p p i n s q for t=O and if is ?he solution of the Schroedinger equation iF qt=Hqt for any t>O with the s-lkm , then is that mapping with the so determined dynamical r%fge the aepropriate 4 . 2) Any vector in the unit sphere of 3 ( H ) belongs to a dynamical range.

Vt,

qt=P

6.’

in the The uniqueness of the differentiable solution =exp(-iHt) = Pofollows from a hndamental tgeorem u strong sense with s-lim of Stone. If ~3 i g 7 ~ l For some positive s, that T(G')cafqj. With a mapping 9 complsx-parameter family of v ~ ~ t ~ ~ ; Rkc z =~) ? O~] , ~ ( z )

322

K. Ingdlfsson

is associated in an one-to-one fashion where each member is the solution of a linear equation, ( z + iH ) q ( z ) . This is then the Laplace transforn: of (3).

-

=Po

q(z)

qt

If the resolvent of the operator H is denoted by R(z,H), one can use the inversed Laplace transform to state that

(yo,R(z,H)qo )

(l)(vo, V t ) = (2ri)-lk\;:xp(zt)

dz

is an identity for all posikLcvalues of t when 6, &H), independentof the positive value of . The convergence 09 the integral and its limit for t+O follows from a well known theorem of Hille and Yosida ( 4 ) . One can therefore claim the identity

1

( 2 ) exp ( -

j/t I ) (qo,Wt)=(2Ti)-’Tdq

exp(i7t)

(qo,(R(i+iq,H) -

-a

for all t when

R(-f+iT,H))qo ) is a fixed positive number.

\'

If the ir-itial state4) is a resonant ( excited ) state, one ex ects for physical reasons a : exponential decay pattern in 1(3 ,q , at least for enough high values of t. Let us pursue the igpligations of this assumption in terms of the quantities entering the relations (1) and (2) abovel The expression exp(-$ t) ( ) is square integrable in t on IR f o r A in the interva (0,z ) . Fourier transanalytic function inLa horizontal strip of form, x(\+iT) is then the width$ and square integrable in in accordance with the theorem of Plancherel. L e t 4 be a nontrivial everywhere dense submanifold in a4 , which is invariantnunder the mapping H from df into itself. When 4, and thereforelany H 4, belongs to /cl , then exp(is analytic in IR for allot. One can express # ( +i$l limit of , R(5.+il)d,H)exp((i(y+H)+~IT)$) ) when T + W , and the Fourier tran2form is i entically zero in the a&ove described strip. This contradicts the propertylv (I =1 stated originally. A similar difficulty appears if one assume2 that the resolvent set/S(H) covers an open interval in for =o. By using the equation (2) one can find the expression

)p

qo,9 tits

\

(9

9

\

00

0

for the Fourier transform?/(f+ when kis small enough. By the may t en conclude that the opening statement.

, which is absolutely less than&, of analytic continuation one zero, which contradicts the

yo

d 9 H ) makes many physically useful The nonexistence of t?dand assumptions on the realization o? quantumdynamical forms inconsistent with the concept of resonant states. For example, it is not possible to consider 4 belonging to a subspace E (2)afor any finite value of the spectrsl parameter and similarly H can not be semibounded. Such a failure may very well be referred to as the "paradox of resonance". Since the following is, however, a discussion about the existence of resonant states we will tperefore have to consider the spectrum of H as everywhere dense in JR and to consider as belonging to &(H) without the same always being true for H n W E where n is a natural number.

9

Resonant States

323

THE DEFINITION OF A RESONANT STATE Let us first state the following as a definition of a resonant state and show why as such it is inconsistent and does not determine a mathematical object: Definition 0: The vector qoca(H) of the general Hilbert space with 114),/= 1 represents a resonant state of the width Ywith respect to the operator H of total energy, when the set of all nonnegative numbers s , fulfilling the inequality

is not empty and bounded above with the supremum f > O . It is obvious that 9 is not a resonant state in accordance with this definition, if the agsolute square to the left i the inequality (3) can take the value 1 for positive t. Therefore can not belong to a subspace assigned to a discrete eigenvalue of $? This is of course what everybody wants. The definition is, however, not consistent with the selfadjointness of the o eratyr H. Let us consider the nonnegative It is continuously differentiable function Ft= L)o,exp(-iHt) in any interval in t around 0. The value of the function is 1 and of its derivative 0, if t is 0. By using the mean value theorem one can find for small enough values of t a contradiction to the inequality (3)whatever positive value s takes. ( "Short time complaint", B.Simon 1oc.cit. (1) )

40)/.

I(

In order to circumnavigate the contradiction which follows from assuming the existence of a resonant state in accordance with definition 0 one can use the function Ft under less constraint:

6.’

ea(H) with 116, 1/=1 represents a resonant Definition 1: The vector state with respect to H , wgen the set ofoall positive s such that the inequality Ft < o( exp(-st)

t large enough

is fulfilled for any .CE (0,l)is not empty and bounded above. The number g=sup{s] is now possibly dependent on A , but even so it does not exist for most physically realistic models. This is for example the case when H is semibounded. ( "Long time complaint", B. Simon 1oc.cit (1) ) Let me show you a slightly more general argument, closely related to the one explained in the introduction. L e t q o , have a nonvanishing projection onto the subspace of singularity with respect to H and assume that the singular spectrum is not everywhere dense in the complement to the absolutely continuous spectrum. Then ( 9 ,4>, ) is the Fourier Transform of a measure, which according to theoPaley-Wiener theorem is at least analytic in a strip around the real line. The resoivent setq(H ) is connected and there exists an open interval in R on which thg measure is 0. Therefore the measure is everywhere 0 according to the principle of analytic continuation. Thus Ft is identically 0 and $5 = 0, which contradicts a preassumption in the definition 1. We wiyl therefore have to assume that the complement to the closure1of the singular spectrum with respect to any finite interval in lR belongs to the absolutely continuous spectrum. Even this situation may lead to contradictions. If there exists, for example, a bounded reduction of the operator H defined onto an invariant singular subspace for which the Schmidt norm is finite, then again no resonant state is consistent with definition 1. This can be proved by using a slightly modified version of the

324

K. Ingblfsson

Weyl- von Neumann perturbation theorem (5). We must therefore assume that there exists no pair {a,b] such that the reduction of H onto the spectral interval la,b) has a Schmidt norm less than C when J b-a I is small enough. The following example seems to be consistent with definition 1: Let (x+iy) be an analytic function in a strip of finite width including tRe real axis and square integrable in x ( i.e. is an analyt2c vector for translations ) . If H means a multiplica?ion by x on ( JR ) , exp(-gt) and there is a 0, for then it follows that )/ 5 k any finite positive valug o? g hdzpendent of H. Obviously some constraint on g is still missing (6).

9

a:

We will finally propose the following definition of a resonant state, which determines such a state, labeled by Y , as a mathematical object: Definition 2: The vectorqO in the domain of the selfadjoint H in the general Hilbert space is a resonant state with respect to H, if there exists a positive number Y , such that the inequality

v

exp(-st) t large enough Ft < is fulfilled for anydtz(0,l) when s < d. Ira Herbst has recently ( 7 ) discussed a model for the Stark fffect, in which the spectrum is absolutely continuous and covers . His resonant state is consistent with our definition 2 . We propose the following example which is somewhat less detailed but maintains the mathematical essence, necessary for our purpose, of the Todel proposed by him: Let V ( k ) be square integrable in k over IR and let the operator H be defined through the equation

when the right side is square integrable. If&(A) is the spectral reso tion of H and the inner product ( $;1dE( ) q o ) is written 1) d J , then one finds that -00

qo,

‘CP

from the posito the real axis, then + ( 2 ) is analytic in the same+egion. Hence exp(- 4 t) F(t) is square integrable over )R because/( +i ) is its (0,g). Fourier transform when

3

THE PHYSICAL RESONANCE The following dynamical form is common to many quantumphysically relevant descriptions of a decay: Let{ Z(t);tlO ]be a one-parameter semigroup of contraction operators defined on-x and let qt = Z(t)po. For example such a semigroup exists, if has the form

qt

(4)

yt \

= (exp(- It) UZ(t) + G ( t )

xl 1

1

Ul(t)

w

)

Po

,

j,o,

are unitary on the complementary orthowhere { U (t) U2 (t) and gonal sukspaces and 8,, w is a partial isometry mapping22 onto and the following side conditions are fulfilled: (i) II I/ 5 1 Y tko, (ii) c a ( G ) Ld;4, with G as the infinitesimal generator,

xl

Wt

vo

Resonant States

(iii) The strong limit of t-’(

qt - 4’,

325

exists as t tends to 0+

)

.

qt

If G ( t ) is a real valued function in t, is it not possible for to be the solution of any Schroedinger equation. The semigroup never = 1 t 2 0 would has a unitary extension. Indeed, the condition contradict (iii). This may again be considered as one aspect of the paradox of resonance. The semigroup can, however, be determined by the equation (4) whenG(t) is the function 1 $xp(- 1 t) I Z(t)W t 0 and U (t) C WU (t)W If -IH and -iH equals W U (t) o n R 2 U it) 1 and 2 are the inginitesimal generators 04 the unigary groups

l[qt/l d

-

z

.

1

{ U 2 (t))

respectively, then the semigroup is completely de4ermined by HI and W through the relations H1 W 3 W H2 r G = (-i H2 ) W t W + (-iH ). W W i + 1 This semigroup is not the only one which could produce the relation (4). On the other hand it is the only such semigroup which can be reduced by H1 and extended in that subspace as a unitary group. The Schroedinger equation is therefore solved in g l w i t h the solution obtained by applying the semigroup to an appropriate vector,plh (H ) 1 We will now consider the range of the projection operator W + (W-1) as the subspace $ of 3 and J as an isometry f r o m g 2 to d? The operators P = ( W t 2 l ) W 2nd P+= W(Wt+l) m a p 3 onto complementgry subspacesdq- and 3, , with respect to These P- are projection . The infinitesimal generator G is reoperators along onto and duced by the subspaces 64; in the sense that G - = PT G c G P-

{

a

-s'

G- = P- ( (-i)H2 G+ = (-i) H1 P,

.

*.

x,

.

t

1

.

4

The s u b s p a c e 2 is ortho onal to 2 and W Jt is a partial isoonto . If HO,fs ae ined as JH Jt2 and cb(Ho) metry mapping fulfills the conaition ( @ , exp(-iH t) )exp(- i t f e in t ?or ( 0 , ?$, !,,then = W 3t q0 is 2 res8nant state in accordance with de inition 2.

9

oq

v,

\!

4

x

y

Equation (4) finally gives us the means to compare the decaying inner products for the "physical decay" Z (t) when ) =. 2 with the solution 2 of the Schroedinger equation. We find that

yo

t

1

.

for any positive less than The fact that the above inequality is not valid for\ = a reflects the "short time complaint’’ of B.Simon. 1

REFERENCES (1) Barry Simon, Int. Journal of Quantum Chem. 14 (1978) 529. (2)

John von Neumann, Mathematical Foundations of Quantum Yechanics, chapter VI ( Princeton University Press, 1955 ) . /

K.Ingolfsson, Helvetica Physica Acta 40 (1967) 237. (4) K.Yosida, Functional Analysis, chapter IX ( Springer, BerlinNew York, 1968 ) .

(3)

326

K. Ingblfsson

(5)

K.Ingolfsson, MRC Technical Summary Report Nr.2212, MRC Univ. of Wisconsin (1981).

(6)

I want to thank Dr.R.M.Kauffman f o r communicating this example. Ira Herbst, Commun. Math. Physics 75 ( 1 9 8 0 ) 1 9 7 .

(7)

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) @ Elsevier Science Publishers B.V.(North-Holland), 1984

321

ASYMPTOTIC BEHAVIOR OF THE SCATTERING AMPLITUDE AT HIGH ENERGIES Hiroshi Isozaki Department of Mathematics Kyoto University Sakyo, Kyoto JAPAN

and

Hitoshi Kitada Department of Mathematics University of Tokyo Meguro, Tokyo JAPAN

1.

INTRODUCTION. In this paper we shall consider an asymptotic behavior of the scattering amplitude at high energies for smooth long-range potentials. Let (1)

Ho =

-

1

?A,

H = H +V. 0

-

Here V = V(x) is a real-valued smooth function on RN satisfying the assumption: For some E E ( 0 , l ) and all a, I a>(x) 1 <= ca<x>- I C 1 1 - E I (2) <x> = J l + / x l ' . Let S be the scattering operator for the pair (HOIH)defined by (14) 2 N in section 2. Then S is a partial isometry in L2 = L (R ) which commutes with Ho. (It is known that S is unitary in L2, but we need not use this fact in what follows.) Let 9 denote the Fourier transform = %S$-'. Then is decomposable with respect to in L2 and set Ho:

<

(3)

2 N-1 where d ( X ) E B(L ( S ) ) is called the Scattering matrix. Our first theorem is Theorem 1 . Let A. > 1 be large enough. Then for X t X o l d ( X ) has a smooth kernel A ( X , w , w ' ) for w # w ' . (d(X,w,w') is called the scattering amplitude.) The order of singularity of ~ ( X , w , w ' ) when ~ w - w l #~ o is small is at most ~ a - a ~ l - ( ~ - ~ ) / ~any - ~ f 6o r> 0.

Remark. For the Coulomb potential V(x) = C/lx( known that the singularity is of order [w--w'I -2 Our second result is Theorem 2. When N 2 2 and 5 # 0, one has

in

R3 ,

it is

.

N-2

/X(w-w*)

=c

where cN = ( 2 7 ) (Ni2)/2i. Furthermore, since T V our assumption, we have

E.

L1) ! R (

under

328

H. Isozaki and H. Kitada

(5)

Namely, the potential is uniquely reconstructed from the scattering amplitude when N 2 2 . Remark. For N = 1 , this is not true. It is known that there are infinitely many potentials which have the same scattering amplitude and give rise to arbitrary bound states when N = 1 . 2.

WAVE OPERATORS AND SCATTERING MATRIX. We first construct the (modified) wave operator in the form: t

(6)

W - = s-lim e J t-tfrn

itH -itHo Je

For this purpose we prepare the following lemma Lemma 1. Let -1 < u o < u < 1, d > 0 and 0 < 6 << 1 . Then 1 there exist a constant R > 0 and Cm funct Om '9 x , C ) and a(x,() such that: i) For 151 2 d, cos(x,5) E [-l,oO] the solution of the eikonal equation

and for any (x,5) e R2N and

ii) equation

For any

and the estimates:

L 2 1

a , B,

y

and 151 2 d,

"

[ul,l]

and 1x1

2

R,

y

satisfies the estimate

a

satisfies the transport

is

329

Scattering Amplitude

and a, can be chosen arbitrarily as far as

In the above a. -1 < o o < o

1-

<

1.

-1 < o - 6 < 0, 0 < o;-s < o < u;

that

GO

We fix one ( y_,a-) corresponding to <+cry + 6 C 0, and another ( y+,a+)

ui

< o1 + d

f cJ=$(RN), Proposition 1 .

(12)

i

w;+

dg

where

=

5; such such that

< 1.

Now we define the Fourier integral operators

for

<

= (2n)

-N

dg

J+

and

J-

by

.

There exist the modified wave operators: s-lim t t-m

e

itH -itH J+e Or

I

s-lim t+icc

e

itH -itHo J-e r

r

-f

Wi

-

=

where r is the projection onto $ - 1 L 2 ( 1 5 ( 2 d). Wi+ and W i 2 the intertwining property, and a;e isometries in TL , satisfy identical: (13)

w;+

=

.

w;-

The scattering operator

s

=

+

(WJ

+

are

)

*WJ-

=

c

is defined by

S

* -

+

(WJ ) WJ

-

+

*

Thus

In order to obtain an expression formula for the scattering matrix $(A), we prepare the following lemma. Set T, = HJ, - J,HO. Lemma 2. Fourier integral operafor: (16)

T+f(x) = j j e

Then

T,

can be written as a

i( 9 , ( X , < ) - Y - ~ ) t+(x,Of(y)dyaS -

.

H. Isozaki and H. Kitada

330

The symbol

t+(x,E)

is given by

Then from (15) we can easily get Proposition 2. be written as

Here

R ( z ) = (H-z)-

(s > 0)

For

X 2 X

for

Im

z

0’

the scattering matrix $(A)

# 0, and

denotes the boundary value

lim

T:R(X+iO)T+

C

*

T-R(A+iE)T+

can

2 2 B(L-s,Ls)

, whose exist-

EJ.0

ence follows from Lemma 3 in the next section. 3.

KEY ESTIMATES. -itH e

We state a key estimate for the fundamental solution the Schradinger equation (211

-I -dtd Lemma 3 .

u(t) + Hu(t) = 0 .

Let

p-(xrc), p + ( x r ~ e )

ern

satisfy on

R

~

~

,

of

33 1

Scattering A mplirude

where

- 1 < a- < a+ < 1.

x(v) =

(23)

i' 0

Then for

for all

For

X 2 lo,

for

p 2 X/2,

for

p <= X/3.

let

x(p)

E

Cm

satisfy

it t 0, one has

L 1 0, s Z 0

Remark.

and

X 2

X0'

This lemma implies the following resolvent estimate:

s 1 0, X 2 h 0 ' Proof of Lemma 3 . Let t t 0. Let E+(t) be the operator defin-=.of Kitada [l] , and write P, = p+ (X,D). Then we have itHp P-x(H)e

for

+

(26)

= P-X (HI{e-itHp+

- ’ -P ~ i -

k-ie(t-

The second summand on the RHS of (26) is where K + ( r ) = (D~+H)E+T ) . 2 4 ) by integration by parts. The first sumeasily seen to satisfy mand can be written as

Thus by Theorem 1.1 and Theorem 2.8 of [ I ] , it follows that ( 2 7 ) also n satisfies (24). The case t i 0 can be treated similarly.

2.

We prepare one more lemma which will be used in proving Theorem m Let Vn(x) = V(x)p(x/n) for n 2 1 , where p E C o such that p(y)

= 1

(for I y l 5 1) and = 0 (for ( y ( 2 2 ) , and set Hn

=

Ho + Vn

and

H Isozaki and H. Kitada

332

Rn ( 2 ) = (Hn-z)-l. Then:

for

Lemma 4. +t L 0

for all

For the same

p+(x,S) and x(u)

as in Lemma 3, one has

~

L 2 0, s 2 0 and A 2 X 0 '

Since the idea of proof is similar to that of Lemma 3, we omit the proof. We remark that (28) implies the resolvent estimate:

for all

s

b 0

and

X 2 X0'

4. PROOF OF THEOREMS.

Proof - of Theorem 1. From Lemmas 2 and 3, the second summand on N- 1 the RHS of (20) has a smooth kernel for all w , w ' E S Hence the (w-w') of A ( A ) - I is completesingularity of the kernel J ( A f o , w ' ) - 6 ly determined by the first summand on the RHS of (20) in Proposition 2. But it is not difficult to see by integration by parts that its order of singularity is at most

.

Proof - of Theorem 2. Let us denote the scattering amplitude of long-range type for Hn = Ho t Vn by gn(X,o,w’). Then it follows from Lemma 4 and (20) in Proposition 2 that for lfi ( w - w ' ) I 2 a(>O) - -N- 2 (30)

(2A)

*

~$(~,w,w’)-~~(h,w,w 1 ’c )ca n- E

On the other hand, since Vn

.

is short-range, there exists the scat-

,$As)

tering amplitude ( A ,o f w ' ) of short-range type (i.e. defined with in the above discussions). Then it is seen that there is a J,,, = I real-valued function Pn(h,wfw’) such that

Scattering A mp litude

333

Furthermore, since Vn has compact support, we can easily see that J ( A ,w ,w I ) satisfies

As)

(fi(u-u’) (32)

c

CN n N + 3 / ~ ,

N+2 cN = ( 2 n ) - 2 i

,

which follows from the so-called limiting absorption principle for Rn(z). (This principle also follows from the time-decay estimate, Theorem 1.1 in [I] , when X is large.) Combining ( 3 0 ) ( 3 2 ) , we get for 6 = fi(w-w’), 151 2 a(>O),

(33)

5 Ca n

-E +

N’

nN+3/fi +

ca

n/fi +

cafL n-L

.

Therefore, we obtain

which proves Theorem 2. Remark. Throughout the paper, we have only considered the high energy part of all relevant operators. But it is also possible to treat the low energy part away from zero in a slightly different way. E.g. using this method, we can prove Theorem 1 for all h > 0. These will be discussed in subsequent publications. Acknowledgements. This work was refined during the stay of H. K. at California Institute of Technology in March and April, 1 9 8 3 . He wishes to express his sincere appreciation to Professors €3. Simon, V. Enss, F. Gesztesy, M. Murata and Dr. P. Perry for their interest in the work and stimulating discussions with him, to California Institute of Technology for its hospitality, and to USNSF under Grant N o . MCS-

H. Isozaki and H. Kitada

33 4

81-20833 f o r f i n a n c i a l s u p p o r t t o him d u r i n g h i s s t a y .

Reference

[I]

Kitada, H.,

Time-decay

of t h e h i g h e n e r g y p a r t o f t h e s o l u t i o n

f o r a Schrodinger e q u a t i o n , t o appear i n J . Fac. S c i . , Tokyo, sec. I A .

(1983).

Univ.

of

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

335

PIILL- AND EALF-BMIGE THEORY OF AN INDEFINITE STUBl-LIOOVILLE PROBLEM1 Hans G. Kaper Mathematics and Computer Science D i v i s i o n Argonne N a t i o n a l Laboratory Argonne, I L 60439

The Sturm-Liouville e i g e n v a l u e problem -u" = Xxu on (-l,l), u(-1) = u(1) = 0, admits a countably i n f i n i t e sequence of p o s i t i v e e i g e n v a l u e s and a countably i n f i n i t e sequence of n e g a t i v e e i g e n v a l u e s . The e i g e n f u n c t i o n s have f u l l - r a n g e , as w e l l as half-range completeness p r o p e r t i e s . 1.

Introduction

In [l] w e announced s e v e r a l r e s u l t s f o r t h e f o l l o w i n g Sturm-Liouville e i g e n v a l u e problem with i n d e f i n i t e weight:

lim

x4-1

u(x) and l i m u ( x ) e x i s t and a r e f i n i t e . x4 1

T h i s e i g e n v a l u e problem arises i n t h e s t u d y of c e r t a i n boundary v a l u e problems The r e s u l t s were p a r t i a l l y p r e l i m i n a r y ; in p a r t i c i n l i n e a r t r a n s p o r t theory. u l a r , t h e s o - c a l l e d h a l f - r a n g e theory w a s n o t w e l l understood by us a t t h e time of t h e announcement; c f . [ l , S e c t i o n VI]. R e c e n t l y , we proved a half-range completeness theorem f o r t h e e i g e n f u n c t i o n s of (l), (2), u s i n g a s y m p t o t i c estimates f o r t h e e i g e n v a l u e s . These r e s u l t s have been submitted f o r p u b l i c a t i o n e l s e w h e r e [2].

In t h i s n o t e w e i l l u s t r a t e t h e method of o u r proof on a problem t h a t is somewhat s i m p l e r , y e t t y p i c a l f o r t h i s class of Sturm-Liouville problems w i t h i n d e f i n i t e weights, v iz. , -u" = Xxu

(-1,l)

on

,

(3)

.

u(-1) = u(1) = 0

(4)

T h i s problem admits a countably i n f i n i t e sequence of e i g e n v a l u e s {An: f2, which can be ordered in such a way t h a t

...}

<

< with limn-Xfn

= fw.

<

0

< x1 < x2 <

n=fl,

... ,

(5)

The a s s o c i a t e d set of e i g e n f u n c t i o n s { $ : n=fl,f2,

has f u l l - r a n g e , a s w e l l as half-range completeness p r o p e r t i e s . were o b t a i n e d by more a b s t r a c t methods by Beals [ 3 , 4 ] .

...]

Similar r e s u l t s

H. G. Kaper

336 2.

Full-Range Theory

Let H = L2(-l,l), and norm ll.Il. Let

w i t h t h e u s u a l d e f i n i t i o n s of t h e i n n e r p r o d u c t

Tu(x) = X U ( X ) , Aufx) = -u"(x)

(-1,l)

,

u

t

H

x c (-1,l)

,

u

t

dom A

x

,

t

( a , * )

,

(6)

,

(7)

where dom A = {u E H: u and u ' a b s o l u t e l y continuous on ( - 1 , l ) , u" E H , u(-1) The o p e r a t o r A i s s e l f a d j o i n t and p o s i t i v e ; i t s i n v e r s e i s t h e = u(1) = O } . i n t e g r a l o p e r a t o r K,

which i s compact i n L ( H ) . The f u n c t i o n a l form of t h e e i g e n v a l u e problem ( 3 ) , ( 4 ) i s

Au = ATu ,

u

dom A

E

.

(9)

The e i g e n v a l u e s c o r r e s p o n d t o t h e c h a r a c t e r i s t i c v a l u e s of t h e p r o d u c t o p e r a t o r KT. S i n c e K and T do n o t commute, KT i s n o t s e l f a d j o i n t i n x ( H ) . L e t HA be t h e completion of dom A w i t h r e s p e c t t o t h e norm II*llA, where IlullA

=

(U,U);/~ with (u.v),

= (Au,~),

u,v

t

dom A

.

(10)

The r e s t r i c t i o n of KT t o HA, which w e d e n o t e by S,

Su = KTu

,

u

,

HA

E

(11)

i s s e l f a d j o i n t and compact i n I ( H A ) . Its spectrum c o n s i s t s of a countably i n f i n i t e sequence of real e i g e n v a l u e s w i t h a n accumulation p o i n t a t t h e o r i g i n . W e w r i t e t h e e i g e n v a l u e e q u a t i o n i n t h e form -1 S$n = An $n

,

n=fl,f2,...

,

(12)

where t h e e i g e n v a l u e s are o r d e r e d as i n d i c a t e d i n ( 5 ) . The e i g e n v a l u e s are simple. The symmetries of t h e problem a r e such t h a t A = -A and $-n(x) = -n $,(-XI. The e i g e n v a l u e e q u a t i o n (12) can be s o l v e d e x p l i c i t l y . v a l u e s are t h e r o o t s of t h e t r a n s c e n d e n t a l e q u a t i o n Ai(X113) - Ai(-*l/3)

.

si(x’/3)- Bi(-A'/3)

'

The p o s i t i v e eigen-

t h e e i g e n f u n c t i o n $n which corresponds t o a p o s i t i v e e i g e n v a l u e X

is

Here, Ai and B i a r e t h e Airy f u n c t i o n s of t h e f i r s t and second k i n d , As n + m w e have t h e a s y m p t o t i c r e l a t i o n s r e s p e c t i v e l y 15, S e c t i o n 10.41.

An Indefinite Sturm-Liouville Problem 1

311

Jx = (n - -)4 2 +

O(e

-2nn)

,

331

n + m ,

n

+-.

Note t h a t t h e e i g e n f u n c t i o n s $n w i t h n > 0 a r e monotone on [-1.01 and o s c i l l a t i n g on [ O , l ] ; f o r t h e e i g e n f u n c t i o n s $n w i t h n < 0 t h e s i t u a t i o n i s reversed. L e t P and Pm be t h e p r o j e c t i o n s which map HA o n t o t h e p o s i t i v e and P n e g a t i v e maximally S - i n v a r i a n t s u b s p a c e s ,

Then I S 1 = SP -SPm. P i n n e r product on HA:

T h i s o p e r a t o r i s p o s i t i v e d e f i n i t e on HA;

i t d e f i n e s a new

The corresponding norm i s II-lls. The completion of HA i n t h e topology d e f i n e d by The t h e S-inner product is a new H i l b e r t s p a c e H s , which e x t e n d s beyond HA. p r o j e c t i o n s Pp and Pm extend uniquely by c o n t i n u i t y t o o r t h o g o n a l p r o j e c t i o n s on The o p e r a t o r S e x t e n d s uniquely by c o n t i n u i t y t o a bounded l i n e a r o p e r a t o r Hs. on Hs; t h e extended o p e r a t o r i s s e l f a d j o i n t and compact i n 1 ( H S ) . TBEOBen 1.

Any u -

E

HS h a s a unique e x p a n s i o n ,

m

where an = ( U , $ ~ ) ~ / ( Q ~ , $T h ~ e)e~x p. a n s i o n converges i n t h e topology o f H S . The topology of HS is weaker t h a n t h a t of HA, and HA i s everywhere dense i n HS (in t h e topology of H s ) , so t h e set n=fl,f2, which i s a b a s i s of HA, i s c e r t a i n l y a fundamental s e t i n Hs. Moreover, t h e $ are m t u a l l y o r t h o g o n a l with r e s p e c t t o t h e S-inner product. 0

PROOF.

...},

I$,.,:

W e adopt t h e n o r m a l i z a t i o n I l + n l l ~ = 1 . Thus, HS i s t o p o l o g i c a l l y isomorphic with L2(Z\{O}). The isomorphism F which maps HS o n t o L2(Z\{O}) is g i v e n by

The t r a n s f o r m a t i o n F d i a g o n a l i z e s t h e o p e r a t o r S on HS. The s p a c e HS can, i n f a c t , be i d e n t i f i e d . Let P+ and P- be t h e p r o j e c t i o n s which map H onto t h e p o s i t i v e and n e g a t i v e maximally T-invariant s u b s p a c e s , P+u(x) = u ( x )

,

x

E

(0,l) ;

P-u(x)

,

x

E

(-1,O)

= u(x)

Then IT1 = TP+-TP-. i n n e r product on H:

;

P+u(x) = 0 P-u(x)

=

0

, ,

x

(-1,O)

E

x

E

(0.1)

;

(21-1)

.

(21-2)

This o p e r a t o r i s p o s i t i v e d e f i n i t e on H; i t d e f i n e s a new

338

H.G. Kaper (U,V)T = ( ) T l U , v )

9

U,V

6

H

(22)

The corresponding norm is 1 l - U ~ . The completion of H in t h e topology d e f i n e d by t h e T-inner product i s a new H i l b e r t s p a c e HT, which e x t e n d s beyond H. The p r o j e c t i o n s P+ and P- extend uniquely by c o n t i n u i t y t o o r t h o g o n a l p r o j e c t i o n s on HT. The o p e r a t o r T e x t e n d s uniquely by c o n t i n u i t y t o a bounded l i n e a r o p e r a t o r on HT; the e x t e n s i o n i s s e l f a d j o i n t in-C(%). TEEOREM 2.

The s p a c e s HS and % are t o p o l o g i c a l l y

isomorphic.

WOOF. The proof proceeds in a number of s t e p s and depends u l t i m a t e l y on t h e asymptotic estimate (15). (i)

From t h e i d e n t i t y 0

0

and t h e e x p r e s s i o n (14) one o b t a i n s t h e estimate

f o r some p o s i t i v e c o n s t a n t Y.

Hence, w i t h (15),

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

(ii)

Given the estimate (23) one shows f i r s t t h a t t h e series 1T11 21:=lanQn

converges in t h e L2-sense

on (-1,O)

Then one shows t h a t t h e c o n d i t i o n

w

i f (a )

n n=l

m

E

I12(R),

with

2

9. (R) i s n e c e s s a r y and s u f f i c i e n t

IT11/211=lanQn

It f o l l o w s t h a t t h e in H. 0 m 2 mapping (an>nP1 ~ l ~ = i s~ bounded a ~ Qand~ boundedly i n v e r t i b l e from 9. ( 8 ) i n t o A s i m i l a r r e s u l t h o l d s f o r ne a t i v e i n d i c e s , so the mapping (a,) H l a n Q n is a continuous imbedding of I12(Z\fO}) i n t o %.

f o r t h e convergence of t h e series

3.

2

( i i i ) Because HS and II ( Z \ { O } ) a r e t o p o l o g i c a l l y isomorphic, t h e r e s u l t of (ii) implies t h a t HS can be imbedded c o n t i n u o u s l y i n %. Then, f o r any u 6 HA, uuu2 = ((P+-PJU,

S

< -

(P -P ) u & P m CR(P+-P-)ullTU(P -P )uII < 4CIlUllTUUllS p m S -

,

i 4ClluU~. The i n e q u a l i t y extends t o HS by c o n t i n u i t y , so HT can a l s o be 1 imbedded c o n t i n u o u s l y i n HS. This proves t h e theorem.

so llulls

COBOLLABY 3.

The s e t {Qn: n=fl,f2, ...} is a b a s i s of

339

An Indefinite Sturm-Liouville Problem T h i s r e s u l t is known as the f u l l - r a n g e completeness p r o p e r t y .

3.

Half-Eange Theory

w.r.t.

The p r o j e c t i o n s P+ and P- d e f i n e t h e d i r e c t sum decomposition ( o r t h o g o n a l t h e T-inner product):

HT = H+ 8 H-

,

H+ = P+(HT)

,

H- = P-(HT)

.

(25)

S i m i l a r l y , t h e p r o j e c t i o n s Pp and Pm d e f i n e the d i r e c t sum decomposition (orthogonal w . r . t . t h e S-inner p r o d u c t ) : HS = H

8 Hm

P

,

Hp = Pp(HS)

,

Hm = Pm(Hs)

.

(26)

The o b j e c t i v e of half-range t h e o r y is t o e s t a b l i s h p a i r w i s e connections between t h e components of (25) and ( 2 6 ) . The c o n n e c t i o n s are e s t a b l i s h e d by means of four connecting transformations, V = P P +P-P + P m ' W=P+P,+P-P

P '

v i/

wt

= P P + P P p + m - ’

(27)

= P P + P P m+’ p -

(28)

Because of Theorem 2, t h e s e are well-defined bounded l i n e a r t r a n s f o r m a t i o n s from HS t o and v i c e v e r s a .

9

TEEOREM 4. Each of t h e c o n n e c t i n g t r a n s f o r m a t i o n s V, W , Vi', b i j e c t i v e mapping of % o n t o i t s e l f .

PROOF.

and W"

defines a

The proof is based on t h e f o l l o w i n g i d e n t i t i e s :

These i d e n t i t i e s can be v e r i f i e d by d i r e c t computations.

0

The theorem i m p l i e s , in p a r t i c u l a r , t h a t P+: Hp + H+ and P-: Hm + H- are b i j e c t i o n s . The i n v e r s e s of t h e s e mappings d e f i n e e x t e n s i o n o p e r a t o r s E+ and E-,

which extend f u n c t i o n s d e f i n e d on t h e half-ranges i n Hp and Hm, r e s p e c t i v e l y .

( 0 , l ) and (-1,O)

..} is a b a s i s of

s e t {P+@n: n=l,2,. THHORPSI 5, {P-$-n: n = 1 , 2 , . . T i s a b a s i s of H-.

PROOF.

For any u

f

E

H*we

H+;

t o functions

the set

have

m

where B+n

= (E+U*,+*)~.

The expansion converges i n t h e topology of HT.

0

H G. Kaper

340

T h i s r e s u l t i s known as the half-range completeness p r o p e r t y . Of c o u r s e , t h e problem of c a l c u l a t i n g t h e expansion c o e f f i c i e n t s in (33) i s s t i l l open, because we d o n ' t have a n o r t h o g o n a l i t y r e l a t i o n f o r t h e f u n c t i o n s P+On and P-4,. In f a c t , a n i n t e r e s t i n g q u e s t i o n i s , whether E+ and E- can indeed be c h a r a c t e r i z e d d i r e c t l y i n terms of boundary v a l u e problems on t h e i n t e r v a l s (0.1) and (-1,O) separately.

References Kaper, H. G., Lekkerkerker, C. G., and Z e t t l , A., L i n e a r T r a n s p o r t Theory and a n I n d e f i n i t e Sturm-Liouville Problem, i n : E v e r i t t , W. N. and Sleeman, B. D. ( e d s . ) , Ordinary and P a r t i a l D i f f e r e n t i a l E q u a t i o n s , L e c t u r e Notes i n Mathematics, Vol. 964 (Springer-Verlag, B e r l i n , 1982). Kaper, H. G., Kwong, M. K., Lekkerkerker, C. G., Sturm-Liouville Problems, p r e p r i n t (1983).

and Z e t t l , A.,

Indefinite

Beals, R., Partial-Range Completeness and E x i s t e n c e of S o l u t i o n s of Two-way D i f f u s i o n E q u a t i o n s , J. Math. Phys. 22 (1981) 954-960. Beals, R., I n d e f i n i t e Sturm-Liouville Problems and Half-Range Completeness, p r e p r i n t (1983).

Handbook of Mathematical F u n c t i o n s , Abramowitz M. and Stegun, I. A. ( e d s . ) , Applied Math. S e r i e s , Vol. 55, Nat'l Bureau of S t a n d a r d s , Washington (1964).

' J o i n t work w i t h M. K. Kwong and A. Z e t t l (Northern I l l i n o i s U n i v e r s i t y ) . This work was supported by t h e Applied Mathematical Sciences Research Program (KC-0402) of t h e O f f i c e of Energy Research of t h e U.S. Department of Energy under Contract W-31-109-Eng-38.

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

341

REMARKS ON HOLOMORPHIC FAMILIES

OF SCHRODINGER AND DIRAC OPERATORS T o s i o Kato Department o f Mathematics U n i v e r s i t y o f C a l i f o r n i a , Berkeley

Holomorphic f a m i l i e s T ( K ) = T + K A o f l i n e a r o p e r a t o r s i n a H i l b e r t space a r e considered, where T and A a r e m-accretive. D e t a i l e d r e s u l t s a r e g i v e n when A i s p o s i t i v e s e l f a d j o i n t . ble a r e i n t e r e s t e d i n t h e case i n which T ( K ) forms a f a m i l y o f t y p e (A) f o r K i n a r e g i o n o f t h e complex p l a n e and o f a weaker t y p e (say t y p e ( B ) ) i n a l a r g e r r e a i o n . D i r a c and Schrodinger o p e r a t o r s w i t h s i n g u l a r p o t e n t i a l s a r e discussed. The a b s t r a c t p a r t c o n s i s t s o f two theorems which a e n e r a l i z e a theorem o f H. Sohr. 1.

INTRODUCTION

I n t h i s paper we a r e concerned w i t h holomorphic f a m i l i e s o f l i n e a r o p e r a t o r s T ( K ) = T + K A i n a H i l b e r t space H, which may have d i f f e r e n t t y p e s a t d i f f e r e n t v a l u e s o f t h e parameter K . We a r e p a r t i c u l a r l y i n t e r e s t e d i n t h e cases i n which T ( K ) i s o f t y p e (A) i n a r e p i o n o f K and o f a more general t y p e i n a l a r g e r r e g i o n . ( F o r d e f i n i t i o n s c f . Kato [2, Chapter V I I ] . )

A s i m p l e example o f t h i s k i n d j s mentioned i n [Z, Example VII-4.151, n o t f u l l y discussed, where t h e Schrodinaer o p e r a t o r (1.1)

T ( K ) = -d2 +

KX-

2,

0

<

x <

m,

though

d = d/dx,

2 i s considered i n H = L ( 0 , m ) . Here T ( K ) i s o f t y p e (A) i f K > 3/4 and o f t y p e ( 6 ) i f K > - 1/4. See Example 1.1 below f o r complex v a l u e s o f K . Another example i s g i v e n i n Kato [3], where t h e D i r a c o p e r a t o r (1.2)

H(K) = i-’a v a r a d + rnB + K V ( X ) ,

3 x € R ,

i s c o n s i d e r e d i n H = (L 2 (R3 ) ) 4 , where m i s r e a l , CY = ( a ,a2,a3) and B a r e t h e usual 4.4 h e r m i t i a n m a t r i c e s , and V ( x ) i s a ( n o t n e c e s i a r i l y h e r m i t i a n ) I f ( V ( x ) I i ( x l - 1 ( t h e m a t r i x norm), i t i s shown t h a t H ( K ) matrix-function. i s holomorphic of t y p e (A) f o r I K I < 1 / 2 and can be c o n t i n u e d t o a f a m i l y of a t y p e analoaous t o t y p e (C) f o r ~ K
T. Kato

342

Ide s t a t e t h e theorems i n s e c t i o n 2, and p r o v e them i n s e c t i o n 3. f o l l o w i n g s e c t i o n s we g i v e s e v e r a l a o p l i c a t i o n s and v a r i o u s remarks. As a t y p i c a l example o f such r e s u l t s , we n o t e of t y p e (A) f o r K o u t s i d e t h e p a r a b o l a d e s c r i b e d and i t can be c o n t i n u e d t o a holomorphic f a m i l y o f c u t alono t h e r a y (--,-1/41. Similar results hold i n h i q h e r dimensions T(K) = -a + K l x l

( 1 -3) considered i n parabola

H

= L2(Rm).

(1.4)

-2

,

x

E

here h a t (1.1) i s holomorphic by n' = 3 - 45 ( K = c + i n ) , t y p e ( B ) on t h e whole K--plane f o r t h e Schrodinger o p e r a t o r m '2,

R"',

(1.3) i s holomorphic o f t y p e ( A ) f o r =

11'

4 - (m - 2)’

-

I n the

K

outside the

4.E,

and i t can be c o n t i n u e d as a holomorphic f a m i l y o f t y p e ( B ) on t h e whole c u t along t h e r a y (1.5)

-m

<

5 c

-

(m

-

2)'/4,

q =

K-plane

0.

I t appears t o be t y p i c a l o f Schrodinqer o p e r a t o r s o f t h e form

(1.6)

-A

+

qg(X)

+

Kqi(x),

x

E

Rm,

t h a t t h e r e g i o n o f t y p e (A) i s t h e e x t e r i o r of a parabola. I t i s i n t e r e s t i n o , however, t h a t i n most cases t h e p a r a b o l a deaenerates i n t o t h e r a y (--,0] so t h a t (1.6) i s a c t u a l l y o f t y p e ( A ) on t h e whole o f t h e K-plane c u t a l o n a t h e n e q a t i v e r e a l a x i s (see s e c t i o n 6 ) . (Thus t h e r e i s no room f o r a r e o i o n o f ( a e n u i n e ) t y p e ( B ) . ) The I x / - 2 - p o t e n t i a l s w i t h nondeqenerate paraboJa (1.4) a r e e x c e p t i o n s , a r e homoaeneous of t h e which may be e x p l a i n e d by t h e f a c t t h a t - A and 1x1same degree -2. I t should be n o t e d t h a t we a r e here u s i n g t h e t e r m t y p e (B) i n a s l i q h t l y g e n e r a l i z e d sense: T(K) need n o t have a semibounded r e a l p a r t , b u t e-i'T(K) w i l l do f o r a s u i t a b l e r e a l number 8 ( w h i c h may depend on K l o c a l l y ) .

lThe O i r a c o p e r a t o r ( l . Z ) , i f V ( x ) i s assumed t o be t h e Coulomb p o t e n t i a l 1x1- , can be c o n t i n u e d t o a l a r o e r r e o i o n of K so as t o become o f t y p e ( A ) f o r I m K > 1/2 and f o r Irn K < -1/2. Thus (1.7) i s holomorphic i n a connected r e a i o n ( t h e u n i o n o f these two h a l f - p l a n e s and t h e u n i t d i s k ) , i n which i t i s p a r t l y o f t y p e (A). The t h r e e p a r t s on which we know t h a t i t i s o f t y p e (A) a r e however d i s j o i n t . A c t u a l l y t h e c e n t r a l I ' k / K I < 1/2 f o r t y p e ( A ) can be enl a r g e d t o a lens-shaped r e g i o n h a v i n g 5 3 1 j 5 / 2 on i t s boundary (see Schmincke[5]), b u t i t i s s t i l l d i s j o i n t f r o m t h e two h a l f - p l a n e s . I f K i s n o t i n t h e t y p e ( A ) r e a i o n , H(K) i s n o t a unique closed extension o f the o r i g i n a l operator (1 . Z ) . Nevertheless, i t i s a d i s t i n o u i s h e d e x t e n s i o n f o r v a r i o u s reasons (see [ 3 1 ) .

F i n a l l y we n o t e t h a t t h e problem c o n s i d e r e d h e r e i s b a s i c a l l y d i f f e r e n t f r o m t h a t o f e s s e n t i a l s e l f a d j o i n t n e s s , a l t h o u q h t h e r e i s a c l o s e r e l a t i o n s h i p between t h e two i n c o n c r e t e cases (see Remark a f t e r Example 7.4 below). Indeed, T + K A i s t r i v i a l l y e s s e n t i a l l y s e l f a d j o i n t i f i t i s s e l f a d . j o i n t , b u t no subset o f H i s i n g e n e r a l s p e c i f i e d on which T + KA m i g h t be e s s e n t i a l l y s e l f a d j o i n t . The r e a d e r i s r e f e r r e d t o [31 r e a a r d i n q t h e p r a c t i c a l use o f t h e holomorphic f a m i l i e s o f t y p e ( A ) and o t h e r t y p e s .

Holomorphic Families of Operators

343

[Supplementary n o t e s ] A c c o r d i n g t o commentslby M. Klaus, t h e t y p e (A) r e g i o n f o r t h e O i r a c o p e r a t o r ( 1 . 2 ) w i t h V = 1x1 i s i n f a c t connected and l a r g r t h a n s t a t e d a?ov,. The boundary o f t h e r e g i o n i s g i v e n by two curves Re(K ) - 3/4 + (Im(K )) , s u g g e s t i n g a c l o s e r e l a t i o n s h i p w i t h t h e S c h r o d i n g e r case (1.4). The t y p e ( C ) r e g i o n i s a l s o l a r g e r . The Author would l i k e t o thank P r o f e s s o r K l a u s f o r t h e s e comments.

5 -

2.

THE M A I N THEOREMS

We c o n s i d e r two m - a c c r e t i v e o p e r a t o r s and c o n s i d e r t h e o p e r a t o r

T + KA,

(2.1)

w i t h domain

0,

T

and

A

i n a H i l b e r t space

H,

= D(T) n D(A),

where K i s a complex parameter. We ask i f T + KA forms a holomorphic f a m i l y o f t y p e ( A ) . An answer i s g i v e n by t h e f o l l o w i n g theorem, which g e n e r a l i z e s Sohr’s theorem (see Sohr [6,7]). THEOREM 2.1. L e t T and A be m - a c c r e t i v e i n H . ( b u t e d n o t be bounded). Assume f u r t h e r t h a t

5

l i m sup Re((A + E)-’v,T*v)

(2.2

-allvll

2

Assume t h a t

A-l

exists

( v E D(T*))

E O ’

(Re E / ~ E ~ L ~ > O ) w i t h a r e a l c o n s t a n t a independent o f

may depend on

(6

v).

( i ) T + K A i s a c l o s e d o p e r a t o r and forms a holomorphic f a m i l y o f t y p e (A) Re K > a, w i t h

for

II(T + KA)uII

(2.3) Re

v

K

(ii) a.

2

(Re

K

- a)llAul[

A l l n e g a t i v e r e a l numbers b e l o n g t o t h e r e s o l v e n t s e t o f

T + KA

for

>

(iii)

T + KA

is

m-accretive f o r r e a l

( i v ) Do i s a c o r e f o r A. T + K A f o r some K > a+, t h e n (v)

If a < 0,

A

K

> a+ = maxIa,OI.

More g e n e r a l l y , i f Boo i s a c o r e f o r

DO0 C Do A.

i s T-bounded w i t h T-bound y ( - a ) - ’

i s a core f o r

so t h a t

Do = D(T).

REMARKS. ( a ) I f A - l i s bounded, t h e l e f t member o f ( 2 . 2 ) e q u a l s Re(A-lv,T*v) and t h e c o n d i t i o n i s i d e n t i c a l w i t h Soht-is., I t s h o u l d be n o t e d t h a t q u a l i t a t i v e l y t h e r e i s n o t much d i f f e r e n c e whether A i s assumed t o be bounded o r n o t , s i n c e one can always c o n s i d e r T + K ( A + E ) i n s t e a d o f T + KA. There i s an e s s e n t i a l q u a n t i t a t i v e d i f f e r e n c e , however, s i n c e t h e v a l u e o f a may be g r e a t l y improved as w i l l be shown by examples below. when we t a k e t h e l i m i t E -,0 i n ( 2 . 2 ) , ( b ) Sohr c o n s i d e r s o n l y nonnegative values o f a i n ( 2 . 2 ) . O f course a negat i v e a i m p l i e s t h a t (2.2) i s t r u e a l s o w i t h a = 0, b u t t h e c o n c l u s i o n s f o r t h e case a < 0 i s s t r o n g (see r e s u l t ( v ) above) and i s w o r t h a s p e c i a l a t t e n t i o n . (c)

Do

need n o t be a c o r e f o r

T.

I n f a c t i n many i n t e r e s t i n g cases i t i s n o t .

( d ) I f A* = A and i f ( 2 . 2 ) i s used w i t h r e a l E > 0, i t i s unchanged when T i s r e p l a c e d by T + i u , LI r e a l . Hence t h e l e f t h a l f - p l a n e C- = {Re z < OI bel o n g s t o t h e r e s o l v e n t s e t o f T + KA, Re K > a.

( e ) P e r t u r b i n g A by a d d i n g a bounded a c c r e t i v e o p e r a t o r does n o t change t h e c h a r a c t e r o f t h e f a m i l y T + KA b e i n g holomorphic o f t y p e (A). But c o n d i t i o n

T. Kato

344

(2.2) i s s e n s i t i v e t o such a p e r t u r b a t i o n . Indeed, i t i s n o t c l e a r whether (2.2) remains t r u e when A i s r e p l a c e d w i t h A + 1 , p o s s i b l y w i t h a d i f f e r e n t a. I t appears t h a t t h i s i s a weak p o i n t i n c o n d i t i o n s o f t y p e (2.2). I n Theorem 2.1, t h e r e g i o n o f K i n which T + KA i s o f t y p e (A) i s r e s t r i c t e d t o a h a l f - p l a n e o f t h e form Re < > a. The n e x t theorem g i v e s a c o n d i t i o n i n which t h e r e g i o n can be l a r g e r and i n c l u d e complex numbers K w i t h a r b i t r a r i l y l a r g e n e g a t i v e r e a l p a r t s . F o r s i m p l i c i t y , however, we s h a l l assume that A i s positive selfadjoint. lThe theorem w i l l i n v o l v e t h e numerical r a n g e o f t h e q u a d r a t i c f o r m -(A- v,T*v). Since t h e numerical ran.ge i s h a r d t o d e t e r m i n e i n p r a c t i c e , we s h a l l use a convex s e t S c o n t a i n i n g i t r a t h e r t h a n t h e n u m e r i c a l range i t s e l f . THEOREM 2.2. L e t T be m - a c c r e t i v e and A p o s i t i v e s e l f a d j o i n t (so t h a t A - l e x i s t s b u t need n o t be bounded). L e t S be a c l o s e d convex subset o f t h e comS contains a t l e a s t a ray o f the form p l e x plane, e x t e n d i n g t o - m ( i . e . ( c - t; 0 5 t c m l w i t h a complex number c ) , such t h a t l i m i n f dist(-((A

(2.4)

+ ~ ) - ~ v , T * v ) / I l v l l ~ , S )= 0

O<E-.O

f o r each K f€

v # 0.

v E [)(T*),

(i)T + KA S, w i t h

Then

i s c l o s e d and forms a holomorphic f a m i l y o f t y p e (A) f o r II(T + KA)uII

for

dist(K,S)llAull

( i i ) A l l n e g a t i v e r e a l numbers b e l o n g t o t h e r e s o l v e n t s e t o f $ S.

T + KA

K

(iii)

T + KA

i s m-accretive i f

( i v ) I f 0 $ S, l/dist(O,S).

then

A

K

$ S

and

Re

K

2 0.

i s T-bounded w i t h T-bound n o t exceeding

( v ) I f , i n p a r t i c u l a r , T i s a l s o s e l f a d j o i n t , t h e r e i s a holomorphic f a m i l y T(K) 2 T + KA o f t y p e (B), d e f i n e d on t h e whole K-plane c u t a l o n g t h e r a y (--,-l/k], w i t h nonempty r e s o l v e n t s e t . Here k i s t h e f o r m T-bound o f T ( K ) i s a unique t h e form A. ( I f A i s n o t f o r m T-bounded, s e t k = -.) a n a l y t i c c o n t i n u a t i o n ( w i t h r e s o l v e n t ) o f t h e f a m i l y T + KA d e f i n e d i n ( i ) . REMARKS. ( a ) A s i m p l e aeornetric c o n s i d e r a t i o n shows t h a t e x c e p t f o r t h e vacuous case i n which S i s t h e whole p l a n e , S can be d e s c r i b e d as t h e s e t of a l l comp l e x numbers K = 6 + i n s a t i s f y i n a an i n e a u a l i t y o f t h e f o r m - m < 5 5 ~ ( r l ) , where 4 i s a r e a l - v a l u e d concave f u n c t i o n d e f i n e d on an i n t e r v a l ( w h i c h may be open o r closed, and may reduce t o a s i n g l e p o i n t . A l l s u p p o r t i n g l i n e s t o S may be assumed t o have normal d i r e c t i o n eei w i t h - ~ / 25 e 5 T / ? . ( b ) I n ( v ) , T ( K ) i s a d i s t i n q u i s h e d c l o s e d e x t e n s i o n o f T + KA i s i n s i d e S b u t n o t on ( - a , - l / k ] (see I n t r o d u c t i o n ) .

3.

PROOF OF THE THEnREIlS To prove Theorem 2.1,

we need a lemma.

LEMMA 3.1. Under t h e assumptions o f Theorem 2.1, l e t T = T ( l + n-’T)-’, n , 2 , . .., be t h e Yosida a p p r o x i m a t i o n s f o r T. Then”we have

(3.1)

Re(T,u,Au)

2 - a l l ( l +n-lT*)-lAu1I2 2 -a+llAull 2

(U E

D(A)),

when

K

345

Holomorphic Families of Operators (3.2)

2 -allAull 2

Re(Tu,Au)

Proof.

Since

E’O, Re

E/

A-l

1 ~ 21

= ( 1 t n-lT*)-’Au

v,

n D(A)).

Do = D(T)

E

i s m - a c c r e t i v e w i t h A, we have u = l i m ( A + E)-’Au 6. F o r each u E D(A), t h e r e f o r e , we have Re(T,u,Au)

Let

(u

Re(Tnu,Au)

which proves ( 3 . 1 ) .

+ E)-~Au,T~Au).

= l i m Re((A

so t h a t

E D(T*),

= l i m Re((A

Tn

Note t h a t

and

as

Au

(1 +n-lT*)vn.

=

Then

+ ~ ) - ‘ ( l + n-’T*)vn,T*vn)

Ti

a r e b o t h bounded and a c c r e t i v e .

(3.2) t h e n f o l l o w s by Re(Tu,Au)

= l i r n Re(Tnu,Au)

n-

1. -a l i m l l v n 112

= -allAull

2

,

n-

F i r s t we n o t e t h a t (2.3) f o l l F w s d i r e c t l y f r o m ( 3 . 2 ) , Proof o f Theorem 2.1. which i m p l i e s t h a t Re((T + KA)u,Au) 1. (Re K - a)llAull

.

Next we p r o v e ( i i i ) . I t i s o b v i o u s t h a t at -> 0. Hence i t s u f f i c e s t o v e r i f y t h a t Re h > 0. K

>

Let f such t h a t

E

H

be a r b i t r a r y .

Since

Tn

t

T T

KA

t t

KA i s a c c r e t i v e f o r K A + h has range H

i s m-accretive, there i s

if unED(A)

(Tn t KA + h)un = f .

(3.3)

As usual t h i s i m p l i e s t h a t II u 1I 5 IIfll/Re A . Moreover, t a k i n g t h e i n n e r p r o d u c t o f (3. w i t h Aun and u s i n g ‘(3.1), we e a s i l y o b t a i n K I I A ~llg)< IIfllIIAu 11 + a, IIAunl12, which i m p l i e s t h a t IIAunll 5 ( K at)-’llfIl. n n Thus IIunll, IIAunll, and IITnunll ( b y ( 3 . 3 ) ) a r e a l l u n i f o r m l y bounded as

-

n

+ m.

to a (T

t

A s t a n d a r d argument shows t h a t un converges, a l o n g a subsequence, weakly w i t h Au, + Au and Tnun + Tu weakly. Then (3.3) shows t h a t

u E Do,

KA + x ) u = f,

showing t h a t

Next we p r o v e ( i ) and ( i i ) . IIA~II 5 ( c

(3.4)

-

f

i s i n t h e range o f

To t h i s end, we f i x

T + KA t X

c > a+

a)-lI1 ( T t C A ) ~ I I

(u

E

as r e q u i r e d .

and n o t e t h a t Do)

by (2.3) ( w h i c h has been p r o v e d ) . Hence A i s T + cA - bounded w i t h bound I ( c - a)-1, so t h a t T + KA = T + cA - ( c - K ) A i s holomorphic o f t y p e ( A ) f o r I c - K ( < c - a, w i t h t h e Neumann s e r i e s f o r h > 0 m

(T + K A + A)-’

(3.5)

= (T

+ cA +

A)-’

1

(c

-

K)’[A(T

+ cA +

n=O

But (3.4) gives IIA(T t cA + A ) -

because 1c - K /

<

1

II

5 ( c - a ) - l I I ( T + c A ) ( T + cA + A)-’Il < ( c - a)-’

T + cA i s m - a c c r e t i v e . Hence t h e r e s o l v e n t ( 3 . 5 ) e x i s t s f o r A > 0 and c - a. B u t e v e r y complex number K such t h a t Re K > a i s i r l c l u d e d i n

T. Kato

346

t h i s r e g i o n , as i s seen b y t a k i n g t h e p o s i t i v e number Thus we have proved (i) and ( i i ) .

c

s u f f i c i e n t l y large.

To p r o v e ( i v ) , l e t DO0 be a c o r e f o r T + KA f o r some K > a;, we have i s dense i n H. Suppose f E ti i s o r t h o g o n a l t o t o show t h a t (A + l)Doo be t h e s o l u t i o n t o ( T + K A + 1 ) u = f (which e x i s t s b y (A + 1)DO0. L e t u E D Since DO0 a c o r e f o r T + KA, t h e r e i s a sequence such (ii)). t h a t (T + KA)u,, + ( T + KA)U and un + u. Since A i s ( T + ItA)%zn%!, it follows t h a t (A + l ) u n -+ (A + 1 ) u . S i n c e f i s o r t h o g o n a l t o (A + l ) u , f i s also orthogonal t o ( A + 1)u. Thus we o b t a i n ( ( T + K A + l ) u , ( A + 1)uT = 0. Using ( 3 . 2 ) 2 a n d t h e f a c t t h a t T + K A and A a r e a c c r e t i v e , we conclude t h a t ( K - a)llAull 5 0 , hence Au = 0, u = 0, f = 0, as we wished t o show.

iP

F i n a l l y we prove ( v ) . I f a < 0, K = 0 i s i n t h e r e g i o n Re K > a, so t h a t T + OA i s c l o s e d w i t h a l l n e g a t i v e numbers i n t h e r e s o l v e n t s e t . Since T 3 T + OA has t h e same p r o p e r t y , we must have D = D(T), T + OA q T . Thus D(T) C D(A) and A i s T-bounded. The T-bound dogs n o t exceed ( - a ) , b y (2.3). P r o o f o f Theorem 2.2. F i x a supporting l i n e by an e q u a t i o n o f t h e f o r m (3.6)

Re e-ieK

= a,

(see Remark a f t e r Theorem 2.2). that liminf[-Re(e-ie(A

(3.7)

L to S,

-Ill2

< 0 <

which may be expressed

n12

I n view o f assumption (2.4),

+ ~ ) - l v , T * v ) ] 5 allvll 2

then, i t i s c l e a r

(v E D(T*))

O < E 4

Hence we can a p p l y Theorem 2.1, i n which A i s r e p l a c e d by eiOA a c c r e t i v e ) and E by ele, (which has argument 0 , 101 < n / 2 ) . T + KA = T + (e-ieK)(eieA)

(3.8)

i s holomorphic o f t y p e (A) f o r s i d e o f L n o t c o n t a i n i n g S. i n g l i n e L , we conclude t h a t I n e q u a l i t y (2.3)

Re e - j e K > a, which i s t h e case i f K i s on t h e Since f o r each K 6 S we can f i n d such a s u p p o r t T + K A i s holomorphic o f t y p e ( A ) f o r K $ s.

e-ieK

-

a)llAull = dist(K,g)llAull.

Given any K @ S. however, we can f i n d a s u p p o r t i n g l i n e dist(K,i?) = dist(K,S). Thus we have proved ( i ) . ( i i ) f o l l o w s b y a p p l y i n g Theorem 2.1, tion,

e

such t h a t

( i i ) , t o (3.8).

I t i s obvious t h a t T + K A i s a c c r e t i v e i f i t i s m - a c c r e t i v e due t o p r o p e r t y ( i i ) .

(iv)

Do)

a p p l i e d t o (3.8) now g i v e s

II(T + KA)UII ,(Re

(3.9)

( w i t h domain

( w h i c h i s mIt f o l l o w s t h a t

Re

K

1. 0.

If

K

d S

i n addi-

can be proved as i n Theorem 2.1.

To prove ( v ) , we have o n l y t o c o n s i d e r t h e q u a d r a t i c f o r m (3.10)

I I T ~ / ~ U+I IK~I I A ~ ’ ~ U I I ~

(u

E

D(T’/~)n D(A’/~)).

T h i s f o r m i s n o t n e c e s s a r i l y semibounded i n i t s r e a l p a r t . B u t i t becomes so, and i n d e e d s e c t o r i a l and closed, when m u l t i p l i e d by e - l @ where 4 i s c l o s e t o ( a r g ~ ) / 2 . I n t h i s sense (3.10) determines an a s s o c i a t e d m - s e c t o r i a l o p e r a t o r T ( K ) 3 T + K A , which forms a f a m i l y o f t y p e (B) because A i s form-T(K)-bounded if K # 0 ( c f . [2, Theorem V I I - 4 . 2 1 ) .

Holomorphic Fumilies of Operators

341

By choosing $ a p p r o p r i a t e l y , we can c o v e r a l l complex numbers k e x c e p t Thus T ( h ) i s h o l o m o r p h i c o f t y p e (6) on t h e whole h f o r the ray (-m,0]. p l a n e c u t a l o n g (-m,O]. I f A i s form-bounded r e l a t i v e t o T, w i t h r e l a t i v e Thus we have proved ( v ) . bound k, t h e n K may be i n t h e i n t e r v a l (-l/k,O]. I t s h o u l d be n o t e d t h a t t h e r e s o l v e n t s e t o f T ( K ) c o n t a i n s t h e e x t e r i o r o f t h e s e c t o r 0 2 a r g z 5 a r g k i f a r g K 1. 0 (and s i m i l a r l y i f a r g K < 0 ) . A l s o we n o t e t h a t T ( K ) i s a unique a n a l y t i c c o n t i n u a t i o n o f T + K A d e f i n e d f o r (see [2, Remark V I I - 1 . 6 3 ) . K 9 S 4.

A FIRST-ORDER DIFFERENTIAL OPERATOR As t h e s i m p l e s t example o f s i n g u l a r d i f f e r e n t i a l o p e r a t o r s , we c o n s i d e r d +

(4.1)

i n t h e H i l b e r t space

o < x < - ,

K/X,

H

=

2

L

d = d/dx,

(0,m).

i s m-accretive. The o p e r a t o r T = d w i t h domain H1 = H1(O -) a l s o m - a c c r e t i v e w i t h i n v e r s e A - l = x. 'Sin& *; = - d w i t h domain p l e computation g i v e s m

(4.2)

l i m Re((A+.)-'v,T*v) E

CO

= lim(1/2) E+O

0

A = l/x i s H I , a sim-

2 ( 1 + ~ x ) - ~ ] v I ~= d(1/2)llvll x 1 (b E H ) .

Thus ( 2 . 2 ) i s t r u e w i t h a = -1/2. A c c o r d i n g t o Theorem 2.1, ( 4 . 1 ) w i t h domain D = H1 forms a holomorphic f a m i l y o f t y p e ( A ) f o r -1/2 < Re K . Since a < 0, i?be18ngs t o case ( i v ) o f Theorem 2.1, w i t h A T-bounded w i t h T-bound < ( - a ) - l = 2. T h i s bound 2 i s sharp, as w e l l as t h e r e g i o n Re K > -1/2. I t may be remarked t h a t (4.2) w i t h a f i n i t e g i v e s o n l y a = 0. Thus t a k i n g t h e l i m i t E 4 0 sharp r e s u l t s .

does n o t g i v e a = -1/2; i s essential f o r getting E

it

I f we exchange T and T* i n t h e above arguments, we o b t a i n a = 1/2. It f o l l o w s t h a t - d + K / X w i t h domain H1 n D ( l / x ) = H A , forms a holornorphic f a m i l y of t y p e (A) f o r Re K > 1/2. I n l o t h e r words, t h e f a m i l y (4.1) i s a l s o o f t y p e ( A ) f o r Re K < -1/2 w i t h domain Ho.

Thus (4.1) g i v e s two s e p a r a t e l f a m i l i e s o f t y p e (A) f o r Re K > -1/2 and f o r -1/2, b o t h w i t h domain Ho. A c t u a l l y t h e second f a m i l y can be c o n t i n u e d Re K a n a l y t i c a l l y across t h e l i n e Re K = - 1 / 2 up t o Re K < 1/2, though i t i s no l o n g e r o f t y p e ( A ) . To see t h i s we have o n l y t o c o n s i d e r t h e a d j o i n t o f t h e f i r s t f a m i l y , w i t h K r e p l a c e d w i t h T. We s h a l l o m i t t h e d e t a i l s . I n t h i s way we can prove THEOREM 4.1.

There a r e two holomorphic f a m i l i e s

T+(K) -

o f r e a l i z a t i o n s o f (4.1).

(i)T + ( K ) = d + K / X , w i t h domain H1 i s o f t y p e ( A ) f o r K > -1/2, and e q u a l s T m i n ( " ) ( t h e c l o s e d minimal r e a l i g i t i o n ) . I t has r e s o l v e n t s e t C- and F o r Re K > 0, T+!K) i s m-accretive. F o r Re K > 1/2, r e s i d u a l spectrum C+. T+(K) i s a l s o equal t o T m a x ( ~ ) ( t h e maximal r e a l i z a t i o n o f ( 4 . 1 ) ) . ( i i ) T - ( K ) i s d e f i n e d f o r Re h < 1/2 and e q u a l s Tmax(;]keALt has r e s o l v e n t s e t C, and p o i n t spectrum C-, w i t h eigenfunctions with 0. For Re h < 0, T - ( k ) i s i n - d i s s i p a t i v e . For Re K < -1/2, T - ( K ) Re i 1 a l s o equals Tmin(") and forms d f a m i l y o f t y p e ( A ) w i t h domain Ho.

T. Kato

348

REMARKS. ( a ) We have T m a X ( ~ =) Tmin(") f o r Re K < -1/2 and Both T+(K) a r e d e f i n e d on t h e s t r i p -1/2 < Re K < 1/2, where = T, C T- = Tmax. Tmni

Re

K

> 1/2.

generates a c o n t r a c t i o n semigroup f o r Re K > 0, and T - ( K ) does Each o f them i s holomorphic i n K i n i t s domain. I t appears t h a t n o r T - ( K ) generates a Co-semigroup f o r o t h e r values o f K .

( b ) -T,(K) f o r Re K < 0. n e i t h e r T+(K)

( c ) The f a m i l y T - ( K ) i s n o t o f t y p e (A) o r o f any f a m i l i a r t y p e f o r - K x x -1/2 2 Re K < 1/2, as i s seen f r o m t h e b e h a v i o r o f i t s e i g e n f u n c t i o n s x e (which become l e s s smooth as Re K grows). 5.

THE DIRAC OPERATOR Let T = a.grad + i m B ,

(5.1) where

a = ( a ,a ,a

)

and

B

A =

XI-^,

( x E R3),

a r e as i n I n t r o d u c t i o n . T i s i - t i m e s t h e f r e e T and -T a r e b o t h m - a c c r e t i v e .

(~elfadjoint)~Di6ac~operator.Thus

A s i m p l e computation analogous t o (4.2) g i v e s (2.2) w i t h a = 1/2. (Note I t f o l l o w s f r o m Theorem 2.1 t h a t t h a t t h e aj and B a r e h e r m i t i a n m a t r i c e s ) . T + K A i s a holomorphic f a m i l y o f t y p e (A) f o r Re K > 1 / 2 and f o r Re K < -1/2. Combined w i t h t h e r e s u l t o f [3] g i v e n i n I n t r o d u c t i o n , we have THEOREM 5.1. There i s a holomorphic, skew a d j o i n t f a m i l y T ( K ) 3 T + KA d e f i n e d on t h e u n i o n o f t h e d i s k < 1 and t h e two h a l f - p l a n e s Re K > 1/2, T ( K ) i s o f t y p e (A) i n these two h a l f - p l a n e s and i n t h e d i s k Re K < -1/2. ( K I < 1/2, w i t h D(T(")) = H1(R3)4 and T ( K ) = T + KA. The r e s o l v e n t s e t o f T(K) c o n t a i n s C- i f Re K > 1/2, C, i f Re K < -1/2, b o t h C, if ' ~ 1: 1/2, dnd a t l e a s t t h e r e a l a x i s ( e x c e p t t h e o r i g i n i f m = 0 ) i f 1

-

REMARK. The s e l f a d j o i n t D i r a c f a m i l y i s g i v e n b y H ( K ) = - i T ( K ) . The r e s u l t s o f Theorem 5.1 a r e n o t sharp. For s t r o n g e r r e s u l t s , see t h e end o f I n t r o d u c t i o n . 6.

A CLASS OF SINGULAR POTENTIALS

I n o r d e r t o a p p l y Theorem 2.2 t o S c h r o d i n g e r o p e r a t o r s , i t i s c o n v e n i e n t t o i n t r o d u c e a c l a s s o f p o t e n t i a l s . We say a p o t e n t i a l q d e f i n e d on Rm i s i n c l a s s ( V ) i f e i t h e r q = 0 i d e n t i c a l l y o r q ( x ) > 0 and

h e r e i t i s understood t h a t measure zero. If q e i ' c a l l y as c

=

+-

i s a l l o w e d on a c l o s e d s e t

Q

C

Rm

of

(V), -+

grad(q + c ) - ' l 2 n umbe r s (6.2)

q(x)

b

l l ( q + c)-1/21, i s f i n i t e i f c > 0 and decreases monoton( T h i s i s seenLiPby n o t i n g t h a t -1 3/2 = ( 1 + cq ) grad q - l l 2 . ) For l a t e r use, we i n t r o d u c e t h e

-.

= b[q]

= l i m II(q + c)-1/211Lip

5 llq'1/211Lip

= b[q]

= b.

C"

The c l a s s ( V ) has s e v e r a l i n t e r e s t i n g p r o p e r t i e s . ( a ) A p o t e n t i a l q E ( V ) may be r e g u l a r o r s i n g u l a r . I f q ( x ) < m everywhere so t h a t Q i s empty, (6.1) i m p l i e s t h a t q i s l o c a l l y L i p s c h i t z i a n on R". If Q i s n o t empty and i f x 9 Q approaches Q. q ( x ) must blow up l i k e , o r

.

3 49

Holomorphic Fundies of Operators

f a s t e r than, c o n s t ( d i s t ( x , Q ) ) - ’ t o s a t i s f y (6.1). I n thij,2ense, potentials i n should look l i k e c l a s s ( V ) a r e i n general h i g h l y s i n g u l a r . The graph o f qa quilt. q , q2 E ( V ) ( b ) The c l a s s ( V ) i s a d d i t i v e and m u l t i p l i c a t i v e , i . e . , I n f a c t , i t i s easy t o l e e t h a t i m p l i e s t h a t q1 + q2 and q1q2 E ( V ) .

( c ) Most p o t e n t i a l s t h a t i n c r e a s e ( e v e n t u a l l y ) m o n o t o n i c a l l y as b e l o n g t o c l a s s ( V ) ; f o r example

(6.4)

q(x) = P(x),

exp[p(x)l,

1x1

+

m

ex~[ex~[~(x)ll,~...~.~

where p ( x ) i s a p o s i t i v e - v a l u e d p o l y n o m i a l . The s i n g u l a r p o t e n t i a l q ( x ) = ! x J - ~ , k > 0, belongs t o c l a s s ( V ) i f and o n l y i f k > 2. Thus modera t e l y s i n g u l a r p o t e n t i a l s (such as Coulomb) a r e excluded. I n Z l l t h e e5amples g i v e n above, i t - i s e a s i l y v e r i f i e d t h a t b = b[q] = 0 e x c e p t f o r 1x1- , f o r which we have b = b = 1. 7.

SCHRODINGER

OPERATORS WITH SINGULAR

POTENTIALS

I n t h i s s e c t i o n we c o n s i d e r S c h r o d i n g e r o p e r a t o r s o f t h e form (7.1)

T

where

T

+

x E Rm,

i s i t s e l f a S c h r o d i n g e r o p e r a t o r g i v e n by T = -A + qo(X),

(7.2)

.ql(x),

40 2 0

( f o r m sum).

By t h i s we mean t h a t T i s a nonnegative s e l f a d j o i n t o p e r a t o r such t h a t and D(T) c H1(Rm) n D(qA/2) = D(T’/‘) (u,Tv)

(7.3)

u

for

E

D(T”2)

and

= ( g r a d u, g r a d v )

+ (qb/2u,qb/2v)

v 6 D(T).

THEOREM 7.1. L e t T be as above and l e t ql b e l o n g t o c l a s s ( V ) i n t r o d u c e d i n t h e p r e c e d i n g s e c t i o n . Then T + K q , forms a holomorphic f a m i l y o f t y p e (A), defined f o r K i n the e x t e r i o r o f the parabolic w i t h domain D(T) n D(q,), region S given by (7.4) (see ( 6 . 2 ) ) . Moreover, T + Kq can be c o n t i n u e d t o a h o l o where bl = b[ql] of t y p e ( B ) d e f i d d on a l l o f t h e K-plane c u t morphic f a m i l y T ( K ) 3 T + Kq a l o n g t h e r a y (-m,-l/k], wheie k i s t h e r e l a t i v e bound o f t h e f o r m q1 w i t h respect t o t h e form T ( s e t k = i f q1 i s n o t f o r m T-bounded).

-

REMARKS. ( a ) The assumption on T i s s a t i s f i e d , a f o r t i o r i , i f qo 0 and This i s true, i n p a r t i c u l a r , - A + qo i s s e l f a d j o i n t w i t h domain H2(Rm) n D(qo). i f qo i s i n c l a s s ( V ) w i t h b[qo] < 1. Indeed, t h e theorem i t s e l f then shows t h a t - A + qo i s s e l f a d j o i n t ( a p p l y t h e theorem w i t h k: = 1, qo = 0, and q1 replaced w i t h qo). ( b ) If bl = 0, which occurs i n many i n t e r e s t i n g cases (see p r e c e d i n g s e c t i o n , ( c ) ) , S reduced t o t h e ( c l o s u r e o f ) n e g a t i v e r e a l a x i s . Thus (7.1) i s o f t y p e (A) on a l l of t h e K-plane c u t a l o n g (-m,O].

(c)

R e s u l t s r e l a t e d t o (7.4) a r e found i n E v e r i t t and G i e r t z [l].

T. Kato

350

EXAMPLES 7.2. (a) -A + 1 x I 2 + ~ 1 x 1i s ~holomorphic o f t y p e (A) f n r a l l K a l o n g (--,O]. Note t h a t 6[q] = 0 f o r b o t h q = 1x1‘ and q = 1xI4 (see section 6,(c)). ( b ) -A + vided k > 2

K [ x ~ - ~i s

of t y p e (A) l i k e w i s e f o r a l l see l o c . c i t ) .

cut

on t h e c u t plane, p r o -

K

-2 ( c ) -A + ~ 1 x 1 i s o f t y p e ( A ) f o r K C $ S, where S i s g i v e n by (7.4) w i t h b = 1. T h i s r e s u l t i s n o t sharp, however. F o r a sharp r e s u l t , see Example 7.4 bilow.

I

P r o o f o f Theorem 7.1. _c > 0 i s a c o n s t a n t . bl i s r e p l a c e d by

We a p p l y Theorem 2.2 t o o u r T and A = c + ql, where L e t S, be t h e p a r a b o l i c r e g i o n o f t h e f o r m ( 7 . 4 ) , where

= 1I ( c + q1 )-1/zil

(7.5)

bl ,c We s h a l l show t h a t f o r

(lim b = bl). ctm 1 - c

Lip v E D(T*) = D(T),

l i m ( - ( A + ~ ) - ’ v , T * v ) = - [ ( c + ql)-’v,Tv)

E Sc, €40 where we have assumed IIvII = 1 w i t h o u t l o s s o f g e n e r a l i t y . Then i t w i l l f o l l o w hence T + Kql t o o , i s h o l o f r o m Theorem 2.2 t h a t T + KA = T + K ( C + ql), m o r p h i c - o f t y p e (A) f o r K $ S., Since, however, S S as c = by we then conclude t h a t T + Kql i s o f t y b e (A) f o r K f! S. bl ,c + bl ,

(7.6)

-f

-f

To p r o v e ( 7 . 6 ) , we s e t (7.7) Since v E D(T) c H1(Rm) n O(qA/‘), t h e same i s t r u e o f ( c + ql)-’v = f % v by fc E Lip(Rm). Thus a s i m p l e computation based on i n t e g r a t i o n b y p a r t s g i v e s (see ( 7 . 3 ) ) 2 -1 v,Tv) 5 - ( g r a d f c v , g r a d v ) = 5, + i q c , - ( ( c + ql) (7.8) where t h e n o t a t i o n 5 means t h a t t h e d i f f e r e n c e i s a n o n p o s i t i v e where = -1lgrad fcvl12

+ Ilv g r a d fcll 2 ,

(7.9)

6,

(7.10)

nc = -2 I m ( g r a d f c v , v g r a d f c ) .

Since (7.11) Hence

= bl,c

IIfcllLip 5,

:n 5 4b:,c(b:,c

<

-

b2 1,c

-

-

number and

b y (7.5) and ( 7 . 7 ) , we o b t a i n ( r e c a l l t h a t 2 IIgrad fcvll 2 , ‘Ic < 4b: ,c ~ i g r a dfcvll .

cc)

so t h a t

cc +

inC E Sc.

IIvII = 1 )

I n view o f (7.8), we have

t h u s proved ( 7 . 6 ) .

As r e a r k e d above, Theorem 7.1 does n o t g i v e a sharp r e s u l t when t h e poteni s i n v o l v e d ( s e e Example 7.2). The f o l l o w i n g theorem d e a l s w i t h t i a l /XI-’ cases o f t h i s k i n d , i n which we a l l o w t h e e x i s t e n c e o f s e v e r a l c e n t e r s o f i n v e r s e square p o t e n t i a l s . THEOREM 7.3.

I n Theorem 7.1,

assume i n a d d i t i o n t h a t

Holomorphic Families of Operators where xJ E Rm, region given by (7.13) where

.

j = l,...,N

-

n2 2 4b:(b:

Then t h e s e t

S

35 1

can be r e p l a c e d by a p a r a b o l i c

N ( 4 e ) - l (m - 2)‘

- <),

e =

I:

“i = l.

bl = b [ q l l .

ej,

-2 EXAMPLE 7.4. ( a ) I n t h e s i n g l e - c e n t e r case q1 = 1x1 , we have 1 1 = 1 , x 1 = 0, el = 1. Hence (7.13) becomes 2 (7.14) r) 2 4 ( - ~< ) , ~ o0 = 1 - (m - 2 ) 2 /4.

bl = 1,

Bo i s a f a m i l i a r number i n t h e t h e o r y of S c h r o d i n g e r o p e r a t o r s . (7.14) i s sharp even f o r r e a l K . A l s o t h e whole f a m i l y T ~ K ) o f t y p e ( B ) i s holomorphic on2the K-plane c u t a l o n g (--,fig - 1 1 = (-m,-(m- 2) /4], s i n c e we have k = 4(m- 2 ) i n t h i s case. Note t h a t i f m 5, Bo i s n e g a t i v e so t h a t t h e p a r a b o l a (7.14) i s e n t i r e l y i n t h e l e f t h a l f - p l a n e ; thus we have t h e s i t u a t i o n o f case ( i v ) i n Theorem 2.2. ( b ) Suppose t h a t q i s e x a c t l y l p 2 u a l t o t h e r i h t - h a n d i d e o f (7.12) (Nc e n t e r p o t e n t i a l ) . SinEe b[q] = e; f o r q = e . 7 ~- x J I - * , i t follows from J (6.3) t h a t bl = b[ql] I m a x ,I1/’ = (min e.)-”‘. Thus - A f Kq i s holomorphic J J o f t y p e (A) f o r K o u t s i d e t h e regJon (7.13) where bl = (min e.!-”‘. On t h e o t h e r hand, we have k = 4e(m - 2 ) as t h e f o r m (-A)-bound f o r J t h e form ql. Hence T ( K ) i s o f t y p e ( B ) on t h e whole K-plane c u t a l o n g (--,-(m - 2)2/4e]. REMARK. I t i s w e l l known (see e.g. Reed and Simon [4]) t h a t - A + KIXI-‘ i s e s s e n t i a l l y s e l f a d j o i n t on CO(Rm\IOJ) i f and o n l y i f K 1. Bo. It i s n o t accid e n t a l t h a t t h i s c o n d i t i o n i s i d e n t i c a l w i t h K 8. S f o r r e a l K . Indeed, i t can be proved t h a t i n Theorem 7.2, C (Rm\tO}) i s a c o r e f o r T ( K ) i f K 8. S. On t h e o t h e r hand, i t s h o u l d be r e c a ? l e d t h a t t h e q u e s t i o n a b o u t t y p e (A) i s b a s i c a l l y d i f f e r e n t from t h a t o f e s s e n t i a l s e l f a d j o i n t n e s s (see I n t r o d u c t i o n ) . P r o o f o f Theorem 7.3. We m o d i f y t h e p r o o f o f Theorem 7.1. Since bl,c be expected t o become s m a l l as c + m f o r p o t e n t i a l s o f t h e f o r m r e v e r s e thP s t r a t e g y and l e t c J 0 t o make t h e n e g a t i v e - s i g n e d t e r m Ilgrad fcvl12 l a r g e . To t h i s end we e x p l o i t t h e well-known i d e n t i t y

ca n o t 1x1- , we

2 j -1 2 1 Ilgrad wl12 = Ilgrad w + gjwl12 + 4- (m - 2) II1x - x I wII ,

(7.15) where

w

(7.17)

Ilgrad fcvl12 = Ilgrad f c v + g . f vl12 + 4-’(rn - 2)2I I I x - x j l - l f c v l l 2 . J C f c and g . a r e r e a l , on t h e o t h e r hand, (7.10) may be w r i t t e n J - - 2 Im(grad f c v f gjfCv,v grad f c ) . r)c -

g. i s the vector-function J g . ( x ) = 2-’(m - 2 ) j x - x j l - 2 ( x - x j ) . (7.16) J (7.15) g i v e s f o r w = f c v

Since

E

Hence, b y

H1(Rm)

\grad

and

fcl 2

bl

( r e c a l l a g a i n t h a t IIvII = 1 ) 2 2 0: 5 4blllgrad f c v + g . f vII J C

.

(7.18)

I t f o l l o w s f r o m (7.18),

(7.19)

b2(4b: qc- 1 <

(7.17), (7.9) t h a t - 4 i c - (m - 2 ) 21I 1x - x J I - ’ f C v l l 2 ) .

B

352

T. Kato

Now we m u l t i p l y (7.19) w i t h

1 eJ. i x -

e.

3

and add f o r

x j I -2 fc2 = ( 1 + c f 2 ) - l

j = l,.-.,N.

1 ejix -

xj/-2f2

Since

1 ej

= e

and

2 (1 + c f 2 ) - l

by (7.7) and (7.12), we o b t a i n , ’ IT

5 b2(4b2 - 45, - e-’(m - 2)‘11(1 + cf2)-1’2vn2).

Ifwe l e t c i 0, then ( 1 + c f 2 ) - ’ / * v + v i n L2 by dominated convergence. from t h e s e t S given by (7.13) tends t o zero as Thus t h e distance o f 5 + i n c J- 0. This v e r i f i e s c 8 n d i t i 8 n ( 2 . 4 ) and completes t h e p r o o f ( r e a d c f o r E ) . Note t h a t t h e a d d i t i o n a l c o n t r i b u t i o n due t o qov i s harmless as i n t h e p r o o f o f Theorem 7.1 . REFERENCES

[l] E v e r i t t , W. N. and G i e r t z , M., I n e q u a l i t i e s and separation f o r Schrodinger type o p e r a t o r s i n L2(Rn), Proc. Roy. SOC. Edinburgh 79A (1978), 257-265. [2]

Kato, T.,

P e r t u r b a t i o n theory f o r l i n e a r operators, Springer 1980.

[3]

Kato, T.,

Holomorphic f a m i l i e s of D i r a c operators, Math. Z. t o appear.

[4]

Reed, M. and Simon, B . , Methods o f modern mathematical physics, Volume 11, Academic Press 1975.

[5]

E s s e n t i a l s e l f a d j o i n t n e s s o f Dirac operators w i t h a Schmincke, U.-W., s t r o n g l y s i n g u l a r p o t e n t i a l , Math. Z. 126 (1972), 71-81.

[6] Sohr, H., Uber d i e S e l b s t a d j u n g i e r t h e i t von Schrodinger-Operatoren,

Math. Z.

160 (1978), 255-261. [7]

Sohr, H., Storungstheoretische Regularitatsuntersuchungen, Math. Z. 179 (1982), 179-192.

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) Q Elsevier Science Publishers B.V.(North-Holland), 1984

353

NECESSARY AND SUFFICIENT CONDITIONS FOR SOLVABILITY OF NON-SOLVABLE LINEAR PARTIAL DIFFERENTIAL EQUATIONS Shmuel Kiro Department o f Mat hema t i c s Rice U n i v e r s i t y Houston, Texas U.S.A.

We c o n s i d e r c o n d i t i o n s one should impose on a d i s t r i b u t i o n f f o r existence of a local solution o f t h e equation P u = f , where P i s a non-solvable 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 ope r a t o r ( e . g . t h e Hans Lewy o p e r a t o r ) .

I NT RO DU CT I ON Let

P

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

i n a neighborhood o f t h e o r i g i n i n IRN ( o r , more g e n e r a l l y , an a n a l y t i c pseudod i f f e r e n t i a l o p e r a t o r ) . For a d i s t r i b u t i o n f we s h a l l d e n o t e by WFAf t h e a n a l y t i c s i n g u l a r spectrum (wave f r o n t s e t ) o f f ( f o r d e f i n i t i o n s e e [ 3 ] o r [ 4 ] ) . We s h a l l s a y t h a t t h e o p e r a t o r f o r a given d i s t r i b u t i o n that (0.1)

f

P

i s micro-locally solvable a t

on R N i f there i s a d i s t r i b u t i o n

(x,c) u

E

T%N\IO} such

on IRN

( x , < ) d wFA(Pu-f).

We s a y t h a t P i s l o c a l l y s o l v a b l e n e a r t h e o r i g i n f o r a given f i f some d i s t r i b u t i o n u s a t i s f i e s t h e e q u a t i o n Pu = f i n a n m E o r h o o d o f t h e o r i g i n U . We assume t h a t P i s n o t s o l v a b l e f o r some f , e . g . P i s the famous Hans Lewy opera t o r (0.2)

S a t o [ s e e 41 and S a t o , Kawai, Kashivara [5] proved n e c e s s a r y and s u f f i c i e n t cond i t i o n s t o be imposed on f f o r e x i s t e n c e o f micro-local s o l u t i o n f o r t h e o p e r a t o r ( 0 . 2 ) and o p e r a t o r s m i c r o l o c a l l y e q u i v a l e n t t o

a

ax,

k

+

ixl

a ax,’

k>Oodd

r e s p e c t i v e l y . G r e i n e r , Kohn and S t e i n [ 2 ] proved n e c e s s a r y and s u f f i c i e n t conditions f o r local s o l v a b i l i t y of operator (0.2) near t h e origin ( f o r the statement o f the r e s u l t s s e e C o r o l l a r y 1 . 3 b e l o w ) . We prove n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r l o c a l s o l v a b i l i t y o f P f o r a given f provided t h a t t h e o p e r a t o r P i s m i c r o l o c a l l y e q u i v a l e n t t o t h e o p e r a t o r s ( 0 . 3 ) a t t h e c h a r a c t e r i s t i c p o i n t s ( x , t ) , where P i s m i c r o l o c a l l y u n s o l v a b l e ( k may depend on ( x , S ) ) . We a l s o prove a s u f f i c i e n t c o n d i t i o n f o r l o c a l s o l v a b i l i t y when P has Cauchy-Kowalevsky normal form. In o r d e r t o o b t a i n l o c a l i n f o r m a t i o n from t h e m i c r o l o c a l o n e , we use t h e t h e o r y o f T o e p l i t z o p e r a t o r s o f B o u t e t de Monvel and Guillemin [ l ] .

354

$1.

S. Kiro STATEMENT OF THE RESULTS

L e t p be t h e p r i n c i p a l symbol o f P, d e f i n e d on p(x,c) = 03 t h e c h a r a c t e r i s t i c v a r i e t y o f P.

N T % ?\{O}

and

E

= {(x,s)

:

Theorem 1.1. Assume t h a t P i s a 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 o p e r a t o r w i t h a n a l y t i c near t h e o r i g i n c o e f f i c i e n t s o f normal Cauchy-Kowalevsky form ( i .e. t h e r e e x i s t s 5 # 0 such t h a t ~ ( 0 , s ) # 0 ) . I f f o r a d i s t r i b u t i o n f WFAf n EIU = 0 f o r some neighborhood o f t h e o r i g i n U t h e n P i s l o c a l l y s o l v a b l e near t h e o r i g i n f o r f. Note t h a t f o r (x,E) 4 E P i s m i c r o - l o c a l l y s o l v a b l e a t (x,E) f o r every f (see [ 3 ] ) . We denote by E, t h e s e t o f (x,E) E E where P i s n o t m i c r o l o c a l l y s o l v a b l e f o r some f . L e t p = p1 + i p , where p j a r e r e a l symbols ( j = 1,2) and denote by H t h e Hamilton f i e l d s j

k

H . = H.H. ... H . . Note t h a t f o r p o i n t s ( x , < ) J J J J k ( j , 8 = 1 , 2 ; j # 8 ) when k i s odd o r H.p ( x , c ) 59. b e l o n g t o E, ( s e e [ 5 ] ) .

Let

E

k E such t h a t H.p (x,c)

# 0 when k

J E i s even

<0

do n o t

Assume t h a t f o r a neighborhood o f t h e o r i g i n U c RN f o r e v e r y t h e r e i s k = k(x,E) an odd i n t e g e r such t h a t

Theorem 1 . 2 . ]x,<) E E o l U

k Hj~,(x,S)

>

(j,f-

0

=

12, j #

f-I.

Then P i s l o c a l l y s o l v a b l e near t h e o r i g i n f o r a g i v e n f WFAf n 6,IV = 0 f o r a neighborhood V c U o f t h e o r i g i n .

i f and o n l y i f

Remark. The c o n d i t i o n s o f t h e Theorem a r e s a t i s f i e d f o r t h e o p e r a t o r s ( 0 . 2 ) on and t h e o p e r a t o r s ( 0 . 3 ) i n R2.

R3

---k

Corollar where C

1 3.

given

i fand o n l y i f

f

=

L e t P be $he Lewy o p e r a t o r ( 0 . 2 ) i n R 3 and C f = l i m C,*f E2+xi+x;+ix,)Then P i s l o c a l l y s o l v a b l e n e a r t h e o r i g i n f o r a

.

Cf

i s a n a l y t i c near t h e o r i g i n .

P r o o f . When IR3 i s i d e n t i f i e d w i t h {(z,z,) E E2 : I m z 1 = 1z21*) t h e n C is The c o n d i t i o n s o f Theorem 1 . 2 a r e s a t i s f i e d ( w i t h t h e z e g o p r o j e c t o r (see [2]). k = 1 ) . The a s s e r t i o n f o l l o w s from t h e f a c t t h a t C i s a m i c r o - l o c a l o p e r a t o r i n t h e r e a l a n a l y t i c c a t e g o r y ( s e e [ l ] and $ 3 b e l o w ) . M i c r o - l o c a l r e s u l t s t h a t we need i n o r d e r t o prove Theorem 1 . 2 a r e , as a m a t t e r o f f a c t , those o f [5]. However, we s k e t c h t h e p r o o f o f t h e r e l a t e d r e s u l t s , s i n c e t h e y g i v e l o c a l s o l u t i o n s w i t h b e s t p o s s i b l e d i f f e r e n t i a b i l i t y p r o p e r t i e s and t h e method i s a p p l i c a b l e t o more general s i t u a t i o n s . The t r a n s i t i o n from m i c r o - l o c a l r e s u l t s t o l o c a l ones i s i l l u s t r a t e d i n t h e s k e t c h o f t h e p r o o f o f Theorem 1 . 2 . The p r o o f o f Theorem 1 . 2 f o l l o w s f r o m r a t h e r obvious m o d i f i c a t i o n o f t h e p r o o f o f Theorem 1.1 and t h e m i c r o l o c a l r e s u l t s , 02% o t h e r w i s e , from p s e u d o - d i f f e r e n t i a l p a r t i o n o f u n i t y . I t can be adapted t o C - c a t e g o r y i n o r d e r t o produce s o l u t i o n s modulo f u n c t i o n s w i t h zero T a y l o r expansion a t t h e o r i g i n .

52.

A CHARACTERIZATION OF WFA

We s e t n = N-1, by ( c , ~ ) . L e t

x = (xl.. .x ) , t = x . The dual v a r i a b l e s t o ( x , t ) we denote r be t h e sgooth s t r r c t l y pseudo-convex boundary o f t h e tube:

355

SoIvability

R

=

~(z,w)

P, _ . .n ) . r

tN : z

E

Im w

6

>

lyl21

z . = x + i y . (j = 1 i s i d e n t i f i e d w i t h En x IR v i a t h e map ~j J (z,t) ( z , t + i \ y I 2 ) . The t a n g e n t i a l Cauchy-Riemann system on r i s generated by t h e o p e r a t o r s

where

-f

Y. = J

Let

OS(r)

a aZj

(j = 1

at

J

... n ) .

be t h e space o f s o l u t i o n s o f t h e system j = 1,

Y.f = 0 J

(2.1)

a + y. -

...

,n

whichmbelong t o t h e Sobolev space H S ( T ) ( s ER) and Om(r) ( O - m ( r ) ) t h e space of C ( d i s t r i b u t i o n r e s p e c t i v e l y ) s o l u t i o n s o f t h e system ( 2 . 1 ) . Performing a p a r t i a l F o u r i e r t r a n s f o r m ( w i t h r e s p e c t t o t ) one gets f o r f E OS(r)

(-+I

= exp

fA(Z,T)

(2.2)

f(z,.c)

I t f o l l o w s t h a t ? ( z , T ) = 0 when T form o f some d i s t r i b u t i o n h, i . e .

i(z ,T)

(2.3)

= (2a)-n’2

{eizg

<0

where and t h a t

a?

,j f(z,T)

...

j = 1,

= 0,

, n.

i s a Laplace t r a n s -

h(5,T)dS.

On t h e o t h e r hand, p e r f o r m i n g t h e F o u r i e r t r a n s f o r m o f (2.1) we o b t a i n a general s o l u t i o n o f t h e system (2.1) which e s t a b l i s h e s an i s o m e t r y w i t h r e s p e c t t o L2norms A

Hg(C,n,T)

(2.4)

TIT)-^'*

=

g

E

exp

L2(RN),

$(C,T)

when

= 0

T

<

0.

I f f = Hg, where H i s o b t a i n e d b y t h e i n v e r s e F o u r i e r t r a n s f o r m o f ( 2 . 4 ) , and h i s d e f i n e d by (2.3) one o b t a i n s e x p l i c i t l y

(2.5)

h ( c , r ) = 2n’2(n~)n’4

exp

[-%I

$(C,T)

(T

> 0)

Since H i s a F o u r i e r i n t e g r a l o p e r a t o r w i t h complex phase f u n c t i o n i t f o l l o w s t h a t H has c o n t i n u o u s r e s t r i c t i o n o r e x t e n s i o n t o Sobolev spaces. I t f o l l o w s from ( 2 . 5 ) : P r o p o s i t i o n 2.1.

f

F(z,w)

(2.6)

E

OS(r)

= (2a)-’

has an a n a l y t i c e x t e n s i o n i n t o

!

eiwT

R

d e f i n e d by

fA(z,-r)d.r.

Moreover ( 2 . 6 ) has an a n a l y t i c e x t e n s i o n t h r o u g h t h e o r i g i n ( i . e . a n a l y t i c near t h e o r i g i n i f and o n l y i f (2.7)

-n/4

( 1 + ( E \ 2 + ~ 2 ) s / e22 k * ( S . T ) ( h ( S , T ) l ’ d ~ d =

f o r a Young’s r e c i p r o c a l

k*(c,-r)

~ I Y ~ ~ k(y,-r) T , > + l ~ 1near~ t~h e

= sup(yE-k(y,T))

f

i s (real)

<

o f a function

k(y,r)

>

origin.

I t f o l l o w s from t h e general t h e o r y o f F o u r i e r I n t e g r a l o p e r a t o r s ( o r i t can be established d i r e c t l y ) t h a t

N

Pro o s i t i o n 2.2. L e t g E HS(R ) ~ ( z , T ) = 0 when T < 0. Then i f :nd n l y i f f = Hg i s a n a l y t i c near t h e p o i n t ( z , t + i I y I 2 ) ( y = - / ~ ) , and, i n p a r t i c u l a r when (2.7) i s v a l i d w i t h z=x-i

e

E

(x.t,S,T) 4 WFAg where k * as i n t h e

r

S. Kiro

356 P r o p o s i t i o n 2 . 1 and

THE TOEPLITZ

53.

d e f i n e d by ( 2 . 5 ) .

h

STRUCTURE ON

r

One o b t a i n s e x p l i c i t l y from ( 2 . 4 ) t h a t H*H = I d and t h a t S = HH* i s t h e Szegb' p r o j e c t o r ( t h e o r t h o g o n a l p r o j e c t o n from L 2 ( r ) + O o ( r ) ) . H and S b e i n g F o u r i e r I n t e g r a l Operators w i t h complex phase f u n c t i o n s a r e d e f i n e d on t h e Sobolev spaces. L e t Z = {(z,t,c,T) E T*r : 5 = s+irl, T > 0, 5 = - y ~ l be t h e charac(1 i s isomorphic as a s y m p l e c t i c m a n i f o l d t e r i s t i c v a r i e t y o f t h e system ( 2 . 1 ) . to

T+IR~\C.~
x

: (x,t,c,-r)

-f

(x

- i 'h ,

151' t + i f7-,

see a l s o Proposi-

t i o n 2 . 1 ) . Then ( S J ) d e f i n e s a T o e p l i t z s t r u c t u r e i n t h e sense o f [ l ] on r. L e t P be a c l a s s i c a l p s e u d o - d i f f e r e n t i a l o p e r a t o r on IRN o f o r d e r m. Then t h e o p e r a t o r T = HPH* : OS(I’) + Os-"'(r) ( s E IR) i s ' a T o e p l i t z o p e r a t o r i n t h e sense o f [ l ] , i . e . t h e r e e x i s t s a p s e u d o - d i f f e r e n t i a l o p e r a t o r Q on r such t h a t T = SQS and hence P = H*QH (see [ l ] , i t a l s o can be e s t a b l i s h e d e x p l i c i t l y i n t h i s case). I t was proved i n [ l ] t h a t T o e p l i t z o p e r a t o r s form an a l g e b r a which i s m i c r o l o c a l l y i s o m o r p h i c t o t h e a1 gebra o f p s e u d o - d i f f e r e n t i a l o p e r a t o r s on RN (on r t h i s correspondence i s e s t a b l i s h e d v i a t h e map

x).

54. MICRO-LOCAL MODEL FOR THE THEOREM 1 . 2 D i v i d i n g p by an a p p r o p r i a t e e l l i p t i c symbol and p e r f o r m i n g s y m p l e c t i c t r a n s f o r m a t i o n we reduce t h e s i t u a t i o n t o an o p e r a t o r o f o r d e r 0 t h a t n e a r t h e p o i n t s f r o f Z o i s m i c r o - l o c a l l y f i q u i v a l e n t t o an o p e r a t o r w i t h t h e symbol x -i(t1/T) near ( 0 , 0 , 0 , 1 ) E T% , k-odd. I f f = Hg and h i s d e f i n e d by ( 2 . 5 ) we o b t a i n t h e f o l l o w i n g e q u a t i o n f o r h

A n a l y s i s o f t h e general s o l u t i o n o f t h i s e q u a t i o n

5'

=

(c2, ...

, )$,

c(<',.r)

= 0

when

T E

[ 1 - ~ , 1 + ~ ] 1, >

E

>0

and P r o p o s i t i o n 2 . 2 show t h a t I f t h e c o n d i t i o n s o f T h e o r e m 1 . 2 a r e s a t i s f i e d t h e n (0.1) i s Proposition 4.1. v a l i d w i t h some u f o r a g i v e n f a t a p o i n t (x,t,c,T) 4 Eo if and o n l y i f (X,t,c,T) 4 WFAf.

From t h i s , by a m o d i f i c a t i o n o f t h e c o n s t r u c t i o n s below, o r by a p s e u d o - d i f f e r e n t i a l p a r t i t i o n o f u n i t y we o b t a i n Theorem 1 . 2 .

55.

TRANSITION TO LOCAL I N THE THEOREM 1.1

SPN.

We e q u i p w i t h a T o e p l i t z s t r u c t u r e , i d e n t i f y i n g i t , f o r example, w i t h To = {z E Ic ] y I 2 = 11 b y ( x , c ) + ( x - i s ) (161 = 1) which i s a c o n t a c t i s o morphism. L e t H and S be o p e r a t o r s w i t h p r o p e r t i e s s i m i l a r t o t h o s e desc r i b e d i n 0 2 and 53 ( s e e [l] o r t h e y can be o b t a i n e d from l o c a l c o n t a c t isomorphi s m To + r ) . L e t T be t h e T o e p l i t z o p e r a t o r d e f i n e d by P ( a s i n 53). L e t K = U x { y : I y I 2 = 13 c T o . Then f o r z E K t h e r e i s a neighborhood V z o f z and a d i s t r i b u t i o n z = x-is

and

u

p(x,c)

E

O-m(ro) such

that

WFA(Tu-Hf)lVZ =

C.

Indeed, i f

# 0 when t h e T o e p l i t z o p e r a t o r i s e l l i p t i c near

z

(see

357

Solvability

[l]) hence t h e a s s e r t i o n f o l l o w s from t h e c o r r e s p o n d i n g r e s u l t s f o r pseudod i f f e r e n t i a l o p e r a t o r s ( s e e [ 4 ] ) . I f p(x,E) = 0 t h e n by t h e c o n d i t i o n o f t h e Theorem and t h e P r o p o s i t i o n 2.2 H f i s a n a l y t i c n e a r z = x - i s . Now, i f p(x,E) f 0 we choose Vz such t h a t V, n { z = x - i S : p(x,S) = 03 = 0 and t h u s , c o v e r i n g t h e r e m a i n i n g p o i n t s o f K by open s e t s where H f i s a n a l y t i c , we o b t a i n a cover { V j l o f K. Consider t h e c o c y c l e uij = ui - u i n Vi n V . j

J:

WFA Tu. . I V i n V = 8 . Hence near t h e p o i n t s where T i s e l l i p 'J j t i c uij i s an a n a l y t i c f u n c t i o n ( s e e [ 4 ] ) and hence has a holomorphic e x t e n s i o n N i n t o a neighborhood o f Vi V . i n t . S i n c e n e a r t h e p o i n t s where T i s n o t J defines a e l l i p t i c uij i s a n a l y t i c by t h e c o n s t r u c t i o n , i t f o l l o w s t h a t u. It follows t h a t

1.j

c o c y c l e o f holomorphic f u n c t i o n s i n some neighborhood o f K i n E , which always can be assumed a domain o f holomorphy. S o l v i n g t h e Cousin problem we o b t a i n an analytic function v on r such t h a t v = ui + v . i s a w e l l - d e f i n e d d i s t r i b u j

t i o n on

K

and

from micro-local

J

WFA(Tv-Hf)IK =

0.

The a s s e r t i o n o f t h e Theorem f o l l o w s t h e n

properties o f

H*

and Cauchy-Kowalevsky Theorem a p p l i e d t o

P.

T o e p l i t z o p e r a t o r s b e i n g m i c r o l o c a l a r e n o t d e f i n e d on CR-functions on z . The l o c a l a c t i o n o f a o e p l i t z o p e r a t o r i s understood, t h e r e f o r e , as TSU" where ii i s some e x t e n s i o n o f t h e f u n c t i o n u t o r o . I n t h i s case t h e r e i s a neighborhood V o f z such h a t WFA(u-SC)Iv = 0 and WFA(Tu-TSG)IV = 0. Remark.

T o i n a neighborhood o f t h e p o i n t

REFERENCES B o u t e t d e M o n v e l , L., G u i l l e m i n , L., The s p e c t r a l t h e o r y o f T o e p l i t z o p e r a t o r s ( P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n , N.J., 1981). G r e i n e r , P.G., Kohn, J.J., and S t e i n , E . , Necessary and s u f f i c i e n t c o n d i t i o n s f o r s o l v a b i l i t y o f t h e Lewi e q u a t i o n , Proc. N a t . Acad. o f S c i . 72 ( 1 9 7 5 ) , 3287-3289. Hormander, L., Uniqueness theorems and wave f r o n t s e t s f o r s o l u t i o n s o f l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h a n a l y t i c c o e f f i c i e n t s , Comm. Pure Appl. Math 24 (1971), 671-704. Sato, M., Kawai, T., and Kashivara, M., M i c r o f u n c t i o n s and pseudod i f f e r e n t i a l e q u a t i o n s , L e c t . Notes i n Math. 287, 265-529. Sato, M., Kawai, T., and Kashivara, M., On t h e s t r u c t u r e o f a s i n g l e l i n e a r p s e u d o - d i f f e r e n t i a l e q u a t i o n , Proc. Japan Acad. 48 ( 1 9 7 2 ) .

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DIFFERENTIAL EQUATIONS

I.W.Knowles and R.T. Lewis (Editors) @Elsevier Science Publishers B.V. (North-Holland), 1984

359

REGULARITY PROPERTIES OF SCHRODINGER OPERATORS ON DOMAINS OF R" Mark A. Kon Department of Mathematics Boston University Boston, Massachusetts, 02215

We extend the study of radial bounds for kernels of resolb(z). V q(z) vents of Schrodinger operators A = -A on R" to operators on domains of R", and obtain consequent regularity information. The technique uses elliptic boundary value theory to study a perturbation series consisting of iterated singular integral operators on the boundary. The procedure in this second order case is an illustration of the approach in higher order ones.

+

+

Although much is known regarding the spectral and regularity theory of Schrodinger operators on R", little of this extends automatically to Schrodinger operators restricted to bounded and unbounded domains of R". The purpose of this article is to examine certain regularity questions regarding such operators; clearly, such an investigation invokes at least some theory of elliptic boundary value problems. Our strategy follows that of 121: the resolvent kernel of the operator is studied explicitly, and further results (e.g., regularity of eigenfunction expansions, essential self-adjointness) follow as consequences. This work represents the first part of a program of study of resolvents and regularity properties of elliptic operators on domains. Some results for the second order case will be formulated more awkwardly than necessary, in order to illustrate their extension t o higher order cases. The formulation here thus provides an example of the approach in these more general situations. Our functional calculus approach to eigenfunction expansions largely bypasses problems inherent in detailed spectral theory for domains, thus making more accessible results in the unbounded case, for which spectral considerations can be complicated. We will consider Schrodinger operators with vector potentials on a possibly unbounded domain R with smooth boundary an = w ; the coefficients b and q are allowed to fall into certain scales of LP spaces on R. The domain P(A) will consist of an appropriate closure of a space of Cm functions on R which satisfy a homogeneous boundary condition

f(.)

cOse

+ a,f(z) sin e = o

(zE u)

where a, represents an outward normal derivative, and 0 may depend on z. In order t o study the resolvent of A: we will first look at that of the leading term, by representing it as a coiivergciit pcrturbation series. A crucial parameter in the cslculatiori will be the Iriaxirnum of the curvature of w R". I lie iterat.cd integrali appearing i u t h e pcrturbation scrim will be non-singular orily due to

c

r >

M. A. Kon

360

the rotation invariance of the Laplacian restricted to w . This symmetry is broken to the extent that w is non-flat, and uniform bounds on the curvature are required for the series to converge. A second perturbation series will be used to represent and bound the full resolvent. The deiailed analysis of resolvent kernels has been used bolh on R" (see, e.g., [l], [2]) and manifolds to analyze spectra of elliptic operators. We remark that the present results have a wide range of applications - including study of semigroup generation and corrcsponding description of boundary behavior of solutions of the heat equation on cylinders, uniform bounding of LP spectrum of elliptic operators, and study of self-adjointness of singular elliptic operators on domains. See [5] for "classical" results on essential self-adjointness of Schrodinger operators; Simon [7] has a thorough survey of Schrodinger semigroup theory on R". Our regularity results on semigroups generalize some of those in [8] on so-called symmetric heat diffusion semigroups. The result in Theorem 1 is new for Schrodinger operators with smooth coefficients, but is profitably expressed in the context of singular (LP) coefficients, given thc forms of Schrodinger operators which often arise in physics. For smooth coefficients, the point of departure of this result from known theory (see [4) for such results on R") is the uniformity in < of the bounds of Theorem 1, which cannot be obtained using standard techniques. b(z) . V V(z) be a We now develop some notation and definitions. Let A = -A Lm Schrodinger operator with a "vector potential" term b(s) = ( b l ( z ) ,. . .,b,(z)) E L"+' and potential V(z) E L%+' L", for some 6 > 0. For notation purposes we also identify bo = V. By L' Lm we mean {f = f 1 + fz : f~ E L', fz E Loo}. Let SZ R" be a domain with C" boundary w . We define H"(R") to be the 15’Sobolev space of order s. Let H"(R) consist of those u E Lz(n)which have extensions 5 E

+

c

+

+

+

HS(R"), ii In= u. If u E H"(R),define

llull, = max

{ IIiiIIH.(R,,)

: ii In= u

+

1

. It is well-

known that if u € HS(R),then the k-th normal derivative b'ku on w restricts to a function E H s - k - - l a(w). We define H " i 2 = H"-?(w) @ H"-s(w), so that ( u ,a,u) €,I H2*' when

fk

u

E H2(n). $1. Bounds on the Laplacian Resolvent

Consider the Laplacian in R,with boundary conditions (1). We will assume that tanO(s1) is bounded in C1norm on w ; this does not exclude Neumann boundary conditions from our results, since they can be handled similarly. We emphasize that the following arguments generalize to higher order constant coefficient homogeneous elliptic operators, and the following should be viewed as an illustration of a more general theory. DEFINiTlON 1: The symbol class S" is defined by

Sm(R") = {f E P ( R " ) : lb'"f(z)l

5 C,(1

+ 1~l)"'-'~' 1 1

where represents a partial derivative of order a,with cy a multiindcx. The constants C, are the symbol class seminorms of f . The rcsolvcnt R: = (< A)-' on all of Rn has a kernel satisfying

+-

( ~ : ( z- d j l

5 ~ h , , , ( . z- d),

(2)

Regularit), Properties of Schr6dinger Operators

36 1

where

with s = n - 2, and t any number greater than n (with suitable adjustment of C). The boiind in (2) m a r z = 5’ follows from the following well-known result. PROPOSITION 1: I f f E S-"(R"), for m > 0 then the Fourier transform o f f satisfies

DEFINITION 2: The boundary w is sparse if (2) its principal cu:vatures are uniformly bounded by a constant K ; (i) its two dimensional measure has bounded concentration in space, i.e., there is a constant M > 0 such that if 7 is n - 1 dimensional Lebesgue measure restricted to w and &(z) is the unit sphere at z, then -y(S u B l ( z ) )5 M (z E R"). We will assume that the boundary w is sparse in order to avoid pathologies. Let R,(z,y) denote the kernel of (c A)--' in R , with boundary conditions (1). We now proceed to estimate R,(zl, y) by assuming that it restricts in z1 on w to a function in H2r2 satisfying (l), leaving the proof of this fact to the end. Formally using Green's identities (in higher order cases certain generalizations of these apply; see [3]), we have the integral equation

+

where a; will henceforth represent outward normal differentiation with respect to the variable labeled i. Initially, we impose (1) on (4) to obtain

W,u(z) =

J,

u ( z l ) { a ~ R ~ (y)z ~ 4-, tan@(zl)R;(zl,y ) } k .

Define the operation S on E12(R) to be rest,riction to N $ ( w ) @ Hf(w);specifically, if E H2(R),then Su = (u,anu)JwEH 2 s 2 . The map S is continuous and onto (see [3]). We note that if w , is that part of R whose distance from w is lcss than E, then 121, is well-defined on functions in I i z ( w c ) . Uy our assuriiption R,(z, 51) is in the domain cf S in this sense, and u

( 5 ) can be written

M. A. Kon

362

where

* denotes complcx conjugate,

c=(

0 -I

I

)

0

is the operator matrix whose blocks correspond to the direct sum decomposition H ~ ( w@) H i ( w ) , and the inner product is component-wise sesquilinear multiplication followed by summation and integration in 21 over w with respect to its inheritcd measure. The operator S is always assumed to operate in the variable 5 1 . If T denotes the projection on EI2v2(w) defined by WO(Z),

fib)) = (fO(4, tano(4fofz)),

then (6) holds with CT replacing C. Note that T'SRf(z,zl)E H2s2,while CTSR,(z, 21) E H f @ H j C H-3 @ H - f ; we thus regard the second component of the inner product as being in the dual space of the first. We now define &(z, 51) to coincide with R,(z, 51) on we, but to be smooth in z1 on the interior of R; k: is deEned similarly. Note that the adjoint

1

(

S' is bounded from li-; @ H - f ( w ) to H-*(R"). Taking adjoints,

(CTSR,(z,zi),SR,O’(zi,y))= (S'CTSR,(z,zi), R,O*(zi,Y)) R" where the objects on the right are in H - 2 and H 2 , respectively. Since the support of S*CTSR,(z,51) is outside n, k:(q,y) may be replaced by (< - A)-'6,, where the delta distribution 6, is in H-f-' for every c > 0. Again taking adjoints, we have

(CTSR,(z, zi),SR ; *( z i,Y),

= (S - A)-'S'CTSR&,

Y),

(7)

where all operators act on the y variable. The operator (<-A)-' here is the resolvent acting on all of R" . We note that (7) is C" in y in the interior of R by standard elliptic theory. We thus have

w,= R;S*CTS It can be shown that W, is in fact bounded in H2(R);see for example [6, Lemma 41. Equation (5) is now

R, = R;

+ W,R,.

(8)

The most direct solution of (8) is it,erative. We first show that

R;

= R: + wSn;+ w : ~+;...

is well-defined, by proving strong convergence via analysis of kernels. Let tan O(sl)R:(zl,s) be defined for z1 E w . The kernel of WFR: is

(9) "(2, zl)

+

= a!,R:(z1, z)

363

Reguluritv Properties of Schrodiriger Operators

DEFINITION 3: If h l and h2 are functions on R" with measurable restrictions to the c R",

surface w

Recall the functions hs,t in (3). LEMMA1: If w is a sparse C" surface of codimension 1 in R", then there is a constant K1 depending on w such that

where s = min(s1, s z ) and t = min(tl, t z ) , whenever t l , t z > n - 1 and ~ 1 , <s n~- 1. Thc constant K1 depends only on the parameters M and K of Definition 2. The lemma is proved using uninteresting technical estimates, based on approximating w by an affine subspace of R". In the latter case, we note only that the Fourier transform preserves the set of radial homogeneous functions, and that is bounded by such functions. If < = p i e l

where 7 denotes Fourier transform. Thus

where

+

and s = n - 2, t = n 2; the singularity in this bound is given by (2) and some calculations of &dependence, and its order is sharp. If z E w , the function V'R,(xl - z) is singular a t 51 = z and is not in L'(w); however, the normal derivative aARI(xl - z) = n(z1) V'R,(x1 - 2) vanishes altogether when w is flat, since R, is spherically symmetric. Since the curvature of w is bounded, this is approximately 60 on a small scale when z1 is near x. An explicit calculation shows that the bounds (12) still hold for aAR(x,,x), as a consequence of our geometric assumptions: LEMMA2: There is a constant C such that, for x 1 , E~ w, if = peie, +

+

where s = n - 2, and t > n 1. C depends only on the sparseness parameters K and M in Def. 2. I m n m a 2 bounds ;dl but the extreme terms of (lo), namely those whose arguments lie on w . Thcse bounds fail to hold uriiforiiily f o r T,fz,x l ) whcn z is near hill riot in w . The

M.A. Kon

364

problem is the term a!,R,(z -- zl); bounding it on w quickly reduces to the same problcm when w is flat; in this case the only bound on a!,R,(z - z1) which is uniform in z and scales correctly with p (as in (13)) is singular in z1 (i.e., not'in L'(w)). Correspondingly, its Fourier transform in 51 on w is uniformly bounded only in the So symbol class. Fourier transforming the integrand of (10) in 5 1 (correct bounds on the integration in the remaining variables exist at this point by Lemmas 1 and 2) and analyzing the symbol classes (in the dual variable to zl)and seminorms of the transform of the first term in (10) and of the remainder, and transforming back with the help of Proposition 1, we have the following bound for k 2. 1 and p 2 1:

where henceforth C denotes a generic constant. We can sum (9) if C p i obtaining

5

lsin

!In+',

where C(s) is bounded outside of a domain of the form (14). Equation (15) shows that (9) converges in the uniform operator topology to R;. At this point we prove that R: is the resolvent R, by showing that the former satisfies the appropriate boundary conditions. To do this one must first verify that, fixing z,the series (9) converges in H'(R); this can be done by termwise differentiation in y, with use of similar estimates to the above. Let P = R:S*C. By (4) and ( 5 )

so that if P+

R: = R:

+ P SR;

R: = R:

+ PTSRY ,

= SP, P+(TSR; - SR;) = 0.

(17)

It can be shown using the identity of lV{ and the integration in (4) (see [6] for details when R is compact) that P+ and its complement P- = I - P+ decompose R3v2(u)= H";(w) @ H " - f ( w ) into bouiidary values of interior and exterior solutions of

365

Regularity Properties of' Sclirodinger Operutors

Namely,

p+H"'2= {f E Ha" : 3~ E H"(R) s.t. (I

+ A).

= 0, S u = f},

and the range of P- is the corresponding set of boundary values of exterior solutions. Equation (17) therefore implies that (TSRY - SR:) is the boundary value of an exterior solution of (18). Since this solution satisfies a self-adjoint Dirichlet boundary condition on w , it is an eigenfunction of a self-adjoint operator corresponding to a complex eigenvalue, and hence vanishes. Therefore RT satisfies (1) in z1. Since it satisfies the resolvent equation (<- A)R,(z,y) = S(z - y), Ry is a right inverse of (( A) with boundary conditions (l), on the dense subset C r ( R ) C H o ( n ) , and thus Ry = R,in L2((n).

+

$2. The Full Operator

+

- +

We use a sccond perturbation series to consider the full operator A = -A b(s) V = bo(z). Let R = b V q(z) be the perturbation. We utilize the resolvent perturbation series (see [2])

- +

q(z), with q(z)

(c -A)-'

= ((

+ A - i9-I

= ((

+ A)-'

m

(-R((

+

(19)

k=O

Let b;(z) E LT1(n),with ro

> 3,

r,

> n, i 2 1, and

d r m a x sup--1, (a2l:

): .

-

Using $1together with bounds and estimates typical of those in [2], we have THEOREM 1: (i) The operator A with boundary conditions (1) can be closed in all LP spaces (1 2 p 5 mint-,). (22) The spectrum of A is contained in the parabolic domain

unifordy in all LP spaces in (i). (iii) If ( 4 R, then the kernel of (I -A)-'

is bounded by

where s = n - 1, t = n 4-1. From above we note that A is bounded below. Let be thc funct,ions in Ifz(n) satisfying (1). Somc irivest,igatiori of thc domain of the leading tcrm and the relative boundedness of t h e perturbation gives the following: C o r t o L L A R Y 1: If A is fornia,ff.v self-ml,joir:t, thcn it is rssentia/ly self-adjoint on fft.c.(n), or on any essential domain of the L a p h i a n -A,,.-..

Hi,,.(n)

M.A. Kon

366

The following extends results on Schrodinger semigroups (see [7], 121) to domains: COROLLARY2: (2) A generates an analytic semigroup e-'A for Re t > 0, in all Lp (1 p 5 minr,); (zi) the semigroup is LP-continuous a t t = 0, for p < 00; (iii) i f f E LP(n) (1 <_ p 5 minr,),

for z in the Lebesgue set off, and in particular a.e. The corollary is proved by representing the semigroup as a contour integral of the resolvent about D. An immediate application is to the perturbed heat equation on R x R + , with homogeneous boundary conditions (I), and n bounded or unbounded. The following extends results of Benzinger in one dimension. COROLLARY3: Let u(z,t ) on R X R+ satisfy

with botindnry conditions u(0, z) = f(z)E L p ,

cos O(z)a,(z)

+ sin O(z)u(z,t ) = 0

(zE w = an),

with tan O(z1) bounded in C1(w). Then

in LP (1 5 p 5 minr,) and for all z on the Lebesgue set o f f . In particular the convergence in (20) occurs almost everywhere in 2. References Agmon, S., Lectures on elliptic boundary value problems (Van Nostrand, 1964). Gurarie, D. and Kon, M., Radial bounds for perturbations of elliptic operators, to appear, Jnl. Funct. Anal. Hormander, L., Pseudo-differential operators and non-elliptic boundary value problems, Ann. Math. 83 (1966) 129-209. Neri, U., The integral kernels of certain pseudo-differential operators, Math. Ann. 186 (1970) 155-162.

Heed, M. and Simon, B., Methods of modern mathematical physics, I! (Academic Press, 1975).

Sreiey, R.T., Singular integrals and boundary vaiuc prublcms, Am. Jnl. Math. 88 (1966), 78 1-808.

Simon, R., Schrodinger scinigroups, Bull. .4m.Math. SCC. 7 (1952) 447-256. Stcin, E.M., Topics in harmonic analysis relat cd t.0 the Littlewooti-Paiey t,heory (Princeton, 1971).

DIFFERENTIALEQUATIONS I.W. Knowles and R.T. Lewis(Editors) 0 Elsevier SciencePublishersB.V. (North-Holland),1984

367

RELATIVE SYMMETRIES OF DIFFERENTIAL EQUATIONS

B. A. Kupershmidt’ Department of Mathematics University of Michigan Arbor, MI 48109, U.S.A.

+ +

AM

m

m

W

Let A : .J u + J n be a differential operator, where J u (resp. J U) is the infinite-jet bundle of the bundle u : F + M (resp. n : E + M). Let :1 be the Cartan submogulg of the module A*(K ) of 1-forms over the ring K = C (J u). Among all derivatigns of K into K along A*, WC! classify those which map ’I into I? They ?urn out to be quasievolution equatiogs.

'

INTRODUCTION Let n : E

M, u

F

M be bundles (smooth, like everything else in the paper). W k : J n + Jan be the corresponding jet bundles, denote J n = o h m w k : Jsw + Jon = E be a bundle lim proj J n, Ks= C (J n) = lim ind C (J n). Let map (over M), which can be thought of a s a differential operator : T(u) + T(n), where T(v) denotes the sheaf of sections of the bundle U : x ( y ) = i-(js(u)(y)), vycr(u), where js = j s ( u ) : T(u) + r(us) denotes the natural lift. Tangent k planes to graphs (j (y)(M)ly&T(u)) form the Cartan distribution in J 0 . Its 1 k annihilator in A (J V ) is the k-th Cartan submodule I ( U ) . The Cartan submodule k 1 I1 (u) in A 1(u) = A 1(J- 0 ) = lim ind A 1(Jkn) is defined by the formula I (U) = lim W a into J U, A : J U + ind Ik(u). Let us denote by A the natural lift of Let nk

m

J n.

:

+

k J n k

+

:

+

M, nk

Then d(IA)C

1

Iu (lemma I1 2 . 1 4 1 3 1 ) . find the set aqeV(A) of all derivations

We consider the following problem: 2 :

-k

Kn. + Ku along the homomorphism A

, which

map I1 n into .’1

1

There are at least

three motivations for this problem: A.

In the case n =

U,

A = id, the set of all such 2 ' s is the set of evolution

derivations aev(n) ; in local coordinates, the equations of trajectories of these evolution derivations are evolution equations (Proposition 1 [ Z ] ; Theorem I 5.6 [ 3 ] ) .

(In the engineering literature, these derivations pass under the

misleading name "Lie-Backlund transformations".) +Supported in part by NSF Grant MCS-8003104 and by the U.S. Department of g!jergy. Currently at the Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, U . S . A .

368

B.A. Kupershmidt

B.

Such Z’s arise in practice as the "generalized sine-Gordon equations" associated with classical simple complex Lie algebras ( [ 4 1 , [ 6 ] ) and even with KacMoody Lie alggbras (111). C. Let U C J n be a closed set considered a s a differential equation: y&T(n) is

U.

a solution if (jk(y))(M)C

Let

fl

C

m

J n be the infinite prolongation of U.

!are those evolution derivations X&DeV(n) which preserve the ideal F(fl) of functions from Kn vanishing on fl. Suppose, however, that m c J u is another equation and A ( v ) C 6 . Then more general symmetries of fi will be those Z’s which map 9 (6) into $(v). That such relative symmetries are

Then the symmetries of

v

useful was demonstrated in a spectacular tour-de-force by Vinogradov and Krasil’shchik who used nonlocal symmetries to compute all (absolute) symmetries of the Korteweg-de Vries equation ([5]). &CLASSIFICATION Denote bya(nm)

n,

m : J E +

m

the K -module of derivations of C (M) into K

along xi, where

M is the natural projection. Note that B(nm) is generated over kn of vector fields on M. If X & B ( n m ) then its lift 2 =

by the Lie algebra B(M)

gn & S ( K , )

into the Lie algebra of derivations of K

= j,(y)*

universal property jm(y)*? X(Cm(M))

c Cm(J2n).

Xjm(y)

The set of all such

n is uniquely defined by the , v y & r ( n ) , where 2 is such that

>t

2's

- and is a Lie

is denoted by&nm)

-

1

The annihilator of 23(nm) in A (K,) algebra and a Kn-module (Theorem I 3 . 6 [3]). 1 is nothing but the Cartan submodule I . [This is the definition of the Cartan

n

submodule; the fact that the corresponding distribution is spanned by the tangent planes of graphs of jets of sections of n is a corollary (Theorem I 4.4 [31).1 If X (

B(M)

then the lifts -L

such that

gu

and

Xn

Obviously, if Xta(nm),

2.13 [3].)

2

are A-related: i U A k = A*gn (Lemma I1

- -

then again there exists a unique

4-

-

2 ea(um)

A" = A Xn; the resulting map a(nm) + .Xj(u,.) is a Lie algebra homo-

morphism. Lemma 2 . 1 .

Let @

:

K1

+

K2 be a homomorphism of commutative rings K1 and

let XI& B(K1) and X2 B(K 2 ) be two @-related derivations. Let%(@) a K -module of derivations of K1 into K2 along @. Then for any Zra(@), 2 (X2Z - ZX,) eE?4(@).

K2,

be

Proof. Obvious. 1 Recall that if w&A (K), X,Z ES(K), then the Lie derivative of w with respect = Z(w(X))-w([Z,X]). to Z is defined by the formula [Z(w)l(X) Lemma 2.2. In the notations of lemma 2.1,3(@) acts by derivations 1 1 1 along @ on A (K1) with values in A (K2). In particular, for w&A (K1) [Z(W)l(X2)

= Z(W(Xl))

-

w(ZX1-X2Z)

,

(2.3) 1

where on the right hand side the pairing between A (K1) andB(@) ,\df,g&K1. naturally : (fdg)(Z) = @(f)Z(g)

is understood

369

Relative Symmetries of Differentid Equations Again, t h e p r o o f i s o b v i o u s .

-

Now w e can h a n d l e t h e problem o f c l a s s i f i c a t i o n o f e l e m e n t s o f Dqev(A). L e t 1 1 1 Z t B q e V ( A ) , t h a t i s , Z ( I ) C Iv. Take any UJ & In = Ann@(nm)). Then Z(u) & _Tr_ 1 t I v = Ann(a(u,)) = Ann@(M),,) i f f , y X & a ( M ) , [Z(UJ)](?,,) = 0. By formula ( 2 . 3 ) , c _

t h i s is equivalent t o 0 = Z(UJ(~ )) 77

-

~ ~ ( 2-i2 2 ) . X f L

But w ( i i % ) = 0 s i n c e w & I

1

n’

1

n -2u 2 ) must b e l o n g t o t h e k e r n e l of In, t h a t i s , w e must have

Thus (2%

7:

(Zin-ivZ)

&

-

KvA$,(M)n, VXEBM).

(2.4)

Every Z&aqeV(A) i s u n i q u e l y d e f i n e d by i t s v a l u e Z e n k Conm,o' &a(n A) i s u n i q u e l y l i f t e d h a ( @ ) t o become

Theorem 2 . 5 .

v e r s e l y , any d e r i v a t i o n

"&O

-L

Z&Bqe"(A),

=

s u c h t h a t 2.n"

2.

- 9 0

__ P r o o f . To s t u d y ( 2 . 4 1 , f i r s t n o t i c e t h a t , l i k e i n t h e a b s o l u t e c a s e (n = V , A = i d ) , one h a s a d i r e c t sum decomposition

-.A’

a ( A ) = B(v,)

@ B(A)vert

,

(2.5)

12-11; = 01, and decomposition ( 2 . 6 ) i s p r o v i d e d by -.7 (Z-IT..yOA'* + [ Z - (Z-n,)U ,*A"]. S i n c e Z - n L ~ I D=)

: *a(p)

where%(A)vert

-L

-%

t h e formula Z =

,

I Cm(M)

satisfied. only

-

t h e n Z1 : = ( Z . r r ~ ) , & ~ v , ) and ( 2 . 4 ) f o r Z = ZIA*

is obviously

vert T h e r e f o r e w e s h a l l r e s t r i c t o u r s e l v e s t o v e r t i c a l Z ' s &%(A)

m { q a l a = 1 , . . . dim E - dim M , a d + ] ID b' b e s t a n d a r d l o c a l c o o r d i n a t e s on J n , and ( p J b = 1, ..., dim F-dim M, U & z y ] b e

Let (x~,. . . ,x ) b e l o c a l c o o r d i n a t e s i n M , m

U

m l o c a l c o o r d i n a t e s on J V .

" -,a as;

Let, locally, Z = UaA"

t o check ( 2 . 4 ) f o r t h e b a s i s v e c t o r f i e l d s X =

a

a Au&K

a &B(M). ax.

.

-

Since

1

a

a +

q:+i

I t i s enough

( u s i n g summation o v e r r e p e a t e d i n d i c e s ) , w e have

(-1a ax .

IT[

=

B. A. Kupershmidt

370

A

This last expression must belong to K A’b(M)n. V

along M, it must vanish, and this happens iff A i + :

Since there are no components

= (Di),

stands for (a/axi),,.

As’s

= (DU),(Aa),

Thus, :A

where (Di)v

( A : ) , 0

(DUIv: = (Di )v 1

1

...

m’ (Di )v m

,

and

are arbitrary.

ITRAJECTORIES Ordinary differential equations are equations of trajectories of vector fields on manifolds. Analogously, evolution equations are equations of trajectories of vertical evolution derivations (Theorem 1 5 . 6 [3]). (The reason for considering only vertical fields is explained in 51 5.3 [3]- for nonvertical. fields, equations become overdetermined.) Now let 2 cBqev(A), and consider 2 to be vertical. A tra’ector of$ is a one-parameler (t) family of sections y = y(t):M + F such tha~[j(v))iy)]~Z = &-[j(n)(Ay)] Let us find a coordinate

.

version of the last equation.

[j(v)(y)leZ

4

-

~-[j(n)(A~)l*

- 0

where DO: = (a/axi)

Let locally 2 = (D’)v(Aa)*A*

a/aq:.

Then 0 =

=

- a .--(a/axi ) m.

Since [a/at,DU] = 0, the above equality

m is reduced to

Thus we obtain the coordinate form of quasievolution equations. Remark 3.2. In contrast to the evolution equations, quasievolution ones need not be formally integrable. Obviously, integrability of a generic 2 depends only upon A. I conjecture that this integrability depends only upon dimensions and codimensions of the finite number of prolongations of the map A : J'V

-*

E.

ACKNOWLEDGMENTS The main theorem 2.5 of this paper was announced in the 1979 preprint IC/79/94 (ICTP, Trieste, Italy). I wish to thank F . Pirani for helpful discussions 1. there. Thanks are due to A . Greenspoon for reading this paper, and to Knowles and R . Lewis for hospitality in Birmingham.

Relative Symmetries o f Differential Equations

37 1

BIBLIOGRAPHY [I] V. G. Drinfel’d and V. V. Sokolov. "Equations of KdV type and simple Lie Algebras," Dokl. Akad. Nauk SSSR 258 (1981), 11-16 (Soviet Math. Dokl. 2 (1981), 457-462). [2] B. A. Kupershmidt. "On geometry of jet manifolds," Uspekhi Mathematicheskikh Nauk XXX:5(1975), 211-212 (in Russian). [3] B. A. Kupershmidt. "Geometry of jet bundles and the structure of Lagrangian and Hamiltonian formalisms," Lect. Notes Math. 775 (19801, 162-218, Springer.

[4] B. A. Kupershmidt and G. Wilson.

"Conservation laws and symmetries of generalized sine-Gordon equations," Comm. Math. Phys. 81 (1981), 189-202.

[5]

A. M. Vinogradov and I. S. Krasil’shchik. "A method of computing higher symmetries of nonlinear evolution equations, and nonlocal symmetries," Dokl. Akad. Nauk SSSR 253 (1980), 1289-1293 (Soviet Math. Dokl. 11 (19801, 235-239).

[6] G. Wilson. "The modified Lax and two-dimensional Toda lattice equations associated with simple Lie algebras," Ergod. Th. & Dynam. Sys. 1(1981), 361-380.

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DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

373

NECESSARY AND SUFFICIENT CONDITIONS FOR OSCILLATIONS OF HIGHER ORDER DELAY DIFFERENTIAL EQUATIONS G. Ladas',

Y .G. Sficas2, and I.P. Stavroulakis' ,2

Department o f Mat hemat ics U n i v e r s i t y o f Rhode I s l a n d Kingston, R I 02881

*Department o f Mathematics U n i v e r s i t y o f Ioannina Ioannina, GREECE

We o b t a i n necessary and s u f f i c i e n t conditions under which a l l s o l u t i o n s o f c e r t a i n n t h order delay d i f f e r e n t i a l equations o s c i l l a t e when n i s odd, and necessary and s u f f i c i e n t cond i t i o n s under which a l l bounded s o l u t i o n s o s c i l l a t e when n i s even. The conditions derived i n v o l v e t h e c h a r a c t e r i s t i c equation o f the delay d i f f e r e n t i a l equation. INTRODUCTION Our aim i n t h i s paper i s t o o b t a i n necessary and s u f f i c i e n t conditions under which a l l s o l u t i o n s o f the n t h order delay d i f f e r e n t i a l equation (1 1 o s c i l l a t e when n i s odd, and necessary and s u f f i c i e n t conditions under which a l l bounded s o l u t i o n s o s c i l l a t e when n i s even. The c o e f f i c i e n t s and t h e delays of the d i f f e r e n t i a l equation are assumed t o be constants such t h a t 0, pi > 0 f o r i = 1,2 ,...,k w i t h k 2 1 and n 2 1. o = T' 0 < T1 < < T ~ po ; * - a

As i s customary, a s o l u t i o n i s said t o o s c i l l a t e i f i t has a r b i t r a r i l y l a r g e zeros. Solutions of (1) are continuous functions x defined on [-T~,-) t h a t s a t i s f y ( 1 ) . As usual, we s h a l l use the term "eventually" t o mean " f o r s u f f i c i e n t l y l a r g e t " . The c h a r a c t e r i s t i c equation o f (1) i s (2) The main r e s u l t i s the f o l l o w i n g . THEOREM 1. ( i ) no r e a l roots.

For n odd, a l l

s o l u t i o n s o f (1) o s c i l l a t e i f and o n l y i f (2)

For

n eyen, a l l bounded s o l u t i o n s o f (1) o s c i l l a t e i f and o n l y i f (2) (ii) has no r e a l r o o t s i n (-m,O]. I t should be noted t h a t f o r n even

F(O)F(+m) < 0. and so equation (2) has always p o s i t i v e r o o t s . That i s , equation ( 1 ) has always unbounded n o n o s c i l l a t o r y s o l u t i o n s . However, the bounded s o l u t i o n s o f (1) may o r

G. Ladas e l al.

314

may n o t o s c i l l a t e .

-

x"(t)

For example, x(t-n) = 0

has the bounded o s c i l l a t o r y s o l u t i o n x ( t ) = s i n t, w h i l e the equation x"(t) x(t -)1 0

-

-

4s

has the bounded n o n o s c i l l a t o r y s o l u t i o n x ( t ) = emt&. PROOF OF THE M A I N RESULT The f o l l o w i n g l e m a w i l l enable us t o r e s t a t e Theorem 1 i n such a way t h a t the odd and even case o f n are proved simultaneously. LEMMA 1. For n odd, ( 1 ) has no unbounded n o n o s c i l l a t o r y s o l u t i o n s and ( 2 ) has no real r o o t s i n 1 0 3 .

Next assume, f o r the sake o f PROOF. Clearly, ( 2 ) has no r e a l r o o t s i n [ O , m ) . c o n t r a d i c t i o n , t h a t ( 1 ) has an unbounded n o n o s c i l l a t o r y s o l u t i o n x ( t ) . Without loss of g e n e r a l i t y we may (and do) assume t h a t x ( t ) i s eventually p o s i t i v e . Then, f o r tl s u f f i c i e n t l y large, x(")(t)

< 0,

x("-')(t)

> 0, and x ' ( t ) > 0 f o r t

2 tl.

I n t e g r a t i n g (1) over [tl,t] we f i n d

Deleting some p o s i t i v e terms and using t h e f a c t t h a t x ( t ) i s increasing, we o b t a i n

which, as t

-+ m ,

leads t o a contradiction.

I n view o f Lemma 1, f o r n odd ever unbounded s o l u t i o n o f (1) i s o s c i l l a t o r y . Hence i t s u f f i c e s t o prove p a r t ( i f o f Theorem 1 only f o r bounded s o l u t i o n s . Therefore, the p r o o f o f Theorem 1 has been reduced t o proving the f o l l o w i n g r e s u l t f o r a r b i t r a r y n. The f o l l o w i n s statements are equivalent.

THEOREM 2 . (a) (b) PROOF.

Every bounded s o l u t i o n o f ( 1 ) o s c i l l a t e s . Equation ( 2 ) has no r e a l roots i n ( - m , O ] . (a)

-

(b).

Otherwise ( 2 ) has a r e a l r o o t Xo 5 0 and so (1) has the bounded

n o n o s c i l l a t o r y s o l u t i o n x ( t ) = exot.

Contradiction. ( b ) -B (a). (We prove i t by c o n t r a d i c t i o n . ) Otherwise (1) has an eventually posit i v e bounded s o l u t i o n x ( t ) . The r e s t o f the p r o o f w i l l be d i v i d e d i n t o a s e r i e s o f s i x lemmas. Throughout the remainder o f t h i s paper x ( t ) stands f o r the p o s i t i v e s o l u t i o n whose existence we have j u s t assumed. LEMMA 2. number

Under the hypothesis t h a t equation ( 2 ) has no r o o t s i n k

rn = min ( * E pie 120 1=0

X T ~

-

ln)

(-m,O],

the (3)

375

e x i s t s and i s a p o s i t i v e c o n s t a n t . LEMMA 3.

For i =

0,1,2

(-l)ix(i)(t) LEMMA 4.

,...,n > 0,

(4)

There i s a c o n s t a n t B > 1 such t h a t x(t-rk) < Bx(t),

LEMMA 5.

eventually.

eventually.

(5)

F o r any

x 2-B

- 1

Tk where B i s as d e f i n e d i n Lemma 4, x ' ( t ) + Xx(t) > 0 and t h e r e f o r e l i m [x(t)e"] tLEMMA 6.

z

0.

Define

xo

= P1l / n ,

and f o r each j = 1,2,

xo(t) = x(t)

... set

n-1 n-1-i i (i) x.(t) = z h (-1) Xj_,(t). J i=o j-1

(9)

Then t h e f o l l o w i n g statements h o l d f o r each j = O y l , 2 , (i)

x j ( t ) i s an e v e n t u a l l y p o s i t i v e s o l u t i o n o f ( 1 ) such t h a t (-l)ixii)(t)

(ii)

>

o for

i = 0,1,2

,...,n

eventually.

(10)

There e x i s t s a p o s i t i v e c o n s t a n t 6 . such t h a t J xj(t-rk)

(iii)

... .

< Bjxj(t)

(11 1

eventually.

( - l ) n + + ' x ! n ) ( t ) + x y x j ( t ) = x! ( t ) J J+1

+

x . x . ( t ) < 0. J J+1

(12)

LEMMA 7. W i t h t h e n o t a t i o n o f Lemma 6 and under t h e h y p o t h e s i s t h a t e q u a t i o n ( 2 ) has no r o o t s i n (-m,O], t h e f o l l o w i n g statements a r e t r u e f o r each j = O , l , Z ¶...

376

G. Ladas et al. Ajt

(ii)

l i m [x(t)e

1

= 0.

t-

The d e s i r e d c o n t r a d i c t i o n i n t h e p r o o f o f Theorem 2 f o l l o w s from ( 7 ) and (14) by choosing A . > A , t h a t i s , by choosing j such t h a t J (pl+jm)l'n > B - 1

-

-

Tk

I n t h e case n = 1 , Lemma 7 i s unnecessary because (6) and (12) a r e c o n t r a d i c t o r y I n f a c t , f o r n = 1, f o r A . > A.

J -

x.(t) = x(t)

for all

J

j

and t h e l a s t i n e q u a l i t y i n (12) reduces t o x ' ( t ) + X x ( t ) < 0. We o m i t t h e p r o o f s o f t h e above Lemmas f o r

l a c k o f space.

REMARKS The case n = 1 was f i r s t i n v e s t i g a t e d by Tramov [ l o ] i n 1975. The same r e s u l t was rediscovered i n 1982 by t h e authors o f t h i s paper 151, by Hunt and Yorke [2] i n an unpublished paper communicated t o the authors, and by Arino, Gy6ri and Jawhari t11* I n the case o f d i f f e r e n t i a l equations with one delay, t h a t i s f o r equations o f t h e form x(n)(t)

+

(-l)ntlpx(t-T)

= 0.

p,T > 0;

n21

(15)

we o b t a i n a necessary and s u f f i c i e n t c o n d i t i o n i n terms o f the c o e f f i c i e n t p and t h e delay T o n l y . I n t h i s case, t h e c h a r a c t e r i s t i c equation o f (15) i s

G(A) :A' + (-l)"+lpe-AT

= 0.

(16)

We prove t h e f o l l o w i n g theorem. THEOREM 3. (a) (b) (c)

For n

[n

even]

the f o l l o w i n g statements a r e e q u i v a l e n t .

u-

A l l s o l u t i o n s o f (15) o s c i l l a t e [ A l l bounded s o l u t i o n s o f (15) late]. Equation (16) has no r e a l r o o t s [Q. (16) has no r e a l r o o t s i n (--,0]]. pl/n > n e

1.

An analogue t o Lemma 1 a l s o holds f o r equation (15). Therefore, the p r o o f o f Theorem 3 i s reduced t o p r o v i n g t h e f o l l o w i n g r e s u l t f o r a r b i t r a r y n. THEOREM 4. (a) (b) (c)

The f o l l o w i n g statements are e q u i v a l e n t . Ever bounded s o l u t i o n o f (15) o s c i l l a t e s . The : h a r a c t e r i s t i c equation (16) has no r e a l r o o t s i n (--,Ol > pl/n I n e

1.

Oscillations o.fDeiajs Diff'crcntiai Equations

PROOF. F i r s t we p r o v e t h a t ( a ) i m p l i e s ( b ) .

377

Otherwise t h e r e e x i s t s a r e a l Xo

such t h a t

50

-XoT

+ (-1)"'pe

= 0.

w h i c h i m p l i e s t h a t x ( t ) = e l o t i s a n o n o s c i l l a t o r y s o l u t i o n of (15), a c o n t r a d i c tion. Next we p r o v e t h a t ( b ) i m p l i e s ( c ) .

-

G ( X ) :(-1)"'

pe-"

Equation (16) i s equivalent t o

= 0.

Since

G(0) = -p < 0 and G(X) has no n e g a t i v e r o o t s i t f o l l o w s t h a t G ( X ) < 0 f o r e v e r y X E Setting

X

=

-

(--,O].

i n t o G(X) we f i n d

F i n a l l y we p r o v e t h a t ( c ) i m p l i e s ( a ) . Otherwise e q u a t i o n ( 1 5 ) has an e v e n t u a l l y n p o s i t i v e bounded s o l u t i o n x ( t ) . As i n Lemma 3, f o r i = 0,1,2,

...,

(-l)ix(i)(t)

>

o

(4)

eventually.

Set q = P''~,

T u = n and

Observe t h a t

= ,(n) ( t ) - q x ( n - - l

1( t - u )

+ q2x(n--2) ( t & )

-.. .+

( - 1 p - 1 q n-1 x ' ( t - ( n - l ) u )

and t h e r e f o r e y ' ( t ) + q y ( t - a ) = x q t ) + (-l)"'px(t-.r) But t h e c h a r a c t e r i s t i c equation o f y ' ( t ) + qy(t-u) = 0

= 0.

378

G. Ladas et al.

is = 0

A t qe-1'

and, i n view o f ( c ) , = ; 1 I n ( q o e ) = 1 ln(pl"'

min (A+qe-")

e ) > 0.

A

T h a t i s , e q u a t i o n (19) has no r e a l r o o t s and t h e r e f o r e ( s e e [ 5 ] ) e v e r y s o l u t i o n y ( t ) o f ( 1 8 ) i s o s c i l l a t o r y . T h i s c o n t r a d i c t s ( 1 7 ) . The p r o o f o f t h e theorem i s complete. The above r e s u l t s have s t r a i g h t f o r w a r d e x t e n s i o n s t o d i f f e r e n t i a l e q u a t i o n s w i t h advanced arguments o f t h e forms X("(t)

-

k

C pix(t+Ti)

= 0

i=O

and x("(t)

-

p x ( t t r ) = 0.

The c h a r a c t e r i s t i c e q u a t i o n s o f ( 1 ) ' and ( 1 5 ) ' a r e r e s p e c t i v e l y

k

f ( X ) E A"

- c

g(A) z A"

-

'=o

Xi.

pie

i=O

(2)'

and (16) '

= 0.

pe"

By s i m i l a r arguments we e s t a b l i s h t h e f o l l o w i n g dual r e s u l t s . THEOREM 1 ' . ( i ) W has no r e a l r o o t s .

n

odd, a l l

s o l u t i o n s o f ( 1 ) ' o s c i l l a t e i f and o n l y i f ( 2 ) '

For even,

(ii) n a l l unbounded s o l u t i o n s o f ( 1 ) ' o s c i l l a t e i f and o n l y i f ( 2 ) ' has no r e a l r o o t s i n [OF). THEOREM 3 ' . (a)

For n odd [ n

3-

t h e f o l l o w i n g statements a r e e q u i v a l e n t .

A l l s o l u t i o n s o f ( 1 5 ) ' o s c i l l a t e [ A l l unbounded s o l u t i o n s o f ( 1 5 ) ' 0shas no r e a l r o o t s [Q.

(c)

( 1 6 ) ' has no r e a l r o o t s i n

[O,m)l.

P

REFERENCES

[l] 0. A r i n o , I . G y o r i and A. Jawhari, O s c i l l a t i o n C r i t e r i a i n Delay Equations, J. D i f f e r e n t i a l Equations ( t o a p p e a r ) .

n

[2] 131

B. R. Hunt and J. A. Yorke, When a l l s o l u t i o n s o f x '

oscillate (preprint).

=

- C

i=l

qi(t)x(t-Ti(t))

T. Kusano, On even o r d e r f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n s w i t h advanced and r e t a r d e d arguments, J . D i f f e r e n t i a l Equations 45 (1982) , 75-84.

Oscillutiotis

01’Drluj, Dijyerentiul Eytiutions

379

141 G. Ladas, Sharp Conditions f o r O s c i l l a t i o n s Caused by Delays, Applicable Anal. 9 (1979), 93-98. [5]

G. Ladas, Y . G. S f i c a s and I . P. Stavroulakis, Necessary and s u f f i c i e n t cond i t i o n s f o r o s c i l l a t i o n s , Amer. Math. Monthly ( t o appear).

[6]

6 . Ladas and I . P. Stavroulakis, On Delay Differential I n e q u a l i t i e s of

[7]

G. Ladas and I . P. Stavroulakis, O s c i l l a t i o n s Caused by Several Retarded and Advanced Arguments, J . Differential Equations 44 (1982), 134-152.

[8]

V. N. Sevelo and N. V. Vareh, Asymptotic Methods in the Theory of Nonlinear O s c i l l a t i o n s , Kiev "Naukova Dumka", 1979 (Russian).

[9]

Y . G. S f i c a s and V. A. Staikos, O s c i l l a t i o n s of Differential Equations with

Higher Order, Canad. Math. Bull. 25 ( 3 ) (1982), 348-354.

Deviating Arguments, Funkcial. Ekvac. 19 (1976), 35-43. [ l o ] M. I . Tramov, Conditions f o r Oscillatory Solutions of First Order Different i a l Equations with a Delayed Argument, Izv. Vys5. UEebn. Zaved., Matematika 19, NO. 3 (1975), 92-96.

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DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) @ Elsevier Science Publishers B.V.(North-Holland), 1984

38 1

BOUNDARY BEHAVIOR OF SOLUTIONS OF DEGENERATE ELLIPTIC EQUATIONS AND GENERATION OF SEMIGROUPS Michel L a n g l a i s

Department of Mathematics Purdue U n i v e r s i t y West L a f a y e t t e , I n d i a n a U.S.A.

The g o a l of t h i s t a l k i s t o s u p p l y a n example of a d e g e n e r a t e e l l i p t i c e q u a t i o n f o r w h i c h the a n a l y t i c a p p r o a c h and t h e p r o b a b i l i s t i c a p p r o a c h y i e l d d i f f e r e n t b u t w e l l - p o s e d boundary v a l u e problems. We c o n s i d e r t h e l i n e a r e q u a t i o n w i t h smooth c o e f f i c i e n t s :

(E+ao)u =

(0.1)

R

herein

=

{x E R

N

The c o n d i t i o n on rp

-

, rp(x)

> 0)

on

2R

,

i

+ aou =

VCp (x) # 0

cp(x)”

means t h a t

R but

t h a t e q . (0.1) i s e l l i p t i c i n

i a ux

+

(Cpaijuxi)xj

f

an

on

fl ;

in

d i s t a n c e from

and:

x

to

.

aR

It f o l l o w s

it degenerates i n t o a f i r s t o r d e r equation

.

The c l a s s i c a l a n a l y t i c t h e o r y of second o r d e r and d e g e n e r a t e e l l i p t i c e q u a t i o n s (FICHERA [31,

OLEINIK and WKEVIC 171, KOHN and NIRENBERG [61) asserts t h a t

2R where t h e v e c t o r

boundary c o n d i t i o n s are t o b e p r e s c r i b e d a t t h o s e p o i n t s o f field

aiaxi

C2 = {x E a R

R:

is directed inside

,

01

.

6 N

.

abxi>

This leads t o

t h e boundary v a l u e problem: 0

(A)

(E+a ) u = f

R

in

;

u = g

on

C a r r y i n g t h e d i f f e r e n t i a t i o n o u t i n (0.1) = - Cpa

ij

i

u ~ . + ~ A. uxi 1 J

-

= ai

c2

*

yields:

+

aijCpxj

0 a u

= f

ij - Cpaxj

in

R

1 s i

From a p r o b a b i l i s t i c v i e w p o i n t (STROOCK and VARADHAN [ 9 ] ) boundary v a l u e s are t o b e p r e s c r i b e d on t h e p o r t i o n of

= CX

E a9

(B)

In g e n e r a l Oax) # 0

,~

Z2

51 , t h i s i s a t

2R where t h e v e c t o r f i e l d A axi i s d i r e c t e d i n s i d e R b > ,01~. We now have t h e boundary v a l u e problem

(Efa 0 ) u = f

on

where t h e d i f f u s i o n c a n e x i t i

t h o s e p o i n t s of.

z;

an

and

20

.

in

R

; u = g

on

c;

.

do n o t c o i n c i d e b e c a u s e

(ai’)

i s e l l i p t i c and

We i n t r o d u c e t h e g e o m e t r i c a l i n v a r i a n t :

:

M. Langlais

38 2 . (0.3)

abXi

z(x) =

2R

on

ij

a cpxpxj Clearly

C2 = tx

E aR , z ( x ) > 0)

1;

while

.

, Z(X) > 1 1

ix E a R

=

(B) a r e d i f f e r e n t i f and only if t h e range of

Hence (A)

intersects (0,l).

z

For sake of s i m p l i c i t y we assume i n t h i s n o t e t h a t :

(0.4)

0 <

C2 = a R

This implies t h a t

C

(5)

on a R

< 1

Z(X)

but

.

*

C2

.

0

=

R of bounded and measurable f u n c t i o n s over R

continuously d i f f e r e n t i a b l e f u n c t i o n i n

;

,

1.

W2 (R)

and

2

t h e space of twice

w i t h norm

Lm(Q)

functions with bounded d e r i v a t i v e s of order

2 C (R)

D ,

i s t h e space of continuous f u n c t i o n s over

i s t h e space

t h e space of bounded

.

R

in

(1 I(

Problem (A) under condition ( 0 . 4 )

Theorem 1.

Given

f

in

C(n)

and

g

s o l u t i o n of (A) say (0.1) and

C(D>

in

C(aR)

on

u = g

t h e r e e x i s t s a unique

u

in

.

aR

This s o l u t i o n s a t i s f i e s t h e e s t i m a t e : IgIL”(a;2)I

111111 5 M ~ X ( ~ ; ’ I I ~ I 9I

and i s nonnegative when Let

A

f

and

*

g

a r e nonnegative.

be t h e unbounded operator i n

D(A) = { u E C(n)

,

C(n)

,

on

u = 0

with domain and

Eu I$ C ( n ) }

Au = Eu

,

u E D(A)

.

W e can summarize t h e foregoing r e s u l t s , v i a t h e HILLE-PHILLIPS-YOSIDA theorem into: Corollary 1.

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 of a F e l l e r semigroup on

(-A)

Proof of Theorem 1.

Any s o l u t i o n

in

u

C(n)

of

(E+ao)u = 0

belongs t o

-

R

.

C2(R)

and uniqueness follows from t h e maximum p r i n c i p l e . To g e t e x i s t e n c e we f i r s t assume t h a t

g = 0

.

For any

E

> 0

let

u‘

be t h e

unique smooth s o l u t i o n o f : -(EST

)aijux.x 1 j

+ ~~u~~ +

0 a u = fE

(1.1) =

where

fE

i s smooth and

fE+f

in

s o l u t i o n of eq. (0.1) continuous i n

Co(n)

R

.

o

in on

When

aR

an E+O

,

uE

converges t o a

.

The problem i s now t o d e r i v e t h e c o n t i n u i t y up t o t h e boundary. follow from the r e s u l t s of [ 7 ] which would r e q u i r e on

C2 , namely

Ahxi>

0

on

I?= (E+ao)cp

Z2 , and t h i s i s p r o h i b i t e d by

This does not t o be p o s i t i v e

z(x) < 1 i n

(0.4).

383

Degericrate Elliptic Equations Lema 1.

Choose

M = M(m)

such t h a t

Define

vE

m

m < z(x)

I

M[cp(x)+Elm

luE(x)

5

l2

Bi

A0 =

+

(z(x)-m)

ij

There e x i s t s

R ,0

f

<

<

E

E~

E~ =

~ ~ ( m ,)

.

r :

and

+ (&+v) ~~v~~ + A0v

aijvxixj

.

E aR

,x

u E ( x ) = [ E t y ( x ) l m v E ( x ) ; by d i r e c t computation and

by t h e r e l a t i o n

for suitable coefficients -(E+cp

,x

i n (0,l) with

r(x)(E+cp)

(c+cp (x)l-mf

=

,

in

R ,

.

rELa(R)

a cpxi’pxj 0 <

T h e r e f o r e provided up w i t h {X f R

Ao(x> 2

,0

n >

< CP(x)<6]

E

.

0

<

E~

and

x

aR

be c l o s e t o

or

6 w e end

0 < Cp(x) <

By a p p l i c a t i o n of t h e maximum p r i n c i p l e i n

we conclude t h a t

vE

is bounded.

I t i s now c l e a r t h a t t h e l i m i t i n g f u n c t i o n i s c o n t i n u o u s and v a n i s h e s a l o n g

When

g

$

0

Remark 1.

we proceed by approximation. The proof of Lemma 1 can b e l o c a l i z e d t o show t h a t f o r any z (XO) -u 0 xo on an u ( x ) = o([cp(x)] ) ,O
u

in

, about

D(A)

2.

.

aR

Problem (B) under c o n d i t i o n ( 0 . 4 )

We f i r s t need a u n i q u e n e s s r e s u l t . Let

Lemma 2.

2 cEC (Q)nW2’”(R)

0 (E+a ) u = 0

satisfy

R

in

.

Then

u = 0

R.

in

.

P r o o f . F i r s t u c a n n o t a t t a i n a nonzero extremum i n R Second on aR w e have i 0 * 2 If u2 a t t a i n s a p o s i t i v e extremum A uxi + a u = o s a y A I ( ~lXi + 2a0u2 = o a t xOEaR , due t o A b x i < 0 on 20 Ai(u 2 lxi(x 0 ) 5 0 and a0 u2 ( x0 ) i 0 ; hence

.

u=o

.

Remark 2.

Given

runs over

C(2R)

f

in

.

a t most one s o l u t i o n of

C(n)

For example l e t

v€C2(D)

u n i q u e s o l u t i o n i n C2(R)nW2y”(Q) of 0 0 s o l u t i o n of (E+a ) u = (E+a ) v i n R

(B)

,

and

with

u = 0

(A)

g

i s smooth when

v$O on aR ; u* = v i s t h e 0 f = (E+a ) v w h i l e t h e u n i q u e

aR

on

i s d i f f e r e n t from

v

and

nonsmooth. Let D(Po)

Po

.

D(Po) = C2(Q)nW2’m(R) b e d e f i n e d by

w i t h domain W e denote

P

t h e minimal c l o s e d e x t e n s i o n of

bounded o p e r a t o r s

C(n)+C(a)

Theorem 2 .

f

(0.1) say

Given (B)

Corollary 2.

in

Po

0 P u = Eu

for

in

.

C(R)

t h e r e e x i s t s a unique

in

u*

D(P)

s o l u t i o n of

. (-P)

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 of a F e l l e r semigroup on

The example i n Remark 2 shows t h a t

D(A)

# D(P)

D(P) # { g C ( n )

and

,

EgC(fi)}

Theorem 2 f o l l o w s from t h e f o l l o w i n g r e s u l t . Theorem 3.

u

i n t h e s e n s e of un-

There e x i s t s

A0 >

0

s u c h t h a t f o r any

A

2

A.

the equation

. .

M. Langlais

384 (E+x)u = f f 2

o

in

implies

R

h a s a unique s o l u t i o n i n D(P)

u 2

o

f o r any

.

; a l s o llull 2 h-lllfll

f

in

C(n)

.

Further

This i s proved i n t h r e e s t e p s .

Assume

Uniqueness. C2(Q)"W2'"(Q)

(E+X)vn = fn+O

n-

in

in

u

with

.

C(n)

Existence f o r

f

in

in

C(n)

when

n-tm

. .

Along t h e l i n e s of Lermna 2

.

C2(n>

-(E+ep)a i juxixj

For any

+ ~~u~~ + A

(2.1) fE

(E+A)u = 0

satisfies

There e x i s t s

(v,)

in

Therefore

((vn[l2 h-lllfnll+O

when

.

and u=O

where

D(P)

, Evn+Eu

vn+u

i s smooth and

i

E>O

let

E

u

~u

=

fE:

in

R

+ xu

=

fE

on

an

fE+f

in

.

C2(n)

be t h e s o l u t i o n o f :

The boundary c o n d i t i o n i s t h e 0 D(P ) s o l u t i o n of (2.1)

n a t u r a l boundary c o n d i t i o n s a t i s f i e d by any f u n c t i o n i n

.

with

E=O

A%xi

i s p o i n t i n g outward along

On t h e o t h e r hand t h i s problem has a unique s o l u t i o n i f and only i f

an

:

t h i s is p r e c i s e l y

z(x) < 1 i n ( 0 . 4 ) .

I n t h e Appendix we o u t l i n e t h e proof o f : (2.2) By l e t t i n g

luEl 0 6 E 5 Eo(h) W2,"(0) E+O w e o b t a i n a s o l u t i o n i n W2’"(Q) ; i t belongs t o

C2(R)

by

e l l i p t i c estimates. Conclusion.

3. 3.1

f

For

in

w e use a d e n s i t y argument.

C(n)

Concluding Remarks.

A s a l r e a d y p o i n t e d o u t i n Remark 2 t h e s o l u t i o n i n

D(P)

of (0.1) i s t h e

smooth s o l u t i o n of (0.1) whenever such a smooth s o l u t i o n e x i s t s . s o l u t i o n is not i d e n t i c a l l y

on

0

3R

I f t h i s smooth

then t h e s o l u t i o n of (0.1) i n D(A)

behaves a s s t a t e d i n Remark 1; t h e one dimensional c a s e shows t h a t t h i s t y p e of behaviour i s t h e b e s t w e can e x p e c t .

3.2

When

(see also

z(x) >

on

SHIMAKURA [ 8] );

continuous on 3.3

1 2

n

C2

s o l u t i o n s of f o r continuous

(A) f

in

H1(n)

are obtained i n

[6]

t h e s e s o l u t i o n s a r e seen t o be

by a d a p t i n g our method.

A s h a r p form of t h e e s t i m a t e i n Remark 1 is d e r i v e d i n GRAHAM I51 f o r t h e N -div(x g r a d u) + z uxN , z p o s i t i v e c o n s t a n t , i n t h e h a l f s p a c e IR+ N " l i m i t i n g case" z=O h a s a f a i r l y d i f f e r e n t s t r u c t u r e see BAOUENDI and

operator The

GOULAOUIC [l] f o r (0.1) when

3.4

ai

= Cpzi

12 i 2 N

Problem (B) i s a t y p i c a l example of t h e

[ 9 ] s e c t i o n 5.

.

B - f i r s t boundary v a l u e problem of

F u r t h e r r e g u l a r i t y p r o p e r t i e s of t h e s o l u t i o n i n

can be deduced from t h e r e s u l t s g i v e n in BOLLEY and CAMUS [2]. i n GOULAOUIC and SHIMAKURA [ 4 ] under t h e c o n d i t i o n s ai = z ( x ) a i b X j

1 6 is N

on

a~

.

z(x) < 1

D(P) (B) on

of

(B)

is treated

aR

and

.

385

Degenerate Elliptic Equations 3.5

The r e s u l t s of s e c t i o n s 1 and 2 a r e e a s i l y extended t o t h e s i t u a t i o n i n

4.

*

c2ucl ,

aR =

which

C2

*

and

c l o s e d and d i s j o i n t

C1

*

Z1

z(x) < 1 on

and

,

Appendix.

We f i r s t prove a weaker form of (2.2);

l/&ll

(4.1)

,j

1s i

for

s N :

I I ( ~ + e P ) u , ~ ~i~ lCl/fElW2'm(W l

,

3R by f i n i t e l y many s m a l l open domains O R

W e cover

using l o c a l coordinates

aR

yl, ...,yN

l i e s on

y =O

1i R 2 L and

14

, in

yN>O in

which

R

noR

.

W e may assume t h a t (2.1) r e a d s :

+

-(E+yN)bijuYiYj (4.2) w i t h-bm=aijCpxiqxj

,b

Nk

b iuyi

+

B

+U

~

Xu = f = ~

f

~

kN=b - 0 , 1 i k s N-1

,

yN > 0

in

yN =

on

, -b

N

z

o , a2 > O

N

-B z a3>0 i n

and

R n oR Let

0

0 =R

'

covering

OR

n

of

.

(€+YN)

W =

We denote by

IJJ

p,

R R w Ci

be a p a r t i t i o n of u n i t y corresponding t o t h e

Setting

2

=

and l e t

R

$ ,O s R i L

and

9 . 2 ~ - 1(vypyp)

z

OR

vR = uE i n

d e f i n e f o r l a r g e constant

6

lJ

+

~ - 1(vyk)

2

and

v

2 2 +

V(v

. v a r i o u s c o n s t a n t independent of

A , p and

V

.

From (4.2) we

have :

11 (E+YN)2 1'$ 1 (VynYn)2 11 5 c211wII + Xc3 .

(4.3)

Next (4.2) viewed a s an o r d i n a r y d i f f e r e n t i a l e q u a t i o n f o r

c4 llwll +

s

(4.4) Il+"vy,)211

AC5

By d i r e c t computation, u s i n g (4.3)-(4.4) i A WXi

(4.5) Choosing

+

-(E+TP) a

X

ij

E

W

i C6

E

(O,€(X))

in

+

+ A%.,,

uXixj

l a r g e enough and

with respect t o 2.Q

2hW

R

(E+yN) $ (vykyN)2 i s bounded.

yields

vyN

E

2 2

X

and C7W

+ xw

s

p,A

+

l a r g e enough:

A(p+v)

(u cg +

c8

on

llwll +

cl0)

s ~ ( h )we conclude t h a t

.

aR,

W

~ t l ~ + v cll )

in

R.

i s bounded uniformly

I n a s i m i l a r f a s h i o n we prove t h a t

This completes t h e proof of (4.1).

Regarding a g a i n (4.2) a s an o r d i n a r y d i f f e r e n t i a l e q u a t i o n f o r

vyN

,a

careful

a n a l y s i s u s i n g (4.1) shows t h a t a c t u a l l y :

R I ?

(4.6)

2

II@ (vYNyN) 11 ' '12"

+

15

5

2

[I

5 N-il+RQGYkYN)

+

Next f o r A l a r g e enough: (4'7)

and f i n a l l y

R

R

(vykyN)

2

11

'i3

(A + hp:wN-l

R

R

(vypyq)

2

11)

1 sp:wN-ll/'$

L a .

(vypyq)211).

M. Langlais

386 (4.8)

(I$

R

R

2

(vyPyq)

I1 5

~ 1 4

The l a s t two estimates are o b t a i n e d by a p p l y i n g t h e o p e r a t o r f u n c t i o n t o estimate and u s i n g ( 4 . 1 ) ,

(4.6).

0 (E+a )

t o the

T h i s g i v e s (2.2).

References

[I]

Baouendi, M.S.

and G o u l a o u i c , C . ,

Re'gularite' e t t h e ' o r i e s p e c t r a l e p o u r ;ne

c l a s s e d ' o p e ' r a t e u r s e l l i p t i q u e s de'ge'ne're's,

Arch. R a t . Mech. Anal. 34-35

(1969) 361-379. [2]

B o l l e y , P. and Camus, J . , Sur une c l a s s e d ' o p g r a t e u r s e l l i p t i q u e s e t de'ge'ne'rbs 2 p l u s i e u r s v a r i a b l e s .

[3]

Fichera, G.,

Bull. S.M. F r a n c e 34 (1973) 55-140.

S u l l e equazioni d i f f e r e n z i a l i l i n e a r i ellittico-paraboliche

del

second0 o r d i n e , A t t i , Acc. Naz. L i n c e i Mem. Ser. 8 , Vol. 5 (1956) 1-30. [41

Goulaouic, C . and Shimakura, N . ,

[5]

Graham, C . R . ,

Rggularit;

Holde'rienne d e c e r t a i n s probl&mes

am l i m i t e s e l l i p t i q u e s dgge'nirbs. The D i r i c h l e t problem f o r t h e Bergman L a p l a c i a n ,

Ph.D. T h e s i s ,

P r i n c e t o n U n i v e r s i t y 1981. [6]

Kohn, J.J. and N i r e n b e r g , L . , D e g e n e r a t e e l l i p t i c p a r a b o l i c e q u a t i o n s of second o r d e r , Comm. P u r e and Appl. Math. 20 (1967) 797-872.

[7]

O l e i n i k , 0. A. and Radkevic, E . V . ,

Second o r d e r e q u a t i o n s w i t h n o n n e g a t i v e

c h a r a c t e r i s t i c form, h e r . Math. S O C . , P r o v i d e n c e , (Plenum P r e s s , N . Y . ,

1973).

[8] Shimakura, N . , Problsme d e D i r i c h l e t pour d e s o p g r a t e u r s e l l i p t i q u e s dgggne're's du second o r d r e , P r o c . J a p a n Acad. 47 (1971) 861-866. [9]

S t r o o c k , D. and Varadhan, S . R . S . ,

On d e g e n e r a t e e l l i p t i c p a r a b o l i c o p e r a t o r s

of second o r d e r and t h e i r a s s o c i a t e d d i f f u s i o n s . 24 (1972) 651-713.

Comm. P u r e and Appl. Math.

DIFFERENTIAL EQUATIONS l.W. Knowles and R.T. Lewis (Editors) @ Elsevier Science Publishers B.V. (North-Holland), 1984

387

THE ZERO DISPERSION LIMIT FOR THE KdV EQUATION

Peter D. Lax Courant Institute of Mathematical Sciences

The equation in question is Ut

- 6uux

+

c 2 uxxx = 0

and the question under discussion is this: u are fixed, u(x, 0; € 1 how does the solution u(x, t; When we set equation

=

u(x)

I

-

6uuX = 0

tends to O?

.

This equation has no solution for all t, =

E

= 0 in the equation, we obtain the reduced

E

Ut

ti

if the initial values of

behave as

E)

I

1

6 Min ux(x) X

< t

<

only in the interval

1

+

= tb ;

6 Max ux(x) X

here u(x) is the initial value of u. It is reasonable to surmise + that for t in (ti, tb), u(x,t, E ) tends as E -+ 0 to the solution of the reduced equation. What happens when t lies outside this interval? Numerical experiments indicate that over some part of the x-axis, u(x,t, E ) is oscillatory. A s E tends to 0, theamplitudes of these oscillations remain finite, their wavelengths are of order E . Clearly, if we can talk of a limiting behavior as E tends to 0, this limit can exist only in the weak sense, e.g. in the sense of distributions. This indeed is the case: the speaker and C. D. Levermore have shown 111 ,121 that u(x.t, E ) tends in the sense of distributions to a limit E, provided that the initial value u(x) is nonpositive and

PD. Lax

388

tends to zero so fast that Ix u(x) dx is finite. These papers show not only the existence of a limit but give a fairly explicit formula for the limit For simplicity we take the case when (x) has a single minimum:

u.

where Q*(x,t) = Min Q ( $ ; x,t). O<$(Q Here Q is a quadratic functional of $:

L the linear integral operator

The functions admissible in the minimum problem are restricted to lie between 0 and 0, where 4 is defined in terms of the initial data as follows:

The function a appearing in the linear term depends linearly on x and t, and is defined as follows:

where e + ( q ) is a function defined in terms of the initial data. 2 Using the KdV equation it follows that also u (x,t;E) has a limit in the distribution sense, and that

7

-

Ut =

2 3ux

.

Multiplying the KdV equation by u and rewriting the resulting equation as a conservation law shows that

exists in the distribution sense, and that

389

The Zero Dispersion Lirnit f o r the K d V Equutioii

-

7x . u2 = 4 u t

-

Combining this with the explicit formulas

form foru leads easily to the

-

, ;7 =

u 2 = 3l axat Q*

1

at2

jTz

Q*

*

The minimum problem defining Q is a so called quadratic programming problem: it turns out that it can be solved explicidy , To see this it is convenient to extend the functions J, admitted in the minimum problem for all real II as odd functions: as a result One Can we may replace the kernel of the operator L by log 1Q-P [ show that L is negative definite, and that it is related to the Hilbert transform H as follows: an*L = H We extend now J, to the upper half of the complex TI plane as a harmonic function that vanishes at m: J, which can be regarded as the real part of an analytic function of Hardy class. The variational condition for the minimum problem can be regarded as prescribing the real and imaginary parts of the function on complementary subsets of the real axis; for details we refer the reader to [ 2 ] . Suffice it here to say that the resulting formulas for show

+

i) for t in (ti , tb), i is a solution of the reduced equation, and that in this time interval the convergence of u to ii takes place not only in the sense of distributions but for each 2 t in the L sense in X.

+

ii) for t outside the interval (ti , tb), ii can be described by Whitham’s averaged equations, or by the more general equations of Flaschka, Forest and McLaughlin based on multiphase averaging. iii) for t tending to

for range ii(x,t) is

0

o(t2).

x/t

-2

m,

Eclecayslike t

4m, where m = max

-

: more precisely

u(x).

Outside this

X

The formula for ii is obtained from Gardner, Greene, Kruskal and Miura’s solution of the KdV equation by the scattering transform.

P.D. Lax

390

We trace carefully the manner in which this solution depends onE , and show that as E. tends to zero, it has a limit in the sense of distributions. The nonpositivity of the initial data makes the GGKM solution of the KdV equation particularly simple. The more difficult case when u is nonpositive has been handled by Stephanos Venakides, I31

.

REFERENCES

[l]

Lax, P. and Levermore, C.D. , The zero dispersion limit for the KdV equation, Proc. Nat. Acad. Sci USA 1979; pp 3602-3606.

[2]

Lax, P. and Levermore, C.D., The small dispersion limit of the KdV equations, Comm. Pure Appl. Math., PattI, Vol 36 1983, p 253-29, Parts 11, I11 to follow.

[3]

Venekides, S., The zero dispersion limit of the KdV Equation, Diss., FGAS., 1982, New York University.

DIFFERENTIALEQUATIONS I.W. Knowles and R.T.Lewis (Editors) 0 Elsevier Science PublishersB.V. (North-Holland),1984

39 1

RECENT MULTIPLICITY RESULTS FOR NONLINEAR BOUNDARY VALUE PROBLEMS

A.C. Lazer University of Miami Coral Gables, F1.

and

P.J. McKenna University o f Florida Gainesville, F1.

In this paper, w e consider recent progress on the multiABSTRACT plicity of the solution set of various nonlinear boundary value problems. W e first consider the semilinear Dirichlet problem and discuss some recent progress o n a conjecture o f the authors arising from a n earlier paper of Ambrosetti and Prodi. We then consider other global bifurcation results for the Dirichlet problem, and finally some results for parabolic and hyperbolic semilinear equations. 1.

T h e elliptic semilinear problem - crossing the first eighnvalue. L e t 51 be a n open bounded domain in Rn w i t h smooth boundary a n

of

class C2+’,

> 1.

y

Let

A

be

a

second

order,

self-adjoint

strongly elliptic operator w i t h smooth real-valued coefficients defined i n il such that -Au > 0 , u f 0, and uI a i l = 0 imply u > 0 on D for example A = A . L e t X 1 < X 2 < --- < X k < --- denote the

---

eigenvalues of -A with Dirichlet boundary conditions, each counted as often as its multiplicity.

Let g

a

<

X1

<

6,

the problem

A u + gfu)

(P)

E

=

P(X)

u = o

C1(R,R)

E a,

Under the assumptions that .fim, X

D

x

an.

-

l.$m u+

and let p g(u) = u

E

C’(Zi).

B exist, and

has been widely studied during the past decade, beginning with a Before Ambrosetti and P r o d i , i n paper of Ambrosetti and Prodi [ Z ] . a classical paper on nonlinear integral equations [ 5 ] Hammerstein Thus showed that if a, 8 < A , , then (P) has a solution for all h. this question was a very natural one. Ambrosetti and Prodi assumed that f"(u) > 0 for all u , 0 < a < hl, and X1 < 8 < h 2 . Under these assumptions,they showed that Cy(x)

E~

=

a

u E2 where E o and E 2 are disjoint open sets 1 and the problem has exactly k solutions for p E E k ,

= Eo u E

E~ = a E 2 ,

0, 1 , 2. The structure of these sets was elucidated by Berger and To describe their results, here and in the remainder Podolak [ 3 ] . of this section, let 4, denote the normalized postitve eigenfunction k

=

corresponding to XI,

<

h,

@, >

=

0 where

and h a smooth function which satisfies

<,>

2

is the L (D) inner product.

Under the same

conditions o n g a s required in [ Z ] , i t is shown i n [ 3 ] that there exists a real number s o = so(h) such that if @ = s @ +~ h , the

A. C. Lazer and P.J. McKenna

392

problem (P) has z e r o , one o r two solutions, depending o n whether s Two natural is less than, equal t o , or greater than so(h). questions arose at this point in the history of the problem. The first was, whether the restriction 0 < a w a s necessary for the The answer to this question was conclusions of [ 2 ] and 1 3 1 to hold. no. Manes and Micheletti [ 1 2 ] showed that the conclusions- still held under the weakened requirement a < X l < < X 2 and f"(u) > 0 . The second question w a s whether the requirement 6 < X 2 w a s necessary. It is relatively easy to construct a nonlinearity of with 6 = A2 and f" > 0 such that ( P ) has infinitely many solutions. However, one could still ask whether the requirement

<

a

A1

<

8 was

enough to guarantee the existence o f 0 , 1 or at least 2 solutions, as in [ 2 ] and [ 3 ] . A partial answer to this question was given by Kazdan and Warner [ 7 1 . They showed that if a < X 1 , < B , and p is as above, then there is no solution if s is less than a certain constant and This at least one solution if s is larger than the same constant. left a clear gap i n the literature. If a < X I < B < X 2 then one had (for

large

two solutions.

6 )

But

<

if X 1

6 is

assumed, then one

could only conclude that there existed one solution. This problem w a s resolved independently by Dancer [ 4 1 and Amman and Hess [ I ] , using essentially the same approach. This was to combine the earlier method of Kazdan and Warner with Leray-Schauder degree theory. The approach of Kazdan and Warner was to show that when s was large and postive, it was possible to construct upper and lower conclude that there exists a solution u solution 1, u , and -to satisfying 2 < u < U . Both [ 2 ] , and 1 4 1 , combined this with the following facts. First there exists a n a priori bound, which depends only on h o w Also, there are no solutions for s large and positive s becomes. large and negative, as can be s e e n by integration by parts. These two remarks can be combined to s h o w that the topological degree o n any large ball is zero. T h e second k e y idea is that the topological degree for the region

{ulu <

u

<

u, U I

an

= 0,

11111

< R}

i n C1+a(i2)

is plus 1. Thus we have a solution i n this region, and i n addition we have that when this region is removed from a large ball, w e have degree -1 for the remainder. This allows u s to deduce the existence of two solutions. This left u s with the following: Let a

=

f(u) s .tim, __

6 =

,t$m, f ( u ) and U

Theormem I Given h , there exists So(hl) Au

(PI

+

f(u)

= s+

1

+

hl

solution if s if s

>

SO,

<

so.

<

XI

<

6.

such that the problem. i n Sl

u = o

has at least two solutions if

a

on s

>

an.

s o , one solution if s = s o .

Moreover, if f "

>

0 and a

there are exactly two solutions.

<

XI

<

6

<

X2,

and no then

393

Nonlinear Boundary Value Problems

T h i s , in turn, gives rise to a natural question. Could there be exactly two solutions for large s i f , s a y , B satisfied A1 < 8 or are more solutions created as the parameter 0 crossed each eigenvalue? T h e authors studied this problem [ 8 ] under the assumptions that A 2 w a s of multiplicity o n e , and that A 2 < 6 < X 3 . The k e y n e w idea of [ 8 ] was to s h o w that f o r large positive s , there existed a large positive solution 2 which was close to the solution of the linear problem Aw + Bw = 6 4

I’

Moreover, the topological degree of the nonlinear map associated with P , on a small ball centered at w was +l. We n o w combine these observations and conclude that there is a large ball with degree z e r o , a small ball with degree +1 and a region contained between and u with degree +l. T h e two regions of degree +1 are shown to be disjoint. Finally w e can conclude (by ecision) that the degree on the remainder of the region is - 2 , from which w e conclude: Theorem I 1 Let a

<

A1,

Az

<

0

<

A3,

and

assume X 2 has

multiplicity

one.

Then f o r large positive s , the problem (P) has at least three (and generically four) solutions. CONJECTURE The authors felt that the fact that one could only prove that generically four solutions existed, was a limitation of their If a < X 1 , proof. Therefore, we made the following: conjecture.

An

<

B

<

An+l

then for sufficiently large positives, the problem P

has at least 2 n solutions. Recently, there has been much progress o n this conjecture, which we n o w discuss. O n e approach might be to attempt to prove the conjecture for the ordinary differential equation, using the addiThey tional techniques available. This the authors did in [ 9 ] . study the equation u*' + f(u) = s Sinx + h(x) O < x < r u(0)

=

u(n)

-

0

under the assumptions a < 1 , n 2 < B < (n+I)’, and s h o w that for large positive 6 , this equation has at least 2n solutions. The proof uses the new large positive, approximately linear, solution and shooting techniques combined with some tricky counting arguments. This gives additional support to the conjecture. and [61. The variational method w a s introduced in 1 1 4 1 , Solimini in [ 1 4 ] showed that if a < A , 0 f A n , then the functional J associated with ( P ) had the property that as u went i n the positive 0, direction, t h e n J(u) + Furthermore, t h e two solutions

--.

explicitly constructed by the authors had the following additional properties; the large negative solution (between the upper and lower solution) was a local minimum f o r J and the large positive (approximately linear) solution w a s non-degenerate of Morse index n, where An < 0 < A n f l . One combined these observations with the mountain pass theorem to s h o w that there existed a mountain pass critical One then had to s h o w that this was distinct from point for J(u). the large positive solution. Solimini did this by observing that if a mountain pass is non-degenerate, it has Morse index o n e , (and thus is distinct) and if it is degenerate, then it is distinct anyway.

394

A. C. Lazer und P.J. McKenna

This showed that if Xu

< 5 <

n

> 2 , then for large s , (P) had

at least three solutions. H e showed Hofer, [ 6 ] independently, went one stage further. that the mountain pass must always have degree - 1 if it is isolated. He concluded that if A n < 5 < then since we had degree zero on the ball, degree fl at the large negative solution, degree - 1 at the mountain pass and degree (-l)n = f 1 at the approximately linear solution, then when these three regions were excised from the large Thus, he concluded for ball, w e had degree fl o n what is left. large s , that (P) had at least four solutions. One problem with these approaches is that both relied very heavily on the existence of the large positive, approximately linear, solution. Thus they could not provide much information The authors, in [lo] provide a partial way round this if 6 = An. difficulty.

X2

<

5 < X3,

They first impose the hypotheses where X2 is assumed simple.

a

< X1,

and

They then use a Lyaunov-

Schmidt method to reduce the problem to one of counting the zeros of a map from R2 to R2 and are able to show, using two-dimensional there are always at least four topology, that for large s , solutions. By combining these methods, with the earlier existence of the large positive approximately linear solution and topological degree in Hilbert spaces, they are then able to show that there exists E > 0 such that if X 3 < 5 < X 3 + , then (P) has at least five, and generically six, solutions. It is natural to suppose that in the near future this result may be proved with only the assumption X j < 5 < X4. The authors have recently combined their techniques with those of [ 6 ] to show that for any n > 2, there exists E such that if An < 5 < then for large 6 , ( P ) has at least five solutions. Again, in the future, we expect it will be possible to prove that in this situation, there are at least solutions. Generalizations: Other Boundary Value Problems Two natural problems arise in conjunction with our earlier discussion. To what extent are these results dependent on selfadjointness, and to what extent are they dependent on compactness. We now discuss parabolic and hyperbolic cases, i n each of which, one of these properties is missing. First we consider solutions of (9) u t - uxx + f(u) = s Sinx + h(x,t) u(0.t) u(x,t+T)

In similar

=

u(n,t)

=

0

= u(x,T).

[13], the second author to that proved for the

and W. Walter elliptic case

Namely, they show that if s is large, a

<

proved a result by the authors. 1 ; and ( 2 n ) 2 < 5 < ( 2 n + l )2 ,

then Q has at least three solutions. In [ l o ] , the authors showed then there exist at that if T is sufficiently small and 4 < f3 < 9 , least four solutions. There is clearly a gap between these two conclusions, with their strikingly different proofs. We consider this example important as variational methods cannot be applied.

395

Nonlinear Boundary Value Problems

(a)

Next we consider the hyperbolic case. utt - ,,u + f(u) = s sin, + h(x,t) u(0,t) u(x,t+ll)

= u(ll,t)

=

We consider

0

= u(x,t>.

Here, the second author, with W. Walter, showed that if 0 < a < 1 , f is monotone, and the interval [ a , B 1 contains an odd number of eigenvalues (counting multiplicities) then for large s, R has at least 3 (generically 4 ) solutions. It is natural to conjecture that a modification of the methods of Hofer could prove the existence of at least four solutions if g + m 2 - n 2. Generalizations: Crossing Higher Eigenvalues A second natural question is to ask what is the situation if X < a < Xj+k < B <j+k+l. This situation appears much j

trickier. There are some results however. The second author 1 1 4 1 showed that if X

<

a

<

.I and if (with some technical conditions) B was closer to

< Xj+l

<

Xj+2, than

a was, then (P) had solutions for any right hand side. In [ l l ] the authors show that if the right hand side p is written as s @ l + hl(x),

and Xj+l

is of multiplicity one, and if B is closer to X

j+l

than a is, then for large negative s , (P) has exactly one solution, whereas if s is large and positive, then (P) has at least three solutions. If a < X < X j + l < B , and a is close to j Xj, B close t o X j + l , and each eigenvalue is of odd multiplicity, then for s large negative there are at least two and for s large positive there are at least two. If f(u) = au- + But, then for = 0 and h = 0, there is exactly one solution. This leads to the X < a < A X < B < Aj+n+l, then following; conjecture if j-1 j j+n there exists k , 1 < k < n such that for s positive there are 2k + 1 solutions and for s large negative, there are at least 2(n-k) + 1 solutions. The authors, with D. Hart have proved a result close to this ordinary differential equation case.

s

A. C Lazer and P.J. McKenna

396

BIBLIOGRAPHY

1. Ama a Cla s of , H., a d Hess, P., A Multiplicity Re ult fo Elliptic Boundary Value Problems, Proc. Royal. SOC. Ed. 8 4 - A ( 1 9 7 9 ) 145-151. 2. Ambrosetti, A., and Prodi, G . , On Differentiable Mappings with Singularties Ann. Mat. Pura. Appl. 9 3 ( 1 9 7 3 ) 2 3 1 - 2 4 7 .

the Inversion of Some Between Banach Spaces,

Berger, M and Podolak, E., O n the Solutions of a Nonlinear 3. Dirichlet Problem, Indiana Univ. Math. Jour. 2 4 ( 1 9 7 5 ) 8 3 7 - 8 4 6 . 4.

Dancer, E.N., On the Ranges of Certain Weakly Nonlinear Elliptic Partial Differential Equations, J. Math. Pures Appl. 5 7 (1978),

351-366.

Hammerstein, A., Nichtlineare Integralgleichungen 5. Anwendung. Act. Math. 5 4 ( 1 9 3 0 ) , 1 1 7 - 1 7 6 . 6.

Hofer, H., Variational and Topological Methods Ordered Hilbert Spaces, Math. Ann. 2 6 1 ( 1 9 8 2 ) , 4 9 3 - 5 1 4 .

i n

nebst

Partially

Kazdan, J., and Warner, F., Remarks o n some Quasilinear 7. Elliptic Equations, Comm. Pure. Appl. Math. 2 8 ( 1 9 7 5 ) , 5 6 7 - 5 9 7 . 8. Lazer A.C., and McKenna, P.J., O n the Number of Solutions of a Nonlinear Dirichlet Problem., J. Math. Anal. Appl. 8 4 ( 1 9 8 1 ) 2 8 2 294.

Lazer, A.C., and McKenna, P.J., O n a Conjecture Related to the 9. Number of Solutions of a Nonlinear Dirichlet Problem, to appear, Proc. Roy. SOC. Edin. 10. Lazer, A.C., and McKenna, P.J., Multiplicity Results for a Class of Semilinear Elliptic and Parabloic Boundary Value Problems, to appear, Jour. Math. Anal. Appl. 11. Lazer, A.C., and McKenna, P.J., O n Global Bifurcation Higher Values for Semilinear Dirichlet Problems, to appear.

at

12. Manes, A., and Micheletti, A.M., Un’ Estensione Della Teoria Variazionale Classica Degli Autovalosi per Operatosi Ellittici del Second0 Ordine, Boll. Un. Mat. Ital. 7 ( 1 9 7 3 ) 2 8 5 - 3 0 1 . 13. McKenna, P.J., and Walter, W., O n the Multiplicity of the Solution Set of some Nonlinear Boundary Value Problems, to appear. 14. McKenna, P.J., O n Nonlinear Pertulations which Cross the Eijenvalues, Nonlinear Analysis, Theory, Methods, and Applications 7,

(1983)

15-19.

15. Solimini, S . , Existence of a Third Solution to a Class of Boundary Value Problems with Jumping Nonlinearity. (preprint).

DIFFERENTIAL EQUATIONS I.W. Knowlea and R.T.Lewis (Editors) @ Elsevier Science Publishers B.V. (North-Holland), 1984

397

NONSTOPPING I T E R A T I O N F O R O R D I N A R Y DIFFERENTIAL OPERATOR EQUATION

Sung J . L e e Department of M a t h e m a t i c s U n i v e r s i t y o f South F l o r i d a Tampa, F l o r i d a 33620

W e g i v e a n o n s t o p p i n g i t e r a t i o n scheme i n f i n d i n g a l e a s t - s q u a r e s o l u t i o n f o r a H i l b e r t space operat o r equation. T h i s o p e r a t o r i s g e n e r a t e d by a reg u l a r f i r s t o r d e r o r d i n a r y d i f f e r e n t i a l s y s t e m and i s nondensely d e f i n e d .

5

1 INTRODUCTION

I n t h i s paper w e w i l l g i v e a nonstopping i t e r a t i o n a l g o r i t h m i n finding a least-square solution f o r a IIilbert space operator equation. T h i s o p e r a t o r i s n o n d e n s e l y d e f i n e d a n d i s g e n e r a t e d by a f i r s t o r d e r A s i m i l a r problem h a s b e e n i n r e g u l a r o r d i n a r y d i f f e r e n t i a l system. v e s t i g a t e d i n [LOC 11 i n t h e c a s e when t h e o p e r a t o r i s d e n s e l y def i n e d and i s s u b j e c t t o u s u a l two p o i n t boundary c o n d i t i o n s and i s g e n e r a t e d by a h i g h e r o r d e r r e g u l a r s c a l a r e x p r e s s i o n . The a b s t r a c t v e r s i o n of t h i s p r o b l e m h a s b e e n c o n s i d e r e d i n [LN 11 f o r a n a r b i t r a r y c l o s e d l i n e a r o p e r a t o r w i t h a c l o s e d range. The main i d e a i s t o t r e a t t h e d i f f e r e n t i a l o p e r a t o r a s a c o n t i n u o u s o p e r a t o r i n i t s graph topology. 52 OPERATOR PART O F LI N EA R MANIFOLD L e t H1, H2 be H i l b e r t s p a c e s . Then

L e t M b e a l i n e a r m a n i f o l d i n H1

t h e graph i n v e r s e of M : M - l N u l l M: Range M: Mfx) : =

= (x/{x,O}~M\,Dom M:

= { ~ x , y ~ ~ ~ y , x ~ ~ M ) , = ( x l { x , y } ~ Mf o r some

= ( y I { x , y } ~ Mf o r some

{ yI

H2.

XI,

XI,

and

{ x , y } ~ M [ (xEDom M I .

A l i n e a r m a n i f o l d R C H l IB H 2 i s c a l l e d an a l g e b r a i c o p e r a t o r p a r t of M i f R i s t h e graph of a l i n e a r o p e r a t o r such t h a t M i s t h e a l g e b r a i c An a l g e b r a i c o p e r a t o r i s c a l l e d d i r e c t sum o f R and M _ : = 1 0 1 @ M ( 0 ) . I n t h e case when M(o) i s c l o s e d r an o p e r a t o r p a r t i f i t i s c l o s e d .

we define Ms:

= (Igry-p(y)

> 1 fg,y>EMI,

where P i s t h e o r t h o g o n a l p r o j e c t i o r o f H 2 o n t o M ( o ) . was p r o v e d i n [ L N 21.

The f o l l o w i n g

S.J. Lee

398

Assume t h a t M is closed.

THEOREM 2.1.

PlS

Then

= MOM,

1

tg,ylltg,yl~M,YEM(O)

=

},

and M i s t h e o p e r a t o r p a r t o f M s u c h t h a t Ms I n paffticular, Dom M = Dom Ms,

i s o r t h o g o n a l t o Mm. -L

(Range M ) fl ( M ( 0 ) )

= Range Ms.

i s c l o s e d i f and o n l y i f Dom M i s c l o s e d .

Moreover, M

Suppose now t h a t M i s c l o s e d . < ' > M by

Then Ms i s a l s o c l o s e d .

D e f i n e a form

< x , Y > ~=: < x , y > + < ~ l s ( x )M , s(y)> f o r a l l x , y i n Dom M I where <,> d e n o t e s t h e i n n e r p r o d u c t H1 o r H2. Then t h e form i s a c o m p l e t e i n n e r p r o d u c t i n Dom M. The t o p o l o g y o n D o m M g e n e r a t e d by t h i s i n n e r p r o d u c t i s c a l l e d t h e g r a p h t o p o l o g y of M Now M i s c o n t i n u o u s i n i t s g r a p h t o p o l o g y and s o i t s a d j o i n t ( s r a p h t o p % l o g y a d j o i n t ) , ( M ) *, i s a c o n t i n u o u s l i n e a r o p e r a t o r o n H 2 i n t o Dom M endowed w i t h t8e t o p o l o g y .

.

The f o l l o w i n g was shown i n [LN 1 1 . THEOREM 2.2. Then M(M*),

+

A s s u m e t h a t MCHl (3 H2 i s a c l o s e d l i n e a r o p e r a t o r . I i s a n one-to-one o p e r a t o r on Dom M(M*), ontq H 2 , a n d M,

= ( M * ) ~ ( M ( M+ *1)1 ~ -l.

In particular, L

Dom M, 13

= 112,

Range M,

= (Dom M)n(Null M)

.

NONDENSELY DEFINED DIFFERENTIAL OPERATOR

L e t [ a , b l b e a bounded i n t e r v a l . L e t L b e t h e H i l b e r t s p a c e of a l l r x 1 column v e c t o r - v a l u e d f u n c t i o n s d e z i n e d o n [ a , b l s u c h t h a t

I Let

T

IyI

I:

= ($y*(x)y(x)dx)1/2

<

W.

be t h e f i r s t o r d e r system (TY)

( x ) : = Q1(x)y'

+

QO(x)y(x) x ~ [ a , b I .

H e r e Q l r Qo are r x r m a t r i x - v a l u e d f u n c t i o n s on [ a r b ] s u c h t h a t i s k t i m e s c o n t i n u o u s l y d i f f e r e n t i a b l e o n [ a r b ] and Q (x) i s i n vertible for a l l x i n [a,bl. L e t T1(') and to(^) b e &he maximal minimal c l o s e d o p e r a t o r s i n L2 g e n e r a t e d by T. W e w i l l define a d e n s e l y d e f i n e d o p e r a t o r T which w i l l b e t h e main o b j e c t o f t h i s per. L e t T b e any c l o s e d l i n e a r o p e r a t o r s u c h t h a t TCT1(') and m: = dim ( T 1 ( ' ) / T ) < - . Then w e c a n show t h a t

Q

k

and nonpa-

A Hilbrrt Spuce Operutor Equation

399

Here

(if W1, W2 are r x d m a t r i x - v a l u e d t h a t whenever f o r some

= {O,O}

{W1B,W2Bj

f u n c t i o n s on [ a , b l s u c h

BE$^,

then B=O,

and whenever

{WIB,

then

B=O.

W 2 B } ~ T O * ( f~o) r s o m e B E $

d

,

( i i ) P l ( m x r ) , P 2 ( m x r ) , R(mxd) a r e c o n s t a n t m a t r i c e s s u c h t h a t

rank

[P1,P2, Rl = m. [ L I ] w e c a n show t h a t

Then using Theorem 1 . 7 of T*: = ( { z

(3.2)

BE$

d

+

W B,T

1

+z +

W 2 B } l z ~Dom

to*(^)

and

such that

Here P 1 ( ( 2 r + d - m ) x r ) , P” ( ( 2 r c o n s t a n t matrices s u c h t h a z rL

+

d - m)xr) and 2 ( ( 2 r

+

d - m ) x d ) are

N o w a n y e l e m e n t o f Dom T* i s t h e sum o f t w o f u n c t i o n s , o n e i s smooth and t h e o t h e r n o t . Thus t o s i m p l i f y o u r d e v e l o p m e n t , w e r e s t r i c t o u r s e l v e s t o t h e cases w h e r e a l l e l e m e n t s of D o m T* a r e s m o o t h . H e r e w e assume t h r o u g h o u t t h i s p a p e r t h a t

(3.4)

Let U ( ( 2 r (3.5)

W1

+

E

Dom T 0 * ( . r )

(Entry-wise).

d - m)xd) b e t h e m a t r i x

U:

= $1 W l ( a )

+

$*

Wl(b)

-

z

R.

L e t (U.)@ d e n o t e t h e d x ( 2 r + d - m ) m a t r i x w h i c h i s t h e Moore-Penrose Then ( 3 . 2 ) c a n be r e w r i t t e n as g e n e r a l i z e d i n v e r s e m a t r i x o f (U.) f 0 1l o w s :

.

where

S.J. Lee

400

14

GRAPH TOPOLOGY A D J O I N T O F T

L e t T b e as ( 3 . 1 ) . Under t h e c o n d i t i o n ( 3 . 4 ) , w e w i l l compute t h e graph topology a d j o i n t , T x r o f T. By Theorem 2 . 2 ,

T,

(4.1)

= (T*)s(T(T*)s

+

I)-'.

Thus t o g e t a f u l l d e s c r i p t i o n o f T*, i t i s n e c e s s a r y t o g e t o n e f o r (T*) s . By Theorem 2 . 1 T*Is =

{ {gfy -

P(y) 1

where P i s t h e o r t h o g o n a l p r o j e c t o r Let

s: = dim T * ( O ) .

(4.2)

I n o r d e r t o g e t t h e o r t h o g o n a l p r o j e c t o r P o f L2 o n t o Then s 5 d. T*(O) it i s n e c e s s a r y t o g e t a b a s i s f o r T*(O). To d o t h i s , l e t E ( d x s ) b e t h e m a t r i x s u c h t h a t r a n k E = s and t h e columns o f E i s a L e t Do(dx(2r+d-m) 1, b a s i s f o r N u l l ( U . ) , where U is d e f i n e d i n ( 3 . 5 ) . Dl(dxd) be t h e m a t r i c e s d e f i n e d by

Then @:

= (W,

- T+ (W1))E

i s a b a s i s f o r T*(O) a n d so P i s d e f i n e d a s f o l l o w s : b b P(y) =

(4.5)

@(jQ * ( x ) Q ( x ) d x ) - l j @ * ( x ) y ( x ) d x ,y & L 2 . a

I g , y l ~ T * . Then u s i n g ( 3 . 6 ) w e see t h a t

Take a n y

for all

g E

Dom ( T * ) s .

This together with (4.1) N o t i c e t h a t T, 16

a

g i v e s a s a t i s f a c t o r y d e s c r i p t i o n f o r T,.

i s an i n t e g r o - d i f f e r e n t i a l o p e r a t o r .

NONSTOPPING ITERATION SCHEME

L e t T be a s ( 3 . 1 ) . L e t gEL2 b e g i v e n . l e a s t - s q u a r e s o l u t i o n of T ( z ) = g i f

Then

YE

Dom T i s c a l l e d a

40 1

A Hilbert Space Operator Equation

i

IT(Y)

-

-

g i 151 l T ( z )

gi

I

f o r a l l z E Dom T. Here 1 1 . 1 1 d e n o t e s t h e norm i n H. It is w e l l known t h a t T ( z ) = g h a s a l e a s t - s q u a r e s o l u t i o n i f and o n l y i f g E Range T 4 N u l l T*. S i n c e Range T i s c l o s e d , T ( z ) = g h a s a l w a y s a least-square solution. I n p a r t i c u l a r , i f g E Range T , t h e n any l e a s t - s q u a r e s o l u t i o n of T ( z ) = g i s a t r u e s o l u t i o n of t h a t e q u a t i o p , . I n t h i s s e c t i o n w e w i l l g i v e a n o n s t o p p i n g i t e r a t i o n scheme f o r a l e a s t - s q u a r e s o l u t i o n i n s u c h a way t h a t w e c a n c h o s e a n a r b i t r a r y s t a r t i n g point. T h i s scheme i s r a t h e r t h e o r e t i c a l t h a n p r a c t i c a l . T h i s shceme i s c l o s e l y r e l a t e d t o t h e s t e e p e s t d e s c e n t method. S i n c e T i s c l o s e d , Dom T becomes a H i l b e r t s p a c e , X , w i t h t h e i n n e r product b b t y ( x ) s ( x ) d t+ (TY) tT(g)dx T : =

1

a

a

f o r a l l y , g E T. Thus t h e t o p o l o g y o f X i s t h e g r a p h t o p o l o g y of T . I t c a n b e shown a s i n [ D S 11 t h a t a s e q u e n c e { y n } i n Dom T c o n v e r g e s t o g i n t h e g r a p h t o p o l o g y o f T i f and o n l y i f g E Dom T and Iy 1 c o n v e r g e s t o g (column-wise) u n i f o r m l y on [ a , b l and I y ’ l converGes n t o g ‘ (column-wise) i n L 2’

The f o l l o w i n g i s t h e main theorem. THEOREM 5.1.

L e t g E L2 b e g i v e n . L e t y o , b e an a r b i t r a r y b u t f i x e d s t a r t i n g p o i n t i n Dom T . L e t a c ( 0 , 2 ) be g i v e n . Define a sequence { y 1 i n Dom T r e c u r s i v e l y as f o l l o w s : n

H e r e T, i s c h a r a c t e r i z e d i n ( 4 . 1 ) t o g e t h e r w i t h ( 4 . 6 ) . Then {y 1 c o n v e r g e s t o a l e a s t - s q u a r e s o l u t i o n , u , o f T ( z ) = g i n t h e grap”h t o T h i s s o l u t i o n i s t h e unique l e a s t - s q u a r e s o l u t i o n of p o l o g y of T. minimum L2-norm i f and o n l y i f y 0 ~ ( N u l TI+ l I n particular, i f g is i n t h e r a n g e of T I t h e n u i s a t r u e s o l u t i o n .

PROOF.

Since

(6.1)

Y , + ~ - yn

-

a T,(T

yn

-

g ) , n = 0,

1, 2 ,

---,

w e c a n show t h a t { y } i s a Cauchy s e q u e n c e i n t h e g r a p h t o p o l o g y o f T. S i n c e T and T, h e c o n t i n u o u s i n t h e g r a p h t o p o l o g y of T I i t f o l l o w s from ( 6 . 1 ) T,(T

that u

-

9) = 0 ,

and h e n c e u i s a l e a s t - s q u a r e s o l u t i o n ( a n d a t r u e s o l u t i o n i f L e t us p u t g E Range T . ) fn: = a T,(T

yn

-

9).

Then

Y,+~

-

-

n

yo

-

C i=O

fi,

( n = 0,

1, 2, - - - )

S.J. Lee

402 E t then follows t h a t


= ,all z

E

N u l l T.

Hence < u , z >= < y o I z > , a l l z

E

N u l l T.

Thus yo

E

Range T* i f a n d o n l y i f

-L

u E (Null T) , i . e . , u i s t h e unique l e a s t - s q u a r e s o l u t i o n of T ( z ) = g w i t h minimum L2-norm. REMARK. The a b o v e t h e o r e m i s r a t h e r t h e o r e t i c a l t h a n p f a c t i c a l . T h i s i s d u e t o t h e i n v o l v e m e n t o f t h e c o m p u t a t i o n of A ( f ) , where A:

= T(T*IS

+

I.

o n t o L 2 , which i s p o s i T h i s o p e r a t o r i s o n e - t o - o n e f r o m Dom T ( T * ) t i v e and s e l f - a d j o i n t . Assumifg t h a t d # 8 , i t would b e i n t e r e s t i n g t o know w h e t h e r o r n o t ( i ) A i s an i n t e g r a l operator, (ii) t h e r e e x i s t s a nonstopping i t e r a t i o n f o r t h e s o l u t i o n of A(y) = f .

R e f e r e n ce s

D u n f o r d a n d J. T . S c h w a r t z , L i n e a r o p e r a t o r s , P a r t 11, I n t e r s c i e n c e ( 1 9 6 3 ) , N e w York.

[ D S 11

N.

[Ll]

S. J . L e e , Boundary c o n d i t i o n s f o r l i n e a r m a n i f o l d s I , J. Math. A n a l . Appl. 7 3 ( 1 9 8 0 ) , 366-380.

[LNl]

S. J. L e e a n d M. Z . N a s h e d , G r a d i e n t method f o r n o n d e n s e l y d e f i n e d c l o s e d unbounded o p e r a t o r , P r o c . Amer. Math. SOC. (to appear).

[LN 21

S. J . L e e a n d M. Z . N a s h e d , G e n e r a l i z e d i n v e r s e s f o r l i n e a r m a n i f o l d s and a p p l i c a t i o n s t o boundary v a l u e problems i n Banach s p a c e s , C. R. Math. Rep. A c a d . S c i . C a n a d a , Vol 4 , n o . 6 ( 1 9 8 2 ) , 347-352.

[Lue 1 1 D . L u e n b e r g e r , O p t i m i z a t i o n by V e c t o r S p a c e Method, W i l e y (1969). [ N 11

Z. Nashed, S t e e p e s t d e s c e n t f o r s i n g u l a r l i n e a r o p e r a t o r e q u a t i o n s , SIAM J . Numer. A n a l . , V o l 7 . N o . 3 ( 1 9 7 0 ) , 358-362.

M.

[LOC 11 J . L o c k e r , Weak s t e e p e s t d e s c e n t f o r l i n e a r b o u n d a r y v a l u e p r o b l e m s , I n d i a n a U n i v e r s i t y Math. J . 2 5 ( 1 9 7 6 ) , 525-530.

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

403

SOME VECTOR FIELD EQUATIONS Elliott

H. L i e b *

Departments o f Mathematics and Physics Princeton U n i v e r s i t y P r i n c e t o n , N. J . 08544

Systems o f t h e t y p e i and g ( u ) = aG/aui

i

= g (u),

-aui

with

u

(ul,*--,un)

=

d

IR , d 1. 2 a r e s t u d i e d .

on

G

s u i t a b l e c o n d i t i o n s on

and

g

Under

i t i s shown t h a t t h e r e

i s a s o l u t i o n o f t h e system t h a t a l s o m i n i m i z e s t h e 2 S ( u ) = ]lVul -/G(u), within the class o f a l l

action,

n o n t r i v i a l f i n i t e a c t i o n s o l u t i o n s o f t h e system. I. INTRODUCTION

A problem t h a t a r i s e s i n v a r i o u s branches o f mathematical p h y s i c s i s t h e following:

1

n

. L e t G: R n + lR be c o n t i n u o u s and i n C ( R \ { O } ) w i t h d e r i v a t i v e g, g ' ( u ) = aG(u)/aui, w i t h d~ = ( u ...., u I , and G(0) = 0, g ( 0 ) E 0. The t o bA c o n s i a e r e d i s system o f e q u a t i o n s on IR where each

ui: R d

-t

IR.

i = 1,

= gi [ u ( x ) ) ,

-Aui(x)

We s h a l l assume

d 22

... ,n

and seek a s o l u t i o n

i.e.,

(1.1) u $ 0.

A s s o c i a t e d w i t h t h e system ( 1 . 1 ) i s t h e A c t i o n S(U)

=

K(u)

=

K(u) - V ( U ) l n 2 - 1 /IVU.I dx 1 2 i-1

1

I

]VU]

2

dx

(1.2)

] G ( u ( x ) ) dx.

(1.3)

I t i s obvious t h a t g e n e r a l l y speaking mum ( t a k e , f o r example G(u) = -u2 + u4). f o l l o w i n g c l a s s f o r which S i s f i n i t e :

S(u) has no f i n i t e minimum o r maxiWe can, however, c o n s i d e r t h e

V(u) =

C =

r u l u s L:oc ( R d ) , 11([iul

V u E L2 (lRd),

> a]) <

m

for all

G(u) E L1 (IRd), CL >

(1 -4)

0)

where u denotes Lebesque measure and where [ l u l > a] denotes e i t h e r t h e s e t { x [ \ u ( x ) I > 31 o r t h e c h a r a c t e r i s t i c f u n c t i o n o f t h i s s e t , a c c o r d i n g t o t h e context. (The l a s t c o n d i t i o n i n ( 1 . 4 ) i s a weak n o t i o n o f u ( x ) + 0 as 1x1 -. I t i s v e r y i m p o r t a n t because w i t h o u t i t we c o u l d e a s i l y have s o l u t i o n s w l t h u ( x ) + c o n s t a n t as 1x1 + m . ) Furthermore, we can c o n s i d e r E C C, where +

E = {u

1

u E

c,

u

s a t i s f i e s (1.1) i n

0'1,

(1.5)

E.H. Lieb

404 and

The t o t a l problem i s : Show t h a t t h e r e i s a Smin

=

-

uE E

such t h a t

S(ii)

(1.7)

O f course some c o n d i t i o n s w i l l have t o be imposed on G so t h a t t h e problem has a s o l u t i o n , b u t f i r s t l e t us b r i e f l y r e v i e w e a r l i e r work on t h e problem. W i t h one e x c e p t i o n n o t e d below a l l p r e v i o u s r e s u l t s f o r t h e problem have been f o r t h e s c a l a r case n = 1. Probablydthe e a r l i e s t general t r e a t m e n t o f e x i s t e n c e o f s o l u t i o n s o f (1.1) ( f o r a l l o f IR , as d i s t i n g u i s h e d f r o m bounded domains) was by Strauss [12]. Coleman, G l a s e r and M a r t i n [8] l a t e r made an i m p o r t a n t c o n t r i b u t i o n t o t h e t o t a l problem (n = 1, d 3) by a s c a l i n g argument, o r " c o n s t r a i n e d minimum method," t o be e x p l a i n e d below. B e r e s t y c k i and P. L. L i o n s [ 4 ] improved t h e r e s u l t s i n [8]. Both [12] and [4] c o n t a i n e x t e n s i v e b i b l i o g r a p h i e s o f r e l a t e d work, so t h e r e i s no need t o r e p e a t i t h e r e . I n [3], B e r e s t y c k i and L i o n s announced t h e e x i s t e n c e o f a r a d i a l s o l u t i o n o f (1.1) f o r n 5 1 which minimizes However, t h e r e i s no reason t o t h e a c t i o n among a l l r a d i a l s o l u t i o n s o f (1.1). b e l i e v e t h a t t h e minimum a c t i o n s o l u t i o n we seek i s n e c e s s a r i l y r a d i a l . I n t h e b e g i n n i n g o f 1983, I a n A f f l e c k encouraged me t o t r y t o e x t e n d these r e s u l t s t o d = 2 and t o n > 1 . He e s p e c i a l l y needed t h e d = 2 case [l]. I was a b l e t o make b o t h e x t e n s i o n s p r o v i d e d t h e a d d i t i o n a l assumption l i m sup G ( u ) / l u 1 2 < 0 was made. T h i s was r e p o r t e d a t t h e conference. However, u+o I d i s c o v e r e d t h a t B e r s t y c k i , G a l l o u e t and #avian ( u n p u b l i s h e d ) a l s o s o l v e d t h e d = 2, n = 1 problem more t h a n a y e a r ago by t h e same method as mine. F u r t h e r more P. L. L i o n s t o l d me i n a r e c e n t l e t t e r t h a t he was a b l e t o make t h e e x t e n s i o n t o n > 1 by u s i n g h i s t h e o r y o f c o n c e n t r a t i o n compactness [lo]. The e x t r a assumption mentioned above seemed u n n e c e s s a r i l y r e s t r i c t i v e . I n c o l l a b o r a t i o n w i t h H. B r e z i s we were a b l e n o t o n l y t o e l i m i n a t e i t , b u t a l s o t o make e s s e n t i a l changes i n t h e p r o o f r e n d e r i n g i t s i m p l e r and s t r o n g e r . Therefore, t h i s e n t i r e work i s now d e f i n i t e l y a j o i n t e f f o r t whose d e t a i l s w i l l be g i v e n i n Here, o n l y an o u t l i n e w i l l be given. a f o r t h c o m i n g a r t i c l e [7]. 11.

THE MAIN RESULT

The d i s t i n c t i o n between d = 2 and d 1. 3 i s v e r y i m p o r t a n t . The assumpt i o n s t o be made about G and g a r e t h e f o l l o w i n g : (The common symbol C w i l l be used throughout t h i s paper t o denote i n e s s e n t i a l p o s i t i v e c o n s t a n t s . ) 1 F i r s t , we assume t h a t G E Co(Rn) and G E C (IR\ {Ol) w i t h d e r i v a t i v e i s done t o a l l o w t h e p o s s i b i l i t y t h a t G(u) 21 - 1 ~ 1 , f o r example, g = vG. ( T h i s n e a r u = 0.) We assume t h a t G ( 0 ) = 0 and d e f i n e g ( 0 ) = 0. d 2 3: --(i) (ii)

G(uo) > 0

f o r some

uo

E

IRn .

l i m sup G(u) U I - ~ . 0 as luJ(;,p p = 2* = 2d/ d - 2 ) . (Note

i

and not

IuI

IG

+ m.

u ) 1 .)

Here (2.2) (2.3)

405

Some Vector Field Equations d = 2: (i) (ii)

G(u,)

>

o

f o r some

uo E IR"

(2.4)

l i m sup G ( ~ ) l u l -5~ 0 as / u 1 m. He e p i s any f i x e d p < m. ( 1 u l - P c o u l d be r e p l a c e d by exp[-C/ul 1 f o r some C s u f f i c i e n t l y

F

+

s m a l l , i f necessary.) (iii)

F o r some

(iv)

Ig(u)I 'C

Remark:

>

E

0,

(2.5)

w

+ clulp-l,

/uI

when

G(u) < 0

(2.6)

< E.

u E R".

(2.7)

(2.3) and (2.7) i m p l y t h a t /G(u)I

C + C/ulP

<

,

W u

E

Rn

(2.8)

Theorem 1: Under t h e f o r e p i n g usswrptions, d 2 2, t h e r e e&sts a u E C t h a t s i l t i s f i e s (1.1) i n D’, so t h a t E # 0. Furthemnore, t h i s u s a t i s f i e s a s given i n ( 1 . 6 ) . S ( U ) = Smin. L a t e r on some p r o p e r t i e s o f s o l u t i o n s t o ( 1 . 1 ) w i l l be g i v e n . 111.

SCALING

To b e g i n t h e p r o o f o f Theorem 1 we i n t r o d u c e t h e s c a l i n g i d e a i n [S]. Form a l l y , i f u € E m i n i m i z e s S(u) t h e n u i s a c r i t i c a l p o i n t o f S. Take A > 0 and l e t u,(x) = ~ ( X X ) . Then d S ( u A ) = 0 a t h = 1. But

K(u,)

2-d

=

V(U,)

-d

= A

K(u),

w

V(u),

u E

c.

(3.1)

T h e r e f o r e , we have t h e v i r i a l theorem:

(d If(3.2)

h o l d s then, f o r

d

- ~)K(u)

= dV(u).

2 d - 2

=

3,

S(u)

=

~

V(U)

2 7 K(u)*

I t t u r n s o u t t h a t (3.2) w i l l h o l d n o t o n l y f o r t h e m i n i m i z i n g u s o l u t i o n t o ( 1 . l ) under somewhat m i l d e r r e s t r i c t i o n s than ( 2 . 1 ) - ( 2 . 7 ) . i s Pohozaev's i d e n t i t y [ll].

Lemma 2.

Suppose

G E C1(IRn\\IOl), dL2.

g=VG

u

E

and

(3.2)

L:~~. g(0)

vu 5

E

0.

(3.3) b u t f o r any This

2, ~ ( u E ) L ~ . suppose G E c 0 ( ~ " ) , Let u s c l t i s f y (1.1) in D’ on IRd,

L

Then, under no f u r t h e r a s s u q t i o n s , (3.2) h o l d s. P r o o f : The f o r m a l i d e a i s t o m u l t i p l y (1.1) b y x . V u . sum on i and i n t e g r a t e . I n t e g r a t i n g by p a r t s y i e l d s ( 3 . 2 ) , f o r m a l l y . T h l s can be t u r n e d i n t o a p r o y e r z r o o f [4], assuming G E C1(lRn)). I t i s s l i g h t l y t r i c k i e r i f G i s o n l y See [ 7 l . i n C (W \ { O l ) . To connect Lemma 2 w i t h t h e p r e v i o u s assumptions on g, and t h e r e b y t o est a b l i s h t h e v a l i d i t y o f ( 3 . 2 ) f o r a l l s o l u t i o n s t o ( 1 . 1 ) i n C, t h e f o l l o w i n g i s needed. Only (2.3) and (2.7) w i l l be used.

E.H. Lieb

406

Suppose ( 1 . l ) hoZds i n 9’

Lemma 3.

In f a c t u

hoZd. Then u E Lyoc. for a l l CY < 1 .

Proof: d = 2: wq <

m *

uE

w‘’q loc

vu

-

L2

E

wq <

u

E

Rd

on

W q

LFloc,

<

m

uE C

with

W2 y q f o r aZZ loc ’

E

q <

* g(u)

E

(2.7)

and (2.3),

and hence i n

m

c;

;:

Lyoc,

m.

’ d,3: K a t o ’ s

inequality states t h a t

nlul

(au)-u/lul.

Thus,

2 g ( u ) * u / l u l 5 C + C l u l P - ’ = C + A / u l where A E L y i E s i n c e , by Sobolev, d By a r e s u l t o f B r e z i s and Kato [5] u E Lqoc, W q < m. Then u E L y o c ( R ).

-A/u/

u E; ; ;w

wq

I

<

0

m.

The s i t u a t i o n , then, i s t h a t any u E C s a t i s f y i n g (1.1) a l s o s a t i s f i e s ( 3 . 2 ) . L e t us i n t r o d u c e t h e m i n i m i z a t i o n problem:

e:T

5

i n f I K ( u ) l u E C,

V(u)

d 2 3:

2

i n f I K ( u ) I u E C,

V(u)

By (2.1),

T

1. 0,

u f 01

11.

(3.4)

(2,4) i t i s easy t o see t h a t t h e m i n i m i z a t i o n s e t i s n o t empty.

Our s t r a t e g y w i l l be t h e f o l l o w i n g : (a)

Show t h a t t h e r e e x i s t s a m i n i m i z i n g

(b)

Show t h a t t h i s y > 0:

u

u f 0

f o r (3.4)

s a t i s f i e s (1.1) b u t w i t h a f i n i t e Lagrange m u l t i p l i e r (3.5)

- A U = vCJ(U).

Then

uh(x) = ~ ( h x ) , w i t h

h2 =

y

s a t i s f i e s (1.1).

I t i s then obvious t h a t t h i s uA m i n i m i z e s S ( u ) i n E : I f w E E , l e t V(w) = a. By (3.2), a = 0 when d = 2 and t h u s w i s a d m i s s i b l e i n ( 3 . 4 ) . (Note t h a t Since S(w) = K(w) then S(w) 2 S(u),) = T, by d e f i n i t i o n . K(uh) = k ( u ) when d = 2.) I f d 5 3 we n o t e t h a t

TCY i n f I K ( u ) s a t i s f i e s (by s c a l i n g

u

-t

ut

I

with

u

E

C,

t2-da =

V(u)

CY

01

(3.6)

1)

Ta = a1 - 2 / d T,

(3.7)

and t h a t minima f o r (3.6) and ( 3 . 4 ) c o i n c i d e under t h e s t a t e d s c a l i n g . then by (3.3) = B, V(u,) 2 B = S ( U ~ )= 2 K(uA) = 2 B 1-2/d T. d-2 d d If

w E E

and

V(w) =

CY >

0

2 a d-2 Comparing (3.8),

If

(3.8)

( b y ( 3 . 2 ) ) then = S(W) =

(3.9) we see t h a t

CY

a2 K ( w ) -> d2

- CY

l - Z / d T.

(3.9)

1. B which i m p l i e s t h a t S ( w ) 2 S ( u A ) .

The i d e a o f i n t r o d u c i n g t h e m i n i m i z a t i o n problem (3.4) i n c o n j u n c t i o n w i t h (3.2) i s i n [S] f o r d 2 3. The new i d e a f o r d = 2 ( w h i c h B e r e s t y c k i , G a l l o u e t and Kavian a l s o had e a r l i e r ) i s t o i n t r o d u c e t h e c o n s t r a i n t u f 0. W i t h o u t it,

407

Some l'ector Field Equations

u = 0 would m i n i m i z e i n ( 3 . 4 ) . With i t , u i s f o r c e d t o s a t i s f y on a s e t o f p o s i t i v e measure, by v i r t u e o f (2.6).

IuI

>

E

The second new i d e a i s t h e method f o r h a n d l i n g t h e v e c t o r case n 1. A l l p r e v i o u s work was r e s t r i c t e d t o r a d i a l f u n c t i o n s , e i t h e r by f i a t o r by n o t i n g t h a t u* ( t h e symmetric d e c r e a s i n g rearrangement o f when n = 1 t h e replacement u u) would l e a v e V(u) i n v a r i a n t w h i l e K(u*) 5 K ( u ) . Thus, i n ( 3 . 4 ) o n l y r a d i a l f u n c t i o n s had t o be considered. For t h i s c l a s s t h e r e i s t h e n i c e compactness I n t h e v e c t o r case, t h e r e i s no reason t o suplemma o f S t r a u s s a v a i l a b l e [12]. pose t h a t t h e m i n i m i z i n g u w i l l be r a d i a l ( u n l e s s G(u) depends o n l y on l u l ) . A new compactness lemma i s needed and i s g i v e n i n t h e n e x t s e c t i o n . -f

IV.

EXISTENCE

OF A MINIMUM FOR (3.4)

Suppose t h a t I u J } i s a m i n i m i z i n g sequence f o r ( 3 . 4 ) . When d = 2 t h e U ( A X ) , A > 0 does n o t a f f e c t K ( u ) . Since u E C we a r e replacement u ( x ) f r e e t o choose (and do so h e n c e f o r t h ) -f

where

i s d e f i n e d i n (2.6).

E

For

d 3, t h e r e e r k t ziriiversaZ constants in a minimizing sequence

Lemma 4. f o r every UJ

FCW

d[luJI

Proof.

K ( u J ) bounded

=)

by a t r u n c a t i o n argument, f o r E

> 0

such t h a t

A = CxlluJ(x)I 1 (IG(uJ)

2 3 we have

d

2

for

L y / l U j I p + /G(uJ) A D

(4.2)

( T h i s i s S o b o l e v ' s i n e q u a l i t y and h o l d s , Let

IuI

y = 1/2C.

By ( 2 . 2 ) , t h e r e e x i s t s

I u / 2 1 / ~ . Let D = [E < I u J l 1 / ~ ] . Then

5 E or

luJ(x)I 5 l / ~ and l

or

such t h a t

€1) 1. 6.

>

I I U J I I ~< C. .P UJ E C.)

G(u) 5 y ! u I p

6 > 0

E,

>++~

( [ ~ U ~ ~ E ] ) S U P ~ ( G ( U ) ~ ~ E ~0 ~ U ~ < ~ / E ~

Given ( 4 . 1 ) o r ( 4 . 2 ) , t h e f o l l o w i n g lemma i s r e l e v a n t .

J

Lemma 5. .Let I U I 1 d > 1 w i t h uJ E Lloc, thzt

u([lujl

>

bs a sdquence of R" v a ~ ud f u n c t i o n s on R v d uniformly bounded i n L' and It vuJll 5 C .

€1) >

6

f o r some

6 ,

Then t h e r e e x i s t s a sequence of t r i i n s l u t i o n s , f i x e d constant CI(S,E,C) > 0 such t h a t u({X E

where

B =

{XI

Proof.

6 >

o

uJ(x)

and +

C~ZZ

5

wJ(x)

B / I W j ( X ) l > ~ / 2 } )> a ( 6 , ~ , c ) ,

and

u

iz

(4.3)

See [9].

E s t i m a t e s on

-f

n 2 I, Suppose

1x1 5 1 1 i s t h e u n i t baZZ.

f i x e d bounded s e t uj

,

j

uJ(x + y . ) J

0

T r a n s l a t i n g uJ as i n Lemma 5 does n o t a f f e c t K ( u j ) s h a l l h e n c e f o r t h assume t h a t ( 4 . 3 ) h o l d s w i t h WJ = uJ.

that

d

uJ:

n,

pointwise

d 3: II uJII ,I (n)

a.e.,

By t h e Sobolev i n e q u a l i t y

C. (Rd)

V(uJ).

or II

uJtI

P

<

C.

Thus, we On any

By compactness, t h e r e i s a subsequence such

and, by Lemmas 4, 5,

u # 0.

By F a t o u ' s

E.H. Lieb

408

W r i t i q g G = G+ - G- we have (as i n t h e p r o o f ~ v l l u J I I P+ / G+(uJ). But u ( [ I u j i > € 1 ) < C s i n c e .p D Therefore G+(uJ) < C. On t h e o t h e r hand, / G - ( u j ) 5 /G+(uj) - 1,

lemna]lG(u)l o f Lemma 4)

I l i m i n f j IG(uJ)l.

i

Gt(uj)

i

< C. P and hence J I G ( u J ) I < C. satisfies

IIujII

Finally

u E Lp

u f 0

(by Fatou).

and

u

E

Thus, t h e l i m i t

u

c.

(4.4)

d = 2 : T h i s i s a b i t t r i c k i e r and we o n l y s k e t c h t h e p r o o f o f ( 4 . 4 ) . By t h e G-rdo-Nirenberg i n e q u a l i t y (which can be extended t o w E c) llwll < C / / V w / . .Apply t h i s t o . wJ = ( I u J I - E)+, w i t h t h e same E as i n (4.1). Not; T h a t l j V w J l < i l V u J l [ l u J I 2 €1. Using t h e Sch a r z i n e q u a l i t y and ( 4 . l ) , 1lwJ11 < C. I n f a c t we can a p p l y i t t o (wJ)’, (wJ)!, e t c . , and i n t e r p o l a t e , t o concfuxe t h a t 11 wJll < C f o r a l l q < m. As i n t h e case d 1. 3, t h e r e i s a q q . uJ u f 0 p o i n t w i s e a.e. By Fatou, ~ ( [ I u l > E l ) 2 1. subsequence such t h a t +

Now

IG+(uj) =,IG+(uj) [ l u j l

>

€1 5 C

/lujlp[lujl

>

€1

I C ,/[lwJ/ + ~ ] ~ [ l u> ~€1 I 5 C ’ . On t h e o t h e r hand /G-(uJ) 5 /Gt(uJ). / I G ( u J ) I < C and, by Fatou, G(u) E L1. Since G(u) < 0 f o r I u l < E G(u) E L1, u E C. Thus (4.4) h o l d s f o r d = 2 as w e l l .

Thus, and

The f a c t t h a t u m i n i m i z e s : I f we knew t h a t V(u) L 1 ( d 2 3) o r V(u) > 0 d =.2 we s h o u l d be f i n i s h e d because, by l o w e r s e m i c o n t i n u i t y , l i m ( K ( u J ) ) = T. Our t a s k w i l l be t o overcome t h i s l a c k o f i n f o r m a t i o n . K(u)

5

Lemma 6. $ E

Lp(B)

d

Let

and h j

E

and p as i n (2.2), ( 2 . 5 ) . Let B be any baZZ. Lp(B) and II hJll C and h j h pointwise. Then 2

G($ + h j )

-

G(hJ)

+

B

Proof.

Since

where

.

\q(u)l 5 C + C l u l P - ’

ME(a) = \ a l p and

C o r o l l a r y 7. Let u f 0 pointwise. V @ E L2. Suppose t h a t +

/

G($ + h )

-

G(h)

(4.5)

B

5 ClbI[l

C l e a r l y N [ $ ) E L1(B) lemma o r ag i n [ 6 ] . uJ

Let

+

/ IG(a + b ) - G ( a ) I

‘

we have t h a t

+ / a l p - ’ + Iblp-’]

2 EME(a) + NE(b) f o r any

NE(b) = C l b l + C E l b l P w i t h and 1I ME(hJ)ll

<

C.

CE

+

-

as

E

+

E

>

0,

0.

The r e s t f o l l o w s f r o m e i t h e r Egorov’s 0

I u J l be t h e aforementioned minimizing sequence w i t h Let $ E Lp be any function of compact support w i t h

Then

P r o o f . L e t supp $ C B :a b a l l . Consider t h e sequence wJ = uJ + $I i n s t e a d n y Lemma 6 ( w i t h hJ = u J ) V(wJ) - V(uJ) + 5. On t h e o t h e r hand, K ( w j ) - K(uj) = jvuJ.o$ + 7 1 / l v @ I 2 + /VU.V$ + 1 / v $ / =~ K(u + $ ) - K(u). (Here

of

u

we assume, w i t h o u t l o s s o f g n e r a l i t y and by p a s s i n g t o a subsequence if necessary, t h a t vuJ + v u weakly i n L ( B ) . ) By (4.6) we see t h a t WJ i s a d m i s s i b l e i n ( 3 . 4 ) f o r d = 2 o r i n (3.6) f o r d 1. 3 w i t h 01 = 1 + 6 ( a t l e a s t f o r j l a r g e

5

409

Some Vector Field Equations

l i m i n f K(wJ)

enough). Since K(uJ) + T and ( f o r d 2 3), (4.7) f o l l o w s .

2T

(for

d = 2)

or

Ta 0

Now t a k e $ = tji w i t h t EIR and $ E Cm and supp $ C B. With Wt = G(u + $) - G(u) - $ . g ( u ) , i t f o l l o w s f r o g dominated convergence t h a t

IW t I: U I

> S] = o ( t ) .

B

On t h e o t h e r hand, Ibltl[u

=

1 lG(tji

01 =

B

B by (2.31,

(2.7).

Iru

01 5 C l t l

=

II * l [ u

=

B

01 +

o(t)

Thus, IV(U+t$)

-

V(u)

I lil

-

t /vg(u)l 5 C l t l

-

K(u) = t J V u * V j i + o ( t ) .

+ o(t).

(4.8)

u=o

Furthermore, K(U+t$)

(4.9)

L e t us f i r s t suppose t h a t ~ ( [ u= 01) = 0, so t h a t we may i g n o r e t h e n e x t t o l a s t term i n (4.8). We have two bounded l i n e a r f u n c t i o n a l s on CE, namely L 1 ( j i ) = I j i * g ( u ) and L 2 ( j i ) = /Vu.V$. When d 2 3 we have, by C o r o l l a r y 7, t h a t f o r small t tL2(ji)

o ( t ) 2 TC(1

+

+

tLl(Ji))

1-2/d

-

11.

(4.10)

This implies t h a t L2($) = (1 When

d = 2

-

(4.11)

2/d)T L l ( j i ) .

we have t h a t >

This i m p l i e s t h a t e i t h e r

L1 E 0

0 * L 2 ( j i ) 2 0.

(4.12)

or

Y L1(*) = L 2 b )

(4.13)

w i t h 0 5 y < -. B u t y > 0, f o r o t h e r w i s e -nu = 0 which i s i m p o s s i b l e w i t h u E C and u $ 0. I f L = 0 we s h o u l d have t h a t g ( u 0. However, 2 OG(u) = V u - g ( u ) i n D' I t h i s makes sense s i n c e v u E LJ 'an? g ( u ) E Lloc). Since Thus G(u) = C = ( c o n s t a n t ) a.e., b u t C = 0 s i n c e G(u) E L G(u) < 0 f o r I u ! < E , we (dx) e i t h e r u(x) = 0 o r + 0 a t m i n t h e weak sense o f l u ( x ) i > E. We know t h a t C. It 7 s n o t hard t o prove f u n c t i o n s have a " ean v a l u e " r o p e r t y : = 01) > O and ~ ( I x T I u ( x ) I 1. > o . Thus P ( { x ~ o 0.

.

EIP

+

For a l l

d 5 2

we have t h a t -AU =

with y = (I - 2/d)T 2 and 3 t o ( 4 . 1 4 ) :

for

d

vg(u)

2 3 and 0 (d - 2)

When d = 2, V(u) = 0. When d 2 3, (3.7). T h i s i m p l i e s t h a t V(u) 2 1.

K(u)

in

<

y <

(4.14)

2)' m

for

d = 2.

= yd V(U).

K(u) = TV(u),

Now a p p l y Lemmas (4.15)

but

K ( u ) 1 TV(u) 1 - 2 / d

by

E.H. Lieb

410

Under t h e assumption t h a t p ( [ u = 01) = 0, t h i s concludes t h e p r o o f t h a t t h e r e i s a m i n i m i z i n g u f o r (3.4). I t a l s o concludes t h e program g i v e n b e f o r e (3.5) and up t o ( 3 . 8 ) . When ~ ( [ u= 01) > 0 some a d d i t i o n a l t e c h n i c a l i t i e s a r e r e q u i r e d . 3. The analogue o f (4.10), (4.11) i s consider d tLp($)

o ( t ) ,T"1

f

i I d J l ) 1-2/d -

tL1($)

-

ltlC

i I+l[u

=

01 5 b

f

First,

11

(4.16)

u=o

Therefore, I L 3 ( @ )I 5 b b = CT(l - 2/d)

and

-

T(l t m d s t o a bounded l i n e a E o p e r a t o r on L 1 L3($) = j h $ w i t h h E L Thus

with

L 3 = L2

.

-Au

-

-

T(l

-

i

2/d)L1.

.

Since

(4.17) (4.17) implies t h a t

Lm i s t h e dual o f

L3 ex1 L ,

2 / d ) g ( u ) = h E Lm.

(4.18) 1

.

BY (4.17) l j h $ l I I$l[u = 01 f o r 1 . E ,C; and hence f r $ E L Thus h ( x ) = 0 when u ( x # 0. B u t (4.18) i m p l i e s t h a t u E C p y a , which i m p l i e s t h a t

-.

u E WzYp, Vp < Then A u = 0 a.e on [u = 01. Since g ( 0 ) = 0, h = 0 a l s o when u = 0. (4.18) i s i d e n t i c a l t o 4.14) and t h e c o n c l u s i o n f o l l o w s as b e f o r e .

( = 2 a l l we know i s t h a t i $ l [ u = 01. The p r o o f t h

When >

-C

L 5

($1 > 0 i m p l i e s t h a t l i m e i s a b i t l o n g e r , b u t we a g a i n r e c o v e r

See [i] f o r details. V.

FURTHER PROPERTIES

OF SOLUTIONS TO (1.1)

The goal s t a t e d i n t h e i n t r o d u c t i o n has been reached-to f i n d a f u n c t i o n i n i n t h e c l a s s o f a l l such f u n c t i o n s . I n t h e process o f d y i n g so i t was shown t h t a l l s o l u t i o n s t o ( 1 . 1 ) i n C s a t i s f y (3.2) and a r e i n W1;2, V q < m and i n C,Y:; WLX < 1 .

C t h a t s a t i s f i e s ( 1 . 1 ) and t h a t m i n i m i z e s S(u)

Here we s h a l l prove t h a t s o l u t i o n s t o ( 1 . 1 ) i n C a c t u a l l y go t o z e r o as i n t n e usual sense and, i n c e r t a i n cases, have compact support. m

1x1

+

-_Lemma

8:

Y

Let

LL

(weak Lr) for some 1 < r < m , A E L r ' with ~ ( S U P A) P < m and f E Lt with t > p. Assume t h a t A, Y, f 2 0. Suppose t h a t f - 1 5 Y * ( A f ) . Then f E Lq(B) for a22 1 5 q < m and f o r a22 B of f i n i t e measure. !Note: t h i s i s stroriger than L?OC* ) See [7] f o r t h e p r o o f . (2.7).

Theorem 9: Let Then l u ( x ) I Let

-Af

0

as

f

f = luIL. f = -2u.A~

Then

- 2 1 0 ~ 1+ ~f 5 2 U - g ( u )

0,

where

S,

since

u E).:;:W

E.

-

+

A f E L:oc

f o r any

Y

s a t i s f y (1.1) in D’ w i t h t h e assumptions (2.1) 1x1 m i n the usuaZ sense.

u E C +

n

Proof:

that

E

>

f

f 5

E

+ C,fP"

i s t h e c h a r a c t e r i s t i c f u n c t i o n of Since

i s t h e Yukawa, o r Bessel, p o t e n t i a l . f

-

u

E

Since

CE 5 Y*A,f

C, f

2 Y*(E

(5.1)

S,

[lul

>

€1.

(Note

+ C E f p / 2 S E ) , where

Y E L1 ,

(5.2)

41 1

Some Vector Field Equations with

AE = CEfPi2-' Y E Lr

u E Lp.

W'

SE. I f d r = p/2, r ' = d/2, since 3, A E L r ' , as i s w e l l known. By Lemma 8, w i t h t = p, f E Lq(B! f o r a l l

B o f f i n i t e measure and 1 5 Y E Lt f o r some 1 < t < r. f < CE + h, and h = Y*AEf hoids f o r a l l E > so t h e

q < m. Therefore, A f E Ls, Ws 1. 1. However, I n $ e r t i n g t h i s i n ( 5 . 2 ) , w i t h s = t ' , we see t h a t i s a f u n c t i o n t h a t goes t o z e r o as 1x1 + m . T h i s theorem i s proved. I f d = 2 we have, by t h e 2 G a g l i a r d o - N i r e n b e r g i n e q u a l i t y and t h e f a c t t h a t v u E L , t h a t f E Lq(B) f o r a l l 1 5 q < m and a l l B o f f i n i t e measure. The remainder o f t h e p r o o f i s t h e same as f o r d 1. 3. 0

6,

F i n a l l y , we t u r n t o t h e q u e s t i o n o f compact s u p p o r t f o r s o l u t i o n s t o (1.1). L e t us assume t h a t g(u).u f o r 1u1 < 6, where 0 < t < 2 7 f o r 1u1

2 -Clulr,

C , 6 > 0. 6.)

with

1

<

r

<

( T h i s i n c l u d e s t h e case

2,

(5.3)

G(u) = - C l u l t ,

2

Theorem 10. L e t and ( 5 . 3 ) hold. Then

u E C s a t i s f y (1.1) in u has compact support.

D’ and assume t h a t (2.1)-(2.7)

P r o o f : By Theorem 9, t h e r e i s an R such t h a t l u ( x ) I 1x1 L r W i t h f = I u l 2 we have, as i n (5.11, f o r 1x1 '> R -Af

5 6 for a l l

5 -2C frj2

(5.4)

With r / 2 < 1 , i t i s easy t o v e r i f y t h a t ( 5 . 4 ) has a r a d i a l s o l u t i o n F ( t h a t i s , F ( x ) depends o n l y on 1x1) s a t i s f y i n g F ( R ) 2 6 and F has compact support. By a comparison argument (maximum modulus p r i n c i p l e ) f ( x ) 5 F ( x ) 1x1 R. The c o n s t r u c t i o n o f F i s e x p l i c i t l y g i v e n i n [2]. 0 REFERENCES A f f l e c k , I.,Two dimensional d i s o r d e r i n t h e presence o f a u n i f o r m magnetic f i e l d , J . Phys. C. ( s u b m i t t e d ) . B e n i l a n , Ph., B r e z i s , H. and C r a n d a l l , M., A s e m i l i n e a r e q u a t i o n i n Ann. Scuola Norm. Pisa, 2 (1975) 523-555.

L 1 (WN ) ,

B e r e s t y c k i H. and L i o n s , P.-L., E x i s t e n c e o f s t a t i o n a r y s t a t e s o f n o n - l i n e a r s c a l a r f i e l d e q u a t i o n s , i n : Bardos, C. and Bessis, D. ( e d s . ) , B i f u r c a t i o n Phenomena i n Mathematical Physics and R e l a t e d Topics, Proc. NATO A S I , Cargese, 1979, ( R e i d e l , 1980). B e r e s t y c k i , H. and L i o n s , P.-L., N o n l i n e a r s c a l a r f i e l d e q u a t i o n s : I - E x i s t e n c e of a Ground S t a t e , Arch. Rat. Mech. Anal. 84(1983) 313-345. See a l s o I 1 - E x i s t e n c e o f i n f i n i t e l y many s o l u t i o n s , E ( m 8 3 ) 347-375. See a l s o An 0. 0. E. a proach t o t h e e x i s t e n c e o f p o s i t i v e s o l u t i o n s f o r s e m i l i n e a r problems i n R ( w i t h L. A. P e l e t i e r ) , I n d i a n a Univ. Math. J . g ( 1 9 8 1 ) 141-157. See a l s o Une methode l o c a l e pour l ' e x i s t e n e de s o l u t i o n s p o s i t i v e s de ’ R 3. Anal. Math B ( 7 9 8 0 ) 144-187. problemes s e m i l i n e a i r e e l l i p t i q u e s dans ,

Fi

B r e z i s , H . and Kato, T., Remarks on t h e Schroedinger o p e r a t o r w i t h s i n g u l a r complex p o t e n t i a l s , J . Math. Pures e t A p p l . 3 (1979) 137-151. B r e z i s , H. and L i e b , E. H . , A r e l a t i o n between p o i n t w i s e convergence o f funct i o n s and convergence o f f u n c t i o n a l s , Proc. Amer. Math. Soc. 88(1983)486-490.

412

E.H. Lieb

[7]

Brezis, H. and Lieb, E. H., Minimum a c t i o n s o l u t i o n s t o some v e c t o r f i e l d equations ( i n p r e p a r a t i o n ) .

[8]

Coleman, S., Glaser, V. and Martin, A., A c t i o n minima among s o l u t i o n s t o a c l a s s o f Euclidean s c a l a r f i e l d equations, Commun. Math. Phys. 58 (1978) 21 1-221.

[9]

Lieb, E. H., On t h e lowest eigenvalue o f the Laplacian f o r t h e i n t e r s e c t i o n o f two domains, I n v e n t . Math. (submitted).

[lo]

Lions, P.-L., P r i n c i p e de concentration-compacit6 en c a l c u l des v a r i a t i o n s , C. R. Acad. S c i . P a r i s 294 (1982) 261-264.

[ l l ] Pohozaev, S. I . , Eigenfunctions o f t h e equation Dokl.

6

Au + h f ( u ) = 0,

Sov. Math.

(1965) 1408-1411.

Existence o f s o l i t a r y waves i n h i g h e r dimensions, Commun. [ 1 2 ] Strauss, W . A., Math. Phys. 55 (1977) 149-162.

*

The author was p a r t i a l l y supported by U. S. N a t i o n a l Science Foundation Grant PHY-8116701 A01.

DIFFERENTIAL EQUATIONS

I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

413

WEIGHT DISTRIBUTIONS AND MOMENTS FOR A CERTAIN CLASS OF ORTHOGONAL POLYNOMIALS

Lance L . L i t t l e j o h n Department of Mathematics, Computer S c i e n c e and Systems Design The U n i v e r s i t y o f Texas a t San Antonio San Antonio, Texas 78285

m

Suppose { ~ m ( x ) l m = i, s an o r t h o g o n a l polynomial sequence o f s o l u t i o n s t o t h e r e a l d i f f e r e n t i a l equation

We s h a l l show how t o c o n s t r u c t a n o r t h o g o n a l i z i n g weight d i s t r i b u t i o n f o r

{t$

m

(x)).

INTRODUCTION I n t h i s p a p e r , w e s h a l l c o n s i d e r t h e c l a s s i f i c a t i o n of a l l real w e i g h t d i s t r i b u t i o n s t h a t have an a s s o c i a t e d o r t h o g o n a l polynomial sequence (OPS) o f s o l u t i o n s t o a d i f f e r e n t i a l e q u a t i o n o f t h e form:

There a r e e i g h t known OPS’s t h a t s a t i s f y d i f f e r e n t i a l e q u a t i o n s of t h e form (1). They a r e l i s t e d i n t h e f o l l o w i n g t a b l e : Order of D . E .

OP s

2

Jacobi

2

Laguerre

2

Hermit e

2

Bessel

4

Legendre t y p e

4

Laguerre t y p e

4

Jacobi type

6

Krall

Real Weight D i s t r i b u t i o n (l-x)a(l+x) B

a -x

x e

-X2

?

a 2

-

+ -12

6(x+1) 1 -~(x) R 1 6(x)

1A. 6(x+1) +

1

+ -12

6(x-1)

+ e-x + (1 - x ) a 6(x-1)

+c

[-1,11

-212 The Bessel polynomials are o r t h o g o n a l w i t h r e s p e c t t o e and c o n t o u r i n t e g r a t i o n around t h e u n i t c i r c l e i n t h e complo-x p l a n e . However, t h e r e i s a r e a l weighr d i s t r i b u t i o n f o r t h e s e p o l y n o m i a l s , as g u a r a n t e e d by a theorem of R. P . Boas [l]:

L. L. Littlejohn

414 Theorem 1

L e t {un3z=o b e any sequence o f real numbers. of bounded v a r i a t i o n on (- -,-)

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

such t h a t : m

vn

x n d u (x)

=

.

-m

We s h a l l l o o k a t t h i s c l a s s i f i c a t i o n problem from a new p o i n t of view, and i n t h e p r o c e s s , o b t a i n some i n t e r e s t i n g new r e s u l t s . A s an a p p l i c a t i o n , we s h a l l f i n d a new OPS t h a t s a t i s f i e s a s i x t h o r d e r d i f f e r e n t i a l e q u a t i o n . M A I N THEORY

...,

Assume { $ ( x ) } i s an OPS and @ ( x ) s a t i s f i e s (l), m = 0 , 1, f o r some m m i n t e g e r n 2 1. Also, suppose t h e r e e x i s t s a f u n c t i o n f(x) such t h a t f ( x ) L (y) 2n is f o r m a l l y s e l f a d j o i n t . The f o l l o w i n g r e s u l t o f H. L. K r a l l [ 3 ] g i v e s us a formula f o r t h e most g e n e r a l f o r m a l l y self a d j o i n t even o r d e r d i f f e r e n t i a l expression :

Theorem 2 The most g e n e r a l f o r m a l l y s e l f a d j o i n t d i f f e r e n t i a l e x p r e s s i o n o f o r d e r 2n w i t h r e a l c o e f f i c i e n t s is g i v e n by:

y ps

n

k=O

b2k(x) y ( 2 k ) ( x )

where B2i

+ 1

2k-1

k = l s=k

( 2 ~ - 2 k + l ) (2k-1) 122s-2k+2 -1 s-k+l B2s-2k+2 b2 s (x) y ( 4

i s t h e B e r n o u l l i number d e f i n e d by:

x= 1 - x + ex -1

B

m

*, xLL 2i

i=1 i

For t h e moment, we s h a l l w r i t e bi(x)

=

1

j = O

L

ij

x j i n (1). The f o l l o w i n g

theorem e a s i l y f o l l o w s from Theorem 2 . Theorem 3 f(x)LZn(y) i s f o r m a l l y s e l f a d j o i n t i f and o n l y i f f ( x ) s a t i s f i e s t h e n homogeneous d i f f e r e n t i a l e q u a t i o n s : n

2s-2k+l

[gE-l]

(2) s=k

-

j=O

b2k-1(x)f(x)

[

1

2~-2k+l 2 j

(2s-2k+l-j) ( j ) 2 s -2 k+2 -1 s-k+l B2s-2k+2 b 2s (XI f (x)

= 0 , k = 1, 2 ,

..., n .

One w i l l n o t i c e t h a t t h e s e e q u a t i o n s are of o r d e r s 1, 3 , 5, p a r t i c u l a r , f ( x ) must s a t i s f y nb

2n

(x)f'(x)

+

Separation of variables y i e l d s

(nb' (x) - b2n-1(x))f(x) 2n

= 0

..., (211-1).

In

415

Orthogonal Polynomials

(3)

However, w e s h a l l b e i n t e r e s t e d i n t h e g e n e r a l d i s t r i b u t i o n a l s o l u t i o n t o t h e e q u a t i o n s i n ( 2 ) . For example, t h e g e n e r a l d i s t r i b u t i o n a l s o l u t i o n t o x 2 f ' ( x ) = 0 is f ( x ) = c1

+ c26(x).

K r a l l and Morton [ 2 ] have s t u d i e d what seems

t o be t h e a p p r o p r i a t e s e t t i n g f o r t h i s d i s t r i b u t i o n a l s o l u t i o n . The s p a c e P of test f u n c t i o n s c o n s i s t s o f a l l i n f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s $ ( x ) , - m < x < m, such t h a t f o r a l l a > 0 , q 4 0 ,

N o t i c e t h a t P c o n t a i n s polynomials. The r e a d e r s h o u l d c o n s u l t [ Z ] f o r t h e a p p r o p r i a t e topology on P. L e t A b e t h e g e n e r a l d i s t r i b u t i o n a l s o l u t i o n t o t h e e q u a t i o n s i n (2) and assume A E P ' , t h e d u a l of P. In view o f t h i s , w e now t h i n k of (2) i n t h e d i s t r i b u t i o n a l sense:

= 0 , k = l , 2,

...,n .

We now s e t o u t t o show t h a t A a c t s as a n o r t h o g o n a l i z i n g weight d i s t r i b u t i o n

It i s wellknown t h a t i f L i s a d i f f e r e n t i a l e x p r e s s i o n and A i s a d i s t r i b u t i o n , t h e n ( L ( A ) , @ ) = ( A , L+(@)) where L+ i s t h e Lagrange a d j o i n t o f L. In o u r case, some r a t h e r t e d i o u s computations g i v e s us the f o l l o w i n g theorem. Theorem 4

'

The d i s t r i b u t i o n A s a t i s f i e s : (5)

(

s=k

2s [2k-l]

k = l , 2, I f we denote

<

22~-2k+2-1 s-k+l B 2s-2k+2 b 2 s (x),(2sT$f+1)+b

..., n ,

A , xm

=

Tk(m) = s=k

J.2= f

and l e t $ ( x ) = x",

j= 0

0 [2&)P(m-2k+l,

'2k-1, j 'm-2k+l+j

where P ( n , k ) = n(n-1)

= 0

e q u a t i o n (5) y i e l d s :

2s-2k+l)

s-k+l

-1

9.

B2s-2k+2 2s,jPm-2s+j

2k-1 +

)

+ E P .

um

2 s -2 k+2 (6)

2k-l(x)$(x)

= 0 , m 2 2 k - 1 , k = 1, 2 ,

..., n .

... ( n - k f l ) .

A t t h i s p o i n t , we r e c a l l H . L. K r a l l ' s 1938 c l a s s i f i c a t i o n theorem [41.

416

L. L. Littlejohn

Theorem 5 I n o r d e r t h a t t h e r e e x i s t a n OPS I @ m ( x )

s a t i s f y i n g t h e d i f f e r e n t i a l equation

(1) i t i s n e c e s s a r y and s u f f i c i e n t t h a t m

t h e moments {um?m=Os a t i s f y t h e n r e c u r r e n c e r e l a t i o n s

i)

s (m)

(7)

m ii)

A

=

1

> 2k+l,

m

$

i - 2 k - l ) L . i,i-u . u m-u = 0 ,

2n [i-~-1)F'(m-2k-l, i=2k+l u=O

k = 0 , 1,

. .. , ( n - l ) , and

*

=

0 , m = 0,1,

...

Vm Pm+l""Uzm

what i s t h e r e l a t i o n s h i p between T (m) and S (m)? k k and c a l c u l a t i o n s r e v e a l , f o r example, t h a t T (m) = Sn-l(m) It i s n a t u r a l t o a s k :

Tn-l(m)

= Sn-2(m)

where

A1 = 1 and

-

n(n-1) (m-2114-3) (m-Zn+2) 2

Sn-l(m).

Easy

In general,

P(m-2nf2k-2i-3,

2j-2i+2)

=

0 , j = 1, 2 ,...,k.

i= 1

Thus, i t i s c l e a r t h a t Tk (m) = 0 i f and o n l y i f Sk (m) = 0 . Assuming t h a t ( 7 ) has a unique s o l u t i o n i n t h e s e n s e t h a t once p o i s known t h e o t h e r urn's are u n i q u e 1 determined, w e can conclude t h a t A a c t s as an o r t h o g o n a l i z i n g weight f o r {$mcX)

7.

By s e t t i n g @ ( x ) = 1 i n ( 5 ) , w e have t h e f o l l o w i n g r a t h e r s u r p r i s i n g c o r o l l a r y . Corollary 6

( A , b2k-1(x))

= 0 , k = 1, 2 ,

..., n.

EXAMPLES i)

The Hermite d i f f e r e n t i a l e q u a t i o n :

-

y"

2xy'

+

2ny = 0 .

In t h i s case, A s a t i s f i e s t h e d i f f e r e n t i a l e q u a t i o n A ' 2

+

2xA = 0 .

.

This has

The r e c u r r e n c e r e l a t i o n t h a t w e o n l y the classical s o l u t i o n A(x) = e-x g e t from ( 6 ) is - 2 ~ ~ + ~ + m u , - ~ =0 . T h i s i s e a s i l y s o l v e d t o y i e l d U ~ ~ + ~ = O

ii)

The Bessel d i f f e r e n t i a l e q u a t i o n : Here,

A satisfies

x2y"

+

2 ( x + l ) y ' - n ( n + l ) y = 0.

417

Orthogonal Polynomials T h i s e q u a t i o n h a s n o t y e t been solved f o r its g e n e r a l d i s t r i b u t i o n a l s o l u t i o n . I t i s n o t even c l e a r what t h e a p p r o p r i a t e s e t t i n g f o r A s h o u l d be. The r e c u r r e n c e r e l a t i o n t h a t w e g e t from (6) i s (m+2)u + 2um = 0. Again, t h i s m+l

um

i s e a s i l y solved t o y i e l d

=

Morton [2] showed t h a t w(x) =

(-1)m2mu'2 (m+l) !

1

.

It s h o u l d be n o t e d t h a t K r a l l and

2m+16(m)(x) ml(m+l)l f o r m a l l y a c t s as a weight d i s t r i -

m= 0 b u t i o n f o r t h e Bessel polynomials. (8).

One can check t h a t w(x) f o r m a l l y s a t i s f i e s

i i i ) The Legendre t y p e d i f f e r e n t i a l e q u a t i o n :

( ~ ~ -2y(4) 1)

+ 8x(x2-l)y"'

+

(4a+12) (x2-l)y"

+ 8axy'

= X

my

For t h i s OPS, A s a t i s f i e s t h e two d i f f e r e n t i a l e q u a t i o n s a)

( x 2 -1)’~’

b)

(x2 - l ) ' d 3 '

=

o

+

+

12x(x2 -1)h"

The g e n e r a l s o l u t i o n t o ( a ) i s A(x) = c 1 t h i s i n t o (b) y i e l d s c 1 = a c 2 = a c 3 . A(x) =

+

$6(x+l)+$&(x-l).

are e a s i l y s o l v e d t o y i e l d u

-

[(24

4a)x'

+ 4aIA'

= 0

+ c26(x-1) + c36(x+1).

I f w e l e t c2 =

t,

S u b s t i t u t i o n of

we find

The moments from t h e two r e c u r r e n c e r e l a t i o n s i n (6) 2m+l

=

0

and

u~~

=

at2m+l 2m+l

A NEW OPS

We now show how t h e t h e o r y developed can b e used t o f i n d t h e d i f f e r e n t i a l e q u a t i o n once t h e w e i g h t d i s t r i b u t i o n i s known. The method t h a t w a s used p r i o r t o t h i s w a s a method c a l l e d S h o r e ' s t e c h n i q u e , which i n v o l v e d the Lagrange i d e n t i t y ( f o r example, see [ 5 1 ) . To i l l u s t r a t e t h i s new a p p r o a c h , w e w i l l f i n d t h e s i x t h o r d e r d i f f e r e n t i a l equat i o n h a v i n g a sequence o f polynomial s o l u t i o n s o r t h o g o n a l on (0,m) w i t h r e s p e c t

1

t o A(x) = - 6(x) +xe-X. A

Suppose, t h e n , t h a t t h i s OPS s a t i s f i e s t h e s i x t h o r d e r

6

e q u a t i o n L6(y) =

1

bi(x)y i=l

formally s e l f a d j o i n t .

(i)

(x) = Amy(x).

It follows t h e n t h a t

W e s h a l l assume t h a t xe-XL6(y) i s

L.L. Littlejohn

418

<

S i n c e i t i s n e c e s s a r y t h a t b . ( x ) i s a polynomial o f d e g r e e

i, w e s h a l l w r i t e

1

i bi(x)

=

1

j=o

..., 6 .

L . . x J , i = 1, 2 ,

I n p a r t i c u l a r , 3bk

3b 6 +-

3b6

must be

=J

<

a polynomial of d e g r e e

5 , from which i t immediately f o l l o w s t h a t L 6 0 = 1 6 6 = 0 .

S i m i l a r l y , one can e a s i l y v e r i f y t h a t : i) ii) iii)

iv) V)

L44 = L64 = L65 = 0 30%62+ 30L61

-

2L40

0

=

L b j = 3L63

- 12L43

-L2*

-

6L4,

.tz0

+

+ 60L63 -

R42

+ 6&41 -

3L,,

=

3L40 - 36011,,

0

+ 180L6,

- 60L61=0.

From Theorem 4 , we f i n d t h a t A s a t i s f i e s t h e t h r e e e q u a t i o n s :

+

( A , 3b,@'

(9) (10)

b5$

= 0, $ E

)

( A , 2b4@' - 5b6$"'

(11) ( A , b2$' - b,+$'"

P

+ b 3 @ )= 0 , @ 6 P + 3b6@(5) + b l @ ) = 0 ,

$ E P

Using i n t e g r a t i o n by p a r t s , w e see t h a t ( 9 ) y i e l d s L 6 1 = 0.

p1

Similarly, a f t e r

i n t e g r a t i o n by p a r t s , (10) becomes 21140 @ ' ( O )

+

-

A

+ j1(-2%,,

f

2Rk0 - 1 2 0 L s 3 A

+

90L,,

]

@ (0)

30L62)e-x@(x)dx = 0 , $I E P.

By choosing @ ( x ) = x 2 , x, 1 w e g e t t h e r e s p e c t i v e e q u a t i o n s : vi) vii) viii)

-

2K0

+ 3 0 Q 6 2=

0

L40 = 0

4L41 - 2 t b O - 120L63

+

=

o

By choosing L63 = 1, w e f i n d 1141 = 30 and L 4 3

=

3.

L a s t l y , i f w e use i n t e g r a t i o n

by p a r t s on (11) and i m i t a t e t h e above argument, w e g e t t h e a d d i t i o n a l e q u a t i o n s : ix)

L20 = 0

X)

2L21 - 1120

xi) X)

kZ2

-

L42

+ 18t4, + 468 - 24 = 0

L42

-

RZ0

+

180

=

- 12A

= 0

0.

One can see t h a t w e have enough e q u a t i o n s t o f i n d t h e c o e f f i c i e n t s of t h e d i f f e r e n t i a l e q u a t i o n . The r e q u i r e d s i x t h o r d e r d i f f e r e n t i a l e q u a t i o n i s :

419

Orthogonal Polynomials (12)

L6(y) = x3y(6)

+

(-3x3

+ 24x’

+

(-x3

+

((-6-6A)x

+

1 2 ~ ' ) ~ ' ~+ ) ( 3 x 3 - 30x2

- 60x)y"' + ( - 6 x 2

+

12A)y' = Xy

+

+

30x)y (4)

(36+6A)x)y"

.

T h i s , o f c o u r s e , i s i n complete agreement w i t h S h o r e ' s t e c h n i q u e . I f w e a p p l y t h e method o f F r o b e n i u s t o ( 1 2 ) , w e f i n d t h a t t h e i n d i c i a 1 It f o l l o w s t h a t t h e r e are f o u r = 0. e q u a t i o n i s r(r-l)(r-2)(r-3)(r+l)(r+2) L'

n

s o l u t i o n s which p u t s us i n t h e l i m i t - f o u r

case.

T h e r e f o r e , one boundary

c o n d i t i o n a t 0 i s needed t o i n s u r e a well-posed boundary problem. N o t i c e t h a t L 6 ( y ) ( 0 ) = 12Ay'(O), so t h e r e q u i r e d boundary c o n d i t i o n is the A-dependent cond i t i o n 12Ay' (0) = Ay(0).

REFERENCES: Boas, R. P . , J r . , "The S t i e l t j e s moment problem f o r f u n c t i o n s o f bounded v a r i a t i o n " , B u l l . Amer. Math. SOC. 45 ( 1 9 3 9 ) , 399-404.

K r a l l , A. M. and Morton, R. D . , " D i s t r i b u t i o n a l weight f u n c t i o n s f o r o r t h o g o n a l polynomials", S.I.A.M. J. Math. Anal. 9 ( 1 9 7 8 ) , 604-626. K r a l l , H. L . , " S e l f - A d j o i n t D i f f e r e n t i a l E x p r e s s i o n s " , h e r . Math. Month., V o l . 67, NO. 9 , 1960, 876-878.

K r a l l , H . L . , " C e r t a i n D i f f e r e n t i a l E q u a t i o n s f o r Tchebycheff P o l y n o m i a l s , "Duke Math. J . , 4 ( 1 9 3 8 ) , 705-718. L i t t l e j o h n , L . L. and S h o r e , S. D . , " N o n c l a s s i c a l o r t h o g o n a l p o l y n o m i a l s as s o l u t i o n s to second o r d e r d i f f e r e n t i a l e q u a t i o n s " , Can. Bull., Vol. 25, 1 9 8 2 , 291-295.

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DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

42 1

ON THE LIMIT-POINT CLASSIFICATION OF A CLASS OF NON-SELF-ADJOINT ORDINARY DIFFERENTIAL OPERATORS Jing-lin Liu Department o f Mathematics U n i v e r s i t y o f Inner-Mongol i a Huhehot, Inner-Mongolia P e o p l e ' s R e p u b l i c o f China

The l i m i t - p o i n t c l a s s i f i c a t i o n o f a c l a s s o f non-symmetric o r d i n a r y d i f f e r e n t i a l e x p r e s s i o n s on h a l f - l i n e i s cons i d e r e d , which i s r e l a t e d t o t h e well-posedness o f c e r t a i n boundary v a l u e problems. We deal w i t h t h e f o l l o w i n g non-symmetric o r d i n a r y d i f f e r e n t i a l e x p r e s s i o n on t h e i n t e r v a l I = [ l , a ) ,

M

where D = d/dx, i = fi, p ( x ) and q s ( x ) a r e r e a l C m - f u n c t i o n s . M i s s a i d t o be l i m i t - p o i n t , i f an8 o n l y i f t h e dimension o f t h e q u o t i e n t space o f t h e domain o f t h e maximal o p e r a t o r o f M modulo t h e domain o f t h e minimal o p e r a t o r o f M equals 2N. dim

(M))/D(To(M))

=

2W.

H a l f o f t h i s dimension i s c a l l e d t h e mean d e f i c i e n c y i n d e x o f M and denoted by U s u a l l y we o n l y d i s c u s s t h e d e f i c i e n c y i n d e x problem f o r f o r m a l l y symmetric d(M). d i f f e r e n t i a l e x p r e s s i o n s , b u t t h e y have a l s o i m p o r t a n t a p p l i c a t i o n s i n t h e nonsymmetric case. As m o t i v a t i o n , c o n s i d e r t h e f o l l o w i n g boun a r y v a l u e problem Q: Given g i n L 2 ( I ) , s o l v e t h e e q u a t i o n M f = g i n L ( I ) , such t h a t D J f ( 1 ) = 0, j = O , l , - . . , N - l . If p k ( x ) a r e non-negative, pN(x) i s p o s i t i v e and p ( x ) i s bounded below away f r o m zero, t h e n i t may be shown t h a t Q is w e l l - p 8 s e d i f and o n l y i f M i s l i m i t - p o i n t . What a r e t h e c o n d i t i o n s on t h e coe f f i c i e n t s o f M w h i c h guarantee t h a t M i s l i m i t - p o i n t ? By u s i n g a method of R. M. Kauffman [l],we can p r o v e t h a t f o r a s p e c i a l c l a s s M w i t h e v e n t u a l l y p o s i t i v e p o l y n o m i a l c o e f f i c i e n t s , M i s u s u a l l y l i m i t - p o i n t , b u t n o t always. F i r s t o f a l l , we g i v e a counter-example.

9

Lemma 1.

Let

N k k k N M = l k Z 0 ( - 1 ) D pk(x)D + i ~ ~ = O ( - l ) s D s q s ( ~ ) D beS a d i f f e r e n -

t i a l e x p r e s s i o n on I , where pk p o i n t , then f o r any f i n domain C:(1

1

and qs a non-negati If M i s i m .i t T1(M), poFP2f and q o v p z f a r e i n L ( I ) .

P r o o f . Suppose f i s i n domain ,-), such t h a t fn + f and M f n

To(M), t h n t h e r e e x i s t s Mf i n L ( I ) . S i n c e

m

(fni i n

5

+

n

"

m

j’

so t h e r e i s some c o n s t a n t K such t h a t f o r e v e r y A > 1, p0lfn/'dx, 1 A 2 f q o l f n l dx 5 K. By u s i n g F a t o u ' s Lemma, p a s s i n g t o t h e l i m i t , we o b t a i n 1 '"f and q0'/'f are i n L2(I). Since M i s l i m i t - p o i n t , domain TI(M) PO

422

J.-L. Liu

d i f f e r s f r o m domain f o l 1ows.

To(M)

o n l y b y compact s u p p o r t f u n c t i o n s , so t h e c o n c l u s i o n

____ Theorem 1. There e x i s t p o s i t i v e numbers M = D~ xy+4 D~ + i o x y i s n o t l i m i t - p o i n t .

and

y

u

such t h a t

i s t h e s o l u t i o n i f and Proof. Consider t h e power s o l u t i o n s o f M f = 0. X’ o n l y i f A(A - l ) ( A + y + 2 ) ( A + y + 1 ) = - i o . L e t A = -1 + i b and el(b) = a r g ( A ) , e2(b) = a r g ( x - 1 1 , e3(b,y) = a r g ( X + y + 2 ) , e4(b,v) = a r g ( A + Y + 1 ) . If t h e r e e x i s t p o s i t i v e numbers b and y such t h a t el(b) + e2(b) + e3(b,v) + e4(b,y) = 3n/2, then t h e r e a r e p o s i t i v e numbers Y and u such t h a t x = -1 + i b i s t h e s o l u t i o n o f t h e a l g e b r a i c equation. It i s e2(b) converge t o n/2 + 0, obvious t h a t when b approaches i n f i n i t y , o,(b), and

e4(b,t)

e3(b,l),

el(bo)

converge t o

+ e2(bo) < 3a/2

H3(bO,Y)

and

and

e4(bo,y)

becomes l a r g e ,

el(bo)

el(bo)

approach

a/2

-

0,

0

as

so we can s e l e c t

+ e4(bo,y)

I t f o l l o w s f r o m c o n t i n u i t y t h a t t h e r e i s some + e2(bo) + e3(bo,y)

el(bo)

+

such t h a t 3a/2.

>

goes t o i n f i n i t y , hence when

y

+ e2(bo) + e3(bo,y)

bo

+ e3(bo,l) + e4(bo,l)

+ e2(bo)

w i l l be less t h a n

But Y 3a/2.

y > 1

such t h a t -l+ibo e4(bo,v) = 3 ~ / 2 . Therefore x i s i n domain

y/2 - l + i b o 2 T](M). B u t y - 2 > -1, x x i s n o t i n L ( I ) , by Lemma 1, M i s n o t l i m i t - p o i n t . T h i s Theorem shows a l t h o u g h t h e r e a l p a r t and t h e i m a g i n a r y p a r t o f M a r e l i m i t - p o i n t , b u t t h e whole t h i n g f a i l s t o be l i m i t - p o i n t . It i s a l s o i n t e r e s t i n g t h a t i n t h e r e a l case, Kauffman proved t h a t a l l f o u r t h - o r d e r symmetric expressions w i t h e v e n t u a l l y p o s i t i v e polynomial c o e f f i c i e n t s are l i m i t - p o i n t . T h i s i s n o t t r u e f o r t h e complex case. B u t we can prove i n t h e second-order case, a l l such non-symmetric e x p r e s s i o n s a r e l i m i t - p o i n t . Lemma 2. and

d,

Let

M =

non-negative.

1Nk=O ( - l ) k c k D

x n(k)Dk

Suppose t h a t

n(k)

-

+ 2k

i I s=o N( - l ) S d S D S x m ( s ) D S , w i t h <

n(j)

-

2j

for

k > j

ck

and

m ( s ) - 2s < m ( t ) - 2 t f o r s > t. I f one o f t h e f o l l o w i n g c o n d i t i o n s i s s a t i s f i e d : ( 1 ) co > 0 and do > 0; ( 2 ) co = 0 o r do = 0 and n ( k ) - 2k # m ( s ) - 2s f o r k # s, t h e n To(M) i s separated. Remark 1. 2.

See [l]f o r t h e p r e c i s e d e f i n i t i o n o f s e p a r a t i o n . The p r o o f o f Lemma 2 i s t o o c o m p l i c a t e d t o be i n d i c a t e d h e r e .

Using t h e same t e c h n i q u e i n [l], we can p r o v e t h e f o l l o w i n g conclusions. Lemma 3. Suppose M s a t i s f i e s t h e c o n d i t i o n s d e s c r i b e d i n Lemma 2. L e t rl = m ’ c k > O } (r2= min{s I d, > 01). Then f o r any k ( s ) such t h a t

k > r1 and ck > 0

5

(a 5

(s

>

r2 and

d,

>

0)

there i s a positive E E n ( k ) - a 2k-k D(To(M)) C D(To(x x D ) ) (D(To(M)) rl

2k

=

Lemma 4. -ck

r 2 = 0,

-

II,

such t h a t C

D(To(x E x m(S)-2D2S-!?1 1 ) .

Suppose M s a t i s f i e s t h e c o n d i t i o n s d e s c r i b e d i n Lemma 2. L e t Suppose one o f t h e f o l l o w i n g con> 01 and r 2 = m i n { s I ds > 01.

ditions i s satisfied: m(r2)

and f o r any n a t u r a l number

2s),

(1)

rl = r2 = 0

and

m(0) 5 0 and n ( r l ) - 2rl < m(0): Then x n f i s i n L2(I)

2r2 < n(0).

n(0) (3)

5 0 or r1 = 0,

f o r any

f

m ( 0 ) 5 0; r2 > 0,

( 2 ) rl

n(0) 5 0

i n k e r T1(M)

>

0,

and

and any

42 3

Limit-Po in t Classification of Non-Self-A djo in t Operators n a t u r a l number

n.

Lemma 5. L e t M = P + iQ, where P and Q a r e 2Nth-order symmetric d i f f e r e n t i a l e x p r e s s i o n s w i t h r e a l c o e f f i c i e n t s . Suppose range To(M) i s c l o s e d t h e Lagrange b i l i n e a r and f o r any f i n k e r T (M) and any g i n k e r T (M'), form [f,g](x) c o n v e r g e i t o z e r o as x approachei i n f i n i t y . Then M i s l i m i t point. Lemma 6. M

n(1) - 2

(2)

n(0)

<

m(1) - 2 < n(O),

+ coxn(o) + i ( - [ d l Dxm(l)D

= -clDxn(')D

Proof. (I)

m(1)

T1(M)

Let

1.

c1 = 0

2.

cldl

2

or

<

(1)

cl,

co, dl,

m(0); ( 3 )

do > 0

Suppose

a r e non-negative;

co > 0,

if

n(1) - 2

then

+ d0 xm(')).

do

do = 0

m(0).

<

then

Then

M

is

i s separated.

+ idlxm('))D

M1 = - D ( c l x n ( l ) M1

M2 = coxn(o) + i d o x m ( o ) .

and

i s always l i m i t - p o i n t .

dl = 0,

t h i s i s a c t u a l l y t h e r e a l case.

n ( 1 ) 2- 2

and

0

To(M1)

-

co = 0,

F i r s t , we prove t h a t

Since

2

and

and i f

l i m i t - p o i n t and

2

M

Let

s a t i s f i e s the f o l l o w i n g conditions:

or

m(1) 2 2.

I n t h i s case, n u l l i t y T 1 ( M 1 ) ( l .

has c l o s e d range ( b y Lemma 2.14 i n [ l ] ) and

2d(M1) = n u l l i t y T1(Ml)

+ n u l l i t y T1(Mt) = 2 n u l l i t y T1(M1) 2 2,

is

M1

1 im i t - p o i n t.

3. n ( 1 ) + 2a

cldl

>

and

0

n(1)

2 2 and m(1) + 2a

2 and m ( 1 )

<

> 2.

Let

2.

<

a > 0

Choose

such t h a t

+ 1 + i)x",

R = x"(M

n(1)+2a + i d x~(~)'*')D, i.1 = S + ( 1 + :)x2". S and ( 1 + i ) x 2 " a r e S = -D(clX 1 I t i s obvious t h a t l i m i t - p o i n t , b y Lemmas4and 5 , W i s l i m i t - p o i n t . R - W = a x n(1)+2a-lD bxn((1)+2n-2 i(cx m(1)+2a-1D dxm(1)+2a-2), According t o t h e L e m a 3 and Lemma 2.20 i n [l], R - W i s r e l a t i v e l y compact w i t h r e s p e c t t o To(W). By Lemma 2.19 i n [l], To(R) has c l o s e d range and +

n u l l i t y T,(R k e r Tl(M1

+

= n u l l i t y T ~ ( w + ) , so

+ 1

i),

n u l l i t y T,(M1

f

1 + i )5 n u l l i t y T1(R)

c l o s e d and

5 2d(M1 + 1 + i )

(11)

Lemma 7. co, dl,

do

m(1) - 2 f n ( o ) , separated. n(o)

>

i s limit-point. =

1.

f

N o t i c e t h a t range

2 n u l l i t y Tl(M1

=

If

i s in

so

i s i n k e r T,(R),

By u s i n g Lemmas 4 and 5,

(111) Since T (M) because t h e do8ain functions .

cl,

x-"f

R

f

2

f

+

To(M,

+ 1 + i ) 5 2,

+

1 + i)

is

i s limit-point.

M1

M i s limit-point.

i s s e p a r a t e d and M i s l i m i t - p o i n t , T (M) i s separated, T1(M) d i f f e r s f r o m t h e domain To(M) d n l y by compact s u p p o r t

Let

M

=

-clDx"(')D

are non-negative. m(1) - 2 # m ( o ) ,

+ c 0xn(') Suppose then

+ n(1)

M

i ( - d 1Dxm(')D + d o x m ( o ) ) ,

-

2 # n(o),

i s l i m i t - p o i n t and

where

-

2 # m(o),

T,(M)

is

n(1)

P r o o f . Suppose a l l t h e c o e f f i c i e n t s a r e p o s i t i v e and max(n(1) - 2, m(1) - 2) m ( o ) (same argument works f o r o t h e r cases).

M0 = -c 1Dxn(')D + C , X ~ ( ~ )- idlDxm(l)D.

By p r e v i o u s Lemma,

Mo

i s limit-point

Let

J.-L. Liu

42 4

and T1(Mo) i s separated. any f i n domain To(Mo),

0 < E < n ( 0 ) - m(0) such t h a t f o r L2 ( I ) . Hence i d o x m ( o ) i s r e l a -

There e x i s t s an xExm(o)f i s i n

t i v e l y compact w i t h r e s p e c t t o To(Mo + 1 ) . By Lemma 2.19 i n [1], + domain To(M) = domain To(Mo), n u l l i t y T1(M + 1 ) = n u l l i t y T o ( M i + 1 ) = 1, M

i s limit-point. Let

Lemma 8. d

, d0 1

Since

To(Mo)

T1(M)

M = - c 1Dxm+'D + coxm + i(-dlDxW2D

a r e non-negative.

l i m i t - p o i n t and

i s separated,

T1(M)

c1 + dl

Suppose

0

>

and

+ doxm), co + do

>

where c1 , co, 0, t h e n M i s

i s separated.

P r o o f . ( I ) Case m .I0. L e t M1 = -D(cl + id,)xm+'D M2 = ( c o + i d o ) x m . F o r any f i n k e r T 1 ( M + l ) , ( M 2 + l ) f

and is in

Mlf. Since M1 Lemma 6, Tl(Ml)

M

(11)

and M 2 + 1 a r e l i m i t - p o i n t , b y Lemma 5, i s separated, so i s T1(M).

Case

m > 0.

Now,

L'(I),

so i s

i s limit-point.

By

has c l o s e d range, so

T1(M)

2d(M) = n u l l i t y T1(M) + n u l l i t y T,(M+) T1(M) = 1,

so

i s a l s o separated.

2 n u l l i t y T1(M). To show n u l l i t y Mf = 0. Mx x = 0 i f and o n l y i f

=

c o n s i d e r t h e power s o l u t i o n s o f

-(cl + i d l ) X ( h + m + 1) + ( c

0

+ i d ) = 0.

Since

0

c1 + dl

>

0,

co + do > 0,

we

always have two d i f f e r e n t s o l u t i o n s A+ and A _ . A s i m p l e computation shows t h a t one of A+ and h- has r e a l p a r t l e s s t h a n -1 and t h e o t h e r ' s r e a l p a r t i s p o s i t i v e , so n u l l i t y T1(M) = 1 . Hence we proved M i s l i m i t - p o i n t , by Lemma 2.33 i n [l], we a l s o o b t a i n e d T1(M) i s separated. Lemma 9 . cl,

co, dl,

Let

do

M = - c 1Dxn(')D

+

a r e non-negative.

+

coxn(o)

M

Then

i(-dlDxm(')D

+

i s l i m i t - p o i n t and

doxm(o)), T1(M)

where is

separated. Proof. (I)

I f n(1)

Suppose a l l t h e c o e f f i c i e n t s a r e p o s i t i v e .

-

2 # m(0)

and

m(1)

-

1.

When n ( 1 ) - 2 # n ( 0 ) f r o m Lemma 7.

2.

When n ( 1 ) - 2 = n ( 0 ) (case f o l l o w i n g t h r e e cases.

and

2 # n(0). m(1)

-

m(1)

2 # m(O),

-

2 = m(0)

the conclusion f o l l o w s i s t h e same), c o n s i d e r

( 1 ) F o r m(1) - 2 < n ( 0 ) and m(0) < n ( O ) , l e t M = - C DXn(0)+2D + coxn ( 0 ) , Mo i s l i m i t - p o i n t and T1(Mo) i s separated. I t 0 1 i s obvious t h a t Mo i s t h e main p a r t o f M. L e t W = i(-dlDxm(')D + doxm(0)) and

ME = Mo + EW,

domain

To(M,)

0

5

equals

E

5 1.

I t can be shown t h a t

domain To(Mo).

To(M,)

i s separated, so

By t h e s t a b i l i t y o f t h e i n d e x o f a

Fredholm o p e r a t o r under s m a l l r e l a t i v e l y bounded p e r t u r b t i o n s , range T (ME + 1 ) i s c l o s e d and t h e d e f i c i e n c y o f range T (M + 1 ) i n L ( I ) i s a c o n s t g n t independent o f E . But t h e d e f i c i e n c y o? r h g e T (M + 1) i n L 2 ( I ) e q u a l s so i s M. n u l l i t y T1(Mr + 1 ) . Hence M + 1 = Mo + 1 + W i! l:mit-point,

h

For a l l t h e cases t h a t we w i l l c o n s i d e r l a t e r , t h i s s o r t o f argument w i l l be used again, so we o m i t t h e d e t a i l s .

Limit-Point Classificutio n of No n-Self-Adjoin t Operators (2) M

m(1) - 2

For

= - C DXn(o)+2D

f

sgparated. 1

<

n(0)

m(0)

and

>

By Lemma 7,

i d xm('). 0

let

n(O),

Mo

425

i s l i m i t - p o i n t and

T1(Mo)

is

( 3 ) F o r m(1) - 2 > n(O), l e t Mo = i(-dlDXm(l)D + d x ~ ( ~ ) ) T. h i s i s a c t u a l l y t h e r e a l case, so Mo i s l i m i t - p o i n t and T1(Mo)O i s separated. (11) I f n(1) - 2 = m(0) f o l l o w i n g cases. 1.

m(0)

>

n(0).

M0 =

- C 1 DXm(o)+2D

(3)

For

(1)

For

+ i d0 xm(').

m(1) - 2 = m(O),

m(1) = 2 = n(0)

(case

m(1)

2

<

m(O),

let

Mo = -clDx m(o)f2D

( 1 ) F o r m(1) - 2 # n(O), M = - C DXm(o)'2D + c xn(') + i ( - d l D x m ( l ) D ) . (2) M0 = co:n(o) + i ( - d l D xon(o)+2D). 2.

consider t h e

F o r m(1) - 2 > m(O),

(2) let

-

i s t h e same),

m(0) < n(0).

+

l e t M = -idlDxm(')D. i( -dl Dxm(o)+BD + d x m ( o ) ) . 0

let For

-

m(1)

2 = n(O),

let

0

3.

m(0) = n ( 0 ) .

M0 = -clDx m(o)f2D

(1)

+ c 0xm(')

For

m(1)

-

+ idoxm(o).

2

<

m(O),

(2)

let

For

m(1)

-

2

= - i d Dxm(')D. ( 3 ) F o r m(1) - 2 = m(O), l e t Mo = M. 0 1 a l l t h e s e Mo a r e l i m i t - p o i n t and T1(Mo) a r e separated.

M

>

m(O),

let

By Lemmas 7 and 8,

F i n a l l y , by u s i n g t h e same argument used i n p r e v i o u s Lemma, we o b t a i n Theorem 2.

Let

M = -Dpl(x)D

+ po(x) + i(-Dql(x)D + qo(x))

o r d e r d i f f e r e n t i a l e x p r e s s i o n , where

p,(x),

p o ( x ) , ql(x),

qo(x)

be a secondare eventually

non-negative and each i s a f i n i t e sum o f r e a l m u l t i p l e s o f r e a l powers o f x. Then M i s l i m i t - p o i n t and T1(M) i s separated. F o r h i g h e r - o r d e r case, u s i n g t h e same idea, we proved Theorem 3.

Let

1

M = l / = o ( - l ) k Dk p k ( x ) Dk + i s=o ( - l ) S D S q s ( x ) D S be a 2Nth-

o r d e r d i f f e r e n t i a l e x p r e s s i o n . Suppose t h a t M s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s : ( 1 ) I f pk and qs a r e n o t i d e n t i c a l l y zero, t h e y a r e f i n i t e sums o f r e a l m u l t i p l e s o f r e a l powers,

p k ( x ) = ckxn(k) + l o w e r o r d e r terms,

q s ( x ) = dsxm(s) + l o w e r o r d e r terms,

ds > 0.

(2)

ck > 0,

n ( k ) = 2k # m(s) - 2s

for

k # s. ( 3 ) L e t k* = m a x I n ( k ) - Z k } and s+ = max{m(s) - 2 s ) . Suppose t h a t t h e r e i s o n l y one k and o n l y one s such t h a t n ( k ) - 2k = k* and m(s) - 2s = s*. Then M i s l i m i t - p o i n t and T1(M) i s separated. Because o f Theorem 1 and t h e counter-example i n [ l ] , c o n d i t i o n ( 2 ) and ( 3 ) a r e needed. Acknowledgement. I w i s h t o express my s i n c e r e thanks t o P r o f e s s o r s R. M. Kauffman and T. T. Read and t h e Department o f Mathematics o f Western Washington U n i v e r s i t y f o r t h e o p p o r t u n i t y t o s t a y t h e r e d u r i n g t h e academic y e a r 1981-82. It i s P r o f e s s o r Kauffman's s u g g e s t i o n t o c o n s i d e r t h i s c l a s s o f n o n - s e l f - a d j o i n t o p e r a t o r s and c o n v e r s a t i o n s w i t h him were v e r y h e l p f u l .

426

J.-L. Liu REFERENCES

[l] Kauffman, R. M., On t h e l i m i t - n c l a s s i f i c a t i o n o f o r d i n a r y d i f f e r e n t i a l o p e r a t o r s w i t h p o s i t i v e c o e f f i c i e n t s , P r x . London Math. SOC. ( 3 ) 35 (1977) 496-526. [2]

Kauffman, R. M., Read, T. T. and Z e t t l , A., The D e f i c i e n c y I n d e x Problem f o r Powers o f O r d i n a r y D i f f e r e n t i a l Expressions, L e c t u r e Notes i n Mathematics No. 621 ( S p r i n g e r - V e r l a g , B e r l i n , 1977).

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) @Elsevier Science Publishers B.V. (North-Holland), 1984

42 7

A M O V I N G BOUNDARY PROBLEM DESCRIBING OXYGEN CONSUMPTION I N SOIL Roger C. McCann Department o f Mathematics and S t a t i s t i c s M i s s i s s i p p i State U n i v e r s i t y Mississippi State, M i s s i s s i p p i

K.

Paul

McConnaughey

Department o f Agronomy Mississippi State University M i s s i s s i ppi State, M i s s i s s i p p i T h i s paper p r e s e n t s some p r e l i m i n a r y r e s u l t s i n a numerical s t u d y o f a moving boundary problem modeling t h e consumption o f oxygen by m i c r o b i o l r e s p i r a t i o n i n s o i l . INTRODUCTION Consider a t h i n l a y e r o f homogeneous s o i l t h a t i s bounded by an impermeable l a y e r a t x = L and i s exposed t o a i r a t x = 0. T h i s l a y e r i s w e t t e d t o s a t u r a t i o n w i t h water. The i n i t i a l oxygen c o n c e n t r a t i o n o f t h e w a t e r and t h e a i r a t t h e s u r f a c e o f t h e s o i l i s Co. F o l l o w i n g w e t t i n g an unmeasurable O2 consumption r a t e F ( t ) a r i s i n g f r o m m i c r o b i o l r e s p i r a t i o n r e s u ! t s i n a measurable oxygen f l u x f ( t ) a t x = 0. T h i s oxygen consumption r e s u l t s i n a d e p l e t i o n o f oxygen t h r o u g h o u t t h e r e g i o n 0 5 x 5 L u n t i l a moving boundary r e p r e s e n t i n g an O2 c o n c e n t r a t i o n o f z e r o moves away f r o m t h e impermeable l a y e r a t x = L. The d i s t a n c e f r o m t h e exposed I n [ 2 ] the s u r f a c e ( x = 0 ) t o t h i s moving boundary a t t i m e t i s n o t e d by x b ( t ) . second a u t h o r has shown t h a t t h e c o n c e n t r a t i o n C ( x , t ) o f oxygen a t a d i s t a n c e x, 0 5 x 5 x b ( t ) 5 L, below t h e s u r f a c e a t t i m e t s a t i s f i e s t h e f o l l o w i n g boundary v a l u e problem: C t = DCxx + F ( t ) ,

C(O,t)

C(x,O)

C0

O < t

= f(t)

0

c0

0 5 X L L

=

-DCx(O,t) =

0 5 x 5 xb(t) 5 L

< t

xb(0) = L O < t

0 5 xb(t) 5 L Cx(xb(t),t) C(xb(t),t)

0

= =

0

0 < t O < t

I n [l]Crank and Gupta c o n s i d e r e d a s i m i l a r problem modeling t h e c o n c e n t r a t i o n o f oxygen i n a medium t h a t absorbs oxygen, such as t i s s u e . I n t h e i r problem F i s a known c o n s t a n t f u n c t i o n and t h e r e i s no p r e s c i b e d v a l u e f o r C ( 0 , t ) . Near t h e moving boundary t h e y approximated t h e f u n c t i o n C(. ,t) by a second degree polynomial u s i n g t h e v a l u e ( z e r o ) o f C ( x b ( t ) , t ) and c a l c u l a t e d values f o r C(nh,t) and C ( ( n - l ) h , t ) where h i s t h e s t e p s i z e i n x and nh < x b ( t ) 5 ( n + l ) h . T h i s gave v e r y good r e s u l t s s o we d e c i d e t o use a s i m i l a r a p p r o x i m a t i o n i n o u r c a l c u l a t i o n s even though t h e i n i t i a l movement o f t h e boundary i n o u r problem i s much f a s t e r t h a n t h e

42 8

R.C.McCann and P. K. McConnaughey

i n i t i a l movement o f t h e boundary i n t h e i r problem. I n s o l v i n g t h e i r problem, Crank and Gupta assumed t h a t b o t h C ( - , t ) and C(x;) a r e decreasing f u n c t i o n s . Due t o t h e unknown n a t u r e o f F we do n o t assume t h a t C(x,.) i s d e c r e a s i n g , b u t do assume t h a t C ( - , t ) i s decreasing. BEFORE THE BOUNDARY MOVES To b e g i n we use a backward d i f f e r e n c e a p p r o x i m a t i o n t o t h e d i f f e r e n t i a l e q u a t i o n a1 ong with fa1 se b o u n d a r i e s t o approximate each o f t h e d e r i v a t i v e boundary condi t i o n s . These a p p r o x i m a t i o n s l e a d t o t h e f o l l o w i n g system o f e q u a t i o n s , where k i s t h e s t e p s i z e i n t, R =Dk/h2,and C . . F . , and f . approximate C ( i h , j k ) , F ( j k ) , and f f j k ) r e s p e c t i v e l y : 13’ J J

+ (1+2R)C. . - RCi+l,j+l -RCi-l ,j+l 1 ,J+1

+ kFj+l

=

C . . i = 0,1,2, 1J

2h ‘-1, j + l = ‘1 , j + l + - Df

...,

N = L/h ( 1 )

j+1

(2) (3)

‘n+l, j + l = ‘n-1, j+l

These equations l e a d t o a system o f N + 1 l i n e a r e q u a t i o n s i n t h e N + 1 unknowns F. Hence s o l o n g as x b ( t . ) = 1 t h e r e i s no d i f f i ‘1, j + 1 * c 2 , j + l ’ * . . ‘n, j+1’ j + 1 . J c u l t y i n a p p r o x i m a t i n g t h e c o n c e n t r a t i o n C and t h e consumption r a t e F. AFTER THE BOUNDARY MOVES Once t h e boundary has moved f r o m x = L, e q u a t i o n ( 3 ) i s no l o n g e r v a l i d . Suppose t h a t x b ( j k ) < 1 and t h a t n i s an i n t e g e r such t h a t nh < x b ( j k ) 5 ( n + l ) h . Using and Cn-l ,j a p p r o x i m a t i n g C ( n h , j k ) and C ( ( n - l ) h , j k ) , r e t h e computed values C n ,j s p e c t i v e l y , we approximate C ( x , j k ) on [ ( n - l ) h , x b ( j k ) ] by a polynomial o f t h e f o r m P(x) = a(x-xb(jk))

2

T h i s a l l o w s us t o f i n d an a p p r o x i m a t i o n x b . f o r x b ( j k ) . J We n e x t o b t a i n a f o r w a r d d i f f e r e n c e a p p r o x i m a t i o n t o t h e e q u a t i o n a t ( n h , j k ) . Assuming t h a t C ( * , j k ) can be expanded i n T a y l o r s e r i e s we have (where A = xb(jk)-nh) 0 = C ( x b ( j k ) , j k ) = C(nh,jk) + Cx(nh,jk)A + C x x ( n h , j k )A~2 +

0 = Cx(xb(jk),jk)A

=

C,(nh,jk)A

+ Cxx(nhh,jk)A2 +

( n h , j k ) k 3 + O(h 4 ) cxxx I f we m u l t i p l y t h e f i r s t e q u a t i o n by a , t h e second by 6, and t h e t h i r d by y, and t h e n sum t h e e q u a t i o n s , where a, 6 , and y a r e chosen so t h a t t h e c o e f f i c i e n t s o f C,(nh,jk) and C x x x ( n h h ’ j k ) i n t h e sum a r e zero, we have (remember ng t h a t

O < A z h )

429

Spectrum of u Humiltoniun System yC((n-l)h,jk)

= (a+y)C(nh,jk)

A‘

t (a2

2 +@A

+ y h2 2 )Cxx(nhh,jk) +

4 O(h )

(4)

One such c h o i c e f o r a , 8 , and y i s a = 3A 2 - h2 , 8 = h2 - A 2, y

E q u a t i o n ( 4 ) a l l o w s us t o approximate Cxx(nh,jk)

= 2~~h

by

We now r e p l a c e A by a = x b . - nh and s i m p l i f y t h i s e x p r e s s i o n t o o b t a i n J

Using t h i s a p p r o x i m a t i o n f o r C

( n h , j k ) and a f o r w a r d d i f f e r e n c e a p p r o x i m a t i o n f o r xx C ( n h , j k ) i n t h e d i f f e r e n t i a l e q u a t i o n we o b t a i n t h e e q u a t i o n t

T h i s e q u a t i o n i s used i n p l a c e o f e q u a t i o n t h e e q u a t i o n s i n ( 1 ) w i t h i = 0,1,2, ..., n unknowns C lj, CZj ,..., C ., F . . Thus when nJ J a l s o a b l e t o approximate t h e c o n c e n t r a t i o n

( 3 ) whenever X j < L . E q u a t i o n ( 5 ) and a r e n t 1 l i n e a r equations i n t h e n + 1 t h e boundary moves from x = 1 we a r e and t h e consumption r a t e .

The above method was r u n on t h e t e s t problem i n which 0 = 1, L = 1, Co = 1, and f ( t ) E 4. The s i t u a t i o n we a r e modeling suggests t h a t F ( t ) + m as t + m f o r some i t can be shown t h a t t h e s o l u c o n s t a n t m. Assuming t h a t t h i s i s i n f a c t t h e cas t h a t xb = .5, and t h a t m = 8. t i o n t o t h e steady s t a t e problem i s C ( x ) = ( 2 x - 1):; T a b l e 1 g i v i n g t y p i c a l c a l c u l a t i o n s when h = .1 and k = .01, k = .001, and k = .0001 shows t h a t x b ( t ) and F ( t ) do converge t o t h e i r steady s t a t e v a l u e s . Our r e s u l t s when K = .0001 a r e d e p i c t e d g r a p h i c a l l y i n F i g u r e 1 . REFERENCES

1.

J. Crank and R. S. Gupta, A moving boundary problem a r i s i n g f r o m t h e d i f f u s i o n o f oxygen i n a b s o r b i n g t i s s u e , J. I n s t . Maths. A p p l i c s . , 10(1972), 19-33.

2.

P. K. McConnaughey, T r a n s i e n t m i c r o s i t e models o f d e n i t r i f i c a t i o n : Theory and experiment, Ph.D. d i s s e r t a t i o n , C o r n e l l U n i v e r s i t y , I t h a c a , New York,1983.

R. C McCann and P.K. McConnaughey

430 h = .1 t

.Ol .02 103 .04 .05 .06 .07 .08 .09 .10 .ll .12 .13 .14 .15

k = F(t)

-35.777 -21.466 -15.741 -12.879 -11.162 -9.995 -9.139 -8.573 -8.280 -8.121 -8.056 -8.021 -8.011 -8.002 -8.002

.Ol

k = .OOl

xb(t)

1 1 1 1 1 .616 .522 .496 .506 .495 .503 .497 .501 .498 .501

F(t)

xb(t)

-25.952 -16.954 -1 3.523 -11.592 -1 0.308 -9.376 -8.6987 -8.295 -8.112 -8.038 -8.012 -8.003 -8.001 -8.000 -8.000 TABLE 1

FIGURE 1

1 1 1 1 1 .552 .510 .502 .501 .500 .500 .500 .500 .500 .500

k = . 0001 F(t) xb(t)

-24.807 -16.598 -13.351 -11.486 -1 0.234 -9.320 -8.653 -8.264 -8.093 -8.030 -8.009 -8.002 -8.000 -8.000 -8.000

1 1 1 1 1 .551 .509 .502 .501 .500 .500 .500 .500 .500 .500

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) OElsevier Science Publishers B.V.(North-Holland), 1984

43 1

Converse BVP for Associated Elliptic and Parabolic Fractional Partial Differential Operators Peter A. McCoy Mathematics Department United States Naval Academy Annapolis, Maryland 21402 For several classes of elliptic and parabolic partial differential equations, the converse boundary and initial value problems that arise in applications yield to summability methods for Jacobi series. A fractional partial differential operator of mixedtype and generalized summation process for Jacobi series produce a unified theory from which applications are drawn. INTRODUCTION A problem of interest in harmonic function theory is the converse to the Dirichlet problem which is to find, in some generalized sense, the boundary function to which the Poisson representation of a harmonic function in an open disk D converges. This problem was solved by I. Gelfand [ 4 1 , G. Johnson [ 6 ] and M. Kothe [81. It is known that the boundary function exists uniquely in the space of hyperfunctions on the aD by constructing an isomorphism between the spaces of harmonic functions in D and hyperfunctions on the aD. An essential feature is the natural map induced by forming the Abel means of the trigonometric Fourier series of the boundary values. The converse to the Dirichlet problem has recent counterparts for elliptic [3,7,10,111 and parabolic [9] partial differential equations where Abel and Weierstrass summability of Fourier-Jacobi series play their respective roles. We find a single fractional differential operator and associated partial differential equation of mixed type that generalizes several of the elliptic and parabolic equations. From the operator, a summation process is defined on the Jacobi series realizations of the BV to induce a natural isomorphism between the spaces of solutions on an open domain and hyperfunctions at its boundary. This identification produces a single theory of converse BVP that includes the original elliptic and parabolic problems viewed in extended form as fractional partial differential equations. BASIC FORMULAE = p~a,6)(~0~e)/~da,B)(i), The normalized Jacobi polynomials $a,b)(cosB): are the eigenfunctions Aia,@) ( a , 6 ) = $a,B) = k(k+a+b+i) % %(a,B), A(a,B) k

of the differential operator [2]

k

E

N,

43 2 and form an orthogonal set

The formal Fourier-Jacobi series of f

X:

E

[o,n]uC[~,P], 2 5 p 5 -,

L

=

P

m

1

f(cose) =

% qa,B)(cose)

ak

k=O P

ak

=

ak(f):

=

JO

f(cos$) $a,B)(cos$)

dp ($), a$

k

E

N

may not converge to f in norm. To induce convergence in the uniform norm,

let the real sequence {cX(k)}XEA,

II 11,

(A is Z+ or N) satisfy the conditions

m

(i>

1

I <

%lc,(k)

and

m

cx(0)

(ii)

=

1, XEA.

k=O This is a factor sequence from which one generates the series m

Cx(f,cose):

=

1

cx(k) ak(f)

y,qa,B)(cose)+f(cose),

X+O+

k=O that is C -sumable t o f.

A positive Lperator for which the series

x

m

converges to f is an operator of factor sequence type associated with the (positive-increasing) factor sequence { +(k) } ksN. The second important summation process,

q’8)(cose),

m

Eqf(cos8): t=t(X),

for

=

F(t,COSB) =

1

e-tq(k)ak(f)wk

k=O

LEA generates classical solutions of the canonical PDE of m

&LLptic-pa/rabofiC

type,

t a m+

B } F

q

=

0, (t,e)

E

t(A) X ( 0 , ~ )

that are C -summable to the initial data F(@,

x

uniform norm. See [2].

cos8)

=

f(cos.0)

Applications of these operators follow.

in the

H

433

FRACTIONAL PDE GENERATED BY SUMMATION PROCESSES PARABOLIC:

Let c

l/t

(k):

=

exp(-


t) and B

*:

=

-

[Aia,B)]a/2, 0

< u<

2,

be a fractional differential operator of order a defined as the operator with factor sequence ( A

Then F is a classical solution of the generalized

k

heat equation

{at -

[A(a9B)1a’2~ F(t,cosB)

e

= 0,

(tS8)E(O,m)

X (0,~)

that is uniformly Weierstrass summable to the initial data F(O+ ELLIPTIC:

Let c (k):= exp [(-1/2 -l/log(r)

,cos8) = f(cos8).

+ Jk) log(r)] and let

B .= A(a*a) be the operator whose eigenfunctions are the generalized axially JI' e symmetric surface harmonics. Then F is a classical solution of the GASP-equation

+

{ar(r2ar)

~ ( a , a )} 8

F(r,cose)

= 0 , (r,e)E[osl)

x

(0,n)

in E 2 d 3 that is uniformly Abel summable to the boundary values F(1-, f(cos9).

When a = (n-3)/2, n=3,4,..

LaPlace equation in En.

And when a

cos8) =

., the equation is the axially symmetric =

-1/2 it is Laplacian on the unit disk in Ez.

REMARK: These examples verify the inclusion of the theory in [6,9-111. Additional examples may be constructed by application of the factor sequence method [ 2 ] to specific problems in [1,5].

THE CONVERSE BVP The fractional partial differential equation and associated BVP are direct

results of the summability process induced by factor sequence operators. The parallel with the classical harmonic function theory [4,6] is clearly established in the following representation theorem. THEOREM 1: The function F is a classical solution of the fractional partial differential equation

+

Iat(A)

B } F(t,cosB) JI

=

0, (t,e)

E

t(A)

X (0,~)

if, and only if, there is a sequence [gn}CCIO,n] such that (i)

1 lim ( 2n! 1 1 g ~ l)l/n

=

o

and, m

(ii)

F(t,cosB)=

1

n=O

[B 1" E gn(cosO).

J I J I

434

P A . filCC0,l

Proof: ( S u f f i c i e n c y ) Apply t h e second f a c t o r sequence o p e r a t o r t o each term This produces a s e t of i n e sequence {gn! t h a t s a t i s f i e s p r o p e r t y (i) functions

.

t o which a p p l i c a t i o n on t h e map [Bg]" y i e l d s a set of p a r t i c u l a r s o l u t i o n s whose sum is t h e formal series s o l u t i o n m

1

F(t,cosB) =

[B$]" E gn(cosO)

6

n=O

, (t,e)E

t(A) X (0,n).

To v a l i d a t e i t s uniform convergence, c o n s i d e r t h e bound m

I[BgIn E g (cos8)

6 n

I

1

I lg,ll

2nI

kn+2dl(e-Et)k

k=n

Y

2E

s

2

N.

Because of p r o p e r t y (i), we f i n d t h e formal series is a classical s o l u t i o n , having a uniformly convergent dominants on compacta of t h e domain. ( N e c e s s i t y ) A classical s o l u t i o n F may be expanded as f o l l o w s m

F(t,cosFl) =

1

anunE6gn(cose)

n=O

5

w i t h l i m ]anI1/n

For E

1.

>

0, l e t use reason as i n [6,10,11] t o f i n d a

sequence {a }n such t h a t k , n k=O

n a

n

=

1

k=O

a

k,n

,

la

k,n

I < (M(E) ~[2a]+~~+~k~~)/(([2a1+2n+l)!).

-

Now, c o n s t r u c t &he sequence of f u n c t i o n s m

g (t,cos9): =

n

for n

[2a] + 1.

0

1

k=n-[Za]-l

(-1)n a

E $as ~)(cos~)/($(k))n

k,n-[Zal-l

E s t i m a t i n g as i n [6,10] g i v e s t h e a p p r a i s a l m

t h a t shows gn(cos8): = gn(O+,cos8) with g (t,cosB):

= E

6

gn(cos8).

E

C[O,n] and l i m ( 2n! I Ig

and r e a r r a n g i n g t h e r e s u l t i n g s e r i e s g i v e s m

k

F i n a l l y , we see t h a t by summing t h e terms

m

OD

11

OD

E

and completes the proof. Interchanging the summation and integration defined by Fourier coefficient formula gives a Poisson type representation. Proceeding directly to the converse BVP, the space H* of hyperfunctions is the strong dual of the space H of analytic functions on [O,n] with the topology in [a]. To characterize the space H* in a compatible manner with the summation process, we reference [lo] for the following result.

6

THEOREM 2: For integers a,

m

A

1. 0 ,

the Fourier-Jacobi series f: = k i ;

converges to a hyperfunction on [O,n] if, and only if, limsup la Illk k

w

R(a~5) k k 1.

This characterization suffices to develop the natural isomorphism along the lines of [6,10,11] by defining the convolution @ : H* X H*

+

H* as

m

h

(f @ The space {H*, 0 , +}

3 )cose

: =

1

a b w R(a*B)(cos I). k k k k

k=O

is a commutative ring with identity element m

$(cosO):

=

1 w R(a,B)(cos8) k=O k k

to which the factor sequence operator is applied, yielding m

e (case): =

Jl

1

e-tJl(k) %$a,B)(cose).

k=O

From the summability properties of factor sequences [ 2 ] we conclude that

E -2(cose)

+

&cose)

J,

The proof of the next theorem follows

,t

+

o+.

[10,11] "quid pro quo".

THEOREM 3 : The function F is a classical solution of the fractional partial differential equation Iatw for intergers a

2$2

0

+

B

J,

} F(t,cosB)

= 0,

(t,e)c

t(A)

X (0,n) A

if, and only if, there is a unique f

E

H* such that

436 REFERENCES [l] K. Bauer 6 S. Ruscheweyh, Differential Operators for Partial Differential Equations 6 Function Theoretic Methods, Lecture Notes in Math., vol. 791 Spring-Verlag, New York, 1980. [21

H. Bravinck, Jacobi Series and Approximation, Mathematical Centre Tracts, No. 39, Mathematisch Centrum, Amsterdam, 1972.

[3] J. Douglas, Jr., Approximate continuation of harmonic 6 parabolic equations, Proc. of the Symposium on the Numerical Solution of Partial Differential Equations, Univ. of Maryland (1965) 353-364.

[4] I. Gelfand

6 G. Shilov, Quelques applications de la theories des fonctions generalisees, J.Math. Pures 6 Appl., vol. 35 (1956) 383-413.

[5] R. P. Gilbert, Constructive Methods of Elliptic Equations, Lecture Notes in Math., vol. 365, Springer-Verlag, New York, 1974.

16) G. Johnson, Jr., Harmonic functions in the unit disk, I J.Math., vol. 12 (1968) 366-396.

6 11,

Illinois

[7] J. Kelingos 6 P. Staples, A characterization of solutions to a perturbed LaPlace equation, Illinois J.Math., vol. 22 (178) 208-216. [8] M. Kothe, Topological Vector Spaces I, Springer-Verlag, New York, (1969).

[9] P. A. McCoy, Converse initial value problems for a class of heat equations, Lecture Notes in Math., vol. 964, Springer-Verlag , New York (1982) 494-502. [lo1

, Characterizations of solutions to the generalization CauchyRiemann system, J.Math.Analysis 6 Appl. (to appear).

[ll] A. Zayed, Hyperfunctions as boundary values of generalized axially symmetric potentials, Illinois J.Math., vol. 25 (1981) 306-317.

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

43 7

BOUNDS FOR CONSTRUCTED SOLUTIONS OF SECOND AND FOURTH ORDER INVERSE EIGENVALUE PROBLEMS J o y c e R . McLaughlin Department o f M a t h e m a t i c a l S c i e n c e s Rensselaer Polytechnic I n s t i t u t e T r o y , N e w York 12181

I n t h i s p a p e r bounds a r e g i v e n f o r c o e f f i c i e n t s , i n d i f f e r e n t i a l o p e r a t o r s , which h a v e b e e n cons t r u c t e d from s p e c t r a l d a t a . The d i f f e r e n t i a l o p e r a t o r s a r e o f s e c o n d and f o u r t h o r d e r and t h e bounds a r e g i v e n i n t e r m s o f d i f f e r e n c e s i n t h e g i v e n s p e c t r a l d a t a and s p e c t r a l d a t a from known (base) d i f f e r e n t i a l operators. INTRODUCTION

The s t u d y o f i n v e r s e e i g e n v a l u e problems o f s e c o n d and f o u r t h o r d e r i s m o t i v a t e d i n p a r t t h r o u g h t h e s t u d y o f s u c h problems as r e c o v e r i n g t h e c h a r a c t e r i s t i c impedance o r d e n s i t y o f an e l a s t i c medium, t h e recovery of t h e c r o s s s e c t i o n a l a r e a of t h e v o c a l t r a c t , o r t h e r e c o v e r y o f d e n s i t y and f l e x u r a l r i g i d i t y of a v i b r a t i n g b e a m f r o m knowledge o f n a t u r a l f r e q u e n c i e s and o t h e r s p e c t r a l d a t a . S e e I51 f o r a r e v i e w of t h e u n d e r l y i n g m o d e l s . The problem b e i n g c o n s i d e r e d h e r e i s t o o b t a i n bounds f o r c o n s t r u c t e d solutions. These bounds c a n b e i n t e r p r e t e d a s y i e l d i n g c o n t i n u i t y r e s u l t s o r s i m p l y a s showing how t h e Lm norm o f t h e d i f f e r e n c e between c o n s t r u c t e d c o e f f i c i e n t s and known ( b a s e ) c o e f f i c i e n t s d e pends on L1 d i f f e r e n c e s between t h e g i v e n s p e c t r a l d a t a and s p e c t r a l d a t a from a known ( b a s e ) problem. Two d i f f e r e n t s e c o n d o r d e r p r o b l e m s w i l l b e c o n s i d e r e d w i t h d i f f e r e n c e s i n t h e e x p e c t e d s o l u t i o n and t y p e o f o b t a i n e d bounds b e i n g noted. I n p a r t i c u l a r , i t i s shown t h a t t h e s o l u t i o n of t h e i n v e r s e problem may depend more ( o r l e s s ) on t h e h i g h e r o r d e r s p e c t r a l d a t a d e p e n d i n g o n t h e form of t h e d e s i r e d d i f f e r e n t i a l o p e r a t o r . This is a phenomenon which h a s b e e n o b s e r v e d n u m e r i c a l l y as w e l l . The method f o r f i n d i n g t h e s o l u t i o n i s t h a t a s s o c i a t e d w i t h a G e l ' f a n d L e v i t a n t y p e of c o n s t r u c t i o n , [ 2 ] ; t h a t i s e i g e n v a l u e s ( n a t u r a l f r e q u e n c i e s ) and L 2 norms ( o r L2 w e i g h t e d norms) of t h e e i g e n f u n c t i o n s w i l l b e g i v e n . A s i s well-known t h i s i s t h e d a t a c o n t a i n e d i n t h e s p e c t r a l f u n c t i o n and it c a n b e o b t a i n e d , a t l e a s t t h e o r e t i c a l l y , from i m p u l s e r e s p o n s e d a t a . T h e s e two problems w i l l b e d i s c u s s e d i n S e c t i o n s 1 and 2 . The bounds f o r o n e f o u r t h o r d e r p r o b l e m w i l l b e g i v e n i n S e c t i o n 3 . The d a t a i n t h i s case i s e i g e n v a l u e s p l u s two "norming" c o n s t a n t s and c a n a t l e a s t i n p r i n c i p a l b e o b t a i n e d from i m p u l s e r e s p o n s e d a t a I t i s a l s o t h e d a t a which i s c o n t a i n e d i n t h e s p e c t r a l m a t r i x f o r t h i s problem and as s u c h i s a n a t u r a l g e n e r a l i z a t i o n o f t h e d a t a g i v e n f o r t h e s e c o n d o r d e r problem.

J. R. McLaughlin

438 S e c t i o n 1:

Second o r d e r p r o b l e m s i n L i o u v i l l e n o r m a l form.

T h e s e c o n d o r d e r p r o b l e m t o be d i s c u s s e d h e r e is t h a t o f f i n d i n g g E L 1 ( O , l ) and r e a l c o n s t a n t s a and b i n t h e e i g e n v a l u e p ro b le m

(1)

YO' + ( X - q ) y = 0,

(2)

~ ' ( 0 )+ a y ( 0 ) = 0 = ~ ' ( 1 +) b y ( 1 ) .

O t x t l

The g i v e n d a t a w i l l be t h e e i g e n v a l u e s ,

h i

< ~2 <

..., a n d

norming

constants. The n o r m i n g c o n s t a n t s are t h e L 2 n o r m s , 1 p i = [ y ( x , X i ) J 2 d x , where y ( x , h ) is t h e s o l u t i o n of (1) w i t h 0 y ( 0 , X ) = 1, y ' ( 0 , X ) = -a. I t s h o u l d be n o t e d t h a t a n e q u i v a l e n t s e t o f n o r m i n g c o n s t a n t s are t h e v a l u e s y ( 0 , X i ) = a i i f y ( x , X i ) i s t h e e i g e n f u n c t i o n f o r ( 1 ) - ( 2 ) c o r r e s p o n d i n g t o X i a n d w i t h ~2 norm, 1 1 [ y ( x , X i I Z d x = 1. 0 I,t is a s s u m e d t h a t a l s o g i v e n a r e q * ( x ) E L 1 [ 0 , 1 ] a n d r e a l c o n s t a n t s a a n d b* f o r t h e ( b a s e ) e i g e n v a l u e p r o b l e m (3)

y" + (X-q*)y = 0 ,

(4)

y'(0)

+

a * y ( O ) = 0 = ~ ' ( 1 +) b * y ( l ) = 0 ,

*

<

To e s t a b l i s h n o t a t i o n l e t A 1

*

*

1

*

12

... be t h e e i g e n v a l u e s f o r ( 3 ) 1 , 2 ,..., w h e r e y * ( x , X ) s a t i s f i e s

<

[y*(x,Xi)12dx, i = 0 ( 3 ) a n d t h e i n i t i a l c o n d i t i o n s y * ( O , X ) = 1, y * ' ( O , X ) = -a*. ( 4 ) and p i =

I t is w e l l - k n o w n

t h a t i f t h e Xi's a n d p i ' s s a t i s f y c e r t a i n a s y m p t o t This derived coeff i c i e n t may be e x p r e s s e d by

i c f o r m s , see Z i k o v [ 7 1 , t h e n q ( x ) e x i s t s [ 2 ] .

where m

and X

y(X,Xi) = y*(x,Xi) +

*

*

y(X,Xi) = y*(X,Xi) +

J

k(X,t)y*(trXi)dt,

0

*

X

J

k(Xrt)y*(t,Xi)dt.

0

T h e bound t h e n o n t h e L m norm o f q

-

qo c a n b e s t a t e d a s follows:

T h e o r e m 1: S u p p o s e t h a t t h e e i g e n v a l u e p r o b l e m ( 3 ) - ( 4 ) and p i > 0 , i = 1,2 t h a t t h e s e q u e n c e s X i < A2 <

...

,...

i s g i v e n and are given

Bounds for Solutions

e x i s t 60, N

>

1 [lAi-X;1 i=l

+

I

-

f; I ( x i )

Then t h e r e

pi

P i

0 s u c h t h a t when 6

.

* 1/2 ] <

m

such t h a t 6 =

439

<

60

m

implies that q

l l q ( x ) - q o ( x ) II C N6. Remark:

The a s s u m p t i o n t h a t 6

<

-

q o is c o n t i n u -

ous. Remark: values.

T h e c o n s t a n t s 6 0 a n d N c a n be g i v e n d e f i n i t e n u m e r i c a l

Remark:

The a s s u m p t i o n t h a t 6

<

-

ensures that

T h e bound is t h e n o b t a i n e d by m a k i n g e s t i m a t e s u s i n g t h e f o r m u l a s

*

*

*

*

*

f o r y ( x , X ) , y ( x , A ) , y ( x , X i ) - y ( x , X i ) , and y ( x , A i ) - y ( x , X i ) i n t e r m s of t h e i n i t i a l v a l u e G r e e n ' s f u n c t i o n and boundary v a l u e G r e e n ' s f u n c t i o n s b o t h f o r ( 3 ) - ( 4 ) a n d (1)-(2). S e c t i o n 2:

Second o r d e r p r o b l e m s i n n o n c a n o n i c a l form.

The d i f f e r e n t i a l e q u a t i o n i n t h e s e c o n d o r d e r p r o b l e m o f S e c t i o n 1 is g i v e n i n L i o u v i l l e normal form. I t is n a t u r a l , h o w e v e r , i n a number o f m a t h e m a t i c a l p r o b l e m s ( e . g . when s e e k i n g t h e c h a r a c t e r i s t i c i m p e d a n c e i n a n e l a s t i c medium) t o s t a t e t h e i n v e r s e p r o b l e m i n w i t h A > 0 a n d A ( 0 ) = 1, t h e f o l l o w i n g form. Find A(x) E C1[O.ll, and r e a l c o n s t a n t b l , i n t h e e i g e n v a l u e problem

+

XAw = 0,

(5)

( A w l ) '

(6)

~ ' ( 0 + ) a l w ( 0 ) = 0 = ~ ' ( 1 +) b l w ( 0 ) .

What is g i v e n a r e t h e r e a l c o n s t a n t a 1 > 0 , 0 < A 1 < A2 < I n a d d i t i o n , i f w(x,A) w h i c h s a t i s f i e s w ( 0 ) = 1, w ' ( 0 ) = - a i r t h e n 1 A{w(x,Ai)12dx = p i , i = 1 , 2 a r e also

... .

0

,...,

and e i g e n v a l u e s is t h e s o l u t i o n o f ( 5 ) t h e w e i g h t e d L2 n o r m s given.

I t is i m p o r t a n t

t o n o t e t h a t a d i f f e r e n c e i n t h i s i n v e r s e p r o b l e m a n d t h a t of S e c t i o n 1 is t h a t h e r e t h e l e f t b o u n d a r y c o n d i t i o n c a n b e s p e c i f i e d a n d is not d e t e r m i n e d s o l e l y by t h e s p e c t r a l d a t a . T h e c o n s t r u c t i v e t e c h n i q u e u n d e r c o n s i d e r a t i o n now is a n a l o g o u s t o t h e G e l ' f a n d - L e v i t a n t e c h n i q u e , and h a s some s i m i l a r i t i e s t o t h a t of S a n t o s a and C a r r o l l [ l ] f o r t h e s e m i - i n f i n i t e l i n e . If A ( x ) E c 2 [ 0 , 1 ) t h e n a s t a n d a r d c h a n g e of v a r i a b l e s w o u l d p u t t h e e q u a t i o n in L i o u v i l l e n o r m a l f o r m a n d t h e t h e o r e m o f S e c t i o n 1 would

J.R. McLaughlin

440

2 With t h e weaker g i v e a bound o n I I d B/BIIm w h e r e [ B ( x ) I 2 = A ( x ) . dx2 s m o o t h n e s s a s s u m p t i o n , h o w e v e r , i t is e s s e n t i a l t o c o n s i d e r t h e above problem s e p a r a t e l y . T h i s s e p a r a t e a p p r o a c h y i e l d s a bound o n I I B ' / B I I ~a n d s h o w s a s m a l l e r c o n t r i b u t i o n t o t h e d e r i v e d f u n c t i o n B’/B f r o m h i g h e r o r d e r s p e c t r a l d a t a ( i . e . f r o m X i , p i f o r l a r g e i ) . I n o r d e r t o s t a t e t h e t h e o r e m w e b e g i n a s b e f o r e w i t h a known ( b a s e ) L e t A*(x) = [ B * ( x ) I 2 F C1[O,l], w i t h A * ( O ) = 1 a n d A* > 0 . problem.

*

0 , b l > 0 be r e a l and l e t t h e e i g e n v a l u e p r o b l e m

L e t a1

+

(7)

(A*w')'

(8)

~ ' ( 0 + ) a l w ( 0 ) = 0 = ~ ' ( 1 +) b l w ( 0 )

XA*w = 0

<

have eigenvalues 0

*

*

*

A 1

< A2 <

w ( 0 , X ) = 1, w ' ( 0 , X ) = - a 1

*

1

A*[w*(x,Xi)]2dx.

... .*

I f w * ( x , A ) s a t i s f i e s ( 7 ) and

*

t h e n l e t p i be t h e w e i g h t e d L 2 norm p i =

T h e f o l l o w i n g t h e o r e m s t a t e s t h e r e s u l t on t h e

0 b o u n d f o r B’/B. T h e o r e m 2: Suppose t h a t t h e e i g e n v a l u e problem t h a t t h e s e q u e n c e s 0 < X 1 < A2 < and p i > 0 ,

...

m

given such t h a t u =

Z

[IXi-Xll

i=l

+

I LP i

.&

d dx

Remark:

log

II

1,2,

...

are

I(X*

P i

a n d t h e r e e x i s t s a v o a n d a n N 1 > 0 s u c h t h a t when u II

8 ) is g i v e n and

< uo

< N1u

B*

The a d d i t i o n a l f a c t o r

1

which a p p e a r s i n u and n o t i n 112

[A;]

t h e bounding t e r m 6 f o r q ( x ) shows t h a t B ( x ) i s l e s s s e n s i t i v e t o changes i n higher o r d e r s p e c t r a l d a t a than q ( x ) . Remark:

The c o n s t r u c t i v e t e c h n i q u e is d e v e l o p e d by w r i t i n g

B ( x ) w ( x ,X )

= B* ( x ) w * ( x , h )

+ , f X k ( x ,t ) B * ( t ) w * ( t ,X ) d t 0

and r e q u i r i n g t h a t w ( x , X i ) be complete and o r t h o g o n a l i n L2 w i t h 1 weighted inner product A ( x ) f ( x ) g ( x ) d xf o r f , g E L 2 ( 0 , 1 ) . 0

Bourids f b r Solutions

44 1

Remark: T h e b o u n d is d e t e r m i n e d by r e w r i t i n g t h e d i f f e r e n t i a l e q u a t i o n s as w"

+

'

B

w

I

+

* '

Xw = 0 a n d w" + (B w B*

I

+ Xw

= 0

a n d u t i l i z i n g r e p r e s e n t a t i o n s of w ( x , X ) , w * ( x , X ) a n d [ w ( x , X i ) -

*

*

w * ( x , ~ i ) i ( J ) ,[ w ( x , x i ) - w * ( x , x i ) l ( I ) , j = 0, 1 , i = 1 , 2 , . . . in terms o f t h e i n i t i a l v a l u e and boundary v a l u e G r e e n ' s f u n c t i o n s . S e c t i o n 3:

A f o u r t h o r d e r problem.

I n t h i s s e c t i o n t h e f o u r t h o r d e r problem to be c o n s i d e r e d w i l l be a g e n e r a l i z a t i o n o f t h e s e c o n d o r d e r p r o b l e m of S e c t i o n 1. F o r ease i n e x p r e s s i n g t h e i d e a s t h e r i g h t hand boundary c o n d i t i o n s are a s s u m e d t o be D i r i c h l e t b o u n d a r y c o n d i t i o n s . I t is a l s o i m p o r t a n t t o n o t e t h a t t h e s p e c t r a l d a t a t h a t is c h o s e n a s known is a n a t u r a l extension of t h a t i n t h e second o r d e r problems. T h a t is, t h e t h r e e g i v e n s e q u e n c e s o f d a t a are c o n t a i n e d i n t h e s p e c t r a l m a t r i x f o r t h e f o u r t h o r d e r problem. To be s p e c i f i c w e s e e k c o e f f i c i e n t s A ( x ) E C ' [ O , l ] , B ( x ) c C [ O , l l a n d r e a l n u m b e r s a , b , c so t h a t t h e s e l f a d j o i n t e i g e n v a l u e p r o b l e m y(4)

+

(Ay(l))(1)

+

By

-

Xy

=

0,

0 = Y ( ~ ) ( O )+ ( b + A ( O ) ) y ( l ) ( O )

+

cy(O),

... .

Two s e t s o f n o r m i n g c o n s t a n t s a r e h a s e i g e n v a l u e s A1 < X2 < a l s o a s s u m e d k n o w n , i . e . i f y ( x , h i ) is t h e e i g e n f u n c t i o n c o r r e s p o n d i n g t o X i w i t h L2 n o r m IIyII = 1 t h e n t h e t w o r e m a i n i n g s e q u e n c e s t o W e also be g i v e n are y ( 0 , X i ) = yi, ; ( 1 ) ( O , X i ) = c i , i = 1,2, a s s u m e k n o w l e d g e of a " b a s e p r o b l e m w i t h c o e f f i c i e n t s A* c C ' [ O , l l , B* E C [ O , l l r e a l c o n s t a n t s a * = b* = c = 0 , a n d e i g e n v a l u e s a n d

....

*

norming c o n s t a n t s X i ,

*

yi,

*

ti.

T h e m e t h o d of c o n s t r u c t i o n f o r t h e s o l u t i o n is a G e l ' f a n d - L e v i t a n V o l t e r r a i n t e g r a l e q u a t i o n a p p r o a c h where A and B are u l t i m a t e l y c o n s t r u c t e d by s o l v i n g a n i n t e g r a l e q u a t i o n w h o s e k e r n e l a n d nonh o m o g e n e o u s term i s c o n s t r u c t e d f r o m s o l u t i o n s o f t h e " b a s e " p r o b lem. T h e m e t h o d o f p r o o f is d e s c r i b e d i n [ 6 ] a n d w i l l b e o m i t t e d here. H o w e v e r , f o r e a s e i n u n d e r s t a n d i n g how t h e b o u n d f o r A , B , d . ( x ) is achieved t h e f o l l o w i n g e x p r e s s i o n s f o r t h e s o l u t i o n of dX t h e i n v e r s e problem are g i v e n .

- 2

3 d dx3

k(x,x)

J. R. McLaughlin

442

*

.. .

i = 1,2,. The f u n c t i o n y ( x , X i ) s a t i s f i e s t h e d e r i v e d d i f f e r e n t i a l e q u a t i o n , t h e t w o d e r i v e d l e f t b o u n d a r y c o n d i t i o n s and a d d i -

tional i n i t i a l conditions

*

*

*

*

y(0,Ai) = yi,y(l)(O,Xi) = ti

+

* y i k(O,O),

i = 1,2,...

.

T h e f o l l o w i n g bound c a n be o b t a i n e d . Theorem: 0

Assume t h e a b o v e m e n t i o n e d b a s e p r o b l e m a n d r e a l s e q u e n c e s

..., y j ,

< A 1 < X2 <

ci,

i = 1,2

,...

are g i v e n such t h a t

m

I y * ( x , A i ) } i = l is c o m p l e t e i n L2(0,1), a n d u

*

*

*

ti = -yi(Xi)

1/41

<

where

* 1-6je

1

* * 3/4 + I y j - ~ j l ( ~ i ) + l ~ i - ~ i l ( x i ) i=l

T h e n t h e r e e x i s t s u o > 0 , N2 > 0 s u c h t h a t when u < 00 t h e s o l u t i o n o f t h e i n v e r s e p r o b l e m d e s c r i b e d a b o v e s a t i s f i e s t h e bound

Bounds for Solutions

IIL(A-A iim, I I B - B I I ~ CN

IIA-A~II~,

dx

Remark:

0

443

u2.

Ag'ain 00 a n d N2 c a n b e g i v e n e x a c t n u m e r i c a l v a l u e s .

Remark: T h e p r o o f u t i l i z e s e s t i m a t e s w h i c h are made i n a s i m i l a r way t o t h o s e o f t h e s e c o n d o r d e r p r o b l e m . I n t h i s c a s e , however, t h e i n t e g r a l r e p r e s e n t a t i o n s t h a t a r e used involve t h e i n i t i a l value G r e e n ' s f u n c t i o n s , t h e G r e e n ' s f u n c t i o n s f o r t h e s e l f a d j o i n t bounda r y v a l u e p r o b l e m s a l r e a d y s t a t e d , and t h e G r e e n ' s f u n c t i o n s f o r t h e n o n s e l f a d j o i n t boundary v a l u e p r o b l e m s where one o f t h e r i g h t hand b o u n d a r y c o n d i t i o n s is o m i t t e d a n d t h e l e f t b o u n d a r y c o n d i t i o n y ( 0 ) = 0 is added.

References C a r r o l l , R . , and S a n t o s a , F . , S c a t t e r i n g t e c h n i q u e s f o r a one d i m e n s i o n a l i n v e r s e p r o b l e m i n g e o p h y s i c s , Math. Meth. i n t h e A p p l . , S c i . 3 , ( 1 9 8 1 ) , 145-171.

G e l ' f a n d , I . M. a n d L e v i t a n , B . M . , On t h e d e t e r m i n a t i o n o f a d i f f e r e n t i a l e q u a t i o n f r o m i t s s p e c t r u m , I z v . Akad. Nauk SSSR S e r . M a t h . 15 ( 1 9 5 1 ) , 3 0 9 - 3 6 0 ; Amer. Math. SOC. T r a n s . , 1 ( 1 9 5 5 ) , 253-304. H a l d , 0. H., T h e i n v e r s e S t u r m - L i o u v i l l e p r o b l e m w i t h s y m m e t r i c p o t e n t i a l s , A c t a . Ma th . , 1 4 1 ( 1 9 7 8 ) , 263-291. L e v i t a n , 8. M . , On t h e d e t e r m i n a t i o n o f a S t u r m - L i o u v i l l e e q u a t i o n by two s p e c t r a , I z v . Akad. Nauk, SSSR S e r . Mat. 28 ( 1 9 6 4 ) 6 3 - 7 8 ; Amer. Math. SOC. T r a n s l . , 6 8 ( 1 9 6 8 ) , 1 - 2 0 . McLaughlin, J o y c e R., A n a l y t i c a l methods f o r r e c o v e r i n g c o e f f i c i e n t s i n d i f f e r e n t i a l e q u a t i o n s from s p e c t r a l d a t a , ( t o appear). McLaughlin, J o y c e R., Higher o r d e r i n v e r s e e i g e n v a l u e problems, O r d i n a r y and P a r t i a l d i f f e r e n t i a l e q u a t i o n s , P r o c e e d i n g s , D u n d e e , S c o t l a n d , 1 9 8 2 , S p r i n g e r - V e r l a g , 503-518. Z i k o v , v. V . , On i n v e r s e S t u r m - L i o u v i l l e p r o b l e m s o n a f i n i t e s e g m e n t , Math. U S S R - I z v e s t i j a , 1 ( 1 9 6 7 ) , 923-934.

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445

DIFFERENCE EQUATION MODELS OF DIFFERENTIAL EQUATIONS HAVING ZERO LOCAL TRUNCATION ERRORS Ronald E. Mickens Department o f Physics Atlanta University A t l a n t a , Georgia 3031 4 U.S.A.

I t i s suggested t h a t t h e r e e x i s t s a “ b e s t f 1 d i f f e r e n c e equation model o f a d i f f e r e n t i a l equation. An e x i s t e n c e theorem i s proven f o r t h e case o f o r d i n a r y d i f f e r e n t i a l equations. The d i f f i c u l t i e s of g e n e r a l i z i n g this result t o p a r t i a l d i f f e r e n t i a l equations a r e discussed. INTRODUCTION Most d i f f e r e n t i a l equations which occur i n s c i e n t i f i c a p p l i c a t i o n s a r e nonlinear and e x a c t a n a l y t i c s o l u t i o n s e i t h e r a r e not known or, i f known, can not be put i n a form s u i t a b l e f o r u s e f u l c a l c u l a t i o n s (Fornberg and Whitham, 1978; Hyman, 1982; P o t t e r , 1973; Whitham, 1974). The s i t u a t i o n i s h a r d l y b e t t e r f o r l i n e a r d i f f e r e n t i a l equations where t h e c o e f f i c i e n t s a r e f u n c t i o n s of t h e independent v a r i a b l e ( s) I n p r a c t i c e , this s i t u a t i o n l e a d s one t o attempt a numerical i n t e g r a t i o n of t h e e q u a t i o n ( s ) o f i n t e r e s t ( C a r r i e r and Pearson, 1976; Hildebrand, 1968). O f t h e many numerical techniques, t h e method of f i n i t e d i f f e r e n c e s i s of p a r t i c u l a r importance ( M i t c h e l l , 1969) s i n c e t h e i r s o l u t i o n s can, i n g e n e r a l , be r e a d i l y obtained by use of high speed computers. The question immediately a r i s e s as t o whether t h e r e i s a ” b e s t ” d i f f e r e n c e equat i o n model o f a d i f f e r e n t i a l equation. I n t h e n e x t s e c t i o n , a d i s c u s s i o n o f such schemes i s presented f o r ordinary d i f f e r e n t i a l equations (ODE’S). For o w purposes, t h e Ilbest” d i f f e r e n c e scheme t u r n s o u t t o be one which has zero l o c a l t r u n c a t i o n e r r o r s . There are a number of reasons why w e would seek such schemes; they include (i)t h e f a c t t h a t t h e s t e p s i z e s can be chosen t o have an a r b i t r a r y value ( f o r a p a r t i c u l a r problem t h e a p p r o p r i a t e s t e p s i z e would be determined by physical c o n s i d e r a t i o n s ) and (ii)q u e s t i o n s of convergence, s t a b i l i t y and cons i s t e n c y (Hildebrand, 1968) would n o t a r i s e . I n t h e t h i r d s e c t i o n , we attempt t o g e n e r a l i z e t h e r e s u l t s obtained f o r ODE’S t o Some fundamental d i f f i c u l t i e s a r i s e . p a r t i a l d i f f e r e n t i a l equations (PDE’ s ) F i n a l l y , i n t h e l a s t s e c t i o n , r e l a t e d a r e a s of i n v e s t i g a t i o n a r e discussed.

.

ORDINARY DIFFERENTIAL EQUATIONS Consider a f i r s t - o r d e r ODE (I)

dt

= f(t,x)

,

x ( t o ) = xo

and a f i r s t - o r d e r o r d i n a r y d i f f e r e n c e equation (OAE)

(2)

Xk+l = g(k,xk)

-

Consider t h e following q u e s t i o n ( P o t t s , 1982; Van d e r Vaart, 1973): Given a n a r b i t r a r y f i r s t - o r d e r ODE, can a f i r s t - o r d e r OAE be determined t h a t has t h e same g e n e r a l s o l u t i o n ? By t h e same general s o l u t i o n , we mean, f o r an a r b i t r a r y

446

R.E. Mickens

constant s t e p - s i z e h. t h e values k' x ( t ) of Eq. ( 1 ) by

s a t i s f y i n g Eq. ( 2 ) a r e r e l a t e d t o t h e s o l u t i o n

(3)

tk = hk.

x(hk) = xk

t

where

+

From a geometrical viewpoint t h i s means t h a t xk s a t i s f y i n g Eq. (2) w i l l be p o i n t s on t h e s o l u t i o n curve x ( t ) of Eq. ( 1 ) a t successive values t = hk, k an i n t e g e r . I f t h e above question can be answered i n t h e a f f i r m a t i v e , then it means t h a t it i s p o s s i b l e t o construct a f i n i t e d i f f e r e n c e model of an ODE t h a t has zero l o c a l t r u n c a t i o n e r r o r . Such a scheme w e can i d e n t i f y as being t h e llbest'l i n f i n i t e difference model of t h e o r i g i n a l ODE. We now show t h a t i f Q. (1) has a well-defined s o l u t i o n , t h e n a f i n i t e d i f f e r e n c e scheme having this property e x i s t s Assume t h a t t h e g e n e r a l s o l u t i o n t o Eq. (1

i s known and i s given by

,

x ( t ) = b(t,to,xo)

(?a)

.

The "group propertyf1 (Nemytski and where xo i s t h e value of x ( t ) a t t = t Stepanov, 1969) of t h e s o l u t i o n allowsothe r e s u l t x ( t t h) = b ( t t h , t , x ( t ) )

(4)

If we now t a k e t = kh, x(hk) = xk, t h e n Eq. ( 4 )

where h i s an a r b i t r a r y constant. be comes

(5)

k'

t 1 = b(h(k

f

,

1 )

hk

,

xk)

and t h i s i s t h e ex l i c i t form of t h e r e q u i r e d OAE which has t h e same general s o l u t i o n as Elq. ( l y . This method can be e a s i l y generalized t o determine a system of nonlinear OAE's t h a t has t h e same general s o l u t i o n as a system of nonlinear

ODE’ s

.

The following two elementary examples i l l u s t r a t e t h e above technique.

'ktl

-

zxk

t 4x

k

g t x =A ;

(7)

xft,

=

(xk 2

' xk-l s i n2 (-)wh 2

-

=

0

A ) exp(-2h)

+ A

m.

From ( 4 ) it i s seen t h a t t h e above procedure l e a d s t o an e x p l i c i t f i n i t e difference model f o r t h e corresponding d i f f e r e n t i a l equation. Further, i t should be c l e a r t h a t a l l we have i s e s s e n t i a l l y an e x i s t e n c e theorem which allows u s t o conclude t h a t such schemes e x i s t , but not how t o c o n s t r u c t them f o r a n a r b i t r a r y ODE. I n general, t h e r e are two problems which a r i s e i n t h e c o n s t r u c t i o n o f f i n i t e They r e l a t e d i r e c t l y t o how one should model t h e , difference models of ODE'S. r e s p e c t i v e , d e r i v a t i v e and nonlinear terms of t h e ODE by f i n i t e d i f f e r e n c e expressions. F i r s t , t h e replacement of a d e r i v a t i v e term of a given o r d e r by a f i n i t e difference expression of a higher o r d e r w i l l , i n g e n e r a l , l e a d t o "chaotic" behavior i n t h e s o l u t i o n t o t h e OAE which i s not p r e s e n t i n t h e ODE (Hildebrand, Our conclusion i s t h a t i n t h e c o n s t r u c t i o n of 1968; Yamaguti and Ushiki, 1981) t h e llbestll OAE corresponding t o an ODE, t h e o r d e r of t h e OAE and ODE must be t h e same. Second, t h e r e i s a nonuniqueness as t o how t h e nonlinear terms i n t h e ODE should be modeled. For example, i n t h e equation

.

(8)

2 d ; t x ? = O

dt

,

Difference Equation Models

447

t h e nonlinear term could, i n p r i n c i p l e , be represented by any one o f t h e following expressions (5)) "k3 9 xk+l ( xE-l) l 2 t (Xk+l 3 xk 3 xk-l) 3 / 3 B

xz ’

'

'

none o f which when combined with standard r e p r e s e n t a t i o n s f o r t h e second-order d e r i v a t i v e term w i l l give t h e Ilbestll d i f f e r e n c e scheme as defined above.

A t t h e p r e s e n t time, t h e way t o proceed i s probably through t h e experimental cons t r u c t i o n of d i f f e r e n c e schemes f o r p a r t i c u l a r c l a s s e s of ODE'S in t h e hope t h a t e v e n t u a l l y a set of r u l e s w i l l emerge f o r o b t a i n i n g t h e Ifbest" OAE models f o r members of t h e s e c l a s s e s . Given an OAE model of a ODE, w e must check it t o see if it i s t h e "best" model. This can be done by c a l c u l a t i n g t h e l o c a l t r u n c a t i o n e r r o r ; i f it i s zero, then the scheme i s t h e " b e s t T f . PARTIAL DIFFERENTIAL EQUATIONS

A major complication i n t r y i n g t o c o n s t r u c t f i n i t e - d i f f e r e n c e models of PDE's i s t h e l a c k of a n unambiguous d e f i n i t i o n of a general s o l u t i o n t o a PDE. This i s , i n p a r t , r e f l e c t e d i n t h e f a c t t h a t one can formally d e r i v e f i n i t e - d i f f e r e n c e schemes which have zero l o c a l t r u n c a t i o n e r r o r s but a r e o f i n f i n i t e order. For example, t h e parabolic equation f o r u( x, t ) (10)

ut = L(t,x,D,D2)u

where L i s a l i n e a r o p e r a t o r and D = a/ax has t h e following Ilbest" f i n i t e - d i f f e r e n c e model ( M i t c h e l l , 1969) (11)

= expw(hm,kn,; 2 sinh-l

u;+l

, (h 2 a i d - ' +12

u:

where t h e x and t g r i d spacings a r e h and k, and 6 i s t h e c e n t r a l d i f f e r e n c e o p e r a t o r i n t h e x - d i r e c t i o n (Hildebrand, 1968). N8te t h a t while this i s a n e x p l i c i t two-level t i n e scheme, it i s formally of i n f i n i t e o r d e r i n t h e d i s c r e t e space v a r i a b l e . However, t h e r e a r e a number o f PDE's f o r which t h e "bestf1 d i f f e r e n c e scheme can be found; they i n c l u d e t h e advective and wave equations:

ODE (12)

Ut

+

u

OAE =

un t l - 'mn-1

0

where i n each case Ax = A t . Again, as i n t h e case f o r ODE'S, we must f o r PDE's "experiment" with p a r t i c u l a r classes of PDE's i n s e a r c h o f p o s s i b l e r u l e s f o r c o n s t r u c t i n g t h e "best" f i n i t e - d i f f e r e n c e models. Recently, w e have begun such a study on t h e two equations (14)

ut t uu

=

o

where a i s a constant.

Ut=

xx

I n t e r e s t i n g and p o s i t i v e r e s u l t s have been obtained.

DISCUSSION The above comments r e a d i l y show t h a t t h e g e n e r a l t a s k of c o n s t r u c t i n g Ifbest" f i n i t e - d i f f e r e n c e models of ODE'S and PDEIs i s d i f f i c u l t . However, t h e d i f f i c u l t y i s lessened somewhat by t h e r e a l i z a t i o n t h a t i n t h e a n a l y s i s of physical problems t h e d i f f e r a n t i a l equations of i n t e r e s t a r e g e n e r a l l y l i m i t e d t o a r a t h e r s m a l l c l a s s of equations ( P o t t e r , 1973). This f a c t a t l e a s t shows u s where t h e i n i t i a l e f f o r t s should be placed. Even p a r t i a l success could g i v e important r e s u l t s . For example, i n t h e c o n s t r u c t i o n of f i n i t e - d i f f e r e n c e schemes f o r nonlinear ODE'S with constant c o e f f i c i e n t s , i f t h e nonlinear term can be considered a small

R.E. Mickens

448

p e r t u r b a t i o n , then a u s e f u l (though not t h e " b e s t " ) model might c o n s i s t o f r e placing t h e l i n e a r part of t h e ODE by i t s "best" f i n i t e - d i f f e r e n c e counterpart and then modeling t h e nonlinear part i n some a p p r o p r i a t e fashion. Thus f o r one-dimens i o n a l nonlinear o s c i l l a t o r s one can i n s u r e t h a t t h e unperturbed period h a s t h e c o r r e c t value; standard methods g i v e f i n i t e - d i f f e r e n c e equations which do not have t h i s property ( P o t t s , 1982). I n view of t h e s e l a s t remarks, it would be of importance t o have p e r t u r b a t i o n techniques a v a i l a b l e f o r t h e c o n s t r u c t i o n of approximate a n a l y t i c s o l u t i o n s t o t h e d i f f e r e n c e equations. To d a t e , t h e s e e f f o r t s have not been very productive s i n c e t h e s o l u t i o n s contain s e c u l a r terms (Huston, 1970; Torng, 1960). Recently, we have begun a d e t a i l e d study of s e r i e s s o l u t i o n s t o a p a r t i c u l a r c l a s s of nonlinear d i f f e r e n c e equations having s m a l l n o n l i n e a r i t i e s . I t w a s shown t h a t uniformly v a l i d asymptotic s e r i e s s o l u t i o n s could be obtained (Mickens, 1982). E f f o r t s t o extend t h e s e c a l c u l a t i o n s t o p a r t i a l d i f f e r e n c e equations a r e p r e s e n t l y underway. F i n a l l y , it should be mentioned t h a t a number of r e s e a r c h e r s have invented methods f o r constructing f i n i t e - d i f f e r e n c e analogues of nonlinear PDE's which e x h i b i t e i t h e r s p e c i a l types of s o l u t i o n s ( e . g . , s o l i t o n s ) o r e x a c t s o l u t i o n s . The r e c e n t paper of Hirota (1982) provides a good summary of t h e r e s u l t s obtained t o date.

ACKNOWLEDCEMIENT This research has been supported i n p a r t by a g r a n t from t h e National Science Foundation. REFERENCFS C a r r i e r , G. F. and Pearson, C. E., P a r t i a l D i f f e r e n t i a l Equations: Theory and Technique (Academic, New York, 1976)

.

Fornberg, B. and W h i t h a m , G. B., A Numerical and Theoretical Study o f Certain Nonlinear Wave Phenomena, P h i l . Trans. of Royal Soc. (London) 289 (1978) 373-404. Greenspan, D.,

D i s c r e t e Models (Addison-Wesley; Reading, MA; 1973).

Hildebrand, F. B., Finite-Difference Equations and Simulations (Prentice-Hall; Englewood C l i f f s , N J ; 1968). Hirota, R., Difference Analogues of Nonlinear Evolution Equations i n Hamiltonian Form, Tech. Rep. No. A-12, Hiroshima University (August 1982). Huston, R. L., Krylov-Bogolijubov Method f o r Difference Equations, Appl. Math. 19 (1970) 334-339.

SIAM J.

Hyman, J. M . , Numerical Methods f o r Nonlinear D i f f e r e n t i a l Equations, LA-8927-MS. Los Alamos National Laboratory (January 1982). Mickens, R. E., A Method f o r Obtaining P e r i o d i c S o l u t i o n s t o A P a r t i c u l a r Class of Nonlinear Difference Equations, J. o f F r a n k l i n I n s t i t u t e (accepted f o r p u b l i c a t i o n , 1982). Mitchell, A. R., Computational Methods i n P a r t i a l D i f f e r e n t i a l Equations (Wiley, New York, 1969). Nemytski, V. V. and Stepanov, V. V., Q u a l i t a t i v e Theory o f D i f f e r e n t i a l Equations (Princeton University P r e s s ; Princeton, N J ; 1969).

DiJjerencc Equation Models

449

(11)

P o t t e r , D.,

(12)

P o t t s , R. B., D i f f e r e n t i a l and Difference Equations, Am. Math. Monthly (1982) 402-407.

(13)

Torng , H. C. , Second-Order Nonlinear Difference Equations Containing Small Parameters, J. of Franklin I n s t i t u t e 269 ( 1960) 97-1 04.

(1 4 )

Van d e r Vaart, A Comparative I n v e s t i g a t i o n of Certain Difference Equations and Related D i f f e r e n t i a l Equations: I m p l i c a t i o n s for Model Building, B u l l . Math. Biology 35 (1973) 195-211.

Computational Physics (Wiley, New York, 1973).

(1 5) Whitham, G. B., 1974). (16)

3,

Linear and Nonlinear Waves (Wiley-Interscience, New York,

Yamaguti, M. and Ushiki, S., Chaos i n Numerical Analysis o f Ordinary D i f f e r e n t i a l Equations, Physica 3D (1981 ) 618-626.

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45 1

A REMARK ON CONTINUUM EIGENFUNCTIONS OF N-BODY SCHRUDINGER OPERATORS P e t e r A. P e r r y * Department o f Mathematics C a l i f o r n i a I n s t i t u t e o f Technology Pasadena, C a l i f o r n i a 91 125

Let

H be a g e n e r a l i z e d N-body S c h r d d i n g e r o p e r a t o r on IRn.

We s t u d y e x p o n e n t i a l decay p r o p e r t i e s o f p o l y n o m i a l l y bounded

s o l u t i o n s t o S c h r d d i n g e r ' s e q u a t i o n Hu = Xu, where X l i e s i n t h e n e g a t i v e e s s e n t i a l spectrum o f H, u s i n g Agmon's method. Our r e s u l t s show t h a t such s o l u t i o n s decay e x p o n e n t i a l l y o u t s i d e an i n f i n i t e , c l o s e d subset o f R n d e p e n d i n g on A and s o r e p r e s e n t bound c l u s t e r s o f p a r t i c l e s . These r e s u l t s part i a l l y c h a r a c t e r i z e n e g a t i v e energy continuum e i g e n f u n c t i o n s o f H. Let

H be a g e n e r a l i z e d N-body S c h r d d i n g e r o p e r a t o r on L 2 ( IRn), H = -A + xr=l Vi(IIix),

where -A i s t h e L a p l a c i a n on IRn, Xi

o f Rn, and t h e Vi

the

p r o j e c t o n t o vi-dimensional

i s t h e L a p l a c i a n on Xi;

1

+ l)-'

t h a t vanish a t

i s a compact o p e r a t o r on L2(Xi) F o r such V i ,

s e l f - a d j o i n t o p e r a t o r w i t h e s s e n t i a l spectrum

C<

subspaces

t h e r e s u l t s we d e s c r i b e a l s o h o l d t r u e f o r

i p o t e n t i a l s w i t h l e s s l o c a l r e g u l a r i t y [9]). that

(11

a r e r e a l - v a l u e d , measurable f u n c t i o n s on IRvi

i n f i n i t y , say i n t h e sense t h a t Vi(-Ax. (here -AX

ni

i. e . ,

&,-)

0 s t r i c t l y here t o avoid t r i v i a l i t i e s ) .

H i s a semibounded,

f o r some

1 <_

0 (we w i l l assume

As i s w e l l known ( c f . [13],gC.5,

f o r e x t e n s i v e d i s c u s s i o n and r e f e r e n c e s ) H has an expansion i n p o l y n o m i a l l y bounded e i g e n f u n c t i o n s .

These e i g e n f u n c t i o n s a r e polynomial l y bounded, d i s t r i b u -

t i o n a l s o l u t i o n s o f t h e e q u a t i o n Hu =

Au f o r

h i n t h e spectrum o f

H.

The d e t a i l e d s t r u c t u r e o f t h e e i g e n f u n c t i o n s i s b o t h q u i t e complex and o f great physical i n t e r e s t .

For

X < C, t h e e i g e n f u n c t i o n s a r e L2 e i g e n f u n c t i o n s o f

H, c o r r e s p o n d i n g t o t o t a l l y bound s t a t e s o f t h e many-body system:

f o r a guide t o

t h e e x t e n s i v e l i t e r a t u r e on t h i s s u b j e c t , see [ 1 3 ] , g C . 3 , and see a l s o [4] r e c e n t work.

For " t y p i c a l "

X

for

e i g e n f u n c t i o n s o f H w i l l n o t be L2 because t h e

many-body system can s e p a r a t e i n t o bound c l u s t e r s moving f r e e l y w i t h r e s p e c t t o each o t h e r ( t h e r e may be e x c e p t i o n a l values o f X

>_Cwhere

L2 e i g e n f u n c t i o n s occur,

b u t under s l i g h t l y s t r o n g e r hypotheses on t h e V i t h a n assumed here, such values form a closed, c o u n t a b l e s e t [3,7,10]).

F o r c e r t a i n 3-body Schrddinger o p e r a t o r s ,

techniques o f time-independent s c a t t e r i n g t h e o r y y i e l d e x p l i c i t e i g e n f u n c t i o n expansions [ 2 , 6 ] , b u t an e x p l i c i t e i g e n f u n c t i o n expansion f o r a l l A has n o t y e t

P.A. Perry

452

been c o n s t r u c t e d f o r t h e l a r g e r c l a s s o f o p e r a t o r s we c o n s i d e r h e r e .

F o r an i n t r o -

d u c t i o n t o N-body systems and f u r t h e r r e f e r e n c e s , see [ll]. The s i t u a t i o n we want t o c o n s i d e r i s t h e case o f " t y p i c a l "

X

E

breakup i n t o c l u s t e r s occurs b u t some p a r t i c l e s must s t i l l be bound.

B,O), where We w i l l

a p p l y Agmon's [l]approach t o e x p o n e n t i a l decay o f s o l u t i o n s o f second-order e l l i p t i c e q u a t i o n s , which h e used t o s t u d y L2 e i g e n f u n c t i o n s o f S c h r d d i n g e r operators w i t h h c with h E [c,O)

C.

We w i l l show t h a t p o l y n n m i a l l y bounded s o l u t i o n s o f Hu = hu

a r e " c o n c e n t r a t e d " on an i n f i n i t e c l o s e d s u b s e t o f R n depending on

h and decay e x p o n e n t i a l l y o u t s i d e t h a t s e t a t an e x p l i c i t r a t e .

To d e s c r i b e o u r r e s u l t , we i n t r o d u c e some i d e a s and n o t a t i o n o f [1,3]. be t h e s e m i l a t t i c e o f subspaces generated by X1, inclusion.

For a d i r e c t i o n w E S"',

orthogonal t o

...,

we l e t X(w) be t h e maximal subspace o f g

and we d e f i n e

0.

+ 1. l : x i C x ( w ) v.1 (nix).

H ( w ) = -A

(2)

That i s , H(w) i s o b t a i n e d f r o m H b y removing t h o s e V i which v a n i s h as 1x1 the w direction.

Note t h a t X(w) i s t h e z e r o subspace f o r a.e. w e S n - l .

C(w) (o(A) i s t h e spectrum of A). = infE(w) :w B

C ( w ) = i n f o(-A)

X<

S"'1 = 0.

-+ m

in

We s e t

= i n f o(H(w)).

C(W) i s a l o w e r semicontinuous f u n c t i o n on Sn-'

t a k i n g values i n a f i n i t e s u b s e t o f [ c , O ]

C

L e t di

p a r t i a l l y o r d e r e d by

Xn,

( t h e "HVZ Theorem,"

( t h e r e a r e f i n i t e l y many H(w)), cf.

[1,12]).

and

Note t h a t f o r a.e.

W,

Agmon r l ] showed t h a t L2 e i g e n f u n c t i o n s o f H w i t h e i g e n v a l u e

decay a t an e x p o n e n t i a l r a t e determined by t h e d i s t a n c e f r o m t h e o r i g i n i n

the metric ds2 = ( c ( x / I x I ) - A ) dx2

(3)

where dx2 is t h e E u c l i d e a n m e t r i c on IRn. Suppose now t h a t u s o l u t i o n o f Hu = h u w i t h

Hioc(@)

B

X

E

C(w) - h i s n o n p o s i t i v e on a subset c l o s e d subset o f S n - l components.

i s a p o l y n o m i a l l y bounded, d i s t r i b u t i o n a l

[c,O).

The m e t r i c ( 3 ) i s no l o n g e r d e f i n e d s i n c e

s(X)o f S"'.

I t i s easy t o see t h a t ?(A)

is a

o f measure z e r o w i t h a f i n i t e number o f closed, connected

On t h e complement S ( h ) o f

s(X),we have C ( w )

p o s i t i v e c o n s t a n t c s i n c e z has range i n a f i n i t e s e t . t h a t u i s "concentrated" i n the i n f i n i t e closed s e t % ( A )

-

h L c for a strictly

The f o l l o w i n g r e s u l t shows = I x B IRn : x = r w f o r some

N

rTO, w E S ( A ) l and decays e x p o n e n t i a l l y i n t h e complementary s e t Q ( h ) . To s t a t e the result,

let C(w;h)

and l e t p(x;h)

= max(h,C(w))

denote t h e d i s t a n c e f r o m 0 t o x i n t h e m e t r i c ds2 = c ( x / \ x l ) - A ) dx2.

Eigenfunctions of N-Body Schrodinger Operators = 0 if x

Note t h a t p(x;A)

and p(x;A)

%(A)

c dist(x,x(X))

= { f 8 L ~ o c ( I R n ) : ~ d x ( l1+x 1 2 ) s l f ( x ) ( 2 <

L e t L:(IRn)

if x E

n(A).

m l w i t h t h e obvious norm.

I f u i s p o l y n o m i a l l y bounded, u E L I S ( I R n ) f o r some s > n/2.

We have:

1 Let u E Hkoc( IRn) be a poZynorniaZZy bounded, d i s t r i b u t i o n a z soZution o f

Theorem. HU = A U

453

with

x

[ c , o ) and

E

s>

choose

o

so t h a t u E L $ ( I R ~ ) ) .

fien for

6 > 0,

e x p [ ( l - G ) ~ ( ~ ; x ) I u ( x ) e L!~(IR~). 2 The p r o o f o f Theorem 1 f o l l o w s b y m d i f y i n g Agmon's t h e o r y t o o b t a i n L-, e s t i m a t e s 2 i n s t e a d o f L e s t i m a t e s : see [9] f o r d e t a i l s . U s i n g s u b s o l u t i o n e s t i m a t e s ( c f . [l]o r [13],

g C . 1 ) one can r e c o v e r a p o i n t w i s e bound l u ( x ) ~5 c 6 ( l + I ~ I ) - ~e x'p ~( - ( l - b ) p ( x ; A ) )

t r u e f o r a.e. x E

n(A)

and a l l 6 > 0.

(4)

We remark t h a t t h e e x p o n e n t i a l r a t e i n ( 4 )

need n o t be o p t i m a l , j u s t as t h e c o r r e s p o n d i n g e s t i m a t e s f o r L2 e i g e n f u n c t ons need

[4,51).

n o t be o p t i m a l (Cf.

i n g physical s i g n i f i c a n c e .

The s e t :(A)

where u i s c o n c e n t r a t e d has t h e f o l 1ow-

n the

A nonzero p o i n t x = r w l i e s i n ? ( ( A ) i f X l i e s

spectrum o f H(o) : H(w) i s a c l u s t e r decomposition H a m i l t o n i a n [ll]t h a t d e s c r i b e s f r e e r e l a t i v e m o t i o n o f t h e c l u s t e r s bound b y t h e p o t e n t i a l s V i w i t h Xi c X(W). The f a c t t h a t A E H(w) means t h a t A i s an a l l o w e d energy f o r such f r e e m o t i o n t o occur.

The system i s bound i n t h e c o o r d i n a t e s o f X ( W ) ( t h e " i n t e r n a l " c o o r d i n a t e s

of bound c l u s t e r s ) and f r e e i n a l l o r t h o g o n a l c o o r d i n a t e s ( t h e c o o r d i n a t e s o f f r e e r e l a t i v e m o t i o n between c l u s t e r s ) .

Hence, one expects e x p o n e n t i a l decay o f u when

t h e c o o r d i n a t e s i n X(w) grow (away f r o m z ( A ) ) b u t n o t when t h e y a r e a l l z e r o (in

:(A)). One m i g h t e x p e c t t h a t t h e s o l u t i o n s we hve c o n s i d e r e d a r e bounded on ?(A).

The f o l l o w i n g example shows t h a t t h i s need n o t be t h e case. Example.

L e t n = ml + m2 and w r i t e x E IRn as (x1,xz) w i t h x i E IRmi,i

L e t - A i be t h e L a p l a c i a n on IRmi,

l e t II(x1,x2)

= ( x i ,0)

= 1, 2.

and choose v E C;(W

m

I ) so

t h a t H1 = -A1 + v ( x 1 ) has a s i n g l e L 2 e i g e n v e c t o r cp(x1) w i t h e i g e n v a l u e Xc 0 . have I V ( X ) I c c6 e - ( l - G ) A l x l f o r

a l l 6 > 0 ([1],[7]).

Now l e t H

=

-A1

-

A2

+

We

V(nx)

on L2(IRn). Then

C

$(xl,x2)

P(x2) s o l v e s Hu = Xu, decays e x p o n e n t i a l l y i n a l l d i r e c t i o n s w

=?(XI)

=

X and i f P ( x 2 ) i s any harmonic p o l y n o m i a l , t h e f u n c t i o n $

w i t h nw # 0, and grows p o l y n o m i a l l y i n some o f t h e d i r e c t i o n s w i t h IIw = 0 .

454

P.A. Perry

REFERENCES. Agmon, S., L e c t u r e s on e x p o n e n t i a l decay o f second-order e l l e p t i c equations: Bounds on e i g e n f u n c t i o n s o f N-body S c h r d d i n g e r o p e r a t o r s ( P r i n c e t o n U n i v e r s i t y Press, P r i n c e t o n , 1982). Faddeev, L. O., Mathematical Aspects o f t h e Three-Body Problem i n Quantum S c a t t e r i n g Theory ( I s r a e l Program f o r S c i e n t i f i c T r a n s l a t i o n , Jerusalem, 1965). Froese, R.,

49 (1982),

and H e r b s t , I . , A new p r o o f o f t h e Mourre e s t i m a t e , Ouke Math. 1075-1085.

J.

Froese, R., and H e r b s t , I . , E x p o n e n t i a l bounds and absence o f p o s i t i v e eigenvalues f o r N-body S c h r d d i n g e r o p e r a t o r s , Comm. Math. Phys. (1983), 429 44 7.

-

F oese, R., Herbst, I . , Hoffmannn-Ostenhof, M., and Hoffmann-Ostenhof, T . , Lf;-exponential l o w e r bounds t o s o l u t i o n s o f t h e Schrddinger e q u a t i o n , Comrn. Math. Phys. 87 (1982), 265-286.

Mercuriev, S. P., On t h e three-body coulomb s c a t t e r i n g problem, Ann. Phys. 395-426.

130 (1980),

Mourre, E., Absence o f s i n g u l a r continuous spectrum f o r c e r t a i n s e l f - a d j o i n t o p e r a t o r s , Comm. Math. Phys. 78 (1981), 391-408. O'Connor, A. J., E x p o n e n t i a l decay o f b o u n d - s t a t e wave f u n c t i o n s , Comm. Math. PhyS. 32 (1973), 319-340. P e r r y , P.,

i n preparation.

Perry, P., S i g a l , I . M., and Simon, B., S p e c t r a l a n a l y s i s o f N-body Schrddinger o p e r a t o r s , Ann. Math. (1981), 519-567. See a l s o Reference 3.

114

Reed, M., and Simon, B . , Methods o f Modern Mathematical Physics, 111: S c a t t e r i n g Theory (Academic Press, New York, London, 1979). Reed, M., and Simon, 8., Methods o f Modern Mathematical Physics, IV: A n a l y s i s o f Operators (Academic Press, New York, London, 1978), Simon, B.,

*

S c h r d d i n g e r semigroups, B u l l . A.M.S.

B a n t r e l l F e l l o w i n Mathematical Physics

1 (1983),

447-526.

DIFFERENTIAL EQUATIONS 1.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

45 5

A CURIOUS SINGULAR PERTURBATION PROBLEM

1

Paul H . Rabinowitz D e p a r t m e n t o f Mat hema t i c s U n i v e r s i t y o f Wisconsin-Madison

In t h i s l e c t u r e w e w i l l d i s c u s s a n u n u s u a l s i n g u l a r p e r t u r b a t i o n problem f o r To b e g i n c o n s i d e r t h e e q u a t i o n

an ordinary d i f f e r e n t i a l equation.

'

where

d dt

denotes

CiIT The o p e r a t o r

and

= Ev E

g

CJ(IR,IR)

is

. '

and

Likewise i f w e d e f i n e

CiT.

v

is a n i s o m o r p h i s m , i . e .

L

t i n u o u s i n v e r s e between and

217

is

t.

ZIT

.

periodic}

Let

is a c o n t i n u o u s l i n e a r map w i t h a cono r more g e n e r a l l y b e t w e e n

c217

HJ

periodic in

Cin

t o b e t h e c l o s u r e of

Cj+2 2rr with

respect t o

then

is a n i s o m o r p h i s m between

L

Hj+’

Hj.

and

Now c o n s i d e r t h e non-

l i n e a r equation Lu

(2) where

h

is

ZIT

+

periodic i n

Eh(t,u,u',u'') t.

Suppose

= 0

i n its arguments.

h E C1

It

i s t h e n a n e a s y e x e r c i s e t o u s e t h e C o n t r a c t i n g Mapping Theorem o r I m p l i c i t F u n c t i o n Theorem t o show ( 2 ) h a s a c u r v e o f s o l u t i o n s E

for

near

E

0

with

u(t;O)

=

u(t;&)

p a r a m e t r i z e d by

0.

The p r o b l e m r e f e r r e d t o i n o u r t i t l e i s t h e f o l l o w i n g .

(3) where

Lu

f

is

( 3 ) f o r small

Zn E.

+

Ef(t,u,u',u",n"')

periodic i n

t

=

0

and we s e e k a

Zrr

periodic s o l u t i o n of

T h i s is a kind of s i n g u l a r p e r t u r b a t i o n problem i n t h a t the

term o f h i g h e s t o r d e r i s m u l t i p l i e d by a small p a r a m e t e r .

However u n l i k e u s u a l

s u c h s i t u a t i o n s , w e do n o t a s s u m e a n y t h i n g o t h e r t h a n t h e s m o o t h n e s s of respect t o the

u"'

f

with

a r g u m e n t and t h e r e f o r e it may n o t b e p o s s i b l e t o c o n v e r t

P.H. Rabinowitz

45 6 ( 3 ) t o a n e q u a t i o n of t h e form

-

N e v e r t h e l e s s w e s h a l l show i f all small

.

= f(E,t,U,U',U")

I11

(4)

i s s u f f i c i e n t l y smooth, (3) h a s a s o l u t i o n f o r

f

T h i s i s a b i t s u r p r i s i n g i n t h a t w e a r e accustomed t o d i f f e r e n t i a l

E.

e q u a t i o n s i n which t h e h i g h e s t o r d e r t e r m s a r e dominating o r a t l e a s t i m p o r t a n t w h i l e h e r e t h e b e h a v i o r of

f

in

is irrelevant.

u'"

Before d e s c r i b i n g how we t r e a t ( 3 ) , i t i s worth n o t i n g t h a t r e p l a c e d by an o p e r a t o r of o r d e r order

j

+

1

provided t h a t

become e v i d e n t l a t e r .

and

j

f

could b e

L

can depend on d e r i v a t i v e s o f

is appropriately coercive i n a sense that w i l l

L

Thus o t h e r examples of t h e s i t u a t i o n w e w i l l d e s c r i b e a r e

(5)

U'

+u +

ch(t,u,u',u")

0

=

and (6)

u

+

cg(t,u,u1) = 0

The t e c h n i c a l i t i e s i n v o l v e d a r e e s s e n t i a l l y t h e s a m e f o r a l l cases.

-

(7)

where

x = (x,,..

.,xn),

d i s c u s s t h e P.D.E.

f

u

n o t e s d e r i v a t i v e s of

Au

+u +

cf(x,D'u)

is periodic i n

of o r d e r

\a\

= 0

15 i

xi,

101

for

5 n,

5 3.

and

Du'

de-

However w e w i l l n o t

case f u r t h e r h e r e .

There i s one s p e c i a l c a s e of (3) t h a t is e a s y t o t r e a t . independent of

In f a c t t h e

a n a l o g u e s of ( 3 ) l i k e

methods we u s e can a l s o b e a p p l i e d t o P.D.E.

Suppose

f

is

Then it is n a t u r a l t o s e e k a t i m e independent s o l u t i o n

t.

U(E)

which t h e r e f o r e s a t i s f i e s (8)

+

u

Ef(U,O,O,O)

=

.

0

The I n t e r m e d i a t e Value Theorem t h e n p r o v i d e s s o l u t i o n s of (8) f o r s m a l l o b s e r v a t i o n does n o t seem t o h e l p i f Since

L

(3) i s a P i c a r d

depends e x p l i c i t l y on

iteration

scheme of t h e form

u

0

Lu = - c f ( t , u n , u 'n' u" n''n n+l

k f

E

C

, u1

E

k+2 C2rr , u2

E

k+l C2n , u3

=

0

and f o r

This

Even i f

f

E

Cm,

1. 0,

.

11')

k E

n

C2T,

etc. so our i t e r a t i o n

l o s s e s a d e r i v a t i v e a t e a c h s t e p and w e cannot i t e r a t e i n d e f i n i t e l y . method f a i l s .

c.

t.

is i n v e r t i b l e , a n a t u r a l approach t o t a k e i n a t t e m p t i n g t o s o l v e

(9) However i f e.g.

f

convergence i s d u b i o u s .

Thus t h i s

F o r t u n a t e l y methods of

Newton o r r a p i d convergence t y p e have been developed by v a r i o u s a u t h o r s , expecia l l y Nash [l] and Moser [ 2 ] ,

t h a t c a n sometimes h a n d l e problems in which t h i s l o s s

457

A Curious Singular Perturbation Problem S e e a l s o Schwartz [ 3 ] , S e r g e r a e r t [ 4 1 ,

of d e r i v a t i v e s phenomenon o c c u r s .

Hamilton [51, Zehnder [ 6 1 , & Hormander [ 7 1 .

W e w i l l show how Moser's method [ 2 ]

can b e a p p l i e d t o o u r s i t u a t i o n . I n 121 and e a r l i e r work, Moser p r e s e n t e d t e c h n i q u e s t o t r e a t l o s s of d e r i v a t i v e and s m a l l d i v i s o r problems. t h e f o l l o w i n g problem: find

near

u

F

t i v e of

uO

I n p a r t i c u l a r , s t a t e d i n o u r s e t t i n g , he s t u d i e d l r s C (H , H ) , F(uo) = q0, and V near

E

q03

such t h a t

F(u)

If

= IP.

F'(uo),

were a n isomorphism between

uo

at

F

For

Hr

and

c i t F u n c t i o n Theorem immediately p r o v i d e s t h e s o l u t i o n . i n t e r e s t , a s i n o u r above example, may e x i s t b u t b e unbounded, e.g.

t h e Frechet derivaHS,

may n o t b e i n v e r t i b l e o r t h e i n v e r s e

f'(uo)

i t may l o s e d e r i v a t i v e s .

A s a f u r t h e r conseu E Hr

quence of t h i s l o s s of d e r i v a t i v e s , one cannot e x p e c t t o f i n d Ip E HS

and n e a r

f o r some

p <

r.

U

=

lu

11 ]Is, 11 lip, 11 u - uOIlp <

( i i ) For

+ v)

meters

=

11 F (u) \ I s

r'

U n Hr.

Stated rather lose-

grows a t most l i n e a r l y i n

F'(IP)v = g

possesses "appropriately"

( i i i ) One can g e t " a p p r o p r i a t e " estimates f o r

smooth "approximate" s o l u t i o n s ; c ( u , v ) = F(u

Hp

i n Moser's c o n s t r u c t i o n .

U

and

1)

u E U r' t h e equation

( i ) For u E U

11 \ I o

and

E HpI

l y , Moser shows i f :

1111 1 ,;

f o r each

e x i s t s a t a l l , it may o n l y belong t o

F-'(q)

If

C o n t r o l of s e v e r a l norms i s t h e r e f o r e r e q u i r e d f o r such pro-

11 I [ ,

blems, e . g . Let

q0.

t h e Impli-

However i n problems of

- F(u) - F ' ( u ) v

for

u E U

and

v

E

( i v ) The p a r a -

Hr;

and a d d i t i o n a l ones t h a t o c c u r i n making t h e above q u a l i f i e r s

p,r,s

" a p p r o p r i a t e " and "approximate" p r e c i s e t i e t o g e t h e r i n an a p p r o p r i a t e f a s h i o n ; and

(v)

F(u)

= 9

uO

F(u) = V ,

i s a good enough approximate s o l u t i o n t o

U.

has a solution i n

of a p p r o x i m a t e s o l u t i o n s t o

then

I n f a c t Moser c o n s t r u c t s a sequence

F(u) = 9

un

which converge t o a s o l u t i o n

um

'r in

HP.

W e w i l l make t h e s e vague s t a t e m e n t s p r e c i s e and show how t h e r e s u l t i n g

theorem g i v e s a s o l u t i o n of ( 3 ) . tion

F(u) = q

for a l l large and

11 u 1 ,

$(MI,

K,

< K.

of o r d e r

FI

there exists

(where

u

a

Q

11 v 1 , II

(12)

and

v

u

v

F * ( U )-~

(11)

11,

W e s a y t h e equa-

(where

11

X

> 0)

if

lpl10

E U such t h a t F(u) < K-' K r F ' ( u ) v = g a d m i t s approximate s o l u t i o n s

> 0) i f t h e r e e x i s t s a c o n s t a n t

II

whenever

X

u = u

We s a y t h e e q u a t i o n

and f o r a l l l a r g e

(10)

Two d e f i n i t i o n s a r e i n o r d e r .

h a s an approximate s o l u t i o n of o r d e r

Q

gl/o

E

HS

5

+(M)KQ-'

c > 0,

a function

satisfying the estimates

,

5 +(M)KQ ,

F' ( u ) v II 0

1) ) I s

z c II v 1l0 )I ]lo

(where X E U g < K-' < K, g 5 MK, and r ' g E HS, I ) u satisfy 5O below). The main d i f f i c u l t y i n a p p l y i n g Moser's r e s u l t

u

P.H Rabinowitz

45 8

is i n f i n d i n g approximate s o l u t i o n s of Theorem 1 3 [ 2 1 :

M

and

p,6,

1"

u

2'

For a i l l a r g e

3"

The e q u a t i o n

E

F

Let

11

implies

U

2-8

5M

~ I V IJo

/I Q(u,v)II, (i)

B E (0,U

(ii)

p <

(iii)

0 <

(iv)

0 <

r

X X+2 6

\Is

5 MK0,

IIv

1,

+

1)

h L ( 1- 2 X+l I;+l p+1

~ p ~c l l ~ m,

)Iu

and

11,

< K,

u,

v E H ~ ,

such t h a t i f

p o s s e s s e s a Cauchy sequence

X.

urn = l i m u n-)ao n'

If

11 P

(un)

u

t h e proof of

E,

provided

E

w e mean t h e map

E

and

Kgh

of approx-

F(u_) = P .

w i l l depend c o n t i n u o u s l y on

urn

(By c o n t i n u o u s l y on

~ ~ 1 1 <,

-

in

depends c o n t i n u o u s l y on a p a r a m e t e r , s a y

i n 3' d o e s .

v

u r'

u E

KO = Ko(M,c,B,u,X) > 0 = Ip

Theorem 1 3 shows t h e s o l u t i o n that

F(u) -

u E U

s) .

F

If

when

for a l l

<

imate s o l u t i o n s of o r d e r Remark 1 4 :

B

1 1 < y(p

F(u)

11

and

< M

c,p,X,

a d m i t s a p p r o x i m a t e s o l u t i o n s of o r d e r

+

Then t h e r e e x i s t s a IP

lo

F ( u ) l l s 5 MK

F'(u)v = g

5"

and suppose t h e r e e x i s t c o n s t a n t s such t h a t F E C 1 (Ur,H 0 )

$(M)

F(u) - p o l

1)

K,

4'

1)

: Hr + HS

and a f u n c t i o n

F'(u)v = g

E

+

u(~,E)

C ( [ - E ~ , E ~, ]I T P ) ) .

belongs t o

We w i l l n o t d e s c r i b e t h e proof of Theorem 1 3 o t h e r t h a n t o s a y t h a t t h e

sequence

(u,)

u = u n+l n

i s constructed inductively with

is a n approximate s o l u t i o n of o r d e r F'(un)vn

=

+ vn

where

v

of

h

+P

-F(u,)

n

K = Kn = KE

with

f o r some

K

E

(1,2)

and

Q = Q(Kn).

Now w e w i l l v e r i f y t h a t Theorem 1 3 c a n b e a p p l i e d t o ( 3 ) .

It w i l l b e seen

t h a t w i t h t h e e x c e p t i o n of 3 " , a l l of i t s hypotheses a r e e a s i l y s a t i s f i e d . begin we make a t e c h n i c a l m o d i f i c a t i o n of ( 3 ) .

(1 (1

for

E

small.

C

arguments a r e l e s s t h a n 1 i n magnitude should m a t t e r . by a c u t o f f f u n c t i o n which e q u a l s 1 i f a l l magnitude and which v a n i s h e s whenever any

Hence

f

when i t s

arguments a r e

"u"

"u"

f

c

2

f in

argument exceeds 1 i n magnif,

w e can assume

and each of i t s d e r i v a t i v e s i s u n i f o r m l y

bounded. Now s e t define

r = k

+

3

and

s = k

where

in

"u"

Thus by m u l t i p l y i n g

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

0

We expect a s o l u t i o n n e a r

T h e r e f o r e o n l y t h e v a l u e s of

To

k

w i l l be chosen l a t e r and

f

45 9

A Curious Singular Perturbation Problem Next set

p =

Thus a l l f u n c t i o n s i n

5.

i . e . t h e r e i s an

such t h a t

R > 0

u

U

a r e u n i f o r m l y bounded i n

U

implies

E

C

4

,

(15)

F

and p = 0.

1

0

C (H~,H ).

E

uo

W e choose

0, p o

=

F(0) = s f ( t , 0 , 0 , 0 , 0 )

=

and t a k e

o f Theorem 13, w e r e q u i r e a "Composition o f

T o h e l p v e r i f y lo and 2'

Functions Inequality". P r o p o s i t i o n 16 [ 2 ] : then

t,

w

F(t,.) with

H'

F(t,E)

If : H ’

Ijw

11

5

Ci(R

2

,R ) ,

1,

j

R,

then

11

F(',w)

2a

and i s

-

Moreover t h e r e e x i s t s

HJ.

-t

E

c =

c(R)

periodic in

such t h a t i f

L

Ilj

iZ(R)(IIwllj

.

+ 1)

A s a n e a s y consequence of C o r o l l a r y 1 6 and (15) w e have Corollary 17:

If

u

U

E

/I

and

f

C

E

k

,

then

( c ( R ) ( / / u / / k + 3 + 1)

f(',u,u',u",u'l')I/k

C o r o l l a r y 1 7 immediately shows t h e r e i s a n is satisfied.

F

For Z 0 , n o t e t h a t

II mil, 5

: U

Hk

-f

+

2llu

f(-,u,u’,u",u"’)

I E l

M

E

Hk

and

-

such t h a t 1 ’ of Theorem 13

and

II f(.,U,U',U'',U"') II k

(18)

< 211ullk+, + F ( R ) ( l l u j l k + 3 + 1)

I E ~5

v i a C o r o l l a r y 1 7 provided t h a t

1

and

V e r i f i c a t i o n of 3" w i l l be saved f o r l a s t .

M

5m

is s u i t a b l y l a r g e .

To g e t 4 " , a n e s t i m a t e i s re-

quired f o r O_(U,W)= F(u Since a l l

u E U

r' f = f(t,x0,x1,x2,x3)

where

f

some

B(t)

v E Hr

(0,l).

are in

C

3

- F(u)

,

- F'(u)v

.

by T a y l o r ' s theorem f o r

we have

is evaluated a t E

+ v)

(t, u

+

Bv, u '

+

e v ' , u"

+

Bv", u"'

+

ev"’)

T h e r e f o r e by above remarks, t h e r e i s a c o n s t a n t

pending on o u r t r u n c a t i o n of

f

such t h a t

al

for de-

P. H. Rabinowitz

460 Taking

L2

norms i n (19) t h e n y i e l d s

II v II II v 113

II Q ( U , V ) 1l0 5

(20)

C

By s t a n d a r d i n t e r p o l a t i o n i n e q u a l i t i e s [ 2 ] : k-1 k+3 (21) Ilc3 5 a3 110

11

11

4 -

k+3 1lk+3

11

and

(22)

Thus f o r

5

=

p o s s i b l y s t i l l l a r g e r (20)-(22)

M

show 4" i s s a t i s f i e d w i t h

7 / (k+3).

u

Setting e.g.

5" h o l d s .

k X = -

4

-

1,

and

k = 19,

i t is e a s y t o check t h a t

T h e r e f o r e assuming 3" f o r t h e moment, Theorem 1 3 p r o v i d e s a

depending on (23)

= k - 1,

M,c,B,h,u

(i) (ii) (iii)

11 P - voIIo < II uoIlk+3 < K~ 1) qIlk 5 m0

hold, (3) has a s o l u t i o n . satisfied.

KiO"

Since

IE~

Choosing

KO > 1

such t h a t i f

[I v

uo = q = 0 ,

( i i ) and ( i i i ) a r e t r i v i a l l y

s m a l l , w e t h e n have

- v0((

=

(1

Ef(t,O,O,O,O)

(lo

.

<

Thus by Theorem 1 3 and Research 1 4 , (3) p o s s e s s e s a c u r v e of s o l u t i o n s u(~,E) 3 E for E near 0 with u ( t , E ) E CZT and u ( t , O ) = 0. p a r a m e t r i z e d by

11 u ( t , E ) 11

In particular

<

1 7

for

E

s m a l l so t h e r a n g e of

t h e r e g i o n i n which

f

Theorem 2 4 :

i s s u f f i c i e n t l y smooth, t h e r e i s a n

f o r each

If E

i s continuous

E

f

'has

[-E~,E~],

in

E

and

lies in

u

not been m o d i f i e d and w e have shown:

(3) h a s a s o l u t i o n u(t,O)

=

E~

3 u ( ~ , E )E C Z T .

> 0

such t h a t

Moreover

u(~,E)

0.

It remains o n l y t o check 3 " , i . e . t o v e r i f y (10)-(1’2).

I n e q u a l i t y (12)

happens t o be v a l i d i n d e p e n d e n t l y of how we choose approximate s o l u t i o n s . S e t af A. A.(t,u) - (t,u,u',u",u"'), i = 0,1,2,3. We w i l l g e n e r a l l y s u p p r e s s t h e 1 1 ax. t dependence of' Ai and o f

Thus

46 1

A Curious Singular Perturbation Problem F'(u)v

+v +

-v"

=

EA(U)V

Now (26)

so (12) holds with

1 -. 4

c =

T h e r e a r e many ways i n w h i c h o n e m i g h t t r y t o f i n d smooth a p p r o x i m a t e s o l u t i o n s of

F'(u)v = g.

T h i s i s t h e most a d h o c s t e p i n t r y i n g t o v e r i f y t h e

h y p o t h e s e s of Theorem 1 3 .

See e.g.

[Z]

f o r a d i s c u s s i o n of t h i s q u e s t i o n .

w i l l f i n d e x a c t s o l u t i o n s of a n e l l i p t i c r e g u l a r i z a t i o n of

t h e r e s u l t i n g functions s a t i s f y (10)-(11).

where

y > 0 , v(m)

m = m(k)

2

>

0

y

(30)

f

If

5

1

E

u

Ck+', <

and

Y ak

i s odd,

m

such t h a t f o r

Moreover i f

where

if

w i l l be chosen s h o r t l y

1

P r o p o s i t i o n 29: E

0

=

/I

v(m)

m 2

=

.

if

k

m

i s e v e n , and

,

t h e r e e x i s t s an k+m+2 (28) h a s a u n i q u e s o l u t i o n v E H g

E

H

is s u f f i c i e n t l y s m a l l ,

11

I/k+m+l +

d e p e n d s o n l y on

/L+2

5

11 g Ilk

+

11 A(u)

Ilk )

k.

Assuming P r o p o s i t i o n 29 f o r t h e moment, t h e v e r i f i c a t i o n o f (10)-(11) be c a r r i e d o u t .

Suppose t h a t

We

a n d show

Consider

and

E

E ~ ,

11 g

F'(u)v

11 g Ilk

5 MK

and

11 ~ / l <~ K+ ~ w h e r e

can

K > 1.

P.H. Rabinowitz

462 Then by (30) and C o r o l l a r y 1 7 ,

and

(32) y < 1.

T h e r e f o r e by i n t e r p o l a t i o n , f o r a l l

1

m-2

1

(33)

-- 1 $(M)

where

= c a (M

o v e r choosing

l k m k

+

2c(R)).

+

1,

Taking

Q = y

m-1

,

(33) y i e l d s (11).

by ( 3 3 ) and t h e d e f i n i t i o n of

u(=k - l),

More-

w e have

so (10) h o l d s . Proof of P r o p o s i t i o n 29:

The proof r e l i e s on i d e a s from [81.

W e must prove t h e

e x i s t e n c e and uniqueness of a s o l u t i o n o f (28) and t h e n o b t a i n t h e estimate ( 3 0 ) .

m

For t h e e x i s t e n c e and u n i q u e n e s s , assume

with of

E

v)

i s even ( o t h e r w i s e w e a r g u e i n a

Note t h a t from (25)-(26) w e have

related fashion).

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

T h i s c o e r c i v i t y (which i m p l i e s t h e u n i q u e n e s s

and a s t a n d a r d Hahn-Banach Theorem t y p e argument

there exists

f o r a l l smooth

v

E

P.

-

see [ 8 , p.23-261

such t h a t

H"

An easy r e g u l a r i t y argument t h e n shows

b o o t s t r a p arguments show

v

E

Hm+2+k

if

g e H

k

.

v

E

m+2 H .

The proof of t h e estimate (30) is somewhat l e n g t h y and t e c h n i c a l . w e w i l l only s k e t c h its proof.

Forming

show

Standard

Therefore

See [ 8 ] o r [ 9 ] f o r r e l a t e d d e m o n s t r a t i o n s .

A Curious Singular Perturbation Problem

-

dj (-. (Lv f E A ( u ) v ) , dt'

(37)

and a r g u i n g i n p a r t a s i n ( 2 5 ) - ( 2 6 )

- v

(j+2)

y

F o r t h e s t e p s t h a t h a v e been o m i t t e d we need

t o be s u f f i c i e n t l y s m a l l .

Corollary 17.

This r e s t r i c t i o n implies

term:

A IIcl + IIA IIv

I] 3,

and IIv

L a s t l y (38) y i e l d s (30)'with

114

IIg

< 1

and

t h e a i d of

Indeed w r i t i n g (35) a s

k

taking t h e

)

( w i t h some f u r t h e r i n t e r p o l a t i o n i n e q u a l i t i e s

r e q u i r e d ) l e a d s i n d u c t i v e l y t o (30) w i t h o u t t h e

p l a y s a r o l e i n t h e form o f ( 3 8 ) .

46 3

-

1

norm of e a c h s i d e of (39) and u s i n g C o r o l l a r y 1 7 and (38)

y i e l d s (30). Remark 40:

Using estimates a s i n (25)-(26) i t i s s t r a i g h t f o r w a r d t o e s t a b l i s h 3 o f t h e s o l u t i o n we have o b t a i n e d .

t h e l o c a l uniqueness i n e.g. Remark 4 1 :

CZT

T o s i o Kato h a s found a n o t h e r a p p r o a c h t o (3) u s i n g t h e s t a t i o n a r y

v e r s i o n of h i s t h e o r y of q u a s i l i n e a r e v o l u t i o n e q u a t i o n s .

His a p p r o a c h g i v e s a

sharper existence statement:

H

f

E

C3

and a s o l u t i o n i n

5

.

REFERENCES : Nash, J . , The embedding of Riemannian m a n i f o l d s , h e r . Math. 63 (1956) 20-63. Moser, J . , A r a p i d l y c o n v e r g i n g i t e r a t i o n method and n o n 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 e q u a t i o n s I & I I ,Ann. S c u o l a Norm.Sup.Pisa

20 (1966) 265-315 &

499-535. S c h w a r t z , J. T . ,

On N a s h ' s i m p l i c i t f u n c t i o n theorem, Comm. P u r e Appl. Math.

1 3 (1960) 509-530. S e r g e r a e r t , F.,

Un theoreme d e f o n c t i o n s i m p l i c i t e s s u r c e r t a i n s e s p a c e s

d e F r e c h e t e t q u e l q u e s a p p l i c a t i o n s , Ann. S c i . E c o l e Norm Sup ( 4 s e r . )

5 (1972) 599-660. Hamilton, R. S . , A.M.S.

The i n v e r s e f u n c t i o n theorem of Nash and Moser, B u l l .

(new s e r i e s ) 7 (1982) 65-222.

Zehnder, E . ,

G e n e r a l i z e d i m p l i c i t f u n c t i o n theorems w i t h a p p l i c a t i o n s t o

some s m a l l d i v i s o r problems I & 11, Comm. P u r e Appl. Math. 28 (1975) 91-141;

29 (1976) 49-113.

P.H. Rabinowitz Hormander, L . ,

I m p l i c i t f u n c t i o n theorems,

S t a n f o r d Univ. l e c t u r e n o t e s ,

1977. Rabinowitz, P. H. P e r i o d i c s o l u t i o n s of n o n l i n e a r h y p e r b o l i c p a r t i a l d i f f e r e n t i a l e q u a t i o n s , 11, Comm. P u r e Appl. Math. 22 (1969) 15-39. Craig, W.,

A b i f u r c a t i o n t h e o r y f o r p e r i o d i c d i s s i p a t i v e wave e q u a t i o n s ,

New York Univ. t h e s i s , 1981, t o a p p e a r Ann. S u c o l a Norm. Sup. P i s a .

FOOTNOTE : ’This

r e s e a r c h w a s s p o n s o r e d i n p a r t by t h e N a t i o n a l S c i e n c e F o u n d a t i o n under

Grant No. MCS-8110556.

R e p r o d u c t i o n i n whole o r i n p a r t i s p e r m i t t e d f o r any

p u r p o s e of t h e U n i t e d S t a t e s Government.

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) @Elsevier Science Publishers B.V. (North-Holland), 1984

465

A SIMPLIFIED CHARACTERIZATION OF THE BOUNDARY CONDITIONS b!HICH DETERMINE J-SELFADJOINT EXTENSIONS OF J-SYMMETRIC (DIFFERENTIAL) OPERATORS David Race Mathematics Department U n i v e r s i t y o f t h e Witwatersrand Jan Smuts Avenue Johannesburg, South A f r i c a

V a r i o u s a u t h o r s , i n c l u d i n q N. A. Z h i k h a r and I.b!. Knowles, have l o o k e d a t t h e problem of c h a r a c t e r i z i n q t h e boundary c o n d i t i o n s which d e t e r m i n e J - s e l f a d j o i n t e x t e n s i o n s o f Jsymmetric operatcrrs such as t h e minimal o p e r a t o r , o e n e r a t e d b y c e r t a i n , o r d i n a r y , l i n e a r , d i f f e r e n t i a l expressions w i t h complex-valued c o e f f i c i e n t s . A l l t h e work t o d a t e has r e q u i r e d t h e r e g u l a r i t y f i e l d o f t h i s J-symmetric o p e r a t o r t o be non-empty. We g i v e here a more g e n e r a l b u t s h o r t e r and s i m p l e r method o f p r o o f , which y i e l d s t h e same r e s u l t s , b u t r e q u i r e s no such assumption. 1.

GENERAL J-SYMMETRIC OPERATORS

I n t h e decade f r o m 1954 onwards a l a r q e number o f papers began t o appear, concerned w i t h nonsymmetric d i f f e r e n t i a l e x p r e s s i o n s . See, f o r example, t h e The o p e r a t o r t h e o r y needed t o develop these ideas, r e f e r e n c e s l i s t e d i n [5], [6]. namely t h e t h e o r y o f J-symmetric and J - s e l f a d j o i n t o p e r a t o r s , was beaun by I. M. Glazman [ Z ] , N. A. Z h i k h a r [ll]and A. G a l i n d o [I]. B e f o r e proceeding f u r t h e r we d e f i n e t h e terms a s s o c i a t e d w i t h t h i s work. D e f i n i t i o n 1.1. An o p e r a t o r J d e f i n e d on a complex H i l b e r t space H i s c a l l e d a c o n j u g a t i o n i f i t i s a c o n j u g a t e - l i n e a r i n v o l u t i o n and (Jx,Jy) = (y,x) for a l l x and y i n H . D e f i n i t i o n 1.2.([2]) I f J i s a c o n j u g a t i o n on a complex H i l b e r t space H , a d e n s e l y d e f i n e d , l i n e a r o p e r a t o r A i n H i s s a i d t o be J-symmetric i f A C JA*J, i n t h e u s u a l sense o f o p e r a t o r i n c l u s i o n , where A* i s t h e a d j o i n t of A. I f A = JA*J t h e n A i s s a i d t o be J - s e l f a d j o i n t . D e f i n i t i o n 1.3. The r e g u l a r i t y f i e l d , n ( A ) , o f a closed, densely d e f i n e d , f o r which l i n e a r o p e r a t o r A i s d e f i n e d t o be t h e s e t o f a l l complex numbers dependin? o n l y on A , and II(A - i I ) x l l > khllxll f o r some p o s i t i v e number k, a l l x i n D ( A ) , t h e domain o f A . D e f i n i t i o n 1.4. ( [ l l ] ) Let A empty. For any h o E v ( A ) t h e t h e dimension o f t h e subspace, denotes t h e domain o f A* and

be a closed, J-symmetric o p e r a t o r w i t h n ( A ) n o t d e f e c t number o f d e f ( A ) , i s d e f i n e d t o be { x E D(A*): (A* - X,oI)x = @I, where D(A*) I i s t h e i d e n t i t y o p e r a t o r on H.

A,

I n [ll],Z h i k h a r o b t a i n e d a p a r t i a l c h a r a c t e r i z a t i o n o f t h e J - s e l f a d j o i n t e x t e n s i o n s o f any q i v e n J-symmetric o p e r a t o r , under t h e assumption t h a t i t s regul a r i t y f i e l d i s n o t t h e empty s e t . I t was a l s o shown t h e r e t h a t t h e above d e f i n i t i o n o f t h e d e f e c t number o f a closed, J-symmetric o p e r a t o r A, i s independent o f t h e c h o i c e o f l o i n n ( A ) . I t i s a f u r t h e r consequence o f t h i s work t h a t f o r a c l o s e d J-symmetric t h e dimension o f t h e q u o t i e n t space, D(JA*J)/D(A), i s t w i c e t h e d e f e c t number o f A, i.e., o p e r a t o r A,

D.Race

466 dim(D(JA*J)/D(A))

= 2

Y

d e f (A)

(1.1)

I n p r a c t i c e , however, i t i s d i f f i c u l t t o check whether v(A) i s empty and hence t o know whether Z h i k h a r ' s c h a r a c t e r i z a t i o n i s a p p l i c a b l e i n any q i v e n s i t u a t i o n . Subsequent work by I. Ld. Knowles [5] (see a l s o [7], [ l o ] ) has extended s i g n i f i c a n t l y t h e r e s u l t s i n [ll].However, t h e c o n d i t i o n t h a t a ( A ) be not empty i s I t i s t h e i n t e n t i o n o f t h e p r e s e n t work t o remove t h i s s t i l l r e q u i r e d i n [5]. c o n d i t i o n c o m p l e t e l y . The f i r s t key t o d o i n a so i s t h e f o l l o w i n g simple observation. Lemma 1.5. I f A i s a c l o s e d , 3-symmetric o p e r a t o r and extension o f A then dim (D(JA*J)/D(A'))

A'

i s any J - s e l f a d j o i n t

= dim ( D ( A ' ) / D ( A ) ) .

The e x i s t e n c e o f such an o p e r a t o r A ' i s auaranteed by a r e s u l t o f A. G a l i n d o [l]which says t h a t e v e r y J-symmetric o p e r a t o r has a J - s e l f a d j o i n t ext e n s i o n . The above lemna can be t h o u o h t o f , as s a y i n g t h a t t h e domain o f such an o p e r a t o r A ' i s "mid-way'' between t h e domains o f A and JA*J. I t now f o l l o w s that dim (D(JA*J)/D(A))

= 2

x

dim ( D ( A ' ) / D ( A ) ) .

(1.2)

Comparing (1.1) and (1.2) i t i s n a t u r a l t o make t h e f o l l o w i n g d e f i n i t i o n . D e f i n i t i o n 1.6. I f A i s any c l o s e d J-symmetric o p e r a t o r , t h e ( g e n e r a l i z e d ) d e f e c t number of A, d e f (A) i s d e f i n e d by def (A) =

x

dim (D(JA*J)/D(A)).

Lemma 1.5 guarantees t h a t even i f n ( A ) were empty, d e f ( A ) i s e i t h e r a non-negative i n t e g e r o r i n f i n i t e . Since ( 1 . l ) was a consequence o f D e f i n i t i o n 1.4, D e f i n i t i o n s 1.4 and 1.6 c o i n c i d e when v(A) i s n o t empty, b u t t h e l a t t e r a l s o covers t h e case when n ( A ) i s empty. I n o r d e r t o d e t e r m i n e t h e Js e l f a d j o i n t e x t e n s i o n s o f a J-symmetric o p e r a t o r A we a l s o need t h e f o l l o w i n q result. Lemma 1.7. I f A i s a c l o s e d , J-symmetric o p e r a t o r w i t h d e f (A) < and i s a J-symmetric e x t e n s i o n o f A t h e n A ' i s J - s e l f a d j o i n t i f and o n l y i f dim (D(A')/D(A)) = d e f ( A ) .

A'

Using U e f i n i t i o n 1.6, t h i s i s an e a s i l y proved consequence o f a r e s u l t of

I . W. Knowles [4] which s t a t e s t h a t t h e J - s e l f a d j o i n t e x t e n s i o n s o f a J-symmetric o p e r a t o r , a r e t h e e x t e n s i o n s which a r e maximal J-symmetric. We a l s o need an e x p l i c i t c o n n e c t i o n between t h e domains o f A and JA*J. F o r t h i s we c o n v e r t D(JA*J) i n t o a H i l b e r t space by d e f i n i n g t h e u s u a l i n n e r product as f o l l o w s : ( o r . ) *

( x , y ) * = (Jx,Jy) + (A*Jx,A*Jy)

= (y,x)

+ (JA*Jy,JA*Jx).

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

U(JA*J)/D(A)

2 D(JA*J)

8 D(A)

.

One may now use s t a n d a r d t e c h n i q u e s t o prove Lemma 1.8. Q(JA*J)

If

f) D(A)

A

i s any closed, J-symmetric o p e r a t o r i n

= Iy E D(A*JA*J):

A*JA*Jy

= -yi

.

ff

then

46 7

J-Sclfadjoint Operators

2.

APPLICATION TO DIFFERENTIAL OPERATORS

We now c o n s i d e r t h e a p p l i c a t i o n o f t h e above r e s u l t s t o o p e r a t o r s a s s o c i a t e d w i t h the d i f f e r e n t i a l expression,

on an i n t e r v a l I on t h e r e a l l i n e . We assume t h a t t h e f u n c t i o n s p-1 ,pl,--.,pn a r e complex-Val ued, Lebesgue measurable o v e r I and Lebesgue i n t e g r a g l e on be r e s p e c t i v e l y t h e minimal and compact subsets of I. We l e t To and

lmaX

maximal o p e r a t o r s i n t h e H i l b e r t space L ( I ) , generated by T (see [5]). It i s w e l l known (see f o r Fxample [5]) t h a t i f J denotes o r d i n a r y complex conjugat i o n o f f u n c t i o n s i n L ( I ) then To i s J-symmetric and T; = J Tmax J. Thus JT*J = Tmax and D e f i n i t i o n 1.6 a p p l i e d t o To g i v e s 0

T h i s d e f i n i t i o n of t h e g e n e r a l i z e d d e f e c t number o f To c o i n c i d e s w i t h t h e d e f i I n t h i s terminology, n i t i o n o f t h e mean d e f i c i e n c y i n d e x f o r T , g i v e n i n L3]. lemma 1.5 may be i n t e r p r e t e d as s a y i n g t h a t t h e mean d e f i c i e n c y i n d e x o f a f o r m a l l y J-symmetric d i f f e r e n t i a l e x p r e s s i o n i s an i n t e g e r . (The n e x t theorem shows t h a t i t cannot be i n f i n i t e . ) T h i s i s n o t t r u e f o r g e n e r a l d i f f e r e n t i a l express i o n s , as i s observed i n [3]. Setting

A = To

i n Lemma 1 . 8 g i v e s t h e f i r s t p a r t o f

Theorem 2.1. ( i ) d e f ( T o ) = h a l f t h e number o f l i n e a r l y independent s o l u t i o n s o f y, T Y E L 2 ( I ) . J T J T Y = -y f o r which (ii) (iii)

0 5 d e f (To) if

I = [a,b),

5 2n. a < b

-m

5" then

n 5 d e f (To) 5 2n.

The second p a r t f o l l o w s f r o m t h e f i r s t , when i t i s observed t h a t JTJT i s a l i n e a r d i f f e r e n t i a l e x p r e s s i o n o f 4nth o r d e r . Theorem 2.1 was p r o v e d i n [ll] under t h e assumption t h a t n ( T o ) i s n o t empty b u t o u r methods have enabled us t o remove t h a t c o n d i t i o n , by u s i n g D e f i n i t i o n 1 . 6 and t h e r e s u l t s g i v e n above. Lemma 1.7 may now be used t o c h a r a c t e r i z e t h e boundary c o n d i t i o n s which determine a l l t h e J - s e l f a d j o i n t e x t e n s i o n s o f To. We denote by [.,.I the form obtained from L a g r a n g e ' s i d e n t i t y f o r T (see [5]) and s t a t e t h e main r e s u l t of t h i s t y p e . I t i s hoped t h a t f u r t h e r d e t a i l s o f t h e s e c h a r a c t e r i z a t i o n s w i l l appear e l sewhere. be a r b i t r a r y f u n c t i o n s b e l o n g i n g Theorem 2.2. L e t m = d e f ( T o ) and w l , - - - , w m t o D(Tmax) which a r e l i n e a r l y independent modulo D(To) and which s a t i s f y t h e relations

[w.,w I ( b ) - [w.,; J k J The s e t o f a l l f u n c t i o n s y i n

j , k = 1;-. ,m. ](a) = 0, k D(Tmax) which s a t i s f y t h e c o n d i t i o n s

_.

1Y,Wkl(b) - [ Y ' i k l ( a )

=

0,

k = 1,

.... ,m

i s t h e domain o f d e f i n i t i o n o f a J - s e l f a d j o i n t e x t e n s i o n o f J - s e l f a d j o i n t e x t e n s i o n s o f To a r e o f t h i s form.

To.

Conversely a l l

The s t a t e m e n t o f these c h a r a c t e r i z a t i o n s i s i d e n t i c a l t o t h a t g i v e n i n [5] b u t w i t h o u t t h e r e q u i r e m e n t t h a t n ( T o ) be n o t empty. The p r o o f s employed here a r e a l s o s h o r t e r and s i m p l e r t h a n those g i v e n i n [5].

D. Race

468 3.

THE CASE

n(To) = fl

We conclude b y c o n s i d e r i n g what happens i n t h e case when T ( T ) i s indeed empty. I t i s p o s s i b l e t o show t h a t i f rr(T ) = @ then d e f (To) O f 2n. T h i s has p a r t i c u l a r s i g n i f i c a n c e when n = 1 an8 T i s r e g u l a r a t one e n d - p o i n t o f I:

Theorem 3.1. I f n = 1, I = [a,b), -- < a < b def(To) = 1 and t h e J - s e l f a d j o i n t e x t e n s i o n s T,

D ( T y ) = {Y TY , where

y = ( y 1 ,y2)

=

TY

5

,

Tmax: r , y ( a )

+

5

and IT(T,) = fl t h e n To a r e p r e c i s e l y given by

m

0;

Y ~ P ~ ( ~ ) Y= ' 01 ( ~ )

Y E m y )

i s an a r b i t r a r y non-zero element o f

(c

2.

T h i s completes t h e g e n e r a l i z a t i o n o f t h e l i m i t - p o i n t , l i m i t - c i r c l e dichotomy f o r S t u r m - L i o u v i l l e e x p r e s s i o n s h a v i n g complex-valued c o e f f i c i e n t s : Theorem 3.2A. (i) (ii) (iii)

(i)

(iii)

on

r

[a,b)

when

n = 1:

d e f (To) = 1. p r e c i s e l y one boundary c o n d i t i o n i s needed t o d e t e r m i n e J - s e l f a d j o i n t e x t e n s i o n s o f To. t h e r e i s a t most one independent s o l u t i o n o f any A E E.

Theorem 3.2B. (ii)

The f o l l o w i n g a r e e q u i v a l e n t f o r

The f o l l o w i n g a r e e q u i v a l e n t f o r

on

T

TY

in

L2[a,b)

when

n = 1:

= hy

[a,b)

for

def (To) = 2. p r e c i s e l y two boundary c o n d i t i o n s a r e needed t o determine J - s e l f a d j o i n t e x t e n s i o n s o f To. t h e r e a r e two independent s o l u t i o n s o f x E t.

TY

=

in

Ay

L 2[a,b)

f o r any

Since Theorems 3.2A, B c o v e r a l l p o s s i b i l i t i e s t h i s would seem t o j u s t i f y r e f e r r i n g t o these as t h e l i m i t - p o i n t and l i m i t - c i r c l e a l t e r n a t i v e s even when t h e c o e f f i c i e n t s a r e complex-valued. F i n a l l y , we g i v e an example i n which " ( T o ) i s empty.

8.

Example 3.3. The f o l l o w i n g example was c o n s i d e r e d by J. McLeod i n [ 8 ] , where i t was shown t h a t t h e r e i s no s o l u t i o n o f T Y = Xy i n L (0,~)f o r any 4 E t. .ry(x) = - y " ( x ) - 2 i e2(1+i)x

y(x)

on

[o,-).

It f o l l o w s (see [9]) t h a t n ( T ) - p1 and t h e c o n t i n u o u s spectrum o f a s s o c i a t e d o p e r a t o r s covers t h e complex pgane. From Theorem 3.1 we may deduce t h a t t h i s example f a l l s i n t o t h e l i m i t - p o i n t case as d e s c r i b e d i n Theorem 3.2A. We t h e r e f o r e now know what t h e a s s o c i a t e d J - s e l f a d j o i n t o p e r a t o r s a r e and t h a t t h e maximal domain i s a two-dimensional e x t e n s i o n o f t h e minimal domain.

J-Selfadjoirit Operators

469

REFERENCES

Galindo, A . , On t h e e x i s t e n c e o f J - s e l f a d j o i n t e x t e n s i o n s o f J-symmetric o p e r a t o r s w i t h a d j o i n t , Comm. Pure and A p p l i e d Math. 15 (1962), 423-425. Glazman, I . M., An analogue o f t h e e x t e n s i o n t h e o r y o f h e r m i t i a n o p e r a t o r s and a non-symmetric one-dimensional boundary-value problem on a h a l f - a x i s , D o k l . Akad. Nauk SSSR 115 (1957), 214-216. Kauffman, R. M., Read T. T. and Z e t t l , A., The d e f i c i e n c y i n d e x problem f o r powers o f o r d i n a r y d i f f e r e n t i a l e x p r e s s i o n s , L e c t u r e n o t e s i n math 621, S p r i n g e r - V e r l ag, B e r l i n , 1977. Knowles, I . W., On J - s e l f a d j o i n t e x t e n s i o n s o f J-symmetric o p e r a t o r s , Proc. Amer. Math. SOC. 79 (1980), 42-44. Knowles, I . W., On t h e boundary c o n d i t i o n s c h a r a c t e r i z i n g J - s e l f j o i n t e x t e n s i o n s o f J-symmetric o p e r a t o r s , J. D i f f . Equations 40 (1981), 193-216. Knowles, I . W. and Race, D., On t h e p o i n t s p e c t r a o f complex S t u r r n - L i o u v i l l e o p e r a t o r s , Proc. Roy. SOC. Edin. 85A (1980) 263-289. Makarova, A. D., On t h e J-symmetry o f o p e r a t o r s w i t h a nondense domain of d e f i n i t i o n , Volzhsk. Matem. Sb., Ser. Funkts. A n a l i z i Teor. Funkts. 10 (1969) 77-83. McLeod, J. B., S q u a r e - i n t e g r a b l e s o l u t i o n s o f a second-order d i f f e r e n t i a l e q u a t i o n w i t h complex c o e f f i c i e n t s , Q u a r t . J. Math. O x f o r d ( 2 ) 13 (1962) 129-1 33. Race, D., S p e c t r a l t h e o r y o f complex S t u r m - L i o u v i l l e o p e r a t o r s , Ph.D. t h e s i s , U n i v e r s i t y o f t h e Witwatersrand, 1980.

[lo]

Raikh, L. M., On t h e e x t e n s i o n o f a J - h e r m i t i a n o p e r a t o r w i t h nondense domain o f d e f i n i t i o n , Mathematical Notes 17 (1975), 439-442.

[ll]Z h i k h a r , N. A., The t h e o r y o f e x t e n s i o n s o f J-symmetric o p e r a t o r s , Ukrain. Mat. Z. 11 (1959), 352-364.

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DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), 1984

47 1

ESSENTIAL SELF-ADJOINTNESS FOR POWERS OF SCHRdDINGER OPERATORS THOMAS T . READ

DEPARTMENT OF MATHEMATICS WESTERN WASHINGTON UNIVERSITY BELLINGHAM, WASHINGTON 98225

C o n d i t i o n s a r e g i v e n which a r e s u f f i c i e n t f o r t h e minimal o p e r a t o r To a s s o c i a t e d w i t h a second o r d e r e l l i p t i c d i f f e r e n t i a l

Rn t o be e s s e n t i a l l y s e l f a d j o i n t , and f o r a l l powers of T o t o be s e l f - a d j o i n t . The c o n d i t i o n s f o r TO which a r e s i m i l a r t o t h e most g e n e r a l cone x p r e s s i o n on

d i t i o n s from t h e one-dimensional c a s e , a l l o w some c o e f f i c i e n t s n o t covered by o t h e r c r i t e r i a and a v o i d some of t h e i r t e c h n i c a l assumptions.

The c o n d i t i o n s

f o r powers of T o a l l o w f a s t e r growth by t h e l e a d i n g terms t h a n p r e v i o u s r e s u l t s . Let

L

be t h e second o r d e r e l l i p t i c e x p r e s s i o n

L

=

( p ( x ) ) - ’ ( - I D . a . (x)Dk t q ( x ) ) , J Jk

D . = a/ax. + i b . d e f i n e d on Rn. Here A = ( a . ) i s symmetric J J J’ t C1(Rn) and i k 6 C1(Rn). Also and p o s i t i v e d e f i n i t e , w i t h a 2 jk J q t Lloc(Rn) and t h e p o s i t i v e weight f u n c t i o n p i s c o n t i n u o u s .

We have t h e f o l l o w i n g r e s u l t on e s s e n t i a l s e l f - a d j o i n t n e s s of d e f i n e d by Tof t h e o p e r a t o r To w i t h domain C c ( R n ) use x * y t o denote t h e usual s c a l a r product i n C n . THEOREM 1.

Let

non-negative,

M

{wm(r)lm=l

2 sequence

of

=

L[f I .

We

compactly s u p p o r t e d ,

l o c a l l y a b s o l u t e l y c o n t i n u o u s f u n c t i o n s of t h e r a d i a l

v a r i a b l e r such t h a t f o r e a c h f i x e d d e c r e a s i n g & m & bounded above.

r,

{wm(r)l

is non-

T. T. Read

472

Suppose that

with

91 (i) (ii)

0 , 42

q3

=

5

q 0,

can be decomposed into a sum and each

q

=

2 n q j € Lloc(R )

so that

5 (1

+

q1 t q2 t q3

div Q a.e.,

A V W ~ - VtW C ~ A - ~ Q --Q q2)wi

for some constants --

6

i

0

- 6)qlwi

KO

K > 0,

(iii)

Then

To

REMARKS. 1. For the one-dimensional expression -1 (-DpD t q) on [ O , 03 ) the hypotheses above become = p

L

(i),

q3

=

Q’ a.e.,

This is very similar to, and in some respects stronger than, the limit-point result in Read [7, Theorem 31 which contains essentially all of the standard limit-point criteria for the interval [ O , D o ) . 2. We will see following Theorem 2 that in many situations the family {wml can be replaced by a single function w. However it should also be noted that a sequence {wml of compactly supported functions can arise very naturally, as in the Corollary below. 3. For p = 1 , the hypotheses are somewhat similar to those of Evans [3] who was also interested in the essential selfadjointness of powers of To. However the hypotheses in [3] imply that lop OD -’l2(r)dr = C o t while Theorem 1 does not put any fixed limit on the rate at which the leading terms can grow. If one supposes, for instance that p(r) 5 ’r and q(x) = ql(x) 2 Cr’, then by letting w = 1 and using a modification of the Proposition following Theorem 2 it can be seen that the hypotheses of Theorem 1 2 are satisfied whenever ,' a - 2 or B = a - 2 and C > ( a - 2) . The case 6 > a - 2 is also covered by a recent result of Kato [ 6 ] , but 13 = a - 2 is not.

Powers of Schrijdinger Operators

473

4 . The p r e s e n c e of t h e d i v e r g e n c e term q 3 a l l o w s q t o have f i n i t e s i n g u l a r i t i e s , s i n c e t h e s e can a r i s e a s t h e d i v e r g e n c e of a bounded o r slowly growing f u n c t i o n .

= ( n - 1)-' d i v [ ( x - x o ) / / x - x o I l . I n Evans' r e s u l t [ 3 ] and a l s o i n a number of e a r l i e r 1 9 1 , 1 1 0 1 , f i n i t e s i n g u l a r i t i e s a r e p r o v i d e d f o r by a

Ix - x 0 1-1

xo 6 Rn,

5. papers [ 4 1 , term

For i n s t a n c e , f o r f i x e d

qs

satisfying

1 I q s ( x ) Idx -< Kr2s, I x I" j / q s ( y - x)Iv(x)dx * 0

and

Ixl5r uniformly f o r y

Rn

in

Y(X) =

r

as

+

0

with

I

n>2,

1 - l o g l x l , n=2.

Such a term could be i n c o r p o r a t e d i n t o Theorem 1 i n much t h e same

twml

way a s i n [ 3 ] p r o v i d e d i t i s assumed t h a t bounded, and t h a t

2

A(x)

i s uniformly

c > 0.

One remarkable consequence of Theorem 1 i s t h a t

T o can b e made t o be e s s e n t i a l l y s e l f - a d j o i n t by r e s t r i c t i n g t h e c o e f f i c i e n t s

o n l y on a c o m p l e t e l y a r b i t r a r y sequence of c o n c e n t r i c a n n u l i . Other r e s u l t s of t h i s t y p e , f o r i n s t a n c e Eastham, Evans, and McLeod [ Z ] , impose, a t l e a s t i m p l i c i t l y , some r e s t r i c t i o n on t h e t h i c k n e s s of t h e a n n u l i , g e n e r a l l y something l i k e

2 1 ym/pm =

t h i c k n e s s o f t h e m-th on t h e m-th a n n u l u s .

p,

COROLLARY.

a n n u l u s , and

where

y,

i s the p(r)

Suppose t h e r e i s a d i s j o i n t sequence of a n n u l i

J,:am 5 r constants

5

then

& essentially self-adjoint.

To

00

i s t h e maximum of

bm, am

+

go

c < 1

Here components

y

Im:aA 5 r I

of

5

, on which

q

0.

I f there exist positive

such t h a t --

b& where

a

m

< a;

< bk

Jm\Im each have t h e p r o p e r t y

b,

and t h e two

T T Read

474

REMARKS.

1.

The proof depends i n an e s s e n t i a l way on t h e p o s s i -

b i l i t y t h a t Iwm} may f a i l t o be uniformly bounded. Thus t h i s r e s u l t does n o t f o l l o w from e s s e n t i a l s e l f - a d j o i n t n e s s theorems such a s t h o s e of Evans 131 o r Kato [ 6 ] . 2 . I f w e w r i t e x = rc, 151 = 1 and L = -A

+

f ( c ) r a s i n ( r b ) where

f(5)

1c

> 0

for

151

=

1,

then by

a n argument i n [ 7 1 t h e hypotheses of t h e C o r o l l a r y a r e s a t i s f i e d provided a > 2~ - 2 . We t u r n now t o e s s e n t i a l s e l f - a d j o i n t n e s s f o r powers of . TO Our r e s u l t i s a s f o l l o w s : THEOREM 2 . I f t h e c o e f f i c i e n t s of L a r e i n C o o ( R n ) , t h e hypoof Theorem 1 s a t i s f i e d , and i n a d d i t i o n ( i v ) wm 5 W f o r some c o n s t a n t W and a l l m , ( v ) f o r each ro > 0 , E > 0 , positive integer M there

=

-theses

is -

-then a l l

s

2 r 0 -and m 1. M

powers of

To

are

such t h a t

essentially self-adjoint.

REMARK. A s noted above, t h e theorem of Evans [ 3 ] which c o n t a i n s previous r e s u l t s of Chernoff [l] and Kato [ 5 ] on powers of To r e q u i r e s fi-’l2 = 00 f o r p = 1. Condition ( v ) does n o t i n 0

p r a c t i c e impose any r e s t r i c t i o n s on t h e r e l a t i v e s i z e s of p and beyond t h o s e a l r e a d y p r e s e n t i n Theorem 1. I n p a r t i c u l a r , f o r qil p = 1, i f p(x) < ra, q(x) 2 Cr’ f o r D > a - 2 , then a l l powers of To a r e e s s e n t i a l l y s e l f - a d j o i n t . The proof of Theorem 2 r e s t s on t h e f o l l o w i n g lemma which permits t h e c o n s t r u c t i o n of s u i t a b l e c u t o f f f u n c t i o n s .

If ( v ) h o l d s , then t h e r e i s m > M and a compactly supported f u n c t i o n v ( r ) [ O , go ) such t h a t

LEMMA.

on

0

5 v 5 1, v(r)

p1/2/v’ I

2

=

1

for

0 2 r 5 r0/2,

Ewm(q;/2wm t P 1 / 2 ) .

By a d a p t i n g t h e proof of t h i s lemma s l i g h t l y we can o b t a i n

Powers of Schriidinger Operators

PROPOSITION. Suppose that w(r) lutely continuous function 10,

g0

475

a non-negative locally abso) such that

(i) pw12 t ( A - ~ Q --Q q2)w2 5 (1 - 6)qlw2 t ~ p ,

or (b) E

< 1

s2(r)

-

E L1(O,

(pII/p)1’2

(1 - &)’I2 with s1

then there is such that Cw,} --

a

00

1,

w

is

bounded, and for some

-there exist increasing +

m

as

r

+

gr,

functions

sl(r)

such that

sequence Ivml of compactly supported functions Cwvm 1 satisfies the hypotheses of Theorem 1.

=

REFERENCES: Chernoff, P.R., Essential self-adjointness of powers of generators of hyperbolic equations, J. Functional Anal. 1 2 (19731, 401-414. Eastham, M.S.P., Evans, W.D., and McLeod, J.B., The essential self-adjointness of Schrbdinger-type operators, Arch. Rat. Mech. Anal. 60 (1976), 185-204. Evans, W.D., On the essential self-adjointness of powers of Schrbdinger-type operators, Proc. Royal SOC. Edinburgh 79A (1977), 61-77. Kato, T., Schrbdinger operators with singular potentials, Israel J. Math. 13 (1972), 135-148. Kato, T., A remark on the preceding paper by Chernoff, J. Functional Anal. 1 2 (1973), 415-417. Kato, T., Remarks on the self-adjointness and related problems for differential operators, in: Knowles, I.W. and Lewis. R.T. eds., Spectral Theory of Differential Operators (NorthHolland, Amsterdam, 1981).

476

T.T Read

[7]

Read, T.T., A limit-point criterion for expressions with oscillatory coefficients, Pacific J. Math., 6 6 ( 1 9 7 6 ) , 243-255.

[8]

Read, T.T., A limit-point criterion for expressions with intermittently positive coefficients, J. London Math. SOC. ( 2 ) 15 ( 1 9 7 7 ) , 2 7 1 - 2 7 6 .

[91

Stummel, F . , SingulHre elliptische Differentialoperatoren in Hilbertschen R a m e n , Math. Ann. 1 3 2 ( 1 9 5 6 1 , 1 5 0 - 1 7 6 .

[lo]

Wienholtz, E,, Halbbeschrznkte partielle Differentialoperatoren Zweiter Ordnung vom elliptischen Typus, Math. Ann. 135 ( 1 9 5 8 ) , 5 0 - 8 0 .

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) OElsevier Science Publishers B.V. (North-Holland), 1984

417

GEOMETRY AND DISCRETE VELOCITY APPROXIMATIONS TO THE BOLTZMANN EQUATION* Michael C. Reed Department of Mathematics Duke U n i v e r s i t y Durham, NC 27706

I would l i k e t o d e s c r i b e some c a l c u l a t i o n s which Reinhard I l l n e r and I have been making on t h e Carleman model, t h e s i m p l e s t d i s c r e t e v e l o c i t y model o f t h e Boltzmann e q u a t i o n . But, perhaps i t i s w o r t h w h i l e t o s t a r t by d e s c r i b i n g how these d i s c r e t e v e l o c i t y models a r i s e . The Boltzmann e q u a t i o n i s an i n t e g r o d i f f e r e n t i a l e q u a t i o n f o r t h e d e n s i t y i n c o n f i g u r a t i o n and momentum space, n!x,p,t), of a l a r g e number o f p a r t i c l e s ( f o r example, t h e molecules o f a gas). Since t h e Boltzmann e q u a t i o n i s n o t o r i o u s l y d i f f i c u l t t o s t u d y , one can i n s t e a d p e r m i t o n l y f i n i t e l y many f i x e d v e l o c i t i e s p1 ,. . . ,pk. One then wishes t o d e t e r mine t h e c o n f i g u r a t i o n space d e n s i t i e s

By analogy w i t h t h e pi(x,t) = n(x,pi,t). Boltzmann e q u a t i o n t h e s e s h o u l d s a t i s f y e q u a t i o n s o f t h e f o r m

The l e f t hand s i d e m e r e l y expresses t h e f a c t t h a t t h e p a r t i c l e s i n t h e d e n s i t y pi a l l have f i x e d v e l o c i t y pi. The r i g h t hand s i d e s a r e ( u s u a l l y ) taken t o be quadratic functions

o f the

pi

w h i c h mimic i n some way t h e i n t e r a c t i o n s i n t h e

Boltzmann e q u a t i o n ; i n p a r t i c u l a r , t h e

qi

s h o u l d be such t h a t t h e expected

q u a n t i t i e s a r e conserved. The Carleman model,

1 (Ut + u x ) = v2 - u -

u(x,O)

= uo(x)

20

1 (Vt -

v(x,O)

= v (x)

20

n

VP

-

v x ) = u2

-

v2

i s t h e s i m p l e s t d i s c r e t e v e l o c i t y model. p a r t i c l e s w i t h v e l o c i t y p l u s one; v ( x , t ) v e l o c i t y minus one. The t o t a l mass

0

u ( x , t ) i s the density a t time t i s the density o f p a r t i c l e s w i t h

of

m

m =

/

u(x,t) + v(x,t)dx

-m

i s conserved. The i n t e r a c t i n g terms on t h e r i g h t guarantee t h a t i f t h e r e i s ( l o c a l l y ) an excess o f u ' s o v e r v ' s , t h e n some u p a r t i c l e s w i l l be t u r n e d i n t o v's. Some y e a r s ago, I l l n e r [ 7 ] showed t h a t t h e s o l u t i o n o f ( 1 ) i s g l o b a l i n time. 1 Suppose t h e i n i t i a l d a t a a r e C , nonThen, a s t a n d a r d c o n t r a c t i o n argument proves

Here i s t h e i d e a i n a s i m p l e case. n e g a t i v e , and have compact s u p p o r t .

*Research s u p p o r t e d b y NSF G r a n t #MCS-8201258

478

M.C.Reed

the existence f o r s h o r t time o f a so u t i o n o f Let B ( t ) = max I u ( x , t ) , v ( x , t

1 ) w i t h t h e same t h r e e p r o p e r t i e s .

}

X

and l e t

xo

be a p o i n t where

u(xo,t) = B ( t )

and

u(xo,t)

>

v(xo,t).

Then

has a maximum a t xo so u x ( x o , t ) = 0. Thus, f r o m t h e d i f f e r e n t i a l au < 0 so t h e maximum w i l l decrease. equation, ( x o , t ) = v(x0,t)' - u(x0,t)' A au s i m i l a r b u t somewhat more t e c h n i c a l p r o o f shows t h a t 5 (x,,t) 5 0 a t points u(x,t)

where b o t h

u(xo,t) = B ( t ) = v(xo,t).

Thus,

B(t)

i s n o n - i n c r e a s i n g so t h e

l o c a l s o l u t i o n o f (1) i s g l o b a l . Two y e a r s ago, I l l n e r and I g o t i n t e r e s t e d i n t h e a s y m p t o t i c p r o p e r t i e s o f t h e Carleman model and proved t h a t [8]

0 2 v(x,t) 5 ;

t 1.1

where c i s a c o n s t a n t depending o n l y on t h e mass. I t i s easy t o see t h a t t h i s t i m e decay i s t h e b e s t one c o u l d e x p e c t ( j u s t observe t h a t 1 - (ut + u x ) 2 -u2 becomes u ' ( s ) -u(s)' on r i g h t w a r d c h a r a c t e r i s t i c s ) . The

Q

f a c t t h a t c depends i n i t i a l data v e r i f i e s mechanics: l o n g t e r m i n i t i a l data b u t o n l y m w i l l disperse very o f t h e same mass m.

o n l y on t h e mass and n o t on any o t h e r p r o p e r t i e s o f t h e ( i n t h i s model) one o f t h e b a s i c i d e a s o f s t a t i s t i c a l b e h a v i o r s h o u l d n o t depend on d e t a i l e d p r o p e r t i e s o f t h e on a few o v e r a l l parameters. Very peaked d e n s i t i e s o f mass r a p i d l y and w i l l have t h e same a s y m p t o t i c s as f l a t d e n s i t i e s

The c a l c u l a t i o n which I want t o s k e t c h today i s f o r t h e Carleman model ( 1 ) i n box. The p a r t i c l e s a r e c o n s t r a i n e d t o s t a y on t h e f i n i t e i n t e r v a l [O,e] by r e f l e c t i n g w a l l s a t z e r o and e. u(0,t)

= v(0,t)

u(e,t)

= V(Q,t)

(2) These boundary c o n d i t i o n s m e r e l y a s s e r t t h a t p a r t i c l e s t h a t a r r i v e a t e w i t h speed p l u s one i m m e d i a t e l y l e a v e w i t h speed minus one and v i c e v e r s a a t zero. Q

u + v dx i s again conserved. L o c a l l y , excesses o f 0 produce v ' s and excesses o f v ' s produce u ' s so t h i s suggests t h a t u s h o u l d approach t h e homogeneous s t a t e as t m. That i s , there should be d as t m. O f course, d = c o n s t a n t d so t h a t u ( x , t ) + d, v ( x , t ) by c o n s e r v a t i o n o f mass. Here i s t h e r e s u l t .

m =

The t o t a l mass

-f

-f

-f

Theorem [9]-

Suppose t h a t

conditions (2).

uo(x)

Then t h e s o l u t i o n

and u, v

vo(x) of

are

C1

u's and v a m/2e

and s a t i s f y t h e boundary

(1) s a t i s f i e s

Approximations to the Boltzmann Equation

where

c

i s a constant depending o n l y on

I want t o sketch t h e main step i n completely elementary, i l l u s t r a t e s t h e t h e geometry o f t h e c h a r a c t e r i s t i c s i s on i n these d i s c r e t e v e l o c i t y models. max

8(t) = a(t) =

m

and

419

e.

t h e p r o o f s i n c e t h e technique, though main p o i n t which I want t o make. Namely, fundamental t o understanding what i s going Set

iu(x,t),v(x,t)l

OLXZP" min { u ( x , t ) , v ( x , t ) }

ozxze

By conservation o f mass, a ( t ) < d < 8 ( t ) and a s i m i l a r a r ument t o t h e one i s non-decreasing. sketched above shows t h a t B ( t ) - is-non-increasing and act! I w i l l sketch t h e main step i n t h e proof t h a t B ( t ) - d 5 c / t . Since B ( t ) i s n = 1,2,... non-increasing i t i s s u f f i c i e n t t o show t h a t B(ne) - d 5 c/n, What we want t o do i s t o show t h a t B((n + lie) - d i s s t r i c t l y s m a l l e r than B(niL) - d. That i s , a f t e r a time s t e p o f l e n g t h e the maximum a c t u a l l y decreases (and o f course we need an e s t i m a t e on the decrease).

.

Define 6 = ( B ( n k ) + d)/2. Then, by conservation of mass t h e r e must be a at s e t o f reasonagly l a r g e measure so t h a t e i t h e r u o r v i s l e s s than 6, t = ne. More p r e c i s e l y ,

where

u

i s Lebesgue measure on [ O , e ] . Thus one o f t h e two terms on t h e l e f t m 7 , suppose i t i s t h e second. Then

must be g r e a t e r than

I n o t h e r words, we have a lower bound on t h e s i z e o f the s e t N a t t = n& on Set N- = N n [O,pG), N+ = N n [ p o t e l where po i s chosen so which v < 8., C

t h a t uW-} ' 7 ( B n - d ) , p{N+> 2~ (B,, - d ) . What we have i s t h a t v i s small ( l e s s than B n ) on a f a i r l y l a r g e s e t a t t = na. We want t o use t h i s t o show t h a t b o t h u and v a r e s t r i c t l y l e s s than ~ ( n e ) a t This i s where the geometry of t h e c h a r a c t e r i s t i c s comes in.

Figure 1

t = ( n + 1)e.

480

M.C. Reed

Let

p

be a p o i n t i n N+ and c o n s i d e r t h e l e f t - w a r d c h a r a c t e r i s t i c f r o m 1 S e t t i n g D- = - ( a t - ax) and l e t t i n g s denote a r c l e n g t h we have

p.

6

and

Thus

v ( 0 ) < B,.

-

5 u2

(D-v)(s)

v(s)

v2 = ( u

*

2 w(s)

where

v ) ( u - v ) 5 2B(ne)(3(ne) - v(s)) w(s)

solves

(D-w)(s) = Z a ( n e ) ( B ( n e ) - w) w(0) = Bn S o l v i n g t h e comparison problem e x p l i c i t l y y i e l d s

-

v ( s ) 5 B(ne) - c(B(ne)

(4) where

c

depends o n l y on

e.

and

m

d)

~

be a p o i n t on AP1 and c o n s i d e r t h e r i g h t w a r d c h a r a c t e r i s t i c , r. L e t M denote t h e p o i n t s on R where v s a t i s f i e s ( 4 ) . 1 utM} 2- c(Bn - d ) . We want t o f i n d a p o i n t on R where u i s

Now l e t q from q t o

R,

We know t h a t small.

n

Suppose B(nL)

f o r a l l p o i n t s on From ( l ) ,

u

2b2

B(l1k)

-

5 (o(ne)

I w i l l e x p l a i n why

R.

u ( r ) = u(q) +

J,

2 u(r),

ufq)

M We know t h a t v2

-

u2

b

b

M

5 B(ni).

d) c a n ' t be t o o c l o s e t o

b

If

B(ne),

i s very close t o

M

i s large.

possible positive contribution o f

But i f

1

v2

-

b

i s very close t o

u2

i s small s i n c e

B(ne)

b < a(ne)

-

-

i n any

When one makes

c(B(ne) - d)'

depends o n l y on

u(wo) 5 b ( n a )

the

v 5 B(nk)

MC case. Thus we g e t a c o n t r a d i c t i o n if b i s t o o c l o s e t o B ( n e ) . t h e e x p l i c i t e s t i m a t e s i n t h e above argument, one f i n d s t h a t

c

then

makes a s u b s t a n t i a l n e g a t i v e c o n t r i b u t i o n because o f ( 4 ) and t h e f a c t

M t h a t t h e measure of

where

~(ne).

- u2.

1 v2

v2 - u2

-

and

m

c ( 0 ( n e ) - d)'.

e . Thus t h e r e e x i s t s wo

on

R

such t h a t

By u s i n g t h e comparison argument which was used

above one concludes f r o m t h i s t h a t (5)

u ( r ) 5 B ( n i ) - c(B(nil) - d)

2

.

~

Notice t h a t t h i s holds f o r

r

in

P2C.

We would l i k e t o show t h a t ( 5 ) h o l d s f o r

r

in

6 too.

Here i s t h e

geometric argument ( o m i t t i n g a l l t h e a n a l y s i s ) . v i s s m a l l on a l a r g e s e t i n ApO. T h e r e f o r e by t h e comparison argument, v i s s m a l l on a l a r g e s e t i n Apl. ~

By t h e boundary c o n d i t i o n s ,

t h e comparison argument

u

F.

u i s t h e r e f o r e small on a l a r g e s e t i n By is s m a l l on a l a r g e s e t o f r i g h t w a r d c h a r a c t e r i s t i c s

Approximations to the Boltzrnunn Equation

48 1

q.

from T h u s , every leftward c h a r a c t e r i s t i c from to must cross a large s e t of points where u i s small. By the argument above, v i s small a t a l l points of Thus, by the boundary conditions, u i s small a l l points of plB. Therefore, by the comparison argument, u i s small on Bp2. This i s how one concludes t h a t ( 5 ) holds f o r a l l r in BC. Similar methods show t h a t ( 5 ) holds i f u i s replaced by v. T h u s ,

p.

~

which imp1 i e s

Iterating t h i s inequality yields

which i s what we wanted t o prove. This gives a sketch of t h e main s t e p of the proof of the theorem. More There t h e theorem i s used t o prove, i n a d d i t i o n , d e t a i l s can be found in [9]. t h a t t h e decay t o d i s a c t u a l l y exponential in the L2 norm. The point of making these c a l c u l a t i o n s here i s t o emphasize t h a t t h e geometry o f the characteri s t i c s i s crucial to t h e Boltzmann-like properties of the Carleman model. This geometry becomes obscured i f one w r i t e s ( 1 ) in standard evolution equation form (by taking the x d e r i v a t i v e to the o t h e r s i d e ) :

Of course, A generates a nice l i n e a r semigroup etA and one can t r y to t r e a t the whole equation by using semigroup theory, Ouhamel's formula, and estimates on the nonlinear map F. B u t , t h i s seems t o me t o be the wrong approach because i t obscures t h e underlying geometry of the problem. This then i s the point o f my l e c t u r e . There has been a tremendous development of functional analysis over t h e l a s t 50 years. I t i s therefore tempting t o r e c a s t a l l i n i t i a l - v a l u e problems i n semi-group form and t o i n v e s t i g a t e properties of solutions by investigating prope r t i e s of the generator. I think i t i s a mistake in hyperbolic problems and other problems where the underlying geometry i s important.

I t i s appropriate t o end t h i s l e c t u r e by suggesting to you a nice unsolved problem. The next simplest d i s c r e t e velocity model i s the Broadwell model. 2

Vt

+ vx = z

Wt

- wX

=

22

Zt

=

2(vw -

- vw

- vw 2

2)

.

Suppose t h a t one s t a r t s with smooth i n i t i a l data (say of compact support) a t time t = 0. I t i s known from the work of Nishida [lo], Crandall-Tartar [ l l ] , t h a t the solution e x i s t s globally and will be smooth. What i s the asymptotic behavior of the solution as t m? Notice t h a t i f the i n i t i a l data i s zero outside of the interval [ - k , a ] , then z and v should be zero f o r x < - 2 and z a n d w should be zero f o r x > P. Thus the i n t e r a c t i o n should be confined t o the s t r i p t > 0. -QZ X 5 ,.P +

482

M.C. Reed

Figure 2 Thus v w i l l be c o n s t a n t on r i g h t w a r d c h a r a c t e r i s t i c s from p o i n t s p on t h e r i g h t edge o f t h e s t r i p and w w i l l be c o n s t a n t on l e f t w a r d c h a r a c t e r i s t i c s f r o m p o i n t s q on t h e l e f t s i d e o f t h e s t r i p . I n o t h e r words, v and w w i l l be t r a v e l l i n g waves t o t h e r i g h t and t h e l e f t of t h e s t r i p r e s p e c t i v e l y . The t o t a l mass,

m =

v + w + z dx,

leak out o f the s t r i p ?

w

i s conserved.

The q u e s t i o n i s , does a l l t h e mass

I b e l i e v e t h a t i t does and t h a t t h e r e a r e f u n c t i o n s

i,

such t h a t

(6)

-

v(x,t)

-+

i(X

t)

w(x,t)

+

i(X + t )

z(x,t)

-+

0

as t m. I a l s o b e l i e v e t h a t one should be a b l e t o c o n s t r u c t a p r o o f u s i n g o n l y c a l c u l u s and t h e geometry of t h e c h a r a c t e r i s t i c s . -f

I should m e n t i o n t h a t a s t u d e n t o f J o e l S m o l l e r (D. Chang) has some numeri c a l evidence f o r t h i s c o n j e c t u r e and t h a t Russ C a f l i s c h and I have a p r o o f i n t h e case where t h e d a t a i s s m a l l . For l a r g e smooth data, t h e problem i s c o m p l e t e l y open. I t i s n o t even known whether t h e s o l u t i o n i s bounded. References

[ll Broadwell, J . E.

Shock s t r u c t u r e i n a s i m p l e d i s c r e t e v e l o c i t y gas, Phys. o f F l u i d s 7 (1964), 1243-1247.

[2]

Cabannes, H. S o l u t i o n g l o b a l e du p r o b l h e de Cauchy en t h i o r i e c i n i t i q u e d i s c r G t e , J. de M6c. 17 (1978), 1-22.

[3]

Cabannes, H. The D i s c r e t e Boltzmann Equation, L e c t u r e Notes g i v e n a t U n i v e r s i t y o f C a l i f o r n i a , B e r k e l e y , 1980.

[4]

Cabannes, H. P r o b l b e s mathbmatiques dans l a t h i o r i e c i n i t i q u e des gaz, Publ. s c i . de l ' i n s t . M i t t a g - L e f f l e r , Uppsala, 1957.

[5]

G a t i n g n o l , R. T h g o r i e c i n i t i q u e de qaz 2 r 6 p a r t i t i o n d i s c r G t e de v i t e s s e s , L e c t u r e Notes i n Phys. 36 ( S p r i n a e r - V e r l a g , New York, 1975).

Approximntions to the Boltzmann Equation

[6]

Godunov, K. and Sultangazin, U. M., On the d i s c r e t e models of k i n e t i c equation o f Boltzmann, Uspekhi Mat. Nauk 26 (1971), 3-51.

[7]

I l l n e r , R . , Global existence f o r two-velocity models o f the Boltzmann equation, Math. Meth. Appl. Sci. 1 (1979), 187-193.

[8]

I l l n e r , R . and Reed, M . The decay of solutions o f t h e Carleman Model, Math. Meth. Appl. Sci. 3 (1981), 121-127.

[9]

I l l n e r , R . and Reed, M. box, p r e p r i n t , 1983.

48 3

Decay t o equilibrium f o r t h e Carleman model in a

[lo] Nishida, T. and Mimura, M. On t h e Broadwell's Yodel f o r a Simple Discrete Velocity Gas, Proc. Japan Acad. 50 (1974), 812-817. [ l l ] T a r t a r , L. Existence globale pour u n systhrne hyperbolique semi-lin6aire de l a t h 6 o r i e c i n h a t i q u e des gaz, Ecole Polytechnique, Sgminaire GoulaouicSchwartz, 28 Octobre, 1975. [12] Temam,,R: S u r l a r6solution exacte e t approchse d ' u n probleme hyperbolique nonlineaire de T. Carleman, Arch. f o r Rational Mech. Anal. 35 (1969), 351 -362.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) @) Elsevier Science Publishers B.V. (North-Holland), 1984

485

DOMAINS IN HYPERBOLIC SPACE AND LIMIT SETS OF KLEINIAN GROUPS P. SARNAK COURANT INSTITUTE Lecture Given At The International Conference In Partial Differential Equations, Birmingham, Alabama March 1983

Though it may not be clear from the title, this talk is concerned with differential equations and in particular the Laplacian on domains in hyperbolic space. I will assume that the audience is more familiar with spectral theory rather than Kleinian groups. The work reported on this lecture is joint work with R. Phillips. We begin with a problem in geometry which seems first to have been posed by Beardon [l]. Consider a finite set of spheres in En, n 2 2, which are mutually exterior to one another. We are interested in the smallest closed subset of Bn which is invariant by inversions in these spheres. If we denote this set by A , the then spheres by S1,S2,...Sk and the inversions by R1,R2...%l clearly A is invariant under the group r = = the group A is called the limit or singular set of generated by R1,R 2...Rk. r. It is easy to see that r is free on R1,R2,...Rk except for the relation Ri2 = 1. One way of describing A is that A is the set of accumulation points of any fixed orbit I’x = {yxlyEr}. It is easy to verify that A so obtained is independent of x. To give an example consider the case of n = 2 and four circles, see figure 1 (a). The region exterior to C1,C2,C3,C4 clearly contains no limit points of I’x, since an application of an inversion in a circle will place one inside that circle. Thus 4 A C.I.’,INT(C-7’. If we now apply inversions in C C C C we 1 1’ 2’ 3’ 4 fini=that the images of EXT(C1,C2,CJ,C4) also contain no limit points. Thus A is contained in the union of the twelve circles shown in figure l.(b). Repeating this procedure we find that A is inside the 3 6 circles of figure l.(C). Continuing this indefinitely we fine that A looks something like figure l.(d). For more such pictures and constructions see [ 4 ] , from where these pictures are borrowed. As is apparent from figure l.d, A is a curve whose dimension should be close to 1, in fact being wiggly, we expect it to be

...

48 6

P. Sarnak

larger than 1. By dimension we mean Hansdorff dimension, we recall the definition: For K C En, E > 0 define 6 : K C U Bi , where Bi are hg (K,E) = inf{Iri i Balls of radius ri, and r < € 1 i -

c3 FIGURE 1

Limit Sets of Kleinian Groups

48 7

Let h (K) = lim h6(K,E), which gives the definition of Hansdorff outer E + O 6-measure. For 6=n this is nothing but Lebesque outer measure. From the definition it follows that there is a 6 such that

We call 6 = 6(K) the dimension of K, and it clearly generalizes our usual notion of dimension. For example, for figure 1 with r1 = r2 = r3 = l,r4 = m, we have found numerically that & ( A ) = 1.0015. We are now in the position to pose the question referred to at the beginning. It may at first seem a little ad-hoc, but its relevance will become clear as we go along. Question 1 [l] In En, can the dimension of A be made arbitrarily close to n by packing in enough spheres? For n 2 3 theorem 1 contains a solution to this problem. For n = 2 the problem remains unsolved, we will discuss it further later. Theorem 1: For n 2 3, there is a Cn < n such that 6 ( A ) 2 Cn independent of the configuration of the spheres. For example if n = 3 , & ( A ) 5 2.99

This, and a number of related such results, follows from an analysis of spectral properties of the Laplacian for domains in hyperbolic space. The point is that our group r is a group of conformal mappings of En, and so is a subgroup of the group G(n) generated by all inversions in hyperplanes and spheres of En. The group G(n) is also the group of isometries of hyperbolic n+l space. This allows one to phrase question 1 in terms of hyperbolic geometry. So we make a slight detour to discuss hyperbolic geometry. Let Hn+l = {(y,x): y > 0, x E En} and equip it with the line This turns Hn+l into element ds2 = (dx12 + dxn2 + dy2 )/y 2 a Riemannian space of constant negative curvature, -1. The The geodesis of this geometry are vertical rays or semi circles perpendicular to En. Every two points of Hn+l are joined by a unique geodesic. Geodesic submanifolds of dimension n (ie

...

.

488

P. Sarnak

submanifolds M of dimension n, for which the geodesic joining any two points of M I lies entirely within M) are simply the n-hemispheres with equator on En or vertical planes of dimensions n.

FIGURE 2

The group of conformal mappings of 1 may be extended to act on Hn+l. Indeed the generators which are the inversions in En, act on Hn+ by simply inversion in the sphere of Bn whose equator is the sphere in question, ie inversions in hyperplanes (geodesic). This action preserves Hn+l and it is an exercize to show that such motions are isometries of Hn+l Hn+l is a symmetric space of rank one. The Lclndamental differential operator for hyperbolic space is its Laplacian A , which generates all invariant differential operators (ie ones which commute with G(n)). In terms of the coordinates introduced

.

489

Limit Sets of Kleinian Groups

The Riemannian volume element in these co-ordinates is xi1 dv = (dxldx2. dxndy)/y

..

and

We are interested in the spectrum of for various self adjoint 2 n+l boundary conditions. For the free space, ie L (H ,dv) the ',=) and is absolutely continuous. This is well spectrum is [ known but it is instructive to see why the spectrum only begins at (n/212.

(t)

.

Let u (x1,x2,. .xn,y) E c;(H"+~) and let D(u) =

lVuI2 dV

, H(u)

=

1

u2 dV

be the Dirichlet and L2 forms respectively. For each fixed (x1,x2, xn) an integration by parts in y yields

...

Integrating ( 3 ) w.r,t.

x gives

Thus from the variational description of the bottom of the spectrum, we have inf = X o = inf (D(u)/H(u)) UEL2 VU L2

2

2

(n/2)

.

(3' 1

Let r be a discrete subgroup of G(n) acting discontinuously on Hn+l, that is r acts on Hn+I so that no orbit of r in Hn+’ has accumulation points in Hn+l. An example of such a r is the

490

P. Surnuk

group of reflections in spheres introduced at the beginning. Corresponding to rone may choose a fundamental domain F. Such a set F, is an open set which has the property that for xI y E F and x = yyI y # identity => x = yI and further more for any If T acts freely z E Hn+l there is a y E r such that YzfF on Hn , which is to say that y E I r y # identity => y has no fixed points in Hn , then one may form the quotient space Hn+ /I , to obtain a hyperbolic manifold M = Hn+l/r. One may think of M as F with its faces glued appropriately. In general F may be chosen to be a convex(in hyperbolic metric) domain bounded by geodesic hyperplanes. For example for r the group generated by inversions in the disjoint circles C1,C2, Ck in the plane, the corresponding F in hyperbolic three space may be chosen as the exterior to the hemispheres

...

FIGURE 3 For simplicity we will restrict ourselves (except for the very end) to geometrically finite groups r I ie ones for which F may be chosen to have only finitely many boundary hyperplanes. The limit set of a discontinuous group r , denoted A r , is defined as before, as the set of accumulation points of any fixed

49 1

Limit Sets of Kleinian Groups

orbit Tx, with x E Hn+l. Thus A r is a closed subset of Sn = 2 ( Hn+I). In the case of the inversion group r in the beginning, the limit set as defined before and now are clearly the same. We turn to the I’-spectral problem for A. Let L2 (Hn+l/r) be the Hilbert space of automorphic functions for which

lfI2

dv <

m

Fr

f(yw) = f w , Y Y E r ) . The Laplacian, A has a unique self adjoint extension to Let k o (r) be the bottom of this spectrum. The following L2 (H"+’/r). remarkable theorem is a key one in our investigation. The theorem is due to SULLIVAN [7], though PATTERSON [5] obtained a number of important special cases. THEOREM (PATTERSON-SULLIVAN) If l' is geometrically finite, and 6 ( r ) >: then (SO

Xo(r)

=

6(n-6)

Here are a few words as to where this theorem comes from. If one projects the unit sphere about x E Hn+l onto the boundary Sn at infinity by geodesic projection, one obtains a line element p(x) on Sn. If 6 is the Hansdorff dimension of hr, we let @(x) be the S-Hansdorff measure of A when computed relative to p(x). That @ is r invariant is not surprising as A is ’I invariant. What is more surprising is that

For proofs of these and more see 171. (i), (ii), (iii) the theorem follows.

It is clear that given

is a hyperbolic three manifold, Thurston [8] Remark: If M = Hn+’/r then this number is a good measure of has shown that if Vol(M) < the complexity of the topology of M. If Vol(M) < m, then clearly X = 0 since @!constant is an eigenfunction with eigenvalue zero, 0 correspondingly 6 = n. In this case r is as large as possible. If Vol(M) = then the number 6 ( r ) or X (Hn+’/r) gives a measure

-

0

P. Sarnak

492

of the size of r. This is one reason for studying 6 ( r ) in detail. Notice that a lower bound for A ( r ) gives an upper bound for 6 ( r ) , 0 X ( r ) may beestimatedby considering the eigenvalue problem for a 0 fundamental domain R with Neumann boundary conditions. More precisely let R be a domain in hyperbolic space and let u ( R ) and X o ( R ) be the spectrum and bottom of the spectrum for A on 52 with free , ie Neumann, boundary conditions. Thus 0 (0) =

inf UEW’ (R)

(u)

Ho

where

Since h (R) corresponds to face conditions it follows that if 0 is a fundamental domain for that

In Neumann bounded Theorem then (i) (ii)

[6] we have made a careful and detailed study of the

spectrum for geometrically finite domains R of Hn+l(ie by finitely many sides) 1: Let 52 be geometrically finite and assume Vol(S2) = a(A) U(A)

is discrete in [ 0, ($)2) is continuous in

1(5)~,-

See also [ 3 ] .

...

.

2) Let h o ( n ) , X l ( R ) Xk R) be the L2 eigenvalues in [0,(-)n 2 If none exist we simply have h o ( R ) = .2I:( That the number (n/2) comes up as the divid ng point should be no surprise in view of (3l). The basic building blocks in our theory are free domains , which we define as follows: Definition is free if h o ( R ) 2 ( 4 2 )2 If R is geometrically finite then by theorem 1 this is equivalent to the fact that R does not carry any L2-eigenfunctions of A subject to any self adjoint boundary conditions on R. The following theorem assures the existence of many free domains

.

Theorem 2:

If R

C

Hn+l is bounded by at most 7 n+2 hyperplanes then

Limit Sets of Kleiniaii Groups

493

R is free. This is sharp in the sense that if we allow one more face then f2 can be made non free. EXAMPLE: In H3 let $2 be the exterior of the three mutually tangent are the circles hemispheres whose intersection with the plane shown in figure 4 (a), then R is free. On the other hand adding a hemisphere as in 4(b), yields an L2 eigenfunction or a.

FIG 4 . (a)

FIG 4 . (b)

Theorem 2 allows one to prove useful continuity properties of In fact one may allow the sides to degenerate inclumpsof size up to n+2 and still have continuity [6],again such a result is sharp. We don’t continue the discussion of the Neumann problem any further here, see [6] for more concerning this. We turn to an outline of the proof of theorem 1. We call a domain R a Schottky domain if f2 is bounded by hyperplanes which do not overlap. A group which has a fundamental domain which is a Schottky domain will be called a Schottky group. This is essentially the classical definition, and is different to some recent definitions [8]. These groups form an important subclass of discrete subgroups of G(n) Notice that if R is Schottky then V o l ( f 2 ) = m. A l s o notice that the original r is Schottky. So the real question is whether geometrically finite Schottky gorups are bounded, ie have 6 ( r ) bounded. In the case of dimension n = 1, that is r I M ( 1 ) PSL(2,B), it is easy to see that such groups are not bounded. Indeed if r is generated by z z + 2 and z -+ -l/z tk A h o ( R ) under various degenerations of the sides of R.

.

-+

P. Sarnak

494

6 ( r ) = 1 since A,. = R. Our result is that for n 2 3 a Schottky group is small. Theorem 1: There is for each n 2 3 , a Cn > 0 such that for any Schottky domain 51

...

’Outline of proof’: Let S1, S2, Sk be thebounding hemispheres rk. Let C1,c2,...Ck be the cylinders with radii rl,r2,

...

where p ,...pk are the centers of S1, 1 Let T be the domain 51 \ C j=l j *

... k respectively. S

Firstly we note that T is free, as an integration as in ( 4 ) immediatly shows. Next we need a uniform lower bound on ho(Ci). Since dilations are isometries of the hyperbolic metric it follows that 1. (C ) = X o (C,) = h o (C) where C is any cylinder. O j Lemma: For n 2 3 , h o ( C ) 2 Cn > 0. For a proof see our paper [6]. The theorem then follows from the simple fact that if fi = U . B . is a disjoint union, and if h (B.) 2 a then h , ( n ) 3 1

0

7

2

a

.

The above argument fails in dimension n = 2 since Xo(C) = 0 for cylinders in H3 , see [6]. To give some idea of what happens in the case n = 2 here are some numerical calculations of 6 ( r ) for I' generated by the inversions in the circles shown. (figure 5) These examples show 6 ( r ) can be made at least as large as 1’58 for Schottky groups. Actually we can make examples (rigorously) for which 6 ( r ) is quite a bit larger. Theorem 3 : For n = 2 , there exist geometrically finite Schottky groups for which 6 ( r ) 2 1.7598. PROOF: The proof makes use of a geometrically non-finite domain. Consider the following Appolomian packing of the square S,

495

Limit Sets of Kleinian Groups 6 FREE, HAS A NULL VECTOR

co

0.998

1.305

1.402

0 0

1.456

1.499

1.557

1.581

FIGURE 5

P. Sarnak

496

Figure 6 The circle C1 is placed inside the square so as to be tangent to the four sides of the square. This divides what remains into four curvilinear triangles. Now place the unique mutually tangent circle to each side of such a triangle, to abtain C2,C3,C4,C5. This leaves us with eight curvilinear triangles and we repeat the process indefinitely. This packing is well known to be solid, that is to say that K 1 ri2 = 1, where ri are radii of the circles i which we order in decreasing sizes. Consider the domain RCH 3 defined by R = {(y,x):

x E S,

(y,x) exterior to all

hemispheres in H3 whose equators are C.1 . 7

Proposition 2:

X,(Q)

2

0’42263

PROOF: We first appeal to a result of Boyd [23 concerning the Appolonian packing. Let 6 be the exponent of convergence of the series

I

Now Boyd shows 1.300197

<

<

1.314534

497

Limit Sets oj' Klrinian Groups

L e t C ( y ) = c r o s s s e c t i o n a l area of il a t h e i g h t y , y > 0. C l e a r l y w e have

A H e r e A ( y ) = 11

1

r

r.
packing only

i s t h e area of S which i s o m i t t e d by t h e r a d i i g r e a t e r t h a n or e q u a l t o y.

i

1

L e t N(y) = #{j: r

> y j -

1

A ( y ) = -II

lo

C ( y ) = -II

:1

r2 d N ( r )

-

J

rs d N ( r )

(8)

Y

Then

1 riS

=

i

r2 dN(r)

and

+ y 2 II N ( y )

(9)

0 E

= lim

[rS N(r)]

E-+O

+

Im

m

E

I

m

= l i m [ EsN(E)

+

T>B,

Sr

s-1

t h e n t h e l e f t hand s i d e of

ET

N ( r ) dr

1

(10)

E

40

If

dr

SrS-l N ( r f

lim E+O

( 1 0 ) c o n v e r g e s and hence

N(E) = O ( 1 )

or N(E)

=

Thus

E

c ( y ) = lirn

(II r 2 N ( r )

Im +

2r N ( r ) d r )

E-tO

+ y2 N ( Y ) = O(Y

'-'+

1;

r r

--T

dr)

P. Sarnak

498

2-’

= O(Y

Thus C(y) =

Let

)

0, (r>B)

0

as y

1

if y 2 1.

+

B < y < 2 be arbitrary, and define

U(Y) = yy/2 Thus

B < T < Y .

for any

.

SO that H(u) <

Furthermore

ie

From (14), (15) u

3

W’(n),

.A

Y 2 5 (7)

.A

Y 2 5 (7)

and so it follows that

Y

Y > B (16)

Using Boyd’s bound (6) it follows that

X

< -42263, completing the proof prop"

0 -

(2)

To conclude the proof of theorem 3 , we observe that by an easy approximation argument one has, for E > O there is Q' such that A,(Q') -< -42263 + E here a ' is bounded by only finitely many F,of the circles

Limit Sets of Kleinian Groups

499

of the aboveAppolonianpacking. Now if we translate enough of F, by the translations z * z + 1, z + z + i, we obtain a Schottky domain " sounded by these circles and their translations, or which x o ( Q " ) 5 .42263 + 2E. The last follows from considerations of such periodic (ie Euclidean translations) configurations (see t61 for more) It follows that there are geometrically finite lchottky r for which Xo(r) 5 042263 + 2 E, with E arbitrarily small. When translated into the language of Hansdorff dimension we get theorem 3. Whether 6(r) for such can be made arbitrarily close to 2, for H3 groups, is as far as we know still unsettled. We end by remarking that an argument similar to that used in the proof theorem 3 above, together with Boyd’s efficient packing of 2 spheres in a 3-cube [ 9 ] , shows that there are Schottky groups r in dimension n = 3 for which 6(Ar) > 2=0078

.

P. Sarnak

5 00

REFERENCES Beardon A.F. Boyd D.W.

Amer JR. of Math’ 88 (1966) 722-736 Aequationes Math 9 (1973) 99-106

Lax. P. and Phillips R. 280-350

Jr. of Funt. Analysis, 46(1982)

Mandelbrodt B. "Fractals" Freeman and Co, San Fransisco 1977 S.J. Patterson Acta Math 136(1976) 241-273

Phillips R and Sarnak "?he Laplacian For Domains in Hyperbolic Space and Limit Sets of Kleinian Groups 'I To Appear D. Sullivan Bull. Am.S. 6 (1982) 57-73 Thurston W. Geometry and Topology of Three Manifolds" Notes from Princeton University 1978 D. Boyd "On the Exponent of An Osculatory Packing" Can J. Math, Vol XXIII, 355-363 (1971)

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V.(North-Holland), I984

50 1

SELFADJOINT OPERATORS, SPECTRAL AND SCATTERING THEORY, VARIATIONAL TECHNIOUES, NON-LINEAR PHENOMENA, LINEAR AND NON-LINEAR PARTIAL DIFFERENTIAL EQUATIONS, AND RELATED TOPICS M a r t i n Schechter Courant I n s t i t u t e of Mathematical Sciences New York U n i v e r s i t y New York, N. Y. 10012

ble d e s c r i b e some r e c e n t r e s u l t s c o n c e r n i n s t h e t o p i c s men t ioned

.

1.

SELFADJOINT REALIZATIONS I N ANOTHER HILBERT SPACE.

L e t H, V be o p e r a t o r s on a H i l b e r t space H, and assume t h a t H i s s e l f a d j o i n t . The o p e r a t o r H ' = H + V may n o t have a s e l f a d j o i n t r e a l i z a t i o n on H . However, i t may have a s e l f a d j o i n t r e a l i z a t i o n on some o t h e r space. By t h i s we mean t h a t t h e r e a r e a n o t h e r t l i l b e r t space H1, a bounded l i n e a r map J from H1 such t h a t H t o H1 and a s e l f a d j o i n t o p e r a t o r H1 on H J = JH'. (1.1) 1 I n t h i s case we may be a b l e t o i n v e s t i g a t e H ' by means o f H1. Even i f H ' does have a s e l f a d j o i n t r e a l i z a t i o n on H , t h e problem may c a l l f o r one t o compare H n o t w i t h H ' , b u t w i t h an o p e r a t o r H1 on H1 s a t i s f y i n g ( l . l ) , where til and J a r e g i v e n . Again t h e q u e s t i o n a r i s e s , does t h e r e e x i s t a s e l f a d j o i n t o p e r a t o r H1 s a t i s f y i n g ( 1 . 1 ) . When H1 i s s e l f a d j o i n t , we c a l l i t a s e l f a d j o i n t r e a l i z a t i o n o f H ' on H1 r e l a t i v e t o J.

We s h a l l p r e s e n t two r e s u l t s i n t h i s c o n n e c t i o n . They b o t h concern a s e l f a d j o i n t o p e r a t o r H on a H i l b e r t space H and a b i l i n e a r f o r m ( V U , ~ ) = (Au,Bv)~,

(1.2)

where K i s a H i l b e r t space and A, B a r e l i n e a r o p e r a t o r s from H t o K such t h a t D(H) C D ( A ) fl D(B). (The use o f a b i l i n e a r f o r m i n p l a c e o f t h e opera t o r V a l l o w s us t o t r e a t more p e n e r a l problems.) !Ale assume t h a t t h e r e i s a complex number z i n t h e r e s o l v e n t s e t p(1.I) o f H such t h a t t h e f o l l o w i n q h o l d f o r z = zo 'and z =

q:

(a)

AR(z), BR(z)

(b)

R([AR(z)]*)

(c)

G(z) z 1

-

B(H,K),

E C

D(B)

and

where

R(z)

=

( z - H)-'

Q ( r ) z B(AR(T))*

has a bounded i n v e r s e on

O(z)

is in

B(K)

k'.

Here B(H,K) denotes t h e bounded l i n e a r o p e r a t o r s f r o m denotes t h e ranse o f H.

H to K

Our f i r s t r e s u l t concerns t h e case when H1 and J a r e a i v e n . such t h a t t h e r e e x i s t a s e l f a d j o i n t o p e r a t o r H1 on H1 (1.3)

(Ju,H,v)

=

(JHu,v) + ( A U , B J * V ) ~ ,

u

E

D(H),

v E D(H1).

and

R(H)

Nhen does

M. Schech ter

502

We have

Theoretn 1.1. There i s a s e Z f a J 3 o i n t operator H on H, such t h a t Jo =lJ*J s a t i s f i e s u(H1) C u(BJ*) and (1.3) hoZds i f and onZy i f

z = z0’

JoT(z)* = T ( F ) J o ,

(1.4)

-

zo

where

(1.5) If (1.5) holds, t h e n (1.6) and

~ ( z =) ~ ( z +) (AR(T))* ~ ( Z 1 - lB R ( ~ ) . H1 s a t i s f i e s R1(z)J = JT(T)*, R1(z) = (z - H1)-'.

I n the second r e s u l t , we a r e not qiven H, such t h a t (1.3) holds. We have

H1

and

J.

We search f o r

H1,

J

Theorem 1.2. There e x i s t a HiZbert space H,, an operator J E B(H,H1) and a s e l f a d j o i n t operator HI on H1 such t h a t D ( H 1 ) C U(BJ*) and (1.3) holds i f and only if t h e r e e z z s t s an operator J o E B ( H ) such t h a t J o 2 0 am! (1.4) hoZds. The followina pives a s u f f i c i e n t condition f o r (1.4) t o hold.

D(BJ,)

and

ImI(Hu,Jou) + (AU,BJ~U),I= 0 ,

u

E

C

U(BJo)

Theorem 1.3. (1)

( 2 ) R(CBJoR(z)I*)

(3) P ( z ) *

=

If D(H)

C

U(B),

C

R([BR(z)l*)

D(H)

BJ~(BR(z))*

where P(Z) =

B(BJ~R(I))*

then ( 1 . 4 ) holds.

Exam l e 1 . H = L2(R) , H = -id/dx, V ( x ) i s a complex-valuer! function in L1(R)*s W(x) 5 Im V(x) = 0, H ' = H + V does not have a s e l f a d j o i n t extension. By Theorem 1.2 i t does have a s e l f a d j o i n t r e a l i z a t i o n in some Hi 1b e r t space. Example 2. With H t h e same, take V = a6(x) where 6 ( x ) i s the Dirac d e l t a "function" and ~1 i s a complex constant. I f CY i s not r e a l , H ' does not have a s e l f a d j o i n t extension on H. By Theorem 1.2, i t does have a s e l f a d j o i n t r e a l i z a t i o n in another Hilbert space provided a # - i . Once t h e existence of a s e l f a d j o i n t r e a l i z a t i o n has been e s t a b l i s h e d , i t i s For instance we have

n o t d i f f i c u l t t o e s t a b l i s h a spectral and s c a t t e r i n g theory.

R -

Theorem 1.4. f

has measure

Assume f u r t h e r t h a t t h e r e i s an open s e t T 0 and hypotheses ( a ) - ( c ) hold i n t h e s e t wT = i z

f o r some

co

>

1 Re

z

E

T,

0 < Im z

<

co)

0. Assume a l s o t h a t lim sup sup a(UAR(s+ ia)1I2 + I I B R ( s + ia)ll 2 :)

a

+

O+

SEI

m

C

R

such t h a t

Spectral and Scattering Theory

503

I c r and t h a t o ( z ) is uniformly continuous i n W I for each such I. Assume finally t h a t t h e r e i s a z1 E p(H) such t h a t BR(z) (AR1 (z))* i s a compact operator on K for every z i n wr. Then t h e Wave operators f o r each compact s e t

W+f = l i m e t-tt-

itHl Je-itH

f

e x i s t and are complete. F o r d e f i n i t i o n s , c f . [l]. 2 3 2 Example 3. H = L ( R ) I H1 = L ( a ) , where n i s t h e h a l f s p a c e x1 > 0. i s t h e r e s t r i c t i o n t o n, H i s t h e s e l f a d j o i n t r e a l i z a t i o n o f - A i n H, K = L2(an), A = -a/an (normal d e r i v a t i v e ) , B = 1, on an. The wave o p e r a t o r s e x i s t and a r e complete. J

Example 4. H = H = L 2 ( R ) , H = -d2/dx2, o p e r a t o r s e x i s t and a r 4 complete. 2.

V

= 6(x),

J = 1.

The wave

DIFFERENTIATION I N ABSTRACT SPACES.

L e t G be a mapping o f an a r b i t r a r y subset V o f a t o p o l o g i c a l v e c t o r space X i n t o R , and suppose we a r e i n t e r e s t e d i n f i n d i n g an element u E f o r which t h e v a r i a t i o n o f G vanishes, i . e . ,

V

h o l d s f o r a l l q i n some dense s e t 0. Many problems i n mathematical p h y s i c s a r e s o l v e d i n t h i s way. I f t h e s e t V i s a l i n e a r m a n i f o l d and G i s bounded f r o m above o r below, one can t r y t o f i n d a maximum o r minimum. However, i f V i s n o t l i n e a r and G i s unbounded b o t h f r o m above and below, s e r i o u s problems p r e s e n t themselves. A l o c a l maximum o r minimum may n o t e x i s t . One s u g g e s t i o n i n a t t a c k i n g such problems i s t o i n t r o d u c e c o n s t r a i n t s . i n s t a n c e , one can c o n s i d e r G on a s e t S o f t h e f o r m (2.2)

S = {v E V

I F(v)

For

= 0)

where F i s some mapping o f V i n t o a t o p o l o g i c a l v e c t o r space Y. One has t o p i c k F i n such a way t h a t one can f i n d an element u i n S such t h a t ( s a y ) (2.3)

G(u) = min G(v). S

T h i s i n i t s e l f i s f a r from a t r i v i a l problem. Moreover, once (2.3) i s achieved, In p a r t i c u l a r , one wishes t o use ( 2 . 3 ) t o o t h e r d i f f i c u l t i e s p r e s e n t themselves. c o n c l u d e t h a t ( 2 . 1 ) h o l d s . B u t i n o r d e r t o do so, one must r e q u i r e u + t q E S f o r t near 0 and q E 0. T h i s would mean F ( u + t q ) = 0 f o r such t and q. T h i s i s t o o much t o ask. The most t h a t one can hope t o o b t a i n i s t h a t f o r each q E Q t h e r e i s a mapping q ( t ) f r o m IR t o Q such t h a t (2.4)

F(u + t q ( t ) ) = 0,

q(t)

+

q

in

Q

and even t h i s i s e x t r e m e l y d i f f i c u l t t o achieve. Moreover, t h e sense i n which q ( t ) converges t o q w i l l depend on F and w i l l n o t n e c e s s a r i l y c o i n c i d e w i t h t h e t o p o l o g y of X. The l i m i t (2.1) i s c l o s e l y r e l a t e d t o t h e d e f i n i t i o n o f d i f f e r e n t i a t i o n i n a b s t r a c t spaces. When V i s a l i n e a r s e t and t h e l i m i t i s a c o n t i n u o u s l i n e a r

504

M. Sehechter

o p e r a t o r , i t i s c a l l e d t h e Gateaux d e r i v a t i v e . O f t h e a p p r o x i m a t e l y t w e n t y - f i v e d e f i n i t i o n s t h a t e x i s t i n t h e l i t e r a t u r e , none o f them i s s u i t a b l e f o r o u r s i t u a t i o n . I n a l l cases one i s a i v e n a mappina f f r o m a t o p o l o g i c a l v e c t o r space X t o another, say Y. The mappina f i s d e f i n e d everywhere and i s s a i d t o have a d e r i v a t i v e A a t x i f A i s a c o n t i n u o u s l i n e a r o p e r a t o r f r o m X t o Y and f ( x + h)

(2.5)

-

f ( x ) = Ah + r ( h ) ,

h € X

where r ( h ) belongs t o a s e t o f " s m a l l " o p e r a t o r s f r o m X t o Y. It i s i n the c h o i c e o f t h e s e t of " s m a l l " o p e r a t o r s t h a t t h e v a r i o u s d e f i n i t i o n s d i f f e r . (Some a t t e m p t s have been made t o c o n s i d e r f n o t d e f i n e d on t h e whole o f X. However, i t i s always t a k e n as t h e r e s t r i c t i o n o f a mappina d e f i n e d everywhere.) We t h e n t u r n e d t o t h e problem o f f i n d i n g a d e f i n i t i o n o f d i f f e r e n t i a t i o n which would be s u i t a b l e f o r o u r c o n s i d e r a t i o n s . llle wanted a d e f i n i t i o n t h a t would t o be a l i n e a r m a n i f o l d

(a)

not require

V

(b)

not require

G'

t o be c o n t i n u o u s

(c)

not require

G'

t o be d e f i n e d on t h e same space as

(d)

have a reasonable c a l c u l u s

(e)

be u s e f u l i n s o l v i n o problems of t h e t y p e mentioned

(f)

i n c l u d e a l l o t h e r known d e f i n i t i o n s .

G

The d e f i n i t i o n which we f i n a l l y developed i s as f o l l o w s . L e t X be a r e a l v e c t o r space, and l e t 0, Y be separated r e a l t o p o l o g i c a l v e c t o r spaces such t h a t Q C X. L e t G be a mappina from a subset V c X t o Y. The d e r i v a t i v e o f G a t a p o i n t u E V w i l l be a l i n e a r o p e r a t o r f r o m X t o Y . I t s domain i s d e t e r mined as f o l l o w s . L e t C(V,O,u) denote t h e s e t o f t h o s e q € 0 f o r which t h e r e e x i s t sequences [ q n ? C 0 , { t n lc lR such t h a t (2.6)

qn

+

q

in

0,

0 # tn

-f

0,

u + tnqn E V

C(V,Q,u) i s a double cone. I t r e p r e s e n t s t h e l i m i t s of t h o s e elements o f 0 f o r be t h e s e t o f a l l which one can form a d i f f e r e n c e q u o t i e n t i n V . L e t E(V,Q,u) It i s t h e s m a l l e s t l i n e a r m a n i f o l d conf i n i t e sums o f elements o f C(V,O,u). t a i n i n g C(V,Q,u). of

G

D e f i n i t i o n . A l i n e a r operator A from X t o Y i s c a l l e d the d e r i v a t i v e a t u w i t h r e s p e c t t o Q and denoted by G ' ( u ) i f

Q

(1)

u E V

(2)

F o r any sequences

(2.7)

and

D(A) = E(V,Q,u) {qnl,

1 t i [G(u + t n q n )

-

I t n } s a t i s f y i n q ( 2 . 6 ) we have

G(uj]

+

Aq

in

Y

as

n

+

m.

Both t h e d e r i v lnlhen i t e x i s t s , G ' ( u ) i s determined u n i q u e l y on E(V,Q,u). a t i v e and i t s domain degend on C! and i t s t o p o l o g y . A l l o f t h e u s u a l theorems o f c a l c u l u s hold, i n c l u d i n g t h e mean v a l u e theorem, t h e c h a i n r u l e , t h e i m p l i c i t f u n c t i o n theorem, e t c . They h o l d under hypotheses no s t r o n a e r (and u s u a l l y weaker) t h a n t h o s e assumed f o r o t h e r d e f i n i t i o n s . Moreover, t h e p r e s e n t d e f i n i t i o n cont a i n s a l l o t h e r s i n t h e sense t h a t a mapping d i f f e r e n t i a b l e i n any o t h e r sense i s a l s o d i f f e r e n t i a b l e w i t h r e s p e c t t o t h i s d e f i n i t i o n . However, t h e f o l l o w i n a has no c o u n t e r p a r t i n o t h e r d e f i n i t i o n s .

Let 11. be a topological vector space continuously embedded in Theorem 2.1. e x i s t s , then GI:,(u) exists and is equal, to t h e restriction of Gb(u) toQ(EIVybI,u).

Q. If 6 ' u

Spectrul und Scuttering Tlzeorj’

505

Returnins to our oriqinal problem (2.11, one can show that if S is given by (2.2),and (2.3) holds, then under suitable conditions (2.8)

G (u)q

=

0

F (u)q

=

0.

Q

for all

q

E

C(Y,Q,u)

such that

(2.9)

Q

It then follows that by pickina F properly we can deduce that (2.8)holds for all q. Example 5. To find a stationary point of the functional . 2 + F1 r- 2 (x2 -l) +r 2V ( Y ) 1+ ~2~9 2 +x2y2-$r2i2-x 2 z 2 Idr G(x,y,z) = 1= [x 0

where x(r),

y(r),

z(r)

defined in 0 y(-)

=

b, z ( m )

<

r 5-

=

c, b2

such that c2

.

V(t) is a continuously differentiable, non-negative function vanishing at b. Here b, c are given constants and i = dx/dr, X = di/dr, etc. The solution to this qives the existence of dyons (particles with both electric and magnetic charges) according to the Yang-Mills-Higgs theory (cf. [ Z ] ) . Example 6. The same problem for m

2 2 2 j 2 + V(x) - 2x y - x y )dr 0 for x(r), y(r), where V(t) is a continuous function satisfying V(t) 2 cot 2 for some co > 0 and x(0) = a, y(0) = b are given.

G(x,y)

=

./ ( i 2-

3. ONE SIDED DERIVATIVES.

one if the any

The use of the restriction ( 2 . 2 ) is only one possibility. As an alternative may wish to use sets defined by means of inequalities. This is easily done Y (the space into which F maps) is finite dimensional. I prefer to use followina procedure which allows Y to be infinite dimensional. Let B be subset of Y ‘ (the space of continuous linear functional5 on Y), and put

S

(3.1)

=

tv

E

V 1 y F(v)

0 VY

E

B}.

For this situation we shall need the definition of a one-sided derivative. Let C+(V,Q,u) denote the set of those q E 9 for which there exist sequences tqn} c Q, {tn} cR, such that (3.2)

q,,

-

-f

q in Q , 0

<

tn

-f

0, u + t,qn

E V.

If F maps V c X Y, we shall write A = FiQ(u) and call A the right hand derivative of F at u with respect to Q if A is a linear operator from X to Y with D(A) = E(V,Q,u) and (3.3)

tnl[F(u + tnqn) - F(u)]

holds for all sequences (3.4)

and

tqn},

Bu

+

Aq

in Y as n

{tnl. satisfying (3.2). =

{y

E

B I y F(u)

=

01

+

Put

M. Schechter

506

R

(3.5)

=

yipBuN ( Y ' ) .

We assume (a)

Y = R @ M w i t h continuous p r o j e c t i o n

(b)

For e v e r y such t h a t

q E C+ (V,Q,u)

P

(c)

Y

t h e r e 'are sequences PF u + t n q n ) =

(3.6)

of

The above hypotheses h o l d t i v e multiple o f i t i s i n

onto

M. Itn1

{qnl,

s a t i s f y i n g (3.2)

PF(u). Bu

f we remove f r o m

any

y'

f o r which no nega-

B.

We have Theorem 3.1. e x i s t s and t h a t

Let

G be a mapping of V

into

G(u) = m'n G(v)

(3.7)

5

S

where

Giq(u) q

(3.8)

(3.9) Example 7. equation

Asswne t h a t

G’ (u) +Q

,

Under hypotheses ( a ) - ( c ),

i s given by ( 3 . 1 ).

f o r a22 q E C+ (V,Q,u)

lR

20

satisfying

t/ y ' E Bu. y'FtO(u)q 2 0 2 TO f i n d a s o l u t i o n i n L ( I ) o f t h e D i r i c h l e t problem f o r t h e

(3.10)

-

u(r)

2u(r) = h'(u)

-

w(r)eu

where I i s any i n t e r v a l (bounded o r unbounded), w ( r ) > 0, h ( u ) > 0, h ( 0 ) = 0, h ' ( u ) i s c o n t i n u o u s and t h e s o l u t i o n u ( r ) of (3.10) t o vanish-at t h e endp o i n t s o f I . Theorem 3.1 i m p l i e s t h a t a s o l u t i o n e x i s t s p r o v i d e d

is

I w(r)dr

(3.11)

2 0.6476103.

I Moreover, t h e s o l u t i o n s a t i s f i e s Example 8.

Same problem f o r t h e e q u a t i o n

(3.12) where

u(r) h,

w

I u I 5 1.5936243.

a r e as above.

-

2u(r) = h ' ( u ) + w(r) s i n u

A s o l u t i o n e x i s t s provided w(r)dr

(3.13)

I Example 9. (3.14)

<

-.

Same problem f o r u(r)

-

2u(r) = h ' ( u )

- co(u 2

+ 2u)eU + g ( r ) .

A s o l u t i o n e x i s t s i f max(co,IIgllp) 5 0.1452526 f o r some I t s a t i s f i e s I u I 5 0.7034674. 4.

p

such t h a t

1 5 p 5 2.

SOME INEQUALITIES

I n t h e s t u d y o f b o t h l i n e a r and n o n - 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 e q u a t i o n s , one o f t e n i s r e q u i r e d t o o b t a i n e s t i m a t e s o f t h e form

Spectral and Scattering Theory II VUll

(4.1) where V i s a f u n c t i o n , space w i t h norm

ti iiq

(4.2)

IIuII

5 YP

5 C(V , s ,p ,q) II UII ,p i s the

u E Hsyp

I

Lq(IRn) norm and

= llF(1 +

507

tisyp

i s t h e Banach

1 ~ 1 ~ ) Full~ P’ ~

( h e r e F denotes t h e F o u r i e r t r a n s f o r m and F denotes i t s i n v e r s e ) . When s i s a p o s i t i v e i n t e g e r , t h e norm (4.2) i s e q u i v a l e n t t o t h e sum o f t h e Lp norms o f a l l d e r i v a t i v e s of u up t o o r d e r s. For s i m p l i c i t y we s h a l l assume t h a t s > O , n > 2 and l < p < - .

In [3],

i t was shown t h a t

(4.3) p r o v i d e d 0 < a < sp ( t h e e x p r e s s i o n I x - 1 o g l x - y l when ~1 = n and by 1 when cx I n [4] i t was shown t h a t depend on V. (4.4)

C(V,s,p,q)

-

yja-n i s t o be r e p l a c e d by n . The c o n s t a n t i n ( 4 . 3 ) does n o t

1

5- C sup( I V ( x ) l q l ~ - ~ l a - dx)’/q n y IX-Yl
provided t h a t

Here we s h a l l improve t h e s e e s t i m a t e s t o i n c l u d e a g r e a t e r range o f s i t u a t i o n s . We i n t r o d u c e t h e f o l l o w i n g f a m i l y o f norms. P u t

(4.7)

We have

with constants

provided

Cj

independent of

V

and

u and C2 a l s o independent of

6

M. Schechter

5 08 0

(4.11)

1 < 1 1 1 <-i-

r - q - p

t

and

< n < l1 + - -1 l - -

(4.12)

'-nr-n

q < t

with

n = sp

required uhen

If (4.11);

C o r o l l a r y 4.2.

and p

,

(4.13)

(4.14)

If V

for

M a,p,q

12q <

(V)

(V)

K

KII UII

+

C(V,s,p,p)

Example 11.

C(V,s,p,q)

Example 12.

The i n e q u a l i t y

~

HsYp.

on-'

I (VU,U) I h o l d s i f 2- < nr -n

-

-

.

, - a< - - snr-n

1 t '

5 C Ma,q,pl(V) i f 2 - < 5 +1 - - 1.

5 C0Ma,.J6

Co

5 CM,,r,m

p ( q 5 r

q

(V)llulll2 ,2

The c o n s t a n t

C(V,s,p,q)

if

r

nq-n

C1Ma,r,t(V)~~u~~2 2

does n o t depend on

(V) and

+

a < ?1 +1- - -

In some cases we can t a k e p IIVull 9 c C 0Ma,q,m,6

does n o t depend on

6,

ilvuifl c C 0Ms,l,-,

(V)llulls,l if

CL

6.

if

nr-n

Co

E

then

min(pa-' ,6"-1)dp)q/p

5 C Ma,r,t(V) o

where

u

9

1x1,

P =

Gn

I ( / I V ( p ) 1'

such that

and

Example 10.

Example 14.

s/n = l / t + l / q ' .

6 * 0

us

,p

'-11

depends onZy

5 C(

1 when

9 6

m,

Example 13.

0

+

5 €11

II VUII

Theorem 4.3.

>

there is a c o n s t a n t

0

>

E

t

p

(4.12) hoZd and

Ma,r t ,6

then f o r every

q

(qs

+

p '

q

= 1.

Thus

CIMa,q,m(V)II~IIl

+ n

-

nq.

I n particular

6(V)ilulls,1 + Cl~s,l,m(V)llulll.

Spectrul and Scattering Theory

5 09

REFERENCES [l]

Schechter, Martin, Operator Methods in Quantum Mechanics (North-Holland,

New York, 1981).

[2]

Schechter, Martin and Weder, Ricardo, A theorem on the existence of dyon s o l u t i o n s , Annals of Physics, 132 (1981) 292-327.

[3]

Schechter, Martin, Spectra of P a r t i a l Differential Operators (North-Holland, Arns terdam, 1971 ) .

[4] -~

Berger, M. S. and Schechter, Martin, Embeddinq theorems and a u a s i - l i n e a r e l l j p t i c boundary value problems f o r unboundei domains, Trans. Amer. Math. S O C . , 175 (1973) 439-467.

[5]

Stummel, F., Singulare e l l i p t i s c h e Differential operatoren in Hilbertschen Raurnen, Math. Ann. 132 (1956) 150-176.

[6]

Balslev, Erik, The e s s e n t i a l spectrum o f e l l i p t i c d i f f e r e n t i a l operators in L p ( R n ) , Trans. Amer. Math. SOC. 116 (1965) 193-217.

[7]

Ljusternik, L. A . , On the extremals of f u n c t i o n a l s , Mat. Sb. 41 (1934) 390-401.

[8]

I o f f e , A. 0. and Tihomirov, V. M . , Theory of Extremal Problems (NorthHolland, Amsterdam, 1979).

[9]

Averbukh, V. I . and Srnolyanov, 0. B . , The theory of d i f f e r e n t i a t i o n in l i n e a r topological spaces, Russian Math. Surveys 22 (1967) 201-258; The various d e f i n i t i o n s of t h e derivative i n l i n e a r topological spaces, i b i d . , 23 (1968) 67-112.

[lo] Berger, M. S . , Nonlinearity and Functional Analysis (Academic Press, New York, 1977).

[11] Federer, Herbert, Geometric Measure Theory (Springer-Verlag, New York, 1969). [12] Ljusternik, L . A. and Sobolev, V. J . , Elements of Functional Analysis ( F . Ungar, New York, 1961).

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T. Lewis (Editors) OElsevier Science Publishers B.V. (North-Holland), 1984

511

ON QUASIUNIQUENESS Vladimir Schuchman Department of Mathematics Texas Tech University Lubbock, Texas 79409

In this paper, is studied the quasiuniqueness o f solutions for degenerate equations of the type: au au t-at = A[,(, a x L , x ) ~ l . Under the assumptions that K(OrX) is a real-valued C1 function and uIx;K(O,x) = 01 = 0 we obtain that quasiuniqueness (or uniqueness in the class of flat functions) takes place at the point t = 0. INTRODUCTION Let us consider the following equation a au t-au (1) at = -[K(t,x)--] ax ax where tEI = [0,1], xER, R is a compact set of R1 or a compact manifold without boundary. If R is a compact set of R1 , then u l a n = 0. K(t,x) is a real-valued C1 function for (t,x)EIxR. Recall that if u = u1+iu2 is a solution of (l), then u1 and u will be solutions of (1) also. Because of this, it suffices to stzdy real-valued solutions of (1) In this paper is studied the behavior of solutions of equation (1) as t +O. More precisely, we study the quasiuniqueness of solutions of equation (1) or uniqueness in the class of flat functions (a function f(t) is called flat at the point t=O if f o r any k>O t-kf (t)+O as t+O). We assume that pIxER: K(0,x) measure on R1 .

=

O } = 0 where

p

is the standard Lebesque

The main result of this paper is the following Theorem. Let us consider equation (1) under the assumptions listed above. Then quasiuniqueness takes place for smooth solutions of (1). Or if u(t,x) is a C2 flat solution of equation (11, then u(t,x) 0 for tEIO,TOl where To>O. The idea of the proof is the following: Let u(t) be a non-trivial flat solution of (1). We fix this solution.

C2

V. Schuchman

512

Under assumptions of the theorem R = Q l U n 2 with R l ! 7 n 2 K(0,x)L 0 on R1 and K(O,x)< 0 on R 2 .

where

=

9 such that

is a scalar product in real Hilbert space L 2 ( R ) .

(.,.)

q(t) = ql(t) + q2 (t)

Then (3)

where

First of all, we prove that ql(t) = 0 and after this, that q2(t) = 0 for our fixed flat C2-solution u(t,x) of (1). We obtain the proof of this theorem in Section 2. collect the lemmas which are used for the proof.

In Section 3, we

We use in the second part of the proof a method which was first used by Agmon-Nirenberg for the study of the Cauchy problem in the nondegenerate case [ 1 , 2 ] . If K(t,x) is non-negative, this theorem will be trivial. In the case when K(O,x)< 0, this theorem follows from results of Alinhac-Baouendi 141, or from results of [ 3 1 , but if K(0,x) changes sign, this result does not follow either from [3] or from [4]. Remark 1. It is easy to see that the same proof as that of the theorem will work if we require only that K(0,x) is real-valued. If K(t,x) is real-valued for (t,x)EIxR, we can add smooth Remark 2. lower order terms to our equation (1). In this case, quasiuniqueness takes place too, but this is not true if p{x;K(t,x) = 0 1 > 0.

Remark 3 . Let us consider equation (1) for IxSl where S1 is the unit circle. 1. Let K(t,x) = 0 for xES1 and tEI. Then the following function (for more details see [ 4 ] ) ,

will be a flat solution of the following equation

1 and 3 = t-.a exp(--) t at 2. If K(0,x) is not real-valued, the statement of the theorem is not true. For example, let us consider the following equation

In ( 5 ) f(t)

=

On Quasiuniqueness

a 'uz ) (~

au t-= at

ie -1 e

513 (7)

where f3€S1, the unit circle. Let f(t)

where

a

=

=

(7).

Remark 4.

1 for t20, then the following function exp(- t)

a

t-at ,

will be a flat function and a solution of equation

Let us consider the non-linear equation

a au au t-at = -[K(~,x,u)-] ax ax with real-valued C1 function K (t,x,u). Then quasiuniqueness takes place for solutions of this equation if p{x;K(O,x,O) = O ) = 0. This result completes results of [5]. Remark 5 .

After changing t by the following formula t

=

e-’

we obtain from our equation (1) the following non-degenerate (by time) equation

where s [l,+-)and our results can be rewritten for this equation as an uniqueness theorem with conditions of exponential type for s+ + m . Remark 6 . By using standard techniques, it is possible to obtain analogous results about quasiuniqueness for more general cases of equations of the second order in the geometric variables and the first order in time. For equations of the second order in time, this type of Remark 7. result for nondegenerate second order elliptic equations with real coefficients in geometric variables is well known as the AronszajnCordes theorem. See, for example, [2,4]. If K ( 0 , x ) is a real-valued analytic non-trivial function, Remark 8. the conclusion of the theorem is true also if smooth lower terms are added.

PROOF OF THE THEOREM is flat Let us note that if q(t) is flat then q1 is flat and q 1x also. Let us take X(t) from Lemma 1 for 6 = R and fix it. We will 1 prove that from the flatness of q (t) follows it’s triviality. By 1x using Lemma 1 we get that q1 is trivial too.

V. Schuchman

514

After taking the scalar product of (1) with X(x)u(t,x) we obtain

= -2

= -2

2

I

K(t,x)X(x) Ql

a 2 121 dx- I K(t,x)X’(x)~u (t,x)dx

au I K(t,x)x(x)lzl

Rl 2 dx+

R1

I

2

[K(t,X)X’(x)lx’u (t,x)dx. (11) R1

The second term in (11) can be estimated as Cql(t) (and as Cq (t) 1X from Lemma 1)and if the first is non-positive, then qlX(t) = 0. This follows from Lemma 2. From the smoothness of K(t,x), we have that K(t,x) = K(O,x)+tKl(t,x) and K(0,x) is non-negative on R, , and on s u p p x (x), K(0,x)> 6. Then there exists T 0 > O such that for any t [O,?Ol, the following estimate holds 2 au 2 t Kl(t,x)X(x) dx(I K(O,x)x(x) 1 ~ dx. 1 (12) R1 Ol IKl(t,x) I We can choose = max 6 O XERl tEI I

121

I

In this case, the first term in (11) will be non-positive for t [O,?o] and q (t) = 0 for tEIO,TO] , as follows from Lemma 2. 1x ql(t) E 0 for tEIO,TO].

-

Then

Our work now is proving that q2(t) = 0 too. As follows from the previous discussion, we can consider now equation (1) on 6, with the a ~ = 2 0. condition

UI

In this case we have

It is easy to see from (13) that q2(t) is a twice differentiable function and

On Quasiuniqueness

It follows from (13)-(14) that

,.

'1

2 =D!L(t).

(18)

It follows from Lemma 3 that for quasiuniqueness it is enough for us to prove that

Since Dq2 is not negative, we can rewrite ( 1 9 ) in the form M(t) + CtDq2

2

0.

(20)

But

For each sufficiently small 61>0 it is possible to divide Q2 into C2iun: with the following conditions: IK(0,x) I > 6 1 on Q; and p ( C 2 ; ) < ~ ( 6 , ) + 0

as 61+0.

The right side of (21) can now be rewritten in the form

Since K(t,x) = K(0,x)

+

tKl(t,x) with continuous Kl(t,x), then

K1 (t,x) with K’ (t,x) will be bounded and we can choose Ti such that t for any tE[O,T’] the first term in (22) will be non-negative for 0 C ,Co > 0, T;t maybe depends on C. Let now C = CO+C1, where we can choose C1 such that

V: Schuchrnan

516

S i n c e u ( t , x ) i s a C‘-solution of e q u a t i o n (1), t h e n f o r t h e same r e a s o n as i n Lemma 1, w e c a n choose E ~ > O such t h a t

W e now f i x 61 such t h a t and Q; = Q 2 \ Q ; . 0 can be chosen a s A and i f T = rnin(Tb,$G1), t h e n f o r

f o r each Q; w i t h p ( Q ; ) C

<

E

E ( ~ ~ ) < E ~ . 1 e a c h t E [ O , T ] , t h e r i g h t s i d e of case w e o b t a i n t h a t

(23) w i l l be non-negative.

In this

D2.L(t) + CtD.L(t)zO (25) w i t h C depending on u ( t , x ) o r k ( t ) i t s e l f . From Lemma 3 , w e o b t a i n T h i s completes t h e t h a t i f q 2 ( t ) w i l l be f l a t , it w i l l be t r i v i a l .

proof of t h e theorem f o r To = min(T0,Tb,+S1). From t h e f i r s t p a r t o f t h e p r o o f , it i s e a s y t o see t h a t w e can add lower terms ( i f p{xEQ, K ( 0 , x ) = O}= 0 ) and t h i s t e r m w i l l b e t h e e s t i m a t e f o r t h e second t e r m i n (11). I n t h e c a s e o f lower t e r m s i n t h e second p a r t o f t h e p r o o f , w e must u s e Lemma 3 w i t h a ( t ) and B ( t ) and n o t o n l y a ( t ) . W e o b t a i n ( 2 5 ) i n a more g e n e r a l c a s e . R e c a l l , t h a t w e c o n s i d e r a s o l u t i o n u ( t , x ) of i t s f l a t n e s s , we obtain i t s t r i v i a l i t y .

(1) and from assuming

LEMMAS USED I N PROOF O F THEOREM

i,

fi

L e t u ( t , x ) be a C 2 f u n c t i o n on I x where I = [0,11, is 1 a compact set of R Then, t h e r e e x i s t s a smooth f u n c t i o n X ( X ) on Q such t h a t ( ~ ( x depends ) on u ( t , x ) ) .

Lemma 1.

.

= 0. i ) o c x ( x ) ( l f o r x~fi,XI ii) q x ( t ) z $ q ( , t ) , f o r t E I .

Here

q ( t ) = ( u ( t , . ) , u ( t , . ) ) and q Proof.

p = (xu(t,.),u(t,.)).

For e a c h t E I , t h e r e e x i s t s E > O such t h a t q E ( t ) z $ q ( t ) f o r any ijEl

where q E ( t ) = ( u ( t ) , u ( t ) ) ; and p ( f i \ G E ) < ~ . E

T h i s f o l l o w s from t h e smoothness of u ( t , x ) . From t h e compactness of I = [0,1] f o l l o w s t h a t i t - i s p o s s i b l e t o choose unique E > O f o r a l l W e must choose now Q E and ~ ( x s)u c h t h a t a Q c R \ Q E and tEI.

a

517

On Q uasiuniqu en ess

x(x) = 1 on

EE

and is smooth on R and x(x) 1 a; J

= 0.

Lemma 2 . Let q(t) be a non-negative flat function which satisfies the following equation: tq’ (t) ’cq(t) with constant 0 0 . Then q(t) is trivial.

(29)

Proof. If q (t) is flat then (t) = is flat too. After substitC tution q(t) = tcG(t) in (29) we get the following equation for q ( t ) C C t tG’(t)t’ cq(t)-ctCG(t) = 0 or tG’ (t) 2 0 and if G ( 0 ) = 0, then G(t) '0 for t > O . And from flatness G(t) follows it’s triviality and the triviality of q(t). Lemma 2 is proved. [ 3 ] Let R(t) be a twice differentiable function in the inLemma 3 . terval I = (0,1], satisfying the following second-order differential inequality D2R(t) + ta(t) IDR(t) 1 + tB(t) 2 0 (30) where a(t) and %(t) are non-negative continuous functions in the interval [0,11.

Then for each solution of ( 3 0 ) there exist constants v , ~such that

with v,p non-negative, v is a constant, depending on a(t), B(t) only, and 1-1 is a constant depending on a(t) , B(t) and the solution 9, (t) itself. Proof.

From ( 3 0 ) it follows that 2 >0 D R (t) + Ctl DR (t)I + Ct We change the variable using the formula

is true.

and for

9,

t = e-T (T) we have R

( T )

+ ce-T112( T ) 1 + Ce-T > o .

From Lemma 1.2 of [ 2 ] we get CeT R

(TI>

(To)

or for R (t) we have

+ min 10, a(T0)}e

O(r-T0)-Ce

CeT0 -r0 e (T-T~)

V. Schuchman

518

+ minC (o,toi(to)3exp(-)ln(,) C

t 0

%)to c

-’In+ t

C t + c exp(r)to ’ln-

= a(to)

+

max{O,-tO~(tO)~exp(-)~nC t

= e(t

+

p(tO )In-

0

where p(tO) = max{O,-tOi (to))exp(,)

0

C

t

+

,v (to) = c

t v(t )InO

exp("tO)

to

.

to-1

0

Lemma 3 is proved. Acknowledgments: This question was posed by Prof. L. Nirenberg in a private discussion on this topic. REFERENCES 1.

Agmon, S. and Nirenberg, L., "Properties of solutions of ordinary differential equations in Banach space", Corn. Pure Appl. Math., 16 (1963), pp. 121-239.

2.

Agmon, S., Unicit6 et convexit6 dans les probldmes diffgrentieles, Montreal, 1966.

3.

Schuchman, V., Dissertation, The Weizmann Institute of Science, Rehovot, 1981.

4.

Alinhac, S. and Baouendi, M.S., Uniqueness for the characteristic Cauchv Droblem and strons unique continuation for hisher order 2. of Math.; 102 (1980), partial-differential inequalities, pp. 179-217.

-.

5.

Schuchman, V., The behavior of solutions of non-linear differential equations in Hilbert space, North Texas State University, Denton, 1981.

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

519

m-Functions and t h e A b s o l u t e l y Continuous Spectrum of One Dimensional Almost P e r i o d i c SchrBdinger Operators B a r r y Simon Departments o f Mathematics and Physics C a l i f o r n i a I n s t i t u t e o f Technology Pasadena, C a l i f o r n i a 91125

Abstract.

We describe a program i n i t i a t e d by Moser [l],and Johnson-Moser [Z],

and then developed by Kotani [3] ( K o t a n i ' s work was extended t o t h e d i s c r e t e case by Simon [4])

and Deift-Simon [5].

We discuss r e l a t i o n s between t h e

d e n s i t y o f s t a t e s , t h e Lyaponov exponent, and t h e c l a s s i c a l in-function o f Weyl. I n p a r t i c u l a r , we o b t a i n K o t a n i ' s r e s u l t s t h a t t h e e s s e n t i a l support o f t h e a b s o l u t e l y continuous spectrum i s p r e c i s e l y t h e s e t where t h e Lyaponov exponent vanishes, and t h a t i n t h e random n o n - d e t e r m i n i s t i c case t h e Lyaponov exponent i s a.e. p o s i t i v e .

We a l s o d e s c r i b e t h e Deift-Simon r e s u l t s t h a t , i n t h e d i s c r e t e

case, t h e Lebesgue measure o f t h e s e t where t h e Lyaponov exponent vanishes i s a t most 4, and t h e c o n s t r u c t i o n o f continuum e i g e n f u n c t i o n s f o r t h e a b s o l u t e l y continuous spectrum. References 1.

J. Moser, An example o f a Schrodinger o p e r a t o r w i t h almost p e r i o d i c p o t e n t i a l and nowhere dense spectrum, Comm. Math. Helv. 56 (1981), 198-224.

2.

R. Johnson and J. Moser, The r o t a t i o n number f o r almost p e r i o d i c p o t e n t i a l s , Comm. Math. Phys. (1982), 403-438.

3.

S. Kotani , Lyaponov i n d i c e s determine a b s o l u t e l y continuous s p e c t r a of s t a t i o n a r y random one-dimensional Schrodi nger operators, Proc. Kyoto Stoch. Conf., 1982.

4.

B. Simon, Kotani t h e o r y f o r one dimensional s t o c h a s t i c Jacobi matrices, t o appear i n Comm. Math. Phys.

5.

P. D e i f t and B. Simon, Almost p e r i o d i c Schrodinger operators, 111. The a b s o l u t e l y continuous spectrum i n one dimension, t o be submitted t o Comm. Math. Phys.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS

I.W.Knowles and R.T. Lewis (Editors)

521

0 Elsevier Science Publishers B.V. (North-Holland), 1984

THE COMPLETE S O L U T I O N SPACE FOR A SYSTEM

OF REACTION-DIFFUSION EQUATIONS J o e l A.

Smoller'

Department of M a t h e m a t i c s The U n i v e r s i t y o f Michigan Ann A r b o r , Michigan U.S.A.

W e show how t h e g e n e r a l i z e d Morse t h e o r y as d e v e l o p e d by C . Conley c a n be u s e d i n conj u n c t i o n w i t h a n a l y t i c methods t o d e t e r m i n e t h e e n t i r e solution space of a p a r t i c u l a r system of r e a c t i o n - d i f f u s i o n e q u a t i o n s .

1.

INTRODUCTION

The p u r p o s e - o f t h i s p a p e r i s t o show how t o combine a n a l y t i c a l and t o p o l o g i c a l t e c h n i q u e s i n o r d e r t o d e t e r m i n e q u a l i t a t i v e p r o p e r t i e s c o n c e r n i n g t h e s p a c e of s o l u t i o n s of r e a c t i o n - d i f f u s i o n equations.

The methods a r e i l l u s t r a t e d by c o n s i d e r i n g a s p e c i f i c

example. I n s t u d y i n g t h e s p a c e o f s o l u t i o n s , One i s l e d n a t u r a l l y t o a c o n s i d e r a t i o n of c e r t a i n t o p o l o g i c a l a s p e c t s .

For t h e

c a s e u n d e r c o n s i d e r a t i o n h e r e , o u r main t o p o l o g i c a l t o o l i s t h e Conley i n d e x ,

[ l ] , a homotopy i n v a r i a n t which g e n e r a l i z e s t h e

c l a s s i c a l Morse i n d e x . I n o r d e r t h a t t o p o l o g i c a l t e c h n i q u e s can be a p p l i e d t o t h e s t u d y of p a r t i a l d i f f e r e n t i a l e q u a t i o n s , i t i s n e c e s s a r y t h a t t h e r e b e s u f f i c i e n t "compactness" a v a i l a b l e .

Furthermore, i n o r d e r t h a t

we c a n a c t u a l l y compute t h e t o p o l o g i c a l i n v a r i a n t s , t h e e q u a t i o n s s h o u l d a d m i t some s o r t of " g r a d i e n t - l i k e s t r u c t u r e . ' '

The compact-

n e s s i s e a s y t o o b t a i n , namely, i t comes from h a v i n g s o - c a l l e d " a t t r a c t i n g r e g i o n s " i n phase s p a c e ; i . e . ,

compact r e g i o n s i n

which a l l s o l u t i o n s e v e n t u a l l y e n t e r as

+

t

+

m .

Such sets can

b e l o c a t e d by u s i n g s t a n d a r d " i n v a r i a n t r e g i o n t e c h n i q u e s " , Chap.

141.

[4,

The g r a d i e n t s t r u c t u r e , on t h e o t h e r h a n d , i s f a r

more r e s t r i c t i v e , and comes, g e n e r a l l y s p e a k i n g , from t h e e x i s t e n c e of a Lyapunov

function f o r the equations.

P e r h a p s t h e main

i d e a i n t h i s paper i s t o demonstrate t h e g r a d i e n t - l i k e s t r u c t u r e w i t h o u t h a v i n g a Lyapunov f u n c t i o n . 'Research

T h i s i s done by showing

s u p p o r t e d i n p a r t by NSF C o n t r a c t # MCS 8 0 - 0 2 3 3 7

J.A. Smoller

522

t h a t the equations a r e "effectively" gradient-like

, in

t h a t a l l solutions tend t o s t a t i o n a r y solutions a s

t

t h e sense

+

+ a .

W e assume t h a t o u r e q u a t i o n s v a r y s m o o t h l y w i t h t h e p a r a m e t e r s i n t h e problem; i n d e e d , t h e n o n l i n e a r f u n c t i o n s themselves can be taken as parameters.

Thus, a s t h e p a r a m e t e r s change i n t h e

e q u a t i o n s , t h e c o r r e s p o n d i n g s p a c e of s o l u t i o n s u n d e r g o c h a n g e s i n s u c h a way t h a t a r e l e v a n t homotopy i n v a r i a n t , t h e Conley i n d e x , d o e s n ' t change.

T h i s i n t u r n i m p l i e s , by a s t a n d a r d p r o c e d u r e i n

a l g e b r a i c t o p o l o g y , t h e e x i s t e n c e of c e r t a i n a l g e b r a i c i n v a r i a n t s a s s o c i a t e d w i t h o u r homotopy i n v a r i a n t s .

These are e x a c t

s e q u e n c e s o f cohomology g r o u p s o b t a i n e d from t h e Morse decomposit i o n s of t h e s o l u t i o n s p a c e ,

[4, Chap. 2 3 5 D I . S i n c e t h e a l g e b r a i c

i n v a r i a n t s a r e " d i s c r e t e " o b j e c t s , t h e y remain c o n s t a n t under s m a l l p e r t u r b a t i o n s , and c a n t h u s b e u s e d t o o b t a i n i n f o r m a t i o n o n t h e s p a c e of s o l u t i o n s .

I n o t h e r w o r d s , w e c a n " c o n t i n u e " t h e equa-

t i o n s t o less c o m p l i c a t e d o n e s , i n which w e c a n e x p l i c i t l y compute the algebraic invariants. T h e s e methods w i l l b e i l l u s t r a t e d by c o n s i d e r i n g a s p e c i f i c s y s t e m of r e a c t i o n - d i f f u s i o n e q u a t i o n s i n which t h e above p r o c e d u r e s are a p p l i c a b l e and e n a b l e u s t o o b t a i n ( q u a l i t a t i v e l y ) ,

W e do n o t claim t h a t

t h e s t r u c t u r e of t h e e n t i r e s o l u t i o n space.

o u r example h a s any r e l e v a n c e o u t s i d e of i t s u s e a s a n i l l u s t r a t i o n

2.

THE EQUATIONS

We c o n s i d e r t h e f o l l o w i n g s y s t e m of r e a c t i o n - d i f f u s i o n equat i o n s i n a s i n g l e space variable:

where

a

and

y

are positive constants.

W e assume t h a t

u

s a t i s f i e s homogeneous Neumann boundary c o n d i t i o n s , a n d t h a t

z

s a t i s f i e s homogeneous D i r i c h l e t b o u n d a r y c o n d i t i o n s ; i . e . ,

(2)

(UX('L,t)

,Z(*L,t))

t > O .

= (OIO),

A s f o r t h e f u n c t i o n s which a p p e a r i n t h e e q u a t i o n s (l), w e r e q u i r e

t h a t t h e i r g r a p h s have t h e q u a l i t a t i v e forms a s d e p i c t e d i n F i g u r e 1.

to of

W e require t h a t f o r the function

g2

,

b

i s s m a l l , and t h a t

f o r t h e function b, t h a t

O ~ Z ' 1 + 5 , 5 > 0 .

f

r

g, t h a t

g1

is close

i s n e a r z e r o and t h e s l o p e

i s c o n s t a n t on some i n t e r v a l

523

A System o f Reaction-Diffusion Equations

Figure 1 Now l e t

b e a g i v e n s m a l l p o s i t i v e number, and c o n s i d e r

> 0

E

the set

where

a

and

C o n c e r n i n g t h i s s e t , w e h a v e t h e f o l l o w i n g lemma.

respectively. If

Lemma 1.

$!

a r e a s d e p i c t e d i n F i g u r e l ( i )and l(iii)

6

/g1-g21

i s s m a l l , and

b

h a s s m a l l slope, t h e n

i s a g l o b a l a t t r a c t i n g region f o r a l l s o l u t i o n s of

,

i n t h e s e n s e t h a t g i v e n any n e i g h b o r h o o d N of

lies i n

(u(x,t),z(x,t)) Proof.

I

while i f

-

-E

-

t

sufficiently large.

F ( u , z ) = ( a ( u ) f ( z ) , b ( u ) g ( z )+ $ ( z ) ) ; t h e n i f

Let

G1 ( u , z ) =

for

N

(l), ( 2 ) ,

every solution

u,

-

VG1

a >

a,

<

E

*

and

VG2

E

, we

Flu=-;

have = -a(-E)f(z)

-

G2 ( u , z ) = u - a

- Flu,;

<

o ,

w e have

I

= a(a)f(z) <

o

.

I t f o l l o w s from t h i s t h a t a l l s o l u t i o n s t e n d t o t h e r e g i o n -E

< u <

a,

[4, Chap. 14,SEI. G3(u,z) = z

and

I n t h i s range, i f

- n,

Q

=

1

+

E,

524

G4

then vG3

-F

OG4

-F

and Z=

-e

= -b(u)g(G)

- @(G)

< 0

,

- 921 i s s m a l l , and b h a s s m a l l s l o p e . Using a 191 r e s u l t i n 4 , Chap. 4 , § E ] , it follows t h a t is a global since

a

0

attractor.

A s a n immediate c o r o l l a r y , w e have

Fact 1 A l l steady-state

( i . e . , t i m e i n d e p e n d e n t ) s o l u t i o n s of

R .

(11,( 2 ) l i e i n

The s t e a d y - s t a t e s o l u t i o n s w i l l b e s t u d i e d i n t h e n e x t section. 3.

Steady-State Solutions I t i s easy t o construct c e r t a i n p a r t i c u l a r steady-state

s o l u t i o n s of a(;)

= 0

(1); namely, i f

u

is a r o o t o f

( s e e F i g u r e l ( i )) , t h e n t h e f u n c t i o n

s t e a d y - s t a t e s o l u t i o n of

(1) , ( 2 )

,

where

z

a(ii): i . e . ,

( u , z ( x ))

is a

satisfies the

equation (3)

+

YZ"

@(z)

+

b(u)g(z) = 0

,

t o g e t h e r w i t h t h e boundary c o n d i t i o n s (4)

Z(+L) = 0

.

I f we set o(z) = @(z)+ b(u)g(z),

t h e n t h e g r a p h of

@

h a s t h e form d e p i c t e d i n F i g u r e 2 , and

t h e c o r r e s p o n d i n g phase p l a n e f o r ( 3 ) can be d e p i c t e d i n F i g u r e 4.

Namely, t h e r e are t h r e e s o l u t i o n s ,

Figure 2

zo,z1,z2,

Figure 3

as

525

A S.vstern of Reaction- Diffusion Equations

shown, b u t or

and

z1

e x i s t only i f

z2

is sufficiently small,

y

i s sufficiently large

L

[51.

Now from t h e r e s u l t s i n [ 4 , Chap. 24,SD], f o r t h e s c a l a r p r o b l e m ( 3 ) , (4), w e know t h a t t h e Conley i n d e x of b o t h z2

of

is

Io, t h e ( p o i n t e d ) z e r o s p h e r e , and t h e Conley i n d e x

z1

is

El,

t h e ( p o i n t e d ) 1-sphere.

and

zo

That i s , both

and

zo

a r e s t a b l e s o l u t i o n s o f t h e a s s o c i a t e d t i m e - d e p e n d e n t problem

z2

z

(5)

while

z1

=

yzxx

+

O(z), Z(+L,t)

1x1 < L , t

= 0,

I f now w e u s e

has a 1-dimensional u n s t a b l e manifold.

the notation

' 0,

t o d e n o t e t h e Conley i n d e x , t h e n i n t h e s e t e r m s

h(-)

w e have h(z ) = h(z ) = ; ’ 1

(6)

h(zl) = C

2

0

1

.

F u r t h e r m o r e , a s shown i n [4, Chap. 2 4 , § E ] , t h e r e e x i s t s o l u t i o n s v

w

and

of

z1

( 5 ) which " c o n n e c t "

to

zo

and

z2

,

in the

sense t h a t

,

l i m v ( x , t ) = z,(x)

lirn v(x,t) = z,(x), t+-m

t++m

and

,

l i m w ( x , t ) = z2(x)

lirn w(x,t) = z,(x), t+-m

t++m

1x1 < L

uniformly f o r

.

Using t h e s e f u n c t i o n s , w e s e e t h a t t h e

corresponding rest points

( i . e . , steady-state)

s o l u t i o n s of

(l), ( 2 ) a r e c o n n e c t e d by o r b i t s of (l), ( 2 ) ; namely, l i m (ii,v(x,t)) = (U,zo(x)), t=

l i m [L,v(x,t)) = (U,zl(x)), t+-m and

l i m (ii,w(x,t)) = (ii,z1(x)) t+-m

1x1 < L

uniformly f o r

.

Now i f w e assume t h a t root

a

O,b, o r

,

,

L >>

1

lim(ii,w(x,t)) t'" or

y << 1

,

=

(U,z,(x))

,

t h e n f o r each

a ( u ) , w e have t h e p h a s e p l a n e s a s i n

of

Figure 3 , together with the corresponding three s o l u t i o n s ; t h a t i s

i)

to

Oo ( z )

three solutions

ii)

yz"

+

to

Q(z)

-

0

(2)

= 0,

Z(?L)

b(b)g(z)

three solutions = 0;

@ ( z ) , there corresponds t h e

zo(x), z , ( x ) , z 2 ( x ) , of t h e p r o b l e m

5

Z(?L)

+

b(O)g(z)

f

+ 5

= 0

;

@ ( z ) , there corresponds the 1

zo,z1,z2, of t h e p r o b l e m

-

yz" + O(z) = 0,

526

J. A. Smoller iii)

-

+

@(z) 5 b(a)g(z)

to

- - _

solutions

zo,z1,z2

,

of t h e p r o b l e m

yz”

+

m(z) = 0 ,

.

= 0

Z(+L)

$ ( z ) , t h e s e corresponds t h e t h r e e

W e t h u s have

Fact 2

If

or

1

L >>

y << 1, t h e s y s t e m

(1) - ( 2 )

has a t

least 9 steady-state solutions,

_ -

(O,zi),

(b,zi),

--

( a , ~ ~i )=, 1 , 2 , 3 ,

and t h e corresponding connecting o r b i t s a t each “u-level.

Now f i x

L > 0,

and t a k e

y > 0

“

s o s m a l l t h a t t h e above 9

steady-state solutions a l l exist. Lemma 2 .

as

t

+

a

If

s o l u t i o n of m

.

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

(1), ( 2 1 , t h e n

u

In particular,

if

tends t o i t s ct

(u,z)

i s any

p a t i a l average

i s l a r g e , t h e above 9

s o l u t i o n s are t h e o n l y s t a t i o n a r y o n e s . Proof.

Let

Then f o r

t

(u,z)

b e any s o l u t i o n of

large,

(u,z) €

e i g e n v a l u e of

on

D2

1x1 < L

,

(1),

X

and i f

2 ) , and s e t

i s t h e f i r s t non-zero

w i t h homogeneous

Neumann

boundary c o n d i t i o n s ,

where w e h a v e u s e d P o i n c a r g ’ s i n q u a l i t y . then

0

> 0

if

CY

> M/X

and w e have

at

T h u s , u s i n g a s t a n d a r d i n e q u a l i t y [ 4 , Th.

1 = aX 2 so t h a t

Now t h e n i f < -54

,

-5

11.111, w e f i n d

- M,

527

A System ofReaction-Diffusion Equations where

v(t)

l v(t) = 2L

Thus i f u(x,t) if

L

t e n d s t o a f u n c t i o n of u ( x ) : U.

,

a(u) = 0

and

a

as

t

so t h a t

y

L > 0 y <<

+

m

.

I t follows t h a t

(O,Zo(X))

,

u ( x ) must b e

i s f i x e d , and t h a t w e h a v e

1, a. >> 1, a n d t h a t w e h a v e

p r e c i s e l y 9 s t a t i o n a r y s o l u t i o n s of Po =

t

S i n c e o = aUxx + a ( K ) f ( z ) = a ( L ) f ( z ) , a n d t h e p r o o f i s c o m p l e t e . [I

Now l e t u s assume t h a t chosen

.

is a s t a t i o n a r y s o l u t i o n , then

(u(x),z(x))

constant, say

u(x,t)dx

i s a s o l u t i o n of ( I ) , ( 2 ) , then

( u ( x , t ), z ( x , t ) )

w e see t h a t

u; i.e.,

i s t h e s p a t i a l a v e r a g e of

(11, ( 2 ) .

P1 = ( 0 , Z l ( X ) )

W e d e n o t e t h e s e by

,

P2 = ( 0 , z 2 ( x ) )

Qo = ( 6 , s z o ( x ) )

, Ql

=

(b,zl(x))

I

Q2 = ( E l i 2 ( x ) )

Ro = ( a , y o ( x )

, R1

=

(a,T,(x))

,

R2 =

,

(a,z,(X)) .

These 9 s t a t i o n a r y s o l u t i o n s , t o g e t h e r w i t h t h e i r c o r r e s p o n d i n g c o n n e c t i n g o r b i t s c a n b e c o n v e n i e n t l y d e s c r i b e i n F i g u r e 4 below.

Fjgure 4

I n o r d e r t o o b t a i n more p r e c i s e i n f o r m a t i o n c o n c e r n i n g t h e u n s t a b l e m a n i f o l d s of t h e s e s t a t i o n a r y s o l u t i o n s , w e s h a l l l i n e a r i z e e q u a t i o n s ( 1 ) . ( 2 ) a b o u t them.

We c o n s i d e r f i r s t t h e

The C o r r e s p o n d i n g l i n e a r i z e d e q u a t i o n s a r e , f o r

P.’s.

i = 0,1,2,

A;

=

A;

= ~ ~ " + ~ ' ( O ) ~ ( Z ~ ( X ) ) ; + ~ ; ) ( Z ~ ( z(?L) X ) ) ~ = , 0

(7)

a;’! + a l ( o ) f ( z i ( x ) ) ;

Suppose f i r s t t h a t

,

; I ( ~ L )=

i = 0 ; then since

o

,

a ' (0) f ( z o ( x ) ) < 0

see from t h e f i r s t e q u a t i o n i n ( 7 ) t h a t

-

A < 0

if

;f

.

0

.

we

.

On t h e o t h e r h a n d , i f u : 0 , ( 6 ) shows t h a t A < 0 I t follows 0 that h(Po) = C I n a s i m i l a r way, w e c a n show t h a t P2,R0

.

and

R2

a r e a l s o non-degenerate

and t h a t

h(P2) = h(Ro) =

528

J. A . Smoller

.

Now consider ( 7 ) when i = 1, i.e., at h(R2) = ’1 pl' Again, if u 4 0 , then X < 0 , and if u 5 0 , then again using ( 6 ) , we find that P1 is non-degenerate and h(P1) = E l ; similarly R1 is non-degnerate, and h(R1) = Z 1

-

5

Now let’s turn our

attention to the

Qi’s

.

.

The relevant

equations are

If 5 0 , then from (6) we find that Qi is non-degenerate, and that the second equation in ( 8 ) has exactly one positive eigenvalue if i = 1, and no positive eigenvalues if i = 0, o r i = 2. But let us note that

L =

+ a’(b)f(O)$)dx

sup @

,

1-L

and that these two expressions are the variational characterizations of the operators aD2

+

a’(b)f(zi(x))

, and

aD2

+

a’(b)f(O)

,

together with homogeneous Neumann boundary conditions. Since is a positive eigenvalue of the second operator, a’ (b)f ( 0 ) (corresponding to the eigenvector

-

u

E

11, we see that

Hence the first equation in(8) has at least one positive eigenvalue Xi If u is the corresponding eigenvector, we shall show that the second equation in ( 8 ) can be solved for z To this end, consider the operator 2 (9) yD - Xi .t s’(ii(x)) ,

.

- .

with homogeneous Dirichlet boundary conditions. Since Xi > 0, this operator is invertible if i = 0, or i = 2, (by ( 6 ) ) . k It follows that h(Qo) = 1 , h(Q2) = Em, and h(Q1) = En , where k 2 1, m 2 1 and n 2 1. We summarize these results in the following proposition.

529

A System of' Reaction-Diffusion Equations

Proposition 3 .

A.

h(R1) = C1: h(Qo) m 2 1, and n 5 1.

that that operators.

h(R2) = Z o ; n h(Q1) = Z m , h(Q2) =I: , where

h(P ) = h(P2) = h(R ko

,

C

=

=

0

h(P1)

=

k 2 1,

B. The P i t s and R i g s are non-degenerate in the sense 0 is not in the spectrum of the corresponding linearized

We next show that the equations (l), (2) are "effectively" gradient-like; in fact we have the following proposition. Proposition 4 .

If

is sufficiently large, and

c1

y

is sufficiently

small, then all solutions of (l), (2) tend to staionary solutions as t + +m

.

Proof.

Let

(u,z) be any solution: then

exponentially to zero as of

u.

t

-f

m ,

1Iu

v

where

-

vII L

tends

(.- L ..L )

is the spatial average

Since v

utdx = l

= -

- -l -

L

cla(v)

+ a(v)k

a(u) -a(v)]dx

2L

=

j-La(~)f(z)dx L

+

1,

f(z)dx

0 e-Ot)

as t + m, where c1 > 0 , hence u , tends to a root x (see 121).

we can conclude from 131 that v , and u of a(u), as t m, uniformly in -f

To complete the proof, we must show that

z

tends to a solu-

tion of yz"

(9)

+

b(a)g(z)

+ @(z)

=

0,

1x1 < L,

z(+L)

=

0.

The problem w t = y w x x + b(u)g(w) + @(w), (10)

W(fL, t) =

o , t

1x1 < L,

0, W(X,O) = Z(X,O),

t > 0, 1x1 < L

w(x) as t + + m , has a unique solution w(x,t), and w(x,t) uniformly on 1x1 < L, where w satisfies (9). Choose E > 0 so small that if Ilw(x,O) - z(x,O) I / m < E , then the corresponding solution of (10) tends to w as t + + m . -f

Now we can find

TE > 0

such that

530

J.A. Smoller wE

Let

be t h e s o l u t i o n of

w

=

wxx + b ( o + ~ ) g ( w+)$ ( w ) = 0 ,

If

2

t

T

t > 0,

= 0,

W(fL,t)

= z(x

W(X,O)

then

E '

-

zt

<

0 = WE

-

ywxx

-

(b(u)

g(z

Y Z ~ , - ~ ( O + E ) ~ ( Z- )@ ( z ) =

b ( a + E g(wE)

-

- b ( a + €1)

@(WE)

,

s o t h a t b y t h e s t a n d a r d c o m p a r i s o n t h e o r e m [ 4 , Th.lO.11, z(x,t) 5 wE(x,t),

(11)

But as

t

+

wE (x,t )

OD,

1x1

(9) as

0,

of

E +

f o r any g i v e n

u n i f o r m l y on

lim z(x,t) t-t M is s u f f i c i e n t l y s m a l l .

E

,

1x1 < L

wE. ( x )

.

of = 0

,

tends t o a solution

w

2

we see from (11) t h a t

1x1 < L ,

6 > 0,

-

if

T~

1x1 < L, w ( + L )

(see [ 6 ] ) . S i n c e

< L;

2

tends t o a solution

y w " + b ( u + ~ ) g ( w+) @ ( z ) = 0 , u n i f o r m l y on

t

if

5 wE(x) 2

Z(x)

Replacing

(u

+ 6 +

E)

(u

by

- E)

in the

above argument g i v e s

These l a s t t w o i n e q u a l i t i e s imply t h a t

t

on

1x1 < L ,

4.

T o p o l o g i c a l Methods

as

l i m z ( x , t ) = Z ( x ) , uniformly

T h i s completes t h e proof.

+ m.

0

W e a r e now i n a p o s i t i o n t o a p p l y t o p o l o g i c a l t e c h n i q u e s t o o u r problem.

Our main t o o l w i l l b e t h e C o n l e y i n d e x , and w e assume

t h a t t h e r e a d e r i s f a m i l i a r w i t h t h e main r e s u l t s i n t h i s t h e o r y ;

see 113 o r [ 4 , P a r t IV] Let points

I.

(resp.1

P o , P1,

and

a

)

. d e n o t e t h e s e t c o n s i s t i n g of t h e t h r e e r e s t

P2 ( r e s p . Ro,

R1

and R 2 )

,

together with t h e i r

c o r r e s p o n d i n g c o n n e c t i n g o r b i t s : see F i g u r e 4 . Lemma 5 .

I.

h(I ) = h ( I - ) 0

a

and =

I-

o a C .

a r e i s o l a t e d i n v a r i a n t s e t s and

A System of Reactioii-Diffusion Equations

Proof.

531

Define L

Y(t)

=

Y’(t)

=

then

I-, I-,

u(x,t)dx

;

L

.

f(z(x,t)) a(u(x,t))dx

Then Y ' < 0 if u(x,t) < 5; 1x1 < L. Since small neighborhoods of I. lie in u < g , any invariant set in such a neighborhood would have to tend to rest points in both time directions. The Hence I. and only such sets are the depicted ones in I. similarly I- are isolated invariant sets. a To compute the index of I0 ' we deform the function a(u) as depicted in the figure below:

.

Figure 5 Under this deformation I. continues to the maximal invariant set in @ ; hence h(Io) = c o by the continuation theorern. In a 0 similar way, h(I-1 = 1 . I] a We can now prove the following result which gives the precise indices of the Qi’s. Lemma 6.

h(Qo)

=

h(Q2) =

c1 ,

and

h(Q1) = 1

2

.

Proof. If I (@ ) denotes the maximal invariant set in , then h(I(&)) = z 0 We now use the continuation theorem and deform @ ( z ) by "pushing down the hill; see Figure 6.

.

Figure 6

J.A. Smoller

532

t h e r e s t p o i n t s R1, R 2 , 0 1 1 Q2 and c o n t a i n s o n l y t h e r e s t p o i n t s R o , Qo

Under t h i s d e f o r m a t i o n , P1, P 2 c a n c e l , a n d @ and

Since

Po.

connecting

and

Ro

t o both

Qo

are a t t r a c t o r s , t h e r e e x i s t o r b i t s

Po Ro

,

and

( s e e [ 4 , Ch. 2 4 ,

Po,

§El), a s

d e p i c t e d below.

Figure 7 PJext, l e t of

M

denote t h e i s o l a t e d i n v a r i a n t set c o n s i s t i n g

a n d t h e o r b i t s c o n n e c t i n g them.

Q o , Po

Then

h(n) =

( t h e homotopy c l a s s of a p o i n t ) , s i n c e w e may d e f o r m " p u l l i n g u p t h e v a l l e y " , whereby

and

Qo

Po

cancel;

a(u)

by

see

F i g u r e 8.

Figure 8 W e now u s e t h e l o n g e x a c t s e q u e n c e o f cohomology g r o u p s (see [ 4 , Ch. 2 3 ,

-t

§D])

Hn-l(h(P4))

-+

Hn-'(h(P0))

-t

H n ( h ( Q o ) )+ Hn(h(P4))

-+

-.-,

or

... + 0

+

En-1

This implies t h a t Similarly

1x0)

+

Hn-l (Co)

h ( Q 2 ) = IL.

down t h e v a l l e y " i n

IIn (Zk)

I$

Hn(Ck)

Since I

+ 0 +

O1

,

and

... . for e a c h n : h e n c e k = 1. 0, c a n c e l wehn w e " p u s h

w e c a n u s e an argument s i m i l a r t o t h e

one j u s t given t o conclude t h a t

h(Q1) = C 2

.

A System o f Reaction-Diffusion Equations

533

At this stage it is useful to depict the rest points in

@

together with the known connecting orbits in Figure 9.

Figure 9 We can now show that there are orbits-running from Qo to and Po as well as orbits running from Q2 to both R2 and P2. We shall give the details only for Q o to R o . Thus, let denote the isolated invariant set containing as rest points I2 the Qi’s and R . ' s , i = 0,1,2, as depicted in Figure 9. By continuation (push down the hill in @ as in Figure 61, this set Ro

continues to the empty set so that h(12) = 5 - If S denotes the isolated invariant set containing Ro and 0 as its only -0 (This is easily seen by pushing rest points, then h ( S ) = 8 . down the hill in a ; see Figure 5. Under this deformation, R1 and R2] as well as Q , and Q2 cancel so S continues to I2 . ) Now we can invoke the connecting orbit theorem [4, Th. 22-23], to conclude that there is an orbit running from Q o to Ro. Similarly, there are orbits running from and P2

.

R2

Qo

to

Po

and from

Q2

to

Finally, we shall show that there are orbits running from Q, to both P1 and R 1 ; we shall give the details only for the case Q, R1. Thus, suppose on the contrary] that there is no orbit running from Q1 to R1. Let T be the isolated invariant set containing Ol and R1 as its only rest points. We claim . that Hn(h(T)) = 0, n = 0,1,2 To see this, let Ptl be the disjoint union of Q2,R2 and the orbit connecting them, together with Q o l R o and the orbit connecting these rest points; see Figure 10. +

,...

J. A. Smoller

5 34

Figure 10 Let fa2 = T; then ( H l , M 2 ) is a Morse decomposition of the following sequence is exact (see [ 4 , Ch. 23, S D I ) :

...

+

H"-’(h(M1))

+

Hn(h(M2))

.

Hn(h(12)) *

-r

12. Thus

Since H " - ~(h(lll)) = 0 = Hn(h(12)) for all n, it follows that Hn(h(M2)) = 0 for all n , thereby proving our claim. Now if there was no a f4orse decomposition of

orbit from T,

to

Q1

R1

then

(U1,R1) is

and using the exactness of the

sequence .-.+Hn-l(h(T)) * Hn-l(h(Q1))

-r'

Hn (h(R1))

+

Hn(h(T))

+

--*,

or 0

+

Hn-l(h(Ql))

--t

Hn(h(R,))

-f

we see that Hn-l (h(Q1)) = Hn(h(R1)) or all n. This is impossible, and thus R1 p1 and Q,) are connected by orbits.

0

+

, H ~ ( C ~ )for ,

Hn-l(C2)

2

and

(as well as

Ql

We may depict (schematically) the maximal invariant set in

@ in the following diagram. That is, there is an invariant 2 dimensional manifold in @ consisting of orbits connecting rest

R,

Figure 11

535

A System of Reaction-Diffusion Equations

p o i n t s as d e p i c t e d .

rest p o i n t s as a l l s o l u t i o n s of

t

-+

A l l o t h e r s o l u t i o n s t e n d t o one of t h e 9 +m

.

T h i s i s t h e complete g l o b a l p i c t u r e of

(1), ( 2 ) .

REFERENCES : C o n l e y , C . , I s o l a t e d I n v a r i a n t S e t s a n d t h e Morse I n d e x , CBMS R e g i o n a l C o n f e r e n c e S e r i e s i n M a t h . , N o . 38, Amer. Math. S O C . , Providence, R . I . , 1978. Conway, E . , D. H o f f , a n d J. S m o l l e r , L a r g e t i m e b e h a v i o r o f s o l u t i o n s of systems of n o n l i n e a r r e a c t i o n - d i f f u s i o n equat i o n s , SIAM J . Appl. M a t h . , 2 ( 1 9 7 8 ) , 1-16. Markus, L . , A s y m p t o t i c a l l y autonomous d i f f e r e n t i a l s y s t e m s , C o n t r i b u t i o n t o t h e Theory o f N o n l i n e a r O s c i l l a t i o n s , v o l . 3, A n n a l s of Math. S t u d i e s , N o . 3 6 , P r i n c e t o n Univ. P r e s s , P r i n c e t o n , 1 9 5 6 , 17-29. S m o l l e r , J . , Shock Waves a n d R e a c t i o n - D i f f u s i o n S p r i n g e r - V e r l a g , New York, 1983.

Equations,

S m o l l e r , J . , and A. Wasserman, G l o b a l b i f u r c a t i o n of s t e a d y s t a t e s o l u t i o n s , J. D i f f . E q u s . , 3 9 ( 1 9 8 1 ) , 269-290. Conway, E . , R . G a r d n e r , a n d J. S m o l l e r , S t a b i l i t y a n d b i f u r c a t i o n o f p r e d a t o r - p r e y s y s t e m s w i t h d i f f u s i o n , Adv. Appl. M a t h . , 3 , ( 1 9 8 2 1 , 288-334.

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DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

5 31

VARIATIONAL PROBLEMS WITH SINGULAR SOLUTIONS

R. Temam

L a b o r a t o r i e D’ Analyse Numerique U n i v e r s i t e de P a r i s - Sud B a t imen t 425 91405 Orsay Cedex France INTRODUCTION V a r i a t i o n a l problems which a r e c o e r c i v e on a n o n r e f l e x i v e Banach space may possess s i n g u l a r s o l u t i o n s . S i n g u l a r v a r i a t i o n a l problems o f t h i s t y p e a r i s e i n mechanics and p h y s i c s , f o r i n s t a n c e i n s o l i d mechanics ( p l a s t i c i t y ) , o r i n problems r e l a t e d t o minimal area s u r f a c e s o r t o s u r f a c e s o f g i v e n mean c u r v a t u r e . The t y p i c a l s i t u a t i o n i s as f o l l o w s : we a r e g i v e n a n o n r e f l e x i v e Banach space V and a l o w e r semi-continuous f u n c t i o n a l 0 f r o m V i n t o IR which i s when IIvllv + + -. The f u n c t i o n i s t h e n bounded coercive, i . e . ~ ( v )+ +

-

from below on V b u t t h e minimum i s n o t n e c e s s a r i l y achieved. I n order t o obtain t h e e x i s t e n c e o f a g e n e r a l i z e d m i n i m i z e r i t i s necessary t o imbed V i n t o a l a r g e r space

i,

t o extend

@

l o o k f o r t h e minimum o f

on

on

as a f u n c t i o n

from

into

IR and then t o

i.

I n t h e a p p l i c a t i o n s mentioned above, V i s s i m i l a r t o a Sobolev space b u i l t 1 i s a space s i m i l a r t o t h e space o f f u n c t i o n s w i t h bounded L (n) w h i l e

v a r i a t i o n , BV(n). F o r such problems t h e e x t e n s i o n o f O t o necessitates t h e s t u d y o f f u n c t i o n a l s o f a measure: t h i s i s o u r aim i n S e c t i o n 1 which p r o v i d e s t h e b a s i c framework f o r S e c t i o n s 2 and 3. S e c t i o n s 2 and 3 g i v e two examples o f v a r i a t i o n a l problems 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 w i t h s i n g u l a r s o l u t i o n s : one of them i s r e l a t e d t o minimal s u r f a c e s , t h e o t h e r one i s a problem encountered i n p l a s t i c i t y . The p l a n i s as f o l l o w s : 1. 2. 3. 1.

Convex f u n c t i o n o f a measure. Time dependent minimal s u r f a c e s . A problem i n t h e p l a s t i c i t y o f p l a t e s .

W E X FUNCTION OF A HEASURE Let

Q

I f ( t ) i 'c a151

(1.1) where an

u

E

1ci

r and l e t f IR, which i s s u b l i n e a r , i.e., has a t most

be an open bounded s e t o f Rn w i t h a smooth boundary

be a c o n t i n u o u s f u n c t i o n f r o m IRE i n t o a l i n e a r growth a t i n f i n i t y : +

v 5 €Re,

b,

i s t h e e u c l i d e a n norm o f

5

i n Re,

and

a, b 2 0.

Then, i f

u

is

L1 f u n c t i o n ( f o r t h e Lebesque measure dx = dxl.. .dxn) f r o m R i n t o R', 1 L ( a ) ' , t h e c m p o s e d f u n c t i o n x t+ f ( u ( x ) ) i s (a.e.) w e l l d e f i n e d on Q and

R. Temam

538 due t o ( l . l ) , f o

u

f o

u t o b e i n L 1 (n),

t o t h e case where f o

E

u

1 L (n)

( a c t u a l l y (1.1) i s necessary and s u f f i c i e n t f o r b' u E L'(Q)'). We want t o e x t e n d t h e d e f i n i t i o n o f f o u a. i s a bounded measure on R, u = u E Ml(n) , i n which case

w i l l be a bounded measure on

1-1

n,

f, 1-1

E

fa o f

We assume t h a t t h e a s y m p t o t i c f u n c t i o n

Then

V

< ER'.

If

5 a\<\,

If,(S)I

and

1-1 E Ml(R)',

f,

then

is

1-1

lPlr

(fmo u b e i n g then independent o f t h e c h o i c e o f

gh

p =

R

exists:

b e i n g e s s e n t i a l l y bounded f o r 1 ~ ;1 more g e n e r a l l y (see f m ( U ) = ( f m o g) A , f o r any p o s i t i v e measure h such t h a t

where u = g - 11-11, g [l],c h a p t e r V . § 5 ) , l ~ l i h ,with

f

N. Bourbaki [ l ] ) :

f,. 1-I = (fmo 9 )

(1.3)

into

n,

i s a bounded measure on

1-1

n a t u r a l l y d e f i n e d (see

F o r a general measure of

IRe

i s p o s i t i v e l y homogeneous from

f,

M1(~).

p E

M,(R)

a.

,

1).

we w r i t e t h e Lebesgue decomposition

U,

S p = h d x + u ,

(1.4) where (1.5)

h E L1(R)' 1-1 =

f o

f o

It is c l e a r t h a t

i s s i n g u l a r , and we d e f i n e f.

and ’1-1 h dx

+

fa,

f. u = f o h

and f. 1-1 f u n c t i o n i n L'(R)'l t h a t f. II depends l o c a l l y on

( o r f o h d x ) when

i s any Bore1 subset o f

B

If

-f

R,

= h

p = lpl

(or

p =

h dx)

is a

Also i t i s easy t o see 1-1, v E M,(n)' and

f. v on 0. The measure and i s 2 0 when f 2 0.

f, u

is

R , one can d e f i n e a f u n c t i o n a l

by s e t t i n g

Fg(1-1) =

11.61

1-1

depends l i n e a r l y on f. 1-1 i n t h e sense t h a t i f

p = v on an open s e t 0 C n, t h e n f. absolutely continuous w i t h respect t o

FB: M1(R)'

as

1-1

S.

J

f o

1-1-

B

f,

The d e f i n i t i o n above o f t h e measure

and t h e f u n c t i o n a l

p

FB,

generalizes

t o t h e nonconvex case, p r e v i o u s d e f i n i t i o n s used b y ti. B r e z i s [2], C. Goffman, J . S e r r i n [7], R. T. R o c k a f e l l a r [S], F. Demengel, R. Temam [5], R. Temam [12]. I n t h e case where f i s convex, two r e s u l t s were proved i n [5]:

-

A d u a l i t y formula:

) $ ( f a u)

(1.7) where

f

*

= Sup{/

V@U

-

1 f*(v)$

n V R R i s t h e Legendre t r a n s f o r m o f

dx},

f,

and t h e supremum i n (1.7) i s taken f o r t h e

v's

i n either

C;(s)'

or

5 39

Variational Problems with Singular Solutions 1 L ( ~ , p ) ' and such t h a t

-

1 f * o v E L (n)'.

An a p p r o x i m a t i o n r e s u l t For every

such t h a t

uj

weakly t o

f o

Ml(n)',

P E

one can c o n s t r u c t (see [5])

u

converges weakly t o

f, u . dx

and

p

+

I

as

j

+

a sequence

while

m,

f

uj

u . E Cg(n)' J converges

U.

f o

R O t h e r r e s u l t s e s t a b l i s h e d i n [5] w i l l be r e c a l l e d when needed. R

2.

J

TIME DEPENDENT MINIMAL SURFACES As b e f o r e , l e t

i s sufficient).

c i e n t l y smooth ( h e r e r.E C1 and i n i t i a l v a l u e problem

+

g

AU =

in

(2.1)

at

(2.2)

u(x,t) = $(x,t),

(2.3)

u(x,O)

where and

QX(O,T)

r

(x,t) E

= uo(x),

x

E

(O,T),

a,

h, 4, uo a r e g i v e n f u n c t i o n s , A

w i t h a boundary r s u f f i We c o n s i d e r t h e f o l l o w i n g boundary

be an open bounded s e t i n lRn

Q

u : nX(0,T)

-f

R

i s t h e unknown f u n c t i o n

i s t h e minimal s u r f a c e o p e r a t o r : Av =

-

grad v

div

41 + ] g r a d v l T Under reasonable assumptions on t h e d a t a , i t i s easy t o show t h a t t h i s problem has a t most one s o l u t i o n , b u t i s n o t e x p e c t e d t o possess i n g e n e r a l a smooth s o l u t i o n . A concept o f p s e u d o - s o l u t i o n was i n t r o d u c e d and s t u d i e d i n R. TemamC111, A. Lichnewsky, R. Temam [8]: assuming f o r s i m p l i c i t y t h a t h = 0, $ i s i n d e 1 2 pendent of t, $(.,t) = $ ( - ) E L ( r ) and uo E L (a), t h e problem ( 2 . 1 ) - ( 2 . 3 ) is shown t o be e q u i v a l e n t t o t h e problem du

(2.4)

-

(2B5)

u(0) =

where

(2.6)

u : [O,T]

( t ) E aF(u(t))

t E (O,T),

UO'

+ H = L

F(v) =

a.e.

f

+

2

(n),

F i s t h e f u n c t i o n d e f i n e d on H by

J1 + / [ w a d v I L

m

if

v

E

+

I r

lv

-

$ldr

if

v E BV(n),

L~(Q)\BV(~),

and a F i s t h e s u b d i f f e r e n t i a l o f F (see [6]). I t follows immediately from t h e n o n l i n e a r semigroup t h e o r y t h a t (2.6) ( 2 . 5 ) possess a u n i q u e s o l u t i o n u ,

(2.7)

u E L2(0,T;D(aF)),

u'

E

L2(0,T;H),

which i s t h e g e n e r a l i z e d s o l u t i o n o f (2.1) - (2.3) and which does n o t s a t i s f y i n g e n e r a l t h e c o n d i t i o n s (2.1) - ( 2 . 3 ) . I n f a c t , t h e f o l l o w i n g was proven ( t h e reader i s referred t o [5] f o r the proofs):

540

R. Temam

Theorem 2.1. be an open bounded s e t

Let

class

Let t h e r e be given

C1.

~i

R n whose boundary r L1 ( r ) and uo E D ( a F ) .

~7f

in

Then t h e r e exists a unique f u n c t i o n Furthermore i f

Vu(t)

g r a d u ( t ) E Ml(n)n

=

k(.,t)

(a.e.

-di v (

(2.8)

dx +

S

~i

u

(t)

t)

t h e n , f o r a.e

, )

+ g = O , d x

i s a manifold of

s a t i s f y i n g ( 2 . 7 ) , (2.4) and ( 2 . 5 ) . i s t h e Lebesyut? decomposition o f t E

[O,T]

- a.e.

in

R

J1 + lkl

(2.10)

41 +

I kl

v E

Sgn($

-

u),

d r - a.e. on

r.

(2)

the unit outward normal on r, Sgn = t h e sign function. FinaZZy the condisupplemented by (2.7) are e q u i v a l e n t t o ( 2 . 4 ) - ( 2 . 7 ) . t i o n s (2.8)(2.9)(2.10)

u

3.

A PROBLEM I N THE PLASTICITY OF PLATES

class

We now assume t h a t 2 C

n i s a bounded open s e t o f lR2

r of

w i t h a boundary

.

Consider a p e r f e c t l y p l a s t i c p l a t e o f shape n whose boundary r i s r i g i d l y f i x e d and which i s loaded o r t h o g o n a l l y t o i t s p l a n e w i t h a s u r f a c e d e n s i t y o f f o r c e s equal t o f ( f E C ( c ) ) . Then under a p p r o p r i a t e p h y s i c a l assumptions, i t s v e r t i c a l displacement u = u ( x ) , x = (xl,x2), i s a solution t o the variational problem

where

v v u i s t h e h e s s i a n o f u,

2 a u ( v v u ) ~= ~ 1

,

1 5 i, j 5 2,

and

4

J

convex f u n c t i o n which grows l i n e a r l y a t i n f i n i t y ,

2

W 6

E

E = space o f symmetric m a t r i c e s o f o r d e r 2,

= {

1

i,j=1

(cij)

2 1/2 }

Since t h i s v a r i a t i o n a l problem i s w e l l s e t i n a n o n r e f l e x i v e space, we c o n s i d e r t h e space HB(n) = I u E L1(n), v u E L ’ ( Q ) ~ , vvu E M 1 ( ~ ; E ) l which i s a Banach space f o r t h e norm

is a

541

Variational Problems with Singular Solutions

We t h e n c o n s i d e r t h e f o l l o w i n g e x t e n s i o n o f problem (3.1)

i s t h e bounded measure on where +(VVu) t h e t e n s o r w i t h elements

R

d e f i n e d i n S e c t i o n 1, and

Fp

is

( F p ) . . = F. . ( p ) = p v i v j , 1J

v =

(v,,~,)

1J

r.

= t h e u n i t outward normal on

The c o n d i t i o n

u

= 0

r

on

makes

sense s i n c e (see F. Demengel [3])

HB(R)

(3.3)

c(E)

c

w i t h a c o n t i n u o u s imbedding. We r e f e r t h e r e a d e r t o F. Demengel [4] and R. Temam [13] f o r t h e q u e s t i o n o f e x i s t e n c e of s o l u t i o n o f (3.2) i n H B ( n ) ( f u n c t i o n s w i t h second d e r i v a t i v e s measures), and f o r t h e r e l a t i o n s o f (3.2) w i t h t h e i n i t i a l problem ( 3 . 1 ) , and we conclude t h i s S e c t i o n w i t h t h e f o l l o w i n g c h a r a c t e r i s a t i o n o f a s o l u t i o n . We i n t r o d u c e t h e space H b ( n ) = { M E Lm(n;E),

V * V - M E M1(n))3

and we have Theorem 3.1.

Under t h e above assumptions, i f u E H B ( n ) i s a soZution of (3.2) and if VVu = h dx + us i s t h e Lebesgue decomposition of t h e kessian of U, then t h e r e e z i s t s M E Hb(n), suck t h a t

+ f

(3.4)

v.v.M

(3.5)

IM(x)]

51

(3.6)

M(x)

E

a+(h(x))

(3.7)

M.U’

= lplS ,

(3.8)

=

n

in

0

a.e.

in

n

f o r a.e.

in

R

x

E

R

(3)

.

Mijvivj

6

i,j=1

Conversely if u E HB(n), s o l u t i o n of ( 3 . 2 ) .

M

- Sgn( % au ) , d r E

Hb(n)

a.e. on

r

(4)

s a t i s f y t h e s e conditions, t h e n

u

is a

We r e f e r t h e r e a d e r t o [ 4 ] f o r t h e p r o o f s and t o R. Temam [13] f o r more g e n e r a l v a r i a t i o n a l problems appearing i n p l a s t i c i t y .

R. Temam

5 42

Footnotes (1)

The measure

rn

us

S

e x i s t s and s a t i s f i e s

IU

I)

so t h a t (2.8) i s n o t t r i v i a l . (2)

S i m i l a r l y t h e t r a c e on

r

< 1

and s a t i s f i e s

( 3 ) Maus

k

o f the function a.e.,

and (2.9)

i s defined as a bounded measure on

i s defined

;iv

dr

a.e.

i s nontrivial.

n such t h a t

1M.v S I 5 Iv S 1

and thus (3.7) i s n o t obvious. 2 (4)

S i m i l a r l y t h e t r a c e of t h e f u n c t i o n

bl(M)

=

1

Mijvivj

i s defined

i,j=1 on

r,

with

Ib,(M)I

5 1

dr

a.e.

and (3.8) i s n o n t r i v i a l .

References

[l] Bourbaki, N . , I n t e g r a t i o n des mesures (Herman, Paris, 1964). [2]

B r e z i s , H., I n t 6 g r a l e s convexes dans l e s espaces de Sobolev, I s r . J . Math 13 (1972), 9-23.

[3]

Demengel, F.,

[4]

Demengel, F., Problemes v a r i a t i o n n e l s en p l a s t i c i t 6 p a r f a i t e des plaques, Num. Funct. Anal. and Optim., t o appear.

[5]

Demengel, F and Temam, R., Convex f u n c t i o n s o f a measure and a p p l i c a t i o n s , Indiana Univ. Math. J o u r n a l , t o appear.

[6]

Ekeland, I . and Temam, R., Convex A n a l y s i s and V a r i a t i o n a l Problems (NorthHolland, Amsterdam, 1976).

[7]

Goffman, C. and S e r r i n , J., Sublinear f u n c t i o n s of measures and v a r i a t i o n a l i n t e g r a l s , Duke Math. J. 31 (19641, 159-178.

[8]

Lichnewsky, A. and Temam, R., Pseudo-solutions o f time dependent minimal surface problems, J. D i f f . Equ. 30 (1978), 340-364.

[9]

R o c k a f e l l a r , R. T., I n t e g r a l f u n c t i o n s , normal integrands and measurable s e l e c t i o n s , i n Nonlinear Operators and t h e Calculus of V a r i a t i o n s , Lect. Notes i n Math., v o l . 543, S p r i n g e r Verlag (1976).

Fonctions a Hessien born&, Ann. I n s t . F o u r i e r , t o appear.

[lo] Temam, R.,

S o l u t i o n s general i s e e s de c e r t a i n e s Gquations du t y p e hypersurfaces minima, Arch. Rat. Mech. Anal. 44 (1971), 121-156.

[ll]Temam, R., A p p l i c a t i o n s de l ' a n a l y s e convexe au c a l c u l des v a r i a t i o n s , i n Nonlinear Operators and t h e Calculus o f V a r i a t i o n s , Lect. Notes i n Math., v o l . 543, Springer-Verl ag ( 1 976). [12] Temam, R., Approximation de f o n c t i o n s convexes s u r un espace de mesures e t a p p l i c a t i o n s , Canad. Math. B u l l , v o l . 25 ( 4 ) , 1982, 392-413.

Vuriutio nu1 Problems with Singular Solutions

[13]

Temam, R., Problemes Mathematiques en Plasticite, MGthodes MathGmatiques de l’Informatique, vol. 12 (Gauthier-Villars, Paris, 1983).

5 43

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DIFFERENTIAL EQUATIONS LW. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

545

BIFURCATION OF SUBHWNIC SOLUTIONS A GEI'ERIC APPROAa

.

A. Vanderbauwhede

I n s t i t u u t voor Theoretische bkchanica Rij ksunive r s i t e i t Gent Krijgslaan 281 B-9000 Gent, Belgium

In t h i s contribution we consider abstract bifurcation problems having a t r i v i a l branch of solutions, and equivariant under a f i n i t e cyclic group action. An example is the bifurcation of subharmonic solutions. We obtain, under a generic a l l y s a t i s f i e d transversality condition, an abstract version of the Arnol'd tongues [ 21 ; when r e s t r i c t i n g t o oneparameter problems one obtains r e s u l t s similar t o those of Iooss and Joseph [ 5 I. 1 . INTRODUCTION

Consider the ordinary d i f f e r e n t i a l equation

x = f(t,x,X) where f : IR x#

,

(1.1)

xlRm +I# i s of class Cp (p> 2) , T-periodic i n the t-variable f o r

some T > 0 , and such t h a t f(t,O,X) = 0 for a l l (t,Xj. Tne problem of bifurcation of kT-periodic solutions of (1.1) from the branch of t r i v i a l solutions x = 0 has been studied by several authors, in p a r t i c u l a r by Iooss and Joseph [ 4 , 5 , 6 ] ; the paper [ 7 1 contains a review of the main r e s u l t s , which can (rather loosely) be described as follows. Let m = 1 , and suppose that the variational equation

x

=

Dxf(t,O,X).x

(1.2)

has a branch a(A) of Floquet multipliers which crosses the unit c i r c l e transversal y f o r A = lo, such t h a t

a(Xolk =

l and a(+,)'

# l for k < k. Then, i f k

=

3,

there is bifurcation of 3T-periodic solutions a t A = l o ; f o r k = 4 there is bifurcation of 4T-periodic solutions only i f an inequality between coefficients appea5 there is only bifurcation of kTring i n the equations is s a t i s f i e d ; for k periodic solutions under very special conditions ("weak resonance"). The problem of bifurcation of subharmonic solutions has an inherent s p e t r y , which has t o do with the f a c t that it x ( t ) i s a kT-periodic solution of ( l . l ) , then so i s y ( t j = x(t+T). This property was used i n [ 81 t o reformulate the problem

A. Vanderbauwhede

546 i n the form of an equation

where M is a mapping between Banach spaces of kT-periodic functions, which is equivariant with respect t o the cyclic group generated by the operator T defined by (Tx)(t) = x(t+T). W e were able t o show t h a t the results of Iooss and Joseph mntioned above are a consequence of this equivariance, and consequently can also be found for other bifurcation problems having similar symetry properties. As an example we just mention certain boundary value problems on domains which are invariant under a rotation over 2n/k. In this paper we w i l l consider abstract bifurcation problems having a t r i v i a l branch of solutions, and equivariant under a cyclic group. The main point i n our approach w i l l be a generically s a t i s f i e d transversality condition f o r the linearization along the t r i v i a l branch. A s a consequence we w i l l see t h a t the r e s u l t s of Iooss and Joseph describe a non-generic situation, since one needs a t l e a s t two scalar parameters i n order t o s a t i s f y the corresponding transversality condition. This is related to the observation made by Arnol'd [ 2 ] that one needs two para-

meters i n order t o s a t i s f y the conditions on the Floquet m l t i p l i e r a(X) for (1.2) i n a generic way. From OUT approach we also find a m r e abstract version of the "narrow tongues" used by Arnol'd [ 1 , 2 1 to describe the bifurcation of subharmonic solutions. 2. AN EQUIVARIANT BIFURCATION PROBLEM

r E L(X) and f E L ( Z ) be such t h a t Iz for some k 2 1. We consider equations of the form :

Let X and Z be real Banach spaces; l e t and

v&re (a)

Fk

=

M(x,X) = 0

,

M : Xxfl

-t

M(O,X)=

o

rk

=

Ix

(2.1) Z is of class Cp (p>O), and s a t i s f i e s the following hypotheses:

,

vx

E

R~ ;

(b) M is equivariant, i.e. we have : M(Tx,X)

=

?M(x,A)

,

V(x,X)

E

Xxfl

.

(2.2)

(c) Lo = DxM(O,O) is a Fredholm operator, with dim N(LO) = codim R(LO) = n 2 1. I t follows from ( 2 . 2 ) that Lo i s equivariant, i . e . we have Lo E Lr,~(X,Z), where

5 47

Bifurcution of Subharmonic Solutions

Then (see [ 81) there e x i s t equivariant projections Po that N(LO) = R(Po) and R(LO) = N ( Q ) .

The action of

r

E

.Cr(X) and Qo E Li;(Z) such

leaves N(LO) invariant,

while R(QO) i s an n-dimensional complement of R(LO) in Z which i s invariant under r. MOreover, i f there is a second projection Q, E d:i;(Z) such that R(LO) = N(Q,), then

maps R(Q,) isomrphically on R(Q), w i t h inverse Q,, and on R(Q,)

cides with

This shows that the action of

Q1o?oQ.

?

? coin-

on an invariant complement of

R(LO) i s independent on the particular choice of t h i s complement. W e can use this

t o formulate a further condition on M : (d) The action of r on N(LO) i s equivalent t o the action of complement of R(LO) i n Z.

?

on a ?-invariant

This means t h a t there are linear isomorphisms q : N(LO) *Rn and 5 : R(Qo) +IF?

such that

We can even choose q and 5 such that

ro E

i s orthogonal (see [ 81). Our final

hypothesis w i l l be a transversality condition; i n order t o formulate it we need some results on Fredholm operators. W e w i l l say that two operators L,

E

there e x i s t linear automorphisms A

E

E(X,Z) and L2

E

% ( X , Z ) are eqlLiude&

if

L ( X ) and B E l ( Z ) such that Lz = BoL,oA. W e

denote by S(L) the equivalence class of L

E

d:(X,Z). Similarly, two operators L,

and L2 i n d: -(X,Z) are (r,?)-equiuaeutX i f they are equivalent via equivariant r,r automrphisms A 6 f y ( X ) and B E d:-,(Z). The corresponding equivalence classes are denoted by E

r , -(L). r

The following lemma's describe the structure of &(Lo) and

Zr,?(Lo) when Lo is a Fredholm operator.

Lemma 1 . Let Lo

E

d:(X,Z) be a Fredholm operator. Then 8(Lo) is the s e t of a l l

Fredholm operators L codim R(LO)

E

.C(X,Z) such that dim N(L) = dim N(LO) and codim R(L) =

.

In one direction the proof is t r i v i a l ; the other direction can be proved by actually constructing appropriate automrphisms A Lemma 2. Let Lo

E

6(X,Z)

E d: (X)

and B

be a Fredholm operator, and l e t Po

E

E d:

(Z)

.

S(X) and Q,

E

d:(Z)

be projections such that R(PO) = N(LO) and N(Qo) = R(LO). Then there e x i s t a neighborhood U of Lo i n d:(X,Z) , and smooth mappings A* : u+.C(x)

5 48

A. Vanderbauwhede

and B’ : U+f(Z), with A* (Lo) for a l l L

=

Ix and B* (Lo)

=

Iz, such t h a t the following holds

E U :

( i ) A*(L) = I ~ + A ( L ) P f o~r sorne smooth ( i i ) B* (L) ( i i i ) Y(L)

=

:=

X

: u+~:(N(L~),N(P ; ~))

I z + g(L) (I-Qo) for some smooth

;

: U+f(R(LO),R(Qo))

B*(L)oLoA*(L) maps NIPo) isomorphically on R(LO) ;

(iv) Q(L), the r e s t r i c t i o n of Y(L) t o N(LO), maps N(LO) i n t o R(Q)

.

This r e s u l t follows from a straightforward application of the implicit function U is a Fredholm operator, w i t h index L = index

theorem. I t implies t h a t each L

E

Lo and dim N(L) = dim N(Q(L))

dim N(LO).

Q

Corollary 1 . Under the assumptions of lemma 2 we have E(Lo) n U = { L E U I @(L)= O } , and &(Lo) i s a submanifold of f(X,Z) with codimension equal t o dim f(N(LO),R(QO)). P r o o f . This follows from the f a c t t h a t

D ~ Q ( .L L ~=) oJl

,

“LO)

sri: Ef(X,Z) ,

(2.5)

so t h a t %@(Lo) E L(f(X,Z),L(N(Lo) ,R(Qo))) is surjective. One can even show t h a t t h e equivalence classes 8 ( L ) form a s t r a t i f i c a t i o n of the open subset of f(X,Z) containing a l l Fredholm operators. For equivariant operators we have similar results :

Lemma 3. Let Lo L

E

Ed:

r,r-(X,Z)

be a Fredholm operator. Then gr,r(Lo) consists of a l l

&(Lo)nd: -(x,Z) such t h a t :

r,r

( i ) the action of

r

on N(L) is equivalent t o the action of

r

( i i ) the action of

?

on an ?-invariant complement of R(L) is equivalent t o the

on N(LO) ;

action of ? on an ?-invariant complement of R(LO). Lema 4. U d e r the conditions of lemma 2 , suppose t h a t Lo E fr,i;(X,Z), Po E f,(X)

ard

%E

Li;(Z). Then we have f o r each L

B*(L) E d:?{Z) and O(L)

E f

E

Unfr,i;(X,Z) t h a t A*(L)

E

fr(X),

r,r-(N(LO),R(Q)).

Corollary 2 . Under the conditions of lemma 4 we have that E r , ~ ( L o ) n U= ILEUnd:r,~(X,Z)I @(L)= O } , and gr,f(L0) is a submanifold of S,,i;(X,Z) w i t h codimension equal to dim d: -(N(LO) ,R(Q)).

r ,r

The foregoing r e s u l t s allow us t o formulate our last hypothesis, which w i l l be s a t i s f i e d f o r generic mappings M i n the c l a s s under consideration. We define

5 49

Bifurcation of Sub harmon ic Solutions

L : fl + 6:,,,(X,Z)

by

,

L(X) =DxM(O,X)

VXEX?

.

We have L(0) = L o , and we w i l l assume : (e) The mapping L defined by (2.6) is a t X Since

g

r,r-(L o )

=

0 transversal t o grr,i;(Lo).

is the s e t of a l l operators having the same "structure" as L o ,

this i s a natural condition t o impose. I n analytic form (e) requires t h a t ), considered as a linear operator from the parmeter space QDXL(0)

fl into

r , r-(N(Lo)

6:

m

2

,R(Qg)),

is surjective. A necessary condition for this i s that

dim 6:r ,r-(N(Lo),R(%)) = dim LrQ@In)

.

(2.7)

For given m and k , t h i s severely r e s t r i c t s the p o s s i b i l i t i e s for n = dim N(LO) and ~ , for ro. Since rok = IRn, the eigenvalues of ro must have the form ( v ~ ) with

..

,k-1 I ; moreover, each of these eigenva'?< = exp(iek) Ok = h / k , and r E {0,1,. lues is semi-simple. Writing ro i n i t s real normal form one can e a s i l y verify

the following :

Lemma 5 . L e t ro E emn) be such that rok = I. For each r E { O , l j...jk-ll l e t vr denote the multiplicity of (pk)r as an eigenvalue of ro. Then we have : k- 1 d i m l y mn) =

1

r=O

w

.

An immediate consequence of (2.8) is that n

< dim 6:r omn) < n'.

Remark also that

~ =- vr,~ since ro is a r e a l operator. In the example of the subharmonic solutions of ( I . 1 ) , vr is the dimnsion of the eigenspace corresponding t o the eigen-

v

value (1~)’ X = h

of the mncdromy matrix of ( 1 . 2 ) for the c r i t i c a l parameter value

0'

In the next section we w i l l discuss some of the bifurcation pictures which a r i s e under our hypotheses when m = 1 o r m = 2. The relations (2.7) and (2.8) w i l l give the corresponding p o s s i b i l i t i e s f o r n and

ro.

3. THX BIFURCATION EQUATIONS

Ender the hypotheses (a)-(d) one can apply a n equivariant Liapunov-Schmidt method ([ 81) and use the isomorphisms n and 5 appearing i n ( 2 . 4 ) t o reduce ( 2 . 1 ) i n a neighborhood of the origin t o a bifurcation equation

550

A. Vanderbauwhede

F(u,X) = 0

where F : Rn xC? ( i ) F(0,X) = 0

, +

Rn is of class Cp, and such t h a t :

,

YX ;

( i i ) DuF(O,O) = 0 ;

,

( i i i ) F(rOu,A) = TOF(u,X)

V(u,X)

.

The hypothesis (e) translates into : (iv) D ~ D ~ F ( O E , OL)@ , E ~ ~ Q R ~ ) is ) surjective. When m = 1 t h i s condition can only be s a t i s f i e d f o r n = 1 , i n which case L @) = To EN). Our hypotheses (except for the equivariance) reduce i n t h a t case t o those of the Crandall-Rabinowitz theorem on bifurcation from simple eigenvalues [-3]. In particular, (iv) becomes the condition on the mixed derivative appearing i n the CR-theorem, and we conclude that this l a s t condition i s i n f a c t a transversality condition of the form described i n section 2. As i n the CR-theorem, (3.1) has precise,ly one branch {(u,X' (u)) I uElR1 of nontrivial solutions. As f o r ro, there are two p o s s i b i l i t i e s : (1) ro = 1. In this case the bifurcating solutions of (2.1) w i l l s a t i s f y Tx = x (See [ 8 1 for d e t a i l s ) . (2) k i s even and r0 = -1. Then it follows from ( i i i ) t h a t A* (-u) = A* (u), and we get a pitchfork bifurcation. The bifurcating solutions remain invariant under Y2, while the two symmetric parts of the bifurcating branch can be obtained one from the other by application of the symmetry operator

r.

L e t us now consider 2 - p a r a t e r problems, i.e. we take m = 2. I t follows from (iv) t h a t we must have e i t h e r n = 1 , o r n = 2 and dim L QR2 ) = 2. For n = 1 we obtain ro the same results as we have j u s t described, except that there is an extra parameter. So we w i l l assume t h a t n = 2 and dim f (IR 2 ) = 2. This implies that e i t h e r To k is even and ro has the eigenvalues +1 and -1, o r that ro has the eigenvalues r = exp(ir'dk) and ikr= exp(-ir'dk), for some r with 0 < r < k/Z. In the f i r s t case a l l bifurcating solutions w i l l be invariant under Y z , and we obtain a situation which describes the interaction between the cases (1) and (2) obtained for m = 1. We w i l l not consider this case here any further. 2 , and that ro has the eigenvalues pkr and Gkr, with r ' / k ' , such that r' and k' have no c o m n divisors. Then k' = I , and a l l bifurcating solutions w i l l s a t i s f y r x = x; replacing k and r

Let us assume that n 0

=

< r < k/2. Write r/k

rk'

m =

=

55 1

Bifurcation of Subharmonic Solutions

by k' and r*, we may therefore assume that k and r have no c o m n divisors. W e w i l l also identify IR2 with the complex place <: (considered as a heal vectorspace!), such that the form

r0

becomes m l t i p l i c a t i o n by

%r.

The bifurcation equation (3.1) takes

where F : CxlR' + C is such t h a t F(O,A1,XZ) = 0 f o r a l l X = ( A l , X 2 ) .

From the

equivariance and kZ + r Z = 2Z we obtain :

This implies t h a t i f we expand F i n the form

then

aja(Xl,Az)

=

, V(X,,hZ),i f

0

j # &+1 (mod k)

.

(3.5)

I t follows from ( i i ) t h a t alO(O,O) = 0 , while the transversality condition implies that a(Re alo,Im alO)

w.,,AZ)

#

0

(3.6)

+

To find the nontrivial solutions of (3.2) we write z = pe

iB

, multiply

(3.2) by

z

and divide by p2; we obtain an equation of the form :

where G = O ( p ) as for x = x * ( p , @ ) = A*(p,@). There is c i e n t l y small z E

p

+

0 , and G(p,B+Bk,X) = G(p,B,h). By (3.6) we can solve (3.7) with X'(0,e) = (0,O) and A*(p,B+Bk) =

(A;(p,e),A;(p,o)),

always bifurcation of nontrivial solutions : for each s u f f i C we can adjust the parameters (A,,XZ) such t h a t (2.1) has a -1 solution x with Pox = TI (2).

To see the relation with the results of Iooss and Joseph mentioned i n the intro2 dh duction, l e t : IR + IR be a smooth mapping, with K(0) = (0,O) and =(O) # (0,O) and consider the r e s t r i c t e d problem

,

A. Vanderbauwhede

552

M(x,i(s))

= 0

,

(3.8)

where M : XxR' + Z s a t i s f i e s the hypotheses of t h i s section. There w i l l be bifurcation f o r (3.8) when for each EO > 0 the set I ( s , z ) E I R x C I I s \ < E O , O < I z I < E i(s) =

~ ,

A* ( 2 ) ) is non-empty; or, wre p i c t u r a l l y , when the path of X intersects the

range of A" i n a r b i t r a r y small points outside the origin. For the purpose of describing the range of A* we identify A with X l + i X z ,

and via a diffeomorphism

(see (3.6)), with a,0. In the case k = 3 we find then from (3.7) :

A*

ID,@)

f o r some b

E

=

bpe-3i@ + O(p2)

(3.9)

C; the range of A" f i l l s out a neighborhood of the o r i g i n i n A-space,

and there w i l l always be bifurcation f o r (3.8). When k

A* ( p , e ) f o r some b,c

= p 2 ( b + ce

E

=

4 we find :

-4i3+ 0 ( ~ 3 )

(3.10)

C. I f Ibl < IcI then the range of A* i s again a full neighborhood

of the origin; i f / b \ 2 / c / then the range of A' w i l l be a conezlike region with dX f i n i t e opening angle. There w i l l be bifurcation f o r (3.8) i f t z ( 0 ) is directed t o the inside of the cone. Finally we have for k

B

5 that

and the range of A* w i l l have the form of a narrow tongue a s described by Arnol'd ([ 1 , 2 I ) . Generically, the path of w i l l miss the tongue, and there w i l l be no bifurcation for (3.8), except under very special conditions. For a d i r e c t treat-

ment of one-parameter problems, see also [ 8 ]. REFERENCES [ 1 1 Arnol'd, V.I.,

Loss of s t a b i l i t y of self-oscillations close t o resonance and versa1 deformations of equivariant vector f i e l d s , Funct. Anal. and its. Appl. 11 (1977) 85-92. [ 2 ] Arnol'd, V.I., Chapitres supplementaires de la theorie des equations different i e l l e s ordinaires (Editions Mir, bbscou, 1980). [ 31 Crandall, M. and Rabinowitz, P., Bifurcation from simple eigenvalue, J. Funct. Anal. 8 (1977) 371-340. [ 4 ] Iooss, G., Bifurcation of maps and applications, Mathematics Studies, vol. 36 (North-Holland, Amsterdam, 1979). [ 5 ] Iooss, G. and Joseph, D., Bifurcation and s t a b i l i t y o f nT-periodic solutions branching from T-periodic solutions a t points of resonance, Arch. Rat. Mech. Anal. 66 (1977) 135-172. [ 6 ] IOOSS,G. and Joseph, D., Elementary s t a b i l i t y and bifurcation theory (Springer-Verlag, New York, 1980). [ 7 1 Joseph, D., Lectures on bifurcation from periodic o r b i t s , Springer Lecture Notes i n Fkth., vol. 898 (1981) 1-12. [ 8 1 Vanderbauwhede, A., Local bifurcation and symetry, Research Notes i n k t h . , vol. 75 (Pitman, London, 1982).

DIFFERENTIAL EQUATIONS I.W.Knowles and R.T.Lewis (Editors) OElsevier Science Publishers B.V. (North-Holland), 1984

553

A CLASS OF ISOPERIMETRIC VARIATIONAL PROBLEMS ON CERTAIN ORLICZ-SOEOLEV SPACES Pierre A. Vuillermot Department of Mathematics The University of Texas Arlington, Texas 76019

a

I. INTRODUCTION In this paper, we present a method for solving a class of isoperimetric nonreflexive variational problems defined on certain -Banach spaces of Orlicz-Sobolev type. The technique used relizs upon the critical point theory of Ljusternik and Schnirelmann anti we show in particular how one can bypass the lack of reflexivit to prove the Palais-Smale compactness property, using monotonicity methods a:ong with new inequalities valid for all C(2)-Young functions which possess convexity properties in the square of their argument. The technique we propose can be applied to variational problems relevant to both nonlinear Sturm-Liouville eigenvalue problems & nonlinear elliptic boundary-value problems with exponential nonlinearities; however, for the sake of clarity, we shall in this paper restrict ourselves to the former case. For alternate formulations of the results, details and complete proofs, see [l] and [ Z ] . These studies were motivated in the first place by problems in elasticity theory and combustion theory [ 6 ] . 11.

Ct<-ITICAI.POINTS OF SADDLE POINT-TYPE FOR A CLASS OF ISOPERIMETRIC V.\?IAYOSAL PR03:EVS T I T 3 ESPOSEKTIAJ. T.IMS. -

Let Y on LR

function, namely an ex-en convex -1 -1 lim t Y(t) = + m and lim t Y ( t ) tifm t+o

be a c(21Young such that

+-valued

(R

C(2)function

= 0.

By

C( )-function, we shall mean a function which is twice continuously differentiable everywhere except possibly at the origin. Throughout this paper, we shall repeatedly use the following definitions. Definition

11.1.

The C(2)-Young

if it is convex in t2 H

on

+

such that

(R

function Y

in the sense of

Y(t)

=

H(y)

with

[3]:

y

=

is said to satisfy property (P) there exists a C( )-function

t2

and

H

convex in y.

Definition 11.2. The C(2)-Young function Y is said to satisfy property (Q) if it satisfies the following Young s condition near infinity: there exist X > 2 and to > 0 such that Y(2t) for each

t

E

D

Definition Ii.3.

such that

It

The c(2)-Young

near infinity if there exists t

%ermanent

1

5

AY(t)

(11.1)

to. function Y

is

said to be superquadratic

> 0 such that the relation t2

address from September 1, 1983.

5

2Y(t)

P.A. Vuillermot

554 t E R

holds f o r each near infinity i f

t

2

such t h a t

2 2Y(t)

It

I 2 to;

f o r each

i t is s a i d t o be subquadratic

t E R satisfying

It1

2

to.

F o r i n s t a n c e , i t i s e a s i l y seen t h a t t h e t h r e e Young f u n c t i o n s

Y(t) = eltl

q

-

y(t) = elt]

,

- 1

-

It1

2

5q

<

+

(11.2)

m

1

a l l s a t i s f y p r o p e r t y (P) and a r e s u p e r q u a d r a t i c n e a r i n f i n i t y , t h e f i r s t one s a t i s f i e s property

(a).

-

Y

Let

b e t h e Legendre t r a n s f o r m of

Y(s) = with p

s E

C()(

E

R

t,o Y

and let

+ 1R; IR )

,

qslt

max

p

E

-1

by 5 , P pdx and b y

EY,P Luxemburg’s norm

’

+ C( R ; IX )

(11.3)

and

fix

-

00

< a < b Y

on [ a , b ] a s s o c i a t e d w i t h

= i n f {k > 0 :

dxp(x)Y(y) Jab

(See [l], 141, [ 5 ] f o r t h e b a s i c d e f i n i t i o n s ) . %,p.

5,p’

and

LY,P

+. L e t < + m. We

IR

E?,p~.-L<,P

5

(11.3)

11

We d e f i n e t h e s p a c e s

similarly.

The c l a s s of i s o p e r i m e t r i c

v a r i a t i o n a l problems w e s h a l l c o n s i d e r i s now d e f i n e d as f o l l o w s : be a let

C(2)-Young be a

Y2

f u n c t i o n which s a t i s f i e s b o t t , p r o p e r t i e s

C(2)-Young

t h e Banach s p a c e z

on

[a,b]

BY1.

f u n c t i o n which s a t i s f i e s p r o p e r t y

(P) (P);

let

Y1

(Q) and consider

c o n s i s t i n g of a l l real-valued f u n c t i o n s

p, p

Y2’

such t h a t

s h a l l denote

and t h e measure

t h e corresponding Orlicz-spaces with respect t o

Ly,P

1 1 ~ ly,p 1

namely

Y,

- Y(t)l

b e i t s monotone i n v e r s e o n

t h e Orlicz-class

while only

z E E Y2’

P

and w i t h d i s t r i b u t i o n a l d e r i v a t i v e

e q u i p p e d w i t h t h e norm

IIzII p’ p t G(x;t) = d 0 BY1.

Y2S

1

2

Y1, Y2’ P. p

=

2 I I z r I I yl,

+

I

IZI

I 2y*,

p

(11.4)

i s s e p a r a b l e b u t not r e f l e x i v e i n g e n e r a l ; l e t

5

F(x;S)

where

F

i s a f u n c t i o n on [ a , b ] x IR

which s a t i s f i e s

t h e following property:

(R)F(x;T) i s odd i n T , s a t i s f i e s t h e CarathGodory c o n d i t i o n s on [ a , b ] and t h e r e e x i s t s r- > 0 s u c h t h a t

F i n a l l y , let

y,

T,

6,

E

R

be given,

# 0,

# 0, and y 2 +

T2 =

62

+i2 =

1

VariationalProblems on Orlicz-Sobolev Spaces and p i c k

p

(1;

&

+

+ -).

We t h e n d e f i n e on

555

t h e two f u n c t i o n a l s

dx G ( x ; z ( x ) )

(11.6)

Ja

and t h e ( i n g e n e r a l unbounded) s e t of c o n s t r a i n t s K

= { z &By

1-I

1’

C

*

Y29

P. P'

= 0)

u(2)

(11.7)

Our main theorem, whose proof w i l l be s k e t c h e d i n the s e q u e l , i s t h e f o l l o w i n g : Assume t h a t

Theorem 11.1.

(P)

s a t i s f i e s property

sgn(Y) = sgn ($)

that

satisfies properties

and t h a t

s a t i s f i e s property

F

6 # 0, t h a t

and i f

(P)

sgn(6) = sgn

(a),

and

that

Y2

y # 0 , assume

(R);

if

(8).

Then t h e

p o s s e s s e s a t l e a s t a c o u n t a b l e i n f i n i t y of d i s t i n c t one-parameter

V

functional

Yl

{? z ( a ) }

antipodal p a i r s

1-I

aEIN

+

of c r i t i c a l p o i n t s on

Ku,

with c r i t i c a l

v a l u e s of t h e form d

inf

=

m

sup

Tm, w i t h m

In (11.8), t h e s e t

balanced and bounded s u b s e t s of in

V(z)

E

K

m

Remark:

The e x p r e s s i o n s (11.6) f o r

(11.8)

growing t e r m s s i n c e

111.

i n s t e a d of

Y2

(but n o t

+

! l , d e n o t e s t h e s e t of a l l c l o s e d ,

w i t h genus no less t h e n

v

written

(11.2)

(11.8)

Acrm Z E A

m.

W e have

a because of p o s s i b l e degeneracy. and

V

C

1-I

allow f o r exponentially f a s t

may be any one of the Young f u n c t i o n s

Y )

1

under t h e c o n d i t i o n s of theorem (11.1). SKETCH OF THE PROOF OF THEOREM (11.1).

Using v a r i a t i o n s on a known theme, one can f i r s t r e d u c e t h e proof t o t h a t of a deformation theorem which r e q u i r e s t h e proof of t h e Palais-Smale p r o p e r t y i n t h e f o l l o w i n g form: D e f i n i t i o n 111.1. property on

K ,,

The f u n c t i o n a l

b

E

W

n

+ m

in

\

such t h a t

(111.1)

and

v'(zn) as

zn

2 i;

V(Z")

f o r some

i s s a i d t o s a t i s f y t h e Palais-Smale

V

i f , f o r any sequence

- X(z")c;(z")

-+

0

i n t h e s t r o n g topology of t h e d u a l s p a c e

t h e r e e x i s t s a subsequence

In (111.2), w e have

X(z)

znj =

(111.2) L(By

s t r o n g l y convergent i n

.

R),

1’ Y2’ P, P ' BYl,

V '(z )(z )

y2, P ,

and

P' V',

C'

u

P.A. Vuillermot

556

V, CU respectively.

stand for the Frhchet derivatives of

One can subdivide the proof of this property into five steps: Step 1. The Uniform Boundedness of the Norms

HZ ’IY1, Y2’

(P) and ( R ) .

This is a consequence of (111.1) through properties The Strong Convergence zn + z

Step 2.

in

Let

EY2, P '

H(’)

usual Sobolev space on [a,b] with respect to the measures

(P),

property

it follows that both

Y1

and

Y2

P. P '

be the

p, p

pdx

and

pdx; from

are superquadratic near

+ Byl’ Y2' P. P where the first embedding is continuous and the second one

infinity; this along with monotonicity arguments imply

-+ Ey2.

HIP, p

p

compact; the assertion then follows from the result of step 1 and the reflexivity This is the first bypass around the lack of reflexivity of of ff(1) P’ P ' BY,, Y2’ p3 p ' Step 3.

The Strong Convergence

(Y1oz a -~~oz,)(z,~-z’)

+

o

in L 1 P

We first prove that

I

b

lim n + m

-

dxp(x)Y;(z’(x))(z’"(x)

= 0

z’(x))

a (~’"(x)

-

z’(x))

The left-hand side limit follows from the fact that and that

EY1. P

u

+

Y;

O

u

maps

EY1, P

(111.3)

zm

+

satisfies property

(a).

weakly in

into the dual space

the right-hand side limit essentially follows from (111.2) Y1

z,

EY1’ P’

the fact that

We then get

b lim n + m

-

dxp(x)(Y;(z’n(x)) a

whence the conclusion since u by the convexity of

- z’(x))

Y;(z’(x)))(z’"(x)

+

Y;

a

yl’

The Mean-Convergence

for each t,

E

A-

is monotone from

(111.4)

into

.

m

Step 4 .

u

= 0

-+

-2 A in

P.

for -

IR; this along with the result of step ( 3 ) implies

EY1, P

Variational Problems on Orlicz-Sobolev Spaces

557 (111.5)

which proves the result. This is the second bypass around the lack of reflexiof because of the following: Step 5 .

If

The Strong Convergence

z,n

zcn

+

does not converge strongly to

that, for each positive integer n

in E Y1

z’

in

z’

p

EYl, P

there is

> 0 such

E

and by convexity (111.6)

-

if n n.

X

This obviously contradicts (111.5) with

= E

and thereby proves

the result. Steps 1

-

Remarks.

5 along with (11.4) proves the Palais-Smale property on K 1.I

step which requires property (Q) for Y1; thus

Step ( 3 ) is the

all terms containing Y1 may grow at most polynomially fast; whether the Palais-

(a)

Smale property, and thereby theorem (II.l),

is an open question.

remains valid without property The nature of this problem is centered around the loss

of boundedness of the functional

into IR

as soon as YI

u

+

rb dxp(x)Y;(u(x))u(x) a

does n o t satisfy property

conjectures regarding this question,

see

from Ey 1

( 2 ) . For more details and

[l].

IV. AN APPLICATION: SUBHARMONIC PERIODIC ORBITS FOR A CLASS OF MATHIEU EQUATIONS WITH EXPONENTIAL NONLINEARITIES. Using the standard methods of the calculus o f variations, theorem (11.1) immediately implies that the

1.I

are classical solutions of the nonlinear

Sturm-Liouville eigenvalue problem {p(x)Y~(z (x))~~

+np(x)Y;(z(x))

=

F(x;z(x))

(IV.1) (IV.1 )

with appropriate eigenvalues na

1.I.

With

Y (t) 1

=

2 t - and Y 2 2

Z

Y,

this allows

one to prove the following result (see [l] for details, open problems and a

P.A. Vuillermot

558 l i s t of r e f e r e n c e s ) :

Consider t h e c l a s s of d i f f e r e n t i a l equations

Theorem I V . 1 .

+ nY’(z(x))

z”(x) where

13

E

R

an d

whe r e

property (P); assume t h a t

IF(.r)l

f o r some

l~

E

(1,+m)

fi >

0

(IV.1)

i s a C(2)-Young f u n c t i o n which s a t i s f i e s

Y

A

T > 0

with (minimal) period

= BA(x)F(z(x))

i s a continuous

and t h a t

F

- y--I(Y(T))

even p e r i o d i c

i s continuous and

f u n c t i o n on

odd i n

T

Lll

and each

and each

parameter f a m i l i e s

{+

o r b i t s with period

kT

k

T E E

\R.

W+,

P ( k,lJ a) I

IR

with

(IV.2) Then f o r each

R

E

IR

,

each

t h e r e e x i s t a countable i n f i n i t y of onec1

,N+

of n o n t r i v i a l , c l a s s i c a l

and a countable i n f i n i t y

parameter f a m i l i e s of n o n t r i v i a l , c l a s s i c a l which s a t i s f y (IV.1) everywhere on R

{*

$ (k,u’ a) a

odd p e r i o d i c

and such t h a t

E

even p e r i o d i c +

LN

of one-

o r b i t s with period

C (z) i 0;

lJ

the

corresponding eigenvalues a r e r e s p e c t i v e l y given by

(IV.3) r e p l a c i n g 5‘”) k,lJ k,P case, and a r e thus represented by curves i n t h e (B-n)-plane.

i n t h e f i r s t c a s e , and by (IV.3) w i t h

i n t h e second

Acknowledgements. The author would l i k e t o thank Professor Ian Knowles and Professor J . Lewis f o r t h e i r very kind i n v i t a t i o n t o t h i s conference. He a l s o thanks M s . Claudia Harrelson f o r her e x c e l l e n t typing of t h e manuscript

kT

Variational Problems o n Orlicz-Sobolev Spaces REFERENCES Vuillermot, P.A. "A Class of Sturm-Liouville Eigenvalue Problems with Polynomial and Exponential Nonlinearities", to appear in "Nonlinear Analysis, Theory, Methods and Applications" (1984). Vuillermot, P.A. "A Class of Elliptic Partial Differential Equations with Exponential Nonlinearities", to be submitted to Mathematische Zeitschrift (1983). Vuillermot, P.A. "A Class of Orlicz-Sobolev Spaces with Applications to Variational Problems Involving Nonlinear Hill’s Equations", Journal of Math. Anal. and Appl. 9 , 1, (1982). Adams, R.A.

"Sobolev Spaces", Academic Press, New York (1975).

Krasnosels’kii, M.A. and Rutickii, Ya.B. "Convex Functions and Orlicz Spaces", P. Noordhoff Ltd., Groningen, tbe Netherlands (1961). Moseley, J . "Asymptotic Solutions for a Dirichlet Problem with an Exponential Nonlinearity", SIAM Journal for Mathematical Analysis, in press (1983).

559

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DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) @Elsevier Science Publishers B.V. (North-Holland), 1984

561

THE RELATION OF SOLUTIONS OF DIFFERENT ODES IS A COMMUTATION RELATION Bernard F. Whiting Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge. England" and Center for Relativity and Center for Theoretical Physics, University of Texas, Austin. Texas. U. S. A.

The nature of the first order operators which relate solutions of different second order equations i s found to be determined by the exiktence of a (weighted) commutation relation for the second order operators. The properties of the corresponding mappings between solutions of these operators are discussed. especially for those forms which deal with spectral transformations. as found in work dating back to Darboux. Flnally. application to a particular problem arising for scattering on the whole line is dealt with.

INTRODUCTION This work arose out of an attempt to relate solutions of separate second order linear differential equations. Motivation to do this often occurs In the use of approxlmating equations and even in the applications of the well known WKB and uniform approximation schemes: and the recurrence relations for special functions are In general recognizable examples of where exactly this alm has been achieved. relativity. there is already a limited degree of success for the wave-functions which characterize perturbations of the Kerr black hole. The next section gives some justification for the direction of the development which follows. Then the basic framework is set up and developed to give a specific representation of the original second order equations. Next, solutions for the coefficients which define the mapping between these equations is dealt with. A brief glimpse at the underlying role of commutation [11 then leads to a discussion of the special case in which one of the mapping coefficients is constant. This is followed by a comparison with the integral form of transmutation [21 and finally. by way of something of an application, an aspect of scattering on the whole line is discussed in the context of a scalar field around at Schwarrschild black hole.

"Address after 1st August, Meudon. Paris France.

Groupe d'Astrophysique.

Observatoire

de Paris.

92

B. I? Whiting

562

BACKGROUND TO THE MAIN DEVELOPMENT In the use of first order linear differential operators to relate the solution spaces of a pair of second order equations. it is found that the appropriate mappings fall into one of the following three classes: i) the whole (two-dimensional) solution space of one equation is mapped on to the whole solution space of the other in a non-degenerate way such that the mapping is Invertible: ii) the solution space of the first equation is mapped on to only one line in the solution space of the other: this mapping can be viewed as a non-invertible proiection. and is a degenerate form o f the previous case: and iii) one line of the first solution space is mapped on to a line in the second solution space in an invertible way. but the rest of the solution space is mapped into some reglon "orthogonal' to the solution space of the second equation.

As it was desired generally to consider transformations which were invertible and could map a whole solution space. this paper w i l l be concerned mainly with mappins of type i) and occasionally with their specialization in type i i ) . In the present context, the disadvantage of type iii) mappings is the excessive freedom they leave in the specificatlon of the flrst order operator. On the other hand, a remarkably useful result found by constraining the mappings to type i) i s that an explicit representation of these mappings carrles over to an explicit representation of the defining second order equations. Part of the purpose of this paper Is to explore this property. BASIC FRAMEWORK Our development is based on the two equations:

y"

+ qy

+

z"

= 0

Qz = 0

(1)

for which a linear relatlon of the form y = az

+

@z*

( 2)

is sought with a and B being functions of the independent variable (&I&: although the equations are in one-dimensional Schroedinger form. the "potentials" here are allowed to be quite general). Consistency of (2) with the assumed form of the equations ( 1 ) demands that the following are also satisfied:

y' 0 =

(a"

=

+ (q-Q)a -

(a’ - B Q ) z

+ ( a + B’)z’

zQS' - Q ' B ) z

+

(B"

+

(3)

(q-Q)B

+

2a’)z’

(4)

Now, for mappings of type i ) , the relation (2) must be general, in that any solution for the equation for z should give a non-zero solution of the equation for y , via (2). Clearly, this cannot be the case if: a) some solution for z satisfies the operator on the right of (2) - this would result In mappings of the type ill above, or b) some z-solutlon satlsfied ( 4 ) for non-zero coefficients - resulting in a mapping type iii) . If these situations are to be avoided then: a) each coefficient in ( 4 ) must be separately zero. i . e .

a * ’ t ( 4 - Q ) a - 2QB' and

-

Q'B

=

o

563

A Commutation Relation for Second-Order Operators so that ( 4 ) is satisfied by all solutions for z while excluded, and b) the first integral. k. of equations (5) and ( 6 1 ,

a

2

+

aB* - a ' B

mappings of

2

+

B Q = k

type iii) are

( 7)

must not be exactly zero. since otherwise a degenerate mapping of type ii) would result. Since k is the determinant of the mapping between ( y,y’) and ( z , z ' ) occurring in (2) and ( 3 ) , it will be seen that for a mapping of type i ) . these equations can be inverted to give: 1

-

z = r;((cr + B’)Y

(8)

By')

(9)

Note that when k=O. the operator on the right in ( 8 ) will still give a type ii) mapping from y to 2 . Finally in this section. we remark that representations of the operators in ( 1 ) :

axx +

q =

(6)

and

a+B* ( a x + (----))(ax B

axx + Q

=

-

( 7 ) can now be used to give

a+B'

k +-

(--,))

(10)

8'

+

( a x - 4a) ( a x

-a)

+k

(11)

8,

More shall be said about these following the next section. SOLUTION FOR B It will be noted that ( 5 ) and ( 6 ) are a pair of GouDled second order linear equations for a and B . and that elimination of a will therefore result in a fourth order linear equation for B (throughout this discussion it will be assumed that (q-Q)#o):

By inspection ( ! ) . or as we shall see below. integral.

this equation also has

k as a first

Now. (12) is just the equation satisfied by a linear sum of terms, each of which is a product of a solution for y with a solution for z. That this is reasonable can be seen by considering specific mappings in terms of some basis for each of the equations. Let yL,yz be a basis of solutlons to the equatlons for y , and z1,z2 be a basis of solutions to the equations for z such that

W Y l , Y 2 )= (Y1Y’, - Y’,Y,) Then consider the solutions for

Define

y

= 1

,

W(Zl’ZZ) = 1

~

(13)

which z1 and z 2 map into:

a y l + by,

=

cyl

= az2

t dy,

azl

+ Bz', + Bz’,

(14) (15)

B. F. Whiting

5 64 The inverse of ( 1 4 ) and ( 1 5 )

then shows that 4 is exactly of the appropriate form and that (the equation for ) (1 can be found similarly. Substituting these results into equation (7) for the flrst integral. k. and using B=SA. we find

k = (-Z1TBY)2

- ( -Z’~BY+ =

+

(-Z’TBY)(ZrT BY t ZTBY’)

QZ~BY )(z~BY)+ Q( Z ~ B )'Y

(18)

( z ~ B Y ) ( z ’ ~ B Y ~-) (z~BY.)(z’~BY)

= Det

((Z,Z.)TB(Y,Yi)T] = D e t ( A )

.

Thus, k represents the determinant of the matrix characterizing a specific mapping between y and 2 . such as has been chosen in equations (14) and ( 1 5 ) . Clearly, Det A = 0 corresponds to a mapping of type i i ) . Having previously chosen a mapping for which equations ( 5 ) and ( 6 ) were to be satisfied, we now find that a specific solution to equation (12) for 4 is sufficient to characterize the mapping completely, since equation ( 5 ) and the derivative of equatlon ( 6 ) can be used to give an expllcit expression for a i n terms of B and Its derivatives:

This leads immediately to an alternatlve expresslon for k:

whlch, as was mentioned earlier, can also be regarded as a first integral of equatlon ( 12) for 4. One of the unexpected results which we shall now obtain is that ( a t most) two solutions to the equation for 4 are sufficient to allow all solutions to be found. The First, we demonstrate that only argument for this proceeds by a number of steps. one solution is sufficient when the corresponding mapping is of type i i ) . For then, by an appropriate choice of our solution basis, we can write

4, (cf. equation 1 7 ) .

=

z1y2

,

a1 *

-

(21)

' lY,

Thus

z1

=

a

ew(- I Z d x )

by integration and substitution.

z2 =

,

y, = Bl/z ,

(22)

then

ZJ>dX1

z1

r

Y,

= -

Y2S-jdX 1

(23)

=2

are given by Wronskian relations, equations ( 1 3 ) , and finally

complete a set of linearly independent solutions to the equation for 4. This first part of the proof shows that when considering

two given solutions.

we

565

A Commutation Relation f o r Second-Order Operators

need treat only the case when neither solution gives rise to a mapping of type ii). Then. let there be matrices A,, and A2 corresponding to the two solutions and characterizing the separate mappings. Now. in order to use the proof we have already, all we need seek is some C=A1 - &A2 such that Det (C)=O. Such a C cannot be found directly since we know neither the matrix coefficients nor the linearly independent solutions to the two second order equations. However. as knowledge of a a is sufficient to allow us to find its corresponding constant of integration via equatlon (20). we can in particular compute the determinant of C. For, let the mapping between 8 and k be given by F ( B ) = k . From equation (20) which gives F . we can also write:

wal)

F(a1+a2)= where G i s a constant.

+

(25)

G(alta2) + F(a2)

bilinear in its arguments. and such that

G ( B , B ) = zF(P)

(26)

(It is a short exercise to show that G ( B , , B , ) gives TrA1.TrA2 proceed, we next consider the sequence of computations

Det(Al)

=

, D e t ( A 2 ) = F ( B 2 ) = k2

F(B1) = k,

D e t ( A 1 + A Z ) = F(Bl+B2)

k,

=

-

TrAIAz).

To

, and

,

(27)

.

(28)

say, which allow the particular intermediate result

G ( 4 , , B 2 ) = k,

-

k,

-

k1 = k,

Whence finally:

- f12)= - ,~k,+ LL 2k,

D e t ( C ) = Det(Al =

k,

( 29)

F(B,-fiB,) = 0

A use of the first part completes the proof.

gives the required equation for f i .

A CONNECTION WITH COMMUTATION Equations

(

10) and ( 1 1 ) for the original operators can be rewritten as

a 2 (ax,+g) 2

a (a,,+~)

=

(aaX +a)(aaX -a-a’) + k

=

(aax-a-a')(8ax+a)

+

= m + k

(30)

k = AB+k

(31)

where

A = 8 a -a-B’ X

,

B = ~a,+a

.

(32)

Then the mappings i n (2) and ( 8 ) can be expressed by the operator identities:

(BA+k)B = B(AB+k)

( 33)

(AB+k)A = A(BA+k)

(34)

It is obvious that these will be true no matter what value k may have. now have mappings between solutions of the operators

P

2

( axx+q)-

x

and

a2 ( axx+Q)-X

so that we (35)

This corresponds to the fact that solutions to equations ( 5 ) and ( 6 ) . and to equation ( 1 2 ) derived from them are invariant under the changes

5 66

B.F. Whiting (36)

With the variable changes u =

y/dB

,v

=

2/dE, w

J-dx 8

=

,

(37)

and the identifications

c = a W- ( r ,

1

~ = W a + & ,C r = a i2- ~ *

(38)

we can Write (30) and (31), (2) and ( 8 ) as

,

(DCtk)u = 0

,

(CD+k)v = 0

u = Dv 1

v

= J;Pu

(k+O)

(39)

constructed so that the appropriate

which are now equatlons in normal form. unlty.

2

is

Delft [11 has discussed "commutation" relations associated with (39) ( o r (35)) and the 'Essentially iso-spectral' property which follows when C and D ( o r similarly A and B) are either bounded, or (dense) closed mutally adjoint operators acting on in (35) and the Hilbert space. In general, because of the multipllcation by B coordinate change used in C and D. we do not have any control over how the spectral parameter X enters into the original equations ( 1 ) except when B is a constant. which we now discuss. THE CASE B ' = O The case B=1 is of considerable interest. because of Its use In spectral transformations. and because probably of the recurrence relations for special functions can be conveniently cast in a form in which this condition is manifest. W e start b y looking at condltions on the potentials which allow B' to be zero. From (12). we note that a constant solution for B can exist whenever the coefficient of ~3 there is zero. i . e .

- ( ~ + Q ) I +~ (q-Q)’ - ~ = 0 (q-Q) In this case. we have immedlately from (19) and (20) (q+Q)"

1

,

a = y(q+Q)’/(q-Q)

k = a

2

-

8' = 0

1

+ ?(q+Q)

(40)

(41)

and the application of Deift's results requires certain conditions on a being satisfied. In the context of iso-spectral operators, we point out that (40) Is not a necessary condition. not even when there are no eigenvalues. as an example of Sarnak [31 demonstrates. The operators 2

-d +

ae

i ax

2

-a -?

and

dx

,

-.o<x<-

dx

both have only a continuous spectrum composed of the positive half-line.

We now suppose that we have some sequence of iso-spectral transformations: ((dx-a)(axta)+X)z = o

( ( a +o!)(d - a ) + A ) y X

X

=

o ,

y

=

( aX + a ) z

(42)

5 67

A Commutation Relation f o r Second-Order Operators

and consider the effect of compounding them:

w =

( aX - r ) ( a X t a ) z

= ( ( (L-Y ) (

ax+a )-A

(44)

)Z

n

Not only do we have B * + o . but we also have depending on the spectral parameter of the particular function of z for which we eliminated the second derivative: Now and generally, both a and B will become functions of A . as also will k. Darboux 141 has the following interesting result: i f a mapping is known for which k is a polynomial of finite degree i n some common spectral parameter contained within q and Q . then there is a specific de-construction procedure which reduces the mapping to a sequence of first order operators, each with 8 = 1. He can undo what we have just been describing. In a closely related manner. Crum [51 has a method for removing bound states from a Strum-Liouville system. For each step in the process of removing the lowest remaining eigenvalue from the spectrum. he constructs both a new potential and a first order mapping from the old eigenfunctions lo the new. Since h e does this by compounding a sequence of simple operators ( B = l ) . his mappings pick up a dependence on the eigenvalue. as we have seen above in ( 4 4 ) . However, for all his compounded operators. we have

since this is imposed by ( 6 ) . regardless of the individual dependence of a and B Crum also considers adding in extra bound states. and Deift [11 injects points into the continuous sprectum. for scattering on the half line. In all these cases which rely on compounded operators, explicit use i s made of solutions given by certain values of the spectral parameter.

on A .

W e end this section with a note on an evolution problem. When ( 4 0 ) is satisfied, suppose we let q depend on some parameter. t . while Q does not. Then we may investigate how q may vary as a function of t so that it preserves the relation. B=i. Using (411 with a = - b ' / b . we can write

Q = k(t)-(k(t)-Q) = k(t)-b"(t)/b(t)

q = k ( t ) - b ( t ) ( b - l ( t ) ) * *= Q+z(Pnb)"

.^

= 2 ( k ( t ) - ( L n b ) - L )- Q

where b ( x , t ) is a solution of

b"

t

(Q-k(t))b = 0

(47)

Then

q = z ( k - Z ( l n b ) ' ( l n b ) ' ' ) = z(Pnb)"' and this last relation can be integrated to solve for ( P n b ) '

k(o)

+

(48) and then q ( x . 0 ) :

X

1

t (b-2)'((b*-b-b*b*)(xo) k(o)lb2dxJ XO

.

(49)

B.F. Whiting

5 68

for which we need only k(o). ( h ’ ~ b ) ~ * ( x ~and , o )b(x)/b(xo) to find q(x.0) for all x. since Q = O . its spectrum does not change, but obviously k has been allowed a t-dependence which can make sense i f k ( t ) < i n f a ( Q ) . The complete 'evolution' equation for q can also be obtained: it involves ( 4 0 ) " ' . is very complicated and not very illuminating.

m:

COMPARISON WITH INTEGRAL TRANSMUTATIONS Here we consider a comparison with the alternative. integral form of 'transmutation' discussed by Carroll t21. An integration of the differential form in (2) readily yields:

where (51) and

It will be noted that the Kerunless k is zero, in whlch case z(xo) Is arbltrary. nel. K. is here a single product rather than a sum over eigenfunction products or a spectral integral as used by Carroll. This highlights the difficulty we had earlier in discussing the spectral properties for ( 1) without a preferred coordinate change unless B was constant. On the other hand, Carroll specifically constructs his Kernels to produce mappings similar to our type iii) above: i. e. ( f o r each value of the eigenparameter) it is only a particular solution for y which is transformed into a solution for 2 . whereas in ( 5 0 ) . any solution for y should yield a solution for z unless k=O. in which case only the non-zero y given by (2) can be used. SCATTERING ON THE WHOLE LINE This will be discussed with reference to scatterlng in the exterior region of a Schwarzschlld black hole which. with an approprlate coordlnate choice. can be regarded as taking place on the whole line. We point out that i n his discussion of perturbations of the Kerr black hole, Chandrasekhar I61 had to develop a formalism to map his primary wave functions into a set more suitable for direct analysis. and he was forced to have very much the same content as we have concerning the properties of his transformations. though in a form more dedicated to his particular problem. For Bosonic fields the relevant potentials are real. satisfying pure wave boundary conditions:

Then. for real

and one typlcally has solutions

k+o these solutions behave as :

(54) They have Wronskian

W = W(fl,f2) =

- zikA(k)

(55)

.I Cornmutatroil Relution for Second-Order Operators

5 69

For real k note that

of which more will be said shortly.

B(-k)

= B-(k)

(56)

In general the potential depends on a number of parameters which. for definiteness. we may characterize by the angular momentum (subscript 1 ) . It will be of interest to find certain propertles of the a,B which can connect solutlons for two different values of such a parameter: in particular. we shall enumerate below the conditions under which ff and 4 can be non-oscillatory i n the asymptotic regions x+*. Let the actual solutions of interest be RP = fl(x,k,e)

S P = f,(x,k,P)

RItl

= af (x,k,P+l) 1

t bf2(x,k,l+i)

+

S L t l = c f (x,k,P+l) 1

af2(x,k,P+1)

(57)

such that

M

= (ad-bc)

( 58)

# 0.

Then. for a mapping between these two sets of solutions. a and usual way:

4

are given in the

Clearly. a problem arises i f W can become zero. for then R and S are linearly dependent. In these sltuatlons it Is usually more convenient to cloose f ( x , k ) and f(x,-k) as basis functions since. for k+o. these are always independent. In fact. for Bosonic fields around a black hole, A can become zero for certain comL It is also more plex values of k. which are known as quasi-normal modes. appropriate to regard k as the parameter then, since the quasi-normal modes form a discrete set for each P. In the black hole case. ,Ai can glso be zero for purely imaginary k and non-physical. purely imaginary Pi-2 = r X . This corresponds to having boynded L, functions which arise as solutions to a typical In the remaining discussion it w i l l be assumed that eigenvalue problem (for X 1 . A is non-zero.

f

The condition that the oscillatory parts of a and expressed by the four equations:

4

which can be satisfied in the following cases.

Either:

vanish asymptotically (for W # O )

is

= o

i) A,AP+l

or b=o and c=O. in which case either ii) ad-o,

or

iii) B ( k ) B p ( - k ) 1

iv) B ( k ) = t

= Bp+l(k)Bptl(-k)

B P( - k )

= Bptl(k)

which has the special case = Bp+l(-k)

=

o

Case i) has b88n dealt with above: case ii) implies M=O and the mapping is for real k. iii) occurs only when the reflection amplitude ( B 1 is singular: independent of the corresponding parameter in the potential. Exactly this is the case for the only know transformations between wave functions around a black hole.

570

B. F: Whiting

Finally, we note that iv) arises for reflectionless potentials. which are of prime interest in the use of inverse scattering methods to solve non-linear evolution equations.

We remark that the situation here is in considerable contrast to the case of scattering by non-singular potentials on the half line. There, with Bessel functions for example. the boundary condition at the origin is controlled by the angular momentum parameter. and the ladder operator has an explicit dependence on the singularity at the origin. Wlth Coulomb wave functions. where the charge weakly influences the singularity at the origin. there are also mappings of the form considered above. and functionally very similar to those for Bessel functions, where the parameter becomes the number of nodes i n the bound state, which then governs the lower order fall-off rate at infinity. However, these mappings are known for & I Coulomb wave functions. not just for the bound states. ACKNOWLEDGEMENTS The author is grateful for the kind hospitality of DAMTP. Cambridge. U. K. , and of the Centers for Relativity and for Theoretical Physics i n Austin, Texas. This work was supported by the S. E . R . C . in Cambridge. and in Austin by the N . S . F . . which also covered travel expenses for the trip to Birmingham. Alabama. Fruitful conversations wlth Carroll. Chandrasekhar. Deift. J. McLaughlin. and Sarnak are gratefully acknowledged. REFERENCES

Ill I21 131 I41

151 I61

Deift. P. A. , Applications of a Commutation Formula, Duke Math. Jour. 45. 267-310 (1978) Carroll. R . , Transmutation. scatterina theorv. and soecial functions, Notas de Mathernatica 8 7 , North-Holland, Amsterdam, 1982 Sarnak. P.. Spectral behaviour of Quasi Periodic Potentials. Comm. Math. Phys. & 377-401 I ( 1982) Darboux. J. G . , Theorig Surfaces. 1 1 . Livre IV. Ch. IX.1409. pp.213-6 [ 1892?) Crum. M .K. , Associated Strum-Liouville Systems, 6. 121-7 (1955) n! &I!=. Clarendon Press, Chandrasekhar. S., Lhe mathemat i c a Oxford, 1983.

mm

DIFFERENTIAL EQUATIONS I.W. Knowles and R.T. Lewis (Editors) @ Elsevier Science Publishers B.V. (North-Holland),1984

571

POINTWISE INITIAL-VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS Joseph Wiener Department of Mathematics Pan American U n i v e r s i t y Edinburg, Texas U.S.A.

D i f f e r e n t i a l e q u a t i o n s w i t h d e l a y s h a v i n g i n t e r v a l s of constancy a r e s t u d i e d . I t i s shown t h a t they are c l o s e l y r e l a t e d t o impulse and loaded e q u a t i o n s and, e s p e c i a l l y , t o d i f f e r e n c e equations. I n f a c t , t h e e q u a t i o n s c o n s i d e r e d i n t h i s paper have t h e s t r u c t u r e of c o n t i n u o u s dynamical systems w i t h i n i n t e r v a l s of u n i t l e n g t h . C o n t i n u i t y of a s o l u t i o n a t a p o i n t j o i n i n g any two c o n s e c u t i v e i n t e r v a l s then i m p l i e s r e c u r s i o n r e l a t i o n s f o r t h e v a l u e s of t h e s o l u t i o n a t such p o i n t s . The e q u a t i o n s a r e t h u s s i m i l a r i n s t r u c t u r e t o t h o s e found i n c e r t a i n " s e q u e n t i a l - c o n t i n u o u s ' ' models of d i s e a s e dynamics.

INTRODUCTION

I n [11 and [Z] a n a l y t i c s o l u t i o n s t o d i f f e r e n t i a l e q u a t i o n s w i t h l i n e a r t r a n s f o r mations of t h e argument a r e s t u d i e d . The i n i t i a l v a l u e s are g i v e n a t t h e f i x e d p o i n t of t h e argument d e v i a t i o n . Profound and c l o s e l i n k s e x i s t between e n t i r e and d i s t r i b u t i o n a l s o l u t i o n s of such e q u a t i o n s . I n t h i s a r t i c l e w e c o n s i d e r equations x'(t) = f ( t , x(t), x(h(t))) w i t h arguments h ( t ) = [ t ] , h ( t ) = [ t - n ] , h ( t ) = t - n [ t ] , and h ( t ) = [t+n] where [ t ] d e s i g n a t e s t h e g r e a t e s t - i n t e g e r f u n c t i o n . A s u b s t a n t i a l t h e o r y of such equat i o n s i s v i r t u a l l y undeveloped [ 3 ] . Connections are e s t a b l i s h e d between d i f f e r e n t i a l e q u a t i o n s w i t h p i e c e w i s e c o n s t a n t d e l a y s and d i f f e r e n c e e q u a t i o n s of an i n t e g e r - v a l u e d argument, t h e t h e o r y of which h a s been v e r y i n t e n s i v e l y developed i n t h e book [ 4 ] and i n numerous subsequent p a p e r s . B o r d e r i n g on d i f f e r e n c e equat i o n s a r e a l s o impulse f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n s w i t h impacts and s w i t c h i n g and loaded e q u a t i o n s ( t h a t i s , t h o s e i n c l u d i n g v a l u e s of t h e unknown s o l u t i o n f o r g i v e n c o n s t a n t v a l u e s of t h e argument). These e q u a t i o n s may be i n c l u d e d i n our scheme t o o . F i r s t , l i n e a r e q u a t i o n s w i t h c o n s t a n t c o e f f i c i e n t s and two d e l a y s , [ t ] and [t-11, a r e c o n s i d e r e d . The i n i t i a l - v a l u e problem i s posed a t t=-1 and t = O , and t h e The e x i s t e n c e and uniqueness of s o l u t i o n and of i t s s c l u t i o n i s sought f o r t > O . backward c o n t i n u a t i o n on (4, 01 i s proved. Furthermore, a n i m p o r t a n t f a c t i s e s t a b l i s h e d t h a t t h e i n i t i a l c o n d i t i o n s may be posed a t any two p o i n t s to-1 and Necessary and s u f f i c i e n t c o n d i t i o n s f o r a s y m p t o t i c to, n o t n e c e s s a r i l y i n t e g r a l . s t a b i l i t y of t h e t r i v i a l s o l u t i o n a r e determined e x p l i c i t l y v i a c o e f f i c i e n t s of t h e g i v e n e q u a t i o n . Then, t h e f o r e g o i n g r e s u l t s are g e n e r a l i z e d f o r e q u a t i o n s w i t h many d e l a y s and systems of e q u a t i o n s . Some c l a s s e s of advanced and n e u t r a l e q u a t i o n s a r e a l s o s t u d i e d . Next, l i n e a r e q u a t i o n s w i t h v a r i a b l e c o e f f i c i e n t s and some n o n l i n e a r e q u a t i o n s a r e i n v e s t i g a t e d . E q u a t i o n s w i t h unbounded d e l a y [51 a r i s e i n c a s e s of s e v e r a l argument d e v i a t i o n s . I n such problems t h e i n i t i a l f u n c t i o n i s p r e s c r i b e d on [-m, 0 ) and t h e s o l u t i o n i s c o n s i d e r e d f o r t > O .

J. Wiener

572

EQUATIONS WITH CONSTANT COEFFICIENTS Consider the scalar initial-value problem x’(t)

= ax(t)+aox(

[tl )+alx( [t-ll), ~ ( - 1 )= cdl, x(0) = co

(1.1)

with constant coefficients. This equation is very closely related to impulse and loaded equations. Indeed, write E q . (1.1) as m

C

x’ (t) = ax(t)+

(agx( i)+alx(i-l) ) ( H ( t-i)-H( t-i-1) ),

i=-m where H(t)=l for t>O and H(t)=O for t
x"(t)

=

C (agx(i)+alx(i-l))(d(

ax’( t)+

t-i)-6( t-i-I)),

i=-m where 6 is the delta functional. This impulse equation contains the values of the unknown solution for the integral values of t. We introduce DEFINITION 1.1. A solution of Eq. (1.1) on 10, m ) is a function x(t) that satisfies the conditions: (i)x(t) is continuous on [0,m ) ; (ii) the derivative x’(t) exists at each point tf[O, m ) , with the possible exception of the points [ t ] ~ [0, m) where one-sided derivatives exist; (iii) Eq. (1.1) is satisfied on each interval In, n+l)C[O, m) with integral ends. Denote bo(t)

and let X1 and

= eat+a-1ao(eat-l), bl(t)

X2 be the roots

=

a-’al(eat-l)

(1.2)

of the equation

X2-bO(l)A-bl(l)

=

0

(1.3)

THEOREM 1.1. Problem (1.1) with al#O has on [0,m) a unique solution x(t) = bo({t})c[tl+bl({t})c[t-l,,

(1.4)

where {tj is the fractional part of t and

Proof. Assuming that x (t) is a solution of E q . (1.1) on the interval nlt
=

ax (t)+aocn+alcn-l.

(1.6)

The general solution of this equation on the given interval is x (t) = ea(t-n)c-a-l(a

ocn+a 1cn-1)’

with an arbitrary constant c. Putting here t=n gives cn

c-a-l(aocn+alcn-l).

=

Hence, c

=

(l+a-la )c +a-la c 0 n 1 n-1

and x (t) If xn-,(t)

=

bo(t-n)c

+b (t-n)c n-1’ n 1

designates the solution of Eq. (1.1) on [n-1, n) satisfying the condi-

573

Point wise Initial- Value Problems tions x(n-1)

= c ~ - ~ x(n-2) ,

xn-,(t)

=

c n-2

= bo(t-n+l)c

then

+bl(t-n+l)cn-2. n-1

Since x (n) = x (n), we obtain the recursion relation n-1 (1.7)

cn = bo( 1 )cn-l+bl( 1)cn-2.

We look for a nontrivial solution of this difference equation in the form c Then

An and X satisfies (1.3). solution of (1.7 is

=

=

An.

bo(l)Xn-1+bl(l)hn-2,

A1 and h 2 of (1.3) are different, the general

If the roots c

=

klXF2X;,

with arbitrary constants kl and k2. In fact, it satisfies (1.7) for all integral n. In particular, for n=-1 and n=O this formula gives

X -1k +X -1k2 1

1

=

c - ~ ,kl+k2

=

c0

2

whence

kl =

1(C0-X 2c- 1 )/(X1-A2),

These results establish (1.5). c

=

If hl

=

k2

=

X2(X1c~,-c0)/(h,-X2~.

X2

=

A, then

An(co(n+l)-Xc-ln),

(1.8)

which is the limiting case of (1.5) as X + A2. the implicit assumption a#O. If a=O, then

Formula (1.4) was obtained with

)(t-n). x (t) = c +(a c +a c n 0 n 1 n-1 which is the limiting case of (1.4) as a + 0. The uniqueness of solution ( 1 . 4 ) on [ O , -) follows from its continuity and from the uniqueness of the problem xn(n) = cn for (1.6) on each interval [n, nil]. COROLLARY. The solution of (1.1) cannot grow to infinity faster than exponentially as t + + m. In ordinary differential equations with a continuous vector field the solution exists to the right and left of the initial t-value. For retarded functional 161. Eq. (1.1) is differential equations (RFDE) this is not necessarily the case a RFDE. Since the solution of (1.1) on [0, a ) involves only the group eat it can be extended backwards on ( 4 , 01. THEOREM 1.2. If al#O, the solution of (1.1) has a unique backward continuation on ( 4 01 given by formulas (1.4) and (1.5). THEOREM 1.3. The solution x=O of Eq. (1.1) is asymptotically stable as t iff lhll < 1, < 1.

Ix21

THEOREM 1.4. The solution x=O of Eq. (1.1) is asymptotically stable as t iff either of the following hypotheses is satisfied: (i)

-a(2+ea)/(ea-1)

< . a

< -aea/(ea-l),

-

-a(ea+a -1(ea-l)ao) 2 /4(ea-1) 5 al < a(ea+l) +ao; ea-1

+

+m

*+

574

J. Wiener (ii)

-aea/(ea-l) < .a -a(ea+a-l(ea-l)ao)

(iii) -a(2+ea)/(ea-1)

< a(2-ea)/(ea-1), 2

/4(ea-1) 5 al < -a-a

0;

< . a

< a(2-ea)/(ea-1),

-a/(ea-l) < al < -a(e a+a -1 (ea-l)ao) 2/4(ea-1). THEOREM 1.5. The problem x(t ) = xo, x(to-l) 0 solution on (*, m) if a,# 0 and

Proof. With the notations mo = bO({tO}), we obtain from (1.4) the equations

xo’

x - ~for Eq. (1.1) has a unique

# 0, i # 1, 2.

Aibo(~to))+bl({to))

mOc~tO1+nOc~tO-ll=

=

no

=

bl({tO}),

(1.9) and since {to-l} = { t o } (1.10)

moc~tO-ll +n 0c [t0-21= x-l’

Let

Then from (1.5) we have [to] = P [tO+llcO+q~to+llc-l’ C[tO-ll = P~tO1cO+q~tO1 c-l’

[t0-21 = P [tO-llCO+q~tO-llC-l~

and substitute these expressions in (1.10): (moP[tO+ll+noP[tOl )co+(mo~[to+ll+~O~[tOl)C-l

=

xo

(mop[tOl+noP[tO-ll )co+(moq [tOl+nO~[to-l])C-l

=

x-l’

This system with the unknowns co and c - ~ has a unique solution if its determinant

A = (hlh2)[t~-11(~l~2m~(~ +A )m n +n 2 ) 1 2 00 0 is different from zero. The condition al#O gives X1X2#0, and A#O iff (Xlmo+n0). (X2mo+nO)#O which is equivalent to (1.9). THEOREM 1.6. The problem x’(t)

= ax(t)+

N 1 aix([t-i]), i=O

has a unique solution on [N, a). than exponentially as t + + m. THEOREM 1.7.

x(i)

=

ci, i = 0 ,

..., N ,

aN # 0

This solution cannot grow to infinity faster

Assume that in the equation x’(t)

=

Ax(t)+Aox(

[t]

)+A~x( [t-lI ), x(-1)

=

c-1,

X(0)

=

co

(1.11)

the matrices A, eA-1, and A1 are nonsingular. Then (1.11) has a unique solution on [ O , a), and this solution cannot grow to infinity faster than exponentially THEOREM 1.8. The scalar problem

575

Pointwise Initial- Value Problems x (t)

ax(t)+aox(

=

[tl )+alx( [t+ll ), x(0) = co

with constant coefficients has on [0, x(t) if bO(l)#O,

=

a unique solution

m)

ltl c 0 , X

(bo({t))+Xbl({t)))X

=

bo(l)/(l-bl(l))

bl(l)#O.

Consider the equation

with constant rxr- matrices A and Ai and r- vector X. If A.=O, for all i) 1, then (1.12) is a retarded equation. For A#O and Ai=O(i 2$, it is of neutral type. And if Ai#O, for some i 2, (1.12) is an advanced equation. THEOREM 1.9.

If the matrices A and B

has on [ O ,

a unique solution

m)

x(t)

E({t))E[tl(l)co,

=

E(t)

=

2

AiAi-l are nonsingular, problem (1.12) i=l I

=

+

(eAt-I)S, S

=

A

-1 -1 B (A+Ao).

(1.13)

Proof. Assuming that xn(t) is a solution of (1.12) on n 5 t < n+l with the ini= cni, i = 0, N, we have tial conditions x(i)(n)

...,

x (t) = Ax (t)

+

N

C Aicni. i-0

(1.14)

Since the general solution eA(t-n)c of the homogeneous equation corresponding to ( 1 . 1 2 is analytic, we may differentiate (1.12) and put t=n. This gives c

nl

From

=

AC no

+

N

C Aicni, cni i=o

he latter relation, n, i=

Ai-l

=

Ac,,

i-l, i

= 2,

..., N.

nl* and from the first it follows that

Hence, cni.

=

A

i-1 -1 B (A+Ao)cn0,

i 1. 1.

The general solution of (1.14) is

with an arbitrary vector c.

For t=n we get c=Scno and xn(t>

solution of (1.12) on [n-1, n) satisfying x(n-1) E(t-n+l)cn-l , 0 ‘ The requirement xn-1 (n) which cnO = En(l)co. EQUATIONS WITH VARIABLE COEFFICIENTS Along with the equation

=

=

=

For the

E(t-n)cnO.

cn-1, 0 we have xn-l(t)

x (n) implies c nO

=

E(1) c

=

~ , -o, ~from

J. Wiener

576 x’(t)

+

A(t)x(t)

Ao(t)x(

[t])

+

Al(t)x’( [t] ), X(0) = CO we also consider x’(t) = A(t)x(t), x(0) = CO. If A(t) is continuous, the problem has a unique solution x(t) = U(t)cn where U(t) is the solution of trix equation U’(t) = A(t)U(t), U ( 0 ) = I. =

(2.1)

latter the ma(2.2)

THEOREM 2.1. If the coefficients of ( 2 . 1 ) are continuous on 0 5 t < m and the matrices Bi = I-Al(i) are nonsingular f o r each i 2 0, then there exists a unique solution of ( 2 . 1 ) given by the formula

where

and P . i Proof.

=

-1

B i (A(i)

+ Ao(i)).

On the interval [n, n+l) we have the equation

+

x’(t) = A(t)x(t) F o r t=n this gives x’(n)

x’(t)

Pnx(n).

=

+

= A(t)x(t)

+

Ao(t)x(n)

Al(t)x’(n).

Hence,

+ Al(t)Pn)x(n).

(Ao(t)

Its solution xn(t) satisfying the condition x(n)

=

cn is given by the expression

t

xn(t)

=

+

Un(t)(I

+

Ui1(s)(A0(s)

A1(s)Pn)ds)cn,

n Un(t) being the solution of Eq. (2.2) such that Un(n) = I. The solution of E q . (2.1) for t e[n-1, n), with the condition x(n-1) = c ~ - is ~ ,

+

xn-1 (t) = Un-l(t)(I From xn-,(n)

= c

+ A1(s)Pn-l)ds)cn-l

JI-;i!l(s)(Ao(s)

we obtain the relation -1 un-l(s)(Ao(s)

+ Jn

cn = un-l(n)(I

n-1

To prove ( 2 . 4 ) , it remains to observe that A0 (t), A 1(t) are continuous on 0 5 t < and the matrices 1 2 m, then solution B. have uniformly bounded inverses for all i 2 0 : I [BY 1 1 THEOREM 2.2.

If A(t),

where a(t) = max I IA(s)l b(t) = ao(t) + mal(t)(a(t)

I,

I IAi(s)l I,

ai(t) = max + ao(t)).

0

5 s 5 t, i

=

0 , 1;

is the solution of E q . (2.2) with the condiProof. The matrix Ui(t) = U(t)U-l(i) tion Ui(i) = I. Therefore, it is the sum of the absolutely convergent series

1

rt

Ui(t)

=

I

+

A(s)ds i

+

1

It

IS

A(sl)dsl

A(s)ds i

Ji

+

...

511

Pointwise Initial- lhlue Problems

Finally,

In particular, if A(t), Ao(t), and Al(t) are bounded on LO, -), then I Ix(t)lI 5 M exp (ut) with some constants M > 0 , o > 0. In this case, the conclusion about the solutions growth is the same as for the equation with constant coefficients x’(t) THEOREM 2.3.

=

Ax(t)

+

+ Alx’([t]),

Aox([t])

~(0) =

CO.

The solution of the scalar equation x’(t)

tends to zero as t

=

ax(t)

+

+m

+

if b(t)

-a(ea

+

E

l)/ea

C[O,

-

m)

(2.6)

(a = const)

x ( 0 ) = co

b(t)x([t]).

and, starting with some to 2 0,

1) < bo 5 BO < - a,

(2.7)

where bo Proof. la

=

inf b(t), BO

=

sup b(t),

t

1. to.

According to ( 2 . 3 ) and ( 2 . 4 ) , the solution of ( 2 . 6 ) is given by the formu-

Let j be a natural number such that j 1. to, then due to ( 2 . 7 ) we have i vi-l(i) = ea + ji-le a(i-s) b(s)ds 5 ea + Bo(ea-l)/a,

v.

(i)

1- 1

ea

+ bo(ea-l)/a,

i

j+ 1

and (i) < ea - a(ea-l)/a v. 1- 1

=

1,

viAl(i) > ea - a(ea+l)(ea-l)/a(ea-l) Hence, equalities

5 q < 1, ’i 5 ea

j

+

+

=

-1.

1. On the other hand, for i 5 j we use the in-

M(ea-l)/a

This estimate holds also for v [tl(t).

= p,

M

=

sup b(t),

t E [O,

m).

To finish the proof, rewrite (2.8) as

J. Wiener

518

5 pj+ q It]- , and here the second factor vanishes as

This gives Ix(t)l THEOREM 2.4. x (t)

=

t +

+a.

The solution of the scalar equation

+

ax(t)

N C

ai(t)x([t-i]),

x(i)

=

ci, i

=

0,

..., N;

(a=const)(2.9)

i=O tends to zero as t for O(i(N.

-+

+

if a < 0, ai(t)

C[N, -),

E

and lim a.(t) = 0 as t

+

+a,

Proof. The solution of (2.9) is given by (2.10)

Take a number 6 > 0 such that ea + (N+1)6 = q < 1. There exists such a number to that lai(t)l 5 6. i = 0, N, for t > to. Then

...,

(2.11) To evaluate c [ ~ we ~ ,employ the method developed in [2] for the study of distributional and analytic solutions to linear functional differential equations. Denote M [tl = max lcjl, 0 5 j 5 [tl. From (2.11) it follows that IcLtlI 5 qMct-ll, t 1. to.

- MIt-ll,and starting with some k,

Hence, M

Mn

%,

=

n,k.

(2.12)

The application of (2.12) to (2.11) successively yields Ick+ll 5 q q , lck+2I 5 qg, Now we put [t]

I

k+N+2

I

k+N+2,

=

’

2

I

’ * * *

lCk+N+11 5 qMk

...,

k+2N+2 and use the latter inequalities to obtain

k+N+3

I

2

q Mk

’*

9

1

k+2N+2

I

2 Mk

Continuation of the iteration process shows that /ck+(j-l)N+il natural j and i=j, j+l, follows now from (2.10).

..., j+N.

This implies c r t l-+ 0 as t

I qjM,, f o r all + a,

and the proof

Consider the scalar problem x (t)

=

f(x(t),

x([tl)),

x(0) =

C0

( 0 5 t <

(2.13)

a).

If in the equation with a parameter x

=

f(x, A),

f(x, A )

E

(2.14)

C(R2)

f(x, A) # 0 everywhere, then there exists the general integral F(x,

A)

=

t

+

g(X),

Pointwise Initial- Value Problems

519

with a n a r b i t r a r y f u n c t i o n g ( X ) . I f x ( t ) can be extended o v e r a l l t 0, then f o r t h e s o l u t i o n x n ( t ) of (2.13) on n I t < n + 1 w e have t h e e q u a t i o n x,!, = f ( x n , c n ) , Hence, i f 1 = c n , t h e n F ( x n , c n ) = t

x n ( n ) = cn. F ( c n , c,)

+ g(cn)

= n

F(xn, c n ) = t

+

Since xn(n

+

+

For t = n t h i s g i v e s

F ( c n , c n ) - n.

+

1, n

(2.15)

+

2) t h a t s a t i s f i e s x(n + l ) = ~ , + ~ .

+

(n + 1 ) we put t = n 1 i n ( 2 . 1 5 ) and g e t F ( C , + ~ , c n ) = n+l T h i s e q u a t i o n h a s a unique s o l u t i o n c ~ =+h ( ~ c n ) because

1) = x

1.

g(cn).

and

Now w e t a k e t h e s o l u t i o n of (2.13) on [n

F(cn, cn)

+

(2.16) and aF/ax

#

0.

Hence, c

(2.17)

= hn(co),

where hn i s t h e n t h i t e r a t i o n of h. All t h e s e i t e r a t i o n s a r e d e f i n e d a t t h e corresponding p o i n t s s i n c e t h e s o l u t i o n x ( t , c n ) of Eq. (2.13) e x i s t s on each i n t e r v a l [n, n 1 1 . Thus, w e may u s e t h e formula c n = x ( n , c n-1 ) r e p e a t e d l y .

+

I f f ( x , 1) E C(R2) i s d i f f e r e n t from z e r o everywhere and t h e s o l u THEOREM 2.5. t i o n s of ( 2 . 1 4 ) can b e extended over [ O , a), t h e n (2.13) h a s a u n i q u e s o l u t i o n g i v e n by formula ( 2 . 1 5 ) f o r t E [n, n + l ] , w i t h F and cn from (2.16) and ( 2 . 1 7 ) . I f f ( c 0 , c,) = 0 and F ( x , C O ) d i v e r g e s as x -f co, t h e n x = c i s t h e unique s o l u t i o n of ( 2 . 1 3 ) . And i f F ( x , c,) converges f o r x -+ C O , then q2.13) has more t h a n one s o l u t i o n .

In c o n c l u s i o n , w e c o n s i d e r t h e e q u a t i o n x ' ( t ) = A(t)x(t)

+

f(t, x(t

-

[t]), x(t

-

2[t],

..., x ( t - N [ t ] ) ) .

(2.18)

DEFINITION 2 . 1 . The f u n c t i o n x ( t ) is c a l l e d a s o l u t i o n of t h e i n i t i a l - v a l u e probl e m f o r ( 2 . 1 8 ) i f t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d : ( i )x ( t ) i s c o n t i n u o u s m ) ; ( i i ) t h e d e r i v a t i v e x ' ( t ) e x i s t s a t each p o i n t t E [ O , m), w i t h t h e on (4, p o s s i b l e e x c e p t i o n of t h e p o i n t s [ t ] where one-sided d e r i v a t i v e s e x i s t ; ( i i i ) on (4, 01, x ( t ) c o i n c i d e s w i t h a g i v e n c o n t i n u o u s f u n c t i o n @ ( t ) and s a t i s f i e s Eq. (2.18) on e a c h i n t e r v a l [n, n + 1). THEOREM 2.6. I f t h e m a t r i x A ( t ) E C[O, m ) and f i s a c o n t i n u o u s f u n c t i o n i n t h e s p a c e of i t s v a r i a b l e s , t h e f u n c t i o n $ ( t ) E C ( 4 , 01 and t h e s o l u t i o n of t h e equat ion x ' ( t ) = A(t)xo(t) 0

+

f ( t , xo(t), xo(t),

..., x o ( t ) ) ,

xo(o) =

$ (o )

e x i s t s on 0 5 t 5 1 and is u n i q u e , t h e n t h e i n i t i a l - v a l u e problem x ( t ) = @(t), t E (4, 01 f o r (2.18) h a s a unique s o l u t i o n

(2.19) where U ( t ) i s t h e s o l u t i o n of ( 2 . 2 ) and f . ( s ) = f ( s , xo(s-i), $(s-2i), Proof.

On n 5 t < n

+

...,

$(s-Ni)).

1 e q u a t i o n ( 2 . 1 8 ) assumes t h e form x ' ( t ) = A ( t ) x ( t ) + f n ( t ) .

J. Wiener

580

I t s s o l u t i o n x ( t ) t h a t s a t i s f i e s t h e c o n d i t i o n x ( n ) = c,

is

t

xn(t) = un(t)(en

-1 and Un(t) = U(t)U ( n ) .

+

Jn U i l ( s ) f n ( s ) d s ) ,

Hence,

I

t

x n ( t ) = U(t)(U-'(n)cn

+

U-'(s)fn(s)ds).

n

The same formula f o r xn-,(t)

c

=

together with x

U ( n ) ( u -1( n - l ) c n - l

+

1"

n-1

(n) = x (n) yields the r e s u l t

U-l(s)fn-l(s)ds)

n-1

which l e a d s t o ( 2 . 1 9 ) .

REFERENCES Shah, S. M. and Wiener, J . , D i s t r i b u t i o n a l and e n t i r e s o l u t i o n s of o r d i n a r y d i f f e r e n t i a l and f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n s , I n t e r n a t . J r n l . Math. & Math. S c i . 6 ( 2 ) (1983) 243-270. Cooke, K . L. and Wiener, J . , D i s t r i b u t i o n a l and a n a l y t i c s o l u t i o n s of funct i o n a l d i f f e r e n t i a l e q u a t i o n s , J r n l . Math. Anal. and Appl. ( t o a p p e a r ) . Myshkis, A. D., On c e r t a i n problems i n t h e t h e o r y of d i f f e r e n t i a l e q u a t i o n s w i t h d e v i a t i n g argument, Uspekhi Mat. Nauk 3 2 ( 2 ) (1977), 173-202. V

Halanay, A. and Wexler, D . , T e o r i a C a l i t a t i v a a S i s t e m e l o r c u I m p u l s u r i (Acad. RSR, B u c u r e s t i , 1968). Corduneanu, C. and Lakshmikantham, V . , l i n e a r A n a l y s i s 4 ( 5 ) (1980) 831-877.

Equations w i t h unbounded d e l a y , Non-

Hale, J. K . , Theory of F u n c t i o n a l D i f f e r e n t i a l Equations (Springer-Verlag, New York, 1977).

DIFFERENTIALEQUATIONS I.W.Knowles and R.T. Lewis(Editors) @Elsevier Science PublishersB.V.(North-Holland),1984

58 1

ASYMPTOTICS AND SPECTRAL THEORY FOR HIGH ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH POWER COEFFICIENTS Alastair D. Wood School of Mathematical Sciences National Institute for Higher Education Dublin 9, Ireland

1.

Introduction

We consider asymptotic solutions for large values of the independent variable x of the linear ordinary differential equation of arbitrary order n with power coefficients. Although such equations arise in physical problems when n = 2, 4 or 6, the motivation in this paper arises from spectral theory of differential operators in the Hilbert space L2(X,m) where X>O. For a symmetric differential operator L, the deficiency indices (N+,N-) are defined to be the maximal number of linearty independent solutions of Ly = Ay in LZ(X,m) for Im(A)>O and
A. D. Wood

582

deciding on the dominant parts. The direct methods, on the other hand, can be applied as a general rule only to equations whose coefficients are powers, or at best polynomials in X. They do, however, provide an explicit solution as as integral representation from which a full asymptotic expansion may be constructed. This contrasts with the matrix methods which in many cases provide information only as an order term. Chapter 4 of Titchmarsh’s book [ 9 ] contains many equations illustrating points in spectral theory of second order differential equations, including several examples based on Bessel’s equation. For higher order eq at’ons there is a distinct lack of examples, apart from the obvious (-l)"yYPnf+ y = 0 and other constant coefficient equations. It is the purpose of this paper to draw attention to a class of higher order equations with power coefficients which can be used to treat borderline cases of existing theorems, provide counter examples to conjectures or suggest the general direction of future theorems. While it is possible to treat these equations by the integral representation methods mentioned above, it was found simpler to make the transformation to some form of the generalised hypergeometric equation which could be solved in terms of the F-function or Meijer’s G-function whose asymptotics were known. This class of equations includes the hyper Bessel equation. 2.

The even order self-adjoint equation

Asymptotic expansions of the solutions of

where m>q>O, a real and @>O have been given by R. B. Paris and the author in [ 6 1 . For compa?ison purposes we outline the main results here. Let

,..., 2q=p)

-Br(r=l ,2

Let fl = n+p and z Then (2.1) with A equation

=

=

b e the zeros of the polynomial in 8

d xu. Denote by 6 the formal differential operator z z . 0 may be transformed into the generalised hypergeometric

vP-"

WhereZ = h(n-p)+l. This has a fundamental set of Meijer G-function solutions which possess distinct algebraic or exponential behavburat infinity. In terms of the original x variable, when = 0 (2.1) has p linearly independent solutions

4

gr(x)

n,

x-@r

as x+-.

r

=

l,~,...,p

(2.3)

and n-p linearly independent solutions of exponential type at infinity, half exponentially large and half exponentially small. This result is not unexpected. If the intermediate derivatives are missing in (2.1), it is an easy application of the Liouville-Green method to show that all solutions are of exponential type at infinity. Conversely, if t h e highest derivative is missing, (2.1) with X = 0 is an Euler equation whose p algebraic solutions are given by the right-hand side of (2.3,). It is shown in [ 6 ] , using variation of parameters and Gronwall’s inequality that when X f 0 in (2.1) the number of L2 solutions is the same as when A = 0. We

Asymptotzcs and Spectral Theory

583

know that $(n-p) of the exponential type solutions are always in L z . The number of algebraic type L2 solutions (2.3) depends on the location of the zeros of one polynomial ( 2 . 2 ) If we rewrite this in the new variable s = 6 + $ @ + $ -@r we obtain the symmetrical form ~ ( s = )

r

q

c (-I)~-% n (s2-akz)

r=o

rk=l

whose zeros occur in sets of four, s = +of it. While the zeros of G ( s ) are symmetrical in the line o = 0, zeros giving rise to L2 solutions lie to the left of the line 0 = J B . When q = 1 or 2, equal numbers of zeros lie on either side of the L2-line 0 = J B . When q = 3 it is possible to obtain five zeros to the left of the Lz-line and one to the right for certain values of @ and a This is proved by the root locus method illustrated below in Figure 1. Whenr a 2 = a I 7 yo = 0 the zeros of G ( s ) in the right hand half-plane lie at al,a2,a3. For s u ficiently large values of 6 , as a is increased from zero, the zeros follow the locus shown by the arrows, crzssing the L2-line.

.

Figure 1:

Locus of the zeros of G ( s ) when q

=

3

t

t2

tl

0

S.l/%t

2

If we take m = q = 3, this establishes the conjecture by J. B. McLeod that the deficiency indices of a symmetric differential operator with positive coefficients can be as high as 2m-1. The cases with q>3 are fully considered in [ 6 ] using the change of argument theorem: when q = 7, for instance, it is possible to have four zeros crossing over into the L*-region, leading to an extra four algebraic type L2 solutions.

3.

The odd order self-adjoint equation

If the coefficients of the differential expression L are allowed to take complex values, then it is known 12, p.2041 that if L is equal to its formal adjoint L+ then n Ly = rz = ~ry

A. D. Wood

584

7):)' ...

where L y = irqn-r(.!.(qn-r(q ) ' , the differentiation being performed r times: with q.E&"’J and (q.br -1 real-valued. Another form is given by Naimark (5,p.7)! but that is’less suitable for our method of comparison with a hypergeometric equation. For the operator of order n = 2m+l, the deficiency indices are known to satisfy either mlN+i2m+l and m+llN-
...)’y

=

iy on

(3.1)

(X,m).

F u l l details of this and other cases are to appear in [ 7 1 . We may rearrange

(3.1) as n d inxJJll(xzx+Br)~= XY r=l

(3.2)

The asymptotic behaviour of solutions of where J.I = B(n+l)-n and B = r(B-l)+l. (3.2) is of three distingt types according as JJ>O, p = 0, or JJ
we obtain the

(3.3)

When we take the f sign according as J.I 0. When JJ>O, by multiplying ( 3 . 3 ) successively by /!J we obtain a fundamental set $I~,$~,...$I given in terms of x by

x-’k

as x-

(3.4)

where * denotes the omission of the factor corresponding to 6 = Bk. Because the leading behaviour (3.4) is independent of h we have in this czse N = N- = N where N, depending on Bk, Tan be made to take all values m+l
.

The most interesting case arises when JJ = 0. Here (3.2) is an Euler equation with n where v are zeros of the polynomial solutions y (x) = x"r, r = 1,

...,

Noting that JJ = 0 when 6 = n/(n+l) and defining the new variable s rewrite G ( 0 ) in the symmetrical form m G ( S ) = s ~ (s2-akz)+(-l)mii i k= 1

=

e+h we may

Asymptotics and Spectml Theory

585

where n = 2m+l and a = k/(n+l). A s in 1 2 the L2 nature of solutions depends on the disposition of tke zeros of G ( s ) in the complex plane. This time the rootlocus analysis is simpler because the L2-line coincides with the symmetry line Re(s) = 0. When h = 0, the zeros of G ( s ) are s = 0 and s=+a (k=1,2 m). A s Ihl4with fixed argument) these zeros move off to infiniky along paths which do not cross the Im(s) axis and which are asymptotic to the rays

,...,

args

71

-=+

=

2kn argh --+n -, k n

=

0,fl ,..,fm.

Thus we obtain deficiency indices (m,m+l), or, changing the sign in (3.1) as before, (m+l,m). We illustrate in Figure 2 below the root locus in the case n = 3, when G ( s ) = s(sz-a12)-Xi.

Figure 2 :

Root locus of G ( s ) when n = 3 and Im(X)>O We have written V for -! argX. 3

4.

A non-self adjoint equation

It was learned at this conference that Eastham has considered in [ 4 ] asymptotic formulae for the nth. order equations (rn-l(...(r 2 (r1 y’)’)

I . . .

)’)’-qy

= 0

(4.1)

where q and rk are complex-valued C z [ X , - ) functi n s which do not vanish on ( X , m ) . The special case r (x) = xak and q(x) = ’x can very easily be treated by k the methods of 93. If we substitute these values i n (4.1) and let p = a +a

+. .+a

Bk

2+...+ak-l-k+l,

1

2

= a +a

1

n-1

--B-n,

k

=

1,2,

...,n,

A D . Wood

586 we obtain the familiar equation xunn (XZX+Bk)Y d = Y. k=1

(4.2)

whose behaviour falls into three distinct categories according to the sign of p . The case p O is Eastham’s case B where he applies Levinson’s asymptotic perturbation theorem to obtain only a o ( ) estimate. In our case we can obtain asymptotic

equivalence

Y~(x)-x-’~

yl(x)-l, through the use of

o

k

= 2,3

,...,n

Fn-1 hyper Bessel function solutions.

In the case 1.1 = 0, which is not considered by Eastham, ( 4 . 2 ) is an Euler equation with solutions y (x) = xvk where the v are zeros of k k n G ( s ) = II (s+Ok)-l. k=l These zeros can be placed anywhere in the complex plane by choice of 6.

5.

Conclusion

Despite the successes of operator theory and matrix methods in the spectral theory of the nth. order equation, useful information can still be obtained by classical analysis of a model scalar equation. References

1.

C. M. Bender and S . Orszag. Advanced Mathematical Methods for Scientists and Engineers (New York: McGraw H i l l , 1 9 7 8 )

2.

E. A . Coddington and N. Levinson. Theory of Ordinary Differential Equations (New York: McGraw H i l l , 1 9 5 5 )

3.

M.S.P. Eastham, Asymptotic theory and deficiency indices for differential equations of odd order. Proc. Roy. SOC. Edinburgh, 90A, 263-279, 1981.

4.

M.S.P. Eastham, Asymptotic formulae of Liouville-Green type for higher order differential equations. Accepted by J. Lond.Math.Soc.

5.

M.A. Naimark, Linear Differential Operators, Part 1 (London: Harrap, 1968)

6.

R. B. Paris and A . D. Wood, On the L2 nature of solutions of nth. order symmetric differential equations and McLeod’s conjecture. Proc.Roy.Soc. Edinburgh, 90A, 209-236, 1981.

7.

R.B. Paris and A.D. Wood, Deficiency indices of an odd order differential operator. Submitted for publication.

Asymptotics and Spectral Theory 8.

587

3 . S . Pinto and A.D. Wood, On the asymptotic behaviour of the Titchmarsh-Weyl

Coefficients for a fourth-order equation. Quaestiones Mathernaticae,4, 337-345, 1981.

9.

E.C. Titchmarsh. Eigenfunction Expansions, Part 1 (Oxford University Press, 1962).

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS

I.W.Knowles and R.T. Lewis (Editors)

5 89

OEisevier Science Publishers B.V. (North-Holland), 1984

LARGE TIME BEHAVIORS OF TIME-PERIODIC QUANTUM SYSTEMS K e n j i Yajirna Department o f Pure and A p p l i e d Sciences U n i v e r s i t y o f Tokyo 3-8-1, Komaba, Meguroku, Tokyo 153 Japan

51

INTRODUCTION

I n t h i s paper we r e v i e w some of t h e r e c e n t r e s u l t s ([6],[9],[10],[14],[15] and [16]) on t h e l a r g e t i m e b e h a v i o r s o f t h e s o l u t i o n s o f t h e time-dependent S c h r o d i n g e r e q u a t i o n s w i t h H a m i l t o n i a n s H ( t ) = - ( 1 / 2 ) A + V(t,x) which a r e p e r i o d i c i n time:

H ( t + T)

iau/at = H(t)u,

(1.1)

=

H(t).

We c o n s i d e r t h e e q u a t i o n ( 1 . 1 ) i n t h e H i l b e r t space assume t h a t t h e p o t e n t i a l V ( t , x ) satisfies (A.l)

V(t,x)

i s r e a l - v a l u e d and

V(t+T,x)

H = L2(Rn), n 2 1

= V(t,x)

f o r some

and

T > 0;

(A.2) Eq. ( 1 . 1 ) generates a unique u n i t a r y p r o p a g a t o r { U ( t , s ) : -m < t, s < m l i n H and t h e s o l u t i o n u ( t ) o f ( 1 . 1 ) w i t h t h e i n i t i a l c o n d i t i o n u ( s ) = u i s We s h a l l s t u d y t h e s p e c t r a l p r o p e r t y o f t h e one p e r i o d g i v e n as u ( t ) = U ( t , s ) u . propagator U(T+s,s), w h i c h i s a l s o c a l l e d t h e P o i n c a r i map, t h e F l o q u e t opera t o r , o r t h e monodromy o p e r a t o r , and show t h a t i t has c l o s e connections w i t h t h e l a r g e time behavior o f t h e s o l u t i o n s U(t,s)u. For a u n i t a r y o r s e l f - a d j o i n t o p e r a t o r A on a H i l b e r t space JC, u p ( A ) , ac(A) and uess(A) a r e r e s p e c t i v e l y

A;

t h e p o i n t , t h e continuous and t h e e s s e n t i a l spectrum of Kac(A)

JCp(A),

3CC(A) and

a r e r e s p e c t i v e l y t h e p o i n t , t h e c o n t i n u o u s and t h e a b s o l u t e l y c o n t i n u o u s

subspace o f

3c

for

A.

We b e g i n w i t h t h e theorem t h a t f o r u E H p / U ( T + s , s ) ) ( o r u E H c ( U ( T + s , s ) ) ) U ( t , s ) u r e p r e s e n t s t h e bound ( o r t h e s c a t t e r i n g ) s t a t e : Theorem 1.

Suppose

V(t,x)

V2 E Lm(IRl, Lp(IRl) n L q ( R l ) ) (1)

u E HD(U(T+s,s)) such t h a t

(2) u (1.2)

E

= Vl(t,x)

f o r some

+ V2(t,x) w i t h V1 E Lm(Rn+') 1 5 p < n/2 < q 5 m. Then

i f and o n l y i f f o r any

infllU(t,s)ull t L2( I x l d l )

>

E

>

0

there exists

>

>

0

0

T h i s i s a d i r e c t e x t e n s i o n o f t h e R u e l l e [12]-Amrein-Georgescu t h e time-independent H a m i l t o n i a n s . V(t,x)

R

( 1 - E)IIUII ;

Hc(U(T+s,s)) i f and o n l y i f f o r any R L 1 lim J IIU(t,s)ult 2 d t = O . L 2 ( I x IN 0 L++m

I n Theorem 1 t h e p o t e n t i a l

and

[2] theorem f o r

i s assumed o n l y t o be bounded a t i n f i n i t y .

K. Yajima

5 90

I f V ( t , x ) -t 0 s u f f i c i e n t l y r a p i d l y as 1x1 + m, t h e s c a t t e r i n g s o l u t i o n w i l l behave l i k e a f r e e s o l u t i o n a t t h e l a r g e time. u E H (U(T+s,s)), U(t,s)u, The n e x t theorem s t a t e s a l i t t l e more than t h i s : Theorem 2 . (1.3)

Suppose t h a t t h e p o t e n t i a l

IV(t,x)I

2 C(l

+ Ixl)-'-'

V(t,x)

i s s h o r t range:

w i t h some c o n s t a n t s

C >

0 and

E >

0.

Then t h e wave o p e r a t o r s w+(s)u = s-Tim U(t,s)-' t+im

(1.4)

exp(-i(t

- s)Ho)u,

Ho = - ( 1 / 2 ) a

e x i s t and a r e complete: R(W+(s)) = Hc(U(T+s,s)).

(1.5)

When V(t,x) i s l o n g range, t h a t i s , i t decays s l o w e r t h a n i n (1.3), t h e l i m i t s (1.4) do n o t i n general e x i s t . Yet i f t h e l o n g range p a r t i s smooth enough, t h e so c a l l e d m o d i f i e d wave o p e r a t o r s do e x i s t and t h e y a r e complete. F i s t h e F o u r i e r t Pans fo rm. Theorem 2 ' .

Suppose t h a t

s a t i s f i e s (1.3) and

V2(t,x)

V(t,x) E

I (a/ax)av2(t,x) I 2

= Vl(t,x)

C(Rn'')

+ V2(t,x)

satisfies for a l l

c a ( i + 1x1) - I a i - E ,

s-lim U(t,s)-l t++-

exp(-iH(t,s,D))

Vl(t,x)

E

C(iR"+')

a,

E > O .

Then w i t h t h e m o d i f i e d f r e e p r o p a g a t o r exp(-jH(t,s,D)) m o d i f i e d wave o p e r a t o r s (1.4)'

where

= F-'exp(-iH(t,s,E))F,

the

= W+(s)

-

e x i s t and a r e complete:

= Hc(U(T+ s , s ) ) . R(W+(s))

(1.5)'

We r e f e r t o Kitada-Yajima [9] f o r t h e p r e c i s e statement and t h e p r o o f o f Theorem 2 ' where t h e Enss time-dependent method i s employed w i t h t h e h e l p o f t h e machinery o f F o u r i e r i n t e g r a l o p e r a t o r s . We s h a l l n o t d i s c u s s h e r e t h e l o n g range case i n d e t a i1

.

F o r a dynamical system t h e s t a b i l i t y o r t h e i n s t a b i l i t y o f c l o s e d o r b i t s i s an i m p o r t a n t and i n t e r e s t i n g q u e s t i o n . This, o f course, i s t h e case a l s o i n t h e quantum systems. I n t h e time-independent case when H = Ho + V has t h e spectrum a(H) = a (H) u ac(H) = I x k < a , k = 1,2,-..1 u [0,-), i t i s well-known [7] t h a t t h e P e i g e n v a l u e s a r e s t a b l e under p e r t u r b a t i o n s . Thus t h e embedded e i g e n v a l u e s - 1 xkT 1 o f t h e F l o q u e t o p e r a t o r U ( T + s , s ) = exp(-iTH) a r e a l s o s t a b l e under Ie time-independent p e r t u r b a t i o n s . We s h a l l i l l u s t r a t e , however, t h a t these eigenv a l u e s o f t h e F l o q u e t o p e r a t o r a r e g e n e r i c a l l y i n s t a b l e under % - p e r i o d i c p e r t u r b a t i o n s by t a k i n g t h e example o f AC-Stark e f f e c t , t h e Schrfidinger e q u a t i o n f o r a p a r t i c l e i n an a l t e r n a t i n g e l e c t r i c f i e l d :

_c

(1.6)

i a u / a t = -(1/2)4u + V(X)U +

Y{(E.X)COS

wtlu.

w > 0 i s t h e frequence o f t h e f i e l d and u 2 0 i t s Here E = (1,0,...,0), E.x cos a t does n o t appear as a bounded p e r t u r b a t i o n i n ( 1 . 6 ) , s t r e n g t h . Though a f t e r a t i me - depen den t un it a r y t r a n s f o rma t ion

591

Large Time Behaviors of Time-Periodic Quantum Systems u(t,x) = T(t)v(t,x)

e q u a t i o n (1.6)

s i n w t / w + i u 2 s i n 2wt/& 2, - i v 2 t/4w 2 ) v ( t , x - U E cos Ww I

= exp(-iHE.x

i s transformed i n t o t h e e q u a t i o n

i a v / a t = - ( 1 / 2 ) A v + V(x

(1 - 7 )

3

f

uE cos

wt/o

2

)V

which o b v i o u s l y can be regarded as a s m a l l p e r t u r b a t i o n o f t h e s t a t i o n a r y system i a v / a t = (-1/2)AV+ V ( x ) v , t h e p e r t u r b a t i o n b e i n g 2 N k k k k (1.8) V(x + UE cos w t / o ) - V(x) = 1 ( a V/axl)(x) cos w t / ( k ! w Z k ) + 0 ( v N + l ) . k=l The e i g e n v a l u e s { A k ) a r e i n s t a b l e under t h e p e r t u r b a t i o n and a f t e r t h e f i e l d i s s w i t c h e d on each X w i l l t u r n i n t o t h e resonance o f which t h e o r d e r of t h e i m a g i n a r y p a r t i s d k t e r m i n e d by t h e number o f t h e ( c l a s s i c a l ) photons i t t a k e s t o i o n i z e t h e bound s t a t e which i s t u r n i n g i n t o t h e resonance; i f two s t a t e s have t h e energy d i f f e r e n c e e x a c t l y equal t o f w , then t h e s o l u t i o n o s c i l l a t e s between two s t a t e s $1, Q~ f o r l o n g t i m e . We w r i t e as

$i,:$$l

u( t ,s , p ) Theorem 3. (1)

a

t h e p r o p a g a t o r f o r (1.7). Suppose t h a t

V(x)

F o r each e i g e n v a l u e

s a t i s f i e s t h e c o n d i t i o n (B) o f A E u (H)

asymptotic s e r i e s (u(t,s,uh,+)

(1.9)

P

=

x

= e- i x ( u ) ( t - s )

with

+ a4u4 +

+ a2: .+

Hj, = A$,

55.

Then:

t h e r e e x i s t s an

...

such t h a t

o(ll)

u n i f o r m l y i n t 2 s. Here I m aZk = 0 f o r x f wk < 0 and > 0, no b e i n g t h e s m a l l e s t i n t e g e r I m azn0 < 0 f o r a l m o s t a l l

x (2)

If

+ n

w > 0. F o r almost a l l w, 0 H Q ~ = ~ . $ .( j = 1 , 2 ) and A J J

LJ

P

~

(U(T+s,s)) -

x

~

=

p for all s ER.

=T h ~e n f. o r

Oit-szC,,

-2 ,

We s h a l l prove t h e theorems 1, 2 and 3 i n t h e subsequent s e c t i o n s . We s h o u l d emphasize h e r e t h a t t h e s e t h r e e theorems a r e proved by u s i n g a s i n g l e u n i f i e d mac h i ne ry . 52'

THE FLOQUET-HOWLAND METHOD.

We p r e s e n t h e r e a r e d u c t i o n f o r m a l i s m f o r s t u d y i n g t h e spectrum o f t h e F l o q u e t o p e r a t o r which we c a l l t h e Floquet-Howland method. For a t i m e - p e r i o d i c s y s t e (1.1) w i t h t h e p e r i o d T > 0, we t a k e an a u x i l i a r y H i l b e r t space K = L t F , f f ) o f H-valued square i n t e g r a b l e f u n c t i o n s o v e r t h e c i r c l e T = R/T Z w i t h t h e usual Lebesgue measure. Then we d e f i n e a one parameter f a m i l y o f opera t o r s { U ( U ) , -m < u < m } on K by t h e e q u a t i o n (2.1) Since (2.2)

U ( u ) f ( t ) = U(t,t H(t+T) = H(t), U(t+T,

- o)f(t - 0)

the propagator

s + T ) = U(t,s)

for

f E K.

{U(t,s)}

for all

(t,s).

satisfies

K. Yajima

592

Thus U(a) i s w e l l - d e f i n e d and i s a s t r o n g l y continuous u n i t a r y group on I t f o l l o w s by S t o n e ' s theorem t h a t 12.3)

K.

U(o) = e x p f - i a K )

w i t h a unique s e l f a d j o i n t o p e r a t o r . By d i f f e r e n t i a t i n g t h e d e f i n i n g e q u a t i o n (2.1) by a and s e t t i n g a = 0, we see K i s g i v e n by t h e e x p l i c i t e x p r e s s i o n K = - i a / a t + H ( t ) , a t l e a s t f o r m a l l y . The f o l l o w i n g s i m p l e lemma i s t h e c l u e t o e v e r y t h i n g i n t h i s method. be t h e u n i t a r y o p e r a t o r on

Lemma 2.1.

Let

(2.4)

(Usf)(t) = U(t,s)f(t)

Us

and b y t h e p e r i o d i c i t y elsewhere.

I n particular,

<

s + T

Then

e x p ( - i T a D ( K ) ) = o D ( U ( T + s,s)),

e x p ( - i T a a c ( K ) ) = aa,(U(T+

Proof. s 5 t

s 5t

d e f i n e d by

e x p ( - i T K j = U s ( l 8 U(T+s,s))U:.

(2.5)

and

for

K

For f E K s+T

and

s,s))

s EIR,

for all

e x p ( - i T o c ( K ) ) = oC(U(T+ s,s))

s EIR.

we have b y ( 2 . 1 ) ,

2.2) and

that f o r

e x p ( - i T K ) f ( t ) = e x p ( - i T K ) f ( t + T) = U(t+T,t)f(t)

= U ( t + T , s + T U(s + T,s

= U s ( l @ U(s+T,s))U:f(t),

pfoving (2.5). L (a).

The r e s t o f t h e lemma f o l l o w s f r o m ( 2 . 5 ) and t h e s e p a r a b i l i t y o f 0

Since t h e s p e c t r a l p r o p e r t i e s o f e x p ( - i T K ) can be deduced f r o m t h o s e o f K, Lemma 2.1 reduces t h e s t u d y o f t h e s p e c t r a l p r o p e r t y o f t h e F l o q u e t o p e r a t o r U(s+T,s) t o t h e one o f K which has t h e e x p l i c i t e x p r e s s i o n K = - i a / a t - ( 1 / 2 ) a + _V(t,x). We s h o u l d remind t h a t i n t h e s t a n d a r d F l o q u e t t h e o r y a " l o g a r i t h m " H, w h j c h i s a l s o c a l l e d t h e F l o q u e t o p e r a t o r , o f U ( T + s , s ) i n t h e sense t h a t U ( t , s ) = U ( t ) e x p ( - i ( t - s)R) w i t h p e r i o d i c U ( t ) i s t h e o b j e c t o f importance. I n a sense Lemma 2.1 a f f o r d s t h i s l o g a r i t h m i n a n a t u r a l f o r m b u t expanding i t s m u l t i p l i c i t i e s . We c a l l t h i s metho t h e l o u e t Howland-method s i n c e t h e o p e r a t o r K and t h e group U(a) on L @I, L (R )) - were f i r s t i n t r o duced by Hgwland [5] i n t h e quantum s c a t t e r i n g t h e o r y and t h e r e d e f i n i t i o n o f and Lemma 2.1 were h i n t e d from t h e s t a n d a r d F l o q u e t theory. them on L [T,L'(R~))) I n what f o l l o w s we s h a l l show how Lemma 2.1 may be used t o p r o v e Theorems 1, 2 and 3.

9

S3

$ 4

PROOF OF THEOREM 1

F o r p r o v i n g Theorem 1 we need t h e f o l l o w i n g l o c a l compactness p r o p e r t y o f K. For R > 0, F ( I x 1 5 R) 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 t h e c h a r a c t e r i s t i c f u n c t i o n o f t h e b a l l 1x1 5 R. We assume t h e c o n d i t i o n s o f Theorem 1 h e r e . F o r any Lemna 3.1. o p e r a t o r on K .

R > 0

and

z E it \iR, F ( l x 1 5 R)(K

-

z)-'

i s a compact

Proof. Since t h e g e n e r a l case can be t r e a t e d by a p e r t u r b a t i o n t e c h n i q u e ( c f . m, we s h a l l p r o v e o n l y t h e case V -ia/at + H Via the Fourier 0, KO transform i n t h e t - v a r i a b l e , F ( l x / 5 R)(KO - z 1 - l i s u n i t a p i l y e q u i v a l e n t t o 5

=

593

Large Time Behaviors of Time-Periodic Quantum Systems

-1 H. F o r each n €22, F ( l x ~ ~ R ) ( H O + n ~ - Z ) n=-' is o b v i o u s l y a compact o p e r a t o r on H ( R e l l j c h ' s theorem) and by t h e well-known e s t i m a t e o f t h e r e s o l v e n t i n t h e w e i g h t e d L -spaces, we have N

@

n=-m

F ( l x l -< R ) ( H 0 + n w - z ) - '

11 F( I XI

on

5 R) (Ho + no - z)-'Il2

= II F( 1 X I 'R)

(Ho + no - z 1 - l (Ho + no -

71-l F( I x I

5R)II

< ~ I m z ~ ~ l I I F ( ~ x ~ ~ R ) ( H O + n ~ - z )+~ 0l F as ( ~ nx + ~ ~+- R ) ~ ~ (see e.g. follows.

Kuroda [ l l ] ) .

F ( [ x [ zR)(KO- z)-'

Thus t h e compactness o f

in

K 0

Once t h e " o n l y i f " p a r t o f t h e theorem 1 i s proved, t h e r e s t f o l l o w s immedia t e l y ( s e e [16]). So we p r o v e t h i s p a r t o n l y . By Lemma 3.1 and Wiener's e r g o d i c theorem on t h e F o u r i e r t r a n s f o r m o f a bounded measure, we have t h a t f o r f E Kc(K) and any R > 0

o

(3.1)

=

lim L++m

l

I0 LI I F ( I X I <- R ) e - i o KfllK2 do L

= lim

L++-

-

0

T+s do{/ s

2 llF( 1x1 z R ) U ( t , t - u ) f ( t - a ) l l H d t )

By Lemma 2.1, f t ) = U ( t , s ) u f o r s < t c s + T belongs t o Kc(K) u E Hc(U(T+s,s)\. P l u g g i n g t h i s f T n t o ( 3 . 1 ) , we o b t a i n T+s L 0 = lim dtt IIF( 1x1 I R ) U ( t + o,s)ul12 do). (3.2) 0 L+h= s

if

I

Note t h a t

lim L+i-

dent o f

t

cl 01LI I F ( l x l > R ) U ( t + o , s ) u l l 2

i f i t e x i s t s f o r some

t.

do

exists for a l l

S i n c e (3.2)

t

and is indepen-

i m p l i e s f o r a.e.t

~ k l l F ( I x : ~ R ) U ( t + o , s ) u do ~ ~=2 0 a l o n g a subsequence Lk, t h e remark lim Lk++- Lk 0 LIIF( 1x1 L R ) U ( t + u , s ) u l 1 2 do = 0 f o r a l l s t a t e d above proves t h a t s

2 t 5 s + T.

54.

The d e s i r e d r e s u l t i s Qbtained by s e t t i n g

t = s.

0

PROOF OF THEOREM 2.

We f i r s t p r e s e n t an e l e m e n t a r y p r o o f o f t h e e x i s t e n c e o f t h e wave o p e r a t o r s

W ( s ) and t h e n o u t l i n e t h e p r o o f o f t h e completeness (1.5) r e f e r r i n g t o [14] f o r t f e details. 4.1. with

P r o o f o f t h e e x i s t e n c e o f t h e wave o p e r a t o r s I t s u f f i c e s t o show t h e e x i s t e s = 0. Set F ( t ) u ( x ) = ( i t ) ' ? '

Since

llexp(-itHo)u - F(t)ull * 0 l i m U(t,O)-'exp(-itH0)u t++-

e o f t h l i m i t s (1.4) f o r exp(ix /2t)(Fu)(x/t).

5

as = lim t+tm

t+f-,

U(t,O)-lF(t)u

.

E

Ci(Rn\{o})

K. Yajima

594

By an e x p l i c i t computation, 11 ( d/d t ) U( t ,O)

(4.1 )

-’

F ( t ) u 11

~ ~ i ~ ( t , ~ ) - ’ [ ( i t ) - ~ ~ ~ e x p ( i x ~ / ~ t ~ ~ ~ ( t , x ) ~ ( x / t ~ - ~ l /

=

< ~ ~ V ( t , t x ) t i ( x ) I+ l (1/2t2)llnull,

ti = Fu.

If E Cm(Rn\IO}) and V s a t i s f i e s (1.3), i t i s obvious t h a t t h e RHS of (4.1) i s i n t e g r g b l e i n t o u t s i d e zero. T h i s proves t h e e x i s t e n c e o f t h e wave operators. 4.2.

P r o o f o f t h e completeness.

s

Since t h e l i m i t s (1.4) e x i s t f o r each

=

lim U ( t , t + o ) e x p ( - i u H o ) f ( t )

we have

= W+- ( t ) f ( t ) .

u++-

The o p e r a t o r s K and KO a r e time-independent i n t h e sense t h a t t h e y do n o t cont a i n t h e t i m e parameter u . Thus i t i s p o s s i b l e t o a p p l y t h e well-known a b s t r a c t I n f a c t we can p r o v e t h e l i m i t i n g a b s o r p t i o n stationary theory o f scattering. principles f o r and

K

KO: I f M(x) = (1 + I x ~ ) - ( ’ + ~ ) ’ ~t h e n M(KO- h+i6)-’14 (A E R , 6 > 0) have t h e l i m i t s i n t h e space o f bounded oper-

and

M(K- A T i 6 ) - ’ V

+

{q

u (K) U ZZI and t h e y a r e compact P o p e r a t o r s i n K f o r 6 1. 0. A c c o r d i n g t o Kato-Kuroda’s a b s t r a c t t h e o r y [S], t h i s i s enough t o conclude t h a t t h e l i m i t s (4.2) e x i s t and t h e y a r e complete:

ators in K

as

6

0

for

A

outside

R(W+) = Kc(K). W+(t) = U ( t , s ) W + ( s ) e x p ( i ( t - s ) H g ) ,

(4.3) Since Kc(K) =

UsKc(l @

desired equation

U(T+T,s))

R(W+(s))

R(W+) = U s R [ l Q W+(s)).

= Us(l B Hc(U(T+sTs))).

By Lemma 2.1

Hence 7 4 . 3 ) i m p l i e s t h e

= Hc[U(T+s,s)).

-

55.

PROOF OF THEOREM 3.

I n t h i s s e c t i o n we a r e concerned w i t h t h e e q u a t i o n (1.7) o n l y and f o r p r o v i n g t h e theorem we assume t h a t t h e p o t e n t i a l V(x) i s v e r y n i c e . Co(IRn) = I f : f

(6). (1)

i s c o n t i n u o u s and

There e x i s t c o n s t a n t s I V ( x ) I 5 C(1 +

(2)

/XI)-’

If(x)I

C > 0 and

Ho

+

0 as 1x1 + - )

and + V

y > 2

with

IIfll = s u p / f ( x ) l .

such t h a t

has no zero energy resonances.

F o r some 0 < a < n / 4 , t h e Co(lRn)-valued f u n c t i o n V(eex + p E ) o f 8 E R can be extended t o a s t r i p C = { z : I I m zl < a3 as an a n a l y t i c funchion ( f o r each f i x e d p E R ) and f o r each f i x e d R E t a i t i s a Co(Rx)-valued C“-function o f P E R.

W e s h a l l prove t h e absence o f t h e e i g e n v a l u e s o f U ( T + s,s) f i r s t . I n f a c t t h e o t h e r statements can be p r o v e d as byproducts. Ry v i r t u e o f emma 2.1, t h i s f o l l o w s f r o m 0 (K) = @, K = - i a / a t - ( 1 / 2 ) ~+ V(x + UE cos w t / w ) . Since we r e g a r d K as a ’ p e r t u r b a t i o n o f - i a / a t + H, we w r i t e t h e p e r t u r b e d o p e r a t o r K as K ( y ) t o make t h e parameter dependence e x p l i c i t . U s i n g t h e F o u r i e r t r a n s f o r m

b

595

Large Time Behaviors of Time-Periodic Quantum Systems m

i n the t-variable,

o(K(0)) = n!-mtn~

i t i s easy t o see t h a t

+ u(H)}

and t h e

eigenvalues appear as embedded eigenvalues. Thus w i t h o u t s e p a r a t i n g o u t t h e eigenvalues from t h e continuous spectrum, t h e r e a r e p r a c t i c a l l y no ways t o t r e a t t h e problem. F o r t h i s reason, we apply t h e so c a l l e d "complex s c a l i n g technique" o f Aguilar-Balslev-Combes 111, [3] t o separate t h e p o i n t from t h e continuous s p e c t r a ( i f t h e p o t e n t i a l i s n o t as n i c e as i n ( B ) , one may s t i l l apply t h e e x t e r i o r comp l e x s c a l i n g o f B. Simon [13] and some o f t h e statements o f Theorem 3 remain v a l i d , see G r a f f i - Y a j i m a [4]): For e E IR, we d e f i n e 2 K(u,e) = - i a / a t - (em2'/2)a + V(eex + VE cos U t / w ) which i s obtained from K(p) by a change o f t h e v a r i a b l e s x + eex. O f course K(u,e) i s u n i t a r i l y equivalent t o K ( v ) f o r any e E lR and t h e s e p a r a t i o n o f u (K(0)) from uess(K(0)) does P n o t t a k e place. However, i f V(x) s a t i s f i e s the c o n d i t i o n ( B ) , K(p,e) can be defined f o r

e

E

:t

e

E

0;

such t h a t

(K(u,e)-

and i s s t r o n g l y continuous i n

Lemma 5.1.

Suppose

V

z)-'

for

0 E (c:

u IR.

+Im z > 0

i s holomorphic i n

We then have

s a t i s f y t h e c o n d i t i o n ( B ) and l e t

H(e) = - ( e - 2 e / 2 ) a + V(eex)

for

e

E Ica.

Then

(1)

aes,(H(e))

(2)

The d i s c r e t e spectrum od(H(e)) = u (H(e)) and op(H(e)) n IR = up(H). P a r g z < 5 2 I m e l and up(H(e)) n ( 0 \ IR) i s contained i n t z : 0 < A E up(H(e)) i s e-stable. i . e . , i f A E U (H(e)) then A E op(H(e')) P f o r e l E ti s u f f i c i e n t l y c l o s e t o e .

(3)

o ( K ( O , e ) ) = oess(K(O,e)) u ess (K(0.e))

= e-2elR+.

u op(K(O,e))

= n=i,,{nu + e - 2 e R + } ,

and

op(K(O,e))

=

+ up(H(e))l.

We r e f e r t h e reader t o t h e c e l e b r a t e d Aguilar-Combes [l]f o r t h e proof o f ( l ) , ( 2 ) . ( 3 ) f o l l o w s from t h e F o u r i e r t r a n s f o r m i n t h e t - v a r i a b l e and an e t i m a t e o f t h e r e s o l v e n t ( H ( B ) - z ) - l o f H ( 0 ) along t h e l i n e s p a r a l l e l t o i s i s o l a t e d from i t s essenThus f o r e E .:0 t h e p o i n t spectrum of K(0,e) t i a l spectrum and we can a p p l y the standard p e r t u r b a t i o n t h e o r y t o l o c a t e t h e p o i n t spectrum o f K(u,e) f o r small LI. What i s i m p o r t a n t i s t h a t i t i s enough t o work w i t h Lemma 5.2.

e (1)

E : 0

t o o b t a i n the i n f o r m a t i o n about 2

V(eex + LIE cos o t / w )

(2)

up(K(p,e))

(3)

up(K(LI,e))

n D1

= op(K(u)),

i s e-stable f o r

is

(-ia/at

e

E

ti.

e

E

ti.

-

u (K(p)): P -2e (e /2)n)-compact.

Statement ( 1 ) can be proved b y a method s i m i l a r t o t h a t f o r Lemma 3.1 and t h e p r o o f s f o r ( 2 ) and ( 3 ) are s i m i l a r t o those f o r Lemma 5 1 ( 2 ) . Lemma 5.2.(1) 6 {nu + e-2e&. i m p l i e s i n p a r t i c u l a r t h a t ueSs(K(u,e)) = n=-m

Me now compute t h e p e r t u r b a t i o n s e r i e s (Kato ['I]) f o r t h e eigenvalues and t h e e i g e n f u n c t i o n s f o r K(,,,e). Since o(K(v,e)) i s o b v i o u s l y i n v a r i a n t under t h e t r a n s l a t i o n by LO and by Lemma 5.1.(3), i t i s enough t o i n v e s t i g a t e t h e unperturbed eigenvalues A E u (H). I f H$ = A$, H(e)$, = W , w i t h P and $,(x) can be regarded as the e i g e n f u n c t i o n of K(O,e) $,(x) = ene"+(eex)

K. Yajima

596 with the eigenvalue the asymptotic s e r i e s vanish:

a2j+l

Im a

that

= 0

= 0

By a s t a n d a r d computation we see t h a t i n A: K(O,e)+, = A$,. A ( w ) = X + alu + a2u2 + a3v3 + t h e odd o r d e r terms

j’s

2j f o r a discrete set o f

With a l i t t l e work ([15]),

X +

with

0

<

j W

I m aZn < 0 0 i s g i v e n as

and

I m a2

F o r example

W.

I m a2 = -(./w

(5.1)

.

( j = 0,1,2,...) for

i t can be proved

(Im

e

>

0)

except

2 )dllE(K)~aV/axl)$ll 2 /dKIK=A+W

which i s a p i e c e w i s e a n a l y t i c f u n c t i o n o f w and I m a2 = 0 f o r A + w < 0 and I m a2 < 0 e x c e p t f o r a d i s c r e t e s e t o f w i f X + w > 0 u n l e s s t h e RHS o f (5.1) vanishes i d e n t i c a l l y . T h i s proves t h e absence o f u,(U(T + s,s)).

To prove (1.9) we compute t h e p e r t u r b e d e i g e n f u n c t i o n $e,,(t,x) f i r s t o r d e r : ~ ~ , ~ =( $te )+ uf, ( t ) w i t h IIf, 1I = O(1) as w

,w

,P K

using the e x p l i c i t representation o f

t o the -+

0.

Here

and t h e r e l a t i o n

fe 31-I

D(K(u,e))

c H1( T,H) c C(

(5.2)

W,H)

for

Im

SyP”fe,u(t)llH = 0 ( 1 ) ,

e #

0, Ll

+

we have i n f a c t 0.

F o r I m 8 L 0, t h e e q u a t i o n K ( u , e ) u = 0 generates a p r o p a g a t o r {U(t,s,u,e): t sl. i n H which i s a n a l y t i c i n B E k+ and i s u n i f o r m l y bounded f o r t 2 s ([15], Lemma 3.7). Since K ( W , ~ ) $ ~( t ) = ? ~ ‘ ~ , ~ ( itm) p l i e s ,lJ

u(t,s,v,e)$e,p(0)

= e -iA(u)(t-s)$e

(t),

we see f r o m (5.2) and t h e boundedness o f

+

= ey ’ i A ( ~ ) ( t - s ) t j e U(t,s,u,e) t h a t U(t,s,u,e)$e Taking t h e i n n e r p r o d u c t s w i t h $-, we see

e

(5.3)

(u(t,s,u,e)$e,$-

By t h e i n v a r i a n c e i n

e

e

1=

e

-iA(p) ( t - s )

o(p)

uniformly i n

he,$-)+ e

t 2 s.

O(U).

o f t h e i n n e r p r o d u c t s , (5.3) i m p l i e s ( 1 . 9 ) .

F i n a l l y t o p r o v e Theorem 3 . ( 3 ) , we proceed as above b u t u s i n i n t h i s case t h e degenerated p e r t u r b a t i o n t h e o r y . S i n c e H ( e ) $ . ( x ) = A $ ( x s , t h e f u n c t i o n s -iwt Je j je $1 ,e ( t , x ) = +l,e(~) and $2,e(t,x) = e $2,e(x) . s a t i s f y K(O.e)$j,e= Al$j,e Using t h e degenerated p e r t u r b a t i o n t h e o r y , we see t h a t t h e e i g e n v a l u e ( j = 1,Z). A s p l i t s i n t o two l e v e l s 2 i l ( w ) = A + (u/2w)($l,(aV/ax1)$2) + O(, 1, (5.4) 2 (5.4)‘ A,(u) = A - ( u / 2 , ) ( * , , ( a V / a ~ ~ ) * ~ ) + O(, 1, (1 = h l ) w i t h corresponding e i g e n f u n c t i o n s (5.5)

$l,e(t,x3p)

= ( 1 / ~ ) ( + ~ , ~ (+x e-iwt$2,e(x)) )

+

~(v),

(5.5)’

$2,e(t,x,u)

= ( ~ / Z ) ( * ~ , . ( X )- e - i w t $2,e(x))

+

oh).

Here

u.

O ( P ) s t a n d f o r H-valued c o n t i n u o u s f u n c t i o n s w i t h supremum norm o f o r d e r We p l u g (5.5) and ( 5 . 5 ) ’ i n t o

(5.6)

U ( t , s ,u

,e

je ( s ,x,u

and s o l v e t h e e q u a t i o n (5.6) w i t h $ . ( x ) i n H, we have JO

1

= exp( - i h

(u 1 ( t - s)) 0 je (t,x,u 1,

j = 1 2

Taking the i n n e r products

Large Time Behuviors of Time-Periodic Quantum Systems

Since t h e LHS o f (5.7) and ( 5 . 7 ) ' t h e equations (1.10).

do n o t depend on

8,

591

(5.4) and ( 5 . 4 ) '

imply 0

REFERENCES Aguil,ar, J. and Combes, J. M., A c l a s s o f a n a l y t i c p e r t u r b a t i o n s f o r one body Schrodinger Hamiltonians, Commun. Math. Phys. 22 (1971) 269-279. Amrein, W. 0. and Georgescu, V., On the c h a r a c t e r i z a t i o n o f bound s t a t e and s c a t t e r i n g s t a t e i n quantum mechanics, Helv. Phys. Acta 46 (1973) 635-657. B a l s l e v , E. and Combes, J. M., S p e c t r a l p r o p e r t i e s o f many body Scht-odinger o p e r a t o r s w i t h d i l a t i o n a n a l y t i c i n t e r a c t i o n s , Commun. Math. Phys. 22 (1971) 280-294. G r a f f i , S. and Yajima, K., E x t e r i o r complex s c a l i n g and t h e AC-Stark e f f e c t i n a Coulomb f i e l d , Comnun. Math. Phys. 39 (1953) 277-301. Howland, J. S., S t a t i o n a r y t h e o r y f o r t i m e dependent Hamiltonians, Math. Ann. 207 (1974) 315-335. Howland, J. S., S c a t t e r i n g t h e o r y f o r Hamiltonians p e r i o d i c i n time, Ind. Univ. Math. J . 28 (1979) 471-494. Kato, T.,

P e r t u r b a t i o n t h e o r y f o r l i n e a r o p e r a t o r s (Springer, New York, 1966).

Kato, T. and Kuroda, S. T., Math. 1 (1971) 121-171.

The a b s t r a c t t h e o r y o f s c a t t e r i n g ,

Rocky M t . 3.

Kitada, H. and Yajima, K., A s c a t t e r i n g theory f o r time-dependent l o n g range p o t e n t i a l s , Duke Math. J. 49 (1982) 341-376.

[lo]

Kitada, H. and Yajima, K., Remarks on o u r paper ' A s c a t t e r i n g theory f o r timedependent l o n g range p o t e n t i a l s ' , Duke Math. J. 50 (1983).

[11] Kuroda, S. T., An i n t r o d u c t i o n t o s c a t t e r i n g theory, L e c t u r e notes. No. 51 Aarhus Univ. (1978).

Series

[12] Ruelle, D., A remark on bound s t a t e s i n p o t e n t i a l s c a t t e r i n g theory, Nuovo Cimento 59A( 1969) 655-662. [13] Simon, B., The d e f i n i t i o n o f molecular resonance curves by t h e method of e x t e r i o r complex s c a l i n g , Phy. L e t t . 71A (1979) 211-214. [14] Yajima, K., S c a t t e r i n g t h e o r y f o r Schrodinger equations w i t h p o t e n t i a l s p e r i o d i c i n time, J. Math. SOC. Jpn. 29 (1977) 729-743. [15] Yajima, K., 331 - 352.

Resonances f o r t h e AC-Stark e f f e c t , Comnun. Math. Phys. 87 (1982)

[16] Yajima, K . and Kitada, H., Bound s t a t e s and s c a t t e r i n g s t a t e s f o r time p e r i o d i c Hamiltonians, Annales I H P , Sec. A. 39 (1983) 145-157.

This Page Intentionally Left Blank

DIFFERENTIAL EQUATIONS LW. Knowles and R.T. Lewis (Editors) 0 Elsevier Science Publishers B.V. (North-Holland), 1984

599

EIGENVALUES OF THE LAPLACIAN: AN EXTENSION TO HIGHER DIMENSIONS (11) Elsayed M.E. Zayed Mathematics Department Faculty of Science Zagazig University Zagazig, Egypt

In the first two decades of this century, the underlying inverse eigenvalue problem of determining the shape of smooth and convex regions and the unknown attendent boundary conditions from a knowledge of the spectrum of the eigenvalues of the Laplacian was rigorously studied in Gottingen by D. Hilbert, R. Courant and H. Weyl.

1. INTRODUCTION IRn be a bounded region with a piecewise smooth boundary a R Let R sequence of eigenvalues 0 < X I Q X 2 Q h 3 Q ...< Am < . . . + m as m - t m

. Let the (1 -1)

be given, which is counted according to multiplicity of the following eigenvalue problem (A, +h)u = 0 U = O

on a n ,

where A, is the Laplace operator in IR" and u E C2(f2) nC(f2) of R as well as the unknown boundary condition. shape

. Determine the

At the beginning of this century, the principal problem was that of investigating the asymptotic behaviour of the eigenvalues (1.1). If N(X) is the number of these eigenvalues 4 A , then

"XI

-

and

"A)

-

-

VOlLJJTIeRn ’A

as

!(44'

+co

(H. Weyl 1912) ( 1 . 3 )

(9)

n

’A n -

n-1 + O ( 7

2og A) as A+-

(R. Courant 1920).(1.4)

We remark, at the outset that a direct study of these eigenvalues reflects only t h e volume of the region R . In order to obtain further information about the geometry of R , one studies certain functions of the spectrum. The most useful to date comes from the heat equation or the wave equation. Accordingly, let -tAn e denote the heat operator, then we can construct the trace function

E.M.E. Zayed

600

which converges f o r a l l positive

t

.

- i t A n4 Suppose t h a t e i s the wave operator, then an alternative t o (1.5) i s t o study the tempered distribution

i(t)=

- itAn'

)

tr(e

m

=

1 m=l

- i t X ms e

The application of (1.6) t o our problem and t o more general ones can be found i n [€I] and the references given there. See, f o r example, L. Hormander [ l ]

.

For t h i s paper, we s h a l l concentrate on a study of the trace function (1.5). I t is easily seen that the trace function O(t) i s j u s t the Laplace transform m

gtx dN(X) , and then Weyl's formula (1.3) transforms into 3

A. P l e i j e l [41 and M. Kac [21 took up the matter of finding corrections t o (1.7) for plane region s2 with a f i n i t e nmber of holes. Kac put things i n the follow-

'

.

ing amusing language: thinking of R 5 IR as a drum and the eigenvalues (1 1) as i t s fundamental tones, is it possible j u s t by listening with a perfect ear to hear the shape of R ? Weyl's estimate (1.7) when n=2 shows t h a t we can hear the area of R Kac proved t h a t f o r R bounded by a broken l i n e 20 area - length O(t) + the sm over the corners 47Tt 4(4lrt)4

.

- -'

of

lT2-a' + 24m

O(1)

as

,

t+O

where O < c c < 2 ~ being the inside-facing angle a t the corner. Therefore, we can hear the perimeter of s2 By making the broken l i n e aR approximate to a smooth curve, Kac was l e t t o conjecture k+l N area R - length 2 0 O(t) C k t 2 4?rt 4(4lTt)% k=O

.

--

+

+

c

.

for plane region R with smooth boundary a R The constant "C" has special s i g nificance. I f R is smooth and convex, then C = 1 , and i f R , has a f i n i t e C = (1-h)In other nmber of smooth convex holes "h" , then words, i f R is smooth and convex then we can hear i t s area, the length of its boundary and also i t s connectivity. The computation of the remaining terms of (1.9) i s quite formidable, but we do know t h a t they involve integrals of powers of the curvature and i t s derivatives. Furthermore, L. Smith [81 has recently calculated i n addition the coefficients C, ,C, and C, using a new method of constructing a parametrix for e l l i p t i c boundary value problems.

.

So f a r , our discussion has been confined t o the Dirichlet boundary condition;

$

= 0 on suppose we have a Nemann boundary condition has shown t h a t for plane region R with smooth boundary

aR , then P l e i j e l [41 2R

,

60 1

Eigenidues of the Luplacian

-

area n + length as2 O(t) 41Tt 4(4.lTt)$. !!+I +0(t2 as t - 0 . +

N k + l +

I: c i k=O

)

t

2

_ I_

Comparison of this expansion with (1.9) shows that the coefficient of t changed sign.

'

(1 .lo)

has

2. DETECTING SHAPES AND BOUNDARY CONDITIONS

Let us return to the question of Kac; namely, "can one hear the shape of a dnnn?". If we interpret this question in a literal sense, it suggests that the only things we know about the drum are its frequencies of vibrations. To explore this question, we look at the simple one-dimensional problem of asking whether the length of a uniform vibrating string and the unknown boundary conditions can be found from a knowledge of its frequencies of vibrations. Mathematically, we have the problem: Suppose the eigenvalues {Am), m = 1 , 2 , . .. are known exactly for the eigenvalue problem

Determine the &own

length "a" and the d o w n coefficients ai, Bi, i = 1,2

.

This problem is fairly easy to analyse by any of the methods described in the next section and gives the following results

(i)

u,=8,=1,

O(t)

(ii)

O(t)

(iii)

a -- -' (4lTt)L

a1 = B,

a2,

=o ,

(Dirichlet problem)

a2=f3,=0 t + o(t2)

as t+O

,

a2 = B, = 1

1 a -7 (4lTt)4 +

+

o ( t ’ )

(Neumann problem) as t-0

8, + 0 and for any a , , 8,

.

(2.3)

(Mixed problem)

602

E.M.E. Zayed

I t i s of i n t e r e s t t o pose a similar problem for the more general ordinary differ e n t i a l equation - (PY')' q(x) Y = XY Y (2.5) together with the boundary conditions of ( 2 . 1 ) and i n which p(x) and q(x) are a l s o unknown s t i l l an open problem. The problem (2.1) has been investigated by myself i n [ l o ] when a1 =cos a , a p =-sin CY , B, =cos B and B, = - s i n B , where Oeacn , O
.

From the experience gained by t h i s r e l a t i v e l y simple one-dimensional example, we are encouraged t o look a t the problem i n higher dimensions. Suppose then we have a smooth and convex domain R i n IR2 and we have the impedance boundary condition au + y u = 0 on a R , where y i s a positive constant. This problem can be analysed rigorously by the methods outlined i n the next section, and we have shown [ 71 that if O
--

as

+ O(1)

and if

y>>1

O(t)

- --R area 4Trt

+

0(1)

+

t+O

(,-

,

(2.6)

- {length an - 2n y-’} 4 (47~1%

as

t+O

1

+6

.

(2.7)

On the other hand, suppose we have a smooth and convex domain i n R 3 and we have au + y u = 0 on S , where S i s i t s bounding surface, the impedance condition then we have shown [ill t h a t if

O
O(t)

-

volume R (47-1

3/2

+

surface area S 16nt

0(1)

as

t+O

,

+ 0(1)

as

t+O

,

+

and R,

+

'

(H-3y)da

are the principal r a d i i of curvature.

Furthermore, suppose n = { ( r , B ) : a < r < b , O<€I
is a c i r c u l a r annulus, then

603

Eigenvalues of the Laplacian

B.C.on r=a

B.C.on r=b

Dirichlet

Dirichlet

Dirichlet

Neumann

The trace formulae

Dirichlet Neumann

Neumann

Finally, suppose Q = ( ( r , e , + ) : a g r g b , O < O < l r , then we have obtained [121 the following r e s u l t s B.C.on r=a

B.C.on r=b

Dirichlet

Dirichlet

O(t)

Dirichlet

Neumann

O(t)

Neumann

Dirichlet

Neumann

NeUmann

0<+<2r 1 i s a spherical s h e l l ,

The trace formulae 41r(b3-a3) -4.ir(a2+bz)+4a(a+b) + O ( t ) , t +o 3(47~t)~” 6lr(4nt)&

-

O(t)

O(t)

4n(b3-a3) ,4n(b2-a2) +4lr(a+b) + o ( t ) , t + O 3(4~rt)~”

16nt

6lr(4~t)%

4n(b3-a3) - 4n(b2-a2) + 4 ~ ( a + b ) + O ( t ) , t +o 3 ( 4 ~ r t ) ~ ” 161rt 6lr(4nt)% 41r(b3-a3) +4n(a2+b2)+4n(a+b) + o ( t ) , t + O 3(4~t)~” 6lr(4nt)%

I t is apparent from the above discussion t h a t a howledge of the eigenvalues alone for our boundary value problems i s i n s u f f i c i e n t t o determine a vibrating membrane uniquely. A conclusion not t o be unexpected i f we r e c a l l the counter example by J. Milnor [31. Nevertheless, it i s of fundamental importance t o ask what conditions should be imposed on the region Q and the boundary conditions t o ensure uniqueness. 3. THE METHODS OF SOLVING THE PROBLEMS We s h a l l assume t h a t R i s smooth and convex and return t o the Dirichlet problem. Neumann problems and impedance problems can be treated i n the same way. Let

G(E,$;t)

be the Green‘s function of the problem

A%

0 = a+

G(X

=

o,

G(-x,$;t)

-+

b(x-F)

F.t) %)d

where

in

% <

R

x

IR+

E a2

as

,

, t+O

I ,

6 represents the Dirac d e l t a function.

As i s well known,

G(x F ’ t ) can be expanded a s the s e r i e s %’<’

(3.1)

E.M.E. Zayed

604

where the set IX,3 are the eigenvalues corresponding to the set of normalized eigenfunctions {@,I . Consequently, setting x = E ; and integrating over R , we % % have

is the "fundamental solution" of the heat equation and x(x,E;;t) is a "regular the appropriate solution", chosen in such a way that G(x F’t) satisfies 'i, boundary condition. Consequently, we %'" have

In what follows we shall use Laplace transforms with respect to ’It" and use " s Z r t as the Laplace transform parameter; thus

1 esZt

co

E(x,E;s') % d

=

G(x,F,;t) dt % %

,

0

and hence the heat equation transforms into the membrane equation

- ~ ( x%- E%, ) . (3.9) The asymptotic expansion of K(t) , for small positive t , may then be deduced directly from the asymptotic expansion of R(sz) , for large positive s , where (An

- s') E(X,<;S') % %

=

(3.10)

In the literature, there are two possible analytic methods of solving (3.9) and obtaining asymptotic expansions for large positive s . The first is due to Pleijel [51 who recasts the problem in terms of integral equations, and the second is due to Waechter 191 who works with integral representation of the Green’s function. 4 . APPLICATIONS TO EIGENVALUE PROBLEMS

In this section, we use the methods of separation variables to determine trace formulae for some shapes together with Dirichlet boundary conditions. Neumann problems can be treated in the same way. If we consider Dirichlet problems, we have the following results.

Eigenvalues o f the Laplacian

605

1 . For a rectangle of sides a,b , (4.1.1) 2. For a cuboid of sides a,b,c , abc - 2(ab+bc+ca) +-- a+b+c B(t) (4nt 3'2 16nt 4(4nt)’

-

4

as t+O

(4.1.2)

3. For a cylinder of finite length "2" whose cross-section has area Is21 and perimeter Ian1 , 2 1 aRi B(t) I R I R - 2 1 0 1+a.iani A (C(u))* d5 (4nt)32' 16nt 4(4Kt)’ 5'27I aR

-

- -112 as t+O

+

,

I

+L

(4.1.3)

where C(u) is the curvature of the boundary an . 4. For a cylindrical shell of finite length "R" whose cross-section is a circular annulus n = { ( r , e ) : a
It is a pleasure to record my thanks to the organizing committee of the UAB conference for this invitation to deliver the present paper. REFERENCES

1 1 I Hormander, L . , The spectral function of an elliptic operator, Acta. Math. 121 (1968) 193-218. [il Kac, M., Can one hear the shape of a drum?, her. Math. Monthly 73 (1966) No. 4 part 11, 1-23. [31 Milnor, J., Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A. 51 (1964) 542. [41 Pleijel, A., A study of certain Green’s functions with applications in the theory of vibrating membranes, Arkiv. Matematik. 2 (1953) 553-569. 151 Pleijel, A., On Green’s functions and the eigenvalue distribution of the three-dimensional membrane equation, XII. Skandinav. Mat. Konger (1954) 222240. [61 Singer, I.M., Eigenvalues of the Laplacian and invariants of manifolds, Proc. Int. Cong. Math. Vancouver 1 (1974) 187-200. [71 Sleeran B.D. and Zayed, E.M.E., An inverse eigenvalue problem for a general convex domain, J. Math. Anal. Applics. (to appear). [ 8 1 Smith, L., The asymptotics of the heat equation for a boundary value problem, Invent. Math. 63 (1981) 467-493. i91 Waechter, R.T., On hearing the shape of a drum: An extension to higher dimensions, Proc. Camb. Phil. SOC. 72 (1972) 439-447. [ l o ] Zayed, E.M.E., An inverse eigenvalue problem for the Laplace operator, Proc. Int. Conference, Dundee (1982). Lecture notes in mathematics, Springer Verlag, 964, 710-726.

606

E.M.E. Zayed

An inverse eigenvalue problem for the three-dimensional membrane equation, Proc. Edinburgh. Math. Soc.(submitted). [ 1 2 ] Zayed, E.M.E., Eigenvalues of the Laplacian: An extension to higher dimensions, IW. J. Applied Mathematics (submitted).

[ l l ] Zayed, E.M.E.,

607

AUTHOR INDEX

Afuwape, A.U., 1 Agmon, S., 7 Alikakos, N.D., 19 Ashbaugh, M., 25 Baras, P., 31 Barnsley, M.F., 37 Ben-Artri, M., 43 Berger, M., 49 Brezis, H., 53 61 Brown, R.C., CZc, N.P., 71 Carmona, R., 77 C a r r o l l , R. , 87 Chafee, N., 105 111 Chan, Y.C., Corduneanu, C., 115 Cosner, C., 123 Crandall, M.G., 131 Daubechies, I.,143 Devinatz, A., 43 Drager, L.D., 149 Eastham, M.S.P., 155 Ekeland, I . , 165 Enss, V., 173 Faierman, M., 205 Fitzgibbon, W.E., 213 Fleckinger, J., 219 Freedman, M.A., 229

Herod, J.V., 285 Hinton, D.B., 301, 307 Hughes, R.J., 313 Ingijlfsson, K. , 321 I s o z a k i , H., 327 Kaper, H.G., 335 Kato, T., 341 K i r o , S., 353 Kitada, H. , 327 Kon, M.A., 359 K r a l l , A.M. , 301 Kupershmidt, B.A.,

367

Ladas, G., 373 Langlais, M., 381 Lax, P.D., 387 Layton, W. , 149 Lazer, A.C. , 391 Lee, S.J., 397 Lieb, E.H., 143, 403 L i t t l e j o h n , L.L., 413 L i u , J.-L., 421 McCann, R.C., 427 McConnaughey, P.K. , 427 McCoy, P.A., 431 McKenna, P.J., 391 McLaughlin, J.R., 437 Mickens, R.E., 445 M i n g a r e l l i , A.B., 219 Perry, P.A.,

265, 451

Goldstein, J.A., 31, 115 Graef, J.R., 237 Grammatikopoulos, M.K., 237 Grimmer, R., 241 Gurarie, D., 249 Gustafson, K., 257

Rabinowitz, P.H., 455 Race, D., 465 Read, T.T., 471 Reed, M.C., 477 Rostamian, R., 19

Hagedorn, G.A., 265 Halasi, K., 257 Halvorsen, S.G., 271 Handy, C.R., 279 Hardin, D.P., 285 Harrington, A.N., 37

Sarnak, P. , 485 Schechter, M., 501 Schuchman, V., 511 Sficas, Y.G., 373 Shaw, J. K. , 301 Simon, B., 519

Author Index

608

Sloan, A., 49 Smoller, J.A., 521 Souganidis, P.E. , 131 Spikes, R.W., 237 Stavroulakis, I.P., 373 Sundberg, C., 25 Teman, R.,

537

Vanderbauwhede, A., 545 Vuillermot, P . A . , 553

Whiting, B.F., 561 Wiener, J. , 571 Wood, A.D., 581 Yajima, K . , 589 Zayed, E.M.E., 599 Zeman, M., 241