M. V. Pedoryuk Asymptotic Analysis Mikhail V

M. V. Pedoryuk Asymptotic Analysis Mikhail V. Fedoryuk Asymptotic Analysis

Linear Ordinary Differential Equations

Translated from the Russian by Andrew Rodick

With 26 Figures

Springer-Verlag Berlin Heidelberg GmbH Mikhail V. Fedoryuk t

Title of the Russian edition: Asimptoticheskie metody dlya linejnykh obyknovennykh differentsial 'nykh uravnenij Publisher Nauka, Moscow 1983

Mathematics Subject Classification (1991): 34Exx

ISBN 978-3-642-63435-2

Library of Congress Cataloging-in-Publication Data. Fedoriuk, Mikhail Vasil'evich. [Asimptoticheskie metody dlia lineinykh obyknovennykh differentsial' nykh uravnenil. English] Asymptotic analysis : linear ordinary differential equations / Mikhail V. Fedoryuk; translated from the Russian by Andrew Rodick. p. cm. Translation of: Asimptoticheskie metody dlia lineinykh obyknovennykh differentsial' nykh uravnenii. Includes bibliographical references and index. ISBN 978-3-642-63435-2 ISBN 978-3-642-58016-1 (eBook) DOI 10.1007/978-3-642-58016-1

1. Differential equations-Asymptotic theory. 1. Title. QA371.F3413 1993 515'.352-dc20 92-5200 This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concemed, specifically the rights of translation. reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplica• tion of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current vers ion, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1993 Originally published by Springer-Verlag Berlin Heidelberg New York in 1993 Softcover reprint of the hardcover 1st edition 1993 Typesetting: Springer TEX in-house system 41/3140 - 5 4 3 210 - Printed on acid-free paper Contents

Chapter 1. The Analytic Theory of Differential Equations .. 1 § 1. Analyticity of the Solutions of a System of Ordinary Differential Equations ...... 1 § 2. Regular Singular Points ...... 5 § 3. Irregular Singular Points ...... 16

Chapter 2. Second-Order Equations on the Real Line ...... 24 § 1. Transformations of Second-Order Equations ...... 24 § 2. WKB-Bounds ...... 28 § 3. Asymptotic Behaviour of Solutions of a Second-Order Equation for Large Values of the Parameter ...... 31 § 4. Systems of Two Equations Containing a Large Parameter. 42 § 5. Systems of Equations Close to Diagonal Form ...... 45 § 6. Asymptotic Behaviour of the Solutions for Large Values of the Argument ...... 50 § 7. Dual Asymptotic Behaviour ...... 57 § 8. Counterexamples ...... 63 § 9. Roots of Constant Multiplicity ...... 66 § 10. Problems on Eigenvalues ...... 68 § 11. A Problem on Scattering ...... 72

Chapter 3. Second-Order Equations in the Complex Plane .. 79 § 1. Stokes Lines and the Domains Bounded by them ...... 79 § 2. WKB-Bounds in the Complex Plane ...... 87 § 3. Equations with Polynomial Coefficients. Asymptotic Behaviour of a Solution in the Large ...... 91 § 4. Equations with Entire or Meromorphic Coefficients 108 § 5. Asymptotic Behaviour of the Eigenvalues of the Operator -d2/dx2 + A2q(x). Self-Adjoint Problems ...... 112 § 6. Asymptotic Behaviour of the Discrete Spectrum of the Operator -y" + A2q(x)y. Non-Self-Adjoint Problems ..... 126 § 7. The Eigenvalue Problem with Regular Singular Points .... 133 VI Contents

§ 8. Quasiclassical Approximation in Scattering Problems 141 § 9. Sturm-Liouville Equations with Periodic Potential ...... 161

Chapter 4. Second-Order Equations with Turning Points 168 § 1. Simple Turning Points. The Real Case ...... 168 § 2. A Simple Turning Point. The Complex Case ...... 182 § 3. Some Standard Equations ...... 188 § 4. Multiple and Fractional Turning Points ...... 191 § 5. The Fusion of a Turning Point and Regular Singular Point ...... 204 § 6. Multiple Turning Points. The Complex Case ...... 207 § 7. Two Close Turning Points ...... 211 § 8. Fusion of Several Turning Points ...... 217

Chapter 5. nth-Order Equations and Systems ...... 227 § 1. Equations and Systems on a Finite Interval ...... 227 § 2. Systems of Equations on a Finite Interval ...... 239 § 3. Equations on an Infinite Interval ...... 250 § 4. Systems of Equations on an Infinite Interval ...... 268 § 5. Equations and Systems in the Complex Plane ...... 288 § 6. Turning Points ...... 298 § 7. A Problem on Scattering, Adiabatic Invariants and a Problem on Eigenvalues ...... 332 § 8. Examples ...... 340

References ...... 352 Subject Index 361 Introduction

In this book we present the main results on the asymptotic theory of ordinary linear differential equations and systems where there is a small parameter in the higher derivatives. We are concerned with the behaviour of solutions with respect to the parameter and for large values of the independent variable. The literature on this question is considerable and widely dispersed, but the methods of proofs are sufficiently similar for this material to be put together as a reference book. We have restricted ourselves to homogeneous equations. The asymptotic behaviour of an inhomogeneous equation can be obtained from the asymptotic behaviour of the corresponding fundamental system of solutions by applying methods for deriving asymptotic bounds on the relevant integrals. We systematically use the concept of an asymptotic expansion, details of which can if necessary be found in [Wasow 2, Olver 6]. By the "formal asymptotic solution" (F.A.S.) is understood a function which satisfies the equation to some degree of accuracy. Although this concept is not precisely defined, its meaning is always clear from the context. We also note that the term "Stokes line" used in the book is equivalent to the term "anti-Stokes line" employed in the physics literature. In Chapter 1 we study briefly the basic facts of the analytic theory of differential equations. In § 2, paragraph 4, and § 3, paragraph 3, we give results obtained in recent years on the moving of the boundary condition from a singular point of the equation to a non-singular one. In Chapter 2 we consider second-order equations on a finite interval and on the half line. We give the asymptotic formulae for the solutions of equa• tions with a small parameter in the highest derivative for the case where the equation has no turning points. We give the asymptotic formulae for large values of the independent variable, and also formulae that are applicable for both small values of the parameter and also large values of the inde• pendent variable (dual asymptotic behaviour). In § 5 analogous results are given for systems of equations of arbitrary order that are close to diagonal form. In § 8 we give examples which show that the existence of formal asymp• totic behaviour does not always imply the existence of solutions having such asymptotic behaviour. VIII Introduction

In Chapter 3 we consider second-order equations in the complex plane which have a large parameter. This part of the asymptotic theory is poorly represented in the literature. Asymptotic formulae for solutions are given in domains not containing turning points or small neighbourhoods of them. We describe the maximal domains for which the asymptotic formulae for the solutions are valid, and we give asymptotic formulae for transition matrices, which allows us to construct the asymptotic behaviour of solutions in the large. A series of applications is considered: the asymptotic behaviour of eigenvalues of equations with analytic coefficients (including non-selfadjoint ones or those having regular singular points), the asymptotic behaviour of the scattering matrix in quasi classical approximation, and the asymptotic behaviour of the width of the gaps in the spectrum of the Sturm-Liouville operator with periodic potential. In Chapter 4 we give the asymptotic formulae for the solutions of second• order equations in a real or complex neighbourhood of a turning point. We consider the cases where there is the fusion of turning points or of turning points and singular points of the equation. In Chapter 5 we give results of the same type as those in Chapters 2 to 4, but for equations and systems of order greater than two. In § 6 we formulate results obtained with the help of the Maslov canonical operator. In § 8 we consider the scattering problem for the Stueckelberg system. In this reference book we do not discuss results relating to the Orr• Sommerfeld equation, nor the method of averaging for equations with rapidly oscillating coefficients. Despite this limitation, we hope that this book will be useful in mathematics, mechanics and physics when the asymptotic methods of the theory of ordinary linear differential equations are needed.

M. V. Fedoryuk

Acknowledgements

The translator would like to thank Dr. R.K. Thomas for mathematical advice, Mr. David Woolley for his technical assistance, and Ms. Terri Moss for her secretarial skills.

A. Rodick

Springer-Verlag would like to thank Professor Boris Vainberg and Professor Michael Eastham for their valuable help with the publication of this book.