STURM-LIOUVILLE OPERATORS AND APPLICATIONS

REVISED EDITION

VLADIMIR A. MARCHENKO

AMS CHELSEA PUBLISHINGAmerican Mathematical Society • Providence, Rhode Island

Sturm-LiouviLLe operatorS and appLicationS

reviSed edition

Sturm-LiouviLLe operatorS and appLicationS

reviSed edition

vLadimir a. marchenko

AMS CHELSEA PUBLISHINGAmerican Mathematical Society • Providence, Rhode Island

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FOUNDED 1888

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2000 Mathematics Subject Classification. Primary 34A55, 34B24, 35Q51, 47E05, 47J35.

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Library of Congress Cataloging-in-Publication Data

Marchenko, V. A. (Vladimir Aleksandrovich), 1922–Sturm-Liouville operators and applications / Vladimir A. Marchenko. — Rev. ed.

p. cm.Rev. ed. of: Sturm-Liouville operators and applications. 1986.Includes bibliographical references.ISBN 978-0-8218-5316-0 (alk. paper)1. Spectral theory (Mathematics) 2. Transformations (Mathematics) 3. Operator theory.

I. Title.

QA320.M286 2011515′.7222—dc22

2010051019

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CONTENTS

PREFACE TO THE REVISED EDITION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Chapter 1 THE STURM-LIOUVILLE EQUATION ANDTRANSFORMATION OPERATORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. Riemann’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Transformation Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73. The Sturm-Liouville Boundary Value Problem on a Bounded

Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264. Asymptotic Formulas for Solutions of the Sturm-Liouville

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505. Asymptotic Formulas for Eigenvalues and Trace Formulas . . . . . . . . . . . . . .67

Chapter 2 THE STURM-LIOUVILLE BOUNDARY VALUE PROBLEM ONTHE HALF LINE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101

1. Some Information on Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1012. Distribution-Valued Spectral Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1173. The Inverse Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344. The Asymptotic Formula for the Spectral Functions of Symmetric

Boundary Value Problems and the Equiconvergence Theorem . . . . . . . . . 153

Chapter 3 THE BOUNDARY VALUE PROBLEM OF SCATTERING THEORY . . 1731. Auxiliary Propositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1732. The Parseval Equality and the Fundamental Equation . . . . . . . . . . . . . . . . 2003. The Inverse Problem of Quantum Scattering Theory . . . . . . . . . . . . . . . . . 2164. Inverse Sturm-Liouville Problems on a Bounded Interval . . . . . . . . . . . . . . 2405. The Inverse Problem of Scattering Theory on the Full Line . . . . . . . . . . . 284

Chapter 4 NONLINEAR EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3071. Transformation Operators of a Special Form . . . . . . . . . . . . . . . . . . . . . . . . . 3072. Rapidly Decreasing Solutions of the Korteweg-de Vries Equation . . . . . . 3223. Periodic Solutions of the Korteweg-de Vries Equation . . . . . . . . . . . . . . . . 3324. Explicit Formulas for Periodic Solutions of the Korteweg-de Vries

Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

Chapter 5 STABILITY OF INVERSE PROBLEMS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3631. Problem Formulation and Derivation of Main Formulas . . . . . . . . . . . . . . . 3632. Stability of the Inverse Scattering Problem. . . . . . . . . . . . . . . . . . . . . . . . . . .370

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3. Error Estimate for the Reconstruction of a Boundary Value Problemfrom its Spectral Function Given on the Set (−∞, N2) Only. . . . . . . . . .380

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Preface to the revised edition

In the first edition of this book the main attention was focused on the methodsof solving the inverse problem of spectral analysis and on the conditions (necessaryand sufficient) which the spectral data must satisfy in order to make it possibleto reconstruct the potential of the corresponding Sturm-Liouville operator. Theseconditions imply that the spectral data (e.g. spectral function or scattering data)must be known for all values of spectral parameter which belong to the spectrumof the operator.

But from the physical meaning of the inverse problem it is obvious that thevalues of spectral data on the whole spectrum are impossible to obtain from anyobservations. For example, in the inverse problem of quantum scattering theorythe energy of the particles acts as the spectral parameter, and in order to find thevalues of scattering data on the whole spectrum one has to conduct an experimentwith the particles of infinitely large energy. But for big enough values of energy thescattering process is not any more described by Schrodinger equation with potentialq(x). Therefore, even allowing, ideally, the possibility to experiment with particlesof arbitrarily large energies, we would obtain, starting from a certain energy, datarelevant to process, which has certainly nothing to do with the equation that wewant to reconstruct. Hence, a principal question is as follows: what informationabout the potential q(x) can be obtained, if the spectral function or scattering dataare known (generally speaking, approximately) only on a finite interval of values ofthe spectral parameter?

The new Chapter 5, devoted to solving this problem, was added to this edition.The convenient formulae are obtained, which allow to estimate the precision withwhich the eigenfunctions and potentials of Schrodinger operator can be restoredwhen the scattering data or spectral function are known only on a finite interval ofvalues of spectral parameter.

V. Marchenko

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