Inverse Problems: Mathematical and Analytical Techniques with Applications to Engineering

INVERSE PROBLEMS

MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING

MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING Alan Jeffrey, Consulting Editor

Published: Inverse Problems A. G. Ramm Singular Perturbation Theory R. S. Johnson

Forthcoming: Methods for Constructing Exact Solutions of Partial Differential Equations with Applications S. V. Meleshko The Fast Solution of Boundary Integral Equations S. Rjasanow and O. Steinbach Stochastic Differential Equations with Applications R. Situ

INVERSE PROBLEMS

MATHEMATICAL AND ANALYTICAL TECHNIQUES WITH APPLICATIONS TO ENGINEERING

ALEXANDER G. RAMM

Springer

eBook ISBN: Print ISBN:

0-387-23218-4 0-387-23195-1

©2005 Springer Science + Business Media, Inc. Print ©2005 Springer Science + Business Media, Inc. Boston All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: and the Springer Global Website Online at:

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To Luba and Olga

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CONTENTS

Foreword Preface

xv

xvii

1. Introduction

1

1.1 Why are inverse problems interesting and practically important?

1

1.2 Examples of inverse problems 2 1.2.1 Inverse problems of potential theory 2 1.2.2 Inverse spectral problems 2 1.2.3 Inverse scattering problems in quantum physics; finding the potential from the impedance function 2 1.2.4 Inverse problems of interest in geophysics 3 1.2.5 Inverse problems for the heat and wave equations 3 1.2.6 Inverse obstacle scattering 4 1.2.7 Finding small subsurface inhomogeneities from the measurements of the scattered field on the surface 5 1.2.8 Inverse problem of radiomeasurements 5 1.2.9 Impedance tomography (inverse conductivity) problem 5 1.2.10 Tomography and other integral geometry problems 5 1.2.11 Inverse problems with “incomplete data” 6 1.2.12 The Pompeiu problem, Schiffer’s conjecture, and inverse problem of plasma theory 7 1.2.13 Multidimensional inverse potential scattering 8 1.2.14 Ground-penetrating radar 8 1.2.15 A geometrical inverse problem 9 1.2.16 Inverse source problems 10

viii

Contents

Identification problems for integral-differential equations 12 Inverse problem for an abstract evolution equation 12 Inverse gravimetry problem 12 Phase retrieval problem (PRP) 12 Non-overdetermined inverse problems 12 Image processing, deconvolution 13 Inverse problem of electrodynamics, recovery of layered medium from the surface scattering data 13 1.2.24 Finding ODE from a trajectory 13 1.2.17 1.2.18 1.2.19 1.2.20 1.2.21 1.2.22 1.2.23

1.3 Ill-posed problems

14

1.4 Examples of Ill-posed problems 15 1.4.1 Stable numerical differentiation of noisy data 15 1.4.2 Stable summation of the Fourier series and integrals with randomly perturbed coefficients 15 1.4.3 Solving ill-conditioned linear algebraic systems 15 1.4.4 Fredholm and Volterra integral equations of the first kind 16 1.4.5 Deconvolution problems 16 1.4.6 Minimization problems 16 1.4.7 The Cauchy problem for Laplace’s equation 16 1.4.8 The backwards heat equation 17 2. Methods of solving ill-posed problems

19

2.1 Variational regularization 19 2.1.1 Pseudoinverse. Singular values decomposition 19 2.1.2 Variational (Phillips-Tikhonov) regularization 20 2.1.3 Discrepancy principle 22 2.1.4 Nonlinear ill-posed problems 23 2.1.5 Regularization of nonlinear, possibly unbounded, operator 24 2.1.6 Regularization based on spectral theory 25 2.1.7 On the notion of ill-posedness for nonlinear equations 26 2.1.8 Discrepancy principle for nonlinear ill-posed problems with monotone operators 26 2.1.9 Regularizers for Ill-posed problems must depend on the noise level 29 2.2 Quasisolutions, quasinversion, and Backus-Gilbert method 30 2.2.1 Quasisolutions for continuous operator 30 2.2.2 Quasisolution for unbounded operators 31 2.2.3 Quasiinversion 32 2.2.4 A Backus-Gilbert-type method: Recovery of signals from discrete and noisy data 32 2.3 Iterative methods

40

2.4 Dynamical system method (DSM) 41 2.4.1 The idea of the DSM 41 2.4.2 DSM for well-posed problems 42 2.4.3 Linear ill-posed problems 45 2.4.4 Nonlinear ill-posed problems with monotone operators 49 2.4.5 Nonlinear ill-posed problems with non-monotone operators 57 2.4.6 Nonlinear ill-posed problems: avoiding inverting of operators in the Newton-type continuous schemes 59 2.4.7 Iterative schemes 62

ix

2.4.8 A spectral assumption 64 2.4.9 Nonlinear integral inequality 2.4.10 Riccati equation 70

65

2.5 Examples of solutions of ill-posed problems 71 2.5.1 Stable numerical differentiation: when is it possible? 71 2.5.2 Stable summation of the Fourier series and integrals with perturbed coefficients 85 2.5.3 Stable solution of some Volterra equations of the first kind 2.5.4 Deconvolution problems 87 2.5.5 Ill-conditioned linear algebraic systems 88 2.6 Projection methods for ill-posed problems

87

89

3. One-dimensional inverse scattering and spectral problems

91

3.1 Introduction 92 3.1.1 What is new in this chapter? 92 3.1.2 Auxiliary results 92 3.1.3 Statement of the inverse scattering and inverse spectral problems 97 3.1.4 Property C for ODE 98 3.1.5 A brief description of the basic results 99 3.2 Property C for ODE 104 3.2.1 Property 104 3.2.2 Properties and

105

3.3 Inverse problem with I-function as the data 108 3.3.1 Uniqueness theorem 108 3.3.2 Characterization of the I-functions 110 3.3.3 Inversion procedures 112 3.3.4 Properties of 112 3.4 Inverse spectral problem 122 3.4.1 Auxiliary results 122 3.4.2 Uniqueness theorem 124 3.4.3 Reconstruction procedure 126 3.4.4 Invertibility of the reconstruction steps 128 3.4.5 Characterization of the class of spectral functions of the Sturm-Liouville operators 130 3.4.6 Relation to the inverse scattering problem 130 3.5 Inverse scattering on half-line 132 3.5.1 Auxiliary material 132 3.5.2 Statement of the inverse scattering problem on the half-line. Uniqueness theorem 137 3.5.3 Reconstruction procedure 139 3.5.4 Invertibility of the steps of the reconstruction procedure 143 3.5.5 Characterization of the scattering data 145 3.5.6 A new Marchenko-type equation 147 3.5.7 Inequalities for the transformation operators and applications 148 3.6 Inverse scattering problem with fixed-energy phase shifts as the data 3.6.1 Introduction 156 3.6.2 Existence and uniqueness of the transformation operators independent of angular momentum 157

156

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Contents

3.6.3 Uniqueness theorem 165 3.6.4 Why is the Newton-Sabatier (NS) procedure fundamentally wrong? 166 3.6.5 Formula for the radius of the support of the potential in terms of scattering data 172 3.7 Inverse scattering with “incomplete data” 176 3.7.1 Uniqueness results 176 3.7.2 Uniqueness results: compactly supported potentials 180 3.7.3 Inverse scattering on the full line by a potential vanishing on a half-line 181 3.8 Recovery of quarkonium systems 181 3.8.1 Statement of the inverse problem 3.8.2 Proof 183 3.8.3 Reconstruction method 185

181

3.9 Krein’s method in inverse scattering 186 3.9.1 Introduction and description of the method 186 3.9.2 Proofs 192 3.9.3 Numerical aspects of the Krein inversion procedure 200 3.9.4 Discussion of the ISP when the bound states are present 201 3.9.5 Relation between Krein’s and GL’s methods 201

3.10 Inverse problems for the heat and wave equations 202 3.10.1 Inverse problem for the heat equation 202 3.10.2 What are the “correct” measurements? 203 3.10.3 Inverse problem for the wave equation 204 3.11 Inverse problem for an inhomogeneous Schrödinger equation 204 3.12 An inverse problem of ocean acoustics 208 3.12.1 The problem 208 3.12.2 Introduction 209 3.12.3 Proofs: uniqueness theorem and inversion algorithm 212 3.13 Theory of ground-penetrating radars 216 3.13.1 Introduction 216 3.13.2 Derivation of the basic equations 217 3.13.3 Basic analytical results 219 3.13.4 Numerical results 221 3.13.5 The case of a source which is a loop of current 222 3.13.6 Basic analytical results 225 4. Inverse obstacle scattering 227 4.1 Statement of the problem 227 4.2 Inverse obstacle scattering problems 234 4.3 Stability estimates for the solution to IOSP 240 4.4 High-frequency asymptotics of the scattering amplitude and inverse scattering problem 243 4.5 Remarks about numerical methods for finding S from the scattering data 245 4.6 Analysis of a method for identification of obstacles 247

xi

5. Stability of the solutions to 3D Inverse scattering problems with fixed-energy data 255 5.1 Introduction 255 5.1.1 The direct potential scattering problem 5.1.2 Review of the known results 256

256

5.2 Inverse potential scattering problem with fixed-energy data 264 5.2.1 Uniqueness theorem 264 5.2.2 Reconstruction formula for exact data 264 5.2.3 Stability estimate for inversion of the exact data 267 5.2.4 Stability estimate for inversion of noisy data 270 5.2.5 Stability estimate for the scattering solutions 273 5.2.6 Spherically symmetric potentials 274 5.3 Inverse geophysical scattering with fixed-frequency data 275 5.4 Proofs 5.4.1 5.4.2 5.4.3 5.4.4 5.4.5 5.4.6 5.4.7 5.4.8

of some estimates 277 Proof of (5.1.18) 277 Proof of (5.1.20) and (5.1.21) Proof of (5.2.17) 283 Proof of (5.4.49) 285 Proof of (5.4.51) 286 Proof of (5.2.13) 287 Proof of (5.2.23) 289 Proof of (5.1.30) 292

278

5.5 Construction of the Dirichlet-to-Neumann map from the scattering data and vice versa 293 5.6 Property C

298

5.7 Necessary and sufficient condition for scatterers to be spherically symmetric 300 5.8 The Born inversion

307

5.9 Uniqueness theorems for inverse spectral problems 6. Non-uniqueness and uniqueness results

312

317

6.1 Examples of nonuniqueness for an inverse problem of geophysics 6.1.1 Statement of the problem 317 6.1.2 Example of nonuniqueness of the solution to IP 318

317

6.2 A uniqueness theorem for inverse boundary value problems for parabolic equations 319 6.3 Property C and an inverse problem for a hyperbolic equation 6.3.1 Introduction 321 6.3.2 Statement of the result. Proofs 321 6.4 Continuation of the data

321

330

7. Inverse problems of potential theory and other inverse source problems 7.1 Inverse problem of potential theory 7.2 Antenna synthesis problems

333

336

7.3 Inverse source problem for hyperbolic equations

337

333

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Contents

8. Non-overdetermined inverse problems 8.1 Introduction

339

8.2 Assumptions

340

8.3 The problem and the result 8.4 Finding 8.5 Appendix

from

339

340 342

347

9. Low-frequency inversion

349

9.1 Derivation of the basic equation. Uniqueness results 9.2 Analytical solution of the basic equation

353

9.3 Characterization of the low-frequency data 9.4 Problems of numerical implementation

349

355

355

9.5 Half-spaces with different properties 356 9.6 Inversion of the data given on a sphere 9.7 Inversion of the data given on a cylinder 9.8 Two-dimensional inverse problems 9.9 One-dimensional inversion

357 358

359

362

9.10 Inversion of the backscattering data and a problem of integral geometry 363 9.11 Inversion of the well-to-well data 9.12 Induction logging problems

364

366

9.13 Examples of non-uniqueness of the solution to an inverse problem of geophysics 369 9.14 Scattering in absorptive medium 9.15 A geometrical inverse problem

371 371

9.16 An inverse problem for a biharmonic equation

373

9.17 Inverse scattering when the background is variable 375 9.18 Remarks concerning the basic equation

377

10. Wave scattering by small bodies of arbitrary shapes

379

10.1 Wave scattering by small bodies 379 10.1.1 Introduction 379 10.1.2 Scalar wave scattering by a single body 380 10.1.3 Electromagnetic wave scattering by a single body 10.1.4 Many-body wave scattering 385

383

10.2 Equations for the self-consistent field in media consisting of many small particles 388 10.2.1 Introduction 388 10.2.2 Acoustic fields in random media 390 10.2.3 Electromagnetic waves in random media 394

xiii

10.3 Finding small subsurface inhomogeneities from scattering data 395 10.3.1 Introduction 396 10.3.2 Basic equations 397 10.3.3 Justification of the proposed method 398 10.4 Inverse problem of radiomeasurements 11. The Pompeiu problem

401

405

11.1 The Pompeiu problem 405 11.1.1 Introduction 405 11.1.2 Proofs 407 11.2 Necessary and sufficient condition for a domain, which fails to have Pompeiu property, to be a ball 414 11.2.1 Introduction 414 11.2.2 Proof 416 Bibliographical Notes References Index

441

425

421

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FOREWORD

The importance of mathematics in the study of problems arising from the real world, and the increasing success with which it has been used to model situations ranging from the purely deterministic to the stochastic, is well established. The purpose of the set of volumes to which the present one belongs is to make available authoritative, up to date, and self-contained accounts of some of the most important and useful of these analytical approaches and techniques. Each volume provides a detailed introduction to a specific subject area of current importance that is summarized below, and then goes beyond this by reviewing recent contributions, and so serving as a valuable reference source. The progress in applicable mathematics has been brought about by the extension and development of many important analytical approaches and techniques, in areas both old and new, frequently aided by the use of computers without which the solution of realistic problems would otherwise have been impossible. A case in point is the analytical technique of singular perturbation theory which has a long history. In recent years it has been used in many different ways, and its importance has been enhanced by it having been used in various fields to derive sequences of asymptotic approximations, each with a higher order of accuracy than its predecessor. These approximations have, in turn, provided a better understanding of the subject and stimulated the development of new methods for the numerical solution of the higher order approximations. A typical example of this type is to be found in the general study of nonlinear wave propagation phenomena as typified by the study of water waves.

xvi

Foreword

Elsewhere, as with the identification and emergence of the study of inverse problems, new analytical approaches have stimulated the development of numerical techniques for the solution of this major class of practical problems. Such work divides naturally into two parts, the first being the identification and formulation of inverse problems, the theory of ill-posed problems and the class of one-dimensional inverse problems, and the second being the study and theory of multidimensional inverse problems. On occasions the development of analytical results and their implementation by computer have proceeded in parallel, as with the development of the fast boundary element methods necessary for the numerical solution of partial differential equations in several dimensions. This work has been stimulated by the study of boundary integral equations, which in turn has involved the study of boundary elements, collocation methods, Galerkin methods, iterative methods and others, and then on to their implementation in the case of the Helmholtz equation, the Lamé equations, the Stokes equations, and various other equations of physical significance. A major development in the theory of partial differential equations has been the use of group theoretic methods when seeking solutions, and in the introduction of the comparatively new method of differential constraints. In addition to the useful contributions made by such studies to the understanding of the properties of solutions, and to the identification and construction of new analytical solutions for well established equations, the approach has also been of value when seeking numerical solutions. This is mainly because of the way in many special cases, as with similarity solutions, a group theoretic approach can enable the number of dimensions occurring in a physical problem to be reduced, thereby resulting in a significant simplification when seeking a numerical solution in several dimensions. Special analytical solutions found in this way are also of value when testing the accuracy and efficiency of new numerical schemes. A different area in which significant analytical advances have been achieved is in the field of stochastic differential equations. These equations are finding an increasing number of applications in physical problems involving random phenomena, and others that are only now beginning to emerge, as is happening with the current use of stochastic models in the financial world. The methods used in the study of stochastic differential equations differ somewhat from those employed in the applications mentioned so far, since they depend for their success on the Ito calculus, martingale theory and the Doob-Meyer decomposition theorem, the details of which are developed as necessary in the volume on stochastic differential equations. There are, of course, other topics in addition to those mentioned above that are of considerable practical importance, and which have experienced significant developments in recent years, but accounts of these must wait until later. Alan Jeffrey University of Newcastle Newcastle upon Tyne United Kingdom

PREFACE

This book can be used for courses at various levels in ill-posed problems and inverse problems. The bibliography of the subject is enormous. It is not possible to compile a complete bibliography and no attempt was made to do this. The bibliography contains some books where the reader will find additional references. The author has used extensively his earlier published papers, and referenced these, as well as the papers of other authors that were used or mentioned. Let us outline some of the novel features in this book. In Chapter 1 the statement of various inverse problems is given. In Chapter 2 the presentation of the theory of ill-posed problems is shorter and sometimes simpler than that published earlier, and quite a few new results are included. Regularization for ill-posed operator equations with unbounded nonlinear operators is studied. A novel version of the discrepancy principle is formulated for nonlinear operator equations. Convergence rate estimates are given for Backus-Gilbert-type methods. The DSM (Dynamical systems method) in ill-posed problems is presented in detail. The presentation is based on the author’s papers and the joint papers of the author and his students. These results appear for the first time in book form. Papers [R216], [R217], [R218], [R220], [ARS3], [AR1] have been used in this Chapter. In Chapter 3 the presentation of one-dimensional inverse problems is based mostly on the author’s papers, especially on [R221]. It contains many novel results, which are described at the beginning of the Chapter. The presentation of the classical results, for example, Gel’fand-Levitan’s theory, and Marchenko’s theory, contains many novel points. The presentation of M. G. Krein’s inversion theory with complete proofs is

xviii

Preface

given for the first time. The Newton-Sabatier inversion theory, which has been in the literature for more than 40 years, and was presented in two monographs [CS], [N], is analyzed and shown to be fundamentally wrong in the sense that its foundations are wrong (cf. [R206]). This Chapter is based on the papers [R221], [R199], [R197], [R196], [R195], [R192], [R185]. One of the first papers on inverse spectral problems was Ambartsumian’s paper (1929) [Am], where it was proved that one spectrum determines the one-dimensional Neumann Schrödinger’s operator uniquely. This result is an exceptional one: in general one spectrum does not determine the potential uniquely (see Section 3.7 and [PT]). Only 63 years later a multidimensional analog of Ambartsumian’s result was obtained ([RSt1]). The main technical tool in this Chapter and in Chapter 5 is Property C, that is, completeness of the set of products of solutions to homogeneous differential equations. For partial differential equations this tool has been introduced in [R87] and developed in many papers and in the monograph [R139]. For ordinary differential equations completeness of the products of solutions to homegeneous ordinary equations has been used in different forms in [B], [L1]. In our book Property C for ODE is presented in the form introduced and developed by the author in [R196]. In Chapter 4 the presentation of inverse obstacle scattering problems contains many novel points. The requirements on the smoothness of the boundary are minimal, stability estimates for the inversion procedure corresponding to fixed-frequency data are given, the high-frequency inversion formulas are discussed and the error of the inversion from noisy data is estimated. Analysis of the currently used numerical methods is given. This Chapter is based on [R83], [R155], [R162], [R164], [R167], [R167], [R171], [RSa]. In Chapter 5 a presentation of the solution of the 3D inverse potential scattering problem with fixed-energy noisy data is given. This Chapter is based on the series of the author’s papers, especially on the paper [R203]. The basic concept used in the analysis of the inverse scattering problem in Chapter 5 is the concept of Property C, i.e., completeness of the set of products of solutions to homogeneous partial differential equations. This concept was introduced by the author ([R87]) and applied to many inverse problems (see [R139] and references therein). An important part of the theory consists of obtaining stability estimates for the potential, reconstructed from fixedenergy noisy data (and from exact data). Error estimates for the Born inversion are given under suitable assumptions. It is shown that the Born inversion may fail while the Born approximation works well. In other words, the Born approximation may be applicable for solving the direct scattering problem, while the Born inversion, that is, inversion based on the Born approximation, may fail. The Born inversion is still popular in applications, therefore these error estimates will hopefully be useful for practitioners. The author’s inversion method for fixed-energy scattering data, is compared with that based on the usage of the Dirichlet-to-Neumann map. The author shows why the difficulties in numerical implementation of his method are less formidable than the difficulties in implementing the inversion method based on the Dirichlet-to-Neumann map.

xix

A necessary and sufficient condition for a scatterer to be spherically symmetric is given ([R128]). In Chapter 6 an example of non-uniqueness of the solution to a 3D problem of geophysics is given. It illustrates the crucial role of the uniqueness theorems in a study of inverse problems. One may try to solve numerically such a problem, by a parameterfitting, which is very popular among practitioners. But if the uniqueness result is not established, the numerical results may be meaningless. Some uniqueness theorems for inverse boundary value problem and for an inverse problem for hyperbolic equations are established in this Chapter. In Chapter 7, inverse problems of potential theory and antenna synthesis are briefly discussed. The presentation of the theory on this topic is not complete: there are books and many papers on antenna synthesis (e.g., [MJ], [ZK], [AVST], [R21], [R26]) including nonlinear problems of antenna synthesis [R23], [R27]). Chapter 8 contains a discussion of non-overdetermined problems. These are, roughly speaking, the inverse problems in which the unknown function depends on the same number of variables as the data function. Examples of such problems are given. Most of these problems are open: even uniqueness theorems are not available. Such a problem, namely, recovery of an unknown coefficient in a Schrödinger equation in a bounded domain from the knowledge of the values of the spectral function on the boundary is discussed under the assumption that all the eigenvalues are simple, that is, the corresponding eigenspaces are one-dimensional. The presentation follows [R198]. In Chapter 9 the theory of the inversion of low-frequency data is presented. This theory is based on the series of author’s papers, starting with [R68], [R77], and uses the presentation in [R83] and [R139]. Almost all of the results in this Chapter are from the above papers and books. Chapter 10 is a summary of the author’s results regarding the theory of wave scattering by small bodies of arbitrary shapes. These results have been obtained in a series of the author’s papers and are summarized in [R65], [R50]. The solution of inverse radiomeasurements problem ([R33], [R65]) is based on these results. Also, these results are used in the solution of the problem of finding small subsurface inhomogeneities from the scattering data, measured on the surface. The solution to this problem can be used in modeling ultrasound mammography, in finding small holes in metallic objects, and in many other applied problems. In Chapter 11 the classical Pompeiu problem is presented following the papers [R177], [R186]. The author thanks several publishers of his papers, mentioned above, for the permission to use these papers in the book. There are many questions that the author did not discuss in this book: inverse scattering for periodic potentials and other periodic objects, such as gratings, periodic objects, (see, e.g., [L] for one-dimensional scattering problems for periodic potentials), the Carleman estimates and their applications to inverse problems ([Bu2], [H], [LRS]), the inverse problems for elasticity and Maxwell’s equations ([RK], [Ya]), the methods based on controllability results ([Bel]), problems of tomography and integral geometry ([RKa], [R139]), etc. Numerous parameter-fitting schemes for solving various

xx

Preface

engineering problems are not discussed. There are many papers published, which use parameter-fitting for solving inverse problems. However, in most cases there are no error estimates for parameter-fitting schemes for solving inverse problems, and one cannot guarantee any accuracy of the inversion result. In [GRS] the concept of stability index is introduced and applied to a parameter-fitting scheme for solving a one-dimensional inverse scattering problem in quantum physics. This concept allows one to get some idea about the error estimate in a parameter-fitting scheme. The applications of inverse scattering to integration of nonlinear evolution equations are not discussed as there are many books on this topic (see e.g., [M], [FT] and references therein).

1. INTRODUCTION

1.1

WHY ARE INVERSE PROBLEMS INTERESTING AND PRACTICALLY IMPORTANT?

Inverse problems are the problems that consist of finding an unknown property of an object, or a medium, from the observation of a response of this object, or medium, to a probing signal. Thus, the theory of inverse problems yields a theoretical basis for remote sensing and non-destructive evaluation. For example, if an acoustic plane wave is scattered by an obstacle, and one observes the scattered field far from the obstacle, or in some exterior region, then the inverse problem is to find the shape and material properties of the obstacle. Such problems are important in identification of flying objects (airplanes missiles, etc.), objects immersed in water (submarines, paces of fish, etc.), and in many other situations. In geophysics one sends an acoustic wave from the surface of the earth and collects the scattered field on the surface for various positions of the source of the field for a fixed frequency, or for several frequencies. The inverse problem is to find the subsurface inhomogeneities. In technology one measures the eigenfrequencies of a piece of a material, and the inverse problem is to find a defect in this material, for example, a hole in a metal. In geophysics the inhomogeneity can be an oil deposit, a cave, a mine. In medicine it may be a tumor, or some abnormality in a human body. If one is able to find inhomogeneities in a medium by processing the scattered field on the surface, then one does not have to drill a hole in a medium. This, in turn, avoids expensive and destructive evaluation. The practical advantages of remote sensing are what makes the inverse problems important.

2

1. Introduction

1.2

EXAMPLES OF INVERSE PROBLEMS

1.2.1

Inverse problems of potential theory

Suppose a body

with a density

generates gravitational potential

Is it possible to find given the potential u (x) for far away from D? A point mass m and a uniformly distributed mass m in a ball of radius a produce the same potential Thus, it is not possible to find uniquely from the knowledge of u in However, if one knows a priori that in D, then it is possible to find D from the knowledge of u (x) in provided that D is, for example, star-shaped, that is, every ray issued from some interior point intersects the boundary of D at only one point. 1.2.2

Inverse spectral problems

Let be the Sturm-Liouville operator defined by the Dirichlet boundary conditions as self-adjoint operator in and be its eigenvalues. To what extent does the set of these eigenvalues determine q (x )? Roughly speaking, one spectrum, that is, the set determines “half of q (x),” in the sense that if q (x) is known on then one spectrum determines uniquely q (x) on A classical result due to Borg [B]and Marchenko [M] says that two spectra uniquely determine the operator i.e., the potential q and the boundary conditions at x = 0 and x = 1 of the type and where and are constants, and one assumes that the two spectra correspond to the same and two distinct The author (see [R196]) asked the following question: if q (x) is known on the segment [b, 1], 0 < b < 1, then what part of the spectrum one needs to know in order to uniquely recover q (x) on [0, b]? It is assumed that q is real valued: and Let be the spectral measure of the self-adjoint operator l. This notion is defined in Chapter 3. The inverse spectral problem is: given find q (x), and the boundary conditions, characterizing Similar problems can be formulated in the multidimensional cases, when a bounded domain D plays the role of the segment [0, 1], the role of the spectral data is played by the eigenvalues and the values on the boundary S of D of the normal derivatives of the normalized eigenfunctions, One may choose other spectral data. 1.2.3

Inverse scattering problems in quantum physics; finding the potential from the impedance function

Consider

the

by function), and by

Dirichlet operator in the Jost function, by

Denote by the Jost solution, the I-function (impedance the scattering data (see Chapter 3).

3

The inverse problem of quantum scattering on the half-axis consists of finding q (x), given It was studied in [M]. The inverse problem of finding q (x), given is of interest in many applications. The I-function has the physical meaning of the impedance function, it is the ratio in the problem of electromagnetic wave falling perpendicularly onto the earth, when the dielectric permittivity and conductivity of the earth depend on the vertical coordinate only. One can prove that coincides with the Weyl function (Chapter 3). It turns out that known determines uniquely q (x), and one can explicitly calculate and given ([R196]). 1.2.4

Inverse problems of interest in geophysics

There are many inverse problems of interest in geophysics. A typical one consists of finding an unknown inhomogeneity in the velocity profile (refraction coefficient) from the scattered acoustic field measured on the surface of the earth and generated by a point source, situated on the surface of the earth at varying positions. Its mathematical formulation (in a simplified form) is:

where

is the known background refraction coefficient, where supp v(x) is the support of v, and v is an inhomogeneity in the refraction coefficient (or in the velocity profile), u is the acoustic pressure, u satisfies the radiation condition at infinity (or the limiting absorption principle). An inverse problem of geophysics consists of finding the function v from the knowledge of the scattered field on the surface of the Earth, that is from known for all at a fixed or for all where is a small number (the case of low-frequency surface data) (cf [LRS], [Ro], [R83], [R139]). Another problem is to find the conductivity of the medium from the measurements of the electromagnetic waves, scattered by a source that moves in a borehole along the vertical line. ([R83], [R139]) 1.2.5

Inverse problems for the heat and wave equations

A typical inverse problem for the heat equation

is to find q (x) from the flux measurements: The extra data (measured data), allow one to find q ( x ) . Another inverse problem is to find the unknown conductivity from boundary measurements. For example, let Can one find given a(t) and

4

1. Introduction

Consider the inverse conductivity problem: let in on where N is the outer unit normal to S, the extra data is the flux g at the boundary. Suppose that the set is known. Can one determine uniquely? Here D is a bounded domain with a sufficiently smooth boundary S, and is the Sobolev space. In applications in medicine, f is the electrostatic potential, which can be applied to a human chest, and g is the flux of the electrostatic field, which can be measured. If one can determine from these measurements then some diagnostic information is obtained ([R157], [R139], [R131], [R103], [Gro], [LRS], [Ro], [Is1]). There are many inverse problem for the wave equation. One of them is to find the velocity c (x) in the equation at t = 0, u = 0 on S, given the extra data on S for a fixed and all t > 0, or for varying on S, and where T > 0 is some number. ([LRS], [Ro], [RRa], [RSj]). 1.2.6

Inverse obstacle scattering

Let be bounded domain with a Lipschitz boundary be the exterior domain, be the unit sphere in The scattering problem consists of finding the scattering solution, i.e., the solution to the problem

The coefficient A is called the scattering amplitude. Existence and uniqueness of the solution to (1.2.2)–(1.2.3) are proved in [RSa] without any assumption on the smoothness of the boundary. If the Neumann boundary condition

is used in place of (1.2.3), then the existence and uniqueness of the solution to (1.2.2)– (1.2.3N) are proved in [RSa] under the assumption of compactness of the embedding where R > 0 is such that and is the Sobolev space. See also [GoR]. The inverse obstacle scattering problem consists of finding S and the boundary condition on S, given in the following cases: (1) either at a fixed for all and all or, (2) at a fixed for all and running through open subsets of (3) for fixed and and all

or,

Uniqueness of the solution of the first inverse problem is proved by M. Schiffer (1964), (see [R83]) of the second by A. G. Ramm (1986) (see [R83]), and the third

5

problem is still open. See also [R154], [R155], [R162], [R159], [R164], [R167], [R171], [CK]. One may consider a penetrable layered obstacle, and ask if the scattering amplitude at a fixed allows one to determine the boundaries of all the layers uniquely, and the constant velocity profiles in each of the layers. See [RPY] for an answer to this question. 1.2.7

Finding small subsurface inhomogeneities from the measurements of the scattered field on the surface

Suppose there are few small, in comparison with the wave-length, holes in the metallic body. A source of acoustic waves is on the surface of the body, and the scattered field is measured on the surface of the body for various positions of the acoustic source, at a fixed frequency. The inverse problem is to find the number of the small holes, their locations, and their volume. A similar problem is important in medicine, where the small bodies are the cancer cells to be found in the healthy tissue of a human’s body. In the ultrasound mammography modeling, one deals with the tissue of a woman’s breast ([R193], [GR1]). 1.2.8

Inverse problem of radiomeasurements

Suppose a complicated electromagnetic field distribution (E, H) exists in the aperture of a mirror antenna. For many practical reasons one wants to know this distribution. Let be the field scattered by a small probe placed at a point x in the aperture of the antenna. Given the shape and electromagnetic constants and of the probe, the inverse problem of radiomeasurements consists of finding (E(x), H(x)) from the knowledge of (See [R65] for a solution to this problem). 1.2.9

Impedance tomography (inverse conductivity) problem

This problem was briefly mentioned in Section 1.2.5. 1.2.10

Tomography and other integral geometry problems

Define

where is the unit sphere in The function is called the Radon transform of f. The function f can be assumed piecewise-continuous and absolutely integrable over every plane so that the Radon transform would be well defined in the classical sense. But in fact, one can define the Radon transform for much larger sets of functions and on distributions [R170], [RKa], [Hel]. Given one can uniquely recover f (x) provided, for example, that or where is the weighted space with the norm Practically interesting questions are: (a) How are singularities of f and related? (b) Given the noisy measurements of f at a grid, how does one find the discontinuities of f ?

6

1. Introduction

A grid is a set of points

where The noisy measurement are where are identically distributed, independent random variables with zero mean value and a finite variance See [R176], [RKa] for a detailed investigation of the above problem. An open problem is: what are the minimal assumptions on the growth of f (x) at infinity that guarantee the injectivity of the Radon transform? There is an example of a smooth function such that for all p, and [RKa]. In many applications one integrates f not over the planes but over some other family of manifolds. The problem of integral geometry is to recover f from the knowledge of its integral over a family of manifolds. For example, if the family of manifolds is a family of spheres of various radii r > 0 and centers s running over some surface S, then are the spherical means of f, and the problem is to recover f from the knowledge of and Conditions on S that guarantee the injectivity of the operator M are given in [R211], where some inversion formulas are also derived. 1.2.11

Inverse problems with “incomplete data”

Suppose that not all the scattering data in Section 1.2.3 are given, for example, is given, but and J are unknown. In general, one cannot recover a from these “incomplete” data. However, if one knows a priori that q (x) has compact support, or then the data alone determine q (x) uniquely. Such type of inverse problems we call inverse problems with “incomplete” data. The “incompleteness” of the data is remedied by the additional a priori assumption about q (x), so, in fact, the data are complete in the sense that q (x) is determined uniquely by these data. Another example of an inverse problem with “incomplete” data, is recovery of q (x) = 0 for x < 0, from the knowledge of the reflection coefficient in the full-axis (full-line) scattering problem:

The coefficients and are reflection and transmission coefficients, respectively. A general cannot be uniquely recovered from the knowledge of alone: one needs to know additionally the bound states and norming constants to recover q uniquely. However if one knows a priori that q (x) = 0 for for example, for x < 0, then q (x) is uniquely determined by alone.

7

1.2.12 The Pompeiu problem, Schiffer’s conjecture, and inverse problem of plasma theory

Let

Assume

where SO(n) is the group of rotations, and is a bounded domain. The problem (going back to Pompeiu (1929)) is to prove that (1.2.4) implies that D is a ball. Originally Pompeiu claimed that (1.2.4) implies that f = 0, but this claim is wrong. References related to this problem are given in [R186], [R177], [Z]. One can prove that (1.2.4) holds iff (= if and only if) for all and some where some

and

Iff

and

then the overdetermined problem

has a solution. The Schiffer’s conjecture is: if D is a bounded connected domain homeomorphic to a ball, and

then D is a ball. The Pompeiu problem in the form (1.2.4) is equivalent to the following conjecture: if

and D is homeomorphic to a ball, then D is a ball. An inverse problem of plasma theory consists of the following. Let

where u is a non trivial solution to (1.2.8), (i.e., if f (0) = 0 then and let the extra data (measured data) be the value Assume that f (u) is an entire function of u. The inverse problem, of interest in plasma theory, is: given can one recover f (u) uniquely. Even for j = 0, 1, the problem is open (cf [Vog]).

8

1. Introduction

1.2.13 Multidimensional inverse potential scattering

Let

where is given, and The coefficient is called the scattering amplitude, and the solution to (1.2.9)–(1.2.10) is called the scattering solution. The direct scattering problem is: given q, and find and, in particular, This problem has been studied in great detail (see, e.g., [CFKS], [[R121], Appendix]) under various assumptions on q ( x ) . We assume that for and often we assume additionally that The inverse scattering problem (ISP) consists of finding q ( x ) , given Consider several cases: (1) A is given for all and all (2) A is given for all and a fixed (3) A is given for a fixed all and all (4) is given for all and all

(back scattering data).

In case (1) uniqueness of the solution to ISP has been established long ago, and follows easily from the asymptotics of A as An inversion formula based on highenergy asymptotics of A is known (Born inversion), (cf [Sai]). In case (2) the uniqueness of the solution to ISP is proved by Ramm [R109], [R100], (see also [R105], [R112], [R114], [R115], [R120], [R125], [R130], [R133], [R140], [R142], [R143], [RSt2], [R203]), an inversion formula for the exact data is derived in [R109], [R143], an inversion formula for the noisy data is derived by in [R143], and stability estimates for the inversion formulas for the exact and noisy data are derived by in [R143], [R203]. In case (3) and (4) uniqueness of the solution to ISP is an open problem, but in the case (4) uniqueness holds if one assumes a priori that q is sufficiently small. A generic uniqueness result is given in [St1]. See also [StU]. 1.2.14

Ground-penetrating radar

Let the source of electromagnetic waves be located above the ground, and the scattered field be observed on the ground. From these data one wants to get information about the properties of the ground. Mathematical modeling of this problem is based on the Maxwell equations

where is the region of the ground,

is the vertical coordinate, is the source,

f (t)

9

describes the shape of the pulse of the current j along a wire going along the axis at the height above the ground. Assume for (in the air), for f (t) = 0 for t < 0 and t > T, and are dielectric and magnetic constants, for Differentiate the second equation (1.2.1) with respect to t, and get Let Then

where for groundpenetrating radar. The ground-penetrating radar inverse problem is: given find and One may use the source which is a current along a loop of wire, is the unit vector in cylindrical coordinates. In this case, one looks for E of the form: and from (1.2.11) one gets:

Let

where is the Bessel function. Set Then u solves (1.2.12), and the inverse problem is the same as above

([R185]). 1.2.15

A geometrical inverse problem

Let

Here D is domain homeomorphic to an annulus, is its inner boundary, outer boundary. The geometrical inverse problem is: given and find

is its

10

1. Introduction

One can interpret the data as the Cauchy data on for an electrostatic potential u, and then is the surface on which the potential is vanishing if or the charge distribution is vanishing if ([R139]). 1.2.16

Inverse source problems

(1) Inverse source problems in acoustics Let in f = 0 for uniformly in

tern A by the formula:

as

satisfies the radiation condition Define the radiation patThen

The inverse source problems are: (i) Given and find f (x). (ii) Given and a fixed find f (x). Clearly, by (1.2.15) problem (i) has at most one solution, but an a priori given function may be not of the form (1.2.15): the right-hand side of (1.2.15) is an entire function of exponential type of the vector Problem (ii), in general, may have many solutions, since may vanish for all at some (2) Inverse source problem in electrodynamics Consider Maxwell’s equations (1.2.11) in and assume j = 0 for radiation condition for (E, H) is:

The

where E and H in (1.2.11) are assumed monochromatic with time dependence, and where are the constant values of and near infinity. The inverse source problem is: given find j. Again, one should specify for what and the function is known. One can derive the relation between A and j. Namely, assuming and constants, and j smooth and compactly supported, one starts with the equations

then gets

11

then

so

Let

Thus

and

Then

where [a, b] is the vector product. So

and

It is now clear, that even if A is known for all and all vector J is not uniquely determined, but only its component orthogonal to is determined. Therefore, the solution to the inverse source problem in electrodynamics is not unique and may not exist, in general. The antenna synthesis problems are inverse source-type problems of electrodynamics. For example, if j is the current along a linear antenna (which is a wire along is the length of the antenna, then

so

is determined uniquely by the data A. Finding which produces the desired diagram is the problem of linear antenna synthesis. There is a large body of literature on this subject. Let for The inverse source problem is: given find f (x, t), The questions mentioned in this subsection were discussed in many papers and books ([AVST], [MJ], [ZK], [R11], [R21], [R26], [R27], [R28], [R73], [Is2]).

12

1. Introduction

1.2.17 Identification problems for integral-differential equations

Consider a Cauchy problem.

where is a closed, linear, densely defined in a Hilbert space H, operator, D(A) is its domain of definition, Assume that h (t) is unknown, and the extra data are given for The inverse problem is: given find h(t). See [LR]. 1.2.18 Inverse problem for an abstract evolution equation

Consider a Cauchy problem with the extra data (measured data) Here A(t) is a one-parametric family of closed, densely defined, linear, operators on a Banach space X, which generates an evolution family is a given function on [0, T] with values in X, and is an unknown scalar function. The inverse problem is: given f(t), and w, find (see [RKo]). In applications may be a control function which should be chosen so that the measured data are reproduced. 1.2.19 Inverse gravimetry problem

Let be the of the gravitational field generated by some masses, located in the region Assume that the values u(x, 0) := f ( x ) are known, in the region and u(x, –h) := g(x). Then is the equation for g. The inverse gravimetry problem is: given f, find g. See [VA], [RSm1]. 1.2.20

Phase retrieval problem (PRP)

Let The PRP consists of finding arg given Clearly, the solution to this problem is not unique: produces the same Under suitable assumptions on f ( x ) and one can get uniqueness results for PRP. See [KST], [R139]. 1.2.21

Non-overdetermined inverse problems

Formally we call an inverse problem non-overdetermined if the unknown function, which is to be found, depends on the same number of variables as the data. For example, problems in cases (1) and (2) in Section 1.2.13 are overdetermined, while in cases (3) and (4) they are not overdetermined. In multidimensional inverse scattering problems uniqueness of the solution is an open problem for most of the non-overdetermined

13

problems. For example, in Section 1.2.13 case (3) is a non-overdetermined problem, and uniqueness of its solution is an open problem. Recently Ramm ([R198]) proved that the spectral data and all determine q (x) uniquely provided that all the eigenvalues are simple. Here in is the kernel of the resolution of the identity of the selfadjoint Neumann operator in The above inverse problem is not overdetermined, because depends on three variables in and q (x) is also a function of three variables in It is an open problem to find out if this result remains valid without the assumption about the simplicity of all the eigenvalues. 1.2.22 Image processing, deconvolution

In many applications one is interested in the following inverse problem: given the properties of a linear device and the output signal, find the input signal. By the properties of a linear device one means its point-spread function (scattering function) or transfer function. For example, Given and f (x), one wants to find In practice the output signal f (x) is noisy, i.e., is given in place of f(x), where the norm depends on the problem at hand. See [BG], [RSm6], [RG]. 1.2.23

Inverse problem of electrodynamics, recovery of layered medium from the surface scattering data

There are many inverse problems arising in electrodynamics. If (1.2.11) are the governing equations,

where e is a constant vector, and are known constants: and respectively, outside a bounded domain and the data are measured on the surface of the Earth for various orientations of e then the inverse problem is to determine and from the above data. See [RSo], [RK]. 1.2.24

Finding ODE from a trajectory

Let and are constants, u = u ( t ) , Can one find and uniquely? In general, the answer is no. A trivial example is u ( t ) = 0. What trajectory allows one to find and uniquely? Suppose

14

1. Introduction

Then the system of equations

for finding and is uniquely solvable, so that (1.2.20) guarantees that u(t), determines and uniquely. More generally, consider a system where is a constant matrix, u (0) = v. If there are points such that the system is a linearly independent system of vectors, then u (t) determines the matrix uniquely. The reader can easily prove this. One can find more details in [Den]. 1.3

ILL-POSED PROBLEMS

Why are ill-posed problems important in applications? How are they related to inverse problems? Let

where X and Y are Banach spaces, or metric spaces, and A is a nonlinear operator, in general. Problem (1.3.1) is called well-posed if A is a homeomorphism of X onto Y . In other words, the solution to (1.3.1) exists for any is unique, and depends on f continuously, so that is a continuous map. If some of these conditions do not hold, then the problem is called ill-posed. Ill-posed problems are important in many application, in which one may reduce a physical problem to equation (1.3.1) where A is not boundedly invertible. For example, consider the equation

If then A is not boundedly invertible: A is injective, its range belongs to the Sobolev space is an unbounded operator in If noisy data are given, then may be not in the range of A. A practically interesting problem is: can one find an operator such that as In other words, can one estimate stably, given and Any Fredholm first-kind integral equation with linear compact operator is of the form (1.3.1). Such an operator in an infinite-dimensional space cannot have closed range and cannot be boundedly invertible. Since many inverse problems can be reduced to ill-posed equations (1.3.1), these inverse problems are ill-posed. That is how Ill-posed problems are related to inverse problems. Methods for stable solution of Ill-posed problems are developed in Chapter 2. The literature on Ill-posed problems is enormous ([IVT], [EHN], [Gro], [TLY], [VV], [VA], [R58]).

15

1.4 EXAMPLES OF ILL-POSED PROBLEMS

1.4.1

Stable numerical differentiation of noisy data

This example has been mentioned in Section 1.3. Methods for stable numerical differentiation of noisy data are given in Chapter 2. In navigation a ship receives a navigation signal which is a univalent function f (x) (that is, a smooth function which has precisely one point of maximum), and the course of the ship is determined by this point. The function f is observed in an additive noise. Given noisy data one wants to find A possible approach to this problem, is to search for a point at which One can see from a simple example that small perturbations of f can lead to large perturbations of let Then No matter how small is, one can choose so large that will take arbitrary large values at some points x. 1.4.2

Stable summation of the Fourier series and integrals with randomly perturbed coefficients

Let be an orthonormal basis of a Hilbert space H and L be a linear system (a linear operator) such that where are some numbers. Due to the inner noise in L, one observes the noisy output where is the noise. Thus, if the input signal is the output is One has noisy Fourier coefficients and one wants to recover the function If one has a Fourier integral, one can formulate a similar problem. To see that this problem is ill-posed, in general, let us take denote and assume that is playing the role of noise, Then one has the problem: given the noisy Fourier coefficients find This problem is ill-posed because the series may diverge. For example, if and then the series diverges. Similarly, if is the Fourier transform of a function and is the noisy data, then the problem is to calculate f (x) with “minimal error”, given noisy data The notion of “minimal error” should be specified. 1.4.3

Solving ill-conditioned linear algebraic systems

Let

be a linear operator such that its condition number is large. Then the linear algebraic system Au = f can be considered practically as an ill-posed problem, because small perturbation of the data f may lead to a large perturbation in the solution u. One has: so that the relative error in the data may result in relative error in the solution. In the above derivation we use the inequality Methods for stable solution of ill-conditioned algebraic systems are given in Chapter 2.

16

1. Introduction

1.4.4

Fredholm and Volterra integral equations of the first kind

If or and is a continuous kernel in D := [a, b] × [a, b], then the operators A and V are compact in these operator are not boundedly invertible in H. Therefore problems Au = f and Vu = f are ill-posed. 1.4.5

Deconvolution problems

These are problems, arising in applications: an input signal u generates an output signal f by the formula

Often one has:

The deconvolution problem consists of finding u, given f and For (1.4.2) the identification problem is of practical interest: given u (t) and f (t), find A (t). The function A(t) characterizes the linear system which generates the output f (t) gives the input u(t). Mathematically the deconvolution problems are the problems from Section 1.4.4. 1.4.6

Minimization problems

Consider the minimization problem Suppose that is the infimum of If f is perturbed, that is, then the infimum of may be not attained, or it may be attained at an element which is far away from Thus the map may be not continuous. In this case the minimization problem is ill-posed. Such problems were studied [Vas]. 1.4.7

The Cauchy problem for Laplace’s equation

Claim 1. The Cauchy problem for Laplace’s equation is an ill-posed problem. Consider the problem:

in the half-plane It is clear that solves the above problem, and this solution is unique (by the uniqueness of the solution to the Cauchy problem for elliptic equations). This example belongs to J. Hadamard, and it shows that the Cauchy data may be arbitrarily small (take while the solution tends to infinity, as at any point Thus, the claim is verified (cf. [LRS], [R139]).

17

1.4.8

The backwards heat equation

Consider the backwards heat equation problem:

Given v(x), one wants to find u (x, 0) := w (x). By separation of variables one finds Therefore, vided that this series converges, in that is, provided that

pro-

This cannot happen unless decays sufficiently fast. Therefore the backwards heat equation problem is ill-posed: it is not solvable for a given v (x) unless (1.4.3) holds, and small perturbations of the data v in may lead to arbitrary large perturbations of the function w(x), but also may lead to a function v for which the solution u(x, t) does not exist for t < T (cf. [LRS], [IVT]).

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2. METHODS OF SOLVING ILL-POSED PROBLEMS

2.1

VARIATIONAL REGULARIZATION

2.1.1

Pseudoinverse. Singular values decomposition

Consider linear Equation (1.3.1). Let be a linear closed operator, D(A) and R(A) be its domain and range, and be Hilbert spaces, N(A) := {u : Au = 0}, is the adjoint operator, the bar stands for the closure, and is the orthogonal sum. If A is injective, i.e., N(A) = {0}, and surjective, i.e. and then the inverse operator is defined on I is the identity operator. A closed, linear, defined on all of operator, is bounded, so A is an isomorphism of onto if it is injective, surjective, and If P is the orthoprojector onto N(A) in and Q is the orthoprojector onto then one defines a pseudoinverse (generalized inverse) Thus, and for The operator is bounded iff If i.e., for some then the problem has a solution every element is also a solution, and there is a unique solution with minimal norm, namely the solution such that If then the infimum of is not attained. If A is bounded and then the element solves the equation and is the minimal norm solution to this equation,

20

2. Methods of solving ill-posed problems

i.e., Conversely, if

Indeed, if and and

and with Thus,

then then if where

One can prove the formula: is a regularization parameter (see Section 2.1.2) and Let us define the singular value decomposition. Let be a linear compact operator, is a compact selfadjoint operator,

are called s -values of A. If

test). If

then Thus,

and

If is arbitrary, then Thus, an element are eigenvalues of then where

(Picard’s Then

If dim then A can be written as where A is an m × n matrix, U and V are unitary matrices (n × n and m × m, respectively), whose columns are vectors and respectively, and S is an m × n matrix with the diagonal elements r is the rank of the matrix A, and other elements of S are zeros. The matrix can be calculated by the formula where is an n × m matrix with diagonal elements and other elements of are zeros. 2.1.2

Variational (Phillips-Tikhonov) regularization

Assume is linear, is not necessarily in R(A). The problem Au = f is assumed ill-posed (cf. Sec. 1.3). Consider the problem:

where

is a parameter.

Theorem 2.1.1. Assume Au = f, and (i) The minimizer (ii) If and

Then:

of (2.1.1) does exist and is unique satisfies the condition where

as

then

21

Proof. Functional (2.1.1) is quadratic. A necessary and sufficient condition for its minimizer is the Euler’s equation:

which has a unique solution has Thus Choose so that so as and that prove that

Claim (i) is proved. One if Below stands for various positive constants. as and let Then This implies and we claim This claim we prove later. Thus Let us One has

Here the estimate and the relation as were used. To prove the first estimate, one uses the formula: and the polar representation of A* yields, where V is an isometry, One has where the spectral representation for T was used. Let us prove the second relation: where P is the orthoprojector onto N(A), and is the resolution of the identity of the self-adjoint operator B (see [KA]). If then Finally, let us prove the claim used above. Lemma 2.1.2. If B is a monotone hemicontinuous operator in a Hilbert space H, D(B) = H, then

In our proof above, hemicontinuous, if

is a linear operator, so B is monotone, that is, Recall that a nonlinear operator A is called is a continuous function of for any

Proof of Lemma 2.1.2. Clearly, If then that is, monotone operator is w -closed. Indeed, if B is monotone, then for any Passing to the limit and using hemicontinuity of B, one gets: This implies so Lemma 2.1.2 is proved. The claim is proved for nonlinear monotone operators in Theorem 2.1.6 below. Theorem 2.1.1 is proved.

2. Methods of solving ill-posed problems

22

2.1.3

Discrepancy principle

Theorem 2.1.1 gives an a priori choice of which guarantees convergence An a posteriori choice of is given by Theorem 2.1.3 below. Theorem 2.1.3. (Discrepancy principle). Assume equation

If

is the root of the

then Proof. First, let us prove that equation (2.1.3) has a unique solution. Write this equation as

where is the resolution of the identity of the selfadjoint operator and the commutation formula was used. One checks this formula easily. If then If then where is the orthoprojector on Indeed, because and since and Thus, if then equation (2.1.3) has a solution. This solution is unique because is a monotone increasing function of for each fixed Now let us prove Since

One has it follows that Therefore (*) If then one can select a weakly convergent sequence as In the proof of Theorem 2.1.1 it was proved that where is the unique minimal-norm solution of the equation By the lower semicontinuity of the norm in H, one has Together with (*), one gets This and the weak convergence imply

Our proof is based on the following useful result. Theorem 2.1.4. If Proof. If Thus, as

then

and

then

Also one has and

23

2.1.4

Nonlinear ill-posed problems

Lemma 2.1.5. Assume that A in (1.3.1) is a closed, nonlinear, injective map. If K is a compactum, then the inverse operator is continuous on A(K). Proof. Since A is injective, Then closed, and A, one has

is well-defined on A(K ). Let where is a subsequence denoted again Since A is imply and, by the injectivity of Lemma 2.1.5 is proved.

Claim: Let us assume that and Then

is monotone, continuous, D(A) = H, A(u) = f,

Proof. Indeed, Multiply this equation by the monotonicity of A, to get Thus, Let a sequence, denoted again by such that as Then Since A is monotone, it is w-closed (see the proof of Theorem 2.1.1), so and is the minimal norm solution to equation (1.3.1).

and use Select

Theorem 2.1.6. If is monotone and hemicontinuous, if D(A) = H, if where is the minimal-norm solution to A(u) = f, and if then the minimal-norm solution to (1.3.1) is unique and Proof. We have Thus, Let Then and Thus, This and the weak convergence imply strong convergence as in Theorem 2.1.3. The minimal norm solution to (1.3.1) is unique if A is monotone and continuous, because in this case the set of solutions N is convex and closed. Its closedness is obvious, if A is continuous. Its convexity follows from the monotonicity of A and the following lemma: Lemma 2.1.7 (Minty). If A is monotone and continuous, then (a) is equivalent to (b) Proof. If (a) holds, then then take Take and get Lemma 2.1.7 is proved.

and (b) holds by the monotonicity of A. If (b) holds, where w is arbitrary, and get Thus, A(u) = f, and (a) holds.

To prove that N is convex, one assumes that and derives that Indeed, if then, by Lemma 2.1.7, Thus, Thus,

2. Methods of solving ill-posed problems

24

To prove uniqueness of the minimal norm element of a convex and closed set N in a Hilbert space, one assumes that there are two such elements, and Then and so that any element of the segment joining and has minimal norm m. Since Hilbert space is strictly convex, this implies Indeed, take t = 1/2. Then So Thus, Theorem 2.1.6 is proved. Consider the equation

Theorem 2.1.8. Assume that A is monotone and continuous, and equation (1.3.1) has a solution. If and as then the unique solution to (2.1.4) converges strongly to u, the unique solution to (1.3.1) of minimal norm. Proof. Because A is monotone and the equation has a solution, and this solution is unique. Let solve (2.1.4), and One has By Theorem 2.1.6, Let us prove We have Multiply this by and use the monotonicity of A to get This implies if Theorem 2.1.8 is proved. 2.1.5

Regularization of nonlinear, possibly unbounded, operator

Assume that : is a closed, injective, possibly nonlinear, map in Banach space X. is a functional such that the set is precompact in X for any constant (3) Equation (1.3.1) has a solution (4) The last assumption can be replaced in some cases when A is an unbounded operator, by the assumption.

(1) (2)

Define the functional

Let equality

where is a parameter, Consider the minimization problem:

where Denote and

is the smallest integer satisfying the inOne has By assumption (2), as one can select a

25

convergent subsequence, denoted again Thus, A(u) = f by the closedness of A, and limit of any subsequence is the same, namely We have proved:

such that by the injectivity of A. Since the it follows that

Theorem 2.1.9. If (1.3.1) has a solution, then, under the assumptions (1)–(4) (or any sequence such that converges strongly to the solution of (1.3.1) as Remark 2.1.10. In the proof of Theorem 2.1.9 we do not need existence of the minimizer of the function (2.1.5). 2.1.6

Regularization based on spectral theory

Assume that A in (1.3.1) is a linear bounded operator, i.e., is the unique minimal-norm solution.

and

Lemma 2.1.11. Solvable equation (1.3.1) with bounded linear operator A is equivalent to the equation

Proof. If u solves (1.3.1), apply A* to (1.3.1) and get (2.1.6), so u solves (2.1.6). If u solves (2.1.6) and (1.3.1) is solvable, i.e., then and This implies so Au = f. Thus u solves (1.3.1).

Equation (2.1.6) is a solvable equation with monotone, continuous operator, so Theorem 2.1.6 is applicable and yields the following theorem: Theorem 2.1.12. If the minimal-norm solution to (1.3.1), then solution of the equation

Lemma 2.1.13. Consider the elements resolution of the identity of B = A* A, on s and piecewise-continuous function. Let

as where

so that

and is is the unique

where is the does not depend and is a as Then

26

2. Methods of solving ill-posed problems

Proof. If

then

and

Thus, where

and

One has

as If the rate of decay of of can be estimated, then a quasioptimal choice of minimizing with respect to for a fixed

Thus and the rate of growth can be made by

Remark 2.1.14. We have used the spectral theorem for a selfadjoint operator B, namely is the resolution of the identity of B, the formula where

Remark 2.1.15. Similarly, one can use the theory of spectral operators in place of the spectral theory of selfadjoint operators, in particular Riesz bases formed by the root vectors. 2.1.7

On the notion of ill-posedness for nonlinear equations

If A is a linear operator, then problem (1.3.1) is ill-posed if either or or R(A) is not closed, i.e. is unbounded. If A is nonlinear and Frechet differentiable, then there are several possibilities. If is boundedly invertible at some u, then A(u) is a local homeomorphism at this point, but it may be not a global homeomorphism. If is not boundedly invertible, this does not imply, in general, that A is not a homeomorphism. For example, a homeomorphism A(u) may have a compact derivative, so its linearization yields an ill-posed problem. On the other hand, A(u) may be compact, so (1.3.1) is an ill-posed problem, but may be a finite-rank operator, so that the range of is closed. In spite of the above, we will often call a nonlinear equation problem (1.3.1) ill-posed if is not boundedly invertible, and well-posed if is boundedly invertible, deviating therefore from the usual terminology. 2.1.8

Discrepancy principle for nonlinear ill-posed problems with monotone operators

Assume that A in (1.3.1) is monotone, i.e., D(A) = H, A is continuous, is unbounded or does not exist, so (1.3.1) is an ill-posed problem, Consider the discrepancy principle for finding assuming that A is nonlinear monotone:

where C = const > 1,

is any element such that where of the regularization parameter We need three lemmas.

and

plays the role

27

Lemma 2.1.16. If A is monotone and continuous, and the set nonempty, then it is convex and closed.

is

Lemma 2.1.17. If A is monotone and continuous, then it is w-closed, that is, and imply A(u) = f, where and stand for the weak and strong convergence in H, respectively. Lemma 2.1.17 in a stronger form (hemicontinuity of A replaces continuity, and it is assumed in this case that the monotone operator A is defined on all of H ) follows from the proof of Lemma 2.1.2. Lemma 2.1.18. If

then

and

Proof of Lemma 2.1.16. If and If A is monotone, A(u) = f and A(v) = f, then and vice versa. Thus, for any

then A(u) = f, so

is closed. and the element

Lemma 2.1.18 is Theorem 2.1.4. Theorem 2.1.19. Assume: (i) A is a monotone, continuous operator, defined on all of H, (ii) equation A(u) = f is solvable, is its minimal-norm solution, and (iii) where C > 1 is a constant. Then: (j) the equation

is solvable for

for any fixed

Here

is any element satisfying inequality

where and and (jj) if

solves (2.1.8), and

then

Remark 2.1.20. The equation is uniquely solvable for any and any If is its solution, and where C = const > 1, then equation (2.1.8) with replaced by is solvable for If is its solution, then If A is injective, and if then where solves the equation If A is not injective, then it is not true, in general, that where is the minimal-norm solution to the equation A(u) = f even if one assumes that A is a linear operator.

28

2. Methods of solving ill-posed problems

Proof of Theorem 2.1.19. If A is monotone, continuous and is defined on all of H, then the set is convex and closed, so it has a unique minimal-norm element To prove the existence of a solution to (2.1.8), we prove that the function is greater than for sufficiently large and smaller than for sufficiently small If this is proved, then the continuity of with respect to on implies that the equation has a solution. Let us give the proof. As we use the inequality:

and, as

we use another inequality:

As

one gets where c > 0 is a constant depending on Thus, by the continuity of A, one obtains As one gets Thus, Therefore equation has a solution Let us now prove that if then From the estimate

and from the equation (2.1.8), it follows that Thus, one may assume that and from (2.1.8) it follows that as By w-closedness of monotone continuous operators (hemicontinuity in place of continuity would suffice), one gets A(U) = f, and from it follows that Because A is monotone, the minimal norm solution to the equation A(u) = f in H is unique. Consequently, Thus, and By Theorem 2.1.4, it follows that Note that because due to the assumption where C > 1. Theorem 2.1.19 is proved.

Proof of Remark 2.1.20. Let solve the equation let solve the equation and Then Multiply this equation by and use the monotonicity of A to get The triangle inequality yields: Note that and as Thus, Therefore Fix and let Then and where is the minimal-norm solution to the equation A(u) = f. If A is injective, then this equation has only one solution Since one gets the inequality

29

Consequently, equation (2.1.8), with w replacing has a solution We claim that in fact, as Indeed, from (2.1.8), with w replacing one gets and we prove below that where This implies We now claim that the limit does exist, that u solves the equation A(u) = f, and It is sufficient to check that where c = const does not depend on as Indeed, if then a subsequence, denoted again, converges weakly to an element and (2.1.8) implies Since A is monotone, it is w–closed, so By the injectivity of A, any subsequence converges weakly to the same element so Consequently, as claimed. The inequality follows from the assumption To prove the inequality note that where Since C > 1, this implies where Thus, where The last statement of Remark 2.1.20 is illustrated by the following example: Example 2.1.21. Let Aw = (w, p ) p , where (q, p) = 0, One has where is the minimal-norm solution to the equation Au = p. Equation has the unique solution Equation (2.1.8) is This equation yields where and we assume 1 (see the second inequality in the assumption (iii) of Theorem 2.1.19). Let Then, and Au = p. Therefore is not p, i.e., u is not the minimal-norm solution to the equation Au = p. Remark 2.1.20 is proved. Remark 2.1.22. It is easy to prove that if conditions (i) and (ii) of Theorem 2.1.19 hold and and if where then where and is the minimal-norm solution to the equation A(u) = f. In particular, if 0 < a < l, then Indeed, where is the unique solution to the equation It is well known that provided that and, clearly, one multiplies the identity by and uses the monotonicity of A and the inequality The result similar to the one in the above remark can be found in [ARy]. 2.1.9

Regularizers for ill-posed problems must depend on the noise level

In this Section we prove the following simple claim: Claim 2. There is no regularizer independent of the noise level to a linear ill-posed problem. If such a regularizer exists, then the problem is well-posed.

2. Methods of solving ill-posed problems

30

Let A be a linear operator in a Banach space X. Assume that A is injective and is unbounded, that equation

is solvable, and g is such that

where is the norm in X and is the noise level. Nothing is assumed about the statistical nature of noise. In particular, we do not assume that the noise has zero mean value or finite variance. Question: Can one find a linear operator R with the property:

for any Answer: no.

where Ran( A) is the range of A, and any

satisfying (2.1.10)?

Proof. If such an R is found, then, taking and using the fact that is arbitrary, one concludes that on the range of A. Secondly, writing where

and w is arbitrary otherwise, one concludes from (2.1.11) and from the fact that that

for any w satisfying (2.1.12). Since R is linear, this implies that R is bounded, which contradicts the equation on Ran(A) and the unboundedness of which is the necessary condition for the ill-posedness of (2.1.9). A similar result one can find in [LY]. 2.2

2.2.1

QUASISOLUTIONS, QUASINVERSION, AND BACKUS-GILBERT METHOD

Quasisolutions for continuous operator

Assume that equation (1.3.1) is solvable, its solution where K is a compactum in a Banach space X, and A is continuous. Consider the problem

31

where is the infimum of the function of and A minimizer for (2.2.1) is called a quasisolution to (1.3.1) with Let be a minimizing sequence for (2.2.1). Since K is a compactum, one may assume that as Thus so is a minimizer for the problem (2.2.1). The above argument shows that if f replaces in (2.2.1), and if equation (1.3.1) is solvable and its solution belongs to K, then any minimizer for (2.2.1) with is a solution to (1.3.1). Let us prove that where is a minimizer for (2.2.1), and u is a solution to (1.3.1), where existence of a solution (1.3.1) is assumed. Indeed, so one may assume that as By continuity of A, one has Thus, The last conclusion follows from the solvability of (1.3.1), which yields f = A(u) and from the inequality We have proved: Theorem 2.2.1. If equation (1.3.1) is solvable, K is a compactum containing all the solutions to (1.3.1), and then (2.2.1) has a minimizer and for every minimizer and some solution u to (1.3.1). Remark. Suppose that X is strictly convex, i.e., if then u = v. For example Hilbert spaces H are strictly convex, the spaces are strictly convex, but and C(D) are not. Suppose that K is a convex compactum, i.e., convex closed compact set. The metric projection of an element onto K is the element such that If X is strictly convex, then is unique, and if K is a convex compactum, then depends continuously on f. If A is injective and closed, not necessarily linear, and K is a compactum, then is continuous on the set AK. Indeed, if and then a subsequence, denoted again converges to u because K is compact, and if A is closed, then A(u) = f, which proves the claim. Therefore, if X is strictly convex and K is a convex compactum, and if A is an injective bounded linear operator, then the quasisolution depends continuously on f in the norm of X. 2.2.2

Quasisolution for unbounded operators

Assume that A is closed, possibly nonlinear, injective, unbounded operator, is possibly, unbounded, equation (1.3.1) is solvable, assumptions (1)–(4) of Section 2.1.5 hold, and K is a compactum containing all the solutions to (1.3.1). Theorem 2.2.2. Under the above assumptions, if K, then where u is a solution to (1.3.1). Proof. One can assume because

as

where

because K is a compactum. Note that Thus,

32

2. Methods of solving ill-posed problems

By closedness of A, one gets A(w) = f, so w is a solution to (1.3.1), and one can denote w by u. 2.2.3

Quasiinversion

Let A be a linear bounded operator in (1.3.1), and (1.3.1) is solvable, consider the equation is a parameter, Q is an operator chosen so that and where u is a solution to (1.3.1). If A is unbounded, a similar idea can be applied to equation (1.3.1): consider the equation where Q is chosen so that The problem is: how does one choose Q with these properties? If A is a linear bounded operator, then Q = I can be used by Theorem 2.1.1. If A is unbounded, some assumptions on its spectrum are needed. See [LL] for details. 2.2.4

A Backus-Gilbert-type method: Recovery of signals from discrete and noisy data

We discuss in this section the following Problem 2. Let D and the bar denote variance and mean value respectively,

Problem 1. Given

estimate

The idea is as follows: the estimate is sought in the form

The problem is to find such that: (1) if then (2) if then where is a certain number. We find optimal, in a certain sense. Namely, if then are found from the requirements:

Note that

If then are found from {(A) and if this problem is solvable and, if not, one increases so that this problem becomes solvable.

33

2.2.4.1. A typical problem we are concerned with is the problem of estimating the spectrum of a compactly supported function from the knowledge of the spectrum at a finite number of frequencies. More precisely, let

Suppose that the numbers:

are given. At the moment we assume that are given exactly, i.e., there is no noise. The case when the data are noisy will be considered below. Problem 2. Given

find an estimate

To be specific let us assume that

where the functions

of

such that

and that the estimate is of the form

will be chosen soon. From (2.2.5) and (2.2.2) it follows that

and

where and the star denotes complex conjugate. Property (2.2.4), convergence, holds if is a delta-sequence, i.e.,

2. Methods of solving ill-posed problems

34

Let

and let

be a sequence of functions such that

where

One can interpret (2.2.11) as the requirement that the estimate is exact for f (x) = const. Given (2.2.11), the smaller Q(x), the better is the quality of the delta-sequence Thus we are led to the optimization problem: Find such a sequence that

Note that the general problem of the type

where is a linearly independent set of functions, and is a bounded domain in can be treated in exactly the same way as before. If the problem (2.2.13) has the unique solution then (2.2.6) is the optimal estimate which, as we prove, has the convergence property (2.2.4). 2.2.4.2. If the data are noisy, that is are given in place of random vector with the covariance matrix

where

35

where the bar denotes the mean value, then the variance

where ( , ) is the inner product in

Let us fix

can be computed

and require that

The optimization problem for finding the vector formulated as follows:

can be

Here Clearly, problem (2.2.18) is not solvable for all We will discuss this important point below. If (2.2.18) is solvable, the solution is unique, and the optimal estimate is given by

This estimate has variance Our arguments so far are close to the usual ones. The new point is our convergence requirement (2.2.4). We prove the convergence property of our estimate and give the rate of convergence. The case when the data are the finite number of moments is treated, and the optimization requirements are introduced. The problem we discuss is of interest in geophysics and many other applications. 2.2.4.3. Here a solution of the estimation problems is given. We start with problem (2.2.13). Let us write Q(x) as a quadratic form

where

2. Methods of solving ill-posed problems

36

and

is a self-adjoint positive definite matrix. Let us write (2.2.13) as

Using the Lagrange multiplier condition for the minimizer

one obtains the standard necessary and sufficient

Therefore

is uniquely defined by (2.2.25), since is positive definite, and the denominator in (2.2.25) does not vanish. The minimum of Q(x) is

We assume that

If (2.2.27) holds, then (2.2.4) holds. Indeed, (2.2.13) and (2.2.27) imply that

In this argument we assume that the function f (x) satisfies the inequality

37

This inequality is satisfied if, for example, the derivative of f (x) exists except at a finite number of points and is uniformly bounded. Let us illustrate the assumption (2.2.27). Let Then

where where is the cofactor corresponding to the element

of the matrix One can show that (2.2.24) holds at any point at which f (x) is differentiable and Indeed, if we do not take the but use then the error of the estimate will be not less that On the other hand, for this choice of h, the kernel (2.2.8) is the Dirichlet kernel. From the theory of the Fourier series one knows that (2.2.9) holds in and at any point at which f (x) is differentiable. In practice it is advisable to choose the system in such a way that tends rapidly to zero. Note that depends only on the system and therefore we can control this quantity to some extent by choosing the system Let us note that one can estimate f (x) at a given point optimally using the same procedure. In this case the convergence condition (2.2.4) will hold for If is fixed, we can choose the system so that

In this case

and we can choose

so that, in addition to (2.2.29), the condition

is satisfied. For example, take

then (2.2.29) reduces to

38

2. Methods of solving ill-posed problems

and one can choose Then

which behave nearly like can be made very large, and

in a small neighborhood of as in (2.2.31).

2.2.4.4. In this section we solve problem (2.2.18). As we have already mentioned, this problem may not be solvable for every because there may be no h which satisfies both restrictions of (2.2.18). Since the set is convex and Q(h) is a strictly convex function of h, it is clear that the solution to (2.2.18) is unique when it exists. For the solution to exist it is necessary and sufficient that the set M of h, which satisfy the restrictions (2.2.18), be not empty. Let us give an analytic solution to problem (2.2.18). If for the optimal h the inequality holds, then the solution to (2.2.18) is the same as the solution to (2.2.13) and is given by formula (2.2.25). Therefore, first one checks if the function (2.2.25) satisfies the inequality

If it does, then it is the solution to (2.2.18). If it does not, then the solution satisfies the equality

By the Lagrange method the necessary condition for the optimal h, for the solution to problem (2.2.18), is

where

and

are the Lagrange multipliers. It follows from (2.2.36) that

Taking the complex conjugate in the first equation (2.2.36) we see that

From (2.2.37) and (2.2.38) one gets

Substituting (2.2.40) into

39

yields an equation for

The roots of Eq. (2.2.42), give by formula (2.2.38), and h by formula (2.2.37). Finally choose the for which (Bh, h) = min. This solves problem (2.2.18). 2.2.4.5. One can simplify the solution to problem (2.2.18) in the following way. If the problem (2.2.18) takes the form

Let us choose the coordinate system so that

and normalize

so that

In this case (2.2.43) can be written as

where

Thus, problem (2.2.43) in with two constraints is reduced to problem (2.2.46) in with one constraint. Problem (2.2.46) can be solved by the Lagrange multipliers method. One has

where

is the Lagrange multiplier. Thus,

40

2. Methods of solving ill-posed problems

Substitute (2.2.49) into the constraint equation (2.2.46) to obtain an equation for If this equation is solved then (2.2.49) gives the corresponding H. If there are several solutions then the is the one that minimizes the quadratic form (2.2.46). 2.3

ITERATIVE METHODS

There is a vast literature on iterative methods [VV], [BG], [R65]. First, we prove the following result. Theorem 2.3.1. Every solvable equation (1.3.1) with bounded linear operator A can be solved by a convergent iterative method. Proof. It was proved in Section 2.1.6 that if A is a bounded linear operator and equation (1.3.1) is solvable, then it is equivalent to the equation (2.1.6). By y we denote the minimal norm solution to (1.3.1), i.e., the solution orthogonal to N(A), the null– space of A. Note that N(B) = N(A). Without loss of generality assume (if then one can divide by equation (2.1.6)). Consider the iterations

where

is arbitrary. Denote

Thus,

and write (2.3.1) as Since

one gets:

where is the resolution of the identity of B. Thus, where P is the orthoprojector onto N(B) = N(A). If one takes and then by induction, and In particular, if then (by induction, since and where A+ is the pseudoinverse of A, defined in Section 2.1.1. Exercise 2.3.2. Prove that if

then

in (2.3.1).

Assume now that is given in place of f, Let us show that if one stops iterations (2.3.1), with in place of f and at then as if is properly chosen, and is defined by (2.3.1). Let Then and Thus, because if and we had assumed which, together with implies Therefore, if is chosen so that then Indeed, We have proved in Theorem 2.3.1

41

that rize the result. Theorem 2.3.3.

as

and

If

Let us summa-

then

where is the minimal norm solution to (1.3.1), is obtained by iterations (2.3.1), with in place of f and

and

A general approach to construction of convergent iterative methods for nonlinear problems is developed in Section 2.4. 2.4

DYNAMICAL SYSTEM METHOD (DSM)

2.4.1

The idea of the DSM

Consider the equation:

We assume in Section 2.4 that

that is,

where

is the minimal-norm solution to (2.4.1), H is a real Hilbert space. Many of our results hold in reflexive Banach spaces X and F : X X*, but we do not go into detail. The element in (2.4.2) will be specified later. In Section 2.4 we will call (2.4.1) a wellposed problem if

and ill-posed if is not boundedly invertible. We assume existence of a solution to (2.4.1) unless otherwise stated, but uniqueness of the solution is not assumed. If (2.4.3) holds, then one can construct Newton-type methods for solving (2.4.1). But if (2.4.3) fails, then it seems that there is no general approach to solving (2.4.1). One of our goals is to develop such an approach, which we call the Dynamical Systems Method (DSM). The DSM consists of finding a nonlinear locally Lipschitz operator such that the Cauchy problem:

has the following three properties:

42

2. Methods of solving ill-posed problems

that is, (2.4.4) is globally uniquely solvable, its unique solution has a limit at infinity and this limit solves (2.4.1). On first motivation is to develop a general approach to solving equation (2.4.1), especially nonlinear and ill-posed. Our second motivation is to develop a general approach to constructing convergent iterative methods for solving (2.4.1). We justify the DSM in the following cases (1) For well-posed problems, (2) For ill-posed linear problems with bounded linear operator A and also for (3) For ill-posed problem with monotone nonlinear A, for

and and also for

(4) For ill-posed problem with nonlinear A assuming some additional condition. We give a general construction of convergent iterative schemes for well-posed nonlinear problems, and also for ill-posed nonlinear problems with monotone and nonmonotone operators. These results are presented in subsections below. 2.4.2

DSM for well-posed problems

Consider (2.4.1), let (2.4.2) hold, and assume

where

is an integrable function,

where

is such that

is a constant. Assume

Remark 2.4.1. Sometimes the assumption (2.4.7) can be used in the following modified form:

where is a constant. The statement and proof of Theorem 2.4.2 can be easily adjusted to this assumption.

Our first basic result is the folowing:

43

Theorem 2.4.2. (i) If (2.4.6)–(2.4.8) hold and

then (2.4.4) has a global solution, (2.4.5) holds, (2.4.1) has a solution and

(ii) If (2.4.6)–(2.4.8) hold, 0 < a < 2, and

where T > 0 is defined by the equation

then (2.4.4) has a global solution, (2.4.5) holds, (2.4.1) has a solution and u(t) = y for (iii) If (2.4.6)–(2.4.8) hold, a > 2, and

where

then (2.4.4) has a global solution, (2.4.5) holds, (2.4.1) has a solution and

as

Let us sketch the proof.

44

2. Methods of solving ill-posed problems

Proof of Theorem 2.4.2. The assumptions about imply local existence and uniqueness of the solution u(t) to (2.4.4). To prove global existence of u, it is sufficient to prove a uniform with respect to t bound on Indeed, if the maximal interval of the existence of u(t) is finite, say [0, T), and is locally Lipschitz with respect to u, then as Assume a = 2. Let Since H is real, one uses (2.4.4) and (2.4.6) to get so and integrating this inequality one gets the second inequality (2.4.10), because Using (2.4.7), (2.4.4) and the second inequality (2.4.10), one gets:

Because it follows from (2.4.10’) that the limit exists, and by (2.4.9). From the second inequality (2.4.10) and the continuity of F one gets so solves (2.4.1). Taking and setting s = t in (2.4.10’) yields the first inequality (2.4.10). The inclusion for all follows from (2.4.9) and (2.4.10’). The first part of Theorem 2.4.2 is proved. The proof of the other parts is similar. There are many applications of this theorem. We mention just a few, and assume that and Example 2.4.3. [Continuous Newton-type method]: Assume that (2.4.3) holds, then (2.4.9) takes the form (*) and (*) implies that (2.4.4) has a global solution, (2.4.5) and (2.4.10) hold, and (2.4.1) has a solution in This result belongs to Gavurin ([Gav]). Example 2.4.4. [Continuous simple iterations method:] Let for all Then and the conclusions of Example 1 hold. Example 2.4.5. [Continuous gradient method:] Let hold, (2.4.9) is (**) conclusions of Example 2.4.3.

and assume (2.4.9) is:

(2.4.2) and (2.4.3) and (**) implies the

Example 2.4.6. [Continuous Gauss-Newton method:] Let (2.4.2) and (2.4.3) hold, (2.4.9) is (***) (***) implies the conclusions of Example 2.4.3. Example 2.4.7. [Continuous modified Newton method:] Let sume and let (2.4.2) hold. Then

and

AsChoose

45

and Thus, if

Then (2.4.9) is

that is, then the conclusions of Example 2.4.3 hold.

Example 2.4.8. [Descent methods.] Let where f = f(u(t)) is a differetiable functional and h is an element of H. From (2.4.4) one gets Thus, where Assume Then Therefore does exist, and c = const > 0. If and then and (2.4.4) is a descent method. For this one has and where is defined in (2.4.3). Condition (2.4.9) is: If this inequality holds, then the conclusions of Example 2.4.3 hold. In Example 2.4.8 we have obtained some results from [Alb]. Our approach is more general than that in [Alb], since the choices of f and h do not allow one, for example, to obtain used in Example 2.4.7. Remark 2.4.9. A method for proving the existence of a solution to equation (2.4.1) can be stated as follows. Consider (2.4.4) with and assume that (2.4.4) is locally solvable and where u(t) solves (2.4.4). Let Then so and Assume that Then does exist and as Therefore F(u(t))= 0, so solves (2.4.1). The proof of Theorem 2.4.2 is given by this method and Theorem 2.4.20 below is an example of many applications of this method. Conditions (2.4.7) and (2.4.9) are essential: if and H is the real line with the usual product of real numbers as the inner product and then condition (2.4.7) is not satisfied and equation (2.4.1), i.e. does not have a solution in H. Exercise (cf. [R222]). Use the above Remark to prove the following result: Assume F : H H, (2.4.2)–(2.4.3) hold, and lim Then the map F is surjective. This result is related to Hadamard's theorem about homeomorphisms and its generalization by Meyer (see [OR], p. 139). 2.4.3

Linear ill-posed problems

We assume that (2.4.3) fails. Consider

Let us denote by (A) the folowing assumption:

46

2. Methods of solving ill-posed problems

(A) : A is a linear, bounded operator in H, defined on all of H, the range R(A) is not closed,

so (2.4.15) is an ill-posed problem, there is a the null-space of A. Let solves

and, if function to assume that

such that

where N is

is the adjoint of A. Every solution to (2.4.15)

then every solution to (2.4.16) solves (2.4.15). Choose a continuous monotonically decaying to zero on Sometimes it is convenient

For example, the functions where and are positive constants, satisfy (2.4.17). There are many such functions. One can prove the following: Claim 3. If is a continuous monotonically decaying function on 0, and (2.4.17) holds, then

In this Section we do not use assumption (2.4.17): in the proof of Theorem 2.4.9 one uses only the monotonicity of a continuous function and (2.4.17’). One can drop assumption (2.4.17’), but then convergence is proved in Theorem 2.4.9 to some element of N, not necessarily to the normal solution that is, to the solution orthogonal to N, or, which is the same, to the minimal-norm solution to (2.4.15). However, (2.4.17) is used (in a slightly weaker form) in the next section. Consider problems (2.4.4) with

where

Without loss of generality one may assume that which we do in what follows. Our main result is Theorem 2.4.9, stated below. It yields the following: Conclusion: Given noisy data every linear ill-posed problem (2.4.15) under the assumptions (A) can be stably solved by the DSM. The result presented in Theorem 2.4.9 was essentially obtained in [R200], but the proof given here is different and much shorter. Theorem 2.4.9. Problem (2.4.4) with from (2.4.18) has a unique global solution u(t), (2.4.5) holds, and Problem (2.4.4) with from (2.4.18) has a unique global

47

solution

This

There exists

such that

can be chosen, for example, as a root of the equation

or of the equation (2.4.20’), see below.

Proof of Theorem 2.4.9. Linear equations (2.4.4) with bounded operators have unique global solutions. If then the solution u to (2.4.4) is

where as is the resolution of the identity corresponding to the selfadjoint operator B, and is a nonexpansive operator, because Actually, (2.4.21) can be used also when B is unbounded, Using L‘Hospital’s rule one checks that

provided only that and From (2.4.21), (2.4.22), and the Lebesgue dominated convergence theorem, one gets where P is the orthogonal projection operator onto the null-space of B. Under our assumptions (A), so If then In general, the rate of convergence of v to zero can be arbitrarily slow for a suitably chosen f. Under an additional a priori assumption on f (for example, the source-type assumptions), this rate can be estimated. Let us describe a method for deriving a stopping rule. One has:

Since

any choice of

gives a stopping rule: for such

one has

such that

48

2. Methods of solving ill-posed problems

To prove that (2.4.20) gives such a rule, it is sufficient to check that

Let us prove (2.4.25). Denote

Then

Integrating (2.4.26), and using the property one gets (2.4.25). Alternatively, multiply (2.4.26) by w, let use and get Thus, exp exp A more precise estimate, used at the end of the proof of Theorem 2.4.10 below, yields:

and the corresponding stopping time

can be taken as the root of the equation:

Theorem 2.4.9 is proved. If the rate of decay of v is known, then a more efficient stopping rule can be derived: is the minimizer of the problem:

For example, if then is the root of the equation that one gets from (2.4.27) with One can also use a stopping rule based on an a posteriori choice of the stopping time, for example, the choice by a discrepancy principle. A method, much more efficient numerically than Theorem 2.4.9, is given below in Theorem 2.4.12 and in Theorem 2.4.10 (see (2.4.29)). For linear equation (2.4.16) with exact data this method uses (2.4.4) with

and for noisy data it uses (2.4.4) with The linear operator is monotone, so Theorem 2.4.12 is applicable. For exact data, (2.4.4) with defined in (2.4.28), yields:

49

and (2.4.5) holds if is monotone, continuous, decreasing to 0 as Let us formulate the result: Theorem 2.4.10. Assume (A), and let B := A* A, q := A* f. Assume to be a continuous, monotonically decaying to zero function on Then, for any problem (2.4.29) has a unique global solution, and is the minimalnorm solution to (2.4.15). If is given in place of f, then (2.4.19) holds, with solving (2.4.29) with q replaced by and is chosen, for example, as the root of (3.4.20’) (or by a discrepancy principle). Proof of Theorem 2.4.10. One has is

where

and the solution to (2.4.29)

where

and

is the resolution of the identity of the selfadjoint operator B. One has

From (2.4.30)–(2.4.32) it follows that where is the minimal-norm solution to (2.4.15), N := N(B) = N(A) is the null-space of B and of A, and is the orthoprojector onto N in H. This proves the first part of Theorem 2.4.10. To prove the second part, denote where we dropped the dependence on in w and g for brevity. Then w (0) = 0. Thus so where the known estimate was used: proved. 2.4.4

Theorem 2.4.10 is

Nonlinear ill-posed problems with monotone operators

There is a large body of literature on equations (2.4.1) and (2.4.4) with monotone operators. In the result we present, the problem is nonlinear and ill-posed, the new technical tool, Theorem 2.4.11, is used, and the stopping rules are discussed. Consider (2.4.33) with monotone F under standard assumptions (2.4.2), and

50

2. Methods of solving ill-posed problems

where A* is its adjoint, is the same as in Theorem 2.4.10, and in Theorem 2.4.12 is further specified, is an element we can choose to improve the numerical performance of the method. If noisy data are given, then, as in Section 3.3, we take

where B is a monotone nonlinear operator, and solves (2.4.4) with in place of To prove that (2.4.33) with the above has a global solution and (2.4.5) holds, we use the following: Theorem 2.4.11. Let exists a positive function

for some real number

If there

such that

where is the initial condition in (2.4.35), then a nonnegative solution g to the following differential inequality:

exists for all

and satisfies the estimate:

for all

where

There are several novel features in this result. First, differential equation, that one gets from (2.4.35) by replacing the inequality sign by the equality sign, is a Riccati equation, whose solution may blow up in a finite time, in general. Conditions (2.4.34) guarantee the global existence of the solution to this Riccati equation with the initial condition (2.4.35). Secondly, this Riccati differential equation cannot be integrated analytically by separation of variables. Thirdly, the coefficient may grow to infinity as so that the quadratic term does not necessarily have a small coefficient, or the coefficient smaller than Without loss of generality one may assume in Theorem 2.4.11. This Theorem is proved in Section 2.4.9. The main result in this Section is new. It claims a global convergence in the sense that no assumptions on the choice of the initial approximation are made. Usually

51

one assumes that is sufficiently close to the solution of (2.4.1) in order to prove convergence. We take in Theorem 2.4.12 because in this theorem does not play any role. The proof is valid for any choice of but then the definition of r in Theorem 2.4.12 is changed. Theorem 2.4.12. If (2.4.2) holds, R = 3r, where and is the (unique) minimal norm solution to (2.4.1), then, for any choice of problem (2.4.4) with defined in (2.4.33), and with some positive constants and specified in the proof of Theorem 2.4.12, has a global solution, this solution stays in the ball and (2.4.5) holds. If solves (2.4.4) with in place of then there is a such that Proof of Theorem 2.4.12. Let us sketch the steps of the proof. Let V solve the equation

Under our assumptions on F, it is easy to prove that: (i) (2.4.38) has a unique solution for every t > 0, and (ii) Indeed, so This implies (ii). If F is Fréchet differentiable, then V is differentiable, and It is also known (see Section 2.1.4) that if where y is the minimal-norm solution to (2.4.1), then

We will show that the global solution u to (2.4.4), with the from (2.4.33), does exist, and This is done by deriving a differential inequality for w := u – V, and by applying Theorem 2.4.11 to Since one obtains (2.4.5). We also check that where for any choice of and a suitable choice of Let us derive the differential inequality for w. One has

and F(u) –F(V) = Aw + K, where and is the constant from (2.4.2). Multiply (2.4.39) by w, use the monotonicity of F, that is, the property and the estimate and get:

52

2. Methods of solving ill-posed problems

where

Inequality (2.4.40) is of the type (2.4.35): Choose

Clearly as Let us check three conditions (2.4.34). One has Take where are constants, and choose these constants so that for example, Then the first condition (2.4.34) is satisfied. The second condition (2.4.34) holds if

One has

Choose

Then

so (2.4.42) holds. Thus, the second condition (2.4.34) holds. The last condition (2.4.34) holds because

By Theorem 2.4.11 one concludes that

when

and

This estimate implies the global existence of the solution to (2.4.4), because if u(t) had a finite maximal interval of existence, [0, T), then u(t) could not stay bounded when which contradicts the boundedness of and from (2.4.45) it follows that We have proved the first part of Theorem 2.4.12, namely properties (2.4.5). To derive a stopping rule we argue as in Section 2.4. One has:

We have already proved that The rate of decay of v can be arbitrarily slow, in general. Additional assumptions, for example, the source-type ones, can be used to estimate the rate of decay of v (t). One derives differential inequality (2.4.35) for and estimates using (2.4.36). The analog of (2.4.40) for contains additional term on the right-hand side. If

53

then conditions (2.4.34) hold, and Let be the root of the Then and because and but the convergence can be slow. See [ARS3], [KNR] for the rate of convergence under source assumptions. If the rate of decay of v(t) is known, then one chooses as the minimizer of the problem, similar to (2.4.27), equation

where the minimum is taken over t > 0 for a fixed small timal stopping rule. Theorem 2.4.12 is proved.

This yields a quasiop-

Let us give another result: Theorem 2.4.13. Assume that F is monotone, rem 2.4.10, and (2.4.17), and (2.4.2) hold. Then (2.4.5) holds.

as in Theo-

Proof of Theorem 2.4.13. As in the proof of Theorem 2.4.12, it is sufficient to prove that where g, w, and V are the same as in Theorem 2.4.12, and u solves (2.4.4) with the defined in Theorem 2.4.13. Similarly to the derivation of (2.4.39), one gets:

Multiply (2.4.48) by w, use the monotonicity of F and the estimate which was used also in the proof of Theorem 2.4.12, and get:

This implies

From our assumptions relation (2.4.17’) follows, and (2.4.50) together with (2.4.17) imply Theorem 2.4.13 is proved. Remark 2.4.14. One can drop the smoothness of F assumption (2.4.2) in Theorem 2.4.13 and assume only that F is a monotone hemicontinuous operator defined on all of H. Claim 4. If (2.4.4) with

= const > 0, then and is any number such that

where u(t) solves

54

2. Methods of solving ill-posed problems

Proof of the claim. One has where solves (2.4.38) with = const > 0. Under our assumptions on F, equation (2.4.38) has a unique solution, and So, to prove the claim, it is sufficient to prove that provided that Let and Because one has the equation: Multiplying this equation by w, and using the monotonicity of F, one gets Therefore provided that The claim is proved. Remark 2.4.15. One can prove claims (i) and (ii), formulated below formula (2.4.38), using DSM version presented in Theorem 2.4.20 below. Claim 5. Assume that F is monotone, (2.4.2) holds, and (ii), formulated below formula (2.4.38), hold.

Then claims (i) and

Proof. First, note that (ii) follows easily from (i), because the assumptions F is monotone, and imply, after multiplying the inequality from which claim (ii) follows. Claim (i) follows from Theorem 2.4.20, proved below. Claim 6. Assume that the operator F is monotone, hemicontinuous, defined on all of H, equation F(u) = 0 has a solution, possibly non-unique, is the minimal-norm element of and where and b are constants. Then (2.4.5) holds for the solution to (2.4.4). Proof. The steps of the proof are: 1) we prove that

for the solution to (2.4.4) with where Inequality implies that from any sequence one can select a subsequence, denoted again such that where is some element. We prove that F(v) = 0. 2) we prove that the solution u (t) to the equation

satisfies the relation: arbitrary fixed number. 3) we prove that 4) passing to the limit as and

as

where h > 0 is an

in equation (*) yields F (v) = 0. We prove that

Let us give the details of the proof. 1) Let

note

where u solves (*).Then Multiply this equation by w, use the monotonicity of F, deand get Because this implies

55

This in-

SO

equality implies 2) Denote

Thus where u solves (*), and

Then Multiply this equation

by and use the monotonicity of F to get: Because one gets and

This implies

Under our assumptions about one can check that where 0 < a := 1 – b < 1. Also

as Thus

Divide (2.4.52) by h and let Then one gets 3) Denote so one has in equation (*), yields F(v) = 0. The limit 4) Passing to the limit exists because and so that Lemma 2.1.2 imSince one gets plies F(v) = 0. Let us prove that then lim If this implies strong convergence and together with the weak convergence

Let us prove that lim and

One has Since F is monotone, Thus,

it follows that because Let us prove that Replacing v by in the above argument yields so Since is the unique minimal-norm solution to (2.4.1) and v solves (2.4.1), it follows that Since the limit is the same for every subsequence for which the weak limit of exists, one concludes that the strong limit Indeed, assuming that for some sequence the limit of does not exist, one selects a subsequence, denoted again for which the weak limit of does exist, and proves as before that this limit is thus getting a contradiction. Claim 6 is proved. For convenience of the reader let us prove the global existence and uniqueness of the solution to (2.4.4) with where F is a monotone, hemicontinuous operator in H (cf [Dei]). Uniqueness of the solution is trivial: if there are two solutions, u and v, then their difference w := u – v solves the problem

56

2. Methods of solving ill-posed problems

Multiply this by w and use the monotonicity of F to get g (0) = 0, where Thus, g = 0, so w = 0, and uniqueness is proved. Let us prove the global existence of the solution to (2.4.4) with Consider the equation:

We wish to prove that

where Our assumptions (monotonicity and hemicontinuity of F) imply demicontinuity of F. Fix an arbitrary T > 0, and let be the ball centered at with radius r > 0. Let Then If then and Define

From (**) one gets:

One has:

Using the monotonicity of F, the estimate one gets:

Therefore

and the estimate

57

This implies

Therefore there exists the strong limit w(t):

This function w satisfies the integral equation:

and solves the Cauchy problem

If F is continuous, then this Cauchy problem and the preceding integral equation are equivalent. If F is demicontinuous, then they are also equivalent, but the derivative in the Cauchy problem should be understood in the weak sense. We have proved the existence of the unique local solution to(***). To prove that the solution to (***) exists for any let us assume that the solution exists on [0, T), but not on a larger interval [0, T + d), and show that this leads to a contradiction. It is sufficient to prove that the finite limit: does exist, because then one can solve locally, on the interval [T, T + d), equation (***) with the initial data and construct the solution to (***) on the interval [0, T + d), thus getting a contradiction. To prove that W exists, consider

One has

Using the monotonicity of F, one gets Thus,

The right-hand side of the above inequality tends to zero as Cauchy test imply the existence of W. The proof is complete. 2.4.5

This, and the

Nonlinear ill-posed problems with non-monotone operators

Assume that F(u) := B(u) — f, B is a non-monotone operator, where I is the identity operator,

is as

58

2. Methods of solving ill-posed problems

in Theorem 2.4.10 and

and is defined similarly, with replacing f and The main result of this Section is:

replacing u.

Theorem 2.4.16. If (2.4.2) holds, u, (that is, is sufficiently small), and R << 1 then problem (2.4.4) has a unique global solution and (2.4.5) holds. If solves (2.4.4) (with in place of then there exists a such that

The derivation of the stopping rule, that is, the choice of presented in Section 2.4.4.

is based on the ideas

Sketch of proof of Theorem 2.4.16. Proof of Theorem 2.4.16 consists of the following steps. First, we prove that satisfies a differential inequality (2.4.35), and, applying (4.4), conclude that as A new point in this derivation (compared with that for monotone operators) is the usage of the source assumption Secondly, we derive the stopping rule. The source assumption allows one to get a rate of convergence [KNR]. Details of the proof are technical and are not included. One can see [KNR] for some proofs. Let us sketch the derivation of the differential inequality for g . Write B(u) – f = B(u) – B(y) = Aw + K, where Then (2.4.53) can be written as

Multiplying (2.4.4), with

Since

one obtains:

where the estimate

and

defined in (2.4.54), by w, one gets:

was used. Clearly,

59

One has:

Let This is possible since transforms (2.4.56) into the following inequality:

Using the above estimates, one

Now, apply Theorem 2.4.11 to (2.4.57), choosing and

Then conditions (2.4.34) are satisfied, and Theorem 2.4.11 yields the estimate:

This is the main part of the proof of Theorem 2.4.16. Remark. The assumption is satisfied under very weak assumption on namely, i.e. the image of a ball = const > 0, intersects a punctured ball where a = const > 0. This implies the existence of an element such that where and The condition is satisfied because can be chosen arbitrarily small. 2.4.6

Nonlinear ill-posed problems: avoiding inverting of operators in the Newton-type continuous schemes

In the Newton-type methods for solving well-posed nonlinear problems, for example, in the continuous Newton method with the difficult and expensive part of the solution is inverting the operator In this section we give a method to avoid inverting of this operator. This is especially important in the ill-posed problems, where one has to invert some regularized versions of and to face more difficulties than in the well-posed problems. Consider problem (2.4.1) and assume (2.4.2), and (2.4.3). Thus, we discuss our method in the simplest well-posed case. Replace (2.4.4) by the following Cauchy problem (dynamical system):

where

T := A* A, and Q = Q(t) is a bounded operator in H.

60

2. Methods of solving ill-posed problems

First, let us state our new technical tool: an operator version of the Gronwall inequality. Theorem 2.4.17. Let

where T(t), G(t), and Q(t) are linear bounded operators on a real Hilbert space H. If there exists such that

then

A simple proof of Theorem 2.4.17 is left to the reader. It can be found in [RSm4]. Let us turn now to the proof of Theorem 2.4.18, formulated at the end of this Section. This theorem is the main result of Section 2.4.6. Applying (2.4.62) to (2.4.59), and using (2.4.2) and (2.4.3), which implies

one gets:

as long as Let K, where (2.4.58) as

Let

and

Since one has is the constant from (2.4.2). Rewrite

Multiply (2.4.65) by w and get

We prove below that

61

From (2.4.66) and (2.4.67) one gets the following differential inequality:

which implies:

provided that

Inequality (2.4.70) holds if is sufficiently close to y. From (2.4.69) it follows that Thus, (2.1.6) holds. The trajectory provided that

This inequality holds if is sufficiently close to y, that is, r is sufficiently small. To complete the argument, let us prove (2.4.67). One has:

and one has

Using (2.4.69) and Theorem 2.4.17, one gets

Thus,

If is sufficiently close to y and is sufficiently close to made arbitrarily small. We have proved:

then

can be

Theorem 2.4.18. If (2.4.2), and (2.4.3) hold, and are sufficiently close to and y, respectively, then problem (2.4.58)–(2.4.59) has a unique global solution, (2.4.5) holds, and u(t) converges to y, which solves (2.4.1), exponentially fast.

In [RSm4] a generalization of Theorem 2.4.18 is given for ill-posed problems. Exercise. 1. Let where is a constant, (2.4.2) holds in where y is a solution to (2.4.1) and R > 0 is a constant. Assume

62

2. Methods of solving ill-posed problems

that there exists a and

Then, if R,

such that

where w is some element,

Assume that:

and w are sufficiently small, then the above Cauchy problem has a

unique global solution

and

Precise meaning of the above

smallness condition is explained in [RSm4]. 2. If F is monotone, then the assumption can be dropped, and, under the assumption is sufficiently small, one derives the existence and uniqueness of the global solution to the Cauchy problem of n. 1 of this Exercise. The method of the proof is the same as in [RSm4]. 2.4.7

Iterative schemes

In this section we present a method for constructing convergent iterative schemes for a wide class of well-posed equations (2.4.1). Some methods for constructing convergent iterative schemes for a wide class of Ill-posed problems are given in [AR1]. There is an enormous literature on iterative methods ([BG], [VV]). Consider a discretization scheme for solving (2.4.4) with so that we assume no explicit time dependence in

One of the results from [AR1], concerning the well-posed equations (2.4.1) is Theorem 2.4.19, formulated below. Its proof is shorter than in [AR1]. Theorem 2.4.19. Assume (2.4.2), (2.4.3), (2.4.6)–(2.4.9) with a = 2, sufficiently small, and which

is sufficiently close to

for Then, if h > 0 is then (2.4.75) produces a sequence for

where

Proof of Theorem 2.4.19. The proof is by induction. For n = 0 estimates (2.4.76) are clear. Assuming these estimates for let us prove them for j = n + 1. Let and let solve problem (2.4.4) on the interval with By (2.4.10) (with and (2.4.76) one gets:

63

One has:

and

where we have used the formula

and the estimate:

From (2.4.77)–(2.4.80) it follows that:

provided that

Inequality (2.4.82) holds if h is sufficiently small and So, the first inequality (2.4.76), with n + 1 in place of n, is proved if h is sufficiently small and Now

Using (2.4.2) and (2.4.79), one gets:

From (2.4.83) and (2.4.84) it follows that:

provided that

Inequality (2.4.86) holds if h is sufficiently small and So, the second inequality (2.4.76) with n + 1 in place of n is proved if h is sufficiently small and Theorem 2.4.19 is proved.

64

2. Methods of solving ill-posed problems

In the well-posed case, if

the discrete Newton’s method

converges superexponentially if is sufficiently close to y. Indeed, if then where Thus, satisfies the inequality: where Therefore and if then the method converges superexponentially. If one uses the iterative method with then, in the well-posed case, assuming that this method converges, it converges exponentially, that is, slower than in the case h = 1. The continuous analog of the above method

where a = const > 0, converges at the rate so

Indeed, if Thus

then

In the continuous case one does not have superexponential convergence no matter what a > 0 is (see [R210]). 2.4.8

A spectral assumption

In this section we introduce the spectral assumption which allows one to treat some nonlinear non-monotone operators. Assumption S: The set where and are arbitrarily small, fixed numbers, consists of the regular points of the operator for all Assumption S implies the estimate:

because and our case,

where s (A) is the spectrum of a linear operator A, is the distance from a point of a complex plane to the spectrum. In and if

Theorem 2.4.20. If (2.4.2) and (2.4.87) hold, and (2.4.38), with is solvable, problem (2.4.4), with and has a unique global solution, and solution to the equation is isolated.

then problem defined in (2.4.33) Every

65

Proof of Theorem 2.4.20. Let solves locally (2.4.4), where is defined in (2.4.33) and

where u = u(t) Then:

so

Thus,

Therefore equation

has a solution in where Every solution to equation (2.4.91) is isolated. Indeed, if and then so where Thus, using (2.4.87), one gets unless Consequently, if is sufficiently small, then Theorem 2.4.20 is proved. 2.4.9

Nonlinear integral inequality

The main result of this section is Theorem 2.4.22, which is used extensively in Sections 2.4.4–2.4.8. The following lemma is a version of some known results concerning integral inequalities. Lemma 2.4.21. Let f (t, w), g (t, u) be continuous on region and if Assume that g (t, u) is such that the Cauchy problem

has a unique solution. If

then

for all t for which u (t) and w (t) are defined.

Proof of Lemma Lemma 2.4.21. Step 1. Suppose first f ( t , w ) < g ( t , u ) , if Since there exists such that u (t) > w (t) on

and Assume

66

2. Methods of solving ill-posed problems

that for some

one has

Then for some

one has

One gets

This contradiction proves that there is no point Step 2. Now consider the case

where

such that if

Define

tends monotonically to zero. Then

By Step 1

Fix an arbitrary compact set

Since g (t, u) is continuous, the sequence is uniformly bounded and equicontinuous on Therefore there exists a subsequence which converges uniformly to a continuous function u (t). By continuity of g (t, u) we can pass to the limit in (2.4.94) and get

Since is arbitrary (2.4.95) is equivalent to the initial Cauchy problem that has a unique solution. The inequality implies If the solution to the Cauchy problem (2.4.92) is not unique, the inequality holds for the maximal solution to (2.4.92). The following theorem was stated earlier as Theorem 2.4.11. For convenience of the reader, we repeat its formulation. Theorem 2.4.22. Let a positive function

for some real number such that

If there exists

67

then a non-negative solution to the following inequalities:

satisfies the estimate:

for all

where

Remark 2.4.23. Without loss of generality one can assume In [Alb1] a differential inequality was studied under some assumptions which include, among others, the positivity of for v > 0. In Theorem 2.4.22 the term (which is analogous to some extent to the term can change sign. Our Theorem 2.4.22 is not covered by the result in [Alb1]. In particular, in Theorem 2.4.22 an analog of for the case is the function This function goes to as v goes to so it does not satisfy the positivity condition imposed in [Alb1]. Lemma 1 in [Alb1] is wrong. Its corrected version is given in [ARS3], where a counterexample to Lemma 1 from [Alb1] is constructed. In [ARS3] the following result is proved: Lemma. Let

and

Assume: 1) for t > 0, 2) as 3) f (0) = 0, f (0) > 0 for u > 0, 4) there exists c > 0 such that for Then as Unlike in the case of Bihari integral inequality ([BB]) one cannot separate variables in the right hand side of the first inequality (2.4.97) and estimate v(t) by a solution of the Cauchy problem for a differential equation with separating variables. The proof below is based on a special choice of the solution to the Riccati equation majorizing a solution of inequality (2.4.97). Proof of Theorem 2.4.22. Denote:

68

2. Methods of solving ill-posed problems

then (2.4.97) implies:

where

Consider Riccati’s equation:

One can check by a direct calculation that the the solution to problem (2.4.102) is given by the following formula [Kam, eq. 1.33]:

Define f and g as follows:

and consider the Cauchy problem for equation (2.4.102) with the initial condition Then C in (2.4.103) takes the form:

From (2.4.96)) one gets

Since fg = –1 one has:

Thus

69

It follows from conditions (2.4.96) and from the second inequality in (2.4.97) that the solution to problem (2.4.102) exists for all and the following inequality holds with v(t) defined by (2.4.99):

From Lemma 2.4.21 and from formula (2.4.105) one gets:

and thus estimate (2.4.98) is proved. To illustrate conditions of Theorem 2.4.22 consider the following examples of functions and satisfying (2.4.96) for Example 2.4.24. Let

where Choose the following conditions

From (2.4.96), (2.4.97) one gets

Thus, one obtains the following conditions:

and

Therefore for such

and

and

a function

with the desired properties exists if

70

2. Methods of solving ill-posed problems

In this case one can choose as one needs the following conditions:

However in order to have

and

Example 2.4.25. If

then conditions (2.4.96), (2.4.97) are satisfied if

Example 2.4.26. Here log stands for the natural logarithm. For some

conditions (2.4.96), (2.4.97) are satisfied if

In all considered examples tends to infinity as and provide a decay of a nonnegative solution to integral inequality (2.4.97) even if tends to infinity. Moreover, in the first and the third examples v(t) tends to zero as when and 2.4.10 Riccati equation

An alternative approach to a study of Riccati equation (2.4.101) with non-negative coefficients a (t) and b (t) is based on the iterative method for solving integral equation Let Then Assume and consider the process By induction, If where T > 0 is an arbitrary number, then exists and solves (2.4.101).

71

Assume

Then

where is a constant estimated below. Inequality (2.4.117) holds for n = 0, and, by induction,

Using (2.4.116) one gets:

If

then (2.4.117) has been proved by induction. Inequality (2.4.118) holds if and 2.5

2.5.1

EXAMPLES OF SOLUTIONS OF ILL-POSED PROBLEMS

Stable numerical differentiation: when is it possible?

In many applications one has to estimate a derivative given the noisy values of the function f to be differentiated. As an example we refer to the analysis of photoelectric response data. The goal of that experiment is to determine the relationship between the intensity of light falling on certain plant cells and their rate of uptake of various substances. Rather than measuring the uptake rate directly, the experimentalists measure the amount of each substance not absorbed as a function of time, the uptake rate being defined as minus the derivative of this function. As for the other example, one can mention the problem of finding the heat capacity of a gas as a function of temperature T. Experimentally one measures the heat content and the heat capacity is determined by numerical differentiation. One can give many other examples of practical problems in which one has to differentiate noisy data. In navigation problems one selects the direction of the motion

72

2. Methods of solving ill-posed problems

of a ship by the maximum of a certain univalent curve, called the navigation characteristic. This direction can be obtained by differentiation of this curve. Since the navigation characteristic is communicated with some errors, one has to differentiate it numerically in order to find its maximum. In [R83, p. 94], the shape of a convex obstacle is found by differentiation of a support function of this obstacle. The support function is found from the experimentally measured scattering data, and by this reason the support function is noisy. In [R121, pp. 81–92], optimal estimates for the derivatives of random functions are obtained. In [RKa, p. 438], numerical differentiation of functions, contaminated by random noise is discussed. The noise has zero mean value and finite variance, and is identically distributed independently of the point x. It is proved that in this case the error of an optimal formula of numerical differentiation can be made where p is the number of observation points and is the error for a noise which is non-random (see the precise formulation of the result in [RKa]). Section 2.5.1 is essentially paper [RSm5]. The differentiation of noisy data is an ill-posed problem: small (in some norm) perturbations of a function may lead to large errors in its derivative. Indeed, if one takes then and so that small in perturbations of f result in large perturbations of in Various methods have been developed for stable numerical differentiation of f given We mention three groups of methods: being a regularization (1) regularized difference methods with a step size parameter, see [R18], where this idea was proposed for the first time, and [R156], [R58], [R168]. As an example of such a method one may consider:

If and where functions whose n-th derivative belongs to

where

is the Sobolev space of then

is an estimation constant: The error in the interval can be estimated slightly better (by a quantity In this paper The right-hand side of (2.5.2) attains the absolute we denote by while if one uses the error estimate minimum

73

for the interval then one gets the absolute minimum at When the function one can modify (2.5.1) near the ends so that it has the order of the error of approximation as and results in an algorithm of order For example one can take

The difference methods use only local values of the function which is natural when one estimates a derivative, and these methods are very simple, which is an advantage. (2) An alternative approach is first to smooth by a mollification with a certain kernel, for example with a Gaussian kernel, or to use a mollification by splines, and then to differentiate the resulting smooth approximation, see e.g. [VA]. If one applies mollification with the Gaussian kernel then

where that

stands for the convolution, with f (0) = f (1) = 0 and

and

Assume One has

From the Cauchy inequality the first term in the right-hand side of (2.5.5) can be estimated as follows:

because one gets:

By a partial integration

75

or

where

> is a regularization parameter. One proves that for a suitable choice of the above functionals have a unique minimizer and as An optimal choice of the regularization parameter in this approach is a nontrivial problem. Some methods for choosing are presented in Section 2.1.

The above methods have been discussed in the literature, and their analysis is not our goal. Our goal is to study two principally different statements of the problem of stable numerical differentiation, and to understand when it is possible in principle to get a stable approximation to given noisy data In Problem I a new notion of regularizer is introduced. Our treatment of the stable differentiation is an example of application of this new notion. In Section 2.1 a regularization method for unbounded linear and nonlinear operators is discussed. Statements of the problem of stable numerical differentiation

First, we recall some standard definitions (cf. Section 1.3). The problem of finding a solution u to the equation

where X and Y are Banach spaces, A is an operator, possibly nonlinear, is well-posed (in the sense of J. Hadamard) if the following conditions hold: (a) for every element there exists a solution (b) this solution is unique; (c) the problem is stable under small perturbations of the initial data in the sense:

If at least one of the conditions (a), (b) or (c) is violated, then the problem is called ill-posed. The problem of numerical differentiation can be written as

We study the cases p = 2 and in detail. Problem (2.5.14) is solvable only if So, condition (a) is not satisfied, condition (c) is not satisfied either, and condition (b) is satisfied. Therefore, problem (2.5.14) is ill-posed. Practically, one does not know f and the only information available for computational processing is together with an a priori information about f, for example, one

76

2. Methods of solving ill-posed problems

may know that

a = 0, a = 1, or 1 <

where

For 1 < a < 2

Therefore given and one has to estimate for any and the problem of stable numerical differentiation has to be understood in the following sense: Problem I: Find a linear or nonlinear operator such that

where is some positive continuous function of and is some number. The traditional formulation of the problem of stable numerical differentiation is different from the above: Problem II: Find a linear or nonlinear operator such that

where or even in a weaker form: Find such that

and

is fixed,

for a fixed and fixed family Note the principal difference in the statements of Problems I and II of stable numerical differentiation: in Problem I the data are f is arbitrary in the set and we wish to find a stable approximation of which is valid uniformly with respect to On the other hand, in Problem II it is assumed that is fixed and the approximation of is either uniform with respect to or holds for a particular family Since in practice we do not know f and we do know only one family Problem I is much more important practically than Problem II. In this Section we show when one can obtain, in principle, a stable approximation of in the sense formulated in Problems I and II, and when it is not possible, in principle, to obtain such an approximation of from noisy data.

77

The main result on the stable numerical differentiation problem in the first formulation is stated in Theorem 2.5.1: Theorem 2.5.1. There does not exist an operator nonlinear, for p = 2 and such that inequality (2.5.17) holds for then

If a > 1 and mple, one can use

then there does exist an operator defined in 2.4.92 with

linear or If

such that 2.5.17 holds. For exa-

The error of the corresponding differentiation formula is

The main result on the stable numerical differentiation problem in the second formulation is stated in Theorem 2.5.2: Theorem 2.5.2. If a = 1, then there exists an operator that inequality (2.5.18) holds.

such

The principal difference is: for a = 1 one cannot differentiate stably in the sense formulated in Problem I. In the sense of Problem II Stable differentiation is possible in principle. However the approximation error, cannot be estimated, and this error may go to zero arbitrarily slowly as This is in sharp contrast with the practically computable error estimate given in (2.5.21). Moreover, no matter how small the error bound is, there exists a function such that not only (with any fixed function defined in (2.5.35), but any other operator linear or nonlinear, will satisfy the inequality where c > 0 does not depend on This follows from (2.5.19). Proofs Proof of Theorem 2.5.1. First, consider the case

Take

78

2. Methods of solving ill-posed problems

Extend from (0, 2h) to (2h, 1) by zero and denote the extended function by again. Then and the norms a = 0 and a = 1 are preserved. Define Note that

Choose

so that

Then for

one has:

Let

One gets

If then If a = 0, then (2.5.23) implies that k = 1, 2, with For any fixed and the constant M in (2.5.22) can be chosen arbitrary. Therefore inequality (2.5.25) proves that (2.4.10) is false in the class and, in fact, as Suppose now that One has

Thus, for given

and

one can take

so that

k = 1, 2, holds, and then take M so that For these and M the functions k = 1, 2. One obtains from (2.5.25) the following inequality

which implies that estimate (2.4.10) is false in the class Now let p = 2. For defined in (2.5.22) one has

79

Extend from (0, 2h) to (2h, 1) by zero and denote the extended function again. Then and fine Choose to satisfy the identity

then

De-

k = 1 , 2 . Thus,

where

Therefore

If a = 0, then (2.5.29) yields k = 1 , 2 , with gets as Given constants and (in the case a = 1), one takes and then takes M so that and M the functions

Finally, consider using the Lagrange formula:

and one so that With this choice of

k = 1, 2, and one obtains from (2.5.30)

For the operator

defined by (2.4.92) one gets

Minimizing the right-hand side of (2.5.32) with respect to

yields

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2. Methods of solving ill-posed problems

The case a = 2 is treated in the introduction (see estimate 2.4.93). So one arrives at 2.4.13–2.4.14. This completes the proof. Proof of Theorem 2.5.2. We give two proofs based on quite different methods. The first proof uses the construction of the regularizing operator defined in (2.4.92). The right-hand side of the error estimate of the type (2.4.93) is now replaced by where as provided that a = 1. Minimizing E with respect to h for a fixed one obtains a minimizer as and the error estimate as Therefore one gets (2.6). Alternatively, if one chooses as such that then The first proof is completed. Remark 2.5.3. The statement of Theorem 2.5.2 with (2.4.11) replaced by (2.4.12) is obvious: since f is fixed, one may take This is, of course, of no practical use because is unknown. The second proof is based on the DSM. The ideas of this proof have an advantage of being applicable to a wide variety of ill-posed problems, and not only to stable numerical differentiation. By this reason we give this proof in detail. In order to show that for a = 1 there exists an operator is a real Hilbert space) such that (2.5.18) holds we will use the DSM (dynamical systems method) (see Section 2.4). This approach consists of the following steps: Step 1. Solve the Cauchy problem:

where A is defined in (2.4.7), p = 2,

Step 2. Calculate as Then

where

is a number such that

and

with given by (2.5.49) below and To verify (2.5.36) consider the problem

and

and

81

Since A is monotone in

for any and admits the estimate

the solution

to (2.5.37) exists, is unique, and

Differentiate (2.5.37) with respect to t (this is possible by the implicit function theorem) and get

where (2.5.39) was used. Denote

From (2.5.37) and (2.5.33) one obtains

Multiply (2.5.42) by

Let

Since

So,

and get

then (2.5.39) and (2.5.43) imply

from (2.5.44) and (2.5.40) it follows that

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2. Methods of solving ill-posed problems

Under assumption (2.5.34), one has

Indeed, (2.5.34) implies and Conclusion (2.5.46) follows. If one chooses so that and as then by (2.5.47’) and (2.5.46), using L’Hospital’s rule one obtains

The existence of the solution to (2.5.33) on is linear with a bounded operator. We claim that

is obvious, since equation (2.5.33)

For convenience of the reader this claim is established below. Equations (2.5.34), (2.5.45), (2.5.47), and (2.5.48) imply:

Let us now prove (2.5.48). Because and one can rewrite (2.5.37) as This and the monotonicity of A imply so, since h(t) > 0, one gets and Thus, w converges weakly in to some element as Because A is monotone, it is w-closed, that is and imply so and The inequality can be written as and because Therefore the claim (2.5.48) is proved and the second proof is completed. Numerical aspects

Figures 1–4 illustrate the impossibility to differentiate stably a function, which does a < 1. If one takes the function not have a bound on

83

and

then

and any formula of numerical differentiation will give error not going to zero as because, by (2.5.27), one has:

In Figure 1 one can see f (x) given by (2.5.50) with

and

84

2. Methods of solving ill-posed problems

Figure 2 presents

Figure 3 shows the case

and

The derivatives are given in Figure 4:

Even if the bound on in some norm is given, one can experience difficulties with stable differentiation. Namely, if is fixed and is very large, then in finite difference scheme (2.4.92) is very small, and practitioners might not have sufficiently many observation points. Another difficulty is: the estimated error in such a case is very big and does not give any information regarding the accuracy of computations. This is illustrated in Table 2.1 below for the function and Figures 5–8 show the exact and computed derivatives of The derivatives of this function were computed in the presence of the noise function

and with different step sizes. One can see in Figure 5 that for the computed derivative is very accurate. However as h grows, the accuracy decreases.

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2.5.2

Stable summation of the Fourier series and integrals with perturbed coefficients

Assume that f (x) is a smooth

where

and

function

86

2. Methods of solving ill-posed problems

Assume that

are given such that

and are not known. The problem is: given

calculate stably

In other words, calculate

such that

The norm here can depend on the particular problem. Let us assume that this is the norm. Let us look for of the form:

One has, using Parseval’s equality,

where the constants and are defined by the last equation. Minimizing the righthand side of (2.5.62) with respect to v > 1, with being fixed, we find the optimal

and the error estimate:

If s = 2, then

Let us summarize the result.

Theorem 2.5.4. Assume that are given such that (2.5.58) and (2.5.59) hold. Define by formula (2.5.61) with where is defined in (2.5.63). Then (2.5.60) holds with defined by (2.5.64).

87

The above arguments are applicable also to Fourier integrals with perturbed Fourier transforms, which play the role of the perturbed coefficients. 2.5.3

Stable solution of some Volterra equations of the first kind

Consider equation (1.3.1) with bounded function, and is a kernel for which then (1.3.1), after a differentiation, yields

If

is a

provided that This is a Fredholm second-kind integral equation, for which the Fredholm alternative is valid in If is not differentiable with respect to x, or V(x, x) may vanish, then equation (1.3.1) with A = V can be solved by the methods discussed in Section 2.1–2.4. A different general approach to stable solution of the equation Vu = f consists of the factorization where S is a compact operator, I is the identity, the null-space N(I + S) = {0} is trivial, so that by Fredholm’s alternative the equation (I + S)u = w can be stably solved by a projection method, and the operator Q is such that w can be stably found from the equation Qw = f. If the noisy data are given in place of f, then a stable solution to the equation is given by a formula and a stable solution of the equation Vu = f with noisy data is given by the formula A numerical implementation of this scheme is given in [RSm6]. If and then the above scheme leads to the operator Q whose stable inversion is equivalent to stable differentiation, A discretization method for stable solution of Volterra integral equations of the first kind is proposed and justified in [RG], where the error estimates for the proposed method are also obtained. 2.5.4

Deconvolution problems

Equation (1.3.1) with tions 2.1–2.4, provided that

can be solved by the methods of SecIn the special case, when the kernel

where Q and P are elliptic operators, is the delta-function, one can use the theory from [R121], [R189], and prove, under suitable assumptions, that the operator is an isomorphism of onto where n = ord Q, m = ord P, is the usual Sobolev space, is its dual with respect to inner product, is, in other words, the closure of in the norm of that is, the subset of the elements of with support

88

2. Methods of solving ill-posed problems

If L is a selfadjoint elliptic operator in ord L := s and and are positive polynomials, deg Q = n, deg then Q := Q(L) and P := P(L) are elliptic operators, and if solves (2.5.1) then Equation Rh = f has a unique solution in for any and is an isomorphism. In [R121], [R189] one finds analytical formulas for the solution h. Under the above assumptions, the equation (*) Rh = f in does not have integrable solutions, in general. It has only distributional solutions of finite order of singularity, in general. Finding a solution of minimal order of singularity (mos solution) is a well posed problem ([R121]). The minimal order of singularity is equal to a. Since R is an isomorphism of onto the problem of solving equation (*) is well-posed. One does not need regularization methods for finding the solution to (*). Example 2.5.1.

In this example

One can see that generically h (x) is not an integrable function, it is a distribution, If and only if and and is Thus, equation

has a solution

which depends on where

continuously in the norm Details one can see in

[R121]. 2.5.5

Ill-conditioned linear algebraic systems

Let in (1.3.1). Then A can be represented by a matrix non-singular, define its condition number

If A is

One has (see (1.4.3)) Thus, if errors

is large then small relative errors

in the data may lead to large relative

in the solution. Solving linear algebraic ill-conditioned is an ill-posed prob-

lem practically. Note that if A is not singular (not degenerate: det are eigenvalues of A* A.

because Also,

where

89

Examples below are taken from [VA]. f = (4.1, 9.7), u = (1, 0), Au = f. Let g =

Example 2.5.2.

(4.11, 9.70). Then v = (0.34, 0.97). Here

249.5.

Example 2.5.3. Hilbert matrix

if n = 6,

if n = 10. 2.6

Let

PROJECTION METHODS FOR ILL-POSED PROBLEMS

be a linear, injective, bounded operator in a Hilbert space is unbounded, so that the problem of solving the equation Au = f is

ill-posed. Let be a sequence of finite-dimensional subspaces of H which is limit-dense in H, that is, for any one has where is the distance from f to Let be the orthoprojection operators on Assume that is given such that The problem consists of finding which solves the equation (*) and such that Equation (*) is an equation of a projection method. A general approach to finding a stable approximation is the following one: let be a regularizer in the sense and define Then where

as as and is chosen suitably. More precisely, for any fixed one has because the sequence is limit-dense. Thus, one can choose so that then and for a fixed one can choose so that as Therefore as so that u is stably approximated. There are many ways to choose the regularizer one can use a Variational Regularization, Quasisolutions, Iterative Regularization, and the DSM. One can also use the method developed in Section 2.5.2 for constructing a convergent projection method for solving (1.3.1). Assume that X = Y = H in (1.3.1), A is a linear compact operator, and (1.3.1) is solvable, i.e, f = Ay for some We assume that that is, is the minimal-norm solution. By Lemma 2.1.11, a solvable equation (1.3.1) is equivalent to equation (2.1.6). Denote := A* f, assume that is given in place of and Let where j = 1, 2, , is an orthonormal system of eigenvectors of the selfadjoint compact operator B, and are the corresponding eigenvalues. Let and The element is a regularizer if is chosen properly, as in Section 2.5.2. In this case one has See more on this topic in [IVT], [VA].

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3. ONE-DIMENSIONAL INVERSE SCATTERING AND SPECTRAL PROBLEMS

Inverse scattering and spectral one-dimensional problems are discussed in this Chapter systematically in a self-contained way. Many novel results due to the author are presented. The classical results are often presented in a new way. Several highlights of the new results include: (1) Analysis of the invertibility of the steps in the Gel’fand-Levitan and Marchenko inversion procedures; (2) Theory of the inverse problem with I-function as the data and its applications; (3) Proof of the Property C for ordinary differential operators, numerous applications of Property C; (4) Inverse problems with “incomplete” data; (5) Spherically symmetric inverse scattering problem with fixed-energy phase shifts: uniqueness result is obtained for the case when part of the phase shifts is known. The Newton-Sabatier (NS) scheme for inversion of fixed-energy phase shifts is analyzed. This analysis shows that the NS scheme is fundamentally wrong in the sense that its foundations are wrong, and this scheme is not a valid inversion method; (6) Complete presentation of the Krein inverse scattering theory is given for the first time. Consistency of this theory is proved; (7) Quarkonium systems are recovered from experimental data; (8) A study of the properties of I-function; (9) Some new inverse problems for the heat and wave equations are studied;

92

3. One-dimensional inverse scattering and spectral problems

(10) A study of an inverse scattering problem for an inhomogeneous Schrödinger equation; (11) A study of inverse problem of ocean acoustics; (12) Theory of ground-penetrating radars is given. 3.1

INTRODUCTION

3.1.1

What is new in this chapter?

There are excellent books [M] and [L], where inverse spectral and scattering problems are discussed in detail. Therefore let us point out the novel points in this Chapter. (1) A new approach to the uniqueness of the solutions to these problems. This approach is based on Property C for Sturm-Liouville operators; (2) the inverse problem with I-function as the data is studied and applied to many inverse problems; (3) a detailed analysis of the invertibility of the steps in Marchenko and Gel’fand-Levitan (GL) inversion procedures is given; (4) inverse problems with “incomplete” data are studied; (5) a detailed presentation of Krein’s inversion method with proofs is given apparently for the first time; (6) a number of new results for various inverse problems are presented. These include, in particular, (a) a uniqueness theorem for recovery of a compactly supported sphericallysymmetric potential from a part of the corresponding fixed-energy phase shifts; (b) a method for finding confining potential (a quarkonium system) from a few experimental data; (c) solution of several new inverse problems for the heat- and wave equations; (d) a uniqueness theorem for finding a potential q from a part of the corresponding fixed-energy phase shifts; and many other results which are taken from [R196], and especially [R221] and [R139]. (e) a solution to an inverse problem of ocean acoustics. An attempt to study a similar problem has been made in [GX]. We show that the method presented in [GX] is invalid; (f) a theory of ground-penetrating radars.

Due to the space limitations, several important questions are not discussed: inverse scattering on the full line, iterative methods for finding potential q : (a) from two spectra [R181], [R221], (b) from S-matrix alone when q is compactly supported [R184], approximate methods for finding q from fixed-energy phase shifts [RSch], [RSm2], property of resonances [R139], inverse scattering for systems of equations, etc. 3.1.2

Auxiliary results

Let

where consists of functions belonging to and overline stands for complex conjugate.

and for any

93

Consider the differential expression with domain of definition where is the set of vanishing in a neighborhood of infinity, If H is the Hilbert space then is densely defined symmetric linear operator in H, essentially self-adjoint, that is, the closure of in H is selfadjoint. It is possible to construct a selfadjoint operator assuming that Such a theory is technically more difficult, because it is not even obvious a priori that the set is dense in H (in fact, it is). Such a theory is presented in [Nai]. If one drops the assumption then is not a domain of definition of since there are functions for which In the future we mean by a self-adjoint operator generated by the differential expression and the boundary condition u(0) = 0. This operator has absolutely continuous spectrum, which fills and discrete, finite, negative spectrum where are the eigenvalues of all of them are simple,

where

are corresponding eigenfunctions, which are real-valued functions, and

The functions

and

are defined as the unique solutions to the problems:

These functions are well defined for any Their existence and uniqueness can be proved by using the Volterra equations for and If then the Jost solution f (x, k) exists and is unique. This solution is defined by the problem:

Existence and uniqueness of f is proved by means of the Volterra equation:

If

then this equation implies that f (x, k) is an analytic function of k in for k > 0. The Jost function is defined as f (k) := f (0, k). It has exactly J simple roots where are the negative eigenvalues of The number k = 0 can be a zero of f (k). If f (0) = 0, then where Existence of is a fine result under the only

94

3. One-dimensional inverse scattering and spectral problems

assumption

(see Theorem 3.1.3 below, and [R139]) and an easy one if The phase shift is defined

by the formula

where the last equation in (3.1.7) follows from (3.1.6). Because for One defines the S-matrix by the formula

one has

The function S(k) is not defined for complex k if but if then f (k) is an entire function of k and S (k) is meromorphic in If q (x) = 0 for x > a, then f (k) is an entire function of exponential type (see Section 3.5.1). If then at the Jost solution is proportional to and both belong to The integral equation for is:

One has:

because the right-hand side of (3.1.10) solves equation (3.1.5) and satisfies conditions (3.1.3) at x = 0. The first condition (3.1.3) is obvious, and the second one follows from the Wronskian formula:

If then as one can derive easily from equation (3.1.6). In fact, If k > 0, then If then is a real-valued function. The function f (x, k) is analytic in but is, in general, not defined for In particular, (3.1.11), in general, is valid on the real axis only. However, if then f (k) is defined on the whole complex plane of k, as was mentioned above. Let us denote for and let be the second, linearly independent, solution to equation (3.1.5) for If then One can write a formula, similar to (3.1.10), for

95

where

For

it is necessary and sufficient that

In fact,

where To prove the second relation in (3.1.13), one differentiates (3.1.5) with respect to k and gets

Existence of the derivative with respect to k in follows easily from equation (3.1.6). Multiply (3.1.13a) by f and (3.1.5) by subtract and integrate over then by parts, put and get:

Thus,

It follows from (3.1.14) that constants:

The numbers

are called the norming

Definition 3.1.1. Scattering data is the triple:

The Jost function f (k) may vanish at k = 0. If f (0) = 0, then the point k = 0 is called a resonance. If then the zeros of f (k) in are called resonances. As we have seen above, there are finitely many zeros of f (k) in these zeros are simple, their number J is the number of negative eigenvalues of the selfadjoint Dirichlet operator If then the negative spectrum of is finite [M]. The phase shift defined in (3.1.17), is related to S(k):

96

3. One-dimensional inverse scattering and spectral problems

so that S (k) and are interchangeable in the scattering data. One has because solves (3.1.3). Therefore

Thus

In Section 3.4.1 the notion of spectral function is defined. It will be proved in Section 3.5.1 for that the formula for the spectral function is:

where are defined in (3.1.18)–(3.1.19). The spectral function is defined in Section 3.4.5 for any Such a q may grow at infinity. On the other hand, the scattering theory is constructed for Let us define the index of S(k):

This definition implies that

Therefore:

because a simple zero k = 0 contributes to the index, and the index of an analytic in function f (k), such that equals to the number of zeros of f (k) in plus half of the number of its zeros on the real axis, provided that all the zeros are simple. This follows from the argument principle. In Section 3.4.2 and Section 3.5.2 the existence and uniqueness of the transformation (transmutation) operators will be proved. Namely,

97

and

and the properties of the kernels A(x, y) and K(x, y) are discussed in Section 3.5.2 and Section 3.4.2 respectively. The transformation operator I + K transforms the solution to the equation (3.1.3) with q = 0 into the solution of (3.1.3), satisfying the same as boundary conditions at x = 0. The transformation operator I + A transforms the solution to equation (3.1.5) with q = 0 into the solution f of (3.1.5) satisfying the same as “boundary conditions at infinity”. One can prove (see [M] and Sec. 5.7) the following estimates

and A(x, y) solves the equation:

By we denote Sobolev spaces The kernel A(x, y) is the unique solution to (3.1.27), and also of the problem (3.5.1)–(3.5.3). 3.1.3

Statement of the inverse scattering and inverse spectral problems

ISP: Inverse scattering problem (ISP) consists of finding tering data (see (3.1.6)). A study of ISP consists of the following:

from the corresponding scat-

(1) One proves that ISP has at most one solution. to be scattering data corre(2) One finds necessary and sufficient conditions for sponding to a (characterization of the scattering data problem). (3) One gives a reconstruction method for calculating from the corresponding

In Section 3.5 these three problems are solved. ISpP: Inverse spectral problem consists of finding q from the corresponding spectral function. A study of ISpP consists of the similar steps: (1) One proves that ISpP has at most one solution in an appropriate class of q: if and from this class generate the same then

98

3. One-dimensional inverse scattering and spectral problems

(2) One finds necessary and sufficient conditions on which guarantee that a spectral function corresponding to some q from the above class. (3) One gives a reconstruction method for finding q (x) from the corresponding 3.1.4

is

Property C for ODE

Denote by operators corresponding to potentials corresponding Jost solutions, m = 1, 2. Definition 3.1.2. We say that a pair is complete (total) in This means that if

and by

has property

iff the set

then

We prove in Section 2.1 that a pair does have property and let correspond to

Definition 3.1.3. We say that a pair is complete in for any b > 0, This means that if

the

has property

if

Let

iff the set

then:

In Theorem 3.2.5 we prove that there is a

for which

for a suitable Therefore Property with hold, in general. Property is defined similarly to Property with functions In Section 3.2 we prove that properties and cations of these properties throughout this work.

does not replacing

hold, and give many appli-

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3.1.5

A brief description of the basic results

The basic results of this work include: (1) Proof of properties and Demonstration of many applications of these properties. (2) Analysis of the invertibility of the steps in the inversion procedures of Gel’fandLevitan (GL) for solving inverse spectral problem:

where

the kernel L = L(x, y) is:

is the spectral function of Levitan equation

with q = 0, and K solves the Gel’fand-

Our basic result is a proof of the invertibility of all the steps in (3.1.31):

which holds under a weak assumption on

where

Namely, assume that

is the set of non-decreasing functions of bounded variation on every interval such that the following two assumptions, and hold. Denote the set of functions vanishing in a neighborhood of infinity. Let and Assumption is:

100

3. One-dimensional inverse scattering and spectral problems

Let

and belong to Assumption is:

and

(see Section 3.4.2).

In order to insure the one-to-one correspondence between spectral functions and selfadjoint operators we assume that q is such that the corresponding is “in the limit point at infinity case”. This means that the equation has exactly one nontrivial solution in If then is “in the limit point at infinity case”. (3) Analysis of the invertibility of the Marchenko inversion procedure for solving ISP:

where

and A(x, y) solves the Marchenko equation

Our basic result is a proof of the invertibility of the steps in (3.1.40):

under the assumption

We also derive a new equation for

101

This equation is:

The function A(y) is of interest because

Therefore the knowledge of A(y) is equivalent to the knowledge of f(k). In Section 3.5.5 we give necessary and sufficient conditions for to be the scattering data corresponding to We also prove that if

and, in particular, if

then S(k) alone determines q (x) uniquely, because it determines under the assumption (3.1.47) or (3.1.48).

and J uniquely

(4) We give a very short and simple proof of the uniqueness theorem which says that the I-function,

determines

uniquely. The I-function is equal to Weyl’s m -function if

We give many applications of the above uniqueness theorem. In particular, we give short and simple proofs of the Marchenko’s uniqueness theorems which say that determines uniquely, and determines q uniquely. We prove that if (3.1.48) (or (3.1.47)) holds, then either of the four functions S(k), f (k), determines q (x) uniquely. This result is applied in Section 3.10 to the heat and wave equations. It allows one to study some new inverse problems. For example, let

102

3. One-dimensional inverse scattering and spectral problems

Assume

and let the extra data (measured data) be

The inverse problem is: given these data, find q (x). Another example: Let

The extra data are

The inverse problem is: given these data, find q (x). Using the above uniqueness results, we prove that these two inverse problems have at most one solution. The proof gives also a constructive procedure for finding q . (5) We have already mentioned uniqueness theorems for some inverse problems with “incomplete data”. “Incomplete data” means the data, which are a proper subset of the classical data, but “incompleteness” of the data is compensated by the additional assumptions on q. For example, the classical scattering data are the triple (3.1.16), but if (3.1.48) or (3.1.47) is assumed, then the “incomplete data” alone, such as S(k), or or f(k), or determine q uniquely. Another general result of this nature, that we prove in Section 3.7, is the following one. Consider, for example, the problem

Other boundary conditions can also be considered. Assume that the following data are given.

103

where 0 < b < 1, and

Assume also

We prove Theorem 3.1.4. Data (3.1.60)–(3.1.61) determine uniquely q (x) on the interval if If (3.1.62) is assumed additionally, then q is uniquely determined if The gives the “part of the spectra” sufficient for the unique recovery of q on [0, b]. For example, if and (3.1.62) holds, then so “one spectrum” determines If then so “half of the spectrum” determines uniquely q on If then of the spectrum” determine uniquely q uniquely q on on If b = 1, then and “two spectra” determine q uniquely on the whole interval [0, 1]. The last result belongs to Borg [B]. By “two spectra” one means where are the eigenvalues of the problem:

In fact, two spectra determine not only q but the boundary conditions as well [M]. (6) Our basic results on the spherically symmetric inverse scattering problem with fixed-energy data are the following. The first result: If q = q (r) = 0 for r > a, a > 0 is an arbitrary large fixed number, and then the data determine q (r) uniquely. Here is the phase shift at a fixed energy is the angular momentum, and is any fixed set of positive integers such that

The second result is: If q = q(x), q = 0 for where then the knowledge of the scattering amplitude at a fixed energy and all determine q (x) uniquely, see Section 5.2. Here are arbitrary small open subsets in and is the unit sphere in The scattering amplitude is defined in Section 3.6.1.

104

3. One-dimensional inverse scattering and spectral problems

The third result is: The Newton-Sabatier inversion procedure (see [CS], [N]) is fundamentally wrong. (7) Following [R197] we present, apparently for the first time, a detailed exposition

(with proofs) of the Krein inversion theory for solving inverse scattering problem and prove the consistency of this theory. (8) We give a method for recovery of a quarkonium system (a confining potential) from a few experimental measurements. (9) We study various properties of the I-function. (10) We study an inverse scattering problem for inhomogeneous Schrödinger equation. (11) We study an inverse problem of ocean acoustics. (12) We develop a theory of ground-penetrating radars. 3.2

PROPERTY C FOR ODE

3.2.1

Property

By ODE in this section, the equation

is meant. Assume Then the Jost solution f (x, k) is uniquely defined. In Section 1.3 Definition 3.1.2, property is explained. Let us prove Theorem 3.2.1. If

j = 1, 2 then property

holds.

Proof. We use (3.1.24) and (3.1.25). Denote

for some

where

Set

Let

and get

105

Thus,

where V* is the adjoint to a Volterra operator,

Using (3.2.3) and (3.2.4) one rewrites (3.2.2) as

The right-hand side is an analytic function of k in vanishing for all k > 0 Thus, it vanishes identically in and, consequently, for k < 0 Therefore

Since A(x, y) and T(x, y) are bounded continuous functions, the Volterra equation (3.2.6) has only the trivial solution h = 0. Define functions asymptotics:

and

Let us denote

and

as the solutions to equation (3.1.5) with the following

Definition 3.2.2. The pair

has property C_ iff the set

Similar definition can be given with As above, one proves: Theorem 3.2.3. If By

replacing

is complete in

j = 1, 2.

j = 1, 2, then property C_ holds for

we mean the set

3.2.2 Properties

and

We prove only property Section 3.1.4. Theorem 3.2.4. If

Property

is proved similarly. Property

j = 1, 2, then property

holds for

is defined in

106

3. One-dimensional inverse scattering and spectral problems

Proof. Our proof is similar to the proof of Theorem 3.2.1. Using (3.1.23) and denoting one writes

Assume:

Then

where

Then

Let

where

and

if

if

107

Therefore

From (3.2.12) and (3.2.13), taking

one gets:

and (using completeness of the system cos(ks), equation:

in

the following

The kernels K, and are bounded and continuous functions. Therefore, if and h(x) = 0 for x > b, (3.2.14) implies:

where c > 0 is a constant which bounds the kernels 2K, 2y = s. From the above inequality one gets

and

from above and

where is sufficiently small so that and Then inequality (3.2.15) implies h(x) = 0 if Repeating this argument, one proves, in finitely many steps, that h(x) = 0, 0 < x < b. Theorem 3.2.4 is proved. The proof of Theorem 3.2.4 is not valid if Let us give a counterexample. Theorem 3.2.5. There exist

and an

The result is not valid either if

such that

Proof. Let and are two potentials in such that have one negative eigenvalue which is the same for and but that Let Let us prove that (3.2.16) holds. One has

and so

108

3. One-dimensional inverse scattering and spectral problems

subtract from the first equation the second and get:

Multiply (3.2.17) by

integrate over

At x = 0 we use conditions (3.2.17), and at and are the same (because (3.2.18) vanishes. Theorem 3.2.5 is proved. 3.3

and then by parts to get

the phase shifts corresponding to and therefore the right-hand side of

INVERSE PROBLEM WITH I-FUNCTION AS THE DATA

3.3.1

Uniqueness theorem

Consider equation (3.1.5) and assume the I-function:

Then f (x, k) is analytic in

Define

From (3.3.1) it follows that I(k) is meromorphic in with the finitely many simple poles Indeed, are simple zeros of f (k) and as follows from (3.1.4). Using (3.1.19), one gets

where and f (0) = 0 then a subtle result.

and exists and

iff f (0) = 0. We prove that if (Theorem 3.3.3 below). This is

Lemma 3.3.1. The I (k) equals to the Weyl function m (k). Proof. The m (k) is a function such that Clearly

where and (3.1.4).

Im k > 0. Thus,

Our basic uniqueness theorem is:

if Im k > 0.

because of (3.1.3)

109

Theorem 3.3.2. If Proof. Let Then one has

Multiply (3.3.3) by

By property

j = 1, 2, generate the same I(k), then be the Jost solution (3.1.5) corresponding to

integrate over

then by parts, using (3.3.3), and get

(Theorem 3.2.1), p(x) = 0.

Remark. If

j = 1, 2, and (*) where then for almost all This result is proved in [GS1] for Our proof is based on (3.3.3), from which, using (*), one gets (**) Note that where V* is the adjoint to a Volterra operator (see the formula below (3.2.4)). Thus, (**) can be written as (***) where Theorem 96 in [Tit] and (***) imply for almost all Since V is a Volterra operator, it follows that p = 0 for almost all as claimed. Theorem 3.3.3. If

and f (0) = 0, then

Proof. Let us prove that Let A(x, y) is defined in (3.1.24) and gets difficulty is to prove that To prove that and with x = 0: existence of is proved, one gets here the existence of the limit from (3.1.24):

and

exists and

and and where by (3.1.25). Integrating by parts, one Thus, The basic If this is done, then exists one uses the Wronskian formula (3.3.2) Divide by k and let Since We have used so The existence of it follows

110

3. One-dimensional inverse scattering and spectral problems

From (3.1.26) one sees that Thus, to complete the proof, one has to prove To prove this, use (3.1.43) with x = 0 and (3.1.41). Since one has Therefore (3.1.43) with x = 0 yields:

Integrate (3.3.7) over

where

to get:

Integrating by parts yields:

Because (3.3.9) imply:

one has

From this equation and from the inclusion as follows. Choose a such that Then (3.3.10) can be written as:

Since

Therefore (3.3.8) and

one derives and let

and it follows that is bounded. Thus The operator has norm Therefore equation (3.3.3) is uniquely solvable in Theorem 3.3.3 is proved. 3.3.2

Characterization of the I-functions

One has

where the Wronskian formula was used with x = 0:

111

From (3.1.20) and (3.3.12) with

one gets:

The I (k) determines uniquely the points as the (simple) poles of I (k) on the imaginary axis, and the numbers by (3.3.2). Therefore I (k) determines uniquely the spectral function by formula (3.1.20). The characterization of the class of spectral functions given in Section 3.4.6, induces a characterization of the class of I-functions. The other characterization of the I-functions one obtains by establishing a oneto-one correspondence between the I-function and the scattering data (3.1.16). Namely, the numbers and J, are obtained from I (k) since are the only poles of I (k) in the numbers are obtained by the formula (see (3.1.15) and (3.3.2)):

if f (k) is found from I (k). Finally, f (k) can be uniquely recovered from I (k) by solving a Riemann problem. To derive this problem, define

and

Assumption (3.3.16), means that Define

Write (3.3.12) as

where in

and (3.3.17) means f (0) = 0.

or

is analytic in in has similar properties in

the closure of

112

3. One-dimensional inverse scattering and spectral problems

g (k) > 0 for k > 0, g (k) is bounded in a neighborhood of k = 0 and has a finite limit at k = 0. From (3.3.20) and the properties of h, one gets:

and

In Section 3.4 we prove:

where

Taking in (3.3.24), one finds step by step all the numbers and J. If then Thus the data (3.1.16) are algorithmically recovered from I (k) known for all k > 0. A characterization of is given in Section 3.5.5, and thus an implicit characterization of I (k) is also given. 3.3.3

Inversion procedures

Both procedures in Section 3.3.2, which allow one to construct either or from I (k) can be considered as inversion procedures because in Section 3.4 and Section 3.5 reconstruction procedures are given for recovery of q (x) from either or All three data, I (k), and are equivalent. Thus, our inversion schemes are:

where (3.1.31) gives the details of the step of the step 3.3.4

and (3.1.40) gives the details

Properties of I(k)

In this section, we derive the following formula for I (k): Theorem 3.3.4. One has

113

where

if and only if f (0) = 0, are the constants defined in (3.3.2), if and if f (0) = 0 and

We prove this result in several steps which are formulated as lemmas. Using (3.1.24) one gets

One has (cf. (3.3.16))

is analytic in

the factor

in (3.3.29) is present if and only if f (0) = 0,

and

Lemma 3.3.5. If

and

Proof. It is sufficient to prove that, for any j,

Since and since (3.1.25)–(3.1.26)), it is sufficient to check that

Note that

then

the function

provided that

(see

114

3. One-dimensional inverse scattering and spectral problems

One has

thus

where

From (3.3.34) one obtains (3.3.33) since

Lemma 3.3.6. If f (0) = 0 and

Lemma 3.3.5 is proved. then (3.3.31) holds.

Proof. The proof goes as above with one difference: if f (0) = 0, then in formula (3.3.27) and in formula (3.3.34) with one has

Thus, using (3.1.25), one gets

where c > 0 is a constant. Similarly one checks that Lemma 3.3.6 is proved. Lemma 3.3.7. Formula (3.3.27) holds. Proof. Write

is present

115

Clearly

By the Wiener-Levy theorem [GeRS], one has

where and

is defined in (3.3.29). Actually, the Wiener-Levy theorem yields However, since one can prove that Indeed, are related by the equation:

which implies

or

where is the convolution operation. the convolution Since and as claimed. entiating (3.3.36) one sees that From the above formulas one gets:

So, differ-

where c is a constant defined in (3.3.39) below, the constants are defined in (3.3.40) and the function is defined in (3.3.41). We will prove that c = 0 (see (3.3.43)). To derive (3.3.37), we have used the formula:

116

3. One-dimensional inverse scattering and spectral problems

and made the following transformations:

where

Comparing (3.3.38) and (3.3.37) one concludes that

To complete the proof of Lemma 3.3.7 one has to prove that c = 0, where c is defined in (3.3.39). This follows from the asymptotics of I (k) as Namely, one has:

From (3.3.42) and (3.3.28) one gets:

From (3.3.43) and (3.3.37) it follows that c = 0. Lemma 3.3.7 is proved. Lemma 3.3.8. One has if f (0) = 0 Proof. From (3.3.2) one gets:

and

if

and

117

If j = 0, then

Here by we mean the right-hand side of (3.3.45) since I (k) is, in general, not analytic in a disc centered at k = 0, it is analytic in and, in general, cannot be continued analytically into By Theorem 3.3.3 the right-hand side of (3.3.45) is well defined and

From (3.1.24) one gets:

Since A(y) is a real-valued function if q (x) is real-valued (this follows from the integral equation (3.1.27), formula (3.3.47) shows that

and (3.3.46) implies

Lemma 3.3.8 is proved. One may be interested in the properties of function a (t) in (3.3.27). These can be obtained from (3.3.41) and (3.3.31) as in the proof of Lemma 3.3.5 and Lemma 3.3.6. In particular, the statements of Theorem 3.3.4 are obtained. Remark 3.3.9. Even if compactly supported.

is compactly supported, one cannot claim that a(t) is

Proof. Assume for simplicity that J = 0 and In this case, if a (t) is compactly supported then I(k) is an entire function of exponential type. It is proved in [R139], p. 278, that if is compactly supported, then f (k) has infinitely many zeros in The function if f (z) = 0. Indeed, if f (z) = 0 and then by the uniqueness of the solution of the Cauchy problem for equation (3.1.5) with k = z. Since one has a contradiction, which proves that if f(z) = 0. Thus, I (k) cannot be an entire function if and q (x) is compactly supported.

118

3. One-dimensional inverse scattering and spectral problems

Let us consider the following question: What are the potentials for which a (t) = 0 in (3.3.27)? In other words, let us assume

and find q (x) corresponding to I-function (3.3.50), and describe the decay properties of q (x) as We give two approaches to this problem. The first one is as follows. By definition

Using (3.3.51) and (3.1.11) one gets [ I (k) – I (–k)] f (k) f (–k) = 2ik, or

By (3.3.44) one can write (see (3.1.20)) the spectral function corresponding to the I-function (3.3.50)

where is the delta-function. Knowing one can recover q (x) algorithmically by the scheme (3.1.31). J=1. Consider an example. Suppose

Then (3.3.53) yields:

Thus, (3.1.33) yields:

119

and, setting

and taking for simplicity

one finds:

where the known formula was used:

Thus,

Equation (3.1.34) with kernel (3.3.59) is not an integral equation with degenerate kernel:

This equation can be solved analytically [R121], but the solution is long. By this reason we do not give the theory developed in [R121], but give the second approach to a study of the properties of q (x) given I (k) of the form (3.3.54). This approach is based on the theory of the Riemann problem [G]. Equations (3.3.52) and (3.3.54) imply

The function

120

3. One-dimensional inverse scattering and spectral problems

Write (3.3.61) as

Thus,

The function in in so Consider (3.3.63) as a Riemann problem. One has

is analytic in

Therefore (see [G]) problem (3.3.63) is uniquely solvable. Its solution is:

as one can check. Thus, by (3.3.62),

The corresponding S-matrix is:

Thus,

and

Equation (3.1.43) implies

so

121

Thus, if by the number

and a (t) = 0 then q (x) decays exponentially at the rate determined

If f (0) = 0, J = 0, and a (t) = 0, then

Let

and Thus, since (3.3.73). One has:

Then equation (3.3.72) implies:

in

and

and So one gets:

Equation (3.1.43) yields:

Solving (3.3.76) yields:

is uniquely determined by the Riemann problem

3. One-dimensional inverse scattering and spectral problems

122

The corresponding potential (3.1.42) is

If case by (3.3.41), the function the details to the reader. 3.4

then a (t) in (3.3.27) decays exponentially. Indeed, in this decay exponentially, so g (t) decays exponentially, and, with h(t) decaying exponentially. We leave

INVERSE SPECTRAL PROBLEM

3.4.1

Auxiliary results

Transformation operators

If and are linear operators in a Banach space X , and T is a boundedly invertible linear operator such that then T is called a transformation (transmutation) operator. If then so that T sends eigenfunctions of into eigenfunctions of with the same eigenvalue. Let j = 1, 2, be selfadjoint in operators generated by the Dirichlet boundary condition at x = 0. Other selfadjoint boundary conditions can be considered also, for example,

Theorem 3.4.1. Transformation operator for a pair exists and is of the form Tf = (I + K) f, where the operator I + K is defined in (3.1.23) and the kernel K(x, y) is the unique solution to the problem:

Proof. Consider for simplicity the case If then (3.4.3) can be written as

If

and

The proof is similar in the case

then

123

Since and f is arbitrary otherwise, (3.4.5) implies (3.4.1), (3.4.2) and (3.4.4). Conversely, if K(x, y) solves (3.4.1), (3.4.2) and (3.4.4), then I + K is the transformation operator. To finish the proof of Theorem 3.4.1 we need to prove existence of the solution to (3.4.1), (3.4.2) and (3.4.4). Let Then (3.4.1), (3.4.2) and (3.4.4) can be written as

Integrate (3.4.6) to get

Integrate (3.4.7) with respect to

over

and get

This is a Volterra integral equation which has a solution, this solution is unique, and it can be obtained by iterations. Theorem 3.4.1 is proved. Spectral function

Consider the problem (3.1.1). The classical result, going back to Weyl, is: Theorem 3.4.2. There exists a monotone increasing function possibly nonunique, such that for every there exists such that

where the limit is understood in sense. If the potential q in (3.1.1) generates the Dirichlet operator in the limit point at infinity case, then is uniquely defined by q, otherwise is defined by q nonuniquely. The spectral function of has the following properties:

The remainder in (4.1.10) can be improved if additional assumptions on q are made. For example, for one can get the remainder Theorem 3.4.3. (Weyl). For any

The function

is analytic in

there exists

and in

such that

124

3. One-dimensional inverse scattering and spectral problems

The function is called Weyl’s function, or m -function, and W is Weyl’s solution. Theorem 3.4.2 and Theorem 3.4.3 are proved in [M]. 3.4.2

Uniqueness theorem

Let be a non-decreasing function of bounded variation on every compact subset of the real axis. Let where is a subset of functions which vanish near infinity. Let

Our first assumption

and

on

is:

This implication should hold for any It holds, for example, if on a set which has a finite limit point: in this case the entire function of vanishes identically, and thus h = 0. Denote by a subset of and assume that if where is defined in (3.4.12), then

Our second assumption

on

and

is:

Let us start with two lemmas. Lemma 3.4.4. Spectral functions at infinity case belong to

of an operator

in the limit-point

Proof. Let

be two spectral functions corresponding to and and Let I + V and I + W be the transformation operators corresponding to and respectively, such that

where

is the regular solution (3.1.1) corresponding to

Condition

implies

125

where, for example,

It follows from (3.4.17) that

where U is a unitary operator in Indeed, U is an isometry and it is surjective because is. To finish the proof, one uses Lemma 3.4.5 below and concludes from (3.4.19) that so V = W, and Since, by assumption, q is in the limit-point at infinity case, there is only one spectral function corresponding to q, so Lemma 3.4.5. If U is unitary and V and W are Volterra operators, then (3.4.19) implies V = W. Proof. From (3.4.19) one gets where

Since U is unitary, one has (I + V) Because V is a Volterra operator, is also a Volterra (of the same type as V in (3.4.18)). Thus, or

The left-hand side in (3.4.20) is a Volterra operator of the type V in (3.4.18), while its right-hand side is a Volterra operator of the type Since they are equal, each of them must be equal to zero. Thus, or or V = W. Theorem 3.4.6. The spectral function determines Proof. If

and

uniquely.

have the same spectral function

then

where

Let I + K

be the Then

transformation

operator

and From (3.4.21) one gets

126

3. One-dimensional inverse scattering and spectral problems

Thus is isometric, and, because is a Volterra operator, the range of is the whole space Therefore is unitary. This implies Indeed, (unitarity) and (Volterra property of Thus so Therefore and so This result was proved by Marchenko (see [M]). Remark 3.4.7. If c = const > 0, then the above argument is applicable and shows that c must be equal to 1, c = 1 and Indeed, the above argument yields the unitarity of the operator which implies c = 1 and Here the following lemma is useful: Lemma 3.4.8. If b I + Q = 0, where b = const and Q is a compact linear operator, then b = 0 and Q = 0. A simple proof is left to the reader. 3.4.3 Reconstruction procedure

Assume that the spectral function corresponding to is given. How can one reconstruct that is, to find q (x)? We assume for simplicity the Dirichlet boundary condition at x = 0, but the method allows one to reconstruct the boundary condition without knowing it a priori. The reconstruction procedure (the GL, i.e., the Gel’fand-Levitan procedure) is given in (3.1.31)–(3.1.34). Its basic step consists of the derivation of equation (3.1.34) and of a study of this equation. Let us derive (3.1.34). We start with the formula

and assume that L(x, y) is a continuous function of x, y in [0, b) × [0, b) for any If

one gets from (3.4.22) the relation:

Using (3.1.23), one gets (3.4.23), one gets

Applying

to

in

127

The right-hand side can be rewritten as:

From (3.4.24) and (3.4.25) one gets, using continuity at y = x, equation (3.1.34). In the above proof the integrals (3.4.23)–(3.4.25) are understood in the distributional sense. If the first inequality (3.4.10) holds, then the above integrals over are well defined in the classical sense. If one assumes that the integral in (3.4.26) converges to a function L (x) which is twice differentiable in the classical sense:

then the above proof can be understood in the classical sense, provided that for any If is a spectral function corresponding to then the sequence satisfies It is known (see [L]) that the sequence

satisfies

and converges to zero.

Lemma 3.4.9. Assume (3.4.13) and suppose that the function

Then equation (3.1.34) has a solution in

for any b > 0, and this solution is unique.

Proof. Equation (3.1.34) is of Fredholm-type: its kernel

is in for any Therefore Lemma 3.4.9 is proved if it is proved that the homogeneous version of (3.1.34) has only the trivial solution. Let

128

3. One-dimensional inverse scattering and spectral problems

Because L(x, y) is a real-valued function, one may assume that h(y) is real-valued. Multiply (3.4.28) by h(y), integrate over (0, x), and use (3.4.12), (3.1.33) and Parseval’s equation to get

From (3.4.13) and (3.4.29) it follows that h = 0. If the kernel K(x, y) is found from equation (3.1.34), then q (x) is found by formula (3.4.4). 3.4.4

Invertibility of the reconstruction steps

Our basic result is: Theorem 3.4.10. Assume (3.4.13), (3.4.14), and suppose of the steps in (3.1.31) is invertible, so that (3.1.35) holds. Proof. 1. Step. is done by formula (3.1.33). Let us prove and corresponding to the same L(x, y), and then

Multiply (3.4.30) by h(x)h(y),

use (3.4.12) and get

By (3.4.14) it follows that v = 0, so

Thus

Then each

If there are

2. Step. is done by solving (3.1.34). Lemma 3.4.9 says that K is uniquely determined by L. Let us do the step Put y = x in (3.1.34), use (3.4.26) and (3.4.27) and get:

or

129

This is a Volterra integral equation for L(x) which has a solution and the solution is unique. Thus the step is done. The functions L(x) and K(x, x) are of the same smoothness. 3. Step. is done by formula (3.4.4), q (x) is one derivative less smooth than K(x, x) and therefore one derivative less smooth than L(x). Thus The step is done by solving the Goursat problem (3.4.1), (3.4.2), (3.4.4) (with or, equivalently, by solving Volterra equation (3.4.8), which is solvable and has a unique solution. The corresponding K(x, y) is in if

Theorem 3.4.10 is proved. Let us prove that the q obtained by formula (3.4.4) generates the function identical to the function K obtained in Step 2. The idea of the proof is to show that both K and solve the problem (3.4.1), (3.4.2), (3.4.4) with the same and This is clear for In order to prove it for K, it is sufficient to derive from equation (3.1.34) equations (3.4.1) and (3.4.2) with q given by (3.4.4). Let us do this. Equation (3.4.2) follows from (3.1.5) because L(x, 0) = 0. Define Apply D to (3.1.34) assuming L(x, y) twice differentiable with respect to x and y, in which case K(x, y) is also twice differentiable. (See Remark 3.4.12). By (3.4.27), DL = 0, so

Integrate by parts the last integral, (use (3.4.2)), and get

where K = K(x, x), L = L(x, y ) , and Subtract from (3.4.34) equation (3.1.34) multiplied by q (x), denote DK(x, y) – q(x)K(x, y) := v(x, y), and get:

provided that which is true because of (3.4.4). Equation (3.4.35) has only the trivial solution by Lemma 3.4.9. Thus v = 0, and equation (3.4.1) is derived. We have proved

130

3. One-dimensional inverse scattering and spectral problems

Lemma 3.4.11. If L(x, y) is twice differentiable continuously or in K(x, y) of (3.1.34) solves (3.4.1), (3.4.2) with q given by (3.4.4).

then the solution

Remark 3.4.12. If a Fredholm equation

in a Banach space X depends on a parameter x continuously in the sense and at equation (3.4.36) has where N(A) = {u : Au = 0}, then the solution u(x) exists, is unique, and depends continuously on x in some neighborhood of If the data, that is, A(x) and f(x), have m derivatives with respect to x, then the solution has the same number of derivatives. Derivatives are understood in the strong sense for the elements of X and in the operator norm for the operator A(x). A simple proof of this known result is left to the reader. Hint. Use the identity which shows that if the operator is bounded, and is sufficiently small, then exists and is bounded. 3.4.5

Characterization of the class of spectral functions of the Sturm-Liouville operators

From Theorem 3.4.10 it follows that if (3.4.13) holds and then Condition (3.4.14) was used only to prove so if one starts with a then by diagram (3.1.35) one gets L(x, y) by formula (3.4.27), where If (3.4.14) holds, then one gets from L(x) a unique Recall that assumption is (3.4.13). Let be the assumption:

Theorem 3.4.13. If holds, and is a spectral function of assumption holds. Conversely, if assumptions and hold, then of

then is a spectral function

Proof. If holds and then by (4.1.4). If and hold, then by (3.1.32), because equation (3.1.34) is uniquely solvable, and (3.1.35) holds by Theorem 3.4.10.

3.4.6

Relation to the inverse scattering problem

Assume in this Section that spectral function is (3.1.20).

Then the scattering data

are (3.1.16) and the

131

Let us show how to get given If is given then finds then is recovered because

and J are known. If one

as follows from (3.1.19) and (3.1.15). To find f (k), consider the Riemann problem

which can be written as (see (3.3.29)):

Note that if The function zeros in and has similar properties in (3.4.40) have unique solutions:

is analytic in and has no Therefore problems (3.4.39) and

and

One can calculate f(x) for k > 0 by taking k = k + i0 in (3.4.43) or (3.4.44). Thus, to find given one goes through the following steps: (1) one finds J, (2) one calculates If then one calculates by formulas (3.4.41), (3.4.43), where is defined in (3.3.29), and by formula (3.4.37), and, finally, by formula (3.1.20). If

then one calculates f (k) by formulas (3.4.42) and (3.4.44), where is an arbitrary number such that If f (k) is found, one

132

3. One-dimensional inverse scattering and spectral problems

calculates by formula (3.4.37), and then by formula (3.1.20). Note that in (3.4.42) depends on but f (k) in (3.4.44) does not. This completes the description of the step Let us show how to get given From formula (3.1.20) one finds J, and If then if Thus, if then is analytic in and vanishes at infinity. It can be found in from the values of its real part by Schwarz’s formula for the half-plane:

If

If

then

so

then the same formula (3.4.46) remains valid. One can see this because is analytic in has no zeroes in tends to 1 at infinity, and

if

Let us summarize the step one finds J, calculates f(k) by formula (3.4.46), and then and are calculated by formula (3.4.37). To calculate f(k) for k > 0 one takes k = k + i0 in (3.4.46) and gets:

3.5

3.5.1

INVERSE SCATTERING ON HALF-LINE

Auxiliary material

Transformation Operators

Theorem 3.5.1. If then there exists a unique operator I + A such that (3.1.24)– (3.1.27) hold, and A(x, y) solves the following Goursat problem:

133

Proof. Equations (3.5.1) and (3.5.2) are derived similarly to the derivation of the similar equations for K(x, y) in Theorem 3.4.1. Relations (3.5.3) follow from the estimates (3.1.25)–(3.1.26), which give more precise information than (3.5.3). Estimates (3.1.25)–(3.1.27) can be derived from the Volterra equation (3.1.27) which is solvable by iterations. Equation (3.1.27) can be derived, for example, similarly to the derivation of equation (3.4.8), or by substituting (3.1.24) into (3.1.6). A detailed derivation of all of the results of Theorem 3.5.1 can be found in [M]. Statement of the direct scattering problem on half-axis Existence and uniqueness of its solution

The direct scattering problem on half-line consists of finding the solution to the equation:

satisfying the boundary conditions at r = 0 and at

where is called the phase shift, and it has to be found. An equivalent formulation of (3.5.6) is:

where

where one gets

Clearly

is defined in (3.1.1), see also (3.1.10). From (3.5.8), (3.1.7) and (3.5.6)

Existence and uniqueness of the scattering solution existence and uniqueness of the regular solution (3.1.9).

follows from (3.5.8) because follows from (3.1.1) or from

Higher angular momenta

If one studies the three-dimensional scattering problem with a spherically-symmetric potential q (x) = q (r), then the scattering solution solves

134

3. One-dimensional inverse scattering and spectral problems

the problem:

Here is the unit sphere in is given, is called the scattering amplitude. If q = q(r), then The converse is a theorem of Ramm [R139], p. 130. The scattering solution solves the integral equation:

It is known that

are orthonormal in spherical harmonics, summation over m in (3.5.13) is understood but not shown, and function. If q = q ( r), then

where

Relation (3.5.16) is equivalent to

and is the Bessel

135

similar to (3.5.8), which is (3.5.18) with If q = q(r), then the scattering amplitude can be written as

while in the general case q = q (x), one has

If q = q (r) then

in (3.5.18) are related to

in (3.5.19) by the formula

In the general case q = q (x), one has a relation between S -matrix and the scattering amplitude:

so that (3.5.21) is a consequence of (3.5.22) in the case q = q (r) : are the eigenvalues of S in the eigenbasis of the spherical harmonics. Since S is unitary, one has so for some real numbers which are called phase-shifts. These numbers are the same as in (3.5.17) (cf. (3.5.18)). From (3.5.21) one gets

The Green function

which solves the equation

can be written explicitly:

and the function

solves the equation:

136

3. One-dimensional inverse scattering and spectral problems

The function is the Wronskian is defined in (3.5.25) and is the solution to (3.5.15) (with q = 0) with the asymptotics

Let

be the regular solution to (3.5.15) which is defined by the asymptotics as

Then

Lemma 3.5.2. One has:

where k > 0 is an arbitrary fixed number. We omit the proof of this lemma. Eigenfunction expansion

We assume that

and

let

be the resolvent kernel of this with respect to

and over

Then and divide by

The function gh is analytic with respect to except for the points because as

Integrate to get

on the complex plane with the cut which are simple poles of g h, and Therefore:

137

One has (cf. (3.1.10)):

Also

are defined in (3.1.15), and

Therefore

This implies (cf. (3.1.20), (3.1.19), (3.1.15)):

We have proved the eigenfunction expansion theorem for is dense in one gets the theorem for Theorem 3.5.3. If then (3.5.33) holds for any converge in sense. Parseval’s equality is:

3.5.2

Since this set

and the integrals

Statement of the inverse scattering problem on the half-line. Uniqueness theorem

In Section 1.2 the statement of the ISP is given. Let us prove the uniqueness theorem.

138

3. One-dimensional inverse scattering and spectral problems

Theorem 3.5.4. If

generate the same data (3.1.16), then

Proof. We prove that the data (3.1.16) determine uniquely I (k), and this implies by Theorem 3.3.2. Claim 1. If (3.1.16) is given, then f(k) is uniquely determined. Assume there are and corresponding to the data (3.1.16). Then

The left-hand side of (3.5.36) is analytic in and tends to 1 as and the right-hand side of (3.5.36) is analytic in and tends to 1 as By analytic continuation is an analytic function in which tends to 1 as Thus, by Liouville theorem, so Claim 2. If (3.1.16) is given, then is uniquely defined. Assume there are and corresponding to (3.1.16). By the Wronskian relation (3.1.11), taking into account that by Claim 1, one gets

Denote

Then:

The function is analytic in and tends to zero as and has similar properties in It follows that so Let us check that is analytic in One has to check that This follows from (3.1.15): if and are given, then are uniquely determined. Let us check that as Using (3.3.5) it is sufficient to check that A(0, 0) is uniquely determined by f(k), because the integral in (3.3.5) tends to zero as by the Riemann-Lebesgue lemma. From (3.1.46), integrating by parts one gets:

Thus

Claim 2 is proved. Thus, Theorem 3.5.4 is proved.

139

3.5.3

Reconstruction procedure

This procedure is described in (3.1.40). Let us derive equation (3.1.43). Our starting point is formula (3.5.34):

From (3.1.10) and (3.1.24) one gets:

Apply to (3.5.41) operator

acting on the functions of y, and get:

From (3.5.42), (3.5.43), and (3.1.24) with

one gets:

One has

From (3.1.41), (3.5.44) and (3.5.45) one gets (3.1.43). By continuity equation (3.1.43), derived for remains valid for Theorem 3.5.5. If solution in

and F is defined by (3.1.41) then equation (3.1.43) has a for any and this solution is unique.

Let us outline the steps of the proof.

140

3. One-dimensional inverse scattering and spectral problems

Step 1. If

where

then F(x), defined by (3.1.41) satisfies the following estimates:

is defined in (3.1.25), and

Step 2. Equation

is of Fredholm type in and in It has only the trivial solution h = 0. Using estimates (3.5.46)–(3.5.48) and the criteria of compactness in one checks that is compact in these spaces for any The space because where We need the following lemma: Lemma 3.5.6. Let h solve (3.5.49). If If then

then

Proof. If h solves (3.5.49), then as Also as So the first claim is proved. Also as If

If

then

where

then

as

Lemma 3.5.7. If

solves (3.5.49) and

then h = 0.

Proof. By Lemma 3.5.6, It is sufficient to give a proof assuming x = 0. The function F(x) is real-valued, so one can assume that h is real-valued. Multiply (3.5.49) by h and integrate over to get

141

where

one gets

Also,

Therefore (3.5.50) implies

and

where

Since h is real valued, one has

of S implies and of (3.5.51), one has equality sign in the Cauchy inequality This means that and (3.1.16) implies

Because and vanishes as as

The unitarity Because

then and if is analytic in Also and vanishes is analytic in Therefore, by analytic continuation, is analytic in and

one has

vanishes as By Liouville theorem, then, by Theorem 3.3.3, works.

so

and If and the above argument

Because is compact in the Fredholm alternative is applicable to (3.5.49), and Lemma 3.5.7 implies that (3.1.43) has a solution in for any and this solution is unique. Note that the free term in (3.1.43) is – and this function of belongs to (cf. (3.5.48)). Because is compact in Lemma 3.5.7 and Lemma 3.5.6 imply existence and uniqueness of the solution to (3.1.43) in for any and for any Note that the solution to (3.1.43) in is the same as its solution in This is established by the argument used in the proof of Lemma 3.5.6. We give a method for the derivation of the estimates (3.5.46)–(3.5.48). Estimate (3.5.48) is an immediate consequence of the first estimate (3.5.46). Indeed,

Let us prove the first estimate (3.5.46). Put in (3.1.43)

142

3. One-dimensional inverse scattering and spectral problems

Thus

From (3.1.25) and (3.5.54) one gets

where c = const > 0 stands for various constants and we have used the estimate

This estimate can be derived from (3.1.25). Write (3.1.43) as

Let us prove that equation (3.5.56) is uniquely solvable for F in p = 1 for all where N is a sufficiently large number. In fact, we prove that the operator in (3.5.56) has small norm in if N is sufficiently large. Its norm in is not more than

because

We have used estimate (3.1.25) above. The function so our claim is proved for Consider the case p = 1. One has the following upper estimate for the norm of the operator in (3.5.56) in Also Thus equation (3.5.56) is uniquely solvable in for all if N is sufficiently large. In order to finish the proof of the first estimate (3.5.46) it is sufficient to prove that This estimate is obvious for (cf. (3.1.41)). Let us prove it for Using (3.3.29), (3.3.31), (3.3.32), one gets

143

where all the Fourier transforms are taken of conclude that if one can prove that function. One has transform of

functions. Thus, one can is the Fourier and

From (3.1.25) it follows that We have proved that Differentiate (3.5.53) to get

or

One has

and

Use

Let us check that (3.1.26) and get The third estimate (3.5.46), cause mate rem 3.5.5 is proved. 3.5.4

The desired estimate is derived. follows from (3.5.60) beand The estifollows similarly from (3.5.53) and (3.1.25). Theo-

Invertibility of the steps of the reconstruction procedure

The reconstruction procedure is (3.1.40). 1. The step

is done by formula (3.1.41). To do the step one takes in (3.1.41) and finds and J . Thus is found and is found. From one finds 1 – S(k) by the inverse Fourier transform. So S(k) is found and the data (see (3.1.16)) is found

144

3. One-dimensional inverse scattering and spectral problems

2. The step is done by solving equation (3.1.43). By Theorem 3.5.5 this equation is uniquely solvable in for all if that is, if F came from corresponding to

To do the step one finds the zeros of f (k) in the number J, are found by formula (3.1.15), where

Thus and by formula (3.1.41). We also give a direct way to do the step Write equation (3.1.43) with

The norm of the operator

in

and

then the numbers The numbers

as

is estimated as follows:

where and estimate (3.1.25) was used. If is sufficiently large then for because as if Therefore equation (3.5.62) is uniquely solvable in for all (by the contraction mapping principle), and so is uniquely determined for all Now rewrite (3.5.62) as

This is a Volterra equation for on the finite interval It is uniquely solvable since its kernel is a continuous function. One can put x = 0 in (3.5.64) and the kernel is a continuous function of v and and the right-hand side of (3.5.64) at x = 0 is a continuous function of Thus is uniquely recovered for all from A(x, y), Step is done.

145

is done by formula (3.1.42). The converse step is done by 3. The step solving Volterra equation (3.1.27), or, equivalently, the Goursat problem (3.5.1)– (3.5.3). We have proved: Theorem 3.5.8. If and are the corresponding data (3.1.16), then each step in (3.1.40) is invertible. In particular, the potential obtained by the procedure (3.1.40) equals to the original potential q. Remark 3.5.9. If and is the solution to (3.1.27), then satisfies equation (3.1.43) and, by the uniqueness of its solution, where A is the function obtained by the scheme (3.1.40). Therefore, the q obtained by (3.1.40) equals to the original q. Remark 3.5.10. One can verify directly that the solution A(x, y) to (3.1.43) solves the Goursat problem (3.5.1)–(3.5.3). This is done as in Section 3.4.4, Step 3. Therefore q (x), obtained by the scheme (3.1.40), generates the same A(x, y) which was obtained at the second step of this scheme, and therefore this q generates the original scattering data. Remark 3.5.11. The uniqueness Theorem 3.5.4 does not imply that if one starts with a computes the corresponding scattering data (3.1.16), and applies inversion scheme (3.1.40), then the q, obtained by this scheme, is equal to Logically it is possible that this q generates data which generate by the scheme (3.1.40) potential etc. To close this loop one has to check that This is done in Theorem 3.5.8, because 3.5.5

Characterization of the scattering data

In this Section we give a necessary and sufficient condition for the data (3.1.16) to be the scattering data corresponding to In Section 3.5.7 we give such conditions on for q to be compactly supported, or Theorem 3.5.12. If

then the following conditions hold:

(1) (3.1.22), index condition; (2) (3) (3.5.47) hold. Conversely, if

satisfies conditions (1)–(3), then

corresponds to a unique

Proof. The necessity of conditions (1)–(3) has been proved in Theorem 3.5.5. Let us prove the sufficiency. If conditions (1)–(3) hold, then the scheme (3.1.40) yields a unique potential, as was proved in Remark 3.5.9. Indeed, equation (3.1.43) is of Fredholm type in for every if F satisfies (3.5.47). Moreover, equation

146

3. One-dimensional inverse scattering and spectral problems

(3.5.49) has only the trivial solution if conditions (1)–(3) hold. Every solution to (3.5.49) in is also a solution in and in and the proof of the uniqueness of the solution to (3.5.49) under the conditions (1)–(3) goes as in Theorem 3.5.5. The role of f (k) is played by the unique solution of the Riemann problem:

which consists of finding two functions and satisfying (3.5.65) such that is an analytic function in and is an analytic function in such that and if ind if ind S(k) = –2J. Existence of a solution to (3.5.65) follows from the non-negativity of ind S(–k) = –indS(k). Uniqueness of the solution to the above problem is proved as follows. Denote and Assume that and solve the above problem. Then (3.5.65) implies

The function is analytic in and tends to 1 at infinity in The function is analytic in and tends to 1 at infinity in Both functions agree on Thus is analytic in and tends to 1 at infinity. Therefore To complete the proof we need to check that q, obtained by (3.1.40), belongs to In other words, that To prove this, use (3.5.59) and (3.5.60). It is sufficient to check that and The first inclusion follows from Let us prove that One has Because and it follows that the limit exists. This limit has to be zero: if as and then Now The last inequality follows from (3.5.53): since it is sufficient to check that One has Here

Note that to

because

is monotonically decreasing and belongs

147

3.5.6

A new Marchenko-type equation

The basic result of this Section is: Theorem 3.5.13. Equation

holds, where A(y) := A(0, y), A(y) = 0 for y < 0, A(x, y) is defined in (3.1.24) and F(x) is defined in (3.1.41). Proof. Take the Fourier transform of (3.5.67) in the sense of distributions and get:

where, by (3.1.41),

Use (3.1.46), the equation

add 1 to both sides of (3.5.48), and get:

From (3.5.70) and (3.5.71) one gets:

where the equation was used. This equation holds because and the product makes sense because is analytic in Equation (3.5.72) holds obviously, and since each of our steps was invertible, equation (3.5.67) holds. Remark 3.5.14. Equation (3.5.67) has a unique solution A(y), such that and A(y) = 0 for y < 0. Proof. Equation (3.5.68) for y > 0 is identical with (3.1.43) because A ( – y ) = 0 for y > 0. Equation (3.1.43) has a solution in and this solution is unique, see Theorem 3.5.5. Thus, equation (3.5.68) cannot have more than one solution, because every solution A(y) = 0 for y < 0, of (3.5.68) solves (3.1.43), and (3.1.43) has no more than one solution. On the other hand, the solution

148

3. One-dimensional inverse scattering and spectral problems

of (3.1.43) does exist, is unique, and solves (3.5.68), as was shown in the proof of Theorem 3.5.13. This proves Remark 3.5.14. 3.5.7

Inequalities for the transformation operators and applications

Inequalities for A and F

The scattering data (1.2.17) satisfy the following conditions: (A) (B) (C)

is a nonpositive integer,

If one wants to study the characteristic properties of the scattering data, that is, a necessary and sufficient condition on these data to guarantee that the corresponding potential belongs to a prescribed functional class, then conditions (A) and (B) are always necessary for a real-valued q to be in the usual class in the scattering theory, or in some other class for which the scattering theory is constructed, and a condition of the type (C) determines actually the class of potentials q. Conditions (A) and (B) are consequences of the selfadjointness of the Hamiltonian, fmiteness of its negative spectrum, and of the unitarity of the S-matrix. Our aim is to derive inequalities for F and A from equation (3.1.43). This allows one to describe the set of q, defined by (3.1.42). Let us assume:

The function is monotone decreasing, Equation (3.1.43) is of Fredholm type in and p = 1. The norm of the operator in (3.1.43) can be estimated:

Therefore (3.1.43) is uniquely solvable in

for any

if

This conclusion is valid for any F satisfying (3.5.75), and conditions (A), (B), and (C) are not used. Assuming (3.5.75) and (3.5.73) and taking let us derive inequalities for A = A(x, y). Define

149

From (3.1.43) one gets:

Thus, if (3.5.75) holds, then

By c < 0 different constants depending on

are denoted. Let

Then (3.1.43) yields

Differentiate (3.1.43) with respect to x and y and get:

and

Denote

Then, using (3.5.79) and (3.5.76), one gets

and using (3.5.78) one gets:

So

150

3. One-dimensional inverse scattering and spectral problems

so

Let y = x in (3.1.43), then differentiate (3.1.43) with respect to x and get:

From (3.5.76), (3.5.77), (3.5.82) and (3.5.83) one gets:

Thus,

provided that (3.5.73) implies ically, then

Assumption decreases monotonas Thus and because due to (3.5.73). Thus, (3.5.73) implies (3.5.76), (3.5.77), (3.5.80), (3.5.81), and (3.5.84), while (3.5.84) and (3.1.42) imply where and satisfies (3.5.75). Let us assume now that (3.5.76), (3.5.77), (3.5.81), and (3.5.82) hold, where and are some positive monotone decaying functions (which have nothing to do now with the function F, solving equation (3.1.43), and derive estimates for this function F. Let us rewrite (3.1.43) as: If

Let x + Y = z, s + y = v. Then,

From (3.5.87) one gets:

and and

151

Thus, using (3.5.77) and (3.5.75), one obtains:

Also from (3.5.87) it follows that:

From (3.5.78) one gets:

Let us summarize the results: Theorem 3.5.15. If

Conversely, if

and (3.5.73) hold, then one has:

and

then

In the next section we replace the assumption in this case is based on the Fredholm alternative.

by

The argument

Characterization of the scattering data revisited

First, let us give necessary and sufficient conditions on for q to be in These conditions are obtained in Section 3.5.5, but we give a short new argument. We assume throughout that conditions (A), (B), and (C) hold. These conditions are known to be

152

3. One-dimensional inverse scattering and spectral problems

necessary for Indeed, conditions (A) and (B) are obvious, and (C) is proved in Theorem 3.5.15 and Theorem 3.5.18. Conditions (A), (B), and (C) are also sufficient for Indeed if they hold, then we prove that equation (3.1.43) has a unique solution in for all This was proved in Theorem 3.5.5, but we give another proof. Theorem 3.5.16. If (A), (B), and (C) hold, then (3.1.43) has a solution in and this solution is unique. Proof. Since to prove that

is compact in

for any

by the Fredholm alternative it is sufficient

implies h = 0. Let us prove it for x = 0. The proof is similar for x > 0. If because If then because Thus, if and solves (3.5.94), then Denote Then,

then

Since F (x) is real-valued, one can assume h real-valued. One has, using Parseval’s equation:

Thus, using (3.5.95), one gets

where we have used real-valuedness of h, i.e., one has where (A) was used. Since Thus, so the equality sign is attained in the Cauchy inequality. Therefore, By condition (B), the theory of Riemann problem guarantees existence and uniqueness of an analytic in function such that

153

and

is analytic in Here the property

in is used.

One has

The function is analytic in and is analytic in they agree on so is analytic in Since and it follows that Thus, and, consequently, h(x)=0, as claimed. Theorem 3.5.16 is proved. The unique solution to equation (3.1.44) satisfies the estimates given in Theorem 3.5.15. In the proof of Theorem 3.5.15 the estimate was established. So, by (3.1.42), The method developed in the previous Section gives accurate information about the behavior of q near infinity. An immediate consequence of Theorem 3.5.15 and Theorem 3.5.16 is: Theorem 3.5.17. If (A), (B), and (C) hold, then q, obtained by the scheme (3.1.40) belongs to Investigation of the behavior of q (x) on requires additional argument. Instead of using the contraction mapping principle and inequalities, one has to use the Fredholm theorem, which says that for any where the operator norm is taken for acting in and and the constant c does not depend on Such an analysis yields: Theorem 3.5.18. If and only if (A), (B), and (C) hold, then Proof. It is sufficient to check that Theorem 3.5.15 holds with To get (3.5.76) with one uses (3.1.44) and the estimate:

where the constant c > 0 does not depend on x. Similarly:

replacing

154

3. One-dimensional inverse scattering and spectral problems

From (3.5.78) one gets:

From (3.5.79) one gets:

Similarly, from (3.5.83) and (3.5.96)–(3.5.99) one gets (3.5.84). Then one checks (3.5.85) as in the proof of Theorem 3.5.15. Consequently Theorem 3.5.15 holds with Theorem 3.5.18 is proved. Compactly supported potentials

In this Section necessary and sufficient conditions are given for if x > a, Recall that the Jost solution is:

Lemma 3.5.19. If F(x + y) = 0 for

then for x > a, A(x, y) = 0 for (cf (3.1.43)), and F(x) = 0 for

Thus, (3.1.43) with x = 0 yields A(0, y) := A(y) = 0 for

The Jost function

is an entire function of exponential type that is, S(k) = f ( – k ) / f ( k ) is a meromorphic function in In (3.5.102) space, and the inclusion (3.5.102) follows from Theorem 3.5.15. Let us formulate the assumption (D):

and is the Sobolev

(D) the Jost function f (k) is an entire function of exponential type Theorem 3.5.20. Assume (A), (B), (C) and (D). Then then (A), (B), (C) and (D) hold.

Conversely, if

Necessity. If then (A), (B) and (C) hold by Theorem 3.5.18, and (D) is proved in Lemma 3.5.19. The necessity is proved. Sufficiency. If (A), (B) and (C) hold, then One has to prove that q = 0 for x > a. If (D) holds, then from the proof of Lemma 3.5.19 it follows that A(y) = 0 for We claim that F (x) = 0 for

155

If this is proved, then (3.1.43) yields A(x, y) = 0 for and so q = 0 for x > a by (3.1.42). Let us prove the claim. Take x > 2a in (3.1.41). The function 1 – S(k) is analytic in except for J simple poles at the points If x > 2a then one can use the Jordan lemma and the residue theorem and get:

Since f (k) is entire, the Wronskian formula

is valid on

because

and at

it yields:

This and (3.5.103) yield

Thus, for x > 2a. The sufficiency is proved. Theorem Theorem 3.5.20 is proved. In [M] a condition on which guarantees that q = 0 for x > a, is given under the assumption that there is no discrete spectrum, that is Square integrable potentials

Let us introduce conditions (3.5.104)–(3.5.106)

Theorem 3.5.21. If (A), (B), (C), and any one of the conditions (3.5.104)–(3.5.106) hold, then Proof. We refer to [R139] for the proof.

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3.6

3. One-dimensional inverse scattering and spectral problems

INVERSE SCATTERING PROBLEM WITH FIXED-ENERGY PHASE SHIFTS AS THE DATA

3.6.1

Introduction

In Subsection 3.5.1 the scattering problem for spherically symmetric q was formulated, see (3.5.15)–(3.5.17). The are the fixed-energy (k = const > 0) phase shifts. Define

where

is a regular solution to

such that

and is the Bessel function. In (3.6.3) is the transformation kernel, I + K is the transformation operator. In (3.6.2) we assume that k = 1 without loss of generality. The is uniquely defined by its behavior near the origin:

For

we will use the known formula ([GR, 8.411.8]):

where is the gamma-function. The inverse scattering problem with fixed-energy phase shifts as the data consists of finding q (r) from these data. We assume throughout this chapter that q (r) is a real-valued function, q (r) = 0 for r > a,

Conditions (3.6.6) imply that In the literature there are books [CS] and [N] where the Newton-Sabatier (NS) theory is presented, and many papers were published on this theory, which attempts to solve the above inverse scattering problem with fixed-energy phase shifts as the data. In Section 3.6.5 it is proved that the NS theory is fundamentally wrong and is not an inversion method. The main results of this Chapter are Theorems 3.6.2–3.6.4, and the proof of the fact that the Newton-Sabatier theory is fundamentally wrong.

157

3.6.2

Existence and uniqueness of the transformation operators independent of angular momentum

The existence and uniqueness of in (3.6.3) we prove by deriving a Goursat problem for it, and investigating this problem. Substitute (3.6.3) into (3.6.2), drop index for notational simplicity and get

We assume first that is twice continuously differentiable with respect to its variables in the region This assumption requires extra smoothness of If q (r) satisfies condition (3.6.6), then equation (3.6.13) below has to be understood in the sense of distributions. Eventually we will work with an integral equation (3.6.40) (see below) for which assumption (3.6.6) suffices. Note that

provided that

We assume (3.6.9) to be valid. Denote

Then

Combining (3.6.7)–(3.6.11) and writing again

in place of u, one gets

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3. One-dimensional inverse scattering and spectral problems

Let us prove that (3.6.12) implies:

This proof requires a lemma. Lemma 3.6.1. Assume that

and

then

Proof. Equations (3.6.15) and (3.6.5) imply:

Therefore

Recall that the Legendre polynomials are defined by the formula

and they form a complete system in Therefore (3.6.17) implies

Equation (3.6.19) implies

If

159

and

Therefore A(r) = 0. Also because the left-hand side of (3.6.20) is an entire function of t, which vanishes on the interval [–1, 1] and, consequently, it vanishes identically, so that and therefore Lemma 3.6.1 is proved. We prove that the problem (3.6.13), (3.6.14), (3.6.9), which is a Goursat-type problem, has a solution and this solution is unique in the class of functions which are twice continuously differentiable with respect to and r, In this section we assume that This assumption implies that is twice continuously differentiable. If (3.6.6) holds, then the arguments in this section which deal with integral equation (3.6.40) remain valid. Specifically, existence and uniqueness of the solution to equation (3.6.40) is proved under the only assumption as far as the smoothness of q (r) is concerned. By a limiting argument one can reduce the smoothness requirements on q to the condition (3.6.6), but in this case equation (3.6.13) has to be understood in distributional sense. Let us rewrite the problem we want to study:

The difficulty in the study of this Goursat-type problem comes from the fact that the coefficients in front of the second derivatives of the kernel are variable. Let us reduce problem (3.6.22)–(3.6.24) to the one with constant coefficients. To do this, introduce the new variables:

Note that

and

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3. One-dimensional inverse scattering and spectral problems

Let

A routine calculation transforms equations (3.6.22)–(3.6.24) to the following ones:

where g (r) is defined in (3.6.23). Here we have defined

and took into account that that Note that

implies

while

implies, for any fixed

for any and B > 0, where c (A, B) > 0 is a constant. To get rid of the second term on the left-hand side of (3.6.29), let us introduce the new kernel by the formula:

Then (3.6.29)–(3.6.31) can be written as:

161

We want to prove existence and uniqueness of the solution to (3.6.36)–(3.6.38). In order to choose a convenient Banach space in which to work, let us transform problem (3.6.36)–(3.6.38) to an equivalent Volterra-type integral equation. Integrate (3.6.36) with respect to from 0 to and use (3.6.37) to get

Integrate (3.6.39) with respect to

from

to

and use (3.6.39) to get

where

Consider the space X of continuous functions such that for any B > 0 and any

defined in the half-plane one has

where is a number which will be chosen later so that the operator V in (3.6.40) will be a contraction mapping on the Banach space of functions with norm (3.6.42) for a fixed pair A, B. To choose let us estimate the norm of V. One has:

where c > 0 is a constant depending on A, B and

Indeed, one has:

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3. One-dimensional inverse scattering and spectral problems

and, using the substitution

one gets:

From these estimates inequality (3.6.44) follows. It follows from (3.6.44) that V is a contraction mapping in the space of continuous functions in the region with the norm (3.6.42) provided that

Therefore equation (3.6.40) has a unique solution

in the region

for any real A and B > 0 if (3.6.45) holds. This means that the above solution is defined for any and any Equation (3.6.40) is equivalent to problem (3.6.36)–(3.6.38) and, by (3.6.35), one has:

Therefore we have proved the existence and uniqueness of that is, of the kernel of the transformation operator (3.6.3). Recall that and are related to and by formulas (3.6.26). Let us formulate the result: Theorem 3.6.2. The kernel of the transformation operator (3.6.3) solves problem (3.6.22)–(3.6.24). The solution to this problem does exist and is unique in the class of twice continuously differentiable functions for any potential If then has first derivatives which are bounded and equation (3.6.22) has to be understood in the sense of distributions. The following estimate holds for any r > 0 :

Proof of Theorem 3.6.2. We have already proved all the assertions of Theorem 3.6.2 except for the estimate (3.6.48). Let us prove this estimate.

163

Note that

Indeed, if and

is fixed, then, by (3.6.26),

Therefore Thus:

The following estimate holds:

where j = 1 , 2 , are arbitrarily small numbers and is defined in formula (3.6.57) below, see also formula (3.6.54) for the definition of Estimate (3.6.51) is proved below, in Theorem 3.6.2. From (3.6.50) and estimate (3.6.61) (see below) estimate (3.6.48) follows. Indeed, denote by I the integral on the right-hand side of (3.6.50). Then, by (3.6.61) one gets:

Theorem 3.6.2 is proved. Theorem 3.6.3. Estimate (3.6.51) holds. Proof of Theorem 3.6.3. From (3.6.40) one gets:

where

(see (3.6.37)), and

It is sufficient to consider inequality (3.6.53) with if to (3.6.53) satisfies (3.6.51) with then the solution with any satisfies (3.6.51) with

and the solution of (3.6.53)

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3. One-dimensional inverse scattering and spectral problems

Therefore, assume that

then (3.6.53) reduces to:

Inequality (3.6.51) follows from (3.6.55) by iterations. Let us give the details. Note that

Here we have used the notation

and the fact that also that Furthermore,

is a monotonically increasing function, since for any

Note

Let us prove by induction that

For n = 1 and n = 2 we have checked (3.6.59). Suppose (3.6.59) holds for some n, then

By induction, estimate (3.6.58) is proved for all n = 1, 2, 3, . . . . Therefore (3.6.55) implies

165

where we have used Theorem 2 from [Lev, section 1.2], namely the order of the entire function and its type is 2. The constant c > 0 in (3.6.51) depends on j = 1, 2. Recall that the order of an entire function is the number where The type of is the number It is known [Lev], that if is an entire function, then its order and type can be calculated by the formulas:

If ved.

then the above formulas yield

3.6.3

and

Theorem 3.6.3 is pro-

Uniqueness theorem

Denote by

any fixed subset of the set

of integers {0, 1, 2, ...} with the property:

Theorem 3.6.4. ([R393]) Assume that q satisfies (3.6.6) and (3.6.63) holds. Then the data determine uniquely. The idea of the proof is based on Property C-type argument. Step 1: If and relation holds for

generate the same data

then the following orthogonality

where is the scattering solution corresponding to j = 1 , 2. Step 2: Define where is the Gamma-function. Check that is holomorphic in and are real numbers, (where N is the Nevanlinna class in that is

where and, by property

vanishes Theorem 3.6.4 is proved.

then

in

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3.6.4

3. One-dimensional inverse scattering and spectral problems

Why is the Newton-Sabatier (NS) procedure fundamentally wrong?

The NS procedure is described in [N] and [CS]. A vast bibliography of this topic is given in [CS] and [N]. Below two cases are discussed. The first case deals with the inverse scattering problem with fixed-energy phase shifts as the data. This problem is understood as follows: an unknown spherically symmetric potential q from an a priori fixed class, say a l = 0, 1 , 2 , . . . . The standard scattering class, generates fixed-energy phase shifts inverse scattering problem consists of recovery of q from these data. The second case deals with a different problem: given some numbers l= 0, 1 , 2 , . . . , which are assumed to be fixed-energy phase shifts of some potential q , from a class not specified, find some potential which generates fixed-energy phase shifts equal to l = 0, 1 , 2 , . . . . This potential may have no physical interest because of its non-physical behavior at infinity or other undesirable properties. Also, in the second case it may be that We first discuss NS procedure assuming that it is intended to solve the inverse scattering problem in case 1. Then we discuss NS procedure assuming that it is intended to solve the problem in case 2. Discussion of case 1:

In [N2] and [N] a procedure was proposed by R. Newton for inverting fixed-energy l = 0, 1 , 2 , . . . , corresponding to an unknown spherically symmetric phase shifts potential q(r). R. Newton did not specify the class of potentials for which he tried to develop an inversion theory and did not formulate and proved any results which would justify the inversion procedure he proposed (NS procedure). His arguments are based on the following claim, which is implicit in his works, but crucial for the validity of NS procedure: Claim N1: The basic integral equation.

is uniquely solvable for all r > 0.

Here

are real numbers, the energy is fixed: is taken without loss of generality, are the Bessel functions. It is assumed in [CS], p. 196, that If equation (3.6.65) is uniquely solvable for all r > 0, then the potential that NS

167

procedure yields, is defined by the formula:

The R. Newton’s ansatz (3.6.65)–(3.6.66) for the transformation kernel K (r, s) of the Schrödinger operator, corresponding to some q (r ), namely, that K (r, s) is the unique solution to (3.6.65)–(3.6.66), is not correct for a generic potential, as follows from our argument below (see the justification of Conclusions). If for some r > 0 equation (3.6.65) is not uniquely solvable, then NS procedure breaks down: it leads to locally non-integrable potentials for which the scattering theory is, in general, not available.

In the original paper [N2] and in his book [N] R. Newton did not study the question fundamental for any inversion theory: does the reconstructed potential generate the data from which it was reconstructed? In [CS, p. 205], there are two claims: Claim (i): generates the original shifts “provided that are not “exceptional” ”, and Claim (ii): the NS procedure “yields one (only one) potential which decays faster than ” and generates the original phase shifts If one considers NS procedure as a solution to inverse scattering problem of finding an unknown potential q from a certain class, for example from the fixed-energy phase shifts, generated by this q , then the proof, given in [CS], of Claim (i) is not convincing: it is not clear why the potential obtained by NS procedure, has the transformation operator generated by the potential corresponding to the original data, that is, to the given fixed-energy phase shifts. In fact, as follows from Proposition 3.6.5 below, the potential cannot generate the kernel K (r, s) of the transformation operator corresponding to a generic original potential Claim (ii) is incorrect because the original generic potential generates the phase shifts and if the potential obtained by NS procedure and therefore not equal to q (r) by Proposition 3.6.5 then one has two different potentials q (r) and which both decay faster than and both generate the original phase shifts contrary to Claim (ii). Our aim is to formulate and justify the following Conclusions: Claim N1 and ansatz (3.6.65)–(3.6.66) are not proved by R. Newton and, in general, are wrong. Moreover, one cannot approximate with a prescribed accuracy in the norm a generic potential by the potentials which might possibly be obtained by the NS procedure. Therefore NS procedure cannot be justified even as an approximate inversion procedure.

Let us justify these conclusions: Claim N1 formulated above (and basic for NS procedure) is wrong, in general, for the following reason:

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3. One-dimensional inverse scattering and spectral problems

Given fixed-energy phase shifts, corresponding to a generic potential either cannot carry through NS procedure because:

one

(a) the system (12.2.5a) in [CS], which should determine numbers in formula (3.6.66), given the phase shifts may be not solvable, or (b) if the above system is solvable, equation (3.6.65) may be not (uniquely) solvable for some r > 0 , and in this case NS procedure breaks down since it yields a potential which is not locally integrable. If equation (3.6.65) is solvable for all r > 0 and yields a potential by formula (3.6.67), then this potential is not equal to the original generic potential as follows from Proposition 3.6.5: Proposition 3.6.5. If equation (3.6.65) is solvable for all r > 0 and yields a potential by formula (3.6.67), then this is a restriction to (0, of a function analytic in a neighborhood of (0, Since a generic potential is not a restriction to (0, of an analytic function, one concludes that even if equation (3.6.65) is solvable for all r > 0, the potential defined by formula (3.6.67), is not equal to the original generic potential and therefore the inverse scattering problem of finding an unknown from its fixed-energy phase shifts is not solved by NS procedure. The ansatz (3.6.65)–(3.6.66) for the transformation kernel is, in general, incorrect, as follows also from Proposition 3.6.5. Indeed, if the ansatz (3.6.65)—(3.6.66) would be true and formula (3.6.67) would yield the original generic q , that is this would contradict Proposition 3.6.5 If formula (3.6.67) would yield a which is different from the original generic q , then NS procedure does not solve the inverse scattering problem formulated above. Note also that it is proved in [R192] that independent of the angular momenta l transformation operator, corresponding to a generic does exist, is unique, and is defined by a kernel K (r , s) which cannot have representation (3.6.66), since it yields by the formula similar to (3.6.67) the original generic potential q , which is not a restriction of an analytic in a neighborhood of (0, function to (0, The conclusion, concerning impossibility of approximation of a generic by potentials which can possibly be obtained by NS procedure, is proved in Claim 7, see proof of Claim 7 below. Thus, our conclusions are justified. Let us give some additional comments concerning NS procedure. Uniqueness of the solution to the inverse problem in case 1 was first proved by A. G. Ramm in 1987 (see [R100], [R109]) for a class of compactly supported potentials, while R. Newton’s procedure was published in [N2], when no uniqueness results for this inverse problem were known. It is still an open problem if for the standard in scattering theory class of potentials the uniqueness theorem for the solution of the above inverse scattering problem holds.

169

We discuss the inverse scattering problem with fixed-energy phase shifts (as the data) for potentials because only for this class of potentials a general theorem of existence and uniqueness of the transformation operators, independent of the angular momenta l , has been proved, see [R192]. In [N2], [N], and in [CS] this result was not formulated and proved, and it was not dear for what class of potentials the transformation operators, independent of l , do exist. For slowly decaying potentials the existence of the transformation operators, independent of is not established, in general, and the potentials, discussed in [CS] and [N] in connection with NS procedure, are slowly decaying. Starting with [N2], [N], and [CS] Claim N1 was not proved or the proofs given (see [CT]) were incorrect (see [R207]). This equation is uniquely solvable for sufficiently small r > 0, but, in general, it may be not solvable for some r > 0, and if it is solvable for all r > 0, then it yields by formula (3.6.67) a potential which is not equal to the original generic potential as follows from Proposition 3.6.5. Existence of “transparent” potentials is often cited in the literature. A “transparent” potential is a potential which is not equal to zero identically, but generates the fixedenergy shifts which are all equal to zero. In [CS, p. 207], there is a remark concerning the existence of “transparent” potentials. This remark is not justified because it is not proved that for the values used in [CS, p. 207], equation (3.6.65) is solvable for all r > 0. If it is not solvable even for one r > 0 , then NS procedure breaks down and the existence of transparent potentials is not established. In the proof, given for the existence of the “transparent” potentials in [CS, p. 197], formula (12.3.5), is used. This formula involves a certain infinite matrix M. It is claimed in [CS, p. 197], that this matrix M has the property MM = I, where I is the unit matrix, and on [CS, p. 198], formula (12.3.10), it is claimed that a vector exists such that Mv = 0. However, then MMv = 0 and at the same time which is a contradiction. The difficulties come from the claims about infinite matrices, which are not formulated clearly: it is not clear in what space M, as an operator, acts, what is the domain of definition of M, and on what set of vectors formula (12.3.5) in [CS] holds. The construction of the “transparent” potential in [CS] is based on the following logic: take all the fixed-energy shifts equal to zero and find the corresponding from the infinite linear algebraic system (12.2.7) in [CS]; then construct the kernel f (r , s) by formula (3.6.66) and solve equation (3.6.65) for all r > 0, finally construct the “transparent” potential by formula (3.6.67). As was noted above, it is not proved that equation (3.6.65) with the constructed above kernel f (r , s) is solvable for all r > 0. Therefore the existence of the “transparent” potentials is not established. The physicists have been using NS procedure without questioning its validity for several decades. Apparently the physicists still believe that NS procedure is “an analog of the Gel’fand-Levitan method” for inverse scattering problem with fixed-energy phase shifts as the data. In fact, the NS procedure is not a valid inversion method. Since modifications of NS procedure are still used by some physicists, who believe that this procedure is an inversion theory, the author pointed out some questions concerning this procedure. This concludes the discussion of case 1.

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3. One-dimensional inverse scattering and spectral problems

Discussion of case 2: Suppose now that one wants just to construct a potential corresponding to some q .

which generates the phase shifts

This problem is actually not an inverse scattering problem because one does not recover an original potential from the scattering data, but rather wants to construct some potential which generates these data and may have no physical meaning. Therefore this problem is much less interesting practically than the inverse scattering problem. However, NS procedure does not solve this problem either: there is no guarantee that this procedure is applicable, that is, that the steps a) and b), described in the justification of the conclusions, can be done, in particular, that equation (3.6.65) is uniquely solvable for all r > 0.

If these steps can be done, then one needs to check that the potential obtained by formula (3.6.67), generates the original phase shifts. This was not done in [N2] and [N]. This concludes the discussion of case 2. The rest of the paper contains formulation and proof of Remark 3.6.6 and Claim 7. It was mentioned in [N3] that if then the numbers in formula (3.6.66) cannot satisfy the condition This observation can be obtained also from the following Remark 3.6.6. For any potential such that equation (3.6.65) is not solvable for some r > 0 and any choice of

the basic such that

Since generically, for one has this gives an additional illustration to the conclusion that equation (3.6.65), in general, is not solvable for some r > 0. Conditions and are incompatible. In [CS, p. 196], a weaker condition is used, but in the examples ([CS, pp. 189–191]), for all so that in all of these examples. Claim 7. The set of the potentials procedure, is not dense (in the norm

which can possibly be obtained by the NS in the set

Let us prove Remark 3.6.6 and Claim 7. Proof of Remark 3.6.6. Writing (3.6.67) as one gets the following relation:

and assuming

If (3.6.65) is solvable for all r > 0, then from (3.6.66) and (3.6.65) it follows that where so that I – K

171

is a transformation operator, where K is the operator with kernel K(r, s), where

where are the phase shifts at k = 1 and prove that Thus, if

and

is the Jost function at k = 1. One can then

If then (3.6.69) contradicts (3.6.68). It follows that if then equation (3.6.65) cannot be uniquely solvable for all r > 0, so that NS procedure cannot be carried through if and This proves Remark 3.6.6. Proof of Claim 7. Suppose that and because otherwise NS procedure cannot be carried through as was proved in Remark 3.6.6. If then there is also no guarantee that NS procedure can be carried through. However, we claim that if one assumes that it can be carried through, then the set of potentials, which can possibly be obtained by NS procedure, is not dense in in the norm In fact, any potential q such that and the set of such potentials is dense in cannot be approximated with a prescribed accuracy by the potentials which can be possibly obtained by the NS procedure. Let us prove this. Suppose that

where the potentials We assume verge in the norm to

are obtained by the NS procedure, so that because otherwise obviously cannot conDefine a linear bounded on functional

where The potentials which can possibly be obtained by the NS procedure, belong to the null-space of f, that is f (v) = 0. If then Since f is a linear bounded functional and one gets: So if then Therefore, no potential with can be approximated arbitrarily accurately by a potential which can possibly be obtained by the NS procedure. Claim 7 is proved.

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3.6.5

3. One-dimensional inverse scattering and spectral problems

Formula for the radius of the support of the potential in terms of scattering data

The aim of this section is to prove formula (3.6.70). Let us make the following assumption. Assumption (A): the potential q (r) , is spherically symmetric, real-valued, and q (r) = 0 for r > a , but on for all sufficiently small The number a > 0 we call the radius of compactness of the potential, or simply the radius of the potential. Let denote the scattering amplitude corresponding to the potential q at a fixed energy Without loss of generality let us take k = 1 in what follows. By the unit vectors in the direction of the scattered, respectively, incident wave, are meant, is the unit sphere in Let us use formulas (3.5.19) and (3.5.20). It is of interest to obtain some information about q from the (fixed-energy) scattering data, that is, from the scattering amplitude or, equivalently, from the coefficients Very few results of such type are known. A result of such type is a necessary and sufficient condition for it was proved [R139, p. 131] (see Section 5.7, Vol. 2), that if and only if Of course, the necessity of this condition was a common knowledge, but the sufficiency, that is, the implication: is a deeper result [R139]. A (modified) conjecture from [R139, p. 356] says that if the potential q (x) is compactly supported, and a > 0 is its radius (defined for non-spherically symmetric potentials in the same way as for the spherically symmetric), then

where are the fixed-energy (k = 1) phase shifts. We prove (3.6.70) for the spherically symmetric potentials q = q (r). If q = q (r) then where depends only on l and k, but not on or Since k = 1 is fixed, depends only on l for q = q (r). Assuming q = q(r), one takes and calculates where

Here we have used

formula (14.4.46) in [RKa, p. 413], and

are the Gegenbauer polynomials (see

[RKa, p. 408]). Since where are the Legendre polynomials (see, e.g., [RKa, p. 409]), one gets: Formula (3.6.70) for q = q (r) can be written as Indeed, as is well known (see, e.g., [MPr], p. 261). Thus and formula for (3.6.70) yields:

173

Note that assumption (A) implies the following assumption: Assumption the potential q (r) does not change sign in some left neighborhood of the point a. This assumption in practice is not restrictive, however, as shown in [R139, p. 282], the potentials which oscillate infinitely often in a neighborhood of the right end of their support, may have some new properties which the potentials without this property do not have. For example, it is proved in [R139], p. 282, that such infinitely oscillating potentials may have infinitely many purely imaginary resonances, while the potentials which do not change sign in a neighborhood of the right end of their support cannot have infinitely many purely imaginary resonances. Therefore it is of interest to find out if assumption is necessary for the validity of (3.6.71). The main result is: Theorem 3.6.7. Let assumption (A) hold. Then formula (3.6.71) holds with by lim.

replaced

This result can be stated equivalently in terms of the fixed-energy phase shift

Below, we prove an auxiliary result: Lemma 3.6.8. If some interval where

is real-valued and does not change sign in and a is the radius of q, then

Below we prove (3.6.72) and, therefore, (3.6.70) for spherically symmetric potentials. Proof of Lemma 3.6.8. First, we obtain a slightly different result than (3.6.73) as an immediate consequence of the Paley-Wiener theorem. Namely, we prove Lemma 3.6.8 with a continuous parameter t replacing the integer and replacing lim. This is done for and without additional assumptions about However, we are not able to prove Lemma 3.6.8 assuming only that Since q (r) is compactly supported, one can write

Let us recall that Paley-Wiener theorem implies the following claim (see [Lev]):

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3. One-dimensional inverse scattering and spectral problems

If and

is the smallest interval containing the support of g (u), then

Thus, using (3.6.74) and (3.6.75), one gets:

Formula (3.6.76) is similar to (3.6.73) with m replaced by t and lim replaced by Remark 3.6.9. We have used formula (3.6.75) with theorem it is assumed that However, for one has:

Thus,

while in the Paley-Wiener on for any

and

Therefore formula (3.6.76) follows. To prove (3.6.73), we use a different approach independent of the Paley-Wiener theorem. We will use (3.6.73) below, in formula (3.6.87). In this formula the role of q (r) in (3.6.73) is played by where Let us prove (3.6.73). Assume without loss of generality that near a. Let We have where is an arbitrary small positive number. Thus, I > 0 for all sufficiently large m, and One has and as Since is arbitrary small, it follows that This completes the proof of (3.6.73). Lemma 3.6.8 is proved. Proof of formula (3.6.72). From (3.5.19) and (3.5.23) denoting

one gets where,

and

175

where and

as are the spherical Bessel functions, solves (3.5.15)–(3.5.17), and the integral

where

and is the Hankel function. It is known [RKa, p. 407] that

and [AlR, Appendix 4]:

It follows from (3.6.81) that is sufficiently large. Define

does not have zeros on any fixed interval (0, a] if l Then (3.6.78) yields

From (3.6.79) and (3.6.81) one gets

Thus

176

3. One-dimensional inverse scattering and spectral problems

This implies that for sufficiently large equation (3.6.83) has small kernel and therefore is uniquely solvable in C(0, a) and one has

uniformly with respect to In the book [N, formula (12.180)], which gives the asymptotic behavior of for large l, is misleading: the remainder in this formula is of order which is much greater, in general, than the order of the main term in this formula. That is why we had to find a different approach, which yielded formula (3.6.87). From (3.6.77), (3.6.80), (3.6.81), and (3.6.87) one has:

Therefore, using (3.6.73), one gets:

Theorem 3.6.7 is proved. Remark 3.6.10. Since as and formulas (3.6.89) and sin imply phase shift at a fixed positive energy. This is formula (3.6.72). 3.7

as where

is the

INVERSE SCATTERING WITH “INCOMPLETE DATA”

3.7.1 Uniqueness results

Consider equation (3.1.3) on the interval [0, 1] with boundary conditions u (0) = u(1) = 1 (or some other selfadjoint homogeneous separated boundary conditions), and Fix Assume q (x) on [b, 1] is known and a subset of the eigenvalues of the operator corresponding to the chosen boundary conditions is known. Here

177

We assume sometimes that

Theorem 3.7.1. If (3.7.1) holds and then the data determine q (x) on [0, b] uniquely. If (3.7.1) and (3.7.2) hold, the same conclusion holds also if The number is “the percentage” of the spectrum of l which is sufficient to determine q on [0, b] if and (3.7.2) holds. For example, if and then “one spectrum” determines q on the half-interval If then “half of the spectrum” determines q on Of course, q is assumed known on [b, 1]. If b = 1, then “two spectra” determines q on the whole interval. By “two spectra” one means the set where is the set of eigenvalues of corresponding to the same boundary condition u(0) = 0 at one end, say at x = 0, and some other selfadjoint boundary condition at the other end, say or The last result is a well-known theorem of Borg ([B]), which was strengthened in [M], where it is proved that not only the potential but the boundary conditions as well are uniquely determined by two spectra. A version of “one spectrum” result was mentioned in [L1, p. 81]. In [L] and [M] there is an algorithm for recovery of q from two spectra. A numerical method for solving this inverse problem is given in [RuS]. Proof of Theorem 3.7.1. First, assume the same data, then as above, one gets

If there are

where

Thus

The function is an entire function of of order and is an entire even function of k of exponential type

The indicator of g is defined by the formula

and

which produce

(see (3.1.11) with One has

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3. One-dimensional inverse scattering and spectral problems

where estimate

Since

one gets from (3.7.5) and (3.7.6) the following

It is known [Lev, formula (4.16)] that for any entire function type one has:

where n(r) is the number of zeros of g(k) in the disk

of exponential

From (3.7.7) one gets

From (3.7.2) and the known asymptotics of the Dirichlet eigenvalues:

one gets for the number of zeros the estimate

From (3.7.8), (3.7.9) and (3.7.11) it follows that

Therefore, if then Theorem 3.7.1 is proved in the case Assume now that and

If

We claim that if an entire function

and (3.7.13) holds, then above.

then, by property

in (3.7.3) of order

p (x) = 0.

vanishes at the points

If this is proved, then Theorem 3.7.1 is proved as

179

Let us prove the claim. Define

and recall that

Since

is entire, of order

the function

Let us use a Phragmen-Lindelöf lemma.

Lemma 3.7.2. [Lev, Theorem 1.22] If an entire function then If, in addition

We use this lemma to prove that Theorem 3.7.1 is proved. The function is entire of order Let us check that

of order as

has the property then

If this is proved then

and that

One has, using (3.7.5), (3.7.15), (3.7.16) and taking into account that

Here we have used elementary inequalities:

and

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3. One-dimensional inverse scattering and spectral problems

with and the assumption (3.7.13). We also used the relation:

Estimate (3.7.20) implies (3.7.18) and (3.7.19). An estimate similar to (3.7.20) has been used in the literature (see [GS]). Theorem 3.7.1 is proved. 3.7.2

Uniqueness results: compactly supported potentials

Consider the inverse scattering problem of Section 3.5.2 and assume

Theorem 3.7.3. If satisfies (3.7.22), then any one of the data S(k), determine q uniquely.

f (k),

Proof. We prove first Note that without assumption (3.7.22), or an assumption which implies that f (k) is an entire function on the result does not hold. If (3.7.22) holds, or even a weaker assumption:

then f (k), the Jost function (3.1.5), is an entire function of k, and S(k) is a meromorphic function on with the only poles in at the points Thus, and J are determined by S(k). Using (3.1.15) and (3.1.11), which holds for all because f (k) and f (x, k) are entire functions of k, one finds Thus all the data (3.1.16) are found from S(k) if (3.7.22) (or (3.7.23)) holds. If the data (3.1.16) are known, then q is uniquely determined, see Theorem 3.5.4. If is given, then so If f (k) is given, then so If is given, then one can uniquely find f(k) from (3.1.11). Indeed, assume there are two f (k), and corresponding to the given Subtract from (3.1.11) with equation (3.1.11) with denote and get (*) Since and one can conclude that w = 0 if one can check that is analytic in The function has at most finitely many zeros in and these zeros are simple. From (*) one concludes that if then because if then (see (3.1.11)). Thus is analytic in Similarly is analytic in These two functions agree on the real axis, so, by analytic continuation, the function is analytic in and vanishes at infinity. Thus it vanishes identically. So w (k) = 0, and f (k) is uniquely determined by Thus Theorem 3.7.3 is proved.

181

In [R181], [R184] a convergent iterative method is given for finding a compactly supported (or decaying faster than exponential) potential q (x) from its S-matrix alone. 3.7.3

Inverse scattering on the full line by a potential vanishing on a half-line

The scattering problem on the full line consists of finding the solution to:

where r (k) and t (k) are, respectively, the reflection and transmission coefficients. The above scattering problem describes plane wave scattering by a potential, the plane wave is incident from in the positive direction of the x-axis. The inverse scattering problem consists of finding q(x) given the scattering data

where are norming constants, and are the negative eigenvalues of the operator It is known [M], that the data (3.7.27) determine uniquely. Assume that

Theorem 3.7.4. If

and (3.7.28) holds, then

determines q uniquely.

Proof. If (3.7.28) holds, then for x < 0, and u = t (k) f (x, k) for x > 0, where f (k, x) is the Jost solution (3.1.5). Thus

Therefore r (k) determines I (k), so by Theorem 3.3.2 q is uniquely determined. 3.8

3.8.1

RECOVERY OF QUARKONIUM SYSTEMS

Statement of the inverse problem

The problem discussed in this Section is: to what extent does the spectrum of a quarkonium system together with other experimental data determines the interquark potential? This problem was discussed in [TQR], where one can find further references. The method given in [TQR] for solving the problem is this: one has few scattering data which will be defined precisely later, one constructs, using the known results of inverse scattering theory, a Bargmann potential with the same scattering data and

182

3. One-dimensional inverse scattering and spectral problems

considers this a solution to the problem. This approach is wrong because the scattering theory is applicable to the potentials which tend to zero at infinity, while our confining potentials grow to infinity at infinity and no Bargmann potential can approximate a confining potential on the whole semiaxis (0, Our aim is to give an algorithm which is consistent and yields a solution to the above problem. The algorithm is based on the Gel’fand-Levitan procedure of Section 3.4.3. A Bargman potential has a rational Jost function. Such potentials decay exponentially at infinity, while the confining potentials grow (see (3.8.2)). Let us formulate the problem precisely. Consider the Schrödinger equation

where q(r) is a real-valued spherically symmetric potential,

The functions these states. We define the resulting equation for

are the bound states, are the energies of which correspond to s-waves, and consider

One can measure the energies of the bound states and the quantities experimentally. Therefore the following inverse problem (IP) is of interest: (IP): given:

can one recover p (r)?

In [TQR] this question was considered but the approach in [TQR] is inconsistent and no exact results are obtained. The inconsistency of the approach in [TQR] is the following: on the one hand [TQR] uses the inverse scattering theory which is applicable to the potentials decaying sufficiently rapidly at infinity, on the other hand, [TQR] is concerned with potentials which grow to infinity as It is nevertheless of some interest that numerical results in [TQR] seem to give some approximation of the potentials in a neighborhood of the origin. Here we present a rigorous approach to (IP) and prove the following result: Theorem 3.8.1. IP has at most one solution and the potential q (r) can be reconstructed from data (3.8.4) algorithmically. The reconstruction algorithm is based on the Gel’fand-Levitan procedure for the reconstruction of q(x) from the spectral function. We show that the data (3.8.4) allow one to write the spectral function of the selfadjoint in operator defined by the differential expression (3.8.3) and the boundary condition (3.8.3) at zero.

183

In Section 3.8.2 proofs are given and the recovery procedure is described. Since in experiments one has only finitely many data the question arises: How does one use these data for the recovery of the potential?

We give the following recipe: the unknown confining potential is assumed to be of the form (3.8.2) and it is assumed that for j > J the data for this potential are the same as for the unperturbed potential In this case an easy algorithm is given for finding q (r). This algorithm is described in Section 3.8.3. 3.8.2

Proof

We prove Theorem 3.8.1 by reducing (IP) to the problem of recovery of q(r) from the spectral function (Section 3.4). Let us recall that the selfadjoint operator L has discrete spectrum since The formula for the number of eigenvalues (energies of the bound states), not exceeding is known:

This formula yields, under the assumption q (r) ~ r as totics of the eigenvalues:

The spectral function

where

Here

the following asymp-

of the operator L is defined by the formula

are the normalizing constants:

and

is the unique solution of the problem:

If then tion to the Volterra integral equation:

The function

is the unique solu-

184

3. One-dimensional inverse scattering and spectral problems

For any fixed r the function is an entire function of E of order that is, where c denotes various positive constants. At where are the eigenvalues of (3.8.3), one has In fact, if a > 0, then for some Let us relate and From (3.8.7) with and from (3.8.3), it follows that

Therefore

Thus data (3.8.4) define uniquely the spectral function of the operator L by the formula:

Given one can use the Gel’fand-Levitan (GL) method (Section 3.4) for recovery of q (r). According to this method, define

where is the spectral function of the unperturbed problem, which in our case is the problem with q(r) = r, then set

where are the eigenfunctions of the problem (3.8.7) with q(r) = r, and solve the second kind Fredholm integral equation for the kernel K(x, y):

The kernel L(x, y) in equation (3.8.14) is given by formula (3.8.13). If K(x, y) solves (3.8.14), then

185

3.8.3

Reconstruction method

Let us describe the algorithm we propose for recovery of the function q(r) from few experimental data Denote by the data corresponding to These data are known and the corresponding eigenfunctions (3.8.3) can be expressed in terms of Airy function Ai(r), which solves the equation and decays at see [Leb]. The spectral function of the operator corresponding to is

Define

and

where

can be obtained by solving the Volterra equation (3.8.9) with and represented in the form:

where K(x, y) is the transformation kernel corresponding to the potential and are the eigenfunctions of the unperturbed problem:

Note that for the functions (3.8.20) do not belong to but We denoted in this section the eigenfunctions of the unperturbed problem by rather than for simplicity of notations, since the eigenfunctions of the perturbed problem are not used in this section. One has: where is the j –th positive root if the equation Ai (– E) = 0 and, by formula (3.8.10), one has These formulas make the calculation of and easy since the tables of Airy functions are available [Leb].

186

3. One-dimensional inverse scattering and spectral problems

The equation analogous to (3.8.14) is:

where

and Equation (3.8.22) has degenerate kernel and therefore can be reduced to a linear algebraic system. If K(x, y) is found from (3.8.22), then

Equation (3.8.14) and, in particular (3.8.22), is uniquely solvable by the Fredholm alternative: the homogeneous version of (3.8.14) has only the trivial solution. Indeed, if then so that, by Parseval equality, Here where are defined by (3.8.20). This implies that for all j = 1, 2, . . . . Since is an entire function of exponential type and since the density of the sequence is infinite, i.e., because as was shown in the beginning of Section 3.8.2, it follows that and consequently h(t) = 0, as claimed. In conclusion consider the case when for all and is the new eigenvalue, with the corresponding data In this case so that equation (3.8.14) takes the form

Thus, one gets:

3.9

3.9.1

KREIN’S METHOD IN INVERSE SCATTERING

Introduction and description of the method

Consider inverse scattering problem studied in Section 3.5.1 and for simplicity assume that there are no bound states. This assumption is removed in Section 3.9.4. This chapter is a commentary to Krein’s paper [K1]. It contains not only a detailed proof of the results announced in [K1] but also a proof of the new results not mentioned in [K1]. In particular, it contains an analysis of the invertibility of the steps in the inversion procedure based on Krein’s results, and a proof of the consistency of this procedure, that is, a proof of the fact that the reconstructed potential generates the scattering data from

187

which it was reconstructed. A numerical scheme for solving inverse scattering problem, based on Krein’s inversion method, is proposed, and its advantages compared with the Marchenko and Gel’fand–Levitan methods are discussed. Some of the results are stated in Theorem 3.9.2–Theorem 3.9.5 below. Consider the equation for a function

Equation (3.9.1) shows that

so

in operator form, and

Let us assume that H(t) is a real-valued even function

Then (3.9.1) is uniquely solvable for any x > 0, and there exists a limit

where

solves the equation

Given H(t), one solves (3.9.1), finds

then defines

where

Formula (3.9.8) gives a one-to-one correspondence between E(x, k) and

188

3. One-dimensional inverse scattering and spectral problems

Remark 3.9.1. In [K1] is used in place of in the definition of By formula (3.9.46) (see Section 3.9.2 below) one has but in general. The theory presented below cannot be constructed with in place of in formula (3.9.8). Note that

where

and

Furthermore,

Note that

where

The function is called the phase shift. One has We have changed the notations from [K1] in order to show the physical meaning of the function (3.9.9): is the Jost function of the scattering theory. The function is the solution to the scattering problem: it solves equation (3.1.3), and satisfies the correct boundary conditions: and as Krein [K1] calls the S -function, and is the S -matrix used in physics. Assuming no bound states, one can solve the inverse scattering problem (ISP): Given find q (x). A solution of the ISP, based on the results of [K1], consists of four steps:

(1) Given S(k), find f (k) by solving the Riemann problem (3.9.62). (2) Given f (k), calculate H(t) using the formula

(3) Given H(t), solve (3.9.1) for

and then find

189

(4) Define

where

and calculate the potential

One can also calculate q(x) by the formula:

Indeed, see (3.9.14), see (3.9.47), and see (3.9.46). There is an alternative way, based on the Wiener-Levy theorem, to do step (1). Namely, given S(k), find the phase shift, then calculate the function and finally calculate The potential generates the S -matrix S(k), with which we started, provided that the following conditions (3.9.18)–(3.9.21) hold:

the overbar stands for complex conjugation, and

where

By the index (3.9.19) one means the increment of the argument of S(k) (when k runs from to along the real axis) divided by (cf (3.1.21)). The function (3.9.7) satisfies equation (3.1.5). Recall that we have assumed that there are no bound states. In Section 3.9.2 the above method is justified and the following theorems are proved:

190

3. One-dimensional inverse scattering and spectral problems

Theorem 3.9.2. If (3.9.18)–(3.9.20) hold, then q (x) defined by (3.9.16) is the unique solution to ISP and this q (x) has S (k) as the scattering matrix. Theorem 3.9.3. The function f (k), defined by (3.9.10), is the Jost function corresponding to potential (3.9.16). Theorem 3.9.4. Condition (3.9.4) implies that equation (3.9.1) is solvable for all and its solution is unique. Theorem 3.9.5. If condition (3.9.4) holds, then relation (3.9.11) holds and is the unique solution to the equation

The diagram explaining the inversion method for solving ISP, based on Krein’s results, can be shown now:

In this diagram denotes step number m. Steps and are trivial. Step is almost trivial: it requires solving a Riemann problem with index zero and can be done analytically, in closed form. Step is the basic (non-trivial) step which requires solving a family of Fredholm-type linear integral equations (3.9.1). These equations are uniquely solvable if assumption (3.9.4) holds, or if assumptions (3.9.18)–(3.9.20) hold. In Section 3.9.2 we analyze the invertibility of the steps in diagram (3.9.23). Note also that, if one assumes (3.9.18)–(3.9.20), diagram (3.9.23) can be used for solving the inverse problems of finding q (x) from the following data: (a) from or (b) from (c) from the spectral function

Indeed, if (3.9.18)–(3.9.20) hold, then a) and b) are contained in diagram (3.9.23), and c) follows from the known formula Then (still assuming (3.9.19)) one has: Note that the general case of the inverse scattering problem on the half-axis, when can be reduced to the case v = 0 by the procedure, described in Section 3.9.4, provided that S(k) is the S-matrix corresponding to a potential Necessary and sufficient conditions for this are conditions (3.9.18)– (3.9.20).

191

Subsection 3.9.3 contains a discussion of the numerical aspects of the inversion procedure based on Krein’s method. There are advantages in using this procedure (as compared with the Gel’fand-Levitan procedure): integral equation (3.9.1), solving of which constitutes the basic step in the Krein inversion method, is a Fredholm convolution-type equation. Solving such an equation numerically leads to inversion of Toeplitz matrices, which can be done efficiently and with much less computer time than solving the Gel’fand-Levitan equation (3.1.34). Combining Krein’s and Marchenko’s inversion methods yields an efficient way to solve inverse scattering problems. Indeed, for small x equation (3.9.1) can be solved by iterations since the norm of the integral operator in (3.9.1) is less than 1 for sufficiently small x, say Thus q (x) can be calculated for by diagram (3.9.23). For one can solve by iterations Marchenko’s equation (3.1.43) for the kernel A(x, y), where, if (3.9.19) holds, the function F(x) is defined by the (3.1.41) with Indeed, for x > 0 the norm of the operator in (3.1.41) is less than 1 and it tends to 0 as Finally let us discuss the following question: in the justification of both the Gel’fandLevitan and Marchenko methods, the eigenfunction expansion theorem and the Parseval relation play a fundamental role. In contrast, the Krein method apparently does not use the eigenfunction expansion theorem and the Parseval relation. However, implicitly, this method is also based on such relations. Namely, assumption (3.9.4) implies that the S-matrix corresponding to the potential (3.9.16), has index 0. If, in addition, this potential is in then conditions (3.9.18) and (3.9.20) are satisfied as well, and the eigenfunction expansion theorem and Parseval’s equality hold. Necessary and sufficient conditions, imposed directly on the function H(t), which guarantee that conditions (3.9.18)–(3.9.20) hold, are not known. However, it follows that conditions (3.9.18)–(3.9.20) hold if and only if H(t) is such that the diagram (3.9.23) leads to a Alternatively, conditions (3.9.18)–(3.9.20) hold (and consequently, if and only if condition (3.9.4) holds and the function f (k), which is uniquely defined as the solution to the Riemann problem

by the formula generates the S-matrix S(k) by formula (3.9.15), and this S(k) satisfies conditions (3.9.18)–(3.9.20). Although the above conditions are verifiable, they are not quite satisfactory because they are implicit, they are not formulated in terms of structural properties of the function H(t) (such as smoothness, rate of decay, etc.). In Subsection 3.9.2 Theorem 3.9.2–Theorem 3.9.5 are proved. In Subsection 3.9.3 numerical aspects of the inversion method based on Krein’s results are discussed. In Subsection 3.9.4 the ISP with bound states is discussed. In Subsection 3.9.5 a relation between Krein’s and Gel’fand–Levitan’s methods is explained.

192

3.9.2

3. One-dimensional inverse scattering and spectral problems

Proofs

Proof of Theorem 3.9.4. If

then

where the Parseval equality was used,

Thus is a positive definite selfadjoint operator in the Hilbert space if (3.9.4) holds. Note that, since one has as (3.9.4) implies

so

A positive definite selfadjoint operator in a Hilbert space is boundedly invertible. Theorem 3.9.4 is proved. Note that our argument shows that

Before we prove Theorem 3.9.5, let us prove a simple lemma. For results of this type, see [K2]. Lemma 3.9.6. If (3.9.4) holds, then the operator

is a bounded operator in For

Proof. Let

one has

One has

193

where we have used the assumption H(t) = H ( – t ) . Similarly,

Finally, using Parseval’s equality, one gets:

where

Since

one gets from (3.9.33) the estimate:

To prove (3.9.30), one notes that

Estimate (3.9.30) is obtained. Lemma 3.9.6 is proved. Proof of Theorem 3.9.5. Define for t or s greater than x. Let Then (3.9.1) and (3.9.6) imply

If condition (3.9.4) holds, then equations (3.9.6) and (3.9.22) have solutions in and, since it is clear that this solution belongs to and consequently to because The proof of Theorem 3.9.4 shows that such a solution is unique and does exist. From (3.9.28) one gets

For any fixed s > 0 one sees that here stands for any of the three norms

as

where the norm Therefore (3.9.36) and

194

3. One-dimensional inverse scattering and spectral problems

(3.9.35) imply

since Also

where

for any fixed s > 0 and

are some constants. Finally, by (3.9.30), one has;

From (3.9.39) and (3.9.41) relation (3.9.11) follows. Theorem 3.9.5 is proved. Let us now prove Theorem 3.9.3. We need several lemmas. Lemma 3.9.7. The function (3.9.8) satisfies the equations

where

and a (x) is defined in (3.9.14).

Proof. Differentiate (3.9.8) and get

We will check below that

195

and

Thus, by (3.9.45),

Therefore (3.9.44) can be written as

By (3.9.46) one gets

Thus

From (3.9.48) and (3.9.50) one gets (3.9.42). Equation (3.9.43) can be obtained from (3.9.42) by changing k to –k. Lemma 3.9.7 is proved if formulas (3.9.45)–(3.9.46) are checked. To check (3.9.46), use H ( – t ) = H(t) and compare the equation for

with equation (3.9.1). Let u = x – y. Then (3.9.51) can be written as

which is equation (3.9.1) for Since (3.9.1) has at most one solution, as we have proved above (Theorem 3.9.4), formula (3.9.46) is proved.

196

3. One-dimensional inverse scattering and spectral problems

To prove (3.9.45), differentiate (3.9.1) with respect to x and get:

Set s = x in (3.9.1), multiply (3.9.1) by compare with (3.9.53) and use again the uniqueness of the solution to (3.9.1). This yields (3.9.45). Lemma 3.9.7 is proved. Lemma 3.9.8. Equation (3.1.5) holds for

defined in (3.9.7).

Proof. From (3.9.7) and (3.9.42)–(3.9.43) one gets

Using (3.9.42)–(3.9.43) again one gets

Lemma 3.9.8 is proved. Proof of Theorem 3.9.3. The function satisfies the conditions

defined in (3.9.7) solves equation (3.1.5) and

The first condition is obvious (in [Kl] there is a misprint: it is written that and the second condition follows from (3.9.7) and (3.9.42):

Let f (x, k) be the Jost solution. Since f (x, k) and f (x, –k) are linearly independent, one has where are some constants independent of x but depending on k. From (3.9.56) one gets Indeed, the choice of and guarantees that the first condition (3.9.56) is obviously satisfied, while the second follows from the Wronskian formula: Comparing this with (3.9.12) yields the conclusion of Theorem 3.9.3. Invertibility of the steps of the inversion procedure and proof of Theorem 1.1

Let us start with a discussion of the inversion steps (1)–(4) described in the introduction.

197

Then we discuss the uniqueness of the solution to ISP and the consistency of the inversion method, that is, the fact that q (x), reconstructed from S(k) by steps 1)–4), generates the original S(k). Let us go through steps l)–4) of the reconstruction method and prove their invertibility. The consistency of the inversion method follows from the invertibility of the steps of the inversion method. Step 1. Assume S(k) satisfying (3.9.18)–(3.9.20) is given. Then solve the Riemann problem

Since one has Therefore the problem (3.9.57) of finding an analytic function in in (and an analytic function in from equation (3.9.57) can be solved in closed form. Namely, define

Then f (k) solves (3.9.57),

Indeed,

by the known jump formula for the Cauchy integral. Integral (3.9.58) converges absolutely at infinity, ln S(– y) is differentiable with respect to y for and is bounded on the real axis, so the Cauchy integral in (3.9.58) is well defined. To justify the above claims, one uses the known properties of the Jost function

where estimates (3.1.25) and (3.1.26) hold and A(y) is a real-valued function. Thus

Therefore

198

3. One-dimensional inverse scattering and spectral problems

Also

Estimate (3.1.25) implies

so that

is bounded for all is differentiable for and ln S(– y) is bounded on the real axis, as claimed. Note that

The converse step

is trivial:

analytic in in holds. Step 2. This step is done by formula (3.9.13):

One has

If

then f (k) is

as

and (3.9.66)

Indeed, it follows from (3.9.61) that

The function

is continuous because as

by (3.1.26), and

since

Thus, Also, Indeed, integrating by parts, one gets from (3.9.67) the and relation: one uses (3.9.60), (3.1.25)– therefore To check that (3.1.26), and (3.9.64)–(3.9.65), to conclude that and, since on and it follows that The inclusion follows from (3.9.60), (3.1.25)–(3.1.26), and (3.9.64)–(3.9.65).

199

By (3.9.67), the function

is the Fourier transform of

and, by (3.9.68), Thus, form is

as behaves, essentially, as

plus a function, whose Fourier trans-

Estimate (3.1.26) shows how

behaves. Equation (3.9.1) shows

that is as smooth as H(t), so that formula (3.9.17) for q (x) shows that q is essentially as smooth as The converse step

is also done by formula (3.9.13): Fourier inversion gives and factorization of yields the unique f (k), since f (k) does not vanish in and tends to 1 at infinity. Step 3. This step is done by solving equation (3.9.1). By Theorem 3.9.4 equation (3.9.1) is uniquely solvable since condition (3.9.4) is assumed. Formula (3.9.13) holds and the known properties of the Jost function are used: as for since The converse step is done by formula (3.9.3). The converse step

constitutes the essence of the inversion method. This step is done as follows:

Given a (x), system (3.9.42)–(3.9.43) is uniquely solvable for E (x, k). Note that the step can be done by solving the uniquely solvable integral equation (3.1.6): with and then calculating f (k) = f (0, k). Step 4. This step is done by formula (3.9.16). The converse step

can be done by solving the Riccati problem (3.9.16) for a (x) given q (x) and the initial condition 2H(0). Given q (x), one can find 2H(0) as follows: one finds f (x, k) by solving equation (3.1.6), which is uniquely solvable if then one gets f (k) := f (0, k), and then calculates 2H(0) using formula (3.9.67) with t = 0:

200

3. One-dimensional inverse scattering and spectral problems

Proof of Theorem 3.9.2. If (3.9.18)–(3.9.20) hold, then, as has been proved in Section 3.5.5, there is a unique which generates the given S-matrix S(k). It is not proved in [K1] that q (x) defined in (1.19) (and obtained as a final result of steps 1)–4)) generates the scattering matrix S(k) with which we started the inversion. Let us now prove this. We have already discussed the following diagram:

To close this diagram and therefore establish the basic one-to-one correspondence one needs to prove This is done by the scheme (3.9.72). Note that the step requires solving Riccati equation (3.9.16) with the boundary condition a (0) = 2H(0). Existence of the solution to this problem on all of is guaranteed by the assumptions (3.9.18)–(3.9.20). The fact that these assumptions imply is proved in Section 3.5.5. Theorem 3.9.2 is proved. Uniqueness theorems for the inverse scattering problem are not given in [K1]. They can be found in Section 3.5.5. Remark 3.9.9. From our analysis one gets the following result: Proposition 3.9.10. If and has no bounds states and no resonance at zero, then Riccati equation (3.9.16) with the initial condition (3.9.15) has the solution a (x) defined for all 3.9.3

Numerical aspects of the Krein inversion procedure

The main step in this procedure from the numerical viewpoint is to solve equation (3.9.1) for all x > 0 and all 0 < s < x, which are the parameters in equation (3.9.1). Since equation (3.9.1) is an equation with the convolution kernel, its numerical solution involves inversion of a Toeplitz matrix, which is a well developed area of numerical analysis. Moreover, such an inversion requires much less computer memory and time than the inversion based on the Gel’fand-Levitan or Marchenko methods. This is the main advantage of Krein’s inversion method. This method may become even more attractive if it is combined with the Marchenko method. In the Marchenko method the equation to be solved is (3.1.43) where F (x) is defined in (3.1.41) and is known if S(k) is known. The kernel A (x, y) is to be found from (3.1.41) and if A(x, y) is found then the potential is recovered by the formula: Equation (3.1.41) can be written in operator form: The operator is a contraction mapping in the Banach space for x > 0. The operator in (3.9.1) is a contraction mapping in for where is chosen to that Therefore it seems reasonable from the numerical point of view to use the following approach:

201

1. Given S(k), calculate f(k) and H(t) as explained in Steps 1 and 2, and also F(x) by formula (3.1.41). 2. Solve by iterations equation (3.9.1) for where is chosen so that the iteration method for solving (3.9.6) converges rapidly. Then find q (x) as explained in Step 4. 3. Solve equation (3.1.43) for by iterations. Find q (x) for by formula (3.1.42). 3.9.4

Discussion of the ISP when the bound states are present

If the given data are (3.9.15), then one defines if and if where is arbitrary, and is chosen so that Then one defines if or if Since and one has The theory of Subsection 3.9.2 applies to and yields From one gets q (x) by adding bound states and norming constants using the known procedure (e.g. see [M]). 3.9.5

Relation between Krein’s and GL’s methods

The GL (Gel’fand-Levitan) method in the case of absence of bound states of the following steps (see Section 3.4, for example): Step 1. Given f(k), the Jost function, find

where Step 2. Solve the integral equation (3.1.34) for K(x, y). Step 3. Find Krein’s function, H(t), see (3.9.13), can be written as follows:

Thus, the relation between the two methods is given by the formula:

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3. One-dimensional inverse scattering and spectral problems

In fact, the GL method deals with the inversion of the spectral foundation of the operator defined in by the Dirichlet boundary condition at x = 0. However, if (in this case there are no bound states and no resonance at k = 0), then (see (3.1.20)):

so

in this case is uniquely defined by 3.10

INVERSE PROBLEMS FOR THE HEAT AND WAVE EQUATIONS

3.10.1

Inverse problem for the heat equation

Consider problem (3.1.55)–(3.1.58). Assume

One can also take where is the delta-function. We prove that the inverse problem of finding from the conditions (3.1.55)–(3.1.58) has at most one solution. If (3.1.58) is replaced by the condition

then q (x), in general, is not uniquely defined by the conditions (3.1.55), (3.1.56), (3.1.57) and (3.10.2), but q is uniquely defined by these data if, for example, or if q (x) is known on Let us take the Laplace transform of (3.1.55)–(3.1.58) and put Then (3.1.55)–(3.1.58) can be written as

and (3.10.2) takes the form

Theorem 3.10.1. The data known on a set of finite positive limit point, determine q uniquely. Proof. Since and

and are analytic in are known for all If

and

which has a

one can assume that is defined in (3.1.3) then so

203

Thus the function is meromorphic in its zeros on the axis are corresponding to the boundary conditions the eigenvalues of and its poles on the axis are the eigenvalues of corresponding to u(0) = u(1) = 0. The knowledge of two spectra determines q uniquely (Section 3.7.1). An alternative proof of Theorem 3.10.1, based on property and generate the same data, solves (3.10.3)–(3.10.4) with and get (*) Multiply (*) by integrate over [0, 1] to get

By property

is: assume that where j = 1, 2, and

it follows from (3.10.7) that p = 0. Theorem 3.10.1 is proved.

Theorem 3.10.2. Data (3.10.3), (3.10.5) do not determine q uniquely in general. They do if q (x) is known on orif Proof. Arguing as in the first proof of Theorem 3.10.1, one concluded that the data (3.10.3), (3.10.5) yields only one (Dirichlet) spectrum of since One spectrum determines q only on “a half of the interval”, see Section 3.7.1. Theorem 3.10.2 is proved. 3.10.2 What are the “correct” measurements?

From Theorem 3.10.1 and Theorem 3.10.2 it follows that the measurements are much more informative than for the problem (3.1.55)– (3.1.57). In this section we state a similar result for the problem

The extra data, that is, measurements, are

which is the flux. Assume:

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3. One-dimensional inverse scattering and spectral problems

is the Sobolev space. Physically, a (x) is the conductivity, u is the temperature. We also consider in place of (3.10.10) the following data:

Our results are similar to those in Subsection 3.10.1: Data determine q(x) uniquely, while data do not, in general, determine a (x) uniquely. Therefore, the measurements are much more informative than the measurements 3.10.3

Inverse problem for the wave equation

Consider inverse problem (3.1.50)–(3.1.54). Our result is Theorem 3.10.3. The above inverse problem has at most one solution. Proof. Take the Fourier transform of (3.1.50)–(3.1.54) and get:

From (3.10.13) one gets where f(x, k) is the Jost solution, and from (3.10.14) one gets and because q = 0 for x > 1. Thus is known. By Theorem 3.7.3, q is uniquely determined. Theorem 3.10.3 is proved. Remark 3.10.4. The above method allows one to consider other boundary conditions at x = 0, such as or h = const > 0, and different data at x = 1, for example, 3.11

INVERSE PROBLEM FOR AN INHOMOGENEOUS SCHRÖDINGER EQUATION

In this Section an inverse problem is studied for an inhomogeneous Schrödinger equation. Most of the earlier studies dealt with inverse problems for homogeneous equations. Let

Assume that q (x) is a real-valued function, q (x) = 0 for Suppose that the data are given.

205

The inverse problem is: (IP) Given the data, find q (x). This problem is of practical interest: think about finding the properties of an inhomogeneous slab (the governing equation is plasma equation) from the boundary measurements of the field, generated by a point source inside the slab. Assume that the self-adjoint operator in has no negative eigenvalues (this is the case when for example). The operator is the closure in of the symmetric operator defined on by the formula Our result is: Theorem 3.11.1. Under the above assumptions IP has at most one solution. Proof of Theorem 11.1: The solution to (3.11.1) is

Here f (x, k) and g(x, k) solve homogeneous version of equation (3.11.1) and have the following asymptotics:

where the prime denotes differentiation with respect to x-variable, and a (k) is defined by the equation

It is known that (see e.g. [M]):

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3. One-dimensional inverse scattering and spectral problems

a (k) in analytic in b (k) in general does not admit analytic continuation from if q (x) is compactly supported, then a (k) and b (k) are analytic functions of The functions

but

are the data, they are known for all k > 0. Therefore one can assume the functions

to be known for all k > 0 because

as follows from the assumption q = 0 if and from (3.11.3). From (3.11.12), (3.11.7) and (3.11.6) it follows that

From (3.11.14) and (3.11.15) it follows:

Eliminating b(–k) from (3.11.16) and (3.11.17), one gets:

or

where

Problem (3.11.19) is a Riemann problem for the pair the function a(k) is analytic in and a ( – k ) is analytic in The

207

functions a (k) and a (–k) tend to one as k tends to infinity in and, respectively, in see equation (3.11.9). The function a(k) has finitely many simple zeros at the points where are the negative eigenvalues of the operator defined by the differential expression in The zeros are the only zeros of a (k) in the upper half-plane k. Define

One has

where J is the number of negative eigenvalues of the operator (3.11.22) and (3.11.20), one gets

and, using (3.11.12),

Since has no negative eigenvalues, it follows that J = 0. In this case ind f (k) = ind g (k) = 0 (see Lemma 1 below), so ind m(k) = 0, and a(k) is uniquely recovered from the data as the solution of (3.11.19) which tends to one at infinity, see equation (3.11.9). If a(k) is found, then b(k) is uniquely determined by equation (3.11.17) and so the reflection coefficient is found. The reflection coefficient determines a compactly supported q(x) uniquely [R196], but we give a new proof. If q (x) is compactly supported, then the reflection coefficient is meromorphic. Therefore, its values for all k > 0 determine uniquely r(k) in the whole complex k-plane as a meromorphic function. The poles of this function in the upper half-plane are the numbers They determine uniquely the numbers which are a part of the standard scattering data where are the norming constants. Note that if then otherwise equation (3.11.6) would imply in contradiction to the first relation (3.11.3). If r (k) is meromorphic, then the norming where constants can be calculated by the formula the dot denotes differentiation with respect to k, and Res denotes the residue. So, for compactly supported potential the values of r(K) for all k > 0 determine uniquely the and standard scattering data, that is, the reflection coefficient, the bound states the norming constants These data determine the potential uniquely. Theorem 3.11.1 is proved.

Lemma 3.11.2. If J = 0 then ind f = ind g = 0. Proof. We prove ind f = 0. The proof of the equation ind g = 0 is similar. Since ind f(k) equals to the number of zeros of f (k) in we have to prove that f (k)

208

3. One-dimensional inverse scattering and spectral problems

does not vanish in If then z = i k, k > 0, and is an eigenvalue of the operator in with the boundary condition u(0) = 0. From the variational principle one can find the negative eigenvalues of the operator in with the Dirichlet condition at x = 0 as consequitive minima of the quadratic functional. The minimal eigenvalue is:

where is the Sobolev space of u(0) = 0. On the other hand, if J = 0, then

satisfying the condition

Since any element u of can be considered as an element of if one extends u to the whole axis by setting u = 0 for x < 0, it follows from the variational definitions (3.11.24) and (3.11.25) that Therefore, if J =0, then and therefore This means that the operator on with the Dirichlet condition at x = 0 has no negative eigenvalues. This means that does not have zeros in ifJ = 0. Thus J = 0 implies ind f(k) = 0. Lemma 3.11.2 is proved. Remark 3.11.3. The above argument shows that in general

so that (3.11.23) implies

Therefore the Riemann problem (3.11.19) is always solvable. It is an open problem to find out if the IP has at most one solution without assuming the absence of negative eigenvalues of the operator 3.12

AN INVERSE PROBLEM OF OCEAN ACOUSTICS

3.12.1 The problem

In this Section the result from [R199] is presented and it is shown that the approach to a somewhat similar problem in the paper [GX] is invalid.

209

Let

where is the deltafunction, n(z) is the refraction coefficient, which is assumed to be a real-valued integrable function, k > 0 is a fixed wavenumber. The solution to (3.12.1)-(3.12.2) is selected by the limiting absorption principle. It is proved that if then n(z) is uniquely determined by the data known 3.12.2

Introduction

In [GX] the following inverse problem is studied:

Here k > 0 is a fixed wavenumber, n(z) > 0 is the refraction coefficient, which is assumed in [GX] to be a continuous real-valued function satisfying the condition the layer models shallow ocean, is the delta-function, is a function satisfying the following conditions (see [GX], p. 127):

The solution to (3.12.3)–(3.12.4) in [GX] is required to satisfy some radiation conditions. It is convenient to define the solution as that is by the limiting absorption principle. We do not show the dependence on k in u(x) since k > 0 is fixed throughout this Section. The function is the unique solution to problem (3.12.3)– (3.12.4) in which equation (3.12.3) is replaced by the equation with absorption:

One defines the differential operator corresponding to differential expression (3.12.3) and the boundary conditions (3.12.4) in as a selfadjoint operator (for example, as the Friedrichs extension of the symmetric operator with the domain consisting of functions vanishing near infinity and satisfying conditions

210

3. One-dimensional inverse scattering and spectral problems

(3.12.4)), and then the function is uniquely defined. By we mean the usual Sobolev space. One can prove that the limit of this function does exist globally in the weighted space and locally in outside a neighborhood of the set provided where are defined in (3.12.10) below. This limit defines the unique solution to problem (3.12.3)–(3.12.4) satisfying the limiting absorption principle if If where is the delta-function, then an analytical formula for can be written:

where is the modified Bessel function (the Macdonald function), and are defined in (3.12.9) below, and and are defined in formula (3.12.10) below. This formula can be checked directly. It is obtained by separation of variables. The known formula was used, and is the Fourier transform defined above formula (3.12.6). From the formula for the known asymptotics for large values of r, the boundedness of as and formula (3.12.11) below, one can see that the limit of as does exist for any and if and only if If for some then the limiting absorption principle holds if and only if If then the limiting absorption principle holds and the solution to problem (3.12.3)–(3.12.4) is well defined. If for some then we define the solution to problem (3.12.3)–(3.12.4) with by the formula:

This solution is unique in the class of functions of the form in where if then satisfies the radiation condition uniformly in directions and if then

as as

The inverse problem (IP) consists of finding n(z) given and assuming that in (3.12.3). By the cylindrical symmetry one has It is claimed in [GX], that the above inverse problem (without the assumption has not more than one solution, and a method for finding this solution is proposed. The arguments in [GX] are erroneous (see Remark 3.12.3 below, where some of the incorrect statements from [GX], which invalidate the approach in [GX], are pointed out).

211

Our aim is to prove that if then n (z) can be uniquely and constructively determined from the data g (r) known for all r > 0. It is an open problem to find all such f (z) for which the IP has at most one solution. Let us outline our approach to IP. Take the Fourier transform of (1.1)–(1.2) with respect to and let

and

Then

IP: The inverse problem is: given for all q (z). The solution to (3.12.6)–(3.12.7) is:

where

and a fixed

find

are the real-valued normalized eigenfunctions of the operator

We can choose the eigenfunctions real-valued since the function is assumed real-valued. One can check that all the eigenvalues are simple, that is, there is just one eigenfunction corresponding to the eigenvalue (up to a constant factor, which for real-valued normalized eigenfunctions can be either 1 or –1). It is known (see e.g. [M], p. 71) that

212

3. One-dimensional inverse scattering and spectral problems

The data can be written as

where are defined in (3.12.9). The series (3.12.12) converges absolutely and uniformly on compact sets of the complex plane outside the union of small discs centered at the points Thus, is a meromorphic function on the whole complex with simple poles at the points Its residue at equals If then proof of the inequality uniquely the set

(see Section 3.12.3 for a and the data (3.12.12) determine

In Section 3.12.3 we prove the basic result: Theorem 3.12.1. If uniquely.

then the data (3.12.13) determine

An algorithm for calculation of q (z) from the data is also described in Section 3.12.3. Remark 3.12.2. The proof and the conclusion of Theorem 3.12.1 remain valid for other boundary conditions, for example, with the data known for all 3.12.3

Proofs: uniqueness theorem and inversion algorithm

Proof of Theorem 3.12.1. The data (3.12.12) with that is, with determine uniquely since are the poles of the meromorphic function which is uniquely determined for all by its values for all (in fact, by its values at any infinite sequence of which has a finite limit point on the real axis). The residues of at are also uniquely determined. Let us show that: (i) (ii) The set (3.12.13) determines

uniquely.

Let us prove (i): If then equation (3.12.10) and the Cauchy data that

which is impossible since

where

imply

213

Let us prove (ii): It is sufficient to prove that the set (3.12.13) determines the norming constants

and therefore the set

where the eigenvalues

and

are defined in (3.12.10),

are the zeros of the equation

The function

is an entire function of

of order 1/2, so that (see [Lev]):

From the Hadamard factorization theorem for entire functions of order < 1 formula (3.12.16) follows but the constant factor remains undetermined. This factor is determined by the data because the main term of the asymptotics of function (3.12.16) for large positive s is cos(s), and the result in [M], p. 243, (see Claim 8 below) implies that the constant in formula (3.12.16) can be computed explicitly:

where are the roots of the equation and the infinite product in (3.12.17) converges because of (3.12.11). A simple derivation of (3.12.17), independent of the result formulated in Claim 8 below, is based on the formula:

For convenience of the reader let us formulate the result from [M], p. 243, which yields formula (3.12.17) as well:

214

3. One-dimensional inverse scattering and spectral problems

Claim 8. The function

where B = const,

admits the representation

and

if and only if

where

are some numbers satisfying the condition: are the roots of the even function and positive roots of

are the

The equality

where is defined in (3.12.17), is easy to prove: if w is the left-hand side and v the right-hand side of the above equality, then w and v are entire functions of the infinite products converge absolutely, and taking the infinite product and using (3.12.17), one concludes that w /v = 1, as claimed. In fact, one can establish formula (3.12.18) and prove that in (3.12.18) is defined by (3.12.17) without assuming a priori that (3.12.17) holds and without using Claim 8. The following assumption suffices for the proof of (3.12.18):

Indeed, if i) holds then both sides of (3.12.18) are entire functions with the same set of zeros and their ratio is a constant. This constant equals to 1 if there is a sequence of points at which this ratio converges to 1. Using the known formula: and the assumption i) one checks easily that the ratio of the left- and right-hand sides of (3.12.18) tends to 1 along the positive imaginary semiaxis. Thus, we have proved formulas (3.12.16)–(3.12.17) without reference to Claim 8. The above claim is used with in our paper. The fact that admits the representation required in the claim is checked by means of the formula for in terms of the transformation operator: and the properties of the kernel are studied in Section 3.4.1. Thus, This is the representation of used in Claim 8.

215

Let us derive a formula for Denote (3.12.14), with replaced by with respect to and get:

by

differentiate

Since q (z) is assumed real-valued, one may assume real-valued. Multiply (3.12.19) and (3.12.14) by subtract and integrate over (0, 1) to get

where the boundary conditions From (3.12.16) with one finds the number

Claim 9. The data (3.12.20) determine uniquely

where

were used.

and equation

Indeed, the numbers are the known numbers from formula (3.12.21). Denote by the quantities known from the data (3.12.13). Then it follows from (3.12.20) that so that

Claim 9 is proved. Thus, the data (3.12.13) determine uniquely and analytically by the above formula, and consequently q (z) is uniquely determined by the following known result (see Section 3.4): The spectral function of the operator L determines q (z) uniquely. The spectral function of the operator L is defined by the formula:

The Gel’fand-Levitan algorithm (see Section 3.4) allows one to reconstruct analytically q (z) from the spectral function and therefore from the data (3.12.13), since, as we have proved already, these data determine the spectral function uniquely. Thus Theorem 1 is proved. Let us describe an algorithm for calculation of q (z) from the data

216

3. One-dimensional inverse scattering and spectral problems

Step 1 : Calculate the Fourier transform of Given find its poles and consequently the numbers then find its residues, and consequently the numbers Step 2 : Calculate the function (3.12.16), and the constant by formulas (3.12.16) and (3.12.17). Calculate the numbers by formula (3.12.21) and by formula (3.12.22). Calculate the spectral function by formula (3.12.23). Step 3 : Use the known Gel’fand – Levitan algorithm to calculate q (z) from This completes the description of the inversion algorithm for IP. Remark 3.12.3. There are inaccuracies, errors, and erroneous arguments in [GX] which invalidate the approach in [GX]. The authors assume (see (C2) on p. 128 in [GX]) that where is a negative decreasing sequence of numbers satisfying formula (3.1)” (a typo, made in [GX] in the above statement, is corrected: in [GX] it was written for the “negative decreasing sequence”, not There are many such sequences. Which one should one choose? The authors do not give any answer to this question. They write on the line below formula (3.1) in [GX], p. 128: “Let be the sequence generated by g(r)”. The function g(r) does not generate a uniquely defined sequence of The authors do not explain how a given function g generates such a sequence. If one chooses a sequence satisfying formula (3.1), then this sequence will not satisfy the problem on lines 2 and 3 below (3.1), unless the authors know a priori the eigenvalues of the problem (3.12.11)–(3.12.13), but these eigenvalues are unknown because is unknown. The authors use the same notation for different objects, and this causes errors in their arguments. Consequently the arguments in [GX] below formula (3.1) are without foundation, they are erroneous. Further arguments in [GX] are also erroneous. For example, the authors have to solve equation (3.13) in [GX] for If one corrects the typo in (3.13), replacing a by where are unknown numbers (since is not known), then the problem of finding from (3.13) means that one has to determine simultaneously a sequence and a function given one function g (r). This is not possible, in general. The authors do not discuss this problem and, apparently, do not see the difficulty. Instead they write that (3.13) is “some kind of H–transform”, forgetting that the numbers are unknown. One can also point out that (3.13) is not the H–transform in the standard sense (i.e., with H the Struwe function, and not a Hankel function, and with the integration over (0, and not over the set used in [GX] but not clearly defined there). 3.13

THEORY OF GROUND-PENETRATING RADARS

3.13.1 Introduction

In many physical and technical applications the problem of determining the inner structure of a material arises. In particular, such problems arise in geophysics when

217

one wants to get information about the medium from the observations of the electromagnetic fields on the surface of the Earth. Let the source of electromagnetic waves be located above the ground. These waves, radiated by the source, penetrate into the ground, interact with it and the resulting electromagnetic field is observed on the surface of the Earth. The present work shows how to get information about the inner structure of the Earth layers from these observations. The proposed algorithm might be useful, for example, in geophysical exploration, detecting the location of nuclear waste, etc. The mathematical model of the above problem is based on Maxwell’s equations for the electromagnetic fields E and H:

written in the Cartesian coordinates (x, y, z). The plane (x, y) is assumed to be parallel to the Earth’s surface, the z-axis is perpendicular to the Earth’s surface and directed into the Earth. In (3.13.1) t is time, and are conductivity, permittivity and magnetic permeability respectively, is the exterior source which is supposed to be a wire located above the Earth’s surface and going along the through the point is the delta-function, f (t) is a function of time which shows the shape of the electromagnetic pulse. It is assumed that

where

is the dielectric constant of free space while is the magnetic permeability of vacuum.

Problem 2 (The Ground-Penetrating Radar Problem (GPR):). Given E on the plane z = 0 for all t > 0, find and for 0 < z < L. 3.13.2 Derivation of the basic equations

Let us differentiate the second equation in (1) with respect to t and substitute by rot E taken from the first equation. One gets

in it

Assuming that the vector is directed along the and depends on t, x and z, one can reduce (3.13.5) to a one-dimensional inverse problem

218

3. One-dimensional inverse scattering and spectral problems

for a differential equation satisfied by

are the Fourier transforms of unit. The characteristic value of u is

where

and f(t) respectively, and i is the imaginary The function u solves the problem:

where the conditions at infinity are written under the assumption that is sufficiently small. In this Section we use only such k. In (3.13.7), and Note that, due to (3.13.2) and (3.13.3), B(z) = 0 outside the interval [0, L]. Equation (3.13.7) may be rewritten as follows

where

and

Equation (3.13.8) is equivalent to the equation

In (3.13.9) g (z, s, k,

is the Green function that solves the equation

219

and satisfies the conditions:

The solution to (3.13.10)–(3.13.11) is

where

and we use the function g assuming that k is sufficiently small, so that and are both positive. Let C(0, L) denote the Banach space of continuous on [0, L] functions with the usual norm. Denote

3.13.3 Basic analytical results

Theorem 3.13.1. Equation (3.13.9) is uniquely solvable in C(0, L) for sufficiently small k; its solution is analytic with respect to k in a neighborhood of the point k = 0 and can be obtained by iterations

This theorem is proved similarly to the proof of Lemma 1 in [R83], p. 219 (see also Chapter 9, Vol. 2).

220

3. One-dimensional inverse scattering and spectral problems

Let u = v + i w, where v and w are the real and imaginary parts of u respectively. The functions w and v solve the equations

Put z = 0. Taking into account the analyticity of v and w in the neighborhood of k = 0, one differentiates equation (3.13.13) with respect to k at k = 0 and gets:

Similarly, one differentiates twice equation (3.13.14) with respect to k and puts k = 0. This yields:

If one introduces the variable and substitutes it into in the above equations instead of then (3.13.15) and (3.13.16) will express the Laplace transforms of the unknown functions and in terms of the data on the plane z = 0. Since an can be uniquely recovered from its Laplace transform, one gets the following uniqueness theorem: Theorem 3.13.2. The functions on the plane z=0.

and

are uniquely determined by the data

Some methods are given in [R139] for recovery of and from their Laplace transforms known for only. Let us rewrite (3.13.15) as where denotes the Laplace transform operator and F is the right-hand side of (3.13.15). A possible numerical inversion method consists of minimizing the functional

with respect to in an appropriate norm (we have chosen norm), where a is a regularization parameter. If one looks for of the form of a linear combination of finitely many linearly independent functions, then (3.13.17) is equivalent to determining the coefficients of the linear combination which minimize (3.13.17). To find the unknown coefficients, one differentiates (3.13.17) with respect to these coefficients and gets a system of linear algebraic equations for them. If is determined, one can

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use its values to recover an analogous procedure.

(and consequently

from equation (3.13.16) using

3.13.4 Numerical results

To test the proposed algorithm a number of calculations were carried out. First, the direct problem (3.13.9) was numerically solved by iterations for some known and To evaluate the integrals in (3.13.9), the Simpson’s method was used. Then, the obtained values of the function u on the plane z = 0 were used to compute the Laplace transforms (3.13.15), (3.13.16) of and respectively. Finally, the functions and considered as unknown ones, were recovered from their Laplace transforms on the basis of (3.13.17), and the computed values of and were compared with given ones to estimate the accuracy of the the obtained results. As basis functions for expansion of and on the interval [0, L], the following piecewise-constant functions were chosen:

where the points form a partition of [0, L] (here N is the number of subintervals of [0, L]). The functions form an orthonormal system in To demonstrate the efficiency of the proposed approach for the recovery of and which are continuous in the interval (0, L), we considered the following example:

The results of the calculations are shown in Figs. 1, 2 in [RSh] for and respectively. In these Figures z-coordinate is taken along the horizontal axis, the vertical axis shows the values of the functions, the solid lines denote the exact values of and respectively, and the black circles indicate the obtained numerical results. It is worth to note that the recovery of and is carried out, in fact, on the basis of the noisy data in (3.13.15) and (3.13.16): indeed, each step of the numerical scheme contributes some computational error. The accuracy of the calculation of the Laplace transforms (3.13.15), (3.13.16) in was about and respectively. The accuracy of the obtained results is about for and approximately for This accuracy is defined by the formula: where f denotes the exact function and f* is the calculated one. Here by f we mean or the Laplace transforms (3.13.15), (3.13.16). To investigate the applicability of our approach for recovery of discontinuous functions, we took

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3. One-dimensional inverse scattering and spectral problems

and for L=0.5, Fig. 3 and Fig. 4 in [RSh] show the values of the recovered and respectively. To recover the discontinuous function an adaptive grid generation procedure was used. This procedure is based on the following coordinate transformation:

where f denotes the unknown function to be recovered, and c is some positive constant. The transformation (3.13.18) is one-to-one. A uniform grid i = 0, . . . , N, along the corresponds to the grid i = 0, . . . , N, along the z-axis with larger density of the nodes in places with large provided that c is sufficiently small. In practice, the procedure for the recovery of is organized as follows. First, the constant c is taken large. This corresponds to a uniform grid along the z-axis. Then minimization (3.13.17) is done. At the next step the value of c is slightly decreased. To get the new distribution of nodes along the z-axis, we integrate (3.13.18) numerically and obtain the nodes j = 0, ..., N, along the Then a uniform grid is constructed along the and the values of are calculated at these points by linear interpolation of the data from the nodes obtained after solving (3.13.18). Using the new distribution of minimization procedure (3.13.17) is done again and the new grid is generated on the basis of (3.13.18) with the smaller value of the constant c. Such steps are repeated until the minimal possible error of the recovery of is attained. In such a way, one can reduce the computational error of the recovery of to (the accuracy of the calculation of the Laplace transform of from the data was about The accuracy of the recovery of is approximately (the accuracy of the calculation of the Laplace transform (3.13.16) of was about The proposed algorithm allows one to recover two coefficients simultaneously using a reasonable a priori information about the unknown functions and 3.13.5 The case of a source which is a loop of current

Consider now the case of a source which is a loop of electric current and prove that the information about the electromagnetic field on the Earth’s surface allows one to recover uniquely both conductivity and permittivity, and analytical recovery from exact data is possible. These parameters may have polynomial growth as the vertical coordinate approaches infinity. Mathematical model of the above problem is based on Maxwell’s equations (3.13.1), written in the cylindrical coordinates (r, z). The plane z = 0 is assumed to be the Earth’s surface, the z-axis is perpendicular to the Earth’s surface and directed into the ground. In (3.13.1) t is time, and are conductivity, permittivity and magnetic permeability respectively, is the exterior source which is supposed to be a loop of a radius located above the Earth’s surface at the point is the delta-function, f (t) is a piecewise-continuos function

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of time which shows the shape of the electromagnetic pulse. The z-axis is supposed to pass through the center of the loop and perpendicular to the plane in which that loop lies. It is assumed that

The functions functions and functions on tributional sense.

and constant are considered to be known. The are assumed to be piecewise-continuous uniformly bounded The integral transforms, used below, are understood in the dis-

Problem 3 (The Ground-Penetrating Radar Problem (GPR)). Given E on the plane z = 0 for all t > 0, find and for Our basic result is formulated in the following theorem. Theorem 3.13.3. Under the above assumption the functions determined by the surface data E(r, t) known for all t > 0, and

and

are uniquely for fixed

Let us derive the basic equations. Differentiate the second equation in (3.13.1) with respect to t and substitute in it by rot E taken from the first equation. One gets

Assume that the vector where is the unit vector of the cylindrical coordinates. In this case equation (3.13.22) takes the form of a onedimensional inverse problem for a differential equation. First one rewrites equation (3.13.22) in cylindrical coordinates taking into account the well-known relation

where

stands for Laplace operator, and One gets

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3. One-dimensional inverse scattering and spectral problems

Here

and

Defining the Fourier transform of E as

and applying it to (3.13.23) one gets

where Hankel-Bessel transform

to equation (3.13.24), where

is the Fourier transform of f (t). Then one applies the

is the Bessel function. Let us denote

The function u solves the problem:

Equation (3.13.25) is equivalent to the equation

In (3.13.26),

is the Green function that solves the equation

and satisfies the condition:

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The solution to (3.13.27)–(3.13.28) is

3.13.6

Basic analytical results

One can prove the existence and uniqueness of the solution to (3.13.26) for sufficiently small k and show its analyticity with respect to k in some neighborhood of the point k = 0. Let us derive an equation from which B(z) can be obtained. Put z = 0 in (3.13.26) and get:

Passing in 3.13.30) to the limit and denoting by function in the left-hand side of (3.1) one gets

Since

the limit of he known

and the function B(z) is known for z < 0, one gets from (3.13.31)

where

is a known function computable from the surface data and the values of Therefore setting

for z < 0.

one gets

So is uniquely determined as the inverse Laplace transform of the known function b(v). The function b(v) is defined by formulas (3.13.33) and (3.13.32), and in (3.13.32) is the limit, as of the left-hand side of (3.13.30).

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3. One-dimensional inverse scattering and spectral problems

If

has been found equation (3.13.26) yields

Therefore

Denote the right-hand side of (3.13.36) by equation (3.13.36) yields

Since

is known for z < 0,

where

Setting

one gets

Hence is uniquely determined as the inverse Laplace transform of the known function a(v). The function a(v) is computable from the surface data, the values of for z < 0 and from the function B(z) calculated by solving equation (3.13.34).

4. INVERSE OBSTACLE SCATTERING

4.1

STATEMENT OF THE PROBLEM

First, let us formulate the direct scattering problem. Assume that is a bounded domain (obstacle) with the boundary S. D may consist of finitely many connected components. By denote the complement of D. Assume that D has finite perimeter relative to By the perimeter of the set D we mean where is the characteristic function of D, and is the space of functions of bounded variation in This space consists of functions u such that (in the sense of distribution theory) is a charge, and the total variation of this charge is If D is a Lipschitz domain, then D has a finite perimeter. The set of domains with finite perimeter is much larger than the set of Lipschitz domains. If is the local equation of S, then Lipschitz boundary is the one for which D has finite-perimeter if S has finite (n – 1)dimensional Hausdorff measure, that is where the infimum is taken over all coverings of S by open balls of radii Let v be a unit vector, We say that v is the normal (in the sense of Federer) to a set E at the point if

where By

is n-dimensional Lebesgue measure, we denote the boundary of E. The set of

for which normals to E

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4. Inverse obstacle scattering

exist is denoted E and is called the reduced boundary of E. The following result is known (e.g. [Maz]): If then the set E is measurable with respect to and if u is a compactly supported Lipschitz function in then

More generally, let D be a bounded domain with finite perimeter, and assume that the normal to D exists everywhere on Then for which has a trace on everywhere, the Green’s formula holds. Let us consider first the direct scattering problem. Let D be a bounded domain in which may have finitely many connected components, is the unit sphere in Let

where

is one of the following boundary conditions:

N is the exterior unit normal to S, is a piecewise-continuous bounded function with finitely many discontinuity curves on S. If S is rough, so that N may be not defined on S, we have to define the meaning of the boundary condition. For the Dirichlet condition this meaning consists of the inclusion where is the closure in the norm of of the functions, and is the set of functions such that R > 0 is such that Equation (4.1.1) we replace by the identity:

where is the set of functions vanishing near infinity. We call u the weak solution of the scattering problem. Thus, problem (4.1.1)–(4.1.4) with is formulated for non-smooth D as the problem of finding of the form where v satisfies (4.1.2), and u satisfies (4.1.5).

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If

then (4.1.5) holds functions vanishing near infinity, and the conditions (4.1.2), hold. If then u satisfies the identity:

is the set of v solves

u is of the form and v satisfies (4.1.2). For (4.1.6) to make sense one has to define the traces of functions from on S. The integration in (4.1.6) over S is with respect to measure. If S is Lipschitz, or D is an extension domain, or D satisfies the cone condition, then the embedding operator from into is compact, so the traces of the elements of on S are well defined. If then we assume that the domain D is bounded; if we assume additionally that the embedding operator is compact; if we assume get additionally that the embedding operator is compact. The role of these assumptions will be made clear later, but our basic idea can be explained now: the left-hand sides of (4.1.5) (or (4.1.6)) is a symmetric, closed, densely defined in bounded from below, quadratic form

This form defines a unique self-adjoint operator L, which is denoted by When we want to emphasize the corresponding boundary condition, we write Consequently, if Im k > 0 and Re k > 0, so that Im then is a bounded operator in Its kernel G(x, y, k) is called the resolvent kernel of the Laplacian corresponding to one of the above boundary conditions. We prove the limiting absorption principle, thus defining G(x, y, k) for k > 0 as the limit Let us first prove that the quadratic forms (4.1.5) (and (4.1.6)) with domains (and corresponding to (and are closed. These forms are symmetric and, under our assumptions on D, bounded from below. Closedness means (see [Kat]) that if in and as then and If a quadratic form is bounded from below, then the form for a suitable constant c > 0. We will assume therefore, without loss of generality, that Then the norm is equivalent to the norm t[u, u]. Since is a complete Hilbert space, the form t with the domain (and with the domain is closed. This covers conditions and Consider condition If is compact, then one has [Rl] the inequality

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4. Inverse obstacle scattering

where

is an arbitrary small number, d s is measure on S, and and R is throughout a sufficiently large number, so that Inequality (4.1.7) guarantees that the form t defined by formula (4.1.6) is bounded from below in and that the norm it defines is equivalent to the norm of Therefore this form with the domain is closed. Actually, one could replace the assumption of compactness of the embedding operator by the assumption of its boundedness, with relative bound < 1, that is, by the inequality where const < 1, and where the constants are independent of The closed symmetric forms generate the corresponding self-adjoint operators which we denote for any of the conditions The scattering problem can be formulated as the equation Let us show how this is done. Take a cut-off function if in a neighborhood of S, and let Then w = v for r > R, Thus we have reduced the scattering problem to the problem of solving the equation with the self-adjoint operator L and Actually This equation has a solution in H if Im and the solution is unique. Define and take n = 3 in what follows. Let us take k > 0, sufficiently small, and prove the limiting absorption principle (LAP): Theorem 4.1.1. For any n = 3, there exists in the norm of The function w(x, k) is the weak solution of the problem

where j = 1, 2, or 3, and w satisfies (4.1.2). Proof. First, let us prove Lemma 4.1.2. If w is a weak solution of (4.1.8) with f = 0 and satisfies (4.1.2), then w = 0. Proof. From (4.1.2) it follows that

From (4.1.5) (or (4.1.6)), taking for 0 for then taking complex conjugate equation, subtracting

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from (4.1.5) its complex conjugate, and taking

one gets:

From (4.1.8) and (4.1.9) it follows that

Any solution to equation (4.1.1) which satisfy (4.1.11), vanishes identically everywhere in its domain of definition. Indeed, any solution w to (4.1.1) can be written as

where are the orthonormal in spherical harmonics, are constants, are spherical Hankel functions, as (4.1.11) one derives for all so w = 0. Lemma 4.1.2 is proved.

From

Let us continue with the proof of Theorem 4.1.1. The steps of the proof are: Step 1. We prove that

c does not depend on

By c we denote various constants.

Step 2. From (4.1.12) it follows that This and (4.1.5) (or (4.1.6)) imply so and in because is compact (or because and are compact, when holds). Thus, one can pass to the limit as in (4.1.5) (or (4.1.6)), and get that w is a weak solution of (4.1.7). From elliptic regularity it follows that in where is any strictly interior bounded subset of This allows one to pass to the limit in the representation

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4. Inverse obstacle scattering

where N is the normal to the sphere get

pointing into

and

It follows that

Let us now