In this post, we will see the book The Inverse Problem of Scattering Theory by Z. S. Agranovich and V. A. Marchenko.

# About the book

In spectral theory, the inverse problem is the usual name for any problem in which it is required to ascertain the spectral data that will determine a differential operator uniquely and a method of constructing this operator from the data. A problem of this kind was first formulated and investigated by V. A. Ambartsumian in 1929. Since 1946, various forms of the inverse problem have been considered by numerous foreign authors, and there now exists an extensive literature on the question.

No attempt is made in this monograph to review the work done on the inverse problem. Instead, merely one of its variants will be treated and solved, namely, the problem arising in connection with the quantum theory of scattering and which is apparently the most interesting from the standpoint of application. The mathematical techniques developed in the solution of the problem may also be applied to related questions.The basic question treated in this book, a translation of the monograph entitled Obratnaya zadacha teorii rasseyaniya, is encountered in many fields. Besides the quantum theoretical problem, there is, for example, the electromagnetic inverse scattering problem, i. e., the problem of determining information about a medium from which an electromagnetic wave is reflected, given a knowledge of the reflection coefficient. The authors Agranovich and Marchenko have presented a comprehensive lucid solution of another such problem arising in the theory of the deuteron. It is based mainly on the considerable amount of work they have done in this and related areas. Moreover, the functional analytic and algebraic methods used should also be of great interest to pure and applied mathematicians.

The book was translated from Russian by B. D. Seckler and was published in 1963.

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# Contents

Introduction 1

## PART ONE

THE BOUNDARY-VALUE PROBLEM WITHOUT SINGULARITIES

## I Particular solutions of the system without singularities 13

§ 1. Preliminary Remarks and Notation 13

§ 2. A Fundamental System of Solutions With a Prescribed Behavior in the Vicinity of Zero 14

§ 3. The Special Solution and Transformation Operator 20

§ 4. A Fundamental System of Solutions With a Prescribed Behavior at Infinity for the Case 𝜆 ≠ 0 27

§ 5. A Fundamental System of Solutions With a prescribed Behavior at Infinity for 𝜆=0 34

## II The spectrum and scattering matrix for the boundary-value problem without singularities 37

§ 1. The Point Spectrum 37

§ 2. Properties of the Matrix E^{-1}(𝜆) 42

8 3. The Scattering Matrix 46

§ 4. Behavior of the Matrix E^{-1}(𝜆) in the Neighborhood of 𝜆 = 0 51

## III The fundamental Equation 57

§ 1. Derivation of the Fundamental Equation 57

§ 2. Properties of the Kernel 62

§ 3. Lemmas on Integral Equations With Kernels Dependent on a Sum 70

§ 4. Existence of Solutions 76

§ 5. Investigation of Homogeneous Equations Constructed from the Scattering Data 84

## IV Parseval’s Equality 89

§ 1. Preliminaries 89

§ 2. Derivation of Parseval’s Equality from the Fundamental Equation 94

§ 3. Derivation of the Fundamental Equation from Parsevals Equality 99

## V The inverse problem 105

§ 1. Statement of the Problem 105

§ 2. Estimates for the Matrix K(x,y) 107

§ 3. Existence of the Derivatives of K(x,y) 110

§ 4. Derivation of the Differential Equation 117

§ 5. Fulfillment of the Boundary Condition 123

§6 . Characteristic Properties of the Scattering Data and Scattering Matrix 129

§ 7. Examples 138

## PART TWO

THE BOUNDARY-VALUE PROBLEM WITH SINGULARITIES

VI Special transformation Operators 147

§ 1. Method of Investigation 147

§ 2. Transformation Operators for Matrix Equations 153

§ 3. Transformation of Parseval’s Equality 157

## VII Spectral analysis of the boundary-value problem with singularities 163

§ 1. Statement of the Problem. Notation 163

§ 2. Particular Solutions 164

§ 3. The First Transformation 169

§ 4. The Second Transformation 177

§ 5. The Third Transformation 188

§ 6. Properties of the Scattering Data (the Case a_{22} =0, a_{12} ≠ 0

§ 7. Behavior of the Scattering Matrix When 𝜆 ⟶ 0. Summary of Results 217

## VIII Reconstruction of the singular boundary-value problem from its scattering data 223

§ 1. Case (a) 223

§ 2. Case (b) 226

§ 3. Case (c) 234

§ 4. Algorithm for Determining the Potential Matrix. Examples 254

*Appendix I On the characteristic properties of the scattering data of the boundary-value problem without singularities 265*

§ 1. Factorization of a Unitary Matrix . 265

§ 2. Indices of S(𝜆) 270

§ 3. A New Characterization of the Scattering Data 280

*Appendix II Refinement of certain inequalities 283*

Bibliography 289

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