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 con structing this oparator 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 (G. Borg, V. Bargmann, N. Levinson, etc.) and Soviet authors (V. A. Marchenko, M. G. Krein, I. M. Gelfand, B. M. Levitan, etc.), 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 quan tum 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 mono graph 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 ref lection 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 consi derable amount of work they have done in this and related areas. Moreover, the functional analytic and algebraic methods used should also be of great interet to pure and applied mathematicians.
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