To the question of the uniqueness of the reduction potential in the inverse problem of the Borg-Levinson

Abstract

Under inverse problems of spectral analysis to understand the problem of reconstructing an operator from its predetermined spectral characteristics. In this study, hesitant recovery uniqueness of the operator from the spectral data, specified for different boundary conditions. Many inverse problems are not the only solution. Thus, for example, Borg showed that Sturm - Liouville generally not uniquely defined by one spectrum. Therefore, one of the most important is the problem of the uniqueness of the potential recovery in the solution which raises the question of the identification of additional conditions to ensure the uniqueness of the inverse problem solution. For the first time the inverse problem for the Sturm - Liouville was set VA Ambartsumian, and in its simplest formulation, it zaklyualas to identify the operator, knowing its spectrum. Further advances in the theory of inverse problems has been achieved as a result of the application to the study of inverse problems of the so-called conversion operators. This method was developed in detail in the works of VA Marchenko, MG Crane, IM Gelfand, BM Levitan, LD Faddeev, MG Gasimov, YM Berezan and others. They were considered by the theorem on the existence of capacity in the inverse problem of spectral analysis for ordinary differential equations, and the formulation of the inverse spectral problem was to find a building for a given spectral function     