An approximate solution of inverse spectral problem for perturbed self-adjoint operator

Abstract

Dynamic measurements are becoming more common in engineering and scientific research. These measurements are associated primarily with the study of regularities of the physical processes in the investigated objects. The relevance of these tasks, their great theoretical interest and practical importance are provided by intensive development of optimal control theory and the theory of inverse and ill-posed problems. Currently, there is a very extensive literature interest to an optimal control theory, inverse and ill-posed problems. However, one cannot claim that it covers all parties in mathematics. It continues to develop intensively in both theoretical and applied aspects. One of perspective directions of development are inverse spectral problems. It is known that the inverse problems have many applications in engineering and physics. This article is devoted to an approximate reconstruction of the potential in the inverse spectral problem for a discrete positive self-adjoint operator. Using the resolvent method and the contraction mapping theorem, the existence of solution of the inverse spectral problem is proved. Formulas for the approximate solutions of the considered problem have been obtained. Research is conducted in the framework of the theory of regularized traces, developed by V. A. Sadovnichy, V. V. Dubrovski and their disciples.