Recovery of the nearest potential field from the [formula omitted] observed eigenvalues

Abstract

A new class of inverse problems is considered. In the context of classical theory, inverse problems are concerned with finding a model that has the observed measurements. It is well known that such problems usually are ill-posed. At the same time, it is often the case when there is some a priori information about the system. This naturally leads to the following inverse optimal problem: find data Fˆ of a model which is the nearest to a priori given data F0 and sufficient to ensure the model has the observed measurements S.In this note, an approach to a complete solution to such a problem is developed. Within the framework of this approach, we consider a model problem of recovering the potential field Vˆ from the m observed eigenvalues of the Schrödinger operator, provided that such potential field is at the minimum distance from a priori given potential Va. In the main result, we establish a new type of relationship between the linear spectral problems and systems of nonlinear differential equations which enables us to find a solution to the inverse optimal spectral problem and obtain novel results on the existence of solutions to nonlinear problems as well.