Inverse spectral problems and closed exponential systems

Abstract

Consider the inverse eigenvalue problem of the Schrödinger operator defined on a finite interval. We give optimal and almost optimal conditions for a set of eigenvalues to determine the Schrödinger operator. These conditions are simple closedness properties of the exponential system corresponding to the known eigenvalues. The statements contain nearly all former results of this topic. We give also conditions for recovering the Weyl-Titchmarsh $m$-function from its values $m(\lambda_n)$.