Uniqueness Theorem for the Eigenvalues’ Function

Abstract

We study the family of Sturm-Liouville operators, generated by fixed potential q and the family of separated boundary conditions. We prove that the union of the spectra of all these operators can be represented as the values of a real analytic function of two variables. We call this function “the eigenvalues’ function” of the family of Sturm-Liouville operators (EVF). We show that the knowledge of some eigenvalues for an infinite set of different boundary conditions is sufficient to determine the EVF, which is equivalent to uniquely determine the unknown potential. Our assertion is the extention of McLaughlin-Rundell theorem.