Inverse spectral problems for discontinuous Sturm–Liouville problems of Atkinson type

Abstract

We investigate inverse spectral problems for discontinuous Sturm–Liouville problems of Atkinson type whose spectrum consists of a finite set of eigenvalues. For given two finite sets of interlacing real numbers, there exists a class of Sturm–Liouville equations such that the two sets of numbers are exactly the eigenvalues of their associated Sturm–Liouville problems with two different separated boundary conditions. The main approach is to give an equivalent relation between Sturm–Liouville problems of Atkinson type and matrix eigenvalue problems, and the theory of inverse matrix eigenvalue problems.