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

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.