Inverse Sturm–Liouville problems with finite spectrum

We study inverse Sturm–Liouville problems of Atkinson type whose spectrum consists entirely of a finite set of eigenvalues. We show that given two finite sets of interlacing real numbers there exists a class of Sturm–Liouville equations of Atkinson type such that the two sets of numbers are the eigenvalues of their associated Sturm–Liouville problems with two different separated boundary conditions. Parallel results are also obtained for real coupled boundary conditions. Our approach is to use the equivalence between Sturm–Liouville problems of Atkinson type and matrix eigenvalue problems and to apply our development of the well-known theory for inverse matrix eigenvalue problems.