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. (C) 2011 Elsevier Inc. All rights reserved.