Matrix Representations of Sturm–Liouville Problems with Finite Spectrum

Abstract.

We identify a class of Sturm–Liouville equations such that any Sturm–Liouville (SL) problem consisting of such an equation and an arbitrary separated or coupled real self-adjoint boundary condition has a representation as an equivalent finite dimensional matrix eigenvalue problem. Conversely, given any matrix eigenvalue problem of certain type and an arbitrary separated or coupled real self-adjoint (SL) boundary condition, we construct a class of Sturm–Liouville problems with the specified boundary condition, each of which is equivalent to the given matrix eigenvalue problem.