Matrix representations of Sturm–Liouville problems with transmission conditions

We identify a class of Sturm–Liouville equations with transmission conditions such that any Sturm–Liouville problem consisting of such an equation with transmission condition and an arbitrary separated or real coupled 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 real coupled self-adjoint boundary condition and transmission condition, we construct a class of Sturm–Liouville problems with this specified boundary condition and transmission condition, each of which is equivalent to the given matrix eigenvalue problem.