Sturm-Liouville eigenvalue characterizations

We study the relationship between the eigenvalues of separated self-adjoint boundary conditions and coupled self-adjoint conditions. Given an arbitrary real coupled boundary condition determined by a coupling matrix K we construct a one parameter family of separated conditions and show that all the eigenvalues for K and -K are extrema of the eigencurves of this family. This characterization makes it possible to use the well known Prufer transformation which has been used very successfully, both theoretically and numerically, for separated conditions, also in the coupled case. In particular, this characterization makes it possible to compute the eigenvalues for any real coupled self-adjoint boundary condition using any code which works for separated conditions.