Sturm-liouville Problems and Discontinuous Eigenvalues

If a Sturm-Liouville problem is given in an open interval of the real line then regular boundary value problems can be considered on compact sub-intervals. For these regular problems, all with necessarily discrete spectra, the eigenvalues depend on both the end-points of the compact intervals, and upon the choice of the real separated boundary conditions at these end-points. These eigenvalues are not, in general, continuous functions of the end-points and boundary conditions. The paper shows the surprising form of these discontinuities. The results have applications to the approximations of singular Sturm-Liouville problems by regular problems, and to the theoretical aspects of the Sleign2 computer program.