Efficient computation of Sturm-Liouville eigenvalues using modified Neumann schemes

We discuss a numerical method, based on a modified Neumann integrator, to solve the general Sturm-Liouville eigenvalue problem. Such a modified integral series method was already shown to be very succesfull for problems in the Schrodinger form. Here we discuss these methods in a more general context and apply them directly on the Sturm-Liouville problem, avoiding the need of a Liouville's transformation to convert the problem into the Schrodinger form. This allows to use the modified integral series methods also for those Sturm-Liouville problems where the Liouville's transformation is problematic or expensive. By applying the presented modified Neumann method in a shooting procedure, the eigenvalues can efficiently be computed upto high accuracy. It is shown that the method presented, is closely related to the second-order Pruess method, but provides for higher order approximations