Solving Sturm–Liouville problems by piecewise perturbation methods, revisited

We present the extension of the successful Constant Perturbation Method (CPM) for Schrödinger problems to the more general class of Sturm–Liouville eigenvalue problems. Whereas the original CPM can only be applied to Sturm–Liouville problems after a Liouville transformation, the more general CPM presented here solves the Sturm–Liouville problem directly. This enlarges the range of applicability of the CPM to a wider variety of problems and allows a more efficient solution of many problems. The CPMs are closely related to the second-order coefficient approximation method underlying the SLEDGE software package, but provide for higher order approximations. These higher order approximations can also be obtained by applying a modified Neumann method. The CPM approach, however, leads to simpler formulae in a more convenient form.