Solving Sturm–Liouville problems by piecewise perturbation methods, revisited

Abstract

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 orginal 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.

Matslise 2.0

“…For problems not in Liouville normal form, however, the (absolute) error of the CP method behaves asymptotically (i.e., for large index) like O(E), see Ledoux and Van Daele [2010]. Moreover, higher-order CP methods are available for Schrödinger problems: We choose to use the CP method of order 16 from Ledoux et al [2004] in MATSLISE 2.0.…”

“…As a consequence, the original software package MATSLISE has a smaller range of applicability in comparison with, for example, the Fortran solver SLEDGE [Pruess and Fulton 1993], which applies a second-order coefficient approximation method directly to an SLP. In Ledoux and Van Daele [2010], the extension of the CP algorithm to the general form of an SLP (1) was described. A sixth-order CP method was presented that can be applied directly to the SLP without the need for a Liouville transformation.…”

“…This means that remarkably large stepsizes can then be taken, and one and the same E-independent mesh can be used to compute all eigenvalues required. The sixth-order CP algorithm from Ledoux and Van Daele [2010] is only used to apply a CP method directly to the SLP when a Liouville transformation may be difficult or expensive to realize, such as, for example, in the presence of jumps in the coefficient functions or a singular endpoint.…”

“…The successor code MATSLISE 2.0 now includes the recent extension of the CP algorithm from problems in Liouville normal form to the general Sturm-Liouville form (see Ledoux and Van Daele [2010]). This extension and some specially adapted algorithms that are applied in a narrow interval around the singularity make MATSLISE 2.0 suitable for a broader class of singular problems.…”

On CP, LP and other piecewise perturbation methods for the numerical solution of the Schrödinger equation

“…The integrals in the series terms are replaced by Filon quadrature which respects high oscillation (see [15]). As shown in [32,33], applying a modified Neumann or Magnus method in combination with a Filon quadrature rule to a problem of the form (1.1), involves the computation of the exact solution of the problem with constant potentialV and the replacement of V (x) by polynomial approximations to allow the evaluation of the resulting series terms. This illustrates that, next to the PPM, modified Magnus or Neumann schemes also form a natural extension of the coefficient approximation idea to higher order methods.…”