Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schr\"odinger Equations

TL;DR

This paper evaluates spectral collocation methods and Chebfun for computing high order eigenvalues of singular and regular Schrödinger problems, highlighting their strengths, limitations, and accuracy in challenging benchmark cases.

Contribution

It introduces specialized techniques within Chebfun for boundary singularities and domain transformations, improving high order eigenvalue computation for challenging Schrödinger problems.

Findings

Chebfun effectively computes high order eigenvalues with domain truncation.

Eigenvalue drift estimation helps assess numerical stability and error.

Methods successfully handle problems with almost multiple and mixed spectra.

Abstract

We are concerned with the study of some classical spectral collocation methods as well as with the new software system Chebfun in computing high order eigenpairs of singular and regular Schrodinger eigenproblems. We want to highlight both the qualities as well as the shortcomings of these methods and evaluate them in conjunction with the usual ones. In order to resolve a boundary singularity we use Chebfun with domain truncation. Although it is applicable with spectral collocation, a special technique to introduce boundary conditions as well as a coordinate transform, which maps an unbounded domain to a finite one, are the special ingredients. A challenging set of \hard"benchmark problems, for which usual numerical methods (f. d., f. e. m., shooting etc.) fail, are analyzed. In order to separate \good"and \bad"eigenvalues we estimate the drift of the set of eigenvalues of interest with respect to the order of approximation and/or scaling of domain parameter. It automatically provides us with a measure of the error within which the eigenvalues are computed and a hint on numerical stability. We pay a particular attention to problems with almost multiple eigenvalues as well as to problems with a mixed spectrum.