Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations

Abstract

We are concerned with the study of some classical spectral collocation methods, mainly Chebyshev and sinc as well as with the new software system Chebfun in computing high order eigenpairs of singular and regular Schrödinger 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, were analyzed. In order to separate “good”and “bad”eigenvalues, we have estimated 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.