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.
Highlights
There is clearly an increasing interest to develop accurate and efficient methods of solution to singular Schrödinger eigenproblems.the first aim of our study is to qualitatively compare the classical and the new Chebfun spectral methods in solving singular eigenproblems
In order to resolve a boundary singularity, we use Chebfun with domain truncation. 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
The main purpose of this paper is to argue that Chebfun, along with the spectral collocation methods, can be a very feasible alternative to these software packages regarding accuracy, robustness as well as simplicity of implementation
Summary
There is clearly an increasing interest to develop accurate and efficient methods of solution to singular Schrödinger eigenproblems. It can be argued that, generally speaking, for solving various differential problems, the Chebfun software provides a greater flexibility than the classical spectral methods. If the interval of definition is semi-infinite or infinite, or is finite and q(x) vanishes at one or both endpoints, or if q is discontinuous, we can not obtain from (1) a regular Sturm–Liouville problem In any such case, the Schrödinger Equation (1) is called singular. The main purpose of this paper is to argue that Chebfun, along with the spectral collocation methods, can be a very feasible alternative to these software packages regarding accuracy, robustness as well as simplicity of implementation These methods can compute the “whole”set of eigenvectors and provide some details on the accuracy and numerical stability of the results provided. We improve on several previous results available in the literature, and present a MATLAB toolbox for solving a wide range of problems
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