The fast and accurate computation of eigenvalues and eigenfunctions of time-independent one-dimensional Schrödinger equations

Abstract

In this paper, we present the basic routines of the C++-program Matslise 3.0, an updated but yet restricted version of the matlab package Matslise 2.0. Matslise 3.0 currently allows the accurate, but in comparison to Matslise 2.0, faster computation of eigenvalues and eigenfunctions of one dimensional time-independent Schrödinger problems. The numerical examples show that speed up factors up to 20 (for the eigenvalues) and 200 (for the eigenfunctions) are obtained. These highly optimized routines will enable us, in the near future, to extend Matslise 3.0 to solve time-independent 2D and 3D as well as time-dependent 1D problems. Program summaryProgram Title: Matslise 3.0CPC Library link to program files:http://dx.doi.org/10.17632/4yn7pp92y4.1Developer’s repository link:https://github.com/twist-numerical/matsliseCode Ocean capsule:https://codeocean.com/capsule/2123682/tree/v1Licensing provisions: MITProgramming language: C++, PythonNature of problem: The time-independent one-dimensional Schrödinger equation is an ordinary differential equation with highly oscillatory solutions. This program calculates the eigenvalues and eigenfunctions for a given potential accurately and efficiently.Solution method: A constant-perturbation method is used. In this method a piecewise constant approximation of the potential is improved by adding perturbation terms. To locate the eigenvalues multiple shooting is employed.Additional comments including restrictions and unusual features: The code submitted also contains routines to solve the two-dimensional time-independent Schrödinger equation. This extra code will be described in a later article.