Matrix methods for radial Schrödinger eigenproblems defined on a semi-infinite domain

TL;DR

Numerical approximation of the eigenvalues of the one-dimensional radial Schrodinger equation posed on a semi-infinite interval based on high order finite difference schemes is discussed.

Abstract

In this paper, we discuss numerical approximation of the eigenvalues of the one-dimensional radial Schrödinger equation posed on a semi-infinite interval. The original problem is first transformed to one defined on a finite domain by applying suitable change of the independent variable. The eigenvalue problem for the resulting differential operator is then approximated by a generalized algebraic eigenvalue problem arising after discretization of the analytical problem by the matrix method based on high order finite difference schemes. Numerical experiments illustrate the performance of the approach.

Figures (3)

• Figure 1: Hydrogen atom equation, $\ell=3$: relative errors in the eigenvalues calculated using TDS, TCII with (\ref{['sceltaxi']}), and ATCII.
• Figure 2: Hulthén potential, $\ell=0,3$, $\alpha=0.02$: relative errors in the eigenvalues calculated using TCII with (\ref{['sceltaxi']}).
• Figure 3: Hulthén potential, $\ell=0,3$, $\alpha=0.02$: relative errors in the eigenvalues calculated using ATCII.