Discrete eigenproblems

TL;DR

This paper investigates discrete Schrödinger eigenproblems, focusing on the properties of Lommel polynomials related to eigenvalues, and compares discrete approximations with continuum solutions, demonstrating good agreement.

Contribution

It reveals the connection between eigenvalues and Lommel polynomials, and analyzes the continuum limit of determinants in discrete Schrödinger problems.

Findings

Lommel polynomials determine eigenvalues in discrete Schrödinger problems.

Discrete approximations closely match continuum eigenvalues even with few vertices.

The continuum limit of the determinant is obtained via a transitional limit of Lommel polynomials.

Abstract

Schrodinger eigenproblems on a discrete interval are further investigated with special attention given to test cases such as the linear and Rosen--Morse potentials. In the former case it is shown that the characteristic function determining the eigenvalues is a Lommel polynomial and considerable space is devoted to these objects. For example it is shown that the continuum limit of the determinant is obtained by a transitional limit of the Lommel polynomials for large order and argument. Numerical comparisons between discrete approximations and (known) continuum values for the ratio of functional determinants with and without the potential are made and show good agreement, even for small numbers of vertices. The zero mode problem is also briefly dealt with.