An ubiquitous three-term recurrence relation

TL;DR

This paper analyzes a three-term recurrence relation arising in physical problems, demonstrating that certain predicted physical quantities are artifacts of the truncation method rather than actual eigenvalues.

Contribution

It provides a comparison between analytical solutions from truncation and numerical eigenvalues, clarifying the nature of physical predictions derived from the recurrence relation.

Findings

Truncation yields eigenvalues that are artifacts, not true physical quantities.

Numerical methods confirm the discrepancy between truncated solutions and actual eigenvalues.

Physical predictions based on truncation should be interpreted with caution.

Abstract

We solve an eigenvalue equation that appears in several papers about a wide range of physical problems. The Frobenius method leads to a three-term recurrence relation for the coefficients of the power series that, under suitable truncation, yields exact analytical eigenvalues and eigenfunctions for particular values of a model parameter. From these solutions some researchers have derived a variety of predictions like allowed angular frequencies, allowed field intensities and the like. We also solve the eigenvalue equation numerically by means of the variational Rayleigh-Ritz method and compare the resulting eigenvalues with those provided by the truncation condition. In this way we prove that those physical predictions are merely artifacts of the truncation condition.