Gross misinterpretation of a conditionally solvable eigenvalue equation

TL;DR

This paper critically examines a commonly used truncation method for solving a specific eigenvalue equation in physics, demonstrating that many physical predictions based on this method are artifacts rather than real solutions.

Contribution

It provides a numerical validation that the truncation-based solutions are not physically meaningful, challenging prior interpretations in the literature.

Findings

Truncation solutions do not correspond to true eigenvalues.

Numerical solutions differ significantly from truncated analytical solutions.

Physical predictions based on truncation are artifacts, not real phenomena.

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.