On some conditionally solvable quantum-mechanical problems

TL;DR

This paper investigates two specific quantum models, analyzing their eigenvalues and eigenfunctions, and clarifies misconceptions in their interpretation, contributing to the understanding of conditionally solvable quantum systems.

Contribution

It provides a detailed analysis of two conditionally solvable quantum models and clarifies common misinterpretations in their spectral properties.

Findings

Distribution diagrams of eigenvalues are presented.

Exact eigenvalues are compared with variational results.

Misinterpretations of eigenvalues and eigenfunctions are addressed.

Abstract

We analyze two conditionally solvable quantum-mechanical models: a one-dimensional sextic oscillator and a perturbed Coulomb problem. Both lead to a three-term recurrence relation for the expansion coefficients. We show diagrams of the distribution of their exact eigenvalues with the addition of accurate ones from variational calculations. We discuss the symmetry of such distributions. We also comment on the wrong interpretation of the exact eigenvalues and eigenfunctions by some researchers that has led to the prediction of allowed cyclotron frequencies and field intensities.