The singular harmonic oscillator revisited

TL;DR

This paper reexamines the quantum singular harmonic oscillator, revealing new insights into its solutions, degeneracy, and the limitations of perturbation theory, challenging previous literature.

Contribution

It demonstrates that the singular oscillator has infinitely many solutions above a critical parameter, and shows solutions cannot be derived from the nonsingular case via perturbation.

Findings

Infinite solutions exist for parameter above critical value

Degeneracy occurs in solutions on the whole line

Perturbation theory cannot derive singular oscillator solutions

Abstract

The one-dimensional Schr\"{o}dinger equation with the singular harmonic oscillator is investigated. The Hermiticity of the operators related to observable physical quantities is used as a criterion to show that the attractive or repulsive singular oscillator exhibits an infinite number of acceptable solutions provided the parameter responsible for the singularity is greater than a certain critical value, in disagreement with the literature. The problem for the whole line exhibits a two-fold degeneracy in the case of the singular oscillator, and the intrusion of additional solutions in the case of a nonsingular oscillator. Additionally, it is shown that the solution of the singular oscillator can not be obtained from the nonsingular oscillator via perturbation theory.