On the quantum-mechanical singular harmonic oscillator

TL;DR

This paper derives eigenvalues and eigenfunctions of the singular harmonic oscillator using the Frobenius method, revealing two eigenvalue branches for negative alpha and discussing conditions for square-integrable solutions.

Contribution

It provides a straightforward derivation of eigenvalues and eigenfunctions for the singular harmonic oscillator, including new insights into eigenvalue branches for negative alpha.

Findings

Eigenvalues and eigenfunctions obtained via Frobenius method

Two branches of eigenvalues identified for negative alpha

Square-integrable solutions exist only for specific discrete energies

Abstract

We obtain the eigenvalues and eigenfunctions of the singular harmonic oscillator $V(x)=\alpha/(2x^2)+x^2/2$ by means of the simple and straightforward Frobenius (power-series) method. From the behaviour of the eigenfunctions at origin we derive two branches for the eigenvalues for negative values of $\alpha$. We discuss the well known fact that there are square-integrable solutions only for some allowed discrete values of the energy.