The bound-state solutions of the one-dimensional pseudoharmonic oscillator

TL;DR

This paper investigates the bound states of a one-dimensional pseudoharmonic oscillator with inverse square interaction, analyzing both singular and regularized potentials through analytical and numerical methods, revealing degeneracies and state behaviors.

Contribution

It provides a comprehensive analysis of the bound states of the pseudoharmonic oscillator, including regularization effects and state degeneracies, which were not fully explored before.

Findings

Degenerate bound states for specific interaction strengths.

Regularized potential yields even and odd eigenfunctions.

Ground state approaches a Dirac delta as cutoff vanishes.

Abstract

We study the bound states of a quantum mechanical system consisting of a simple harmonic oscillator with an inverse square interaction, whose interaction strength is governed by a constant $\alpha$. The singular form of this potential has doubly-degenerate bound states for $-1/4\leq\alpha<0$ and $\alpha>0$; since the potential is symmetric, these consist of even and odd-parity states. In addition we consider a regularized form of this potential with a constant cutoff near the origin. For this regularized potential, there are also even and odd-parity eigenfunctions for $\alpha\geq-1/4$. For attractive potentials within the range $-1/4\leq\alpha<0$, there is an even-parity ground state with increasingly negative energy and a probability density that approaches a Dirac delta function as the cutoff parameter becomes zero. These properties are analogous to a similar ground state present in the regularized one-dimensional hydrogen atom. We solve this problem both analytically and numerically, and show how the regularized excited states approach their unregularized counterparts.