  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.
