A Perturbation of the Dunkl Harmonic Oscillator on the Line

Let $J_\sigma$ be the Dunkl harmonic oscillator on ${\mathbb{R}}$ ($\sigma>-1/2$). For $0<u<1$ and $\xi>0$, it is proved that, if $\sigma>u-1/2$, then the operator $U=J_\sigma+\xi|x|^{-2u}$, with appropriate domain, is essentially self-adjoint in $L^2({\mathbb{R}},|x|^{2\sigma} dx)$, the Schwartz space ${\mathcal{S}}$ is a core of $\overline U^{1/2}$, and $\overline U$ has a discrete spectrum, which is estimated in terms of the spectrum of $\overline{J_\sigma}$. A generalization $J_{\sigma,\tau}$ of $J_\sigma$ is also considered by taking different parameters $\sigma$ and $\tau$ on even and odd functions. Then extensions of the above result are proved for $J_{\sigma,\tau}$, where the perturbation has an additional term involving, either the factor $x^{-1}$ on odd functions, or the factor $x$ on even functions. Versions of these results on ${\mathbb{R}}_+$ are derived.