A Perturbation of the Dunkl Harmonic Oscillator on the Line

Abstract

Let $J_{\sigma }$ be the Dunkl harmonic oscillator on $\mathbb{R}$ ($\sigma >-\frac{1}{2}$). For $0<u<1$ and $\xi >0$, it is proved that, if $\sigma >u-\frac{1}{2}$, 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 dif\/ferent 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.

We correct the second main theorem of the previous version, by the first two authors. The corrections concern mainly certain estimates, which were also improved by adding more methods

References in corpus (1)

• Imaginary Powers of the Dunkl Harmonic Oscillator

Cited by in corpus (1)

• On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings