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.

arXiv commentWe 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 (5)

• Generalized Hermite Polynomials and the Heat Equation for Dunkl Operators
• Imaginary Powers of the Dunkl Harmonic Oscillator
• Symmetric Functions and B_N-invariant Spherical Harmonics
• Embedding Theorems for the Dunkl Harmonic Oscillator on the Line
• Application of the method of Bonan-Clark to the generalized Hermite polynomials

Cited by in corpus (1)

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