### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 059, 47 pages      arXiv:1412.4655      https://doi.org/10.3842/SIGMA.2015.059

### A Perturbation of the Dunkl Harmonic Oscillator on the Line

Jesús A. Álvarez López a, Manuel Calaza b and Carlos Franco a
a) Departamento de Xeometría e Topoloxía, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
b) Laboratorio de Investigación 2 and Rheumatology Unit, Hospital Clinico Universitario de Santiago, Santiago de Compostela, Spain

Received February 19, 2015, in final form July 20, 2015; Published online July 25, 2015; Corrected June 28, 2017

Abstract
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.

Key words: Dunkl harmonic oscillator; perturbation theory.

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