A Perturbation of the Dunkl Harmonic Oscillator on the Line

Symmetry Integrability and Geometry-Methods and Applications

article

Jesús A. Álvarez López¹  Manuel Calaza²  Carlos Franco¹ 

[1] Universidade de Santiago de Compostela;Laboratorio de Investigación 2 and Rheumatology Unit, Hospital Clinico Universitario de Santiago

关键词: Dunkl harmonic oscillator;    perturbation theory;   

DOI  :  10.3842/SIGMA.2015.059

来源: National Academy of Science of Ukraine

【 摘 要 】

Let $J_\sigma$ be the Dunkl harmonic oscillator on ${\mathbb{R}}$ ($\sigma$>$-1/2$. For $0$ $0$, 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.

【 授权许可】

Unknown