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Symmetry, Integrability and Geometry: Methods and Applications, 2015, Volume 11, 059, 47 pp.
DOI: https://doi.org/10.3842/SIGMA.2015.059
(Mi sigma1040)
 

This article is cited in 2 scientific papers (total in 2 papers)

A Perturbation of the Dunkl Harmonic Oscillator on the Line

Jesús A. Álvarez Lópeza, Manuel Calazab, Carlos Francoa

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
References:
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.
Keywords: Dunkl harmonic oscillator; perturbation theory.
Received: February 19, 2015; in final form July 20, 2015; Published online July 25, 2015
Bibliographic databases:
Document Type: Article
MSC: 47A55; 47B25; 33C45
Language: English
Citation: Jesús A. Álvarez López, Manuel Calaza, Carlos Franco, “A Perturbation of the Dunkl Harmonic Oscillator on the Line”, SIGMA, 11 (2015), 059, 47 pp.
Citation in format AMSBIB
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\by Jes\'us~A.~\'Alvarez L\'opez, Manuel~Calaza, Carlos~Franco
\paper A Perturbation of the Dunkl Harmonic Oscillator on the Line
\jour SIGMA
\yr 2015
\vol 11
\papernumber 059
\totalpages 47
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\crossref{https://doi.org/10.3842/SIGMA.2015.059}
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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