A Perturbation of the Dunkl Harmonic Oscillator on the Line
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Title: | A Perturbation of the Dunkl Harmonic Oscillator on the Line |
Author: | Álvarez López, Jesús Antonio Calaza Cabanas, Manuel Franco, Carlos |
Affiliation: | Universidade de Santiago de Compostela. Departamento de Xeometría e Topoloxía Universidade de Santiago de Compostela. Departamento de Matemática Aplicada |
Subject: | Dunkl harmonic oscillator | Perturbation theory | |
Date of Issue: | 2015 |
Publisher: | National Academy of Science of Ukraine |
Citation: | Álvarez López, J. A., Calaza, M., and Franco, C. (2015). A perturbation of the Dunkl harmonic oscillator on the line. Symmetry, Integrability and Geometry: Methods and Applications. |
Abstract: | Let Jσ be the Dunkl harmonic oscillator on R (σ>−1/2. For 0<u<1 and ξ>0, it is proved that, if σ>u−1/2, then the operator U=Jσ+ξ|x|−2u, with appropriate domain, is essentially self-adjoint in L2(R,|x|2σdx), the Schwartz space S is a core of U¯¯¯¯1/2, and U¯¯¯¯ has a discrete spectrum, which is estimated in terms of the spectrum of Jσ¯¯¯¯¯. A generalization Jσ,τ of Jσ is also considered by taking different parameters σ and τ on even and odd functions. Then extensions of the above result are proved for Jσ,τ, 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 R+ are derived |
Publisher version: | https://doi.org/10.3842/SIGMA.2015.059 |
URI: | http://hdl.handle.net/10347/21868 |
DOI: | 10.3842/SIGMA.2015.059 |
E-ISSN: | 1815-0659 |
Rights: | © Author(s) 2015. This work is licensed under a Creative Commons Attribution-ShareAlike (CC BY-SA) license |
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