Álvarez López, Jesús AntonioCalaza Cabanas, ManuelFranco, Carlos2020-04-292020-04-292015Álvarez 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.http://hdl.handle.net/10347/21868Let 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 derivedeng© Author(s) 2015. This work is licensed under a Creative Commons Attribution-ShareAlike (CC BY-SA) licensehttps://creativecommons.org/licenses/by-sa/3.0/Dunkl harmonic oscillatorPerturbation theoryjournal article10.3842/SIGMA.2015.0591815-0659open access