A Perturbation of the Dunkl Harmonic Oscillator on the Line

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