A Perturbation of the Dunkl Harmonic Oscillator on the Line

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 L(R, |x|dx), 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.