A Perturbation of the Dunkl Harmonic Oscillator on the Line

Let J(sigma) be the Dunkl harmonic oscillator on 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 vertical bar x vertical bar(-2u), with appropriate domain, is essentially self-adjoint in L-2 (R,vertical bar x vertical bar(2 sigma)dx), the Schwartz space S is a core of (U) over bar (1/2) and (U) over bar has a discrete spectrum, which is estimated in terms of the spectrum of (J) over bar (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 R+ are derived.