Rational extensions of the Dunkl oscillator in the plane and exceptional orthogonal polynomials

Abstract

It is shown that rational extensions of the isotropic Dunkl oscillator in the plane can be obtained by adding some terms either to the radial equation or to the angular one obtained in the polar coordinates approach. In the former case, the isotropic harmonic oscillator is replaced by an isotropic anharmonic one, whose wavefunctions are expressed in terms of $X_m$-Laguerre exceptional orthogonal polynomials. In the latter, it becomes an anisotropic potential, whose explicit form has been found in the simplest case associated with $X_1$-Jacobi exceptional orthogonal polynomials.