Solvable Rational Potentials and Exceptional Orthogonal Polynomials in Supersymmetric Quantum Mechanics

Abstract

New exactly solvable rationally-extended radial oscillator and Scarf I potentials are generated by using a constructive supersymmetric quantum mechanical method based on a reparametrization of the corresponding conventional superpotential and on the addition of an extra rational contribution expressed in terms of some polynomial g. The cases where g is linear or quadratic are considered. In the former, the extended potentials are strictly isospectral to the conventional ones with reparametrized couplings and are shape invariant. In the latter, there appears a variety of extended potentials, some with the same charac- teristics as the previous ones and others with an extra bound state below the conventional potential spectrum. Furthermore, the wavefunctions of the extended potentials are con- structed. In the linear case, they contain ( + 1)th-degree polynomials with = 0,1,2,..., which are shown to beX1-Laguerre orX1-Jacobi exceptional orthogonal polynomials. In the quadratic case, several extensions of these polynomials appear. Among them, two different kinds of ( + 2)th-degree Laguerre-type polynomials and a single one of ( + 2)th-degree Jacobi-type polynomials with = 0,1,2,... are identified. They are candidates for the still unknown X2-Laguerre and X2-Jacobi exceptional orthogonal polynomials, respectively.