Two-step rational extensions of the harmonic oscillator: exceptional orthogonal polynomials and ladder operators

Abstract

The type III Hermite Xm exceptional orthogonal polynomial family is generalized to a double-indexed one (with m1 even and m2 odd such that m2 > m1) and the corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics. The new polynomials are proved to be expressible in terms of mixed products of Hermite and pseudo-Hermite ones, while some of the associated potentials are linked with rational solutions of the Painlevé IV equation. A novel set of ladder operators for the extended oscillators is also built and shown to satisfy a polynomial Heisenberg algebra of order m2 − m1 + 1, which may alternatively be interpreted in terms of a special type of (m2 − m1 + 2)th-order shape invariance property.