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

The type III Hermite X-m exceptional orthogonal polynomial family is generalized to a double-indexed one X-m1,X- m2 (with m(1) even and m(2) odd such that m(2) > m(1)) 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 Painleve IV equation. A novel set of ladder operators for the extended oscillators is also built and shown to satisfy a polynomialHeisenberg algebra of order m(2) - m(1) + 1, which may alternatively be interpreted in terms of a special type of (m(2) - m(1) + 2)th-order shape invariance property.