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

Abstract

The type III Hermite \(X_m\) exceptional orthogonal polynomial family is generalized to a double-indexed one \(X_{m_1,m_2}\) (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 Painlev\'e 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 \(m_2-m_1+1\), which may alternatively be interpreted in terms of a special type of \((m_2-m_1+2)\)th-order shape invariance property.