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

TL;DR

This paper introduces a new class of exceptional orthogonal polynomials and constructs rational extensions of the harmonic oscillator using supersymmetric quantum mechanics, revealing new algebraic structures and connections to Painlevé equations.

Contribution

It generalizes type III Hermite exceptional polynomials to a double-indexed family and develops associated ladder operators with polynomial Heisenberg algebra.

Findings

New double-indexed exceptional Hermite polynomials expressed via Hermite and pseudo-Hermite polynomials.

Constructed rational extensions of the harmonic oscillator with novel ladder operators.

Linked some potentials to rational solutions of Painlevé IV equation.

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.