Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials

TL;DR

This paper generalizes third-order ladder operators and constructs new Hamiltonians related to the fourth Painlevé transcendent, revealing connections to generalized Okamoto and exceptional Hermite polynomials with implications for eigenfunction structures.

Contribution

It extends the construction of shape-invariant Hamiltonians to the most general case involving third-order operators and links them to generalized Okamoto and exceptional Hermite polynomials.

Findings

Decomposition of eigenfunctions into three disjoint solution sequences

Identification of three zero-modes via generalized Okamoto polynomials

Connection between eigenfunctions and exceptional Hermite polynomials

Abstract

We extend and generalize the construction of Sturm-Liouville problems for a family of Hamiltonians constrained to fulfill a third-order shape-invariance condition and focusing on the "$-2x/3$" hierarchy of solutions to the fourth Painlev\'e transcendent. Such a construction has been previously addressed in the literature for some particular cases but we realize it here in the most general case. The corresponding potential in the Hamiltonian operator is a rationally extended oscillator defined in terms of the conventional Okamoto polynomials, from which we identify three different zero-modes constructed in terms of the generalized Okamoto polynomials. The third-order ladder operators of the system reveal that the complete set of eigenfunctions is decomposed as a union of three disjoint sequences of solutions, generated from a set of three-term recurrence relations. We also identify a link between the eigenfunctions of the Hamiltonian operator and a special family of exceptional Hermite polynomial.