Third-order ladder operators, generalized Okamoto and exceptional orthogonal 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 "" 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.
Keywords
Cite
@article{arxiv.2101.12313,
title = {Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials},
author = {Véronique Hussin and Ian Marquette and Kevin Zelaya},
journal= {arXiv preprint arXiv:2101.12313},
year = {2022}
}