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

• URL: http://arxiv.org/abs/2101.12313v2
• Date: Mon, 15 Feb 2021 09:16:31 GMT
• Title: Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials
• Authors: V\'eronique Hussin, Ian Marquette, and Kevin Zelaya
• Abstract summary: We extend and generalize the construction of Sturm-Liouville problems for a family of Hamiltonians constrained to fulfill a third-order shape-in condition. We identify a link between the eigenfunctions of the Hamiltonian operator and a special family of exceptional Hermite.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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