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arXiv · 2101.12313

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

Véronique Hussin·Ian Marquette·Kevin Zelaya
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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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Véronique Hussin, Ian Marquette, Kevin Zelaya. 2021-01-28. Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials. https://doi.org/10.1088/1751-8121%2Fac43cc

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