Third-order ladder operators, generalized Okamoto and exceptional orthogonal polynomials
- 6 January 2022
- journal article
- research article
- Published by IOP Publishing in Journal of Physics A: Mathematical and Theoretical
- Vol. 55 (4), 045205
- https://doi.org/10.1088/1751-8121/ac43cc
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 Painlevé 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.Funding Information
- Australian Research Council (FT180100099)
- Consejo Nacional de Ciencia y Tecnología (A1-S-24569)
- Physicists on the move II (CZ.02.2.69/0.0/0.0/18 053/0017163)
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