Painlevé VI superintegrability and Heun-Racah algebras
Abstract
We study the two fourth-order quantum superintegrable systems in the Euclidean plane that separate in polar coordinates and whose angular potentials are determined by the general four-parameter sixth Painlevé transcendent. One system has an oscillator radial term and the other an extended-Coulomb radial term. For both systems we give the fourth-order quantum integral explicitly, derive the cubic symmetry algebra and its Casimir, and construct a deformed-oscillator realization. At fixed energy the two algebras are specializations of the Heun--Racah algebra. Their structure functions contain the same quartic angular factor, determined by the Painlevé~VI parameters, while different radial factors distinguish oscillator and Coulomb dynamics. The Coulomb integral is obtained by transforming the Cartesian Hermitian symmetry at the operator level. The lower-order terms generated by the polar coordinate change are essential for commutation with the Hamiltonian. The two quantum systems are also related by the Levi-Civita transformation and coupling-constant metamorphosis, which gives a direct correspondence between their Painlevé parameters, algebras, Casimirs and representations. Earlier exceptional-Jacobi models arise at special rational parameter values, whereas the results here apply to the full four-parameter Painlevé~VI family. We also compare the quantum results with the corresponding classical Poisson algebras.
Related
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BibTeX
@misc{marquette2026painlevé,
title = {{Painlevé VI superintegrability and Heun-Racah algebras}},
author = {Ian Marquette and Sarah Post},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00322},
eprint = {2611.00322},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00322},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Painlevé VI superintegrability and Heun-Racah algebras
AU - Ian Marquette
AU - Sarah Post
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00322
UR - https://arxiv.org/abs/2611.00322
AB - We study the two fourth-order quantum superintegrable systems in the Euclidean plane that separate in polar coordinates and whose angular potentials are determined by the general four-parameter sixth Painlevé transcendent. One system has an oscillator radial term and the other an extended-Coulomb radial term. For both systems we give the fourth-order quantum integral explicitly, derive the cubic symmetry algebra and its Casimir, and construct a deformed-oscillator realization. At fixed energy the two algebras are specializations of the Heun--Racah algebra. Their structure functions contain the same quartic angular factor, determined by the Painlevé~VI parameters, while different radial factors distinguish oscillator and Coulomb dynamics. The Coulomb integral is obtained by transforming the Cartesian Hermitian symmetry at the operator level. The lower-order terms generated by the polar coordinate change are essential for commutation with the Hamiltonian. The two quantum systems are also related by the Levi-Civita transformation and coupling-constant metamorphosis, which gives a direct correspondence between their Painlevé parameters, algebras, Casimirs and representations. Earlier exceptional-Jacobi models arise at special rational parameter values, whereas the results here apply to the full four-parameter Painlevé~VI family. We also compare the quantum results with the corresponding classical Poisson algebras.
ER -