Two-dimensional superintegrable systems from operator algebras in one dimension

We develop new constructions of 2D classical and quantum superintegrable Hamiltonians allowing separation of variables in Cartesian coordinates. In classical mechanics we start from two functions on a one-dimensional phase space, a natural Hamiltonian H and a polynomial of order N in the momentum p...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2019-03, Vol.52 (11), p.115202
Main Authors: Marquette, Ian, Sajedi, Masoumeh, Winternitz, Pavel
Format: Article
Language:English
Subjects:
ladder operators
Painlevé transcendents
separation of variables
superintegrable systems
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Summary:We develop new constructions of 2D classical and quantum superintegrable Hamiltonians allowing separation of variables in Cartesian coordinates. In classical mechanics we start from two functions on a one-dimensional phase space, a natural Hamiltonian H and a polynomial of order N in the momentum p . We assume that their Poisson commutator vanishes, is a constant, a constant times H, or a constant times K. In the quantum case H and K are operators and their Lie commutator has one of the above properties. We use two copies of such pairs to generate two-dimensional superintegrable systems in the Euclidean space E2, allowing the separation of variables in Cartesian coordinates. Nearly all known separable superintegrable systems in E2 can be obtained in this manner and we obtain new ones for N  =  4.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8121/ab01a2