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arXiv · 1810.05793

Two-dimensional superintegrable systems from operator algebras in one dimension

Ian Marquette·Masoumeh Sajedi·Pavel Winternitz
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Abstract

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 $\{H,K\}$ 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 $(H,K)$ pairs to generate two-dimensional superintegrable systems in the Euclidean space $E_2$, allowing the separation of variables in Cartesian coordinates. All known separable superintegrable systems in $E_2$ can be obtained in this manner and we obtain new ones for $N=4.$

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Ian Marquette, Masoumeh Sajedi, Pavel Winternitz. 2018-10-13. Two-dimensional superintegrable systems from operator algebras in one dimension. https://doi.org/10.1088/1751-8121%2Fab01a2

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