Two-dimensional superintegrable systems from operator algebras in one dimension

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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Cited by in corpus (9)

• Classical Superintegrable Systems in a Magnetic Field that Separate in Cartesian Coordinates
• Cylindrical type integrable classical systems in a magnetic field
• Higher Order Quantum Superintegrability: a new "Painlevé conjecture"
• New infinite families of $N$th-order superintegrable systems separating in Cartesian coordinates
• Polynomial algebras of superintegrable systems separating in Cartesian coordinates from higher order ladder operators
• Higher order superintegrability, Painlevé transcendents and representations of polynomial algebras
• Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras
• Recurrence relations and general solution of the exceptional Hermite equation
• Representations of quadratic Heisenberg-Weyl algebras and polynomials in the fourth Painlevé transcendent