Classification of quantum superintegrable systems with quadratic integrals on two dimensional manifolds

There are two classes of quantum integrable systems on a manifold with quadratic integrals, the Liouville and the Lie integrable systems as it happens in the classical case. The quantum Liouville quadratic integrable systems are defined on a Liouville manifold and the Schr\"odinger equation can be solved by separation of variables in one coordinate system. The Lie integrable systems are defined on a Lie manifold and are not generally separable ones but the can be solved. Therefore there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems. The quantum analogues of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion. There are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones. Analytic formulas for the involved integrals are calculated in all the cases. All the known quantum superintegrable systems are classified as special cases of these six general classes. The coefficients of the associative algebra of the general cases are calculated. These coefficients are the same as the coefficients of the classical case multiplied by $-\hbar^2$ plus quantum corrections of order $\hbar^4$ and $\hbar^6$.Comment: LaTeX file, 25 page

Generalized five-dimensional Kepler system, Yang-Coulomb monopole, and Hurwitz transformation

The 5D Kepler system possesses many interesting properties. This system is superintegrable and also with a su(2) non-Abelian monopole interaction (Yang-Coulomb monopole). This system is also related to an 8D isotropic harmonic oscillator by a Hurwitz transformation. We introduce a new superintegrable Hamiltonian that consists in a 5D Kepler system with new terms of Smorodinsky-Winternitz type. We obtain the integrals of motion of this system. They generate a quadratic algebra with structure constants involving the Casimir operator of a so(4) Lie algebra. We also show that this system remains superintegrable with a su(2) non-Abelian monopole (generalized Yang-Coulomb monopole). We study this system using parabolic coordinates and obtain from Hurwitz transformation its dual that is an 8D singular oscillator. This 8D singular oscillator is also a new superintegrable system and multiseparable. We obtained its quadratic algebra that involves two Casimir operators of so(4) Lie algebras. This correspondence is used to obtain algebraically the energy spectrum of the generalized Yang-Coulomb monopole

Structure results for higher order symmetry algebras of 2D classical superintegrable systems

Recently the authors and J.M. Kress presented a special function recurrence relation method to prove quantum superintegrability of an integrable 2D system that included explicit constructions of higher order symmetries and the structure relations for the closed algebra generated by these symmetries. We applied the method to 5 families of systems, each depending on a rational parameter k, including most notably the caged anisotropic oscillator, the Tremblay, Turbiner and Winternitz system and a deformed Kepler-Coulomb system. Here we work out the analogs of these constructions for all of the associated classical Hamiltonian systems, as well as for a family including the generic potential on the 2-sphere. We do not have a proof in every case that the generating symmetries are of lowest possible order, but we believe this to be so via an extension of our method.Comment: 23 page