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

Abstract

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 Schrödinger 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 $-{\hslash }^2$ plus quantum corrections of order ${\hslash }^4$ and ${\hslash }^6$.

LaTeX file, 25 pages

References in corpus (13)

• Completeness of superintegrability in two-dimensional constant curvature spaces
• Deformed oscillator algebras for two dimensional quantum superintegrable systems
• Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems
• Superintegrability with third order invariants in quantum and classical mechanics
• Path Integral Discussion for Smorodinsky-Winternitz Potentials: I.\ Two- and Three Dimensional Euclidean Space
• Superintegrable Systems in Darboux spaces
• Superintegrability in a two-dimensional space of non-constant curvature
• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold
• Integrable and superintegrable quantum systems in a magnetic field
• Superintegrability on the two dimensional hyperboloid II
• On the Drach superintegrable systems
• Subgroup type coordinates and the separation of variables in Hamilton-Jacobi and Schrődinger equations

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