DEFORMED OSCILLATOR ALGEBRAS FOR 2-DIMENSIONAL QUANTUM SUPERINTEGRABLE SYSTEMS

Quantum superintegrable systems in two dimensions are obtained from th eir classical counterparts, the quantum integrals of motion being obta ined from the corresponding classical integrals by a symmetrization pr ocedure. For each quantum superintegrable system a deformed oscillator algebra, characterized by a structure function specific for each syst em, is constructed, the generators of the algebra being functions of t he quantum integrals of motion. The energy eigenvalues corresponding t o a state with finite-dimensional degeneracy can then be obtained in a n economical way from solving a system of two equations satisfied by t he structure function, the results being in agreement to the ones obta ined from the solution of the relevant Schrodinger equation. Applicati ons to the harmonic oscillator in a hat space and in a curved space wi th constant curvature, the Kepler problem in a hat or curved space, th e Fokas-Lagerstrom potential, the Smorodinsky-Winternitz potential, an d the Holt potential are given. The method shows how quantum-algebraic techniques can simplify the study of quantum superintegrable systems, especially in higher dimensions.