Quadratic Poisson algebras for two dimensional classical superintegrable systems and quadratic associative algebras for quantum superintegrable systems

摘要

The integrals of motion of the classical two dimensional superintegrablesystems with quadratic integrals of motion close in a restrained quadraticPoisson algebra, whose the general form is investigated. Each classicalsuperintegrable problem has a quantum counterpart, a quantum superintegrablesystem. The quadratic Poisson algebra is deformed to a quantum associativealgebra, the finite dimensional representations of this algebra are calculatedby using a deformed parafermion oscillator technique. It is shown that, thefinite dimensional representations of the quadratic algebra are determined bythe energy eigenvalues of the superintegrable system. The calculation of energyeigenvalues is reduced to the solution of algebraic equations, which areuniversal for all two dimensional superintegrable systems with quadraticintegrals of motion.