Classification of the quantum two-dimensional superintegrable systems with quadratic integrals and the Staeckel transforms

The two-dimensional quantum 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 with quadratic integrals are classified as special cases of these six general classes. The coefficients of the quadratic associative algebra of integrals are calculated and they are compared to the coefficients of the corresponding coefficients of the Poisson quadratic algebra of the classical systems. The quantum coefficients are similar to the classical ones multiplied by a quantum coefficient -ℎ2 plus a quantum deformation of order ℎ4 and ℎ6. The systems inside the classes are transformed using Staeckel transforms in the quantum case as in the classical case. The general form of the Staeckel transform between superintegrable systems is discussed.