Quantum-algebraic description of quantum superintegrable systems in two dimensions

An alternative method for the description of quantum superintegrable systems in two dimensions through the use of quantum algebraic techniques is introduced. It is suggested that such systems can be described in terms of a generalized deformed oscillator, characterized by a structure function specific to the system. The energy eigenvalues corresponding to a state with finite-dimensional degeneracy can then be determined directly from the properties of the relevant structure function. The validity of the method is demonstrated in the case of the isotropic harmonic oscillator in a space with constant curvature. The method can be used for constructing the quantum versions of several classical superintegrable systems, the Holt potential being given as an example.