Deformed oscillator algebras for two dimensional quantum superintegrable systems

TL;DR

This paper introduces a method to analyze two-dimensional quantum superintegrable systems using deformed oscillator algebras, simplifying the calculation of energy spectra through algebraic techniques.

Contribution

It constructs specific deformed oscillator algebras for quantum superintegrable systems, enabling efficient energy eigenvalue computation and demonstrating the utility of algebraic methods.

Findings

Deformed oscillator algebras are constructed for each system.

Energy eigenvalues can be obtained algebraically from the structure function.

Results agree with traditional Schrödinger equation solutions.

Abstract

Quantum superintegrable systems in two dimensions are obtained from their classical counterparts, the quantum integrals of motion being obtained from the corresponding classical integrals by a symmetrization procedure. For each quantum superintegrable systema deformed oscillator algebra, characterized by a structure function specific for each system, is constructed, the generators of the algebra being functions of the quantum integrals of motion. The energy eigenvalues corresponding to a state with finite dimensional degeneracy can then be obtained in an economical way from solving a system of two equations satisfied by the structure function, the results being in agreement to the ones obtained from the solution of the relevant Schrodinger equation. The method shows how quantum algebraic techniques can simplify the study of quantum superintegrable systems, especially in two dimensions.