Deformed oscillator algebra for quantum superintegrable systems in two-dimensional Euclidean space and on the complex two-sphere

In this work, we study the superintegrable quantum systems in two-dimensional Euclidean space and on the complex two-sphere with second-order constants of the motion. We show that these constants of motion satisfy the deformed oscillator algebra. Then, we easily calculate the energy eigenvalues in an algebraic way by solving of a system of two equations satisfied by its structure function. The results are in agreement to the ones obtained from the solution of the relevant Schrödinger equation.