Algebraic derivation of the Energy Eigenvalues for the quantum oscillator defined on the Sphere and the Hyperbolic plane

We give an algebraic derivation of the eigenvalues of energy of a quantum harmonic oscillator on the surface of constant curvature, i.e. on the sphere or on the hyperbolic plane. We use the method proposed by Daskaloyannis for fixing the energy eigenvalues of two-dimensional (2D) quadratically superintegrable systems by assuming that they are determined by the existence of finite-dimensional representation of the polynomial algebra of the motion integral operators. The tool for realizing representations is the deformed parafermionic oscillator. The eigenvalues of energy are calculated and the result derived by us algebraically agrees with the known energy eigenvalues calculated by classical analytical methods. This assertion which is the main result of this article is demonstrated by a detailed presentation. We also discuss the qualitative difference of the energy spectra on the sphere and on the hyperbolic plane.