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We give an algebraic derivation of the eigenvalues of energy of a quantumharmonic oscillator on the surface of constant curvature, i.e. on the sphere oron the hyperbolic plane. We use the method proposed by Daskaloyannis for fixingthe energy eigenvalues of two-dimensional (2D) quadratically superintegrablesystems by assuming that they are determined by the existence offinite-dimensional representation of the polynomial algebra of the motionintegral operators. The tool for realizing representations is the deformedparafermionic oscillator. The eigenvalues of energy are calculated and theresult derived by us algebraically agrees with the known energy eigenvaluescalculated by classical analytical methods. This assertion which is the mainresult of this article is demonstrated by a detailed presentation. We alsodiscuss the qualitative difference of the energy spectra on the sphere and onthe hyperbolic plane.

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