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 energy eigenvalues for the two-dimensional(2D) quantum harmonic oscillator on the sphere and the hyperbolic plane in the context of the method proposed by Daskaloyannis for the 2D quadratically superintegrable systems. We provide the special form of the quadratic Poisson algebra for the classical harmonic oscillator system and deformed it into the quantum associative algebra of the operators. We express the corresponding Casimir operator for this algebra in terms of the Hamiltonian and provide the finite-dimensional representations for this quantum associative algebra by using the deformed parafermionic oscillator technique. The calculation of the energy eigen-values is then reduced to finding the solution of the two algebraic equations whose form is universal for all the 2D quadratically superintegrable systems. The result derived algebraically agrees with the energy eigenvalues obtained by solving the Schrodinger equation.