No abstract

We give an algebraic derivation of 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 [J. Math. Phys. 42, 1100–1119 (2001)] for fixing the energy eigenvalues of two-dimensional quadratically superintegrable systems by assuming that they are determined by the existence of a finite-dimensional representation of the polynomial algebra of 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 using 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.

We give an algebraic derivation of the energy eigenvalues for the twodimensional(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 eigenvalues 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.