The quantum harmonic oscillator on the sphere and the hyperbolic plane

A nonlinear model of the quantum harmonic oscillator on two-dimensional space of constant curvature is exactly solved. This model depends on a parameter λ that is related with the curvature of the space. First, the relation with other approaches is discussed and then the classical system is quantized by analyzing the symmetries of the metric (Killing vectors), obtaining a λ-dependent invariant measure dμ_λ and expressing the Hamiltonian as a function of the Noether momenta. In the second part, the quantum superintegrability of the Hamiltonian and the multiple separability of the Schrödinger equation is studied. Two λ-dependent Sturm-Liouville problems, related with two different λ-deformations of the Hermite equation, are obtained. This leads to the study of two λ-dependent families of orthogonal polynomials both related with the Hermite polynomials. Finally the wave functions Ψ_m,n and the energies E_m,n of the bound states are exactly obtained in both the sphere S² and the hyperbolic plane H².