Curvature-dependent formalism, Schr\"odinger equation and energy levels for the harmonic oscillator on three-dimensional spherical and hyperbolic spaces

A nonlinear model representing the quantum harmonic oscillator on the three-dimensional spherical and hyperbolic spaces, $S_{\text{k}}^3$ ($\kappa >0$) and $H_k^3$ ($\kappa <0$), is studied. The curvature $\text{k}$ is considered as a parameter and then the radial Schr"odinger equation becomes a $\text{k}$-dependent Gauss hypergeometric equation that can be considered as a $\text{k}$-deformation of the confluent hypergeometric equation that appears in the Euclidean case. The energy spectrum and the wavefunctions are exactly obtained in both the three-dimensional sphere $S_{\text{k}}^3$ ($\kappa >0$) and the hyperbolic space $H_k^3$ ($\kappa <0$). A comparative study between the spherical and the hyperbolic quantum results is presented.