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

journal of physics. a, mathematical and theoretical

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