The quantum harmonic oscillator on the sphere and the hyperbolic plane (Q2382653)

The results of the preceding paper [\textit{J. F. Cariñena, M. F. Rañada} and \textit{M. Santander}, Ann. Phys. 322, No. 2, 434--459 (2007; Zbl 1119.81098)] on exactly solvable one-dimensional nonlinear oscillator are generalized onto the case of the quantum version of a two-dimensional nonlinear system. A non-linear quasi-harmonic oscillator is considered on the two-dimensional space of constant curvature (sphere or pseudo-sphere). The approach developed in the above-cited paper is applied after comparison it with other approaches. The classical system is quantized by analyzing symmetries of the metric (Killing vectors). Invariant measure is derived and the Hamiltonian is expressed as a function of the Noether momenta. Dependence of the theory of the parameter \(\lambda\) characterizing curvature of the background space. Two \(\lambda\)-dependent Sturm-Liouville problems, related with two different \(\lambda\)-deformations of the Hermite equation, are obtained.