The quantum harmonic oscillator on the sphere and the hyperbolic plane

TL;DR

This paper exactly solves a nonlinear quantum harmonic oscillator model on curved two-dimensional spaces, analyzing its classical and quantum properties, symmetries, and deriving explicit wave functions and energy spectra for spherical and hyperbolic geometries.

Contribution

It introduces a parameter-dependent model of the quantum harmonic oscillator on curved spaces, providing exact solutions and exploring its superintegrability and polynomial eigenfunctions.

Findings

Exact wave functions and energy spectra for the oscillator on S^2 and H^2

Development of $ u$-dependent orthogonal polynomial families related to Hermite polynomials

Analysis of symmetries and invariant measures for the curved space quantum system

Abstract

A nonlinear model of the quantum harmonic oscillator on two-dimensional spaces of constant curvature is exactly solved. This model depends of a parameter $\la$ that is related with the curvature of the space. Firstly 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 $\la$-dependent invariant measure $d\mu_\la$ 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 Schr\"odinger equation is studied. Two $\la$-dependent Sturm-Liouville problems, related with two different $\la$-deformations of the Hermite equation, are obtained. This leads to the study of two $\la$-dependent families of orthogonal polynomials both related with the Hermite polynomials. Finally the wave functions $\Psi_{m,n}$ and the energies $E_{m,n}$ of the bound states are exactly obtained in both the sphere $S^2$ and the hyperbolic plane $H^2$.