On the harmonic oscillator on the Lobachevsky plane

TL;DR

This paper studies the quantum harmonic oscillator on the Lobachevsky plane, deriving the eigenvalue problem involving spheroidal functions and analyzing eigenvalues and eigenfunctions numerically for zero angular momentum.

Contribution

It introduces a new formulation of the harmonic oscillator on hyperbolic geometry and analyzes its spectral properties using spheroidal functions.

Findings

Eigenvalue equation reduces to spheroidal functions.

Numerical analysis performed for zero angular momentum case.

Eigenvalues and eigenfunctions characterized for specific parameters.

Abstract

We introduce the harmonic oscillator on the Lobachevsky plane with the aid of the potential $V(r)=(a^2\omega^2/4)sinh(r/a)^2$ where $a$ is the curvature radius and $r$ is the geodesic distance from a fixed center. Thus the potential is rotationally symmetric and unbounded likewise as in the Euclidean case. The eigenvalue equation leads to the differential equation of spheroidal functions. We provide a basic numerical analysis of eigenvalues and eigenfunctions in the case when the value of the angular momentum, $m$, equals 0.