On the harmonic oscillator on the Lobachevsky plane

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.

arXiv commentto appear in Russian Journal of Mathematical Physics (memorial volume in honor of Vladimir Geyler)

References in corpus (4)

• The quantum harmonic oscillator on the sphere and the hyperbolic plane
• Effect of the surface curvature on the magnetic moment and persistent currents in two-dimensional quantum rings and dots
• Spectral properties of a short-range impurity in a quantum dot
• Zero modes in a system of Aharonov--Bohm solenoids on the Lobachevsky plane

Cited by in corpus (1)

• The quantum free particle on spherical and hyperbolic spaces: A curvature dependent approach