On the harmonic oscillator on the Lobachevsky plane

We introduce the harmonic oscillator on the Lobachevsky plane with the aid of the potential V (ϱ) = (a ² ω ²/4) sinh(ϱ/a)², where a is the curvature radius and ϱ is the geodesic distance from a chosen center. Thus, the potential is rotationally symmetric and unbounded, 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 provided that the value of the angular momentum, m, is equal to 0. Dedicated to the memory of V. A. Geyler