Discrete coherent states for higher Landau levels

TL;DR

This paper studies the properties of discrete coherent states related to higher Landau levels in Euclidean and Hyperbolic geometries, extending classical results through advanced mathematical frameworks.

Contribution

It extends the understanding of completeness of discrete coherent states for higher Landau levels using methods from Gabor frames and automorphic forms.

Findings

Completeness of Euclidean Landau level coherent states linked to Gabor frame theory.

Hyperbolic Landau level states analyzed using Fuchsian groups and automorphic forms.

Partial extension of classical results by Perelomov and others.

Abstract

We consider the quantum dynamics of a charged particle evolving under the action of a constant homogeneous magnetic field, with emphasis on the discrete subgroups of the Heisenberg group (in the Euclidean case) and of the SL(2, R) group (in the Hyperbolic case). We investigate completeness properties of discrete coherent states associated with higher order Euclidean and hyperbolic Landau levels, partially extending classic results of Perelomov and of Bargmann, Butera, Girardello and Klauder. In the Euclidean case, our results follow from identifying the completeness problem with known results from the theory of Gabor frames. The results for the hyperbolic setting follow by using a combination of methods from coherent states, time-scale analysis and the theory of Fuchsian groups and their associated automorphic forms.