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.Comment: Revised for Annals of Physic