Coherent states on the group SL(2, R). series of coherent states and automorphic forms; Mellin transforms

Abstract

Coherent states corresponding to different series of representations of the group SL(2, R) are introduced. The infinite sums of such coherent states may lead to automorphic forms on a discontinuous subgroup Γ s ⊂ SL(2, R) if certain conditions are fulfilled. The obtained forms may be considered as “fixed” states, and the group Γ s as the corresponding stationary group. Starting with this composite “fixed” state, “composite” coherent states are defined in the usual manner. Mellin transforms of the coherent states, interesting from the physical point of view are examined. As a first example, it is shown that, under certain circumstances, the Mellin transform of a first kind coherent state is a Veneziano amplitudelike function. A second example starts with a two-dimensional theta wave packet (i.e. a two-dimensional free wave packet with Gaussian structure in the momentum space). The Mellin transform is an Epstein ζ-function; if the uncertainties of the momenta are satisfying the relation Δk 2 1Δk 2 2 = (Δk 1Δk 2) 2 , the Mellin transform is an Eisenstein series. The Maass series are obtained as Mellin transforms of an adequately redefined theta wave packet.