Hilbert spaces of analytic functions for the SL(2,R) coherent states and a class of associated integral transforms

Abstract

It is shown that the Bargmann–Segal Hilbert spaces of entire analytic functions and Bargmann’s canonical Hilbert spaces of functions analytic within the open unit disc are connected by simple integral transform pairs. A further consequence of this analysis is the analog of the Bargmann transform for the SL(2,R) [or SU(1,1)] group.