The coherent states of the SU(1,1) group and a class of associated integral transforms

The coordinate representation of the SU(1, 1) boson operators for the discrete series of representations used extensively by Moshinsky, Wolf and co-workers, is shown to be essentially the space of SU(1, 1) wave functions in the E(1) representation that span an L-k(2)(R+) space with a suitnble scalar product. On the other hand, the SU(1, 1) wave functions in the coherent state representation, used in the study of the pair-coherent states, span a subspace B-k(C) of the Bargmann-Segal space invariant under SU(1, 1). It is shown that there exists a simple mapping of the Hilbert spaces L-k(2)(R+) and B-k(C) Onto Bargmann's canonical carrier space, H-k, consisting of functions analytic within the open unit disc. The three Hilbert spaces are shown to be connected by simple integral transform pairs.