Analytic representations based on SU(1,1) coherent states and their applications

We consider two analytic representations of the SU(1,1) Lie group: the representation in the unit disk based on the SU(1,1) Perelomov coherent states and the Barut-Girardello representation based on the eigenstates of the SU(1,1) lowering generator. We show that these representations are related through a Laplace transform. A ``weak'' resolution of the identity in terms of the Perelomov SU(1,1) coherent states is presented which is valid even when the Bargmann index $k$ is smaller than one half. Various applications of these results in the context of the two-photon realization of SU(1,1) in quantum optics are also discussed.