2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/187338We 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.LaTeX, 15 pages, no figures, to appear in J. Phys. A. More information on http://www.technion.ac.il/~brif/science.htmlQuantum Physicstext