THE BARUT-GIRADELLO COHERENT STATES

It is shown that the completeness problem of the SL(2,R) coherent states proposed by Barut and Girardello leads to a moment problem, not a Mellin transform. This moment problem, which also appears in the theory of para-Bose oscillators, has been solved following the Sharma-Mehta-Mukunda-Sudarshan solution of the problem. The matrix element of finite transformation in the coherent state basis is shown to satisfy a "quasiorthogonality" condition analogous to the orthogonality condition of the matrix element in the canonical basis. Finally, the Barut-Girardello "Hilbert space of entire analytic functions of growth (1,1)" turns out to be only a subspace of Bargmann's well-known Hilbert spare of analytic functions. This subspace, which has been called "the reduced Bargmann space" in a previous paper, is an invariant subspace of SL(2,R). With this identification the generators of the group in this realization turn out to be the well-known boson operators of Holman and Biedenharn.