Abstract
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 space 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.
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
