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arXiv:1111.0889 · Journal of Physics A Mathematical and Theoretical

Generalized Bargmann functions, their growth and von Neumann lattices

Apostol Vourdas, K. A. Penson, Gérard Duchamp, A. I. Solomon

Published May 30, 2012• 4 citations
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Abstract

Generalized Bargmann representations which are based on generalized coherent states are considered. The growth of the corresponding analytic functions in the complex plane is studied. Results about the overcompleteness or undercompleteness of discrete sets of these generalized coherent states are given. Several examples are discussed in detail.

DOI
10.1088/1751-8113/45/24/244031

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Generalized Bargmann representations which are based on generalized coherent states are considered. The growth of the corresponding analytic functions in the complex plane is studied. Results about the overcompleteness or undercompleteness of discrete sets of these generalized coherent states are given. Several examples are discussed in detail.
The growth of Bargmann functions is intimately connected to the density of the zeros of these functions and to the completeness of sequences of coherent states. Using these ideas we find the least density that a sequence of coherent states must have in order to be overcomplete within the space of Bargmann functions of an order not exceeding (and of a type not exceeding if of order ). These results generalize known results on the completeness of von Neumann lattices. The practical significance of this formalism in the context of quantum optics is also discussed.
Various Euclidean, hyperbolic and elliptic analytic representations are introduced and relations among them are discussed. The Bargmann analytic representation in the complex plane is considered and its relation to other phase-space methods for the harmonic oscillator is reviewed. The general theory that relates the growth of analytic functions with the density of their zeros is applied to Bargmann functions and it leads to theorems on the completeness of sequences of Glauber coherent states. Two hyperbolic analytic representations in the unit disc, based on SU (1, 1) coherent states and also on phase states are introduced. A third analytic representation in the complex plane based on Barut–Girardello states is also considered and transformations which relate it to the other ones are studied. In the case of systems with finite-dimensional Hilbert space, an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced. The Berezin formalism in the Euclidean, hyperbolic and elliptic cases is discussed. Contour analytic representations in these three cases are also presented.
We investigate the completeness of coherent states subsystems with respect to several classes of states on rational lattices. Our main results show that the phase-space shifts of every finite linear combination of Hermite functions with respect to a rational lattice are complete in L^2(R), thus generalizing a remark of von Neumann (and proved by Bargmann, Perelomov et al.). An analogous result is proven for windows that factor into certain rational functions and the Gaussian. The results are also interesting from a conceptual point of view since they show a vast difference between the completeness and the frame property of a discrete set of coherent states.
Generalized coherent states which are associated with a generalization of the harmonic oscillator commutation relation are investigated. It is shown that these states form an overcomplete basis in a Hilbert space of analytic functions. The generalized creation and annihilation operators are bounded except in a limit in which they reduce to the usual boson creation and annihilation operators. In this limit the Hilbert space of analytic functions reduces to the Bargmann–Segal Hilbert space of entire functions and in another limit it reduces to the Hardy–Lebesgue space.