Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

The growth of Bargmann functions and the completeness of sequences of coherent states

Apostol Vourdas

Published Jul 7, 1997• 21 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/30/13/034

Related papers

The completeness of sequences of generalized coherent states

Apostol Vourdas

Analytic representations based on generalized coherent states are studied. The growth of the analytic functions is intimately connected to the completeness of the sequences of these generalized coherent states. The least density that such sequences must have in order to be overcomplete is calculated. The results generalize known results on the completeness of von Neumann lattices for the standard coherent states to other sets of coherent states.

Journal of Optics B Quantum and Semiclassical Optics•Jun 1, 2003•7 citations

Generalized Bargmann functions, their growth and von Neumann lattices

Apostol Vourdas, K. A. Penson, Gérard Duchamp +1 more

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.

arXiv:1111.0889•Journal of Physics A Mathematical and Theoretical•May 30, 2012•4 citations

Completeness of coherent state subsystems

Karlheinz Gröchenig, Antti Haimi, José Luis Romero

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.

•Jul 8, 2015•1 citation

The growth and zeros of Bargmann functions

Apostol Vourdas

The Bargmann formalism is presented with emphasis on the growth and zeros of Bargmann functions. Hadamard's theorem is expressed in a physical language and it is used to construct quantum states, with given zeros in their Bargmann function. Various examples are discussed. The time evolution of Bargmann functions and the motion of their zeros, for various Hamiltonians, is studied.

Journal of Physics Conference Series•Mar 1, 2010•1 citation

Off-Center Coherent-State Representation and an Application to Semiclassics

Fernando Parisio

By using the overcompleteness of coherent states we find an alternative form of the unit operator for which the ket and the bra appearing under the integration sign do not refer to the same phase-space point. This defines a new quantum representation in terms of Bargmann functions, whose basic features are presented. A continuous family of secondary reproducing kernels for the Bargmann functions is obtained, showing that this quantity is not unique for representations based on overcomplete sets. We illustrate the applicability of the presented results by deriving a semiclassical expression for the Feynman propagator that generalizes the well-known van Vleck formula and seems to point a way to cope with long-standing problems in semiclassical propagation of localized states.

arXiv:1003.2639•Progress of Theoretical Physics•Jul 1, 2010•1 citation

Composite von Neumann lattice

Itzhack Dana

A new general subset of coherent states, which is complete and overcomplete by just one element, is introduced. It is the union of $N$ arbitrary nonintersecting von Neumann lattices whose unit-cell areas have a harmonic mean of $\mathrm{Nh}$. First applications are given in the problems of crystal electrons in rational magnetic fields, incommensurate structures in a magnetic field, and quantum dynamics.

Physical Review A•Oct 1, 1983•3 citations
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.
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.
By using the overcompleteness of coherent states we find an alternative form of the unit operator for which the ket and the bra appearing under the integration sign do not refer to the same phase-space point. This defines a new quantum representation in terms of Bargmann functions, whose basic features are presented. A continuous family of secondary reproducing kernels for the Bargmann functions is obtained, showing that this quantity is not unique for representations based on overcomplete sets. We illustrate the applicability of the presented results by deriving a semiclassical expression for the Feynman propagator that generalizes the well-known van Vleck formula and seems to point a way to cope with long-standing problems in semiclassical propagation of localized states.
A new general subset of coherent states, which is complete and overcomplete by just one element, is introduced. It is the union of NNN arbitrary nonintersecting von Neumann lattices whose unit-cell areas have a harmonic mean of Nh\mathrm{Nh}Nh. First applications are given in the problems of crystal electrons in rational magnetic fields, incommensurate structures in a magnetic field, and quantum dynamics.