Bargmann Space Research Articles

Generalized hypergeometric coherent states

We introduce a large class of holomorphic quantum states by choosing their normalization functions to be given by generalized hypergeometric functions. We call them generalized hypergeometric states in general, and generalized hypergeometric coherent states in particular, if they allow a resolution of unity. Depending on the domain of convergence of the generalized hypergeometric functions, we distinguish generalized hypergeometric states on the plane, the open unit disk, and the unit circle. All states are eigenstates of suitably defined lowering operators. We then study their photon number statistics and phase properties as revealed by the Husimi and Pegg-Barnett phase distributions. On the basis of the generalized hypergeometric coherent states we introduce generalized hypergeometric Husimi distributions and corresponding phase distributions as well as new analytic representations of arbitrary quantum states in Bargmann and Hardy spaces.

Bargmann representation for Landau levels in two dimensions

We present a formulation of the quantum mechanics of an electron gas confined to two dimensions in a strong magnetic field within the framework of the Hilbert space of analytic functions (Bargmann's space). Our approach extends the representation introduced by Girvin and Jach for the ground state to arbitrary Landau levels and to the regime of coupling between Landau levels. By projecting out the rapid cyclotron motion, the quantum mechanics of the slow guiding centre motion is converted into a system of coupled-channel equations describing the coupling between Landau levels due to the (disorder) potentials. In the limit of strong fields, the coupled-channel equations can be solved perturbatively. For the single-channel case we derive a WKB-like quantization condition for the one-dimensional motion along equipotential lines for arbitrary Landau levels. Two applications of this formalism are discussed: the weak-levitation problem in quantum Hall systems and a two-electron quantum dot in a strong magnetic field.

Resolution of Unity of Generalized Bargmann Spaces by means ofDisplaced Fock States

We resolve the unity of generalized Bargmann spaces in Cn [J.Math. Phys. 41 (2000) 3507] by means of displaced Fock states.

Invariant Subspaces, Quasi-invariant Subspaces, and Hankel Operators

In this paper, using the theory of Hilbert modules we study invariant subspaces of the Bergman spaces on bounded symmetric domains and quasi-invariant subspaces of the Segal–Bargmann spaces. We completely characterize small Hankel operators with finite rank on these spaces.

Unconditional Basis in Bargmann Space-New Proof of Gröcheni-Walnut Theorem

In 1992 K. Gröchenig and D. Walnut [12] constructed an explicit unconditional basis in Bargmann spaces. The author gives a short and self–contained proof of this theorem. It omits the notion of Zak transform used originally in many parts of the proof. The geometric structure of the problem is revealed.

Chaoticité de l'opérateur de Gribov dans l'espace de Bargmann

Inspiring by Gulisashvili and MacCluer [14] result we show that the limit of Gribov's operator is chaotic in the Bargmann space. Our method can be applied for obtaining the chaoticity of the annihilation operator in his entier vectors space.

Explicit formulas for reproducing kernels of generalized Bargmann spaces on Cn

We introduce a class of generalized Bargmann spaces on Cn for which we establish explicit formulas of their reproducing kernels. Some applications of the obtained formulas are given.

Monogenic functions and representations of nilpotent Lie groups in quantum mechanics

We describe several different representations of nilpotent step two Lie groups in spaces of monogenic Clifford-valued functions. We are inspired by the classic representation of the Heisenberg group in the Segal–Bargmann space of holomorphic functions. Connections with quantum mechanics are described. Copyright © 1999 John Wiley & Sons, Ltd.

On an Unconditional Basis of Generalized Eigenvectors of the Nonself-adjoint Gribov Operator in Bargmann Space

In this article we prove that the system of generalized eigenvectors of the Gribov operator is an unconditional basis in Bargmann space. We also give a generalized diagonalization (in the Abel sense) of the semigroups, respectively, associated to this operator and to its square root. Finally, some open problems are pointed out.

Analyse de Scattering d'un Opérateur Cubique de Heun dans l'Espace de Bargmann

The boundary conditions at infinity are used in a description of all maximal dissipative extensions in Bargmann space of the minimal Heun's operator \(\). The characteristic functions of the dissipative extensions are computed. Completeness theorems are obtained for the system of generalized eigenvectors.

Fock–Bargmann space and SU(3) models

The aim of this review is to present an approach to unitary models that has been developed during the last ten years. The approach uses wave functions and operators defined in Fock–Bargmann space. A solution for the SU(3) shell model is introduced, as well as some extensions that are related to a classical treatment of SU(3) theory.

Microscopic SUπ(3)⊗SUν(3) interpretation of a two triaxial rotor model

A description of the relative motion of neutrons and protons in strongly deformed even-even nuclei is reduced to that of two coupled triaxial rotors, one representing the neutrons and the other the protons, by using generating function methods. The two rotor theory is interpreted as a generalization of the Elliott SU(3) model with a neutron–proton direct product SUν(3)×SUπ(3) substructure. The second order Casimir operator for this system is constructed and some of its eigenfunctions are found in explicit form in the Fock–Bargmann space representation. Analytical results for isovector M1 transitions probabilities are obtained and the results compared with experimental data and pseudo-SU(3) calculations. Graphs picturing the relative orientation of neutron-proton rotors in Ne22 are presented.

Espaces de Bargmann généralisés et formules explicites pour leurs noyaux reproduisants

We establish explicit formulas for the reproducing kernels of a class of generalized Bargmann spaces on C n.

Vecteurs d'état cohérent et image de la transformée de Husimi

A coherent state vector for an operator is defined similarly to the way analytic vectors are defined. We study the properties of the coherent state vectors with respect to the Bargmann space and the Heisenberg 's uncertainty principle. The coherent state vectors permit a characterization of the range of the Husimi transformation.

On the complete basis of Pauli-allowed states of three-cluster systems in the Fock–Bargmann space

In the Fock–Bargmann space, the complete harmonic-oscillator basis for the three-cluster system is constructed. The indices of the reduction U(6)⊇U(2)×U(3) are chosen as the quantum numbers. The Pauli-forbidden states are eliminated by the orthogonal transformation of basis functions. The basis states are obtained in terms of hypergeometric functions and the spherical Wigner functions. Their simple form allows one to solve the problem of calculating the matrix elements of the microscopic Hamiltonian needed for the study of three-cluster systems within the algebraic version of RGM.

On resolving the multiplicity of the branching rule GL(2k+1,C) down arrow SO(2k+1,C)

We consider the multiplicity problem of the branching rule GL(2k+1,C) down arrow SO(2k+1,C). Finite-dimensional irreducible representations of GL(2k+1,C) are realized as right translations on subspaces of the holomorphic Hilbert (Bargmann) spaces of q*(2k+1) complex variables. Maps are exhibited which carry an irreducible representation of SO(2k+1,C) into these subspaces. An algebra of commuting operators is constructed. Eigenvalues and eigenvectors of certain of these operators can then be used to resolve the multiplicity in the branching rule.

A Possible Generalization of the Bargmann Space11The project supported in part by National Natural Science Foundation of China

By virtue of the technique of integration within an ordered product of operators we show that the ordinary Bargmann space basis function can be extended to two-variable Hermite polynomials, the latter can span a generalized Bargmann space. The corresponding two-mode Hermite polynomial states are thus introduced and their properties are investigated.

On resolving the multiplicity of the branching rule GL(2k,C) down arrow Sp(2k,C)

We consider the multiplicity problem of the branching rule GL(2k,C) down arrow Sp(2k,C). Finite-dimensional irreducible representations of GL(2k,C) are realized as right translations on subspaces of the holomorphic Hilbert (Bargmann) spaces of q*2k complex variables. Maps are exhibited which carry an irreducible representation of Sp(2k,C) into these subspaces. An algebra of commuting operators is constructed and it is shown how eigenvalues of certain of these operators can be used to resolve the multiplicity.

Asymptotic invertibility of Bergman and Bargmann space Toeplitz operators

Asymptotic invertibility of Bergman and Bargmann space Toeplitz operators

On resolving the multiplicity of tensor products of irreducible representations of symplectic groups

The multiplicity problem of r-fold tensor products of irreducible representations of Sp(2k, C) are considered. The arbitrary r-fold tensor product is shown to be isomorphic to a subspace of the holomorphic Hilbert (Bargmann) space of n*2k complex variables. Maps are constructed which carry an irreducible representation of Sp(2k, C) into this subspace. An algebra of commuting operators is constructed and it is shown how eigenvalues and eigenvectors of certain of these operators can be used to resolve the multiplicity.