Bargmann Space Research Articles

Semiclassical Properties of Coherent States for L 2(S n ), n = 2, 3, 5

We show semiclassical concentration of a system of coherent states for L 2(S n ), n = 2, 3, 5 in both configuration and Bargmann space representation. We obtain asymptotics for both the integral kernel defining an associated Bargmann transform and the inner product of two coherent states. In particular, we obtain the main asymptotic term for the phase function appearing in that inner product.

THE μ-DEFORMED SEGAL–BARGMANN TRANSFORM IS A HALL TYPE TRANSFORM

We present an explanation of how the μ-deformed Segal–Bargmann spaces, that are studied in various articles of the author in collaboration with Angulo, Echavarría and Pita, can be viewed as deserving their name, that is, how they should be considered as a part of Segal–Bargmann analysis. This explanation relates the μ-deformed Segal–Bargmann transforms to the generalized Segal–Bargmann transforms introduced by B. Hall using heat kernel analysis. All the versions of the μ-deformed Segal–Bargmann transform can be understood as Hall type transforms. In particular, we define a μ-deformation of Hall's "Version C" generalized Segal–Bargmann transform which is then shown to be a μ-deformed convolution with a μ-deformed heat kernel followed by analytic continuation. Our results are generalizations and analogues of the results of Hall.

Fast rotating Bose–Einstein condensates in an asymmetric trap

We investigate the effect of the anisotropy of a harmonic trap on the behaviour of a fast rotating Bose–Einstein condensate. This is done in the framework of the 2D Gross–Pitaevskii equation and requires a symplectic reduction of the quadratic form defining the energy. This reduction allows us to simplify the energy on a Bargmann space and study the asymptotics of large rotational velocity. We characterize two regimes of velocity and anisotropy; in the first one where the behaviour is similar to the isotropic case, we construct an upper bound: a hexagonal Abrikosov lattice of vortices, with an inverted parabola profile. The second regime deals with very large velocities, a case in which we prove that the ground state does not display vortices in the bulk, with a 1D limiting problem. In that case, we show that the coarse grained atomic density behaves like an inverted parabola with large radius in the deconfined direction but keeps a fixed profile given by a Gaussian in the other direction. The features of this second regime appear as new phenomena.

(q, ν)-deformation of generalized basic hypergeometric states

We provide with a (q, ν)-deformation of the generalized hypergeometric coherent states defined such that the normalization function is given by generalized basic hypergeometric functions. These states are eigenstates of suitably defined deformed lowering operators. We study the domain of convergence of the corresponding normalization function. On the basis of these states, we investigate generalized basic hypergeometric Husimi distributions and corresponding phase distributions as well as new analytic basis representations of arbitrary quantum states in Bargmann and Hardy spaces. The quantum statistical properties of the states, such as photon-counting statistics and quadrature squeezing are analytically and numerically discussed in the framework of conventional quantum optics.

Two integral operators in Clifford analysis

Segal–Bargmann space F 2 ( C n ) and monogenic Fock space M 2 ( R n + 1 ) are introduced first. Then, with the help of exponential functions in Clifford analysis, two integral operators are defined to connect F 2 ( C n ) and M 2 ( R n + 1 ) together. The corresponding integral properties are studied in detail.

Berezin–Toeplitz quantization on Lie groups

Let K be a connected compact semisimple Lie group and K C its complexification. The generalized Segal–Bargmann space for K C is a space of square-integrable holomorphic functions on K C , with respect to a K-invariant heat kernel measure. This space is connected to the “Schrödinger” Hilbert space L 2 ( K ) by a unitary map, the generalized Segal–Bargmann transform. This paper considers certain natural operators on L 2 ( K ) , namely multiplication operators and differential operators, conjugated by the generalized Segal–Bargmann transform. The main results show that the resulting operators on the generalized Segal–Bargmann space can be represented as Toeplitz operators. The symbols of these Toeplitz operators are expressed in terms of a certain subelliptic heat kernel on K C . I also examine some of the results from an infinite-dimensional point of view based on the work of L. Gross and P. Malliavin.

The Mellin Transform and Spectral Properties of Toric Varieties

In this paper we apply the results of [W] on the twisted Mellin transform to problems in toric geometry. In particular, we use these results to describe the asymptotics of probability densities associated with the monomial eigenstates, zk, \( k \in \mathbb{Z}^{d} \), in Bargmann space and prove an “upstairs” version of the spectral density theorem of [BGU]. We also obtain for the zk’s, “upstairs” versions of the results of [STZ] on distribution laws for eigenstates on toric varieties.

Chiral bosons on Bargmann space associated with Ar statistics

We consider a large collection of particles obeying Ar statistics. The system behaves like a quantum droplet characterized by a constant Husimi distribution. We show that the excitations of this system live on the boundary of the droplet and they are described by an effective chiral boson action generalizing the Wess–Zumino–Witten theory in two dimension. Our analysis is based on the Fock–Bargmann analytical representations associated with Ar statistics. The quantization of the theory describing the dynamics on the edge is achieved. As a by product, we prove that the edge excitations are given by a tensorial product of r abelian bosonic fields.

Real Bargmann spaces, Fischer decompositions, and sets of uniqueness for polyharmonic functions

In this paper a positive answer is given to the following question of W.K. Hayman: if a polyharmonic entire function of order k vanishes on k distinct ellipsoids in the euclidean space R n then it vanishes everywhere. Moreover a characterization of ellipsoids is given in terms of an extension property of solutions of entire data functions for the Dirichlet problem answering a question of D. Khavinson and H.S. Shapiro. These results are consequences from a more general result in the context of direct sum decom- positions (Fischer decompositions) of polynomials or functions in the algebra A(BR) of all real-analytic functions defined on the ball BR of radius R and center 0 whose Taylor series of homogeneous polynomials converges compactly in BR. The main result states that for a given elliptic polynomial P of degree 2k and suciently

Eigenfunctions of the complex fractional Fourier transform obtained in the context of quantum optics

We employ the recently established basis (the two-variable Hermite-Gaussian function) of the generalized Bargmann space (BGBS) [Phys. Lett. A303, 311 (2002)] to study the generalized form of the fractional Fourier transform (FRFT). By using the technique of integration within an ordered product of operators and the bipartite entangled-state representations, we derive the generalized generating function of the BGBS with which the undecomposable kernel of the two-dimensional FRFT [also named complex fractional Fourier transform (CFRFT)] is obtained. This approach naturally shows that the BGBS is just the eigenfunction of the CFRFT.

DIRECT AND REVERSE LOG-SOBOLEV INEQUALITIES IN μ-DEFORMED SEGAL–BARGMANN ANALYSIS

Both direct and reverse log-Sobolev inequalities, relating the Shannon entropy with a μ-deformed energy, are shown to hold in a family of μ-deformed Segal–Bargmann spaces. This shows that the μ-deformed energy of a state is finite if and only if its Shannon entropy is finite. The direct inequality is a new result, while the reverse inequality has already been shown by the authors but using different methods. Next the μ-deformed energy of a state is shown to be finite if and only if its Dirichlet form energy is finite. This leads to both direct and reverse log-Sobolev inequalities that relate the Shannon entropy with the Dirichlet energy. We obtain that the Dirichlet energy of a state is finite if and only if its Shannon entropy is finite. The main method used here is based on a study of the reproducing kernel function of these spaces and the associated integral kernel transform.

Sectorial forms and unbounded subnormals

AbstractWe use the theory of sectorial sesquilinear forms to characterize the closure of the Creation Operator of Quantum Mechanics in the classical set-up. Further, we bring that theory to bear upon the class of unbounded cyclic subnormal operators that admit analytic models; in particular, we provide a sufficient condition for the existence of complete sets of eigenvectors for certain sectorial operators related to unbounded subnormals. The relevant theory is illustrated in the context of a class of analytic models of which the classical Segal–Bargmann space is a prototype. The framework of sectorial sesquilinear forms is also shown to be specially useful for treating questions related to the existence, uniqueness and stability of certain parabolic evolution equations naturally associated with such analytic models.

Lowest Landau level functional and Bargmann spaces for Bose–Einstein condensates

A fast rotating Bose–Einstein condensate can be described by a complex valued wave function minimizing an energy restricted to the lowest Landau level or Fock–Bargmann space. Using some structures associated with this space, we study the distribution of zeroes of the minimizer and prove in particular that the number of zeroes is infinite. We relate their location to the combination of two problems: a confining problem producing an inverted parabola profile and the Abrikosov problem of minimizing an energy on a lattice, using Theta functions.

Spectral properties of the Cauchy transform on [formula omitted

Let h m , p ( z ) , ( m , p ) ∈ Z + × Z + , be the Landau orthogonal basis of the Hilbert space on L 2 ( C , e − | z | 2 d λ ( z ) ) where λ ( z ) is the usual Lebesgue measure on the complex plane. In this paper we give some spectral properties of the Cauchy transform on the orthogonal complement of Bargmann space Λ 0 ( C ) in L 2 ( C , e − | z | 2 d λ ( z ) ) . In particular for m fixed, we consider the orthogonal projection operator on the Hilbert subspace spanned by h m , p ( z ) , p = 0 , 1 , 2 , … , and we give explicitly the sequence of singular values of its composition with the Cauchy transform in L 2 ( C , e − | z | 2 d λ ( z ) ) . As application of these of the Cauchy transform we get some identities for special functions which could be of independent interest.

The Spectral Theory of Toeplitz Operators Applied to Approximation Problems in Hilbert Spaces

The solution to a particular constrained approximation problem, in an abstract Hilbert space setting, may be interpreted in terms of a generalised Toeplitz operator. We consider concrete versions of this problem, in settings which involve generalised Hardy spaces, Paley–Wiener spaces and the Segal–Bargmann space, and derive spectral representations of the associated Toeplitz operators.

Chaotic weighted shifts in Bargmann space

This paper deals with the unilateral backward shift operator T on a Bargmann space F ( C ) . This space can be identified with the sequence space ℓ 2 ( N ) . We use the hypercyclicity criterion of Bès, Chan, and Seubert and the program of K.-G. Grosse-Erdmann to give a necessary and sufficient condition in order that T be a chaotic operator. The chaoticity of differentiation which correspond to the annihilation operator in quantum radiation field theory is in view, since the Bargmann space is an infinite-dimensional separable complex Hilbert space.

Approximation of the semigroup generated by the Hamiltonian of Reggeon field theory in Bargmann space

The Reggeon field theory is governed by a non-self adjoint operator constructed as a polynomial in A, A * , the standard Bose annihilation and creation operators. In zero transverse dimension, this Hamiltonian acting in Bargmann space is defined by H λ ′ , μ = λ ′ A * 2 A 2 + μ A * A + i λ A * ( A * + A ) A , where i 2 = − 1 , λ ′ , μ and λ are real numbers and the operators A , A * satisfy the commutation relation [ A , A * ] = I . As the quantum mechanical system described by H λ ′ , μ has a velocity-dependent potential containing powers of momentum up to the fourth, the problem of existence of Hamiltonian path integral for the evolution operator e − t H λ ′ , μ of this theory is of interest on its own. In particular, can we express e − t H λ ′ , μ as a limit of “integral” operators? In this article one considerably reduces the difficulty by studying the Trotter product formula of H λ ′ , μ to reach two objectives: • The first objective is to prove a very specific error estimate for the error in a Trotter product formula in trace-norm for H viewed as the sum of the operators λ ′ A * 2 A 2 and μ A * A + i λ A * ( A * + A ) A . • The second objective of this work is to give a approximation of the semigroup generated by H λ ′ , μ when H λ ′ , μ is split in the sum of λ ′ A * 2 A 2 + μ A * A and i λ A * ( A * + A ) A . We note that this case is entirely different. In fact, the usual Trotter product formula is not defined, because the interaction operator A * ( A * + A ) A is not the infinitesimal generator of a semigroup on Bargmann space. For λ ′ > 0 and ɛ > 0 , we choose an approximation operator θ ɛ = [ I − ɛ i λ A * ( A * + A ) A ] e − ɛ ( λ ′ A * 2 A 2 + μ A * A ) and we give a connection between θ ɛ and e − ɛ H λ ′ , μ . This choice allows us to give in [A. Intissar, Note on the path integral formulation of Reggeon field theory, preprint] a “generalized Trotter product formula” for T μ = μ A * A + i λ A * ( A + A * ) A , i.e., for limit case as λ ′ = 0 and answers to the above question.

Q-fermionic numbers and their roles in some physical problems

The q -fermion numbers emerging from the q -fermion oscillator algebra are used to reproduce the q -fermionic Stirling and Bell numbers. New recurrence relations for the expansion coefficients in the ‘anti-normal ordering’ of the q -fermion operators are derived. The roles of the q -fermion numbers in q -stochastic point processes and the Bargmann space representation for q -fermion operators are explored.

6He + 6He Clustering of 12Be in a Microscopic Algebraic Approach

The norm kernel of the A=12 system composed of two 6He clusters, and the L=0 basis functions (in the SU(3) and angular momentum-coupled schemes) are analytically obtained in the Fock--Bargmann space. The norm kernel has a diagonal form in the former basis, but the asymptotic conditions are naturally defined in the latter one. The system is a good illustration for the method of projection of the norm kernel to the basis functions in the presence of SU(3) degeneracy that was proposed by the authors. The coupled-channel problem is considered in the Algebraic Version of the resonating-group method, with the multiple decay thresholds being properly accounted for. The structure of the ground state of 12Be obtained in the approximation of zero-range nuclear force is compared with the shell-model predictions. In the continuum part of the spectrum, the S-matrix is constructed, the asymptotic normalization coefficients are deduced and their energy dependence is analyzed.

Q-fermionic numbers and their roles in some physical problems

The q-fermion numbers emerging from the q-fermion oscillator algebra are used to reproduce the q-fermionic Stirling and Bell numbers. New recurrence relations for the expansion coefficients in the ‘anti-normal ordering’ of the q-fermion operators are derived. The roles of the q-fermion numbers in q-stochastic point processes and the Bargmann space representation for q-fermion operators are explored.