Bargmann Fock Space Research Articles

Sampling and interpolation in Bargmann–Fock spaces of polyanalytic functions

Using Gabor analysis, we give a complete characterization of all lattice sampling and interpolating sequences in the Fock space of polyanalytic functions, displaying a “Nyquist rate” which increases with n, the degree of polyanaliticity of the space. Such conditions are equivalent to sharp lattice density conditions for certain vector-valued Gabor systems, namely superframes and Gabor super-Riesz sequences with Hermite windows, and in the case of superframes they were studied recently by Gröchenig and Lyubarskii. The proofs of our main results use variations of the Janssen–Ron–Shen duality principle and reveal a duality between sampling and interpolation in polyanalytic spaces, and multiple interpolation and sampling in analytic spaces. To connect these topics we introduce the polyanalytic Bargmann transform, a unitary mapping between vector-valued Hilbert spaces and polyanalytic Fock spaces, which extends the Bargmann transform to polyanalytic spaces. Motivated by this connection, we discuss a vector-valued version of the Gabor transform. These ideas have natural applications in the context of multiplexing of signals. We also point out that a recent result of Balan, Casazza and Landau, concerning density of Gabor frames, has important consequences for the Gröchenig–Lyubarskii conjecture on the density of Gabor frames with Hermite windows.

Minimal length uncertainty and generalized non-commutative geometry

A generalized formulation of non-commutative geometry for the Bargmann–Fock space of quantum field theory is presented. The analysis is related to the symmetry of the simplistic space and a minimal length uncertainty.

Gabor (super)frames with Hermite functions

We investigate vector-valued Gabor frames (sometimes called Gabor superframes) based on Hermite functions H n . Let h = (H 0, H 1, . . . , H n ) be the vector of the first n + 1 Hermite functions. We give a complete characterization of all lattices $${\Lambda \subseteq \mathbb{R} ^2}$$ such that the Gabor system $${\{ {\rm e}^{2\pi i \lambda _{2} t}{\bf h} (t-\lambda _1): \lambda = (\lambda _1, \lambda _2) \in \Lambda \}}$$ is a frame for $${L^2 (\mathbb{R} , \mathbb{C} ^{n+1})}$$ . As a corollary we obtain sufficient conditions for a single Hermite function to generate a Gabor frame and a new estimate for the lower frame bound. The main tools are growth estimates for the Weierstrass σ-function, a new type of interpolation problem for entire functions on the Bargmann–Fock space, and structural results about vector-valued Gabor frames.

From random matrices to random analytic functions

We consider two families of random matrix-valued analytic functions: (1) G1−zG2 and (2) G0+zG1+z2G2+⋯, where Gi are n×n random matrices with independent standard complex Gaussian entries. The random set of z where these matrix-analytic functions become singular is shown to be determinantal point processes in the sphere and the hyperbolic plane, respectively. The kernels of these determinantal processes are reproducing kernels of certain Hilbert spaces (“Bargmann–Fock spaces”) of holomorphic functions on the corresponding surfaces. Along with the new results, this also gives a unified framework in which to view a theorem of Peres and Virág (n=1 in the second setting above) and a well-known result of Ginibre on Gaussian random matrices (which may be viewed as an analogue of our results in the whole plane).

Noncommutative Grassmannian U(1) sigma model and a Bargmann-Fock space

We consider a Grassmannian version of the noncommutative U(1) sigma model specified by the energy functional E(P) = ‖[a, P]‖ 2HS , where P is an orthogonal projection operator in a Hilbert space H and a: H → H is the standard annihilation operator. With H realized as a Bargmann-Fock space, we describe all solutions with a one-dimensional range and prove that the operator [a, P] is densely defined in H for a certain class of projection operators P with infinite-dimensional ranges and kernels.

Norm expansion along a zero variety

The reproducing kernel function of a weighted Bergman space over domains in C d is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of (complex) codimension 1, which also leads to a series expansion of the reproducing kernel in terms of reproducing kernels defined on the subvariety. The problem of finding the reproducing kernel is thus reduced to the same kind of problem when one of the two entries is on the subvariety. A complete expansion of the reproducing kernel may be achieved in this manner. We carry this out in dimension d = 2 for certain classes of weighted Bergman spaces over the bidisk (with the diagonal z 1 = z 2 as subvariety) and the ball (with z 2 = 0 as subvariety), as well as for a weighted Bargmann–Fock space over C 2 (with the diagonal z 1 = z 2 as subvariety).

SUFFICIENT CONDITIONS FOR INTERPOLATION AND SAMPLING HYPERSURFACES IN THE BERGMAN BALL

We give sufficient conditions for a closed smooth hypersurface W in the n-dimensional Bergman ball to be interpolating or sampling. As in the recent work [5] of Ortega-Cerdà, Schuster and the second author on the Bargmann–Fock space, our sufficient conditions are expressed in terms of a geometric density of the hypersurface that, though less natural, is shown to be equivalent to Bergman ball analogs of the Beurling-type densities used in [5]. In the interpolation theorem we interpolate L2 data from W to the ball using the method of Ohsawa–Takegoshi, extended to the present setting, rather than the Cousin I approach used in [5]. In the sampling theorem, our proof is completely different from [5]. We adapt the more natural method of Berndtsson and Ortega-Cerdà [1] to higher dimensions. This adaptation motivated the notion of density that we introduced. The approaches of [5] and the present paper both work in either the case of the Bergman ball or of the Bargmann–Fock space.

Convergence of the Wick Star Product

We construct a Frechet space as a subspace of $${C^\omega(\mathbb{C}^n)}$$ where the Wick star product converges and is continuous. The resulting Frechet algebra $${\mathcal{A}}$$ ħ is studied in detail including a *-representation of $${\mathcal{A}}$$ ħ in the Bargmann-Fock space and a discussion of star exponentials and coherent states.

Gabor frames with Hermite functions

We investigate Gabor frames based on a linear combination of Hermite functions H n . We derive sufficient conditions on the lattice Λ ⊆ R 2 such that the Gabor system { e 2 π i λ 2 t H n ( t − λ 1 ) : λ = ( λ 1 , λ 2 ) ∈ Λ } is a frame. An example supports our conjecture that our conditions are sharp. The main tools are growth estimates for the Weierstrass σ-function and a new type of interpolation problem for entire functions on the Bargmann–Fock space. To cite this article: K. Gröchenig, Y. Lyubarskii, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Interpolation and sampling hypersurfaces for the Bargmann-Fock space in higher dimensions

We study those smooth complex hypersurfaces W in Open image in new window having the property that all holomorphic functions of finite weighted Lp norm on W extend to entire functions with finite weighted Lp norm. Such hypersurfaces are called interpolation hypersurfaces. We also examine the dual problem of finding all sampling hypersurfaces, i.e., smooth hypersurfaces W in Open image in new window such that any entire function with finite weighted Lp norm is stably determined by its restriction to W.

Generalized Calogero model in arbitrary dimensions

We define a new multispecies model of Calogero type in D dimensions with harmonic, two-body and three-body interactions. Using the underlying conformal SU(1,1) algebra, we indicate how to find the complete set of the states in Bargmann–Fock space. There are towers of states, with equidistant energy spectra in each tower. We explicitely construct all polynomial eigenstates, namely the center-of-mass states and global dilatation modes, and find their corresponding eigenenergies. We also construct ladder operators for these global collective states. Analysing corresponding Fock space, we detect the universal critical point at which the model exhibits singular behavior. The above results are universal for all systems with underlying conformal SU(1,1) symmetry.

Compactness criteria in function spaces

The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency do- main. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for L 2 (R d ), and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin spaces, modulation spaces and Bargmann-Fock spaces.

On completeness of random exponentials in the Bargmann–Fock space

We study the completeness/incompleteness properties of a system of exponentials EΛ={eπλz; λ∈Λ}, viewed as elements of the Bargmann–Fock space of entire functions. We assume that the index set Λ is a realization of a random point field in ℂ (the support of a random measure). We prove that the properties are determined by the density of the field, i.e., by the mean number of the field points per unit area. We also discuss certain implications and motivations of our results, in particular, the jumps of the integrated density of states of the Landau Hamiltonian with the random potential, equal to the sum of point scatters.

On the Fourier Series of Unbounded Harmonic Functions

The Fourier series of the elements in the generalized Bergman spaces bp, q of harmonic functions over D and over C (as well as those of holomorphic functions) is analysed. It is shown that the trigonometric system Ω = {r∣k∣eikφ}k∈Z is never a basis of b1, 1 and b∞, 0 for any weighted L1-norm and L∞-norm over D. The same result holds in the special case of Bargmann–Fock space over C (with respect to the weighted L1-norms and L∞-norms) which answers a question of Garling and Wojtaszczyk. On the other hand examples are given of weighted L1-norms and L∞-norms over C where Ω is indeed a basis of b1, 1 and b∞, 0. Moreover, using similar methods, a weight is constructed on D where b∞, ∞ is not isomorphic to l∞ which shows that there are weighted spaces whose Banach space classifications differ completely from those which have been characterized so far.

Interpolating sequences for Bargmann-Fock spaces in [formula omitted

We study interpolating sequences for the Bargmann-Fock space consisting of entire functions f in C n , n > 1, such that fe − builtα 2 ¦z¦ 2 ϵ L p ( C n), α > 0 and 0 < p ≤ ∞. We first study the relationship between interpolating sequences for various spaces and then give a sufficient condition for interpolation. As a consequence, we see that the qualitative behaviour of these sequences is as in one variable, according to the characterizing density condition given by Seip [8] and Seip and Wallstén [9]. We complement these results by pointing out that the same proofs as in one variable give also, when n > 1, a necessary and a sufficient density condition which, although quite different, are both formally analogous to the 1-dimensional one.

Two-dimensional boson and W-symmetry in the quantum Hall effect

We perform consistently the Gupta–Bleuler–Dirac quantization for a two-dimensional boson with parameter (α) on the circle, the boundary of the circular droplet. For α=1, we obtain the chiral (holomorphic) constraints. Using the representation of Bargmann–Fock space and the Schrodinger picture, we construct the holomorphic wave function. In order to interpret this function, we construct the coherent state representation by using the infinite-dimensional translation (W∞) symmetry for each Fourier (edge) mode. The α=1 chiral wave function explains the neutral edge states for integer quantum Hall effect very well. In the case of α=−1, we obtain a new wave function which may describe the higher modes (radial excitations) of edge states. The charged edge states are described by the |α|≠1 wave function. Finally, the application of our model to the fractional quantum Hall effect is discussed.

Linear Quantum Transform Theory in Bargmann–Fock Space and Its Preliminary Application11The project supported by National Natural Science Foundation of China

A general theory for the linear transform in Bargmann–Fock space of n-mode boson system is presented. The group structure of transform is analyzed and the explicit expressions of transform operations in terms of normal ordering are given. As applications, a series of useful identities are given, the decoupling problem for general two-boson interaction and the normal product calculation of boson exponential quadratic operators are studied.

A q-analogue of the supersymmetric oscillator and its q-supercoherent states

A q-analogue of the supersymmetric oscillator is constructed out of q-boson and ordinary fermion creation and annihilation operators. q-supercoherent states are explicitly obtained for the q-deformed supersymmetric oscillator. They are shown to be eigenstates of both q-boson and fermion annihilation operators and to satisfy a completeness relation. The representation of the q-deformed superalgebra in super Bargmann-Fock space is also discussed by means of the q-supercoherent states.

General theory of linear quantum transformation of Bargmann-Fock space

A general theory for the linear transformation in Bargmann-Fock space ofn-mode boson system is presented. It is pointed out that the unbounded operators should not be excluded absolutely and the use of the non-unitary transformations is irresistible in many applications. The group structure of the transformation is analysed and the explicit expressions of the transformation operators in terms of normal ordering are given. As some applications, a series of useful operator identities are given and the decoupling problem for general two-boson interaction is studied.

Quantum group symmetric Bargmann–Fock space: Integral kernels, Green functions, driving forces

Raising and lowering operators that transform under the SUq(n)-quantum group get deformed commutation relations. They are represented as adjoint operators on a Hilbert space of noncommutative holomorphic functions. Through the algebraically defined integral on this function space, every operator on the Fock space can also be represented as an integral kernel. The Green function for free harmonic oscillators and spin-1/2’s in a constant magnetic field is given. Further on it is studied how such systems react on the switching on of a driving force. Calculating the vacuum–vacuum transition amplitude, it is found that the deviations from the undeformed case grow with the strength of the driving force. Throughout all calculations the bosonic and the fermionic cases are considered simultaneously.