Bargmann Space Research Articles

Analytic Solutions of Two-photon Rabi Model Based on Bargmann Space

The quantum Rabi model (QRM) is identified as the simplest model to study the interaction between light and matter. In recent years, Braak proposed an alternative more efficient method to obtain the exact solvability and spectrum of the QRM in the Bargmann space, which is appropriate frame for extended squeezed coherent states. Based on this feasible method, some study for the two-photon QRM is concentrated. In this paper we diagonalized the Hamiltonian and obtain the Spectrum of the system of two-photon QRM in the special case that the level splitting of the atom was equal to 0, and compared with Travĕnec’s results without diagonalizing the Hamiltonian. Finally, we confirm the validity of diagonalization for the Hamiltonian in the two-photon QRM.

Level crossings and new exact solutions of the two-photon Rabi model

An infinite family of exact solutions of the two-photon Rabi model was found by investigating the differential algebraic properties of the Hamiltonian. This family corresponds to energy level crossings not covered by the Juddian class, which is given by elemetary functions. In contrast, the new states are expressible in terms of parabolic cylinder or Bessel functions. We discuss three approaches for discovering this hidden structure: factorization of differential equations, Kimura transformation, and a doubly-infinite, transcendental basis of the Bargmann space.

S-Polyregular Bargmann Spaces

We introduce two classes of right quaternionic Hilbert spaces in the context of slice polyregular functions, generalizing the so-called slice and full hyperholomorphic Bargmann spaces. Their basic properties are discussed, the explicit formulas of their reproducing kernels are given and associated Segal--Bargmann transforms are also introduced and studied. The spectral description as special subspaces of $L^2$-eigenspaces of a second order differential operator involving the slice derivative is investigated.

Bicomplex Analogs of Segal–Bargmann and Fractional Fourier Transforms

We consider and discuss some basic properties of the bicomplex analog of the classical Bargmann space. The explicit expression of the integral operator connecting the complex and bicomplex Bargmann spaces is also given. The corresponding bicomplex Segal–Bargmann transform is introduced and studied as well. Its explicit expression as well as the one of its inverse are then used to introduce a class of two-parameter bicomplex Fourier transforms (bicomplex fractional Fourier transform). This approach is convenient in exploring some useful properties of this bicomplex fractional Fourier transform.

Positivity, complex FIOs, and Toeplitz operators

We establish a characterization of complex linear canonical transformations that are positive with respect to a pair of strictly plurisubharmonic quadratic weights. As an application, we show that the boundedness of a class of Toeplitz operators on the Bargmann space is implied by the boundedness of their Weyl symbols.

Construction of some classes of commutative Banach and C⋆-algebras of Toeplitz operators

In the present paper, we construct commutative algebras generated by Toeplitz operators on the Segal–Bargmann space Hs2(Cn) and on the true-k-Fock spaces. Analogous to the commutative Banach algebras constructed by N. Vasilevski for the case of the unit ball, we obtain a commutative algebra on Hs2(Cn) formed only by Toeplitz operators and a composition formula is obtained. Employing a natural extension for the notion of Toeplitz operators, we introduce “true-k-Toeplitz operators” acting on the true-k-Fock spaces. We provide a commutative C⋆-algebra generated by such operators, whose symbol depends on the real and imaginary parts of the complex variable in a certain sense.

The $\partial $-complex on the Segal–Bargmann space

The $\partial $-complex on the Segal–Bargmann space

Holomorphic Hermite polynomials in two variables

Generalizations of the Hermite polynomials to many variables and/or to the complex domain have been located in mathematical and physical literature for some decades. Polynomials traditionally called complex Hermite ones are mostly understood as polynomials in $z$ and $\bar{z}$ which in fact makes them polynomials in two real variables with complex coefficients. The present paper proposes to investigate for the first time holomorphic Hermite polynomials in two variables. Their algebraic and analytic properties are developed here. While the algebraic properties do not differ too much for those considered so far, their analytic features are based on a kind of non-rotational orthogonality invented by van Eijndhoven and Meyers. Inspired by their invention we merely follow the idea of Bargmann's seminal paper (1961) giving explicit construction of reproducing kernel Hilbert spaces based on those polynomials. "Homotopic" behavior of our new formation culminates in comparing it to the very classical Bargmann space of two variables on one edge and the aforementioned Hermite polynomials in $z$ and $\bar{z}$ on the other. Unlike in the case of Bargmann's basis our Hermite polynomials are not product ones but factorize to it when bonded together with the first case of limit properties leading both to the Bargmann basis and suitable form of the reproducing kernel. Also in the second limit we recover standard results obeyed by Hermite polynomials in $z$ and $\bar{z}$.

Holomorphic Hermite Functions in Segal–Bargmann Spaces

We study systems of holomorphic Hermite functions in the Segal–Bargmann spaces, which are Hilbert spaces of entire functions on the complex Euclidean space, and are determined by the Bargmann-type integral transform on the real Euclidean space. We prove that for any positive parameter which is strictly smaller than the minimum eigenvalue of the positive Hermitian matrix associated with the transform, one can find a generator of holomorphic Hermite functions whose annihilation and creation operators satisfy canonical commutation relations. In other words, we find the necessary and sufficient conditions so that some kinds of entire functions can be such generators. Moreover, we also study the complete orthogonality, the eigenvalue problems and the Rodrigues formulas.

Mehler's formulas for the univariate complex Hermite polynomials and applications

We give 2 widest Mehler's formulas for the univariate complex Hermite polynomials , by performing double summations involving the products and . They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level m. The second Mehler's formula generalizes the one appearing as a particular case of the so‐called Kibble‐Slepian formula. The proofs we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.

On an unconditional basis with parentheses for generalized subordinate perturbations and application to Gribov operators in Bargmann space

In this article, we give a new definition which generalizes the notion of subordination between two operators. Moreover, we give a description of the changed spectrum and we establish different conditions in terms of the spectrum to prove the existence of unconditional basis with parentheses. We apply this work to some block operators matrix. The obtained results are of importance to be applied to Gribov operators in Bargmann space.

Carroll symmetry of plane gravitational waves

The well-known 5-parameter isometry group of plane gravitational waves in 4 dimensions is identified as Lévy-Leblond’s Carroll group in dimensions with no rotations. Our clue is that plane waves are Bargmann spaces into which Carroll manifolds can be embedded. We also comment on the scattering of light by a gravitational wave and calculate its electric permittivity considered as an impedance-matched metamaterial.

Vanishing Bergman Kernels on the Disk

We discuss topics related to zeroes of the Bergman kernels, and present a method for generating Bergman kernels with arbitrarily, but finitely, many zeroes. It is also shown that a Bergman kernel induced by a radial weight on the unit disk cannot have infinitely many zeroes. Similar questions for the Segal–Bargmann spaces of the complex plane are briefly discussed.

Holomorphic Hermite functions and ellipses

ABSTRACTIn 1990 van Eijndhoven and Meyers introduced systems of holomorphic Hermite functions and reproducing kernel Hilbert spaces associated with the systems on the complex plane. Moreover they studied the relationship between the family of all their Hilbert spaces and a class of Gelfand–Shilov functions. After that, their systems of holomorphic Hermite functions have been applied to studying quantization on the complex plane, combinatorics, and etc. On the other hand, the author recently introduced systems of holomorphic Hermite functions associated with ellipses on the complex plane. The present paper shows that their systems of holomorphic Hermite functions are determined by some cases of ellipses, and that their reproducing kernel Hilbert spaces are some cases of the Segal–Bargmann spaces determined by the Bargmann-type transforms introduced by Sjöstrand.

Composition operators on Hilbert spaces of entire functions with analytic symbols

Composition operators with analytic symbols on some reproducing kernel Hilbert spaces of entire functions on a complex Hilbert space are studied. The questions of their boundedness, seminormality and positivity are investigated. It is proved that if such an operator is bounded, then its symbol is a polynomial of degree at most 1, i.e., it is an affine mapping. Fock's type model for composition operators with linear symbols is established. As a consequence, explicit formulas for their polar decomposition, Aluthge transform and powers with positive real exponents are provided. The theorem of Carswell, MacCluer and Schuster is generalized to the case of Segal–Bargmann spaces of infinite order. Some related questions are also discussed.

On Chaoticity of the Sum of Chaotic Shifts with Their Adjoints in Hilbert Space and Applications to Some Weighted Shifts Acting on Some Fock–Bargmann Spaces

The aim in this paper is to give sufficient conditions for an operator of the form \(\mathbb {T} + \mathbb {T}^{*}\) to be chaotic (in the sense of Devaney). Here \(\mathbb {T}\) is assumed to be a weighted chaotic shift operator on a Hilbert space and \(\mathbb {T}^{*}\) is its adjoint. Sufficient conditions on the weight sequence are given and then applied to some particular bakward shift unbounded operators realized as differential operators acting on some Fock–Bargmann spaces.

Regularized trace formula of magic Gribov operator on Bargmann space

In this article, we obtain a regularized trace formula for magic Gribov operator H=λ″G+Hμ,λ acting on Bargmann space whereG=a⁎3a3 and Hμ,λ=μa⁎a+iλa⁎(a+a⁎)a Here a and a⁎ are the standard Bose annihilation and creation operators and in Reggeon field theory, the real parameters λ″ is the magic coupling of Pomeron, μ is Pomeron intercept, λ is the triple coupling of Pomeron and i2=−1. By applying some abstract results of Sadovnichii–Podol'skii (2002) [23], we give the number of corrections sufficient for the existence of finite formula of the trace of concrete magic Gribov's operator.

Dynamics of two-cluster systems in phase space

We present a phase-space representation of quantum state vectors for two-cluster systems. Density distributions in the Fock–Bargmann space are constructed for bound and resonance states of 6,7Li and 7,8Be, provided that all these nuclei are treated within a microscopic two-cluster model. The density distribution in the phase space is compared with those in the coordinate and momentum representations. Bound states realize themselves in a compact area of the phase space, as also do narrow resonance states. We establish the quantitative boundaries of this region in the phase space for the nuclei under consideration. Quantum trajectories are demonstrated to approach their classical limit with increasing energy.

Weighted Shifts on Directed Semi-Trees: an Application to Creation Operators on Segal–Bargmann Spaces

We extend the notion of a weighted shift on a directed tree to the case of a more general graph which we call a directed semi-tree. Some basic properties of such operators are investigated. It is shown that a generalized creation operator on the Segal–Bargman space is unitarily isomorphic to a weighted shift on a directed semi-tree of a particular form.

Toeplitz Operators with Uniformly Continuous Symbols

Let Tf be a Toeplitz operator on the Segal–Bargmann space or the standard weighted Bergman space over a bounded symmetric domain \({\Omega \subset {\bf C}^n}\) with possibly unbounded symbol f. Combining recent results in Bauer et al. (J. Funct. Anal. 259:57–78, 2010), Bauer et al. (J. reine angew. Math. doi:10.1515/crelle-2015-0016), Issa (Integr. Equ. Oper. Theory 70:569–582, 2011) we show that in the case of uniformly continuous symbols f with respect to the Euclidean metric on Cn and the Bergman metric on \({\Omega}\), respectively, the operator Tf is bounded if and only if f is bounded. Moreover, Tf is compact if and only if f vanishes at the boundary of \({\Omega.}\) This observation substantially extends a result in Coburn (Indiana Univ. Math. J. 23:433–439, 1973).