Bargmann Space Research Articles

Pullback operators on Bargmann spaces

We characterize boundedness and compactness of pullback operators under holomorphic maps between Bargmann spaces of entire holomorphic functions with quadratic strictly plurisubharmonic exponential weights, extending the result of Carswell, MacCluer, and Schuster [Acta Sci. Math. (Szeged) 69 (2003), pp. 871–887], obtained in the case of the radial quadratic weight. We also show that the pullback operator between Bargmann spaces is compact precisely when it is of trace class, with sub-exponentially decaying singular values.

The three-dimensional harmonic oscillator and solid harmonics in Bargmann space

The three-dimensional harmonic oscillator is solved in Bargmann space. The treatment is pedagogically more transparent than the standard ones, at the price of introducing the Bargmann transform in the context of the one-dimensional oscillator. The standard solid harmonics are similarly derived with minimal technical effort, amounting to a complete self-contained exposition suitable for introductory courses in quantum mechanics or mathematical methods of physics. It provides an early exposure to wavelets, with important contemporary applications in signal analysis and quantum optics.

Spectral continuum in the Rabi–Stark model [Invited

The Rabi–Stark model is a nonlinear generalization of the quantum Rabi model including the dynamical Stark shift as a tunable term, which can be realized via quantum simulation on a cavity QED platform. When the Stark coupling becomes equal to the mode frequency, the spectrum changes drastically, a transition usually termed “spectral collapse” because numerical studies indicate an infinitely degenerate ground state. We show that the spectrum extends continuously from a threshold value up to infinity. A set of normalizable states is embedded in the continuum, which furnishes an unexpected analogy to the atomic Stark effect. Bound states and continuum can be obtained analytically through two equally justified but different confluence processes of the associated differential equation in Bargmann space. Moreover, these results are obtained independently using a method based on adiabatic elimination of the spin degree of freedom and corroborated through large-scale numerical checks.

Spectral properties of weighted Cauchy singular integral transform on S-poly-Barmgann spaces

We investigate some spectral properties of the weighted quaternionic Cauchy transform when acting on the right quaternionic Hilbert space of Gaussian integrable functions. We study its boundedness, compactness, and memberships to the k-Schatten class, and we identify its range. This is done by means of its restriction to the n-th S-polyregular Bargmann space of the second kind, for which we provide an explicit closed expression for its action on the quaternionic Itô–Hermite polynomials constituting an orthogonal basis. We also exhibit an orthogonal basis of eigenfunctions of its n-Bergman projection leading to the explicit determination of its singular values. The provided results extend to the quaternionic setting those obtained by A. Intissar and A. Intissar in (2006) [32] for the weighted Cauchy transform on the complex plane.

Boundedness of metaplectic Toeplitz operators and Weyl symbols

We study Toeplitz operators on the Bargmann space, whose Toeplitz symbols are exponentials of complex inhomogeneous quadratic polynomials. Extending a result by Coburn–Hitrik–Sjöstrand [7], we show that the boundedness of such Toeplitz operators implies the boundedness of the corresponding Weyl symbols, thus completing the proof of the Berger–Coburn conjecture in this case. We also show that a Toeplitz operator is compact precisely when its Weyl symbol vanishes at infinity in this case.

Characterizing Boundedness of Metaplectic Toeplitz Operators

Abstract We study Toeplitz operators on the Bargmann space, with Toeplitz symbols given by exponentials of complex quadratic forms. We show that the boundedness of the corresponding Weyl symbols is necessary for the boundedness of the operators, thereby completing the proof of the Berger–Coburn conjecture in this case. We also show that the compactness of such Toeplitz operators is equivalent to the vanishing of their Weyl symbols at infinity.

A Note on the Completeness of Generalized Eigenfunctions of the Hamiltonian of Reggeon Field Theory in Bargmann Space

A Note on the Completeness of Generalized Eigenfunctions of the Hamiltonian of Reggeon Field Theory in Bargmann Space

Integral transforms and [formula omitted]-symmetric Hamiltonians

Motivated by the fact that twice the Fourier transform plays the role of a parity operator. We systematically study the integral transforms of PT-symmetric Hamiltonians. First, we obtain a closed analytical formula for the exponential Fourier transform of a general PT-symmetric Hamiltonian. Using the Segal–Bargmann transform, we investigate the effect of the Fourier transforms on the eigenfunctions of the original Hamiltonian. As an immediate application, we comment on the holomorphic representation of non-Hermitian spin chains. In this case, the Hamiltonian is written in terms of analytical phase space coordinates and their partial derivatives in the Bargmann space instead of matrices in the vector Hilbert space. Finally, we discuss the effect of integral transforms in the study of the Swanson Hamiltonian.

Complex FIOs and composition of Toeplitz operators

We study Toeplitz operators on the Bargmann space, with Toeplitz symbols that are exponentials of complex quadratic forms, from the point of view of Fourier integral operators in the complex domain. Sufficient conditions are established for the composition of two such operators to be a Toeplitz operator.

Cross-Toeplitz operators on the Fock–Segal–Bargmann spaces and two-sided convolutions on the Heisenberg group

We introduce an extended class of cross-Toeplitz operators which act between Fock–Segal–Bargmann spaces with different weights. It is natural to consider these operators in the framework of representation theory of the Heisenberg group. Our main technique is representation of cross-Toeplitz by two-sided relative convolutions from the Heisenberg group. In turn, two-sided convolutions are reduced to usual (one-sided) convolutions on the Heisenberg group of the doubled dimensionality. This allows us to utilise the powerful group-representation technique of coherent states, co- and contra-variant transforms, twisted convolutions, symplectic Fourier transform, etc. We discuss connections of (cross-)Toeplitz operators with pseudo-differential operators, localisation operators in time–frequency analysis, and characterisation of kernels in terms of ladder operators. The paper is written in a detailed and reasonably self-contained manner to be suitable as an introduction into group-theoretical methods in phase space and time–frequency operator theory.

On a Riesz basis of diagonally generalized subordinate operator matrices and application to a Gribov operator matrix in Bargmann space

In this paper, we study the change of spectrumand the existence of Riesz bases of specific classes of n?n unbounded operator matrices, called: diagonally and off-diagonally generalized subordinate block operator matrices. An application to a n ? n Gribov operator matrix acting on a sum of Bargmann spaces, illustrates the abstract results. As example, we consider a particular Gribov operator matrix by taking special values of the real parameters of Pomeron.

Segal–Bargmann Transforms Associated to a Family of Coupled Supersymmetries

The Segal–Bargmann transform is a Lie algebra and Hilbert space isomorphism between real and complex representations of the oscillator algebra. The Segal–Bargmann transform is useful in time-frequency analysis as it is closely related to the short-time Fourier transform. The Segal–Bargmann space provides a useful example of a reproducing kernel Hilbert space. Coupled supersymmetries (coupled SUSYs) are generalizations of the quantum harmonic oscillator that have a built-in supersymmetric nature and enjoy similar properties to the quantum harmonic oscillator. In this paper, we will develop Segal–Bargmann transforms for a specific class of coupled SUSYs which includes the quantum harmonic oscillator as a special case. We will show that the associated Segal–Bargmann spaces are distinct from the usual Segal–Bargmann space: their associated weight functions are no longer Gaussian and are spanned by stricter subsets of the holomorphic polynomials. The coupled SUSY Segal–Bargmann spaces provide new examples of reproducing kernel Hilbert spaces.

Semiclassical Gevrey operators and magnetic translations

We study semiclassical Gevrey pseudodifferential operators acting on the Bargmann space of entire functions with quadratic exponential weights. Using some ideas from time frequency analysis, we show that such operators are uniformly bounded on a natural scale of exponentially weighted spaces of holomorphic functions, provided that the Gevrey index is greater than or equal to 2.

Gravitational waves and conformal time transformations

Recent interest in the “memory effect” prompted us to revisit the relation of gravitational wavesto oscillators. 50 years ago Niederer (1973) found that an isotropic harmonic oscillator with a constant frequency can be mapped onto a free particle. Later Takagi (1990) has shown that “time-dependent scaling” extends the oscillator versus free particle correspondence to a time-dependent frequency when the scale factor satisfies a Sturm–Liouville equation. More recently Gibbons (0000) pointed out that time redefinition is conveniently studied in terms of the Schwarzian derivative. The oscillator versus free particle correspondence “Eisenhart-Duval lifts” to a conformal transformation between Bargmann spaces (Eisenhart, 1928; Duval et al., 1985; Duval et al., 1991; Cariglia et al., 2016). These methods are extended to spacetimes which are not conformally flat and have a time-dependent profile, and can then be applied to the geodesic motion in a plane gravitational wave.

Propagation and Reduction of Coherent States in Bargmann Spaces

Propagation and Reduction of Coherent States in Bargmann Spaces

Time-Dependent Conformal Transformations and the Propagator for Quadratic Systems

The method proposed by Inomata and his collaborators allows us to transform a damped Caldirola–Kanai oscillator with a time-dependent frequency to one with a constant frequency and no friction by redefining the time variable, obtained by solving an Ermakov–Milne–Pinney equation. Their mapping “Eisenhart–Duval” lifts as a conformal transformation between two appropriate Bargmann spaces. The quantum propagator is calculated also by bringing the quadratic system to free form by another time-dependent Bargmann-conformal transformation, which generalizes the one introduced before by Niederer and is related to the mapping proposed by Arnold. Our approach allows us to extend the Maslov phase correction to an arbitrary time-dependent frequency. The method is illustrated by the Mathieu profile.

Parameter-dependent unitary transformation approach for quantum Rabi model

Quantum Rabi model has been exactly solved by employing the parameter-dependent unitary transformation method in both the occupation number representation and the Bargmann space. The analytical expressions for the complete energy spectrum consisting of two double-fold degenerate sub-energy spectra are presented in the whole range of all the physical parameters. Each energy level is determined by a parameter in the unitary transformation, which obeys a highly nonlinear equation. The corresponding eigenfunction is a convergent infinite series in terms of the physical parameters. Due to the level crossings between the neighboring eigenstates at certain physical parameter values, such the degeneracies could lead to novel physical phenomena in the two-level system with the light–matter interaction.

On the semiclassical spectrum of the Dirichlet–Pauli operator

This paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set with Dirichlet conditions on the boundary. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Segal–Bargmann and Hardy spaces associated with the magnetic field.

Reduction and coherent states

We apply a quantum version of dimensional reduction to Gaussian coherent states in Bargmann space to obtain squeezed states on complex projective spaces. This leads to a definition of a family of squeezed spin states with excellent semi-classical properties, governed by a symbol calculus. We prove semiclassical norm estimates and a propagation result.

Coherent states for the Manin plane via Toeplitz quantization

In the theory of Toeplitz quantization of algebras, as developed by the second author, coherent states are defined as eigenvectors of a Toeplitz annihilation operator. These coherent states are studied in the case when the algebra is the generically non-commutative Manin plane. In usual quantization schemes, one starts with a classical phase space and then quantizes it in order to produce annihilation operators and then their eigenvectors and eigenvalues. However, we do this in the opposite order, namely, the set of the eigenvalues of the previously defined annihilation operator is identified as a generalization of a classical mechanical phase space. We introduce the resolution of the identity, upper and lower symbols, and a coherent state quantization, which in turn quantizes the Toeplitz quantization. We thereby have a curious composition of quantization schemes. We proceed by identifying a generalized Segal–Bargmann space SB of square-integrable, anti-holomorphic functions as the image of a coherent state transform. Then, SB has a reproducing kernel function, which allows us to define a secondary Toeplitz quantization, whose symbols are functions. Finally, this is compared with the coherent states of the Toeplitz quantization of a closely related non-commutative space known as the paragrassmann algebra.