Complex Hermite Polynomials Research Articles

Triply extended group of translations of as defining group of NCQM: relation to various gauges

The role of the triply extended group of translations of , as the defining group of two-dimensional noncommutative quantum mechanics (NCQM), has been studied in Chowdhury and Ali (2013 J. Math. Phys. 54 032101). In this paper, we revisit the coadjoint orbit structure and various irreducible representations of the group associated with them. The two irreducible representations corresponding to the Landau and symmetric gauges are found to arise from the underlying defining group. The group structure of the transformations, preserving the commutation relations of NCQM, has been studied along with specific examples. Finally, the relationship of a certain family of unitary irreducible representations of the underlying defining group with a family of deformed complex Hermite polynomials has been explored.

Operational formulae for the complex Hermite polynomials Hp, q(z, z¯)

We give operational formulae of the Burchnall type involving complex Hermite polynomials. Short proofs of some known formulae are given and new results involving these polynomials, including Nielsen's identities and Runge addition formula, are derived.

Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated coherent states

We define quaternionic Hermite polynomials by analogy with two families of complex Hermite polynomials. As in the complex case, these polynomials constitute orthogonal families of vectors in ambient quaternionic L2-spaces. Using these polynomials, we then define regular and anti-regular subspaces of these L2-spaces, the associated reproducing kernels and the ensuing quaternionic coherent states.

Quantizations from reproducing kernel spaces

The purpose of this work is to explore the existence and properties of reproducing kernel Hilbert subspaces of L2(C,d2z/π) based on subsets of complex Hermite polynomials. The resulting coherent states (CS) form a family depending on a nonnegative parameter s. We examine some interesting issues, mainly related to CS quantization, like the existence of the usual harmonic oscillator spectrum despite the absence of canonical commutation rules. The question of mathematical and physical equivalences between the s-dependent quantizations is also considered.

Coherent states and Hermite polynomials on quaternionic Hilbert spaces

Coherent states, similar to the canonical coherent states of the harmonic oscillator, labeled by quaternions are established on the right and left quaternionic Hilbert spaces. On the left quaternionic Hilbert space reproducing kernels are established. As was claimed by Adler and Millard (J. Math. Phys. 1997 38 2117–26) it is proved that these coherent states cannot be realized as a group action on a quaternionic Hilbert space. As an extension of the complex Hermite polynomials, quaternionic Hermite polynomials are defined and these polynomials are associated with an operator as eigenfunctions.

Complex and real Hermite polynomials and related quantizations

It is known that the anti-Wick (or standard coherent state) quantization of the complex plane produces both canonical commutation rule and quantum spectrum of the harmonic oscillator (up to the addition of a constant). In this work, we show that these two issues are not necessarily coupled: there exists a family of separable Hilbert spaces, including the usual Fock–Bargmann space, and in each element in this family there exists an overcomplete set of unit-norm states resolving the unity. With the exception of the Fock–Bargmann case, they all produce non-canonical commutation relation whereas the quantum spectrum of the harmonic oscillator remains the same up to the addition of a constant. The statistical aspects of these non-equivalent coherent state quantizations are investigated. We also explore the localization aspects in the real line yielded by similar quantizations based on real Hermite polynomials.

An interesting modular structure associated to Landau levels

We analyze here, from a von Neumann algebraic point of view, the well known problem of a charged particle, moving on the infinite two-dimensional plane under the influence of a constant magnetic field, perpendicular to this plane. The energy levels of this system, known in the literature as Landau levels, are those of the harmonic oscillator, with each level being infinitely degenerate. There is an interesting algebraic property associated with this problem, that we discuss here. It arises from the fact that there are two mutually commuting von Neumann algebras associated to this system, which correspond to the two possible directions of the constant magnetic field. This commuting pair has a modular structure, in the sense of the Tomita-Takesaki theory, leading to the existence of KMS states for the system with physically interesting properties. An example of such a state is the Gibbs state related to the interaction part of the Landau Hamiltonian, which in this case is invariant under the modular automorphism group, generated by this Hamiltonian. Also associated to the problem is a family of orthogonal polynomials, the complex Hermite polynomials, which form a basis of the underlying Hilbert space.

A Novel Predistorter Design for Nonlinear Power Amplifier with Memory Effects in OFDM Communication Systems Using Orthogonal Polynomials

Orthogonal frequency division multiplexing (OFDM) signals have high peak-to-average power ratio (PAPR) and cause large nonlinear distortions in power amplifiers (PAs). Memory effects in PAs also become no longer ignorable for the wide bandwidth of OFDM signals. Digital baseband predistorter is a highly efficient technique to compensate the nonlinear distortions. But it usually has many parameters and takes long time to converge. This paper presents a novel predistorter design using a set of orthogonal polynomials to increase the convergence speed and the compensation quality. Because OFDM signals are approximately complex Gaussian distributed, the complex Hermite polynomials which have a closed-form expression can be used as a set of orthogonal polynomials for OFDM signals. A differential envelope model is adopted in the predistorter design to compensate nonlinear PAs with memory effects. This model is superior to other predistorter models in parameter number to calculate. We inspect the proposed predistorter performance by using an OFDM signal referred to the IEEE 802.11a WLAN standard. Simulation results show that the proposed predistorter is efficient in compensating memory PAs. It is also demonstrated that the proposal acquires a faster convergence speed and a better compensation effect than conventional predistorters.

Complex Hermite polynomials: from the semi-circular law to the circular law

We study asymptotics of orthogonal polynomial measures of the form |HN| 2 d∞ where HN are real or complex Hermite polynomials with re- spect to the Gaussian measure ∞. By means of dierential equations on Laplace transforms, interpolation between the (real) arcsine law and the (complex) uniform distribution on the circle is emphasized. Suitable aver- ages by an independent uniform law give rise to the limiting semi-circular and circular laws of Hermitian and non-Hermitian Gaussian random matrix models. The intermediate regime between strong and weak non-Hermiticity is clearly identified on the limiting dierential equation by means of an addi- tional normal variable in the vertical direction.

A class of generalized complex Hermite polynomials

A class of generalized complex polynomials of Hermite type, suggested by a special magnetic Schrödinger operator, is introduced and some related basic properties are discussed.

Integrable vortex dynamics in anisotropic planar spin liquid model

The problem of magnetic vortex dynamics in an anisotropic spin liquid model is considered. For incompressible flow the model admits reduction to saturating Bogomolny inequality analytic projections of spin variables, subject the linear holomorphic Schrödinger equation. It allows us to construct N vortex configurations in terms of the complex Hermite polynomials. Using complex Galilean boost transformations, the interaction of the vortices and the vortex chain lattices (vortex crystals) is studied. By the complexified Cole–Hopf transformation, integrable N vortex dynamics is described by the holomorphic Burgers equation. Mapping of the point vortex problem to N-particle problem, the complexified Calogero–Moser system, showing its integrability and the Hamiltonian structure, is given.

Simple representations for Hermite polynomials

Hermite polynomials have applications in almost every area of electrical and electronic engineering. Simple (hitherto unknown) representations are derived for the Hermite polynomials. For simplicity, the univariate case is considered. The same result holds for the multivariate and complex Hermite polynomials.

Hermite and Laguerre polynomials with complex matrix arguments

This paper defines and discusses the complex Hermite and Laguerre polynomials associated with the complex matrix-variate normal and Wishart distributions, respectively. Various properties of these polynomials are investigated, including generating functions, Rodrigues formulae (differential and integral versions), and series expressions. These polynomials are also discussed from the viewpoint of the multivariate complex Meixner distributions. We present applications in asymptotic distribution theory on the complex Stiefel manifold. The theory of complex zonal polynomials is of great use in the derivations.

Applications of the theory of the metaplectic representation to quadratic Hamiltonians on the two-dimensional Euclidean space

The spectra of the quadratic Hamiltonians on the twodimensional Euclidean space are determined completely by using the theory of the metaplectic representation. In some cases, the corresponding heat kernels are studied in connection with the well-definedness of the Wiener integrations. A proof of the Lévy formula for the stochastic area and a relation between the real and complex Hermite polynomials are given in our framework.

Quadratic Hamiltonians and Associated Orthogonal Polynomials

In connection with the quadratic quantum Hamiltonians, we define and study a family of the systems of orthogonal polynomials with two variables. The family includes the usual real and the complex Hermite polynomials as special examples. Some recurrence formulae and the eigenfunction expansion for the heat kernel of the corresponding quantum Hamiltonian are shown.

Complex Gaussian noise moments

The problem of the computation of moments of nonzero mean circularly complex Gaussian noise is treated. The noise need not be symmetric about the carrier frequency. In particular, the second-order moments are computed, and expansions are given. The necessary univariate and bivariate complex Hermite polynomials are discussed. The means of some rational functions useful in FM theory are given. This paper extends work of Rice, Middleton, and Zadeh to complex Gaussian noise with nonzero mean and nonsymmetrical power spectrum.

Circularly complex Gaussian noise--A price theorem and a Mehler expansion (Corresp.)

In this correspondence, the Price theorem for n -dimensional circularly complex Gaussian noise is given and examples of its use are presented. The Mehler expansion for two-dimensional circularly complex Gaussian noise is derived. Some properties of the complex Hermite polynomials required in this expansion are presented. The use of the complex Hermite polynomials is illustrated. These results ease some of the calculations involved with bandpass signals and noise.