Natig M. Atakishiyev

Fractional Fourier–Kravchuk transform

We introduce a model of multimodal waveguides with a finite number of sensor points. This is a finite oscillator whose eigenstates are Kravchuk functions, which are orthonormal on a finite set of points and satisfy a physically important difference equation. The fractional finite Fourier–Kravchuk transform is defined to self-reproduce these functions. The analysis of finite signal processing uses the representations of the ordinary rotation group SO(3). This leads naturally to a phase space for finite optics such that the continuum limit (N→∞) reproduces Fourier paraxial optics.

The Wigner function for general Lie groups and the wavelet transform

Abstract. We build Wigner maps, functions and operators on general phase spaces arising from a class of Lie groups, including non-unimodular groups (such as the affine group). The phase spaces are coadjoint orbits in the dual of the Lie algebra of these groups and they come equipped with natural symplectic structures and Liouville-type invariant measures. When the group admits square-integrable representations, we present a very general construction of a Wigner function which enjoys all the desirable properties, including full covariance and reconstruction formulae. We study in detail the case of the affine group on the line. In particular, we put into focus the close connection between the well-known wavelet transform and the Wigner function on such groups.

On a one-parameter family of q-exponential functions

We examine the properties of a family of q-exponential functions, which depend on an extra parameter . These functions have a well defined meaning for both the 0 1 cases if only . It is shown that any two members of this family with different values of the parameter are related to each other by a Fourier - Gauss transformation.

CONTRACTION OF THE FINITE ONE-DIMENSIONAL OSCILLATOR

The finite oscillator model of 2j + 1 points has the dynamical algebra u(2), consisting of position, momentum and mode number. It is a paradigm of finite quantum mechanics where a sequence of finite unitary models contract to the well-known continuum theory. We examine its contraction as the number and density of points increase. This is done on the level of the dynamical algebra, of the Schrodinger difference equation, the (Kravchuk) wave functions, and the Fourier–Kravchuk transformation between position and momentum representations.

Finite two-dimensional oscillator: II. The radial model

A finite two-dimensional radial oscillator of (N + 1)2 points is proposed, with the dynamical Lie algebra so(4) = su(2)x⊕su(2)y examined in part I of this work, but reduced by a subalgebra chain so(4)⊃so(3)⊃so(2). As before, there are a finite number of energies and angular momenta; the Casimir spectrum of the new chain provides the integer radii 0≤ρ≤N, and the 2ρ + 1 discrete angles on each circle ρ are obtained from the finite Fourier transform of angular momenta. The wavefunctions of the finite radial oscillator are so(3) Clebsch-Gordan coefficients. We define here the Hankel-Hahn transforms (with dual Hahn polynomials) as finite-N unitary approximations to Hankel integral transforms (with Bessel functions), obtained in the contraction limit N→∞.

A simple difference realization of the Heisenberg q‐algebra

A realization of the Heisenberg q‐algebra whose generators are first‐order difference operators on the full real line is discussed herein. The eigenfunctions of the corresponding q‐oscillator Hamiltonian are given explicitly in terms of the q−1‐Hermite polynomials. The nonuniqueness of the measure for these q‐oscillator states is also studied.

Quantum algebraic structures compatible with the harmonic oscillator Newton equation

We study some of the algebraic structures that are compatible with the quantization of the harmonic oscillator through its Newton equation. Examples of such structures are given; they include undeformed and q-deformed oscillators, as well as the SU(2) and the deformed SUq(2) Lie algebras, which appear in a variety of physical models.

On the Rogers-Szego polynomials

An orthogonality relation on the full real line for the Rogers-Szego polynomials is discussed. It is argued that Fourier transformation with the standard exponential kernel exp(ixy) relates the Rogers-Szego and Stieltjes-Wigert functions.

Big q-Laguerre and q-Meixner polynomials and representations of the quantum algebra Uq(su1,1)

Diagonalization of a certain operator in irreducible representations of the positive discrete series of the quantum algebra Uq(su1,1) is studied. Spectrum and eigenfunctions of this operator are explicitly found. These eigenfunctions, when normalized, constitute an orthonormal basis in the representation space. The initial Uq(su1,1) basis and the basis of these eigenfunctions are interconnected by a matrix with entries expressed in terms of big q-Laguerre polynomials. The unitarity of this connection matrix leads to an orthogonal system of functions, which are dual with respect to big q-Laguerre polynomials. This system of functions consists of two separate sets of functions, which can be expressed in terms of q-Meixner polynomials Mn(x; b, c; q) either with positive or negative values of the parameter b. The orthogonality property of these two sets of functions follows directly from the unitarity of the connection matrix. As a consequence, one obtains an orthogonality relation for the q-Meixner polynomials Mn(x; b, c; q) with b < 0. A biorthogonal system of functions (with respect to the scalar product in the representation space) is also derived.

On the phase space description of quantum nonlinear dynamics

Abstract We analyze the difference between classical dynamics (geometric optics) and quantum dynamics (wave optics) by calculating the time history of the Wigner function for the simplest nonlinear Hamiltonians which are fourth-degree polynomials in p and q . It is shown that the moments of the Wigner function carry important information about the state of a system and can be used to distinguish between quasiclassical and quantum evolution.