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Natig M. Atakishiyev

Publications and source records attributed to Natig M. Atakishiyev.

7 recordsLinked to original sources
arXiv2017

Interbasis expansions in the Zernike system

The differential equation with free boundary conditions on the unit disk that was proposed by Frits Zernike in 1934 to find Jacobi polynomial solutions (indicated as I), serves to define a classical and a quantum system which have been found to be superintegrable. We have determined two new orthogonal polynomial solutions (indicated as II and III) that are separable, and which involve Legendre and Gegenbauer polynomials. Here we report on their three interbasis expansion coefficients: between the I--II and I--III bases they are given by $_3F_2(\cdots|1)$ polynomials that are also special su($2$) Clebsch-Gordan coefficients and Hahn polynomials. Between the II--III bases, we find an xpansion expressed by $_4F_3(\cdots|1)$'s and Racah polynomials that are related to the Wigner $6j$ coefficients.

math-ph↗
arXiv2010

SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms

We propose a group-theoretical approach to the generalized oscillator algebra Ak recently investigated in J. Phys. A: Math. Theor. 43 (2010) 115303. The case k > or 0 corresponds to the noncompact group SU(1,1) (as for the harmonic oscillator and the Poeschl-Teller systems) while the case k < 0 is described by the compact group SU(2) (as for the Morse system). We construct the phase operators and the corresponding temporally stable phase eigenstates for Ak in this group-theoretical context. The SU(2) case is exploited for deriving families of mutually unbiased bases used in quantum information. Along this vein, we examine some characteristics of a quadratic discrete Fourier transform in connection with generalized quadratic Gauss sums and generalized Hadamard matrices.

quant-ph↗
arXiv2007

Discrete quantum model of the harmonic oscillator

We construct a new model of the quantum oscillator, whose energy spectrum is equally-spaced and lower-bounded, whereas the spectra of position and momentum are a denumerable non-degenerate set of points in [-1,1] that depends on the deformation parameter q from (0,1). We provide its explicit wavefunctions, both in position and momentum representations, in terms of the discrete q-Hermite polynomials. We build a Hilbert space with a unique measure, where an analogue of the fractional Fourier transform is defined in order to govern the time evolution of this discrete oscillator. In the limit q to 1, one recovers the ordinary quantum harmonic oscillator.

math-ph↗
arXiv2004

Finite q-oscillator

The finite q-oscillator is a model that obeys the dynamics of the harmonic oscillator, with the operators of position, momentum and Hamiltonian being functions of elements of the q-algebra su_q(2). The spectrum of position in this discrete system, in a fixed representation j, consists of 2j+1 "sensor"-points x_s=(1/2)[2s]_q, s=-j, -j+1,..., j, and similarly for the momentum observable. The spectrum of energies is finite and equally spaced, so the system supports coherent states. The wave functions involve dual q-Kravchuk polynomials, which are solutions to a finite-difference Schrodinger equation. Time evolution (times a phase) defines the fractional Fourier-q-Kravchuk transform. In the classical limit q -> 1 we recover the finite oscillator Lie algebra, the N=2j -> infinity limit returns the Macfarlane-Biedenharn q-oscillator and both limits contract the generators to the standard quantum-mechanical oscillator.

math-ph↗
arXiv1998

Meixner Oscillators

Meixner oscillators have a ground state and an `energy' spectrum that is equally spaced; they are a two-parameter family of models that satisfy a Hamiltonian equation with a {\it difference} operator. Meixner oscillators include as limits and particular cases the Charlier, Kravchuk and Hermite (common quantum-mechanical) harmonic oscillators. By the Sommerfeld-Watson transformation they are also related with a relativistic model of the linear harmonic oscillator, built in terms of the Meixner-Pollaczek polynomials, and their continuous weight function. We construct explicitly the corresponding coherent states with the dynamical symmetry group Sp(2,$\Re$). The reproducing kernel for the wavefunctions of these models is also found.

math-ph↗