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George S. Pogosyan

Publications and source records attributed to George S. Pogosyan.

8 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↗
arXiv2017

New separated polynomial solutions to the Zernike system on the unit disk and interbasis expansion

The differential equation proposed by Frits Zernike to obtain a basis of polynomial orthogonal solutions on the the unit disk to classify wavefront aberrations in circular pupils, is shown to have a set of new orthonormal solution bases, involving Legendre and Gegenbauer polynomials, in non-orthogonal coordinates close to Cartesian ones. We find the overlaps between the original Zernike basis and a representative of the new set, which turn out to be Clebsch-Gordan coefficients.

math-ph↗
arXiv2017

Superintegrable classical Zernike system

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, as if it were a classical Hamiltonian with a non-standard potential. The trajectories turn out to be closed ellipses. We show that this is due to the existence of higher-order invariants that close into a cubic Higgs algebra. The Zernike classical system thus belongs to the class of superintegrable systems. Its Hamilton-Jacobi action separates in three vertical projections of polar coordinates of a sphere, polar and equidistant coordinates on half-hyperboloids, and also in elliptic coordinates on the sphere.

math-ph↗
arXiv2017

Quantum superintegrable Zernike system

We consider the differential equation that Zernike proposed to classify aberrations of wavefronts in a circular pupil, whose value at the boundary can be nonzero. On this account the quantum Zernike system, where that differential equation is seen as a Schrödinger equation with a potential, is special in that it has a potential and boundary condition that are not standard in quantum mechanics. We project the disk on a half-sphere and there we find that, in addition to polar coordinates, this system separates in two additional coordinate systems (non-orthogonal on the pupil disk), which lead to Schrödinger-type equations with Pöschl-Teller potentials, whose eigen-solutions involve Legendre, Gegenbauer and Jacobi polynomials. This provides new expressions for separated polynomial solutions of the original Zernike system that are real. The operators which provide the separation constants are found to participate in a superintegrable cubic Higgs algebra.

math-ph↗
arXiv2015

Harmonic Oscillator on the ${\rm SO}(2,2)$ Hyperboloid

In the present work the classical problem of harmonic oscillator in the hyperbolic space $H_2^2$: $z_0^2+z_1^2-z_2^2-z_3^2=R^2$ has been completely solved in framework of Hamilton-Jacobi equation. We have shown that the harmonic oscillator on $H_2^2$, as in the other spaces with constant curvature, is exactly solvable and belongs to the class of maximally superintegrable system. We have proved that all the bounded classical trajectories are closed and periodic. The orbits of motion are ellipses or circles for bounded motion and ultraellipses or equidistant curve for infinite ones.

math-ph↗
arXiv2006

Path Integral Approach for Superintegrable Potentials on Spaces of Non-constant Curvature: I. Darboux Spaces DI and DII

In this paper the Feynman path integral technique is applied for superintegrable potentials on two-dimensional spaces of non-constant curvature: these spaces are Darboux spaces D_I and D_II, respectively. On D_I there are three and on D_II four such potentials, respectively. We are able to evaluate the path integral in most of the separating coordinate systems, leading to expressions for the Green functions, the discrete and continuous wave-functions, and the discrete energy-spectra. In some cases, however, the discrete spectrum cannot be stated explicitly, because it is either determined by a transcendental equation involving parabolic cylinder functions (Darboux space I), or by a higher order polynomial equation. The solutions on D_I in particular show that superintegrable systems are not necessarily degenerate. We can also show how the limiting cases of flat space (constant curvature zero) and the two-dimensional hyperboloid (constant negative curvature) emerge.

quant-ph↗
arXiv2002

Wigner functions for curved spaces I: On hyperboloids

We propose a Wigner quasiprobability distribution function for Hamiltonian systems in spaces of constant curvature --in this paper on hyperboloids--, which returns the correct marginals and has the covariance of the Shapiro functions under SO(D,1) transformations. To the free systems obeying the Laplace-Beltrami equation on the hyperboloid, we add a conic-oscillator potential in the hyperbolic coordinate. As an example, we analyze the 1-dimensional case on a hyperbola branch, where this conic-oscillator is the Poschl-Teller potential. We present the analytical solutions and plot the computed results. The standard theory of quantum oscillators is regained in the contraction limit to the space of zero curvature.

quant-ph↗