G. S. Pogosyan

Finite two-dimensional oscillator: I. The Cartesian model

A finite two-dimensional oscillator is built as the direct product of two finite one-dimensional oscillators, using the dynamical Lie algebra su(2)x⊕su(2)y. The position space in this model is a square grid of points. While the ordinary `continuous two-dimensional quantum oscillator has a symmetry algebra u(2), the symmetry algebra of the finite model is only u(1)x⊕u(1)y, because it lacks rotations in the position (and momentum) plane. We show how to `import an SO(2) group of rotations from the continuum model that transforms unitarily the finite wavefunctions on the fixed square grid. We thus propose a finite analogue for fractional U(2) Fourier transforms.

Generalized KS transformation: from five-dimensional hydrogen atom to eight-dimensional isotropic oscillator

It is shown that non-bijective quadratic transformation generated by the Kelly matrix changes the problem of a five-dimensional hydrogen atom into the problem of an eight-dimensional isotropic oscillator.

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→∞.

HIDDEN SYMMETRY, SEPARATION OF VARIABLES AND INTERBASIS EXPANSIONS IN THE TWO-DIMENSIONAL HYDROGEN ATOM

Expansions for each fundamental basis of the hydrogen atom over two others are found and an additional integral of motion corresponding to an elliptic basis is determined. Rcpresentations of the elliptic basis as a superposition of polar and parabolic states are obtained. Certain interesting limiting cases are investigated.

On the hidden symmetry of a one-dimensional hydrogen atom

The Fock method is applied to the problem of a one-dimensional hydrogen atom. Integral Fock equations are obtained in discrete and continuous spectra; the case of zero energy is studied and wavefunctions and normalisation constants are calculated in the momentum representation.

Hydrogen atom as indicator of hidden symmetry of a ring-shaped potential

The idea of an analogy between a ring-shaped potential and a Coulomb potential is advanced. It is shown that the expansion of the parabolic basis with respect to the spherical basis in the problem of a ring-shaped potential is determined by the Clebsch-Gordan coefficients of the group SU(2) continued to the region of arbitrary real indices. The connection between these coefficients and the functions 3F2 is found, and it is shown that they have a symmetry property under substitution of the parabolic quantum numbers.

Non-relativistic Coulomb problem in a one-dimensional quantum mechanics

It is shown that, unlike the model based only on the potential singularity U(x)=- mod x mod -1, the group of hidden symmetry explains not only the double degeneration of the energy spectrum but also the explicit form of the spectrum, wavefunctions and an extra constant of motion analogous to the Runge-Lenz vector.

Elliptic basis for a circular oscillator

This paper presents a complete set of mutually commuting operators determining an elliptic basis for a quantum circular oscillator. The elliptic basis of the circular oscillator is introduced and three-term recursion relations generating it are found. The elliptic corrections to the polar and Cartesian bases of the circular oscillator are calculated.

Spheroidal corrections to the spherical and parabolic bases of the hydrogen atom

This paper introduces the bases of the hydrogen atom and obtains recursion relations that determine the expansion of the spheroidal basis with respect to its parabolic basis. The leading spheroidal corrections to the spherical and parabolic bases are calculated by perturbation theory.

Two-dimensional hydrogen atom. Reciprocal expansions of the polar and parabolic bases of the continuous spectrum

An expansion of the parabolic basis of the two-dimensional hydrogen atom with respect to the polar basis and its inverse expansion are found in the region of the continuous spectrum. The connection between these expansions and the expansions corresponding to them in the discrete spectrum is traced. The group-theoretical significance of the two-dimensional Coulomb phase shift is established.