On a generalized Kepler–Coulomb system: Interbasis expansions

Abstract

This paper deals with a dynamical system that generalizes the Kepler-Coulomb system and the Hartmann system. It is shown that the Schrödinger equation for this generalized KeplerCoulomb system can be separated in prolate spheroidal coordinates. The coefficients of the interbasis expansions between three bases (spherical, parabolic and spheroidal) are studied in detail. It is found that the coefficients for the expansion of the parabolic basis in terms of the spherical basis, and vice-versa, can be expressed through the Clebsch-Gordan coefficients for the group SU(2) analytically continued to real values of their arguments. The coefficients for the expansions of the spheroidal basis in terms of the spherical and parabolic bases are proved to satisfy three-term recursion relations.

Exactly complete solutions of the Schrödinger equation with a spherically harmonic oscillatory ring-shaped potential

“…Recently, the non-central potentials have been the subject of some studies in various fields since the advance of the Coulomb ringshaped potential and the oscillatory ring-shaped potential [1][2][3][4][5][6][7][8][9][10][11]. This is because the occurrence of 'accidental' degeneracy and 'hidden' symmetry in those non-central potentials and possible applications in quantum chemistry and nuclear physics, which may be used to describe ring-shaped molecules like benzene and the interactions between deformed pair of nuclei.…”

Superintegrability in three-dimensional Euclidean space

“…3,8, and 10 and investigated from different points of view in the last decade. [8][9][10][11][12][13] Note also that superintegrable potentials in spaces of constant curvature were introduced in Refs. 14-16. In previous articles [17][18][19] we have looked at potentials in two-dimensional Euclidean space and the two-dimensional sphere and hyperboloid, for which the Schrödinger equation is maximally superintegrable.…”

Solutions of the Schrödinger equation in the tridiagonal representation with the noncentral electric dipole plus a novel angle-dependent component

“…In most but not all cases, this recursion relation is solved easily by correspondence with those for well-known orthogonal polynomials. Moreover, the representation equation (5) clearly shows that the discrete spectrum is easily obtained by diagonalization which requires that b n = 0 and a n − z = 0,…”

“…In recent years, the noncentral potentials have been in the focus of some studies in various fields since the advance of the Coulomb ring-shaped potential and the oscillatory ring-shaped potential [2][3][4][5][6][7][8][9][10][11][12]. This partly due to the occurrence of 'accidental' degeneracy and 'hidden' symmetry in those noncentral potentials.…”