Quantum Kepler problem for spin 1/2 particle in spaces on constant curvature. I. Pauli theory

TL;DR

This paper derives exact solutions and energy spectra for a spin-1/2 particle in curved spaces with Coulomb interaction, extending the Kepler problem to spaces of constant positive and negative curvature.

Contribution

It presents a nonrelativistic Pauli equation in curved spaces and constructs exact solutions for the quantum Kepler problem in both spherical and hyperbolic geometries.

Findings

Exact solutions in terms of Hein functions for both spaces.

Derived energy spectra for the Coulomb problem in curved geometries.

Extended the classical Kepler problem to non-Euclidean spaces.

Abstract

Transition to a nonrelativistic Pauli equation in Riemann space of constant positive curvature for a Dirac particle in presence of the Coulomb field is performed in the system of radial equations, exact solutions are constructed in terms of Hein functions, the energy spectrum is derived. The same is done for the Kepler quantum problem in hyperbolic Lobachevsky space, solutions are constructed in terms of Hein functions, the energy spectrum is obtained.