Parabolic Coordinates and the Hydrogen Atom in Spaces H3 and S3.

Abstract

The Coulomb problem for the Schrödinger equation is examined in spaces of constant curvature, Lobachevsky H3 and Riemann S3 models, on the base of generalized parabolic coordinates. Opposite to the hyperbolic case, in spherical space S3 such parabolic coordinates turn to be complex-valued, with additional constraint imposed on them. The technique of the use of such real and complex coordinates in two space models within the method of separation of variables in the Schrödinger equation with Coulomb potential is developed in detail. The energy spectra and corresponding wave functions for bound states have been constructed in an explicit form for both spaces; connections with Runge-Lenz operators in both curved space models are described.