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The Coulomb problem for Schr\"{o}dinger equation is examined, in spaces ofconstant curvature, Lobachevsky H_{3} and Riemann S_{3} models, on the base ofgeneralized parabolic coordinates. In contrast to the hyperbolic case, inspherical space S_{3} such parabolic coordinates turn to be complex-valued,with additional constraint on them. The technique of the use of such real andcomplex coordinates in two space models within the method of separation ofvariables in Schr\"{o}dinger equation with Kepler potential is developed indetail; the energy spectra and corresponding wave functions for bound stateshave been constructed in explicit form for both spaces; connections withRunge-Lenz operators in both curved space models are described.

Parabolic Coordinates and the Hydrogen Atom in Spaces H_{3} and S_{3}