Nonlinear Phenomena in Complex Systems, 2021, Vol.24, No.3, pp.260

NONLINEAR PHENOMENA IN COMPLEX SYSTEMS
An Interdisciplinary Journal

2021, Vol.24, No.3, pp.260 - 271

Dirac Particle in the Coulomb Field on the Background of Hyperbolic Lobachevsky Model

E. M. Ovsiyuk, A. D. Koral'kov, A. V. Chichurin, and V. M. Red'kov

The known systems of radial equations describing the relativistic hydrogen atom on the base of the Dirac equation in Lobachevsky hyperbolic space is solved. The relevant 2-nd order differential equation has six regular singular points, its solutions of Frobenius type are constructed explicitly. To produce the quantization rule for energy values we have used the known condition for determination of the transcendental Frobenius solutions. This defines the energy spectrum which is physically interpretable and similar to the spectrum arising for the scalar Klein-Fock-Gordon equation in Lobachevsky space. In the present paper, exact analytical solutions referring to this spectrum are constructed. Convergence of the series involved is proved analytically and numerically. Squared integrability of the solutions is demonstrated numerically. It is shown that the spectrum coincides precisely with that previously found within the semi-classical approximation.

Key words: Dirac particle, Coulomb field, Lobachevsky space, Frobenius solutions, convergence of the series, transcendency conditions, energy spectrum, numerical study, squared integrability

DOI: https://doi.org/10.33581/1561-4085-2021-24-3-260-271

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