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Double-perturbation approach to the relativistic hydrogenic atom in a static and uniform magnetic field

Double-perturbation theory is developed for the case where relativity is one perturbation and the second perturbation describes the interaction with an external magnetic field. Relativity is treated according to the direct relativistic perturbation theory developed by Rutkowski (1986 Journal of Physics B 19 149, 3431, 3443) and Kutzelnigg (1989 Zeitschrift für Physik D 11 15; 1990 Zeitschrift für Physik D 15 27). For hydrogen-like atoms the pertinent perturbation parameters are and . The relevant radial perturbation equations and formulae for corrections to the energy are derived. For the ground state up to the first order in a magnetic field all the differential equations arising are solved directly to infinite order in relativity. The general solutions of the radial equations are obtained by expansion in appropriate Laguerre polynomials. Exact energy corrections for zeroth and first order as well as very accurate results for second and third order in relativity are presented up to 12 orders in a magnetic field.