Quadratic Zeeman effect in hydrogen Rydberg states: Rigorous error estimates for energy eigenvalues, energy eigenfunctions, and oscillator strengths

A variational method, based on some results due to T. Kato [Proc. Phys. Soc. Jpn. 4, 334 (1949)], and previously discussed is here applied to the hydrogen atom in uniform magnetic fields of tesla in order to calculate, with a rigorous error estimate, energy eigenvalues, energy eigenfunctions, and oscillator strengths relative to Rydberg states up to just below the field-free ionization threshold. Making use of a basis (parabolic Sturmian basis) with a size varying from 990 up to 5050, we obtain, over the energy range of -190 to -24 ${\mathrm{cm}}^{\mathrm{-}1}$, all of the eigenvalues and a good part of the oscillator strengths with a remarkable accuracy. This, however, decreases with increasing excitation energy and, thus, above ∼-24 ${\mathrm{cm}}^{\mathrm{-}1}$, we obtain results of good accuracy only for eigenvalues ranging up to ∼-12 ${\mathrm{cm}}^{\mathrm{-}1}$.