Quadratic Zeeman effect in hydrogen Rydberg states: Rigorous bound-state error estimates in the weak-field regime

Applying a method based on some results due to Kato [Proc. Phys. Soc. Jpn. 4, 334 (1949)], we show that series of Rydberg eigenvalues and Rydberg eigenfunctions of hydrogen in a uniform magnetic field can be calculated with a rigorous error estimate. The efficiency of the method decreases as the eigenvalue density increases and as γ${\mathit{n}}^3$→1, where γ is the magnetic-field strength in units of 2.35×${10}^9$ G and n is the principal quantum number of the unperturbed hydrogenic manifold from which the diamagnetic Rydberg states evolve. Fixing γ at the laboratory value 2×${10}^{\mathrm{-}5}$ and confining our calculations to the region γ${\mathit{n}}^3$<1 (weak-field regime), we obtain extremely accurate results up to states corresponding to the n=32 manifold.