Relativistic Ritz Approach to Hydrogen-like Atoms*

The Rydberg formula along with the Ritz quantum defect ansatz has been a standard theoretical tool used in atomic physics since before the advent of quantum mechanics, yet this approach has remained limited by its nonrelativistic foundation. Here I present a long-distance relativistic effective theory describing hydrogenlike systems with arbitrary mass ratios, thereby extending the canonical Ritz-like approach. Fitting the relativistic theory to the hydrogen energy levels predicted by bound-state QED indicates that it is superior to the canonical, nonrelativistic approach. An analytic analysis reveals nonlinear consistency relations within the bound-state QED level predictions that relate higher-order corrections to those at lower order, providing guideposts for future perturbative calculations as well as insights into the asymptotic behavior of the Bethe logarithm. Applications of the approach include fitting to atomic spectroscopic data, allowing for the determination of the Rydberg constant from large spectral data sets and also to check for internal consistency of the data -- all independently from bound-state QED.