Perturbativity and nonperturbativity in large-$Z$ effects for hydrogen-like atoms

The effects induced by the interaction $\Delta U_{AMM}$ of the anomalous component of the electron magnetic moment with the Coulomb field of a nucleus at Zα > 1 are examined for the lower levels of a hydrogenlike atom within purely perturbative and essentially nonperturbative in α/π approaches. Owing to the specific features of the $\Delta U_{AMM}$ operator for a point-like Coulomb source, the distribution of the charge over the volume of the nucleus is set in the nonperturbative case at the quark structure level using valence (constituent) u and d quarks. It is demonstrated that such nonperturbative analysis turns out to be nontrivial as well as the manner in which its results are altered in the transition from the central problem with spherical symmetry to the inclusion of the peripheral contribution. It is also demonstrated that these methods yield matching (with an accuracy no worse than 0.1%) results for the shift of levels$1s_{1/2}$ and $2p_{1/2}$ due to $\Delta U_{AMM}$ at all values of Z through to critical ones. Thus, it is confirmed that the perturbation theory in α/π provides correct results for superheavy atoms if the complete dependence of the wave function on Zα is taken into account from the beginning.