Modern high-precision experiments with highly charged ions require equivalent accuracy intheoretical calculations within bound-state quantum electrodynamics (QED). Measurements of the $g$-factor in hydrogen-like ions demonstrate record accuracy up to 10^-11 [1], necessitating the account of quantum-electrodynamic corrections to atomic characteristics.

Among QED corrections, the self-energy contribution plays a key role, being critical for calculating energy levels, the g-factor, hyperfine splitting, and the quadratic Zeeman effect. Theoretical calculations for highly charged ions must consider all orders of the electron–nucleus coupling parameter, with the main uncertainty arising from the slow convergence of partial-wave expansions [2, 3]. Green's function of an electron in a central potential, computed through solutions of systems of differential equations, serves as an effective tool for high-precision calculations [4, 5].

In this work, Green's function method is applied to analyze single and double interactions with an external field. General expressions for radial Green functions with potentials that change angular quantum numbers have been obtained. The developed approach allows for the calculation of one- and two-potential self-energy corrections. The method is applied to hyperfine structure calculations, showing its high accuracy compared to alternative computational methods. The quadratic Zeeman effect in hydrogen-like ions is also calculated.

This research was supported by the Theoretical Physics and Mathematics Advancement Foundation ``BASIS'' (Grant No. 24-1-2-74-3).

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[2] V. A. Yerokhin et al., Phys. Rev. A 60, 3522 (1999).
[3] V. A. Yerokhin et al., Phys. Rev. A 69, 052503 (2004).
[4] P. Mohr et al., Phys. Rep. 293, 227 (1998).
[5] V. A. Yerokhin, A. V. Maiorova, Symmetry 12, 800 (2020).