Nonperturbative effects caused by the radiative component of the electron magnetic moment in hydrogen-like atoms

The lower levels of the discrete spectrum of a hydrogen-like atom are calculated within the point-like nucleus approximation with nonperturbative consideration for the Schwinger interaction of the radiative component of the magnetic moment of a free electron with the Coulomb field of a nucleus. The behavior of the 1s $1s_{1/2}$, $2s_{1/2}$, $2p_{1/2}$ and $2p_{3/2}$ levels is investigated depending on the nuclear charge values, including the range of ${Z>137}$, where the Dirac Hamiltonian continues to be self adjoint in the presence of the Schwinger term. It is shown that the Schwinger interaction for large $Z$ causes significant changes in the properties of the discrete spectrum; in particular, the first level that reaches the threshold of a negative continuum is $2p_{1/2}$ and this occurs at ${Z=147}$. The behavior of the $g$-factor of an electron for the $1s_{1/2}$ and $2p_{1/2}$ states as a function of $Z$ is considered as well and it is shown that for extremely large charges the correction to the $g$-factor due to the Schwinger term becomes a very significant effect.