New method for solving the `$\bf Z > 137$' problem and determining hydrogen-like energy levels

A new method for including finite nuclear size effects is suggested to overcome the `$Z > 137$ catastrophe' encountered in solving the Dirac equation for an electron in the field of a point charge $Ze$. In this method, the boundary condition for the numerical solution of the equations for the Dirac radial wave functions is taken so that the components of the electron current density are zero at the boundary of the nucleus. As a result, for all of the nuclei of the Periodic Table the calculated energy levels practically coincide with those obtained in a standard way from the Dirac equation for a Coulomb pointlike charge potential. For $Z > 105$, the calculated energy level functions $E(Z)$ prove to be smooth and monotonic. The ground energy level reaches $E = -mc^2$ (i.e., the electron drops onto the nucleus) at $Z_c = 178$. The proposed method of accounting for the finite size of nuclei can be useful in numerically simulating the energy levels of many-electron atoms.