A new method for solving the Z>137 problem for energy levels of hydrogen-like atoms

The "catastrophe" in solving the Dirac equation for an electron in the field of a point electric charge, which emerges for the charge numbers Z > 137, is removed in this work by effective accounting of finite dimensions of nuclei. For this purpose, in numerical solutions of equations for Dirac radial wave functions, we introduce a boundary condition at the nucleus boundary such that the components of the electron current density is zero. As a result, for all nuclei of the periodic table the calculated energy levels practically coincide with the energy levels in standard solutions of the Dirac equation in the external field of the Coulomb potential of a point charge. Further, for Z > 105, the calculated energy level functions E(Z) are monotone and smooth.The lower energy level reaches the energy E=-mc^{2} (the electron "drop" on a nuclei) at Zc = 178. The proposed method of accounting of the finite size of nuclei can be easily used in numerical calculations of energy levels of many-electron atoms.