Peculiarities of electron energy spectrum in Coulomb field of super heavy nucleus

Abstract: Just after the Dirac equation was established, a number of physicists tried to comment on and solve the spectral problem for the Dirac Hamiltonian with the Coulomb field of arbitrarily large charge $Z$, especially with $Z$ that is more than the critical value $Z_{\mathrm{c}}=\alpha^{-1}\simeq137,04$, making sometimes contradictory conclusions and presenting doubtful solutions. It seems that there is no consesus on this problem up until now and especially on the way of using corresponding solutions of the Dirac equation in calculating physical processes. That is why in the present article, we turn once again to discussing peculiarities of electron energy spectrum in the Coulomb field of superheavy nucleus. In the beginning, we remind the reader of a long story with a wrong interpretation of the problem in the case of a point nucleus and its present correct solution. We then turn to the spectral problem in the case of a regularized Coulomb field. Under a specific regularization, we derive an exact spectrum equation determining the point spectrum in the energy interval $(-m,m)$ and present some of its numerical solutions. We also derive an exact equation for charges $Z$ providing bound states with energy $E=-m$. Its analytical and numerical analysis shows that there exists an infinite number of such charges; in this connection , we discuss the notion of supercritical charge.