  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.
