  We consider the quantum-mechanical problem of a relativistic Dirac particle
moving in the Coulomb field of a point charge $Ze$. In the literature, it is
often declared that a quantum-mechanical description of such a system does not
exist for charge values exceeding the so-called critical charge with $%
Z=\alpha ^{-1}=137$ based on the fact that the standard expression for the
lower bound state energy yields complex values at overcritical charges. We show
that from the mathematical standpoint, there is no problem in defining a
self-adjoint Hamiltonian for any value of charge. What is more, the transition
through the critical charge does not lead to any qualitative changes in the
mathematical description of the system. A specific feature of overcritical
charges is a non uniqueness of the self-adjoint Hamiltonian, but this non
uniqueness is also characteristic for charge values less than the critical one
(and larger than the subcritical charge with $Z=(\sqrt{3}% /2)\alpha
^{-1}=118$). We present the spectra and (generalized) eigenfunctions for all
self-adjoint Hamiltonians. The methods used are the methods of the theory of
self-adjoint extensions of symmetric operators and the Krein method of guiding
functionals. The relation of the constructed one-particle quantum mechanics to
the real physics of electrons in superstrong Coulomb fields where multiparticle
effects may be of crucial importance is an open question.
