The Dirac Hamiltonian with a superstrong Coulomb field

We consider the quantum mechanical problem of a relativistic Dirac particle moving in the Coulomb field of a point charge $Ze$. It is often declared in the literature that a quantum mechanical description of such a system does not exist for charge values exceeding the so-called critical charge with $Z=\alpha^{-1}=137$ because 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 charge value. Furthermore, 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 nonuniqueness of the self-adjoint Hamiltonian, but this nonuniqueness is also characteristic for charge values less than critical $($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. We use 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 crucially important is an open question.