  Creation of charged fermion pair from a vacuum in the so-called supercritical
Coulomb potential is examined for the case when created pair moves in one
plane. In which case the quantum dynamics of charged massive or massless
fermions can be described by the two-dimensional Dirac Hamiltonians with a
Coulomb potential. These Hamiltonians are singular and require the additional
definition in order for them to be treated as self-adjoint quantum-mechanical
operators. We construct the self-adjoint two-dimensional Dirac Hamiltonians
with a Coulomb potential and determine the quantum-mechanical states for such
Hamiltonians in the corresponding Hilbert spaces of square-integrable
functions. We determine the scattering amplitude in which the self-adjoint
extension parameter is incorporated and then obtain the equations implicitly
defining the possible discrete energy spectra of the self-adjoint Dirac
Hamiltonians with a Coulomb potential. It is shown that the quantum system in
the presence of a supercritical Coulomb potential becomes unstable which
manifests in the appearance of quasi-stationary states in the lower (negative)
energy continuum. The energy spectrum of these states is quasi-discrete,
consists of broadened levels whose width is related to the inverse lifetime of
the quasi-stationary state as well as the creation probability of charged
fermion pair by supercritical Coulomb field. Explicit analytical expressions
for the creation probabilities of charged (massive or massless) fermion pair
are obtained in a supercritical Coulomb field.
