  Solutions to the Dirac equation are constructed for a massless charged
fermion in Coulomb and Aharonov--Bohm potentials in 2+1 dimensions. The Dirac
Hamiltonian on this background is singular and needs a one-parameter
self-adjoint extension, which can be given in terms of self-adjoint boundary
conditions. We show that the virtual (quasistationary) bound states emerge in
the presence of an attractive Coulomb potential when the so-called effective
charges become overcritical and discuss a restructuring of the vacuum of the
quantum electrodynamics when the virtual bound states emerge. We derive
equations, which determine the energies and lifetimes of virtual bound states,
find solutions of obtained equations for some values of parameters as well as
analyze the local density of states as a function of energy in the presence of
Coulomb and Aharonov--Bohm potentials.
