Fermion pair creation by a strong Coulomb field in $2+1$ dimensions

The creation of charged fermion pairs by a strong external Coulomb field in a space with two dimensions is investigated. Exact solutions to the Dirac equation are found for the Coulomb external field in $2+1$ dimensions. The equation for determining the critical charge is obtained and is numerically solved for a simplified model. The critical charge for $2+1$ dimensions is much less than the critical charge for the similar model with $3+1$ dimensions. The influence of the vacuum polarization on the critical charge is studied in the one-loop approximation to the $(2+1)$-dimensional quantum electrodynamics.