Fermions in strong external fields in $2+1$ and $1+1$ dimensions

The creation of electron-positron pairs from a vacuum by an external Coulomb field is examined within $(2+1)$-dimensional quantum electrodynamics. If the electromagnetic coupling constant exceeds $0.62$ $(Z=85)$, then in a simple model with a finite-size nucleus, the lower electron level crosses the boundary of the negative-energy continuum (i.e., Dirac sea), and a hole (i.e., positively charged fermion) appears in the negative-energy continuum. An equation is obtained that describes the levels of the ground and excited electron states in a strong Coulomb field of the nucleus. The critical nucleus charge is found for a few lowest electron states. The critical charge in $2+1$ dimensions is significantly smaller than in $3+1$ dimensions. The problem is reduced to the case of a bounded Coulomb field in $1+1$ dimensions without a magnetic field. The interaction of a fermion and an external scalar field in $2+1$ and $1+1$ dimensions is investigated.