Spontaneous fermion creation in the Coulomb field and Aharonov–Bohm potential in $2+1$ dimensions

We find exact solutions of the Dirac equation and the fermion energy spectrum in the Coulomb (vector and scalar) potential and Aharonov–Bohm potential in $2+1$ dimensions taking the particle spin into account. We describe the fermion spin using the two-component Dirac equation with the additional (spin) parameter introduced by Hagen. We consider the effect of creation of fermion pairs from the vacuum by a strong Coulomb field in the Aharonov–Bohm potential in $2+1$ dimensions. We obtain transcendental equations implicitly determining the electron energy spectrum near the boundary of the lower energy continuum and the critical charge. We numerically solve the equation for the critical charge. We show that for relatively weak magnetic flows, the critical charge decreases (compared with the case with no magnetic field) if the energy of interaction of the electron spin magnetic moment with the magnetic field is negative and increases if this energy is positive.