Fermion bound states in the Aharonov–Bohm field in $2+1$ dimensions

We find exact solutions of the Dirac equation that describe fermion bound states in the Aharonov–Bohm potential in $2+1$ dimensions with the particle spin taken into account. For this, we construct self-adjoint extensions of the Hamiltonian of the Dirac equation in the Aharonov–Bohm potential in $2+1$ dimensions. The self-adjoint extensions depend on a single parameter. We select the range of this parameter in which quantum fermion states are bound. We demonstrate that the energy levels of particles and antiparticles intersect. Because solutions of the Dirac equation in the Aharonov–Bohm potential in $2+1$ dimensions describe the behavior of relativistic fermions in the field of the cosmic string in $3+1$ dimensions, our results can presumably be used to describe fermions in the cosmic string field.