Fermions in scalar Coulomb and Aharonov-Bohm potentials in 2+1 dimensions

The quantum-mechanical problem of constructing the self-adjoint Hamiltonians is physically rigorously solved for a Dirac Hamiltonian with a Coulomb scalar potential and an Aharonov-Bohm potential in 2+1 dimensions by taking into account a fermion spin. It is found that the Dirac Hamiltonian on this background requires the additional specification of a one-parameter self-adjoint extension, which can be given in terms of the physically acceptable boundary conditions. We derive equations that determine the spectra of the self-adjoint radial Dirac Hamiltonians for various parameter values. We discuss the role of a particle spin as the physical reason of the existence of bound fermion states in a pure Aharonov-Bohm potential and show that the particle and antiparticle states with zero energy exist only owing to the interaction of the fermion spin magnetic moment with the magnetic field. The energy levels of particles and antiparticles are intersected what may signal on the instability of a quantum system.