Electron in the Aharonov–Bohm potential and in the Coulomb field in $2{+}1$ dimensions

We obtain exact solutions of the Dirac equation in $2{+}1$ dimensions and the electron energy spectrum in the superposition of the Aharonov–Bohm and Coulomb potentials, which are used to study the Aharonov–Bohm effect for states with continuous and discrete energy spectra. We represent the total scattering amplitude as the sum of amplitudes of scattering by the Aharonov–Bohm and Coulomb potentials. We show that the gauge-invariant phase of the wave function or the energy of the electron bound state can be observed. We obtain a formula for the scattering cross section of spin-polarized electrons scattered by the Aharonov–Bohm potential. We discuss the problem of the appearance of a bound state if the interaction between the electron spin and the magnetic field is taken into account in the form of the two-dimensional Dirac delta function.