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Free and bound spin-polarized fermions in the fields of Aharonov--Bohm kind

The scattering of electrons by an Aharonov--Bohm field is considered from the viewpoint of quantum-mechanical problem of constructing a self-adjoint Hamiltonian for the Pauli equation. The correct domain for the self-adjoint Hamiltonian, which takes into account explicitly the electron spin is found. A one-parameter self-adjoint extension of the Hamiltonian for spin-polarized electrons in the Aharonov--Bohm field is selected. The correct domain of the self-adjoint Hamiltonian can contain regular and singular (at the point ${\bf r}=0$) square-integrable functions on the half-line with measure $rdr$. We argue that the physical reason of the existence of singular functions is the additional attractive potential, which appear due to the interaction between the spin magnetic moment of fermion and Aharonov--Bohm magnetic field. The scattering amplitude and cross section are obtained for spin-polarized electrons scattered by the Aharonov--Bohm field. It is shown that in some range of the extension parameter there appears a bound state. Since the Hamiltonian of the nonrelativistic Dirac--Pauli equation for a massive neutral fermion with the anomalous magnetic moment (AMM) in the electric field of a linear charge aligned perpendicularly to the fermion motion has the form of the Hamiltonian for the Pauli equation in the Aharonov--Bohm flux tube, we also calculate the scattering amplitude and cross section for the neutral fermion.