  The quantum-mechanical problems of electron scattering by an infinitely thin
solenoid and by a half of an infinitely thin solenoid are examined from the
viewpoint of constructing a self-adjoint Hamiltonian. It is demonstrated that
in both problems there exist unique self-adjoint operators with a
``non-singular'' domain, that, due to physical reasons, are identified with the
corresponding Hamiltonians. In the case of quantized values of magnetic flow
along the solenoid, the electron does not experience any scattering by the
string. It is shown that the scattering amplitude and wave function of an
electron in the problem of scattering by an infinitely long solenoid of radius
a in the limit a->0 turn into the corresponding expressions for the problem of
an infinitely thin solenoid. In particular, at a quantized value of magnetic
flow along the solenoid, scattering disappears at a->0.
