We study the Aharonov–Bohm effect from the point of view of time-dependent inverse scattering theory. As this three-dimensional problem is invariant under translations along the vertical axis, it reduces to a problem in ². We first consider an unshielded magnetic field that has a singular part produced by a tiny solenoid and a regular part. The wavefunction is zero at the location of the solenoid. We then consider the case where the singular part of the magnetic field is shielded inside a cylinder whose transverse section is a compact set K, and there is also a regular magnetic field. In this case the magnetic field inside K is quite general. In fact, the only condition is that the magnetic flux across K has to be finite. Moreover, the wavefunction is defined in Ω : = ² K and it is zero on ∂K.

Assuming that K is convex, we prove that in the unshielded case the scattering operator determines uniquely the regular magnetic field and that in the shielded case it determines uniquely the magnetic field in Ω. Moreover, in the unshielded case the scattering operator determines the magnetic flux of the solenoid modulo 2 and in the shielded case it determines the magnetic flux across K modulo 2. Our results follow from a reconstruction formula with an error term.