Semiclassical Analysis for Magnetic Scattering by Two Solenoidal Fields: Total Cross Sections

The fact that vector potentials have a direct significance to quantum particles moving in magnetic fields is known as the Aharonov–Bohm effect (A–B effect). We study this quantum effect through the semiclassical analysis on total cross sections in the magnetic scattering by two solenoidal (point-like) fields with total flux vanishing in two dimensions. We derive the asymptotic formula with first three terms. The system with two parallel fields seems to be important in practical aspects as well as in theoretical aspects, because it may be thought of a toroidal solenoid with zero cross section in three dimensions under the idealization that the two fields connect at infinity in their direction. The corresponding classical mechanical system has the trajectory oscillating between two centers of fields. The special emphasis is placed on analyzing how the trapping effect from classical mechanics is related to the A–B quantum effect in the semiclassical asymptotic formula.