
Physics 2007
HighVelocity Estimates for the Scattering Operator and AharonovBohm Effect in Three DimensionsDOI: 10.1007/s0022000805791 Abstract: We obtain highvelocity estimates with error bounds for the scattering operator of the Schr\"odinger equation in three dimensions with electromagnetic potentials in the exterior of bounded obstacles that are handlebodies. A particular case is a finite number of tori. We prove our results with timedependent methods. We consider highvelocity estimates where the direction of the velocity of the incoming electrons is kept fixed as its absolute value goes to infinity. In the case of one torus our results give a rigorous proof that quantum mechanics predicts the interference patterns observed in the fundamental experiments of Tonomura et al. that gave a conclusive evidence of the existence of the AharonovBohm effect using a toroidal magnet. We give a method for the reconstruction of the flux of the magnetic field over a crosssection of the torus modulo $2\pi$. Equivalently, we determine modulo $2\pi$ the difference in phase for two electrons that travel to infinity, when one goes inside the hole and the other outside it. For this purpose we only need the highvelocity limit of the scattering operator for one direction of the velocity of the incoming electrons. When there are several tori or more generally handlebodies the information that we obtain in the fluxes, and on the difference of phases, depends on the relative position of the tori and on the direction of the velocities when we take the highvelocity limit of the incoming electrons. For some locations of the tori we can determine all the fluxes modulo 2$\pi$ by taking the highvelocity limit in only one direction. We also give a method for the unique reconstruction of the electric potential and the magnetic field outside the handlebodies from the highvelocity limit of the scattering operator.
