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Aharonov-Bohm Effect and High-Velocity Estimates of Solutions to the Schrödinger Equation

The Aharonov-Bohm effect is a fundamental and controversial issue in physics. At stake are what are the fundamental electromagnetic quantities in quantum physics, if magnetic fields can act at a distance on charged particles and if the magnetic potentials have a real physical significance. From the experimental side the issues were settled by the remarkable experiments of Tonomura et al. in 1982 and 1986 with toroidal magnets that gave a strong experimental evidence of the physical existence of the Aharonov-Bohm effect, and by the recent experiment of Caprez et al. in 2007 that shows that the results of these experiments can not be explained by a force. The Aharonov-Bohm Ansatz of 1959 predicts the results of the experiments of Tonomura et al. and of Caprez et al. In 2009 we gave the first rigorous proof that the Aharonov-Bohm Ansatz is a good approximation to the exact solution for toroidal magnets under the conditions of the experiments of Tonomura et al.. In this paper we prove that our results do not depend on the particular geometry of the magnets and on the velocities of the incoming electrons used on the experiments, and on the gaussian shape of the wave packets used to obtain our quantitative error bound. We consider a general class of magnets that are a finite union of handle bodies. We formulate the Aharonov-bohm Ansatz that is appropriate to this general case and we prove that the exact solution to the Schroedinger equation is given by the Aharonov-Bohm Ansatz up to an error bound in norm that is uniform in time and that decays as a constant divided by $v^ρ, 0 < ρ<1$, with $v$ the velocity. The results of Tonomura et al., of Caprez et al., our previous results and the results of this paper give a firm experimental and theoretical basis to the existence of the Aharonov-Bohm effect and to its quantum nature.