  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
$v^\rho, 0 < \rho <1$, 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.
