### Abstract

The seminal paper of Aharonov and Bohm [Phys. Rev.115, 485 (1959)]10.1103/PhysRev.115.485 is at the origin of a very extensive literature in some of the more fundamental issues in physics. They claimed that electromagnetic fields can *act at a distance* on charged particles even if they are identically zero in the region of space where the particles propagate, that the fundamental electromagnetic quantities in quantum physics are not only the electromagnetic fields but also the circulations of the electromagnetic potentials; what gives them a real physical significance. They proposed two experiments to verify their theoretical conclusions. The magnetic Aharonov-Bohm effect, where an electron is influenced by a magnetic field that is zero in the region of space accessible to the electron, and the electric Aharonov-Bohm effect where an electron is affected by a time-dependent electric potential that is constant in the region where the electron is propagating, i.e., such that the electric field vanishes along its trajectory. The Aharonov-Bohm effects imply such a strong departure from the physical intuition coming from classical physics that it is no wonder that they remain a highly controversial issue after more than fifty years, in spite of the fact that they are discussed in most of the text books in quantum mechanics. The magnetic case has been studied extensively. The experimental issues were settled by the remarkable experiments of Tonomura *et al.* [Phys. Rev. Lett.48, 1443 (1982); Phys. Rev. Lett.56, 792 (1986)] with toroidal magnets, that gave a strong evidence of the existence of the effect, and by the recent experiment of Caprez *et al.* [Phys. Rev. Lett.99, 210401 (2007)]10.1103/PhysRevLett.99.210401 that shows that the results of the Tonomura *et al.* experiments cannot be explained by the action of a force. The theoretical issues were settled by Ballesteros and Weder [Commun. Math. Phys.285, 345 (2009)10.1007/s00220-008-0579-1; J. Math. Phys.50, 122108 (2009)10.1063/1.3266176; Commun. Math. Phys.303, 175 (2011)]10.1007/s00220-010-1166-9 who rigorously proved that quantum mechanics predicts the experimental results of Tonomura *et al.* and of Caprez *et al.* The electric Aharonov-Bohm effect has been much less studied. Actually, its existence, that has not been confirmed experimentally, is a very controversial issue. In their 1959 paper Aharonov and Bohm proposed an ansatz for the solution to the Schrödinger equation in regions where there is a time-dependent electric potential that is constant in space. It consists in multiplying the free evolution by a phase given by the integral in time of the potential. The validity of this ansatz predicts interference fringes between parts of a coherent electron beam that are subjected to different potentials. In this paper we prove that the exact solution to the Schrödinger 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 the velocity. Our results give, for the first time, a rigorous proof that quantum mechanics predicts the existence of the electric Aharonov-Bohm effect, under conditions that we provide. We hope that our results will stimulate the experimental research on the electric Aharonov-Bohm effect.

Received 23 November 2010
Accepted 25 April 2011
Published online 26 May 2011

Acknowledgments:
This research is partially supported by CONACYT under Project No. CB-2008-01-99100.

Article outline:

I. INTRODUCTION
II. PRELIMINARY RESULTS
A. The tube *K*
B. The Electric potential
C. The unitary propagator
D. Propagation estimates
E. The wave and scattering operators
III. HIGH-VELOCITY ESTIMATES
A. High-velocity solutions to the Schrödinger equation
B. The Aharonov-Bohm ansatz
C. Uniform estimates for the exact solution to the Schrödinger equation
D. High-velocity estimates of the wave and the scattering operators

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