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High-Velocity Estimates for the Scattering Operator and Aharonov-Bohm Effect in Three Dimensions

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Abstract

We obtain high-velocity estimates with error bounds for the scattering operator of the Schrödinger 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 time-dependent methods. We consider high-velocity 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 conclusive evidence of the existence of the Aharonov-Bohm effect using a toroidal magnet. We give a method for the reconstruction of the flux of the magnetic field over a cross-section of the torus modulo 2π. Equivalently, we determine modulo 2π 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 high-velocity 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 high-velocity limit of the incoming electrons. For some locations of the tori we can determine all the fluxes modulo 2π by taking the high-velocity 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 high-velocity limit of the scattering operator.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, Gustaf Hallstromin katu 2b, Helsinki, FI-00014, Finland

    Miguel Ballesteros & Ricardo Weder

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  1. Miguel Ballesteros
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  2. Ricardo Weder
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Correspondence to Ricardo Weder.

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Communicated by I.M. Sigal

Research partially supported by CONACYT under Project P42553­F.

On leave of absence from Departamento de Métodos Matemáticos y Numéricos. Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas. Universidad Nacional Autónoma de México. Apartado Postal 20-726, México DF 01000. Ricardo Weder is a Fellow of the Sistema Nacional de Investigadores.

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Ballesteros, M., Weder, R. High-Velocity Estimates for the Scattering Operator and Aharonov-Bohm Effect in Three Dimensions. Commun. Math. Phys. 285, 345–398 (2009). https://doi.org/10.1007/s00220-008-0579-1

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  • DOI: https://doi.org/10.1007/s00220-008-0579-1

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