HAL CCSD
High-velocity estimates for the scattering operator and Aharanov-Bohm effect in tree dimensions
Ballesteros, Miguel
Weder, Ricardo
Propagation des Ondes : Étude Mathématique et Simulation (POEMS) ; Inria Saclay - Ile de France ; Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Unité de Mathématiques Appliquées (UMA) ; École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-École Nationale Supérieure de Techniques Avancées (ENSTA Paris)-Centre National de la Recherche Scientifique (CNRS)
International audience
ISSN: 0010-3616
EISSN: 1432-0916
Communications in Mathematical Physics
Springer Verlag
hal-00976023
https://ensta-paris.hal.science/hal-00976023
https://ensta-paris.hal.science/hal-00976023
Communications in Mathematical Physics, 2009, 285 (1), pp.345-398
en
info:eu-repo/semantics/article
Journal articles
We obtain high-velocity estimates with error bounds for the scattering operator of the Schrï¿½dinger 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 2p. Equivalently, we determine modulo 2p 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 2p 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.
2009-01