Publications

2009

• Exact boundary conditions for time-harmonic wave propagation in locally perturbed periodic media
  
  • Fliss Sonia
  • Joly Patrick
  Applied Numerical Mathematics: an IMACS journal, Elsevier, 2009, 59 (9), pp.2155-2178. We consider the solution of the Helmholtz equation with absorption − u(x)−n(x)2(ω2 + ıε)u(x) = f (x), x = (x, y), in a 2D periodic medium Ω = R2. We assume that f (x) is supported in a bounded domain Ωi and that n(x) is periodic in the two directions in Ωe = Ω \ Ωi . We show how to obtain exact boundary conditions on the boundary of Ωi ,ΣS that will enable us to find the solution on Ωi . Then the solution can be extended in Ω in a straightforward manner from the values on ΣS . The particular case of medium with symmetries is exposed. The exact boundary conditions are found by solving a family of waveguide problems. © 2008 IMACS. (10.1016/j.apnum.2008.12.013)
  DOI : 10.1016/j.apnum.2008.12.013
• Numerical analysis of the generalized Maxwell equations (with an elliptic correction) for charged particle simulations
  
  • Ciarlet Patrick
  • Labrunie Simon
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2009, 19 (11), pp.1959-1994. When computing numerical solutions to the Vlasov--Maxwell equations, the source terms in Maxwell's equations usually fail to satisfy the continuity equation. Since this condition is required for the well-posedness of Maxwell's equations, it is necessary to introduce generalized Maxwell's equations which remain well-posed when there are errors in the sources. These approaches, which involve a hyperbolic, a parabolic and an elliptic correction, have been recently analyzed mathematically. The goal of this paper is to carry out the numerical analysis for several variants of Maxwell's equations with an elliptic correction. (10.1142/S0218202509004017)
  DOI : 10.1142/S0218202509004017
• Space-time mesh refinement for discontunuous Galerkin methods for symmetric hyperbolic systems
  
  • Ezziani Abdelaâziz
  • Joly Patrick
  Journal of Computational and Applied Mathematics, Elsevier, 2009, 234 (6), pp.1886-1895. We present a new non-conforming space-time mesh refinement method for the symmetric first order hyperbolic system. This method is based on the one hand on the use of a conservative higher order discontinuous Galerkin approximation for space discretization and a finite difference scheme in time, on the other hand on appropriate discrete transmission conditions between the grids. We use a discrete energy technique to drive the construction of the matching procedure between the grids and guarantee the stability of the method. (10.1016/j.cam.2009.08.094)
  DOI : 10.1016/j.cam.2009.08.094
• High-velocity estimates for the scattering operator and Aharanov-Bohm effect in tree dimensions
  
  • Ballesteros Miguel
  • Weder Ricardo
  Communications in Mathematical Physics, Springer Verlag, 2009, 285 (1), pp.345-398. We obtain high-velocity estimates with error bounds for the scattering operator of the Schr�dinger 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.
• The Aharonov-Bohm effect and Tonomura experiments: Rigorous results
  
  • Ballesteros Miguel
  • Weder Ricardo
  Journal of Mathematical Physics, American Institute of Physics (AIP), 2009, 50 (12), pp.122108. The Aharonov-Bohm effect is a fundamental issue in physics. It describes the physically important electromagnetic quantities in quantum mechanics. Its experimental verification constitutes a test of the theory of quantum mechanics itself. The remarkable experiments of Tonomura ["Observation of Aharonov-Bohm effect by electron holography," Phys. Rev. Lett 48, 1443 (1982) and "Evidence for Aharonov-Bohm effect with magnetic field completely shielded from electron wave," Phys. Rev. Lett 56, 792 (1986)] are widely considered as the only experimental evidence of the physical existence of the Aharonov-Bohm effect. Here we give the first rigorous proof that the classical ansatz of Aharonov and Bohm of 1959 ["Significance of electromagnetic potentials in the quantum theory," Phys. Rev. 115, 485 (1959)], that was tested by Tonomura, is a good approximation to the exact solution to the Schrödinger equation. This also proves that the electron, that is, represented by the exact solution, is not accelerated, in agreement with the recent experiment of Caprez in 2007 ["Macroscopic test of the Aharonov-Bohm effect," Phys. Rev. Lett. 99, 210401 (2007)], that shows that the results of the Tonomura experiments can not be explained by the action of a force. Under the assumption that the incoming free electron is a Gaussian wave packet, we estimate the exact solution to the Schrödinger equation for all times. We provide a rigorous, quantitative error bound for the difference in norm between the exact solution and the Aharonov-Bohm Ansatz. Our bound is uniform in time. We also prove that on the Gaussian asymptotic state the scattering operator is given by a constant phase shift, up to a quantitative error bound that we provide. Our results show that for intermediate size electron wave packets, smaller than the ones used in the Tonomura experiments, quantum mechanics predicts the results observed by Tonomura with an error bound smaller than 10-99. It would be quite interesting to perform experiments with electron wave packets of intermediate size. Furthermore, we provide a physical interpretation of our error bound. © 2009 American Institute of Physics. (10.1063/1.3266176)
  DOI : 10.1063/1.3266176