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On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging

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Abstract

We analyse a nonlinear Schrödinger equation for the time-evolution of the wave function of an electron beam, interacting selfconsistently through a Hartree–Fock nonlinearity and through the repulsive Coulomb interaction of an atomic nucleus. The electrons are supposed to move under the action of a time dependent, rapidly periodically oscillating electromagnetic potential. This can be considered a simplified effective single particle model for an X-ray free electron laser. We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding solutions of a Schrödinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential.

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Authors and Affiliations

  1. CEREMADE, Université de Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France

    Paolo Antonelli

  2. Department of Applied Mathematics, University of Crete, 71409, Heraklion, Greece

    Agisillaos Athanassoulis

  3. College of Science Riyadh, King Saud University (KSU), Riyadh, Kingdom of Saudi Arabia

    Hichem Hajaiej

  4. MCSE Division, King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Kingdom of Saudi Arabia

    Peter Markowich

Authors
  1. Paolo Antonelli
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  2. Agisillaos Athanassoulis
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  3. Hichem Hajaiej
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  4. Peter Markowich
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Corresponding author

Correspondence to Peter Markowich.

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Communicated by C. Dafermos

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Antonelli, P., Athanassoulis, A., Hajaiej, H. et al. On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging. Arch Rational Mech Anal 211, 711–732 (2014). https://doi.org/10.1007/s00205-013-0715-8

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  • DOI: https://doi.org/10.1007/s00205-013-0715-8

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