1209.6089 (Paolo Antonelli et al.)

On the XFEL Schroedinger Equation: Highly Oscillatory Magnetic
Potentials and Time Averaging    [PDF]

Paolo Antonelli, Agisillaos Athanassoulis, Hichem Hajaiej, Peter Markowich

We analyse a nonlinear Schr\"odinger 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 (XFEL). We prove the existence and uniqueness for the Cauchy problem and the convergence of wave-functions to corresponding solutions of a Schr\"odinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential.

View original: http://arxiv.org/abs/1209.6089