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

Antonelli, Paolo; Athanassoulis, Agisillaos; Hajaiej, Hichem; Markowich, Peter A.

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

Type

Article

Authors

Antonelli, Paolo
Athanassoulis, Agisillaos
Hajaiej, Hichem
Markowich, Peter A.

KAUST Department

Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Applied Mathematics and Computational Science Program

Preprint Posting Date

2012-09-26

Online Publication Date

2014-01-14

Print Publication Date

2014-03

Date

2014-01-14

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. © 2014 Springer-Verlag Berlin Heidelberg.

Citation

Antonelli, P., Athanassoulis, A., Hajaiej, H., & Markowich, P. (2014). On the XFEL Schrödinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging. Archive for Rational Mechanics and Analysis, 211(3), 711–732. doi:10.1007/s00205-013-0715-8

Acknowledgements

P. Antonelli and A. Athanassoulis would like to thank the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge for its hospitality and support during the preparation of this work. This research was supported by the Investigator Award Nr. KUK-I1-007-43 funded by the King Abdullah University of Science and Technology. H. Hajaiej extends his appreciation to the deanship of scientific research at King Saud University through the research group project RGP-VPP124.