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

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

doi:10.1007/s00205-013-0715-8

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

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

Research output: Contribution to journal › Article › peer-review

19 Scopus citations

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.

Original language	English (US)
Pages (from-to)	711-732
Number of pages	22
Journal	Archive for Rational Mechanics and Analysis
Volume	211
Issue number	3
DOIs	https://doi.org/10.1007/s00205-013-0715-8
State	Published - Jan 14 2014

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
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.

ASJC Scopus subject areas

Mechanical Engineering
Analysis
Mathematics (miscellaneous)

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@article{98e236d573f048028d0ec7d0d7aeb37a,

title = "On the XFEL Schr{\"o}dinger Equation: Highly Oscillatory Magnetic Potentials and Time Averaging",

abstract = "We analyse a nonlinear Schr{\"o}dinger 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 Schr{\"o}dinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential. {\textcopyright} 2014 Springer-Verlag Berlin Heidelberg.",

author = "Paolo Antonelli and Agisillaos Athanassoulis and Hichem Hajaiej and Markowich, {Peter A.}",

note = "KAUST Repository Item: Exported on 2020-10-01 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.",

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doi = "10.1007/s00205-013-0715-8",

language = "English (US)",

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AU - Antonelli, Paolo

AU - Athanassoulis, Agisillaos

AU - Hajaiej, Hichem

AU - Markowich, Peter A.

N1 - KAUST Repository Item: Exported on 2020-10-01 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.

PY - 2014/1/14

Y1 - 2014/1/14

N2 - 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.

AB - 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.

UR - http://hdl.handle.net/10754/563339

UR - http://arxiv.org/abs/arXiv:1209.6089v1

UR - http://www.scopus.com/inward/record.url?scp=84895903084&partnerID=8YFLogxK

U2 - 10.1007/s00205-013-0715-8

DO - 10.1007/s00205-013-0715-8

M3 - Article

SN - 0003-9527

VL - 211

SP - 711

EP - 732

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 3

ER -

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

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