### 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.

Original language | English |
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Pages (from-to) | 711-732 |

Number of pages | 22 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 211 |

Issue number | 3 |

Early online date | 14 Jan 2014 |

DOIs | |

Publication status | Published - Mar 2014 |

### Keywords

- Weak Convergence
- Admissible Pair
- Uniform Bound
- Strichartz Estimate
- Electromagnetic Potential

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## Cite this

*Archive for Rational Mechanics and Analysis*,

*211*(3), 711-732. https://doi.org/10.1007/s00205-013-0715-8