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

TL;DR

This paper studies a nonlinear Schrödinger equation modeling electron beams in XFELs, proving existence, uniqueness, and convergence of solutions under highly oscillatory electromagnetic potentials to a time-averaged model.

Contribution

It introduces a rigorous analysis of the high-frequency limit for the Schrödinger equation with oscillatory potentials in the context of XFELs, establishing convergence to a time-averaged Coulomb potential.

Findings

Proved existence and uniqueness of solutions for the nonlinear Schrödinger equation.

Established convergence of solutions to a time-averaged model in the high-frequency limit.

Provided mathematical foundation for simplified XFEL electron beam models.

Abstract

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.