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Abstract
We analyse a nonlinear Schrodinger 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 Schrodinger equation with a time-averaged Coulomb potential in the high frequency limit for the oscillations of the electromagnetic potential.
Highlights
Introduction and Statement of the MainResultsIn this paper we investigate the following nonlinear Schrodinger equation i ∂tψ = (i ∇ − A)2ψ + c 1 |x| ψ C1(|
In this paper we are interested in studying the case when A(t) is rapidly oscillating and we investigate the asymptotic behaviour of solutions of (1.3) in the highly oscillating regime
We use the Fourier transform for the slow time variable. For this purpose we extend the function g from [0, T ]×R×R3 to R×R×R3, such that it is smooth in R×R×R3 and it vanishes outside the slab (−1, T +1)×R×R3
Summary
The idea for the proof is as in [2] and can be explained in the following way: if we consider the Duhamel’s formula for equation (1.6), the oscillating potential (1.5) appears inside the time integral, the weak convergence for (1.5) can be improved to the strong one for {uω}. This is possible thanks to the uniform bounds in ω we have for {uω}
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