Averaging in scattering problems

TL;DR

This paper analyzes the asymptotic behavior of solutions and scattering amplitudes in periodic scattering problems as the period parameter approaches zero, using an averaged potential to describe leading-order behavior.

Contribution

It introduces a method to approximate solutions and scattering amplitudes in periodic scattering problems via an averaged potential as the period tends to zero.

Findings

Solutions are asymptotically described by an averaged potential.

Scattering amplitudes converge to those of the averaged problem.

Provides explicit asymptotic formulas for solutions and amplitudes.

Abstract

We consider the scattering that is described by the equation $(-\Delta_x + q(x,\frac{x}{\epsilon}) - E)\psi= f(x), \psi = \psi(x,\epsilon) \in \C, x \in \R^d, \epsilon > 0, E > 0,$ where $q(x,y)$ is a periodic function of $y$, $q$ and $f$ have compact supports with respect to $x$. We are interested in the solution satisfying the radiation condition at infinity and describe the asymptotic behavior of the solution as $\epsilon \to 0$. In addition, we find the asymptotic behavior of the scattering amplitude of the plain wave. Either of them (the solution and the amplitude) in the leading orders are described by the averaged equation with the potential $$\hat{q}(x) = \frac{1}{|\Omega|}\int_{\Omega}q(x,y)dy.$$