Averaging in scattering problems

Details Averaging in scattering problems Averaging in scattering problems

2009-02-19

arxiv.org/abs/0902.3269v1

Physics

Mathematical Physics

16 pages

Scientific paper

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 $$\hat{q}(x) = \frac{1}{|\Omega|}\int_{\Omega}q(x,y)dy.$$

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