Averaging in scattering problems

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$q(x,y)$isaperiodicfunctionof$y$,$q$and$f$havecompactsupportswithrespectto$x$. We are interested in the solution satisfying the radiation condition at infinity and describe the asymptotic behavior of the solution as $ϵ\to 0$. 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 $$\widehat{q}(x)=\frac{1}{\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }}{\int }_{\mathrm{\Omega }}q(x,y)dy\mathrm{.}$$