Homogenization in the Scattering Problem

The scattering problem is studied, which is described by the equation $(-\Delta_x+q(x,x/\varepsilon)-E)\psi=f(x)$, where $\psi=\psi(x,\varepsilon)\in\mathbb{C}$, $x\in\mathbb{R}^d$, $\varepsilon>0$, $E>0$, the function $q(x,y)$ is periodic with respect to $y$, and the function $f$ is compactly supported. The solution satisfying radiation conditions at infinity is considered, and its asymptotic behavior as $\varepsilon\to0$ is described. The asymptotic behavior of the scattering amplitude of a plane wave is also considered. It is shown that in principal order both the solution and the scattering amplitude are described by the homogenized equation with potential
$$ \hat{q}(x)=\frac1{|\Omega|}\int_\Omega q(x,y)\,dy. $$