Homogenization in the scattering problem

The scattering problem is studied, which is described by the equation (-Δ _x +q(x,x/ɛ)−E)ψ = f(x), where ψ = ψ (x,ɛ) ∈ ℂ, x ℂ ℝ ^d , ɛ > 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 ɛ → O 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) = \frac{1} {{\left| \Omega \right|}}\int_\Omega {q(x,y)dy} . $ \hat q(x) = \frac{1} {{\left| \Omega \right|}}\int_\Omega {q(x,y)dy} .
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