## Homogenization in the scattering problem.(English. Russian original)Zbl 1271.35058

Funct. Anal. Appl. 44, No. 4, 243-252 (2010); translation from Funkts. Anal. Prilozh. 44, No. 4, 2-13 (2010).
Summary: 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 $$\varepsilon\to O$$ 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 $\hat q(x)=\frac{1}{|\Omega|}\int_\Omega q(x,y)dy.$

### MSC:

 35P25 Scattering theory for PDEs
Full Text:

### References:

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