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

Buslaev, V. S.; Pozharskii, A. A.

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

Keywords:

scattering problem for the Schrödinger equation; two-scale dependence of potential on coordinates; homogenization; static load model

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