Averaging in scattering problems

arXiv0902.3269MaRDI QIDQ6212814

V. S. Buslaev, A. A. Pozharskii

Publication date: 19 February 2009

Abstract: We consider the scattering that is described by the equation $(-Delt{a}_x+q(x,fracxepsilon)-E)psi=f(x),psi=psi(x,epsilon)inC,xin{R}^d,epsilon>0,E>0,$ where $q(x,y)$ is a periodic function of $y$, $q$ and $f$ have compact supports with respect to $x$. We are interested in the solution satisfying the radiation condition at infinity and describe the asymptotic behavior of the solution as $epsilono0$. 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 ${\displaystyle }$hat{q}(x) = frac{1}{|Omega|}int_{Omega}q(x,y)dy.${\displaystyle }$

Mathematics Subject Classification ID

Scattering theory for PDEs (35P25) Spectrum, resolvent (47A10) Schrödinger operator, Schrödinger equation (35J10) Scattering theory of linear operators (47A40)

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