散射问题中的平均
Averaging in scattering problems
摘要：
我们考虑由方程$(-Δ_x + q(x,\frac{x}ε) - E)ψ= f(x), ψ= ψ(x,ε) \in \C, x \in \R^d, ε> 0, E > 0,$描述的散射，其中$q(x,y)$是$y$的周期函数，$q$和$f$相对于$x$具有紧支撑。我们感兴趣的是满足无穷远辐射条件的解，并将解的渐近行为描述为$ε\to 0$。此外，我们还发现了平面波散射振幅的渐近行为。其中任何一个（解和振幅）在前几阶都可以用具有势$$\hat{q}(x) = \frac{1}{|Ω|}\int_Ωq(x,y)dy.$$的平均方程来描述
英文摘要：
We consider the scattering that is described by the equation $(-Δ_x + q(x,\frac{x}ε) - E)ψ= f(x), ψ= ψ(x,ε) \in \C, x \in \R^d, ε> 0, E > 0,$ 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 $ε\to 0$. 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 $$\hat{q}(x) = \frac{1}{|Ω|}\int_Ωq(x,y)dy.$$
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