与俘获射线有关的散射核的奇异性
Singularities of the scattering kernel related to trapping rays
摘要：
一个障碍物$K \subset \R^n,\: n \geq 3,$ $n$奇数，本文研究了在$Ω= \R^n \setminus K.$中波动方程Dirichlet问题的散射核$s(t, θ, ω)$（定义为与Dirichlet问题相关的散射振幅$a(λ, θ, ω)$的傅里叶变换）的奇异性，证明了如果$K$是陷波的，$γ(t)$是陷波的非简并的，则存在具有逗留时间$T_m \to +\infty$为$m \to \infty$的反射光$(ω_m, θ_m)$ -射线$δ_m,\: m \in \N,$，使得$-T_m \in {\rm sing}\:{\rm supp}\: s(t, θ_m, ω_m),\: \forall m \in \N$。我们应用这一性质来研究$\C$中散射振幅的行为。
英文摘要：
An obstacle $K \subset \R^n,\: n \geq 3,$ $n$ odd, is called trapping if there exists at least one generalized bicharacteristic $γ(t)$ of the wave equation staying in a neighborhood of $K$ for all $t \geq 0.$ We examine the singularities of the scattering kernel $s(t, θ, ω)$ defined as the Fourier transform of the scattering amplitude $a(λ, θ, ω)$ related to the Dirichlet problem for the wave equation in $Ω= \R^n \setminus K.$ We prove that if $K$ is trapping and $γ(t)$ is non-degenerate, then there exist reflecting $(ω_m, θ_m)$-rays $δ_m,\: m \in \N,$ with sojourn times $T_m \to +\infty$ as $m \to \infty$, so that $-T_m \in {\rm sing}\:{\rm supp}\: s(t, θ_m, ω_m),\: \forall m \in \N$. We apply this property to study the behavior of the scattering amplitude in $\C$.
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