Singularities of the scattering kernel related to trapping rays

Abstract

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 $\C$.