Singularities of the scattering kernel related to trapping rays

Abstract

An obstacle $K\subset {\mathbb{R}}^n,n\ge 3,$ $n$ odd, is called trapping if there exists at least one generalized bicharacteristic $\gamma (t)$ of the wave equation staying in a neighborhood of $K$ for all $t\ge 0.$ We examine the singularities of the scattering kernel $s(t,\theta ,\omega )$ defined as the Fourier transform of the scattering amplitude $a(\lambda ,\theta ,\omega )$ related to the Dirichlet problem for the wave equation in $\Omega ={\mathbb{R}}^n\setminus K.$ We prove that if $K$ is trapping and $\gamma (t)$ is non-degenerate, then there exist reflecting $({\omega }_m,{\theta }_m)$-rays ${\delta }_m,m\in \mathbb{N},$ with sojourn times $T_m\to +\infty$ as $m\to \infty$, so that $-T_m\in \mathrm{sing}\mathrm{supp}s(t,{\theta }_m,{\omega }_m),\forall m\in \mathbb{N}$. We apply this property to study the behavior of the scattering amplitude in $\C$.