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 $\gamma(t)$ of the wave equation staying in a neighborhood of $K$ for all $t \geq 0.$ We examine the singularities of the scattering kernel $s(t, \theta, \omega)$ defined as the Fourier transform of the scattering amplitude $a(\lambda, \theta, \omega)$ related to the Dirichlet problem for the wave equation in $\Omega = \R^n \setminus K.$ We prove that if $K$ is trapping and $\gamma(t)$ is non-degenerate, then there exist reflecting $(\omega_m, \theta_m)$-rays $\delta_m,\: m \in \N,$ with sojourn times $T_m \to +\infty$ as $m \to \infty$, so that $-T_m \in {\rm sing}\:{\rm supp}\: s(t, \theta_m, \omega_m),\: \forall m \in \N$. We apply this property to study the behavior of the scattering amplitude in $\C$.