Singularities of the scattering kernel related to trapping rays

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 $\Omega = \R^n \setminus K.$Weprovethatif$K$istrappingand$\gamma(t)$ is non-degenerate, then there exist reflecting $({\omega }_m,{\theta }_m)$-rays ${\delta }_m,m\in \mathbb{N},$ with sojourn times $T_m\to +\mathrm{\infty }$ as $m\to \mathrm{\infty }$, 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 $\text{C}$.