Singularities of the scattering kernel related to trapping rays

TL;DR

This paper investigates how trapping obstacles in odd-dimensional space cause specific singularities in the scattering kernel, revealing links between trapped rays and the kernel's support structure.

Contribution

It establishes a connection between trapping rays and the singular support of the scattering kernel, particularly showing how non-degenerate trapped rays lead to specific singularities.

Findings

Existence of reflecting rays with infinite sojourn times in trapping obstacles.

Singular support of the scattering kernel contains points related to these rays.

Application to the behavior of the scattering amplitude in complex domains.

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 $\C$.