Singularities of the scattering kernel and scattering invariants for several strictly convex obstacles

Abstract:

Let $\Omega \subset {{\mathbf {R}}^n}$ be a domain such that ${{\mathbf {R}}^n}\backslash \Omega$ is a disjoint union of a finite number of compact strictly convex obstacles with ${C^\infty }$ smooth boundaries. In this paper the singularities of the scattering kernel $s(t,\theta ,\omega )$, related to the wave equation in ${\mathbf {R}} \times \Omega$ with Dirichlet boundary condition, are studied. It is proved that for every $\omega \in {S^{n - 1}}$ there exists a residual subset $\mathcal {R}(\omega )$ of ${S^{n - 1}}$ such that for each $\theta \in \mathcal {R}(\omega ),\theta \ne \omega$ \[ {\text {singsupp}} s(t,\theta ,\omega ) = {\{ - {T_\gamma }\} _\gamma },\] where $\gamma$ runs over the scattering rays in $\Omega$ with incoming direction $\omega$ and with outgoing direction $\theta$ having no segments tangent to $\partial \Omega$, and ${T_\gamma }$ is the sojourn time of $\gamma$. Under some condition on $\Omega$, introduced by M. Ikawa, the asymptotic behavior of the sojourn times of the scattering rays related to a given configuration, as well as the precise rate of the decay of the coefficients of the main singularity of $s(t,\theta ,\omega )$, is examined.