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

Abstract

Let Ω ⊂ R n \Omega \subset {{\mathbf {R}}^n} be a domain such that R n ∖ Ω {{\mathbf {R}}^n}\backslash \Omega is a disjoint union of a finite number of compact strictly convex obstacles with C ∞ {C^\infty } smooth boundaries. In this paper the singularities of the scattering kernel s ( t , θ , ω ) s(t,\theta ,\omega ) , related to the wave equation in R × Ω {\mathbf {R}} \times \Omega with Dirichlet boundary condition, are studied. It is proved that for every ω ∈ S n − 1 \omega \in {S^{n - 1}} there exists a residual subset R ( ω ) \mathcal {R}(\omega ) of S n − 1 {S^{n - 1}} such that for each θ ∈ R ( ω ) , θ ≠ ω \theta \in \mathcal {R}(\omega ),\theta \ne \omega \[ singsupp s ( t , θ , ω ) = { − T γ } γ , {\text {singsupp}}\,s(t,\theta ,\omega ) = {\{ - {T_\gamma }\} _\gamma }, \] where γ \gamma runs over the scattering rays in Ω \Omega with incoming direction ω \omega and with outgoing direction θ \theta having no segments tangent to ∂ Ω \partial \Omega , and T γ {T_\gamma } is the sojourn time of γ \gamma . Under some condition on Ω \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 , θ , ω ) s(t,\theta ,\omega ) , is examined.