Scattering kernel for trapping obstacles
Vesselin M. Petkov · 2016
Let Ω be a closed domain in Rn, n ≥ 2, with bounded complement K and smooth boundary ∂Ω. This chapter shows that if the obstacle K = Rn \ Ω° is trapping and K ∈ K, then there exists a sequence of ordinary reflecting non-degenerate (ωm, θm)-rays γm in Ω with sojourn times Tγm →∞such that −Tγm ∈ sing supp s(t, θm, ωm). Here K is the class of obstacles introduced in the beginning of the chapter. The chapter shows the representation of the scattering amplitude a(λ, θ, ω), introduced by the cut-off outgoing resolvent of the Dirichlet Laplacian. From this the chapter deduces a meromorphic continuation of a(λ, θ, ω) in C for n odd and in the logarithmic covering of C for n even. Finally, the chapter examines the estimate of the scattering amplitude if there exists at least one such ray.