Singularities of the scattering kernel
Vesselin M. Petkov · 2016
In this chapter a formula is proved for the leading singularity of the scattering kernel at −Tγ, where Tγ is the sojourn time of an ordinary non-degenerate reflecting (ω, θ)-ray satisfying some additional assumptions. A special emphasis is given to three-dimensional generic domains. The chapter proves that for such domains, the (ω, θ)-rays of mixed type disappear and any singularity of the scattering kernel has the form −Tγ for some reflecting (ω, θ)-ray. It shows that for generic domains Ω, for any k ≥ 1 the glancing ω-rays with k vertices in Ω form a discrete subset of a certain manifold. Let Ω be an arbitrary domain with smooth compact boundary X = ∂Ω and bounded complement in R3, and let ω = θ be two unit vectors in R3. Considering appropriate convergence subsequences, the chapter assumes that there exists δ(t) = limm→∞ δm(t) for every t.