Nonexistence of generalized scattering rays and singularities of the scattering kernel for generic domains in $\textbf {R}^ 3$

Abstract:

It is proved for fixed unit vectors $\omega \ne \theta$ in ${\mathbb {R}^3}$ and generic bounded open domains $\mathfrak {D} \subset {\mathbb {R}^3}$ that there do not exist generalized $(\omega ,\theta )$-rays in $\Omega = {\mathbb {R}^3}\backslash \mathfrak {D}$ containing nontrivial geodesies on $\partial \Omega$. Consequently, for generic domains the sojourn times of reflecting $(\omega ,\theta )$-rays completely describe the set of singularities of the scattering kernel $s(t,\theta ,\omega )$.