Singularities of the Scattering Kernel and Scattering Invariants for Several Strictly Convex Obstacles

• Published in: Transactions of the American Mathematical Society 1989-03, Vol.312 (1), p.203-235
• Main Authors: Petkov, Vesselin M., Stojanov, Luchezar N.
• Format: Article
• Language: English
• Subjects: Billiards; Exact sciences and technology; Hyperplanes; Linear transformations; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations, miscellaneous problems; S matrix theory; Sciences and techniques of general use; Symmetry; Tangent planes; Tangents; Wave equations
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-1989-0929661-5

Let $\Omega \subset R^n$ be a domain such that $R^n \backslash \Omega$ 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 $R \times \Omega$ with Dirichlet boundary condition, are studied. It is proved that for every $\omega \in S^{n-1}$ there exists a residual subset $\mathscr{R}(\omega)$ of $S^{n-1}$ such that for each $\theta \in \mathscr{R}(\omega), \theta \neq \omega$, $$\operatorname{singsupp} s(t, \theta, \omega) = \{-T_\gamma\}_\gamma,$$ where $\gamma$ runs over the cattering 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.