Singularities of the scattering kernel for two convex obstacles
The author considers the scattering problem for the wave equation with two finite convex obstacles \({\mathcal O}_ 1\) and \({\mathcal O}_ 2\). The main purpose of the paper is to investigate the location of singularities of the scattering kernel S(s,\(\theta\),\(\omega)\). Set \(r_ i(\omega)=\min \{x\cdot \omega\); \(x\in {\mathcal O}_ i\}\), \(i=1,2\). The first main result says that if \({\mathcal O}_ 1\cap \{x+\ell \omega\); \(x\in {\mathcal O}_ 2\), \(\ell \in {\mathbb{R}}\}=\emptyset\), then S(\(\cdot,-\omega,\omega)\) has only two singularities in the interval [\(\min_{i=1,2}(-2r_ i(\omega)),\infty)\), namely \(-2r_ 1(\omega)\) and \(-2r_ 2(\omega).\) For more restricted \(\omega\) the author describes all singularities of S(s,\(\theta\),\(\omega)\). In case \({\mathcal O}_ i\), \(i=1,2\), are balls similar statements have been obtained by the author and \textit{H. Soga} [J. Math. Soc. Japan 40, No.2, 205-220 (1988; Zbl 0638.35068)].
scientific article; zbMATH DE number 4146674
scientific article