On the distribution of the poles of the scattering matrix for two strictly convex obstacles
The author considers the scattering matrix \({\mathcal S}(z)\) for the acoustic problem \(\partial^ 2u/\partial t^ 2-\Delta u=0\) in the exterior of a region \(O=O_ 1\cup O_ 2,\quad \bar O_ 1\cap \bar O_ 2=\emptyset, O_ 1\) and \(O_ 2\) being open subsets of \({\mathbb{R}}^ 3\), with u satisfying the Dirichlet condition \(u=0\) on the frontier \(\Gamma_ 1\cup \Gamma_ 2\). It is assumed that the Gaussian curvatures of \(\Gamma_ j\), \(j=1,2\) are non-vanishing. It is an important question to examine the relationship between the geometrical properties of the obstacles and the location of the poles of the scattering matrix. In previous work by Bardos, Guillot and Rolston the notion of pseudo-poles \(\alpha_{m,\bar m}\) was introduced and used to establish the existence of infinitely many poles in a regio\(n\{z:Im z\leq \epsilon \log(| z| +1)\) for any \(\epsilon>0\}.\) In the present paper the author shows that the pseudo-poles approximate the actual poles of \({\mathcal S}(z)\) under the additional geometrical condition that at the points \(a_ j\), \(j=1,2\) with \(| a_ 1-a_ 2| =dist(O_ 1,O_ 2), \Gamma_ j\) are umbilical. The result is established by means of a precise analysis of the periodic properties in t of asymptotic solutions whose superpositions are used to approximate the fundamental solution for the acoustic problem.
scientific article; zbMATH DE number 3861675
scientific article