Trapping obstacles with a sequence of poles on the scattering matrix converging to the real axis (Q1821962)

The scattering of the acoustic equation by bounded obstacles is considered. In the book of \textit{P. D. Lax} and \textit{R. S. Phillips} [''Scattering theory'' (1967; Zbl 0186.163)], they conjectured that, when the obstacle is trapping, the scattering matrix \({\mathcal S}(z)\) has a sequence of poles converging to the real axis. Although it is believed since a long time that this conjecture is correct, we didn't find even one of such examples. The purpose of this paper is to give such an example of obstacle. Suppose that \({\mathcal O}\) satisfies: (i) \({\mathcal O}={\mathcal O}_ 1\cup {\mathcal O}_ 2\), where \({\mathcal O}_ 1\) and \({\mathcal O}_ 2\) are convex and bounded in \({\mathbb{R}}^ 3\) such that \(\bar {\mathcal O}_ 1\cap \bar {\mathcal O}_ 2=\phi.\) (ii) Let \(a_ j\in \Gamma_ j=\partial {\mathcal O}_ j\), \(j=1,2\), be the points such that \(| a_ 1-a_ 2| =dis({\mathcal O}_ 1,{\mathcal O}_ 2)\). Then it holds that \[ C| x-a_ j|^ e \geq \kappa_{j\ell}(x)\geq C^{-1}| x-a_ j|^ e\text{ for all } x\in \Gamma_ j \] for some \(2\leq e<\infty\), where \(\kappa_{j\ell}\), \(\ell =1,2\) are the principal curvatures of \(\Gamma_ j\) at \(\kappa\). Then the scattering matrix \({\mathcal S}(z)\) has a sequence of the poles \(\{z_ j\}^{\infty}_{j=1}\) such that Im \(z_ j\to 0\), \(j\to \infty\).