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On the poles of the scattering matrix for two strictly convex obstacles: An addendum. (English) Zbl 0559.35061
The object of the paper is to clarify the relationship between an obstacle $${\mathcal O}$$ in $${\mathbb{R}}^ 3$$ and the poles of the associated scattering matrix $${\mathcal S}(z)$$ (in the sense of Lax and Phillips). Assume $${\mathcal O}={\mathcal O}_ 1\cup {\mathcal O}_ 2$$, $$\bar {\mathcal O}_ 1\cap \bar {\mathcal O}_ 2=\emptyset$$ and $${\mathcal O}_ i$$ strictly convex open bounded sets with smooth boundaries $$\Gamma_ i$$. Let $${\mathcal S}(\sigma)$$ ($$\sigma\in {\mathbb{R}})$$ be the scattering matrix associated to the acoustic problem $$\square u(x,t)=0$$ in $$({\mathbb{R}}^ 3\setminus \bar {\mathcal O})\times {\mathbb{R}}$$ and $$u(x,t)=0$$ on $$(\Gamma_ 1\cup \Gamma_ 2)\times {\mathbb{R}}$$. Hence $${\mathcal S}(\sigma)$$ is a unitary operator in $$L^ 2(S^ 2)$$ for all $$\sigma\in {\mathbb{R}}$$. It is known that $${\mathcal S}$$ extends to an operator valued function $${\mathcal S}(z)$$ analytic in Im z$$<0$$ and meromorphic on the whole plane. It is shown that there are strictly positive constants $$c_ 0$$, c such that, denoting $$z_ j=ic_ 0+\pi \cdot j[dist({\mathcal O}_ 1,{\mathcal O}_ 2)]^{-1},$$ (j$$\in {\mathbb{Z}})$$, $${\mathcal S}$$ has at least one pole in $$\{z\in {\mathbb{C}}| | z-z_ j| \leq c(1+| j|)^{-1/2}\}$$ for all large $$| j|$$.
Reviewer: V.Georgescu

##### MSC:
 35P25 Scattering theory for PDEs 47A40 Scattering theory of linear operators 78A45 Diffraction, scattering 76Q05 Hydro- and aero-acoustics 35L05 Wave equation
##### Keywords:
obstacle; poles; scattering matrix; acoustic problem
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