On the poles of the scattering matrix for two strictly convex obstacles: An addendum. (English) Zbl 0559.35061

Ikawa, Mitsuru

J. Math. Kyoto Univ. 23, 795-802 (1983).

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