On the existence of the poles of the scattering matrix for several convex bodies. (English) Zbl 0704.35113

Let \(\theta_ j\), \(j=1,2,...,j\) be open bounded sets in \(R^ 3\) with smooth boundary \(\Gamma_ j\). We assume:
(H.1) Every \(\theta_ j\) is strictly convex, that is, the Gaussian curvature of \(\Gamma_ j\) is positive everywhere.
(H.2) For all \(\{j_ 1,j_ 2,j_ 3\}\in \{1,2,...,J\}^ 3\) such that \(j_{\ell}\neq j_ h\) if \(\ell \neq h\), the convex hull of \({\bar \theta}{}_{j_ 1}\) and \({\bar \theta}{}_{j_ 2}\) has no intersection with \({\bar \theta}{}_{j_ 3}.\)
Set \(\theta =\cup^{J}_{j=1}\theta_ j\) and \(\Omega =R_ 3-{\bar \theta}.\)
Consider the following acoustic problem \[ \square u(x,t)=\partial^ 2u/\partial t^ 2-\Delta u=0\text{ in } \Omega \times (-\infty,\infty) \]
\[ u(x,0)=f_ 1(x);\quad (\partial u/\partial t)(x,0)=f_ 2(x) \] u satisfies either Dirichlet or Neumann conditions on \(\Gamma\times (- \infty,\infty).\)
The modified Lax-Phillips conjecture states that when \(\theta\) is trapping, there exists \(\alpha >0\) such that a subdomain \(\{\) z; Im z\(<\alpha \}\) contains an infinite number of poles of the scattering matrix. The author proves theorems which imply among other things that in the case of the Neumann condition, the modified Lax and Phillips conjecture holds for \(\theta\) satisfying (H.1) and (H.2).
Reviewer: R.Guenther


35P25 Scattering theory for PDEs
35L05 Wave equation
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