Precise informations on the poles of the scattering matrix for two strictly convex obstacles. (English) Zbl 0637.35068

The author considers the scattering problem \[ \square u=u_{tt}-\Delta u=0\quad in\quad \Omega \times (-\infty,\infty),\quad u=0\quad on\quad \partial \Omega \times (-\infty,\infty), \] where \(\Omega\) is the complement of (the closure of) two bounded, strictly convex open sets \({\mathcal O}_ 1\), \({\mathcal O}_ 2\) in \({\mathbb{R}}^ 3\) with smooth boundaries and such that \(\bar {\mathcal O}_ 1\cap \bar {\mathcal O}_ 2=\emptyset.\)
His mean result complements some of his previous results in this direction and give precise informations on the location of the poles of the scattering matrix \({\mathcal S}(z)\) of the above problem. In fact, by proving a suitable representation for the solution operator \(g\mapsto U(\mu)g\) of the problem \((\mu^ 2-\Delta)u=0\) in \(\Omega\), \(u=g\) on \(\partial \Omega\) (where Re \(\mu\) \(>0\), \(g\in C^{\infty}(\partial \Omega))\), and by relating the poles of \({\mathcal S}(z)\) and U(\(\mu)\), he shows that, for large \(| j| \in {\mathbb{N}},\)
(a) \({\mathcal S}(z)\) has exactly one pole \(p_ j\) in \(B_ j=\{z|| z-z_ j| \leq C(1+| j|)^{-}\}\), where \(z_ j=(\pi /d)j+ic_ 0\), \(d=dist({\mathcal O}_ 1,{\mathcal O}_ 2)\), \(c_ 0>0;\)
(b) \(p_ j\) has an asymptotic expansion of the form \(p_ j\sim z_ j+\beta_ 1j^{-1}+\beta_ 2j^{-2}+..\). where the \(\beta_ k's\) are constants determined by \({\mathcal O}_ 1,{\mathcal O}_ 2;\)
(c) \({\mathcal S}(z)\) has the following representation, for \(z\in B_ j\), \(f\in L^ 2(S^ 2):\) \({\mathcal S}(z)f=(n_ j/(z-p_ j))(f,\psi_ j)+{\mathcal H}_ j(z)f,\)
where \(\eta_ j,\psi_ j\in L^ 2(S^ 2)\) are nonzero and \({\mathcal H}_ j(z)\in {\mathcal L}(L^ 2(S^ 2),L^ 2(S^ 2))\) depends holomorphically on \(z\in B_ j\).
Reviewer: D.G.Costa


35P25 Scattering theory for PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
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