## 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

### MSC:

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