Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes. (Asymptotic properties of poles of the scattering matrix for two strictly convex obstacles). (French) Zbl 0622.35054

Sémin., Équations Dériv. Partielles 1985-1986, Exposé No. 14, 9 p. (1986).
The author reports on the asymptotic properties of the poles of the scattering matrix S(\(\lambda)\) associated with acoustic scattering from an obstacle \(\Omega\), formed by two disjoint convex bodies \(\Omega =\Omega_ 1\cup \Omega_ 2\) with \(C^{\infty}\)-boundaries \(\Gamma_ j\), \(j=1,2\), in \(R^{n+1}\), n even, \(n\geq 2\). Let \(d=dist(\Omega_ 1,\Omega_ 2)\). Studying the poles of S(\(\lambda)\) is equivalent, via standard micro-local analysis envolving the billiard map \(\chi\) of \(T^*(\Gamma_ 1)\to T^*(\Gamma_ 1)\), to the Grushin problem for \(I-M_ 0\) where \(M_ 0\) is given by \[ M_ 0u(x,\lambda)=(\lambda /2\pi)^ n\int e^{-i\lambda (A^{-1}\cdot x-y\cdot \theta)-i\lambda 2d} b_ 0u(y,\lambda)dy d\theta =b_ 0u(A^{-1} x,x)e^{-i\lambda 2d} \] and \(A^{-1}\) has the form of a canonical Jordan matrix with values \(\nu_ 1^{-1},...,\nu_ n^{-1}\) and \(b_ 0=(\nu_ 1,...,\nu_ n)^{-}\). For \(\alpha \in N^ n\) set \(K_{\alpha}=\nu^{- \alpha}\times b_ 0\). For a fixed value \(K_ 0\) of K, set \(J=\{\alpha \in N^ n:\) \(K_{\alpha}=K_ 0\}\) and \(N=card J\). Let \(\lambda_ j=-i \log (K_ 0/2d)+j(\pi /\alpha)\), \(j\in Z.\)
In fact, one observes that the eigenvectors of \(M_ 0\) are \(x^{\alpha}\) with eigenvalues \(b_ 0e^{-i\lambda 2d} \nu^{- \alpha}\) so that the pseudo-poles \(\lambda_ j\) are exactly the values of \(\lambda\) such that \(1-e^{-i\lambda 2d} b_ 0\nu^{-\alpha}=0\). For each value of \(K_ 0\) there exists an asymptotic expansion \[ \lambda_{\ell}(j)=\lambda_ j+\sum^{\infty}_{k=1}a_{k,\ell}(\lambda_ j)^{-k/2a\ell} \] corresponding to an asymptotic expansion in \(\lambda\) of an \(N\times N\) matrix depending on \(\lambda\). Let \(p_{\ell}\) be the asymptotic multiplicity of \(\lambda_{\ell}(j).\)
Then the principal result states that there are exactly \(p_{\ell}\) poles of S(\(\lambda)\) asymptotic to \(\lambda_{\ell}(S)\).
Reviewer: M.Thompson


35P25 Scattering theory for PDEs
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
47A40 Scattering theory of linear operators
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