On the poles of the scattering matrix for two convex obstacles. (English) Zbl 0587.35057

Ikawa, Mitsuru

Journ. Équ. Dériv. Partielles, Saint-Jean-De-Monts 1985, No. 1, Exp. No. 5, 14 p. (1985).

This paper is a contribution to the discussion of the relation between the scatterer and the poles of the corresponding scattering matrix. The three-dimensional wave equation in the exterior of two smooth, disjoint, strictly convex obstacles \(O_ 1\), \(O_ 2\) under the assumption of a Dirichlet boundary condition is considered. The author not only narrows down the location of the poles considerably but also provides information about the order of the singularity for certain poles. It is shown that the scattering matrix is holomorphic in \(\{\) \(z|\) Im \(z\leq c_ 0+c_ 1\}-\cup^{\infty}_{j=-\infty}D_ j\) where \[ D_ j=\{z| \quad | z-z_ j| \leq C(1+| j|)^{-1/2}\},\quad z_ j=ic_ 0+(\pi /d)^ j,\quad d=dist(O_ 1,O_ 2). \] Moreover, for large j there is only one simple pole in \(D_ j\). These poles \(\phi_ j\) are asymptotically (for large j) closer and closer to \(z_ j\), they are simple and there is an asymptotic expansion of the form \[ \phi_ j\sim z_ j+\beta_ 1j^{-1}+\beta_ 2j^{-2}+...\quad as\quad | j| \to \infty \] where the \(\beta_ k\), k positive integer, are complex numbers determined by the obstacle.

Reviewer: R.Picard

MSC:

35L05	Wave equation
35P25	Scattering theory for PDEs
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

scattering matrix; wave equation; obstacles; order of the singularity; asymptotic expansion