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Scattering poles near the real axis for two strictly convex obstacles. (English) Zbl 1124.47009
Summary: To study the location of poles for the acoustic scattering matrix for two strictly convex obstacles with smooth boundaries, one uses an approximation of the quantized billiard operator $$M$$ along the trapped ray between the two obstacles. Using this method, C. Gérard [“Asymptotique des pôles de la matrice de scattering pour deux obstacles strictement convexes” (Bull. Soc. Math. Fr. 116, No. 1 (Suppl.), Mem. Soc. Math. Fr. (N.S.) 31; Soc. Math. Fr., Paris) (1988; Zbl 0654.35081)] obtained complete asymptotic expansions for the poles in a strip $$\operatorname{Im} z \leq c$$ as $$\operatorname{Re} z$$ tends to infinity. He established the existence of parallel rows of poles close to $$\frac{\pi k}{d}+ ij\delta$$, $$k \in {\mathbb{Z}}$$, $$j \in {{\mathbb{Z}}_{+}}$$. Assuming that the boundaries are analytic and the eigenvalues of Poincaré map are non-resonant we use the Birkhoff normal form for $$M$$ to improve his result and to get the complete asymptotic expansions for the poles in any logarithmic neighborhood of real axis.
Reviewer: Reviewer (Berlin)

##### MSC:
 47A40 Scattering theory of linear operators 35P25 Scattering theory for PDEs 47N50 Applications of operator theory in the physical sciences
##### Keywords:
obstacle scattering; scattering poles; resonances
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