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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)

47A40 Scattering theory of linear operators
35P25 Scattering theory for PDEs
47N50 Applications of operator theory in the physical sciences
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