Asymptotic behaviour of the scattering phase for non-trapping obstacles

Petkov, Veselin ; Popov, Georgi

Annales de l'Institut Fourier, Tome 32 (1982), p. 111-149 / Harvested from Numdam

Résumé

Soit $S(\lambda )$ la matrice de diffusion, associée à l’équation des ondes dans l’extérieur d’un obstacle non-captif ${\cal O} \subset {\bf R}^n$, $n\ge 3$ avec condition de Dirichlet ou Neumann sur $\partial {\cal O}$. La fonction $s( \lambda )$, dite phase de diffusion, est déterminée par l’égalité $e^{-2\pi is(\lambda )} = {\rm det} S(\lambda )$. On démontre que $s(\lambda )$ admet un développement asymptotique $s(\lambda ) \sim \sum ^\infty _{j=0} c_j \lambda ^{n-j}$ et on calcule les trois premiers coefficients. Notre résultat prouve la conjecture de Majda et Ralston pour des obstacles non-captifs.

Abstract

Let $S(\lambda )$ be the scattering matrix related to the wave equation in the exterior of a non-trapping obstacle ${\cal O} \subset {\bf R}^n$, $n\ge 3$ with Dirichlet or Neumann boundary conditions on $\partial {\cal O}$. The function $s(\lambda )$, called scattering phase, is determined from the equality $e^{-2\pi is(\lambda )} = {\rm det} S(\lambda )$. We show that $s(\lambda )$ has an asymptotic expansion $s(\lambda ) \sim \sum ^\infty _{j=0} c_j \lambda ^{n-j}$ as $\lambda \rightarrow +\infty $ and we compute the first three coefficients. Our result proves the conjecture of Majda and Ralston for non-trapping obstacles.

Détail

Publié le : 1982-01-01

MR 85c:35070 | Zbl 0476.35014 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.882

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