Asymptotic behaviour of the scattering phase for non-trapping obstacles
Annales de l'Institut Fourier, Volume 32 (1982) no. 3, p. 111-149

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

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

@article{AIF_1982__32_3_111_0,
author = {Petkov, Veselin and Popov, Georgi},
title = {Asymptotic behaviour of the scattering phase for non-trapping obstacles},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
volume = {32},
number = {3},
year = {1982},
pages = {111-149},
doi = {10.5802/aif.882},
zbl = {0476.35014},
mrnumber = {85c:35070},
language = {en},
url = {http://www.numdam.org/item/AIF_1982__32_3_111_0}
}

Petkov, Veselin; Popov, Georgi. Asymptotic behaviour of the scattering phase for non-trapping obstacles. Annales de l'Institut Fourier, Volume 32 (1982) no. 3, pp. 111-149. doi : 10.5802/aif.882. http://www.numdam.org/item/AIF_1982__32_3_111_0/

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