Asymptotic behaviour of the scattering phase for non-trapping obstacles

Veselin Petkov; Georgi Popov

Annales de l'institut Fourier (1982)

  • Volume: 32, Issue: 3, page 111-149
  • ISSN: 0373-0956

Abstract

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Let S ( λ ) be the scattering matrix related to the wave equation in the exterior of a non-trapping obstacle 𝒪 ⊂ R n , n ≥ 3 with Dirichlet or Neumann boundary conditions on ∂ 𝒪 . The function s ( λ ) , called scattering phase, is determined from the equality e - 2 π i s ( λ ) = det S ( λ ) . We show that s ( λ ) has an asymptotic expansion s ( λ ) ∼ ∑ j = 0 ∞ c j λ n - j as λ → + ∞ and we compute the first three coefficients. Our result proves the conjecture of Majda and Ralston for non-trapping obstacles.

How to cite

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Petkov, Veselin, and Popov, Georgi. "Asymptotic behaviour of the scattering phase for non-trapping obstacles." Annales de l'institut Fourier 32.3 (1982): 111-149. <http://eudml.org/doc/74543>.

@article{Petkov1982,
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.},
author = {Petkov, Veselin, Popov, Georgi},
journal = {Annales de l'institut Fourier},
keywords = {asymptotics; scattering phase; wave equation in the exterior of a non- trapping obstacle; Dirichlet problem; non-homogeneous Neumann problem; Fourier transform},
language = {eng},
number = {3},
pages = {111-149},
publisher = {Association des Annales de l'Institut Fourier},
title = {Asymptotic behaviour of the scattering phase for non-trapping obstacles},
url = {http://eudml.org/doc/74543},
volume = {32},
year = {1982},
}

TY - JOUR
AU - Petkov, Veselin
AU - Popov, Georgi
TI - Asymptotic behaviour of the scattering phase for non-trapping obstacles
JO - Annales de l'institut Fourier
PY - 1982
PB - Association des Annales de l'Institut Fourier
VL - 32
IS - 3
SP - 111
EP - 149
AB - 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.
LA - eng
KW - asymptotics; scattering phase; wave equation in the exterior of a non- trapping obstacle; Dirichlet problem; non-homogeneous Neumann problem; Fourier transform
UR - http://eudml.org/doc/74543
ER -

References

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Citations in EuDML Documents

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  1. V. M. Petkov, La distribution des pôles de la matrice de diffusion
  2. D. Robert, Approximation semi-classique de la phase de diffusion pour un potentiel (d'après un travail de D. Robert et H. Tamura)
  3. D. Robert, Asymptotique de la phase de diffusion à haute énergie pour des perturbations du laplacien
  4. Gilles Carron, Déterminant relatif et la fonction Xi
  5. Xue Ping Wang, Time-delay operators in semiclassical limit, finite range potentials
  6. D. Robert, Asymptotique de la phase de diffusion à haute énergie pour des perturbations du second ordre du laplacien

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