Recovering Asymptotics at Infinity of Perturbations of Stratified Media

Tanya Christiansen; Mark S. Joshi

Journées équations aux dérivées partielles (2000)

  • page 1-9
  • ISSN: 0752-0360

Abstract

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We consider perturbations of a stratified medium x n - 1 × y , where the operator studied is c 2 ( x , y ) Δ . The function c is a perturbation of c 0 ( y ) , which is constant for sufficiently large | y | and satisfies some other conditions. Under certain restrictions on the perturbation c , we give results on the Fourier integral operator structure of the scattering matrix. Moreover, we show that we can recover the asymptotic expansion at infinity of c from knowledge of c 0 and the singularities of the scattering matrix at fixed energy.

How to cite

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Christiansen, Tanya, and Joshi, Mark S.. "Recovering Asymptotics at Infinity of Perturbations of Stratified Media." Journées équations aux dérivées partielles (2000): 1-9. <http://eudml.org/doc/93399>.

@article{Christiansen2000,
abstract = {We consider perturbations of a stratified medium $\mathbb \{R\}^\{n-1\}_x\times \mathbb \{R\}_y$, where the operator studied is $c^2(x,y) \Delta $. The function $c$ is a perturbation of $c_0(y)$, which is constant for sufficiently large $|y|$ and satisfies some other conditions. Under certain restrictions on the perturbation $c$, we give results on the Fourier integral operator structure of the scattering matrix. Moreover, we show that we can recover the asymptotic expansion at infinity of $c$ from knowledge of $c_0$ and the singularities of the scattering matrix at fixed energy.},
author = {Christiansen, Tanya, Joshi, Mark S.},
journal = {Journées équations aux dérivées partielles},
language = {eng},
pages = {1-9},
publisher = {Université de Nantes},
title = {Recovering Asymptotics at Infinity of Perturbations of Stratified Media},
url = {http://eudml.org/doc/93399},
year = {2000},
}

TY - JOUR
AU - Christiansen, Tanya
AU - Joshi, Mark S.
TI - Recovering Asymptotics at Infinity of Perturbations of Stratified Media
JO - Journées équations aux dérivées partielles
PY - 2000
PB - Université de Nantes
SP - 1
EP - 9
AB - We consider perturbations of a stratified medium $\mathbb {R}^{n-1}_x\times \mathbb {R}_y$, where the operator studied is $c^2(x,y) \Delta $. The function $c$ is a perturbation of $c_0(y)$, which is constant for sufficiently large $|y|$ and satisfies some other conditions. Under certain restrictions on the perturbation $c$, we give results on the Fourier integral operator structure of the scattering matrix. Moreover, we show that we can recover the asymptotic expansion at infinity of $c$ from knowledge of $c_0$ and the singularities of the scattering matrix at fixed energy.
LA - eng
UR - http://eudml.org/doc/93399
ER -

References

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  1. [1] I. Beltiţă, Inverse scattering in a layered medium, C.R. Acad. Sci Paris Sér. I Math 329 (1999), no. 10, 927-932. Zbl0941.35134MR2000i:35156
  2. [2] A. Boutet de Monvel-Berthier and D. Manda, Spectral and scattering theory for wave propagation in perturbed stratified media, J. Math. Analysis and Applications 191 (1995), 137-167. Zbl0831.35119MR96d:35102
  3. [3] T. Christiansen, Scattering theory for perturbed stratified media. Journal d'Analyse Mathématique 76 (1998), 1-44. Zbl0926.35106MR2000a:35189
  4. [4] S. DeBièvre and D.W. Pravica, Spectral analysis for optical fibres and stratified fluids II: absence of eigenvalues, Commun. Partial Differential Equations 17 (1&2) (1992), 69-97. Zbl0850.35067MR93c:35114
  5. [5] Y. Dermenjian and J.-C. Guillot, Théorie spectrale et la propagation des ondes acoustiques dans un milieu stratifié perturbé, J. Differential Equations 62 No. 3 (1986), 357-409. Zbl0611.35063MR87h:35291
  6. [6] C. Gérard, H. Isozaki, and E. Skibsted, Commutator algebra and resolvent estimates, volume 23 of Advanced studies in pure mathematics, p. 69-82, 1994. Zbl0814.35086MR95h:35154
  7. [7] J.-C. Guillot and J. Ralston, Inverse scattering at fixed energy for layered media, J. Math. Pures Appl. (9) 78 (1999), 27-48. Zbl0930.35117MR99m:35262
  8. [8] S. Helgason, Groups and Geometric Analysis, Academic Press, Orlando, 1984. Zbl0543.58001MR86c:22017
  9. [9] H. Isozaki, Inverse scattering for wave equations in stratified media, J. Differential Equations 138 (1997), 19-54. Zbl0878.35084MR98f:35156
  10. [10] M.S. Joshi, Explicitly Recovering Asymptotics of Short Range Potentials, to appear in Communications on Partial Differential Equations. Zbl0963.35148
  11. [11] M.S. Joshi and A. Sá Barreto, Recovering Asymptotics of Short Range Potentials, Comm. Math. Phys. 193 (1998), 197-208. Zbl0920.58052MR99e:58195
  12. [12] M.S. Joshi and A. Sá Barreto, Recovering Asymptotics of Metrics from Fixed Energy Scattering Data, Invent. Math. 137 (1999) 127-143. Zbl0953.58025MR2000m:58052
  13. [13] R.B. Melrose, Spectral and Scattering Theory for the Laplacian on Asymptotically Euclidean spaces, in Spectral and Scattering Theory (M. Ikawa, ed), p. 85-130, Marcel Dekker, New York, 1994. Zbl0837.35107MR95k:58168
  14. [14] R.B. Melrose and M. Zworski, Scattering Metrics and Geodesic Flow at Infinity, Invent. Math. 124 (1996), 389-436. Zbl0855.58058MR96k:58230
  15. [15] A. Vasy, Asymptotic behavior of generalized eigenfunctions in N-body scattering, J. Funct. Anal. 148 (1997), no. 1, 170-184. Zbl0884.35110MR98f:81344
  16. [16] A. Vasy, Structure of the resolvent for three-body potentials, Duke Math. J. 90 (1997), no. 2, 379-434. Zbl0891.35111MR98k:81295
  17. [17] R. Weder, Spectral and scattering theory for wave propagation in perturbed stratified media, Springer-Verlag, New York, 1991. Zbl0711.76083MR91j:35198
  18. [18] R. Weder, Multidimensional inverse problems in perturbed stratified media, J. Differential Equations 152 (1999), no. 1, 191-239. Zbl0922.35184MR99k:76133
  19. [19] C. Wilcox, Sound Propagation in Stratified Fluids, Applied Mathematical Sciences 50. Springer-Verlag, New York, Berlin, Heidelberg. Zbl0543.76107MR85i:76040

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