Recovering Asymptotics at Infinity of Perturbations of Stratified Media
Christiansen, Tanya ; Joshi, Mark S.
Journées équations aux dérivées partielles, (2000), p. 1-9 / Harvested from Numdam
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We consider perturbations of a stratified medium ℝ x n-1 ×ℝ 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.

@article{JEDP_2000____A2_0,
     author = {Christiansen, Tanya and Joshi, Mark S.},
     title = {Recovering Asymptotics at Infinity of Perturbations of Stratified Media},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     year = {2000},
     pages = {1-9},
     mrnumber = {2001g:35261},
     zbl = {01808692},
     language = {en},
     url = {http://dml.mathdoc.fr/item/JEDP_2000____A2_0}
}

Christiansen, Tanya; Joshi, Mark S. Recovering Asymptotics at Infinity of Perturbations of Stratified Media. Journées équations aux dérivées partielles,  (2000), pp. 1-9. http://gdmltest.u-ga.fr/item/JEDP_2000____A2_0/

[1] I. Beltiţă, Inverse scattering in a layered medium, C.R. Acad. Sci Paris Sér. I Math 329 (1999), no. 10, 927-932. | MR 2000i:35156 | Zbl 0941.35134

[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. | MR 96d:35102 | Zbl 0831.35119

[3] T. Christiansen, Scattering theory for perturbed stratified media. Journal d'Analyse Mathématique 76 (1998), 1-44. | MR 2000a:35189 | Zbl 0926.35106

[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. | MR 93c:35114 | Zbl 0850.35067

[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. | MR 87h:35291 | Zbl 0611.35063

[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. | MR 95h:35154 | Zbl 0814.35086

[7] J.-C. Guillot and J. Ralston, Inverse scattering at fixed energy for layered media, J. Math. Pures Appl. (9) 78 (1999), 27-48. | MR 99m:35262 | Zbl 0930.35117

[8] S. Helgason, Groups and Geometric Analysis, Academic Press, Orlando, 1984. | MR 86c:22017 | Zbl 0543.58001

[9] H. Isozaki, Inverse scattering for wave equations in stratified media, J. Differential Equations 138 (1997), 19-54. | MR 98f:35156 | Zbl 0878.35084

[10] M.S. Joshi, Explicitly Recovering Asymptotics of Short Range Potentials, to appear in Communications on Partial Differential Equations. | Zbl 0963.35148

[11] M.S. Joshi and A. Sá Barreto, Recovering Asymptotics of Short Range Potentials, Comm. Math. Phys. 193 (1998), 197-208. | MR 99e:58195 | Zbl 0920.58052

[12] M.S. Joshi and A. Sá Barreto, Recovering Asymptotics of Metrics from Fixed Energy Scattering Data, Invent. Math. 137 (1999) 127-143. | MR 2000m:58052 | Zbl 0953.58025

[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. | MR 95k:58168 | Zbl 0837.35107

[14] R.B. Melrose and M. Zworski, Scattering Metrics and Geodesic Flow at Infinity, Invent. Math. 124 (1996), 389-436. | MR 96k:58230 | Zbl 0855.58058

[15] A. Vasy, Asymptotic behavior of generalized eigenfunctions in N-body scattering, J. Funct. Anal. 148 (1997), no. 1, 170-184. | MR 98f:81344 | Zbl 0884.35110

[16] A. Vasy, Structure of the resolvent for three-body potentials, Duke Math. J. 90 (1997), no. 2, 379-434. | MR 98k:81295 | Zbl 0891.35111

[17] R. Weder, Spectral and scattering theory for wave propagation in perturbed stratified media, Springer-Verlag, New York, 1991. | MR 91j:35198 | Zbl 0711.76083

[18] R. Weder, Multidimensional inverse problems in perturbed stratified media, J. Differential Equations 152 (1999), no. 1, 191-239. | MR 99k:76133 | Zbl 0922.35184

[19] C. Wilcox, Sound Propagation in Stratified Fluids, Applied Mathematical Sciences 50. Springer-Verlag, New York, Berlin, Heidelberg. | MR 85i:76040 | Zbl 0543.76107