Scattering on stratified media: the microlocal properties of the scattering matrix and recovering asymptotics of perturbations  [ Diffusion dans un milieu stratifié : les propriétés microlocales de la matrice de diffusion et l’obtention du comportement asymptotique des perturbations ]
Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 565-624.

On définit la matrice de diffusion dans un milieu stratifié perturbé. Pour une classe de perturbations, on démontre que la partie principale est un opérateur intégral de Fourier sur la sphère à l’infini. On développe un principe d’absorption limite raffiné. Dans de nombreux cas, le symbole de la matrice de diffusion détermine le comportement asymptotique des perturbations.

The scattering matrix is defined on a perturbed stratified medium. For a class of perturbations, its main part at fixed energy is a Fourier integral operator on the sphere at infinity. Proving this is facilitated by developing a refined limiting absorption principle. The symbol of the scattering matrix determines the asymptotics of a large class of perturbations.

DOI : https://doi.org/10.5802/aif.1953
Classification : 35P25,  81U40,  35S30
Mots clés: milieu stratifié, matrice de diffusion, problèmes d’inversion, principe d’absorption limite
@article{AIF_2003__53_2_565_0,
     author = {Christiansen, Tanya and Joshi, M. S.},
     title = {Scattering on stratified media: the microlocal properties of the scattering matrix and recovering asymptotics of perturbations},
     journal = {Annales de l'Institut Fourier},
     pages = {565--624},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {2},
     year = {2003},
     doi = {10.5802/aif.1953},
     mrnumber = {1990007},
     zbl = {01940705},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2003__53_2_565_0/}
}
Christiansen, Tanya; Joshi, M. S. Scattering on stratified media: the microlocal properties of the scattering matrix and recovering asymptotics of perturbations. Annales de l'Institut Fourier, Tome 53 (2003) no. 2, pp. 565-624. doi : 10.5802/aif.1953. https://aif.centre-mersenne.org/item/AIF_2003__53_2_565_0/

[1] I. Beltiţă Inverse scattering in a layered medium, C.R. Acad. Sci Paris, Sér. I Math, Tome 329 (1999) no. 10, pp. 927-932 | Article | MR 1728010 | Zbl 0941.35134

[2] I. Beltiţă Inverse scattering in a layered medium, Comm. Partial Differential Equations, Tome 26 (2001) no. 9-10, pp. 1739-1786 | Article | MR 1865944 | Zbl 01721831

[3] M. Ben; - Artzi; Y. Dermenjian; J.-C. Guillot Acoustic waves in perturbed stratified fluids: a spectral theory, Comm. Partial Differential Equations, Tome 14 (1989) no. 4, pp. 479-517 | MR 989667 | Zbl 0675.35065

[4] A. Boutet de; Monvel; - Berthier; D. Manda Spectral and scattering theory for wave propagation in perturbed stratified media, J. Math. Anal. Appl., Tome 191 (1995), pp. 137-167 | MR 1323768 | Zbl 0831.35119

[5] T. Christiansen Scattering theory for perturbed stratified media, Journal d'Analyse Mathématique, Tome 76 (1998), pp. 1-44 | Article | MR 1676944 | Zbl 0926.35106

[6] T. Christiansen; M.S. Joshi Higher order scattering on asymptotically Euclidean manifolds, Canadian J. Math, Tome 52 (2000) no. 5, pp. 897-919 | Article | MR 1782333 | Zbl 0984.58019

[7] T. Christiansen; M.S. Joshi Recovering asymptotics at infinity of perturbations of stratified media, Équations aux Dérivées Partielles (La Chapelle sur Erdre, 2000), Tome Exp. No. II (2000), pp. 9 pp. | Numdam | Zbl 01808692

[8] A. Cohen; T. Kappeler Scattering and inverse scattering for steplike potentials in the Schrödinger equation, Indiana Univ. Math. J, Tome 34 (1985), pp. 127-180 | Article | MR 773398 | Zbl 0553.34015

[9] H.L. Cycon; R.G. Froese; W. Kirsch; B. Simon Schrödinger operators with application to quantum mechanics and global geometry, Springer-Verlag, Berlin, 1987 | MR 883643 | Zbl 0619.47005

[10] S. DeBièvre; D.W. Pravica Spectral analysis for optical fibres and stratified fluids I: the limiting absorption principle, J. Functional Analysis, Tome 98 (1991), pp. 404-436 | Article | MR 1111576 | Zbl 0731.35069

[11] S. DeBièvre; D.W. Pravica Spectral analysis for optical fibres and stratified fluids II: Absence of eigenvalues, Comm. Partial Differential Equations, Tome 17 (1992) no. 1-2, pp. 69-97 | Article | MR 1151257 | Zbl 0850.35067

[12] P. Deift; E. Trubowitz Inverse scattering on the line, Commun. Pure Appl. Math, Tome 32 (1979), pp. 121-251 | Article | MR 512420 | Zbl 0388.34005

[13] Y. Dermenjian; J.-C. Guillot Théorie spectrale de la propagation des ondes acoustiques dans un milieu stratifié perturbé, J. Differential Equations, Tome 62 (1986) no. 3, pp. 357-409 | Article | MR 837761 | Zbl 0611.35063

[14] C. Gérard; H. Isozaki; E. Skibsted Commutator algebra and resolvent estimates, Advanced Studies in Pure Mathematics, Tome 23 (1994), pp. 69-82 | MR 1275395 | Zbl 0814.35086

[15] J.-C. Guillot; J. Ralston Inverse scattering at fixed energy for layered media, J. Math. Pures Appl (9), Tome 78 (1999), pp. 27-48 | Article | MR 1671219 | Zbl 0930.35117

[16] S. Helgason Groups and Geometric Analysis, Academic Press, Orlando, 1984 | MR 754767 | Zbl 0543.58001

[17] B. Helffer; J. Sjöstrand; H. Holden and A. Jensen, eds. Equation de Schrödinger avec champ magnétique et équation de Harper, Schrödinger Operators (Lecture Notes in Phys.) Tome vol. 345, pp. 118-197 | Zbl 0699.35189

[18] L. Hörmander The Weyl calculus of pseudo-differential operators, Comm. Pure Appl. Math, Tome 32 (1979), pp. 359-443 | Article | MR 180740 | Zbl 0388.47032

[19] L. Hörmander The analysis of linear partial differential operators II, Springer-Verlag, Berlin, 1983 | MR 705278 | Zbl 0521.35002

[20] H. Isozaki Inverse scattering for wave equations in stratified media, Journal of Differential Equations, Tome 138 (1997), pp. 19-54 | Article | MR 1458455 | Zbl 0878.35084

[21] M.S. Joshi Recovering asymptotics of Coulomb-like potentials from fixed energy scattering data, S.I.A.M. J. Math. Anal., Tome 30 (1999) no. 3, pp. 516-526 | MR 1677941 | Zbl 0927.58016

[22] M.S. Joshi Explicitly recovering asymptotics of short range potentials, Comm. Partial Differential Equations, Tome 25 (2000) no. 9 \& 10, pp. 1907-1923 | Article | MR 1778785 | Zbl 0963.35148

[23] M.S. Joshi; A. Sá; Barreto Recovering asymptotics of short range potentials, Comm. Math. Phys, Tome 193 (1998), pp. 197-208 | Article | MR 1620321 | Zbl 0920.58052

[24] M.S. Joshi; A. Sá; Barreto Recovering asymptotics of metrics from fixed energy scattering data, Invent. Math, Tome 137 (1999), pp. 127-143 | Article | MR 1703335 | Zbl 0953.58025

[25] M.S. Joshi; A. Sá; Barreto Determining asymptotics of magnetic potentials from fixed energy scattering data, Asymptotic Analysis, Tome 21 (1999) no. 1, pp. 61-70 | MR 1718632 | Zbl 0934.35203

[26] R.B. Melrose; M. Ikawa, ed Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces, Spectral and Scattering Theory (1994), pp. 85-130 | Zbl 0837.35107

[27] R.B. Melrose; M. Zworski Scattering metrics and geodesic flow at infinity, Invent. Math., Tome 124 (1996), pp. 389-436 | Article | MR 1369423 | Zbl 0855.58058

[28] A. Vasy Asymptotic behavior of generalized eigenfunctions in N-body scattering, J. Funct. Anal, Tome 148 (1997) no. 1, pp. 170-184 | Article | MR 1461498 | Zbl 0884.35110

[29] A. Vasy Structure of the resolvent for three-body potentials, Duke Math. J, Tome 90 (1997) no. 2, pp. 379-434 | Article | MR 1484859 | Zbl 0891.35111

[30] A. Vasy Propagation of singularities in Euclidean many-body scattering in the presence of bound states, Journées Équations aux Dérivées Partielles (Saint-Jean-de-Monts, 1999), Tome Exp. No. XVI (1999), pp. 20 pp. | Numdam

[31] R. Weder The limiting absorption principle at thresholds, J. Math. Pures et Appl, Tome 67 (1988), pp. 313-338 | MR 978574 | Zbl 0611.76090

[32] R. Weder Spectral and Scattering Theory for Wave Propagation in Perturbed Stratified Media, Springer-Verlag, New York, 1991 | MR 1082152 | Zbl 0711.76083

[33] R. Weder Multidimensional inverse problems in perturbed stratified media, J. Differential Equations, Tome 152 (1999) no. 1, pp. 191-239 | Article | MR 1672028 | Zbl 0922.35184

[34] C. Wilcox Sound Propagation in Stratified Fluids, Applied Mathematical Sciences, Tome 50, Springer-Verlag, New York, Berlin, Heidelberg | MR 742932 | Zbl 0543.76107