Scattering on stratified media: the microlocal properties of the scattering matrix and recovering asymptotics of perturbations
Annales de l'Institut Fourier, Volume 53 (2003) no. 2, p. 565-624
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.
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.
DOI : https://doi.org/10.5802/aif.1953
Classification:  35P25,  81U40,  35S30
Keywords: stratified media, scattering matrix, inverse problems, limiting absorption principle
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {2},
     year = {2003},
     pages = {565-624},
     doi = {10.5802/aif.1953},
     mrnumber = {1990007},
     zbl = {01940705},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2003__53_2_565_0}
}
Scattering on stratified media: the microlocal properties of the scattering matrix and recovering asymptotics of perturbations. Annales de l'Institut Fourier, Volume 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), Univ. Nantes, Nantes, 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, Springer-Verlag, New York (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, Marcel Dekker, New York (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), Univ. Nantes, Nantes, 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, Springer-Verlag, New York, Berlin, Heidelberg, Applied Mathematical Sciences, Tome 50 | MR 742932 | Zbl 0543.76107