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Recovering Asymptotics at Infinity of Perturbations of Stratified Media

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Recovering Asymptotics at Infinity of Perturbations of Stratified Media JOURNÉES ÉQUATIONS AUX DÉRIVÉES PARTIELLES TANYA CHRISTIANSEN MARK S. JOSHI Recovering Asymptotics at Infinity of Perturbations of Stratified Media Journées Équations aux dérivées partielles (2000), p. 1-9. <http://www.numdam.org/item?id=JEDP_2000____A2_0> © Journées Équations aux dérivées partielles, 2000, tous droits réservés. L’accès aux archives de la revue « Journées Équations aux dérivées partielles » (http://www. math.sciences.univ-nantes.fr/edpa/), implique l’accord avec les conditions générales d’utili- sation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression sys- tématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fi- chier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Journees Equations aux derivees partielles Nantes, 5-9 juin 2000 GDR 1151 (CNRS) Recovering Asymptotics at Infinity of Perturbations of Stratified Media T. CHRISTIANSEN M.S. Josm Abstract We consider perturbations of a stratified medium R^"1 x Ry, where the operator studied is c2{x^ ^/)A. The function c is a perturbation of co(^/), which is constant for sufficiently large \y\ and satisfies some other conditions. Un- der 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 Co and the singularities of the scattering matrix at fixed energy. 1. Introduction In these lecture notes we describe the problem of recovering a sound speed c which is a perturbation of a stratified sound speed CQ. That is, Co depends on the variable y G R but the main operator one studies is c^, where A = — Z^ i -^-2- and z = {x,y) G R71"1 x R. The wave equation (—^- — c^)v = 0 models the propaga

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