Inverse scattering at fixed energy for layered media

In this article we show that exponentially decreasing perturbations of the sound speed in a layered medium can be recovered from the scattering amplit ude at fixed energy. We consider the unperturbed equation u(tt) = c(0)(2)(x (n))Delta u in R x R-n, where n greater than or equal to 3. The unperturbed sound speed, c(0)(x(n)), is assumed to be bounded, strictly positive, and constant outside a bounded interval on the real axis. The perturbed sound s peed, c(x), satisfies \c(x) - c(0)(x(n))\ < C exp(-delta\x\) for some delta > 0. Our work is related to the recent results of H. Isozaki (J. Diff. Eq. 138) on the case where co takes the constant values c(+) and c(-) on the p ositive and negative half-lines, and R. Weder on the case c(0) = c(+) for x (n) > h, c(0) = c(h) for 0 < x(n) < h, and c(0) = c(-) for x(n) < 0 (IIMAS- UNAM Preprint 70, November, 1997). (C) Elsevier, Paris.