98-224 Guillot J. C., Ralston J.

Inverse Scattering at Fixed Energy for Layered Media (64K, AMSTeX) Mar 24, 98

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Abstract. \magnification=1200 \noindent {\bf Inverse Scatterng at Fixed Energy for Layered Media} \medskip In this article we show that exponentially decreasing perturbations of the sound speed in a layered medium can be recovered from the scattering amplitude at fixed energy. We consider the unperturbed equation $u _{tt} = c_0^2(x_n)\Delta u$ in $ R \times R^n$, where $n \geq 3$. 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 speed, $c(x)$, 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.{\bf 138}) on the case where $c_0$ takes the constant values $c_+ $ and $c_-$ on the positive 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). \end

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