The inverse scattering problem for the acoustic equation in a half-space

In this paper we consider the acoustic equation c(2)(x)Deltau + k(2)u = 0, x is an element of R-+(n), n greater than or equal to 3, where 0 < c(1) less than or equal to c(2)(x) less than or equal to c(2), c(x) - 1 = O(e(-delta\x\)), delta > 0 and R-+(n) {(x',x(n)) is an element of R-n : x(n) greater than or equal to 0}. This equation can be rewritten in the form -Deltau-k(2)u+q(x)u = 0, where q(x) = k(2)(1-c(-2)(x)) = O(e(-delta\x\)). Assuming that u(x) satisfies Robin boundary conditions on x(n) = 0, we prove that the scattering amplitude at fixed energy determines the potential function q(x).