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

Author: Karamyan G.

Source: Inverse Problems, Volume 18, Number 6, 2002 , pp. 1673-1686(14)

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Abstract:

In this paper we consider the acoustic equation c² (x)u + k²u 0, x₊ⁿ, n 3, where 0 < c₁c² (x) c₂, c(x) - 1 O(e^-|x|), > 0 and ₊ⁿ {(x', x_n) ⁿ : x_n 0}. This equation can be rewritten in the form -u - k²u + q(x)u 0, where q(x) k² (1 - c^-2 (x)) O(e^-|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).