Inverse scattering problem with data at fixed energy and fixed incident direction

Let A(q) (alpha', alpha, k) be the scattering amplitude, corresponding to a local potential q(x), x epsilon R-3, q(x) = 0 for vertical bar x vertical bar > a, where a > 0 is an arbitrary large fixed number, alpha', alpha epsilon S-2 are unit vectors, S-2 is the unit sphere in R-3, alpha is the direction of the incident wave, k(2) > 0 is the energy. We prove that given an arbitrary function f(alpha') epsilon L-2(S-2), an arbitrary fixed alpha(0) epsilon S-2, an arbitrary fixed k > 0, and an arbitrary small epsilon > 0, there exists a potential q(x) epsilon L-2(D) where D subset of R-3 is a bounded domain such that parallel to A(q)(alpha', alpha(0,) k) - f(alpha')parallel to(L2(S2)) < epsilon. The potential q, for which (*) holds, is not unique. We give a method for finding such a q, and a formula for this q. (C) 2007 Elsevier Ltd. All rights reserved.