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

Let Aq(α′,α,k) be the scattering amplitude, corresponding to a local potential q(x), x∈R3, q(x)=0 for |x|>a, where a>0 is an arbitrary large fixed number, α′,α∈S2 are unit vectors, S2 is the unit sphere in R3, α is the direction of the incident wave, k2>0 is the energy. We prove that given an arbitrary function f(α′)∈L2(S2), an arbitrary fixed α0∈S2, an arbitrary fixed k>0, and an arbitrary small ε>0, there exists a potential q(x)∈L2(D) where D⊂R3 is a bounded domain such that (∗)‖Aq(α′,α0,k)−f(α′)‖L2(S2)<ε. The potential q, for which (∗) holds, is not unique. We give a method for finding such a q, and a formula for this q.