Inverse scattering problem with data at fixed energy and fixed incident direction Original Research Article

Let Aq(α′,α,k)Aq(α′,α,k) be the scattering amplitude, corresponding to a local potential q(x)q(x), x∈R3x∈R3, q(x)=0q(x)=0 for |x|>a|x|>a, where a>0a>0 is an arbitrary large fixed number, α′,α∈S2α′,α∈S2 are unit vectors, S2S2 is the unit sphere in R3R3, αα is the direction of the incident wave, k2>0k2>0 is the energy. We prove that given an arbitrary function f(α′)∈L2(S2)f(α′)∈L2(S2), an arbitrary fixed α0∈S2α0∈S2, an arbitrary fixed k>0k>0, and an arbitrary small ε>0ε>0, there exists a potential q(x)∈L2(D)q(x)∈L2(D) where D⊂R3D⊂R3 is a bounded domain such that equation(∗∗ Turn MathJax on ) ‖Aq(α′,α0,k)−f(α′)‖L2(S2)<ε.‖Aq(α′,α0,k)−f(α′)‖L2(S2)<ε. Turn MathJax on The potential qq, for which (∗) holds, is not unique. We give a method for finding such a qq, and a formula for this qq.