Completeness of the set of scattering amplitudes

Let f∈L2(S2) be an arbitrary fixed function on the unit sphere S2, and D⊂R3 be an arbitrary fixed bounded domain. Let k>0 and α∈S2 be fixed. It is proved that there exists a potential q∈L2(D) such that the corresponding scattering amplitude A(α′)=Aq(α′)=Aq(α′,α,k) approximates f(α′) with arbitrary high accuracy: ‖f(α′)−Aq(α′)L2(S2)‖⩽ε where ε>0 is an arbitrarily small fixed number. This means that the set {Aq(α′)}∀q∈L2(D) is complete in L2(S2). The results can be used for constructing nanotechnologically “smart materials”.