Let f∈L2(S2) be an arbitrary fixed function with small norm 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(\alpha')-A_q(\alpha')_{L^2(S^2)}\|\leq\ve where
\ve>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"