Completeness of the set of scattering amplitudes

Abstract

Let $f\in L^2(S^2)$ be an arbitrary fixed function with small norm on the unit sphere $S^2$, and $D\subset \R^3$ be an arbitrary fixed bounded domain. Let $k>0$ and $α\in S^2$ be fixed. It is proved that there exists a potential $q\in L^2(D)$ such that the corresponding scattering amplitude $A(α')=A_q(α')=A_q(α',α,k)$ approximates $f(α')$ with arbitrary high accuracy: $\|f(α')-A_q(α')_{L^2(S^2)}\|\leq\ve$ where $\ve>0$ is an arbitrarily small fixed number. This means that the set $\{A_q(α')\}_{\forall q\in L^2(D)}$ is complete in $L^2(S^2)$. The results can be used for constructing nanotechnologically "smart materials".