Completeness of the set of scattering amplitudes

Details Completeness of the set of scattering amplitudes Completeness of the set of scattering amplitudes

2006-04-14

arxiv.org/abs/math-ph/0604030v2

Physics

Mathematical Physics

Scientific paper

10.1016/j.physleta.2006.07.054

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 $\alpha\in S^2$ be fixed. It is proved that there exists a potential $q\in L^2(D)$ such that the corresponding scattering amplitude $A(\alpha')=A_q(\alpha')=A_q(\alpha',\alpha,k)$ approximates $f(\alpha')$ 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 $\{A_q(\alpha')\}_{\forall q\in L^2(D)}$ is complete in $L^2(S^2)$. The results can be used for constructing nanotechnologically "smart materials".

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