Completeness of the set of scattering amplitudes

TL;DR

The paper proves that the set of scattering amplitudes generated by potentials in a bounded domain is complete in the space of square-integrable functions on the sphere, enabling high-accuracy approximations of arbitrary functions.

Contribution

It demonstrates the existence of potentials that produce scattering amplitudes capable of approximating any function in L^2(S^2) arbitrarily closely, establishing completeness.

Findings

The set of scattering amplitudes is complete in L^2(S^2).

Any function in L^2(S^2) can be approximated with arbitrary accuracy by scattering amplitudes.

Results have potential applications in designing nanotechnological materials.

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 $\alpha\in S^2$ 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".