Completeness of the set $\{e^{ik\beta \cdot s}\}{|}_{\forall \beta \in S^2}$

Abstract

It is proved that the set $\{e^{ik\beta \cdot s}\}{|}_{\forall \beta \in S^2}$, where $S^2$ is the unit sphere in ${\mathbb{R}}^3$, $k>0$ is a fixed constant, $k^2$ is not a Dirichlet eigenvalue of the Laplacian in $D$, $s\in S$, is total in $L^2(S)$. Here $S$ is a smooth, closed, connected surface in ${\mathbb{R}}^3$.