Completeness of the set \(\{e^{ikβ · s}\}|_{\forall β \in S^2}\) - Ramm, Alexander G. (Global Journal of Mathematical Analysis, 2017)
Let Â \(S^2\) be the unit sphere in \(\mathbb{R}^3\), Â \(k>0\) be a fixed constant, \(s\in S\), and \(S\) is a smooth, closed, connected surface, the boundary of a bounded domain \(D\) in \(\mathbb{R}^3\). It is proved that the set \(\{e^{ikβ · s}\}|_{\forall β \in S^2}\) is total in \(L^2(S)\) if and only if \(k^2\) is not a Dirichlet eigenvalue of the Laplacian in \(D\).