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