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

Alexander G. Ramm
2017 Global Journal of Mathematical Analysis  
Let \(S^2\) be the unit sphere in \(\mathbb{R}^3\), \(k>0\) 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 \(\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}\) is total in \(L^2(S)\) if and only if \(k^2\) is not a Dirichlet eigenvalue of the Laplacian in \(D\).
doi:10.14419/gjma.v5i2.7975 fatcat:guhjigq2anbnnhpsfrslb3yb6u