Completeness of the set \(\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}\)
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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 \(D\).
2019 ◽
Vol 22
(5)
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pp. 1414-1436
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2006 ◽
Vol 11
(4)
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pp. 323-329
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2022 ◽
Vol 63
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pp. 103397
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