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

  • Authors

    • Alexander G. Ramm Mathematics Department, Kansas State University, CW 207, Manhattan, KS 66506-2602, USA
    2017-07-15
    https://doi.org/10.14419/gjma.v5i2.7975
  • Completeness, Scattering Theory

  • Abstract

    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 \(D\).

  • References

    1. [1] A.G.Ramm, Scattering by obstacles, D.Reidel, Dordrecht, 1986.

      [2] A.G.Ramm, Inverse problems, Springer, New York, 2005.

      [3] A.G.Ramm, Solution to the Pompeiu problem and the related symmetry problem, Appl. Math. Lett., 63, (2017), 28-33.

      [4] A.G.Ramm, Perturbation of zero surfaces, Global Journ. of Math. Analysis, 5, (1), (2017), 27-28.

      [5] A.G.Ramm, Uniqueness of the solution to inverse obstacle scattering with non-over-determined data, Appl. Math. Lett., 58, (2016), 81-86.

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  • How to Cite

    Ramm, A. G. (2017). Completeness of the set \(\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}\). Global Journal of Mathematical Analysis, 5(2), 43-43. https://doi.org/10.14419/gjma.v5i2.7975

    Received date: 2017-06-12

    Accepted date: 2017-07-07

    Published date: 2017-07-15