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

Alexander G. Ramm

doi:10.14419/gjma.v5i2.7975

Article Details

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

Authors:

Alexander G. Ramm

Publisher:

Science Publishing Corporation

Journal:

Global Journal of Mathematical Analysis Vol.5 Issue 2, pp.43-43

Document Type:

Article

Publication Date:

7/2017

Document Language:

English

Document ID (Org No.):

2001385550

pISSN:

2307-9002

DOI:

10.14419/gjma.v5i2.7975

Keywords:

Completeness

Abstract:

 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 \(\{e^{ik\beta \cdot s}\}|_{\forall \beta \in S^2}\) is total in \(L^2(S)\) if and only if \(k^2\) is not a Dirichlet eigenvalue of the Laplacian in \(D\).

Article Details

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

Authors: Alexander G. Ramm

Publisher: Science Publishing Corporation

Journal: Global Journal of Mathematical Analysis Vol.5 Issue 2, pp.43-43

Document Type: Article

Publication Date: 7/2017

Document Language: English

Document ID (Org No.): 2001385550

pISSN: 2307-9002

DOI: 10.14419/gjma.v5i2.7975

Completeness

Issues

Authors:

Alexander G. Ramm

Publisher:

Science Publishing Corporation

Journal:

Global Journal of Mathematical Analysis Vol.5 Issue 2, pp.43-43

Document Type:

Article

Publication Date:

7/2017

Document Language:

English

Document ID (Org No.):

2001385550

pISSN:

2307-9002

DOI:

10.14419/gjma.v5i2.7975