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A reflection principle for solutions to the Helmholtz equation and an application to the inverse scattering problem†

Published online by Cambridge University Press:  18 May 2009

David Colton
David Colton
Affiliation:
University of Strathclyde, Glasgow, Scotland
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A classical result in potential theory is the Schwarz reflection principle for solutions of Laplace's equation which vanish on a portion of a spherical boundary. The question naturally arises whether or not such a property is also true for solutions of the Helmholtz equation. This has been answered in the affirmative by Diaz and Ludford ([4]; see also [10]) in the limiting case of the plane. It is the purpose of this paper to show that a reflection principle is also valid for spheres of finite radius. As an application of this result we shall study the problem of the analytic continuation of solutions to the Helmholtz equation defined in the exterior of a bounded domain in three-dimensional Euclidean space ℝ3 We shall show that through the use of the reflection principle derived in this paper, this problem can be reduced to the problem of the analytic continuation of an analytic function of two complex variables, which in turn can be performed through a variety of known methods (cf. [7]).

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 18 , Issue 2 , July 1977 , pp. 125 - 130
Copyright
Copyright © Glasgow Mathematical Journal Trust 1977

References

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