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2019 | OriginalPaper | Chapter

10. Transmission Eigenvalues

Authors : David Colton, Rainer Kress

Published in: Inverse Acoustic and Electromagnetic Scattering Theory

Publisher: Springer International Publishing

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Abstract

The transmission eigenvalue problem was previously introduced in Sect. 8.​4 where it was shown to play a central role in establishing the completeness of the set of far field patterns in \(L^2(\mathbb {S}^2)\). It was then shown in Sect. 8.​6 that the set of transmission eigenvalues was either empty or formed a discrete set, thus leading to the conclusion that except possibly for a discrete set of values of the wave number k > 0, the set of far field patterns is complete in \(L^2(\mathbb {S}^2)\). In this chapter we return to the subject of transmission eigenvalues and consider further topics of interest. In particular, we begin by showing the existence of transmission eigenvalues and then deriving a monotonicity result for the first positive transmission eigenvalue. We then proceed to describe a boundary integral equation approach to the transmission eigenvalue problem, the existence of complex transmission eigenvalues in the case of a spherically stratified medium, and the inverse spectral problem for the case of such a medium. We conclude this chapter by considering a modified transmission eigenvalue problem in which the wave number k > 0 is kept fixed and the eigenparameter is now an artificial coefficient introduced through the use of a modified far field operator. Our analysis is restricted to the case of acoustic waves.

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previous chapter Electromagnetic Waves in an Inhomogeneous Medium
next chapter The Inverse Medium Problem
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Metadata
Title
Transmission Eigenvalues
Authors
David Colton
Rainer Kress
Copyright Year
2019
Book
Inverse Acoustic and Electromagnetic Scattering Theory

Print ISBN: 978-3-030-30350-1

Electronic ISBN: 978-3-030-30351-8

Copyright Year: 2019

DOI
https://doi.org/10.1007/978-3-030-30351-8_10

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