Transmission Eigenvalues

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 L2(𝕊2). 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 L2(𝕊2). 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.