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Complex eigenvalues and the inverse spectral problem for transmission eigenvalues

We continue our investigation of complex eigenvalues of the interior transmission problem for spherically stratified media (Leung and Colton 2012 Inverse Problems 28 075005). In this paper we show that if complex transmission values exist for a spherically stratified medium with (normalized) support in {x: |x| ⩽ 1} then they must lie in a strip containing the real axis. We also give a new and shorter proof of the result of Aktosun et al (2011 Inverse Problems 27 115004), showing that a knowledge of all the transmission eigenvalues (real and complex) uniquely determine the index of refraction provided 0 < η(r) < 1 for 0 < r < 1 and η(0) is known.