Robbiano, Luc Spectral analysis of the interior transmission eigenvalue problem. (English) Zbl 1296.35105 Inverse Probl. 29, No. 10, Article ID 104001, 28 p. (2013). The author studies the following interior transmission eigenvalue problem \[ \begin{aligned} \Delta w+ k^2n(x)w= 0\quad &\text{in }\Omega,\\ \Delta v+ k^2v= 0\quad &\text{in }\Omega,\\ w=v\quad &\text{in }\partial\Omega,\\ \partial_\nu w=\partial_\nu v\quad &\text{on }\partial \Omega,\end{aligned} \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^n\). In application of the spectral theory of Hilbert-Schmidt operators it is proved that the spectrum is a discrete countable set and the generalized eigenfunctions span a dense subspace in the range of resolvent under weak conditions for the refraction index. Lower bounds of the resolvent are obtained by using coresponding results for the pseudospectrum by N. Dencker et al. [Commun. Pure Appl. Math. 57, No. 3, 384–415 (2004; Zbl 1054.35035)]. The semiclassical pseudodifferential calculus (which is partly presented within an Appendix) is used deeply. Reviewer: Wolfgang Sprößig (Freiberg) Cited in 38 Documents MSC: 35P25 Scattering theory for PDEs 35P15 Estimates of eigenvalues in context of PDEs 35R20 Operator partial differential equations (= PDEs on finite-dimensional spaces for abstract space valued functions) 35S05 Pseudodifferential operators as generalizations of partial differential operators Keywords:spectral theory; lower eigenvalue estimations; interior transmission problems; Hilbert-Schmidt operators; semiclassical pseudodifferential calculus Citations:Zbl 1054.35035 PDFBibTeX XMLCite \textit{L. Robbiano}, Inverse Probl. 29, No. 10, Article ID 104001, 28 p. (2013; Zbl 1296.35105) Full Text: DOI arXiv