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Spectral analysis of the interior transmission eigenvalue problem. (English) Zbl 1296.35105

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.

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

Citations:

Zbl 1054.35035
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