Transmission Eigenvalues and Upper Triangular Compactness
Seminar Room 1, Newton Institute
The interior transmission eigenvalue problem can be formulated as a 2x2 system of pdes, where one of the two unknown functions must satisfy too many boundary conditions, and the other too few. The system is not self-adjoint and the resolvent is not compact.
Under the hypothesis that the contrast satisfies a coercivity condition on the boundary of the domain, we show that the corresponding operator has Upper Triangular Compact Resolvent and that the analytic Fredholm theorem holds for such opertors.
As corollaries, we can show that the set of (complex) interior transmission eigenvalues is a (possibly empty) discrete set which depends contunuously on the contrast, and that eigenfunctions must be linearly independent. This is different from previous results because the contrast need not have a constant sign (or be real valued) in the interior of the domain.