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| It is well-known that the interior eigenvalues of a bounded domain share connections to scattering problems in the exterior of this domain. For instance, certain boundary integral equations for exterior scattering problems fail at interior eigenvalues.
Similar connections also exist for inverse exterior scattering problems - for instance, if zero is an eigenvalue of the far-field operator at a fixed wave number, then the squared wave number is an interior eigenvalue. Despite it is in general wrong that interior eigenvalues correspond to zero being an eigenvalue of the far field operator, one can prove a pretty direct characterization of interior eigenvalues via the behavior of the phases of the eigenvalues of the far-field operator.
We present such a characterization of transmission eigenvalues for penetrable media governed by the H-mode equations $\mathrm{div}(A \nabla u) + k^2 u = 0$ or the Maxwell's equations $\mathrm{curl} (A \mathrm{curl} u) - k^2 u = 0$. |
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