We study the high-frequency behaviour of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a nonempty smooth boundary. We show that far from the real axis it can be approximated by a simpler operator. We use this fact to get new results concerning the location of the transmission eigenvalues on the complex plane. In some cases we obtain optimal transmission eigenvalue-free regions.
"High-frequency approximation of the interior Dirichlet-to-Neumann map and applications to the transmission eigenvalues." Anal. PDE 11 (1) 213 - 236, 2018. https://doi.org/10.2140/apde.2018.11.213