Abstract
We study the semiclassical microlocal structure of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a nonempty smooth boundary. We build a new, improved parametrix in the glancing region compaired with that one built in [Vodev, Commun. Math. Phys. 2015;336(3):1141–1166; Vodev, Asymptotic Anal. 2018;106:147–168]. We also study the way in which the parametrix depends on the refraction index. As a consequence, we improve the transmission eigenvalue-free regions obtained in (Vodev, Asymptotic Anal. 2018;106:147–168) in the isotropic case when the restrictions of the refraction indices on the boundary coincide.
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