Improved parametrix in the glancing region for the interior Dirichlet-to-Neumann map

We study the semi-classical microlocal structure of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a non-empty smooth boundary. We build a new, improved parametrix in the glancing region compaired with that one built in [9], [12]. 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 [12] in the isotropic case when the restrictions of the refraction indices on the boundary coincide.