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

TL;DR

This paper develops an improved semi-classical parametrix for the Dirichlet-to-Neumann map in the glancing region, enhancing understanding of its microlocal structure and dependence on the refraction index.

Contribution

It introduces a new, refined parametrix in the glancing region and analyzes its dependence on the refraction index, improving previous results on transmission eigenvalue-free regions.

Findings

Enhanced parametrix in the glancing region compared to previous models

Detailed analysis of the parametrix's dependence on the refraction index

Improved transmission eigenvalue-free regions in the isotropic case

Abstract

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.