# High-frequency approximation of the interior dirichlet-to-neumann map and applications to the transmission eigenvalues

## TL;DR

This paper investigates the high-frequency approximation of the Dirichlet-to-Neumann map on Riemannian manifolds and uses this to analyze the distribution of transmission eigenvalues, achieving optimal eigenvalue-free regions.

## Contribution

It introduces a novel high-frequency approximation method for the Dirichlet-to-Neumann map on arbitrary Riemannian manifolds and applies it to determine transmission eigenvalue locations.

## Key findings

- Approximation of the Dirichlet-to-Neumann map away from the real axis.
- Identification of eigenvalue-free regions in the complex plane.
- Enhanced understanding of transmission eigenvalues on Riemannian manifolds.

## Abstract

We study the high-frequency behavior of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a non-empty 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.