Semiclassical parametrix for the Maxwell equation and applications to the electromagnetic transmission eigenvalues

TL;DR

This paper develops a semiclassical parametrix for the Maxwell equation, enabling the approximation of the Dirichlet-to-Neumann map and identifying regions free of transmission eigenvalues.

Contribution

It introduces a pseudodifferential operator approximation for the Maxwell Dirichlet-to-Neumann map and computes its principal symbol, advancing spectral analysis techniques.

Findings

Derived a pseudodifferential operator approximation for the Maxwell Dirichlet-to-Neumann map

Computed the principal symbol of the operator

Identified a parabolic region free of transmission eigenvalues

Abstract

We introduce an analog of the Dirichlet-to-Neumann map for the Maxwell equation in a bounded domain. We show that it can be approximated by a pseudodifferential operator on the boundary with a matrix-valued symbol and we compute the principal symbol. As a consequence, we obtain a parabolic region free of the transmission eigenvalues associated to the Maxwell equation.