Spectral theory for Maxwell's equations at the interface of a metamaterial. Part I: Generalized Fourier transform

TL;DR

This paper develops a generalized Fourier transform to analyze the spectral properties of Maxwell's equations at a metamaterial-vacuum interface, facilitating future proofs of absorption and amplitude principles.

Contribution

It introduces an explicit generalized Fourier transform that diagonalizes the Maxwell Hamiltonian at the interface, a novel approach for spectral analysis in this context.

Findings

Constructed a generalized Fourier transform for the interface problem.

Diagonalized the Maxwell Hamiltonian using this transform.

Laid groundwork for proving absorption and amplitude principles.

Abstract

We explore the spectral properties of the time-dependent Maxwell's equations for a plane interface between a metamaterial represented by the Drude model and the vacuum, which fill respectively complementary half-spaces. We construct explicitly a generalized Fourier transform which diagonalizes the Hamiltonian that describes the propagation of transverse electric waves. This transform appears as an operator of decomposition on a family of generalized eigenfunctions of the problem. It will be used in a forthcoming paper to prove both limiting absorption and limiting amplitude principles.