Dispersive Estimates for Maxwell's Equations in the Exterior of a Sphere

TL;DR

This paper establishes high frequency dispersive estimates for Maxwell's equations outside a conducting sphere, introducing new eigenfunctions and analyzing decay rates for electromagnetic wave propagators.

Contribution

It constructs novel generalized eigenfunctions for Maxwell's propagator and analyzes decay rates, revealing polarization-dependent scattering behavior.

Findings

Electric field propagator decays at a specific rate in $L^1-L^{ {infty}}$ norm.

Some polarizations scatter similarly to wave operators, others do not.

Dirichlet Laplacian estimates do not fully apply to Maxwell's equations in this setting.

Abstract

The goal of this article is to establish general principles for high frequency dispersive estimates for Maxwell's equation in the exterior of a perfectly conducting ball. We construct entirely new generalized eigenfunctions for the corresponding Maxwell propagator. We show that the propagator corresponding to the electric field has a global rate of decay in $L^1-L^{\infty}$ operator norm in terms of time $t$ and powers of $h$. In particular we show that some, but not all, polarizations of electromagnetic waves scatter at the same rate as the usual wave operator. The Dirichlet Laplacian wave operator $L^1-L^{\infty}$ norm estimate should not be expected to hold in general for Maxwell's equations in the exterior of a ball because of the Helmholtz decomposition theorem.