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

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.