A Rellich type theorem for discrete Maxwell operators

TL;DR

This paper establishes a Rellich type theorem and unique continuation results for discrete Maxwell operators with constant anisotropic media on the integer lattice, showing that solutions vanishing outside a compact set must vanish near infinity.

Contribution

It extends Rellich type theorems and unique continuation principles to discrete Maxwell operators with anisotropic media, including perturbed cases on ${f Z}^3$.

Findings

Rellich type theorem holds for discrete Maxwell operators with anisotropic media.

Unique continuation theorem is established for perturbed Maxwell operators.

Solutions satisfying certain conditions vanish near infinity if they vanish outside a compact set.

Abstract

We study the Rellich type theorem (RT) for the Maxwell operator $\hat H^D=\hat D\hat H_0$ on ${\bf Z}^3$ with constant anisotropic medium, i.e. the permittivity and permeability of which are non-scalar constant diagonal matrices. We also study the unique continuation theorem for the perturbed Maxwell operator $\hat H^{D_p}=\hat {D}_p\hat H_0$ on ${\bf Z}^3$ where the permittivity and permeability are locally perturbed from a constant matrix on a compact set in ${\bf Z}^3$. It then implies that, if $\hat H^D- \lambda$ satisfies (RT), then all distributions $\hat u$ in Besov space $\mathcal B_0^{\ast}({\bf Z}^3;{\bf C}^3)$ satisfying the equation $(\hat H^{D_p} - \lambda) \hat u = 0$ outside a compact set vanish near infinity.