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\begin{document}

\title{Localization of Classical Waves II:\\
Electromagnetic Waves.}
\author{Alexander Figotin\thanks{%
This author was supported  by the U.S. Air Force Grant F49620-94-1-0172.} \\
Department of Mathematics\\
University of North Carolina at Charlotte\\
Charlotte, NC 28223\\
figotin@mosaic.uncc.edu
 \and Abel Klein
\thanks{This author was supported in part by the NSF Grant DMS-9500720 .} \\
Department of Mathematics\\
University of California, Irvine\\
Irvine, CA 92697-3875\\
aklein@math.uci.edu}

\date{}
\maketitle

\begin{abstract}
We consider electromagnetic  waves in a medium  described by a  position dependent dielectric
constant $\varepsilon (x)$. We assume that $\varepsilon (x)$ is a  random perturbation
of a periodic function $\varepsilon_{0}(x)$ and   that the periodic
Maxwell operator ${\bf M}_{0} =\nabla^{\times} \frac 1{\varepsilon_0 (x)}\nabla^{\times} $ 
has a gap in the spectrum, were $\nabla^{\times} \Psi =\nabla {\times} \Psi $. We prove the existence of localized waves, i.e., finite energy solutions
of Maxwell's equations with the property that
almost all of the wave's energy remains in a fixed bounded region of space at all times. 
Localization of electromagnetic waves is a consequence of Anderson localization for the   self-adjoint operators
${\bf M} =\nabla^{\times} \frac 1{\varepsilon(x)}\nabla^{\times}$.  
We prove that, in the random medium
described by $\varepsilon(x)$, the random operator ${\bf M}$ exhibits Anderson localization
 inside the gap in the spectrum of  ${\bf M}_{0}$ .  This is shown even in situations  when the gap is totally
 filled by the spectrum of the random operator;  we can  prescribe random environments that ensure 
 localization in almost the whole gap. 
\end{abstract}


\tableofcontents

\section{Introduction}

This is the second of a series of papers on the localization of classical waves.  In
 the first paper  we discussed some general aspects of the localization
 of classical waves, and proved the existence of localized acoustic waves in  appropriate
 random media \cite{FKl3}.  The present  paper is concerned with the localization of
electromagnetic waves.   This phenomenon  arises from coherent multiple scattering and
interference, when the scale of the coherent multiple scattering reduces to
the wavelength itself. It has numerous potential applications (e.g., \cite
{DE,J91,J93,VP,JMW}), for instance, the optical transistor, which explain the recent
interest in the localization of light. 

Although the localization of light has a lot in common with the
localization of acoustic waves, the vector nature of electromagnetic waves
poses additional problems for the appropriate arguments, let alone their
numerical implementation. (For a discussion of the failure of standard
arguments to work for classical waves see \cite{A1}.) In this paper we
develop adequate tools in order to prove the  localization of
electromagnetic waves, in a randomly perturbed, lossless periodic dielectric
 medium with a gap in the spectrum. These tools include interior estimates for the
intensity of the electromagnetic field components, properties of an
electromagnetic analog of Dirichlet problems in finite domains, bounds on
 traces of the Green's functions associated with the relevant Maxwell
operators, existence of polynomially bounded generalized eigenfunctions,
 exponential decay of the Green's functions of the underlying
periodic medium if the frequency falls in a spectral gap, Wegner-type estimates
of the density of states, and more. After all these preparations the proof of
 localization goes along the same guidelines as in the case of acoustic
waves \cite{FKl3}. The multiscale analysis developed
 in \cite{FKl3}, based on studies of Anderson localization for random Schr\"odinger
operators \cite{FS,FMSS,DK,S,CH}, is  extended to the case of electromagnetic waves,
 using the new technical tools.  As far as the essence of the localization
phenomenon is concerned, it remains the same. 
As in the case of electron waves, a strong enough single defect in a periodic
dielectric medium with a spectral gap generates exponentially localized
eigenmodes \cite{FKl4}. If we have a random array of such defects then, under some
natural conditions, the electromagnetic wave tunneling becomes inefficient
(that is the main result of the multiscale analysis) and, hence,
Anderson localization of electromagnetic waves occurs in spectral gaps of
the underlying periodic medium. 

To create an environment which would favor
localization, one considers first a perfectly periodic dielectric medium
 (a ``photonic crystal", e.g., \cite{JMW}), such that
 the associated spectrum  has band
gap structure;  the most significant manifestation of  coherent
multiple scattering is the rise of a gap in the spectrum (``photonic band gaps"). If  such a
periodic medium with a gap in the spectrum is slightly randomized, eigenvalues 
 with exponentially localized eigenfunctions should arise in the gap. If the disorder
is increased further within some limits the localized states can fill the
gap completely. {\em This is exactly the medium in which we  study
 electromagnetic  waves;  we 
assume an underlying periodic dielectric medium with a gap in the spectrum}.
  We will slightly randomize 
such periodic media with a gap in the spectrum and show that, under pretty reasonable hypotheses,  
Anderson localization  occurs in a vicinity of the edges of the gap. (The existence of periodic dielectric media
exhibiting gaps in the spectrum has been proved rigorously for $2D$-periodic
dielectric structures \cite{FKu,FKu1}.)

We previously considered these questions and media in a lattice approximation, both for classical waves
 \cite{FKl2} and for Schr\"odinger operators \cite{FKl1}.  The strategy of this paper and of \cite{FKl3}  is the same one  
 we used in  \cite{FKl2}, the main differences are of technical nature and due to working on the continuum
instead of the lattice.  Acoustic waves were similarly treated in  \cite{FKl3}.  Localization created 
by (non-random) local defects was studied in \cite{FKl4}.   


\subsection{Maxwell's equations and localization}
 
In a linear, lossless dielectric medium Maxwell's equations are given by %@maxeq
\begin{equation}
\begin{array}{cc}
\frac 1c\mu\frac{\partial}{\partial t} {\bf H} =
-\nabla \times {\bf E} \qquad &\qquad \nabla \cdot \mu {\bf H}=0 \\ [2ex]
\frac 1c\varepsilon \frac{\partial}{\partial t} {\bf E}=\nabla \times {\bf H}   \qquad &\qquad 
\nabla \cdot \varepsilon {\bf E}=0, 
 \end{array} \label{maxeq}
\end{equation}
where ${\bf E} ={\bf E}(x,t)$ is the electric field, ${\bf H} ={\bf H}(x,t)$ is the magnetic field,  
$\varepsilon=\varepsilon ( x) $ is the position dependent dielectric constant, and $\mu =\mu (x) $
is the magnetic permeability. We take units in which the speed of light $c=1$.

The energy density ${\cal E}(x,t)={\cal E}_{{\bf H},{\bf E}}(x,t)$ and  the (conserved) energy 
${\cal E}={\cal E}_{{\bf H},{\bf E}}$ 
of a solution $({\bf H},{\bf E})$
of the Maxwell's equations (\ref{maxeq}) are given by
\begin{equation}
{\cal E}(x,t) =  
\frac 12\left[ \varepsilon(x) | {\bf E}(x,t)|^2+\mu(x) |{\bf H}(x,t)|^2\right],
 \;\;{\cal E}=\int_{\Bbbs{R}^3}{\cal E}(x,t)\,dx .
\label{energy}
\end{equation}

 Maxwell's equations  can be recast as a Schr\"odinger-like equation (i.e., a first order conservative linear 
equation): %@maxeq2
\begin{equation}
-i \frac{\partial}{\partial t} { \bf \Psi}_t=   \Bbb{M}{ \bf \Psi}_t, \label{maxeq2}
 \end{equation}
with
\begin{equation}
{ \bf \Psi}_t = \left(\begin{array}{c} {\bf H}_t \\{\bf E}_t \end{array}\right)  \in \Bbb{H}_\varepsilon, \;\;\;\;\;\;\;\;
\Bbb{M}=\left[\begin{array}{cc}
0 &\frac{i }{\mu} \nabla^\times\\
\frac{-i }{\varepsilon}\nabla^\times & 0
\end{array}\right],  
\end{equation}
where 
$\Bbb{H}_\varepsilon =\Bbb{S}_\mu \oplus \Bbb{S}_\varepsilon  $ is  the Hilbert space  of finite energy 
solutions; for a given $\varrho=\varrho(x) > 0 $, bounded from above and away from $0$, we set $\Bbb{S}_\varrho$ 
to be the closure in 
$L^2(\Bbb{R}^3;\,\Bbb{C}^3, \varrho(x) dx)$ of the linear
subset of functions $\Psi$ with 
$\varrho\Psi \in C^1_0(\Bbb{R}^3;\,\Bbb{C}^3), \;\nabla\cdot\varrho\Psi=0$.
 The matrix operator  $ \Bbb{M}$, where $\nabla^\times$ denotes the operator given by 
$\nabla^\times \Psi = \nabla \times \Psi= {\rm curl} \,\Psi$, 
has a natural definition as a self-adjoint operator on $\Bbb{H}_\varepsilon$.  The solution to (\ref{maxeq2}) is then
given by $ {\bf \Psi}_t = {\rm e}^{it\Bbbs{M}}  { \bf \Psi}_0$, it has  energy  
 \begin{equation}
{\cal E} = \frac12 \|{\bf \Psi}_t\|_{\Bbbs{H}_\varepsilon}^2=
\frac12 \|{\bf \Psi}_0\|_{\Bbbs{H}_\varepsilon}^2.
\end{equation}

A localized electromagnetic wave can be characterized as a finite energy solution of Maxwell's equations with 
the property that almost all of the wave's energy remains in a fixed bounded region of space at all times, e.g.,
%@locwave
\begin{equation}
\lim_{R \to \infty} \inf_t   \frac1 {\cal E}\int_{ |x| \le R} {\cal E}(x,t) \, dx = 1 . \label{locwave}
\end{equation}

If the operator $\Bbb{M}$ has an eigenvalue $\omega$ with eigenmode $ {\bf \Psi}_\omega$,
i.e., $\Bbb{M}{\bf \Psi}_\omega= \omega {\bf \Psi}_\omega$, with $ {\bf \Psi}_\omega \in \Bbb{H}_\varepsilon$,
${\bf \Psi}_\omega  \not= 0$, then $ {\bf \Psi}_{\omega, t} = {\rm e}^{it\omega}  { \bf \Psi}_\omega$ is a localized
electromagnetic wave, i.e., it satisfies (\ref{maxeq2}) and  (\ref{locwave}).  Notice that in this case   
  $-\omega$ is also an eigenvalue of $\Bbb{M}$ with eigenmode
 $\overline{\bf \Psi}_\omega$, so $\overline{\bf \Psi}_{\omega, t}= {\rm e}^{-it\omega} \overline{ \bf \Psi}_\omega$ 
is also a localized wave,  since if ${\bf J}$ denotes the 
antiunitary involution
corresponding to  complex conjugation on $\Bbb{H}_\varepsilon$, i.e., ${\bf J}{\bf \Psi}= \overline{\bf \Psi}$, we have
${\bf J} \Bbb{M} {\bf J} = - \Bbb{M}$.  It also follows that the spectrum of  
$\Bbb{M}$ is symmetric, i.e.,
$\sigma(\Bbb{M})= -  \sigma(\Bbb{M})$, with ${\bf J} \Bbb{M}_+ {\bf J} =\Bbb{M}_-$, $\Bbb{M}_\pm$ being the
 the positive and negative parts of $\Bbb{M}$.
In addition, linear combinations of
 eigenmodes of $\Bbb{M}$ give raise to localized electromagnetic waves.    

If $ {\bf \Psi}_t$ is a solution of the equation (\ref{maxeq2}), it must satisfy the second order equation
 $\frac{\partial^2}{\partial t^2} { \bf \Psi}_t=-   \Bbb{M}^2{ \bf \Psi}_t$, so the magnetic and electric fields
 satisfy the  second order equations
\begin{eqnarray}
\frac{\partial^2}{\partial t^2} { \bf H}_t  &=& -  \frac 1{\mu} \nabla^{\times} \frac 1{\varepsilon }\nabla^{\times} { \bf H}_t,\;\;
 { \bf H}_t  \in \Bbb{S}_\mu\\ [1ex]
\frac{\partial^2}{\partial t^2} { \bf E}_t  &=& 
- \frac 1{\varepsilon }\nabla^{\times}\frac 1{\mu} \nabla^{\times} { \bf E}_t,\;\;
{ \bf E}_t  \in \Bbb{S}_\varepsilon. 
\end{eqnarray}

The Maxwell operators ${\bf M}_{\bf H} =  \frac 1{\mu}\nabla^{\times} \frac 1{\varepsilon }\nabla^{\times}$ and 
${\bf M}_{\bf E} = \frac 1{\varepsilon }\nabla^{\times} \frac 1{\mu} \nabla^{\times}$ have natural definitions as nonnegative
self-adjoint operators on $\Bbb{S}_\mu$ and $ \Bbb{S}_\varepsilon$, respectively.  The two Maxwell operators 
 are unitarily equivalent, more precisely %@mtilde
\begin{equation}
{\bf M}_{\bf E} = U {\bf M}_{\bf H} U^*,  \label{mtilde}
\end{equation}
 where 
$U: \  \Bbb{S}_\mu \to \Bbb{S}_\varepsilon$ is the unitary operator given by %@U
\begin{equation}
U {\bf H} = \frac{-i }{\varepsilon}\nabla^\times{\bf M}_{\bf H}^{- \frac12}{\bf H}, \;\;
 {\bf H} \in {\rm Ran} \, {\bf M}_{\bf H}^{ \frac12}.  \label{U}
\end{equation}

Thus $\sigma(\Bbb{M})= \sigma( {\bf M}_{\bf H}^{ \frac12}) \cup [- \sigma( {\bf M}_{\bf H}^{ \frac12})]$.   We 
 obtain  solutions of  (\ref{maxeq2}) by setting %@sol 
\begin{equation}
{\bf \Psi}_{\pm, t} =
\left(  {\rm e}^{\pm it{\bf M}_{\bf H}^{ \frac12}} { \bf H}_0,
 \pm U  {\rm e}^{\pm it{\bf M}_{\bf H}^{ \frac12}} { \bf H}_0 \right), \;\;\;\; { \bf H}_0 \in \Bbb{S}_\mu.
\label{sol}
\end{equation}
Conversely, any solution of (\ref{maxeq2}) can be written as a linear combination of at most 
four solutions of this form. 

It follows that to find all eigenvalues and eigenmodes for $\Bbb{M}$, it is necessary and sufficient to find all  eigenvalues
 and eigenmodes for $ {\bf M}_{\bf H}$.  For if $ {\bf M}_{\bf H} {\bf H}_{\omega^2} = \omega^2  {\bf H}_{\omega^2}$, 
with $\omega > 0$, $ {\bf H}_{\omega^2} \in \Bbb{S}_\mu$, $ {\bf H}_{\omega^2} \not=0$,
we have %@U2
\begin{equation}
 U {\bf H}_{\omega^2} =
\frac{-i }{\omega\varepsilon}\nabla^\times {\bf H}_{\omega^2} \label{U2}
\end{equation}
 and %@eigen 
 \begin{equation}
\Bbb{M} \left(  {\bf H}_{\omega^2}, \pm \frac{-i }{\omega\varepsilon}\nabla^\times {\bf H}_{\omega^2}\right)=
 \pm \omega \left(  {\bf H}_{\omega^2}, \pm \frac{-i }{\omega\varepsilon}\nabla^\times {\bf H}_{\omega^2}\right).
\label{eigen}
\end{equation}
Conversely, if   
 $\Bbb{M} ( {\bf H}_{\pm\omega},  {\bf E}_{\pm\omega})= \pm\omega  ( {\bf H}_{\pm\omega},  {\bf E}_{\pm\omega})$,
with $\omega > 0$,  $ ( {\bf H}_{\pm\omega},  {\bf E}_{\pm\omega}) \in \Bbb{H}_\varepsilon$, not $0$, it follows that
$ {\bf M}_{\bf H} {\bf H}_{\pm\omega} = \omega^2 {\bf H}_{\pm\omega}$ and 
$ {\bf E}_{\pm\omega}=  \pm U {\bf H}_{\pm\omega}
=\pm\frac{-i }{\omega\varepsilon}\nabla^\times  {\bf H}_{\pm\omega}$.

{\em Our strategy for proving the existence of localized electromagnetic waves is the following:
 first the operator ${\bf M}_{\bf H}$ is shown to have 
pure point spectrum in some closed interval $I \subset (0,\infty)$,  with all the corresponding eigenfunctions being 
exponentially decaying (in the sense of having exponentially decaying local $L^2$-norms).  
 For this operator we prove that
the curl of an exponentially decaying eigenfunction is also exponentially decaying, so it follows from
\eq{mtilde} and \eq{U2} that the operator ${\bf M}_{\bf E}$ has also pure point spectrum  in
the closed interval $I$, with all the corresponding eigenfunctions being exponentially decaying.
In addition, it ensues from \eq{eigen} that the operator $\Bbb{M}$ has  pure point spectrum  in
$ \{\omega \in \Bbb{R};\;\; \omega^2 \in I\}$, with all the corresponding eigenfunctions
 being exponentially decaying, so the energy densities  of the corresponding solutions of 
\eq{maxeq2}    are also 
 exponentially decaying, uniformly in the time $t$. 
 If $\chi_{{I}}({\bf M}_{\bf H})$ is the corresponding spectral projection, then any solution
of   (\ref{maxeq2})   given by (\ref{sol}), with ${\bf H}_0$  
 in the range of $\chi_I({\bf M}_{\bf H})$, satisfies (\ref{locwave}). 

The localization of electromagnetic waves is thus a consequence of Anderson localization for operators
${\bf M}_{\bf H} =  \frac 1{\mu}\nabla^\times\frac 1 \varepsilon \nabla^\times$ on $\Bbb{S}_\mu$, i.e., the existence of 
closed intervals where these operators have pure point spectrum with exponentially 
decaying eigenfunctions. }  


\subsection{Statement of results}

In this article we study electromagnetic waves in a linear, lossless dielectric medium described by a position
dependent dielectric constant $\varepsilon=\varepsilon (x)$.  For most dielectric materials of interest, the magnetic
permeability $\mu(x)$ is close to one (e.g., \cite{JMW}), so we set $\mu(x) \equiv 1$.
\medskip 

{\em We always assume that $\varepsilon (x)$ is a measurable real
valued function satisfying} %@bound 
\begin{equation}
0<\varepsilon _{-}\leq \varepsilon (x)\leq \varepsilon _{+}<\infty \;\;\;
\mbox{a.e. for some constants }\;\varepsilon _{-}\;\mbox{ and }\;\varepsilon _{+} . \label{bound}
\end{equation}
Such general conditions on $\varepsilon (x)$, particularly the lack of smoothness, are required on
physical grounds. In practice
only a few materials are used in the  fabrication of periodic and
disordered media,  in which case $\varepsilon (x)$ takes just a
finite number of values, so $\varepsilon (x)$ is piecewise constant, hence discontinuous. 
The abrupt changes in the  medium produce  discontinuities in 
$\varepsilon (x)$, which favor and enhance multiscattering
and, hence, localization. 

 In such a medium electromagnetic  waves  are described by the  formally 
self-adjoint Maxwell operator %@Max   
\begin{equation}
{\bf M}={\bf M}(\varepsilon)={\bf M}_{\bf H} =  \nabla^{\times} \frac 1{\varepsilon }\nabla^{\times},  \label{Max}
\end{equation}
acting on the Hilbert space
\begin{equation}
\Bbb{S} =\overline{\{\Psi \in L^2(\Bbb{R}^3;\,\Bbb{C}^3);\;\; \Psi \in C^1_0(\Bbb{R}^3;\,\Bbb{C}^3) 
\;\;\mbox{with}\;\; \nabla\cdot\Psi=0 \}} .
\end{equation}
For the rigorous definition, we start by defining the unrestricted Maxwell operator    %@Max2
\begin{equation}
M= M(\varepsilon)=  \nabla^{\times} \frac 1{\varepsilon }\nabla^{\times},  \label{Max2}
\end{equation}
 as the nonnegative self-adjoint operator on $ L^2(\Bbb{R}^3;\,\Bbb{C}^3)$,
uniquely defined by the nonnegative quadratic form given as the closure of 
\begin{equation}
\mathcal{M}(\Psi,\Phi )=\langle \nabla  \times  \Psi ,\frac 1{\varepsilon}\nabla  \times  \Phi \rangle,
 \;\;\Psi,\Phi \in C_0^1(\Bbb{R}^3;\,\Bbb{C}^3). 
\label{calM}
\end{equation}

By Weyl's decomposition (see \cite{BS}), we have 
\begin{equation}
L^2(\Bbb{R}^3;\,\Bbb{C}^3) = \Bbb{S} \oplus \Bbb{G}, \label{weyl}
\end{equation}
where 
 $\Bbb{G}$, the space of potential fields,  is the closure in $L^2(\Bbb{R}^3;\,\Bbb{C}^3)$ of the linear
subset $\{\Psi \in C^1_0(\Bbb{R}^3;\,\Bbb{C}^3); \;\;\Psi=\nabla \varphi\;\;\mbox{with}\;\;
\varphi \in  C^1_0(\Bbb{R}^3)\}$.  
The spaces $\Bbb{S}$ and $\Bbb{G}$ are left invariant by  $M$, with  
$\Bbb{G} \subset \mathcal{D}\left( M\right) $ and  $\left.M\right| _{\Bbbs{G}} =0$.  We define
 ${\bf M}$ as the restriction of $M$ to ${\Bbb{S}}$, i.e.,
$\mathcal{D}\left({\bf M}\right) =\mathcal{D}\left( M\right) \cap \Bbb{S}$ and 
${\bf M}=\left. M\right| _{{\cal D}\left( M\right) \cap \Bbbs{S}}$.  Thus
\begin{equation}
{\bf M} =    P_{\Bbbs{S}}MI_{\Bbbs{S}}=MI_{\Bbbs{S}},   \label{PS}
\end{equation}
 with $ P_{\Bbbs{S}} $ the
 orthogonal projection onto $\Bbb{S}$ and $ I_{\Bbbs{S}}:\Bbb{S} \to L^2(\Bbb{R}^3;\,\Bbb{C}^3)  $
 the restriction of the  identity map.   
 Notice that $M =  {\bf M} \oplus 0_{\Bbbs{G}}$ and $0 \in \sigma({\bf M})$, so %@1spectra 
\begin{equation}
\sigma({\bf M}) = \sigma(M).  \label{1spectra}
\end{equation}
 We can thus work with $M$ to answer questions about the
 spectrum of ${\bf M}$.

In the special case of a homogeneous medium with $\varepsilon(x) \equiv 1$, we will use the notation
\begin{equation}
\Xi = M(1)= ( \nabla^{\times})^2, \;\;\;\; {\bf \Xi} = {\bf M}(1)=
\left. ( \nabla^{\times})^2\right| _{{\cal D}\left(( \nabla^{\times})^2\right)
 \cap \Bbbs{S}} .
\end{equation}




In this article we consider electromagnetic waves in random
media obtained by random perturbations of a periodic medium.  The properties
of the medium are described by the position dependent quantity $\varepsilon(x)$, 
which we will take to always satisfy the following assumptions.
%@aeu

\begin{assumption}[The Random Media]
\label{aeu}
$\varepsilon _g(x)=\varepsilon _{g,\omega }(x)$ is a random function of the
form 
\begin{equation}
\varepsilon _{g,\omega }(x)=\varepsilon _0(x)\gamma _{g,\omega }(x)\;\;,\mbox{with}\;\;
\gamma _{g,\omega }(x)=1+g\sum_{i\in \Bbbs{Z}^3}\omega _iu_i(x),
\end{equation}
where
\begin{enumerate}
\item[(i)]  $\varepsilon _0(x)$ is a measurable real valued function which
is $q$-periodic for some $q\in \Bbb{N}$, i.e., 
$\varepsilon_0(x)=\varepsilon _0(x+qi)$ for all $x\in \Bbb{R}^3$ and ${i\in \Bbb{Z}^3}$, with 
\begin{equation}
0<\varepsilon _{0,-}\le \varepsilon _0(x)\le \varepsilon _{0,+}<\infty \;\;\mbox{for a.e.}
\,x\in \Bbb{R}^3,
\end{equation}
for some constants $\varepsilon _{0,-}$ and $\varepsilon _{0,+}$.

\item[(ii)]  $u_i(x)=u(x-i)$ for each $i\in \Bbb{Z}^3$, $u$ being a
nonnegative measurable real valued function with compact support, say $u(x)=0
$ if $\Vert x\Vert _\infty \le r_u$ for some $ r_u<\infty $, such that
%%@muM 
\begin{equation}
0<U_-\le U(x)\equiv \sum_{i\in \Bbbs{Z}^3}u_i(x)\le U_+<\infty \;\;\mbox{for a.e.}\,
\mbox{ }x\in \Bbb{R}^3 ,  \label{muM}
\end{equation}
for some constants $U_-$ and $U_+$.

\item[(iii)]  $\omega =\{\omega _i;\,i\in \Bbb{Z}^3\}$ is a family of
independent, identically distributed random variables taking values in the
interval $[-1,1]$, whose common probability distribution $\mu $ has a
bounded density $\rho >0$ a.e. in $[-1,1]$.

\item[(iv)]  $g$, satisfying $0\le g<\frac 1{U_+}$, is the disorder
parameter.
\end{enumerate}
\end{assumption}

For electromagnetic waves $\varepsilon _{g,\omega }(x)$ is the random position
dependent dielectric constant of the medium.
Notice that Assumption \ref{aeu} implies that each $\varepsilon_{g,\omega }$ satisfies 
(\ref{bound}) with %@bound1
\begin{equation}
\varepsilon_\pm =\varepsilon_{g,\pm}= \varepsilon_{0,\pm}(1 \pm g U_+). \label{bound1}
\end{equation}
For later use we set %%@delta
\begin{equation}
\delta_\pm(g) = \frac{U_\pm}{ 1 \mp gU_+ },
\;\;\mbox{with}\;\;0\le g<\frac 1{U_+}.
\label{delta}
\end{equation}


The periodic operators associated with
the coefficient $\varepsilon _0(x)$ will carry the subscript $0$, i.e., $M_0 =M(\varepsilon _0)$, 
${\bf M}_0 ={\bf M}(\varepsilon _0)$. We will study the random operators (see \cite[Appendix A]{FKl3} for the definition)

%%@At 
\begin{equation}
M_g=M_{g,\omega }= M(\varepsilon_{g,\omega });\;\; \;\;
{\bf M}_g={\bf M}_{g,\omega }= {\bf M}(\varepsilon_{g,\omega }). \label{At}
\end{equation}

It is a consequence of  ergodicity (measurability follows from \cite[Theorem 38 ]{ FKl3})
that there exists a nonrandom set $\Sigma _g$,
such that $ \sigma({\bf M}_{g,\omega }) = \sigma(M_{g,\omega })=\Sigma _g$ with probability one.
In addition, the
decompositions of $\sigma({\bf M}_{g,\omega })$ and $\sigma(M_{g,\omega })$ into pure point spectrum,
absolutely continuous spectrum and singular continuous spectrum are also
independent of the choice of $\omega $ with probability one \cite{KM1,PF}.

In this article we are interested in the phenomenon of localization. 
According to the philosophy of Anderson localization we will assume that the
operator ${\bf M}_0$ has at least one gap in the spectrum. 
%@agap

\begin{assumption}[The gap in the spectrum]
\label{agap}There is a gap in the spectrum of the operator ${\bf M}_0$. More
precisely, there exist $0\le \hat{a}<a<b<\hat{b}$ such that 
\[
\sigma ({\bf M}_0)\bigcap [\hat{a},\hat{b}]=[\hat{a},a]\bigcup [b,\hat{b}] ,
\]
so the interval $(a,b)$ is a gap in $\sigma ({\bf M}_0)$. 
\end{assumption}


The following theorem gives information on the location of $\Sigma_g$, the (nonrandom) spectrum  of
the random Maxwell operator ${\bf M}_g$.
%@tlct

\begin{theorem}[Location of the Spectrum]
\label{tlct} Let the random operator ${\bf M}_g$ defined by (\ref{At}) satisfy
Assumptions \ref{aeu} and \ref{agap}.  There exists $g_0 $, with
\begin{equation}
\frac{1}{U_+} \left(1 - \left(\frac{a}{b}\right)^{\frac{1}{2}}\right) \le g_0 
\le\frac{1}{U_+}\min\left\{1, \left(\left(\frac{b}{a}\right)^{\frac{U_+}{2U_-}}  -1\right)        \right\},  \label{g0}
\end{equation}
and strictly increasing, Lipschitz continuous real valued functions $a(g)$ and $-b(g)$ on the interval
 $[0, \frac{1}{U_+})$, with $a(0)=a$, $b(0)=b$ and  $a(g) \le b(g)$, such that: 
\begin{description}

\item[(i)] 
%%@abgs 
\begin{equation}
\Sigma _g\bigcap [\hat{a},\hat{b}]=[\hat{a},a(g)]\bigcup [b(g),\hat{b}]\;. 
\label{abgs}
\end{equation}

\item[(ii)]  For $g < g_0$, we have $a(g) < b(g)$ and $( a(g) , b(g))$ is a gap in the spectrum of the 
random operator  ${\bf M}_g$,
located inside the gap $(a,b)$ of the unperturbed periodic operator ${\bf M}_0$.  Moreover, we have  %@abgaa
\begin{equation}
a \le a(1 + g U_+)^{\frac{U_-}{U_+ }} \le a(g) \le \frac{a}{1 - g U_+}
\label{abgaa}
\end{equation}
and %@abgbb
\begin{equation}
b(1 - g U_+) \le b(g) \le  \frac{b}{(1 + g U_+)^{\frac{U_-}{U_+ }} } \le b.
\label{abgbb}
\end{equation}
  
\item[(iii)]  If $g_0< \frac{1}{U_+}$,  we have $a(g)=b(g)$ for all $g \in [g_0, \frac{1}{U_+})$, and the 
random operator ${\bf M}_g$ has no gap inside the gap $(a,b)$ of the unperturbed periodic operator $A_0$,
i.e., $  [\hat{a},\hat{b}] \subset \Sigma _g$.   
\end{description}
\end{theorem}


\begin{definition}[Exponential localization] We say
that the random operator ${\bf M}_g$ exhibits {\em localization} in an 
interval $I \subset \Sigma_g $, if ${\bf M}_g$ has {\em only} pure point
spectrum in $I$ with probability one. We have {\em exponential localization}
in $I$ if we have localization and, with probability one, all the
eigenfunctions  corresponding to eigenvalues in $I$ are exponentially decaying 
(in the sense of having exponentially decaying local $L^2$-norms).
\end{definition}

\begin{remark} 
The curls of exponentially decaying eigenfunctions of  ${\bf M}_g$  always have exponentially 
decaying local $L^2$-norms (Corollary \ref{cGT}).  Thus the 
corresponding energy densities (see  (\ref{energy}) ) also have exponentially 
decaying local $L^2$-norms, uniformly in the time $t$.     
\end{remark}

Our main results show that random perturbations create exponentially
localized eigenfunctions near the edges of the gap.  Our method requires low probability
of extremal values for the random variables; the following two theorems achieve this
 in different ways.     The results are formulated for the left edge of the gap,  with similar
results holding at the right edge. %@loc1

\begin{theorem}[Localization at the edge] \label{loc1}
Let the random operator ${\bf M}_g$ defined by (\ref{At}) satisfy Assumptions \ref
{aeu} and \ref{agap}, with %%@ext
\begin{equation}
\mu\{(1- \gamma,1] \} \le K \gamma^\eta \;\;\mbox{for}\;\;0\le\gamma \le1,
 \label{ext}
\end{equation}
where $ K< \infty$ and $\eta > 3 $. 
 For any $g < g_0$ there exists $\delta(g) >0$, depending only on the constants 
$g, \,q, \,\varepsilon_{0,\pm},\,U_\pm, \,r_u,\,K,\,\eta$,  an upper bound on 
$\Vert \rho \Vert _\infty $, and on $a$, $b$, such that
the random operator ${\bf M}_g$ exhibits exponential localization in the interval 
$[a(g) - \delta(g) ,a(g)]$.
\end{theorem} 


\begin{theorem}[Localization in a specified interval]  \label{loc2}
Let the random operator ${\bf M}_g$ defined by (\ref{At}) satisfy Assumptions \ref
{aeu} and \ref{agap}.  For any $g < g_0$,  given  $a < a_1 < a_2 < a(g)$, with
 $a(g)-a_{1}\le b(g)-a(g)$,  there exists $p_{1}>0$,
 depending only on the constants 
$g$, $q$, $\varepsilon_{0,\pm}$, $U_\pm,\,r_u,\,a$, an upper bound on 
$\Vert \rho \Vert _\infty $ and
on the given  $a_{1}$, $a_{2 }$, such that if 
\begin{equation}
\mu \left( \left( \frac{g_{1}}g,1\right] \right) <p_{1},
\end{equation}
where $g_1$ is defined by $a(g_{1})=a_{1}$,
the random operator ${\bf M}_g$ exhibits exponential localization in the interval 
$[a_2,a(g)]$.
\end{theorem}


 Theorems \ref{loc1} and  \ref{loc2} can be extended
 to the situation when the gap is totally filled by the spectrum of the random operator, we then establish
the existence of an interval (inside the original gap) where the random Maxwell operator exhibits
 exponential localization.   
Notice that the extension of Theorem \ref{loc2} says that we can arrange for  localization in as 
much of the gap as we want.   

\begin{theorem}[Localization at the meeting of the edges] \label{loc10}
Let the random operator ${\bf M}_g$ defined by (\ref{At}) satisfy Assumptions \ref
{aeu} and \ref{agap}, with %%@ext
\begin{equation}
\mu\{(1- \gamma,1] \},\;\;\mu\{[-1,-1+\gamma) \}\le K \gamma^\eta \;\;\mbox{for}\;\;0\le\gamma \le1,
 \label{ext11}
\end{equation}
where $ K< \infty$ and $\eta > 3 $. 
Suppose $g_0 < \frac{1}{U_+}$ (e.g., if $\left(\frac{b}{a}\right)^{\frac{U_+}{2U_-}}  <2$), so the random
operator  ${\bf M}_g$ has no gap inside $(a,b)$ for  $g \in [g_0,\frac{1}{U_+})$.
Then there exist $0<\epsilon<\frac{1}{U_+} - g_0 $ and $\delta >0$, depending only on the constants 
$q, \,\varepsilon_{0,\pm},\,U_\pm, \, r_u,\, K,\,\eta$,  an upper bound on 
$\Vert \rho \Vert _\infty $, and on $a$, $b$,  such that
the random operator ${\bf M}_g$ exhibits exponential localization in the interval 
$[a(g_0) - \delta ,a(g_0) + \delta]$ for all $g_0 \le g <g_0 + \epsilon$.
\end{theorem} 


\begin{theorem}[Localization in a specified interval in the closed gap]  \label{loc20}
Let the random operator ${\bf M}_g$ defined by (\ref{At}) satisfy Assumptions \ref
{aeu} and \ref{agap}.  Suppose $g_0 < \frac{1}{U_+}$
 (e.g., if $\left(\frac{b}{a}\right)^{\frac{U_+}{2U_-}}  <2$), so the random
operator ${\bf M}_g$ has no gap inside $(a,b)$ for  $g \in [g_0,\frac{1}{U_+})$.
Let $a < a_1 < a_2 < a(g_0) =b(g_0)< b_2 < b_1 <b$ be given.  For any $g \in [g_0,\frac{1}{U_+})$ 
  there exist $p_{1},p_{2} >0$,
 depending only on the constants 
$g$, $q$, $\varepsilon_{0,\pm}$, $U_\pm,\,r_u,\,a,b$, an upper bound on 
$\Vert \rho \Vert _\infty $ and
on the given  $a_{1}$, $a_{2 }$, $b_1$, $b_2$, such that if 
\begin{equation}
\mu \left( \left( \frac{g_{1}}g,1\right] \right) <p_{1},\;\; 
\mu \left( \left[-1, - \frac{g_{2}}g \right) \right) <p_{2},
\end{equation}
where $g_1$ and $g_2$  are defined by $a(g_{1})=a_{1}$ and $b(g_{2})=b_{1}$ (notice $0< g_1, \ g_2 < g_0\le g$),
the random operator ${\bf M}_g$ exhibits exponential localization in the interval 
$[a_2,b_2]$.
\end{theorem}

Theorems \ref{loc10} and \ref{loc20} are proved exactly as Theorems \ref{loc1} and \ref{loc2}, respectively,
taking into account both edges of the gap. 



\begin{remark}  
Theorems  \ref{loc1} and  \ref{loc10} should be true without the extra  hypotheses (\ref{ext}) and 
(\ref{ext11}).  They are used in conjunction with a Combes-Thomas argument to obtain the starting 
hypothesis for the multiscale analysis, in the proof of localization.  One may expect estimates 
similar to Lifshitz tails (e.g., \cite{PF})  for the density of states inside the gap, which would replace  (\ref{ext}) and 
(\ref{ext11}) in the proofs.  This is how the starting hypothesis is obtained for random Schr\"odinger
operators at the bottom of the spectrum \cite{HM}.

Combes and Hislop have announced an improved Combes-Thomas argument inside a gap; they obtain a
 decay rate proportional to the square root of the product of the distances to the edges of the gap.  With 
this result we would only need $\eta > \frac{3}{2}$ in Theorem  \ref{loc1}, but we would still need to
require $\eta >3$ in Theorem  \ref{loc10}.    
\end{remark}

Theorem \ref{tlct} is proved in Section \ref{slocation}; the proof requires periodic operators and 
periodic boundary condition, studied in Section \ref{sper}.  
Theorems  \ref{loc1} and  \ref{loc2} are proved in Section \ref{sloc} by  multiscale analyses.  
Dirichlet boundary condition, used in the proofs, is discussed in Section \ref{sdirich}. 
 The required 
Wegner-type estimate is in Section \ref{swegner}.  The starting hypotheses are proved first for finite volume
Maxwell operators with periodic boundary condition, using a Combes-Thomas argument
 for operators with periodic boundary condition (Subsection \ref{ssctt}) and Theorem \ref{tlct}.     
We collect properties of Maxwell operators needed for the proof of localization in Section \ref{genmax},
they include an interior estimate for curls and existence of polynomially bounded generalized eigenfunctions.

\subsection{Notation}

We adopt the following definitions and notations:

\begin{itemize}
\item 
 For $x=\left( x_1,x_2 ,x_3\right) \in \Bbb{R}^3$ we let
$ \left| x\right| _p = ( x_1^p +x_2^p +x_3^p)^{1/p}$ for $1 \le p<\infty$, and  
  $\left| x\right| _\infty = \max_{1\leq j\leq 3} \left| x_j\right|  $.  We set 
$\left| x\right|  =  \left| x\right| _2$ and $\left\| x\right\|  = \left| x\right| _\infty$.

\item $\Lambda_L(x) = \{y \in \Bbb{R}^3; \;\|y-x\| < \frac{L}{2}\}$ is the (open) cube of side $L$
centered at $x  \in \Bbb{R}^3$;  $\bar{\Lambda}_L(x) $ is the closed cube, and
$\breve{\Lambda}_L(x)  =
 \{y \in \Bbb{R}^3; \; -\frac{L}{2}\le y_i-x_i < \frac{L}{2}, \, i=1,2,3\}$ the
 half-open/half-closed cube. 

\item $\chi_\Lambda$ is the characteristic function of the set $\Lambda$; we write
 $\chi_{x,L} = \chi_{\Lambda_L(x)}$.

\item  A function $f$ on $ \Bbb{R}^3$ is called $q$-periodic for some $q > 0$ if
$f(x+q i) = f(x)$ for all $x  \in \Bbb{R}^3$ and $ i \in  \Bbb{Z}^3$.      

\item  A domain $\Omega $ is an open connected subset of $\Bbb{R}^3$; 
 its boundary is denoted by $\partial \Omega $.

\item  $L^p(\Omega;\, \Bbb{C}^d )$ is the space of $\Bbb{C}^d $ measurable functions $u: \Omega \to \Bbb{C}^d$
 with the norm $\left\| u\right\| _p=\left\| u\right\| _{p,\Omega
}=\left[ \int_\Omega \left| u(x)\right| ^p\,dx\right] ^{1/p}$. We will often
use the space $L^2(\Omega;\, \Bbb{C}^d  )$ and in this case we will write 
$\left\|u\right\|_\Omega  $ for $\left\|u\right\|_{2,\Omega } $.  If $\Omega = \Bbb{R}^d$
we may omit it from the subscript.  We  write  $L^p(\Omega)$ if $d=1$. 

\item  $C^n( \Omega;\, \Bbb{C}^d ) $ is the linear space of $n$-times continuously differentiable
 functions $u: \Omega \to \Bbb{C}^d$, $C^n_0 ( \Omega;\, \Bbb{C}^d) $ is the subspace of functions with 
compact support.  We  write   $C^n( \Omega ) $ if $d=1$.

%\item  $L^2_{loc}( \Bbb{R}^3) $ is the Frechet  space of locally square integrable 
%functions $u$ on  $\Bbb{R}^3$, the topology being given by the  seminorms
%$\{ \left\|u\right\|_{2,\Omega };\; \mbox{$\Omega$ bounded domain} \}$.     

\item  The domain, spectrum and adjoint of a linear operator $A$  are denoted by $\mathcal{D}(A)$,
 $\sigma (A)$ and $A^{*}$, respectively .

\item If $\mathcal{A}$ is the   quadratic form  associated with an operator $A$, its domain
 will be denoted   by either 
$\mathcal{Q}(\mathcal{A})$ or $\mathcal{Q}({A})$ .  We also write $\mathcal{A}[\Psi]$ for 
$\mathcal{A}(\Psi,\Psi)$.

\item $\mathcal{B}(\mathcal{X},\mathcal{Y})$ is the Banach space of bounded operators
from the normed space $\mathcal{X}$ to  the normed space $\mathcal{Y}$; 
 $\mathcal{B}(\mathcal{X}) =  \mathcal{B}(\mathcal{X},\mathcal{X})$.


\item  For a complex number $z$ its conjugate is denoted by $z^{*}$.

\end{itemize}

\section{Properties of Maxwell operators} \label{genmax}

\subsection{An interior estimate}

Consider the formally self-adjoint operator %@SS
\begin{equation}
S=  D^*  \Gamma D \;\; \;\;\mbox{on}\;\;\;\; L^2(\Bbb{R}^d; \Bbb{C}^\nu),   \label{SS}
\end{equation}
where $D =\{ D_{\alpha,\beta}\}_{\alpha,\beta =1,\dots,\nu}$ with each 
$ D_{\alpha,\beta}= a_{\alpha,\beta} \cdot \nabla$ for some $a_{\alpha,\beta} \in \Bbb{C}^d$, $D^*$ is the formal adjoint of $D$,
and $\Gamma = \Gamma(x)$ is a measurable function on $\Bbb{R}^d$ whose values are
 $\nu \times \nu $ complex matrices with 
\begin{equation}
0 \le \Gamma(x) \le \Gamma_+ I_\nu  \;\;\;
\mbox{a.e. for some constant }\; \Gamma _{+} < \infty , \label{boundg}
\end{equation}
 $ I_\nu $ being the  $\nu \times \nu $ identity matrix. 

 Given an open set  $\Omega \subset  \Bbb{R}^d$, we define 
 $S_\Omega$ as the restriction of $S$ to $\Omega$
with Dirichlet boundary condition, i.e., $S_\Omega$ is the nonnegative self-adjoint operator on
 $L^2(\Omega; \Bbb{C}^\nu)$ 
 defined by the closure
 of the  nonnegative quadratic form 
\begin{equation}
\mathcal{S}_{\Omega}(\Psi, \Phi )=\langle D  \Psi ,\Gamma D \Phi \rangle_\Omega,
\;\;\Psi, \Phi \in C_0^1(\Omega;\,\Bbb{C}^\nu). 
\label{calMS}
\end{equation}
Notice that
\begin{equation}
0 \le S_\Omega \le \Gamma_+ (D^* D)_\Omega .
 \label{bounds}
\end{equation}
We extend $D|_{ C_0^1(\Omega;\,\Bbbs{C}^\nu)}$ to  a closed operator, which we will call $D_\Omega$, with domain
 ${\cal D}( D_\Omega) ={\cal Q}((D^* D)_\Omega) \subset  {\cal Q}(S_\Omega) $.  
If $\Omega ^{\prime }\subset \Omega$, it is easy to see that if $u \in  {\cal D}( D_\Omega)$, then
$u|_{\Omega ^{\prime }} \in  {\cal D}( D_{\Omega ^{\prime }})$ with 
$D_{\Omega ^{\prime }} u|_{\Omega ^{\prime }} =\left( D_\Omega u \right)|_{\Omega ^{\prime }}$, so we 
will simply write $Du$ to denote the function  $D_\Omega u$.


  We say that a  function $u\in {\cal D}( D_\Omega)$ is a weak solution for the equation $S u= f$ in $\Omega$,
 where $ f \in L^2(\Omega; \Bbb{C}^\nu)$,
 if $\langle D  \Psi ,\Gamma D u \rangle_\Omega = \langle   \Psi , f \rangle_\Omega$ for all 
$\Psi \in  C^2_0(\Omega;\,\Bbb{C}^\nu)$.

\begin{theorem} \label{interior}
Let $S$ be an  operator of the form (\ref{SS})
with (\ref{boundg}).   For any $\delta >0$    
there exists a constant 
$\xi_\delta=\xi(d, \nu,\{ a_{\alpha,\beta} \}_{\alpha,\beta=1,\ldots,\nu},\delta ) <\infty$,
depending only on the indicated parameters, such that  if
 $u \in  {\cal D}( D_\Omega)$ is a weak solution for the equation $S u= f$ in an open 
 subset $\Omega$  of $\Bbb{R}^d$,   
 with $ f \in L^2(\Omega; \Bbb{C}^\nu)$,
we have %@Xi
 \begin{equation}
\langle D  u ,\Gamma D u \rangle_{\Omega ^{\prime }}  \leq \xi_\delta\left[
\Gamma_+\left\| u\right\| _{\Omega }^2+{\frac 1{\Gamma_+}}\left \| f\right\| _{\Omega}^2\right]
  \label{Xi}
\end{equation}
 for any $\Omega ^{\prime }\subset \Omega$ with 
 $\mbox{\rm dist}(\Omega^{\prime } ,\partial \Omega ) \ge\delta $.  
\end{theorem}

\proof  We consider first the case when $\Omega$ and $\Omega ^{\prime }$ are open cubes, say 
$\Omega = \Lambda_L(x_0)$, $\Omega^\prime = \Lambda_{L-2\delta}(x_0)$,
 for some $x_0 \in \Bbb{R}^d$, $L > 2\delta$.  We fix $\phi \in   C_0^1(\Bbb{R}^d)$ such that
$0 \le \phi(x) \le 1$, $\phi(x) \equiv 1$ in $\Omega ^{\prime }$, $\phi(x) \equiv 0$ in 
$\Bbb{R}^d  \backslash \Lambda_{L-\frac \delta 2}(x_0)$,
 and $|(\nabla \phi)(x)| \le \frac{2 \sqrt{d}}{\delta}$.  (Such a function always exists.)  We set
$D\phi =\{ D_{\alpha,\beta}\phi \}_{\alpha,\beta =1,\dots,\nu} = 
\{a_{\alpha,\beta} \cdot \nabla \phi\}_{\alpha,\beta =1,\dots,\nu}$.
 

Since $\phi^2 u \in {\cal D}( D_\Omega)$, we have
\begin{eqnarray}
\langle D \phi^2 u ,\Gamma D u \rangle_{\Omega }  = \langle \phi^2 u , f \rangle_{\Omega },
\end{eqnarray} 
so it follows that
\begin{eqnarray}
\lefteqn{0\le\langle D  u ,\phi^2\Gamma D u \rangle_{\Omega } =\langle \phi^2 u , f \rangle_{\Omega } -
2\langle( D\phi)  u ,\phi \Gamma D u \rangle_{\Omega }}  \qquad \qquad\\
&& \le\|u\|_\Omega  \|f\|_\Omega   +
2 \langle( D\phi)  u , \Gamma ( D\phi)  u\rangle_{\Omega }^{\frac 1 2}
\langle D  u ,\phi^2\Gamma D u \rangle_{\Omega }^{\frac 1 2} \\
&&\le  \left({\frac {\Gamma_+}{ 2}}\|u\|_\Omega^2 + {\frac 1{2\Gamma_+}} \|f\|_\Omega^2\right) +
 \left(2\Gamma_+ C_\delta\|u\|_\Omega^2   
+{\frac 1 2}\langle D  u ,\phi^2\Gamma D u \rangle_{\Omega }\right),
 \end{eqnarray}
where we used the elementary inequality $ab \le r^2 a^2 + s^2 b^2$, for any $a,b\ge0$, $r,s >0$ with
$2rs =1$,
and  $ C_\delta = C(d, \nu,\{ a_{\alpha,\beta} \}_{\alpha,\beta=1,\ldots,\nu}, \delta)  < \infty$ is a constant
depending only on the indicated parameters.

Thus,
\begin{equation}
\langle D  u ,\Gamma D u \rangle_{\Omega^\prime }  \le\langle D  u ,\phi^2\Gamma D u \rangle_{\Omega }
\le  {\frac 1{\Gamma_+}} \|f\|_\Omega^2 +(1 + 4  C_\delta     )\Gamma_+ \|u\|_\Omega^2,  \label{Xi2}
\end{equation}
which implies \eq{Xi} when $\Omega$ is an open cube.

We now consider the general case:  let $\Omega$ and $\Omega^\prime$ be as in the theorem,
and let 
\begin{equation}
\Omega^\prime_\delta =
\{ x \in \frac{\delta}{2}  \Bbb{Z}^d;
 \;\;\Lambda_ \frac{\delta}{2} \cap \Omega^\prime \not=  \emptyset  \}.
\end{equation}
Using \eq{Xi2}, we get
 \begin{eqnarray}
\langle D  u ,\Gamma D u \rangle_{\Omega^\prime }  &\le&
\sum_{x \in \Omega^\prime_\delta}\langle D  u ,
\Gamma D u \rangle_{\Lambda_ \frac{\delta}{2}(x) } \\
 &\le&  \sum_{x \in \Omega^\prime_\delta} \left(  
 \frac 1{\Gamma_+}\|f\|_{\Lambda_ {\delta}(x) }^2 +(1 + 4  C_\frac{\delta}{4})
\Gamma_+\|u\|_{\Lambda_{\delta}(x) }^2\right)  \\
 &\le& (2d +1)\left(  
  \frac 1{\Gamma_+}\|f\|_ {\Omega}^2 +(1 + 4 C_\frac{\delta}{4} )\Gamma_+ 
\|u\|_{\Omega}^2\right), 
\end{eqnarray}
from which \eq{Xi} follows.  {{$\; \Box$}}
\bigskip

Theorem \ref{interior} has the following immediate corollaries for Maxwell operators. 
 In this case $D = \nabla^\times$, i.e., $D \Psi= \nabla\times \Psi$, $D^*D = \Xi$.  Notice that
$ {\cal D}(\nabla^\times|_\Omega)=  {\cal Q}(\Xi_\Omega)$.

\begin{corollary} \label{cint} 
 Let the operator $M$ on $L^2(\Bbb{R}^3; \Bbb{C}^3)$
 be given by \eq{Max2} with \eq{bound}.
 For any $\delta >0$    
there exists $\Theta_\delta <\infty$,
depending only on $\delta$, such that  if
 $\Psi \in {\cal D}(\nabla^\times|_\Omega)$ is a weak solution for the equation $M\Psi=F$ in an open 
 subset $\Omega$  of $\Bbb{R}^3$,   
 with $F \in L^2(\Omega; \Bbb{C}^3)$,
we have %@Theta
 \begin{equation}
\|\nabla \times \Psi \|_{\Omega ^{\prime }}  \leq \Theta_\delta\sqrt{\varepsilon_+} \left[\frac 1 {\sqrt{\varepsilon_-}}
\left\| \Psi\right\| _{\Omega }+\sqrt{\varepsilon_-}\left\| F\right\| _{\Omega}\right]
  \label{Theta}
\end{equation}
 for any $\Omega ^{\prime }\subset \Omega$ with 
 $\mbox{\rm dist}(\Omega^{\prime } ,\partial \Omega ) \ge\delta $.   
\end{corollary}

\begin{corollary} \label{cint2} 
 Let the operator ${ M}$ 
 be given by \eq{Max2} with \eq{bound}.
Let $\Psi \in  L^2(\Bbb{R}^3; \Bbb{C}^3)$ be such that $\nabla \times\Psi$ is locally in $L^2$, i.e.,  
 $\Psi|_\Omega \in {\cal D}(\nabla^\times|_\Omega)$ for any bounded  open 
$\Omega \subset \Bbb{R}^3$.  Then,   
if $\Psi$ is a weak solution for the equation ${\bf M}\Psi=F$ in $\Bbb{R}^3$,
  with $F \in  L^2(\Bbb{R}^3; \Bbb{C}^3)$,
we have %@Theta2
 \begin{equation}
\|\nabla \times \Psi\|  \leq \Theta_\infty \sqrt{\varepsilon_+}\left[\frac 1 {\sqrt{\varepsilon_-}}
\left\| \Psi\right\| +\sqrt{\varepsilon_-}\left\| F\right\| \right],
  \label{Theta2}
\end{equation}
with $\Theta_\infty = \inf_{\delta >0} \Theta_\delta$.
\end{corollary}


Corollary \ref{cint} gives exponential decay for the curl of an exponentially decaying eigenfunction of a 
Maxwell operator.


\begin{corollary} \label{cGT}
Let ${\bf M}$ be an  operator of the form \eq{Max} satisfying the bounds (\ref{bound}), and let  $\Psi$ be an
 eigenfunction for ${\bf M}$.  Suppose $\Psi$ has exponentially decaying local $L^2$-norms, i.e.,
$ \left\|\chi_{x,\ell}\Psi\right\| _{2 }$  decays exponentially as $\|x\| \to \infty$  for some $\ell >0$.
Then $\nabla \times \Psi$ also has exponentially decaying local $L^2$-norms. 
\end{corollary}






\subsection{A Combes-Thomas argument} \label{scombes}

Let the operator $M$ be given by (\ref{Max2}). If $z\notin \sigma (M)$, we
write $R(z)=(M-z)^{-1}$.

%%@lct1

\begin{lemma}
\label{lct1}Let the operator $M$ be given by (\ref{Max2}) with (\ref{bound}).
Then for any $z\notin \sigma (M)$, $n \in  \Bbb{N}$  and $\ell >0$  we have
%%@ct1 
\begin{equation}
\Vert \chi _{x, \ell}R(z)^n\chi _{y, \ell}\Vert \;\;\le 
\left(\frac{9}{\eta} \right)^n e^{(\sqrt{3}\ell/4)} e^{-m_z|x-y|}\;\;\;
\mbox{for all}\;\;x,y\in \Bbb{R}^3,  \label{ct1}
\end{equation}
with
%%@mz 
\begin{equation}
m_z=\frac \eta {4\left[\varepsilon_-^{-1}+|z|+\eta \right] },
\label{mz}
\end{equation}
where $\eta =\mbox{\rm dist}(z,\sigma (M))$.
\end{lemma}

\proof The lemma is proved in the same way as \cite[Lemma 12]{FKl3}, with the obvious modifications 
to take into account that in this lemma we have {\em curls} instead of {\em gradients}.   
{{$\; \Box$}}
\bigskip

The next lemma gives an exponential estimate for the curl of the
resolvent.

%%@lgrR
\begin{lemma}
\label{lgrR} 
Let the operator $M$ be given by (\ref{Max2}) with (\ref{bound}), and let
 $z\notin \sigma (M)$ with $\eta,\; m_z$  as in Lemma \ref{lct1}.   Then 
$\nabla^\times  R(z)$ is a bounded operator on $ L^2(\Bbb{R}^3, \Bbb{C}^3) $ 
 with
\begin{equation}
\left\|\nabla^\times  R(z)\right\|  \le  
\Theta_1 \sqrt{ \varepsilon_+} \left(\sqrt{ \varepsilon_-} + \frac {1} {\sqrt{ \varepsilon_-}}  \right)
\left( \frac{(1 + |z|)}{\eta} + 1 \right), \label{grR00}
\end{equation}
 where
$\Theta_1$  is given in (\ref{Theta}).
Furthermore, for each $\ell >0$  we have
%%@grR0 
\begin{equation}
\left\| \chi _{x, \ell}\nabla^\times  R(z)\chi _{y, \ell}\right\| 
\leq  \Theta_1 \sqrt{ \varepsilon_+} \left(\sqrt{ \varepsilon_-} + \frac {1} {\sqrt{ \varepsilon_-}}  \right)
(1 + |z|) \frac{9}{\eta} e^{(3\sqrt{3}\ell/4)}e^{-m_z|x-y|}
\label{grR0}
\end{equation}
for all $ x,\;y\in \Bbb{R}^3$ with $|x - y| \ge 2 \ell$.
\end{lemma}

\proof  This lemma is proven in the same way as \cite[Lemma 13]{FKl3}, using Corollaries 
\ref{cint} and \ref{cint2}, and Lemma \ref{lct1}.
{{$\; \Box$}}

\subsection{Generalized eigenfunctions}  \label{ageneig}

 Let $M$ be an  operator of the form (\ref{Max2})
satisfying the bounds (\ref{bound}).  Given $z \in  \Bbb{C}$, a measurable function
 $\Psi:\Bbb{R}^3 \to \Bbb{C}^3 $ will be called a
{\em generalized eigenfunction} for $z$ if both $\Psi$ and $\nabla \times \Psi$ are locally in $L^2$, i.e.,
$\Psi|_\Omega \in {\cal D}(\nabla^\times|_\Omega)$ for 
all open bounded subsets $\Omega$ of $ \Bbb{R}^3$, and $\Psi$ is  a weak solution for 
the equation $M\Psi= z\Psi $ on $ \Bbb{R}^3$, i.e., %@weakly
\begin{equation}
\langle \nabla \times \Phi ,\frac 1{\varepsilon}\nabla \times \Psi \rangle =
 z \langle \Phi , \Psi\rangle \;\; \mbox{for all} \;\;\Phi \in C_0^2( \Bbb{R}^3; \, \Bbb{C}^3 ). \label{weakly}
 \end{equation}
 
 %@pBer

\begin{theorem}
\label{pBer}Let $M$ be an  operator of the form (\ref{Max2})
satisfying the bounds (\ref{bound}), $\rho \left( d\lambda \right) $ its spectral measure.
Let  $w(x)=\left( |x|^{p}+1\right)^{-1}$ with $p > 3$.  Then, for
 $\rho \left( d\lambda \right) $-almost all $\lambda >0 $, $M$ has  
 a  generalized eigenfunction $\Psi _\lambda $  
 satisfying
%%@Ber1 
\begin{equation}
\int_{\Bbb{R}^3}|\Psi _\lambda (x)|^2w(x)\,dx<\infty,   \label{Ber1}
\end{equation}
so for any $\ell\in \Bbb{N}$ we have %@Ber10
\begin{equation}
\|\chi_{x,\ell} \Psi_\lambda\| \le C_\ell \left( |x|^{p}+1\right)
 \;\mbox{for all}\; x \in\ell \Bbb{Z}^3,   \label{Ber10}
\end{equation}
 for some constant $C_\ell < \infty$ depending only on $\ell$,  $\varepsilon_\pm$ 
and the LHS of (\ref{Ber1}). 
 \end{theorem}

\proof  Let
\begin{equation}
F(t) = \left\{
\begin{array}{ll}
(t +1)^{-1},   &  \mbox{if $t >0$}; \\
0,   &  \mbox{if $t  \le 0$}.
\end{array}
\right.
\end{equation}
$F$ is a bounded measurable function on the real line, continuous on $(0,\infty)$, such that
\begin{equation}
F(M) = ({\bf M} + I_{\Bbbs{S}})^{-1} \oplus 0_{\Bbbs{G}} \label{fm}
\end{equation}
with respect to Weyl's decomposition \eq{weyl}.

The operator $F(M){W}^{ \frac 1 2}$ is Hilbert-Schmidt by Theorem \ref{ths} below,  $W$
being the operator given by multiplication by the function $w(x)$.  The existence of generalized eigenfunctions
satisfying  (\ref{Ber1}), for $\rho \left( d\lambda \right) $-almost all $\lambda >0 $, now follows from
\cite[Subsections  V.4.1-V.4.2]{B}.

The estimate (\ref{Ber10}) is an immediate consequence of (\ref{Ber1}).  {{$\; \Box$}}  





\subsection{Estimates on traces}
  
\begin{theorem} \label{ths}
Let $M$ be an  operator of the form (\ref{Max2}) with (\ref{bound}), and let $V$ denote the bounded operator 
given by multiplication by the bounded measurable function $v(x)$,  with $v(x) \ge 0$ and  %@vv
\begin{equation}
\sum_{x \in \Bbbs{Z}^3}  \| \chi_{x,1} v^2\|_\infty  < \infty.  \label{vv}
\end{equation}
Then the operator
\begin{equation}
P_{\Bbbs{S}} (M + I)^{-1} V = \left[ ({\bf M} + I_{\Bbbs{S}})^{-1} \oplus 0_{\Bbbs{G}}\right] V \label{hso}
\end{equation}
is Hilbert-Schmidt.
\end{theorem}

Theorem \ref{ths} was used in the proof of Theorem \ref{pBer} with 
$v(x) = [w(x)]^{\frac 1 2}= \left( |x|^{p}+1\right)^{-\frac 1 2} $. 

To prove the theorem we will introduce a modified Maxwell operator $\widetilde{M}$ which is  elliptic.  Formally,
\begin{equation}
\widetilde{M} = \widetilde{M}(\varepsilon) =  
 M  + Y   ,  \label{Max5}
\end{equation}
with $Y = - \nabla  \frac 1{\varepsilon } \nabla{\cdot }$, i.e., 
$Y\Psi = - \nabla \left\{ \frac 1{\varepsilon }[ \nabla{\cdot }\Psi]\right\}$. $\widetilde{M}$  is rigorously defined
 as the nonnegative self-adjoint operator on $ L^2(\Bbb{R}^3;\,\Bbb{C}^3)$
given by the closure of the nonnegative quadratic form 
\begin{equation}
\widetilde{\mathcal{M}}[\Psi ]= {\mathcal{M}}[\Psi] +
 \int_{\Bbbs{R}^3}  \frac1{\varepsilon(x) } \left| [\nabla{\cdot }\Psi](x)\right|^2 \, dx
,\;\;\Psi \in C_0^1(\Bbb{R}^3;\,\Bbb{C}^3). 
\label{calM5}
\end{equation}
The operator $\widetilde{M}$ is diagonal with respect to Weyl's decomposition \eq{weyl}, with
$\widetilde{M} = {\bf M} \oplus {\bf Y}$ for the appropriate operator ${\bf Y}$ on $\Bbb{G}$. 

 If $\varepsilon(x) \equiv 1$, we have
\begin{equation}
\widetilde{\Xi} \equiv \widetilde{M}(1)=  - \Delta \otimes I_3 , \label{wxi}
\end{equation} 
where  $\Delta$ is the Laplacian in $ L^2(\Bbb{R}^3)$ and $ I_3$ is the identity operator on $\Bbb{C}^3$.

Since
\begin{equation}
 ({\bf M} + I_{\Bbbs{S}})^{-2} \oplus 0_{\Bbbs{G}} \le 
 ({\bf M} + I_{\Bbbs{S}})^{-2} \oplus  ({\bf Y} + I_{\Bbbs{G}})^{-2} = \left(  \widetilde{M} + I \right)^{-2} ,
\end{equation}
 Theorem \ref{ths} is an immediate consequence of the following theorem.%@ths1

 \begin{theorem} \label{ths1}
Let $\widetilde{M}$ be as in (\ref{Max5}) with (\ref{bound}), and let $v(x)$ and $V$ be as in Theorem \ref{ths}.
Then the operator
$\left(  \widetilde{M} + I \right)^{-1} V $
is Hilbert-Schmidt.
\end{theorem}

\proof
We  set $ \chi_x = \chi_{x,1}$ for  $x \in \Bbb{R}^3$ and
\begin{equation}
\widetilde{R} = \left( \widetilde{M} + 1 \right)^{-1}, \;\; 
\widetilde{S}(\mu) = \left( \widetilde{\Xi} + \mu\right)^{-1} \;\;\mbox{for} \;\; \mu > 0;
\end{equation}
\begin{equation}
\widetilde{R}_{x,y} = \chi_x\widetilde{R}\chi_y , \;\; 
\widetilde{S}(\mu)_{x,y} =  \chi_x\widetilde{S}(\mu)\chi_y \;\;\mbox{for} \;\; x, y \in \Bbb{R}^3.
\end{equation}
It follows from \eq{bound} and \eq{tra2} that
\begin{equation}
\varepsilon_-\widetilde{S}(\varepsilon_-) \le \widetilde{R} \le \varepsilon_+\widetilde{S}(\varepsilon_+). 
\label{tra21}
\end{equation}
We let 
\begin{equation} 
\widehat{S}=\varepsilon_+\widetilde{S}(\varepsilon_+), \;\; \label{tra211}
 \widehat{S}_{x,y} =  \chi_x\widehat{S}(\mu)\chi_y.
\end{equation}
We also set  $\chi_{x,y} = \max\{\chi _x,\chi _y\}$ for $x,y \in \Bbb{R}^3$, 
notice $\chi_{x,y}^2=\chi_{x,y} $ and $\chi_{x,x}=\chi_{x}$.


%@ltrL
\begin{lemma} \label{ltrL}
Let $p >\frac 3 2$ and $\mu >0$. Then there exists a constant $%
c_1=c_1\left(p ,\mu \right) < \infty $, depending only on the indicated parameters,
such that 
%@trL1 
\begin{equation}
{\rm Tr} \,\left\{\chi_{x,y}\left[\widetilde{S}(\mu)\right] ^{p}\chi_{x,y} \right\}\leq c_1  \label{trL1}
\end{equation}
for all $x,y \in \Bbb{R}^3$.
\end{lemma}

\proof  It follows from \eq{wxi} that it suffices to show that
%@trL11
\begin{equation}
{\rm Tr} \,\left\{\chi _{x,y}\left( - \Delta + \mu\right) ^{-p}\chi_{x,y} \right\}\leq c  \label{trL11}
\end{equation}
for all $x,y \in \Bbb{R}^3$, the trace now being calculated in $L^2 (\Bbb{R}^3)$.  But this is a consequence
of the fact that the operator $\left( - \Delta + \mu\right) ^{-p}$ has a bounded kernel;  it is
taken into multiplication by an integrable function by the  Fourier transform. {{$\; \Box$}} 
\bigskip

 We recall some general results.  Given a compact operator $A$ on a Hilbert space, we set $s_j(A) = \lambda_j(|A|)$, where  
$\lambda _1\left( |A|\right) \geq \lambda _2\left(
|A|\right) \geq \ldots $ are the strictly positive eigenvalues of $|A|$, repeated according to their
multiplicity.  For such $A$ we have (e.g.,  \cite{GK}):
\begin{equation}
\|A\|_p^p \equiv {\rm Tr}\,\left( |A|^p\right) = \sum_j [s_j(A)]^p , \;\; 1 \le p < \infty;  \label{tra1}
\end{equation}
\begin{equation}
s_j\left( A\right) =s_j\left( A^{*}\right) \;\;\mbox{for any $j$},  \;\; \mbox{so} \;\;
\|A\|_p= \|A^*\|_p;\label{GK1}
\end{equation}
\begin{equation}
s_j\left( BA\right) ,s_j\left( AB\right) \leq \left\|
B\right\| s_j\left( A\right)  \;\;\mbox{for any bounded operator  $B$}. \label{GK2}
\end{equation}
If $A$ and $B$ are self-adjoint operators and $A\geq B\geq 0$, we have
%@tra2 
\begin{equation}
{\rm Tr} \,A^2\geq {\rm Tr} \,B^2;\quad A^{-1}\leq B^{-1};\quad A^\beta
\geq B^\beta ,\ 0<\beta \leq 1  .\label{tra2}
\end{equation}

We will also need the following general statement.
%@ltr2

\begin{lemma}
\label{ltr2} Let $A$ be a nonnegative
 bounded operator and $P$  an orthogonal projection on a  Hilbert space ${\cal H}$. For any $\gamma
\geq 1$ we have

\begin{equation}
{\rm Tr}\,[PAP]^\gamma\leq {\rm Tr}\, PA^\gamma P . \label{papt}
\end{equation}

\end{lemma}

{\proof}  
 Let  $B$ be a  nonegative compact operator on ${\cal H}$ .  By the 
mini-max principle, we get 
\begin{equation}
\lambda _j(B)=
\max_{\{F\subset{\cal H};\, {\rm dim} \,F=j\}}\min_{\{\varphi \in F;\,\left\| \varphi \right\| =1\}}\left\langle \varphi ,B\varphi \right\rangle .  \label{pap1}
\end{equation}
If $\gamma\geq 1$, it  follows from Jensen's inequality that  
 for any $\varphi \in{\cal H}$ with $\left\| \varphi
\right\| =1$ we have
\begin{equation}
\left\langle \varphi ,B\varphi \right\rangle^\gamma \le \left\langle \varphi ,B^\gamma \varphi \right\rangle. 
\label{pap2}
\end{equation}

Without loss of generality we can assume  ${\rm Tr}\, PA^\gamma P < \infty$. In this case we claim that 
\begin{equation}
[\lambda _j (PAP)]^\gamma\leq \lambda _j(PA^\gamma P) \;\;\mbox{for any $j$},  \label{pap}
\end{equation}
so \eq{papt} follows.  Indeed, using \eq{pap2} 
 and (\ref{pap1}) we obtain, with ${\cal F}=P{\cal H}$,
\begin{eqnarray*}
[\lambda _j (PAP)]^\gamma &=&
\left[ \max_{\{F\subset {\cal F}; \,{\rm dim} \,F=j\}}\min_{\{\varphi \in F;\,\left\| \varphi \right\| =1\}}\left\langle \varphi ,B\varphi
\right\rangle \right] ^\gamma =
\max_{\{F\subset {\cal F}; \,{\rm dim} \, F=j\}}\min_{\{\varphi \in F;\,\left\| \varphi \right\| =1\}}\left\langle \varphi ,B\varphi
\right\rangle ^\gamma  \\
&\leq &\max_{\{F\subset {\cal F}; \,{\rm dim} \,F=j\}}\min_{\{\varphi \in F;\,\left\| \varphi \right\| =1\}}\left\langle \varphi ,B^\gamma \varphi \right\rangle
=\lambda _j(PA^\gamma P).
\end{eqnarray*}
 {{$\; \Box$}} 
\bigskip 



%@ltr1
\begin{lemma}
\label{ltr1} There exists a constant $c_2=c_2\left( \varepsilon_+\right) < \infty $,
 depending only on   $\varepsilon_+ $,  such that
%@tra10 
\begin{equation}
{\rm Tr}\,\left| {\widetilde{R}}_{x,y}\right| ^2=
{\rm Tr}\,{\widetilde{R}}_{x,y}^{*}{\widetilde{R}}_{x,y}\leq c_2 \;\;
\mbox{for all $x,y\in \Bbb{R}^3$}.  \label{tra10}
\end{equation}
In particular, the operators ${\widetilde{R}}_{x,y}$ are compact.
\end{lemma}

{\proof} We have
\begin{eqnarray}
{\rm Tr}\,{\widetilde{R}}_{x,y}^{*}{\widetilde{R}}_{x,y} &=&{\rm Tr}\,\chi _y%
{\widetilde{R}}\chi _x{\widetilde{R}}\chi _y
 \leq {\rm Tr}\,\chi _y{\widetilde{R}}\chi_{x,y} {\widetilde{R}}\chi _y
\label{tra11} \\
&=&{\rm Tr}\,\chi_{x,y} {\widetilde{R}}\chi _y{\widetilde{R}}\chi_{x,y}
 \leq {\rm Tr}\,\chi_{x,y}{\widetilde{R}}\chi_{x,y}{\widetilde{R}}\chi_{x,y} 
={\rm Tr}\,\left( \chi_{x,y} {\widetilde{R}}\chi_{x,y}\right) ^2 .  \nonumber
\end{eqnarray}
On the other hand using (\ref{tra2}), (\ref{tra21}), (\ref{tra211}) and \eq{trL1}  we obtain 
\begin{eqnarray}
{\rm Tr}\,\left(\chi_{x,y} {\widetilde{R}}\chi_{x,y} \right) ^2 &\leq &
{\rm Tr}\,\left(\chi_{x,y}  {\widehat{S}}\chi_{x,y} \right) ^2={\rm Tr}\,\chi_{x,y} 
 {\widehat{S}}\chi_{x,y} {\widehat{S}}\chi_{x,y}  \label{tra12} \\
&\leq &{\rm Tr}\,\chi_{x,y}  {\widehat{S}}^2\chi_{x,y} 
\le  \varepsilon_+^2 c_1(2, \varepsilon_+). 
 \nonumber
\end{eqnarray}
The inequalities (\ref{tra11}) and (\ref{tra12}) 
imply (\ref{tra10}). {{$\; \Box$}} 



%@ltr5
\begin{lemma} \label{ltr5}
There exists a constant $c_3=c_3\left(\varepsilon_\pm\right) < \infty$, depending only
 on $\varepsilon_\pm$,  such that
 %@trR1 
\begin{equation}
{\rm Tr}\,\chi _x{\widetilde{R}}^2\chi _x\leq c_3 \;\;\mbox{for all $x\in \Bbb{R}^3$}.
   \label{trR1}
\end{equation}
\end{lemma}

{\proof}  We have
%@tra13 
\begin{equation}
{\rm Tr}\,\chi _x{\widetilde{R}}^2\chi _x=
\sum_{y \in \Bbbs{Z}^3}{\rm Tr}\,\chi _x{\widetilde{R}}\chi _y{\widetilde{R}}\chi _x
=\sum_{y \in \Bbbs{Z}^3}{\rm Tr}\,\left| {\widetilde{R}}_{x,y}\right|.
^2  \label{tra13}
\end{equation}
In addition,
if $0\leq \alpha <1$, we also have
%@tra14 
\begin{eqnarray}
{\rm Tr}\,\left| {\widetilde{R}}_{x,y}\right| ^2 ={\rm Tr}\,\left\{
\left|{\widetilde{R}}_{x,y}\right| ^{1-\frac \alpha 2}\left| {\widetilde{R}}_{x,y}\right|
^\alpha \left| {\widetilde{R}}_{x,y}\right| ^{1-\frac \alpha 2} \right\} 
\leq \left\|  {\widetilde{R}}_{x,y}\right\| ^\alpha {\rm Tr}\,
\left| {\widetilde{R}}_{x,y}\right| ^{2-\alpha }, \label{tra14}
\end{eqnarray}
so %@trR2
\begin{eqnarray}
{\rm Tr}\,\chi_x{\widetilde{R}}^2\chi _x &\leq  & \sum_{y \in \Bbbs{Z}^3}
\left\| {\widetilde{R}}_{x,y}\right\| ^\alpha {\rm Tr}\,\left| {\widetilde{R}}_{x,y}\right|
^{2-\alpha }  \nonumber\\
&\leq  &\left[\sup_{y \in \Bbbs{Z}^3}  {\rm Tr}\,\left| {\widetilde{R}}_{x,y}\right|
^{2-\alpha }\right]  \sum_{y \in \Bbbs{Z}^3} \left\| {\widetilde{R}}_{x,y}\right\| ^\alpha. 
 \label{trR2} 
\end{eqnarray}

>From Lemma \ref{lct1}, which  holds exactly as stated with $\widetilde{M}$ substituted for 
$M$, we get that %@ctct
\begin{equation}
\sum_{y \in \Bbbs{Z}^3} \left\| {\widetilde{R}}_{x,y}\right\| ^\alpha \le b(\varepsilon_-, \alpha) 
\label{ctct}
\end{equation}
for some constant $ b(\varepsilon_-,\alpha) < \infty$, which depends only on $\varepsilon_-$
 and $\alpha$.
To estimate the other term, notice that
using (\ref{GK1}), (\ref{GK2}), (\ref{tra2}),  (\ref{tra21}) and  (\ref{tra211}), we
obtain 
\begin{eqnarray}
\left[s_j\left( {\widetilde{R}}_{x,y}\right)\right]^2  &=&s_j\left( \left| {\widetilde{R}}_{x,y}\right| ^2\right) 
=s_j\left( \chi _y{\widetilde{R}}\chi _x{\widetilde{R}}\chi_y\right) 
\leq s_j\left( \chi _y{\widetilde{R}}\chi_{x,y} {\widetilde{R}}\chi _y\right) 
\nonumber \\
&=&s_j\left( \chi _y\chi_{x,y} {\widetilde{R}}\chi_{x,y} {\widetilde{R}}\chi_{x,y} \chi _y\right) \leq
s_j\left( \chi_{x,y} {\widetilde{R}}\chi_{x,y} {\widetilde{R}}\chi_{x,y} \right) 
=s_j\left( \left[\chi_{x,y} {\widetilde{R}}\chi_{x,y} \right] ^2\right)   \nonumber \\
&=& \left[s_j\left( \chi_{x,y} {\widetilde{R}}\chi_{x,y}\right)\right]^2
\le \left[s_j\left( \chi_{x,y} {\widehat{S}}\chi_{x,y}\right)\right]^2
= s_j\left(\left[ \chi_{x,y} {\widehat{S}}\chi_{x,y} \right] ^2\right)
\nonumber \\
&= & s_j\left( \chi_{x,y} {\widehat{S}}\chi_{x,y} {\widehat{S}}\chi_{x,y} \right)
=\left[s_j\left( \chi_{x,y} {\widehat{S}}\chi_{x,y} \right) \right]^2. \label{sRx3} 
\end{eqnarray}
Taking  $\alpha \in \left( 0,1/2\right) $ so
 $2 - \alpha  > \frac 3 2$, we use \eq{papt} and \eq{trL1} to get
\begin{eqnarray}
{\rm Tr}\,\left| {\widetilde{R}}_{x,y}\right| ^{2-\alpha }
&=&\sum_j \left[s_j\left( {\widetilde{R}}_{x,y}\right)\right]^{2-\alpha } \leq
\sum_j \left[s_j\left( \chi_{x,y} {\widehat{S}}\chi_{x,y} \right)\right]^{2-\alpha }\nonumber \\
&=&{\rm Tr}\,\left[ \chi_{x,y} {\widehat{S}}\chi_{x,y} \right]^{2-\alpha } 
\le {\rm Tr}\, \chi_{x,y} {\widehat{S}}^{2-\alpha } \chi_{x,y} \nonumber\\
&\leq& c_1\left(2-\alpha,\varepsilon_+ \right). \label{trR3}
\end{eqnarray}

The lemma is proved, since (\ref{trR1}) follows from   (\ref{trR2}), \eq{ctct} and (\ref{trR3}). {{$\; \Box$}}
\bigskip 


We can now finish the proof of Theorem \ref{ths1}.  Using \eq{trR1} and   \eq{vv}, we get 
\begin{eqnarray}
{\rm Tr}\, V {\widetilde{R}}^2V& =& {\rm Tr}\,{\widetilde{R}} V^2 {\widetilde{R}}=
\sum_{x \in \Bbbs{Z}^3} {\rm Tr}\,{\widetilde{R}} \chi_x V^2 {\widetilde{R}} 
 \le   \sum_{x \in \Bbbs{Z}^3} \| \chi_x v^2\|_\infty {\rm Tr}\,{\widetilde{R}} \chi_x {\widetilde{R}} \nonumber \\
& =& \sum_{x \in \Bbbs{Z}^3} \| \chi_x v^2\|_\infty {\rm Tr}\,{\chi_x\widetilde{R}}^2 \chi_x 
\le c_3 \sum_{x \in \Bbbs{Z}^3} \| \chi_x v^2\|_\infty  < \infty,
\end{eqnarray}
so  ${\widetilde{R}}V$ is a Hilbert-Schmidt operator.  {{$\; \Box$}}



\section{Periodic Maxwell operators and periodic boundary condition} \label{sper}

The (non-random) spectrum of a random Maxwell operator 
 can be represented as the union of the spectra of relevant periodic Maxwell operators, which in turn
are given as  the union of the spectra of  finite volume Maxwell operators with 
 periodic boundary condition. This is analogous to the situation for random Schr\"odinger operators \cite{KM2} and random 
acoustic operators \cite{FKl3}.   

In this section we study  Maxwell operators in periodic media.  We say that the  operators ${\bf M}$, $M$, given
  by (\ref{Max}), (\ref{Max2}) with (\ref{bound}), are 
$q$-periodic for some $q>0$, if  $\varepsilon(x)$ is a $q$-periodic function.  In this section we work with a
given period $q>0$  and $q$-periodic operators  ${\bf M}$ and $M$.

\subsection{Periodic boundary condition} 

We start by defining the restriction of such $M$ to a cube with periodic boundary condition.
Given a cube $\Lambda = \Lambda_\ell(x)$, where $x \in \Bbb{R}^3$ and 
$\ell >0$, we 
will denote by $\stackrel{\circ }{\Lambda}$ the torus we obtain by identifying the  edges
 of the closed cube $\bar{\Lambda}$ in the usual way.  We introduce the usual distance in the torus:
%%@noru 
\begin{equation}
\stackrel{\circ }{d}(x,y)\equiv\min_{m\in\ell\Bbbs{Z}^3}|x-y +m| 
\le \frac{\sqrt{3}\ell}{2}  \;\mbox{for all}\;
x,y \in\bar{\Lambda}.  \label{noru}
\end{equation}
We will identify functions on 
$\stackrel{\circ }{\Lambda}$ with their $\ell$-periodic extensions to $\Bbb{R}^3$; for example,
$C^1\left( \stackrel{\circ }{\Lambda}; \,\Bbb{C}^3\right)$ will be identified with the space of continuously 
differentiable, $\ell$-periodic, $\Bbb{C}^3$-valued functions on $\Bbb{R}^3$.  We define 
$W^{1,2}\left(\stackrel{\circ }{\Lambda};\,\Bbb{C}^3\right) $ as the closure of $C^1\left( \stackrel{\circ }{\Lambda};\,\Bbb{C}^3\right)$
in  $W^{1,2}\left({\Lambda};\,\Bbb{C}^3\right) $.

We will always take $\ell \in q\Bbb{N}$ and  define  $\stackrel{\circ }{M}_{\Lambda}$,  the  restriction
 of $M$ to $\Lambda$ with periodic boundary condition, 
 as the unique nonnegative self-adjoint operator on 
$L^2\left(\stackrel{\circ }{\Lambda};\Bbb{C}^3\right) \cong L^2\left({\Lambda};\Bbb{C}^3\right)$,  
 defined by  the nonnegative densely defined closed
quadratic form %%@MpC  
\begin{equation}
\stackrel{\circ }{\mathcal{M}}_{\Lambda}\left(\Psi,\Phi \right)=
 \langle \nabla  \times  \Psi ,\frac 1{\varepsilon}\nabla  \times  \Phi \rangle,\;\;\mbox{with}\;\;\Psi,\Phi \in
W^{1,2}\left(\stackrel{\circ }{\Lambda};\Bbb{C}^3\right)_\Lambda\;,  \label{MpC}
\end{equation} 
the inner product being in $L^2\left(\Lambda;\Bbb{C}^3\right)$. 

We also have a corresponding Weyl's decomposition in the torus:  
$L^2\left(\Lambda;\Bbb{C}^3\right) = 
{\stackrel{\circ }{\Bbb{S}}}_\Lambda \oplus {\stackrel{\circ }{\Bbb{G}}}_\Lambda$, where
\begin{eqnarray}
{\stackrel{\circ }{\Bbb{S}}}_\Lambda &=&
\overline{\left\{\Psi \in L^2\left(\Lambda;\Bbb{C}^3\right);\;\; \Psi \in C^1\left(\stackrel{\circ }{\Lambda};\,\Bbb{C}^3\right) 
\;\;\mbox{with}\;\; \nabla\cdot\Psi=0 \right\}}, \\[.1in]
{\stackrel{\circ }{\Bbb{G}}}_\Lambda &=&
\overline{\left\{\Psi \in L^2\left(\Lambda;\Bbb{C}^3\right);\;\; \Psi= \nabla\varphi 
\;\;\mbox{with}\;\; \varphi \in  C^1\left(\stackrel{\circ }{\Lambda}\right) \right\}} .
\end{eqnarray}
The spaces ${\stackrel{\circ }{\Bbb{S}}}_\Lambda$ and ${\stackrel{\circ }{\Bbb{G}}}_\Lambda$ are left invariant by
  $\stackrel{\circ }{M}_{\Lambda}$, with  
${\stackrel{\circ }{\Bbb{G}}}_\Lambda \subset \mathcal{D}\left( \stackrel{\circ }{M}_{\Lambda}\right) $ and 
 $\left.\stackrel{\circ }{M}_{\Lambda}\right| _{{\stackrel{\circ }{\Bbbs{G}}}_\Lambda} =0$.  We define
 ${\stackrel{\circ }{\bf M}}_{\Lambda}$ as the restriction of $\stackrel{\circ }{M}_{\Lambda}$ to ${{\stackrel{\circ }{\Bbb{S}}}_\Lambda}$,
 i.e.,
$\mathcal{D}\left({\stackrel{\circ }{\bf M}}_{\Lambda}\right) =
\mathcal{D}\left( \stackrel{\circ }{M}_{\Lambda}\right) \cap {\stackrel{\circ }{\Bbb{S}}}_\Lambda$ and 
${\stackrel{\circ }{\bf M}}_{\Lambda}=
\left. \stackrel{\circ }{M}_{\Lambda}\right| _{{\cal D}\left( \stackrel{\circ }{M}_{\Lambda}\right) 
\cap {\stackrel{\circ }{\Bbbs{S}}}_\Lambda}$. 
 Thus
$
{\stackrel{\circ }{\bf M}}_{\Lambda} =    P_{{\stackrel{\circ }{\Bbbs{S}}}_\Lambda}\stackrel{\circ }{M}_{\Lambda} I_{{\stackrel{\circ }{\Bbbs{S}}}_\Lambda}=
\stackrel{\circ }{M}_{\Lambda} I_{{\stackrel{\circ }{\Bbbs{S}}}_\Lambda}  
$, with $ P_{{\stackrel{\circ }{\Bbbs{S}}}_\Lambda} $ the
 orthogonal projection onto ${\stackrel{\circ }{\Bbb{S}}}_\Lambda$ and
 $ I_{{\stackrel{\circ }{\Bbbs{S}}}_\Lambda}:{\stackrel{\circ }{\Bbb{S}}}_\Lambda \to L^2\left(\Lambda;\,\Bbb{C}^3\right)  $
 the restriction of the  identity map.   
 Notice that $\stackrel{\circ }{M}_{\Lambda} = 
 {\stackrel{\circ }{\bf M}}_{\Lambda} \oplus 0_{{\stackrel{\circ }{\Bbbs{G}}}_\Lambda}$,
and $0$ is easily seen to be an eigenvalue of  $\stackrel{\circ }{{\bf M}}_{\Lambda}$ with
 multiplicity three, 
 so   %@2spectra 
\begin{equation}
\sigma\left({\stackrel{\circ }{\bf M}}_{\Lambda}\right) = 
\sigma\left(\stackrel{\circ }{M}_{\Lambda}\right).   \label{2spectra}
\end{equation} 

If $\varepsilon(x) \equiv 1$ we write $ {\stackrel{\circ }{ \Xi}}_{\Lambda}$, $ {\stackrel{\circ }{\bf \Xi}}_{\Lambda}$
for ${\stackrel{\circ }{ M}}_{\Lambda}$, ${\stackrel{\circ }{\bf M}}_{\Lambda}$, respectively.  Since
$ {\stackrel{\circ }{\bf \Xi}}_{\Lambda}$ has compact resolvent  (its eigenvalues and eigenfunctions 
can be explicitly computed), and 
  $ {\stackrel{\circ }{\bf M}}_{\Lambda}  \ge \frac{1}{\varepsilon_+} {\stackrel{\circ }{\bf \Xi}}_{\Lambda}$
by \eq{bound}, we can  conclude that 
 ${\stackrel{\circ }{\bf M}}_{\Lambda}$ has compact resolvent. 

 \subsection{A Combes-Thomas argument for the torus} \label{ssctt}

 If $z\notin \sigma (\stackrel{\circ }{M}_{\Lambda})$, we
write $ \stackrel{\circ }{R}_{\Lambda}(z)=(\stackrel{\circ }{M}_{\Lambda}-z)^{-1}$.

\begin{lemma} \label{ctt}
Let the operator $M$  given  by (\ref{Max2}) with (\ref{bound}) be $q$-periodic, and let
 $\Lambda = \Lambda_\ell(x_0)$ for some $x_0 \in \Bbb{R}^3$ and  $\ell \in q\Bbb{N}$, 
$\ell > 2r + 8$, where $r > 0$.
Then for any $z\notin  \sigma (\stackrel{\circ }{M}_{\Lambda})$ and $n \in \Bbb{N}$  we have
%%@ctt1 
\begin{equation}
\Vert \chi _{x,r} \stackrel{\circ }{R}_{\Lambda}(z)^n\chi _{y,r}\Vert \;\;\le 
\left(\frac{9}{\eta} \right)^n e^{\frac{\sqrt{3} r \stackrel{\circ }{m}_{z,r,\ell}}{2}} e^{-\stackrel{\circ }{m}_{z,r,\ell} \stackrel{\circ }{d}(x,y)}\;\;\;
\mbox{for all}\;\;x,y\in \stackrel{\circ }{\Lambda},  \label{ctt1}
\end{equation}
with
%%@mztt
\begin{equation}
\stackrel{\circ }{m}_{z,r,\ell}=
\frac \eta {4 \left(\frac{ 2\sqrt{3}}{1-\frac{2r + 8}{\ell} } +1\right)
 \left[\varepsilon_-^{-1}+|z|+\eta \right] },
\label{mztt}
\end{equation}
where $\eta =\mbox{ \rm dist}(z, \sigma (\stackrel{\circ }{M}_{\Lambda}))$.
\end{lemma}

\proof The lemma is proved in the same way as \cite[Lemma 18]{FKl3}, with the obvious modifications 
to take into account that in this lemma we have {\em curls} instead of {\em gradients}.  
{$\; \Box$}


   

\subsection{Floquet theory and the spectrum of periodic operators}


If $k,n \in \Bbb{N}$, we say that $k \preceq n$ if $n \in k\Bbb{N}$ and that $k \prec n$  
if $k \preceq n$ and $k \not= n$.  The main result of this section is the following theorem.
%@tMp1

\begin{theorem}
\label{tMp1} Suppose the operator ${\bf M}$  given  by (\ref{Max}) with (\ref{bound}) is 
$q$-periodic.  Let $\{\ell_n;\; n =0,1,2,\ldots\}$ be a sequence in $\Bbb{N}$ such that $\ell_0 =q$
and $\ell_n  \prec \ell_{n+1}$ for each $n =0,1,2,\ldots$. 
 Then 
%@MC1a 
\begin{equation}
\sigma \left( \stackrel{\circ }{{\bf M}}_{\Lambda_{\ell_n}(0)}\right) \subset
\sigma \left( \stackrel{\circ }{{\bf M}}_{\Lambda_{\ell_n+1}(0)}\right) \subset \sigma ({\bf M})\;
\; \mbox{for all}\;\; n =0,1,2,\ldots,
\label{MC1a}
\end{equation}
and
%@MC1b 
\begin{equation}
\sigma ({\bf M})=
\overline{\bigcup_{n\geq 1}\sigma \left( \stackrel{\circ }{{\bf M}}_{\Lambda_{\ell_n}(0)}\right) }.
\label{MC1b}
\end{equation}
\end{theorem}


Related  results for periodic
Schr\"odinger operators can be found in \cite{Ea}, where  Floquet theory is used.
 For the  nonsmooth coefficients we are interested in some aspects of the Floquet theory
have to be revised.  Periodic acoustic operators are treated in \cite[Theorem 14]{FKl3}, with a proof that does not use 
Floquet theory.  In this subsection we will develop an appropriate Floquet theory for our Maxwell operators, and use it
to prove Theorem \ref{tMp1}.  We refer to \cite[Section XIII.6]{RS4} for the definitions and notations of direct
integrals of Hilbert spaces. 

Let $Q = \breve{\Lambda}_q(0)$ be the basic period cell, $\tilde{Q} = \breve{\Lambda}_{\frac {2\pi}{q}}(0)$ the dual basic cell.  We
 define the Floquet transform
\begin{equation}
{\cal F}:\;\; L^2\left(\Bbb{R}^3; \Bbb{C}^3\right) \to \int_{\tilde{Q}}^\oplus  L^2\left(Q;\Bbb{C}^3\right) \, dk \equiv
 L^2\left(\tilde{Q}; L^2\left(Q;\Bbb{C}^3\right)\right)
\end{equation}
by
\begin{equation}
({\cal F}\Psi)(k,x) =
 \left(\frac{q}{2\pi}\right)^{\frac 3 2} \sum_{m \in q\Bbbs{Z}^3} {\rm e}^{ik\cdot (x-m)} \Psi(x-m),
\;\;\ x \in Q, \ k \in \tilde{Q},
\end{equation}
if $\Psi$ has compact support; it extends by continuity to a unitary operator.

The $q$-periodic operator $M $ is decomposable in this direct integral representation, more precisely,
\begin{equation}
{\cal F} M {\cal F}^* = \int_{\tilde{Q}}^\oplus  \stackrel{\circ }{M}_{Q}(k) \,  dk,
\end{equation}
where for each $k \in  \Bbb{R}^3$ we define  
$ \stackrel{\circ }{M}_{Q}(k)$ to be the  operator 
$ (\nabla - ik)^\times \frac1 \varepsilon (\nabla - ik)^\times$
 on $L^2\left({Q};\Bbb{C}^3\right)$ with periodic boundary condition; $ \stackrel{\circ }{M}_{Q}(k)$ is rigorously
defined as a self-adjoint operator by the appropriate quadratic form 
$\stackrel{\circ }{\mathcal{M}}_{Q}(k)$ as in \eq{MpC}.  As before $(\nabla - ik)^\times$ denotes the operator
$(\nabla - ik)^\times \Phi =(\nabla - ik) \times \Phi $.  We also have Weyl's decompositions for each
$k \in   \Bbb{R}^3$:  
$L^2\left(Q;\Bbb{C}^3\right) = {\stackrel{\circ }{\Bbb{S}}}_Q(k) \oplus{\stackrel{\circ }{\Bbb{G}}}_Q(k)$, where
\begin{eqnarray}
{\stackrel{\circ }{\Bbb{S}}}_Q(k) &=&
\overline{\left\{\Psi \in L^2\left(Q;\Bbb{C}^3\right);\;\; \Psi \in C^1\left(\stackrel{\circ }{Q};\,\Bbb{C}^3\right) 
\;\;\mbox{with}\;\; (\nabla - ik)\cdot\Psi=0 \right\}}, \\[.1in]
{\stackrel{\circ }{\Bbb{G}}}_Q(k) &=&
\overline{\left\{\Psi \in L^2\left(Q;\Bbb{C}^3\right);\;\; \Psi= (\nabla - ik)\varphi 
\;\;\mbox{with}\;\; \varphi \in  C^1\left(\stackrel{\circ }{Q}\right) \right\}} .
\end{eqnarray}
The spaces ${\stackrel{\circ }{\Bbb{S}}}_Q(k)$ and ${\stackrel{\circ }{\Bbb{G}}}_Q(k)$ are left invariant by  $\stackrel{\circ }{M}_{Q}(k)$, with  
${\stackrel{\circ }{\Bbb{G}}}_Q(k) \subset \mathcal{D}\left( \stackrel{\circ }{M}_{Q}(k)\right) $ and 
 $\left.\stackrel{\circ }{M}_{Q}(k)\right| _{ {\stackrel{\circ }{\Bbbs{G}} }_{Q}(k)} =0$.  We define
 ${\stackrel{\circ }{\bf M}}_{Q}(k)$ as the restriction of $\stackrel{\circ }{M}_{Q}(k)$ to ${{\stackrel{\circ }{\Bbb{S}}}_Q(k)}$,
 i.e.,
$\mathcal{D}\left({\stackrel{\circ }{\bf M}}_{Q}(k)\right) =
\mathcal{D}\left( \stackrel{\circ }{M}_{Q}{k}\right) \cap {\stackrel{\circ }{\Bbb{S}}}_Q(k)$ and 
${\stackrel{\circ }{\bf M}}_{Q}(k)=
\left. \stackrel{\circ }{M}_{Q}(k)\right| _{{\cal D}\left( \stackrel{\circ }{M}_{Q}(k)\right) 
\cap {{\stackrel{\circ }{\Bbbs{S}}}_{Q}(k)}}$. 
 Thus
$
{\stackrel{\circ }{\bf M}}_{Q}(k) = 
   P_{ { \stackrel{\circ }{\Bbbs{S}} }_{Q}(k) }\stackrel{\circ }{M}_{Q}(k) I_{ {\stackrel{\circ }{\Bbbs{S}}}_{Q}(k) }=
\stackrel{\circ }{M}_{Q}(k) I_{ { \stackrel{\circ }{\Bbbs{S}} }_{Q}(k) } 
$,
 with $ P_{ { \stackrel{\circ }{\Bbbs{S}} }_{Q}(k) } $ the
 orthogonal projection onto ${\stackrel{\circ }{\Bbb{S}}}_Q(k)$ and
 $ I_{ { \stackrel{\circ }{\Bbbs{S}} }_{Q}(k) }:{\stackrel{\circ }{\Bbb{S}}}_Q(k) \to L^2\left(Q;\,\Bbb{C}^3\right)  $
 the restriction of the  identity map.   
 Notice $\stackrel{\circ }{M}_{Q}(k) =  {\stackrel{\circ }{\bf M}}_{Q}(k) \oplus 0_{  {\stackrel{\circ }{\Bbbs{G}} }_{Q}(k)  }$, so 
$\sigma\left({\stackrel{\circ }{\bf M}}_{Q}(k)\right) = 
\sigma\left(\stackrel{\circ }{M}_{Q}(k)\right)$.  Each $\stackrel{\circ }{{\bf M}}_{Q}(k) $ has compact
resolvent.   We have
\begin{equation}
{\cal F}{\Bbb{S}}= \int_{\tilde{Q}}^\oplus {{\stackrel{\circ }{\Bbb{S}}}}_Q(k) \, dk, \;\;\;\;
{\cal F} {\bf M} {\cal F}^* = \int_{\tilde{Q}}^\oplus  \stackrel{\circ }{{\bf M}}_{Q}(k) \,  dk.
\end{equation}

In addition, if for each $p \in \frac{2 \pi}{q} \Bbb{Z}^3$ we let $U_p$ denote the unitary operator on 
$ L^2\left(Q;\Bbb{C}^3\right)$ given by multiplication by the function ${\rm e}^{-i p \cdot x}$,
then for all $k \in  \Bbb{R}^d$ and $p \in \frac{2 \pi}{q} \Bbb{Z}^3$ we have
\begin{equation}
\stackrel{\circ }{M}_{Q}(k + p) =U_p^*\stackrel{\circ }{M}_{Q}(k)U_p,  \label{mper}
\end{equation}
and, since 
$U_p{\stackrel{\circ }{\Bbb{S}}}_Q(k+p)={\stackrel{\circ }{\Bbb{S}}}_Q(k)$, we can also think of $U_p$ as a unitary operator from 
$ {\stackrel{\circ }{\Bbb{S}}}_Q(k+p)$ to ${\stackrel{\circ }{\Bbb{S}}}_Q(k)$, with
 \begin{equation}
\stackrel{\circ }{{\bf M}}_{Q}(k + p) =U_p^*\stackrel{\circ }{{\bf M}}_{Q}(k)U_p \label{mbper}
.\end{equation}

 
%@lperres

\begin{lemma} \label{lperres}
\begin{description}
\item[(i)]
The mapping
 \begin{equation}
k \in  \Bbb{R}^3 \; \;\longmapsto\;  \; \stackrel{\circ }{R}_{Q}(k) \equiv
\left(\stackrel{\circ }{M}_{Q}(k)  + I \right)^{-1} \in {\cal L}\left(L^2\left(Q;\Bbb{C}^3\right)  \right)
\end{equation}
 is operator norm continuous.
\item[(ii)]  We have
\begin{equation}
\sigma(M)= \overline{\bigcup_{k \in \tilde{Q}} \sigma\left(\stackrel{\circ }{M}_{Q}(k)\right)}
\;\;\;\;{\rm and}\;\;\;\;
\sigma({\bf M})=
 \overline{\bigcup_{k \in \tilde{Q}} \sigma\left(\stackrel{\circ }{{\bf M}}_{Q}(k)\right)}.
\label{spper}
\end{equation}
\end{description}
\end{lemma} 

\proof 
 Let $k, h \in \Bbb{R}^3$, $\Psi \in  L^2\left(Q;\Bbb{C}^3\right)  $, we have
\begin{eqnarray}
\lefteqn{\stackrel{\circ }{\mathcal{M}}_{Q}(k+h)[\Psi] -\stackrel{\circ }{\mathcal{M}}_{Q}(k)[\Psi]=}\\
&&\langle h \times \Psi, {\frac 1 \varepsilon}  h \times \Psi \rangle_Q  +  
i\langle h \times \Psi, {\frac 1 \varepsilon} (\nabla - ik) \times \Psi \rangle_Q  
-i\langle (\nabla - ik) \times \Psi, {\frac 1 \varepsilon}  h \times \Psi \rangle_Q.
 \nonumber
\end{eqnarray}
Using the Cauchy-Schwarz  inequality and \eq{bound} we get
 (see \cite[Proof of Lemma 12]{FKl3} for a similar argument)
\begin{equation}
\left|\stackrel{\circ }{\mathcal{M}}_{Q}(k+h)[\Psi] -\stackrel{\circ }{\mathcal{M}}_{Q}(k)[\Psi]\right| \le
|h|\stackrel{\circ }{\mathcal{M}}_{Q}(k)[\Psi]+ |h|\left(1 + |h|  \right)  {\frac 1 \varepsilon_-}\|\Psi\|_Q^2.
\end{equation}
If $|h|<1$ we have
\begin{eqnarray}
\Vert (|h|(1+|h|)\varepsilon_-^{-1}+|h|\stackrel{\circ }{{M}}_{Q}(k))\stackrel{\circ }{R}_{Q}(k)\Vert \le
|h| \left( \left(1 + |h|  \right)  {\frac 1 \varepsilon_-} +2  \right) \le  2\left(  {\frac 1 \varepsilon_-} +1  \right)|h|  .
\end{eqnarray} 
If we now require $2\left(  {\frac 1 \varepsilon_-} +1  \right)|h| \le \frac 1 2$,  we can use 
\cite[Theorem VI.3.9]{Ka} to conclude that 
\begin{equation}
\Vert \stackrel{\circ }{R}_{Q}(k+h)-\stackrel{\circ }{R}_{Q}(k)\Vert \leq 
{32\left(  {\frac 1 \varepsilon_-} +1  \right)|h| }.
\end{equation}
Part (i) of the lemma is proved; part (ii) follows from (i) by standard arguments.  {{$\; \Box$}}
\bigskip

If $\ell \in q\Bbb{Z}^3$, similar considerations apply to the operators
$ \stackrel{\circ }{{ M}}_{\Lambda_{\ell}(0)}$ and $ \stackrel{\circ }{{\bf M}}_{\Lambda_{\ell}(0)}$,
which are $q$-periodic on the torus $\stackrel{\circ }{\Lambda}_{\ell}(0)$.  The Floquet transform 
\begin{equation}
{\cal F}_\ell:\;\; L^2\left(\stackrel{\circ }{\Lambda}_{\ell}(0); \Bbb{C}^3\right) \to 
\bigoplus_{k \in\frac{2\pi}{\ell}\Bbbs{Z}^3 \cap\tilde{Q}}  L^2\left(Q;\Bbb{C}^3\right)  
\end{equation}
is a unitary operator now defined by
\begin{equation}
({\cal F}_\ell\Psi)(k,x) =
 \left(\frac{q}{\ell}\right)^{\frac 3 2} \sum_{m \in q\Bbbs{Z}^3 \cap {\breve{\Lambda}}_{\ell}(0) } 
{\rm e}^{ik\cdot (x-m)} \Psi(x-m),
\end{equation}
where $ x \in Q, \ k \in \frac{2\pi}{\ell}\Bbb{Z}^3 \cap \tilde{Q}, 
\Psi \in L^2\left(\stackrel{\circ }{\Lambda}_{\ell}(0); \, \Bbb{C}^3\right)$,
 $ \Psi(x-m)$ being properly interpreted in the torus $\stackrel{\circ }{\Lambda}_{\ell}(0)$.   We also have
\begin{equation}
{\cal F}_\ell \stackrel{\circ }{{ M}}_{\Lambda_{\ell}(0)} {\cal F}_\ell^* = 
\bigoplus_{k \in\frac{2\pi}{\ell}\Bbbs{Z}^3 \cap\tilde{Q}}  \stackrel{\circ }{M}_{Q}(k),
\end{equation}
and
\begin{equation}
{\cal F}_\ell {{\stackrel{\circ }{\Bbb{S}}}}_{\Lambda_{\ell}(0)}= 
\bigoplus_{k \in\frac{2\pi}{\ell}\Bbbs{Z}^3 \cap\tilde{Q}} {{\stackrel{\circ }{\Bbb{S}}}}_Q(k) , \;\;\;\;
{\cal F}_\ell\stackrel{\circ }{{\bf M}}_{\Lambda_{\ell}(0)} {\cal F_\ell}^* =
\bigoplus_{k \in\frac{2\pi}{\ell}\Bbbs{Z}^3 \cap\tilde{Q}} \stackrel{\circ }{{\bf M}}_{Q}(k).
\end{equation}
Thus we have
\begin{equation}
\sigma( \stackrel{\circ }{{ M}}_{\Lambda_{\ell}(0)})= 
{\bigcup_{k \in\frac{2\pi}{\ell}\Bbbs{Z}^3 \cap\tilde{Q}} \sigma\left(\stackrel{\circ }{M}_{Q}(k)\right)}
\;\;\;\;{\rm and}\;\;\;\;
\sigma(\stackrel{\circ }{{\bf M}}_{\Lambda_{\ell}(0)})=
 {\bigcup_{k \in\frac{2\pi}{\ell}\Bbbs{Z}^3 \cap\tilde{Q}} \sigma\left(\stackrel{\circ }{{\bf M}}_{Q}(k)\right)}.
\label{spper2}
\end{equation}

Theorem \ref{tMp1} is an immediate consequence of \eq{spper2} and Lemma \ref{lperres}.


\section{Location of the spectrum of random operators} \label{slocation}

In this section we prove Theorem \ref{tlct}.  Since  we already proved 
Theorem \ref{tMp1}, the proof proceeds almost exactly as in \cite[Section 4]{FKl3}, so we will 
only outline the key steps.  

  In order to investigate the samples of the random
 quantity $\varepsilon_{g,\omega}(x)$, for a fixed $g$,   we set
\begin{equation}
{\cal T}_g=\{\tau:\tau=\{\tau_i,i\in \Bbb{Z}^3\},-g\leq \tau_i\leq
g\}, 
\end{equation}
\begin{equation}
{\cal T}_g^{(n)}=\{\tau\in {\cal T}:\tau_{i+nj}=\tau_i \;\;\mbox{for all}
\;\;i,j\in \Bbb{Z}^3\}, \;\; n \in \Bbb{N}, 
\end{equation}
and
\begin{equation}
{\cal T}_g^{(\infty)}=\bigcup_{n\succeq q}{\cal T}_g^{(n)}.
\end{equation}
For $\tau \in{\cal T}_g $ we let 
\begin{equation}
\varepsilon_\tau(x)=\varepsilon _0(x)\left[
1+\sum_{i\in \Bbbs{Z}^3}\tau_iu(x-i)\right] 
\end{equation}
and %%@atau
\begin{equation}
M(\tau)= M(\varepsilon_\tau), \;\; {\bf M}(\tau)= {\bf M}(\varepsilon_\tau). \label{atau}
\end{equation}
We recall \eq{1spectra}.

To approximate Maxwell operators by periodic operators, given  $\tau \in {\cal T}_g$,
  $n \in \Bbb{N}$ and $x \in \Bbb{R}^3$, we specify
$\tau_{\Lambda_n(x)} \in  {\cal T}_g^{(n)}$ by requiring
$\left(\tau_{\Lambda_n(x)}\right)_i=\tau_i$ for all 
$i\in {\breve{\Lambda}_n(x)}\cap \Bbb{Z}^3$, and define %%@perapp
\begin{equation} 
M_{\Lambda_n(x)}(\tau)=M(\tau_{\Lambda_n(x)}). \label{perapp}
\end{equation}


The following lemma shows that the (nonrandom) spectrum of the random Maxwell operator 
$M_g$ is determined by the spectra of the periodic Maxwell operators $ M(\tau)$, 
$\tau\in {\cal T}_g^{(\infty)}$.  The analogous  result for random Schr\"odinger operators
was proven by Kirsch and Martinelli \cite[Theorem 4]{KM2}.  

%%@lsAe
\begin{lemma}
\label{lsAe}
Let the random operator $M_g$ defined by (\ref{At}) satisfy Assumption \ref{aeu},
and let
%%@sgAe 
\begin{equation}
\Sigma _g=\overline{\bigcup_{\tau\in {\cal T}_g^{(\infty)} }\sigma\left( M(\tau)\right) } .
 \label{sgAe}
\end{equation}
Then  $\sigma (M_g)=\Sigma _g$ with probability one.
\end{lemma}

\proof Same proof as \cite[Lemma 19]{FKl3}.  {$\; \Box$} 
\bigskip

 Given a real number $h; \ |h| < \frac1{U_+}$, let %%@hhh
\begin{equation}
M(h)= M(\varepsilon_h)\;\;, {\bf M}(h)= {\bf M}(\varepsilon_h) \;\;\mbox{with}\;\;
\varepsilon_h(x)=\varepsilon _0(x)\left[1+ h U(x)\right].  \label{hhh}
\end{equation}
If $|h| \le g$, and we define $\tau(h) \in  {\cal T}_g$ by 
$\tau(h)_i  =h$ for all $i\in \Bbb{Z}^3$,  we have 
$\varepsilon_h =\varepsilon_{\tau(h)}$ and $ M(h)=M(\tau(h))$, 
$ {\bf M}(h)={\bf M}(\tau(h))$. 

%%@llco
\begin{lemma} \label{llco}  Let  $M(h)$, $ |h| < \frac1{U_+}$, be given by (\ref{hhh}), with 
$\varepsilon _0$ and  $U$ given in Assumption \ref{aeu}. Let $\Lambda = \Lambda_\ell(x_0)$ for some $x_0 \in \Bbb{R}^3$ and 
$\ell\succeq q$.  The positive self-adjoint operator 
$\stackrel{\circ }{{\bf M}(h)}_\Lambda$ has compact 
resolvent and $0$ as an eigenvalue, so  let $0< \mu _1(h)\leq
\mu _2(h)\leq \ldots $ be its nonzero eigenvalues, repeated according to their (finite) multiplicity.
  Then each $\mu _j(h)$, $j= 1,2,\ldots$, is a
Lipschitz continuous, strictly decreasing function of $h$, with
%%@gmh1 
\begin{equation}
\delta_-(g)\max_{l=1,2}\{\mu _j(h_l)\}\leq \frac{\mu _j(h_1)-\mu _j(h_2)}{%
h_2-h_1}\leq\delta_+(g)\min_{l=1,2}\{\mu _j(h_l)\}\label{gmh1}
\end{equation}
for any $h_1,h_2 \in (-g,g)$, $0<g <\frac1{U_+}$, where $\delta_\pm(g)$ are given in (\ref{delta}).
\end{lemma}


\proof Same proof as \cite[Lemma 20]{FKl3}.  {$\; \Box$} 
\bigskip

The following corollary follows immediately from Theorem \ref{tMp1}, 
Lemmas \ref{lsAe} and \ref{llco}, and the min-max principle.

%%@cAta
\begin{corollary} \label{cAta} Let the random operator $M_g$ defined by (\ref{At}) satisfy
 Assumption \ref{aeu}, and let $\{\ell_n;\; n =0,1,2,\ldots\}$ be a 
sequence in $\Bbb{N}$ such that $\ell_0 =q$
and $\ell_n  \prec \ell_{n+1}$ for each $n =0,1,2,\ldots$.  Then   
%%@sgCn 
\begin{equation}
\Sigma_g =\overline{\bigcup_{h \in [-g,g]}\sigma \left(M(h)\right) }
=\overline{\bigcup_{h \in [-g,g]}\bigcup_{n\geq 1}
\sigma \left( \stackrel{\circ }{M(h)}_{\Lambda_{\ell_n}(0)}\right) }.
\label{sgCn}
\end{equation}
In particular, $\Sigma_g$ is increasing in $g$.
\end{corollary}



 Theorem \ref{tlct} is now proven as in \cite[Subsection 4.2]{FKl3}, using Theorem \ref{tMp1},
 Lemma \ref{llco}  and Corollary \ref{cAta}, and taking \eq{1spectra} and \eq{2spectra}   
into account.

  


\section{Dirichlet boundary condition for Maxwell operators} \label{sdirich}

Given an open cube $\Lambda $ in $\Bbb{R}^3$ and $M$ as in \eq{Max2}, we will denote by $M_{\Lambda}$
 the restriction of $M$ to $\Lambda $ with Dirichlet boundary condition, i.e., $M_{\Lambda}$ is the 
 nonnegative self-adjoint operator on $ L^2(\Lambda;\,\Bbb{C}^3)$,
uniquely defined by the nonnegative quadratic form given as the closure of 
\begin{equation}
\mathcal{M}_{\Lambda}(\Psi,\Phi )=\langle \nabla  \times  \Psi ,\frac 1{\varepsilon}\nabla  \times  \Phi \rangle,
 \;\;\Psi,\Phi \in C_0^1(\Lambda;\,\Bbb{C}^3), 
\label{calMd}
\end{equation}
the inner product being in $ L^2(\Lambda;\,\Bbb{C}^3)$. 

 If $\varepsilon(x) \equiv 1$, we write $\Xi_\Lambda$
for $M(1)_{\Lambda}$. $\Xi_\Lambda$ has an operator core consisting of functions which are $C^2$ up to 
$\partial\Lambda$ and whose tangential component vanishes on $\partial\Lambda$. (For a discussion of 
boundary conditions for Maxwell operators in bounded domains see \cite{BS}.)  We will need this last description
to find all eigenvalues for $\Xi_\Lambda$.  This is all given in the next theorem. 

Some notation. If 
$\Psi \in C(\bar{\Lambda};\,\Bbb{C}^3)$, we use $\Psi_\nu$ and $\Psi_\tau$ to denote its (outer) normal and
 tangential components on $\partial\Lambda$.
%@tdirichlet


\begin{theorem} \label{tdirichlet}
Let  $\Lambda $ be an open cube of side $L$  in $\Bbb{R}^3$  .
\begin{description}

\item[(i)]  The dense linear subset 
\begin{equation}
{\cal D}_\Lambda^D = \left\{ \Psi \in C^2(\bar{\Lambda};\,\Bbb{C}^3); \, \Psi_\tau\equiv 0      
 \right\}
\end{equation}
is an operator core for $\Xi_\Lambda$, with 
$\Xi_\Lambda \Psi = \nabla \times  \nabla \times \Psi$ for $\Psi \in  {\cal D}_\Lambda^D$.  \label{core}

\item[(ii)] 
The operator $\Xi_\Lambda$ has an orthogonal basis of eigenfunctions
\begin{equation}
{\bf \Psi}=\left\{\Psi_{\mu,j} \in {\cal D}_\Lambda^D; \,\mu \in 
\frac{\pi}{L}
\left(\Bbb{N}^3 \cup\left[ \{0\} \times \Bbb{N}^2 \right]\cup 
 \left[ \Bbb{N}\times\{0\} \times \Bbb{N}  \right] \cup
\left[ \Bbb{N}^2 \times \{0\} \right]\right), \ j=0,1,2 \right\},  \label{basis}
\end{equation}
 with %@basis0
\begin{eqnarray}
   \nabla \times \Psi_{\mu,0}  &=&0,\;\;\Psi_{\mu,0}= \nabla \varphi_{\mu,0} \;\;\mbox{with}\;\;
\varphi_{\mu,0} \in  C^\infty_0(\bar{\Lambda});  \label{basis01} \\
\nabla \times  \nabla \times \Psi_{\mu,j}  &=&
 |\mu|^2 \Psi_{\mu,j},\;\;  \nabla\cdot \Psi_{\mu,j} =0,\;\; j=1,2.  \label{basis02}
\end{eqnarray}
\end{description}
More precisely, if $\Lambda = \Lambda_L(x_0)  $, we can take 
\begin{eqnarray}
\Psi_{\mu,j}(x)&=&\Phi_{\mu,j}\left(x- x_{0} + \frac L 2(1,1,1)\right),   \label{basis1}\\[.1 in]
\Phi_{\mu,j}(x) &= &\left[    \begin{array}{c}
 a^{(\mu,j)}_1 \cos (\mu_1x_1) \sin(\mu_2x_2) \sin(\mu_3x_3)  \\[.08 in]
 a^{(\mu,j)}_2 \sin (\mu_1x_1) \cos(\mu_2x_2) \sin(\mu_3x_3) \\[.08 in]
 a^{(\mu,j)}_3 \sin (\mu_1x_1) \sin(\mu_2x_2) \cos(\mu_3x_3)
\end{array}\right],  \nonumber
\end{eqnarray}
where for each 
$\mu \in \frac{\pi}{L}
\left(\Bbb{N}^3 \cup\left[ \{0\} \times \Bbb{N}^2 \right]\cup 
 \left[ \Bbb{N}\times\{0\} \times \Bbb{N}  \right] \cup
\left[ \Bbb{N}^2 \times \{0\} \right]\right)$ we set $a^{(\mu,0)}= \mu$ and pick 
$a^{(\mu,1)}, a^{(\mu,2)} \in \Bbb{R}^3$ such that $ \{a^{(\mu,j)}; \, j=0,1,2\}$ is an orthogonal basis for  
$\Bbb{R}^3$. 
\end{theorem}  

\proof  
Let the operator $\Gamma_\Lambda$ be defined by 
$\Gamma_\Lambda \Psi=\nabla \times  \nabla \times \Psi$ for
$\Psi \in {\cal D}_\Lambda^D$. To see that it is a symmetric operator on its domain, notice that for
 $\Phi,\Psi  \in  C^1(\bar{\Lambda};\,\Bbb{C}^3)$ we have
\begin{equation}
\langle \nabla  \times\Phi , \Psi \rangle - \langle \Phi , \nabla  \times\Psi \rangle= 
\int_\Lambda \nabla\cdot (\bar{\Phi} \times\Psi) d^3 x =
 \int_{\partial \Lambda} (\bar{\Phi} \times\Psi)_\nu dS, \label{gauss}
\end{equation} 
where the inner products are in $L^2\left(\Lambda;\Bbb{C}^3\right)$, $dS$ is the surface measure,
and we used Gauss' Theorem.  If $\Phi_\tau \equiv 0$, we must have
 $ (\bar{\Phi} \times\Psi)_\nu \equiv 0$, so we can conclude that the surface integral in \eq{gauss} 
equals $0$. 

We proceed as in \cite[Proof of Proposition 1 in Section XIII.15]{RS4}. 
To  show that the symmetric operator $\Gamma_\Lambda$ is essentially self-adjoint, it suffices to 
 exhibit an orthogonal basis of eigenfunctions in its domain ${\cal D}_\Lambda^D$. 
Since $\left\{\cos (nx); \, n \in  \frac{\pi}{L}(\{0\} \cup \Bbb{N})\right\}$ and 
$\left\{\sin (nx); \, n \in  \frac{\pi}{L}\Bbb{N}\right\}$ are both orthogonal bases for $L^2((0,L))$, it follows that
${\bf \Psi} =\left\{ \Psi_{\mu,j} \right\}$, given in  \eq{basis1}, 
is an orthogonal basis for $ L^2(\Lambda;\,\Bbb{C}^3)$. Since
\begin{equation}
\Phi_{\mu,0} = \nabla [\sin (\mu_1x_1) \sin(\mu_2x_2) \sin(\mu_3x_3)] ,  \label{grad1} 
\end{equation}
we clearly have  \eq{basis01}. 
 It is straightforward to check that
$ {\bf \Psi} \subset {\cal D}_\Lambda^D$ 
and $ {\bf \Psi}$ also satisfies  \eq{basis02},
 so it is an orthogonal basis of eigenfunctions for the operator $\Gamma_\Lambda$.

To finish the proof of the theorem, it suffices to show that 
$\Xi_\Lambda$ is the closure $\overline{\Gamma}_\Lambda$ of $\Gamma_\Lambda$. To do that, 
notice that 
$C^2_0({\Lambda};\,\Bbb{C}^3) \subset \  {\cal D}_\Lambda^D \subset
 {\cal Q}(\overline{\Gamma}_\Lambda) $, where for a self-adjoint operator $A$ we use $ {\cal Q}(A)$
to denote the domain of the corresponding quadratic form.  As quadratic forms, we clearly have
$ \Xi_\Lambda[\Psi] = \overline{\Gamma}_\Lambda  [\Psi]$ for
 $\Psi  \in C^2_0({\Lambda};\,\Bbb{C}^3)$, which is a form core for $ \Xi_\Lambda$ as a quadratic
 form, hence $ {\cal Q}( \Xi_\Lambda) \subset {\cal Q}(\overline{\Gamma}_\Lambda)$.
Since $  {\cal D}_\Lambda^D$ is a form core for $\overline{\Gamma}_\Lambda$  as a quadratic form, 
 to finish the proof of the theorem, it is enough to show that 
$ {\cal D}_\Lambda^D  \subset {\cal Q}( \Xi_\Lambda) $, so
$ {\cal Q}(\overline{\Gamma}_\Lambda) \subset {\cal Q}( \Xi_\Lambda) $. 

Thus, given $\Psi \in {\cal D}_\Lambda^D$, it suffices to find 
$\Psi_n \in C^1_0({\Lambda};\,\Bbb{C}^3)$  such  that
 \begin{equation}
\| \Psi - \Psi_n\| + \| \nabla \times(\Psi - \Psi_n)\| \to 0.  \label{psin} 
\end{equation}
Translating and scaling, if necessary, we can assume that $\Lambda = \Lambda_2(0) = (-1,1)^3$.  For each
$n=1,2,\dots$ we select a function $\eta_n \in C^2([-1,1])$, $0 \le\eta_n(t)\le 1$, such that $ \eta_n(t)=1$ for
$|t| \le \frac{n}{n +1}$ and   $ \eta_n(t)=0$ for
$ \frac{n +\frac 1 2}{n +1} \le |t| \le 1$.  We set $ \Phi_n(x) =\bar{\eta}_n(x) \Theta_n(x)$,
where $\bar{\eta}_n(x) =  \eta_n(x_1) \eta_n(x_2) \eta_n(x_3) $ and
\begin{equation}
\Theta_n(x) = \left\{
\begin{array}{ll}
\Psi(\frac{n+1}{n} x),   &  \mbox{if $|x_1|,|x_2|,|x_3| \le \frac{n}{n +1}$}; \\
\Psi(x_1, x_2, \pm1),   & \mbox{if $|x_1|,|x_2| \le \frac{n}{n +1}$, $\frac{n}{n +1} <\pm x_3\le 1$};\\
\Psi(x_1,  \pm1, x_3 ),   & \mbox{if $|x_1|,|x_3| \le \frac{n}{n +1}$, $\frac{n}{n +1} <\pm x_2\le 1$};\\
\Psi( \pm1,x_2, x_3 ),   & \mbox{if $|x_2|,|x_3| \le \frac{n}{n +1}$, $\frac{n}{n +1} <\pm x_1\le 1$};\\
0,   &  \mbox{otherwise}.
\end{array}
\right.
\end{equation}
We have  $\Phi_n \in   C_0({\Lambda};\,\Bbb{C}^3)$, 
 and  $\Phi_n $ is piecewise 
$C^1$ with  bounded partial derivatives, so $\nabla \times \Phi_n  \in  L^2({\Lambda};\,\Bbb{C}^3)$.  In addition,
\begin{equation}
\nabla \times \Phi_n = \bar{\eta}_n(\nabla \times \Theta_n )+ (\nabla \bar{\eta}_n )\times \Theta_n =
\bar{\eta}_n(\nabla \times \Theta_n ), 
\end{equation}
since $(\nabla \bar{\eta}_n )\times \Theta_n  =0$ by our construction as $ \Psi_\tau \equiv 0 $. 
 If each $\Phi_n$ was a $C^1$-function, instead of only piecewise $C^1$,  we would be done, since     
 $\Psi_n =\Phi_n $ clearly satisfies \eq{psin}.  To repair that we set  $\Psi_n =\gamma_n *\Phi_n$, where 
$\{\gamma_n\}$ 
is a suitably chosen approximate identity, i.e., $\gamma_n(x) = n^3\gamma (nx)$ for some positive $C^\infty$
function $\gamma$ on $\Bbb{R}^3$ with support on 
$\Lambda_1(0)$ and $\int \gamma(x) dx=1$, so $\Psi_n  \in   C^1_0({\Lambda};\,\Bbb{C}^3)$, 
$\nabla \times \Psi_n = \gamma_n *(\nabla \times \Psi_n)$, and \eq{psin} is satisfied.   {{$\; \Box$}}
\bigskip

The Weyl decomposition corresponding to Dirichlet boundary condition is given by 
 $L^2({\Lambda};\,\Bbb{C}^3) = \Bbb{S}_{\Lambda} \oplus \Bbb{G}_{\Lambda}$, where 
 $\Bbb{G}_{\Lambda}$ and $\Bbb{S}_{\Lambda}$  are the closed subspaces spanned by $\{\Psi_{\mu,0}\}$ and
  $\{\Psi_{\mu,j}, \, j=1,2\}$, respectively, where  $\{\Psi_{\mu,j}, \, j=0,1,2\}$ is the orthogonal basis  
given in \eq{basis}.  It is easy to see that
\begin{eqnarray}
\Bbb{G}_{\Lambda}& =&\overline{ \{\Psi \in C^1_0(\Lambda;\,\Bbb{C}^3); \;\;\Psi=\nabla \varphi\;\;\mbox{with}\;\;
\varphi \in  C^1_0(\Lambda)\}        }, \\
\Bbb{S}_{\Lambda}& =&{ \{\Psi \in L^2({\Lambda};\,\Bbb{C}^3); \;\;\nabla\cdot\Psi=0
\;\;\mbox{weakly}\;\;\}    } 
\end{eqnarray}
 The spaces $\Bbb{S}_{\Lambda}$ and $\Bbb{G}_{\Lambda}$ are left invariant by  $M_{\Lambda}$, with  
$\Bbb{G}_{\Lambda} \subset \mathcal{D}\left( M_{\Lambda}\right) $ and 
 $\left.M_{\Lambda}\right| _{{\Bbbs{G}}_{\Lambda}} =0$.  We define
 ${\bf M}_{\Lambda}$ as the restriction of $M_{\Lambda}$ to ${\Bbb{S}}_{\Lambda}$, i.e.,
$\mathcal{D}\left({\bf M}_{\Lambda}\right) =\mathcal{D}\left( M_{\Lambda}\right) \cap \Bbb{S}_{\Lambda}$ and 
${\bf M}_{\Lambda}=\left. M_{\Lambda}\right| _{{\cal D}\left( M_{\Lambda}\right) \cap {\Bbbs{S}}_{\Lambda}}$.  
 Notice $M_{\Lambda} =  {\bf M}_{\Lambda} \oplus 0_{{\Bbbs{G}}_{\Lambda}}$, 
$0 \notin \sigma({\bf M}_{\Lambda})$,    so
 $\sigma({\bf M}_{\Lambda}) = \sigma(M_{\Lambda}) \backslash \{0\}$.

 ${\bf M}_{\Lambda}$ and $M_{\Lambda}$ will be called Dirichlet Maxwell operators. We write 
${\bf \Xi}_{\Lambda}$ for ${\bf M}_{\Lambda}(1)$. Notice that ${\bf \Xi}_{\Lambda} $ is a strictly positive operator 
with discrete spectrum; the same being true of 
 ${\bf M}_{\Lambda}$ in view of \eq{bound}.  

\begin{corollary} \label{cxi} Let  $M$ be as in \eq{Max2} with \eq{bound}, and let $\Lambda$ be an open cube 
in ${\Bbb{R}}^3$. Then
\begin{description}
\item[(i)] ${\bf M}_{\Lambda} $ has compact resolvent; in fact  
${\rm Tr}\left\{ ({\bf M}_{\Lambda} + I)^{-p}   \right\} < \infty$ for any 
$ p > \frac 3 2$. 
\item[(ii)] For any $E > 0$ let $n_{\varepsilon, \Lambda}(E)$ denote the number of eigenvalues of 
${\bf M}_{\Lambda}$  
less than $E$, each eigenvalue counted as many times as its multiplicity. There exists a finite constant $C_0 $,
independent of $\Lambda$ and $\varepsilon$, such that
\begin{equation}
n_{\varepsilon, \Lambda}(E) \le C_0 \varepsilon_+^{\frac 3 2}|\Lambda| E^{\frac 3 2} . \label{n0}
\end{equation}
  \end{description}
\end{corollary}

\proof We clearly have $ {\bf M}_{\Lambda} \ge \frac{1}{\varepsilon_+} {\bf \Xi}_{\Lambda}$, so it suffices to prove
the corollary for ${\bf \Xi}_{\Lambda}$.

 It follows from Theorem \ref{tdirichlet}(ii) that
 the spectrum of ${\bf \Xi}_{\Lambda}$ consists of eigenvalues whose multiplicity can be read from \eq{basis},
so an explicit calculation gives  ${\rm Tr}\left\{ ({\bf \Xi}_{\Lambda} + I)^{-p}   \right\} < \infty$ for any
$ p > \frac 3 2$.  A similar calculation gives \eq{n0}.  {{$\; \Box$}} 

\begin{remark}  $n_{\varepsilon, \Lambda}(E) $ is also equal to  the number of {\em strictly positive} eigenvalues of 
${ M}_{\Lambda}$  less than $E$, each eigenvalue counted as many times as its multiplicity. 
\end{remark}



\section{A Wegner-type estimate } \label{swegner}



Given an open cube $\Lambda $ in $\Bbb{R}^3$, we will denote by $M_{g,\Lambda}=M_{g,\omega ,\Lambda }$ 
the restriction of the random operator $M_{g,\omega }$ to $\Lambda $
with Dirichlet boundary condition. Notice that
 ${ M}_{g,\omega ,\Lambda }$ is a random operator on 
$L^2(\Lambda )$,  measurability follows from  \cite[Theorem 38 ]{ FKl3}. Each ${\bf M}_{g,\omega ,\Lambda }$ 
has compact resolvent by Corollary \ref{cxi}(i).  For any $E > 0$ we  define $n_{g,\Lambda
}(E)=n_{g,\omega ,\Lambda }(E)$ as the number of {\em strictly positive} eigenvalues of $M_{g,\omega
,\Lambda }$ less than $E$.
 Notice that $n_{g,\omega ,\Lambda }(E)$ is the distribution function of
the measure $n_{g,\omega ,\Lambda }(dE)$ on $(0,\infty)$ given by 
\begin{equation}
\int h(E)n_{g,\omega ,\Lambda }(dE)={\rm Tr}(h(M_{g,\omega ,\Lambda }))={\rm Tr}(h({\bf M}_{g,\omega ,\Lambda }))
\end{equation}
for positive continuous functions $h$ with compact support in  $(0,\infty)$.

We will say that the random operator $M_{g}$ defined by (\ref{At}) satisfies
Assumption {\ref{aeu}}$^\prime$, if it satisfies all of Assumption {\ref{aeu}}
 with the exception of the requirement that $\varepsilon _0(x)$ be a $q$
-periodic function.



We have the following ``a priori'' estimate, which is an immediate consequence of
Corollary \ref{cxi}(ii), \eq{bound1} and Assumption {\ref{aeu}}(iv) . 

\begin{lemma}
Let the random operator $M_g$ defined by (\ref{At}) satisfy Assumption {\ref
{aeu}}$^{\prime }$. There exists a finite constant $C_1$, depending only on  $\varepsilon _{0.+}$,
such that we have 
\begin{equation}
n_{g,\omega ,\Lambda }(E)\le C_1|\Lambda |E^{\frac 32}  \label{nest}
\end{equation}
for all $\omega \in [-1,1]^{\Bbb{Z}^3}$, for all $E> 0$ and all open cubes $%
\Lambda $ in $\Bbb{Z}^3$.
\end{lemma}

\begin{theorem}[Wegner-type estimate]  \label{twegner}
Let the random operator $M_g$ defined by (\ref{At}) satisfy Assumption \ref
{aeu}$^{\prime }$. There exists a constant $Q<\infty $, depending only on
 the constants $ r_u$ and $\varepsilon_{0,+}$, such that 
\begin{equation}
\Bbb{P}\left\{ \mbox{ \rm dist}(\sigma (M_{g,\omega ,\Lambda }),E)\le \eta
\right\} \le Q\frac{U_-+2U_+}{gU_+(1-gU_+)U_-}\Vert \rho \Vert _\infty |E|^{{%
\frac 1 2}}\eta |\Lambda |^2  \label{wegner}
\end{equation}
for all  $E>0$, open cubes $\Lambda $ in $\Bbb{R}^3$, and all $\eta \in
[0,E)$.
\end{theorem}

\proof The proof is exactly the same as the proof of \cite[Theorem 23]{FKl3}, with the proviso
that we only integrate $n_{g,\omega ,\Lambda }(E)$ against  positive continuous functions 
 with compact support in  $(0,\infty)$.   {{$\; \Box$}} 


\section{Localization}
 \label{sloc}
 Theorems \ref{loc1} and \ref{loc2} are proved exactly as in \cite{FKl3}, applying  a multiscale 
analysis appropriate for random perturbations of periodic operators on $ \Bbb{R}^3$ 
\cite[Theorems 29 and 35]{FKl3} to operators $M_g$ as in  (\ref{At}).  
  

Let the operator $M$ be as in (\ref{Max2}) with (\ref{bound}).  Given an open cube
 $\Lambda $ in $\Bbb{R}^3$,  $M_{\Lambda}$ is the restriction of $M$ to 
$\Lambda $ with Dirichlet boundary condition (see Section \ref{sdirich}). 
Each $M_{\Lambda }$ is a nonnegative self-adjoint operator on $L^2(\Lambda; \, \Bbb{C}^3 )$ 
with compact resolvent $R_{\Lambda }(z)=(M_{\Lambda }-z)^{-1}$.  If 
$\Lambda = \Lambda_L(x) $,  we will write 
$M_{x,L }=M_{\Lambda_L(x) }$ and  $R_{x,L }(z)= R_{\Lambda_L(x) }(z)$.  The  norm in
 $L^2(\Lambda; \, \Bbb{C}^3 )$ and also the corresponding operator norm will both be denoted 
by $\| \; \|_{x,L }$.   If
 $\Lambda_1 \subset \Lambda_2  $ are open cubes, 
$J_{\Lambda_1}^{\Lambda_2}:  L^2(\Lambda_1; \, \Bbb{C}^3) \to L^2(\Lambda_2; \, \Bbb{C}^3)$ 
is the canonical injection.
If $\Lambda_i = \Lambda_{L_i}(x_i)$, $i=1,2$, we write
$\| \; \|_{x_1,L_1}^{x_2,L_2}$ for the (operator) norm in 
$\mathcal{B}\left(L^2(\Lambda_{L_1}(x_1); \, \Bbb{C}^3), L^2(\Lambda_{L_2}(x_2); \, \Bbb{C}^3)\right)$ and
 $J_{x_1,L_1}^{x_2,L_2} =J_{\Lambda_{L_1}(x_1)}^{\Lambda_{L_2}(x_2)}$. If  
$\varphi \in L^\infty ( \Lambda) $,  we also use $\varphi$ to denote the
 operator   on  $L^2(\Lambda; \, \Bbb{C}^3)$ given by  multiplication by  $\varphi$;
 if $\Phi \in   L^\infty ( \Lambda; \, \Bbb{C}^3)$  we  write 
$\Phi^\times$ for the operator $\Phi\times$, i.e.,
 $\Phi^\times \Psi=\Phi\times \Psi$. 


\subsection{The basic technical tools}

The results of \cite[Subsections 6.1 and 6.3]{FKl3} are valid for the Maxwell operator $M$, with the obvious modifications.
We state the key results for completeness.  We start with the {\em smooth resolvent identity} (SRI), which is used
 to relate resolvents in different scales.
 %@lsri

\begin{lemma}[SRI] \label{lsri} Let the operator $M$ be given by (\ref{Max2}) 
with (\ref{bound}),
 let  $\Lambda_1 \subset \Lambda_2  $ be open cubes in $\Bbb{R}^3$,  and let 
$\varphi_1 \in C^1_0 ( \Lambda_1) $.  Then, for any 
$z \notin \sigma (M_{\Lambda_1}) \cup \sigma (M_{\Lambda_2})$ we have    %%@sri
\begin{eqnarray}
\lefteqn{R_{\Lambda_2}(z)J_{\Lambda_1}^{\Lambda_2} \varphi_1 =} \label{sri} \\
&& J_{\Lambda_1}^{\Lambda_2}\varphi_1  R_{\Lambda_1}(z) + 
R_{\Lambda_2}(z)\left( -J_{\Lambda_1}^{\Lambda_2}
 (\nabla\varphi_1)^\times\frac 1{\varepsilon } \nabla^\times + 
\nabla^\times J_{\Lambda_1}^{\Lambda_2}\frac 1{\varepsilon}(\nabla\varphi_1)^\times \right)
 R_{\Lambda_1}(z) \nonumber
\end{eqnarray}
as quadratic forms on $L^2(\Lambda_2; \, \Bbb{C}^3 ) \times L^2(\Lambda_1; \, \Bbb{C}^3 )$.
 \end{lemma}

\proof The lemma follows immediately from \cite[Lemma 24]{FKl3} and the definition of
Dirichlet boundary condition. {$\; \Box$}
\bigskip 


{\em To take into account the periodicity of the background medium, 
 $q \in \Bbb{N}$ being the period,
  (see Assumption \ref{aeu}), we  work with boxes
 $\Lambda_L(x)$ with
$x \in q\Bbb{Z}^3$ and $L \in 2q \Bbb{N}$, so the background is the same in all boxes in a
 given scale $L$.}  For such boxes (with $L\ge 4q$) we set
\begin{equation}
\Upsilon_L(x) = \{y \in  q\Bbb{Z}^3; \;\; \|y -x\| = \frac{L}{2} - q\} 
\end{equation}
and
\begin{equation}
\tilde{\Upsilon}_L(x) = \Lambda_{L-q}(x) \backslash \overline{\Lambda}_{L-3q}(x) ,\;\;
\hat{\Upsilon}_L(x) = \Lambda_{L-\frac{3q}{2}} (x) \backslash 
\overline{\Lambda}_{L-\frac{5q}{2} }(x)  .
\end{equation}
We also set 
\begin{equation}
\chi_x = \chi_{x,q} \;\;\mbox{and}\;\; \Gamma_{x,L} = \chi_{\tilde{\Upsilon}_L(x)},\;\;
\hat{\Gamma}_{x,L} = \chi_{\hat{\Upsilon}_L(x)}.
\end{equation}
Notice %%@gam 
\begin{equation}
 \Gamma_{x,L} = \sum_{{y \in \Upsilon_L(x)}} \chi_y \;\; \mbox{a.e.}  \label{gam}
\end{equation}
and %%@supsilon
\begin{equation}
| \Upsilon_L(x) | \le 3 (L -2q +1)^{2}. \label{supsilon}
\end{equation}


In addition each $\Lambda_L(x)$ will be equipped with a function $\Phi_{x,L}$ constructed
 in the following way:  we  fix an even function  $\xi \in C^1_0( \Bbb{R})$ with 
$0 \le \xi(t) \le 1$ for all $t \in \Bbb{R}$,   such that 
$\xi (t) = 1$ for $|t| \le  \frac{q}{4}  $, $\xi (t) = 0$ for 
$|t| \ge   \frac{3q}{4}$, and   
$|\xi^\prime (t)| \le  \frac{3}{q} $
for all $t \in \Bbb{R}$.  (Such a function always exists.)  We define 
\begin{equation}
\xi_{L}(t) = \left\{ 
\begin{array}{ll}
1,  &\mbox{ if $|t| \le  \frac{L}{2} -  \frac{5q}{4}$} \\  
\xi\left(|t|-  \left(\frac{L}{2} -  \frac{3q}{2}\right) \right),
&\mbox{ if  $|t| \ge \left(\frac{L}{2} -  \frac{3q}{2}\right)$  }  
\end{array}
\right.
\end{equation}
and set
\begin{equation}
\Phi_{x,L}(y) = \Phi_{L}(y-x) \;\;\mbox{for}\;\; y \in  \Bbb{R}^3,
 \;\;\mbox{with}\;\; \Phi_{L}(y)=  \prod_{i=1}^3 \xi_{L}(y_i).
\end{equation}
We have $\Phi_{x,L} \in  C^1_0( \Lambda_L(x))$, $0\le \Phi_{x,L} \le 1$,  %%@phi
\begin{equation}
\chi_{x,\frac{L}{2} -  \frac{5q}{4}} \Phi_{x,L} = \chi_{x,\frac{L}{2} -  \frac{5q}{4}},\;\;\;\;
\chi_{x,\frac{L}{2} -  \frac{3q}{4}}\Phi_{x,L} =\Phi_{x,L},  \label{phi}
\end{equation}
and %%@nphi
\begin{equation}
\hat{\Gamma}_{x,L}\left(\nabla \Phi_{x,L}\right) =\nabla \Phi_{x,L}, \;\;\;\;
|\nabla \Phi_{x,L}| \le \frac{3\sqrt{3}}{q}.   \label{nphi}
\end{equation}
%%@lsli

We can now state a  {\em Simon-Lieb-type inequality} (SLI); it is used to obtain decay in a larger scale 
from decay in a given scale. 

\begin{lemma}[SLI] \label{lsli} Let the operator $M$ be given by (\ref{Max2}) with (\ref{bound}).
 Then for any 
$\ell,L\in 2q\Bbb{N}$ with $4q \le \ell < L -3q$,   $x,y \in  q\Bbb{Z}^3$ with 
$2\|y-x\| \le L -  \ell -3q$ (so $\Lambda_\ell(y) \subset \Lambda_{L - 3q}(x)$), and
 $z \notin \sigma(M_{x,L})\cup  \sigma(M_{y,\ell})$,  we have %%@sli
\begin{equation}
\| \Gamma_{x,L} R_{x,L}(z) \chi_y\|_{x,L} \le \gamma_z  \ell^{2}
\| \Gamma_{y,\ell} R_{y,\ell}(z) \chi_y\|_{y,\ell} 
\| \Gamma_{x,L} R_{x,L}(z) \chi_{y^\prime}\|_{x,L} \label{sli}
\end{equation}
for some $y^\prime \in \Upsilon_{y,\ell}$, with %%@gath
\begin{equation}
\gamma_z= \frac{18 \sqrt{3}}{q\varepsilon_- }
\Theta_{\frac{q}{4}}\sqrt{ \varepsilon_+} 
\left(\sqrt{ \varepsilon_-} + \frac {1} {\sqrt{ \varepsilon_-}} \right)  (1 + |z|),  \label{gath}
\end{equation}
where $\Theta_{\frac{q}{4}}$ is the constant given in 
Corollary \ref{cint}. 
\end{lemma}

\proof  The lemma is proved as \cite[Lemma 26]{FKl3}, using Lemma \ref{lsri} and
Corollary \ref{cint}.  {$\; \Box$}
\bigskip  

The {\em eigenfunction decay inequality} (EDI) is used 
to obtain decay for generalized eigenfunctions
from decay of local resolvents. %%@leigdec

\begin{lemma}[EDI] \label{leigdec}
 Let the operator $M$ be given by (\ref{Max2}) with (\ref{bound}), and let $\Psi$ be a generalized 
eigenfunction for a given  $z \in  \Bbb{C}$.  For any $x \in  q\Bbb{Z}^3$ and
$\ell\in 2q\Bbb{N}$ with $\ell \ge 4q$, such that 
 $z \notin \sigma(M_{x,\ell})$, we have  %%@eigdec
\begin{equation}
\| \chi_x \psi \| \le  \gamma_z \ell^{2}
\| \Gamma_{x,\ell} R_{x,\ell}(z^*) \chi_x\|_{x,\ell} 
\|  \chi_{y} \psi\| \label{eigdec}
\end{equation}
for some $y \in \Upsilon_{y,\ell}$, with $\gamma_z$ as in (\ref{gath}).
\end{lemma}

\proof Same proof as \cite[Lemma 27]{FKl3}.  {$\; \Box$}
\bigskip  

The starting hypothesis  for the multiscale analysis \cite[(P1) in Theorem 29 and (H1) in Theorem  35]{FKl3} is formulated for operators with
Dirichlet boundary condition.  But under the hypotheses of Theorems \ref{loc1} and \ref{loc2}  
the natural starting hypothesis is the analogue of either (P1) or (H1) for {\em periodic} boundary condition.  
The following lemma enable us to go from periodic boundary condition to Dirichlet boundary
 condition.

For  $M_g$ be as in (\ref{At}) satisfying 
Assumption \ref{aeu},  $x \in q\Bbb{Z}^3$ and $L \in 2q \Bbb{N}$, we set (with
 the notation of (\ref{perapp})) 
\begin{equation}
\stackrel{\circ }{M}_{g, \omega,x,L} = 
 (\stackrel{\circ }{M}((g\omega)_{\Lambda_L(x)}))_{\Lambda_L(x)},
\end{equation}
which is a random operator by \cite[Theorem 38]{FKl3}. We write 
$\stackrel{\circ }{R}_{g, \omega,x,L}(z)$ for its resolvent.           %%@lperd


\begin{lemma} \label{lperd}
Let  $M_g$ be as in (\ref{At}) satisfying Assumption \ref{aeu}. Let $E>0$,
$x \in  q\Bbb{Z}^3$ and $L \in 2q \Bbb{N}$,  $L\ge 4q$; set 
$\hat{L} = L + [2 r_u]_{2q} + 2q$.  If $\omega$ is such that
$ E \notin \sigma(M_{g, \omega,x,L} \cup  \sigma(\stackrel{\circ }{M}_{g, \omega,x,\hat{L}})$, 
then %@perd
\begin{eqnarray}
\lefteqn{\| \Gamma_{x,L} R_{g, \omega,x,L}(E) \chi_x\|_{x,L} \le} \label{perd} \\
&& \left(1 + \frac{3\sqrt{3}}{q\varepsilon_- } 
\left(1 +2(1+E) \| R_{g, \omega,x,L}(E)\|_{x,L}\right)\right)
\| \Gamma_{x,L}\stackrel{\circ }{R}_{g, \omega,x,\hat{L}}(E) \chi_x\|_{x,\hat{L}}.  \nonumber
\end{eqnarray}
\end{lemma}

\proof Same proof as \cite[Lemma 37]{FKl3}.  {$\; \Box$}
   


\subsection{The proofs of localization}

Theorems  \ref{loc1} and  \ref{loc2} can now be proved exactly as in  \cite{FKl3},
using Theorems \ref{tlct}, \ref{tMp1}, \ref{twegner}, and
Lemmas \ref{ctt}, \ref{lsAe},  \ref{lsli},  \ref{leigdec},  \ref{lperd}, so we refer the reader to
 \cite[Subsections 6.4 and 6.5]{FKl3}.
  

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