%latex %\documentstyle[12pt]{article} \documentclass[12pt]{article} \setlength{\oddsidemargin}{.0001in} \setlength{\evensidemargin}{.0001in} \setlength{\textwidth}{6.5in} \setlength{\textheight}{8.5in} \setlength{\topmargin}{.0001in} %%%%%%%%%%%% \newtheorem{theorem}{Theorem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newfont{\msbm}{msbm10 scaled\magstep1}%blackboardbold \newfont{\msbms}{msbm7 scaled\magstep1} %blackboardbold subscript \def\Box{QED} %\newcommand{\mathcal}[1]{\mbox{$\cal #1$}} \newcommand{\Bbb}[1]{\mbox{$\mbox{\msbm #1}$}} \newcommand{\Bbbs}[1]{\mbox{$\mbox{\msbms #1}$}} \newcommand{\proof}{\noindent{\bf Proof. \ }} \newcommand{\eq}[1]{(\ref{#1})} \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 $, such that %%@muM \begin{equation} 00$ 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} 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) 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 $g_0 \le g 0$, 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) \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 $h_1,h_2 \in (-g,g)$, $0 \frac 3 2$. \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}. \begin{thebibliography}{CHST} \bibitem[An2]{A1} P. W. Anderson, {\sl A Question of Classical Localization. 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