Content-Type: multipart/mixed; boundary="-------------0010022247717"

This is a multi-part message in MIME format.

---------------0010022247717
Content-Type: text/plain; name="00-388.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="00-388.keywords"

Periodic operator, elliptic operator,absolutely continuous spectrum,  
Schr\"{o}dinger operator, magnetic and electric potentials
---------------0010022247717
Content-Type: application/x-tex; name="kuch_lev.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="kuch_lev.tex"

%% This document created by Scientific Word (R) Version 2.5
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{exercise}{Exercise}
\newtheorem{remark}{Remark}


\documentclass[12pt]{article}
\usepackage{amsfonts}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newtheorem{problem}{Problem}
\def\stackunder#1#2{\mathrel{\mathop{#2}\limits_{#1}}}
\newtheorem{condition}[theorem]{Condition}
\begin{document}

\author{Peter Kuchment \\
%EndAName
Mathematics and Statistics Department\\
Wichita State University\\
Wichita, KS 67260-0033, USA \\
kuchment@twsu.edu
\and Sergei Levendorski\^{i} \\
%EndAName
Mathematics Department\\
Rostov State Academy of Economics\\
Rostov-on-Don, Russia \\
leven@ns.rnd.runnet.ru}
\title{On the structure of spectra of periodic elliptic operators}
\date{}
\maketitle

\begin{abstract}
The paper discusses the problem of absolute continuity of spectra of
periodic elliptic operators. A new result on absolute continuity for a
matrix operator of Schr\"{o}dinger type is obtained. It is shown that all
types of operators for which the absolute continuity has previously been
established can be reduced to this one. It is also discovered that some
natural generalizations stumble upon an obstacle in the form of
non-triviality of a certain analytic bundle on the two-dimensional torus.
\end{abstract}

\section{Introduction}

\footnotetext{\textit{1991 Mathematics Subject Classification}. Primary
35P99, Secondary 35J10
\par
\textit{Key words and phrases}. Periodic operator, elliptic operator,
absolutely continuous spectrum, Schr\"{o}dinger operator, magnetic and
electric potentials}Elliptic differential operators with periodic
coefficients arise naturally in many areas of mathematical physics. One can
mention among the most prominent examples quantum solid state theory, where
the main operator of interest is the stationary Schr\"{o}dinger operator 
\begin{equation}
-\Delta +V(x)  \label{Schred}
\end{equation}
with a potential $V(x)$ periodic with respect to a lattice in $\Bbb{R}^d$
(see for instance \cite{AM}, \cite{RS}).

Another example that has gained importance due to recent advances in the
theory of photonic crystals (e.g., \cite{JMW}, \cite{K3}) is the periodic
Maxwell operator, one of realizations of which (in fact, of its square) can
be written as 
\begin{equation}
\frac 1{\mu (x)}\nabla \times \frac 1{\varepsilon (x)}\nabla \times
\label{Max}
\end{equation}
defined on zero-divergence vector fields in $\Bbb{R}^3$. Here the periodic
functions $\varepsilon (x)$ and $\mu (x)$ represent the electric
permittivity and magnetic permeability of the medium.

Scalar counterparts of this operator are 
\begin{equation}
-\nabla \cdot \frac 1{\varepsilon (x)}\nabla  \label{Div}
\end{equation}
and 
\begin{equation}
-\frac 1{\mu (x)}\Delta  \label{Weight}
\end{equation}
(in the latter case the operator must be considered in a weighted $L_2$%
-space in order to be self-adjoint). These scalar operators also arise in
studying periodic acoustic media.

Among other operators of interest one can mention the divergence type
operators 
\begin{equation}
\sum \partial _ia_{ij}(x)\partial _j  \label{Diverg}
\end{equation}
and magnetic Schr\"{o}dinger operators 
\begin{equation}
(\frac 1i\nabla -A(x))^2+V(x),  \label{Magn}
\end{equation}
where the scalar functions $a_{ij}(x)$ and $V(x)$ and the vector magnetic
potential $A(x)$ are assumed to be periodic with respect to a lattice.

Another extensively studied case is the one of a periodic Dirac operator 
\[
\frac 1i\sum\limits_{j=1}^d\frac \partial {\partial x_j}\alpha _j+V(x), 
\]
where $\alpha _j$ are self-adjoint matrices satisfying the anti-commutation
relations 
\[
\alpha _j\alpha _i+\alpha _i\alpha _j=2\delta _{ij}I 
\]
(Dirac or Pauli matrices) and the periodic matrix function $V$ is the
potential.

In all these cases the structure of the spectrum of the corresponding
operator is of major interest. Consider for instance the periodic
Schr\"{o}dinger operator (\ref{Schred}) in $\Bbb{R}^d$. Assuming that the
potential is real and satisfies some mild conditions one can prove
self-adjointness of this operator (see \cite{RS}). It is well known that the
spectrum of such an operator has the so called band-gap structure, i.e. it
consists of the union of finite closed segments $[a_j,b_j]$ such that $%
a_j\rightarrow \infty $ (e.g. \cite{AM}, \cite{E}, \cite{JMW}, \cite{Ka}, 
\cite{K2}, \cite{RS}, \cite{Sk}). It has been known to physicists for a long
time (see for instance \cite{AM}) that the spectrum of this operator in $L_2(%
\Bbb{R}^d)$ does not contain any eigenvalues. Put in different terms, this
means that none of the spectral bands $[a_j,b_j]$ collapses into a point.
One can easily prove that there are no eigenvalues of finite multiplicity 
\cite{E}, however the proof of the general statement had to wait for a long
time until the celebrated Thomas' theorem was proven in 1973 \cite{T}. In
fact, the paper \cite{T} contained the proof of a more general statement:
the spectrum of a self-adjoint periodic operator (\ref{Schred}) in $3D$ is
absolutely continuous. One should notice that the statement on absence of
eigenvalues holds also in the non-self-adjoint case. In the self-adjoint
situation the so called Floquet theory enables one to easily deduce absolute
continuity from the absence of eigenvalues, since one can easily prove
absence of the singular continuous spectrum (see for instance \cite{K2}, 
\cite{RS}, and \cite{Sj}). Although the proofs of this implication in \cite
{K2}, \cite{RS}, and \cite{Sj} are formulated for the Schr\"{o}dinger case
only, the derivation is very general and holds for any self-adjoint periodic
elliptic operator with sufficiently regular coefficients (see for instance
the proof of Theorem 4.5.9 in \cite{K2}). In fact, the statement on absence
of the singular continuous spectrum holds \textbf{with the same proof} for
operators in periodic electro-magnetic fields in the case of a non-zero 
\textbf{rational} magnetic flux through the unit cell (i.e., when the
magnetic potential is not periodic). Then one has an abelian group of the so
called ``magnetic translations'' (or Zak group, e.g. \cite{DN1}, \cite{DN2}, 
\cite{Nov}, and \cite{Zak}), which enables one to carry through the proof.
In this situation, however, the point spectrum can arise (Landau levels).

The question arises on whether absence of eigenvalues (absolute continuity
of spectrum in the self-adjoint case) is a property that all periodic
elliptic differential operators enjoy. It is known, however (see p. 135--136
in \cite{K2}) that this is in general not true for elliptic operators of
order four. On the other hand, the common belief is that periodic elliptic
operators of second order do not have eigenvalues (or, in the self-adjoint
case, have absolute continuous spectra), although it is rarely discussed why
this must be true. Let us describe why the authors believe in this. Assuming
that a periodic elliptic operator has an eigenvalue, one can derive from
this existence of an eigenfunction of a super-exponential decay (see
Theorems 4.1.5 and 4.1.6 in \cite{K2} and Theorem 12 in \cite{K1}).
Existence of such fast decaying solutions is possible for fourth order
equations, where even compactly supported solutions can exist (see p.
135--136 in \cite{K2} and the original paper \cite{Plis}). In the case of
equations of second order, such fast decay should probably violate
uniqueness of continuation at infinity and hence be impossible.
Unfortunately, the appropriate uniqueness theorems that prohibit super fast
decay at infinity for equations of the type we consider are not known yet in
full generality. The only applicable results of \cite{FHHO} and \cite{Me}
deal with the Schr\"{o}dinger operator with an electric potential only.
Thus, due to absence of appropriate theorems of possible decay rate of
solutions, the absolute continuity problem since the Thomas' paper \cite{T}
has been approached in a different way. It is reminded to the reader in
Section \ref{Prelim}. Let us list the known results. Thomas' theorem \cite{T}
was extended to all dimensions and a broader class of periodic
Schr\"{o}dinger operators in \cite{RS}. Papers \cite{Da1}--\cite{Da5}
contained the proof of the absolute continuity of the spectrum for the Dirac
operator with a periodic scalar electric potential. Thomas' result for the
Schr\"{o}dinger operator implies the similar statement for the operator 
\begin{equation}
H_{\rho ,V}=-\nabla \cdot \rho ^2(x)\nabla +V(x)  \label{isotr}
\end{equation}
with periodic functions $\rho \neq 0$ and $V$ if $\rho $ is smooth enough,
due to the well known transformation 
\begin{equation}
\rho ^{-1}H_{\rho ,V}\rho ^{-1},  \label{chge}
\end{equation}
which reduces the operator to the form (\ref{Schred}). The disadvantage of
this transform is that it requires smoothness of $\rho $, the condition
which is probably not needed for the result to hold and which is violated in
many applications (for instance, to photonic crystals). Thomas' theorem also
handles the operator (\ref{Weight}). However, the case of the magnetic
Schr\"{o}dinger operator (\ref{Magn}) had resisted attempts to prove
absolute continuity of the spectrum until the paper \cite{HH} where this was
done for the case of small magnetic potentials. Finally, in the remarkable
papers \cite{BSu1} and \cite{So} the full strength statement about the
magnetic Schr\"{o}dinger operator was proven first in dimension two \cite
{BSu1} and then in arbitrary dimension \cite{So}. The elegant algebraic
approach of the paper \cite{BSu1} works only in dimension two. This paper
lead to a proof in arbitrary dimension \cite{So}. Papers \cite{BSu1} and 
\cite{So} triggered an avalanche of publications that significantly advanced
the known results (see \cite{BSu2}--\cite{BSu6}, \cite{Da6}--\cite{Da9}, 
\cite{KL1}, \cite{Mo1}, \cite{Mo2}, \cite{Sh1}--\cite{Sh4}, \cite{So2}, and 
\cite{Su}). Some major questions, however (for instance, how to handle the
operators with variable coefficients in the higher order terms) remain
unanswered. One can find a rather comprehensive survey in \cite{BSu5} and 
\cite{Su}.

The goal of this paper is to provide a new absolute continuity result for a
matrix Schr\"{o}dinger operator such that all operators treated so far can
be transformed into this form. The proof uses standard microlocal analysis
techniques and employs the idea of \cite{So} in a simplified form. We were
also influenced by the papers \cite{NSU} and \cite{Su}, although this might
not be obvious from the text. We believe that this approach clarifies and
unifies the know results. At the same time it singles out a problem of
triviality of a certain analytic vector bundle on the torus, an obstruction
which has not been noticed before. We would also want to notice that our
main results do not require self-adjointness. This happens to be useful even
for some self-adjoint problems, which can sometimes be transformed to
simpler non-self-adjoint ones, as it is explained in Section \ref{Reduction}%
. A limited version of the main result was announced in \cite{KL1}.

We want to make it clear that we did not even attempt to achieve optimal
conditions on the coefficients of the equations, but rather to encompass a
wider set of types of equations. Finding the best possible conditions on the
coefficients is an important task (especially taking into account that in
many applications, for instance to photonic crystals \cite{JMW}, \cite{K3},
the coefficients in the leading terms are piecewise constant). This was,
however, not among our goals, and so our conditions on coefficients are in
most cases not optimal. One can find optimal or close to optimal conditions
for some cases in the recent works \cite{BSu5}, \cite{Sh1}, \cite{Sh2}.

Let us describe the structure of the paper. The next section provides a
preliminary discussion of the problem, in particular an exposition of the
original Thomas' approach \cite{T}. Section \ref{Main} contains the
formulation of the main result of the paper. The next section was written
for convenience of the reader and provides a brief sketch of the proof.
Section \ref{Proof} delivers the proof of the main result in the absence of
electric potential. In the next section, the known technique of including
the electrical potential is indicated with appropriate references. Section 
\ref{Reduction} describes the algebraic reduction of all studied so far
types of operators to the form contained in our main result. The paper ends
with the sections containing concluding remarks and acknowledgments.

\section{\label{Prelim}Formulation of the problem and preliminary discussion}

Let $P(x,D)$ be an elliptic second order scalar or matrix differential
operator in $\Bbb{R}^d$ with coefficients periodic with respect to a lattice 
$\Gamma $ in $\Bbb{R}^d$. Here $D=-i\nabla $. For simplicity of presentation
we will consider in the most of the text the case of the integer lattice $%
\Bbb{Z}^d$. It is easy, however, to modify the proofs in order to cover an
arbitrary lattice. We will not yet specify any conditions on smoothness of
the coefficients of $P$. The reader can assume so far the coefficients to be
as smooth as needed. The natural domain of definition of the operator will
then be the Sobolev space $H^2(\Bbb{R}^d)$ (or its vector analog in the case
of systems, although we will not distinguish these cases in the notations
for the spaces). Our goal is to prove nonexistence of non-zero $L_2$%
-solutions of the equation $Pu=\lambda u$. As it was mentioned in the
introduction, in the self-adjoint case standard considerations show that
this implies absolute continuity of the spectrum of $P$ in $L_2(\Bbb{R}^d)$.
We also note that the spectral parameter $\lambda $ can be included in the
operator, and hence we will assume from now on that $\lambda =0$.

We will describe now the approach of paper \cite{T} to the proof of absolute
continuity of the spectrum. The reader should be aware that we will adhere
to the language of multi-dimensional analyticity, although the original
paper \cite{T} and most of other publications on the subject use
one-dimensional directional analyticity. The language of several complex
variables is not only natural, but often unavoidable in theory of periodic
PDEs, as one can conclude for instance from \cite{KT}, \cite{K1}, \cite{KP} 
\cite{KV1}, \cite{KV2}, and \cite{Pa2}. This is in parallel with the theory
of PDEs with constant coefficients, where complex analysis in several
variables is often the language of choice (see for instance \cite{Eh} and 
\cite{Pa1}). Using the technique of analytic Fredholm operator functions in
several complex variables (see \cite{ZK}, \cite{K2}) makes the methods more
flexible, allowing for a wide class of equations, including non-self-adjoint
ones.

The main technique of the so called Floquet theory (see e.g. \cite{E}, \cite
{K1}, \cite{K2}, \cite{RS}, \cite{Sh}, \cite{Sj}, and \cite{Sk}) is the
decomposition of the operator $P$ in the space $L_2(\Bbb{R}^d)$ into the
direct integral of operators on the torus $\Bbb{T}^d=\Bbb{R}^d/\Bbb{Z}^d$: 
\[
P=\int\limits_B^{\oplus }P(k)dk, 
\]
where 
\[
B=\left\{ k\in \Bbb{R}^d|\,k_j\in [-\pi ,\pi ],j=1,...,d\right\} 
\]
is the so called \textit{Brillouin zone}, and the operator 
\[
P(k)=P(x,D+k) 
\]
acts on $\Bbb{Z}^d$-periodic functions, i.e. on functions on the torus 
\[
\Bbb{T}^d=\Bbb{R}^d/\Bbb{Z}^d. 
\]
The vector $k$ is usually called \textit{quasimomentum}. This direct
integral decomposition provides an isometry between $L_2(\Bbb{R}^d)$ and 
\[
\int\limits_B^{\oplus }L_2(\Bbb{T}^d)dk. 
\]
Let us observe now that one can define the operator family $P(k)$ for all
values $k\in \Bbb{C}^d$. Then $P(k)$ becomes an analytic (in fact,
polynomial) function on $\Bbb{C}^d$ with values in the space of bounded
Fredholm operators acting from $H^2(\Bbb{T}^d)$ into $L_2(\Bbb{T}^d)$.
Consider now the following set that we will call the \textit{Fermi surface}: 
\[
F=\left\{ k\in \Bbb{C}^d|\,\exists u\neq 0,P(k)u=0\right\} \subset \Bbb{C}%
^d. 
\]
It follows from the general theory of Fredholm operator functions (see for
instance \cite{K2}, \cite{ZK} and references therein) that the Fermi surface
is an analytic set in $\Bbb{C}^d$ (see details and references in \cite{KT}
and \cite{K2}). In fact, it is the set of all zeros of an entire function $%
f(k)$ of a finite order in $\Bbb{C}^d$. Assume now that there exists a
non-zero solution $u\in L_2(\Bbb{R}^d)$ of the equation $Pu=0$. After
expanding into the direct integral, we conclude immediately that the set of
quasimomenta $k\in B$ for which the operator $P(k)$ has a non-trivial kernel
must have positive measure. On the other hand, this set is the intersection $%
F\cap B$. We come to the conclusion that the function $f(k)$ is equal to
zero on a set of a positive measure in the real subspace $\Bbb{R}^d\subset 
\Bbb{C}^d$. Due to standard uniqueness theorems for analytic functions, this
implies that $f(k)$ is identically equal to zero, and hence $F=\Bbb{C}^d$.
Looking at the definition of the Fermi surface we see that this leads to
existence of a nontrivial solution of the equation 
\[
P(k)u=0 
\]
on $\Bbb{T}^d$ for arbitrary $k\in \Bbb{C}^d$. So, we come to the following
conclusion:

\begin{theorem}
\label{Thom}(see Theorem 4.1.5 in \cite{K2}) If there exists a vector $k\in 
\Bbb{C}^d$ such that the elliptic operator $P(k)$ on torus $\Bbb{T}^d$ has
zero kernel, then the equation $Pu=0$ in $\Bbb{R}^d$ has no non-trivial $L_2$%
-solutions.
\end{theorem}

\begin{remark}
\begin{enumerate}
\item  For the case of a periodic Schr\"{o}dinger operator this statement
(in a little bit different form) was established by L. Thomas \cite{T}. In
the current form and even for hypoelliptic operators it is proven in \cite
{K1}, \cite{K2}.

\item  If in the self-adjoint case one applies this theorem to the operators 
$P-\lambda $ with arbitrary real $\lambda $, one concludes the absolute
continuity of the spectrum of the operator $P$ in $\Bbb{R}^d$ (e.g., Theorem
4.5.9 in \cite{K2}).
\end{enumerate}
\end{remark}

Now the task, according to Theorem \ref{Thom}, becomes to find a
quasimomentum $k\in \Bbb{C}^d$ such that the operator $P(k)$ on torus $\Bbb{T%
}^d$ is invertible. We will remind the standard procedure of choosing an
appropriate vector $k$ in the case of the periodic Schr\"{o}dinger operator (%
\ref{Schred}) (see \cite{BSu5}, \cite{K2}, \cite{RS}, \cite{T}). It can be
chosen as $k=(\pi +i\rho ,0,...,0)$ with a large real $\rho $. Indeed, then
the operator $P(k)$ is 
\[
(D+k)^2+V(x). 
\]
Assume for simplicity that $V\in L_\infty (\Bbb{R}^d)$ (this assumption can
be significantly weakened, as shown for instance in \cite{Bert}, \cite{BSu5}%
, \cite{K1}, \cite{RS}, \cite{Sh1}, \cite{Sh2}, \cite{T}). We will show that 
\begin{equation}
\left| \left| (D+k)^2u\right| \right| _{L_2(\Bbb{T}^d)}\geq \pi \rho \left|
\left| u\right| \right| _{L_2(\Bbb{T}^d)}  \label{ineq}
\end{equation}
for any $u\in H^2(\Bbb{T}^d)$. In order to derive the inequality (\ref{ineq}%
), we use Fourier expansion 
\[
u(x)=\sum_{m\in \Bbb{Z}^d}\widehat{u}_me^{2\pi im\cdot x}. 
\]
Then 
\[
(D+k)^2u=\sum_{m\in \Bbb{Z}^d}(2\pi m+k)^2\widehat{u}_me^{2\pi im\cdot x}. 
\]
Due to the choice of $k$, we get 
\[
\left| (2\pi m+k)^2\right| \geq \left| Im(2\pi m+k)^2\right| =\left| (2\pi
m_1+\pi )\rho \right| \geq \pi \left| \rho \right| , 
\]
which proves (\ref{ineq}). Now, choosing $\rho $ such that $\pi \rho >\left|
\left| V\right| \right| _{L_\infty }$, one concludes that the equality 
\[
(D+k)^2u+V(x)u=0 
\]
is impossible for $u\neq 0$. Then according to the Theorem \ref{Thom} we
obtain that the equation 
\[
-\Delta u+Vu=0 
\]
in $\Bbb{R}^d$ has no non-zero $L_2$-solutions. This proves the following
result:

\begin{theorem}
\cite{T} The periodic Schr\"{o}dinger operator $H$ in $\Bbb{R}^d$ has no
eigenvalues, and hence in the self-adjoint case its spectrum is absolutely
continuous.
\end{theorem}

\begin{remark}
The statement on absence of eigenvalues does not require self-adjointness,
so it holds even in the case of a complex periodic potential.
\end{remark}

\section{\label{Main}The main result}

The main object of study in this paper is the following matrix
generalization of a periodic magnetic Schr\"{o}dinger operator (\ref{Magn})
in $\Bbb{R}^d$, $d\geq 2$: 
\begin{equation}
H(\mathbf{a},V)=-\sum_{j,k=1}^d\frac \partial {\partial x_j}g_{jk}\frac
\partial {\partial x_k}-2i\sum_{j=1}^da_j(x)\frac \partial {\partial
x_j}+V(x),  \label{Mop1}
\end{equation}
where $G=\{g_{jk}\}$ is a constant positive definite $d\times d$ matrix, the 
$n\times n$ matrix function $V(x)$ is the ``electric potential'' (in fact,
it corresponds to a combination of the magnetic and electric potentials in (%
\ref{Magn})) and the $n\times n$ matrix functions $a_j(x)$ form an analog $%
\mathbf{a}=\{a_j(x)\}$ of a magnetic potential. Both potentials $\mathbf{a}$
and $V$ are assumed to be periodic with respect to a lattice $\Gamma $ in $%
\Bbb{R}^d$. A linear change of variables transforms $\Gamma $ into the
integer lattice $\Bbb{Z}^d$ without changing the form of the operator, so we
will assume from now on that $\Gamma =\Bbb{Z}^d$. Although the matrix $G$
cannot be diagonalized without destroying the equality $\Gamma =\Bbb{Z}^d$,
the proofs will be provided for simplicity for the case when $G=I$ only, and
then can be rather automatically adjusted to arbitrary positive definite $G$%
. So, we will concentrate for the sake of simplicity on the operator 
\begin{equation}
H(\mathbf{a},V)=-\Delta -2i\sum_{j=1}^da_j(x)\frac \partial {\partial
x_j}+V(x)  \label{Mop}
\end{equation}
with $\Bbb{Z}^d$-periodic potentials only, while the results will hold for
any operator (\ref{Mop1}) with arbitrary lattice of periods.

In order to simplify the notations, we will use the standard functional
spaces notations like $H^s(\Bbb{T}^d)$ or $L_2(\Bbb{T}^d)$ for vector and
matrix valued functions, meaning that each of their entries belongs to the
corresponding space.

According to the Theorem \ref{Thom}, if one desires to show absence of
eigenvalues (or absolute continuity of the spectrum in the self-adjoint
case), then the task should be to show existence of a quasimomentum $k\in 
\Bbb{C}^d$ such that the operator 
\[
H(k,\mathbf{a},V)=(D+k)^2+2\mathbf{a}(x)\cdot (D+k)+V 
\]
has zero kernel on the torus $\Bbb{T}^d$. We will prove the following

\begin{theorem}
\label{Estim}Let the magnetic potential $\mathbf{a}$ belong to $H^s(\Bbb{T}%
^d)$ for some $s>3d/2-1$ and satisfy the following conditions:

\begin{enumerate}
\item  The average values of the components of $\mathbf{a}$ are equal to
zero: 
\begin{equation}
\int\limits_{\Bbb{T}^d}a_j(x)dx=0,\,j=1,...,d.  \label{aver}
\end{equation}

\item  For any $j,k=1,...,d$ and $x,y\in \Bbb{R}^d$ the values of the matrix
functions $a_j$ satisfy the commutation relation 
\begin{equation}
[a_j(x),a_k(y)]+[a_k(x),a_j(y)]=0.  \label{commut}
\end{equation}
\end{enumerate}

Then there exist a constant $C>0$, a vector $e\in \Bbb{R}^d$, and a positive
number $\beta \in \Bbb{R}$, such that 
\begin{equation}
\left| \left| H(k,\mathbf{a},0)u\right| \right| _{L_2}\geq C(\left| \rho
\right| \left| \left| u\right| \right| _{L_2}+\left| \left| u\right| \right|
_{H^1})  \label{inequal}
\end{equation}
for any $k=(\beta +i\rho )e\in \Bbb{C}^d$ with sufficiently large $\rho \in 
\Bbb{R}$ and for any $u\in H^2(\Bbb{T}^d)$.
\end{theorem}

\begin{remark}
\begin{enumerate}
\item  Notice that self-adjointness of the matrices $a_j$ is not required.

\item  The commutation relation (\ref{commut}) implies in particular
commutativity of the values of each of the functions $a_j(x)$, but not
necessarily commutation between $a_j(x)$ and $a_k(y)$. Although
commutativity of all the matrices $a_j(x)$ for different values of $j$ and $x
$ is sufficient for (\ref{commut}), it is not necessary for the proof, and
in fact the example of the Dirac operators leads to the case when such
commutativity does not hold, while (\ref{commut}) still does.

\item  We presume that the condition (\ref{commut}) is probably not
necessary for the Theorem to hold. However, the attempts to prove the
statement without it stumble upon an analytic obstacle in the form of
non-triviality of a vector bundle. So, one cannot rule out possibility of
surprises. See further discussion in Section \ref{Remarks} of the paper.
\end{enumerate}
\end{remark}

Theorem \ref{Estim} leads to the immediate conclusion that the equation $H(k,%
\mathbf{a},0)u=0$ has no non-zero solutions on $\Bbb{T}^d$ for large values
of $\rho $. Then, as in the previous section, Thomas' Theorem \ref{Thom}
implies that the operator $H(\mathbf{a},0)$ has no eigenvalues (and hence
has absolutely continuous spectrum in the self-adjoint case). Including an
electric potential $V$ that satisfies appropriate conditions is now a matter
of technique (see \cite{BSu5} and \cite{So}). For instance, using techniques
described in \cite{RS}, \cite{K2}, and \cite{So} this easily leads to the
following result.

\begin{theorem}
\label{noeigen}Assume that the conditions of Theorem \ref{Estim} are
satisfied and that $V\in L_{p,loc}(\Bbb{R}^d)$ with $p=2$ for $d=2$ and $%
p>d-1$ for $d\geq 3$. Then the periodic operator $H(\mathbf{a},V)$ has no
eigenvalues in $L_2(\Bbb{R}^d)$.
\end{theorem}

Notice that self-adjointness of any of the potentials is not assumed. In the
self-adjoint case this implies

\begin{theorem}
\label{abscont}If the conditions of Theorem \ref{noeigen} are satisfied and
the matrices $a_j(x)$ and $V(x)$ are self-adjoint for all $j$ and $x$, then
the spectrum of the operator $H(\mathbf{a},V)$ is absolutely continuous.
\end{theorem}

In fact, the conditions imposed on the magnetic and electric potentials are
too restrictive. Using techniques of Section 5 in \cite{BSu5} one can extend
the above results to the case when $V\in L_{p,loc}$ with $p>1$ when $d=2$, $%
p=d/2$ for $d=3$ and $4$, and $p=d-2$ when $d\geq 5$. In fact, for $d\geq 3$
the Lorentz spaces $L_{p,\infty }^0$ suffice \cite{BSu5}. We will not
provide the proof of this extension here. One can refer to Section 5 in \cite
{BSu5} for the method of its deduction from Theorem \ref{Estim}. One can
find the optimal conditions on the electric potential $V$ in terms of
different functional classes in \cite{BSu5} and \cite{Sh1}--\cite{Sh4}.
Although it is not clear yet how to optimize the conditions on the magnetic
potential $\mathbf{a}$ for $d>2$, for $d=2$ the optimal conditions in $L_p$%
-classes are known, see the corresponding results and conjectures in \cite
{BSu5}, some new advances in \cite{Da9}, and discussion at the end of the
paper.

\section{Sketch of the proof}

The approach we use is rather standard in the microlocal analysis:
microlocal reduction of the problem to a standard one, treating this model
case, and construction of a parametrix using a partition of unity. This
finally leads to the invertibility of the operator for large values of $\rho 
$. Due to periodicity, the dual space will be discrete.

Fourier series expansion transfers the consideration to functions defined on
the dual lattice $2\pi \Bbb{Z}^d$. The Fourier coefficients of a function $f$
on the torus $\Bbb{T}^d$ will be denoted by $\widehat{f}(m)$. We will need
the function 
\[
\Lambda (m,\rho )=(m^2+\rho ^2)^{1/2} 
\]
and the corresponding (pseudo-differential of first order) operator $\Lambda
_{\,\rho }(D)$ on the torus $\Bbb{T}^d$ acting as 
\[
\left( \Lambda _{\,\rho }(D)f\right) ^{\symbol{94}}(m)=\Lambda (m,\rho )%
\widehat{f}(m). 
\]
Notice that 
\[
\left| \left| \Lambda _{\,\rho }(D)f\right| \right| _{L_2}\sim \left| \rho
\right| \left| \left| u\right| \right| _{L_2}+\left| \left| u\right| \right|
_{H^1}. 
\]

Consider the case of zero electric potential first. It is known (see \cite
{BSu5}, \cite{BSu6}, \cite{So}, and Section \ref{Electric}) how to include
then the electric potential.

Let us introduce some notations: 
\[
H=-\Delta +2\mathbf{a}(x)\cdot D, 
\]
\[
H_0=-\Delta , 
\]
\[
H(k)=(D+k)^2+2\mathbf{a}(x)\cdot (D+k), 
\]
and 
\[
H_0(k)=(D+k)^2. 
\]

We consider $H(k)$ and $H_0(k)$ as operators on the torus $\Bbb{T}^d$. Let $%
k=2\pi (i\rho +\beta )e\in \Bbb{C}^d$, where $\beta \in \Bbb{R}$ is fixed, $%
\rho \in \Bbb{R}$ is arbitrarily large, and $e\in \Bbb{R}^d$. The idea
behind the proof of the estimate (\ref{inequal}) is the following. We will
construct a family of operators $R_{\,\rho }$ such that 
\[
R_{\,\rho }H(k)\Lambda _{\,\rho }^{-1}=I+T_{\,\rho },
\]
where 
\[
\left| \left| T_{\,\rho }\right| \right| _{L_2\rightarrow L_2}\rightarrow
0,\;\rho \rightarrow \infty 
\]
and 
\[
\left| \left| R_{\,\rho }\right| \right| _{L_2\rightarrow L_2}\leq
const,\;\rho \rightarrow \infty .
\]
This for large $\rho $ would imply (\ref{inequal}): 
\[
\left| \rho \right| \left| \left| u\right| \right| _{L_2}+\left| \left|
u\right| \right| _{H^1}\sim \left| \left| \Lambda _{\,\rho }(D)f\right|
\right| _{L_2}=\left| \left| \left( I+T_{\,\rho }\right) ^{-1}R_{\,\rho
}H(k)\Lambda _{\,\rho }^{-1}\Lambda _{\,\rho }(D)f\right| \right| _{L_2}
\]
\[
\leq C\left| \left| H(k)f\right| \right| _{L_2}.
\]
Construction of the approximate inversion operators $R_{\,\rho }$ is done
locally and then pasted together using an appropriate partition of unity.
Let us add some details. The symbol of the operator $H_0(k)$ is 
\[
H_0(k,m)=(2\pi m+k)^2=4\pi ^2\left[ (m+\beta e)^2-\rho ^2+2i\rho e\cdot
(m+\beta e)\right] .
\]
We consider this as the principal symbol of the full operator $H(k)$. Notice
that we include some lower order differential terms with parameter $k$ into
the principal symbol, which is rather standard when working with operators
with parameters (see for instance \cite{AV}). The main difficulty is that
the resulting problem with a parameter is not elliptic in the sense of \cite
{AV}. The zero set of this symbol is 
\[
Z_{\,\rho }=\left\{ m|\;(m+\beta e)^2=\rho ^2,\,e\cdot (m+\beta e)=0\right\}
\subset \Bbb{R}^d.
\]
Now the situation looks differently away from the set $Z_{\,\rho }$ and
close to it. Namely, sufficiently far away from this set the principal part
will dominate the magnetic one, hence an approximate inverse can be
constructed by neglecting the magnetic part. One ends up using the estimates
of Section \ref{Prelim}. However, close to $Z_{\,\rho }$ the magnetic
potential part becomes of comparable strength with the principal one. Then
one chooses a suitable finite covering of $Z_{\,\rho }$ by sets of diameter $%
\rho ^{\;\delta }$ with an appropriately chosen $\delta \in (0,1)$. In each
of the open sets of this covering the operator $H(k)\Lambda _{\,\rho }^{-1}$
is split into two parts by linearizing its symbol at a point. The linear
term is a model first order differential operator. It happens to be the
generalized Cauchy-Riemann operator 
\begin{equation}
\frac \partial {\partial \overline{z}}+F(z)  \label{model}
\end{equation}
on the complex plane, where the plane arises as a rational plane in $\Bbb{R}%
^d$, and the matrix function $F(z)$ is periodic and commutative. The
remainder can be estimated away in the sense that approximate inversion of
the model operator immediately leads to an approximate inverse for the whole
sum. So, locally the problem is reduced to inverting the operator (\ref
{model}) on the torus. The nice trick that does exactly that was invented by
A. Sobolev \cite{So}. The idea is that under an appropriate choice of the
vector $e$ used above for defining the quasimomentum $k$, one can gauge away
most of the magnetic potential $a$ in (\ref{model}). Hence the problem is
reduced to the Cauchy-Riemann operator on the torus, which is an easily
treatable case. As the result of this procedure, we have a set of local
approximate inverse operators $R_{k,j}$ with suitable estimates. Then an
appropriate partition of unity $\sum \phi _j=1$ in the dual space finishes
the job. Namely, let 
\[
\psi _j=\sum \phi _k,
\]
where the sum runs over all values of $k$ such that the supports of $\phi _j$
and $\phi _k$ intersect, and define the operator 
\[
R(k)=\sum_j\phi _j(k)R_{k,j}\psi _j(k).
\]
It is constructed in such a way that 
\[
R(k)H(k)\Lambda _{\,\rho }^{-1}=I+T(k),
\]
where 
\begin{equation}
\lim_{\rho \rightarrow \infty }\left| \left| T(k)\right| \right| =0.
\label{zero}
\end{equation}
This proves (\ref{inequal}) and all the theorems stated in the previous
section.

We claim that all types of periodic operators for which absolute continuity
of the spectrum has been proven so far (Schr\"{o}dinger, Dirac, Maxwell,
etc.) can be reduced to the case treated in Theorem \ref{noeigen}. This will
be shown in the Section \ref{Reduction}. However, the case of a magnetic
potential $\mathbf{a}$ not satisfying (\ref{commut}) is also of importance
and still remains a challenge. We expect that the absolute continuity must
also hold without assuming (\ref{commut}), see however discussion in Section 
\ref{Remarks}.

\section{\label{Proof}Proof of the main estimate}

We will now present the proof of Theorem \ref{Estim}.

Let us denote by $H_0$ the principal part $-\Delta $ of the operator $%
H=-\Delta +2\mathbf{a}(x)\cdot D$ and by $H_0(m)$ its symbol $4\pi ^2m^2$.
As before, the Fourier coefficients of a function $f$ on the torus will be
denoted by $\widehat{f}(m)$. We use the function $\Lambda (m,\rho
)=(m^2+\rho ^2)^{1/2}$ introduced above and the corresponding
(pseudo-differential of first order) operator $\Lambda _{\,\rho }(D)$ on the
torus.

Let $n\in \Bbb{Z}^d$ be a primitive integer vector, i.e. such that $\lambda
n\notin \Bbb{Z}^d$ for $\lambda \in (0,1)$. The particular choice of $n$
will be specified later. We denote 
\[
e=\frac n{\left| n\right| }\in \Bbb{R}^d 
\]
and 
\[
\beta =\frac 1{2\left| n\right| }. 
\]
Consider the quasimomentum $k=2\pi (\beta +i\rho )e\in \Bbb{C}^d$, where $%
\rho >0$. This choice of the quasimomentum guarantees invertibility of the
operator $H_0(D+k)$ on the torus. This follows from the direct calculation
of the symbol: 
\[
H_0(2\pi m+k)=(2\pi m+k)^2=4\pi ^2\left[ (m+\beta e)^2-\rho ^2+2i\rho e\cdot
(m+\beta e)\right] . 
\]
Due to the specific choice of $e$, we now get 
\[
\left| H_0(2\pi m+k)\right| \geq \left| ImH_0(2\pi m+k)\right| 
\]
\[
=8\pi ^2\rho \left| n\right| ^{-1}\left| n\cdot m+0.5\right| \geq 4\pi
^2\rho \left| n\right| ^{-1}. 
\]
This proves invertibility. Besides, the norm in $L_2(\Bbb{T}^2)$ of the
inverse operator is bounded from above by $\left| n\right| \left( 4\pi
^2\rho \right) ^{-1}$.

The characteristic set, or the set of degeneration of the symbol is
introduced as in the previous section: 
\[
Z_{\,\rho }=\left\{ m|\;(m+\beta e)^2=\rho ^2,\,e\cdot (m+\beta e)=0\right\}
. 
\]
It can be represented as the intersection of the sphere 
\[
S_{\,\rho }=\left\{ m|\;(m+\beta e)^2=\rho ^2\right\} 
\]
with the hyperplane 
\[
P=\left\{ m|\;e\cdot (m+\beta e)=0\right\} . 
\]

\subsection{\label{char}Inverse operator away from the characteristic set}

Let us fix a number $\delta \in (0,1/2)$. Consider the ``good'' set 
\begin{equation}
U_{\,\rho ,0}=\left\{ m|\;dist(m,Z_{\,\rho })\geq \rho ^{\,\delta }\right\} .
\label{goodset}
\end{equation}
We call this set ``good'', since one can show that on this set the principal
part dominates the magnetic terms. Namely, the following simple lemma holds.

\begin{lemma}
\label{1}There exist operators $R_{\,\rho ,0}$ and $T_{\,\rho ,0}$ such that
for any function $\psi \in L_2(\Bbb{T}^d)$ with support of $\widehat{\psi }$
in $U_{\,\rho ,0}$ one has 
\[
R_{\,\rho ,0}(k)H(k)\Lambda _{\,\rho }^{-1}\psi =\psi +T_{\,\rho ,0}\psi ,
\]
where for $\rho \rightarrow \infty $ one has 
\begin{equation}
\left| \left| R_{\,\rho ,0}\right| \right| _{L_2\rightarrow L_2}\leq
C,\,\,\left| \left| T_{\,\rho ,0}\right| \right| _{L_2\rightarrow
L_2}\rightarrow 0.  \label{inest}
\end{equation}
\end{lemma}

\textbf{Proof of the lemma}. Consider the following representation: 
\[
H(k)\Lambda _{\,\rho }(D)^{-1}=H_0(k)\Lambda _{\,\rho }(D)^{-1}+A(k),
\]
where 
\[
A(k)=2\mathbf{a}\cdot (D+k)\Lambda _{\,\rho }(D)^{-1}.
\]
Let us notice that the magnetic potential is bounded, so there is an uniform
with respect to all parameters involved estimate 
\begin{equation}
\left| \left| A(k)\right| \right| _{L_2\rightarrow L_2}\leq const.
\label{above}
\end{equation}
Let us now look at the constant coefficients part $H_0(D+k)\Lambda _{\,\rho
}(D)^{-1}$. Its symbol is 
\[
4\pi ^2\left[ (m+\beta e)^2-\rho ^2+2i\rho e\cdot (m+\beta e)\right]
(m^2+\rho ^2)^{-1/2}.
\]
We need to estimate this symbol from below on the set $U_{\,\rho ,0}$.
Pulling $\rho $ out of the symbol, we reduce the problem to estimating from
below the function 
\[
\left[ q^2-1+2ie\cdot q\right] (q^2+1)^{-1/2}
\]
when $q=\rho ^{-1}(m+\beta e)$ is at a distance at least $\rho ^{\delta -1}$
from the set $Z_1$. We claim that in this case 
\begin{equation}
\left| \left[ q^2-1+2ie\cdot q\right] (q^2+1)^{-1/2}\right| \geq C\rho
^{\delta -1}.  \label{below}
\end{equation}
Indeed, either $q$ is at a distance at least $\rho ^{\delta -1}$ from the
unit sphere centered at the origin, or it is at a distance at least $\rho
^{\delta -1}$ from the hyperplane $e\cdot q=0$ being at the same time at a
distance no more than $\rho ^{\delta -1}$ from the unit sphere. In the first
case the real part of our expression can be estimated from below by $\rho
^{\delta -1}$. In order to show this, it is sufficient to rewrite the real
part as 
\[
(\left| q\right| -1)(\left| q\right| +1)(q^2+1)^{-1/2},
\]
which gives a lower bound of the kind we need. In the second case we will
estimate the imaginary part. Namely, we know that $\left| q\right| \in
(1-\rho ^{\,\delta -1},1+\rho ^{\,\delta -1})$ and $\left| e\cdot q\right|
\geq \rho ^{\,\delta -1}$. Then in the expression 
\[
2\left| e\cdot q\right| (q^2+1)^{-1/2}
\]
the term $(q^2+1)^{-1/2}$ behaves as $1$, and the rest is estimated from
below by $\rho ^{\delta -1}$. The estimate (\ref{below}) implies that one
can choose an operator $R_{\rho ,0}$ as follows: 
\[
R_{\rho ,0}=\left( H_0(k)\Lambda _{\,\rho }^{-1}\right) ^{-1}\chi (D),
\]
where $\chi $ is a function which is equal to $1$ on $U_{\rho ,0}$ and is
supported in a slightly larger set of the same type. Then on functions $\psi 
$ supported in $U_{\rho ,0}$ one has 
\[
R_{\rho ,0}H(k)\Lambda _{\,\rho }^{-1}\psi =\psi +T_{\rho ,0}\psi ,
\]
where 
\[
T_{\rho ,0}=\left( H_0(k)\Lambda _{\,\rho }^{-1}\right) ^{-1}\chi A(k).
\]
The estimates (\ref{inest}) now follow from (\ref{above}) and (\ref{below}).
This proves the lemma.

\subsection{Local behavior on the characteristic set}

Now we have to deal with the ``bad'' set, i.e. the complement to $U_{\rho
,0} $. Let us choose a point $\xi $ on the characteristic set $Z_{\,\rho }$: 
\begin{equation}
(\xi +\beta e)^2=\rho ^2,\;e\cdot (\xi +\beta e)=0.  \label{charact}
\end{equation}
The conditions (\ref{charact}) are equivalent to the representation of the
vector $\xi $ in the form 
\begin{equation}
\xi =-\beta e+\rho l,  \label{repr}
\end{equation}
where $l\in \Bbb{R}^d$ is a unit vector orthogonal to $e$. We will restrict
consideration only to the rational points $l$, so the line $tl$ contains an
integer point. This reduces the choice of $\xi $ to a dense subset in $%
Z_{\,\rho }$, which will be sufficient for our purpose. Let us also choose a
neighborhood $V_{l,\rho }$ of the point $\xi $ such that its diameter is
bounded by $C\rho ^{\,\delta }$ with an absolute constant $C$ and a finite
set of such neighborhoods covers the complement of $U_{\rho ,0}$ with a
finite (independent on $\rho $) multiplicity. The exact choice of these sets
will be specified later. Consider the operator $H(k)\Lambda _{\,\rho
}(D)^{-1}$. Its symbol is 
\[
h_k(m,x)=(m+k)^2(m^2+\rho ^2)^{-1/2}+2\mathbf{a}(x)\cdot (m+k)(m^2+\rho
^2)^{-1/2}. 
\]
This operator acts as $Op(h_k)$. This means that in order to apply this
operator to a periodic function one expands the function into the Fourier
series (with $m$ as the dual variable), then multiplies its terms by $%
h_k(m,x)$, and sums the series.

We are now going to linearize the symbol at the point $m=\xi $ in the
neighborhood $V_{l,\rho }$.The linearization can be done by applying the
Taylor expansion, or alternatively by a simple algebraic calculation that we
are going to perform. In $V_{l,\rho }$ the function $(m^2+\rho ^2)^{-1/2}$
behaves as $C\rho ^{-1}$. Any point $m$ in this neighborhood can be
represented as $m=-\beta e+\rho l+\nu $, where $\left| \nu \right| \leq
C\rho ^{\,\delta }$. Using this representation and equality $k=(\beta +i\rho
)e$, we get 
\[
(m+k)^2+2\mathbf{a}(x)\cdot (m+k) 
\]
\[
=(\rho (ie+l)+\nu )^2+2(\rho (ie+l)+\nu )\cdot \mathbf{a} 
\]
\[
=2\rho (ie+l)\cdot (\nu +\mathbf{a}(x))+\Phi =2\rho (ie+l)\cdot (m+\beta
e-\rho l+\mathbf{a}(x))+\Phi , 
\]
where the symbol 
\[
\Phi (m,x)=\nu ^2+2\nu \cdot \mathbf{a} 
\]
satisfies in $V_{l,\rho }$ the estimate 
\[
\left| \Phi (m,x)\right| \leq C\rho ^{2\delta }. 
\]
Hence, in the same neighborhood 
\[
\left| \Phi (m,x)(m^2+\rho ^2)^{-1/2}\right| \leq C\rho ^{2\delta -1}. 
\]
Now the symbol $h_k(m,x)$ can be represented as 
\[
h_k(m,x)=\sqrt{2}(ie+l)\cdot (m+\beta e-\rho l+\mathbf{a}(x))+\Psi (m,x), 
\]
where the symbol 
\[
\Psi =2\rho (ie+l)\cdot (m+\beta e-\rho l+\mathbf{a}(x))\left( (m^2+\rho
^2)^{-1/2}-1/\left( \sqrt{2}\rho \right) \right) +\Phi (m,x)(m^2+\rho
^2)^{-1/2} 
\]
satisfies for large values of $\rho $ the estimate 
\[
\left| \Psi (m,x)\right| \leq C\rho ^{2\delta -1}. 
\]
The constants in the last two estimates are absolute.

As the result, we can conclude that the operator $H(k)\Lambda _{\,\delta
}(D)^{-1}$ can be represented on functions $\psi $ whose Fourier
coefficients are supported in $V_{l,\rho }$ as 
\begin{equation}
H(k)\Lambda _{\,\delta }(D)^{-1}=\sqrt{2}H_{\,\xi }+\Psi ,  \label{expand}
\end{equation}
where $H_{\,\xi }=Op(h_{\,\xi })$ has the symbol 
\begin{equation}
h_{\,\xi }(m)=(ie+l)\cdot (m+\beta e-\rho l+\mathbf{a}(x))  \label{models}
\end{equation}
and $\Psi =Op(\Psi (m,x))$ obeys the estimate 
\begin{equation}
\left| \Psi (m,x)\right| \leq C\rho ^{2\delta -1}  \label{sym-est}
\end{equation}
on $V_{l,\rho }$. We remind the reader that here $\xi =-\beta e+\rho l$.

Let us notice that the symbol (\ref{models}) corresponds to the first order
differential operator on the torus of the form $\partial /\partial \overline{%
z}+F(z)$, if we identify the $l$-$e$ plane with the complex plane $\Bbb{C}$.
Here $F(z)=(ie+l)\cdot \mathbf{a}(x)$ is a periodic matrix function with
zero average. This is the model operator that we will have to deal with.

\begin{lemma}
\label{Lemma_commut}The values of the matrix function $F=(ie+l)\cdot \mathbf{%
a}(x)$ commute.
\end{lemma}

\textbf{Proof of the lemma}. Let us denote $(ie+l)=(s_1,...,s_d)$. Then 
\[
(ie+l)\cdot \mathbf{a}(x)=\sum_js_ja_j(x). 
\]
Now direct calculation shows that the condition (\ref{commut}) implies
commutativity of these matrices for different values of $x$. Indeed, 
\[
\lbrack F(x),F(y)]=[\sum_js_ja_j(x),\sum_ks_ka_k(y)]=\sum_{1\leq j,k\leq
d}s_js_k[a_j(x),a_k(y)] 
\]
\[
=\sum_{1\leq j\leq d}s_j^2[a_j(x),a_j(y)]+\sum_{1\leq j<k\leq d}s_js_k\left(
[a_j(x),a_k(y)]+[a_k(x),a_j(y)]\right) . 
\]
Now all the terms in the last two sums are equal to zero due to (\ref{commut}%
).

The following lemma enables us to neglect the remainder term $\Psi $ in (\ref
{expand}) if $\delta $ is small enough.

\begin{lemma}
\label{2}There exists a constant $C$ such that for any function $v$ which
has its Fourier series supported in $V_{l,\rho }$ the following estimate
holds: 
\begin{equation}
\left| \left| \Psi v\right| \right| _{L_2}\leq C\rho ^{(d+4)\delta
/2-1}\left| \left| v\right| \right| _{L_2}.  \label{estim}
\end{equation}
In particular, if 
\begin{equation}
\delta <2/(d+4),  \label{condition}
\end{equation}
then the operator norm of $\Psi $ on the subspace of such functions tends to
zero when $\rho \rightarrow \infty $.
\end{lemma}

The \textbf{proof of the lemma} is straightforward. Indeed, 
\[
\left| \left( \Psi v\right) (x)\right| ^2=\left| \sum_{m\in V_{l,\rho }}\Psi
(m,x)\widehat{v}(m)e^{2\pi im\cdot x}\right| ^2. 
\]
Now the Cauchy-Schwarz inequality together with (\ref{sym-est}) and an
estimate of the size of $V_{l.\rho }$ by $\rho ^{d\delta }$ finish the job.

This lemma and expansion (\ref{expand}) show that it is sufficient for our
purpose to invert the model operator $H_{\,\xi }$ with uniform estimates on
the norm of $\left( H_{\,\xi }\right) ^{-1}$ on functions Fourier supported
in $V_{l,\rho }$.

\subsection{Inversion of the model operator}

In this section our goal is to construct the inverse of the model
inhomogeneous $\overline{\partial }$-operator described above and to get
estimates for it. This will be used to construct a local approximate inverse
operator in $V_{l,\rho }$ and then a global approximate inverse. In order to
make the consideration better understandable, we will start with the
simplest case and move toward the most general one. In each of these cases
we will have to make an appropriate choice of vector $e$, which so far has
been arbitrary.

\subsubsection{Zero magnetic potential}

In this case $\mathbf{a}=0$ (which is the situation considered by L. Thomas 
\cite{T}). Then the model operator (which we will denote by $H_{\,\xi }^0$)
has constant coefficients. The imaginary part of its symbol 
\[
(ie+l)\cdot (m+\beta e-\rho l) 
\]
is equal to 
\[
e\cdot (m+\beta e) 
\]
(recall that vectors $e$ and $l$ are orthogonal). Let us now choose any
non-zero vector $n\in \Bbb{Z}^d$ and denote $e=n/\left| n\right| $. We
assume, as before that $\beta =1/\left( 2\left| n\right| \right) $. Then we
have an estimate of the symbol from below: 
\[
\left| (ie+l)\cdot (m+\beta e-\rho l)\right| \geq \left| e\cdot (m+\beta
e)\right| 
\]
\[
=\frac 1{\left| n\right| }\left| m\cdot n+\frac 12\right| \geq \frac
1{2\left| n\right| }, 
\]
and hence 
\begin{equation}
\left| \left| H_{\,\xi }^0\psi \right| \right| _{L_2}\geq \frac 1{2\left|
n\right| }\left| \left| \psi \right| \right| _{L_2}.  \label{below2}
\end{equation}
This means that we can use as an approximate inverse the operator $R_{l,\rho
}=\frac 1{\sqrt{2}}\left( H_{\,\xi }^0\right) ^{-1}$ on functions with the
Fourier support inside $V_{l,\rho }$. Indeed, we see that $\left| \left|
R_{l,\rho }\right| \right| \leq const$ and 
\[
\frac 1{\sqrt{2}}\left( H_{\,\xi }^0\right) ^{-1}H(D+k)\Lambda _{\,\delta
}(D)^{-1}=\left( H_{\,\xi }^0\right) ^{-1}\left( H_{\,\xi }^0+\Psi \right)
=I+\left( H_{\,\xi }^0\right) ^{-1}\Psi . 
\]
Here the operator $T=R\Psi =\frac 1{\sqrt{2}}\left( H_{\,\xi }^0\right)
^{-1}\Psi $ obviously has the norm tending to zero for large values of $\rho 
$.

\subsubsection{Small magnetic potential}

Let us denote $\left| \left| \mathbf{a}\right| \right| =\max_{x\in \Bbb{T}%
^d,j=1,...,d}\left| \left| a_j(x)\right| \right| $, where $\left| \left|
a_j(x)\right| \right| $ is the operator norm of the matrix $a_j(x)$ in the
Euclidean space $\Bbb{C}^n$. We will assume that $\left| \left| \mathbf{a}%
\right| \right| $ is sufficiently small (it will be clear from what follows
below how small it should be). This is the situation considered in \cite{HH}%
. Let us choose $n=e=(1,0,...,0)$ and $\beta =1/2$. We can consider our
model operator $H_{\,\xi }+\Psi $ as a perturbation of the free one: 
\[
H_{\,\xi }+\Psi =H_{\,\xi }^0+Op(2\pi (ie+l)\cdot \mathbf{a})+\Psi .
\]
If $\left| \left| \mathbf{a}\right| \right| $ is small enough and $\rho $ is
large enough, we conclude that 
\[
\left| \left| T\right| \right| =\left| \left| \left( H_{\,\xi }^0\right)
^{-1}\left( Op(2\pi (ie+l)\cdot \mathbf{a})+\Psi \right) \right| \right| <1,
\]
and hence we can define the inverse operator 
\[
R_{l,\rho }=(I+T)^{-1}\left( H_{\,\xi }^0\right) ^{-1}.
\]
It is clear that uniform norm estimates from above hold for $R_{l,\rho }$.

\subsubsection{2D case}

The two dimensional case was settled in \cite{BSu1}. We will show here how
it fits into the general scheme. Let us assume that $d=2$. In this case the
only vector $m\in \Bbb{R}^d$ such that $(ie+l)\cdot m=0$ is $m=0$. Since the
magnetic potential has zero average and the values of $(ie+l)\cdot \mathbf{a}
$ commute, one can gauge the magnetic potential away completely. Namely, let 
$\phi (x)$ be a commutative matrix function of $x\in \Bbb{R}^2$ periodic
with respect to the lattice $\Bbb{Z}^2$ that also commutes with all values
of $(ie+l)\cdot \mathbf{a}$. We can commute the model operator $H_{\,\xi
}=(ie+l)\cdot (D+\beta e-\rho l)+(ie+l)\cdot \mathbf{a}$ with the exponent $%
e^{i\phi (x)}$: 
\[
e^{-i\phi (x)}\circ H_{\,\xi }\circ e^{i\phi (x)}. 
\]
This, due to commutativity of the values of all matrix functions involved
produces the operator 
\[
(ie+l)\cdot (D+\beta e-\rho l)+(ie+l)\cdot (\mathbf{a}+\nabla \phi ). 
\]
Assume now that we choose a function $\phi $ such that 
\[
(ie+l)\cdot \nabla \phi =-(ie+l)\cdot \mathbf{a}. 
\]
(One can easily identify this equality with an inhomogeneous $\overline{%
\partial }$-equation with respect to $\phi $ in the $e-l$ plane considered
as the complex plane $\Bbb{C}$.) This is possible if $\mathbf{a}$ has zero
average (which we do assume). Namely, the Fourier coefficients of the
function $\phi $ can be found as 
\begin{equation}
\widehat{\phi }(m)=-\frac{(ie+l)\cdot \widehat{\mathbf{a}}(m)}{(ie+l)\cdot m}
\label{division}
\end{equation}
(we use the fact that $\widehat{\mathbf{a}}(0)=0$). It is clear from the
construction of $\phi (x)$ and commutativity of matrices $(ie+l)\cdot 
\widehat{\mathbf{a}}(m)$ that the matrix function $\phi $ is commutative and
commutes with all matrices $(ie+l)\cdot \mathbf{a}(x)$. This enables us to
apply the gauge transform as above, after which the magnetic potential goes
away and we are again in the situation of the zero magnetic potential. In
other words, 
\[
\left( H_{\,\xi }\right) ^{-1}=e^{i\phi (x)}\left( H_{\,\xi }^0\right)
^{-1}e^{-i\phi (x)}. 
\]
The formula (\ref{division}) also provides an uniform (with respect to $\rho 
$ and $l$) estimate 
\[
\left| \left| \phi \right| \right| _{L_\infty (\Bbb{T}^2)}\leq
C_{\,\varepsilon }\left| \left| \mathbf{a}\right| \right| _{H^{\,\varepsilon
}(\Bbb{T}^2)} 
\]
for any positive $\varepsilon $. This gives uniform estimates on the $L_2$%
-norm of the gauge transform. Using now the considerations of the case when $%
\mathbf{a}=\mathbf{0}$, we end up with an uniform estimate from above for $%
R_{l,\rho }=\left( H_{\,\xi }\right) ^{-1}$.

\subsubsection{General case\label{gencase}}

The situation in dimensions higher than two (first considered in the scalar
self-adjoint case in \cite{So}) is definitely much trickier. One can see
where the difficulty lies in comparison with $2D$. Namely, when $d>2$, the
division problem (\ref{division}) cannot be resolved for arbitrary potential 
$\mathbf{a}$ with zero average. The reason is that the denominator in (\ref
{division}) has many zeros besides $m=0$. This means that the magnetic
potential cannot be gauged away completely. The clever trick invented by A.
V. Sobolev is to use a gauge transform for ''almost'' eliminating the
magnetic potential, rather than for its complete removal like it was done
for $d=2$. Let us describe this method in detail.

One needs to find an integer vector $n$ (which will determine the vector $%
e=n/\left| n\right| $ and the number $\beta =1/2\left| n\right| $ as before)
and a large positive number $L$ such that if we keep in the Fourier
expansion of the magnetic potential $\mathbf{a}$ only the finite sum up to $L
$, then the resulting approximation to the potential 
\begin{equation}
\mathbf{a}_L(x)=\sum_{\left| m\right| \leq L}\widehat{\mathbf{a}}(m)e^{2\pi
im\cdot x}  \label{define}
\end{equation}
can be gauged away according to (\ref{division}). If additionally the norm
of the remainder $\mathbf{a}-\mathbf{a}_L$ were sufficiently small, then we
would have reduced the problem to the case of a small magnetic potential.
So, now we need to find a vector $e=n/\left| n\right| $ and a number $L>0$
such that

i) $(ie+l)\cdot m\neq 0$ for any rational vector $l$ orthogonal to $e$ and $%
m\neq 0,$ $\left| m\right| <L$;

ii) $\left| \left| \mathbf{a}-\mathbf{a}_L\right| \right| <\theta /\left|
n\right| $, where $\theta $ is a given small constant. We would also like to
have uniform control over the norm $\left| \left| \mathbf{a}_L\right|
\right| _{H^{\,\varepsilon }}$.

If this is done, then we can find according to (\ref{division}) a gauge
transform with the phase $\phi (x)$ such that it eliminates the $\mathbf{a}%
_L $ part of the magnetic potential. Besides, $\left| \left| \phi \right|
\right| _\infty \leq C(L,\mathbf{a})$, where $C(L,\mathbf{a})$ depends on $L$
and $\mathbf{a}$ only (and in particular does not depend on $\rho $ and $l$%
). This reduces the problem to the case of a small potential $\mathbf{a}-%
\mathbf{a}_L$, which was already treated before. Let us notice that
``small'' means less than $C\beta =\theta /\left| n\right| $ with some
absolute constants $C$ and $\theta $. In other words, choosing a longer
integer vector $n$ we impose much stronger requirement on the part of the
potential that must be gauged away.

Let us implement this program. The idea is that for an appropriately chosen
integer direction $n$ all integer vectors $m$ that are orthogonal to $n$
have a large norm. This means that the plane $e\cdot m=0$ (and hence $%
(ie+l)\cdot m=0$) hits the points of the dual lattice very far from the
origin (i.e., for $\left| m\right| >L$ with a large $L$). If this is true,
then the ``tail'' $\mathbf{a}-\mathbf{a}_L$ will be small. This requires,
however, some estimates, which we are going to present. The first step is to
discuss the relation between $\left| n\right| $ and $L$.

\begin{lemma}
\label{number}\cite{So} For any integer $L>0$ the equation 
\[
m_1+m_2L+...+m_dL^{d-1}=0
\]
has no non-zero integer solutions $m=(m_1,...,m_d)$ with $\left| m_j\right|
<L/2$. Moreover, if $m\neq 0$ and $\left| m_j\right| <L/2$, then the
following estimate holds: 
\begin{equation}
\left| m_1+m_2L+...+m_dL^{d-1}\right| \geq 1.  \label{est_one}
\end{equation}
\end{lemma}

\textbf{Proof of the lemma}. Let $m\neq 0$ and $\left| m_j\right| <L/2$. Let 
$m_k$ ($k\geq 1$) be the first non-zero component among $m_1,...,m_d$. Then 
\[
\left| m_1+m_2L+...+m_dL^{d-1}\right| =\left|
m_kL^{k-1}+...+m_dL^{d-1}\right| 
\]
\[
=\left| L^k(\frac{m_k}L+\left( m_{k+1}+...+m_dL^{d-k}\right) )\right| \geq
L^k\frac 1L=L^{k-1}\geq 1. 
\]
This proves both statements of the lemma.

We choose now the integer vector $n=(1,L,L^2,...,L^{d-1})$, where $L$ is a
large integer $L$ (note that $\left| n\right| \sim L^{d-1}$). Then the lemma
above shows that all non-zero integer solutions of the equation $m\cdot n=0$
lie outside the ball of radius $L/2\sim C\left| n\right| ^{1/(d-1)}$.

Consider the Fourier expansion of the magnetic potential: 
\[
\mathbf{a}(x)=\sum \widehat{\mathbf{a}}(m)e^{2\pi im\cdot x} 
\]
and assume that $\sum \left| \left| \widehat{\mathbf{a}}(m)\right| \right|
\left| m\right| ^{\,\mu }<\infty $ for some $\mu $. Then 
\[
\sum_{\left| m\right| >L}\left| \left| \widehat{\mathbf{a}}(m)\right|
\right| \leq L^{-\mu }\sum_{\left| m\right| >L}\left| m\right| ^{\,\mu
}\left| \left| \widehat{\mathbf{a}}(m)\right| \right| =o(L^{-\mu
}),\,\,L\rightarrow \infty . 
\]
If now $\left| n\right| =O(L^{\,\mu })$, we see that 
\[
\left| n\right| \sum_{\left| m\right| >L}\left| \left| \widehat{\mathbf{a}}%
(m)\right| \right| \rightarrow 0,\,\,L\rightarrow \infty . 
\]
Now the last lemma shows that if $\sum \left| \left| \widehat{\mathbf{a}}%
(m)\right| \right| \left| m\right| ^{d-1}<\infty $, in particular if $%
\mathbf{a}\in H^s(\Bbb{T}^d)$ for some $s>3d/2-1$, our requirement ii) on
smallness of the residual $\mathbf{a}_r=\mathbf{a}-\mathbf{a}_L$ is
satisfied. Here $\mathbf{a}_L$ is defined as in (\ref{define}).

Our goal now is to gauge away the part $(ie+l)\cdot \mathbf{a}_L$ of the
potential in the model operator. This is possible if the vector $e=n/\left|
n\right| $ is chosen as above. Namely, let us again look for a periodic
commutative (and commuting with the matrices $(ie+l)\cdot \mathbf{a}$)
matrix function $\phi (x)$ such that 
\[
(ie+l)\cdot \nabla \phi =-(ie+l)\cdot \mathbf{a}_L. 
\]
This is possible according to the Lemma \ref{Lemma_commut} and formula (\ref
{division}), since we assume that $\widehat{\mathbf{a}}_L(0)=\widehat{%
\mathbf{a}}(0)=0$ and that $\widehat{\mathbf{a}}_L(m)=0$ whenever $%
(ie+l)\cdot m=0$. Besides, the estimate (\ref{est_one}) together with the
definition of the vector $e$ imply that 
\[
\left| \left| \phi \right| \right| _{L_\infty }\leq C_{\,\varepsilon }\left|
\left| \mathbf{a}\right| \right| _{H^{d/2+\varepsilon }} 
\]
for any positive $\varepsilon $, where the constant $C_{\,\varepsilon }$
does not depend on $l$ and $\rho $ (although it does depend on $n$).

We get now the representation 
\[
e^{-i\phi }H_{\,\xi }e^{i\phi }=H_{\,\xi }^0+(ie+l)\cdot \mathbf{a}_r. 
\]
Choosing $L$ sufficiently large, we achieve smallness of $\mathbf{a}_r$ and
hence, as it was explained in the case of a small potential, invertibility
of $H_\xi ^0+(ie+l)\cdot \mathbf{a}_r$ with a uniform estimate of the
inverse. Then the operator 
\[
R_{l,\rho }=e^{i\phi }\left( H_{\,\xi }^0+(ie+l)\cdot \mathbf{a}_r\right)
^{-1}e^{-i\phi } 
\]
inverts the operator $H_{\,\xi }$ and satisfies an uniform (with respect to $%
\rho $ and $l$) estimate from above.

Let us summarize here that the maximal smoothness of $\mathbf{a}$ required
so far in the case $d>2$ is $\mathbf{a}\in H^s(\Bbb{T}^d)$ for some $%
s>3d/2-1 $.

\subsection{Global inverse operator and completion of the proof}

The last step of the proof requires to paste the local approximate inverse
operators together, which is a rather standard procedure. Namely, let $\psi
_0\in C_0^\infty (\Bbb{R}^d)$ be such that $\psi _0\geq 0$, $\psi _0(\xi )=1$
for $\xi \in [-0.5,0.5]^d$, and $\psi _0(\xi )=0$ for $\xi $ outside $%
[-1,1]^d$. Now set $\psi _j(\xi )=\psi _0(\xi -j)$ for $j\in \Bbb{Z}^d$ and $%
\chi _j=\psi _j\left( \sum \psi _i\right) ^{-1}$. Then $\{\chi _j\}$ forms a
unity partition, diameter of the support of $\chi _j$ does not exceed $2%
\sqrt{d}$, and the multiplicity of the covering of the whole space by the
supports of these functions is finite.

Let us now fix $\delta \in (0,2/(d+4))$ and define the stretched functions 
\[
\chi _{\rho j}^1(\xi )=\chi _j(\rho ^{-\delta }\xi ). 
\]
This new set of functions still forms a unity partition and besides 
\begin{equation}
\left| D^{\,\alpha }\chi _{\rho j}^1\right| \leq C_{\,\alpha }\rho ^{-\delta
\left| \alpha \right| }.  \label{deriv_est}
\end{equation}
The support of $\chi _{\rho j}^1$ belongs to the cube 
\[
Q_{\rho j}=\{\xi |\;\rho ^{-\delta }\xi -j\in [-1,1]^d\} 
\]
Now we need a set of cut-off functions with a larger support. Namely, we
define 
\[
\chi _{\rho j}^2=\sum_s\chi _{\rho s}^1, 
\]
where the sum is taken over all values of $s$ for which the supports of
functions $\chi _{\rho s}^1$ and $\chi _{\rho j}^1$ intersect. We obviously
have estimates for $\chi _{\rho j}^2$ similar to those for $\chi _{\rho j}^1$%
. Besides, 
\[
\chi _{\rho j}^1\chi _{\rho j}^2=\chi _{\rho j}^1. 
\]

Consider the set of all values of $j$ such that the support of $\chi _{\rho
j}^2$ is disjoint with the characteristic set $Z_{\,\rho }$. Then obviously
the supports of the corresponding $\chi _{\rho j}^1$of belong to the set $%
U_{\rho ,0}$ as defined in Section \ref{char}. Hence, according to that
section, in each of the corresponding cubes $Q_{\rho j}$ there exists an
operator $R_{\rho j}$ such that $\left| \left| R_{\rho j}\right| \right| <C$
uniformly with respect to $\rho ,$ 
\[
R_{\rho j}H(k)\Lambda _\rho ^{-1}\psi =\psi +T_{\rho j}\psi 
\]
on functions $\psi $ whose Fourier series is supported in $Q_{\rho j}$, and 
\[
\left| \left| T_{\rho j}\right| \right| \rightarrow 0,\,\,\rho \rightarrow
\infty 
\]
uniformly in $\rho $.

Consider now the values of $j$ for which the support of $\chi _{\rho j}^2$
does touch the characteristic set. Then, according to the considerations of
the previous section, on each of the corresponding cubes $Q_{\rho j}$ there
also exist operators $R_{\rho j}$ and $T_{\rho j}$ with the same properties.

We are ready to construct an approximate inverse for the whole operator $%
H(k)\Lambda _{\,\rho }^{-1}$. We define it as follows:
\[
R_{\,\rho }=\sum_j\chi _{\rho j}^1R_{\rho j}\chi _{\rho j}^2,
\]
where we denote the multiplier of the Fourier series by the function $\chi
_{\rho j}^1$ by the same expression $\chi _{\rho j}^1$. Let us show that for
large values of $\rho $ this is indeed an approximate inverse for $H(k)$.

\begin{lemma}
\label{global}For sufficiently large values of $\rho $ the following
relation holds: 
\[
R_{\,\rho }H(k)\Lambda _{\,\rho }^{-1}=I+T_{\,\rho },
\]
where $\left| \left| T_{\,\rho }\right| \right| _{L_2\rightarrow L_2}<1$.
\end{lemma}

As it was explained before, this would prove the Theorem \ref{Estim} and the
consequent results.

Proof of the lemma. Let us compute the product $R_{\,\rho }H(k)\Lambda
_{\,\delta }^{-1}$: 
\[
R_{\,\rho }H(k)\Lambda _{\,\rho }^{-1}=\sum_j\chi _{\rho j}^1R_{\rho j}\chi
_{\rho j}^2H(k)\Lambda _{\,\rho }^{-1} 
\]
\[
=\sum_j\chi _{\rho j}^1R_{\rho j}H(k)\Lambda _{\,\rho }^{-1}\chi _{\rho
j}^2+\sum_j\chi _{\rho j}^1R_{\rho j}\left[ \chi _{\rho j}^2,H(k)\Lambda
_{\,\rho }^{-1}\right] . 
\]
Here $[A,B]$ denotes the commutator of the operators $A$ and $B$. The first
sum in this expression is 
\[
\left( \sum_j\chi _{\rho j}^1\right) I+\sum_j\chi _{\rho j}^1T_{\rho j}\chi
_{\rho j}^2=I+T_{\,\rho }. 
\]
Due to finiteness of the multiplicity of the covering by the supports of $%
\chi _{\rho j}^2$, we conclude that 
\[
\left| \left| T_{\,\rho }\right| \right| _{L_2\rightarrow L_2}\leq
C\max_j\left| \left| T_{\rho j}\right| \right| _{L_2\rightarrow
L_2}\rightarrow 0 
\]
when $\rho \rightarrow \infty $. So, the only remaining task is to show that 
\[
\left| \left| \left[ \chi _{\rho j}^2,H(k)\Lambda _{\,\rho }^{-1}\right]
\right| \right| _{L_2\rightarrow L_2}\rightarrow 0 
\]
uniformly with respect to $j$ when $\rho \rightarrow \infty $. Let us
compute the commutator explicitly, taking into account that 
\[
H(k)=(D+k)^2+2\mathbf{a}\cdot (D+k). 
\]
The second order term $(D+k)^2$ has constant coefficients, and so commutes
with $\chi _{\rho j}^2$, which is itself a constant coefficient operator.
Hence the commutator can be expressed as follows: 
\[
2[\chi _{\rho j}^2,\mathbf{a}]\cdot (D+k)\Lambda _{\,\rho }^{-1}. 
\]
The operator $(D+k)\Lambda _{\,\rho }^{-1}$ is obviously bounded in $L_2$
uniformly with respect to $\rho $. So, the question is whether 
\[
\left| \left| \lbrack \chi _{\rho j}^2,\mathbf{a}]\right| \right|
_{L_2\rightarrow L_2}\rightarrow 0 
\]
uniformly with respect to $j$ when $\rho \rightarrow \infty $. Consider the
block matrix of this operator in the basis of harmonics $e^{2\pi im\cdot x}$%
. The matrix of the operator of multiplication by $\mathbf{a}$ has entries $%
\{\widehat{\mathbf{a}}_{m-m^{\prime }}\}$, where as before $\widehat{\mathbf{%
a}}_m$ are the Fourier coefficients of $\mathbf{a}$. Then the matrix of $%
B=[\chi _{\rho j}^2,\mathbf{a}]$ is 
\[
b_{m,m^{\prime }}=\left( \chi _{\rho j}^2(m)-\chi _{\rho j}^2(m^{\prime
})\right) \widehat{\mathbf{a}}_{m-m^{\prime }}=\nabla \chi _{\rho
j}^2(m^{\prime }+\theta (m-m^{\prime }))\cdot (m-m^{\prime })\widehat{%
\mathbf{a}}_{m-m^{\prime }}. 
\]
Using the estimate (\ref{deriv_est}), we conclude that 
\[
\left| \left| b_{m,m^{\prime }}\right| \right| \leq C\rho ^{-\delta }\left|
m-m^{\prime }\right| \left| \left| \widehat{\mathbf{a}}_{m-m^{\prime
}}\right| \right| . 
\]
If now we assume that 
\begin{equation}
\sum_m\left| m\right| \left| \left| \widehat{\mathbf{a}}_m\right| \right|
<\infty ,  \label{a_estim}
\end{equation}
this would lead to the estimate of $l_1$- and $l_\infty $- operator norms of
the matrix $B=\{b_{m,m^{\prime }}\}$ from above by $C\rho ^{-\delta }$.
This, by interpolation, implies that its $l_2$- norm is estimated by the
same number, and hence uniformly with respect to $j$ tends to zero when $%
\rho \rightarrow \infty $. The estimate (\ref{a_estim}), in turn, follows
for instance from the assumption that $\mathbf{a}\in H^{d/2+1+\varepsilon }$.

\section{\label{Electric}Inclusion of an electric potential}

In this section we will explain how to derive Theorem \ref{noeigen} from
Theorem \ref{Estim}. Assume that $V$ belongs to the space $L_{p,loc}(\Bbb{R}%
^d)$ with $p=2$ for $d=2$ and $p>d-1$ for $d\geq 3$. The way one can include
the electric potential under these conditions is rather standard (see the
proofs of Theorem XIII.10 in \cite{RS}, Theorem 4.1.8 in \cite{K2}, and
Sections 3.5 and 7.2 in \cite{So}). Namely, Young and H\"{o}lder
inequalities easily lead to the following estimate between the electric
potential and free terms: 
\begin{equation}
\left| \left| Vu\right| \right| _{L^t(\Bbb{T}^d)}\leq C\rho
^{-1+(d-1)/p}\left| \left| H(k,0,0)u\right| \right| _{L^t(\mathbf{T}^d)}
\label{Young}
\end{equation}
for any $u\in H^2(\Bbb{T}^d)$ (see the simple details in the proofs of
Theorem 4.1.8 in \cite{K2} and Lemma 7.2 in \cite{So}). Besides, one has an
obvious estimate 
\[
\left| \left| \left( H(k,0,0)-H(k,\mathbf{a},0)\right) u\right| \right|
_{L^t(\mathbf{T}^d)}\leq C\left| \left| \Lambda _\rho (D)u\right| \right|
_{L^t(\mathbf{T}^d)}. 
\]
This inequality, (\ref{Young}), (\ref{inequal}), and conditions on $p$ imply
that 
\[
\left| \left| Vu\right| \right| _{L^t(\Bbb{T}^d)}\leq \varepsilon \left|
\left| H(k,0,0)u\right| \right| _{L^t(\mathbf{T}^d)} 
\]
\[
\leq \varepsilon \left( \left| \left| H(k,\mathbf{a},0)u\right| \right|
_{L^t(\mathbf{T}^d)}+\left| \left| \left( H(k,0,0)-H(k,\mathbf{a},0)\right)
u\right| \right| _{L^t(\mathbf{T}^d)}\right) 
\]
\[
\leq \varepsilon \left( \left| \left| H(k,\mathbf{a},0)u\right| \right|
_{L^t(\mathbf{T}^d)}+C\left| \left| \Lambda _\rho (D)u\right| \right| _{L^t(%
\mathbf{T}^d)}\right) 
\]
\[
\leq C\varepsilon \left| \left| H(k,\mathbf{a},0)u\right| \right| _{L^t(%
\mathbf{T}^d)}, 
\]
where $\varepsilon =C\rho ^{-1+(d-1)/p}$ tends to zero when $\rho $ goes to
infinity. This implies that for large $\rho $ the equation 
\[
H(k,\mathbf{a},V)u=H(k,\mathbf{a},0)u+Vu=0 
\]
has no non-zero solutions on the torus, which concludes the proof of the
theorem.

As we already remarked before, it is possible to weaken the conditions on
the electric potential $V$ to those as in \cite{BSu5}. This, however,
requires a more delicate analytic technique rather than usage of (\ref{Young}%
). This method is described in \cite{BSu5}, and we will not present it here.

\section{\label{Reduction}Other types of operators}

In this section we will show that all the types of periodic operators for
which absolute continuity has been proven, in spite of their seemingly
different appearance, can be reduced to the form (\ref{Mop}). Thus, Theorem 
\ref{noeigen} applies. Speaking of the \textbf{types} of operators, we
neglect the question about the optimal conditions on the coefficients. In
some of the results that we quote these conditions are more relaxed than
ours. We did not pursue the goal of obtaining optimal classes of
coefficients for which our methods and results would hold.

The following remark is due. With the exception of the trivial case of an
isotropic operator (\ref{isotr}), the only situations when the absolute
continuity has been proven for an operator with variable coefficients in the
leading terms, where the ones considered in \cite{Mo1} and \cite{Mo2}. There
the result was established for the Schr\"{o}dinger operator with variable
periodic anisotropic metric in $2D$ and for the Maxwell operator in a
periodic isotropic medium. Albeit these operators appear to have truly
variable coefficients in the leading terms, it will be shown below, that in
fact they also boil down to (\ref{Mop}). In other words, there has not been
any true success achieved yet for the variable coefficient case. This is not
to diminish the role of the remarkable papers \cite{Mo1} and \cite{Mo2},
which not only contained important new results, but also introduced some
innovative analytic tricks that might be useful in more general situations.

\subsection{Schr\"{o}dinger operator}

The standard scalar periodic Schr\"{o}dinger operator 
\[
\left( D-A\right) ^2+V=\sum \left( \frac 1i\frac \partial {\partial
x_j}-a_j(x)\right) ^2+V(x) 
\]
is obviously a particular case of (\ref{Mop}) for $n=1$. Hence, the results
of Theorems \ref{noeigen} and \ref{abscont} apply.

\subsection{Dirac operator}

After the absolute continuity of the spectra of the Schr\"{o}dinger operator
with a periodic electric potential was established in \cite{T}, the similar
result was obtained for the periodic Dirac operator in \cite{Da1}--\cite{Da4}%
. This study of the Dirac operator was continued in \cite{BSu2}, \cite{BSu},
and \cite{Da5}--\cite{Da9}.

Let us recall what the periodic Dirac operator in $\Bbb{R}^d$ is. Let $%
\alpha _j$ $(j=1,2,...,d)$ be Hermitian $m\times m$ matrices for some $m\in 
\Bbb{N}$ satisfying the relations 
\begin{equation}
\alpha _j\alpha _l+\alpha _l\alpha _j=2\delta _{jl}I,  \label{anticom}
\end{equation}
where $\delta _{jl}$ is the Kronecker delta, and $I$ is the unit $m\times m$
matrix. Consider the operator 
\[
\mathcal{D}=\frac 1i\sum\limits_{j=1}^d\frac \partial {\partial x_j}\alpha
_j+V(x) 
\]
acting on vector-functions from $\Bbb{R}^d$ to $\Bbb{C}^m$. Here $V(x)$ is a
function with values in $m\times m$ matrices periodic with respect to a
lattice $\Gamma $ in $\Bbb{R}^d$. Let us assume that $V(x)$ belongs to $%
L_{r,loc}$ with $r>2$ when $d=2$ and $V$ is continuous in higher dimensions.
If additionally $V$ is self-adjoint, then the operator $\mathcal{D}$ is
self-adjoint in $L_2(\Bbb{R}^d)$ with the domain $H^1(\Bbb{R}^d)$.

\begin{conjecture}
\label{C1}The periodic self-adjoint Dirac operator $\mathcal{D}$ has
absolutely continuous spectrum.
\end{conjecture}

This conjecture has not been proven in full generality even for smooth
matrix potentials $V$. As it was explained before, in order to establish
absolute continuity, it is sufficient to prove absence of eigenvalues. This
must hold also in the non-selfadjoint case.

\begin{conjecture}
\label{C2}Without the assumption of self-adjointness of the potential $V$
the periodic Dirac operator still has no $L_2$-eigenfunctions.
\end{conjecture}

We will now explain under what condition one can reduce study of this
problem to the one for an operator (\ref{Mop}) and hence prove absolute
continuity. We will show then that all cases when the conjecture has been
proven fall into this category. We will do this first without paying
attention to the conditions on the potential, At this stage the reader may
think that $V$ is of the class $C^1$. The requirements on $V$ will be
specified a little bit later.

Assume that there exists an $L_2$-eigenfunction $u$ of the Dirac operator,
i.e. 
\begin{equation}
\mathcal{D}u=\frac 1i\sum\limits_{j=1}^d\frac{\partial u}{\partial x_j}%
\alpha _j+Vu=\lambda u.  \label{DiracOp}
\end{equation}
Ellipticity enables is to conclude that then $u\in H^2$. This leads to the
equality 
\[
\mathcal{D}^2u=\lambda ^2u, 
\]
or 
\[
-\Delta u-i\sum_j\left( V(x)\alpha _j+\alpha _jV(x)\right) \frac{\partial u}{%
\partial x_j}+\left( V^2-i\sum\limits_j\alpha _j\frac{\partial V}{\partial
x_j}\right) u=\lambda ^2u. 
\]
It is clear that we are in the situation of an operator (\ref{Mop}).
However, Theorems \ref{Estim} and \ref{noeigen} require the ``magnetic
potential'' matrices $a_j(x)=V(x)\alpha _j+\alpha _jV(x)$ to satisfy (\ref
{commut}). This leads, according to Theorems \ref{noeigen} and \ref{abscont}
to the following statement:

\begin{theorem}
\label{Dirac}Assume that

\begin{enumerate}
\item  The average values of the matrix functions $a_j(x)=V(x)\alpha
_j+\alpha _jV(x)$, $j=1,...,2$ are equal to zero.

\item  The matrices $V(x)\alpha _j+\alpha _jV(x)$ for different values of $x$
and $j$ satisfy the commutation relation (\ref{commut}).

\item  $\left( V(x)\alpha _j+\alpha _jV(x)\right) \in H_{loc}^s(\Bbb{R}^d)$
with $s>\frac 32d-1$.

\item  $\left( V^2-\sum\limits_j\alpha _j\frac{\partial V}{\partial x_j}%
\right) \in L_{p,loc}(\Bbb{R}^d)$ with $p=2$ for $d=2$ and $p>d-1$ for $d>2$.
\end{enumerate}

Then the operator (\ref{DiracOp}) has no $L_2$-eigenfunctions. In
particular, in the self-adjoint case (i.e., when the matrix $V(x)$ is
self-adjoint) the spectrum is absolutely continuous.
\end{theorem}

The main (and probably unnecessary) restriction is the commutation condition
(\ref{commut}) on the matrices $\left( V(x)\alpha _j+\alpha _jV(x)\right) $.
We want to point out, however, that in all cases known to the authors when
absolute continuity of the periodic Dirac operator has been proven so far,
this condition is satisfied. Indeed, all papers \cite{BSu2}, \cite{BSu}, and 
\cite{Da1}-\cite{Da9} assume a very specific types of the potentials.
Namely, it is assumed that the operator has the following form: 
\[
\sum\limits_{j=1}^d\left( \frac 1i\frac \partial {\partial x_j}-c_j\right)
\alpha _j+V_0\alpha _{d+1}+V_1I, 
\]
where $c_j$, $V_0$, and $V_1$ are scalar functions, and the family $\{\alpha
_1,...,\alpha _{d+1}\}$ satisfies the same anti-commutativity condition (\ref
{anticom}) that has been assumed so far for $\{\alpha _1,...,\alpha _d\}$.
In this case 
\begin{equation}
V(x)=-\sum\limits_{j=1}^dc_j\alpha _j+V_0\alpha _{d+1}+V_1I  \label{reprDir}
\end{equation}
and hence for $l\leq d$%
\[
2a_l(x)=\left( V(x)\alpha _l+\alpha _lV(x)\right) 
\]
\[
=-\sum\limits_{j=1}^dc_j(x)\alpha _j\alpha _l+V_0(x)\alpha _{d+1}\alpha
_l+V_1(x)\alpha _l-\sum\limits_{j=1}^dc_j\alpha _l\alpha _j+V_0\alpha
_l\alpha _{d+1}+V_1\alpha _l 
\]
\[
=-2c_l(x)I+2V_1(x)\alpha _l. 
\]
Then 
\[
\lbrack a_l(x),a_j(y)]+[a_j(x),a_l(y)]=V_1(x)V_1(y)\left( [\alpha _l,\alpha
_j]+[\alpha _j,\alpha _l]\right) =0, 
\]
so the condition (\ref{commut}) is satisfied.

In some cases \cite{Da9} potentials of a little bit more general type than (%
\ref{reprDir}) are considered: 
\[
V(x)=-\sum\limits_{j=1}^dc_j\alpha _j+V_0+V_1, 
\]
where the matrix function $V_0$ commutes and $V_1$ anti-commutes with the
matrices $\alpha _1,...\alpha _d$. One can check as in the calculation above
that (\ref{commut}) still holds.

\subsection{\label{Iso}2D operators with variable metric and global
isothermal coordinates}

Most of the known results on absolute continuity of periodic elliptic
operators deal with the operators with constant coefficients in the
principal part (modulo the isotropic operator (\ref{isotr}), where the
metric conformal to the flat one is eliminated by the substitution (\ref
{chge})). The only exceptions were the innovative papers \cite{Mo1} and \cite
{Mo2} by A. Morame. We will consider the operators treated in these papers
in this and the next sub-sections in order to show that they in fact boil
down to a Schr\"{o}dinger operator (\ref{Mop}), and hence belong to the
realm of the operators with the constant coefficient leading terms.

The paper \cite{Mo1} contains the proof of absolute continuity of the
spectrum of the periodic Schr\"{o}dinger operator with variable metric in $%
2D $: 
\[
\sum\limits_{1\leq j,k\leq 2}\left| g\right| ^{-1/2}\left(
D_{x_j}-a_j(x)\right) g^{j,k}(x)\left| g\right| ^{1/2}\left(
D_{x_k}-a_k(x)\right) +V(x),\,x\in \Bbb{R}^2. 
\]
Here the metric $g(x)$ (a positive definite matrix function) is assumed to
be periodic with respect to a lattice $\Gamma $ and of the class $C^\infty $%
. The magnetic potential $a$ is $\Gamma $-periodic and belongs to $C^\infty $%
, and the electric potential $V$ is $\Gamma $-periodic and belongs to $%
L_\infty $. The operator, in order to be self-adjoint, needs to be
considered in the appropriate $L_2$-space, namely in $L_2(\Bbb{R}^2;\left|
g\right| ^{1/2}dx)$.

We will show here that even under much milder conditions on the coefficients
this operator can be reduced to a scalar periodic Schr\"{o}dinger operator.
Existence of such a reduction does not diminish the relevance of the result
and its proof provided in \cite{Mo1}. Besides being the first paper where
the absolute continuity problem in presence of variable coefficient second
order terms was successfully treated, \cite{Mo1} also contains new ingenious
analytic tricks that might be useful in other cases.

Consider the following more general second order periodic operator in $2D$: 
\[
H=D^{*}g(x)D+a(x)\cdot \nabla +V(x) 
\]
\begin{equation}
=-\sum\limits_{1\leq j,k\leq 2}\frac \partial {\partial x_j}g_{j,k}(x)\frac
\partial {\partial x_k}+\sum\limits_{1\leq j\leq 2}a_j(x)\frac \partial
{\partial x_j}+V(x),  \label{2D_op}
\end{equation}
where we do not yet specify conditions on the coefficients $g$, $a$, and $V$
except that $g$ is a positive matrix function, and all the coefficients are
periodic with respect to a lattice $\Gamma $ in $\Bbb{R}^2$. Let first $g(x)$
be a scalar, i.e. $g(x)=b(x)I_2$ where $b(x)$ is a sufficiently smooth
positive scalar function and $I_2$ is the $2\times 2$ unit matrix. We call
such an operator isotropic. Then, as it was explained in the introduction,
the problem of absolute continuity of the spectrum of such an operator can
be transformed to the same problem for the Schr\"{o}dinger operator $%
b^{-1/2}H\circ b^{-1/2}$. Our goal in this section is to show that in $2D$
one can always transform the operator $H$ to an isotropic one by a periodic
change of coordinates, and hence reduce the absolute continuity problem to
the well studied Schr\"{o}dinger case.

First of all, the reader is probably aware of existence of local coordinate
changes that do this. Such coordinates are called isothermal or conformal
and are used for local study of surfaces in $\Bbb{R}^3$ (e.g., Section 13 of
Ch. 2 in \cite{DNF} and p. 376 in \cite{Tay}), for reduction of second order
elliptic PDEs in two variables to the canonical form (\cite{CH}), and in
complex analysis (\cite{Al}, \cite{V}). While \cite{DNF} and \cite{Tay}
assume for simplicity significant smoothness (and even analyticity) of the
metric $g$, one can find in \cite{Al}, \cite{CH}, and \cite{V} mild
conditions under which such coordinates exist.

\begin{lemma}
\label{iso_local}(\$8 in Ch. IV of \cite{CH} and \cite{V}). Let $m$ be a
non-negative integer and $\alpha \in (0,1)$. Assume that the entries of the
matrix function $g(x)$ in (\ref{2D_op}) belong to $C^{m+\alpha }(\Bbb{R}^2)$%
. Then in a neighborhood of any point $x_0\in \Bbb{R}^2$ there exists a
change of variables $y(x)\in C^{m+1+\alpha }(\Bbb{R}^2)$ that transforms the
operator (\ref{2D_op}) to a one with the leading term 
\[
-\nabla \cdot \sigma (y)\nabla 
\]
with a scalar function $\sigma $ that belongs to $C^{m+\alpha }$ in a
neighborhood of $y(x_0)$.
\end{lemma}

So, locally the anisotropy of the operator can be eliminated by switching to
the isothermal variables $y$. The interesting observation is that such
variables can be chosen globally for a periodic operator. Before going into
the details, we want to mention that it was pointed to the first author by
L. Friedlander and S. Novikov that the uniformization theorem for Riemann
surfaces (see for instance Section 11G in Ch. III of \cite{AS}) implies
existence of global periodic isothermal coordinates that do the reduction.
Namely, the following statement holds:

\begin{proposition}
\label{isothermal}Under the same conditions on the matrix $g(x)$ as in Lemma 
\ref{iso_local}, there exists a global change of variables $\Psi :\Bbb{R}%
^2\rightarrow \Bbb{R}^2$ that transforms the operator into an isotropic one
and such that $\Psi (0)=0$ and for any integer vector $\mathbf{n}%
=(n_1,n_2)\in \Bbb{Z}^2$ 
\begin{equation}
\Psi (x+\mathbf{n})=\Psi (x)+n_1\Psi ((1,0))+n_2\Psi ((0,1)).  \label{change}
\end{equation}
The smoothness of the global change $\Psi $ is the same as of the local one.
\end{proposition}

The equality (\ref{change}) means that $\Psi $ transforms the original $\Bbb{%
Z}^2$-periodic operator (\ref{2D_op}) into an isotropic one which is
periodic with respect to the lattice generated by the basis vectors $\Psi
((1,0))$ and $\Psi ((0,1))$. In particular, the operator is transformed into
the one for which the absolute continuity problem can be resolved by
reduction to the Schr\"{o}dinger case. This implies the result of \cite{Mo1}
under much milder conditions on the coefficients than the ones in \cite{Mo1}
(see for instance discussion in \cite{BSu6}).

\textbf{Proof of Proposition \ref{isothermal}}. The local isothermal
coordinates that exist according to Lemma \ref{iso_local} can be chosen
orientation preserving. They provide an atlas of local coordinate charts on
the torus $\Bbb{T}^2=\Bbb{R}^2/\Bbb{Z}^2$. We can think that they map local
coordinate patches onto the unit disk in $\Bbb{C}$ (identified with $\Bbb{R}%
^2$). Then transforms between the intersecting local charts are orientation
preserving and conformal, and hence holomorphic. This introduces a structure
of a Riemannian surface $\mathcal{R}$ on $\Bbb{T}^2$. Since $\mathcal{R}$ is
a compact Riemannian surface of genus $g=1$, there exists unique up to a
constant factor holomorphic differential (differential 1-form) $\omega $,
since the dimension of the vector space of such differentials is equal to
the genus (e.g., Theorem 10-3 in \cite{Spr} and Section 24A in Ch. V of \cite
{AS}). Holomorphy means that in the local isothermal coordinates $x,y$ the
differential looks like $f(z)dz$ with a holomorphic function $f(z)$, where $%
z=x+iy$. This differential has no zeros (i.e., $f(z)$ has no zeros in each
local chart), since otherwise we could create by shifts zeros at an
arbitrary point, which would contradict to the one-dimensionality of the
space of such differentials. Let us consider the covering mapping 
\[
\Bbb{R}^2\rightarrow \Bbb{R}^2/\Bbb{Z}^2=\mathcal{R} 
\]
and pull back this differential $\omega $ to a differential $\Omega $ on the
whole $\Bbb{R}^2$. Let us also fix a point (for instance, the origin $0$) in 
$\Bbb{R}^2$. Then integrating $\Omega $ from this point to a variable point $%
x$ in the plane, we get the mapping 
\[
\Psi (x)=\int\limits_0^x\Omega 
\]
from $\Bbb{R}^2$ into the complex plane. It is easy to see that this mapping
is a homeomorphism of $\Bbb{R}^2$. It maps the lattice of periods
isomorphically onto another lattice and gives the global periodic isothermal
coordinates. Notice that due to analyticity of the differential there was no
loss of smoothness in the process of moving from local to global isothermal
coordinates. This finishes the proof.

\subsection{Maxwell operator}

Consider a $3D$ medium described by positive electric permittivity $%
\varepsilon (x)$ and magnetic permeability $\mu (x)$. Both functions are
scalar (thus the medium is isotropic), periodic with respect to a lattice $%
\Gamma $ in $\Bbb{R}^3$, measurable, and bounded away from zero and
infinity: 
\[
0<c<\varepsilon (x),\mu (x)<C<\infty . 
\]
Later on in this section it will be assumed that these functions are of the
class $C^1$. We will denote by $\mathbf{E}(x)$ and $\mathbf{H}(x)$ the
electric and magnetic fields. Consider the Hilbert space $\mathcal{L}=L_2(%
\Bbb{R}^3;\Bbb{C}^3,\varepsilon )\oplus L_2(\Bbb{R}^3;\Bbb{C}^3,\mu )$ of
pairs $(\mathbf{E,H})$. Here $L_2(\Bbb{R}^3;\Bbb{C}^3,s)$ denotes the $L_2$%
-space with respect to the measure $s\,dx$ of vector functions on $\Bbb{R}^3$
with values in $\Bbb{C}^3$. The subspace $L\subset \mathcal{L}$ consists of
pairs $(\mathbf{E,H})$ that satisfy the zero divergence conditions 
\begin{equation}
\nabla \cdot \mu \mathbf{H=0,\,\,}\nabla \cdot \varepsilon \mathbf{E=0.}
\label{zerodiv}
\end{equation}
The Maxwell operator $M$ acts in $L$ as follows: 
\begin{equation}
M\left( 
\begin{array}{l}
\mathbf{E} \\ 
\mathbf{H}
\end{array}
\right) =\left( 
\begin{array}{ll}
0 & i\frac 1\varepsilon \nabla \times \mathbf{H} \\ 
-i\frac 1\mu \nabla \times \mathbf{E} & 0
\end{array}
\right) ,  \label{Maxwell}
\end{equation}
with the domain consisting of fields $(\mathbf{E,H})$ such $\nabla \times 
\mathbf{E}$ and $\nabla \times \mathbf{H}$ belong to $L_2(\Bbb{R}^3;\Bbb{C}%
^3)$. This operator is self-adjoint \cite{BSo}. The reader can find the
spectral theory of Maxwell operator developed in high generality in \cite
{BSo}.

It is believed that the spectrum of the Maxwell operator $M$ in a periodic
medium is absolutely continuous. This question is relevant, for instance for
the theory of photonic crystals \cite{K3}. This belief was justified for the
case of $C^\infty $ functions $\varepsilon $ and $\mu $ in \cite{Mo2}. The
goal of this section is to show that the essence of what was done in \cite
{Mo2} boils down to a reduction of the problem to a non-self-adjoint matrix
Schr\"{o}dinger operator of the type (\ref{Mop}), so then Theorem \ref
{noeigen} applies. This was noticed by T. Suslina \cite{Su}.

The crucial observation of the paper \cite{Mo2} was the following statement
that can be verified by a direct calculation:

\begin{lemma}
\label{Morame}Let the functions $\varepsilon $ and $\mu $ be of the class $%
C^2$and $\mathbf{u}\in H_{loc}^2(\Bbb{R}^3,\Bbb{C}^3)$ be such that $\nabla
\cdot \varepsilon \mathbf{u=0}$. Then 
\begin{equation}
\nabla \times \frac 1\mu \nabla \times \mathbf{u=-}\frac 1{\mu \sqrt{%
\varepsilon }}\Delta \left( \sqrt{\varepsilon }\mathbf{u}\right) +\left(
\varepsilon \nabla \left( \frac 1{\mu \varepsilon }\right) \right) \times
\left( \nabla \times \mathbf{u}\right) \mathbf{+}W\mathbf{u},
\label{identity}
\end{equation}
where $W$ is a $3\times 3$ matrix, whose entries are expressible in terms of 
$\varepsilon $ and $\mu $, and the equality holds in the distributional
sense.
\end{lemma}

Suppose now that we have an $L_2$-eigenfunction $(\mathbf{E,H})$ of the
Maxwell operator. This means that 
\begin{equation}
\left\{ 
\begin{array}{lll}
i\frac 1\varepsilon \nabla \times \mathbf{H} & = & \lambda \mathbf{E} \\ 
-i\frac 1\mu \nabla \times \mathbf{E} & = & \lambda \mathbf{H}
\end{array}
\right. .  \label{eigenMax}
\end{equation}
In this case further application of the $curl$ operator is possible, which
leads to 
\begin{equation}
\left\{ 
\begin{array}{lll}
\nabla \times \frac 1\mu \nabla \times \mathbf{E}= & i\lambda \nabla \times 
\mathbf{H=} & \lambda \varepsilon \mathbf{E} \\ 
\nabla \times \frac 1\varepsilon \nabla \times \mathbf{H}= & -i\lambda
\nabla \times \mathbf{E=} & -\lambda \mu \mathbf{H} \\ 
\nabla \cdot \varepsilon \mathbf{E} & \mathbf{=0} &  \\ 
\nabla \cdot \mu \mathbf{H} & \mathbf{=0} & 
\end{array}
\right. .  \label{max_red}
\end{equation}
Using now the Lemma for the first two equations, we conclude that 
\[
\left\{ 
\begin{array}{lll}
\mathbf{-}\frac 1{\mu \sqrt{\varepsilon }}\Delta \left( \sqrt{\varepsilon }%
\mathbf{E}\right) +\left( \varepsilon \nabla \left( \frac 1{\mu \varepsilon
}\right) \right) \times \left( \nabla \times \mathbf{E}\right) \mathbf{+}W_1%
\mathbf{E-}\lambda \varepsilon \mathbf{E} & = & 0 \\ 
\mathbf{-}\frac 1{\varepsilon \sqrt{\mu }}\Delta \left( \sqrt{\mu }\mathbf{H}%
\right) +\left( \mu \nabla \left( \frac 1{\mu \varepsilon }\right) \right)
\times \left( \nabla \times \mathbf{H}\right) \mathbf{+}W_2\mathbf{H+}%
\lambda \mu \mathbf{H} & = & 0
\end{array}
\right. , 
\]
where $W_j$ are $3\times 3$ matrices. Using the equations (\ref{eigenMax})
again to eliminate the $curl$ terms, we obtain the system 
\begin{equation}
\left\{ 
\begin{array}{lll}
\mathbf{-}\frac 1{\mu \sqrt{\varepsilon }}\Delta \left( \sqrt{\varepsilon }%
\mathbf{E}\right) +i\lambda \mu \left( \varepsilon \nabla \left( \frac 1{\mu
\varepsilon }\right) \right) \times \mathbf{H+}W_1\mathbf{E-}\lambda
\varepsilon \mathbf{E} & = & 0 \\ 
\mathbf{-}\frac 1{\varepsilon \sqrt{\mu }}\Delta \left( \sqrt{\mu }\mathbf{H}%
\right) -i\lambda \varepsilon \left( \mu \nabla \left( \frac 1{\mu
\varepsilon }\right) \right) \times \mathbf{E+}W_2\mathbf{H+}\lambda \mu 
\mathbf{H} & = & 0
\end{array}
\right. .  \label{reduct}
\end{equation}
Let us introduce a new six-dimensional vector-function $\mathbf{v}=(\sqrt{%
\varepsilon }\mathbf{E,}\sqrt{\mu }\mathbf{H})$. It belongs to $L_2$, due to
the assumptions that $(\mathbf{E,H})$ is in $L_2$ and the functions $%
\varepsilon $ and $\mu $ are bounded. For this vector-function we rewrite (%
\ref{reduct}) as the equation 
\begin{equation}
-\Delta \mathbf{v}+V\mathbf{v}=0,  \label{reducedMax}
\end{equation}
where $V$ is a $6\times 6$ periodic matrix-function with bounded entries
obtained from $\varepsilon $, $\mu $, and $\lambda $ according to the above
procedure. This equation is an eigenvalue problem for the matrix operator $%
-\Delta +V$, which is obviously of the type (\ref{Mop}). This completes our
reduction. Notice that the matrix $V$ is in general not self-adjoint, in
spite the fact that we have started with a self-adjoint operator $\dot{M}$.
According to Theorem \ref{noeigen}, this is irrelevant, and (\ref{reducedMax}%
) has no non-zero $L_2$-solutions.

\begin{theorem}
\label{Max_ac}Under the conditions stated in this section, the Maxwell
operator $M$ has absolutely continuous spectrum.
\end{theorem}

\section{\label{Remarks}Concluding remarks}

\begin{enumerate}
\item  It remains a challenge to prove the absolute continuity of the
spectrum of the general second order periodic elliptic operator 
\[
-\sum\limits_{1\leq j,k\leq 2}\frac \partial {\partial x_j}g_{j,k}(x)\frac
\partial {\partial x_k}+\sum\limits_{1\leq j\leq 2}a_j(x)\frac \partial
{\partial x_j}+V(x)
\]
in dimensions higher than two even for smooth metric $\{g_{j,k}(x)\}$. The
trick with isothermal coordinates works only in $2D$. Similarly, absolute
continuity has not been proven for the periodic Maxwell operator in
anisotropic media, i.e. when the material tensors $\varepsilon (x)$ and $\mu
(x)$ are matrices rather than scalars. One can assume arbitrary smoothness
of $\varepsilon $ and $\mu $. It looks like new approaches are needed in
order to treat these problems. One of them could be to obtain appropriate
theorems on allowed rate of decay of solutions that generalize those of \cite
{FHHO} and \cite{Me}. If one could prohibit super-exponential decay of
eigenfunctions, then results of \cite{K1}, \cite{K2} would imply the
absolute continuity (see the corresponding discussion in the Introduction).
The other obvious possibility is to try to push through the Thomas'
approach. This, however, might happen to be too difficult in the presence of
variable coefficients in the leading terms. As this paper makes clear, all
known results essentially deal with the constant coefficient principal part
only.

\item  In the previous remark we listed the problems that remain unresolved
in the case of smooth coefficients. For some other operators, the
sufficiently smooth coefficients do not pose a challenge anymore, but
reducing conditions on the coefficients is an important task. The optimal
conditions on the electric potential in terms of standard functional spaces
have already been approached in \cite{BSu5}, \cite{Su}, and \cite{Sh1}--\cite
{Sh4} (see also \cite{BSu6} and \cite{Ka1}, where the potential is allowed
to contain parts supported on a graph, or delta-type terms). This is not so
for the magnetic potential. Conditions imposed in \cite{So} and in the
current paper are definitely unnecessarily strong. One can find in \cite
{BSu5} a reasonable conjecture about the optimal conditions on $a$, which
has been justified in $d=2$, and some recent advances in \cite{Da9}. The
situation is even worse with the coefficients in the leading terms. Namely,
the reduction (\ref{chge}) of the isotropic operator (\ref{isotr}) to the
Schr\"{o}dinger case works only when the metric $\rho (x)$ is sufficiently
smooth (for instance, belongs to $H_{loc}^2$). This, however, is far from
what the applied problems request, since many applications work with
composite media, where the coefficient $\rho (x)$ is piecewise constant. The
same remark applies to the discussion in the Section \ref{Iso}, where
isothermal coordinates are used. Analogously, applications to photonic
crystals (\cite{JMW}, \cite{K3}) require considering piecewise constant
dielectric permittivity $\varepsilon (x)$ and magnetic permeability $\mu (x)$
. There is little doubt that the spectra must be absolute continuous in
these cases, but proving this might require new techniques.

\item  We are tempted to say that the commutation condition (\ref{commut})
looks irrelevant and should not be necessary. There are only two
observations that stop us short of doing so. First of all, as it is shown
above, all known results relate to the ``commutative'' situation only.
Secondly, the simplest ``non-commutative'' case of the Dirac operator with
arbitrary smooth \textbf{matrix} potential $V$ has not been covered, in
spite of significant attention the Dirac operator has attracted. So, one
cannot exclude the possibility of surprises. Let us now take a closer look
at the condition (\ref{commut}). In its absence the proof stalls at the
inversion of the model operator $\frac \partial {\partial \overline{z}}+F(z)$
on the torus, when the matrix function $F(z)$ is not commutative. In order
for the process of gauging away most of the magnetic potential to go
through, one needs to find an invertible matrix function $B(z)$ (the
replacement of the exponent $\exp i\phi (z)$ that was used before) on the
torus $\Bbb{T}^2\Bbb{=R}^2/\Bbb{Z}^2$ such that 
\[
\frac{\partial B}{\partial \overline{z}}=BF.
\]
This problem is known in complex analysis in several variables. It is known
(see, for instance \cite{Ma}) that its solution always exists locally. It is
straightforward to check that the ratios of local solutions are holomorphic.
This means that the local solutions determine an analytic vector bundle on
the torus $\Bbb{T}^2$. Thus, existence of the needed global solution is
equivalent to analytic triviality of this bundle. One wanders whether a
version of Sobolev's trick (\cite{So} and Section \ref{gencase} of the
current paper) could let one to eliminate this analytic obstruction and to
carry through gauging away most of the magnetic potential. In the
commutative case this is not hard to do, since one can describe the obstacle
as non-vanishing of some Fourier coefficients of $F$, which is probably
impossible in general.
\end{enumerate}

\section{Acknowledgments}

The authors express their gratitude to Professors M. Birman, L. Danilov, L.
Friedlander, R. Novikov, S. Novikov, A. Sobolev, Z. Shen, Z. Sun, T.
Suslina, A. Tumanov, A. Pankov, and V. Palamodov for information and
comments. We are thankful to Professor T. Suslina for her numerous important
remarks about the first version of the manuscript.

This research was partly sponsored by the NRC through a COBASE Grant, NSF
through the Grant DMS 9610444, and by the Department of Army, Army Research
Office, and state of Kansas through a DEPSCoR Grant. The authors thank the
NRC, NSF, ARO, and the state of Kansas for this support. The content of this
paper does not necessarily reflect the position or the policy of the federal
government, and no official endorsement should be inferred.

\begin{thebibliography}{99}
\bibitem{AV}  M.S. Agranovich and M.I. Vishik, Elliptic problems with a
parameter and parabolic problems of general type, Russian Math. Surveys
19(1964), no.3, 53- 157.

\bibitem{Al}  L. V. Ahlfors, \textit{Lectures on Quasiconformal Mappings},
Van Nostrand Mathematical Studies, No. 10 D, Van Nostrand Co., Inc.,
Toronto, Ont.--New York--London 1966.

\bibitem{AS}  L. V. Ahlfors and L. Sario, \textit{Riemann Surfaces,}
Princeton Univ. Press, Princeton NJ 1960.

\bibitem{AM}  N.W. Ashcroft and N.D. Mermin, \textit{Solid State Physics,}
Holt, Rinehart and Winston, New York-London 1976.

\bibitem{AKK}  W. Axmann, P. Kuchment, and L. Kunyansky, Asymptotic methods
for thin high contrast 2D PBG materials, Journal of Lightwave Technology,
17(1999), no.11, 1996--2007.

\bibitem{Bert}  A. Berthier, On the point spectrum of Schr\"{o}dinger
operators, Ann. Scient. Ec. Norm. Super., 4 serie, 15(1982), 1--15.

\bibitem{BSo}  M. Birman and M. Solomyak, $L_2$-Theory of the Maxwell
operator in arbitrary domains, Russian Math. Surveys \textbf{42(1987)}, no
6: 75-96.

\bibitem{BSu1}  M. Sh. Birman and T.~A.~Suslina, Two-dimensional periodic
magnetic Hamiltonian is absolutely continuous. (Russian) Algebra i Analiz
9(1997), no. 1, 32--48; translation in St. Petersburg Math. J. 9(1998), no.
1, 21--32.

\bibitem{BSu2}  M. Sh. Birman and T. A. Suslina, The periodic Dirac operator
is absolutely continuous, Integr. Equat. and Operator Theory 34(1999),
377--395.

\bibitem{BSu3}  M. Sh. Birman and T. A. Suslina, Absolute continuity of the
two-dimensional periodic magnetic Hamiltonian with discontinuous
vector-valued potential. (Russian) Algebra i Analiz 10(1998), no. 4, 1--36;
translation in St. Petersburg Math. J. 10(1999), no. 4, 579--601.

\bibitem{BSu4}  M. Sh. Birman and T.~A.~Suslina, Two-dimensional periodic
Pauli operator. The effective masses at the lower edge of the spectrum, in 
\textit{Mathematical Results in Quantum Mechanics (QMath7, Prague, June
22-26,1998),} J.~Dittrich, P.~Exner, et al (Editors), 13-31, Operator
Theory, Adv. and Appl., v. 108, Birkh\"{a}user, Basel, 1999.

\bibitem{BSu5}  M. Sh. Birman and T.~A.~Suslina, Periodic magnetic
Hamiltonian with a variable metric. The problem of absolute continuity,
Algebra i Analiz 11(1999), no.2. English translation in St. Petersburg Math
J. 11(2000), no.2.

\bibitem{BSu}  M. Sh. Birman and T.~A.~Suslina, On the absolute continuity
of the periodic Schr\"{o}dinger and Dirac operators with magnetic potential,
in \textit{Differential Equations and Mathematical Physics, (Proc. Internat.
Conf., Univ. of Alabama, Birmingham, March 16--20, 1999)} 2000, 17--25.

\bibitem{BSu6}  M.~Sh.~Birman, T.~A.~Suslina, and R. G. Shterenberg,
Absolute continuity of the spectrum of a two-dimensional Schr\"{o}dinger
operator with potential supported on a periodic system of curves, Preprint
ESI no. 934, http://www.esi.ac.at, 2000.

\bibitem{CH}  R. Courant and D. Hilbert, \textit{Methods of Mathematical
Physics}, Vol. II. Partial differential equations. John Wiley \& Sons, Inc.,
New York, 1989.

\bibitem{Da1}  L. Danilov, Zone structure of the spectrum of the Dirac
operator with periodic potential, in: Nonlinear oscillations and control
theory, Ustinov 1985, 83-90.

\bibitem{Da2}  L. Danilov, On the spectrum of the Dirac operator with
periodic potential, Acad. Sci. USSR, Ural Branch, preprint, Sverdlovsk, 1987.

\bibitem{Da3}  L. Danilov, On the spectrum of the Dirac operator in $\Bbb{R}%
\mathbf{^n}$ with periodic potential. (Russian) Teoret. Mat. Fiz. 85 (1990),
no. 1,41--53; translation in Theoret. and Math. Phys. 85 (1990), no. 1,
1039--1048 (1991).

\bibitem{Da4}  L. Danilov, The spectrum of the Dirac operator with periodic
potential, I, preprint, no.4588-B91, VINITI, Moscow 1991, 36 pp.

\bibitem{Da5}  L. Danilov, Resolvent estimates and the spectrum of the Dirac
operator with a periodic potential. (Russian) Teoret. Mat. Fiz. 103 (1995),
no. 1, 3--22; translation in Theoret. and Math. Phys. 103 (1995), no. 1,
349--365.

\bibitem{Da6}  L. Danilov, On the spectrum of the two-dimensional periodic
Dirac operator. (Russian) Teoret. Mat. Fiz. 118 (1999), no. 1, 3--14;
translation in Theoret. and Math. Phys. 118 (1999), no. 1, 1--11.

\bibitem{Da7}  L. Danilov, Absolute continuity of the spectrum of the
periodic Dirac operator, Differenc. Uravneniya, 36(2000), no.2, 233-240.

\bibitem{Da8}  L. Danilov, On the spectrum of the periodic Dirac operator,
Theor. Math. Phys. 124(2000), no.1, 3-17.

\bibitem{Da9}  L. Danilov, On absolute continuity of spectra of periodic
Schr\"{o}dinger and Dirac operators. I, preprint, no. 1683-B00, VINITI,
Moscow, 2000.

\bibitem{DNF}  B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, \textit{%
Modern Geometry -- Methods and Applications}, Springer Verlag, New York,
1992.

\bibitem{DN1}  B. A. Dubrovin and S. P. Novikov, Ground states in a periodic
field. magnetic Bloch functions and vector bundles, Soviet Math. Dokl.
22(1980), 240--244.

\bibitem{DN2}  B. A. Dubrovin and S. P. Novikov, Ground states of a
two-dimensional electron in a periodic magnetic field, Soviet Physics JETF
52(1980), 511--516.

\bibitem{E}  M. S. P. Eastham, \textit{The Spectral Theory of Periodic
Differential Equations,} Scottish Acad. Press, Edinburgh-London, 1973.

\bibitem{Eh}  L. Ehrenpreis, \textit{Fourier Analysis in Several Complex
Variables}, Wiley-Interscience, New York-London 1970.

\bibitem{FHHO}  R. Froese, I. Herbst, M. Hoffmann-Ostenhof, and T.
Hoffmann-Ostenhof, $L_2$-lower bounds to solutions of one-body
Schr\"{o}dinger equations, Proc. Royal Soc. Edinburgh, 95A(1983), 25-38.

\bibitem{HH}  R. Hempel and I. Herbst, Strong magnetic fields, Dirichlet
boundaries, and spectral gaps, Comm. Math. Phys. 164(1995), 237--259.

\bibitem{JMW}  J.D. Joannopoulos, R.D. Meade, and J. N. Winn, \textit{%
Photonic Crystals. Molding the Flow of Light,} Princeton Univ. Press,
Princeton, NJ, 1995.

\bibitem{Ka1}  Yu. Karpeshina, Spectrum and eigenfunctions of
Schr\"{o}dinger operator with zero-range potential of the homoeneous lattice
type in three dimensional space, Ter. i Mat. Fiz. 57(1983), no.2, 304-313;
Engl. ransl. in Theor. and Math. Phys. 57(1983), 1156-1162.

\bibitem{Ka}  Yu. Karpeshina, \textit{Perturbation Theory for the
Schr\"{o}dinger Operator with a Periodic Potential}. Lecture Notes in
Mathematics, v. 1663. Springer-Verlag, Berlin, 1997.

\bibitem{KT}  H. Kn\"{o}rrer and E. Trubowitz, A directional
compactification of the complex Bloch variety. Comment. Math. Helv. 65
(1990), no. 1, 114--149.

\bibitem{K1}  P. Kuchment, Floquet Theory for Partial Differential
Equations, Russian Math. Surveys, \textbf{37}, no.4, 1-60, \textbf{(}1982).

\bibitem{K2}  P. Kuchment, \textit{Floquet Theory for Partial Differential
Equations,} Birkh\"{a}user Verlag, Basel 1993.

\bibitem{K3}  P. Kuchment, Mathematics of photonic crystals, to appear in
''Mathematical Modeling in Optical Science'', SIAM, 2000.

\bibitem{KK1}  P. Kuchment and L. Kunyansky, Spectral properties of
high-contrast band-gap materials and operators on graphs, Experimental
Mathematics 8(1999), no.1, 1--28.

\bibitem{KK2}  P. Kuchment and L. Kunyansky, Spectral Properties of High
Contrast Band-Gap Materials and Operators on Graphs, II, in preparation

\bibitem{KL1}  P. Kuchment and S.~Levendorski\^{i}, On the absolute
continuity of spectra of periodic elliptic operators, in \textit{%
Mathematical Results in Quantum Mechanics (QMath7, Prague, June 22-26,1998)}%
, J.~Dittrich, P.~Exner, et al (Editors), 291-297, Operator Theory, Adv. and
Appl., v. 108, Birkh\"{a}user, Basel, 1999.

\bibitem{KP}  P. Kuchment and Y. Pinchover, Integral representations and
Liouville theorems for solutions of periodic elliptic equations, Preprint \#
math/0007051 in xxx.lanl.gov, July 2000.

\bibitem{KV1}  P. Kuchment and B. Vainberg, On embedded eigenvalues of
perturbed periodic Schr\"{o}dinger operators, in \textit{Spectral and
Scattering Theory (Newark, DE, 1997)}, 67--75, Plenum, New York,1998.

\bibitem{KV2}  P. Kuchment and B. Vainberg, On absence of embedded
eigenvalues for Schr\"{o}dinger operators with perturbed periodic
potentials, Commun. PDE 25(2000), no. 9-10, 1809 - 1826.

\bibitem{Ma}  B. Malgrange, \textit{Lectures on the theory of functions of
several complex variables}, Tata Institute of Fundamental Research, Bombay,
Springer-Verlag, Berlin 1958.

\bibitem{Me}  V. Meshkov, On the possible rate of decay at infinity of
solutions of second order partial differential equations, Mat. Sbornik,
182(1991), no.3, 364-383. English translation in Math. USSR Sbornik
72(1992), no.2, 343--351.

\bibitem{Mo1}  A. Morame, Absence of singular spectrum for a perturbation of
a two-dimensional Laplace-Beltrami operator with periodic electro-magnetic
potential, J. Phys. A: Math. Gen. 31(1998), 7593--7601.

\bibitem{Mo2}  A. Morame, The absolute continuity of the spectrum of Maxwell
operator in periodic media, Preprint \#99-308 in the Texas Math Physics
archive http://www.ma.utexas.edu/mp\_arc

\bibitem{NSU}  G. Nakamura, Z. Sun, and G. Uhlmann, Global identifiability
for an inverse problem for the Schr\"{o}dinger equation in a magnetic field.
Math. Ann. 303 (1995), no. 3, 377--388.

\bibitem{Nov}  S.Novikov, Two-dimensional Schr\"{o}dinger operators in the
periodic fields, in Current Problems in Mathematics, vol 23, VINITI, Moscow,
1983, 3-32.

\bibitem{Pa1}  V. P. Palamodov, \textit{Linear Differential Equations with
Constant Coefficients}, Springer-Verlag, Berlin e.a. 1970.

\bibitem{Pa2}  V. Palamodov, Harmonic synthesis of solutions of elliptic
equations with periodic coefficients, Ann. Inst. Fourier 43(1993). 751-768.

\bibitem{Plis}  A. Plis, Non-uniqueness in Cauchy's problem for differential
equations of elliptic type, J. Math. Mech. 9(1960), 557--562.

\bibitem{RS}  M. Reed, B. Simon, \textit{Methods of Modern Mathematical
Physics, Vol.IV: Analysis of Operators,} Academic Press, New York, 1978.

\bibitem{Sh1}  Z.~Shen, On absolute continuity of the periodic
Schr\"{o}dinger operators, Preprint ESI 597, 1998, http://www.esi.ac.at and
\#99-189 in the Texas Math Physics archive, 1999,
http://www.ma.utexas.edu/mp_arc.

\bibitem{Sh2}  Z.~Shen, The periodic Schr\"{o}dinger operator with
potentials in the C. Fefferman--Phong class, Preprint \#99-15, Math. Dept.,
Univ. of Kentucky, 1999 and \#99-455 in the Texas Math Physics archive, 1999
http://www.ma.utexas.edu/mp_arc.

\bibitem{Sh3}  Z.~Shen, The periodic Schr\"{o}dinger operator with
potentials in the Morrey-Campanato class, Preprint 1999.

\bibitem{Sh4}  Z.~Shen, Absolute continuity of periodic Schr\"{o}dinger
operators with potentials in the Kato class, Preprint \#00-294 in the Texas
Math Physics archive, 2000 http://www.ma.utexas.edu/mp_arc.

\bibitem{Sh}  M. Shubin, Spectral theory and the index of elliptic operators
with almost periodic coefficients, Uspekhi Mat. Nauk 34(1979), no.2, 95-135.
English translation In Russian Math. Surveys.

\bibitem{Sj}  J. Sjostrand, Microlocal analysis for the periodic magnetic
Schr\"{o}dinger equation and related questions, in : \textit{Microlocal
Analysis and Applications}, Lect. Notes in Math., v.1495, 237-332,
Springer-Verlag, Berlin, 1991.

\bibitem{Sk}  M.M. Skriganov, \textit{Geometric and arithmetic methods in
the spectral theory of multidimensional periodic operators}, Proc. of the
Steklov Inst. of Math. 171(1985), 1-117. Engl. transl. in Proc. Steklov
Inst. Math. 1987, no. 2.

\bibitem{Spr}  G. Springer, \textit{Introduction to Riemann Surfaces},
Chelsea, New York 1981.

\bibitem{Su}  T. Suslina, Absolute continuity of the spectrum of periodic
operators of mathematical physics, Journees Equations aux Derivees
Partielles, Nantes, 5-9 Juin 2000, GDR 1151 (CNRS).

\bibitem{Sun}  Z. Sun, An inverse boundary value problem for Schr\"{o}dinger
operators with vector potentials. Trans. Amer. Math. Soc. 338 (1993), no. 2,
953--969.

\bibitem{So}  A. Sobolev, Absolute continuity of the periodic magnetic
Schr\"{o}dinger operator, Inventiones mathematicae 137(1999), Issue 1, pp
85-112

\bibitem{So2}  A. Sobolev and J. Walthoe, Absolute continuity in periodic
waveguides, Preprint, September 2000.

\bibitem{Tay}  M. Taylor, \textit{Partial Differential Equations. I. Basic
Theory}, Applied Mathematical Sciences, v. 115. Springer-Verlag, New York,
1996.

\bibitem{T}  L. E. Thomas, Time dependent approach to scattering from
impurities in a crystal, Comm. Math. Phys. 33(1973), 335-343.

\bibitem{V}  I. N. Vekua, \textit{Generalized Analytic Functions}, Pergamon
Press, London; Addison Wesley Pub. Co., Inc., Reading, Mass. 1962

\bibitem{ZK}  M. Zaidenberg, S. Krein, P. Kuchment, and A. Pankov, Banach
bundles and linear operators, Russian Math. Surveys \textbf{30}, no.5,
115-175, (1975)

\bibitem{Zak}  J. Zak, Magnetic translation group. I, II, Phys. rev.
134(1964), no. 6A, A1602--A16011
\end{thebibliography}

\end{document}
---------------0010022247717--
