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%preliminary version 09.08.01, 13.09.01, 05.10.01, 22.01.02
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\begin{document}

\begin{titlepage}

\begin{center}
  
  {\bf HOLDER CONTINUITY OF THE INTEGRATED DENSITY OF STATES \\ FOR
    SOME RANDOM OPERATORS AT ALL ENERGIES}
  
  \vspace{0.3 cm}
  
  \setcounter{footnote}{0}
  \renewcommand{\thefootnote}{\arabic{footnote}}
  
  {\bf Jean-Michel Combes \footnote{CPT, CNRS Marseille} }
  
  \vspace{0.1 cm}
  

  {D{\'e}partement de Math{\'e}tiques \\
    Universit{\'e} de Toulon et du Var \\
    83130 La Garde, France}
  
  \vspace{0.2 cm}
  
  {\bf Peter D.\ Hislop \footnote{Supported in part by NSF grant
      DMS-9707049.}}
  
  \vspace{0.1 cm}
  

  {Department of Mathematics \\
    University of Kentucky \\
    Lexington, KY 40506--0027 USA}
  
  \vspace{0.2 cm}
  
  {\bf Fr{\'e}d{\'e}ric Klopp \footnote{Supported in part by FNS 2000
  ``Programme Jeunes Chercheurs''.}}
  
  \vspace{0.1 cm}
  

  {L.A.G.A, Institut Galil{\'e}e\\
    Universit{\'e} Paris-Nord \\
    F-93430 Villetaneuse, FRANCE}

\end{center}

\vspace{0.2 cm}

\begin{center}
  {\bf Abstract}
\end{center}

\noindent
We prove that the integrated density of states of random \Schr\ 
operators with Anderson-type potentials on $L^2 ( \R^d)$, for $d \geq
1$, is locally H{\"o}lder continuous at all energies.  The single-site
potential $u$ must be nonnegative and compactly supported, and the
distribution of the random variable must be absolutely continuous with
a bounded, compactly supported density.  We also prove this result for
random Anderson-type perturbations of the Landau Hamiltonian in
two-dimensions under a rational flux condition.


\vspace{0.5 cm}

\noindent
\today

\end{titlepage}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\setcounter{chapter}{1} \setcounter{equation}{0}

\section{Introduction and Main Results} 

There are now many results on the local continuity of the integrated
density of states (IDS) for random \Schr\ operators $H_\omega (\lambda
) = H_0 + \lambda V_\omega$, on $L^2 ( \R^d)$, for $d \geq 1$, and
related operators describing wave propagation in random media, on
certain energy intervals. The regions of the real axis where
continuity has been proved lie outside the spectrum of the
unperturbed, deterministic operator $H_0$.  The first main result of
this paper is the proof of the H{\"o}lder continuity of the IDS at all
energies for the random operators. This applies to rather general
\Schr\ operators with random Anderson-type potentials $V_\omega$
constructed with nonnegative single-site potentials, and random
coupling constants having absolutely continuous probability densities
of bounded support. This result requires that the background operator $H_0$
have an IDS that is locally H{\"o}lder continuous.
Although this establishes H{\"o}lder
continuity at all energies, the H{\"o}lder exponent of continuity, $q$,
must be rather small, satisfying approximately $0 < q < 1/3$.  In our
second main result, we improve this estimate on the H{\"o}lder exponent of
continuity.  Under the hypothesis that $H_0$ is a periodic operator,
we prove that the IDS is H{\"o}lder continuous with any exponent $q$ of
H{\"o}lder continuity satisfying $0 < q < 1$.  Finally, our third result
concerns the H{\"o}lder continuity of the IDS for Landau Hamiltonians in
two-dimensions. We prove that if the flux through the unit square is
rational, then the IDS for $H_\omega ( \lambda )$, for $\lambda \neq
0$, is H{\"o}lder continuous at all energies with any exponent $0 < q <
1$.  These results improve all known results about the continuity of
the IDS for random \Schr\, and related, operators on $\R^d$.  We
remark that under the strong condition that $\mbox{supp} \; u$ is
sufficiently large, the IDS is known to be Lipschitz continuous at all
energies \cite{[CH1],[CHKN]}.

The family of \Schr\ operators $H_\omega ( \lambda )$ on $L^2 ( \R^d
)$, is constructed from a deterministic, background operator $H_0 = (
- i \nabla - A_0 )^2 + V_0$. We assume that this operator is
self-adjoint with operator core $C_0^\infty ( \R^d )$.  We consider an
Anderson-type potential $V_\omega$ constructed from the single-site
potential $u$ as
\begin{equation}
V_\omega (x) = \Sum_{j \in \Z^d  } \; \lambda_j ( \omega ) u ( x
- j ) .
\end{equation}
The family of random \Schr\ operators is given by
\begin{equation}
H_\omega ( \lambda ) = H_0 + \lambda V_\omega .
\end{equation}
As we will show below, we can normalize $V_\omega$ so that $\lambda >
0$ is a measure of the disorder.  The main results are independent of
the disorder (provided it is nonzero for Theorem 1.2 and
in the Landau case).  The results
of this paper also apply to the random operators describing acoustic
and electromagnetic waves in randomly perturbed media, and we refer
the reader to \cite{[CHT],[FK1],[FK2]}.

We need to define local versions of the Hamiltonians and potentials
associated with bounded regions in $\R^d$.  By $\Lambda_l (x)$, we
mean the open cube of side length $l$ centered at $x \in \R^d$.  For
$\Lambda \subset \R^d$, we denote the lattice points in $\Lambda$ by
${\tilde \Lambda } = \Lambda \cap \Z^d$.  Let $\Lambda \subset \R^d$
be a cube that is a union of translated unit cubes $\Lambda_1 (j) =
\Lambda_1 (0) + j$, $j \in \Z^d$, that is, $\Lambda = \cup_j \Lambda_1
(j)$.  For a given $\Lambda$, we take $H_0^\Lambda$ and
$H_\omega^\Lambda$ to be the restrictions of $H_0$ and $H_\omega$,
respectively, to the region $\Lambda$, with self-adjoint boundary
conditions on the boundary of $\Lambda$, $\partial \Lambda$. We denote
by $E_0^\Lambda ( \cdot )$ and $E_\Lambda ( \cdot )$ the spectral families for
$H_0^\Lambda$ and $H_\omega^\Lambda$, respectively.  Furthermore, for
$\Lambda \subset \R^d$, let $\chi_\Lambda$ be the characteristic
function for $\Lambda$.  The local potential $V_\Lambda$ is defined by
\begin{equation}
V_\Lambda (x) = \Sum_{j \in {\tilde \Lambda} } \; \lambda_j ( \omega ) u ( x
- j ) \chi_\Lambda (x).
\end{equation}
We also let ${\tilde V}_\Lambda$ denote the sum of the single-site
potentials in $\Lambda$, that is
\begin{equation}
{\tilde V}_\Lambda (x) = \sum_{j \in {\tilde \Lambda } } ~u(x-j)
\chi_\Lambda (x) .
\end{equation}
The spectral family $E_\Lambda ( \cdot )$, and the local potential
$V_\Lambda$ depend on the random variables in $\Lambda$, but we
suppress this in the notation.  We note that although $V_\omega |
\Lambda $ differs from $V_\Lambda$ in a neighborhood of the boundary
of $\Lambda$, the difference does not affect the results, cf.\ 
\cite{[CHM]}.

We will always make the following two assumptions:
%\vspace{.1in}
\begin{description} 
\item[(H1).] The single-site potential $u \neq 0$ is nonnegative with
  compact support, $u \in L^\infty ( \R^d )$, and $\|u \|_\infty \leq
  1$.
\item[(H2).] The random coupling constants $\{ \lambda_j ( \omega ) \;
  | \; j \in \Z^d \}$, are independent and identically distributed.
  The distribution has a density $h_0$ with $\mbox{supp} \: h_0
  \subset [0 , 1 ]$.
\end{description}
%\vspace{.1in}
\noindent
We make three important comments on these hypotheses:
\begin{enumerate} 
\item For a given density $h_0$ of compact support $[m , M_1]$, we can
  always add the periodic potential $m \sum_j u_j$, with $u_j (x) =
  u(x-j)$, to the background potential $V_0$, so that the random
  coupling constants take their value in an interval $[0, M]$, with $M
  = M_1 -m$.
  
\item Because of the explicit disorder parameter $\lambda > 0$, we can
  rescale the coupling constants so that, without loss of generality,
  the support of $h_0$ is included in the interval $[0, 1]$.
  
\item Hypotheses (H1) and (H2) imply the following.  There exists a
  finite constant $C_1 (u , M , d) >0$, depending only on the
  single-site potential $u$, and the dimension $d \geq 1$, so that for
  all $\Lambda \subset \R^d$,
\begin{equation}
0 \;  \leq \; V_\Lambda^2 \; \leq  \; C_1 (u , d) {\tilde V}_\Lambda ,
\end{equation}
where $V_\Lambda$ and ${\tilde V}_\Lambda$ are defined in (1.3) and
(1.4), respectively.  This simple inequality is used in the proof of
Theorem 1.1.

\end{enumerate}

We define the IDS $N(E)$ for $H_\omega$ using the counting function
for $H_\omega^\Lambda$. Let $N_\Lambda (E)$ be the number of
eigenvalues of $H_\omega^\Lambda$, with self-adjoint boundary
conditions, less than or equal to $E$. This function depends on the
realization $\omega$. We define
\begin{equation}
N ( E) = \lim_{| \Lambda | \rightarrow \infty } \; \frac{
N_\Lambda (E) }{ | \Lambda | } .
\end{equation}
It is known that this limit exits and is independent of the
realization $\omega$ almost surely.  Furthermore, it is known that the
IDS is independent of the boundary conditions taken on the finite
volumes $\Lambda$, cf.\ \cite{[DIM],[Kirsch1], [Nakamura]}.
%Similarly, we define the IDS $N_0 (E)$ for the operator $H_0$. 

\vspace{.1in}
\noindent
{\bf Theorem 1.1.} {\it In addition to hypotheses (H1) and (H2), we
  assume that the operator $H_0$ admits a density of states
  $N_0 (E)$ that is locally H{\"o}lder continuous on $\R$ 
  for some exponent $0 < q_1 \leq 1$, i.\ e.\ that satisfies
  \begin{equation}
    |N_0(E')-N_0(E)|\leq C_0 |E'-E|^{q_1}
  \end{equation}
  where the finite constant $C_0 > 0$ is locally uniform in
  energy. Then, the IDS for $H_\omega (\lambda)$ is H{\"o}lder
  continuous on $\R$ for any exponent $0 < q < q_1 / ( q_1 + 2 )$.}

\vspace{.1in}

For the second result, we take $H_0 = (- i \nabla - A_0)^2 + V_0$ to
be a periodic \Schr\ operator with a real-valued, periodic, potential
$V_0$, and a periodic vector potential $A_0$. We assume that $V_0$ and
$A_0$ are sufficiently regular so that $H_0$ is essentially
self-adjoint on $C_0^\infty ( \R^d)$. We assume that both $V_0$ and
$A_0$ are periodic with respect to the group $\Gamma = \Z^d$ because
of the form of the Anderson-type potential (1.1). We note that we
could work with a nondegenerate lattice $\Gamma$, by defining a
corresponding Anderson-type potential, but we will explicitly treat
the case $\Gamma = \Z^d$. Let $\Gamma^*$ denote the dual lattice, that
is, $\Gamma^* = \{ \gamma' \; | \; \gamma \cdot \gamma' \in 2 \pi \Z,
~\mbox{for all} ~\gamma \in \Gamma \}$.  We let $\T^d = \R^d / \Z^d$
be the torus, and $(\T^d)^* = (\R^d)^* / (\Z^d)^*$ be the dual torus.
We denote by $C_0$ the unit cell for $\Gamma = \Z^d$, and by $C_0^*$
the unit cell for $\Gamma^* = (\Z^d)^*$.  The Floquet decomposition of
$H_0$ yields a family of operators $H_0 ( \theta)$, for $\theta \in
(\R^d)^*$. Each operator $H_0 (\theta)$ is self-adjoint on $L^2 ( \T^d
)$, and has a compact resolvent. We denote the eigenvalues of $H_0
(\theta)$ by $E_n ( \theta )$. The spectrum of $H_0$ is given by
\begin{equation}
\sigma ( H_0 ) =
\bigcup_{\theta \in (\T^d)^* } \sigma ( H_0 (\theta) ) = 
\bigcup_{n \in \N }
\; \bigcup_{\theta \in (\T^d)^* } \; E_n (\theta ) .
\end{equation}
For a closed, bounded interval $\Delta \subset \R$, we let $E_0 (
\Delta , \theta )$ denote the spectral projector of $H_0 ( \theta )$
onto the eigenspace of $H_0 ( \theta )$ spanned by its eigenfunctions
with eigenvalues $E_n ( \theta ) \in \Delta$.  Because of the
discreteness of the spectrum of $H_0 ( \theta )$, the dimension of
$RanE_0 ( \Delta ,\theta )$ is finite and locally constant.  

We consider Hamiltonian $H_\omega ( \lambda )$ restricted to the cube
$\Lambda$ with periodic boundary conditions (PBC). If the cube
$\Lambda$ has side length $n$, denoted by $\Lambda_n$, we can consider
it as a fundamental unit cell for the lattice $n \Z^d$ and take a
Floquet decomposition relative to it. We let $H_0^n ( \theta )$ and
$E_j^{(n)} ( \theta )$ denote the corresponding fibred operators and
their eigenvalues.  Note that the operator $H_0^{\Lambda_n}$, with PBC
on $\partial \Lambda_n$, coincides with the fibred operator $H_0^n (
\theta=0 )$.  We define the IDS for $H_\omega ( \lambda )$ as above
using the counting function for $H_\omega^\Lambda$ with PBC.

\vspace{.1in}
\noindent
{\bf Theorem 1.2.} {\it Assume hypotheses (H1) and (H2), and that 
  $H_0$ is $\Z^d$-periodic. Let ${\tilde
  \Delta} \subset \R$ be a bounded, closed interval.  
  Suppose that for all ~$\theta \in (\T^d)^*$ there
  exists a finite constant $C({\tilde \Delta} , u)>0$, so that
  the single-site potential $u$ satisfies
\begin{equation}
E_0 ( {\tilde \Delta} , \theta ) u E_0 ( {\tilde \Delta} , \theta ) \; \geq \;
C({\tilde \Delta}, u) E_0 ({\tilde \Delta} , \theta ) .
\end{equation}
Then, the IDS for the random family $H_\omega ( \lambda ) = H_0 + \lambda
V_\omega$, for $\lambda \neq 0$,
is H{\"o}lder continuous on $\R$, for any H{\"o}lder exponent $0 < q <
1$.}

\vspace{.1in}

Theorem 1.2 requires no condition on the IDS for the background
operator $H_0$ as in (1.7). Note that, in general, for a periodic
magnetic potential and no electric potential, the IDS $N_0 (E)$ is not
H{\"o}lder continuous at the bottom of the spectrum
(see~\cite{[DN1],[DN2]}). This is analogous to what happens in the
constant magnetic field case (see the discussion in section 6). The
abstract condition (1.9) involves the single-site potential and the
eigenfunctions of the background, periodic operator $H_0$.  Concerning
the abstract condition (1.9), we have the following result.

\vspace{.1in}

\noindent
{\bf Proposition 1.3.} {\it We assume hypotheses (H1) and (H2).  In
  any dimension $d \geq 1$, if $H_0$ has the unique continuation
  property and $u > 0$ on a nonempty open set, then condition (1.9)
  holds.  }

Here, we say that $H_0$ has the unique continuation property if for
any $E\in\R$ and any function $\varphi$ in the domain of $H_0$, if
$\varphi$ is a solution to $(H_0-E)\varphi=0$ and $\varphi$ vanishes
on some open set, then $\varphi$ vanishes identically. It is well
known that $H_0$ has the unique continuation property if $A_0$ and
$V_0$ are suffciently regular; e.g. in dimension $d\geq3$, $V_0\in
L^{d/2}_{loc}(\R^d)$, $A_0\in L^{d}_{loc}(\R^d)$ and $\nabla A_0\in
L^{d/2}_{loc}(\R^d)$ are sufficient to ensure that $H_0$ has the
unique continuation property (see e.g.~\cite{[Wolff]} and references
therein).

Proposition 1.3 and Theorem 1.2 establish the continuity of the IDS
under rather general conditions.

\vspace{.1in}
\noindent
{\bf Corollary 1.4.} {\it We assume hypothesis (H2) and that $H_0$ is
  $\Z^d$-periodic and has the unique continuation property.  If $d
  \geq 1$, then the IDS is H{\"o}lder continuous with any exponent $0 < q
  < 1$ at all energies provided (H1) holds and $u > 0$ on a nonempty
  open set.  } \vspace{.1in}

We make some comments on the distinction between Theorems 1.1 and 1.2. 
For any real function $f\in C_0 ( \R)$, the density of states measure $dN$
satisfies
$$
\int f( s ) dN(s) = \lim_{|\Lambda| \rightarrow \infty} \frac{1}{| \Lambda |}
\; Tr ( \chi_\Lambda f(H) \chi_\Lambda ),
$$
whenever it exists. It then follows that if $H_1$ and $H_2$ are two
self-adjoint Schr{\"o}dinger operators so that $V = H_2 - H_1$ satisfies
the relative trace-class condition,
$V (H_1 + i )^{-m} \in {\cal I}_1$, for some integer $m > 0$ sufficiently large,
then the IDS $N_i ( \lambda )$, for $H_i$, $i=1,2$ are equal: $N_1 ( \lambda)
= N_2 ( \lambda )$. 
One consequence of this observation is the following. 
Let $H_0$ be a periodic operator and let $V_\omega$
be an Anderson-type random potential, as in (1.1), so that (H1), (H2), and (1.9)
are satisfied. Then, by Theorem 1.2, the IDS for $H_\omega (\lambda )$, 
as in (1.2), is H{\"o}lder continuous on $\R$. Now let us 
consider a perturbation of $H_\omega (\lambda)$ by another
deterministic potential $V_1$ satisfying the relative trace-class 
condition given above. We then have that the IDS for $H_\omega (\lambda)
+ V_1$ equals the IDS for $H_\omega ( \lambda)$.
This construction provides a large family of random operators with
H{\"o}lder continuous IDS. 

This construction raises the question of the existence of operators $H_0$
that are not periodic operators, nor perturbations of periodic operators
by relatively trace-class perturbations, so that Theorem 1.2 cannot be
applied, and for which the assumption
(1.7) of Theorem 1.1 is satisfied. We offer two classes of examples.
The first concern Anderson-type perturbations of one-dimensional random
operators for which the H{\"o}lder continuity or better of the IDS is known
by other methods. Many of these models are described in detail in the book
of Pastur and Figotin \cite{[PF]}.  
Of perhaps more interest are certain background operators $H_0$
with magnetic fields. As an explicit
example, let $H_0 = (-i \nabla - A)^2$, with vector potential $A = (B / 2)
( - x_2 , x_1 , 0)$ be the Hamiltonian describing a particle in $\R^3$
in the presence of a constant magnetic field in the $x_3$-direction with
strength $B > 0$. This Hamiltonian can be as written
$H_0 = H_{(1,2)} \otimes I_3 + I_{(1,2)} \otimes H_3$, 
where $H_{(1,2)}$
is the two-dimensional Landau Hamiltonian (see (1.10)), 
and $H_3 = - \partial^2 / \partial_3^2$ is the free Laplacian in one-dimension.
The IDS for $H_0$ is the convolution $N_0 ( \lambda )
= \int N_3 ( \lambda - \mu ) d N_{(1,2)} ( \mu ) $. Each IDS is known 
explicitly: $N_3 ( \lambda ) = C_1 ( \lambda )_+^{1/2}$, and
$dN_{(1,2)} ( \lambda ) = 
\sum_{j \geq 0} \delta ( \lambda - (2j+1)B) d \lambda$.
As simple computation of $N_0 ( \lambda )$ shows that it is 
H{\"o}lder continuous for any exponent $1/2 < q \leq 1$.
Note that $H_0$ is not a relatively trace-class perturbation of
the free Laplacian on $\R^3$.
Consequently, Theorem 1.1 can be applied to show that $H_0 + V_\omega$
has a H{\"o}lder continuous IDS with no flux condition on $B$ (see the
discussion below and section 5).
In particular, if $B$ does not satisfy a rational flux condition, then
Theorem 1.2 cannot be applied.

Our third main result concerns the IDS for random Anderson-type
perturbations of Landau Hamiltonians. The unperturbed operator $H_L$
on $L^2 ( \R^2 )$ has the form
\begin{equation}
H_L = ( - i \nabla - A)^2, ~~\mbox{where} ~~ A(x_1 , x_2 ) = \frac{B}{2}
( - x_2 ,  x_1 ),
\end{equation}
where $B > 0$ is the magnetic field strength.  The spectrum is pure
point and consists of an increasing sequence of degenerate, isolated
eigenvalues $\{ E_j (B) = (2j + 1)B \; | \; j = 0 , 1, \ldots , \}$ of
infinite multiplicity.  The IDS for this model is a piecewise
constant, monotone increasing function (cf.\ the example in
\cite{[Nakamura]}).  Consequently, it does not satisfy condition
(1.7).  The perturbed family of operators is
\begin{equation}
H_\omega = H_L + \lambda V_\omega ,
\end{equation}
where $V_\omega$ is the Anderson-type random perturbation given in
(1.1).  It is known that $N(E)$ is locally Lipschitz continuous in the
following sense. Given an $N > 0$, there is a $B_N >0$ so that for $B
> B_N$, the IDS $N(E)$ is Lipschitz continuous on $(0 , 2(N+1)B )
\backslash \{ E_j (B) \; | \; j = 0 , 1, \ldots , N \}$
\cite{[CH2],[Wang1]}.  Under some additional conditions, Wang
\cite{[Wang2]} also proved that $N(E)$ is smooth outside of a given
Landau level for sufficiently large magnetic field strength.  There
has been some discussion as to the behavior of the IDS at the Landau
energies $E_j (B)$.  If the single-site potential $u$ in (1.1)
satisfies $u | \Lambda_1 (0) > \epsilon \chi_{\Lambda_1 (0)} > 0$, for
some $\epsilon > 0$, then the IDS is locally Lipschitz continuous at
all energies.  We prove the following general theorem that greatly
extends these known results.

\vspace{.1in}

\noindent
{\bf Theorem 1.5.} {\it Let $H_L$ be the Landau Hamiltonian (1.11)
  with magnetic field $B \neq 0$. In addition to hypotheses (H1) and
  (H2), suppose that the magnetic field strength satisfies the {\it
    rational flux condition} for the unit square $\Lambda_1 (0)$,
\begin{equation}
  \label{eq:1}
B \in 2 \pi Q .
\end{equation}
Then, the IDS for the family $H_\omega = H_L + \lambda V_\omega$, for
$\lambda \neq 0$, is H{\"o}lder continuous at all energies with any H{\"o}lder
exponent $0 < q < 1$.  }

\vspace{.1in}
We do not believe that condition~\eqref{eq:1} is actually necessary;
it is an artifact of our method of proof.

\vspace{.1in}
An immediate consequence of the proof of this theorem is the following
extension.

\vspace{.1in}

\noindent
{\bf Corollary 1.6.} {\it Let $V_0 \in L^2_{loc} ( \R^2)$ 
be a real-valued, $\Z^2$-periodic
function. Then, under the hypotheses of Theorem 1.5,
the IDS for $H_\omega = H_L + V_0 + \lambda V_\omega$
is H{\"o}lder continuous at all energies with any H{\"o}lder exponent $0 < q <
1$.
}

\vspace{.1in}
 
There are very few general results on the IDS for continuous random
\Schr\ operators in higher-dimensions for all energies. One of the
interesting result concerns the case when $V_\omega$ is a Gaussian
random processes. Fischer, Hupfer, Leschke, M{\"u}ller \cite{[FHLM]} 
proved that the IDS is Lipschitz continuous at all
energies for potentials given by certain Gaussian processes. We remark
that this is very similar in spirit to the case $u | \Lambda_1 (0) >
\epsilon \chi_{\Lambda_1 (0) } > 0$. Extensions of this work to magnetic
\Schr\ operators with certain unbounded random potentials is given in
\cite{[HLMW]}. Tip \cite{[Tip]}
proved the absolute continuity of the IDS for the quantum Lorentz gas
for certain repulsive single-site potentials.

The global regularity properties of the IDS for lattice models are
much better understood, and we refer the reader to the monographs by
Carmona and Lacroix \cite{[CL]}, and by Pastur and Figotin
\cite{[PF]}.

The contents of this paper are as follows.  We prove Theorem 1.1 in
section 2. In section 3, we prove Theorem 1.2 under the assumption
(1.9). We study assumption (1.9) for periodic operators $H_0$ in
section 4. In section 5, we give the proof of Theorem 1.5 for random
Landau Hamiltonians. Finally, in section 6, we give a proof of a
technical estimate on the trace of the spectral projectors.

\vspace{.1in}

\noindent
{\bf Acknowledgement.} We thank the organizers, Volker Enss and
Christian G{\'e}rard, of the workshop on \Schr\ operators in May 2001 at
Oberwolfach, where this work was begun.
 
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\section{Proof of Theorem 1.1.}

We prove Theorem 1.1, the H{\"o}lder continuity of the IDS at all energies
provided $q > 0$ is small enough, under the main assumption (1.7) on
the existence and H{\"o}lder continuity of the IDS for the unperturbed
operator $H_0$, and under the assumptions (H1)--(H2) on random potential.

Recall that $\Lambda_1 (0)$ denotes the unit cube centered at the origin,
and that $\Lambda_1 (j)$ denotes its translate to the point $j \in \R^d$. Let
$\Lambda = \cup_j \Lambda_1 (j)$ be a cube in $\R^d$.  We denote by
${\tilde \Lambda} = \Lambda \cap \Z^d$, the lattice points in
$\Lambda$. By the operators $H_0^\Lambda$ and $H_\omega^\Lambda$, we
mean the operators $H_0$ and $H_\omega$ restricted to $\Lambda$ with
self-adjoint boundary conditions.  For a region $\Lambda \subset
\R^d$, let ${\tilde V}_\Lambda \equiv \sum_{j \in {\tilde \Lambda}}
u_j \chi_\Lambda$.  As this quantity is independent of the random
variables, it is bounded by a constant independent of the disorder.

Finally, we recall that hypotheses (H1)--(H2), and the normalization
discussed in section 1, imply that the potential satisfies condition
(1.5),
\begin{equation}
0 \; \leq \; V_\Lambda^2 \; \leq \; C_1 ( u , d) {\tilde V}_\Lambda .
\end{equation}
To prove continuity of the IDS near $E_0 \in \R$, we use (1.7). We fix
$\tilde\Delta = [ E_- , E_+ ]$, 
a nonempty, closed interval. The estimate (1.7) and
the definition of the density of states imply that for $\Lambda$
sufficiently large (depending on $\tilde\Delta$)
\begin{equation}
  Tr E_0^\Lambda({\tilde\Delta})\; \leq
  \; C_2 ( E_+ , d)  |\tilde\Delta|^{q_1} |\Lambda| ,  
\end{equation}
where the finite constant $C_2 ( E_+ , d) > 0$
depends on the constant $C_0 > 0$ in (1.7), the dimension $d$, and is
${\cal O} (E^{d/2})$ as $E \rightarrow \infty$.
To see this, let $N_{0,\Lambda}(E)$ be the number of eigenvalues of
$H_0^\Lambda$ less than $E$. Then, by the
definition of the density of states, one has
%
\begin{gather*}
 N_0(E_+)(1-|E_+-E_-|^{q_1})\leq
 \frac{N_{0,\Lambda}(E_+)}{|\Lambda|}\leq
 N_0(E_+)(1+|E_+-E_-|^{q_1}),\\
 N_0(E_-)(1-|E_+-E_-|^{q_1})\leq
 \frac{N_{0,\Lambda}(E_-)}{|\Lambda|}\leq
 N_0(E_-)(1+|E_+-E_-|^{q_1}).
\end{gather*}
%
Using (1.7), for $\Lambda$ sufficiently large, we get
%
\begin{equation*}
  \begin{split}
  Tr E_0^\Lambda({\tilde\Delta})&\leq (N_{0,\Lambda}(E_+)-
  N_{0,\Lambda}(E_-))\\&\leq |\Lambda|[N_0(E_+)(1+|E_+-E_-|^{q_1})-
  N_0(E_-)(1-|E_+-E_-|^{q_1})]\\&\leq \{C_0 + 2 N_0 (E_+) \} 
  |\Lambda||E_+-E_-|^{q_1}\\
  & \leq C_2 (E_+ , d) |\Lambda||\tilde\Delta|^{q_1}.    
  \end{split}
\end{equation*}
%
We used the monotonicity of the IDS and
the bound $N_0 ( E ) \leq C(E)$, where the finite constant $C(E)>0$
satisfies $\lim_{E \rightarrow \infty} C(E) E^{-d/2} = C_d$, and $C_d >0$
is a finite constant depending only on the dimension $d$.

\vspace{.1in}
\noindent
{\bf Proof of Theorem 1.1.}  \\
\noindent
1. Consider an interval $\Delta \equiv [ \Delta_- , \Delta_+ ] \subset
\R$, with $| \Delta | < 1$.  Let $\alpha \in \R$ be an index
satisfying $0 < \alpha < 1$. Let ${\tilde \Delta}$ be the interval
${\tilde \Delta } = [ \Delta_- - | \Delta |^\alpha , \Delta_+ + |
\Delta|^\alpha ]$, so that $\Delta \subset {\tilde \Delta}$ and $d_0 =
\mbox{dist} \: ( \Delta , {\tilde \Delta}^c ) = | \Delta |^\alpha $.
We decompose the spectral projector $E_\Lambda ( \Delta )$ relative to
$H_0^\Lambda$ as
\begin{equation}
Tr E_\Lambda ( \Delta ) = Tr E_\Lambda ( \Delta ) E_0^\Lambda (
{\tilde \Delta } ) + Tr E_\Lambda ( \Delta ) E_0^\Lambda (
{\tilde \Delta }^c ) .
\end{equation}
We estimate the first term on the right in (2.3) using the assumption
(2.2) on $H_0^\Lambda$ and the positivity of the projectors. For
$\Lambda$ sufficiently large (depending on ${\tilde \Delta}$, this gives
\begin{equation}
Tr E_\Lambda ( \Delta ) E_0^\Lambda ( {\tilde \Delta } ) \; \leq
\; C_2 | \Delta  |^{\alpha q_1} | \Lambda | ,
\end{equation}
where we write $C_2$ for the constant $C_2 ( E_+ , d)$ appearing in (2.2).

\noindent
2. To deal with the second term in (2.3), we pick an energy
$E\in\Delta$ and write
\begin{eqnarray}
   Tr E_\Lambda (
\Delta ) E_0^\Lambda ({\tilde \Delta }^c ) & = & Tr E_\Lambda ( \Delta
) ( H_\omega^\Lambda - E ) E_0^\Lambda ({\tilde
  \Delta }^c ) ( H_0^\Lambda - E )^{-1} \nonumber \\
& & - \lambda Tr E_\Lambda ( \Delta ) V_\Lambda E_0^\Lambda ({\tilde
  \Delta }^c ) ( H_0^\Lambda - E )^{-1} \nonumber \\
& = & (i) + (ii) .
\end{eqnarray}
 For the first term $(i)$, we use the estimate
$\| E_0^\Lambda ( {\tilde \Delta }^c ) ( H_0^\Lambda - E )^{-1} \|
\leq | \Delta |^{-\alpha} $ and easily obtain:
\begin{equation}
| (i) | \; \leq  \; | \Delta |^{1- \alpha } \; Tr
E_\Lambda ( \Delta ) .
\end{equation}
As long as $\alpha < 1$, this term can be absorbed into the left side
of (2.3). For the term $(ii)$ in (2.5), we insert another factor of
$(H_0^\Lambda - E )(H_0^\Lambda - E )^{-1}$, and obtain, \bea (ii)
& = & - \lambda Tr (H_\omega^\Lambda - E ) E_\Lambda ( \Delta )
V_\Lambda E_0^\Lambda({\tilde \Delta }^c ) ( H_0^\Lambda - E )^{-2}
\nonumber \\
& & + \lambda^2 Tr V_\Lambda E_\Lambda ( \Delta ) V_\Lambda
E_0^\Lambda({\tilde \Delta }^c ) ( H_0^\Lambda - E )^{-2}
\nonumber \\
& = & (iii) + (iv) .  \eea The term $(iii)$ is estimated just as term
$(i)$ in (2.6), and we obtain
\begin{equation}
| (iii) | \; \leq \;  \lambda | \Delta |^{1 - 2 \alpha } \; \|
{\tilde V}_\Lambda \| \; Tr E_\Lambda ( \Delta ) .
\end{equation}
For the term $(iv)$ in (2.7), we obtain
\begin{equation}
| ( iv) | \; \leq \: \lambda^2 | \Delta |^{- 2 \alpha } \; Tr V_\Lambda
E_\Lambda ( \Delta ) V_\Lambda .
\end{equation}
This last term is estimated using the condition (1.5) satisfied by
$V_\Lambda$,
\begin{equation}
Tr V_\Lambda E_\Lambda ( \Delta ) V_\Lambda \; \leq \; C_1 (u , d) Tr
{\tilde V}_\Lambda E_\Lambda ( \Delta ).
\end{equation}
We now take the expectation.  From the appendix, we have
\begin{equation}
\E \{ Tr ( \lambda {\tilde V_\Lambda } E_\Lambda ( \Delta ) )\}  \; \leq
\; C_3 ( q_2, u ) | \Delta |^{q_2} | \Lambda | ,
\end{equation}
for any $0 < q_2 < 1$, so that
\begin{equation}
\E \{ | ( iv) | \} \; \leq \; \lambda
| \Delta |^{q_2 - 2 \alpha } C_1 (u , d) C_3
( q_2, u ) | \Lambda | .
\end{equation}

\noindent
3. Gathering the estimates from (2.4), (2.6), (2.8), and (2.12), we
get \bea \lefteqn{ \{ 1 - | \Delta |^{1 - \alpha } - \lambda | \Delta
  |^{1 - 2 \alpha }
  \| {\tilde V}_\Lambda \|  \} \; \E \{ Tr E_\Lambda ( \Delta ) \} } \nonumber \\
& \leq & ( \lambda | \Delta |^{q_2 - 2 \alpha } C_1 (u , d) C_3 ( q_2,
u ) + C_2 | \Delta |^{\alpha q_1} ) | \Lambda | .  \eea It is clear
that if $\alpha < 1/2$ and $| \Delta |$ is taken sufficiently small,
relative to the disorder $\lambda > 0$ (recall that $\| {\tilde
  V}_\Lambda \| = {\cal O}(1)$), then the prefactor on the left side
of (2.13) is invertible.  Finally, we require that $ \alpha < q_2 / (
q_1 + 2) < 1/2$ in order that the right side of (2.13) be bounded
above by $| \Delta |^q$, for $0 < q < q_1 q_2 / ( q_1 + 2 )$. As $q_2
< 1$ is arbitrary, the bound
\begin{equation*}
   \frac{1}{|\Lambda|}\E\{Tr E_\Lambda(\Delta)\}\leq C|\Delta|^q
\end{equation*}
holds for any $0<q<q_1/(q_1+2)$, where $0<q_1\leq1$ is the H{\"o}lder
exponent of continuity for $N_0$, and for $\Lambda$ sufficiently
large. Taking the limit $|\Lambda|\to+\infty$, we complete the proof
of Theorem 1.1. $\Box$

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\section{Proof of Theorem 1.2.}

We now turn to the proof of Theorem 1.2, improving the H{\"o}lder exponent
given in Theorem 1.1.  Let us recall that $\lambda > 0$ is a measure
of the disorder, so that $\| V_\Lambda \| \; \leq \; \| {\tilde
  V}_\Lambda \| $, which is bounded by a constant depending only on
$u$ and independent of the volume $|\Lambda|$.  The main assumption
(1.9) implies that for PBC ($\theta = 0$), there exists a constant $C
({\tilde \Delta}, u ) > 0$, independent of $\Lambda$, so that
\begin{equation}
E_0^\Lambda ({\tilde \Delta }) {\tilde V}_\Lambda 
E_0^\Lambda ({\tilde \Delta })
\; \geq C ({\tilde \Delta}, u ) E_0^\Lambda ({\tilde
\Delta }) .
\end{equation}
Although the background operator $H_0$ is required to be periodic, no
explicit information is needed on the IDS for the operator $H_0$, as
was the case in the Theorem 1.1.  For a fixed, but arbitrary, $E_0 \in
\R$, let $E_0 \in \Delta \subset {\tilde \Delta}$ be 
two closed, bounded intervals
centered on $E_0$, and let $d_0 \equiv ~\mbox{dist} ~( E_0 , {\tilde
  \Delta}^c )$.

\vspace{.1in}

\noindent
{\bf Proof of Theorem 1.2.}  \\
\noindent
1. As in the proof of Theorem 1.1, we need to estimate
\begin{equation}
\E \{ Tr E_\Lambda ( \Delta) \} .
\end{equation}
We begin with a decomposition relative to the spectral projectors
$E_0^\Lambda ( \cdot )$ for the operator $H_0^\Lambda$. We write
\begin{equation}
Tr E_\Lambda ( \Delta) = Tr E_\Lambda ( \Delta ) E_0^\Lambda 
( {\tilde \Delta } ) + Tr E_\Lambda ( \Delta ) E_0^\Lambda 
( {\tilde \Delta }^c ).
\end{equation}
We will choose the intervals $\Delta$ and ${\tilde \Delta}$ below.

\noindent
2. The term involving ${\tilde \Delta }^c$ is easily estimated as
follows. We write \bea Tr E_\Lambda ( \Delta ) E_0^\Lambda ( {\tilde
  \Delta }^c ) & = & Tr E_\Lambda ( \Delta ) (H_\omega^\Lambda - E_0 )
( H_0^\Lambda - E_0 )^{-1}
E_0^\Lambda ( {\tilde \Delta }^c ) \nonumber \\
& & - \lambda Tr E_\Lambda ( \Delta ) V_\Lambda ( H_0^\Lambda - E_0
)^{-1} E_0^\Lambda ({\tilde \Delta }^c ).  \eea 
>From the positivity of
the projector $E_\Lambda ( \Delta )$, we obtain for the first term on
the right of (3.4),
\begin{equation}
| Tr E_\Lambda ( \Delta ) (H_\omega^\Lambda - E_0 ) ( H_0^\Lambda -
E_0 )^{-1} E_0^\Lambda ( {\tilde \Delta }^c )| \; \leq \;
\frac{ | \Delta | }{ 2 d_0 } \; Tr E_\Lambda ( \Delta ) .
\end{equation}
As for the second term of (3.4), we expand the trace in terms of the
finitely-many eigenfunctions $\psi_j$ in the range of the projection
$E_\Lambda ( \Delta )$, 
\bea 
\lefteqn{ Tr E_\Lambda ( \Delta ) (
  \lambda V_\Lambda ) ( H_0^\Lambda - E_0 )^{-1} E_0^\Lambda ({\tilde
    \Delta }^c )
}  \nonumber  \\
& = & \sum_j \; \langle \psi_j , ( \lambda V_\Lambda ) ( H_0^\Lambda -
E_0 )^{-1}
E_0^\Lambda ({\tilde \Delta }^c ) \psi_j \rangle \nonumber \\
& \leq & \sum_j \; \frac{1}{d_0} \; \| E_0^\Lambda ({\tilde \Delta
}^c) \psi_j \| \cdot 
\| ( \lambda V_\Lambda ) \psi_j \| \nonumber \\
& \leq & \frac{1}{2d_0} Tr E_\Lambda ( \Delta ) E_0^\Lambda ( {\tilde
  \Delta }^c ) + \frac{1}{2d_0} Tr E_\Lambda ( \Delta ) ( \lambda^2
V_\Lambda^2 ) .  
\eea 
In order to move the first term on the right in (3.6) to the left side
of (3.4), we must choose ${\tilde \Delta}$ so that
\begin{equation}
1/2 d_0 < 1 /2 , ~~\mbox{or} ~~ d_0 > 1.
\end{equation}
This being done, we find
\begin{equation}
Tr E_\Lambda ( \Delta ) E_0^\Lambda ( {\tilde \Delta }^c ) \leq
\frac{| \Delta |}{d_0} Tr E_\Lambda ( \Delta ) + \frac{1}{d_0}
Tr E_\Lambda ( \Delta ) (\lambda^2 V_\Lambda^2 ) .
\end{equation}
 
\noindent
3. As for the first term on the right in (3.3), we use the fundamental
assumption (3.1). We have 
\bea 
Tr E_\Lambda ( \Delta ) E_0^\Lambda (
{\tilde \Delta } ) & \leq & \frac{1}{C ({\tilde \Delta} , u)}
\left\{ Tr E_\Lambda ( \Delta ) E_0^\Lambda ( {\tilde \Delta } )
  {\tilde V}_\Lambda E_0^\Lambda ({\tilde \Delta} ) E_\Lambda ( \Delta
  )
\right\}  \nonumber \\
& \leq & \frac{1}{C ({\tilde \Delta} , u)} \left\{ Tr E_\Lambda (
  \Delta ) E_0^\Lambda ( {\tilde \Delta }) {\tilde V}_\Lambda
\right. \nonumber \\
& & \left. - Tr E_\Lambda ( \Delta ) E_0^\Lambda ( {\tilde \Delta })
  {\tilde V}_\Lambda E_0^\Lambda ( {\tilde \Delta }^c ) \right\}
\nonumber \\
& \leq & \frac{1}{C ({\tilde \Delta} , u)} \left\{ Tr E_\Lambda (
  \Delta ) {\tilde V}_\Lambda - Tr E_\Lambda ( \Delta ) E_0^\Lambda (
  {\tilde \Delta }^c ) {\tilde V}_\Lambda \right.
\nonumber \\
& & \left. - Tr E_\Lambda ( \Delta ) E_0^\Lambda ( {\tilde \Delta } )
  {\tilde V}_\Lambda E_0^\Lambda ( {\tilde \Delta }^c )
\right\} \nonumber \\
& \leq & \frac{1}{C ({\tilde \Delta} , u)} \left\{ Tr E_\Lambda (
  \Delta ) {\tilde V}_\Lambda - Tr E_\Lambda ( \Delta ) E_0^\Lambda (
  {\tilde \Delta }^c ) {\tilde V}_\Lambda E_0^\Lambda ( {\tilde \Delta
  } ) \right.
\nonumber \\
& & \left. - Tr E_\Lambda ( \Delta ) E_0^\Lambda ( {\tilde \Delta } )
  {\tilde V}_\Lambda E_0^\Lambda ( {\tilde \Delta }^c ) \right\} ,
\eea where we dropped the positive term $Tr E_\Lambda ( \Delta )
E_0^\Lambda ( {\tilde \Delta }^c ) {\tilde V}_\Lambda E_0^\Lambda (
{\tilde \Delta }^c ) E_\Lambda ( \Delta ) $ since it occurs with a negative
sign. We estimate the second and the third
terms on the right in (3.9).
As they are similar, we explicitly
estimate the second term. Let $\| A \|_{HS}$ denote the
Hilbert-Schmidt norm of an operator $A$.  Using the H{\"o}lder inequality
for trace norms, we have, for any $\mu_0 > 0$, 
\bea \lefteqn{ | Tr
  E_\Lambda ( \Delta) E_0^\Lambda ( {\tilde \Delta }^c ) {\tilde
    V}_\Lambda E_0^\Lambda ( {\tilde \Delta } ) | } \nonumber \\
& \leq & \; \| E_\Lambda ( \Delta) E_0^\Lambda ( {\tilde \Delta }^c )
\|_{HS} \; \| {\tilde V}_\Lambda E_0^\Lambda ( {\tilde
  \Delta } ) E_\Lambda ( \Delta) \|_{HS} \nonumber \\
& \leq & \frac{1}{2 \mu_0 } Tr E_0^\Lambda ( {\tilde\Delta }^c ) E_\Lambda
( \Delta) + \frac{ \mu_0 }{2} Tr E_\Lambda ( \Delta) E_0^\Lambda ( {\tilde
  \Delta } ) {\tilde V}_\Lambda^2 E_0^\Lambda
( {\tilde \Delta } )E_\Lambda ( \Delta) . \nonumber \\
& & 
\eea 
We next estimate the second term on the right in (3.10).
There exists a constant $D_0$, depending
only on $u \geq 0$, so that
${\tilde V}_\Lambda^2 \; \leq \; D_0 {\tilde V}_\Lambda$.
Using this, we find that for any $\mu_1 > 0$, 
\bea
\lefteqn{Tr E_\Lambda ( \Delta) E_0^\Lambda ( {\tilde \Delta } ) 
 {\tilde V}_\Lambda^2 E_0^\Lambda ( {\tilde \Delta } ) E_\Lambda ( \Delta)} 
  \nonumber \\
  & \leq & D_0 \| E_\Lambda ( \Delta) E_0^\Lambda ( {\tilde
  \Delta } ) {\tilde V}_\Lambda \|_{HS} ~\|E_0^\Lambda ( {\tilde \Delta } )
  E_\Lambda ( \Delta) \|_{HS} \nonumber \\
 & \leq & \frac{D_0 \mu_1}{2} Tr E_\Lambda ( \Delta) E_0^\Lambda ( {\tilde
  \Delta } ) {\tilde V}_\Lambda^2 E_0^\Lambda ( {\tilde \Delta } )
  E_\Lambda ( \Delta) + \frac{D_0}{ 2 \mu_1} Tr E_\Lambda ( \Delta) 
   E_0^\Lambda ( {\tilde \Delta } ) . \nonumber \\
 & &  
\eea
We choose $\mu_1 = 1 / D_0 > 0$ so that $(1 - D_0 \mu_1 / 2 ) = 1/2 $.
Consequently, we obtain
\beq
Tr E_\Lambda ( \Delta) E_0^\Lambda ( {\tilde
  \Delta } ) {\tilde V}_\Lambda^2 E_0^\Lambda ( {\tilde \Delta } )
   E_\Lambda ( \Delta)
  \; \leq \; D_0^2 Tr E_\Lambda
( \Delta) E_0^\Lambda ( {\tilde \Delta } ) .
\eeq
Inserting this into (3.10), we find
\beq
| Tr E_\Lambda ( \Delta) E_0^\Lambda ( {\tilde \Delta }^c ) 
{\tilde V}_\Lambda E_0^\Lambda ( {\tilde \Delta } ) |
 \; \leq \; \frac{1}{2 \mu_0}
Tr E_0^\Lambda ( {\tilde\Delta }^c ) E_\Lambda
( \Delta) + \frac{ \mu_0 D_0^2}{2}Tr E_\Lambda ( \Delta)
   E_0^\Lambda ( {\tilde \Delta } ) .
\eeq 
A similar calculation for the last term on the right in (3.9) yields
an upper bound identical to the right side of (3.13). 
We substitute (3.13) back into the right side of (3.9)
to obtain the bound, 
\bea 
\lefteqn{ \left\{ 1 - \frac{\mu_0 D_0^2}{C ({\tilde \Delta} , u)} 
 \right\} Tr E_\Lambda ( \Delta ) E_0^\Lambda ( {\tilde \Delta } ) } 
  \nonumber \\
& \leq & \frac{1}{C ({\tilde \Delta} , u)} \left\{  Tr E_\Lambda
  ( \Delta ) {\tilde V}_\Lambda +  \frac{1}{ \mu_0} 
   Tr E_\Lambda ( \Delta)
  E_0^\Lambda ({\tilde \Delta}^c ) \right\} .  
\eea
We take $\mu_0 = C ({\tilde \Delta} , u) / 2 D_0^2$ 
and obtain, 
\beq
Tr E_\Lambda ( \Delta ) E_0^\Lambda ( {\tilde \Delta } )  \leq \frac{2}{C
({\tilde \Delta} , u)} Tr E_\Lambda( \Delta ) {\tilde V}_\Lambda 
+ \frac{4D_0^2}{C ({\tilde \Delta} , u)^2} Tr E_\Lambda ( \Delta)
  E_0^\Lambda ({\tilde \Delta}^c ) .
\eeq

\noindent
4. We now combine the two results (3.8) and (3.15) to obtain 
\bea 
Tr E_\Lambda ( \Delta ) & = & Tr E_\Lambda ( \Delta ) E_0^\Lambda (
{\tilde \Delta } ) + Tr E_\Lambda ( \Delta ) E_0^\Lambda
( {\tilde \Delta }^c ) \nonumber \\
& \leq & \left\{ \frac{| \Delta |}{d_0} + \frac{4 D_0^2 | \Delta|}{d_0 
  C ({\tilde \Delta} , u)^2}  \right\} Tr E_\Lambda ( \Delta) \nonumber \\
& & + \left\{ \frac{2}{C ({\tilde \Delta} , u)} + \frac{ C_1 (u,d)
\lambda^2}{d_0} \left( 1 + \frac{4D_0^2}{C ({\tilde \Delta} , u)^2}
 \right) \right\} Tr E_\Lambda ( \Delta ) {\tilde V}_\Lambda . \nonumber \\
& & 
\eea 
We move the first term on the right in (3.16) to the left by
choosing $d_0$ and $| \Delta |$ as follows. For any $E_0 \in \R$, we
choose $\Delta$ to be an interval centered on $E_0$ with $| \Delta | <
1 $. We choose ${\tilde \Delta }$ so that $d_0 = \mbox{dist} ( E_0 ,
{\tilde \Delta}^c ) >> \mbox{max} ( 4 , 16 D_0^2 )$, 
with $D_0$ defined in (3.11) (this implies, of course, condition
(3.7)).  Then, we have the upper bound on the coefficient
\begin{equation}
\left( \frac{| \Delta |}{d_0}  + \frac{4 D_0^2 | \Delta|}{ d_0
  C ( {\tilde \Delta} , u)^2}  \right)
\leq 
\left( \frac{1}{4} + \frac{ | \Delta | }{ 4 C ( {\tilde \Delta}, u)^2}
\right) .
\end{equation}
Keeping ${\tilde \Delta }$ fixed, we shrink $| \Delta |$, if
necessary, so that $| \Delta | / C ( {\tilde \Delta}, u)^2 < 1 $. In
this way, we arrive at the bound
\begin{equation}
Tr E_\Lambda ( \Delta ) \leq  \; 2 \left\{  
 \frac{2}{\lambda C ( {\tilde \Delta}, u)} + 
 \frac{ \lambda C_1(u,d)}{d_0} \left( 1 + \frac{4 D_0^2 }{C (
{\tilde \Delta} , u)^2} \right)  \right\} 
Tr E_\Lambda ( \Delta ) ( \lambda {\tilde V}_\Lambda ) .
\end{equation}

\noindent
5. It remains to estimate the term on the right in (3.18).  For this,
we use the result of the appendix (as in (2.11)), that for any $0 <
q_2 < 1$, there exists a finite constant $C_3 ( E_0, u ,q_2 ) > 0$, so
that
\begin{equation}
\E \{ Tr E_\Lambda ( \Delta ) ( \lambda {\tilde V}_\Lambda ) \} \;
\leq C_3 ( E_0  , u, q_2 ) \; | \Lambda | \; | \Delta |^{q_2} .
\end{equation}
Inserting this into the right side of (3.18), we obtain the upper
bound 
\bea
\lefteqn{ \E \{ Tr E_\Lambda ( \Delta ) \} } \nonumber \\
& \leq & 2 \left\{ \frac{2}{\lambda C ( {\tilde \Delta}, u)} +
 \frac{ \lambda C_1(u,d)}{d_0} \left( 1 + \frac{4 D_0^2 }{C (
{\tilde \Delta} , u)^2 } \right)
       \right\} C_3 ( E_0 , u, q_2 ) \; | \Lambda | 
 \; | \Delta |^{q_2} , \nonumber \\
& & 
\eea 
provided $| \Delta |$ is sufficiently small as specified
above.  This proves the H{\"o}lder continuity of the IDS with any exponent
$0 < q < 1$.  $\Box$

 
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\section{A Lower Bound Estimate for Periodic Operators}  

The goal of this section is the proof of the following theorem
concerning spectral projections of periodic \Schr\ operators related
to condition (1.9).  Let $\Gamma$ be a non-degenerate lattice in
$\R^d$. Let $V_0: \R^d\to\R$ be a real-valued, $\Gamma$-periodic
potential.  We consider the $\Gamma$-periodic Schr{\"o}dinger operator
$H_0= - \Delta +V_0$.  We assume that $V_0 \in L^p ( \R^d / \Gamma )$,
with $p = 2$ for $d \leq 3$, and $ p > d/2$ for $d \geq 4$.  It is
known that this implies that $H_0$ is essentially self-adjoint on
$C_0^\infty ( \R^d)$, and self-adjoint on the domain $H^2 ( \R^d)$.

We recall the Floquet decomposition of $H_0$ described in section 1.
Let $H_0 ( \theta )$, for $\theta \in ( \R^d )^* / \Gamma^*$ be the
fibred operator acting on $L^2 ( \R^d / \Gamma )$, with
$\theta$-quasi-periodic boundary conditions. We also need to consider
$H_0$ as an $n\Gamma$-periodic operator for any $n\in\N$.  Let
$H_0^n(\theta)$ be the operator $H_0$ restricted to the torus
$\R^d/(n\Gamma)$, with $\theta$-quasi-periodic boundary conditions.
For $I\subset\R$, an interval, let ${\mathbf E}^n_0(I,\theta)$ denote
the spectral projection onto the interval $I$ for $H^n_0(\theta)$.  We
remark that in section 2 we took $I = {\tilde \Delta}$.  Finally, for
any $\Gamma$-periodic function $g$, we write $g^{(n)}$ for the same
function understood as an $n \Gamma$-periodic function.

We remark that we can also treat the case of a periodic background
operator $H_0 = ( - i \nabla - A_0 )^2 + V_0$, for $V_0$ as above, and
a real-valued, $\Gamma$-periodic vector potential $A_0: \R^d \to
\R^d$.  As presented in section 6, the Floquet analysis of $H_0$ used
in the proof of the following theorem is also possible in the case of
a constant magnetic field provided the flux associated with the
magnetic field $B = d A$ through a unit cell of $\Gamma$ is $2 \pi$ times
a rational.
 
\vspace{.1in}
\noindent
{\bf Theorem 4.1.} {\it Let $V: \R^d\to\R$ be a bounded, real-valued,
  $\Gamma$-periodic function.  Consider a bounded interval
  $I\subset\R$. Then, if there exists a finite constant $C ( I , V)
  >0$ such that, for all $\theta\in\R^d$, one has
\begin{equation}
{\mathbf E}_0(I,\theta) V {\mathbf E}_0(I,\theta)
\geq C(I , V) {\mathbf E}_0(I,\theta), 
\end{equation}
then, for all $n\geq1$ and all $\theta\in\R^d$, one has, with the same
constant $C(I , V)$,
\begin{equation}
{\mathbf E}^n_0(I,\theta) V^{(n)} {\mathbf E}^n_0(I,\theta)
 \geq C(I, V) {\mathbf E}^n_0(I,\theta),
\end{equation}
where $V$ is considered as a $\Gamma$-periodic function in (4.1), and
$V^{(n)}$ represents the corresponding
$n \Gamma$-periodic function in (4.2). }

\vspace{.1in}
\noindent{\bf Proof.} \\   
1. The spectrum of $H_0^n(\theta)$ is discrete for any $n$ and
$\theta$. Moreover, the spectral decomposition of $H_0^n(\theta)$ is
obtained from the one of $H_0^1(\theta)$ in the following way. Let
$E_1(\theta)\leq E_2(\theta)\leq \dots\leq E_j(\theta)\leq\dots$ be
the sequence of Floquet eigenvalues of $H_0^1(\theta)$ ordered
increasingly. Let $\{ x\in\R^d/\Gamma\mapsto\varphi_j(x,\theta)
\}_{j\geq1}$ be the associated normalized eigenfunctions. One has
\begin{equation}
  H_0 ( \theta) \varphi_j ( \cdot, \theta) =
  E_j (\theta) \varphi_j (\cdot , \theta) ,
\end{equation}
with the eigenfunctions $\varphi_j$ satisfying the
$\theta$-quasi-periodic boundary conditions,
\begin{equation}
\varphi_j(x+\gamma,\theta)=
  e^{i\gamma\theta}\varphi_j(x,\theta), ~~~\mbox{for all} ~\gamma
\in \Gamma.
\end{equation}
Then, the eigenvalues of $H_0^n(\theta)$ are given by the sequence
\begin{equation}
 \{ E_{j,\gamma^*} ( \theta ) \}_{j\geq1,\ \gamma^*\in\Gamma^*/(n\Gamma^*)}
= \{ E_j(\theta+\gamma^*/n) \}_{j\geq1,\ \gamma^*\in\Gamma^*/(n\Gamma^*)} 
\end{equation}
where $\Gamma^*=\{\gamma^*\in\R^d;\ \forall\gamma\in\Gamma,\ \gamma
\cdot \gamma^*\in2\pi\Z\}$. The corresponding eigenfunctions are given by
$\{ x\in\R^d/\Gamma\mapsto\varphi_j(x,\theta+\gamma^*/n) \}_{j\geq1}$.
To normalize these functions, we divide them by $n^{d/2}$ and call
them $\{ \varphi_{j,\gamma^*} \}_{j\geq1,\ \gamma^*\in
  \Gamma^*/(n\Gamma^*)}$.

\noindent
2. By definition, one has
\begin{equation}
  \mathbf{E}^n_0(I,\theta) = \sum_{ \stackrel{j\geq1, \gamma^*\in
  \Gamma^*/(n\Gamma^*) }{s.\ t.\ E_{j,\gamma^*} \in I}} \;
   | \varphi_{j,\gamma^*} \rangle \langle \varphi_{j,\gamma^*}|.
\end{equation}
This gives
\begin{equation}
 \mathbf{E}^n_0(I,\theta)V^{(n)} \mathbf{E}^n_0(I,\theta)=
  \sum_{\stackrel{ \stackrel{ j\geq1,\ 
        \gamma^*\in \Gamma^*/(n\Gamma^*) }{ s.\ t.\
          E_{j,\gamma^*}\in I}}{ \stackrel{ \tilde\jmath\geq1,\ 
        \tilde\gamma^*\in \Gamma^*/(n\Gamma^*) }{ s.\ t.\
          E_{\tilde\jmath,\tilde\gamma^*}\in I}}} \; 
    \langle\varphi_{j,\gamma^*},V^{(n)} \varphi_{\tilde
        \jmath,\tilde\gamma^*}\rangle_n \;  
    |\varphi_{j,\gamma^*}\rangle \langle\varphi_{\tilde
        \jmath,\tilde\gamma^*}|.
\end{equation}
%
Here the scalar product $\langle,\rangle_n$ is the usual scalar
product in the $x$-variable over $\R^d/(n\Gamma)$.\\

\noindent
3. Using the periodicity of $V$ and the definition of the
eigenfunctions, we compute \bea \langle\varphi_{j,\gamma^*},V^{(n)}
\varphi_{\tilde\jmath,\tilde\gamma^*} \rangle_n &=&
\int_{\R^d/(n\Gamma)} V^{(n)}
(x)\varphi_{j,\gamma^*}(x)\overline{\varphi_{\tilde
    \jmath,\tilde\gamma^*}(x)}dx \nonumber \\
&=&\sum_{\Gamma/(n\Gamma)}\int_{\R^d/\Gamma}
V(x+\gamma)\varphi_{j,\gamma^*}(x+\gamma)\overline{\varphi_{\tilde
    \jmath,\tilde\gamma^*}(x+\gamma)}dx \nonumber \\
&=& \left( \sum_{\gamma\in\Gamma/(n\Gamma)}
  e^{i\gamma(\gamma^*-\tilde\gamma^*)} \right)
\int_{\R^d/\Gamma}V(x)\varphi_{j,\gamma^*}(x)\overline{\varphi_{\tilde
    \jmath,\tilde\gamma^*}(x)}dx \nonumber \\
&=& \left\{
    \begin{array}{ll} 
      0 & \mathrm{ if }\gamma^*\not=\tilde\gamma^*,\\
      \int_{\R^d/\Gamma}V(x)\varphi_j(x,\theta+\gamma^*/n)
      \overline{\varphi_{\tilde\jmath}(x,\theta+\gamma^*/n)}dx, &
       \mathrm{otherwise}
    \end{array} \right. \nonumber \\ 
  &= & \delta_{\gamma^*,\tilde\gamma^*} \langle
  V\varphi_j(\theta+\gamma^*/n),
  \varphi_{\tilde\jmath}(\theta+\gamma^*/n)\rangle_1 \eea Hence, (4.6)
  becomes
\begin{equation}
  {\mathbf E}^n_0(I,\theta)V^{(n)}{\mathbf E}^n_0(I,\theta)=
          \sum_{\stackrel{ j\geq1,\
          \tilde\jmath\geq1,\  \gamma^*\in
          \Gamma^*/(n\Gamma^*) }{ \mathrm{s.t. 
          }E_{j,\gamma^*}\in I \mathrm{\ and\ }E_{\tilde
        \jmath,\gamma^*}\in I}}
  \; \langle V\varphi_j(\theta+\gamma^*/n),
      \varphi_{\tilde\jmath}(\theta+\gamma^*/n)\rangle_1
    |\varphi_{j,\gamma^*}\rangle \langle\varphi_{\tilde
        \jmath,\gamma^*}|.
\end{equation}
So we see that if, for some $C(I,V) > 0$, one has
\begin{equation}
  \forall\gamma^*\in\Gamma^*,\quad \left(\left(\langle
      V\varphi_j(\theta+\gamma^*/n),
      \varphi_{\tilde\jmath}(\theta+\gamma^*/n)\rangle_1 
  \right)_{j,\tilde\jmath} \right) \geq C(I, V) \: Id,
\end{equation}
where the inequality is meant in matrix sense, then (4.8) implies
(4.2). On the other hand, equation (4.9) is nothing but condition
(4.1) taken at the point $\theta+\gamma^*/n$ and rewritten in the
basis $(\varphi_j(\theta+\gamma^*/n))_j$. This completes the proof of
Theorem 4.1. $\Box$

We next show that, in fact, condition (4.1) holds for a wide family of
periodic potentials $V$. We remind the reader that in the
applications, we will take the potential $V$ appearing in (4.1) to be
the single-site potential $u$, restricted to the unit cell $\Lambda_1
(0)$, viewed as a $\Gamma$-periodic function. The $n\Gamma$-periodic
function appearing in (4.2) is ${\tilde V} (x) = \sum_j u(x-j)$,
restricted to $\Lambda^{(n)}$, where $\Lambda^{(n)}$ is the basic $n
\Gamma$-periodic cell.

\vspace{.1in}
\noindent
{\bf Theorem 4.2.} {\it Let $V: \R^d\to\R$ be a bounded,
  $\Gamma$-periodic, nonnegative function. Suppose that $V>0$ on some
  open set and $H_0$ has the unique continuation property. Then,
  condition (4.1) holds for any compact interval $I \subset \R$ with a
  finite constant $C(I , V) > 0$.  }

\vspace{.1in}

In order to prove Theorem 4.2, we need the following lemma.  This
lemma was proved in~\cite{[KW]}.  We note that the proof given there
holds in all dimensions.

\vspace{.1in}
\noindent
{\bf Lemma 4.3. \cite{[KW]} {\it Fix $\theta_0\in\T^*= (\R^d)^* /
    \Gamma^*$, and let $I$ be a compact interval in $\R$. Then, there
    exists $N_{\theta_0}$, an open neighborhood of $\theta_0$ in
    $\T^*$, $J$, a bounded, open interval $I\subset J$ such that
\begin{enumerate}
\item There exists a finite constant $p ( \theta_0 ) \geq 0$ so that
  for any $\theta\in N_{\theta_0}$, the number of Floquet eigenvalues
  $ E_n ( \theta ) \in J$ is a constant equal to $p ( \theta_0 )$;
\item There exists $p ( \theta_0)$ real analytic functions $\theta\in
  N_{\theta_0}\mapsto v_j(\cdot ,\theta)\in {\mathcal H}_\theta$, $j = 1 ,
  \ldots , p( \theta_0 )$, forming an orthonormal system, such that,
  for $\theta\in N_{\theta_0}$, the spectral projector ${\mathbf
    E}_0(J,\theta)$ is the orthogonal projector on the subspace
  spanned by $\{ v_j( \cdot ,\theta) \; | \; 1 \leq j \leq p(\theta_0) \}$.
\end{enumerate}   }
} \vspace{.1in}

\noindent
{\bf Proof of Theorem 4.2.}  We fix $\theta_0\in\T^*$ and let
$W_{\theta_0}$ be an open neighborhood of $\theta_0$ such that
$\overline{W_{\theta_0}}\subset N_{\theta_0}$ (here, $N_{\theta_0}$ is
given by Lemma 4.3.). Let $J$, $I\subset J$, be the open interval
fixed by Lemma 4.3. Notice that, as ${\mathbf E}_0(J,\theta){\mathbf
  E}_0(I,\theta)={\mathbf E}_0(I,\theta)$, the inequality ${\mathbf
  E}_0(J,\theta) V {\mathbf E}_0(J,\theta) \geq C {\mathbf
  E}_0(J,\theta)$ implies ${\mathbf E}_0(I,\theta) V {\mathbf
  E}_0(I,\theta) \geq C {\mathbf E}_0(I,\theta)$, Let us show that
there exists $C_0>0$ such that ${\mathbf E}_0(J,\theta) V {\mathbf
  E}_0(J,\theta) \geq C {\mathbf E}_0(J,\theta)$ holds for all
$\theta\in\overline{W_{\theta_0}}$. Estimate ${\mathbf E}_0(J,\theta)
V {\mathbf E}_0(J,\theta) \geq C {\mathbf E}_0(J,\theta)$ is
equivalent to saying that the matrix
$M(\theta)=((m_{ij}(\theta)))_{1\leq i,j\leq p}$ where
%
\begin{equation}
  m_{ij}(\theta)=\int_{\R^d/\Gamma}V(x)
  v_i(x,\theta)\overline{v_j(x,\theta)}dx 
\end{equation}
%
is positive. The mapping $\theta\mapsto M(\theta)$ being continuous
and $\overline{W_{\theta_0}}$ being compact, the point-wise positivity
of $M$ implies its uniform positivity in $\overline{W_{\theta_0}}$.\\
%
Assume there exists $a=(a_j)_{1\leq j\leq p}$, such that one has
%
\begin{equation}
  \int_{\R^d/\Gamma}\left|\sum_{j=1}^p a_j v_j(x,\theta)\right|^2
  V(x)dx=0.
\end{equation}
%
As $V>0$ on some open set $O\subset \R^d/\Gamma$, one has
%
\begin{equation}
  \label{eq:7}
  \sum_{j=1}^p a_j v_j(x,\theta)=0
\end{equation}
%
By point (2) of Lemma 4.3, there exists an invertible $p\times p$
matrix, say $P$, such that $P(v_j)_j=(u_j)_j$ where $(u_j)_j$ is the
sequence of eigenvectors of $H_0(\theta)$ spanning ${\mathbf
  E}_0(J,\theta)$. So that (4.12) implies that, on $O$, one has
%
\begin{equation}
  \label{eq:8}
  \sum_{j=1}^p (P^{-1}a)_j u_j(x,\theta)=0
\end{equation}
%
To fix ideas, assume $H_0(\theta)u_j(\theta)=E_j(\theta)u_j(\theta)$.
Applying $N$ times the differential operator $H_0(\theta)$ to (4.13),
one obtains that, on $O$, one has
%
\begin{equation}
  \forall N\in\N,\quad\sum_{j=1}^p (P^{-1}a)_j  E_j(\theta)^N
  u_j(x,\theta)=0.
\end{equation}
%
This implies that, for every $E\in\{E_j(\theta);\ 1\leq j\leq p\}$, on
$O$, one has
%
\begin{equation}
  \sum_{j; E_j(\theta)=E} (P^{-1}a)_j u_j(x,\theta)=0
\end{equation}
%
But, by construction, $\sum_{j; E_j(\theta)=E} (P^{-1}a)_j
u_j(x,\theta)$ is an eigenvector of $H_0(\theta)$ associated to the
eigenvalue $E$. Hence, as $H_0$ has the unique continuation property
(see e.g.~\cite{[Wolff]}), we get that $\sum_{j; E_j(\theta)=E}
(P^{-1}a)_j u_j(x,\theta)$ vanishes identically on $\R^d/\Gamma$. The
$(u_j)_j$ being linearly independent, we obtain that for all $1\leq
j\leq p$, $(P^{-1}a)_j=0$. As $P$ is invertible, this implies $a=0$.
So the kernel of $M(\theta)$ is reduced to the vector $0$; as
$M(\theta)$ is obviously nonnegative, it is positive.  This completes
the proof of Lemma 4.2. $\Box$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
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\setcounter{chapter}{5} \setcounter{equation}{0}

\section{The Integrated Density of States for Perturbed Landau Hamiltonians}

The proof of Theorem 1.5 consists in reducing the problem via magnetic
Floquet theory to the situation treated in section 4.  The unperturbed
Landau Hamiltonian $H_L$ has the form $H_L = (p - A)^2$, where $p = -i
\nabla$ and $A = (B/2) ( -x_2 , x_1 )$.  The magnetic translations for
the two-dimensional, constant, transverse magnetic field problem are
defined as follows.  For any field strength $B \in \R$, any vector
$\alpha \in \R^2$, and $f \in C_0^\infty ( \R^2)$, we define the
magnetic translation by $\alpha$ by
\begin{equation}
U_\alpha^B f(x) \equiv e^{ \frac{iB}{2} ( x_1 \alpha_2 - x_2 \alpha_1 )} 
f(x + \alpha ) .
\end{equation}
For $(\alpha , \beta) \in \R^2$, we have the relations:
\begin{equation}
U_\alpha^B U_\beta^B = e^{i B (\alpha_1 \beta_2 - \alpha_2 \beta_1 )}
U_\beta^B U_\alpha^B .
\end{equation}
In a standard way, the family $\{ U_\alpha^B \; | \; \alpha \in \R^2
\}$, defined in (5.1)--(5.2), extends to a projective unitary
representation of $\R^2$ on $L^2 ( \R^2)$.  We note that
\begin{equation}
U_\alpha^B H_L  U_{- \alpha}^B  = H_L .
\end{equation}

We define two fundamental vectors in $\R^2$,
\begin{equation} 
\alpha_1 = ( L_1 , 0 ) , ~~~\mbox{and} ~~\alpha_2 = ( 0, L_2 ) ,\
\end{equation}
for any $L_1 , L_2 \in \R$, and write $U_j^B$, for $j = 1 , 2$, for
the corresponding unitaries as in (5.1).  If follows from (5.2) that
we have the commutation rules:
\begin{equation}
U_1^B U_2^B = e^{ i B L_1 L_2 } U_2^B U_1^B .
\end{equation}
It is clear that $\{ U_1^B, U_2^B \}$ generate a two-parameter Abelian
group provided the following {\it integer flux condition for the
  square} $[0,L_1] \times [0, L_2]$ is satisfied:
\begin{equation}
B L_1 L_2 \in 2 \pi \Z .
\end{equation}
More generally, suppose that only the {\it rational flux condition}
for the square $[0,L_1] \times [0, L_2]$ is satisfied:
\begin{equation}
B L_1 L_2 = 2 \pi p / q, ~~~\mbox{for} ~~p,q \in \Z ,
\end{equation}
with $p,q$ having no common divisors.  Then, the unitary operators $\{
(U_1^B)^q, U_2^B \}$ satisfy the commutation relations
\begin{equation}
(U_1^B)^q U_2^B = e^{ i q B L_1 L_2 } U_2^B (U_1^B)^q ,
\end{equation}
so it is clear that the pair generate an Abelian group.

Let us now suppose

\vspace{.1in}
\noindent
{\bf (H3).} {\it The rational flux condition for a unit square $[0,1]
  \times [0,1]$ holds for the magnetic field strength $B$.}

\vspace{.1in}
\noindent
That is, we suppose (5.6) with $L_1 = L_2 = 1$.  We have $[ (U_1^B)^q
, U_2^B ] = 0$, and $[ (U_1^B)^q , H_L ] = 0 = [ U_2^B , H_L ]$. Let
$\Gamma ' \subset \Gamma = \Z^2$ be the sublattice corresponding to
these translations, that is
\begin{equation}
\Gamma ' = \{ ( n_1 q , n_2 ) \; | \; n_1 , n_2 \in \Z \} .
\end{equation}
We define the dual lattice to be $(\Gamma' )^*$ given by
\begin{equation}
( \Gamma ')^* = \{ ( 2 \pi m_1/ q , 2 \pi m_2 ) \; | \; m_1 , m_2 \in \Z \} .
\end{equation}
If $\gamma^* \in ( \Gamma ')^*$, and $\gamma \in \Gamma'$, then we
have $\gamma^* \cdot \gamma \in 2 \pi \Z$.  For any $\gamma = ( n_1 q , n_2
) \in \Gamma'$, define a phase by
\begin{equation}
\Theta_q ( \gamma ) = e^{i B n_1 n_2 q / 2} \in \{ -1 , +1 \} .
\end{equation}
This allows us to define a unitary representation of the sublattice
$\Gamma '$ by
\begin{equation}
W_{q , \gamma}^B = \Theta_q ( \gamma ) U_\gamma^B .
\end{equation}
It is easy to check that
\begin{equation}
W_{q , \gamma}^B W_{q , \gamma'}^B = W_{q , \gamma + \gamma' }^B , 
~~\gamma , \gamma' \in \Gamma' .
\end{equation}

We define the transformation $T^B$ on smooth functions by
\begin{equation}
( T^B f) ( x , \theta ) = \sum_{\gamma \in \Gamma'} \; 
e^{i \theta \cdot \gamma }  
( W_{q , \gamma}^B f ) (x) , ~~ \theta \in (\R^2)^* / (\Gamma ' )^* .
\end{equation}
Again, a simple calculation shows that
\begin{equation}
( W_{q , \gamma}^B T^B f) ( x , \theta ) = e^{-i \theta \cdot \gamma } ( T^B
f) ( x , \theta ) .
\end{equation}
We define a function space ${\cal H}_{B, \theta}$ by
\begin{equation}
{\cal H}_{B, \theta } = \{ v \in L_{loc}^2 ( \R^2 ) \; | \; 
W_{q , \gamma}^B v = e^{-i \theta \cdot \gamma } v \} .
\end{equation}
It then follows that $T^B$ extends to a unitary map
\begin{equation} T^B : L^2 ( \R^2 ) \rightarrow L^2 (
(\R^2)^* / (\Gamma ' )^* ) , {\cal H}_{B, \theta} ).
\end{equation}

Given this structure, it is clear that the Hamiltonian $H_L$ admits a
direct integral decomposition (see e.g.~\cite{[ReedSimon4]}) over
$(\R^2)^* / (\Gamma ' )^*$, so that
\begin{equation}
H_L = \int^{\oplus}_{(\R^2)^* / (\Gamma ' )^*} \; H_L ( \theta) \; d
\theta.
\end{equation}
The operator $H_L ( \theta)$ is self-adjoint on the Sobolev space
${\cal H}_{B , \theta}^2$, the local Sobolev space of order two of
functions in ${\cal H}_{B , \theta}$. This operator has compact
resolvent. Consequently, the spectrum is discrete and consists of
finite multiplicity eigenvalues $E_j ( B , \theta), j = 1 , 2 ,
\ldots$, labeled in increasing order {\it not} including multiplicity.
Unlike the case of a \Schr\ operator with a periodic potential, since
\begin{equation}
\bigcup_{\theta \in (\R^2)^* / (\Gamma ' )^* } E_j ( B , \theta)
= E_j (B),
\end{equation}
the $j^{th}$-Landau level, the eigenvalues $E_j ( B , \theta)$ are
constants independent of $\theta$.

To prove Theorem 1.5, we follow the proof of Theorem 1.2 given in
section 3. We work with increasing sequences of regions
$\Lambda^{(q)}$ that are integer multiples and $\Z^2$-translates of
the basic cell $\Lambda_0^{(q)} = [0 , q] \times [0 , 1]$. For the
IDS, we consider $H_L$ restricted to $\Lambda^{(q)}$ with periodic
boundary conditions (PBC). We note that functions satisfying PBC form
the reduced Hilbert space ${\cal H}_{B, \theta = 0}$.  Let us consider
the Hamiltonian $H_L ( \theta )$ corresponding to the unit cell
$\Lambda_0^{(q)}$.  For any $I \subset \R$ and $\theta \in (\R^2)^* /
(\Gamma ' )^*$, we define the spectral projector for $H_L (\theta)$
and $I$ by $E_0 ( I , \theta )$. From the above discussion, it follows
that
\begin{equation}  
E_0 ( I ,\theta ) = 0, ~~\mbox{if} ~~I \cap \{ E_j (B) \; | \; \mbox{for any} 
~j = 0 , 1 , \ldots \} = \emptyset ,
\end{equation} 
and \bea E_0 ( I ,\theta ) & = & \Sum_{j=1}^M ~\Sum_{l = 1}^K |
\phi_{j,l} (\theta) \rangle \langle \phi_{j,l} (\theta) |, \\ 
\nonumber & \mbox{if} & E_j ( B) \in I, ~\mbox{for} ~j=1, \ldots, M.
\eea In keeping with the notation of section 4, we let $H_L^n ( \theta
)$ be the family of Hamiltonians on the region $\Lambda_n^{(q)} = [ 0
, nq ] \times [0 , n]$. We denote by $E_0^n ( I ,\theta )$ the
corresponding spectral projectors.

In the Landau context, assumption (3.1) means the following.  Let
${\tilde \Lambda}_n^{(q)} = \Lambda_n^{(q)} \cap \Z^2$ be the lattice
points in $\Lambda_n^{(q)}$. We define the local potential ${\tilde
  V}_{\Lambda_n^{(q)}} = \sum_{j \in \Lambda_n^{(q)}} u_j
\chi_{\Lambda_n^{(q)}}$. For any $\theta \in \Gamma'$, let
$E_L^{\Lambda_n^{(q)}} ( I ) = E_0^n ( I ,\theta )$.  Then, there
exists a finite constant $C ( I , u ) > 0$ so that for all $n \in \N$,
\begin{equation}
E_L^{\Lambda_n^{(q)}} ( I ) {\tilde V}_{\Lambda_n^{(q)}}
E_L^{\Lambda_n^{(q)}} ( I )
\; \geq \; C ( I , u ) E_L^{\Lambda_n^{(q)}} ( I ) .
\end{equation}
With this formulation of (3.1), we can now apply all the results of
section 4 to these operators.

Next, we examine the proof of the lower bound in the Landau context,
as described in section 4. Theorem 4.1 is identical in this case since
it is based on the Floquet decomposition.  The proof of Theorem 4.2
requires addition information on the eigenfunctions as given in Lemma
4.3.  The multiplicity of any eigenvalue $E_j ( B , \theta )$ is
locally constant due to the continuity of the spectral projectors.  We
note that in the case that $I$ contains only one Landau level $E_j
(B)$, which is the case when $B$ is sufficiently large, the actual
proof of the analog of Theorem 4.2 is simpler since the eigenvectors
$\phi_{j,l} ( \theta)$, for a given $j$, all correspond to the same
eigenvalue $E_j (B)$. Hence, the unique continuation argument can be
applied directly to the analog of (4.12).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%6%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Appendix: A Local Trace Estimate}

We prove estimate (2.11) on the expectation of the trace of $\lambda
{\tilde V}_\Lambda E_\Lambda ( \Delta )$. This is based on the
estimate on the spectral shift function proved in \cite{[CHN]}.  The
$L^p$-theory of the spectral shift function for $p \in [1 , \infty ]$
can be viewed as an interpolation between the two well-known cases of
$p = 1$ and $p = \infty$.  The $L^1$ and $L^\infty$-theory can be
found in the review paper of Birman and Yafaev \cite{[BY]}, and the
book of Yafaev \cite{[Yafaev]}.  Suppose that $H_0$ and $H$ are two
self-adjoint operators on a separable Hilbert space $\cal H$ having
the property that $V \equiv H - H_0$ is in the trace class. We denote
by $\| V \|_1$ the trace norm of $V$. Under these conditions, we can
define the Krein spectral shift function (SSF) $\xi ( \lambda; H , H_0
)$ through the perturbation determinant. Let $R_0 ( z ) = ( H_0 -
z)^{-1}$, for $Im \; z \neq 0$. We then have
\begin{equation}
\xi ( \lambda; H , H_0 ) \equiv \frac{1}{\pi} \lim_{ \epsilon
\rightarrow
0^+
} \; \mbox{arg} \; \mbox{det} \; ( 1 + V R_0 ( \lambda + i
\epsilon )).
\end{equation}

We need to consider the case when $H - H_0 \in {\cal I}_{1/p}$, for
the cases $1 < p < \infty$ (cf.\ \cite{[Simon2]}).  Let $A$ be a
compact operator on $\cal H$ and let $\mu_j (A)$ denote the $j^{th}$
singular value of $A$. We say that $A \in {\cal I}_{1/p}$, for some $p
\geq 1$, if
\begin{equation}
\Sum_{j} \; \mu_j (A)^{1/p} < \infty .
\end{equation}
We define a nonnegative functional on the ideal ${\cal I}_{1/p}$ by
\begin{equation}
\| A \|_{1/p} \equiv \left( \Sum_{j} \; \mu_j (A)^{1/p}
\right)^p.
\end{equation}
For $p > 1$, this functional is not a norm but satisfies
\begin{equation}
\| A + B \|_{1/p}^{1/p} \; \leq \| A \|_{1/p}^{1/p} + \| B
\|_{1/p}^{1/p}.
\end{equation}
If we define a metric $\rho_{1/p} ( A , B ) \equiv \| A - B
\|_{1/p}^{1/p}$ on ${\cal I}_{1/p}$, then the linear space ${\cal
  I}_{1/p}$ is a complete, separable linear metric space.  The finite
rank operators are dense in ${\cal I}_{1/p}$ (cf.\ \cite{[BS]}).

Since ${\cal I}_{1/p} \subset {\cal I}_1$, for all $p \geq 1$, we
refer to $A \in {\cal I}_{1/p}$ as being super-trace class.
Consequently, we can define the SSF for a pair of self-adjoint
operators $H_0$ and $H$ for which $V = H - H_0 \in {\cal I}_{1/p}$.
The main theorem on the SSF in \cite{[CHN]} is the following.

\vspace{.1in}

\noindent
{\bf Theorem 6.1. \cite{[CHN]}} {\it Suppose that $H_0$ and $H$ are
  self-adjoint operators so that $V = H - H_0 \in {\cal I}_{1/p}$, for
  some $p \geq 1$. Then, the SSF $\xi ( \lambda; H , H_0 ) \in L^p (
  \R )$, and satisfies the bound
\begin{equation}
\| \xi ( \cdot \; ; H , H_0 ) \|_{L^p } \; \leq \; \| V
\|_{1/p}^{1/p} .
\end{equation}
}

\vspace{.1in}
\noindent
We will use this in the proof of the estimate (2.11).

\vspace{.1in}

\noindent
{\bf Proof of (2.11).} We fix a closed, bounded interval $\Delta = [
\Delta_- , \Delta_+ ] \subset \R$.  Let $f_\Delta \geq 0$ be a
monotone increasing $C^1$-function having so that $f_\Delta ( t ) =
1$, for $ t > \Delta_+ + | \Delta |$, $f_\Delta ( t ) = 0$, for $t <
\Delta_- - | \Delta |$, and such that $f_\Delta ' = {\cal O} ( 1 / |
\Delta |)$. It follows that, for some $C>0$, as operators
\begin{equation}
E_\Lambda ( \Delta ) \; \leq C | \Delta | \; f_\Delta' ( H_\Lambda ) .
\end{equation}
Here, $H_\Lambda$ is $H$ restricted to $\Lambda$ with self-adjoint
boundary conditions. Consequently, as the potential ${\tilde
  V}_\Lambda \geq 0$, we have
\begin{equation}
Tr ( \lambda {\tilde V}_\Lambda E_\Lambda ( \Delta ) ) \; \leq \; C
| \Delta | \; Tr f_\Delta' ( H_\Lambda ) ( \lambda {\tilde V}_\Lambda ) .
\end{equation}
We now take the expectation of (6.8). We use the fact that
\begin{equation}
\sum_{k \in {\tilde \Lambda} } 
\frac{\partial}{\partial \lambda_k } \left\{ Tr f_\Delta (H_\Lambda )
\right\} \; = \; Tr f_\Delta' (H_\Lambda ) (\lambda {\tilde V}_\Lambda ) ,
\end{equation}
so that
\begin{equation}
\E \{ Tr (\lambda {\tilde V}_\Lambda ) E_\Lambda ( \Delta ) \} \leq C
\sum_{k \in{\tilde \Lambda} } ~ | \Delta | \E \left\{ \frac{\partial}{\partial
\lambda_k } \; Tr f_\Delta (H_\Lambda ) \right\} .
\end{equation}
We evaluate the expectation on the right side of (6.9) with respect to
one random variable, say $\lambda_k$, using (H2):
\[ 
\int_0^1 h_0 ( \lambda_k ) ~d \lambda_k \left\{
  \frac{\partial}{\partial \lambda_k } \; Tr f_\Delta (H_\Lambda )
\right\}
\]
\begin{equation}
\leq \; \|h_0\|_\infty  \; Tr \{ f_\Delta (H_\Lambda ) (
\lambda_k = 1) - f_\Delta (H_\Lambda ) ( \lambda_k = 0 ) \} . 
\end{equation}
The trace on the right in (6.10) can be expressed in terms of the
spectral shift function for $H_1 \equiv H_\Lambda ( \lambda_k = 1 )$
and $H_0 \equiv H_\Lambda ( \lambda_k = 0)$ by means of the
Birman-Krein formula,
\begin{equation}
Tr \{ f_\Delta (H_1 ) - f_\Delta (H_0 ) \} \; = \; - \int f_\Delta '
( t ) \xi ( t ; H_1 , H_0 ) ~dt .
\end{equation}
We recall that the SSF for the pair $(H_0 , H_1)$ is defined through
bounded functions of these operators since the difference is not in a
trace ideal. Both $H_0$ and $H_1$ are lower semibounded by, say $- M_0
> - \infty$. We define $g( s ) = ( s + M_0 )^{-k}$, for $s > - M_0$.
It follows as in \cite{[CHN]} that the difference $g(H_1) - g(H_0) \in
{\cal I}_{1/p}$, for any $p > 1$, provided $k > pd/2 + 2$. We have the
identity for the SSF:
\begin{equation}
\xi ( t ; H_1 , H_0 ) = sgn ( g') \xi ( g (s); g(H_1), g(H_0) ).
\end{equation}
We substitute this into the Birman-Krein trace formula and estimate
the integral using H{\"o}lder's inequality as in \cite{[CHN]}. We use the
result in Theorem 6.1 to obtain an upper bound on the $L^p$-norm of
the SSF in terms of $\| g(H_1) - g(H_0) \|_{1/p}^{1/p}$.  This latter
norm is bounded by a constant independent of $| \Lambda |$ and $|
\Delta |$. We find that
\begin{equation}
Tr \{ f_\Delta (H_1 ) - f_\Delta (H_0 ) \} \leq C_p | \Delta|^{p-1},
\end{equation}
for any $0 < p < 1$. Combining this with (6.10), giving an additional
factor of $|\Delta|$, and with (6.11), giving a factor of $| \Lambda
|$, we obtain the result (2.11). $\Box$

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