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density of states, random Landau Hamiltonians, Wegner estimate
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%submitted version to CPDE 05.06.03
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%
\title{Global Continuity of the Integrated Density of States\\
for Random Landau Hamiltonians
 \\
{\tiny {\em Dedicated to Leonid Pastur on the Occasion of his $65^{th}$
Birthday}}}
%
%
\author{Jean Michel Combes}
\address[Jean Michel Combes]{D\'epartement de Math\'ematiques,
 Universit\'e de Toulon et du Var,
 83130 La Garde, France}
\email{combes@cpt.univ-mrs.fr}  
%
\author{Peter D.\ Hislop}
\address[Peter Hislop]{ Department of Mathematics,  
    University of Kentucky,  
    753 Patterson Office Tower,  
    Lexington, Kentucky  40506-0027, USA }
\email{hislop@ms.uky.edu}
%
\author{Fr{\'e}d{\'e}ric Klopp}
\address[Fr{\'e}d{\'e}ric Klopp]{LAGA, U.M.R. 7539 C.N.R.S, 
  Institut Galil{\'e}e,
  Universit{\'e} de Paris-Nord, 99 Avenue J.-B. Cl{\'e}ment, F-93430
  Villetaneuse, France}
\email{klopp@math.univ-paris13.fr}
%
\author{Georgi Raikov} \address[Georgi Raikov]{Departamento de
  Matem{\'a}ticas, Faculdad de Ciencias, Universidad de Chile, Las
  Palmeras, 3425 Casilla 653, Santiago 29504 Chile}
\email{graykov@uchile.cl}
%
\thanks{PDH supported in part by NSF DMS-0202656}
\thanks{FK supported in part by FNS 2000 ``Programme Jeunes Chercheurs''
and by the Chilean Science Foundation {\it Fondecyt} under Grant 7020737}
\thanks{GR supported in part by Chilean Science Foundation {\it Fondecyt}
under Grants 1020737 and 7020737}
%
\keywords{}
\subjclass{}
%
\begin{document}
%
\begin{abstract}
We prove that the integrated density of states (IDS)
for the randomly perturbed
Landau Hamiltonian is H\"older continuous at all energies with any H\"older
exponent $0 < q < 1$. 
The random Anderson-type potential
is constructed with a nonnegative, compactly supported single-site
potential $u$. The distribution of the {\it iid} 
random variables must be absolutely continuous with
a bounded, compactly supported density.
This extends a previous result \cite{[CHK1]}
that was restricted to magnetic fields having rational
flux through the unit square. Furthermore, we prove that the IDS is
H\"older continuous as a function of the nonzero
magnetic field strength, and that
the density of states is in $L^p_{loc} ( \R)$, for any $1 \leq p <
\infty$.
\vskip.5cm\noindent
%\textsc{R{\'e}sum{\'e}.}

\end{abstract}
%
%\setcounter{section}{-1}
\maketitle
%
\today
%
\section{Introduction}
\label{intro}
%
In this note, we prove a new result on the H\"older continuity
of the integrated density of states
(IDS) for Landau Hamiltonians with random, Anderson-type potentials.
The unperturbed Landau Hamiltonian $H_L(B)$
on $L^2 ( \R^2 )$ has the form
\begin{equation}
\label{landau}
H_L (B)  = ( - i \nabla - A)^2, ~~\mbox{where} ~~ A(x_1 , x_2 ) =
\frac{B}{2}
( - x_2 ,  x_1 ),
\end{equation}
where $B \neq 0$ is the magnetic field strength.  The spectrum of $H_L (B)$
is pure point and consists of an increasing sequence of degenerate,
isolated eigenvalues $\{ E_j (B) = (2j + 1)|B| \; | \; j = 0 , 1, \ldots ,
\}$, each of infinite multiplicity. As the spectrum is independent of the
sign of $B$, we will always assume $B > 0$.
The perturbed family of Landau Hamiltonians is given by
\begin{equation}
\label{full}
H_\omega (B) = H_L (B) + V_\omega .
\end{equation}
The random potential
$V_\omega$ is the Anderson-type random perturbation of the form
\begin{equation}
\label{pot1}
V_\omega (x) = \Sum_{j \in \Z^2  } \; \omega_j  u ( x
- j ) .
\eeq
The single-site potential function $u$,  
and the random coupling constants
$\{ \omega_j  \; | \; j \in \Z^2 \}$,
satisfy the following two hypotheses:
\begin{description}
\item[(H1)] The single-site potential $u$
  is a nonnegative, compactly-supported
  function with $u \in L^\infty ( \R^2 )$, satisfying $u > u_0 > 0$ 
  on some nonempty open set, for some constant $u_0 > 0$.
\item[(H2)] The random coupling constants $\{ \omega_j  \;
  | \; j \in \Z^2 \}$, are independent and identically distributed.
  The distribution has a compactly-supported density $h_0 \in L^{\infty}
(\R)$
  with $\mbox{supp} \: h_0 \subset [-m , M ]$, for $0 \leq m , M < \infty$.
\end{description}
We note that there is no limitation in taking $u$ normalized so that
$\| u \|_\infty \leq 1$, that we assume from now on,
as the results are uniform with respect to $\| V_\omega \|_\infty$.

The traditional manner to
define the integrated density of states
(IDS) $N_0(E; B)$ for $H_0 = H_L (B)$, and $N(E; B)$ for $H_\omega$,
is to use the local Hamiltonians associated with finite-area regions
$\Lambda \subset \R^2$.  For a regular, bounded region $\Lambda \subset \R^2$, we define the local
Hamiltonians $H_0^\Lambda$ and $H_\omega^\Lambda$
to be the self-adjoint operators on $L^2 (\Lambda)$ obtained by restricting
$H_0$ and $H_\omega$ to $\Lambda$, with self-adjoint boundary conditions.
We assume that these local operators are lower bounded with a uniform,
finite lower bound independent of $\Lambda$.
The IDS is defined using the eigenvalue counting function
for the local Hamiltonian.
Let $N_0^\Lambda (E;B)$, respectively,
$N_\omega^\Lambda (E;B)$, be the number of
eigenvalues of $H_0^\Lambda$, respectively $H_\omega^\Lambda$,
less than or equal to $E$. The function $N_\omega^\Lambda (E;B)$ depends on
the realization $\omega$. The integrated density of states for $H_\omega$
is defined by
\begin{equation}
N ( E ; B) = \lim_{| \Lambda | \rightarrow \infty } \; \frac{
N_\omega^\Lambda (E;B) }{ | \Lambda | } ,
\end{equation}
when this limit exists.
Similarly, we define $N_0(E; B)$ using the counting function $N_0^\Lambda
(E;B)$
for $H_0^\Lambda$.
Because $N_0(E;B)$ and $N(E;B)$ are monotonic functions of $E$,
each has at most a countable number of discontinuities.
We assume that each has been defined to be right
continuous.

It will be convenient for us to use an equivalent formulation of the IDS as
presented, for example, in \cite{[PF]}.
Let $\Lambda \subset \R^2$ be a finite-area region as above. Let
$\chi_\Lambda$ be the characteristic function for the region $\Lambda$. For
any bounded function $f$ of compact support, the operator $\chi_\Lambda
f(H_\omega)$ is a trace class operator. Let $J \subset \R$ be a
bounded Borel set. The density of states measure $\nu$ is defined by
\beq
\label{dos}
\nu ( J ) \equiv \lim_{| \Lambda | \rightarrow \infty} \frac{1}{ |
\Lambda| } \; Tr ( \chi_\Lambda E_\omega ( J) \chi_\Lambda ) ,
\eeq
when this limit exists.
The IDS is given by $N(E;B) = \nu ( ( - \infty , E])$, when $E$ is a point of
continuity of $N$.

The IDS for both $H_0$ and $H_\omega$ exist under the
hypotheses given above, and we refer to \cite{[DIM],[HLMW1],[Nakamura]} for
proofs of these facts (we mention \cite{[CL],[Kirsch1],[PF]} for
discussions of the IDS without magnetic fields).
Because of $\Z^2$-ergodicity with respect to the magnetic translation group
(see section 2), the IDS $N(E;B)$ is almost surely independent
of the realization $\omega$.
The IDS $N_0 (E;B)$ for the unperturbed Landau Hamiltonian $H_L (B)$
(\ref{landau}) is a piecewise
constant, monotone increasing function (cf.\ the example in
\cite{[Nakamura]}) given by
\beq
\label{ids0}
N_0 (E;B) = \frac{B}{2 \pi} \left\{ \begin{array}{cc}
           0 & ~~\mbox{if} ~~~E < B \\
           \ell & ~~\mbox{if}  ~~~(2 \ell - 1)B \leq E < (2 \ell + 1)B
   \end{array}
  \right.
\eeq
Consequently, it is not locally H\"older continuous at all energies.
The effect of the random potential $V_\omega$ is to improve the regularity
of the IDS almost surely. Let us also note that because of gauge invariance,
the IDS for $H_0$ and $H_\omega$ are independent
of the choice of the vector potential $A$ provided $B = \partial_1 A_2 -
\partial_2 A_1$.
{\it We will often write $N_0 (E)$ and $N(E)$, for $N_0(E;B)$ and $N(E;B)$,
respectively, when the explicit dependence on $B$ is not 
needed.}

\subsection{The Main Results on the IDS}

It is known that $N(E)$ for the random family $H_\omega$
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 $(0 , 2(N+1)B )
\backslash \{ E_j (B) \; | \; j = 0 , 1, \ldots , N \}$ almost surely
\cite{[CH2],[HLMW2],[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 (depending on
the level).  There
has been some discussion as to the behavior of the IDS at the Landau
energies $E_j (B)$, when $B$ is not necessarily large relative to the
level $j$. If the single-site potential $u$ in (\ref{pot1})
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 for any $B \neq 0$. More recently, some of us in \cite{[CHK1]}
extended these results and proved global H\"older continuity of the IDS at
all energies, provided the magnetic field strength $B$
satisfies $B \in 2 \pi \Q $, the {\it rational flux condition} for the unit 
square $\Lambda_1 (0)$. 
This condition allows the problem to be treated using the method of Floquet
decomposition. 
The case of irrational flux was not resolved in that paper because of lack
of explicit control on how a constant
that enters from an application of the unique
continuation principle depends on $B$. In this paper, 
we compute this constant explicitly, and, 
by using a new method, avoid the use of the Floquet 
decomposition. These two advances  
allow us to modify the argument of \cite{[CHK1]}
and prove the following general theorem that
extends these known results. This more general technique will be thoroughly 
explored in \cite{[HK1]}. 
%
\begin{Th}
Let $H_L (B)$ be the Landau Hamiltonian
(\ref{landau})
with magnetic field $B \neq 0$, and assume hypotheses (H1)--(H2). Then
the IDS $N(E)$
is locally uniformly H\"older
continuous at all energies with any exponent $0 < q < 1$.
That is, for any exponent $0 < q < 1$,
and for any bounded interval $I \subset \R$, there is a finite constant
$C_{I,q} > 0$, locally uniformly bounded in $B$ for $B \neq 0$,
so that for any $E, E' \in I$, we have
\beq
| N(E) - N(E') | \; \leq C_{I,q} |E - E' |^q .  
\eeq
\end{Th} 
%
\noindent
This result follows from 
a regularity property of the density of states
(DOS) measure $\nu$ defined in (\ref{dos}) above.
%
\begin{Th}
Under the hypotheses (H1)--(H2), the DOS measure $\nu$
defined in (\ref{dos}) for $H_\omega$ exists
and is globally absolutely continuous with respect to Lebesgue measure.
Furthermore, for any exponent $0 < q < 1$,
and for any bounded interval $I \subset \R$, there is a finite
constant $C_{I,q} > 0$ so that for any Borel $J \subset I$, we have 
\beq
\label{holdernu}
\nu (J) \leq C_{I,q} |J|^q .
\eeq
The density of the measure, i.e. the Radon-Nikodym derivative
$(d \nu / d \lambda)$,
belongs to $L^p_{loc} (\R)$, for any $1 \leq p < \infty$.
\end{Th}
%
\noindent
We also prove a H\"older continuity 
result on the IDS as a function of $B$. 
We do not believe
that the H\"older exponent in the following theorem is optimal.
%
\begin{Th}
Under hypotheses (H1)--(H2), the
IDS $N_B (E)$ is 
a H\"older continuous function of $B$, for $B \neq 0$.
That is, for any exponent $0 < \alpha < 1/4$,
and any bounded interval $J \subset \R \backslash \{ 0 \}$,
there is a finite constant $C_{( \alpha, J, E )} > 0$ so that
for any $B, B_0 \in J$, we have
\beq
\label{lip1}
| N( E; B) - N (E; B_0) | \; \leq C_{(\alpha, J ,E)} | B - B_0|^\alpha.
\eeq
\end{Th}
%
In the proof of this theorem, we use the regularity of $N(E;B)$
as a function of $E$. This idea was used by
B.\ Simon \cite{[Simon3]} who proved that the IDS for $H_L(B) +
V_{per}$, where $V_{per}$ is a $\Z^2$-periodic potential, 
is continuous in the field strength $B$ at any energy $E$ that is a point
of continuity for the IDS.
As a consequence of Theorems 1.1 and 1.3, we have the following corollary
on joint continuity in $E$ and $B$.
%
\begin{Cor}
Under hypotheses (H1)--(H2),
the IDS is jointly H\"older continuous on any compact subset of $\R \times
\R \backslash \{0\}$ in the $(E , B)$ energy-magnetic field plane.
\end{Cor}
%
\subsection{Contents of this Article}

In section 2, we review the group of magnetic translations and the
ergodicity of the perturbed Landau Hamiltonians.
In section 3 we prove a Wegner estimate that will imply the main result,
Theorem 1.1, given a certain positivity estimate.
This key positivity estimate is proved for Landau Hamiltonians in section
4. In section 5, we prove the continuity of the IDS with respect to the
magnetic field. In section 6, the appendix, we collect various technical
estimates needed in the proofs.
%
\section{The Magnetic Translation Group and Ergodicity}
%
The magnetic translations for
the constant magnetic field problem in two-dimensions are
defined as follows. Much of this discussion applies to Schr\"odinger
operators with
constant magnetic fields in higher dimensions (cf.\ \cite{[Sjostrand]}).
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 $\alpha$ as
\begin{equation}
\label{magtr1}
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}
\label{magtr2}
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 (\ref{magtr1})--(\ref{magtr2}), extends to a projective
unitary representation of $\R^2$ on $L^2 ( \R^2)$.  We note that
for any magnetic field strength $B$,
\begin{equation}
\label{invar1}
U_\alpha^B H_L (B)  U_{- \alpha}^B  = H_L (B).
\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 (\ref{magtr1}).  It follows from
(\ref{magtr2}) that we have the commutation rules:
\begin{equation}
\label{comm}
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}
\label{intflux}
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}
\label{ratflux}
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, assuming (\ref{ratflux}),
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 = U_2^B (U_1^B)^q ,
\end{equation}
so the pair $\{ (U_1^B)^q, U_2^B \}$ generate an Abelian group.
Under the rational flux condition,
we used Floquet theory relative to this Abelian group in order to study
the IDS for $H_\omega$ in \cite{[CHK1]}.

Let us now consider the transformation properties of $H_\omega$
with respect to the magnetic translations by $\Z^2$.
The mapping $\gamma = ( n_1 , n_2 ) \in \Z^2
\rightarrow U_\gamma^B$ of $\Z^2$ into a family of unitary operators
defined in (\ref{magtr1}) is
a strongly continuous, unitary
ray representation of the additive group $\Z^2$. For
$\gamma, \gamma' \in \Z^2$, we have
\beq
U_\gamma^B U_{\gamma'}^B = e^{i \frac{B}{2} \gamma \wedge \gamma'}
U_{\gamma + \gamma'}^B,
\eeq
where $\gamma \wedge \gamma' \equiv n_1 n_2' - n_2 n_1'$.
In order to compute
the action of $U_\gamma^B$
on $H_\omega$, we recall the invariance of $H_0$ in (\ref{invar1}),
and consider a related
action by $\Z^2$ on the probability space $\Omega$.
For each $\gamma = ( n_1 , n_2) \in \Z^2$, we define a map
$T_\gamma: \Omega \rightarrow \Omega$ whose action on the random coupling
constants satisfies
\beq
\label{regact1}
(T_\gamma \omega )_{(j_1 , j_2)} =
\omega_{(j_1 + n_1 , j_2 + n_2)} .
\eeq
It is clear, under (H2), that this action is the usual ergodic
group of translations on the probability space $\Omega$.
It follows from this remark, the identity (\ref{invar1}),
and the form of the random potential (\ref{pot1}), that
\beq
\label{eq:ergodic}
U_\gamma^B H_\omega U_{-\gamma}^B = H_{T_\gamma \omega } ,
\eeq
for all $\gamma \in \Z^2$.
From this identity, the general theory of ergodic \Schr\ operators
developed by Kirsch and Martinelli \cite{[KM1],[KM2]} states
that there is a set
$\Sigma \subset \R$ so that $\sigma ( H_\omega ) = \Sigma$ with probability
one. Furthermore, the IDS exists and is independent of $\omega$ almost
surely (for the details, cf.\ \cite{[DIM],[HLMW1],[Nakamura]}).
%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of a Wegner Estimate}

The proof of the H\"older continuity of the DOS measure, Theorem 1.2,
follows from a locally uniform estimate on the following quantity
\beq
\label{wegner1}
\frac{1}{| \Lambda |} \E \{ Tr ( \chi_\Lambda E_\omega ( \cdot) 
\chi_\Lambda ) \} , 
\eeq
where we denote the spectral family of $H_\omega$
by $E_\omega ( \cdot )$, and $\E ( \cdot )$ is the expectation with respect
to the random variables $\omega_j, ~~j \in \Z^2$. Such an estimate
is related to the Wegner estimate \cite{[Wegner]}, except that the Hamiltonian $H_\omega$
in (\ref{wegner1}) is the infinite-area operator (1.2). 
We will prove the following result in this section, and then use it to prove Theorem 1.2.

\begin{Th}
Let $q$ satisfy $0 < q < 1$, and let $I \subset \R$
be a bounded interval. There is a finite constant $ C_{I , q} > 0$
so that for any subinterval $\mathcal{O} \subset I$,
there is a length scale $L_{\mathcal{O}}$, such that for all $L > L_{\mathcal{O}}$,
we have
\beq
\label{wegner2}
\E \{ Tr ( \chi_L E_\omega ( \mathcal{O}) \chi_L ) \} \; 
\leq C_{I,q} | \mathcal{O} |^q L^2 ,  
\eeq
where $\chi_L$ is the characteristic function of the square $\Lambda_L$ of
side length $L$.
\end{Th}

The proof of this estimate will follow the general ideas 
of the proof of the Wegner estimate
given in \cite{[CHK1]}. 
One difference is that the length scale $L_\mathcal{O}$ depends on the
interval $\mathcal{O}$. Of course, this does not affect the infinite area limit, and, consequently,
the properties of the DOS, as will
be shown in the proof of Theorem 1.2 below.
The main positivity estimate that we need to prove Theorem 3.1 is the following.
We will prove this estimate in section 4.

\begin{Th}
Let $H_L (B)$ be the Landau Hamiltonian in (\ref{landau}) and 
let $\Pi_n$ be the projector onto
the infinite-dimensional eigenspace for $H_L(B)$
corresponding to the eigenvalue $E_n(B)$. 
Let $u \geq 0$
be a single-site potential satisfying hypothesis (H1), and define the
potential $\tilde{V}$ by
\beq
\tilde{V} (x) \equiv \sum_{j \in \Z^2} u(x-j) .
\eeq
Then, there exists a
finite constant $0 < C_n (B, u) < \infty$, so that
\beq
\label{positive1}
\Pi_n \tilde{V} \Pi_n \; \geq \; C_n ( B , u ) \Pi_n.
\eeq
\end{Th}

\noindent
{\bf Proof of the Wegner Estimate (\ref{wegner2}) given Theorem 3.2.}\\
%
\noindent
1. We begin with the observation that it suffices to prove the theorem for
bounded intervals $\Delta \subset \R$ sufficiently small so that 
they contain {\it at most one Landau level for $H_L (B)$}. 
Since the separation of the eigenvalues of
$H_0 \equiv H_L (B)$ is $2B$, this can be done provided $|\Delta| <
2B$. We note that it follows from the proof
that the constant $C_{I,q}$ in (\ref{wegner2})
is uniform with respect to any subintervals on $I$.
Suppose that $\Delta$ is centered on an arbitrary, but fixed, energy
$E_0$. For $\Delta \subset \R$, let
$f_\Delta ( \lambda ) \in C_0^\infty ( \R)$ be a smooth, nonnegative
function of compact support with $f_\Delta | \Delta = 1$, $0 \leq
f_\Delta \leq 1$, and $| \mbox{supp} ~f_\Delta | < 2 | \Delta |$, so
that $\mbox{supp} ~f_\Delta$ contains at most one Landau level, the
same as $\Delta$. We then have the operator inequality
$E_\omega ( \Delta ) \leq f_\Delta (H_\omega)$.
We begin with an estimate on 
\beq
\label{smoothweg}
\E \{ Tr ( \chi_\Lambda E_\omega ( \Delta ) \chi_\Lambda ) \} \leq
\E \{ Tr ( \chi_\Lambda f_\Delta ( H_\omega )^2 \chi_\Lambda ) \} .
\eeq
First, if $\Delta$ contains one Landau level, $E_n (B)$,
we write a decomposition of the trace on the right in (\ref{smoothweg})
as
\bea
\label{decomp1}
Tr  \chi_\Lambda f_\Delta ( H_\omega )^2 \chi_\Lambda 
 & =  &Tr \chi_\Lambda f_\Delta ( H_\omega ) \Pi_n^\perp f_\Delta ( H_\omega
)\chi_\Lambda \nonumber \\
 &  & + Tr \chi_\Lambda f_\Delta ( H_\omega ) \Pi_n f_\Delta ( H_\omega
)\chi_\Lambda .
\eea
We will use the fact that $H_0 - E_0$ is boundedly invertible on
the range of $\Pi_n^\perp \equiv 1 - \Pi_n$. 
Second, if $\Delta$ does not contain a Landau level,
we know directly that $H_0 - E_0$ is boundedly invertible.
Let $d_0 \equiv \min \{ \mbox{dist} ~( E_0 , E_q(B) ) \; | \;
E_q (B) \not\in \Delta \}$, so $d_0 > 0$.
We begin with the first case when $E_n (B) \in \Delta$.
It is easy to estimate the term in (\ref{decomp1}) involving
$\Pi_n^\perp$. 
We write
\bea
\label{perp1}
Tr \chi_\Lambda f_\Delta ( H_\omega ) \Pi_n^\perp 
 f_\Delta ( H_\omega)\chi_\Lambda  
& = & Tr \chi_\Lambda f_\Delta (H_\omega) (H_\omega - E_0) (H_0 - E_0)^{-1}
\Pi_n^\perp f_\Delta (H_\omega)\chi_\Lambda 
 \nonumber \\
& & - Tr \chi_\Lambda f_\Delta (H_\omega) V_\omega \Pi_n^\perp (H_0 -
E_0)^{-1} f_\Delta (H_\omega)\chi_\Lambda  .
\eea
To estimate the second term in (\ref{perp1}), we expand the trace in the
eigenfunctions $\psi_j$ of the nonnegative, compact operator $f_\Delta
(H_\omega)\chi_\Lambda f_\Delta (H_\omega)$. 
Applying a quadratic inequality, this results in the bound,
\bea
\label{perp2}
\lefteqn{ | Tr \chi_\Lambda f_\Delta (H_\omega) V_\omega \Pi_n^\perp (H_0 -
E_0)^{-1} f_\Delta (H_\omega)\chi_\Lambda |} \nonumber \\
 & \leq 
\frac{1}{2 d_0^2} Tr \chi_\Lambda f_\Delta V_\omega^2 f_\Delta \chi_\Lambda  + 
\frac{1}{2} Tr \chi_\Lambda f_\Delta \Pi_n^\perp f_\Delta \chi_\Lambda ,
\eea
where we write $f_\Delta$ for $f_\Delta (H_\omega)$, when convenient.
The first term in (\ref{perp1}) is easily seen to be bounded above by
\beq
\label{perp3}
| Tr \chi_\Lambda f_\Delta (H_\omega) (H_\omega - E_0) (H_0 - E_0)^{-1}
\Pi_n^\perp f_\Delta (H_\omega)\chi_\Lambda | \leq \frac{| \Delta | }{d_0}
Tr \chi_\Lambda f_\Delta^2 \chi_\Lambda .
\eeq
Bringing the second term on the right in (\ref{perp2}) to the left in
(\ref{perp1}), we obtain an upper bound for the complementary term 
\beq
\label{perp4}
Tr \chi_\Lambda f_\Delta ( H_\omega ) \Pi_n^\perp
 f_\Delta ( H_\omega)\chi_\Lambda \leq \frac{1}{d_0^2} Tr \chi_\Lambda
f_\Delta V_\omega^2 f_\Delta \chi_\Lambda + \frac{2|\Delta|}{d_0} Tr
\chi_\Lambda f_\Delta^2 \chi_\Lambda .
\eeq

\noindent
2. We use the lower bound (\ref{positive1}) in order to estimate the second 
term on the right in (\ref{decomp1}).
Substituting this into the trace for $\Pi_n$,
we obtain
\bea
\label{direct1}
Tr \chi_\Lambda f_\Delta \Pi_n f_\Delta \chi_\Lambda 
 & \leq  & \frac{1}{C_n ( B , u)}  Tr
  \chi_\Lambda f_\Delta \Pi_n \tilde{V} \Pi_n f_\Delta \chi_\Lambda \nonumber \\
 & \leq & \frac{1}{C_n ( B , u)} \left\{ Tr \chi_\Lambda f_\Delta \tilde{V}
  f_\Delta \chi_\Lambda \right. \nonumber \\
 & & \left. - Tr \chi_\Lambda f_\Delta \Pi_n^\perp \tilde{V} \Pi_n f_\Delta
\chi_\Lambda - Tr \chi_\Lambda f_\Delta \Pi_n \tilde{V} \Pi_n^\perp f_\Delta
\chi_\Lambda \right\} ,
\eea
where we dropped the positive term
$Tr \chi_\Lambda f_\Delta \Pi_n^\perp {\tilde V} \Pi_n^\perp 
f_\Delta \chi_\Lambda $ since it occurs with a negative
sign. As in \cite{[CHK1]}, the two terms on the last line of
(\ref{direct1}) can be estimated in the same manner. 
We note that it follows from hypothesis (H1)
that there exists a finite constant
$D_0 >  0$ , depending only on $u$, so that $\tilde{V}^2 \leq D_0
\tilde{V}$.
Using this constant, the
second term on the last line in (\ref{direct1}) is bounded above by 
\bea
\label{est1}
| Tr \chi_\Lambda f_\Delta \Pi_n^\perp \tilde{V} \Pi_n f_\Delta
\chi_\Lambda | & \leq &
\frac{D_0^2}{C_n}  Tr \chi_\Lambda f_\Delta \Pi_n^\perp f_\Delta
\chi_\Lambda \nonumber \\
 & & + \frac{C_n}{4} Tr \chi_\Lambda f_\Delta \Pi_n f_\Delta \chi_\Lambda ,
\eea
where $C_n \equiv C_n (B, u)$ is the constant occurring in (\ref{positive1}). 
We then obtain for the direct term
\beq
\label{direct2}
Tr \chi_\Lambda f_\Delta \Pi_n f_\Delta \chi_\Lambda  \leq  
\frac{2}{C_n} Tr \chi_\Lambda f_\Delta \tilde{V} f_\Delta \chi_\Lambda
 + \frac{4 D_0^2 }{ C_n^2} Tr \chi_\Lambda f_\Delta \Pi_n^\perp
 f_\Delta\chi_\Lambda .
\eeq

\noindent
3. We now combine (\ref{perp4}) and (\ref{direct2})
and obtain
\bea
\label{mainest}
Tr \chi_\Lambda f_\Delta ( H_\omega )^2 \chi_\Lambda & = & Tr \chi_\Lambda
 f_\Delta ( H_\omega ) \Pi_n f_\Delta ( H_\omega)  \chi_\Lambda 
 + Tr \chi_\Lambda
 f_\Delta ( H_\omega ) \Pi_n^\perp f_\Delta ( H_\omega)  \chi_\Lambda
\nonumber   \\
& \leq &  \frac{2 | \Delta |}{d_0}\left\{ 1 + \frac{4 D_0^2}{C_n^2} \right\} 
   Tr \chi_\Lambda f_\Delta^2 \chi_\Lambda  \nonumber \\
& & + \frac{2}{C_n} Tr \chi_\Lambda f_\Delta \tilde{V} f_\Delta
\chi_\Lambda + \left\{ \frac{1}{d_0^2} + \frac{4D_0^2}{C_n^2 d_0^2}
\right\} Tr \chi_\Lambda f_\Delta V_\omega^2 f_\Delta\chi_\Lambda.
\eea
We first choose $| \Delta |$ sufficiently small so that
the coefficient of $Tr \chi_\Lambda f_\Delta ( H_\omega )^2 \chi_\Lambda$ 
in (\ref{mainest}) satisfies
\beq
\frac{2 | \Delta |}{d_0}\left\{ 1 + \frac{4 D_0^2}{C_n^2} \right\} < 1/2.
\eeq
Hence, this term can be moved to the left in (\ref{mainest}).
Second, in order to evaluate the last term in (\ref{mainest}),
we note that hypotheses (H1) and (H2) 
imply that there exists a finite constant $C_1 ( u , h_0 ) > 0$ so that
$V_\omega^2 \leq C_1(u, h_0) \tilde{V}$.
Using this inequality,
we can combine the two remaining terms on the right in
(\ref{mainest}).
Upon choosing $| \Delta|$ smaller, if necessary,
we find that there exists a finite constant $C_2 > 0$ so that
\beq
\label{mainest2}
Tr \chi_\Lambda E_\omega ( \Delta ) \chi_\Lambda 
 \leq C_2 Tr \chi_\Lambda f_\Delta \tilde{V} f_\Delta \chi_\Lambda ,
\eeq
where
\beq
C_2 \equiv 2 \left\{ \frac{2}{C_n} + C_1 \left(
\frac{1}{d_0^2} + \frac{ 4D_0^2
C_1}{C_n^2 d_0^2} \right) \right\}.
\eeq
In the second case, when $\Delta$ does not contain a Landau level,
we find by an analogous, but simpler, computation 
\beq
\label{mainest3}
Tr \chi_\Lambda E_\omega ( \Delta ) \chi_\Lambda
 \leq \frac{ C_1}{d_0} 
Tr \chi_\Lambda f_\Delta \tilde{V} f_\Delta \chi_\Lambda .
\eeq

\noindent
4. In order to estimate the trace on the right in (\ref{mainest2}) and
(\ref{mainest3}),
we must replace the potential $\tilde{V}$ by a local potential using the
localization due to the characteristic function $\chi_\Lambda$.
Let us suppose that $\Lambda =
\Lambda_L = [- L/2 , L/2 ] \times [- L/2 , L/2]$ is
a square of side length $L>0$. 
For any $0 < \epsilon < 1$,
we denote by $\Lambda_\epsilon \subset \Lambda_L$ 
the square of side length $(1 - \epsilon )L$.
We write $\partial\Lambda_\epsilon$ 
for the annular region $\Lambda_{(1+\epsilon)L}
\backslash \Lambda_{(1 - \epsilon)L}$, so that the area of this annular
region is $4 \epsilon L^2$.
Finally, we denote the complement $\R^2 \backslash \Lambda_{(1+ \epsilon)L}$
by $\Lambda_\epsilon^c$. As above, we use the notation $\chi_X$ 
for the characteristic function of the set $X \subset \R^2$. 
The sets defined here provide a partition of $\R^2$ as
\beq
1_{\R^2} = \chi_{\Lambda_\epsilon} + \chi_{\partial \Lambda_\epsilon} +
\chi_{\Lambda_\epsilon^c} .
\eeq
Using this partition of unity, we write
\beq
\label{part1}
\chi_\Lambda f_\Delta \tilde{V} f_\Delta \chi_\Lambda  =  \chi_\Lambda
f_\Delta \tilde{V} \chi_{\Lambda_\epsilon} f_\Delta \chi_\Lambda 
 + \chi_\Lambda f_\Delta \tilde{V} \chi_{\partial \Lambda_\epsilon} 
f_\Delta \chi_\Lambda  
 + \chi_\Lambda f_\Delta \tilde{V} \chi_{\Lambda_\epsilon^c} f_\Delta
\chi_\Lambda .
\eeq
The first term on the right in (\ref{part1}) is the main term. We estimate
the expectation of the trace of this term
using the estimate on the
spectral shift function proved in \cite{[CHN]}.
The two remainder terms in (\ref{part1})
are dealt with using either the smallness of the
boundary term or the decay of the kernel of the
operator $\chi_\Lambda f_\Delta ( H_\omega) \chi_{\Lambda_\epsilon^c}$ that
follows from the fact that the regions $\Lambda$ and $\Lambda_\epsilon^c$
are disjoint with a positive distance $\epsilon L /2$ between them. 

\noindent
5. We calculate the trace of the
first term on the right in (\ref{part1}) as follows.
Let $g (\lambda) \in \C^\infty (\R)$ be a smooth function with support of
$g'$
containing $\Delta$, and so that $f_\Delta ( \lambda) \leq - 2 | \Delta | g'
(\lambda)$. We can then write
\bea
\label{main1}
Tr \chi_\Lambda f_\Delta \tilde{V} \chi_{\Lambda_\epsilon} f_\Delta
\chi_\Lambda & \leq & Tr f_\Delta \tilde{V} \chi_{\Lambda_\epsilon}
\nonumber \\
& \leq & - 2 | \Delta | Tr g'(H_\omega) \tilde{V} \chi_{\Lambda_\epsilon} .
\eea
A key formula in this setting is
\beq
\sum_{j \in \tilde{\Lambda_\epsilon} } \frac{\partial}{\partial \omega_j}
Tr (g(H_\omega) - g(H_0) ) \chi_{\Lambda_\epsilon} =  Tr g' (H_\omega)
\tilde{V} \chi_{\Lambda_\epsilon} .
\eeq
We can now follow the calculation in \cite{[CHN]} and \cite{[CHK1]}
using the spectral shift function. As a result, we find
that for each $0 \leq q < 1$, there exists a finite, positive constant
$C_{q,1} > 0$, so that
\beq
\label{part2}
\E \{ Tr \chi_\Lambda f_\Delta \tilde{V} \chi_{\Lambda_\epsilon} f_\Delta
\chi_\Lambda \} \leq C_{q,1} | \Delta |^q  (1 - \epsilon )^2 L^2.
\eeq

\noindent
6. We now calculate the two remainder terms in (\ref{part1}).
The boundary term is easily seen to be bounded above by
\beq
\label{part3}
Tr \chi_\Lambda f_\Delta \tilde{V} \chi_{\partial \Lambda_\epsilon}
f_\Delta \chi_\Lambda \leq 4 \epsilon L^2 \left( \sup_{j \in
\partial \Lambda_\epsilon} \| \chi_\Lambda f_\Delta (H_\omega) u_j^{1/2}
\|_{HS}^2 \right) .
\eeq
The term involving the unbounded complementary region is bounded above by
\beq
\label{part4}
Tr \chi_\Lambda f_\Delta \tilde{V} \chi_{\Lambda_\epsilon^c} f_\Delta
\chi_\Lambda \leq \| \chi_\Lambda f_\Delta (H_\omega)
\chi_{\Lambda_\epsilon^c} \|_{HS}^2 .
\eeq
In the appendix, section 6,
we prove two estimates on the Hilbert-Schmidt norms
of the localized operators
appearing in (\ref{part3}) and (\ref{part4}).
We assume that $f_\Delta$ is obtained from a function $f$
by scaling, so the derivatives of $f$ are order one.
In this way, the $k^{th}$-derivative $f_\Delta$ is of
order $|\Delta|^{-k}$.
For the unscaled function $f$, we introduce the following notation that is
used in the estimates of the appendix:
\beq
\label{norm1}
||| f |||_k \equiv 
\sum_{l=0}^k d_l \int_\R | f^{(l)} ( \xi ) |  ~d \xi  ,
\eeq
where the finite constants $d_j > 0$ depend only on the function $\sigma$
used in the construction of the almost analytic extension given in
(\ref{alanext}).
First, we prove that there is a finite constant $C_0  > 0$, depending on $B$
and independent
of $j$, so that
\beq
\| \chi_\Lambda f_\Delta (H_\omega) u_j^{1/2}\|_{HS} \leq C_0 
|||f|||_3 ~|\Delta|^{-2}.
\eeq
Consequently, for the boundary term (\ref{part3}), we obtain
\beq
\label{part5}
Tr \chi_\Lambda f_\Delta \tilde{V} \chi_{\partial \Lambda_\epsilon}
f_\Delta \chi_\Lambda \leq C_2 \epsilon L^2 |\Delta|^{-4}.
\eeq
Second, we consider
$f_\Delta (H_\omega)$ localized between two disjoint regions.
Let $\chi_j$, for $j=1,2$ 
be characteristic functions on two disjoint regions, at least one
of which is bounded, and let $d_{12} \equiv \mbox{dist} ( \mbox{supp} \chi_1 , 
\mbox{supp} \chi_2)$. For any $M \in \N$, and for any $f \in C_0^{k+1} ( \R )$,
with $k \geq M + 1$, there is a constant $0 < B_M < \infty$, 
independent of $f$ and $d_{12}$, so that
\beq
\label{decay1}
\| \chi_1 f_\Delta (H_\omega) \chi_2 \|_{HS} \leq B_M d_{12}^{-M}
 |||f|||_{k + 1} .
\eeq
Hence, setting $k = M + 1$, and taking into account the scaling of $f_\Delta$,
we obtain a bound for the term involving the interior and the disjoint exterior 
(\ref{part4}), 
\beq
\label{part6}
Tr \chi_\Lambda f_\Delta \tilde{V} \chi_{\Lambda_\epsilon^c} f_\Delta
\chi_\Lambda \leq  B_M^2 ||| f |||_{M + 2}^2 (\epsilon L)^{-2M} 
|\Delta|^{-2M - 2 } .
\eeq

\noindent
7. To complete the proof, 
we combine the estimates (\ref{part2}), (\ref{part5}), 
and (\ref{part6}), to obtain
\beq
\label{part7}
\E \{ Tr \chi_\Lambda E_\Delta \chi_{\Lambda} \} \leq C_{q,1} 
| \Delta |^q  (1 - \epsilon )^2 L^2
+ C_2 \epsilon L^2 |\Delta|^{-4} + C_3  (\epsilon L)^{-2M} |\Delta|^{-2M -2},
\eeq
where the constants depend on $f$ and its derivatives, 
but are independent of $L$ and $|\Delta|$.
For any given $\Delta$, we choose $\epsilon = |\Delta|^5$ and $M = 2$.
Then, for all $L > L_\Delta \equiv | \Delta |^{- 9/2}$, we have
\beq
\E \{ Tr \chi_\Lambda E_\Delta \chi_{\Lambda} \} \leq C_q |\Delta|^q |\Lambda| ,
\eeq
proving the first part of the theorem.  We next need to check the
uniformity of the constants with respect to subintervals of $\Delta$.
To see this, notice that the constant $C_2$  in (3.17) depends on $\Delta$ only
through $d_0$ and if $U \subset \Delta$, then $d_0$ increases, so that $C_2$ is
uniform with respect to all subintervals of $\Delta$. 
The same result holds for the constant
in (3.18).  Similarly, the constant $C_{q,1}$ in (3.23) depends only on 
the uniform norm $\|g\|_\infty$
of the function $g$ in (3.21). This norm is 
is independent of $\Delta$. In the same
manner, one checks that the constants 
$C_2$ and $C_3$ in (3.31) depend only on the
norm $|||f|||_3$ of the {\it unscaled} function $f$.
Thus, these constants are independent
of $\Delta$. It follows that the constant $C_{I,q}$ in (3.2) is uniform with
respect to all subintervals of $I$. 
This completes the proof of Theorem 3.1. $\Box$

\vspace{.1in}

We now give a proof of Theorem 1.2, given Theorem 3.1. 
The key fact we use is the uniformity of constant 
$C_{I,q}$ in the Wegner estimate with respect to subintervals.

\vspace{.1in}

\noindent
{\bf Proof of Theorem 1.2 given Theorem 3.1.} \\
\noindent
We fix a bounded interval $I \subset \R$.  
Let $U \subset I$ be an open interval. From Theorem 3.1,
there is a length scale $L_U$ so that for $L > L_U$, we have
\beq
\label{borel1}
\E \{ Tr ( \chi_L E_\omega (U ) \chi_L ) \} \leq C_{I,q} L^2 |U|^q ,
\eeq
where $\chi_L$ is the 
characteristic function on the square $\Lambda_L$. 
Dividing by $L^2$ 
and taking the limit $L \rightarrow \infty$, 
we find that the DOS measure $\nu$ satisfies  
\beq
\label{borel3}
\nu (U) \leq C_{I,q} |U|^q.
\eeq
Next, let $\mathcal{O} \subset I$ be an arbitrary open subset. 
There is a countable collection
$\{ \mathcal{O}_n \}$ of disjoint, open intervals so that
$ \mathcal{O} = \cup_{n} \mathcal{O}_n$. 
From (\ref{borel3}), it follows that
\beq
\label{borel4}
\nu ( \mathcal{O} ) \leq C_{I,q} 
~\sum_n |\mathcal{O}_n|^q = C_{I,q} |\mathcal{O}|^q.
\eeq
We now consider an arbitrary Borel subset $J \subset I$. For any
$\epsilon > 0$, there is an open set $\mathcal{O}_\epsilon$ containing $J$ 
so that $|\mathcal{O}_\epsilon \sim J| \leq 
\epsilon$. It follows from part 1 of the proof that
the constant constant $C_{I,q} > 0$ is independent of $\epsilon$, and we have  
\beq
\label{borel5}
\nu ( J ) \leq \nu ( \mathcal{O}_\epsilon ) \leq C_{I,q} |\mathcal{O}_\epsilon |^q \leq C_{I,q} ( |J| + \epsilon)^q.
\eeq
Since $\epsilon >0$ is arbitrary, we find that
\beq
\label{final1}
\nu (J) \leq C_{I ,q} |J|^q ,
\eeq
which is the main estimate of Theorem 1.2. The fact that (\ref{final1})
implies that the density $(d \nu / d \lambda)$ 
is in $L_{loc}^p (\R)$, for $1 \leq p < \infty$ is given, for
example, in \cite{[CHKN]}. $\Box$ 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Positivity Estimates for Landau Projectors}
%\label{Forma}
%
The key local positivity estimate (\ref{positive1})
follows from an explicit calculation of a lower bound for the left side
of (\ref{positive1}) employing explicit formulas for the infinite-dimensional 
projectors for the Landau levels. 
In this case, we are able to make a fairly explicit calculation
of the constant appearing in (\ref{positive1}).
This provides a quantitative version 
of the unique continuation principle, as used in 
\cite{[CHK1]}.
Recall that $\Pi_n =\Pi_n(B)$, $n \in {\mathbb Z}_+$  
is the orthogonal projection on eigenspace corresponding to 
the $n$-th Landau level for the Landau Hamiltonian $H_0=H_L(B)$ 
with constant magnetic field $B >0$, 
and that the potential $\tilde{V}$ is defined by
\beq
\tilde{V}(x) \equiv \sum_{j \in \Z^2} u(x-j) ,
\eeq
for a single-site potential $u$ satisfying (H1).
%
\begin{Th}
\label{posth2}
For any $n \in \Z_+$, for a 
single-site potential $u$ satisfying (H1), and for
any $B > 0$, there is a finite, positive constant $C_n ( B , u)$,
so that the potential $\tilde{V}$ satisfies
\beq
\label{positive2}
\Pi_n \tilde{V} \Pi_n \geq C_n (B, u) \Pi_n, \quad n \in {\mathbb Z}_+.
\eeq
\end{Th}
%
\noindent
In order to prove Theorem \ref{posth2}, we need several auxiliary assertions. 
The basic technical observation comes from the work of Raikov and Warzel 
\cite{[RW]}. Lemma 4.1 summarizes Lemmas 3.2 and 3.3, and Remark 3.2 of
this paper \cite{[RW]}. 
%
\begin{Le} \label{le1} Let $V \in L^p({\mathbb R}^2)$, $p > 1$. 
Then, the operator $\Pi_n V \Pi_n$, $n \in {\mathbb Z}_+$, is selfadjoint 
and compact in $\Pi_n L^2({\mathbb R}^2)$. 
Moreover, if $V$ is radially symmetric, then the eigenvalues 
$\mu_{n,k}(V)$,  $k \in {\mathbb Z}_+$, of $\Pi_n V \Pi_n$ 
are equal to $\langle V \varphi_{n,k},  \varphi_{n,k} \rangle$, 
where $\langle .,. \rangle$ denotes the scalar product 
in $L^2({\mathbb R}^2)$, and the functions $\varphi_{n,k}$ are given by 
$$
\varphi_{n,k}(x) : = \sqrt{\frac{n!}{\pi k!}} 
\left(\frac{B}{2}\right)^{(k-n+1)/2} (x_1 + ix_2)^{k-n} {\rm L}^{(k-n)}_n 
\left(B|x|^2/2\right)e^{-B|x|^2/4}, \quad x \in {\mathbb R}^2, 
$$ 
where ${\rm L}^{(k-n)}_n$ are the generalized Laguerre polynomials
given by   
$$
{\rm L}^{(k-n)}_n(\xi) : = 
\sum_{l={\rm max}\{0,n-k\}}^n \binom{k}{n-l} \frac{(-\xi)^l}{l!}, 
\quad \xi \geq 0, \quad n \in {\mathbb Z}_+, \quad k \in {\mathbb Z}_+. 
$$
Finally, the normalized eigenfunctions of $\Pi_n V \Pi_n$ corresponding to 
the eigenvalues $\mu_{n,k}(V)$, $k \in {\mathbb Z}_+$, 
are equal to $\varphi_{n,k}$. In particular, the eigenfunctions
are independent of $V$.
\end{Le} 
%
\noindent
{\em Remark}: Note that in \cite{[RW]} the eigenvalues 
$\langle V \varphi_{n,k},  \varphi_{n,k} \rangle$ 
and the eigenfunctions $\varphi_{n,k}$ are enumerated 
starting with $k = -n$, and not with $k = 0$, as is done here. 

\vspace{.1in}

We will apply this result by noting that hypothesis (H1)
guarantees that the single-site potential $u$ is strictly positive
on some open set.
We denote by $D(x,R)$ the disk of radius $R>0$, 
centered at $x \in {\mathbb R}^2$. 
We set $\nu_{n,k} (R) : = \mu_{n,k}(\chi_{D(0,R)})$. 
%
\begin{Cor} \label{f1} 
The eigenvalues $\nu_{n,k} (R)$, $k \in {\mathbb Z}_+$, 
of the operator $\Pi_n \chi_{D(0,R)} \Pi_n$, $n \in {\mathbb Z}_+$,
can be written as 
\begin{equation} 
\label{1}
\nu_{n,k}(R)  = 
\frac{n!}{k!} \int_0^{\varrho} \xi^k \, 
{\rm L}^{(k-n)}_n(\xi)^2 \, e^{-\xi} \, d\xi, 
\end{equation}
where $\varrho = \varrho(R) : = BR^2/2$. 
Moreover, the asymptotic relation 
\begin{equation} 
\label{2}
\nu_{n,k}(R) = 
\frac{e^{-\varrho} \varrho^{-n+1}}{n!} \frac{k^{2n-1}\varrho^k}{k!} (1 + o(1)), \quad k \to \infty, 
\end{equation}
holds uniformly with respect to 
$R \in [R_1, R_2]$, $0 < R_1 < R_2 < \infty$. 
\end{Cor} 

\vspace{.1in}
\noindent
{\bf Proof.} Using Lemma \ref{le1}, 
passing to polar coordinates $(r,\theta)$ 
in the integral $\langle 
\chi_{D(0,R)} \varphi_{n,k},  \varphi_{n,k} \rangle$, 
and changing the variable $Br^2/2 = \xi$, we immediately obtain \eqref{1}. 
In order to check \eqref{2}, 
we assume first of all that $k \geq n$. In this case, we have 
\begin{equation} 
\label{3} 
\nu_{n,k}(R) = 
\frac{n!}{k!} \sum_{l=0}^n \sum_{m=0}^n (-1)^{l+m} 
\frac{1}{m!l!}\binom{k}{n-l}  \binom{k}{n-m} \int_0^{\varrho(R)} 
e^{-\xi} \xi^{k-n+m+l} d\xi. 
\end{equation} 
We note that  
\begin{equation} 
\label{4} 
\lim_{k \to \infty} 
k^{s-n} \binom{k}{n-s} = \frac{1}{(n-s)!}, \quad s \in \{0, \ldots, n\},
\end{equation}
(see e.g. \cite[Eq. 6.1.46]{[AS]}). Moreover, the asymptotic relation 
\begin{equation} \label{5} 
\int_0^{\varrho} e^{-\xi} \xi^{k+\alpha} d\xi 
= e^{-\varrho} \varrho^{\alpha + 1} \frac{\varrho^k}{k} (1 + o(1)), 
\quad k \to \infty, 
\end{equation} 
holds for any fixed $\alpha \in {\mathbb R}$ 
uniformly with respect to $\varrho 
\in [\varrho_1, \varrho_2]$ with $0 < \varrho_1 < \varrho_2 < \infty$. 
Combining \eqref{4} and \eqref{5}, we find that 
the leading asymptotic term of the sum in \eqref{3} 
corresponds to $m=l=0$. Employing \eqref{3}, \eqref{4} with $s=0$, 
and \eqref{5} with  
$\alpha = -n$, we get \eqref{2}. $\Box$  

\vspace{.1in}
%
\begin{Le} \label{le2} 
Let $n \in {\mathbb Z}_+$. 
For each $0 < \varepsilon < R$, $\lambda > 1$, and $\eta > 0$, 
there exists a constant $C_0 = C_0(n, \varepsilon, R, \eta) > 0$, such that 
\begin{equation} 
\label{6b}
\Pi_n \chi_{D(0,\varepsilon)} \Pi_n \geq 
C_0 \left(\Pi_n \chi_{D(0,R)}\Pi_n - \eta \; \Pi_n 
\chi_{D(0,\lambda R)} \Pi_n\right). 
\end{equation}
\end{Le}
%
\vspace{.1in}
\noindent
{\bf Proof.} We fix $\delta \in (0,1)$, 
and, bearing in mind \eqref{2}, pick $K$ such that $k \geq K$ implies 
\begin{equation} \label{6a}
(1-\delta) \frac{e^{-\varrho(R_0)} 
\varrho(R_0)^{k-n+1}}{n!} \frac{k^{2n-1}}{k!} \leq \nu_{n,k}(R_0) 
\leq (1+\delta) 
\frac{e^{-\varrho(R_0)} \varrho(R_0)^{k-n+1}}{n!} \frac{k^{2n-1}}{k!}, 
\quad R_0 \in [R/2, 2\lambda R].  
\end{equation}
We will show that if $0 < \varepsilon < R$, $\lambda > 1$, and $k_0 \geq K$, 
then the operator inequality 
\begin{equation} 
\label{7} 
\Pi_n \chi_{D(0,\varepsilon)} \Pi_n \geq 
C_1 \left(\Pi_n \chi_{D(0,R)} \Pi_n - C_2  \Pi_n 
\chi_{D(0,\lambda R)} \Pi_n\right)
\end{equation}
holds with 
\begin{equation} 
\label{6c}
C_1 = C_1(n, \varepsilon, R, k_0) : 
= \min_{k \in \{0,\ldots,k_0\}} \frac{\nu_{n,k}(\varepsilon)}{\nu_{n,k}(R)},
\end{equation}
and 
\begin{equation} 
\label{6d}
C_2 : = \frac{1+\delta}{1-\delta} 
\lambda^{-2(k_0 - n + 1)} e^{-\varrho(R) + \varrho(\lambda R)}.
\end{equation}
By Lemma \ref{le1}, the operators $\Pi_n \chi_{D(0,\varepsilon)} 
\Pi_n$, $\Pi_n \chi_{D(0,R)} \Pi_n$, and $\Pi_n 
\chi_{D(0,\lambda R)} \Pi_n$, are diagonalizable in the same 
basis $\{\varphi_{n,k}\}_{k \in {\mathbb Z}_+}$. 
Hence, in order to prove \eqref{7}, 
it suffices to check that the numerical inequalities 
\begin{equation} 
\label{8} 
\nu_{n,k}(\varepsilon) 
\geq C_1 \left(\nu_{n,k}(R) - C_2 \;\nu_{n,k}(\lambda R) \right)
\end{equation}
hold for each $k \in {\mathbb Z}_+$.  
If $k \leq k_0$, then \eqref{8} 
is valid because in this case \eqref{6c} implies 
$\nu_{n,k}(\varepsilon) \geq C_1 \nu_{n,k}(R)$ 
which obviously entails \eqref{8}. 
If $k > k_0$, by \eqref{6c} and \eqref{6d} we have 
$$
\nu_{n,k}(R) - C_2 \nu_{n,k}(\lambda R)  
$$
$$
\leq (1+\delta) \frac{e^{-\varrho(R)} \varrho(R)^{-n+1}}{n!} 
\frac{k^{2n-1}\varrho(R)^k}{k!} - \frac{1+\delta}{1-\delta} 
\lambda^{-2(k_0 - n+1)} e^{-\varrho(R) + \varrho(\lambda R)} 
(1-\delta) \frac{e^{-\varrho(\lambda R)} 
\varrho(\lambda R)^{-n+1}}{n!} \frac{k^{2n-1}\varrho(\lambda R)^k}{k!}  
$$
$$
= (1+\delta) \frac{e^{-\varrho(R)}} {n!} \frac{k^{2n-1}
\varrho(\lambda R)^{k-n+1}}{k!} 
\lambda^{2(n-1)} \left( \lambda^{-2k} - \lambda^{-2k_0}\right).
$$
Since $\lambda > 1$, 
we find that $\nu_{n,k}(R) - C_2 \nu_{n,k}(\lambda R) \leq 0$, if $k > k_0$, 
which again implies \eqref{8}.
To complete the proof of the lemma,
we fix $\eta > 0$. Taking into account 
\eqref{6d}, we choose $k_0 = k_0(\eta) \geq K$ so large that  
$C_2 \leq \eta$. 
The main estimate 
\eqref{6b}, with $C_0 = C_1(n, \varepsilon, R, k_0(\eta))$, now  
follows from \eqref{8}. $\Box$
 
\vspace{.1in} 
%
\noindent 
{\em Remark}: If $n = 0$, then \eqref{1} implies 
$$
e^{-\varrho(R)} \frac{\varrho(R)^{k+1}}{(k+1)!} \leq \nu_{0,k}(R) \leq \frac{\varrho(R)^{k+1}}{(k+1)!}, 
\quad \forall k \in {\mathbb Z}_+.
$$
Because of this improved
estimate, we also have \eqref{8} with the constants 
$$
\tilde{C}_1 : = 
e^{-\varrho(R)} \left(\frac{\varrho(\varepsilon)}{\varrho(R)}\right)^{k_0+1} = 
 e^{-\varrho(R)} \left(\frac{\varepsilon}{R}\right)^{2(k_0+1)}, \quad 
\tilde{C}_2 : = e^{\varrho(\lambda R)- \varrho(R)} \lambda^{-2(k_0+1)},
$$
which are more explicit that the ones given by \eqref{6c} -- \eqref{6d}.\\ 
%
\vspace{.1in}

\noindent
We now give the proof of the positivity result, Theorem 4.1.
%
\vspace{.1in}

\noindent
{\bf Proof.} For $j \in \Z^2$, the invariance of $H_L(B)$
with respect to group of magnetic translations (\ref{invar1}) gives
\beq
U_j^B \Pi_n \chi_{D(0,\varepsilon)} \Pi_n U_{-j}^B =
\Pi_n \chi_{D(j,\varepsilon)} \Pi_n .
\eeq
Hypothesis (H1) on the single-site potential $u$ guarantees that there
exists an $\epsilon > 0$ so that
\beq
\tilde{V} (x) \geq \sum_{j \in \Z^2} u(x-j) \geq \sum_{j \in \Z^2}
\chi_{D(j,\varepsilon)} (x) .
\eeq
We now choose $R > \sqrt{2} / 2$ so that $\sum_{j \in \Z^2}
\chi_{D(j,R)} \geq \chi_{\R^2} $. 
We fix $\lambda >1$. Then, there is a finite constant $K_0 > 0$
so that $\sum_{j \in \Z^2} \chi_{D(j, \lambda R)} \leq K_0$.
It now follows from Lemma 4.2 that for any $0 < \eta < 1$, 
there is a constant $C_0$ so that 
\bea
\Pi_n \tilde{V} \Pi_n & \geq & \sum_{j \in \Z^2} \Pi_n \chi_{D(j,\varepsilon)}
\Pi_n \nonumber \\
 & \geq & \sum_{j \in \Z^2} C_0 \left( \Pi_n \chi_{D(j,R)} \Pi_n
  - \eta \Pi_n \chi_{D(j,\lambda R)} \Pi_n \right) \nonumber \\
 & \geq & C_0 ( 1 - \eta K_0 ) \Pi_n .
\eea
We now choose $\eta = 1 / 2 K_0 $. The resulting constant depends on the
Landau level index $n$,
the function $u$, and the field strength $B$. $\Box$ 
%
\section{Continuity of the IDS with respect to the Magnetic Field}
%
The proof of the continuity of the IDS
with respect to magnetic field strength
follows from the vague convergence of the density of
states measures with respect to the magnetic field strength.
The proof of this theorem, given below, relies on the following
representation of the density of states (DOS) measure:
\beq
\label{repre1}
\nu_B (f) \equiv
\int f(\lambda) d\nu_B ( \lambda ) = \E \{ Tr [ \chi_0 f( H_\omega (B) )
 \chi_0 ] \} ,
\eeq
where $\chi_0$ is the characteristic function on the unit cube.
This representation follows from the definition of the DOS and the
ergodicity of the Hamiltonian, expressed in
(\ref{eq:ergodic}). A proof can be found in \cite{[HLMW1]}.
Using this representation, we have the following proposition on the vague
convergence of the density of states measures with respect to the magnetic
field.
%
\begin{Pro}  For any $f \in C_0^5 (\R)$,
there is a finite constant $C(f^{(5)}) > 0$, depending on $f^{(j)}$,
with $j = 1 , \ldots 5$, so
that for any $B, B' \in \R^+$, we have
\beq
\label{eq:smooth}
\left| \int f(\lambda) ( d\nu_B ( \lambda ) - d \nu_{B'} (\lambda) )
\right|
\; \leq \; C(f^{(5)}) {\tilde \delta}(B ,B') ,
\eeq
where
\beq
\label{delta}
{\tilde \delta}(B ,B') \; = \; \mbox{max} \; ( |B - B'| , | (B')^2 -
B^2 | ).
\eeq
If we use a scaled function $f(s / \lambda)$, for $\lambda > 0$,
so that $ |f^{(j)}| \sim \lambda^{-j}$, for $j = 1 , \ldots , 5$, 
then $C(f^{(5)}) = \mathcal{O}( \lambda^{-4})$. 
\end{Pro} 
%
\vspace{.1in}
\noindent
{\bf Proof.}
\newline 
1. We prove the theorem using the
Helffer-Sj\"ostrand formula (\ref{hsj1}) of the appendix,
section 6, and the representation formula (\ref{repre1}).
We write the left side of (\ref{eq:smooth})
using the representation formula (\ref{repre1}) as
\beq
\label{differ1}
\int f(\lambda) ( d\nu_B ( \lambda ) - d \nu_{B'} (\lambda))
= \E \{ Tr [ \chi_0 ( f( H_\omega (B) ) - f(H_\omega (B') )) \chi_0 ] \} .
\eeq
We fix any $\lambda_0$
satisfying $- \infty < \lambda_0 < ~2 \mbox{min} ~ ( 0 , \inf \sigma
(H_\omega))$, and define a function $F(s) = ( - \lambda_0 + s ) f(s)$.
This function $F$ is in $C_0^{5} ( \R)$ like $f$ (note that it is
sufficient to consider those $f$ with support in $[ \mbox{min}
~ ( 0 , \lambda_0 / 2) , \infty)$).
Let $\tilde{F}$ be a compactly supported
almost analytic extension of order $5$ of
$F$, as constructed in (\ref{alanext}). 
We write $z = \xi + i \eta \in \C$.
An extension $\tilde{F}$ of order $5$, as in (\ref{alanext})),
satisfies the estimate
$| \partial_{\overline{z}} \tilde{F} (\xi +i \eta ) |
\sim | \eta|^{4} | F^{(5)} (\xi) |$, as $|\eta| \rightarrow 0$.
With these choices, the Helffer-Sj\"ostrand formula
for the operator $f(H)$ has the form
\beq
\label{hsj2}
f(H) = \frac{1}{\pi} \int_{\C} \partial_{\overline{z}} \tilde{F} (z) (H -
\lambda_0 )^{-1} (H - z)^{-1} ~d \xi ~d\eta .
\eeq
The extra power of the resolvent helps in the calculation of the trace norm
in our case.
Let us denote by $R_\omega^B(z)$
the resolvent $(H_\omega (B) - z)^{-1}$, and by $R_0^B(z)$
the resolvent $(H_L(B) - z)^{-1}$.
We write the right side of (\ref{differ1})
using the Helffer-Sj\"ostrand formula (\ref{hsj2}) as 
\beq
\label{differ2}
f( H_\omega (B) ) - f(H_\omega  (B') ) = \frac{ 1}{\pi}
\int_\C \partial_{\overline{z}} \tilde{F} (z) [ R_\omega^B ( 
\lambda_0 ) R_\omega^B (z) -
 R_\omega^{B'} (\lambda_0) R_\omega^ {B'} (z) ]
~d \xi ~d\eta  .
\eeq
This integral is {\it a priori} convergent because of
the vanishing of $\partial_{\overline{z}} \tilde{F}$ as
$| \eta | \rightarrow 0$, and since $\tilde{F}$ has 
compact support.

\noindent
2. From (\ref{landau}),
the difference of the two perturbed Landau Hamiltonians is
\beq
\label{hdiffer}
\delta (B , B') \equiv
H_\omega (B) - H_\omega (B') =
(B' - B )\hat{A} \cdot p    + (1/4) ( B^2 - (B')^2) \hat{A} \cdot \hat{A} ,
\eeq
where the normalized vector potential $\hat{A}$ is given by
$\hat{A} = ( - x_2 , x_1)$, and $p = -i \nabla$.
The operators $(p_k - A_k)^j$, for $j = 1 , 2$,
are relatively-$H_L(B)$ bounded, and satisfy the norm estimate, 
\beq
\label{relbdd}
\| (p_k - A_k)^j R_0^B (z) \| \; \leq \; C_0 |\eta|^{-1}.
\eeq
In Lemma 6.1 of the appendix, we prove the two following the basic estimates,
using the boundedness of the potential
$V_\omega$, and the relative-bound estimate (\ref{relbdd}). 
Let $B^*$ denote either $B$ or $B'$, and let $\chi$ be any smooth
function $\chi$ of compact support. Then, we have 
\beq
\label{relbdd2}
\| \delta (B , B') R_\omega^{B^*} (z) \chi \|
\; \leq \; C_0 \tilde{\delta} (B,B')
| \eta |^{-3} ,
\eeq
and 
\beq
\label{relbdd3}
\| \delta (B , B') R_\omega^{B^*} (z) R_\omega^{B^*} (
\lambda_0 ) [ H_\omega (B^*) , \chi] \| \; \leq \; C_1
\tilde{\delta} (B,B') |\eta |^{-3} ,
\eeq
where the constants $C_0$ and $C_1$ depend on $B^*$.

\noindent
3. We expand the difference of operators on the right side of
(\ref{differ2}) as follows:
\bea
\label{differ3}
\lefteqn{ \chi_0 ( R_\omega^B ( \lambda_0 ) R_\omega^B (z) - R_\omega^{B'}
(\lambda_0) R_\omega^{B'} (z) ) \chi_0 } \nonumber \\
 &=& - \chi_0 R_\omega^B ( \lambda_0 ) R_\omega^B (z) \delta( B, B') 
  R_\omega^{B'} (z) ) \chi_0 \nonumber \\
 & & - \chi_0 R_\omega^B ( \lambda_0 )  \delta( B, B')
R_\omega^{B'}(\lambda_0) R_\omega^{B'} (z) ) \chi_0 \nonumber \\
 &=& I + II .
\eea
To compute the trace norm of I, 
we use commutators to push the cut-off function through to the left so that
we can use estimate (\ref{differ2}) and the H\"older inequality for trace
ideals (cf.\ \cite{[Simon2]}) in order to
express the trace norm in terms of Hilbert-Schmidt norms. In what follows,
$\chi_{0,j}$ will denote smooth functions with compact support so that
$\chi_{0,j-1} \chi_{0,j} = \chi_{0,j}$. After standard
manipulations, we find
\bea 
\label{differ4}
\| I \|_1 & \leq & \left\{ \| \chi_0 R_\omega^B (\lambda_0) \|_{HS} \; 
            \| \chi_{0,1} R_\omega^B ( z) \|_{HS} \right. \nonumber \\
 & & +\| \chi_0 R_\omega^B (\lambda_0) \|_{HS}  \| \chi_{0,1} 
 R_\omega^B ( z) \|_{HS} \; \| [ H_\omega (B) , \chi_{0,2} ] R_\omega^B
(\lambda_0) \| \nonumber \\
 & & \left.
  +  \| \chi_0 R_\omega^B (\lambda_0) \|_{HS} \; \| \chi_{0,3} R_\omega^B
(\lambda_0) \|_{HS} \; \| R_\omega^B (z) \| \; \| [ H_\omega (B) ,
\chi_{0,1} ] R_\omega^B (\lambda_0) [ H_\omega (B) , \chi_{0,2} ] \|
 \right\} \nonumber \\
 & & \times  \| \delta (B , B') R_\omega^{B'} (z) \chi_0 \| .
\eea
Similarly, the second term II in (\ref{differ3}) can be bounded as
\bea
\label{differ5}
\| II \|_1 & \leq & \| \chi_0 R_\omega^B (\lambda_0) \|_{HS} \; \| 
R_\omega^{B'} (\lambda_0) \chi_0 \|_{HS} \left\{ \| \delta (B , B')
R_\omega^{B'} (z) \chi_0 \| \right. \nonumber \\
 & & + \left. \| \delta (B , B') R_\omega^{B'} (z) R_\omega^{B'} (
\lambda_0 ) [ H_\omega (B') , \chi_{0,1} ] \| \right\}.
\eea
We use the estimates (\ref{relbdd2}) and (\ref{relbdd3})
in order to estimate the terms involving the operator norm in
(\ref{differ4}) and (\ref{differ5}).
We see that it suffices to estimate the 
Hilbert-Schmidt norms $\| \chi_0 R_\omega^B
(\lambda_0) \|_{HS}$ and $\| \chi_0 R_\omega^B (z)\|_{HS}$. For the latter,
we use the resolvent formula and write
\beq
\label{resolv2}
\| \chi_0 R_\omega^B (z)\|_{HS} \leq \frac{C_0}{|\eta|} \|\chi_0 R_\omega^B
(\lambda_0) \|_{HS}.  
\eeq
 

\noindent
4. Hence, it suffices to estimate the Hilbert-Schmidt norm of 
$\chi_0 R_\omega^B ( \lambda_0)$. Squaring this norm, we write
$\| R_\omega^B ( \lambda_0 )\chi_0 \|_{HS}^2 = \| \chi_0 | R_\omega^B
( \lambda_0 )|^2 \chi_0 \|_1$ (cf.\ \cite{[Simon2]}).
In order to estimate this trace norm, we need 
some facts about the resolvent $R_L^B(z)
\equiv ( H_L (B) - z)^{-1}$ of the Landau Hamiltonian presented
in the appendix, section 6.
It follows from (\ref{kernel1})--(\ref{projkernel}) that
\beq
\label{trace}
\| \chi_0 R_L^B(z) \|_{HS}^2
= \| \chi_0 | R_L^B(z)|^2 \chi_0 \|_1 = \frac{B}{2 \pi}
\sum_{n \geq 0} \frac{1}{|E_n(B) - z|^2 }.
\eeq
Note that this is summable for any $z \neq E_m(B)$, for some $m \in \N$.
We will use this for $z = \lambda_0$, so there is no divergence coming from
this term.
Now, returning to the estimate of $\|R_\omega^B ( \lambda_0)\chi_0 \|_{HS}$,
as the random potential $V_\omega$ is bounded,
we can replace $R_\omega^B(\lambda_0 )$ by $R_L^B(\lambda_0)$ using the second
resolvent formula:
\beq
\label{second1}
R_\omega^B(\lambda_0 ) =  R_L^B(\lambda_0) - R_L^B(\lambda_0)
  V_\omega R_\omega^B(\lambda_0 ),
\eeq
and $\| V_\omega R_\omega^B (\lambda_0 ) \|
\leq C(h_0, u, \lambda_0 ) $, for some finite
constant $C(h_0 , u , \lambda_0 ) > 0$,
depending only on $\mbox{supp} ~h_0$, the energy $\lambda_0$,
and the single-site
potential $u$.
Consequently, we can write
\bea
 \chi_0 ( R_\omega^B(\lambda_0))^2 \chi_0 & = & \chi_0 ( R_L^B(\lambda_0))^2 
\chi_0 \nonumber \\
 && - \chi_0 R_L^B(\lambda_0) R_\omega^B(\lambda_0)  V_\omega
   R_L^B(\lambda_0) \chi_0 \nonumber \\
 && - \chi_0 R_L^B(\lambda_0) V_\omega R_\omega^B(\lambda_0) 
 R_L^B(\lambda_0) \chi_0 \nonumber \\
 && + \chi_0 R_L^B(\lambda_0) V_\omega (R_\omega^B(\lambda_0))^2  V_\omega
 R_L^B(\lambda_0) \chi_0 . 
\eea
Using the estimate (\ref{trace}), and this identity, we obtain
\beq
\label{trace2}
\| \chi_0 R_\omega^B(\lambda_0) \|_{HS}^2 = \| \chi_0 |
R_\omega^B(\lambda_0) |^2 \chi_0
\|_1 \leq C_1 .
\eeq

\noindent
5. We return to the right side of (\ref{differ1}).
The trace of (\ref{differ2}) is estimated by writing (\ref{differ3})
and using estimate (\ref{relbdd2}) and (\ref{trace2}).
We see that the singularity
of the trace on the real axis behaves like $|\eta|^{-4}$. As the function
$\partial_{\overline{z}} \tilde{F} (\xi + i \eta)| \sim |\eta|^{4}$, as
$\eta$ vanishes, we see that the integral over $\eta$ is convergent.
This completes the proof.  $\Box$ 

\vspace{.1in}
%
\begin{Th}
For any $E > 0$, the IDS $N (E; B)$ is a
H\"older continuous function of $B$, for $B \neq 0$. 
That is, for any exponent $0 < \alpha < 1/4$,
there is a finite constant $C_{( \alpha, B_0, E )} > 0$, locally uniform
on compact subsets of $E$, so that
for $B_0, B \neq 0$, with $|B_0 - B|$ sufficiently small,
\beq
\label{lip2}
| N (E; B) - N (E; B_0) | \; \leq C_{(\alpha, B_0,E)} | B - B_0|^\alpha.
\eeq
\end{Th} 
%
\vspace{.1in}
\noindent
{\bf Proof.}
\newline 
1. Let $\Sigma_0 \equiv \; \mbox{inf} \; \sigma (H_\omega (B))$, almost
surely. For any $K \subset \R$, let $\chi_K$ be the characteristic 
function for the set $K$. For any $E \geq  \Sigma_0$, 
and for any $0 < \eta < 1$,
let $f_E^+ \in C_0^5 (\R)$ be a nonnegative
function with $0 \leq f_E^+ \leq 1$, and 
such that $f_E^+$
is monotone decreasing on $[ \Sigma_0 - 1 , E + \eta ]$,  
with $\chi_{[\Sigma_0 , E]} f_E^+ = \chi_{[\Sigma_0 , E]}$.
We define $f_E^- ( x ) = f_E^+ ( x - \eta )$
so that $\chi_{[\Sigma_0 -1, E - \eta ]} f_E^- = \chi_{[\Sigma_0 -1, E -
\eta ]}$. 
In this way, we have $0 \leq f_E^+ - f_E^- \leq 1$, and the difference
$(f_E^+ - f_E^-) \chi_{[\Sigma_0, \infty]}$
is supported in $[ E - \eta , E + \eta ]$.

\noindent
2. We fix $E > 0$ and take any $B_0 > 0$, without loss
of generality, since $N(E;B)$ is an even function of $B$.
We consider any $B > 0$ with $|B  - B_0 | < < 1$.
If $N(E;B) > N(E;B_0)$, then monotonicity implies that
\beq
\label{diff1}
0 \leq N(E;B) - N(E;B_0) \leq \nu_B (f_E^+) - \nu_{B_0} (f_E^-) ,
\eeq
whereas, if $N(E;B) < N(E;B_0)$, we have
\beq
\label{diff2}
0 \leq N(E;B_0) - N(E;B) \leq \nu_{B_0} (f_E^+) - \nu_B (f_E^-).
\eeq
In the case of (\ref{diff1}), we have
\bea
\label{cont1}
0 \leq N(E;B) - N(E; B_0) & \leq & \nu_B (f_E^+) - \nu_{B_0} (f_E^-)
\nonumber \\
 & \leq &
| \nu_B (f_E^+) - \nu_{B_0} (f_E^+) | +  \nu_{B_0} (f_E^+ - f_E^-)
\nonumber \\
 & \leq & C((f_E^+)^{(5)}) {\tilde \delta} (B, B_0) + \nu_{B_0} (f_E^+
-f_E^-),
\eea
where we used Proposition 5.1
to evaluate the first term on the right in (\ref{cont1}).
To study the second term on the right of the last line in
(\ref{cont1}), we recall that $\mbox{supp} ~(
f_E^+ - f_E^-) = [ E - \eta , E + \eta ]$,
and that $0 \leq f_E^+ - f_E^- \leq 1$.
We know 
from Theorem 4.3 that the IDS is H\"older continuous in energy at $B_0$, so for 
any $0 < q < 1$, there is a constant $C_{(q,B_0, E)} > 0$, locally uniform in
the energy, so that
\beq
\label{ids1}
| \nu_{B_0} ( f_E^+ - f_E^- ) | \leq C_{(q,B_0, E)}  \eta^q.
\eeq
We recall from Proposition 5.1
that $C( (f_E^+)^{(5)}) \sim \eta^{-4}$. We take $\eta
 = |B - B_0 |^\sigma $, for an
appropriate small constant $\sigma >0 $.
Since we assume that $|B - B_0 | < 1$,
we have ${\tilde \delta} (B, B_0) = |B - B_0 |$.
Hence, for H\"older continuity, we want $1 - 4 \sigma > 0$,
or $0 < \sigma < 1/4$. This implies that
$0 < q \sigma < 1/4$, since $0 < q < 1$ is arbitrary.
If we have the case in (\ref{diff2}),
we find a similar calculation yields
\bea
\label{cont2}
0 < N(E;B_0) - N(E;B) & \leq &  \nu_{B_0} (f_E^+) - \nu_B (f_E^-)
\nonumber \\
 & \leq &  \nu_{B_0} (f_E^+ - f_E^- )  + | \nu_{B_0} (f_E^-) - \nu_B
(f_E^-) |
\nonumber \\
 & \leq &  \nu_{B_0} ( f_E^+ - f_E^- )  +
C((f_E^-)^{(j)}) {\tilde \delta }(B , B_0 ) \nonumber \\
  & \leq & C_{(\alpha, B_0 ,E)} | B - B_0 |^\alpha,
\eea
for some $0 < \alpha < 1/4$, as above.
We used $\eta = | B -B_0|^\sigma$,
for any $0 < \sigma < 1/4$, Proposition 5.1, and (\ref{ids1}).
For both cases, we find
that for $|B - B_0 | < 1$,
and for any $0 < \alpha < 1/4$,
there is a finite constant $C_{(\alpha, B_0, E)} > 0$
so that
\beq
\label{cont3}
| N(E;B) - N(E;B_0) |  \leq   C_{(\alpha, B_0 , E)} |B -B_0|^\alpha.
\eeq
This proves the result. $\Box$ 
%
\section{Appendix: Some Technical Estimates}
%
We briefly recall the Helffer-Sj\"ostrand formula for functions of
self-adjoint operators, and refer to Davies' book \cite{[Davies]}
for more details.
Let $f \in C_0^{(k+1)}$ be a real-valued function. We construct
an almost analytic
extension of $f$ of order $k$ with the aid of a smooth cut-off function
$\sigma ( \eta )$ satisfying $\mbox{supp} ~ \sigma = [-1, 1]$
and $\sigma | [ -1/2, 1/2] = 1$. An almost analytic extension of 
$f$, denoted by $\tilde{f} (z)$, with $z = \xi + i \eta$, is constructed as
\beq
\label{alanext}
\tilde{f} (z) = \sum_{j=0}^k \frac{f^{(j)} (\xi ) }{j !} (i \eta )^j \sigma
(\eta).
\eeq
This compactly-supported
function satisfies $\tilde{f} ( \xi ) = f(\xi)$, and
$|\partial_{\overline{z} } \tilde{f} ( \xi + i \eta )| \sim | \eta|^{k}
|f^{(k+1)} ( \xi )|$, as $|\eta| \downarrow 0$. 
The Helffer-Sj\"ostrand formula for $f(H)$, where $H$
is a self-adjoint operator and function $f$ is as above, is
\beq
\label{hsj1}
f(H) = \frac{i}{2\pi} \int_{\C} ~ \partial_{\overline{z} } \tilde{f} (z)
(H-z)^{-1} ~d\xi ~d\eta .
\eeq

The resolvent of the Landau Hamiltonian $R_0(z) = (H_L(B) -z)^{-1}$ has an
integral kernel given by
\beq
\label{kernel1}
R_0 (z)(x, y)
 = \sum_{n \geq 0} \frac{\Pi_n (x, y) }{ E_n(B) - z} ,
\eeq
for $(x,y) \in \R^2 \times \R^2$.
This sum converges in the strong operator sense for $\Im z \neq 0$.
Let $\Pi_n$ be the projection onto the infinitely-degenerate
subspace corresponding to the Landau level $E_n(B) = (2n+1)B$.
The projector $\Pi_n$ has an explicit kernel given by 
\beq
\label{projkernel}
\Pi_n (x, y) = L_n ( \frac{B}{2} | x - y|^2) \Pi_0 (x,y),
\eeq
where $L_n$ is the $n^{th}$-Laguerre polynomial normalized so that $L_n (0)
= 1$ (see the definition of $L_n = L_n^{(0)}$ in Lemma 4.1).
The projectors satisfy
\beq
\label{projkernel2}
\Pi_n (x , x) = L_n (0) \Pi_0 (x,x) = \frac{B}{2\pi} .
\eeq
The ground state projector $\Pi_0$ has the explicit kernel
\beq
\label{pi0}
\Pi_0 (x, y) = \left( \frac{B}{2\pi} \right)
e^{- \frac{B}{4} |x-y|^2 } e^{-i \frac{B}{2}
x \wedge y } .
\eeq

\vspace{.1in}

\noindent
{\bf Proposition 6.1.} {\it Let $\chi_j$, for $j=1,2$, be two bounded,
real-valued functions of compact support, and suppose that 
$f \in C_0^3(\R)$ is real-valued.
Then, the operator $\chi_1 f(H_\omega)
\chi_2$ is
in the Hilbert-Schmidt class, and there exists a finite, positive constant
$C_2 > 0$, independent of $B$ and $f$, so that 
\bea
\label{hsest1}
\| \chi_1 f( H_\omega ) \chi_2 \|_{HS} & \leq & 
 C_2  (1+B^2)^{1/2} B^{-1/2} \| \chi_2
\|_\infty ~ | \mbox{supp} ~\chi_2|^{1/2} \nonumber \\
 & & \times  
 \left( \| \chi_1 \|_\infty | \mbox{supp} ~\chi_1 |^{1/2} | \mbox{supp}
~f |^{1/2}  \| f \|_\infty + 
\sum_{j=0}^3 d_j \int | f^{(j)} (\xi ) | d \xi \right),
\eea
where the finite constants $d_j > 0$, for $j=0,1,2,3$ depend only on $\sigma$
appearing in (\ref{alanext}).}

\noindent
{\bf Proof.} 
\newline
\noindent
1. From the resolvent formula for $R_\omega (z)$ and $R_0 (z)$,
and Helffer-Sj\"ostrand formula (\ref{hsj1}), 
we have
\beq
\label{hsj3}
f(H_\omega) = f(H_L(B)) - \frac{1}{\pi} \int \int \partial_{\overline{z}}
\tilde{f} (z) R_\omega (z) V_\omega
 R_0 (z) ~d \xi ~d \eta ,
\eeq
where we write $z = \xi + i \eta$.
It follows that it suffices to estimate the Hilbert-Schmidt norm of 
\beq
\label{hslandau}
\chi_1 f(H_L(B)) \chi_2 ,
\eeq
and of
\beq
\label{hsmain}
\chi_1 R_\omega (z) V_\omega R_0 (z) \chi_2 .
\eeq
\noindent
2. To estimate the Hilbert-Schmidt (HS) norm of (\ref{hslandau}), the
compactness of the support of $f$ means that it contains at 
most finitely-many Landau
levels $E_k(B)$, for $k = i_1, \ldots, i_L$. If $\mbox{supp} ~f$
does not contain a Landau level, the operator $f(H_L(B))$ vanishes.
In the former case, we have $f(H_L(B)) = \sum_{k=i_1}^{i_L} f(E_k(B)) \Pi_k$,
and we can use the the explicit form of the projector $\Pi_k$ given in
(\ref{projkernel}) to calculate the HS norm, 
\bea
\label{int0}
\|\chi_1 f(H_L(B)) \chi_2 \|_{HS}^2 & \leq & \sum_{k=i_1}^{i_L} 
|f(E_k(B))|^2 ~ \int \int
\chi_1(x)^2 | L_k ( \frac{B}{2} |x-y|^2 )
\Pi_0 (x,y)|^2 \chi_2(y)^2 ~d^2x ~d^2y \nonumber \\
  & \leq & \left( \frac{B}{2\pi} \right) ~\sum_{k=i_1}^{i_L}
 |f(E_k(B))|^2 ~| \mbox{supp} ~\chi_1|
\|\chi_1\|_\infty^2 \| \chi_2\|_\infty^2 ,
\eea
where we used the 
fact that $\int_0^\infty e^{-w} L_k^2 (w) ~dw = 1$ (cf.\  \cite{[AS]}).   

\noindent
3. As for the main term (\ref{hsmain}), 
we use the fact that the random potential satisfies $\| V_\omega \| \leq
M_0 < \infty$, and the norm bound
\beq
\| R_\omega (z) \| \leq 1/ |\eta| ,
\eeq
with $z = \xi + i \eta $. 
Since $\| A \|_{HS}^2 = \| A^* A \|_1 $, we have to estimate
\beq
\| R_0(z) \chi_2 \|_{HS}^2 = \| \chi_2 |R_0(z)|^2 \chi_2 \|_1.
\eeq
The operator $\chi_2 |R_0(z)|^2 \chi_2$ is a positive operator
with an explicit kernel $K_0 (x, y ; z)$ given by
\beq
K_0 (x,y;z) = \sum_{n=0}^\infty \frac{ \chi_2(x) \Pi_n (x,y) \chi_2 (y) }{
|E_n(B) - z|^2} = \sum_{n=0}^\infty \frac{ \chi_2(x) \Pi_n (x,y) \chi_2 (y)
}{ |(2n+1)B - z|^2},
\eeq
that converges in the strong operator sense for $\Im z \neq 0$.
For the diagonal part of the kernel, it follows from (\ref{projkernel2}) that
\beq
K_0 (x,x;z) = \frac{B \chi_2 (x)^2}{2 \pi}
\sum_{n=0}^\infty\frac{1}{( (2n+1)B - \xi )^2 + \eta^2 }.
\eeq
It is easy to see that 
\beq
K_0 (x,x;z) \leq 
\left( \frac{B \|\chi_2 \|_\infty^2 }{2 \pi} \right) 
\left\{ \frac{1}{\eta^2} + \frac{\pi^2}{4B^2} \right\}.
\eeq
Consequently, the trace norm is bounded as
\beq
\| \chi_2 |R_0(z)|^2 \chi_2 \|_1 \leq \| \chi_2\|_\infty^2 ~| \mbox{supp}
~\chi_2| \left( \frac{B}{2\pi} \right)
\left\{  \frac{1}{\eta^2} + \frac{\pi^2}{4B^2} \right\}.
\eeq
We substitute this estimate into the second term of 
(\ref{hsj3}) and obtain 
\bea
\label{int1}
\lefteqn{ \| \int \int \partial_{\overline{z}} \tilde{f} (z) \chi_1
R_\omega (z) V_\omega R_0 (z) \chi_2 ~d\xi ~d\eta \|_{HS} } \nonumber \\
 & \leq & C_1 M_0 \int \int \frac{ |\partial_{\overline{z}} \tilde{f} (z) |
}{ |\eta|} \left\{  \frac{1}{\eta^2} + \frac{\pi^2}{4B^2} \right\}^{1/2} 
~d \eta d \xi ,
\eea
where the constant $C_1$ is
\beq
C_1 = \| \chi_2 \|_\infty |\mbox{supp} ~ \chi_2|^{1/2} ( B / 2\pi)^{1/2} .
\eeq

\noindent
4. In order to evaluate the integral (\ref{int1}), we take an almost
analytic extension $\tilde{f}$ of order $3$, as in (\ref{alanext}).
We note that 
$\partial_{\overline{z}} \tilde{f}$ has one term depending on
$f^{(3)}$, and another term that is multiplied by $\sigma'(\eta)$. This latter
term vanishes in a neighborhood of the real axis $\eta = 0$, so it can be
estimated directly.
As for the contribution of the former term to the integral (\ref{int1}),
as the coefficient of this term vanishes as $| \eta|^2$, the integral is
finite.
Combining (\ref{int0}) and (\ref{int1}), we obtain the bound
(\ref{hsest1}),  
proving the proposition. $\Box$

\vspace{.1in}

\noindent
{\bf Proposition 6.2.} {\it Let $\chi_j$, for $j=1,2$, be two bounded,
real-valued functions with disjoint supports,
so that $\mbox{dist} ~( \mbox{supp} ~\chi_1 , \mbox{supp} ~\chi_2 ) \equiv
d_{12} > 0$, and
suppose that $\chi_1$ has compact support. For any positive integer
$M$, let $f \in C_0^{k+1} (\R)$, for $k
\geq M + 1$, be a real-valued function. 
Then, there exists a constant $C_M > 0$, 
independent of $f$ and $d_{12}$, so that
\beq
\label{hsnorm1}
\| \chi_1 f(H_\omega) \chi_2 \|_{HS} \; \leq  \; \left( 
\frac{ C_M}{ d_{12}^M } \right) \; \left(
\sum_{j=0}^{k+1} d_j \int_\R | f^{(j)} ( \xi  ) | ~d \xi \right),
\eeq
where the finite constants $d_j$ depend only on $\sigma$ appearing in
(\ref{alanext}).}

\vspace{.1in}
\noindent
{\bf Proof.}
\newline
\noindent
1. The proof of this proposition is based on the polynomial decay
of the operator $f(H_\omega)$ when localized between two functions of
disjoint support.
This result depends on the decay of the localized resolvent $\chi_1 
R_\omega (z) \chi_2$ that in turn follows from the 
Combes-Thomas method \cite{[CT]}.
These results are well-known,
see, for example, \cite{[GK2]} or \cite{[Klopp]}. Here, we
note that
we can obtain the estimate in the Hilbert-Schmidt norm under the above
hypotheses.  
The basic Combes-Thomas estimate for resolvent of $H_\omega$ has the form
\beq
\label{ctresolv}
\| \chi_1 R_\omega (z) \chi_2 \|_{HS} \leq \frac{C_1}{ ( \mbox{dist} ~(z,
\sigma (H_\omega) )) } e^{- C_2 \mbox{dist} ~(z,\sigma (H_\omega) )
d_{12} },
\eeq
where the constants can be chosen independent of $z$.  
For our purposes, it suffices to use the following simple estimate,
\beq
e^{-t} \leq \frac{b_M}{t^M}, ~~\mbox{for} ~t \geq 0, ~~\mbox{with} ~~b_M =
e^{-M} M^M, 
\eeq
so that 
\beq
\label{hsnorm2}
\| \chi_1 R_\omega (z) \chi_2 \|_{HS} \leq \left( \frac{1}{d_{12}^M } \right) 
 \; \left( \frac{ C_1 b_M  }{C_2^M} \right) \; \left( \frac{1}{ \mbox{dist}
~(z,\sigma (H_\omega) )^{1 + M} } \right).
\eeq

\noindent
2. We use the estimate (\ref{hsnorm2}) and the Helffer-Sj\"ostrand formula
(\ref{hsj1}) to calculate the norm (\ref{hsnorm1}).
For any integer $M$, there is a finite constant $C_M > 0$ so that
if the order $k$ of the almost analytic extension of $f$ satisfies $k \geq
M + 1$, then
\bea
\label{hsnorm3}
\| \chi_1 f(H_\omega) \chi_2 \|_{HS} & \leq & C_1 \int \int |
\partial_{\overline{z}} \tilde{f} (z) | ~ \| \chi_1 R_\omega (z) \chi_2
\|_{HS} ~d \xi ~d \eta \nonumber \\
& \leq& \frac{C_M}{d_{12}^M} \int \int |
\partial_{\overline{z}} \tilde{f} (z) | | \eta|^{k -(M+1)} |
f(\xi)^{(k+1)} (\xi)| ~d \xi ~d \eta \nonumber \\
 & & + ( \mbox{finite terms depending on} ~f^{(j)}, j=1, \ldots, k),
\eea
giving the main result. $\Box$ 

\vspace{.1in}
\noindent
Finally, we prove two technical bounds used in section 5.
Let $\chi$ 
be a smooth function with compact support.

\vspace{.1in}
\noindent
{\bf Lemma 6.1.} {\it For any $B, B' \neq 0$,
we define $\delta (B , B' ) \equiv H_\omega (B) - H_\omega (B')$, and define
$\tilde{\delta} ( B,B') = \mbox{max} ~( |B-B'|, |(B')^2 - B^2 |)$.  
We then have
\beq
\label{relbdd4}
\| \delta (B , B' ) R_\omega^{B^*} (\xi + i \eta) \chi_0 \| \leq C_0(B^*)
\tilde{\delta} (B, B') | \eta|^{-3} ,
\eeq
where $B^*$ is either $B$ or $B'$, 
the constant $C_0 (B^*) > 0$ depends on $B^*$ and $\chi$.
Furthermore, we have
\beq
\label{relbdd5}
\| \delta (B , B') R_\omega^{B^*} (z) R_\omega^{B^*} (
\lambda_0 ) [ H_\omega (B^*) , \chi ] \| \; \leq \; C_1 (B^*) 
\tilde{\delta} (B,B') |\eta|^{-3} ,
\eeq
where the constant $C_1(B^*)$ depends on $B^*$ and $\chi$.
}

\vspace{.1in}
\noindent
{\bf Proof.} It is slightly more convenient to do the computation in the
Landau gauge $A = B ( 0 , x_1)$.
We write the difference $\delta ( B, B')$ as
\bea   
\label{bound1}
\delta ( B , B') & = & ( B^2 - (B')^2) x_1^2 + 2 ( B' - B) x_1 p_2  \nonumber \\
 &=& \{( B^2 - (B')^2) + 2 B^* (B' - B) \} x_1^2 + 2 (B' - B)x_1 V_2^* , 
\eea
where $V_2^* = p_2 - B^*x_1$. 
To prove (\ref{relbdd4}), it suffices to estimate $x_1^2 R_\omega^{B^*} 
\chi$, and $x_1 V_2^* R_\omega^{B^*} \chi$.
Since $\| p_1 R_\omega^{B^*} \| = \| V_2^* R_\omega^{B^*} \|
= \mathcal{O} ( | \eta |^{-1} )$, and 
$\|x_1^2 \chi \|$ is bounded,
we easily obtain the estimates by commuting the powers of $x_1$ to the right. 
The proof
of (\ref{relbdd5}) is similar. 
Here, we make use of the fact that one of the resolvents is evaluated 
at the fixed value $\lambda_0$,
and hence does not contribute to the singularity in $\eta$. The proof
is again obtained by commuting the powers of $x_1$ to the right. $\Box$
   






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