Content-Type: multipart/mixed; boundary="-------------0211141123143"

This is a multi-part message in MIME format.

---------------0211141123143
Content-Type: text/plain; name="02-462.keywords"
Content-Transfer-Encoding: 7bit
Content-Disposition: attachment; filename="02-462.keywords"

random operators, density of states, Wegner estimate
---------------0211141123143
Content-Type: application/x-tex; name="idsuab1.tex"
Content-Transfer-Encoding: 7bit
Content-Disposition: inline; filename="idsuab1.tex"

%Submitted paper to the Proceedings of UAB 2002 (03.10.02)
%revised version 13.11.02
\documentclass{conm-p-l}

\usepackage{amssymb}
\usepackage{graphics}

\def\F{I\kern-.30em{F}}
\def\P{I\kern-.30em{P}}
\def\E{I\kern-.30em{E}}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{lem}[theorem]{Lemma}
\newtheorem{prop}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{defn}[theorem]{Definition}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\numberwithin{equation}{section}

\newcommand{\T}{\mathbb{T}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\N}{{\mathbb N}}
\newcommand{\D}{{\mathcal D}}
\newcommand{\un}{{\mathbf 1}}
\newcommand{\Z}{\mathbb{Z}}

\newcommand{\Schr}{Schr{\"o}dinger}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}}
\newcommand{\Sup}{\displaystyle \sup}
\newcommand{\Sum}{\displaystyle \sum }

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

\newcommand{\demi}{\ensuremath{\frac{1}{2}}}
\newcommand{\demip}{\ensuremath{1 \slash 2}}
\newcommand{\troisdemip}{\ensuremath{3 \slash 2}}
\newcommand{\cinqdemip}{\ensuremath{5 \slash 2}}
\newcommand{\quart}{\ensuremath{\frac{1}{4}}}
\newcommand{\quartp}{\ensuremath{1 \slash 4}}
\newcommand{\expo}[1]{\ensuremath{{\rm e}^{#1}}}
\newcommand{\pscal}[3]{\ensuremath{\left( #2, #3 \right)_{#1}}}
\newcommand{\Lp}[2][2]{\ensuremath{L^{#1}(#2)}}
\newcommand{\Lps}[3][2]{\ifthenelse{\equal{#2}{0}}
                                   {\ensuremath{L^{#1}(#3)}}
                                   {\ensuremath{L^{#1}_{#2}(#3)}}}
\newcommand{\Hp}[2][1]{\ensuremath{H^{#1}(#2)}}
\newcommand{\Hps}[3][1]{\ensuremath{H^{#1}_{#2}(#3)}}
\newcommand{\Hun}[1][\R]{\ensuremath{\Hp{#1}}}
\newcommand{\Un}{\ensuremath{{\rm 1\mkern-10mu 1\mkern0mu}}}
\def\C{{\rm C\mkern-8.8mu l\mkern4mu}}

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


\begin{document}

\title{Local and Global Continuity of the Integrated Density of States }
\author{J.-M.\ Combes}
\address{J.-M.\ Combes:
Centre de Physique Th\'eorique, CNRS-Luminy, case 907, 13288
Marseille, FRANCE, and 
PHYMAT, Universit\'e de Toulon et du Var, 83130 La Garde, FRANCE}
\email{combes@cpt.univ-mrs.fr}
\author{P.\ D.\ Hislop}
\address{P.\ D.\ Hislop:
Mathematics Department, University of Kentucky, Lexington,
Kentucky 40506-0027 USA}
\email{hislop@ms.uky.edu}
\author{F.\ Klopp}
\address{F.\ Klopp: L.A.G.A, Institut Galil{\'e}e, Universit{\'e} Paris-Nord, 
    F-93430 Villetaneuse, FRANCE}
\email{klopp@math.univ-paris13.fr} 
\thanks{}


%\subjclass[2000]{}
\date{September, 2002} \keywords{}
\subjclass{}
\date{}

\begin{abstract}
The integrated density of states (IDS) $N(E)$ is the distribution function of a
nonnegative measure $\nu$, the density of states measure (DOS).
This measure usually obtained as the weak infinite-volume
limit of the local eigenvalue counting function for the system restricted to
a finite-volume region. For Schr\"odinger operators with random potentials,
the eigenvalue counting function for the finite-volume system satisfies an
estimate called a Wegner estimate. We present new local and 
global-in-energy Wegner estimates for random Schr\"odinger operators
with Anderson-type random potentials.
These estimates are strong enough to imply that the DOS measure is
absolutely continuous with a density in $L^q_{loc} ( \R)$, for any $1 \leq q <
\infty$. The IDS is also proved to be locally or globally H\"older continuous
with H\"older exponent $1 / q$, for any $q > 1$.
\end{abstract}

\maketitle


\section{Introduction: Models and Main Results} \label{ch-intro}

This paper presents some new results
on the local continuity properties of the
integrated density of states (IDS), and on the local
regularity of the density of states (DOS) measure, for some random operators.
The proofs of these local-in-energy results rely on some recent progress on
continuity properties of the IDS {\it at all energies} by the authors.
We show how the methods of \cite{[CHK1]} can be localized to energy
intervals that provide new results on the local properties of the IDS for
some random models, especially those with constant magnetic fields. 
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 will consider the
family of \Schr\ operators $H_\omega ( \lambda )$ on $L^2 ( \R^d
)$, 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 $C_0^\infty ( \R^d )$, and that $H_0
\geq  - M_0 > - \infty$, for some finite constant $M_0$. We consider an
Anderson-type random
potential $V_\omega$ constructed from the single-site
potential $u$ as
\begin{equation}
\label{potential}
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}
\label{model}
H_\omega ( \lambda ) = H_0 + \lambda V_\omega .
\end{equation}
By re-scaling the random variables, if necessary,
we can normalize $V_\omega$ so that $\| V_\omega \|_\infty \leq 1$. In this
way, the parameter $\lambda >
0$ becomes a measure of the disorder. The main results are independent of
the disorder (provided it is nonzero for Theorem 1.3 and
in the Landau case).

We need two hypotheses throughout the
paper on the single-site potential $u$, and on the distribution of the
random variables.
We denote the cube of side length $L > 0$ centered at $x \in \R^d$ by
$\Lambda_L (x)$.
\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 $h_0 \in L^{\infty} (\R)$
  with $\mbox{supp} \: h_0
  \subset [0 , 1 ]$.
\end{description}
In this partially expository note, we will replace hypothesis (H1) with the
following stronger one whenever it simplifies the discussion
without further mention:
\begin{description}
\item[(H1a)] The single-site potential $u \neq 0$ is nonnegative with
  compact support with $\mbox{supp} ~u \subset \Lambda_1 (0)$,
  $u \in L^\infty ( \R^d )$, and $\|u \|_\infty \leq
  1$.
\end{description}
The hypothesis (H1a) on the single-site potential is stronger than
needed, but it simplifies the discussion here by eliminating contributions
near the boundary of $\Lambda$ coming from the potential outside of
$\Lambda$. These terms can be dealt with in a routine manner as 
presented in \cite{[CHK1]}. 
We note that the case of small support is the most interesting one.
If the support of $u$ is large, that is, if $\Lambda_1 (0)
\subset \mbox{supp} ~u$, then the proof of the Lipschitz continuity of the
IDS is much easier, cf.\ \cite{[CH1],[KS]}.

In order to discuss the IDS, we need Hamiltonians $H_\Lambda$ that describe
the system constrained to a finite-volume region $\Lambda \subset \R^d$.
For $\Lambda \subset \R^d$, we denote the lattice points in $\Lambda$ by
${\tilde \Lambda } = \Lambda \cap \Z^d$, and we let $\chi_\Lambda$ be the
characteristic function for $\Lambda$.  
For a given bounded $\Lambda \subset \R^d$, 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 assume,
without loss of generality, that
the boundary conditions have been chosen so that these operators are
bounded from below
by a constant, independent of $\Lambda$, for all large $\Lambda$.
We denote by $E_0^\Lambda ( \cdot )$ and $E_\Lambda ( \cdot )$
the spectral families for
$H_0^\Lambda$ and $H_\omega^\Lambda$, respectively.  
Under the simplifying hypothesis (H1a),
the local potential $V_\Lambda$ is given by
\bea
V_\Lambda (x) &=& V_\omega ( x) \chi_\Lambda (x) \nonumber \\
  &=&  \Sum_{j \in \tilde{\Lambda} } \; \lambda_j ( \omega ) u ( x - j )
       \chi_\Lambda (x).
\eea
When $\Lambda$ is the union of unit cubes, our hypothesis (H1a) implies that
$\mbox{supp} ~V_\Lambda \subset \Lambda$.

We define the IDS $N(E)$ for $H_\omega$ using the eigenvalue
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 $\omega$, and on the boundary conditions. 
The integrated density of states is defined
by
\begin{equation}
\label{distr1}
N ( E) = \lim_{| \Lambda | \rightarrow \infty } \; \frac{
N_\Lambda (E) }{ | \Lambda | } ,
\end{equation}
when this limit exists.
When it exists, the IDS $N(E)$ is a monotonic function.
We assume that $N(E)$ has been defined to be right
continuous. It has at most a countable number of discontinuities.
{\it We will always assume that the IDS $N(E)$ exists almost surely
for the random family of operators
considered here.}
For example, if the family $H_\omega$ is an ergodic family of
random Schr\"odinger operators with a reasonable random potential, 
it is known that this limit exists 
and is independent of the
realization $\omega$ almost surely (cf.\ \cite{[CL],[Kirsch1],[PF]}).
Furthermore, it is known that the
IDS is independent of the 
boundary conditions taken on the finite
volume $\Lambda$, cf.\ \cite{[DIM],[Kirsch1],[Nakamura]}.
We will give a common condition on $V_\omega$ in section 4 that guarantees
the existence of the IDS.

The IDS $N(E)$ is the distribution function of a nonnegative measure $\nu$
called the {\it density of states} (DOS) measure.
The DOS measure is defined via the Riesz-Markov Theorem and the following
expression, whenever the limit exits:
For any real function $f\in C_0 ( \R)$, the density of states measure $d\nu$
satisfies
\begin{eqnarray} 
\label{vague0}
\int f( s ) d \nu (s) & = &
  \lim_{|\Lambda| \rightarrow \infty} \frac{1}{| \Lambda
|} \; Tr ( \chi_\Lambda f(H_\omega^\Lambda) \chi_\Lambda ) \nonumber \\
 & = & \lim_{|\Lambda| \rightarrow \infty} \int f(s) d \nu_\Lambda (s).
\end{eqnarray}
One usually proves the vague convergence of the measures $\nu_\Lambda$ to
the DOS measure $\nu$ by
proving that the limit of the distribution functions on 
the right side of (\ref{distr1}) 
exists at points of continuity of the limit function.
Properties of the IDS follow from estimates on the expectation of the
eigenvalue counting function for local Hamiltonians known as {\it Wegner
estimates}, after Wegner who first considered them \cite{[Wegner]}.

\subsection{Local-in-Energy Results on the IDS and the DOS}

It is interesting and useful to note that the main results of \cite{[CHK1]}
can be localized in the energy. We give two results, Theorem 1.1 and
Theorem 1.3, that are the analogs
of Theorem 1.1 and Theorem 1.2 from \cite{[CHK1]}, respectively. We also
give results on the behavior of the DOS measure.
Our first theorem deals with the simple case for which the background
operator $H_0$ has a H\"older continuous IDS on some interval.
Although the proof of this theorem is rather simple, the main 
limitation is that
the resulting H\"older exponent is considerably smaller than expected. 

\vspace{.1in}
\noindent
\begin{theorem}
In addition to hypotheses (H1) and (H2), we
assume that the operator $H_0$ admits an integrated density of states
$N_0 (E)$ that is locally H{\"o}lder continuous on an open interval $I
\subset \R$ for some exponent $0 < q_1 \leq 1$. That is, there
exists a finite, positive constant $C_0 (I) > 0$ so that
the IDS satisfies
\begin{equation}
|N_0(E')-N_0(E)| \leq C_0 (I) |E'-E|^{q_1}, ~~\mbox{for} ~~E, E' \in I.
\end{equation}
Then, the IDS for $H_\omega (\lambda)$ is H{\"o}lder
continuous on $I$ for any exponent $0 < q < q_1 / ( q_1 + 2 )$.
\end{theorem}

\vspace{.1in}

We can conclude from this theorem that the DOS measure $\nu$ restricted to
the interval $I$ is absolutely continuous with respect to Lebesgue measure.

\vspace{.1in}
\noindent
\begin{cor}
Under the assumption of Theorem 1.1, the DOS measure $\nu | I$ is
absolutely continuous with respect to Lebesgue measure. The local density
$f_I \in L^p_{loc} (I)$, for any $1 \leq p < 1 + \frac{q_1}{2}$.
\end{cor}

We can improve the H\"older exponent considerably with more work and an
additional assumption that the background operator $H_0$ is
periodic. 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 an open, relatively compact 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 fibered operators and
their eigenvalues.  Note that the operator $H_0^{\Lambda_n}$, with PBC
on $\partial \Lambda_n$, coincides with the fibered operator $H_0^n (
\theta=0 )$.  We define the IDS for $H_\omega ( \lambda )$ as above
using the eigenvalue counting function for $H_\omega^\Lambda$ with PBC.
We define another, deterministic potential associated with a bounded region
$\Lambda$ by 
\beq
{\tilde V}_\Lambda (x) \equiv \sum_{j \in {\tilde \Lambda}} ~u(x-j)
~\chi_\Lambda (x).
\eeq
We define a constant $D_0 > 0$ as
\beq
\label{defd0}
D_0 \equiv \inf \{ D > 0 \; | \; {\tilde V}_\Lambda^2 (x) \leq D {\tilde
V}_\Lambda, \forall x \in \R^d \} .
\eeq
Of course, we have $D_0 \sim \| u \|_\infty$, and 
under hypothesis (H1), $D_0 \leq 1$. We are interested in this
constant because it is a measure the strength of the
perturbation when we take $\lambda = 1$. 
We will see that it plays an important role in the local regularity
of the IDS.

\vspace{.1in}

\begin{theorem}
Assume hypotheses (H1) and (H2). Let $E _0 \in \R$ and let $\tilde{\Delta}
\subset \R$ be an interval centered on $E_0$ and of length $16 D_0^2$, where
$D_0$ is defined in (\ref{defd0}).
Suppose that there exists a finite constant
$C(\tilde{\Delta} , u) > 0$ so that,
for all $\theta \in (\T^d)^*$, the single-site potential $u$ satisfies
\beq
\label{positive1}
E_0 (\tilde{\Delta} , \theta ) u E_0 (\tilde{\Delta} , \theta ) \; \geq 
\; C( \tilde{\Delta} , u) E_0 (\tilde{\Delta} , \theta ) ,
\eeq
Then, for any interval $\Delta \subset \R$ such that $E_0 \in \Delta
\subset \tilde{\Delta}$, with
\[
| \Delta | \; \leq \; \mbox{min} ~( 2 D_0^2, C^2 ( \tilde{\Delta} , u) ),
\]  
the IDS $N(E)$ for $H_\omega$ is H\"older continuous on $\Delta$ with any
H\"older exponent $0 < q < 1$.
\end{theorem}

\vspace{.1in}

Let us note that as $D_0 \sim \| u \|_\infty$, it is clear from the
positivity condition (\ref{positive1}) that the lower bound $C({\tilde
\Delta} , u ) \rightarrow 0$ as
$D_0 \rightarrow 0$. Since we have made no assumption on the IDS
$N_0 (E)$ for the background operator $H_0$, we expect that the interval
$\Delta$ must also shrink as the random potential vanishes. 
Although the periodicity of the operator $H_0$ implies that the IDS
is absolutely continuous (cf.\ \cite{[ReedSimon4]}), 
we do not know anything else about the regularity of its
derivative $d N_0 / d \lambda$.
We have the analog of Corollary 1.2.

\vspace{.1in}
\noindent
\begin{cor}
Under the assumptions of Theorem 1.3, 
the DOS measure $\nu | \Delta$ is
absolutely continuous with respect to Lebesgue measure. The local density
$f_I \in L^p_{loc} ( \Delta )$, for any $1 \leq p < \infty$.
\end{cor}

\vspace{.1in}

We give a condition for the positivity condition (\ref{positive1}) to hold.
We say that $H_0$ has the {\it unique continuation property on an interval
$J \subset \R$} if the following condition holds. For
any $E\in J \subset \R$ and any function $\varphi$ in the domain of $H_0$, if
$\varphi$ satisfies $(H_0-E)\varphi=0$, and $\varphi$ vanishes
on some open set, then $\varphi$ vanishes identically.
It is well-known that a large class of Schr\"odinger operators satisfy the
unique continuation property for $J = \R$.
For example,
the operator $H_0$ has the unique continuation property if, 
in dimension $d\geq3$, $V_0\in L^{d/2}_{loc}(\R^d)$, 
the vector potential $A_0\in L^{d}_{loc}(\R^d)$, and $\nabla A_0\in
L^{d/2}_{loc}(\R^d)$. We refer to \cite{[Wolff]}, and references
therein.

\vspace{.1in}

\noindent
\begin{prop}
We assume hypotheses (H1) and (H2).  In
any dimension $d \geq 1$, if $H_0$ has the unique continuation
property on the interval ${\tilde \Delta}$ of Theorem 1.3,
and $u > 0$ on a nonempty open set, then condition
(\ref{positive1}) holds.  
\end{prop}



\subsection{Global-in-Energy Results on the IDS and the DOS}

For comparison, we state the two main results of \cite{[CHK1]}.
We use the same notation as above.
The first result treats the case when the IDS $N_0(E)$ for $H_0$ is known
to be H\"older continuous on $\R$.
\vspace{.1in}

\begin{theorem}
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 $0 < q < q_1 / ( q_1 + 2 )$.
\end{theorem}

\vspace{.1in}
Similarly, if the background operator $H_0$ is periodic, then we have the
following regularity result that improves the H\"older exponent.
\vspace{.1in}
\begin{theorem}
Assume hypotheses (H1) and (H2), and that
  $H_0$ is $\Z^d$-periodic. Let ${\tilde
  \Delta} \subset \R$ be any 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 $0 < q < 1$.
If the dimension $d =1$, and $H_0 = -d^2 / dx^2$,
then the IDS for $H_\omega (\lambda)$ is Lipschitz
  continuous at all energies.
\end{theorem}

\vspace{.1in}
The DOS measure is correspondingly absolutely continuous with respect to
Lebesgue measure.

\vspace{.1in}

\begin{cor}
Under the assumptions of either Theorem 1.6,
or Theorem 1.7, the DOS measure $\nu$ is
absolutely continuous with respect to Lebesgue measure. The local density
$f \in L^p_{loc} ( \R )$, for any $1 \leq  p_0$, where for Theorem 1.6,
we have $p_0 = 1 + \frac{q_1}{2}$, and for Theorem 1.7, we have $p_0 = \infty$.
\end{cor}

\vspace{.1in}

All of these results can be improved to local and global
{\it Lipschitz continuity}
for the one-dimensional case, with $H_0 = - d^2 / dx^2$,
using better estimates on the spectral shift function
\cite{[Robert1],[Robert2]}. 
Furthermore, the exponent of continuity of the DOS measure can be taken to
be $q=1$ in Corollaries 1.2, 1.4, and 1.8.
The importance of the spectral shift function
to these estimates is discussed further in section 5.
Similar results for the one-dimensional case
were recently obtained by E.\ Giere \cite{[Giere]} using different methods.


\subsection{Outline of Contents}

We will sketch the proof of Theorem 1.1 in section 2, and the proof of
Theorem 1.3 in section 3. In section 4, we prove Corollaries 1.2, 1.4, and
1.8. In section 5, we discuss some related problems, including the question
of Lipschitz continuity of the IDS, 
an application to the continuity of the IDS for Landau Hamiltonians, and
regularity of the IDS.

\section{Preservation of Good Behavior of the IDS in an Interval}
\label{ch-idsbackground}

In this section, we outline the
proof Theorem 1.1. The proof follows closely the proof of Theorem 1.1 in
\cite{[CHK1]}. It is natural to expect that if the
IDS for the unperturbed operator $H_0$ is well-behaved on an interval
${\tilde \Delta} = [E_- , E_+]
\subset \R$, then the IDS for $H_\omega$ will also be
well-behaved on a slightly small interval.  
We first note that for $| \Lambda |$ sufficiently large, depending on
$\tilde\Delta$, the hypothesis of
Theorem 1.1 implies that \cite{[CHK1]},  
\begin{equation}
\label{localtr}
  Tr E_0^\Lambda({\tilde\Delta})\; \leq
  \; C_1 ( E_+ , d)  |\tilde\Delta|^{q_1} |\Lambda| ,
\end{equation}
where the finite constant $C_1 ( E_+ , d) > 0$
depends on the constant $C_0 > 0$ in (1.4), the dimension $d$, and the
energy $E_+$.
To prove result (\ref{localtr}), 
let $N_{0,\Lambda}(E)$ be the number of eigenvalues
of $H_0^\Lambda$ less than or equal to $E$. Let us first assume that
$N_0 (E_-) \neq 0$. 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 (1.4), for $\Lambda$ sufficiently large, we get
%
\begin{equation*}
\label{localtr2}
  \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_1 (E_+ , d) |\Lambda||\tilde\Delta|^{q_1}.
  \end{split}
\end{equation*}
%
We used the monotonicity of the IDS and
the bound $N_0 ( E ) \leq C(E)$, for a finite constant $C(E)$.
If it happens that $N_0 (E_-) = 0$, then
(\ref{localtr2}) follows from the fact that
$\frac{N_{0,\Lambda}(E_+)}{|\Lambda|}\leq
 |E_+-E_-|^{q_1}$, for $|\Lambda|$ sufficiently large.

\begin{proof}
\noindent
1. Suppose that the interval $I = ( E_1 , E_2)$, and that $E_0 \in I$.
Consider a bounded interval $\Delta \equiv
[ \Delta_- , \Delta_+ ] \subset I$, with $E_0 \in \Delta$, and 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 choose $|\Delta|$ small enough so that 
$(1/2) | \Delta | + | \Delta |^\alpha < \min ( E_0 - E_1, E_2 - E_0 )$. 
We decompose the spectral projector $E_\Lambda ( \Delta )$ relative to
$H_0^\Lambda$ as
\begin{equation}
\label{decomp1}
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 (\ref{decomp1}) using the assumption
(\ref{localtr}) on $H_0^\Lambda$ and the positivity of the projectors. For
$\Lambda$ sufficiently large (depending on ${\tilde \Delta})$, this gives
\begin{equation}
\label{est1}
Tr E_\Lambda ( \Delta ) E_0^\Lambda ( {\tilde \Delta } ) \; \leq
\; C_1 | \Delta  |^{\alpha q_1} | \Lambda | ,
\end{equation}
where we write $C_1$ for the constant $C_1 ( E_+ , d)$ appearing in
(\ref{localtr}).

\noindent
2. To deal with the second term in (\ref{decomp1}), we write
\begin{eqnarray}
\label{decomp2}
   Tr E_\Lambda (
\Delta ) E_0^\Lambda ({\tilde \Delta }^c ) & = & Tr E_\Lambda ( \Delta
) ( H_\omega^\Lambda - E_0 ) E_0^\Lambda ({\tilde
  \Delta }^c ) ( H_0^\Lambda - E_0 )^{-1} \nonumber \\
& & - \lambda Tr E_\Lambda ( \Delta ) V_\Lambda E_0^\Lambda ({\tilde
  \Delta }^c ) ( H_0^\Lambda - E_0 )^{-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_0 )^{-1} \|
\leq | \Delta |^{-\alpha} $ and easily obtain:
\begin{equation}
\label{est2}
| (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 (\ref{decomp1}). For the term $(ii)$ in (\ref{decomp2}), 
we insert another factor of
$(H_0^\Lambda - E_0 )(H_0^\Lambda - E_0 )^{-1}$, and obtain, 
\bea 
\label{decomp3}
(ii) & = & - \lambda Tr (H_\omega^\Lambda - E_0 ) E_\Lambda ( \Delta )
V_\Lambda E_0^\Lambda({\tilde \Delta }^c ) ( H_0^\Lambda - E_0 )^{-2}
\nonumber \\
& & + \lambda^2 Tr V_\Lambda E_\Lambda ( \Delta ) V_\Lambda
E_0^\Lambda({\tilde \Delta }^c ) ( H_0^\Lambda - E_0 )^{-2}
\nonumber \\
& = & (iii) + (iv) .  
\eea 
The term $(iii)$ is estimated just as term
$(i)$ in (\ref{est2}), and we obtain
\begin{equation}
\label{est3}
| (iii) | \; \leq \;  \lambda | \Delta |^{1 - 2 \alpha } \; \|
{\tilde V}_\Lambda \| \; Tr E_\Lambda ( \Delta ) .
\end{equation}
For the term $(iv)$ in (\ref{est3}), we obtain
\begin{equation}
\label{est4}
| ( iv) | \; \leq \: \lambda^2 | \Delta |^{- 2 \alpha } \; Tr V_\Lambda
E_\Lambda ( \Delta ) V_\Lambda .
\end{equation}
In order to estimate the last term, we note that hypothesis (H1)
implies that $0 \leq V_\Lambda^2 \leq {\tilde V}_\Lambda$, where
${\tilde V}_\Lambda (x) = \sum_{j \in {\tilde \Lambda}} u(x-j)$.
We then have 
\begin{equation}
Tr V_\Lambda E_\Lambda ( \Delta ) V_\Lambda \; \leq \; Tr
{\tilde V}_\Lambda E_\Lambda ( \Delta ).
\end{equation}
We now take the expectation.  Using the result on the spectral shift
function \cite{[CHN]} discussed in section 5, 
we have
\begin{equation}
\E \{ Tr ( \lambda {\tilde V_\Lambda } E_\Lambda ( \Delta ) )\}  \; \leq
\; C_2 ( q_2, u ) | \Delta |^{q_2} | \Lambda | ,
\end{equation}
for any $0 < q_2 < 1$, and $|\Lambda|$ large enough, so that
\begin{equation}
\label{est5}
\E \{ | ( iv) | \} \; \leq \; \lambda
| \Delta |^{q_2 - 2 \alpha } C_2 ( q_2, u ) | \Lambda | .
\end{equation}

\noindent
3. Gathering the estimates from (\ref{est1}), (\ref{est2}), (\ref{est3}), 
and (\ref{est5}), we
get 
\bea 
\label{est6}
\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_2 ( q_2,
u ) + C_1 | \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 $\| {\tilde V}_\Lambda \|_\infty = \mathcal{O}(1)$), 
then the prefactor on the left side
of (\ref{est6}) is invertible.  Finally, we require that $ \alpha < q_2 / (
q_1 + 2) < 1/2$ in order that the right side of (\ref{est6}) 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.  
\end{proof}

\section{IDS for Periodic $H_0$} \label{ch-periodic} 

In this section, we sketch the proof of Theorem 1.3. The main assumption of
Theorem 1.3 implies the following positivity condition \cite{[CHK1]}:
\beq
\label{positive2}
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}
where we have taken $\theta = 0$ in (1.9) corresponding to periodic
boundary conditions on $\partial \Lambda$. 

\noindent
\begin{proof}
\noindent
1. We begin with a decomposition of the spectral projector as in
(\ref{decomp1}),
\beq
\label{decomp4}
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}
where $E_0 \in \Delta \subset {\tilde \Delta}$ as in the theorem.
Let $d_0 \equiv \mbox{dist} ~( E_0 , {\tilde \Delta}^c)$.
In this case, it is easy to estimate the term involving ${\tilde
\Delta}^c$. Using standard estimates, we obtain
\beq
\label{compest}
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}
%We mention that we can remove the constraint on $d_0$ appearing in (3.7) of
%\cite{[CHK1]} by keeping $d_0$ only in the last term on the right in (3.6).
%Consequently, we have no constraint of $d_0$.

\noindent
2. We use the lower bound (\ref{positive2}) in order to estimate the first
term on the right in (\ref{decomp4}). 
Substituting this into the trace for $E_0^\Lambda ( {\tilde \Delta})$, we
obtain
\bea
\label{direst}
Tr E_\Lambda ( \Delta ) E_0^\Lambda (
{\tilde \Delta } ) &  \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.
Following the calculation in \cite{[CHK1]}, we arrive at
\beq
\label{direst2}
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
3. We now combine (\ref{compest}) and (\ref{direst2})
and obtain 
\bea
\label{mainest}
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 . 
\eea
Since $|{\tilde \Delta }| = 16 D_0$, we have $d_0 = 8 D_0$.
The condition on $\Delta$ implies that $|\Delta | / d_0 \leq 1/4$,
and the second term $4 D_0^2 | \Delta | / d_0 C( {\tilde \Delta} , u )^2 <
1/2 $. So the
coefficient of $Tr E_\Lambda ( \Delta)$ on the right side of (\ref{mainest})
is less than $3/4$, so this term can be moved to the right.
In order to estimate the second term, $Tr E_\Lambda ( \Delta ) {\tilde
V}_\Lambda$, we must take the expectation and use the estimate on the
spectral shift function proved in \cite{[CHN]}.
\end{proof}


There is an alternate manner to express the result of Theorem 1.3.
Estimate (\ref{compest}) can also be written as
\beq
\label{compest1}
Tr E_\Lambda ( \Delta ) E_0^\Lambda ( {\tilde \Delta }^c ) \leq
| \Delta | Tr E_\Lambda ( \Delta ) + \frac{1}{d_0^2}
Tr E_\Lambda ( \Delta ) (\lambda^2 V_\Lambda^2 ) .
\end{equation} 
The effect of keeping $d_0$ with the second term is that the coefficient of 
$Tr E_\Lambda ( \Delta)$ on the right side of (\ref{mainest})
can be written as
\beq
\label{coef1}
\left( \frac{| \Delta |}{1 - | \Delta |} \right) \; \left(
 \frac{4 D_0^2 }{ C ({\tilde \Delta} , u)^2}  \right) .
\eeq
In order to complete the proof, we must fix ${\tilde \Delta}$ and then
adjust $| \Delta|$ so that this is less than one. 

\section{The Density of States Measure}

Let us consider a general random Schr\"odinger operator $H_\omega
= H_0 + V_\omega$, where $V_\omega$ is some random process, not necessarily
of Anderson-type. 
It is well-known that under mild regularity assumptions 
on the random potential
$V_\omega$ the 
IDS $N (E)$ exists almost surely at all points of continuity.
For example (cf.\ \cite{[CL],[Kirsch1],[PF]}), if $p > ~\mbox{max} ~( d/2 , 1)$,
and $r = ( dp/2 )( p - d/2 )^{-1}$, and if $V_\omega$ satisfies
\beq
\label{pot}
\E \{ \int_{\Lambda_1 (0)} | V_\omega (x)|^{1/pr} \} < \infty ,
\eeq
then the IDS exists almost surely at its points of continuity. 
The IDS $N(E)$ is monotonic, right continuous,
and the points of discontinuity form a countable set. 
For Anderson-type random potentials satisfying hypotheses (H1)--(H2), 
condition (\ref{pot}) is satisfied, so that the
IDS exists as described above. As a consequence, 
the DOS $\nu$ measure exists. It is also the vague 
limit of the finite-volume point 
measures $\nu_\Lambda$, corresponding to the
local Hamiltonians $H_\omega^\Lambda$, as in (\ref{vague0}). 
These random point measures $\nu_\Lambda$
are constructed from the eigenvalues of $H_\omega^\Lambda$.
If $\{ E_j ( \Lambda ; \omega ) \; | \; j \in \Z \}$ is the set of random 
eigenvalues of $H_\omega^\Lambda$, 
then we define a random point measure by
\beq
\label{point1}
d \nu_\Lambda ( \lambda ) = \frac{1}{| \Lambda|}
\sum_{j \in \Z} \delta ( \lambda - E_j ( \Lambda
; \omega )) ~d \lambda .
\eeq
For any continuous function $f$ of compact support, we have
\beq
\label{vague}
\lim_{ | \Lambda | \uparrow \infty } \int f (\lambda ) d \nu_\Lambda (
\lambda ) = \int f(\lambda) d \nu ( \lambda) .
\eeq
Furthermore, for any Borel set $A \subset \R$, we have
\beq
\label{vague2}
\nu (A) \leq \liminf_{| \Lambda | \uparrow \infty } \E \{ \nu_\Lambda (A)
\},
\eeq
although pointwise convergence in (\ref{vague2}) does not necessarily hold.
We now turn to the properties of the DOS measure.

\vspace{.1in}

\begin{theorem}
Under the hypotheses (H1)--(H2), the DOS measure for $H_\omega$ exists
and is locally or globally
absolutely continuous with respect to Lebesgue measure. For any
Borel set $A \subset \R$, as stated in Corollaries 1.2, 1.4, or 1.8, we have 
\beq
\label{holdernu}
\nu (A) \leq C |A|^q .
\eeq
For the cases of Corollary 1.2 and Theorem 1.6, the exponent $q$
satisfies $0 < q < q_1 / (q_1 + 2)$. For
the cases of Corollary 1.4 and Theorem 1.7,
the exponent $q$ satisfies 
$0 < q < 1$. The density of the measure
belongs to $L^p_{loc} (\R)$, for any $1 \leq p < p_0$,
where $p_0 = 1 + \frac{q_1}{2}$
in the cases of Corollary 1.2 and Theorem 1.6, and $p_0 = \infty$, in the
case of Corollary 1.4 and Theorem 1.7.
\end{theorem}

\begin{proof}
\noindent
1. We first note that the proofs of Theorems 1.1 and 1.3 can be extended
to include arbitrary Borel sets. We unify notation as follows.
Let the set $J \subset \R$ denote the interval $\Delta \subset I$ in 
the proof of Theorem 1.1 for which the Wegner estimate
holds, or let it denote the interval $\Delta$ in Theorem 1.3.
For an arbitrary Borel set $A \subset \R$, we have
\beq
\E \{ Tr E_\Lambda (A \cap J ) \} \; \leq \; C_J | A \cap J |^q \; |\Lambda|,
\eeq
where the notation $|B|$ means the Lebesgue measure of $B$, and the finite
positive constant $C_J > 0$ is the constant in the
Wegner estimate. The proof
of Theorem 1.1 is identical if we replace $\Delta$ by $\Delta \cap J$ 
throughout. Similarly, the proof of Theorem 1.3 remains the same if we 
make the same replacement.      
Hence, if $A$ is such that $| J \cap A | = 0$, we have that
$\E \{ Tr E_\Lambda ( J \cap A ) \} = 0$, for all large $\Lambda$. 
To relate this finite-volume estimate to the DOS measure $\nu$, we use the
upper bound (\ref{vague2}). We have for
any Borel set $A \subset \R$,
\beq
\label{holder}
\nu (A \cap J ) \; \leq \; \liminf_{|\Lambda| \uparrow \infty} \frac{ \E \{
Tr E_\Lambda ( A \cap J ) \} }{ | \Lambda | } \leq C_J | A \cap J|^q,  
\eeq
for any $q$ in the ranges stated in the theorem. 
This proves the H\"older bound (\ref{holdernu}).
We also conclude from this that for any Borel set $A \subset \R$ of Lebesgue
measure zero, the DOS measure $\nu$
satisfies $\nu ( J \cap A ) = 0$, proving the local absolute
continuity of the measure.

\noindent
2. Let $f_J$ be the local density of the measure $\nu$.
By the general theory, the density $f_J \in L_{loc}^1 ( \R)$.
Since the IDS $N(E)$ is locally
H\"older continuous for any exponent $q >1$, the
density is better behaved.
For any $\lambda \in \R$, let $F_\lambda$ be the subset of $\R$
defined by $F_\lambda \equiv \{ t \in \R \; | \;
f_J (t ) > \lambda \}$. The $\nu$-measure of this set is bounded
from below by 
\beq
\label{meas1}
\nu (F_\lambda ) = \int_{F_\lambda} f_J (t) ~dt > \lambda | F_\lambda |,
\eeq
while from the H\"older Continuity of the measure (\ref{holder}), we have
\beq
\label{meas2}
\nu ( F_\lambda ) \leq C_J | F_\lambda |^q,
\eeq
for any $0 < q < 1$. 
We conclude from (\ref{meas1}) and (\ref{meas2}) that 
\beq
\label{meas3}
| F_\lambda | \leq C_1 | \lambda|^{- \frac{1}{1-q}}. 
\eeq
The $L^p$-norm of the density, for $p \geq 1$, is estimated by
\bea
\label{dens1}
\| f_J \|_{L^p_{loc}}^p & = & \int f_J ( \lambda )^p ~d\lambda \nonumber \\
 & \leq & \sum_{n \in \Z} 2^{p(n+1)} ~ | \{ t \; | \;  2^n < 
   f_J(t) < 2^{n+1} \} | \nonumber \\
 & \leq & \sum_{n \in \Z} 2^{p(n+1)} | F_{2^n } | \nonumber \\
 & \leq & 2^p ~\sum_{n \in \Z} 2^{n ( p - 1/(1-q) ) } .
\eea
For the cases of Corollary 1.2 and Theorem 1.6, we have
$1 < 1/ (1-q) < 1 + \frac{q_1}{2}$, so that the sum in (\ref{dens1})
is finite for any 
$1 \leq p < 1 + \frac{q_1}{2}$. In the other cases of Corollary 1.4 and
Theorem 1.7, we have $1 < 1/(1-q) < \infty$, so the sum in (\ref{dens1})
is finite for any $p \geq 1$. This proves the local result. 
\end{proof}  

We remark that we first proved that if the IDS satisfies a H\"older
continuity condition with exponent $0 < q < 1$, then the density is in
weak-$L^p$, for  $p = 1 / (1 - q)$. 
Let us note that under the best condition of Lipschitz continuity 
of the IDS, the density $f_J$ is given by
the derivative $dN / d \lambda$, and this
is in $L^\infty_{loc}$. This corresponds to $q = 1$. On the other hand,
if we merely know that the DOS $\nu$ is absolutely continuous with
respect to Lebesgue measure, then we are in the $q = 0$ case and
the density $f_J$ is in $L^1_{loc}$.

\section{Concluding Remarks}

\noindent
We make some comments about the results.

\vspace{.1in}

\noindent
1. {\bf Lipschitz Continuity.} It is believed that under hypotheses (H1)--(H2)
the IDS is Lipschitz continuous. The above proofs provide
for Lipschitz continuity provided 
there is an improved estimate on the spectral shift
function (SSF) (for background on the SSF cf.\ \cite{[BY],[Yafaev]}). 
In \cite{[CHN]}, the following estimate on the SSF is proved. Suppose
that $H_0$ and $H$ are two
self-adjoint operators on a separable Hilbert
space with $V \equiv H -H_0 \in \mathcal{I}_{1/p}$, for any $p \geq 1$.
The Schatten-von Neumann ideals $\mathcal{I}_q$ are the classes of bounded
operators $A$ having singular values $\mu_j (A)$ 
satisfying $\| A \|_q^q \equiv \sum_j \mu_j (A)^q <
\infty$, cf.\ \cite{[Simon2]}.
Under these conditions,
the SSF $\xi ( \lambda ; H , H_0 ) \in L^p ( \R)$,
and satisfies the estimate
\beq
\label{SSF}
\| \xi ( \cdot; H , H_0 ) \|_{L^p} \; \leq \; \|V\|_{1/p}^{1/p}.
\eeq
For Schr\"odinger operators $H_0$ and $H$,
the SSF is defined through the resolvents $R_0 (z)$ and $R(z)$.
It is not difficult to show that $V_{eff} \equiv R_0(z)^k - R(z)^k \in
\mathcal{I}_{1/p }$,
for any $p \geq 1$, provided $k > pd/2 + 2$, and provided the perturbation $V =
H - H_0$ has bounded support.
One can then show that the SSF is bounded by $\|V_{eff} \|_{1/p}^{1/p}$.
In the proofs of Theorems 1.1 and 1.3, one considers the expectation
with respect to one random variable only. In particular, choosing
$\lambda_k$, and for a function $f_\Delta$ with support equal to $\Delta$,
one has
\bea
\label{ssf1}
\lefteqn{\int_0^1 h_0 ( \lambda_k ) d \lambda_k 
\left\{ \frac{\partial}{\partial \lambda_k} Tr f_\Delta (H_\omega^\Lambda) 
  \right\}}
 && \nonumber \\ 
 & \leq &  \|h_0 \|_\infty | Tr \{ f_\Delta (H_\omega^\Lambda)(\lambda_k = 1) 
  - f_\Delta ( H_\omega^\Lambda ) ( \lambda_k = 0) \} | \nonumber \\
 & \leq & \|h_0 \|_\infty \left| \int f_\Delta' (t) \xi ( t ; h_1 , h_0 ) ~dt 
\right| .
\eea
We use H\"older's inequality to estimate the integral on the right in
(\ref{ssf1}). Note that the difference of
$ h_1 = H_\omega^\Lambda(\lambda_k = 1)$ and $h_0 =  H_\omega^\Lambda  (
\lambda_k = 0)$ is proportional to $u( \cdot - k)$.
Hence, the SSF for $(h_0, h_1)$ is well-defined
and its $L^P$-norm is bounded by
$\| V_{eff} \|_{1/p}^{1/p} < \infty$, independent of $|\Lambda  |$.
From this use of H\"older's inequality, 
we obtain a fractional power of $|\Delta|$. An improved
pointwise bound on the SSF (or the averaged SSF)
would eliminate the use of the
H\"older inequality and yield Lipschitz continuity of the IDS. This, in
turn, implies that the density $dN / d \lambda$ is locally bounded.
There are two cases for which Lipschitz
continuity of the IDS is known for continuous models in arbitrary
dimension. The simplest case is the Anderson-type random potential as in
(1.1) when the support of the single-site distribution contains the unit
cube. The second case is when $V_\omega$ is a Gaussian process
\cite{[FHLM],[HLMW]}. Finally, Tip \cite{[Tip]} proved that the IDS for the
Poisson model is absolutely continuous at high energy under a repulsive
condition on the single-site potential.  

\vspace{.1in}
\noindent
2. {\bf Landau Hamiltonians.}
A nice application of the method of proof of Theorem 1.1
is to the H\"older continuity of the IDS
for the random Landau Hamiltonian outside of a discrete set.
The unperturbed operator $H_L$
on $L^2 ( \R^2 )$ has the form
\begin{equation}
\label{landau1}
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 infinitely degenerate,
isolated eigenvalues $\{ E_j (B) = (2j + 1)B \; | \; j = 0 , 1, \ldots ,
\}$. The IDS $N_0 (E)$ for this model is a piecewise
constant, monotone increasing function (cf.\ e.\ g.\ \cite{[Nakamura]}).
In particular, the IDS $N_0 (E)$ is constant on any open interval of the
form $I_n \equiv ( E_n(B) , E_{n+1} (B) )$.
Consequently, if ${\tilde \Delta} \subset I_n$, the local estimate
(\ref{localtr}) becomes
\beq
\label{localtr3}
Tr E_0^\Lambda({\tilde\Delta})\; \leq \; 2 N_0 (E_+ )
|\tilde\Delta|^{q_1} |\Lambda|,
\eeq
for any $q_1 \geq 0$.

The perturbed family of Landau operators is
\begin{equation}
\label{landau2}
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 $(0 , (2N+1)B )
\backslash \{ E_j (B) \; | \; j = 0 , 1, \ldots , N \}$
\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.  
By applying a slight extension of Theorem 1.1, possible in this case since
(\ref{localtr3}) holds for any $q_1 \geq 0$,
we are able to prove the
following proposition that improves these known results.

\vspace{.1in}
\begin{prop}
The IDS for the random Landau Hamiltonian $H_\omega$, defined in
(\ref{landau1})--(\ref{landau2}) is H\"older continuous on each interval
$I_n = ( E_n(B) , E_{n+1} (B) )$ with any H\"older exponent $0 < q < 1$.
\end{prop}

\vspace{.1in}

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 \cite{[CH2]}.  In \cite{[CHK1]},
we proved a general theorem on the global H\"older continuity of the IDS
under assumptions (H1)--(H2),
and under the assumption that the magnetic
flux through a unit square is $2 \pi$ times a rational
number.
The proof of this result relies on the methods of the proof of Theorem 1.6.
In the absence of the rational flux assumption, we have been able to prove
that this IDS is continuous at all energies \cite{[CHK2]}. 
It is still an open problem to
prove the H\"older continuity of the IDS at all energies with no flux
condition.

\vspace{.1in}
\noindent
3. {\bf Regularity.} 
Very little is known about the regularity of the IDS for the models
discussed here. Based on results for lattice models (cf.\
\cite{[BCKP],[CL],[CraigSimon],[PF]},
it is clear that the smoothness of the IDS is related
to the smoothness of the probability density $h_0$.
Consequently, one must strengthen hypothesis (H2) and require some
smoothness on the distribution $h_0$. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%% BIBLIOGRAPHY %%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\bibliographystyle{amsalpha}
\begin{thebibliography}{A}

\bibitem{[BY]} M.\ S.\ Birman, D.\ R.\ Yafaev: The spectral shift
  function: The work of M.\ G.\ Krein and its further development,
  {\it St.\ Petersburg Mathematics Journal} {\bf 4}, 833--870 (1992).

\bibitem{[BCKP]} A.\ Bovier, M.\ Campanino, A.\ Klein, F.\ Perez:
Smoothness of the density of states in the Anderson model at high disorder,
{\it Commun.\ Math.\ Phys.} {\bf 114}, 439--461 (1988).

\bibitem{[CL]} R. Carmona, J. Lacroix, {\it Spectral theory of random
    Schr{\"o}dinger operators}, Boston: Birkha{\"u}ser, 1990.

\bibitem{[CH1]} J.\ M.\ Combes, P.\ D.\ Hislop: Localization for some
  continuous, random Hamiltonians in d-dimensions, {\it J.\ Funct.\
    Anal.} {\bf 124}, 149 - 180 (1994).

\bibitem{[CH2]} J.\ M.\ Combes, P.\ D.\ Hislop: Landau Hamiltonians
  with Random Potentials: Localization and the Density of States, {\it
    Commun.\ Math.\ Phys.} {\bf 177}, 603--629 (1996).

\bibitem{[CHK1]} J.\ M.\ Combes, P.\ D.\ Hislop, F.\ Klopp, H\"older
continuity of the integrated density of states for some random operators at
all energies, to appear in {\it International Mathematical Research
Notices}, 2002.

\bibitem{[CHK2]} J.\ M.\ Combes, P.\ D.\ Hislop, F.\ Klopp: A note on the
global continuity of the integrated density of states for random Landau
Hamiltonians, in
preparation.

\bibitem{[CHKN]} J.\ M.\ Combes, P.\ D.\ Hislop, F.\ Klopp, S.\
  Nakamura: The Wegner estimate and the integrated density of states
  for some random operators, {\it Proc.\ Indian Acad.\
    Sci.\ (Math.\ Sci.)} {\bf 112}, 31--53 (2002).

\bibitem{[CHN]} J.\ M.\ Combes, P.\ D.\ Hislop, S.\ Nakamura: The
  $L^p$-theory of the spectral shift function, the Wegner estimate,
  and the integrated density of states for some random operators, {\it
    Commun.\ Math.\ Phys.} {\bf 218}, 113--130 (2001).

\bibitem{[CHT]} J.\ M.\ Combes, P.\ D.\ Hislop, A.\ Tip: Band edge
  localization for acoustic and electromagnetic waves in random media,
  {\it Ann.\ Inst.\ H.\ Poincar{\'e}} {\bf 70}, 381--428 (1999).

\bibitem{[CraigSimon]} W.\ Craig, B.\ Simon: Log-H\"older continuity of the
integrated density of states for stochastic Jacobi matrices, {\it Commun.\
Math.\ Phys.} {\bf 90}, 207--218 (1983).

\bibitem{[DIM]} S.\ Doi, A.\ Iwatsuka, T.\ Mine: The uniqueness of the
  integrated density of states for the \Schr\ operator with magnetic
  field, to appear in {\it Math.\ Z.} 2001.

\bibitem{[FK1]} A. Figotin, A. Klein: Localization of classical waves
  I: Acoustic waves, {\it Commun.\ Math.\ Phys. } {\bf 180}, 439--482
  (1996).

\bibitem{[FK2]} A. Figotin, A. Klein: Localization of classical waves
  II.: Electromagnetic Waves, {\it Commun.\ Math.\ Phys. } {\bf 184},
  411--441 (1997).

\bibitem{[FHLM]} W.\ Fischer, T.\ Hupfer, H.\ Leschke, P.\ M{\"u}ller:
  Existence of the density of states for multi-dimensional continuum
  \Schr\ operators with Gaussian random potentials, {\it Commun.\
    Math.\ Phys.}  {\bf 190}, 133--141 (1997).

\bibitem{[Giere]} E.\ Giere: Wegner estimate in one dimension for
nonoverlapping single site potentials, preprint 2002.

\bibitem{[HLMW]} T.\ Hupfer, H.\ Leschke, P.\ M{\"u}ller, S.\ Warzel: The
  absolute continuity of the integrated density of states for magnetic
  \Schr\ operators with certain unbounded random potentials, {\it
    Commun.\ Math.\ Phys.} {\bf 221}, 229--254 (2001).

\bibitem{[Kirsch1]} W.\ Kirsch: Random \Schr\ operators: A course, in
  {\it \Schr\ operators, Sonderborg DK 1988}, ed.\ H.\ Holden and A.\
  Jensen, Lecture Notes in Physics {\bf 345}, Berlin: Springer 1989.

\bibitem{[KS]} S.\ Kotani, B.\ Simon: Localization in general one
  dimensional systems. II, {\it Commun.\ Math.\ Phys.} {\bf 112},
  103-120 (1987).

\bibitem{[Nakamura]} S.\ Nakamura: A remark on the Dirichlet-Neumann
  decoupling and the integrated density of states, {\it J.\ Func.\
    Anal.} {\bf 179}, 136--152 (2001).

\bibitem{[PF]} L. Pastur, A. Figotin: {\it Spectra of random and
    almost-periodic operators}, Berlin: Springer-Verlag, 1992.

\bibitem{[ReedSimon4]} M.\ Reed, B.\ Simon: {\it Methods of Modern
    Mathematical Physics, vol. IV: {A}nalysis of {O}perators}.
  Academic Press, 1978.

\bibitem{[Robert1]} D.\ Robert: Semiclassical asymptotics for the spectral
shift function, in Differential Operators and Spectral Theory, V.\ Buslaev,
M.\ Solomyak, D.\ Yafaev, eds., American Mathematical Society Translations
{\bf 189}, Providence, RI: American Mathematical Society, 1999.

\bibitem{[Robert2]} D.\ Robert: Semiclassical approximation in quantum
mechanics. A survey of old and recent mathematical 
results, {\it Helv.\ Phys.\ Acta}
{\bf 71}, 44--116 (1998).

\bibitem{[Simon2]} B.\ Simon: {\it Trace ideals and their
    applications}.  London Mathematical society Lecture Series {\bf
    35}, Cambridge: Cambridge University Press, 1979.

\bibitem{[Tip]} A.\ Tip: Absolute continuity of the integrated density of
states for the quantum Lorentz gas for a class of repulsive potentials,
{\it J.\ Phys.\ A} {\bf 27}, 1057--1069 (1994).

\bibitem{[Wang1]} W-M.\ Wang: Microlocalization, percolation, and
  Anderson localization for the magnetic Schr{\"o}dinger operator with a
  random potential, {\it J.\ Funct.\ Anal.} {\bf 146}, 1--26 (1997).

\bibitem{[Wang2]} W-M.\ Wang: Supersymmetry and density of states of the
magnetic Schr{\"o}dinger operator with a random potential revisited,
{\it Commun.\ Part.\ Diff.\ Eqns.} {\bf 25}, 601--679 (2000).

\bibitem{[Wegner]} F.\ Wegner: The density of states for disordered
  systems, {\it Zeit.\ Phy.\ B} {\bf 44}, 9--15 (1981).

\bibitem{[Wolff]} T.\ Wolff: Recent work on sharp estimates in
  second-order elliptic unique continuation problems, in {\em Fourier
    analysis and partial differential equations (Miraflores de la
    Sierra, 1992)}, pages 99--128. CRC, Boca Raton, FL, 1995.

\bibitem{[Yafaev]} D.\ R.\ Yafaev: Mathematical Scattering Theory:
  General Theory, Translations of Mathematical Monographs {\bf 105},
  Providence, RI: American Mathematical Society, 1992.

\end{thebibliography}



\end{document}
---------------0211141123143--
