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Scattering Theory, Random Schr dinger Operators, Extended States
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%TCIDATA{Created=Mon Nov 29 15:07:17 1999}
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\begin{document}


\title{Scattering Theory for Sparce Random Potentials\\}
\author{Werner Kirsch\\Institut f\"{u}r Mathematik and SFB 237\\Ruhr-Universit\"{a}t Bochum\\D 44780 Bochum, Germany\\[0.3cm]\\\medskip werner.kirsch@mathphys.ruhr-uni-bochum.de}
\date{}
\maketitle



\begin{center}
\bigskip
\end{center}

\underline{Abstract:} We prove the existence of wave operators and thus of
absolutely continuous spectrum for a huge class of random Schr\"{o}dinger
operators with randomness decaying at infinity in dimension $d\geq3$.

\bigskip

\section{Introduction}

In scattering theory one usually considers potentials that decay at infinity
with a specified rate. A typical result is the following: If the potential $V$
decays faster then the Coulomb potential (i.e. $\left|  V(x)\right|  \leq
C\left(  1+\left|  x\right|  \right)  ^{-(1+\varepsilon)})$ and is locally not
too singular, then the wave operators $\Omega_{\pm}=s-\lim e^{itH}e^{-itH_{0}%
}$ exist. Here, as usual, we set $H_{0}=-\Delta$ and $H=H_{0}+V$ (see
virtually any textbook on scattering theory, for example \cite{RS},
\cite{CFKS} or \cite{Yaf}). For definiteness let's assume that $(1+\left|
x\right|  )^{1+\varepsilon}V(H_{0}+1)^{-1}$ is a compact operator. The
existence of the wave operators implies that $\sigma_{ac}(H_{0})\subset
\sigma_{ac}(H)$ (with $\left[  0,\infty\right)  =\sigma(H_{0})=\sigma
_{ac}(H_{0})$ in this case). The above assumptions imply that $V$ is
relatively compact, so $\sigma_{ess}(H)=\sigma_{ess}(H_{0})=\left[
0,\infty\right)  $. Similar results hold for quantum mechanics on the lattice
$\mathbb{Z}^{d}$, i.e. for $H_{0}=-\Delta$ on $L^{2}(\mathbb{R}^{d})$ replaced
by the finite difference Laplacian on $\ell^{2}\left(  \mathbb{Z}^{d}\right)
$. In this case $H_{0}$ is given by $H_{0}u\left(  n\right)  =-\sum_{\left|
i\right|  =1}\left(  u(n+i)-u(n)\right)  $ (see \cite{Rieb} or \cite{CFKS}).

It was Krishna (\cite{K1} and \cite{K2}) who observed that for a
\textsl{random} potential $V_{\omega}$ on $\ell^{2}\left(  \mathbb{Z}%
^{d}\right)  $ it is sufficient that a certain \textsl{average} of $V_{\omega
}$ goes to zero at infinity rather than $V_{\omega}$ itself. This leaves place
for potentials which are not relatively compact, as was observed in
\cite{KKO}. In fact, there examples of random potentials with ''decaying
randomness'' are constructed, which exhibit a transition from pure point to
absolutely continuous spectrum.

In this paper we extend Krishna's result about the existence of absolutely
continuous spectrum to a large class of potentials with decaying randomness
giving a very simple argument.

\section{Main Result}

Denote by $H_{0}$ the free Schr\"{o}digeroperator on $L^{2}(\mathbf{R}^{d})$
(resp. $\ell\left(  \mathbf{Z}^{d}\right)  $). Suppose $V_{\omega}(x)$ is a
random potential on $\mathbf{R}^{d}$ (resp. on $\mathbf{Z}^{d}$) such that the
function%
\[
W(x):=E\left(  V_{\omega}(x)^{2}\right)  ^{\frac{1}{2}}%
\]
is finite for (Lebesque-) almost all $x\in\mathbf{R}$ (resp. all
$x\in\mathbf{Z}^{d}$).

\begin{definition}
$V_{\omega}(x)$ satisfies the\emph{\ averaged Cook condition} if
\[
\int_{T}^{\infty}\left\|  W\;e^{\mp\;itH_{0}}\;\varphi\right\|  dt<\infty
\]
for a $T\geq0$ and for all $\varphi$ in a dense set $D_{0}$ in $L^{2}$.
\end{definition}

\begin{theorem}
If $V_{\omega}$ satisfies the averaged Cook condition then the wave operators
$\Omega_{\pm}$  for the random potential $V_{\omega}$ exist for almost all
$\omega$. Consequently%
\[
\sigma(H_{0})\subset\sigma_{ac}(H_{\omega})
\]
almost surely.
\end{theorem}

\begin{proof}
By Cook's method (see e.g.\cite{CFKS}) is suffices to prove that%
\[
\int_{T}^{\infty}\left\|  V_{\omega}\;e^{\mp\;itH_{0}}\;\varphi\right\|
dt<\infty
\]
for almost all $\omega$, wich is certainly true if the expectation of this
quantity is finite.%
\begin{align*}
&  E\left(  \int_{T}^{\infty}\left\|  V_{\omega}\;e^{\mp\;itH_{0}}%
\;\varphi\right\|  dt\right)  \\
&  =\int_{T}^{\infty}E\left(  \int\left|  V_{\omega}(x)\right|  ^{2}\;\left|
e^{\mp\;itH_{0}}\;\varphi(x)\right|  ^{2}dx\right)  ^{\frac{1}{2}}dt\\
&  \leq\int_{T}^{\infty}\left(  \int\left(  E\left(  V_{\omega}(x)^{2}\right)
\right)  \;\left|  e^{\mp\;itH_{0}}\;\varphi(x)\right|  ^{2}dx\right)
^{\frac{1}{2}}dt\\
&  =\int_{T}^{\infty}\left\|  W\;e^{\mp\;itH_{0}}\;\varphi\right\|  dt<\infty
\end{align*}
\end{proof}

\section{Examples}

We give a few examples which illustrate the theorem.

\begin{description}
\item [ a)]Set
\[
V_{n}=a_{n}\ q_{n}%
\]
with $q_{n}$ independent, identically distributed random variables
with\ $E(q_{n}^{2})<\infty$\ and\ $a_{n}\geq0$\ a (deterministic) sequence
with $a_{n}<C_{1}\left(  1+\left|  n\right|  \right)  ^{-\alpha}$\ for
an\ $\alpha>1$.

Then
\[
W_{n}=E\left(  V_{n}^{2}\right)  ^{\frac{1}{2}}\leq C_{2}\left(  1+\left|
n\right|  \right)  ^{-\alpha}\text{\ .}%
\]
By a result of \cite{Rieb}\ $W_{n}$ satisfies the Cook condition.\medskip

\item[ b)] Consider
\[
V_{\omega}(x)=\sum q_{n}(\omega)f_{n}(x)\;\text{.}%
\]
Here again $q_{n}$ are independent, identically distributed random variables
with $E(q_{n}^{2})<\infty$ and $f_{n}$ are functions such that%
\[
\sum_{n\in\mathbf{Z}^{d}}\left|  f_{n}(x)\right|  \leq C\left(  1+\left|
x\right|  \right)  ^{-\alpha}\;\;,\;\;\alpha>1\;\text{.}%
\]
Then
\[
V_{0}(x)=\sum E(q_{n})\;f_{n}(x)
\]
satisfies the Cook criterion. To prove that $V_{\omega}$ satisfies this
criterion, we may therefore assume that $E(q_{n})=0\;$.

Then, due to the independence of the $q_{n}$:%
\begin{align*}
W(x) &  =E\left(  V_{\omega}(x)^{2}\right)  ^{\frac{1}{2}}\\
&  =\left(  \sum E\left(  q_{n}^{2}\right)  \;f_{n}(x)^{2}\right)  ^{\frac
{1}{2}}\\
&  \leq E(q_{0}^{2})^{\frac{1}{2}}\sum\left|  f_{n}(x)\right|  \\
&  \leq C^{\prime}\left(  1+\left|  x\right|  \right)  ^{-\alpha}\;\;\text{.}%
\end{align*}

\item[ c)] Let $V_{n}$ be independent random variables with%
\[
P\left(  V_{n}=1\right)  =p_{n}\;,\;P\left(  V_{n}=0\right)  =1-p_{n}\;\text{,
}%
\]
\[
W_{n}=\left(  E\left(  V_{n}^{2}\right)  \right)  ^{\frac{1}{2}}=p_{n}%
^{\frac{1}{2}}\;\text{.}%
\]
So $V_{n}$ satisfies the averaged Cook criterion if%
\[
p_{n}\leq C\left(  1+\left|  n\right|  \right)  ^{-\beta}\;\text{for }%
\beta>2\;\text{.}%
\]

\item[ d)] Let $\xi_{n}$ be random variables as in (c) and%
\[
V_{n}=q_{n}\ \xi_{n}%
\]
with $q_{n}$ independent, identically distributed random variables independent
of the $\xi_{n}$ and $E(q_{n}^{2})<\infty$. Then $V_{n}$ satisfies the
averaged Cook criterion if for some $\beta>2$%
\[
p_{n}\leq C\left(  1+\left|  n\right|  \right)  ^{-\beta}%
\]
Similar models can be considered on $\mathbf{R}^{d}\;$.

\item[ e)] Let $\mu_{\omega}$ be a Poisson random measure with intensity
measure $\rho$. This means that%
\[
P(\mu_{\omega}(A)=k)=\frac{\rho(A)^{k}}{k!}\;e^{-\rho(A)}\;\text{.}%
\]
Set
\[
V_{\omega}(x)=\int f(x-y)\;\mu_{\omega}\left(  dy\right)  \;\text{,\ }%
\]
then
\begin{align*}
&  E\left(  \int\left|  f(x-y)\right|  \;\mu_{\omega}\left(  dy\right)
\right)  \\
&  =\int f(x-y)\;\rho\left(  dy\right)  \;\text{.}%
\end{align*}
Hence $V_{\omega}(x)$ exists almost surely if the latter integral is finite
for almost all $x$.\medskip\newline Moreover
\begin{align*}
&  E\left(  \left(  \int f(x-y)\;\mu_{\omega}\left(  dy\right)  \right)
^{2}\right)  \\
&  =\int\left|  f(x-y)\right|  ^{2}\;d\rho\left(  dy\right)  +\left(  \int
f(x-y)\;\rho\left(  dy\right)  \right)  ^{2}%
\end{align*}
So, the wave operators exist almost surely if
\[
\int\left|  f(x-y)\right|  ^{2}\rho(dy)+\left(  \int f(x-y)\;\rho(dy)\right)
^{2}\leq C\left(  1+\left|  x\right|  \right)  ^{-\alpha}\;\text{.}%
\]
\end{description}

\section{Discussion and Extensions}

It can be shown that for the Anderson model as well as for the alloy type
models b) and d) there is pure point spectrum outside the spectrum of the free
Hamiltonian provided the distribution of the appropriate random values is
smooth enough. For the Anderson model (i.e. on the lattice) this can be done
as in \cite{KKO}. For the continuous models this will be done by a suitable
modification of the multiscale analysis in a subsequent publication \cite{K}.

While the above argument proves the existence of absolutely continuous
spectrum we can not exclude whether this part of the spectrum is mixed with
pure point or singular continuous spectrum. The remarkable paper by Jaksic and
Last \cite{JL} settles this question for the Anderson model. These authors
prove that the spectrum is purely absolutely continuous in the region under
consideration. Unfortunately their methods relies on the technique of rank one
perturbations and is therefore not applicable to the continuous case.

Above we assumed that our random potential is decaying - in an averaged sense
- in all directions. However, it is enough to ensure absolutely continuous
spectrum that the potential ''decays'' in ''enough'' directions \cite{K}. This
will allow us to treat potentials living in a half space thus proving and
extending results of \cite{Carm} as well as potentials concentrated near a
surface (see \cite{PaMo}, \cite{JL2} and references therein)..

Finally we remark that the statements of this paper are valid ''almost
everywhere'', thus giving no hint for a given deterministic potential. However
such deterministic results will be given in \cite{K} as well.

\bigskip

{\large Acknowlegment:\quad} It is a great pleasure to thank Krishna Maddaly,
Stanislav Molchanov and Michael Demuth for valuable discussions on the subject.\bigskip

\begin{thebibliography}{99}
\bibitem{K1}Krishna M.: Anderson model with decaying randomness existence of
extended states, Proc. Indian Academy of Sciences (Math. Sci.) 100, 285 - 294 (1990)

\bibitem {K2}Krishna M. : Absolutely continuous spectrum for sparse
potentials; Proc. Indian Academy of Sciences (Math. Sci.) 103, 333 - 339 (1993)

\bibitem {KKO}Kirsch W., Krishna M., Obermeit J.: Anderson Model with decaying
randomness: Mobility Edge, to appear in Math Z.

\bibitem {CFKS}Cycon H., Froese R., Kirsch W., Simon B.:
\textit{Schr\"{o}dinger Operators}, Springer 1987

\bibitem {Rieb}Riebling C. : Streutheorie f\"{u}r Anderson-artige Modelle,
Diploma thesis, Bochum

\bibitem {RS}Reed M., Simon B.:\textit{ Methods of Modern Mathematical Physics
III}, Plenum 1979

\bibitem {Yaf}Yafaev D.: \textit{Mathematical Scattering Theory: General
Theory}, American Mathematical Society 1992

\bibitem {K}Kirsch W.: in preparation

\bibitem {JL}Jaksic V., Last Y.: Spectral Structure of Anderson Type
Hamiltonians, Preprint

\bibitem {Carm}Carmona R.: One-dimensional Schr\"{o}dinger Operators with
Random or Deterministic Potentials: New Spectral Types, J. Funct. Anal. 51,
229 (1983)

\bibitem {PaMo}Jaksic V., Molchanov S., Pastur L.: On the propagation
properties of surface waves, in: Wave propagation in complex media, IMA MAth
Appl. 96, 13 (1998)

\bibitem {JL2}Jaksic V., Last Y.: Corrugated Surfaces and A.C. Spectrum, preprint
\end{thebibliography}
\end{document}
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