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%%%%%%%%%%%%%%%%%%%% Title and abstract %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\title{Towards quantum localisation in Gaussian random potentials}
\author{Werner Fischer, Hajo Leschke and Peter M\"uller}
\address{Institut f\"ur Theoretische Physik, Universit\"at
Erlangen-N\"urnberg, Staudtstra{\ss}e 7, D-91058 Erlangen, Germany}
\shorttitle{Letter to the Editor}
\pacs{72.15.Rn, 02.30.Sa, 02.50.Cw}
\date[15 August 1995]
\beginabstract
At sufficiently low energies there is no absolutely continuous spectrum
for a quantum-mechanical particle moving in multi-dimensional
Euclidean space under the influence of certain Gaussian random
potentials.
\endabstract
%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%% Body of paper %%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
Since the pioneering work of Anderson [1] models of a single
quantum-mechanical particle moving under the
influence of a random potential have attracted much
attention from both physicists [2,3] and mathematicians
[4,5]. The central issue is to understand the spectral characteristics
of the associated random Schr\"odinger operators in order to predict physically
interesting quantities such as the (integrated) density of states or
transport coefficients. Considerable progress has been
achieved over the years in proving rigorously the existence of
so-called localised states of one- and higher-dimensional lattice
operators and of one-dimensional continuum operators.
Localisation properties of random Schr\"odinger
operators in $d$-dimensional Euclidean space $ \rz^{d},\, d\ge2 $,
have been derived, however, almost exclusively for random
potentials with an underlying lattice structure, see [6] for the early
work and [7,8] for recent results and further references.
To our knowledge, the only works
devoted to the localisation problem with a truly continuum random
potential deal with a Poisson potential where an additional randomness in
the single-impurity coupling constant had to be assumed [9,7]. Our goal
here is to report on a localisation result for another type of a
truly continuum random potential. More precisely, we will outline a proof for
the absence of the absolutely continuous component in the low-energy
spectrum of Schr\"odinger operators with certain Gaussian random
potentials.
Let $ (\Omega, {\cal F}, P) $ be a complete probability space and let
$ V:\, \Omega \times \rz^{d} \rightarrow \rz,\;
(\omega ,x)\mapsto V_{\omega }(x) $ be a real-valued homogeneous Gaussian
random field indexed by $ \rz^{d} $ with zero
mean, $ \int \d P(\omega )\, V_{\omega }(x) = 0 $, and covariance function
$ C(x):= \int \d P(\omega )\, V_{\omega }(x)V_{\omega }(0) $ obeying, of
course, $ 0 < C(0) < \infty $.
In the following we will need three
assumptions on $ C $. For the formulation of the third one we introduce some
notation. Given a Borel subset $ \Lambda \subset \rz^{d} $, we
denote by $ |\Lambda | := \int_{\Lambda }\d^{d}x $ its $d$-dimensional
Lebesgue measure and by $ {\cal F}_{\Lambda }\subset {\cal F} $ the
sub-sigma-algebra of events generated by the set of random variables
$\{ V_{\cdot}(x):\, x\in \Lambda\} $. For two Borel subsets
$ \Lambda ,\Lambda ' \subset \rz^{d} $ let
$ {\rm dist}(\Lambda ,\Lambda '):=\inf\left\{ |x-x'|_{\infty }:\,
x\in\Lambda ,x'\in\Lambda '\right\} $ be their distance measured in the
maximum norm $ |x|_{\infty } :=\max\{ |x_{j}|:\, 1\le j\le d\} $
of $ x =: (x_{1},\ldots , x_{d}) \in \rz^{d} $. Next we introduce the
quantity $ \kappa (\Lambda ,\Lambda '):=
\sup\left\{ |P(A\cap A') - P(A)P(A')|:\, A\in {\cal F}_{\Lambda }, A'\in
{\cal F}_{\Lambda '}\right\} \le 1/4\, $ which measures the stochastic
dependence of the restrictions of the random field $ V $ to $ \Lambda $
and $ \Lambda ' $, respectively. Finally, for each pair of
reals $ L,G > 0 $ the so-called strong-mixing coefficient [10] of
$ V $ is defined as follows
%
$$
\alpha (L,G):= \sup\left\{ \kappa (\Lambda ,\Lambda '):\,\Lambda ,
\Lambda '\subset \rz^{d};\, {\rm dist}(\Lambda ,\Lambda ') \ge \lambda L;\,
|\Lambda |,|\Lambda '| \le \lambda ^{d} G \right\} \,.
$$
%
Here $ \lambda >0 $ denotes some arbitrary but fixed length scale,
which serves to make $ L $ and $ G $ dimensionless.
Now we are in a position to formulate the above-mentioned\\[2ex]
{\em Assumptions on the covariance.}
\begin{itemize}
\item[(A1)] $ C(x)\ge 0 \; $ for all $ x \in \rz^{d}\,$.
\item[(A2)] There exists a pair of constants $b,\beta >0 $ such that
$$ C(0) - C(x) \le b\, |x|_{\infty }^{\beta } $$
for all $ x $ in a neighbourhood of the origin.
\item[(A3)] There exists a triple of constants $ a,\gamma ,\delta >0 $ obeying
$ \gamma < 1+ \delta /[\delta + 4(d-1)] $ such that
$$ \alpha (l^{\gamma }/4,l^{d})\le a l^{-\delta } \qquad {\rm for~all}\;\;
l>1 \,. $$
\end{itemize}
%
{\em Remarks.}
\begin{itemize}
\item[\em i)]
The H\"older condition at the origin (A2) implies [11, Thm 4.1.1]
the continuity of the realisations of $ V $ almost surely with respect
to the probability measure $ P $.
\item[\em ii)]
The algebraic bound (A3) implies the strong-mixing property [10] of $ V $.
More precisely, $ \lim_{L\to\infty } L^{\mu}\,\alpha (L,G )=0 $ for all
$ G >0 $ and $ \mu < \delta /\gamma $. The strong-mixing property
is considerably stronger than the usually required
ergodicity. It seems to be an open problem to characterise the
covariance functions of strongly mixing Gaussian random fields indexed
by $ \rz^{d}\, ,\;\,d\ge 2 $.
\item[\em iii)]
Taken together, (A2) and (A3) require a compromise between local
dependence and global independence of $ V $.
\item[\em iv)]
In one dimension an example fulfilling (A1) -- (A3) is
$ C(x) = C(0) \exp\{ -|x|/\xi\} $, $ \xi > 0 $. While the first
two assumptions are obviously satisfied, the third one follows from the
general theory developed in [12, ch VI, Thm 6]. A simple class of allowed
covariance functions for all $ d\ge 1 $
may be constructed as follows. Pick a
function $ U:\; \rz^{d}\rightarrow \rz_{+} $ which is
non-negative, compactly supported and H\"older continuous of order
$ \beta > 0 $. Then define $ C(x):= \int \d^{d}x'\, U(x+x')U(x') $.
Again, it is easy to see that the first two assumptions are satisfied.
The third one follows from the fact that $ C $ has compact support.
\end{itemize}
Given a Gaussian random field $ V $ of the above type, the associated
random Schr\"odinger operator -- acting in the Hilbert space $ L^{2}
(\rz^{d}) $ of complex-valued functions on $ \rz^{d}$
which are square-integrable with respect to $ \d^{d}x $ -- is then given as
%
\begin{equation} \label{hamilton}
H_{\omega } := - \frac{1}{2} \Delta + V_{\omega }\,.
\end{equation}
%
Here $ \Delta := \sum_{j=1}^{d} \partial^{2}/\partial x_{j}^{2} $ is
the $ d $-dimensional Laplacian and $ V $ appears as an additive random
potential which acts as a multiplication operator.
Note that we have chosen physical units,
where the square of Planck's constant equals the mass of the point
particle whose dynamics is described by (\ref{hamilton}).
According to standard arguments
[13,4,5], the Schr\"odinger operator (\ref{hamilton}) enjoys
the following properties:
\begin{itemize}
\item[\em i)]
It is almost surely essentially self-adjoint on the dense subspace
$ {\cal C}_{0}^{\infty }(\rz^{d}) \subset L^{2}(\rz^{d}) $, consisting
of arbitrarily often differentiable
complex-valued functions with compact support in $ \rz^{d} $.
\item[\em ii)]
Its spectrum $ \sigma (H_{\omega }) $ equals almost surely the real line
$ \rz $. The spectral components, the absolutely continuous, the singular
continuous and the pure point spectrum, arising in the Lebesgue
decomposition $ \sigma (H_{\omega }) = \sigma_{\rm ac} (H_{\omega })\cup
\sigma_{\rm sc} (H_{\omega })\cup \sigma_{\rm pp} (H_{\omega }) $ are
also almost surely non-random sets.
\item[\em iii)] Its integrated density of states exists
almost surely as a non-random distribution function with
a Gaussian low-energy tail. The topological support of the corresponding
measure is the entire real line $ \rz $.
\end{itemize}
%
Now we are prepared to state the main result.\\[2ex]
%
\begin{samepage}
{\em Theorem.}\\
For the multi-dimensional random Schr\"odinger operator $ H_{\omega } $
given by
(\ref{hamilton}) with a Gaussian random potential satisfying assumptions
(A1) -- (A3), there exists an energy $ E_{0}<0 $ such that
%
$$
\sigma_{\rm ac} (H_{\omega })\; \cap\; ]-\infty ,\,E_{0}] = \emptyset
\qquad {\rm almost~surely.}
$$
\end{samepage}
%
\noindent{\em Outline of proof.}\\
The heart of the proof is a multi-scale analysis in the spirit of
[14]. Its technical realisation is patterned after
[15] and [7] in order to cope with a correlated
random potential and a continuous space, respectively. Here, we will
illustrate only the key points where special properties of Gaussian
random fields require an adaption of the usual strategy and how
assumptions (A1) -- (A3) enter.
%
\begin{itemize}
\item[\em i)] Assumption (A2) enables us to
control the unbounded fluctuations of the Gaussian random potential
restricted to a box $ \Lambda _{l}(x):= \{ x'\in \rz^{d}:
|x' -x|_{\infty } \le \lambda l/2\} $ centred at $ x\in \rz^{d} $
with edges of length $ \lambda l>0 $. For sufficiently large $ l $
one has [11, Thm 4.1.1]
%
\begin{equation}
P\left(\left\{\omega \in\Omega : \sup\limits_{x\in\Lambda _{l}(0)}
|V_{\omega }(x)| \ge v \right\}\right) \le 4^{d+1}\;\exp\left\{
-\,\frac{v^{2}}{2C(0)\left( 1 + 20\sqrt{\ln l}\right)^{2}}\right\}
\end{equation}
%
for all $ v \ge 0 $. This L\'evy-type of
inequality is needed in the base clause of the multi-scale analysis to estimate
the probability that
the norm of the resolvent of the operator $ H_{\omega } $ restricted to
a box of initial length scale $ \lambda l_{0} $
exceeds some given value. It is needed again in the
recursion clause to bound a similar probability involving
the commutator term which arises from the geometric
resolvent equation.
%
\item[\em ii)] The r\^ole of the important
weak-dependence assumption (2.2) in [15] is taken over by our
assumption (A3) together with the estimate
%
\begin{equation}\label{strongmix}
P( A \cap A' ) \le
P( A ) P( A' )
+ \alpha \left({\rm dist}(\Lambda ,\Lambda '),
{\rm max}\{ |\Lambda | , |\Lambda '| \} \right)
\end{equation}
%
for the probability of the joint occurrence of two events
$ A \in {\cal F}_{\Lambda } $ and $ A' \in {\cal F}_{\Lambda '} $.
%
\item[\em iii)] The third ingredient to make the multi-scale analysis
work is a Wegner-type of estimate. For a given triple of constants
$ E_{w}\in \rz $, $ \varepsilon _{w}>0 $ and $ l_{w}>0 $ there exists a
constant $ 0< W <\infty $ such that
%
\begin{equation} \label{wegner}
P\left(\left\{\omega \in\Omega : {\rm dist}\left( \sigma (H_{\omega ,%
\Lambda _{l}(0)}), \{ E\}\right)\le\varepsilon \right\}\right)
\le \varepsilon l^{2d} W
\end{equation}
%
for all $ E\le E_{w}$, $ 0\le \varepsilon \le\varepsilon _{w} $ and
$ l\ge l_{w} $. Here $ H_{\omega ,\Lambda } $ denotes the Dirichlet
restriction of $ H_{\omega } $ to a box $ \Lambda \subset \rz^{d} $.
The first steps in our derivation of (\ref{wegner}) are borrowed from
Wegner's original work for lattice models with independent random
potentials [16], see also, for example, [6].
Due to (A1) the occurring average of the
partial functional derivative $ \delta /\delta V_{\omega }(x) $ of the
(mollified) integrated density of states of $ H_{\omega ,\Lambda_{l}(0) } $
may be bounded from above in such a way that a functional integration by
parts with respect to the Gaussian probability measure [17, Thm 6.3.1] can be
performed. A final appeal to the existence and non-randomness of the
integrated density of states then leads to (\ref{wegner}).
\end{itemize}
%
Having adapted the multi-scale analysis, we follow the proof of
[7, Thm 2.1] to bound the infinite-volume resolvent locally in the usual
operator norm $ ||\cdot || $. More precisely, we get that there exists a
constant $ E_{0}< 0 $ such that
%
\begin{equation}
\sup\limits_{\varepsilon >0} || (H_{\omega } -E -
\i\varepsilon )^{-1} \one_{\Lambda } || < \infty
\end{equation}
%
for all compact $ \Lambda \subset \rz^{d} $ and $ \d E \otimes \d P
(\omega ) $-almost all $ (E,\omega )\in\, ]-\infty , E_{0}]\times\Omega $.
The outline of the proof of the Theorem is now completed by referring to the
de La Vall\'ee-Poussin theorem, see,
for example, [4, pp 16,17] and [5, Thm A.10].\\[2ex]
%
\noindent {\em Remarks.}
\begin{itemize}
\item[\em i)] Unfortunately, we have not yet been able to rule out the
singular continuous spectrum for sufficiently low energies, even though we can
show an algebraic decay in $ {\rm dist}(\Lambda ,\Lambda ') $ of
$ \sup\limits_{\varepsilon >0} || \one_{\Lambda '}
(H_{\omega } -E -\i\varepsilon )^{-1} \one_{\Lambda } || $
for $ \d E \otimes \d P (\omega ) $-almost all $ (E,\omega )\in\,
]-\infty , E_{0}]\times\Omega $.
%
\item[\em ii)] The above Theorem remains true for the case where
$ H_{\omega } $ is generalised to include a periodic potential and/or a
constant magnetic field. For more stringent localisation results in
these cases with, however, a random potential having an underlying
lattice structure, see [8,18--20]. The Theorem should also remain true
in the presence of a homogeneous and ergodic random vector potential which
is sufficiently smooth. The almost-sure constancy of the spectral
components, the existence and certain asymptotic properties of the
integrated density of states in this case have recently been proven
in [21,22].
\end{itemize}
%
%%%%%%%%%%%%%%%%%%%%%%%%%% Acknowledgements %%%%%%%%%%%%%%%%%%%%%%%
%
\ack{This work was partially supported by the Deutsche
Forschungsgemeinschaft and by the Human Capital and Mobility
Programme ``Polarons, bi--polarons and excitons. Properties and
occurrence in new materials.'' of the European Community.}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%% References %%%%%%%%%%%%%%%%%%%%%%%%%
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