Random Schrödinger Operators: Universal Localization, Correlations, and Interactions (09w5116)

Objectives

The theory of random Schrodinger operators has been a very active
field of research during the last few years. Since the workshop {it
Order, Disorder, and Transport: Advances in the theory of
Schr"odinger operators}, held at BIRS in September 2005, many
advances have been made and new topics explored. We propose a
workshop for 2009 in order to capitalize on these advances and
discoveries and to help conserve the current strong momentum in the
field. As done at the 2005 workshop, the proposed meeting will bring
together leading international researchers to share and discuss
recent progress as well as highlight the work of young researchers
in the field. The latter are indicated by an asterisk($^*$) in the
list of possible participants below.

In the following we describe some of the recent advances and their
relation to the objectives of the workshop. The section concludes
with an outline of the main research topics which will be addressed.

The recent paper of Bourgain and Kenig (Invent. Math. 2005) on
localization at the bottom of the spectrum for random
Schr"o-dinger operators with a singular Bernoulli distributions
was a break-through that has subsequently lead to several new
results. The Bourgain-Kenig paper introduced a notion of free sites,
a scale-dependent Wegner estimate, and a new single-energy
multiscale analysis that allows one to prove localization under very
weak conditions. This paper was the stimulus for the proof of
localization for random Schr"odinger operators with Poisson
potentials by Germinet, Hislop, and Klein (JEMS 2007) and recently
announced work on universal Anderson localization. One
of the goals of the workshop will be to further explore the
consequences of these works that allow one to better understand the
large-scale effects of disorder.

There has been recent progress in the understanding of correlations
and transport for random operators. The first proof of nontrivial
transport for two-dimensional Landau Hamiltonians with random
potentials was given by Germinet, Klein, and Schenker (Ann. of
Math. 2007). There has also been recent analysis of the
current-current and higher-order correlation functions.

The ground breaking work of Molchanov and Minami in the 80s and 90s
opened the door to the explorations of energy-level statistics for
random Schr"o-dinger operators. Although energy-level statistics
have been long studied for various random matrix ensembles, and,
recently, for CMV matrices, very little is known for random
Schr"odinger operators. It is expected that for energies in the
localization regime, the eigenvalue spacings are uncorrelated, that
is, there is no repulsion between energy levels. This is quantified
in the statement that the rescaled energy levels near a fixed energy
are distributed according to a Poisson distribution in the
infinite-volume limit. Molchanov proved this for a class of
one-dimensional random models at any energy. Minami also proved
Poisson statistics for the Anderson model on the lattice in any
dimension provided the energy is in the localization regime where
one has a bound on the expectation of a fractional moment of the
Green's function. There have been recent new proofs of Minami's main
correlation result, yet any extension to continuum models is still
unproven. The nature of eigenvalue statistics for other energy
regimes remains an open problem. One expects a Dyson-Wigner
distribution at energies for which there is transport.

The study of higher-order correlation functions has long been
advocated by Pastur. An understanding of these measures is essential
for studying the transport properties of random Schr"odinger
operators. There has been a lot of work on the first correlation
function, that is the density of states. One remaining open problem
is the regularity of the density of states. The fundamental question
is how the Laplacian smooths the density of states of the random
potential. Regularity for higher-order correlation functions is
largely unknown.

The Kubo formula for AC and DC conductivity has been proven recently
in various circumstances. The DC conductivity was established in a
controlled derivation of linear response theory for time-reversal
invariant systems and those with magnetic fields. When the Fermi
level is in the localization regime, the DC conductivity vanishes.
There has been much work on the Liouvillian and on models with
dissipation. The AC conductivity was studied and an upper bound on
the AC conductivity, similar to a behavior predicted by Mott, was
proved by Klein, Lenoble and M"uller (Ann. of Math., to appear).

Recently, Chulaevsky and Suhov gave a proof of the persistence of
localization for two interacting electrons on the one-dimensional
lattice with random potentials. Wegner estimates for Anderson models
with finitely many electrons in any dimension were provided. One of
the main mathematical challenges is understanding the new
correlations that occur because of the lack of independence due to
the interaction of electrons.

vspace{.1in}

noindent
AN OUTLINE OF RESEARCH TOPICS

begin{enumerate}
item UNIVERSALITY OF ANDERSON LOCALIZATION

begin{enumerate}
item Random Schr"odinger Operators with Singular Probability Distributions

noindent
Topics include: exploiting results of Bourgain and Kenig on Anderson
localization, applications to discrete models, role of unique continuation

item Eigenvalue Statistics

noindent Topics include: universality of Poisson statistics in the
localization regime, band matrices, transition in energy-level
statistics for dimer models and CMV matrices, relations with random
matrix theory

end{enumerate}

item CORRELATIONS

noindent Topics include: current-current correlation measure, its
regularity and properties on the diagonal, regularity of the
higher-order measures

item TRANSPORT PROPERTIES

noindent Topics include: Kubo formula for conductivity, Mott
formula, linear response theory, dissipation theory

item INTERACTIONS

noindent Topics include: Wegner estimates for multiparticle
systems, fixed density systems, correlations

end{enumerate}

vspace{.1in}

noindent SUGGESTED WORKSHOP DATES: Preference for September,
otherwise anytime from April to mid July and from September to
November.