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

Arriving Sunday, April 19 and departing Friday, April 24, 2009

Organizers: Jean Bellissard (Georgia Institute of Technology), Peter Hislop (University of Kentucky), Abel Klein (University of California, Irvine), Gunter Stolz (University of Alabama at Birmingham).

Confirmed Participants

Press Release: Random Schrodinger Operators: Universal Localization, Correlations, and Interactions

Information for Participants

Schedule and Abstracts (PDF file)

Mailing List

Final Report (PDF file)

Workshop Videos

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.

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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.