Challenges in the Theory of Random Schrodinger Operators

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Challenges in the theory of random Schrodinger operatorsPeter D. HislopUniversity of KentuckyAbstractThe spectral and transport properties of random Schrodinger operators have been the object of intense study. Anderson localization, the occurrence of dense pure point spectrum almost surely, has been proved for many models at band-edges and at the bottom of the spectrum. Refined estimates give precise information about the decay of the eigenfucntions and the dynamical localization of the system. Conductivity properties of the system are described through the second-order current-current correlation function.Study of these correlation functions reveal information about Mott conductivity, the density of states, and eigenvalue statistics. There are many open questions about the regularity and bounds on these functions. A lower bound on the current-current correlation function implies delocalization, for example. Another new tool for the study of these systems is the use of random matrix theory. This promises to give insight into the density of states in the delocalized regime.Random Schrodinger operators provide a model for the propagation of electrons in perfect crystalline structures that are corrupted by impurities randomly distributed in the medium. It is hoped that the study of these models reveals the mechanisms for finite conductivity at low temperatures and the integer quantum Hall effect. New advances allow one to investigate the transport properties of these models as expressed through correlation functions. These correlations functions describe physically measurable quantities such as the density of states and the Mott conductivity.