Challenges in the Theory of Random Schrodinger Operators

This proposal concerns continuing investigations of the principal investigator into the spectral and transport properties of random media. The basic questions center on understanding the effect of randomness on the physical characteristics of the system, such as the density of states, the current-current correlation function, and the Mott conductivity. The principal investigator proposes to describe these observable properties of the random system in various parameter regimes, such as the strong localization regime and the weak disorder regime. The main aspects of the proposal include discussions and conjectures concerning: 1. Regularity of the Integrated Density of States; 2. Mott Conductivity and the Higher-Order Correlation Functions; 3. Extended States, Edge Currents and Conductance in the Quantum Hall Effect; 4. Random Matrix Theory and the Anderson Model; 5. Schr¨odinger Operators with Random Magnetic Fields. A new method for estimating the expectation of the spectral shift function is proposed that will imply the Lipschitz continuity of the integrated density of states. This opens the way for the study of regularity properties of the density of states. We also propose the study of some new models, homogeneous random potentials, that allow randomly distributed impurities. The treatment of these models relies on a quantitative unique continuation principle. We study transport through the correlation functions of the system, studying their analyticity and their diagonal behavior as functions of the energy parameters. The goal is to understand the nature of Mott conductivity. We also continue to examine the relation between the geometry of confining barriers and the quantum Hall effect. These models of quantum devices display interesting transport and spectral properties.