Topics in the theory of random Schrodinger operators

This proposal describes the continuing investigations of the principal investigator (PI) into the spectral and transport properties of random Schr¨odinger operators. Intellectual merits of the proposed activity: The basic questions addressed in the proposal center on understanding the effects of randomness on the physical characteristics of the system, such as the density of states, conductivity and the current-current correlation function, and eigen- value statistics. The PI will explore 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. Beyond the localization regime: the density of states and the Minami estimate 2. Eigenvalue statistics: the weak disorder limit 3. Conductivity and correlation measures for random Schr¨odinger operators 4. Density of states for random Schr¨odinger operators: regularity, weak disorder, and N-body systems A new method for decoupling finite-volume systems using the Dirichlet-to-Neumann map is proposed that promises to give lower bounds on the density of states for random Schr¨odinger op- erators and the ac conductivity at zero temperature in agreement with the Mott formula. Current investigations have centered on studying phase transitions for random Schr¨odinger operators. One way these are manifest is through transitions in the local and global eigenvalue statistics. Various aspects of the eigenvalue statistics for random Schr¨odinger operators are described, such as the behavior in the weak disorder limit where numerical evidence suggests a transition. Eigenvalue statistics in the localization regime depend upon the Minami estimate and extensions of this esti- mate to finite-rank models are described. Transport in disordered systems is described through the correlation functions of the system. The PI proposes continuation of studies of their analyticity and their diagonal behavior as functions of the energy parameters. The goal is to understand the nature of Mott conductivity and its relation to the localization length. The weak disorder limit of the density of states is described for one-dimensional systems in terms of the invariant measure and for high-dimensional systems through a suitably-rescaled Wegner estimate. The PI also de- scribes approaches to higher-order regularity for the density of states and Lifschitz continuity for the integrated density of states for N-body random Schr¨odinger operators. Broader impact of the proposed research: The theory of disordered systems is of importance in condensed matter physics. The fundamental questions addressed in the proposal clarify the role of disorder in electronic transport. The PI is active in doctoral and post-doctoral education, and in promoting the field through summer school and conference participation and organization. Two of the PI’s doctoral students recently completed their degrees and funds are requested for summer support for the PI’s current doctoral students.