Correlations and Transport for Random Schrodinger Operators

This proposal concerns continuing investigations of the principal investigator (P1) into the spectral and transport properties of random Schrodinger operators. Intellectual Merits of Proposed Activity: The basic questions center on understanding the effect of randomness on the physical characteristics of the quantum system, such as the den- sity of states, the current-current correlation function, and the DC conductivity. Transport is studied through the correlation functions of the system, including their regularity and diagonal behavior as functions of the energy parameters. The P1 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. Correlation Measures for Random Schrddinger Operators 2. Bounds for, and Regularity of, the Density of States; 3. Eigenvalue Statistics for Random Operators 4. Related Topics: Random Quantum Graphs and Nonequilibrium Currents A systematic study of the correlation measures associated with random Schrodinger oper- ators is proposed, with an emphasis on the first and second moments. The properties of the current-current correlation measure are essential for understanding conductivity. The second moment is also related to the question of eigenvalue level statistics. The P1 discusses a new proof of a technical correlation estimate and its application to certain random matrix models. In a continuation of work on the integer quantum Hall effect and edge currents, the P1 proposes models in nonequilibrium statistical physics that provide an understanding of the nature of equilibrium currents and their dependence on the random potential. Broader Impacts of the Proposed Activity: The proposed work has implications for nanotech- nology and semiconductor devices. It is important for design work to understand the effects of disorder on the function of such devices. Some of the proposed research touches on mod- elling questions. The PT is active in disseminating his work through lectures and graduate education. The proposed grant would provide funds for research visitors who interact with the the PDE group and graduate students in the Department. The P1 currently has two doctoral students and requests summer support for them.