## Grants and Contracts Details

### Description

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.

Status | Finished |
---|---|

Effective start/end date | 9/15/11 → 8/31/16 |

### Funding

- National Science Foundation: $199,093.00

## Fingerprint

Explore the research topics touched on by this project. These labels are generated based on the underlying awards/grants. Together they form a unique fingerprint.