This research concerns the spectral and transport properties on one-and many-particle quantum Hamiltonians with random potentials. The goal is to understand the behavior of the correlation functions for these models in the localized and transport regimes. The first-order correlation function, the density of states, has been extensively studied. One part of the project is to prove its smoothness and a lower bound on the density. Of special interest is the second-order correlation function called the current-current correlation measure. One component of the project is to prove the existence of a density for this measure and to study its properties. This is important as the diagonal values of the density give the conductivity. Other second- and higher-order correlation functions arise in the study of eigenvalue statistics for random Schrodinger operators. These correlation functions and related ones for certain random matrix models will also be explored. The integer quantum Hall effect is closely related to these ideas and the project involves a study of quantum Hall systems using nonequilibrium stationary states.
The propagation of electrons in solids has long been an important focus of condensed matter physics. Quantum mechanics has been successful in providing an understanding of metals, semiconductors, and insulators. The nature of finite conductivity remains elusive. It is believed that the natural disorder of crystals, due to dislocations and defects that are observed in nature, is partially responsible for the transport properties of crystals. The focus of this research is to understand the mathematical and physical basis for finite conductivity by modeling crystals as an ordered array with random impurities. Experimentally observable quantities, such as the density of states and the conductivity, can be estimated in these models.
