Topics in the theory of random Schrodinger operators

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One- and many-particle random operators describe the interaction of electrons and acoustic waves with randomly perturbed media. The primary example is the propagation of electrons in a crystal with impurities. The impurities are modeled by a random process and the expectations of observable quantities, such as diffusivity, conductivity, and the density of states, represent the average over these quantities over many measurements. Several models are studied: the impurities may be located at the lattice points of the crystal representing alloy-type models, or they may be localized at interstitial sites whose location is given by a Poisson process. In a perfect crystal, the electrons propagate freely provided their energies lay in fixed energy bands determined by the crystalline structure. The spectrum is absolutely continuous and the conductivity is infinite. The addition of random impurities changes this description. If the disorder is sufficiently large, all electron states are localized in space and the conductivity is zero. At weak disorder, and in dimensions greater than two, it is believed that there is finite conductivity at certain energies, whereas there are localized states at other energies near the edges of the bands. This is an open conjecture and one of the main motivations for the proposed research. The main projects of this proposal concern spectral and transport aspects of random one- and many-body random Schrodinger operators. The regularity of the density of states and higher-order correlation functions, including the conductivity measure and the inter-band light absorption coefficients, will be thoroughly investigated. The behavior of these quantities in the weak disorder regime is particularly interesting. Very little is known for even the simplest correlation functions, such as the density of states. The dynamical properties of random Schrodinger operators are also manifest in the spectral statistics of various models. For many one-particle Schrodinger operators, the local spectral statistics in the disorder regime is Poissonian. This is expected for many-particle Schrodinger operators and is part of the proposed research. The propagation of electrons and waves in random media is a hallmark of the physical world. Random impurities in semiconductors impact the electronic properties of these devices. Although the one-electron model is a good description of many phenomena, a complete picture requires the use of multi-particle Schrodinger operators. The behavior of these random models is just beginning to be studied and is one of the components of this research proposal. The investigator, together with several doctoral students at the University of Kentucky, will investigate transport and spectral phenomena of several random models. One of the main goals is to understand correlations between various particles and how the disorder influences them.