Phenomena in random Schrodinger operators

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Random Schrödinger operators describe an electron moving in a medium with random impurities. In the widely accepted picture, in three or more dimensions there exists a transition from an insulator region, characterized by localized states, to a very different metallic region, characterized by extended states, while in one or two dimensions there are only localized states. This proposal aims to further the mathematical understanding of this picture and of related topics. Arguably the most important open question is the existence of delocalization in three or more dimensions. Since localization has only been proved by a multiscale analysis (or by the fractional moment method, if applicable), in practice the region of localization is the spectral region where the multiscale analysis can be performed. We propose to make progress on delocalization by proving the existence of a spectral region where the multiscale analysis breaks down, i.e., some consequence of the multiscale analysis is violated, and by showing delocalization properties (e.g., transport) in this region. We will continue our study of the ac-conductivity in linear response theory for the Anderson model to obtain insight on localization and delocalization. We will investigate the continuity of the density of states for Schrödinger operators in four or more dimensions, and for Schrödinger operators with a magnetic field in two or more dimensions. We will extend the bootstrap multiscale analysis to the multi-particle continuous Anderson Hamiltonian, an interacting multi-particle random Schrödinger operator, obtaining Anderson localization with finite multiplicity of eigenvalues, dynamical localization, decay of eigenfunction correlations, etc. We will also investigate localization for multi-particle continuous Anderson Hamiltonians with no assumptions on the single site probability distribution except for compact support, allowing for Bernoulli and other singular single site probability distributions. We will investigate localization for the (discrete) Anderson model in two or more dimensions when the single-site potential is a Bernoulli random variable, a longstanding open problem.Random Schrödinger operators describe an electron moving in a medium with random impurities. In the presence of impurities, a material that normally acts like a metal (i.e., it conducts electric current) will exhibit localization and behave like an insulator for electric currents. The impurities create a metal-insulator transition with important practical consequences. This research will contribute to the understanding of electronic phenomena in condensed matter physics, such as Anderson localization and the quantum Hall effect. Some of the topics of research are suitable for PhD theses, and will be used for the training of future researchers.