A wide array of mathematical methods (from functional analysis and operator theory, ordinary and partial differential equations, probability and stochastic processes, complex and harmonic analysis) will be used to increase the understanding of random Schrodinger operators. In particular, level repulsion between the eigenvalues of these operators will be studied. Absence of level repulsion for insulating disordered materials will be shown for larger classes of models than previously possible. Also, a model will be investigated which shows the transition from absence to presence of level repulsion. For the random displacement model, a Schrodinger operator modelling lattice fluctuations in crystals, the low energy states will be determined. Work will be done to better understand the stability of the localization phenomenon in the presence of exterior deterministic forces, leading to non- stationary random media.
Random Schrodinger operators are used as mathematical models to study electron transport of disordered media such as crystals with impurities, alloys, and amorphous materials. The effects arising from the presence of disorder in the structure of a material are central to distinguish between conductors and insulators. The phenomena which are observed, such as localization and extended states, provide mathematical foundations for electronics, but also apply to acoustic, electro-magnetic and elastic waves in disordered media. This project will further develop tools from a broad array of mathematical disciplines which are required for a rigorous understanding of these phenomena. Particular attention will be given to level statistics of electronic energies which allow to characterize insulators and conductors through properties of small material samples. The work will include multiple collaborative efforts and contribute to the training of three PhD students. Results will be widely disseminated through publications and at national and international conferences.
