Spectral Theory of Periodic and Quasiperiodic Quantum Systems

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The main goals of this research project are to develop and study mathematical models of quantum particles in periodic and quasiperiodic media, such as crystals or quasicrystals, based on spectral theory of Schrodinger operators. Periodic media correspond to crystalline structures, such as metals or semiconductors, which can conduct electrons freely at certain energies. It is proposed to study mathematically rigorous models of electron transport near the edges of the "forbidden zones" and develop new approaches to the effective mass approximation. Quasiperiodic operators are examples of disordered systems which, depending on the regime, can look like pure crystals, or crystals with random impurities, while being completely deterministic. One of the models under study demonstrates random-like behavior at arbitrarily small disorder and can potentially be a suitable replacement for a random environment without having to employ a large parameter space. Special emphasis will be given to multi-dimensional and multi-particle models, with possible applications to quantum spin systems and quantum information theory. The project provides research opportunities for undergraduate and graduate students.The activities of this research project fall into several groups distinguished by the classes of the operators under study and the types of their spectra. In the area of Anderson localization for quasiperiodic operators ("random-like behavior"), the project studies multi-particle models with analytic potentials at perturbatively large disorder and low regularity models, with the latter results expected to be non-perturbative. The methods here include operator theory, harmonic analysis, real algebraic geometry, and large deviation theorems for subharmonic or piecewise-monotonic functions. In the area of absolutely continuous spectrum ("crystalline behavior"), the project investigates the relation between low regularity reducibility of Schrodinger cocycles and strong ballistic transport for the corresponding Schrodinger operators, which, in turn, is related to transport properties of quantum spin systems. In the area of periodic operators, it is intended to study possible singularities of the Bloch varieties at the edges of spectral bands, both in 2D and 3D cases. Finally, on the more abstract side, the project aims to develop a quantitative classification of almost commuting matrices in topologically non-trivial cases, which demonstrates connections both with Cantor spectra for quasiperiodic operators and with some quantum spin systems.