Spectral Theory and Integrable Systems

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This project studies problems in spectral theory, the mathematical theory that describes physical notions such as energy levels of quantum systems and vibration frequencies of mechanical systems. The mathematical models considered are in regimes where there are competing influences from a disorder in the interaction and some long-range order, such as spatially slowly decaying interactions and quasi-periodic interactions. One focus is on applications of spectral theory to explain conservation laws in certain nonlinear systems and find otherwise hidden predictability in their behavior, described through the mathematical notion of integrability. The project focuses on Schrodinger operators, central to quantum mechanics, and the mathematical methods developed have the potential to illuminate other mathematical models and physical applications, such as electron conductivity in disordered materials and signal transmission using solitons. One main focus of this project is the spectral theory of almost periodic Schrodinger operators and integrability of the Korteweg-de Vries equation with almost periodic initial data. While a Lax pair representation formally rewrites the equation as an isospectral flow, rigorous characterizations of integrability, such as construction of an inverse scattering transform, are highly dependent on the phase space under consideration and require and motivate deep new results in direct and inverse spectral theory. Other topics considered in this project include estimates for the size and continuity of the solution in terms of spectral data, Schrodinger operators with slowly decaying potentials, higher-order Szego theorems, and transport properties of Schrodinger operators and quantum spin systems.