CAREER: Quantum Systems with Deterministic Disorder

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Disordered quantum systems are fundamental objects in modern mathematical physics, since, due to presence of heat, any real-life system has some level of noise in it. Systems with noise/disorder are often studied by probabilistic methods and assume that the noise is random. The main scope of the project is to study systems where the noise has additional structure and has deterministic (non-random) nature. In physics, strong random disorder often implies that the system becomes an insulator and prevents electrons in it from moving freely. Would it also be true for some systems with non-random disorder? If yes, which properties make non-random systems behave like random and can it be measured? Are there any additional effects if the disorder is not strong? The main goal of the project is to address these questions on many levels, which will include work with undergraduate and graduate students, development of graduate courses on dynamics and spectral theory, and developing a course for high school students that would illustrate connections between basic linear algebra and physics, providing them skills and motivation for possible further education in STEM.This project incorporates teaching and research activities on the analysis of a class of models of mathematical quantum physics, including developing abstract techniques of operator theory and establishing rigorous results on more concrete systems. All proposed models involve disorder, however, unlike usual probabilistic view on disordered systems, the main goal will be studying the disorder in a completely deterministic setting, or with a very small number of random/ergodic parameters. An example of such system would be a Schrodinger operator with dynamically-defined potential, where the underlying dynamical system has small dimension and low degree of mixing (for example, irrational rotation). Typically, quantum systems with large random disorder tend to prevent electrons from moving freely (Anderson localization). To answer even basic questions about electron transport in deterministic disordered systems, one must replace usual probabilistic methods by methods of number theory, ergodic theory, semi-algebraic geometry, and other deep areas. The main directions of the project involve analysis of localization/delocalization for systems of interacting quasiperiodic particles and the effect of interaction, perturbative methods for single-particle operators with rough potentials, perturbation properties for spectral bands of periodic operators, and abstract methods of operator theory applied to almost commuting operators and matrices, with applications to quantum systems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.