Dynamics, Spectral Theory and Arithmetic in Quantum Chaos

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This proposal seeks to investigate the relationships between dynamical systems, spectral theory, mathematical physics, and number theory, in the context of quantum chaos. The general question is: if a dynamical system exhibits chaotic behavior, how is this reflected in the quantum mechanical picture? The "correspondence principle" suggests that properties of the classical dynamics should be reflected in the quantum system as the uncertainty tends to 0. The Quantum Unique Ergodicity (QUE) Conjecture states that, for Riemannian manifolds of negative sectional curvature - for which the geodesic flow is highly chaotic - pure states should become equidistributed in the semiclassical limit. However, the connection between classical dynamics and high-frequency spectral data remains very mysterious. In fact, various models of quantum chaos show conflicting phenomena in this semiclassical limit: QUE is now known to hold in many cases under certain arithmetic assumptions, but for simpler "toy models" of quantum chaos that are not covered by the QUE conjecture, it is known that some pure states can fail to equidistribute. It seems that some of the finer properties of the dynamics and/or spectrum are responsible for this behavior, and isolating these key properties is a main focus of this research. In addition, it has become clear that methods from many different fields are required to shed light on this problem, and in doing so this research also aims to fortify the bridges connecting these disciplines.For much of the last century, connections between some of these these fields (eg., between dynamical systems and number theory, or between geometry and spectral theory) have provided insight for many difficult problems, and promise to continue to do so for a long time to come. The recent infusion of ideas from mathematical physics and ergodic theory has led to a great deal of progress in the type of difficult spectral problems described above, and being at the intersection of so many different areas of active research, the questions of quantum chaos are particularly well positioned to make contributions to a wide variety of endeavors, both in purely theoretical mathematics and in problems of an applied nature. This also makes the projects attractive to graduate students and post-docs of different backgrounds, and part of this program is to add to the rapid growth of diverse researchers in the field through courses, seminars, and collaborations. Thus this research aims not only to study the specific problems surrounding the QUE conjecture, but also to contribute to the rapidly growing network of connections between these important subjects of active research, and ultimately to a wide range of advances throughout the sciences.