Dynamics, Spectral Theory and Arithmetic in Quantum Chaos

这一建议试图在量子混沌的背景下研究动力系统、光谱理论、数学物理和数论之间的关系。一般的问题是：如果一个动力系统表现出混沌行为，这在量子力学图景中是如何反映的？对应原理认为，当不确定度趋于0时，经典动力学的性质应该反映在量子系统中。量子唯一遍历性(QUE)猜想指出，对于具有负截面曲率的黎曼流形--其测地线流是高度混沌的--纯态应该在半经典极限中均匀分布。然而，经典动力学和高频光谱数据之间的联系仍然非常神秘。事实上，量子混沌的各种模型在这一半经典极限中显示出相互矛盾的现象：现在已知在某些算术假设下，量子混沌在许多情况下都成立，但对于不被Que猜想覆盖的更简单的量子混沌“玩具模型”，我们知道一些纯态可能无法均匀分布。似乎动力学和/或光谱的一些更好的属性对这种行为负责，分离这些关键属性是本研究的主要重点。此外，很明显，需要来自许多不同领域的方法来阐明这个问题，在这样做的过程中，这项研究也旨在加强连接这些学科的桥梁。在上个世纪的大部分时间里，这些领域之间的联系(例如，动力系统和数论之间，或者几何和谱论之间)为许多困难的问题提供了洞察力，并承诺在未来很长一段时间内继续这样做。最近来自数学物理和遍历理论的思想的注入导致了上述类型的困难光谱问题的巨大进展，并且由于处于如此多不同活跃研究领域的交叉点，量子混沌问题特别适合为各种努力做出贡献，无论是在纯粹的理论数学中，还是在应用性质的问题中。这也使得项目对不同背景的研究生和博士后具有吸引力，该项目的一部分是通过课程、研讨会和合作增加该领域不同研究人员的快速增长。因此，这项研究的目的不仅是研究围绕Que猜想的具体问题，而且还旨在促进这些活跃研究的重要主题之间迅速增长的联系网络，并最终在整个科学领域取得广泛的进展。

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.