Dynamics, Spectral Theory and Arithmetic in Quantum Chaos

量子混沌中的动力学、谱论和算术

• 批准号: 1101596
• 负责人: Mikhail Lyubich
• 金额: $ 9.6万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-15 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1101596&HistoricalAwards=false
• 关键词: Dynamics; Spectral; Theory; Arithmetic; Quantum

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.

该提案旨在研究量子混沌背景下动力系统、谱论、数学物理和数论之间的关系。 一般问题是：如果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”表明，当不确定性趋于 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，对于负截面曲率的黎曼流形 - 其中测地流是高度混乱的 - 纯态应该在半经典极限内均匀分布。 然而，经典动力学和高频频谱数据之间的联系仍然非常神秘。 事实上，各种量子混沌模型在这个半经典极限中表现出相互矛盾的现象：现在已知 QUE 在某些算术假设下在许多情况下都成立，但对于 QUE 猜想未涵盖的更简单的量子混沌“玩具模型”，它众所周知，一些纯粹的国家可能无法均匀分配。 似乎动力学和/或谱的一些更精细的属性造成了这种行为，并且隔离这些关键属性是本研究的主要焦点。 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，为此，本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，其中一些学科之间的联系这些领域（例如，动力系统和数论之间，或几何学和谱论之间）为许多难题提供了见解，并有望在未来很长一段时间内继续这样做。 最近数学物理学和遍历理论的思想的注入使得上述困难的光谱问题类型取得了很大的进展，并且处于许多不同的活跃研究领域的交叉点，量子混沌问题尤其重要。能够为纯理论数学和应用性问题等各种领域做出贡献。 这也使得这些项目对不同背景的研究生和博士后有吸引力，该计划的一部分是通过课程、研讨会和合作促进该领域多元化研究人员的快速成长。 因此，这项研究的目的不仅是研究围绕 QUE 猜想的具体问题，而且还致力于促进这些活跃研究的重要主题之间快速增长的联系网络，并最终促进整个科学领域的广泛进步。