Number Theory Related to Quantum Chaos

PAR KurlbergABSTRACT：在这个项目中，联合首席研究员将研究一些与数学物理相关的理论问题。量子混沌是关于经典系统中的混沌如何以量子力学的形式表现出来的。例如，当经典系统被量子化时，是否可以在能级的统计行为中检测到混沌或没有混沌？为了研究“盒式谐振子”这个问题，这是一个简单的系统，它的能级显示出有趣的间距行为，这使得合作-P.I.考虑了以高度合成整数为模的正方形之间的间距。这个项目的第一个目标是将关于平方间距的结果推广到更一般的多项式的值之间的间距，包括模高合成整数和模素数。在这种情况下，研究这些多项式映射的像的基数分布也是很有趣的。量子混沌的另一个方面是经典遍历的量子力学类比。“猫映射”是混沌系统的一种简单模型，其量化具有很强的算术特性。利用这些特殊性质，可以用指数和来表示猫映射的量子遍历性。该方案的第二个目标是利用指数和的均匀分布结果证明量子猫映射的量子唯一遍历性。该方案的主要目的是探索量子混沌与数论之间的联系。量子混沌试图回答物理学中有关经典混沌的量子力学类似物的重要问题。该课题应用于微电子学，因此引起计算机和通信行业的兴趣。数论研究整数的性质，是数学中最古老的分支之一。在很长一段时间里，人们主要是因为美学的原因而对它进行研究，但在过去的几十年里，该领域对密码学来说已经变得至关重要

Par KurlbergABSTRACT:In this project, the Co-Principal Investigator will study some number theoretical questions related to mathematical physics. Quantum Chaos is concerned with how chaos in classical systems manifests itself in terms of quantum mechanics. For instance, can chaos, or lack thereof, be detected in the statistical behavior of the energy levels when a classical system is quantized? Pursuing this question for the ``boxed harmonic oscillator'', a simple system whose energy levels show interesting spacing behavior, led the co-P.I. to consider spacings between squares modulo highly composite integers. The first goal of this project is to generalize the results on spacings of squares to spacings between values of more general polynomials, both modulo highly composite integers, as well as modulo primes. In this context it is also interesting to study the distribution of the cardinality of the images of these polynomial maps. Another aspect of Quantum Chaos is the quantum mechanical analogue of classical ergodicity. The ``cat map'' is a simple model of a chaotic system, and its quantization has strong arithmetical features. Using these special features, it is possible to express Quantum Ergodicity for the cat map in terms of exponential sums. A second goal of this proposal is to prove quantum unique ergodicity for the quantized cat map using equidistribution results for exponential sums.The main objective of this project is to explore the connection between Quantum Chaos and Number Theory. Quantum Chaos tries to answer important questions in Physics regarding quantum mechanical analogues of classical chaos. The subject has applications to microelectronics, and is hence of interest to the computer and communications industry. Number Theory is concerned with properties of whole numbers and is one of the oldest branches of mathematics. For a long time it was studied mostly because of aesthetic reasons, but in the last few decades the field has been of fundamental importance to cryptography