Quantum Dynamics: Geometry and Spectrum

摘要奖：DMS-0071358首席研究员：史蒂夫·泽尔迪奇我们建议将量子动力学概念应用到计量学、逆谱理论和数学物理中。第一个是众所周知的问题：你能听到分析鼓的形状吗？我们构造了拉普拉斯绕着弹跳球轨道的‘量子Birkhoff范式’，并证明了它是一个谱不变量。我们将研究有界单连通解析平面域在多大程度上由它在一个或多个弹跳轨道上的正规形式所决定。我们还将研究频谱和该领域的X射线断层扫描之间的新联系。第二组问题涉及最近发展的关于Kahler流形上的线丛的全纯截面的概率论的应用。目前，我们已经确定了小球上方部分的大部分局部统计数据。我们还把结果推广到辛流形上。我们现在的目标是将这些结果应用于确定空穴概率、横截概率以及在复杂和辛几何中出现的其他问题。第三类问题属于量子混沌。我们引入了一种新的随机量子映射模型，它是通过对随机哈密顿流进行量子化而得到的。我们建议为这个模型研究量子混沌的主要问题。20世纪的许多物理学都围绕着粒子和场的量子力学，特别是围绕薛定谔方程，未来的物理和工程将更多地是量子力学。量子系统需要非直观的东西，理解它们的最好方法之一是将它们与它们量子化的经典系统进行比较。例如，一个在强磁场中的氢原子可以被认为是一个在力场中运动的小粒子。随着磁场的增强，粒子的运动变得更加复杂和混乱。关于量子粒子，这说明了什么？根据二十年的数值研究，混沌量子粒子的行为应该是随机的，统计数据类似于随机矩阵的统计。为什么薛定谔方程的解应该知道关于随机矩阵的任何事情？我们的方法是引入一种新的混沌量子系统模型，它允许人们同时考虑大范围的系统。从某种意义上说，我们试图证明一个“随机”量子系统的行为类似于一个“随机”矩阵，尽管这些系统存在差异。这样的模型应该能够解释从核共振能级到化学反应速率的各种观测。

AbstractAward: DMS-0071358Principal Investigator: Steve ZelditchWe propose to apply quantum dynamical notions to problems ingeometry, inverse spectral theory and mathematical physics.First is the well-known problem, `Can you hear the shape of ananalytic drum?' We have constructed a `quantum Birkhoff normalform' of the Laplacian around a bouncing ball orbit and haveproved that it is a spectral invariant. We will investigate theextent to which a bounded simply connected analytic plane domainis determined by its normal forms at one or more bouncing ballorbits. We will also investigate a new connection between thespectrum and X-ray tomography of the domain. The second group ofproblems involves applications of a recently developedprobabalistic theory of holomorphic sections of line bundles overKahler manifolds. We have at this time determined much of thelocal statistics of sections over small balls. We have alsogeneralized the results to symplectic manifolds. We now aim toapply these results to determine hole probabilities,probabilities of transversality, and other matters which arise incomplex and symplectic geometry. The third group of problemsbelongs to Quantum Chaos. We have introduced a new model ofrandom quantum maps, obtained by quantizing stochasticHamiltonian flows. We propose to investigate the main problemsof Quantum Chaos for this model.Much of physics of the twentieth century has revolved around thequantum mechanics of particles and fields, in particular aroundthe Schrodinger equation, and the physics and engineering of thefuture will be even more quantum mechanical. Quantum systems arequite un-intuitive and one of the best ways to understand whatthey do is to compare them to the classical systems which theyquantize. For instance, a hydrogen atom in a strong magneticfield can be thought of as a small particle moving in a forcefield. As the magnetic field gets stronger, the particle'smotion becomes more complex and chaotic. What does this say aboutthe quantum particle? According to twenty years of numericalstudies, the behaviour of a chaotic quantum particle should berandom, with statistics similar to those for random matrices.Why should solutions of the Schrodinger equation know anythingabout random matrices? Our approach is to introduce a new modelof chaotic quantum systems which allows one to consider largefamilies of systems at once. In a sense, we are trying toshow that a `random' quantum system behaves similarly to a`random' matrix, despite the differences in these systems.Such a model should be able to explain observations ranging fromnuclear resonance levels to chemical reaction rates.