Quantum Dynamics: Geometry and Spectrum

AbstractAward：DMS-0071358首席研究员：Steve Zelditch我们建议将量子动力学的概念应用到测量学、逆谱理论和数学物理学的问题中。首先是众所周知的问题，“你能听到一个解析鼓的形状吗？”“我们构造了一个围绕弹跳球轨道的拉普拉斯算子的‘量子Birkhoff标准形’，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一个或多个弹跳球轨道上的正规形所决定的程度。我们还将研究一个新的连接之间的thespectrum和X射线断层扫描的域。 第二组问题涉及应用最近发展起来的probabalistic理论的全纯部分线丛在Kahler流形。 此时，我们已经确定了小球上截面的许多局部统计量。 我们还将结果推广到辛流形。我们现在的目标是应用这些结果来确定洞的概率，概率的横截性，以及其他事项所产生的复杂和辛几何。 第三类问题属于量子混沌。本文介绍了一种新的随机量子映射模型，它是通过量子化随机哈密顿流而得到的。 20世纪世纪的大部分物理学都围绕着粒子和场的量子力学，特别是薛定谔方程，未来的物理学和工程学将更加量子化。 量子系统是非常不直观的，理解它们的最好方法之一就是将它们与它们所描述的经典系统进行比较。例如，在强磁场中的氢原子可以被认为是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。这说明量子粒子是什么？ 根据20年的数值研究，混沌量子粒子的行为应该是随机的，其统计特性与随机矩阵的统计特性相似。为什么薛定谔方程的解应该知道随机矩阵的任何信息？ 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系统族。 从某种意义上说，我们试图证明一个“随机”量子系统的行为与一个“随机”矩阵相似，尽管这些系统之间存在差异。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.