CAREER: Classical and Quantum Chaos

本研究项目涉及表征物理系统振荡特征频率的特征值以及共振，共振是表征能量可以逃逸从而振荡伴随着衰变的系统的广义特征值的复数。共振现象的一个例子是我们敲钟后听到的声音：它有振荡的频率（音调）和衰减的速度（我们多久听不到声音），这两者都取决于钟的形状。我们听到的频率和衰减率对应于钟的最小衰减共振的实部和虚部。特征值和共振在物理和工程中有无数的应用。这个项目的一个主要目标是了解特征值和共振的分布如何依赖于系统（例如，钟的形状如何决定它发出声音的时间）；虽然这一主题最近取得了进展，但仍有许多重要的悬而未决的问题。本项目包括研究生研究课题和本科生数值实验课题。该项目采用微局部分析，这是解释经典/量子或粒子/波对应的数学理论。例如，在最简单的钟模型中，共振的高能分布与钟内台球轨迹的经典动力系统有关：在有限时间内，波近似沿着台球轨迹传播。然而，在很长一段时间内，这种近似会变得更糟，并最终失效。一个特别有趣的情况是当经典系统具有混沌行为时；研究了本征函数和共振在量子混沌领域的意义。项目的一部分是围绕分形不确定性原理，这是一个超越经典/量子对应的新工具，通过使用谐波分析、分形几何和组合学建立起来的。分形不确定性原理已经有了一些应用，包括对所有强混沌开放系统都有指数波衰减的猜想的进展。该项目的另一部分是使用微局部方法研究经典或波利科特-鲁埃尔共振，这是经典/量子对应逆转的罕见例子。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This research project concerns eigenvalues, which represent the characteristic frequencies of oscillation of physical systems, as well as resonances, which are complex numbers generalizing eigenvalues that represent systems where energy can escape and thus oscillation is accompanied by decay. An example of the resonance phenomenon is the sound we hear after striking a bell: it has a frequency of oscillation (the tone) and a rate of decay (how soon we stop hearing the sound), which both depend on the shape of the bell. The frequency and the decay rate that we hear correspond to the real and imaginary part of the least-decaying resonance of the bell. Eigenvalues and resonances have innumerable applications in physics and engineering. A major goal of this project is to understand how the distribution of eigenvalues and resonances depends on the system (for example, how the shape of the bell determines how long it will sound); while this topic has seen recent progress, there remain many important open questions. The project includes research projects for graduate students and numerical experiments for undergraduate students. The project employs microlocal analysis, which is the mathematical theory explaining the classical/quantum, or particle/wave, correspondence. For instance, the high-energy distribution of resonances in the simplest model of a bell is related to the classical dynamical system of billiard ball trajectories inside the bell: for bounded times, waves approximately propagate along billiard ball trajectories. However, for long times this approximation becomes worse and eventually breaks down. An especially interesting situation is when the classical system has chaotic behavior; the implications for eigenfunctions and resonances are studied in the field of quantum chaos. Part of the project is centered around the fractal uncertainty principle, which is a new tool beyond the classical/quantum correspondence, established through use of harmonic analysis, fractal geometry, and combinatorics. The fractal uncertainty principle has already seen several applications, including progress towards the conjecture that all strongly chaotic open systems have exponential wave decay. Another part of the project is the study of classical, or Pollicott-Ruelle, resonances, using microlocal methods -- a rare example of the reversal of the classical/quantum correspondence.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.