CAREER: Classical and Quantum Chaos

本研究项目涉及代表物理系统振荡特征频率的本征值，以及共振，共振是代表能量可以逃逸的系统的本征值的复数，因此振荡伴随着衰减。共振现象的一个例子是我们在敲钟后听到的声音：它有一个振荡频率（音调）和一个衰减率（我们停止听到声音的时间），这两者都取决于钟的形状。我们听到的频率和衰减率，对应于钟的衰减最小的共振的真实的和虚部。本征值和共振在物理学和工程学中有无数的应用。该项目的一个主要目标是了解本征值和共振的分布如何取决于系统（例如，钟的形状如何决定它的声音长度）;虽然这个主题最近取得了进展，但仍然存在许多重要的未决问题。该项目包括研究生的研究项目和本科生的数值实验。该项目采用微局域分析，这是解释经典/量子或粒子/波对应关系的数学理论。例如，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学系统有关：对于有界时间，波近似地沿着沿着台球轨迹传播。然而，在很长一段时间里，这种近似变得更糟，最终崩溃。一个特别有趣的情况是当经典系统具有混沌行为时，在量子混沌领域研究本征函数和共振的影响。该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，通过使用调和分析，分形几何和组合学建立。分形不确定性原理已经有了一些应用，包括猜想所有强混沌开放系统都有指数波衰减。该项目的另一部分是使用微局域方法研究经典或Pollicott-Ruelle共振--经典/量子对应反转的罕见例子。该奖项反映了NSF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.