CAREER: Classical and Quantum Chaos

职业：经典和量子混沌

• 批准号: 1749858
• 负责人: Semyon Dyatlov
• 金额: $ 42.5万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1749858&HistoricalAwards=false
• 关键词: CAREER; Classical; Quantum; Chaos; CAREER; Classical; Quantum; Chaos

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.

该研究项目涉及特征值，它代表了物理系统振荡的特征频率以及共鸣，这些频率是代表能量可以逃脱的系统的复杂数量的复杂数量，因此振荡伴随着衰减。共振现象的一个例子是我们在敲响铃声后听到的声音：它具有振荡频率（音调）和衰减速率（我们多久地停止听到声音），这两者都取决于铃铛的形状。我们听到的频率和衰减率对应于钟声最不确定共鸣的真实和虚构部分。特征值和共鸣在物理和工程中都有无数的应用。该项目的一个主要目标是了解特征值和共振的分布如何取决于系统（例如，钟形的形状如何确定听起来的时间）；尽管该主题最近取得了进展，但仍有许多重要的开放问题。该项目包括针对研究生的研究项目和本科生的数字实验。该项目采用微局部分析，这是解释经典/量子或粒子/波的对应关系的数学理论。例如，在铃铛的最简单模型中共振的高能分布与铃铛内台球球轨迹的经典动力学系统有关：在有界的时期，波浪沿台球球轨迹近似传播。但是，很长时间以来，这种近似变得更糟，并最终崩溃了。一个特别有趣的情况是，何时经典系统具有混乱的行为；在量子混乱的领域研究了对本征函数和共振的影响。该项目的一部分集中在分形不确定性原理上，该原理是通过使用谐波分析，分形几何形状和组合学建立的经典/量子对应的新工具。分形的不确定性原理已经看到了几种应用，包括朝着所有强烈混乱的开放系统都具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古典或Pollicott-ruelle的共鸣，这是经典/量子通信逆转的罕见例子。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力和更广泛影响的评估来通过评估来获得支持的。