The nonlinear Schrodinger equation, its physical origins, and the spectral measures of random matrices

非线性薛定谔方程、其物理起源以及随机矩阵的谱测度

• 批准号: 1265868
• 负责人: Rowan Killip
• 金额: $ 23.6万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1265868&HistoricalAwards=false
• 关键词: nonlinear; Schrodinger; equation; its; physical

The main purpose of the proposed research is to further the mathematical understanding of several standard simple physical models: the nonlinear Schrodinger equation and its connections to many-body quantum mechanics; the scattering resonances of a random black-box system; and the spectral measure of CMV matrices with random decaying coefficients. A major theme of the NLS portion of the project is to investigate the effect of obstacles. This is a concrete and challenging model case for the investigation of NLS in general geometries. Examples of dispersive equations that are scaling-critical but on the very cusp of super-criticality will also be investigated; this is an ideal proving ground for existing energy- and mass-critical techniques. Two classes of physically motivated NLS models with combined attractive and repulsive nonlinearities will be investigated. These models have already started to reveal interesting and unusual variational and dynamical structures that warrant further investigation. Additionally, the distribution of resonances associated to finite dimensional quantum systems (chosen uniformly at random, respecting time-reversal symmetry) will be investigated. Related methods will be employed to investigate the fine spectral properties for discrete Dirac equations with decaying random potentials.The theory of random matrices is driven by empirical data as surely as any physical science: Its roots lie in statistical analysis (specifically, ANOVA) and the analysis of experimental energy-level data. More recently, random matrix statistics have been observed in the vibrations of drums and the behaviour of the prime numbers. The researches of this project are aimed at helping to elucidate and explain the results of these truly mathematical experiments. Although the nonlinear Schrodinger equation is used as a simple effective model for several physical phenomena, the goal of this project is to further our understanding of the behaviour of solutions to this equation principally as a model for general evolution equations. As a step towards the vicissitudes of laboratory science/engineering, we will consider evolution in the presence of obstacles and other irregularities as well as incorporating more complicated nonlinear effects that cannot be described by a simple power law. The training of graduate students as educators and researchers in the mathematical sciences is also a significant portion of the project.

拟议的研究的主要目的是进一步的数学理解几个标准的简单的物理模型：非线性薛定谔方程及其连接到多体量子力学;随机黑箱系统的散射共振;和随机衰减系数的CMV矩阵的频谱测量。该项目的NLS部分的一个主要主题是调查障碍物的影响。 这是一个具体的和具有挑战性的模型的情况下，在一般的几何形状的NLS的调查。色散方程的缩放临界，但在超临界的尖端的例子也将被调查，这是现有的能量和质量临界技术的理想试验场。本文将研究两类具有吸引和排斥非线性的物理激励的NLS模型。 这些模型已经开始揭示有趣和不寻常的变化和动力学结构，值得进一步研究。此外，将研究与有限维量子系统（随机均匀选择，尊重时间反演对称性）相关的共振分布。 随机矩阵理论与任何物理科学一样，都是由经验数据驱动的：它的根源在于统计分析（特别是方差分析）和对实验能级数据的分析。最近，在鼓的振动和素数的行为中观察到了随机矩阵统计。该项目的研究旨在帮助阐明和解释这些真正的数学实验的结果。虽然非线性薛定谔方程被用作几种物理现象的简单有效模型，但本项目的目标是进一步了解该方程的解的行为，主要是作为一般演化方程的模型。 作为迈向实验室科学/工程的变迁的一步，我们将考虑存在障碍和其他不规则性的进化，以及结合更复杂的非线性效应，不能用简单的幂律来描述。 培训研究生成为数学科学的教育工作者和研究人员也是该项目的重要组成部分。