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

本研究的主要目的是进一步加深对几种标准简单物理模型的数学理解：非线性薛定谔方程及其与多体量子力学的联系；随机黑箱系统的散射共振；随机衰减系数CMV矩阵的谱测度。该项目的NLS部分的一个主要主题是调查障碍的影响。这是研究一般几何中NLS的一个具体而具有挑战性的模型案例。我们还将研究尺度临界但处于超临界边缘的色散方程的例子；这是现有的能量临界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驱动NLS模型。这些模型已经开始揭示值得进一步研究的有趣和不寻常的变分和动态结构。此外，与有限维量子系统相关的共振分布（随机均匀选择，尊重时间反转对称性）将被研究。相关方法将用于研究具有衰减随机势的离散狄拉克方程的精细谱性质。随机矩阵理论和任何物理科学一样，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）和实验能级数据的分析。最近，随机矩阵统计在鼓的振动和素数的行为中被观察到。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理解该方程解的行为，主要是作为一般演化方程的模型。作为实验室科学/工程变迁的一步，我们将考虑存在障碍和其他不规则性的进化，以及合并更复杂的非线性效应，这些效应不能用简单的幂律来描述。培养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。

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.