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

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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.