Disorder Effects on Quantum Spectra and Dynamics

无序对量子光谱和动力学的影响拟议研究摘要Michael Aizenman该研究将解决无序对量子系统动力学中起作用的算子的光谱和动力学的影响。 结合甚至弱，但广泛的障碍是已知的，导致在离域的至少一些正常模式，并在创建纯点光谱制度，其特征在于由一个密集的收集对应于良好分离的本征函数的本征值。 拟议的研究的重点是在不太了解的问题的存在下，在无序的扩展状态，以及相应的勒贝格绝对连续谱。还将调查的是一个可能的关系的频谱间隙统计的本地运营商纳入混乱与已知的随机矩阵合奏的特征值统计。由于随机性的影响是维度相关的，最初的目标是澄清高维中的一些问题，其中循环效应通常更可控。在相反的方向上，我们设想了一种新的基于涨落的方法来推导和分析低维中的局域化，在低维中，无序效应对随机算符模型的量子系统的传导特性起着关键的作用，并且与量子网络直接相关。这项工作将利用PI与合作者最近取得的进展，以及他过去在安德森本地化和波动的其他影响方面的工作，例如低维中一阶相变的舍入，即所谓的Imry-Ma效应。 该学科为涉及数学分析的技术与来自统计力学的思想提供了交汇点，统计力学中的科学是由无序组成的。揭示物理问题的挑战也丰富了数学领域。

Disorder Effects on Quantum Spectra and Dynamics Abstract of Proposed ResearchMichael AizenmanThe research will address the effects of disorder on spectra and dynamics of operators which play a role in the dynamics of quantum systems. The incorporation of even weak but extensive disorder is known to result in delocalization of at least some of the normal modes, and in the creation of pure-point spectral regimes, characterized by a dense collection of eigenvalues corresponding to well separated eigenfunctions. The focus of the proposed research is on the less understood question of the existence of extended states in the presence of disorder, and the corresponding Lebesgue absolutely continuous spectra. Also to be investigated is a possible relation of the spectral gap statistics of local operators incorporating disorder with the known eigenvalue statistics of random matrix ensembles. As the effects of randomness are dimension dependent, the initial goal is to clarify some of the issues in high dimensions, where loop effects are in general more controllable. In the converse direction, a new -fluctuation based- method is envisioned for the derivation and analysis of localization in low dimensions, where the effect is most drastic.Disorder effects play a key role in the conduction properties of quantum systems modeled by random operators, and are of direct relevance for quantum networks. The work will capitalize on recent progress made by the PI with collaborators, and on his past work on the Anderson localization and on other effects of fluctuations, such as the rounding of fist-order phase transitions in low dimensions, in what is known as the Imry-Ma effect. The subject provides the meeting grounds for techniques involving mathematical analysis with ideas drawn from statistical mechanics, where science is made of disorder. The challenge of shedding light on issues of physics enriches also fields of mathematics.