Mathematical Sciences: Singular Continuous Spectum and Localization Type Effects if Disordered Systems

数学科学：无序系统的奇异连续谱和局域化效应

• 批准号: 9501265
• 负责人: Svetlana Jitomirskaya
• 金额: $ 4万
• 依托单位: University of California-Irvine
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9501265&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Singular; Continuous; Spectum

9501265 Jitomirskaya This research involves Schrodinger operators with singular continuous spectrum, localization type effects for Sahrodinger operators and for quantum spin systems, percolation and contact processes in disordered environments. The main objective of the research related to the singular continuous spectrum is to study the relation between the spectral properties and longtime behavior of the generalized eigenfunctions, in particular between certain growth properties of generalized eigenfunctions, the Hausdorff dimension of the spectral measure, and stability or instability of singular continuous spectrum with respect to rank one perturbations. Another goal of the proposed research is to study singular continuous spectrum for models where singular spectrum may appear for the critical values (intervals) of the parameters, serving as a transition from absolutely continuous to pure point spectrum. The main objective of the research related to localization is to develop methods of proving localization for ergodic families of operators (disordered systems) with deterministic disorder. Also general uniformity properties of localization and relation to quantum dynamics will be studied. The proposed research is centered around the fundamental properties of disordered and singular systems that are used in modeling of many micro and macro effects: from quantum localization to earthquake theory. The proposed topics include studying properties of highly disordered systems of Quantum ald Statistical Mechanics and of systems at critical levels of disorder, that demonstrate highly unstable behavior. ***

9501265吉托米尔斯卡娅本研究涉及具有奇异连续谱的薛定谔算符，沙罗丁格算符和量子自旋系统的局域型效应，无序环境中的渗流和接触过程。与奇异连续谱相关的研究的主要目的是研究广义特征函数的谱性质与长期行为之间的关系，特别是广义特征函数的某些增长性、谱测度的Hausdorff维度与奇异连续谱关于一阶扰动的稳定性或不稳定性之间的关系。这项研究的另一个目的是研究模型的奇异连续谱，其中奇异谱可能出现在参数的临界值(区间)内，作为从绝对连续谱到纯点谱的过渡。与局部化相关的研究的主要目的是发展证明具有确定性无序的遍历算子族(无序系统)的局部化的方法。此外，还将研究局域化的一般均匀性质及其与量子动力学的关系。这项拟议的研究围绕无序和奇异系统的基本性质展开，这些系统被用来模拟许多微观和宏观效应：从量子局域化到地震理论。拟议的主题包括研究量子统计力学中高度无序的系统的性质，以及处于无序的临界水平的系统的性质，这些系统表现出高度不稳定的行为。***