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

小行星9501265 研究了具有奇异连续的Schrodinger算子 Sahrodinger算子和 无序量子自旋系统、渗流和接触过程 环境.本研究的主要目的是探讨 连续光谱是研究光谱之间的关系， 性质和长期行为的广义本征函数，在 特别是在某些广义增长性质之间 特征函数，谱测度的Hausdorff维数，以及 奇异连续谱关于 秩一扰动拟议研究的另一个目标是研究 可能出现奇异谱模型的奇异连续谱 对于参数的临界值（区间）， 从绝对连续到纯点谱的过渡。主要 与本地化相关的研究目标是开发 证明遍历算子族的局部化（无序 系统）与确定性无序。还有一般的均匀性 局域化和量子动力学的关系将被研究。 拟议的研究是围绕基本性质， 无序和奇异系统，用于许多微观建模 宏观效应：从量子局域化到地震理论。的 提议的课题包括研究高度无序系统的性质 量子力学和统计力学以及处于临界水平的系统 表现出高度不稳定的行为。 ***

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. ***