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

本研究涉及具有奇异连续谱的薛定谔算符、萨定谔算符和量子自旋系统的局域化效应、无序环境中的渗透和接触过程。奇异连续谱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生长性质、谱测度的Hausdorff维数以及奇异连续谱在一级扰动下的稳定性或不稳定性之间的关系。本文研究的另一个目标是对参数的临界值（区间）可能出现奇异谱的模型进行奇异连续谱的研究，作为从绝对连续到纯点谱的过渡。定位研究的主要目的是发展具有确定性失序的遍历算子族（无序系统）的定位证明方法。此外，还将研究局域化的一般均匀性及其与量子动力学的关系。提出的研究是围绕无序和奇异系统的基本性质，用于许多微观和宏观效应的建模：从量子局域化到地震理论。提议的主题包括研究量子和统计力学的高度无序系统和系统在无序的临界水平，表现出高度不稳定的行为的性质。＊＊＊

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. ***