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Resonant delocalisation in random operators

随机算子中的共振离域

基本信息

批准号：
267136260
负责人：
Professorin Dr. Simone Warzel
金额：
--
依托单位：
Lehrstuhl für Analysis (M7)
依托单位国家：
德国
项目类别：
Research Grants
财政年份：
2014
资助国家：
德国
起止时间：
2013-12-31 至 2017-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/267136260?language=en
关键词：
Resonant delocalisation random operators

项目摘要

The goal of this project is to address fundamental issues in the analysis of random quantum many-particle operators. This in particular requires to develop mathematical tools for the analysis. Of particular interest is the possibility of a localization-delocalization transition in random systems of interacting particles. This phase transition, which in physical models causes vanishing or non-vanishing of conduction, is rather well understood in random single-particle models. However, the effects of interaction on this phenomenon still lie rather in the dark. The present work will build on recent progress together with Michael Aizenman which showed that in certain situations extended states emerge through tunnelling which is facilitated in systems with an exponentially growing configuration space by resonances between well separated localization centers. This is the phenomenon of resonant delocalisation which shall be further explored here. In relation to the above mentioned localization-delocalization transition it works on the side of delocalisation. We propose to study this phenomenon in effective mean-field models such as the hierarchical Anderson model. From the mathematical perspective, the proposed research concerns topics which present challenges and opportunities for probability theory and analysis. In particular, a method for analysing random matrix-valued Herglotz-Nevanlinna-Pick functions will have to be developed.

该项目的目标是解决随机量子多粒子算子分析中的基本问题。这尤其需要开发用于分析的数学工具。特别令人感兴趣的是相互作用粒子的随机系统中定位-离域转变的可能性。这种相变在物理模型中会导致传导消失或不消失，但在随机单粒子模型中可以很好地理解。然而，相互作用对这种现象的影响仍然是未知的。目前的工作将建立在与 Michael Aizenman 合作的最新进展的基础上，该进展表明，在某些情况下，扩展状态通过隧道效应出现，这在具有指数增长配置空间的系统中通过良好分离的定位中心之间的共振而得到促进。这就是共振离域现象，这里将进一步探讨。就上述本地化-非本地化转变而言，它在非本地化方面发挥作用。我们建议在有效的平均场模型（例如分层安德森模型）中研究这种现象。从数学角度来看，拟议的研究涉及为概率论和分析带来挑战和机遇的主题。特别是，必须开发一种用于分析随机矩阵值 Herglotz-Nevanlinna-Pick 函数的方法。