Resonant delocalisation in random operators

这个项目的目标是解决随机量子多粒子算符分析中的基本问题。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，在相互作用的随机粒子系统中发生局域-离域转变的可能性。这种相变在物理模型中会导致导电性的消失或不消失，而在随机单粒子模型中，这种相变是相当清楚的。然而，相互作用对这一现象的影响仍然相当隐秘。目前的工作将建立在与Michael Aizenman一起的最新进展的基础上，后者表明在某些情况下，通过隧道效应来产生扩展态，这在具有指数级增长的组态空间的系统中通过良好分离的局域中心之间的共振来促进。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，它是在非本地化方面起作用的。我们建议在有效平均场模型中研究这一现象，例如分层Anderson模型。从数学的角度来看，拟议的研究涉及的主题给概率理论和分析带来了挑战和机遇。特别是，必须开发一种分析随机矩阵值Hergltz-Nevanlinna-Pick函数的方法。

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.