Resonant delocalisation in random operators

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。这特别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局域化-离域化转变的可能性。这种相变在物理模型中导致传导消失或不消失，在随机单粒子模型中得到了很好的理解。然而，相互作用对这一现象的影响仍然处于黑暗之中。目前的工作将建立在最近的进展与迈克尔Aizenman一起表明，在某些情况下，扩展状态出现通过隧穿，这是促进在系统中具有指数增长的配置空间之间的共振分离的本地化中心。这是共振离域现象，将在这里进一步探讨。关于上述的本地化-离域化过渡，它在离域化一侧工作。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。从数学的角度来看，拟议的研究关注的主题，目前的挑战和机遇的概率理论和分析。特别是，随机矩阵值的Herglotz-Nevanlinna-Pick函数的分析方法将不得不开发。

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.