Random Schrödinger operators with breather potentials as a paradigmatic model for non-linear influence of randomness

在这个项目中，我们研究了具有随机呼吸势的薛定谔算符的谱、动力学和统计性质。这类模型的特征是对独立的、同分布的随机变量的规范序列的非线性依赖。与具有线性随机参数的模型相比，这对数学分析和物理理解提出了额外的挑战。例如，合金型薛定谔算符或它的离散晶格表亲安德森模型。我们研究的一个驱动问题是，与线性模型相比，这种非线性模型的典型局域化特征(在适当的无序/能量区域中)的持久性。具体地说，我们想要追问的问题是，无论所考虑的模型是否是线性的，足够的无序是否会导致局域化，从而澄清安德森局域化现象的普遍性的一个方面。作为我们分析的一个好处，我们期望不仅能更好地理解局部化的物理机制，而且还能识别和开发对分析随机微分算子至关重要的新的数学工具。我们之所以选择研究随机呼吸薛定谔算子，是因为这类算子一方面是有形的，便于显式计算，并且包含{可解模型，另一方面它具有许多具有非线性参数影响的随机势的聚合性质：分析(不同构型的)随机势的水平集的体积和几何结构是理解复杂谱性质的核心。呼吸模型还有一个特点，即它们具有不同的困难：点状单调势、平均单调势和真正的非单调势。我们打算建立随机薛定谔算子的Lifshitz尾、初始尺度估计、Wegner估计、局部化的多尺度分析/证明、(去)相关性估计以及谱统计。我们的团队已经有了分析随机薛定谔算子的经验，既有线性参数相关的，例如符号变化的合金类型模型，也有非线性参数相关的，例如随机量子波导。

In the project we study spectral, dynamical, and statistical properties of Schroedinger operators with random breather potential.Such models feature a non linear dependence on a canonical sequence of independent, identically distributed random variables.This poses an additional challenge for the mathematical analysis and physical understanding compared to models with linear random parameters as, for instance the alloy type Schroedinger operator, or its discrete lattice cousin, the Anderson model.A driving question for our research is the persistence of typical signatures of localization (in appropriate disorder/energy regimes) for such non linear models, compared to linear ones. In particular, we want to pursue the question whether sufficient disorder leads to localization, independently of whether the considered model is linear or not, and thus clarify one aspect of universality of the phenomenon of Anderson localization. As a benefit of our analysis we expect not only a better understanding of the physical mechanism of localization, but identification and development of novel mathematical tools crucial for the analysis of random differential operators. We choose to study random breather Schroedinger operators because this class is on one hand tangible enough to be amenable to explicit calculations, and includes {solvable models, on the other hand it has paradigmatic properties common to many random potentials with non linear parameter influence: The analysis of volume and geometric structure of level sets of (different configurations of) random potentials is at the core of understanding intricate spectral properties.Breather models have the additional trait that they come in different varieties of difficulty: pointwise monotone potentials, on average monotone potentials, and truly non monotone ones. We intend to establish Lifshitz tails, initial scale estimates, Wegner estimates, multi scale analysis/proofs of localization, (de)correlation estimates, as well as spectral statistics for the random Schroedinger operators considered. Our group has already experience with the analysis of random Schroedinger operators, both with linear parameter dependence, e.g. sign changing alloy type models, as well as with non linear parameter dependence, e.g. random quantum waveguides.