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

具有呼吸势的随机薛定谔算子作为随机性非线性影响的范例模型

• 批准号: 394221243
• 负责人: Professor Dr. Ivan Veselic
• 金额: --
• 依托单位: Lehrstuhl IX: Analysis, Mathematische Physik und Dynamische Systeme
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2018
• 资助国家: 德国
• 起止时间: 2017-12-31 至 2020-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/394221243?language=en
• 关键词: Random; Schr; dinger; operators; breather

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.

在该项目中，我们研究具有随机呼吸势的薛定谔算子的谱、动力学和统计特性。此类模型具有对独立、同分布随机变量的规范序列的非线性依赖性。这对数学分析和物理分析提出了额外的挑战。与具有线性随机参数的模型（例如合金型薛定谔算子或其离散晶格表兄弟安德森模型）相比，可以理解。我们研究的一个驱动问题是局部化典型特征的持续性（在适当的无序/能量下）与线性模型相比，这种非线性模型的机制）。特别是，我们想要探究足够的无序是否会导致定位的问题，而与所考虑的模型是否是线性的无关，从而阐明安德森定位现象的普遍性的一个方面。作为我们分析的一个好处，我们不仅期望更好地理解定位的物理机制，而且期望识别和开发对于随机微分算子分析至关重要的新颖数学工具。我们选择研究随机呼吸薛定谔算子，因为这一类一方面足够有形，足以进行显式计算，并且包括可解模型，另一方面它具有许多具有非线性参数影响的随机势所共有的范式属性：对随机势（不同配置的）水平集的体积和几何结构的分析是理解复杂光谱特性的核心。呼吸模型具有不同难度的附加特征：平均而言，点状单调势单调势和真正的非单调势。我们打算建立 Lifshitz 尾部、初始尺度估计、Wegner 估计、多尺度分析/定位证明、（去）相关估计以及所考虑的随机薛定谔算子的谱统计。我们的小组已经具有随机薛定谔算子分析的经验，两者都具有线性参数依赖性，例如符号变化的合金类型模型，以及非线性参数依赖性，例如随机量子波导。