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.

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对于我们的研究是，与线性模型相比，这种非线性模型的定位典型签名（在适当的疾病/能量状态下）的持续存在。特别是，我们要提出一个问题，是否有足够的混乱导致本地化，而与所考虑的模型是否是线性的，因此阐明了安德森本地化现象的普遍性的一个方面。作为我们分析的好处，我们期望对定位的物理机理有更好的了解，而且还希望对随机差分运算符分析至关重要的新型数学工具的识别和开发。我们选择研究随机的呼吸schroedinger操作员，因为一方面，该课程足以适合显式计算，包括{可溶解模型，另一方面，它具有许多无线性参数影响的许多随机电位的范式属性，这些特性具有非线性参数影响：元素属性的属性分析（不同的属性）的分析（不同的属性）的核心范围是核心的核心范围。出现不同的难度：平均单调电位和真正的非单调的单调电位。我们打算建立LIFSHITZ尾巴，初始规模估计，Wegner估计，多量表分析/定位证明，（DE）相关估计以及所考虑的随机Schroedinger操作员的光谱统计数据。我们的小组已经在随机施罗丁格运营商的分析方面具有经验，既有线性参数依赖性，例如符号更改合金类型模型以及非线性参数依赖性，例如随机量子波导。