Scattering theoretic methods in the mathematics of disordered media

将研究建立无序介质的物理相关模型的局部化现象，现有的方法在最相关的三维情况下广泛失败。具体的模型进行调查是theBernoulli-Anderson模型，泊松模型，和随机位移模型。主要的障碍是这些模型缺乏单调性，这导致了与研究得更成功的安德森模型的显著差异。在以前的工作中，PI和他的合作者已经证明了散射理论的方法可以用来建立一维非单调模型的局部化。这些方法将在一维得到进一步的结果，并在更高的维度上提供新的工具，如利用单格点的总散射截面来确定扩展态存在的能量。一些相关的目标是建立Wegner估计的特征值的有限盒哈密顿和Lifshitz尾渐近的积分态密度的搜索模型，以及了解的影响，相关性所产生的远程单网站scatersers. the数学理论的无序介质的目的是了解光谱和散射理论性质的不规则固体，如，例如，晶体与杂质，合金，具有晶格偏差的材料和无定形介质。不同的随机算子模型被用来描述不同类型的无序。这提供了一个数学框架来决定导电性：如果能显示出局域态，那么固体就是电绝缘的，而扩展态的存在则是导电材料的特征。PI的研究旨在为具有高度物理相关性的模型建立这些属性，这些属性迄今为止一直抵制严格的数学处理。例如，合金的真实模型的局部化特性还没有被理解。从统计物理学和固体物理学的观点出发，结合散射理论、光谱理论和谐波分析的方法，取得进展。

Research will be done to establish the phenomenon of localizationfor physically relevant models of disordered media where existingmethods have widely failed in the most relevant case of dimensionthree. Particular models to be investigated are theBernoulli-Anderson model, the Poisson model, and the RandomDisplacement model. The main obstacle to overcome is the lack ofmonotonicity properties for these models, which causes significantdifferences to the more successfully studied Anderson model. Inprevious work, the PI and his collaborators have shown thatmethods from scattering theory can be used to establishlocalization for one-dimensional non-monotonic models. Thesemethods will lead to further results in one dimension and providenew tools in higher dimension such as using the total scatteringcross section of single sites to identify energies where extendedstates exist. Some related goals are to establish Wegner estimatesfor the eigenvalues of finite box hamiltonians and Lifshitz tailasymptotics for the integrated density of states in the abovemodels, as well as to understand the effects of correlationsarising from long range single site scatterers.The mathematical theory of disordered media aims at understandingthe spectral and scattering theoretic properties of irregularsolids such as, for example, crystals with impurities, alloys,materials with lattice deviations and amorphous media. Differentmodels of random operators are used to describe the various typesof disorder. This provides a mathematical framework to decide onconductivity properties: If localization of states can beshown, then the solid is electrically insulating, while theexistence of extended states characterizes a conducting material.The PI's research aims at establishing these properties for modelsof high physical relevance, which have so far resisted a rigorousmathematical treatment. For example, localization properties arenot yet understood for realistic models of alloys. Ideas fromstatistical physics and solid state physics will be combined withmethods from scattering theory, spectral theory and harmonicanalysis to make progress.