Scattering theoretic methods in the mathematics of disordered media

将进行研究以建立无序介质的物理相关模型的定位现象，其中现有方法在最相关的三维情况下普遍失败。要研究的特定模型是伯努利-安德森模型，泊松模型和随机位移模型。要克服的主要障碍是这些模型缺乏单调性，这导致了与更成功的安德森模型的显著差异。在之前的工作中，PI和他的合作者已经证明，散射理论的方法可以用来建立一维非单调模型的定位。这些方法将在一维上得到进一步的结果，并在高维上提供新的工具，例如利用单个点的总散射截面来识别存在扩展态的能量。一些相关的目标是建立有限盒哈密顿量特征值的Wegner估计和上述模型中状态的积分密度的Lifshitz尾渐近，以及理解远程单点散射相关化的影响。无序介质的数学理论旨在理解不规则固体的光谱和散射理论特性，例如，含杂质的晶体、合金、有晶格偏差的材料和非晶介质。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能：如果可以显示状态的局部化，那么固体是电绝缘的，而扩展状态的存在表征了导电材料。PI的研究旨在为高物理相关性的模型建立这些属性，这些模型迄今为止还无法进行严格的数学处理。例如，合金实际模型的局部化特性尚未被理解。统计物理学和固体物理学的思想将与散射理论、光谱理论和谐波分析的方法相结合，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.