Localization of waves in random resonant media

项目简介及总体目标：本项目以捕获和定位波浪为主题。该项目的目标是表明，如果它是一种复合（即非均质）介质，在微观几何形状和材料特性之间具有特殊的相互作用，那么通过无界域传播的波可以被捕获。更准确地说，我们将考虑一个具有两个关键特征的空间：(1)在扩展子集上，方程的系数取大值，以及(2)在该子集之外的随机景观。波的局部化是物理界感兴趣的话题，因为它与我们的日常经验相反，例如，声音（声波）在开放空间中传播很远。由于这个原因，这一现象在20世纪50年代首次出现在量子物理文献中时受到了广泛关注。然而，对这种“安德森局部化”现象的数学严谨研究仅适用于少数特定模型，因此工作远未完成。从数学上讲，我们是在一个随机的环境中工作。这个项目的新颖之处在于考虑加入(1)，期望新的环境表现出“更强”的波捕获行为。研究方法及初步目标：本计划的主题大致包括：概率论、线性分析、散射理论及数学物理。具体来说，这个项目将研究一个随机微分算子的谱性质。在设置方面，我们将采用一个已被证明具有安德森局部化行为的模型，并将其修改为具有具有系数的周期区域，我们将其取为无穷大。修改是模拟(1)。最初的目标是证明(1)中固定系数的局部化和(2)中概率测度的方便选择。在证明方法方面，我们将寻求修改用于证明安德森定位的两种主要方法中的任何一种（多尺度方法，分数矩m方法）。

Introduction and overall objective: This project fall under the theme of trapping and localizing waves. The goal of theproject is to show that waves travelling through an unbounded domain can be trapped if it is a composite (i.e.heterogeneous) medium with a special interplay between the microgeometry and the material properties. Moreprecisely, we will consider a space with two key features: (1) on an extended subset, the coefficients of the equationtake large values, and (2) a random landscape outside this subset.The topic of wave localization is of interest in the physics community because it is contrary to our everydayexperience, for instance, sound (acoustic waves) travel very far in open spaces. For this reason, this phenomenonreceived much attention when it first appeared in the quantum physics literature in the 1950s. However, amathematically rigorous study of this 'Anderson localization' phenomenon is only available for a few specific models,so the work is far from complete. Mathematically, we are working with a random environment. The novelty of theproject is to then consider adding in (1), with the expectation that the new setting exhibits 'stronger' wave trappingbehavior.Methodology and initial aims: Broadly, the subjects of this project include: probability theory, linear analysis,scattering theory, and mathematical physics. Concretely, this project will study the spectral properties of a randomdifferential operator. In terms of the setup, we will take one of the models that has been proven to exhibit Andersonlocalization behaviour, and modify it to have periodic regions with a coefficient that we will take to infinity. Themodification is to simulate (1). The initial aim is to prove localization for a fixed coefficient in (1) and a convenientchoice of probability measure in (2). In terms of the proof methods, we will look to modify either of the two mainmethods used to prove Anderson localization (multi-scale method, fractional moments m ethod.)