Schrodinger Operators, Integrable Systems, and Other Simple Models in Mathematical Physics

DMS-0401277PI: Rowan Killip， UCLATitle：薛定谔算子、可积系统和数学物理中的其他简单模型【摘要】该项目致力于促进对某些简单物理模型的数学理解：(a)将通过逆散射/谱方法研究初始数据缓慢减少的KdV方程的长时间谱学，并以仅平方可积势的这类算子的谱理论的最新发展为指导。特别有趣的是，在孤子与孤立的特征值相关的方式中，嵌入的奇异谱可以归因于什么行为的问题。(b)随机势的薛定谔方程（或安德森模型）及其与唯一延拓的联系，从而与不确定性原理的联系（特别是费弗曼所提倡的形式）。这也与辛几何有关。(c)在所有温度下的经典库仑气体，或等价地，在一般$\ β $下的随机矩阵。这将通过Dumitriu和Edelman开创的具有随机递归系数的正交多项式的研究来实现。(d)广义薛定谔算子在粗糙远程（如平方可积）扰动下的绝对连续谱的稳定性。通过研究简单的物理模型，可以集中精力解决基本问题，而不会被技术细节所阻碍。为这些简单模型开发的方法，也许更重要的是透视图，然后通知那些更接近应用程序的工作人员。本项目中的三个例子如下：(a)通过研究一般逆温度下的随机矩阵，人们希望更好地理解最有趣的情况：当β等于2时。这个例子很有趣，因为它（目前主要是经验的）与黎曼ζ函数的零点有关。当然，解析数论在当今社会有很多贡献，特别是在密码学方面；虽然这个项目并没有直接解决这些问题，但人们必须小心记住，构成这条大河的许多支流。(b)虽然近几十年来对可积哈密顿偏微分方程进行了大量的研究，但注意力主要集中在周期性或迅速减少的初始数据的情况。这回避了一个非常自然的问题，即Lax算子的嵌入奇异谱的存在导致了什么行为。众所周知，孤子是孤立特征值的结果。这项工作的潜在意义是非线性介质中新的准粒子模式的预测。(c)从具有粗糙和缓慢衰减势的一维薛定谔方程的研究中发现的对反散射的更好理解可能导致遥感技术的改进。

Proposal DMS-0401277PI: Rowan Killip, UCLATitle: Schroedinger operators, integrable systems, and othersimple models in mathematical physicsABSTRACTThe project is devoted to the furtherance of the mathematicalunderstanding of certain simple physical models: (a) The long-timeasymptotics of the KdV equation will be studied for slowly decreasinginitial data via the inverse scattering/spectral method, with recentdevelopments in the spectral theory of such operators with merelysquare-integrable potentials leading the way. Of particular interest isthe question of what behaviours can be attributed to embedded singularspectrum in the way that solitons are related to isolated eigenvalues. (b)The Schrodinger equation with random potentials (or Anderson model) andits connections to unique continuation and thence to the uncertaintyprinciple (particularly in the form advocated by Fefferman). This alsomakes links to symplectic geometry. (c) The classical Coulomb gas at alltemperatures, or equivalently, random matrices at general $\beta$. Thiswill be pursued through the study of orthogonal polynomials with randomrecurrence coefficients as pioneered by Dumitriu and Edelman. (d) Thestability of the absolutely continuous spectrum of general Schrodingeroperators under rough long-range (say square-integrable) perturbation.By studying simple physical models, it is possible to concentrate onessential difficulties, without being waylaid by technicalities. Themethods and perhaps more importantly, perspectives that developed forthese simple models then inform those working closer to applications.Three examples taken from this project are the following: (a) By studyingrandom matrices at general inverse temperature, beta, one hopes to betterunderstand the most interesting case: when beta equals two. This case isso interesting because of its (currently mostly empirical) connection tothe zeros of the Riemann zeta function. Of course, analytic number theoryhas much to offer society at the present particularly in terms ofcryptography; while this project does not address these questionsdirectly, one must be careful to remember the many tributaries that make amighty river. (b) While integrable Hamiltonian PDEs have receivedintensive study in recent decades, attention has mostly been directed tothe cases of periodic or rapidly-decreasing initial data. This side-stepsthe very natural question of what behaviours are attributable to theexistence of embedded singular spectrum for the Lax operator. As is wellunderstood, solitons are a consequence of isolated eigenvalues. Apotential implication of this work is the prediction of new quasi-particlemodes in non-linear media. (c) The better understanding of inversescattering found from the study of the one-dimensional Schrodingerequation with rough and slowly decaying potentials may lead toimprovements in remote sensing technologies.