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

建议DMS-0401277PI：Rowan Killip，Uclatitle：数学物理中的薛定谔算子，可积系统和其他简单模型摘要本项目致力于促进对某些简单物理模型的数学理解：(A)将研究KdV方程的长期测量符号学，以通过反散射/谱方法缓慢减少初始数据，这类算子的谱理论的最新进展是以小平方可积势为主导的。特别令人感兴趣的问题是，什么行为可以归因于嵌入的奇异谱，就像孤子与孤立本征值相关的方式一样。(B)具有随机势的薛定谔方程(或Anderson模型)及其与唯一延拓和不确定原理(特别是以Fefferman所倡导的形式)的联系。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，在一般情况下的随机矩阵。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进行。(D)一般Schrodinger型算子的绝对连续谱在粗糙的长程(比方可积)扰动下的稳定性。通过研究简单的物理模型，可以专注于本质的困难，而不受技术上的阻碍。这些方法，也许更重要的是，为这些简单模型开发的观点，然后通知那些更接近应用的工作。这个项目中的三个例子如下：(A)通过研究广义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。这个例子非常有趣，因为它(目前主要是经验的)与Riemann Zeta函数的零点有关。当然，解析数论在当今社会有很大的贡献，特别是在密码学方面；虽然这个项目没有直接解决这些问题，但我们必须小心地记住，形成一条河流的支流很多。(B)虽然可积的哈密顿偏微分方程组在最近几十年得到了深入的研究，但人们的注意力大多集中在周期性或快速递减的初始数据的情况上。这里绕过了一个非常自然的问题，即什么行为可以归因于Lax算子嵌入的奇异谱的存在。众所周知，孤子是孤立本征值的结果。这项工作的潜在含义是对非线性介质中新的准粒子模型的预测。(C)通过研究具有粗糙和缓慢衰减势的一维薛定谔方程，可以更好地理解逆散射，这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.