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

提案DMS-0401277 PI：Rowan Killip，UCLATITE：Schroedinger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深入理解：（a）将通过逆散射/谱方法研究初始数据缓慢减少时KdV方程的长期渐近性，与最近的发展，在光谱理论的这种运营商与merelysquare-integrable潜在的领导方式。 特别感兴趣的是什么行为可以归因于嵌入的奇异谱的方式孤立本征值相关的问题。（B）具有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，以及由此与不确定性原理（特别是在费曼所提倡的形式中）的联系。 这也与辛几何有关。(c)所有温度下的经典库仑气体，或者等价地，一般$\beta$下的随机矩阵。这将是追求通过研究正交多项式与randomrecurrence系数率先由杜米特留和埃德尔曼。(d)一般Schrodinger算子的绝对连续谱在粗糙长程（比如平方可积）扰动下的稳定性通过研究简单的物理模型，可以集中于本质上的困难，而不受技术性的阻碍。 这些方法，也许更重要的是，为这些简单的模型开发的观点，然后通知那些工作更接近应用程序。从这个项目中采取的三个例子如下：（a）通过研究一般逆温度下的随机矩阵，β，人们希望更好地理解最有趣的情况：当β等于2。 这种情况是如此有趣，因为它（目前主要是经验）连接到黎曼zeta函数的零点。 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;虽然这个项目没有直接解决这些问题，但人们必须小心记住，许多支流构成了一条大河。 (b)虽然可积Hamilton偏微分方程在近几十年来得到了广泛的研究，但人们的注意力主要集中在周期或快速减少的初始数据的情况下。 这侧步非常自然的问题，什么行为是由于存在嵌入奇异谱的Lax算子。 众所周知，孤子是孤立本征值的结果。 这项工作的一个潜在的意义是在非线性介质中预测新的准粒子模式。(c)通过对具有粗糙和缓慢衰减势的一维薛定谔方程的研究，可以更好地理解逆散射现象，这可能会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.