INTEGRABLE SYSTEMS OF PDE WITH QUASI-PERIODIC INITIAL DATA

具有准周期初始数据的偏微分方程可积系统

• 批准号: RGPIN-2015-05140
• 负责人: Goldstein, Michael
• 金额: $ 1.46万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=613377
• 关键词: INTEGRABLE; SYSTEMS; PDE; QUASI; PERIODIC

The study of the conductance of electrons lies at the heart of condensed-matter physics. The classical theory of electronic conductivity was built on the idea of free electrons scattered by positive ions. A key concept was the mean free path, the average length an electron travels before it collides with an ion. According to classical theory, electronic conductivity should be directly proportional to the mean free path. Quantum mechanics explained why electrons apparently do not scatter from ions that occupy regular lattice sites; the wave character of an electron causes the electron to diffract from an ideal crystal. Resistance appears only when electrons scatter from imperfections in the crystal. With that quantum mechanical revision, the classical model can still be used, but in the new picture an electron is zigzagging between impurities. The more the impurities, the smaller the mean free path and the lower the conductivity. Mathematically, the phenomenon was described via the fundamental Schrodinger Equation. It turned out, however, that to develop a mathematically correct theory of "localization" of electrons, caused by a high level of impurities, posed a deep problem -- one that renowned physicist Phil Anderson successfully confronted in the late 1950s. Anderson’s discovery, which carries his name, helped earn him the 1977 Nobel Prize. Mathematicians realized, starting with ground-breaking works by Yakov G. Sinai, that the theory of Anderson Localization relates to many mathematical structures, and has deep roots in the problems of modern mathematics. It was understood that the phenomenon exhibits itself not only in the presence of random impurities, but also for different other types of structures such as quasi-periodic ones. The latter leads to an analysis of spectrum of the quasi-periodic Schrodinger Equation. The analysis relies on a number of classical fundamental domains of mathematics such as complex variables, Fourier transform, and Dynamical systems. The theory of quasi-periodic Schrodinger Equations has been extensively developed in the last forty years by mathematicians working at Princeton University, the Institute for Advanced Study, the University of Chicago, Caltech, Irvine University, several Paris universities, ETH Zurich, and universities in Brazil, Japan, and Israel. One of applications of the theory addresses the so-called completely integrable non-linear differential equations. In the late 1960s, mathematicians discovered the connection between these non-linear equations and the spectral theory of linear differential equations. The discovery allowed mathematicians to integrate the former with various initial data. However, no theory was developed for quasi-periodic initial data, i.e. data composed from several periodic functions with different periods.  The main objective of the proposal is to develop methods of integration for such initial data.

电子电导的研究是凝聚态物理学的核心。电子导电性的经典理论是建立在自由电子被正离子散射的概念上的。一个关键的概念是平均自由程，即电子在与离子碰撞前的平均距离。 根据经典理论，电子电导率应该与平均自由程成正比，量子力学解释了为什么电子显然不会从占据规则晶格位置的离子中散射;电子的波动特性导致电子从理想晶体中散射。只有当电子从晶体中的缺陷散射时，电阻才会出现。经过量子力学的修正，经典模型仍然可以使用，但在新的图像中，电子在杂质之间徘徊。杂质越多，平均自由程越小，电导率越低。在数学上，这种现象通过基本薛定谔方程来描述。然而，事实证明，要发展一个数学上正确的电子“局域化”理论，由高水平的杂质引起，提出了一个深层次的问题-著名物理学家菲尔安德森在20世纪50年代末成功地面对。安德森的发现，以他的名字命名，帮助他赢得了1977年的诺贝尔奖。数学家们意识到，从雅科夫G。Sinai认为，安德森局部化理论涉及到许多数学结构，并且在现代数学问题中有着深刻的根源。据了解，这种现象不仅表现在随机杂质的存在下，而且还表现在不同的其他类型的结构，如准周期结构。后者导致准周期薛定谔方程的谱分析。分析依赖于一些经典的数学基本领域，如复变量，傅立叶变换和动力系统。准周期薛定谔方程的理论在过去的四十年里被普林斯顿大学、高等研究院、芝加哥大学、加州理工学院、欧文大学、几所巴黎大学、苏黎世联邦理工学院以及巴西、日本和以色列的大学的数学家们广泛地发展。 该理论的一个应用是解决所谓的完全可积非线性微分方程。在20世纪60年代后期，数学家发现了这些非线性方程和线性微分方程谱理论之间的联系。这一发现使数学家能够将前者与各种原始数据结合起来。然而，没有理论被开发用于准周期初始数据，即由具有不同周期的多个周期函数组成的数据。 该提案的主要目标是为这些初始数据制定综合方法。