INTEGRABLE SYSTEMS OF PDE WITH QUASI-PERIODIC INITIAL DATA

研究电子的导电性是凝聚态物理的核心。经典的电子传导性理论是建立在自由电子被正离子散射的思想之上的。一个关键的概念是平均自由程，即电子与离子碰撞前的平均行程。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绕射。只有当电子从晶体中的缺陷散射时，才会出现电阻。有了量子力学的修正，经典模型仍然可以使用，但在新的图像中，电子在杂质之间呈之字形移动。杂质越多，平均自由程越小，电导率越低。在数学上，这一现象可用基本薛定谔方程来描述。然而，事实证明，发展一个数学上正确的理论，即由高水平杂质引起的电子的“局部化”，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年代末成功地遇到了这个问题。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。数学家们从雅科夫·G·西奈的开创性工作开始认识到，安德森本地化理论与许多数学结构有关，并深深植根于现代数学问题。据了解，这种现象不仅表现在存在随机杂质的情况下，而且还表现在其他不同类型的结构中，如准周期结构。后者导致了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础领域，如复变量、傅里叶变换和动力系统。准周期薛定谔方程理论在过去的四十年里得到了广泛的发展，这些数学家在普林斯顿大学、高级研究所、芝加哥大学、加州理工大学、欧文大学、几所巴黎大学、苏黎世理工学院以及巴西、日本和以色列的大学工作。20世纪60年代末，数学家们发现了这些非线性方程与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初始数据结合起来。然而，对于准周期初始数据，即由几个不同周期的周期函数组成的数据，还没有发展出理论。*该提案的主要目标是为此类初始数据制定整合方法。*目标**

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. *** **