Nonperturbative methods for quasiperiodic discrete Schroedinger equations on the line

摘要：本文讨论了一维确定性势格点上离散薛定谔方程的各个方面。到目前为止，在与Jean Bourgain和Michael Goldstein的合作中，作者已经考虑了由环面上的遍历移位给出的拟周期势，通过两个环面上的偏斜移位得到的势，以及根据强混合动力学定义的势，例如圆上的加倍映射或两个环面上的双曲自同构。在这些情况下，积极的李雅普诺夫指数，积分态密度的规律性，和安德森本地化进行了研究。在这一点上，我们计划解决几个剩余的问题，包括以下几个：1）在斜移势的情况下，李雅普诺夫指数是否为正 对于小的混乱？2)是否有可能获得关于 本征函数在准周期的情况下，假设只有积极的 李雅普诺夫指数实际上，非微扰技术 允许定义中所述的基本支持 西奈半岛和Froehlich，斯宾塞，Wittwer的扰动政权？ 这些问题与Y密切相关。G.西奈半岛最近的工作， “准周期介质中的异常传输”，并将提供更好的 更精确的信息的次扩散行为的随机 行走在西奈山上。3)是否有可能将非微扰方法扩展到条带，或 二维平面？4)积分态密度保持器连续的情况下， 几个频率还是偏移5)关于本征能级间距的统计， 值的情况下，偏移？从历史上看，随机薛定谔算符的研究始于20世纪50年代末PhilAnderson的工作，他因此获得了诺贝尔奖。在他的工作之前，人们认为晶体中的小随机杂质不会显著改变其电导。然而，安德森指出，情况并非如此：在每个晶格位置独立出现的微小随机杂质将导体变成绝缘体。由于他的工作，这是不严格的数学，一个精确的理论的发展“安德森本地化”一直追求许多数学家。事实证明，这与几个数学领域的深层结果有关。例如，Fuerstenberg的定理产品的随机矩阵是一个重要的工具，在发展的理论。这些作品吸引了物理学家的注意，特别是统计力学专家。直到今天，数学家和物理学家在这个问题上的思想交流仍然活跃而富有成效。事实上，近年来，物理学对随机现象和方法的兴趣显著增强，由于统计力学提出的许多重要问题已被证明是相当深刻的数学挑战，其解决方案已导致概率技术的重大进步。我们希望本提案中提出的项目将进一步推进我们对统计力学模型的洞察力，并提供有用的为在遍历理论、分析和数学物理方面工作的数学家提供的工具。

ABSTRACT:This proposal deals with various aspects of discrete Schroedinger equationson the one dimensional lattice with deterministic potentials. So far, in collaboration with Jean Bourgain and Michael Goldstein, the author has considered quasi-periodic potentials given by ergodic shifts on tori, potentials obtained by means of the skew-shift on the two torus, as well as potentials defined in terms of strongly mixing dynamics, such as the doubling map on the circle or hyperbolic automorphisms on the two torus. In each of these cases, positivity of the Lyapunov exponent, regularity of the integrated density of states, and Anderson localization were studied. At this point, we are planning to address several remaining questions, including the following ones:1) Is the Lyapunov exponent positive in case of skew-shift potentials for small disorder ? 2) Is it possible to obtained detailed information on the nature of the eigenfunctions in the quasi-periodic case assuming only positivity of the Lyapunov exponent ? In fact, do the non perturbative techniques allow the definition of the essential support as described in the perturbative regime by Sinai and Froehlich, Spencer, Wittwer ? These questions are intimately linked with Y. G. Sinai's recent work on "anomalous transport in quasi-periodic media", and would provide better and more precise information on the subdiffusive behavior of the random walk considered by Sinai. 3) Is it possible to extend the nonperturbative methods to strips, or the two-dimensional plane ?4) Is the integrated density of states Holder continuous in the case of several frequencies or the skew-shift ? 5) What can be said about the statistics of the level-spacings of the eigen values for the case of the skew-shift ?Historically, the study of random Schroedinger operators started with PhilAnderson's work in the late 1950's, for which he received the Nobel prize.Before his work it was believed that small random impurities in a crystalwould not significantly change its conductance. Anderson, however, showed that this is not the case: Arbitrarily small random impurities occurringindependently at each lattice site turn a conductor into an insulator. Sincehis work, which was not mathematically rigorous, the development of a precisetheory of "Anderson localization" has been pursued by many mathematicians. It turned out that there were connections with deep results from several areas of mathematics. For example, Fuerstenberg's theorem on products of random matrices was a crucial tool in the development of the theory. These works attracted the attention of physicists, particularly experts in statistical mechanics. To this day, there is an active and fruitful exchange of ideas between mathematicians and physicists in this subject. In fact, the interest in random phenomena and methods has intensified quite notably in physics in recent years, as many important problems posed by statistical mechanics have proved to be rather deep mathematical challenges whose solution has lead to significant advances of probabilistic techniques.It is our hope that the projects set forth in this proposal will furtheradvance our insight into the models of statistical mechanics as well as providing useful tools for mathematicians working in ergodic theory, analysis, and mathematical physics.