Nonperturbative methods for quasiperiodic discrete Schroedinger equations on the line

摘要：这一建议涉及一维晶格上具有确定势的离散薛定谔方程的各个方面。到目前为止，作者与Jean Bourain和Michael Goldstein合作，已经考虑了由环面上的遍历位移给出的拟周期势，通过两个环面上的斜移得到的势，以及由强混合动力学定义的势，如圆上的倍化映射或两个环面上的双曲自同构。在每种情况下，研究了Lyapunov指数的正性、态积分密度的正规性和Anderson局域化。在这一点上，我们计划解决几个剩余的问题，包括以下几个问题：1)在小无序的斜位移势的情况下，Lyapunov指数是正的吗？2)在只假设Lyapunov指数为正的情况下，是否有可能获得关于准周期情况下本征函数的性质的详细信息？事实上，非微扰技术允许西奈和弗罗伊利希、斯宾塞、维特维尔在微扰制度中所描述的基本支撑力的定义吗？这些问题与Y.G.Sinai最近关于“准周期介质中的反常输运”的工作密切相关，并将为Sinai所考虑的随机游动的次扩散行为提供更好和更精确的信息。3)是否有可能将非微扰方法推广到条带或二维平面？4)在多个频率或斜移的情况下，积分的态密度是连续的吗？5)对于斜移的情况，本征值的能级间距的统计是什么？历史上，对随机薛定谔算符的研究始于菲尔安德森在20世纪50年代末的工作，S因此而获得诺贝尔奖。在他的工作之前，人们认为晶体中的微小随机杂质不会显著改变其电导。然而，安德森证明了情况并非如此：在每个晶格点上独立出现的任意微小的随机杂质将导体变成绝缘体。由于他的工作在数学上并不严谨，因此许多数学家一直在追求发展一种精确的“安德森局部化”理论。事实证明，这与几个数学领域的深刻结果有联系。例如，福尔斯滕伯格关于随机矩阵乘积的定理是该理论发展过程中的一个重要工具。这些工作引起了物理学家，特别是统计力学专家的注意。时至今日，数学家和物理学家在这个课题上进行了积极而卓有成效的思想交流。事实上，近年来物理学对随机现象和方法的兴趣明显增强，因为统计力学提出的许多重要问题被证明是相当深刻的数学挑战，其解决方案导致了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促进我们对统计力学模型的理解，并为从事遍历理论、分析和数学物理工作的数学家提供有用的工具。

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.