Quasiperiodic Schroedinger operators with well-approximated frequencies

准周期算符的研究是现代分析、谱理论、数学和理论物理的中心课题。它在获得诺贝尔奖的大卫·索利斯的工作中发挥了作用，在获得菲尔兹奖的阿图尔·阿维拉的工作中也发挥了作用。过去的几十年见证了一个美丽的数学理论的创造，它与数学的许多部分有联系，也在物理学中有应用。准周期算子的一个有趣的特征是，它们的性质不仅取决于结构参数的大小，还取决于它们的算术性质（例如，是否可以用理数近似）。这种现象在20世纪70年代就被发现了；最近，人们对它进行了详细的研究，主要集中在谱型上，主要是针对Almost Mathieu算子的特殊情况。当前项目的目标是探索同一现象的一些较少研究的表现形式。特别地，我们将把谱的结构看作一个集合，也将把它看作由算符定义的量子动力学（后者对于理论物理中的应用是至关重要的）。我们计划开发一种不局限于Almost Mathieu算子的稳健方法；我们将发展的方法可能用于进一步研究概周期算子的谱性质。

The study of quasiperiodic operators is a central subject in modern analysis, spectral theory, mathematical and theoretical physics. It has played a role in the Nobel Prize winning work of David Thouless, as well as in the Fields medal winning work of Artur Avila. The last decades witnessed the creation of a beautiful mathematical theory with connections to numerous parts of mathematics as well as applications in physics. One of the intriguing features of quasiperiodic operators is that their properties depend not only on the magnitude of the structural parameters but also on their arithmetic properties (for example, being well or poorly approximable by rationals). This phenomenon has been discovered in the 1970's; recently, it has been investigated in great detail, with main focus on the spectral type and mainly for the special case of the Almost Mathieu operator.The goal of the current project is to explore a number of less-investigated manifestations of the same phenomenon. Particularly, we shall look at the structure of the spectrum as a set and also at the quantum dynamics defined by the operator (the latter is crucial for applications in theoretical physics). We plan to develop a robust approach that will not be confined to the Almost Mathieu operator; the methods that we shall develop may be of use in further investigations of the spectral properties of almost periodic operators.