Quasiperiodic Schroedinger operators with well-approximated frequencies

准周期算子的研究是现代分析、谱理论、数学和理论物理的中心课题。它在大卫莱斯的诺贝尔奖获奖作品中发挥了作用，也在阿图尔阿维拉的菲尔兹奖获奖作品中发挥了作用。在过去的几十年里，人们创造了一个美丽的数学理论，它与数学的许多部分以及物理学中的应用都有联系。拟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，而且还取决于它们的算术性质（例如，是否可以很好地或很差地被有理数逼近）。这一现象是在20世纪70年代被发现的;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Almathieu算子的特殊情况。特别地，我们将把光谱的结构看作一个集合，也将研究由算符定义的量子动力学（后者对理论物理学的应用至关重要）。我们计划开发一个强大的方法，将不局限于几乎马蒂厄运营商，我们将开发的方法可能会用于进一步调查的谱性质的几乎周期运营商。

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.