Quasiperiodic Schroedinger operators with well-approximated frequencies

点击翻译按钮获取中文摘要

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.