本项目主要研究刘维尔频率的拟周期薛定谔算子的谱理论及Sarnak猜测。拟周期薛定谔算子与量子霍尔效应等现象相关，获得了数学家和物理学家的广泛关注，用动力系统的方法研究它的谱是近期研究热点。由于小分母问题，现有结果主要集中在特殊算子或丢番图频率的拟周期薛定谔算子中。我们拟对刘维尔频率的拟周期薛定谔算子谱进行研究，利用定量几乎可约性理论证明稠密刘维尔频率的薛定谔算子态密度的Hölder连续性及谱的齐次性，并构造超刘维尔频率，证明此时算子有绝对连续谱。Sarnak猜测由于与素数定理的联系也吸引了一批著名学者如沃尔夫奖、菲尔兹奖得主等参与研究。该猜测在一些具体零熵系统中被证明，但主要集中在正则系统或一维频率可交换系统中。我们拟创新性的用定量几乎可约研究不可交换的非正则系统——拟周期薛定谔cocycle投影流的Möbius正交性，并构造高维刘维尔频率使得一类可交换的拟周期斜积流是Möbius正交的。

This project aims to study the spectral theory of quasi-periodic Schrödinger operator with Liouvillean frequency and Sarnak’s conjecture. Quasi-periodic Schrödinger operator is related to the phenomena in physics such as quantum Hall effect, and it attains a lot of attentions from mathematicians and physicians. Using the methods in dynamical systems to study the spectrum of quasi-periodic Schrödinger operators is a hotspot of recent research. However, due to the small divisor problem, the known results are mainly on special operators or operators with Diophantine frequencies. We are planning to study the spectrum of quasi-periodic Schrödinger operators with Liouvillean frequencies. Applying quantitative almost reducibility theory, we want to prove there exist dense Liouvillean frequencies that the integrated density of states of the corresponding quasi-periodic Schrödinger operator is Hölder continuous and the spectrum of the operator is homogeneous. Moreover, we want to construct a kind of super-Liouvillean frequencies such that the Schrödinger operator has absolutely continuous spectrum. Sarnak’s conjecture has a close relationship with prime number theory, and many outstanding scholars including some Wolf and Fields medal prize winners are involved in this subject. This conjecture has been proved in some specific dynamical systems with zero topological entropy. However, these systems are mainly regular or commutative. We innovatively using quantitative almost reducibility to study the Möbius disjointness for the projective action of quasi-periodic Schrödinger cocycles, which are noncommutative and irregular. Moreover, we want to construct some high-dimensional Liouvillean frequencies such that a special kind of commutative quasi-periodic skew-product flow is also Möbius disjoint.