遍历薛定谔算子在量子力学和固体物理中具有基本的重要性。安德森局域化和谱结构是其中心课题。一维拟周期格点薛定谔算子是最典型的遍历算子之一。它可以定义cocycle，具有动力系统结构。近年来动力系统的方法应用在单频位势薛定谔算子的安德森局域化和谱结构的研究中，取得了巨大的成就，但多频位势情形的研究成果相对较少，有待进一步发展。本项目拟对一维多频拟周期薛定谔算子的安德森局域化、李亚普诺夫指数的恒正性和谱的拓扑结构等几个重要问题进行深入研究。

Ergodic Schrödinger operators have basic significance in quantum mechanics and solid state physics. Anderson localization and spectra structure are central topics in the fields. 1D quasi-periodic Schrödinger operator is one of the most typical ergodic one. With it one can define a cocycle which possesses a dynamical system structure. Recently the applications of dynamical system methods to Anderson localization and spectrum structure of single frequecy quasi-periodic Schrödinger has made a great progress, but few for multi-frequency one. Further development is needed. This project aims at a deep study of some important problems on Anderson localization, positivity of Lyapunov exponents and topological structure of spectrum for 1D multi-frequecy quasi- periodic Schrödinger operators.