拟周期薛定谔算子谱理论是近年来物理和数学研究的热点，吸引了一批诺贝尔奖，沃尔夫奖和菲尔茨奖得主等著名学者参与研究。其中一维情形可表述为拟周期薛定谔cocycle动力系统，从薛定谔cocycle的性质出发可以得到薛定谔算子谱的大量信息，其中李亚普诺夫指数的性质尤为重要。在沃尔夫奖得主Y.Sinai提出的cos-type模型中，我们提出了新方法，得到了有关指数的一系列性质和算子谱的Cantor结构。本项目我们将进一步发展我们的方法以改进拟周期薛定谔算子谱理论，包括 (i)公开问题：对指定的丢番图频率，证明解析算子的Anderson局域化;(ii) 菲尔兹奖得主A.Avila的公开问题：解析条件下不同指数加速子的共存性;(iii)对一般光滑情形研究指数的正性和正则性以及谱集合的拓扑结构,(iv) 单调位势情形中指数的正则性和算子的Anderson局域化；(iv) 指数性质上余弦位势的唯一性。

The spectral theory of quasiperiodic Schrodinger operators is a hotspot in recent research on physics and mathematics, and many outstanding scholars, including some Noble, Wolf and Fields medal prize winners are involved in the field. One dimensional quasiperiodic Schrodinger operators is equivalent to a dynamical system, named quasiperiodic Schrodinger cocycle. From properties of Schrodinger cocycle, many information can be obtained on spectrum of Schrodinger operators, and the knowledge on Lyapunov exponent are most important. For C2-cos-type model proposed by Wolf Prize winner Y. Sinai, we proposed a new method, with which we obtained a series of properties on Lyapunov exponent as well as Cantor structure of spectrum. In this program, we will continue to study the properties of Lyapunov exponent and then improve the spectral theory of quasiperiodic Schrodinger operators. More precisely, we will study the following problems: (i)(open problem) For any fixed Diophantine frequency, Anderson localization occurs under an analytic condition; (ii) (An open problem by Fields metal winner A. Avila) The coexistence of various accelerations under an analytic conditions; (iii) Study the positivity and regularity of Lyapunov exponent and topological structure of spectrum under a general smooth condition; (iv) Study the regularity of Lyapunov exponent and Anderson localization under a monotonic condition; (v) The uniqueness of cosine potential in the sense of the special property of Lyapunov exponent.