In last decades, various methods have been developed in the study of one-dimensional (discrete) quasiperiodic Schr¨odinger operators (QPSO) under an analytic condition. Based on them, a lot of deep results on spectrum of analytic QPSO. However, some of them are not as good as expected and seem difficult to be improved by the original methods. Moreover, these methods depend heavily on analytic conditions and are difficult to be extended to smooth situations. Thus new methods are needed. In 2015, a new method modified from L.S.Young’s works was proposed by Y. Wang and Z. Zhang. Based on it, a series of positive results for Sinai’s model (QPSO with a C 2 cos-type potential and a large coupling) were obtained. Some of them are new even in analytic topology. Recently, we obtain a series of sharp results on this model. They include a sharp estimate on the regularity of Lyapunov exponents (which is even new for Almost Mathieu operator with a cosine potential), the dry version of Cantor spectrum, homogenous spectrum gap and absolute continuity of IDS.
星期四, 2018/10/11 - 从 15:00 到 16:30