Subordinacy theory for long-range operators and its applications

主讲人：许地生 大湾区大学副教授

时间：2025年10月28日16：30

地点：徐汇校区三号楼332会议室

举办单位：数理学院

主讲人介绍：许地生，大湾区大学副教授，研究方向：动力系统，分形几何，谱理论，数学教育，在Inventiones, Duke Math Journal, J.E.M.S., Annales ENS等顶尖期刊发表学术论文，动力系统国际会议Beyond Uniform Hyperbolicity 学术委员会成员，2021，2025年IMO中国国家集训队教练组成员。

内容介绍：This is a joint work with Zhenfu Wang and Qi Zhou. We introduce a comprehensive framework for subordinacy theory applicable to long-range operators on ?^2(Z), bridging dynamical systems and spectral analysis. We establish a correspondence between the dynamical behavior of partially hyperbolic (Hermitian-)symplectic cocycles and the existence of purely absolutely continuous spectrum, resolving an open problem posed by Jitomirskaya.

Our main results include the first rigorous proof of purely absolutely continuous spectrum for quasi-periodic long-range operators with analytic potentials and Diophantine frequencies—in particular, the first proof of the all-phases persistence for finite-range perturbations of subcritical almost Mathieu operators—among other advances in spectral theory of long-range operators.

The key novelty of our approach lies in the unanticipated connection between stable/vertical bundle intersections in geodesic flows—where they detect conjugate points—and their equally fundamental role in governing (de-)localization for Schr¨odinger operators. The geometric insight, combined with a novel coordinate-free monotonicity theory and adapted analytic spectral and KAM techniques, enables our spectral analysis of long-range operators.