Subordinacy theory for long-range operators and its applications

报告题目

Subordinacy theory for long-range operators and its

applications

报 告 人

大湾区大学

许地生 副教授

报告时间

2025年10月28日（周二）下午16：30-18：30

报告地点

上海师范大学徐汇校区西区3号楼3楼会议室332

报告摘要

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 ℓ^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.