Subordinacy theory for long-range operators: hyperbolic geodesic flow insights and monotonicity theory

TL;DR

This paper develops a unified subordinacy theory for long-range operators, linking dynamical systems and spectral analysis, and proves the existence of purely absolutely continuous spectrum in new classes of quasi-periodic operators.

Contribution

It introduces a novel geometric and monotonicity framework for analyzing long-range operators, resolving open problems and extending spectral theory techniques.

Findings

Proved purely absolutely continuous spectrum for quasi-periodic long-range operators with analytic potentials.

Established the first proof of all-phases persistence for finite-range perturbations of subcritical almost Mathieu operators.

Connected stable/vertical bundle intersections in geodesic flows to localization and delocalization phenomena.

Abstract

We introduce a comprehensive framework for subordinacy theory applicable to long-range operators on $\ell^2(\mathbb Z)$, bridging dynamical systems and spectral analysis. For finite-range operators, 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. For infinite-range operators-where traditional cocycle methods become inapplicable-we characterize absolutely continuous spectrum through the growth of generalized eigenfunctions, extending techniques from higher-dimensional lattice models. 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 for general bundles (including its preservation under center-bundle restrictions) and adapted analytic spectral and KAM techniques, enables our spectral analysis of long-range operators.