Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2506.23098

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

Zhenfu Wang·Disheng Xu·Qi Zhou
View original source ↗Read source document ↗Export metadata ↓

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 Schrödinger 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.

Explore related subjects

math.DSmath-phmath.MPmath.SP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhenfu Wang, Disheng Xu, Qi Zhou. 2025-06-29. Subordinacy theory for long-range operators: hyperbolic geodesic flow insights and monotonicity theory. https://arxiv.org/abs/2506.23098

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Pivoting technique for the circle homeomorphism group

We adapt Gou{ë}zel's pivoting technique to the circle homeomorphism group. As an application, we give different proofs of Gilabert Vio's probabilistic Tits alternative and Malicet's exponential synchronization.

math.DS↗
arXiv2026

Asymmetry of a class of Mellin transforms

We introduce the quantity $μ_η$, defined for every complex $s$ in the critical strip, as a transformation of the Mellin transform associated to the functions $η$. We establish a sufficient condition on $η$ under which $μ_η(s)$ and $μ_η(1-s)$ cannot both vanish outside the critical line. An application is given to the case in which $η$ is the fractional part function, and the zeros of $μ_η$ coincide with the zeros of the Riemann zeta function.

math.DS↗
arXiv2026

Infinite Set of Resonances in the Linear Damped Oscillator Subject to Harmonic Forcing with Non-standard Frequency Modulation

It is shown that harmonic signals incorporating a type of weak non-standard frequency modulation (wNSFM) have interesting spectral properties, namely, time-dependent bandwidths that become increasingly broader with increasing time. As such, they represent a class of signals with frequency-time coupling in their spectra. Specifically, the weakly damped oscillator exhibits always two transient resonance captures involving two distinct harmonics possessing relatively high amplitudes over finite time intervals, while the overall response decays as $~t^{-1/2}$ as $t\rightarrow\infty$. Considering the undamped oscillator, it possesses two types of resonances, referred to as simple and non-simple resonances. Simple resonances correspond to finite-amplitude steady-state responses caused by two sustained resonance captures, in the form of two distinct modulated quasi-periodic responses, which, however are "activated" at different time instances. The necessary and sufficient conditions for non-simple resonances are given in the form of a theorem which predicts the existence of resonant harmonics and specifies the special phase conditions that the resonant harmonics must satisfy for constructive interference; the resulting undamped non-simple resonance grows unboundedly as $~t^{-1/2}$ as $t\rightarrow\infty$, in contrast to the classical resonance growth of the linear resonator with unmodulated harmonic excitation whose response grows as $~t$ as $t\rightarrow\infty$. These resonant responses are persistent to changes in the parameters of the wNSFM. Our results reveal interesting infinite sets of resonances in linear SDOF resonators under frequency-modulated excitations.

math.DS↗