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Green's function estimates for quasi-periodic operators on $\mathbb{Z}^d$ with power-law long-range hopping

Yunfeng Shi, Li Wen

TL;DR

The paper studies discrete quasi-periodic Schrödinger operators on $\mathbb{Z}^d$ with power-law long-range hopping and analytic cosine-type potentials, formulating the model $\mathcal{H}(\theta)=\varepsilon \mathcal{W}_{\phi}+v(\theta+\mathbf n\cdot\boldsymbol\omega)\delta_{\mathbf n,\mathbf n'}$. It develops a quantitative Green's function framework via a multi-scale analysis that handles non-self-adjointness and slow (power-law) decay, establishing robust off-diagonal decay estimates and left-inverse control. Consequences include an arithmetic localization result, finite-volume Hölder continuity of the integrated density of states, and absence of eigenvalues for the Aubry dual operator, extending prior Bourgain-CSZ methods to more general potentials and long-range hopping. These results provide a rigorous bridge between Anderson localization techniques and KAM-type approaches in the setting of analytic cosine-type quasi-periodic potentials with power-law hopping, with potential implications for higher-dimensional QP systems and related PDE contexts.

Abstract

We establish quantitative Green's function estimates for a class of quasi-periodic (QP) operators on $\mathbb{Z}^d$ with power-law long-range hopping and analytic cosine type potentials. As applications, we prove the arithmetic version of localization, the finite volume version of $(\frac12-)$-Hölder continuity of the IDS, and the absence of eigenvalues (for Aubry dual operators).

Green's function estimates for quasi-periodic operators on $\mathbb{Z}^d$ with power-law long-range hopping

TL;DR

The paper studies discrete quasi-periodic Schrödinger operators on with power-law long-range hopping and analytic cosine-type potentials, formulating the model . It develops a quantitative Green's function framework via a multi-scale analysis that handles non-self-adjointness and slow (power-law) decay, establishing robust off-diagonal decay estimates and left-inverse control. Consequences include an arithmetic localization result, finite-volume Hölder continuity of the integrated density of states, and absence of eigenvalues for the Aubry dual operator, extending prior Bourgain-CSZ methods to more general potentials and long-range hopping. These results provide a rigorous bridge between Anderson localization techniques and KAM-type approaches in the setting of analytic cosine-type quasi-periodic potentials with power-law hopping, with potential implications for higher-dimensional QP systems and related PDE contexts.

Abstract

We establish quantitative Green's function estimates for a class of quasi-periodic (QP) operators on with power-law long-range hopping and analytic cosine type potentials. As applications, we prove the arithmetic version of localization, the finite volume version of -Hölder continuity of the IDS, and the absence of eigenvalues (for Aubry dual operators).
Paper Structure (22 sections, 28 theorems, 575 equations)

This paper contains 22 sections, 28 theorems, 575 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Arithmetic localization
  4. $\left(\frac{1}{2}-\right)$-Hölder continuity of the IDS
  5. Absence of eigenvalues
  6. Ideas of the proof and new ingredients
  7. Structure of the paper
  8. The notation
  9. Preliminaries
  10. Tame property
  11. Smoothing property
  12. Perturbation argument
  13. Hadamard type estimate
  14. Off-diagonal decay of Green's functions
  15. Quantitative Green's function estimates
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\bm\omega\in DC_{\tau,\gamma}$. Fix $d<\alpha_0<\tau$ and $\alpha_1>2200\tau$. Let Then there is some $\varepsilon_0=\varepsilon_0(\alpha_1,\alpha_0,d,\tau,\gamma,v,R,\phi)>0$ so that for $0<|\varepsilon|\leq \varepsilon_0$ and $E\in v(\mathbb D_{R/2})$, there exists a sequence with the following properties: Fix any $\theta\in \mathbb{T}$, if a subset $\Lambda\subset\mathbb Z^d$ is $s$-${\r

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 3.1: Tame property
  • proof
  • Lemma 3.2: Smoothing property
  • proof
  • Lemma 3.3: Rows estimate
  • ...and 50 more