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Hölder continuity of the integrated density of states for quasi-periodic Jacobi block matrices

Rui Han, Wilhelm Schlag

TL;DR

This work proves Hölder continuity of the integrated density of states for discrete quasi-periodic Jacobi block operators with Diophantine frequencies by linking global zero counts of finite-volume determinants to local zero behavior via a novel center-shifting scheme. The approach blends transfer-matrix/symplectic Lyapunov theory, Avila’s acceleration, and robust large-deviation bounds with careful Green’s function (Poisson) analysis for both Schrödinger and Jacobi-block cases, yielding a Hölder exponent β valid for any 0<β<1/(2κ^d(ω,E0)). The results extend prior scalar Schrödinger theory to block-valued systems, accommodate a broader class of frequencies, and reveal how the acceleration κ^d governs IDS regularity; when symmetries are present, sharper exponents may be obtained. The findings advance understanding of IDS regularity in quasi-periodic operators and have implications for spectral theory and localization phenomena in higher-rank, block-structured models.

Abstract

In this paper, we prove Hölder continuity of the integrated density of states for discrete quasiperiodic Jacobi $d\times d$ block matrices with Diophantine frequencies. The Hölder exponent is shown to be any $β$ such that $0<β<1/(2κ^d)$, where $κ^d$ is the acceleration, i.e., the slope of the sum of the top $d$ Lyapunov exponents in the imaginary direction of the phase. This generalizes the Hölder continuity results in the Schrödinger operator setting in \cites{GS2,HS1}, and also strengthens them in that setting by covering more Diophantine frequencies. The proof is built on a new scheme for obtaining a local zero count for finite-volume characteristic polynomials from a global one.

Hölder continuity of the integrated density of states for quasi-periodic Jacobi block matrices

TL;DR

This work proves Hölder continuity of the integrated density of states for discrete quasi-periodic Jacobi block operators with Diophantine frequencies by linking global zero counts of finite-volume determinants to local zero behavior via a novel center-shifting scheme. The approach blends transfer-matrix/symplectic Lyapunov theory, Avila’s acceleration, and robust large-deviation bounds with careful Green’s function (Poisson) analysis for both Schrödinger and Jacobi-block cases, yielding a Hölder exponent β valid for any 0<β<1/(2κ^d(ω,E0)). The results extend prior scalar Schrödinger theory to block-valued systems, accommodate a broader class of frequencies, and reveal how the acceleration κ^d governs IDS regularity; when symmetries are present, sharper exponents may be obtained. The findings advance understanding of IDS regularity in quasi-periodic operators and have implications for spectral theory and localization phenomena in higher-rank, block-structured models.

Abstract

In this paper, we prove Hölder continuity of the integrated density of states for discrete quasiperiodic Jacobi block matrices with Diophantine frequencies. The Hölder exponent is shown to be any such that , where is the acceleration, i.e., the slope of the sum of the top Lyapunov exponents in the imaginary direction of the phase. This generalizes the Hölder continuity results in the Schrödinger operator setting in \cites{GS2,HS1}, and also strengthens them in that setting by covering more Diophantine frequencies. The proof is built on a new scheme for obtaining a local zero count for finite-volume characteristic polynomials from a global one.

Paper Structure

This paper contains 17 sections, 22 theorems, 213 equations.

Table of Contents

  1. Introduction
  2. Lyapunov exponents, large deviations, and the Green's function
  3. Transfer matrices
  4. Lyapunov exponents
  5. Avila's acceleration
  6. Large deviation estimates for monodromy matrices
  7. Green's function and Poisson formula
  8. Schrödinger case
  9. Jacobi block case
  10. Global zero counts
  11. Schrödinger case
  12. Jacobi block case
  13. Analysis of local zeros for the Schrödinger case
  14. Hölder continuity of the Schrödinger case
  15. Analysis of local zeros for the Jacobi block case
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $E_0\in \mathbb R$ be a fixed energy for which $L_d(\omega,E_0)>0$. For any $\omega\in \mathrm{DC}$, the integrated density of states of $H_{\omega,\theta}$ satisfies for any $E,E'\in I_{E_0}$, a neighborhood of $E_0$, and any Hölder exponent $0<\beta<1/(2\kappa^d(\omega,E_0))$.

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • ...and 23 more