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Anderson localization for 1-d quasi-periodic Schrödinger operators with degenerate weights

Yingdu Dong, Haoxuan Liu, Zuhong You, Xiaoping Yuan

Abstract

We establish Anderson localization for 1-d discrete Schrödinger operators with positive weights. The distinctive feature of this work lies in the degeneracy of the weights, with both the potentials and weights assumed to be analytic and quasi-periodic. Operators of this kind originate from distinct mathematical physics problems, which include the Frenkel-Kontorova model with impurities, the discretization of singular Sturm-Liouville operators, and the Fisher-KPP lattice equation in heterogeneous media.

Anderson localization for 1-d quasi-periodic Schrödinger operators with degenerate weights

Abstract

We establish Anderson localization for 1-d discrete Schrödinger operators with positive weights. The distinctive feature of this work lies in the degeneracy of the weights, with both the potentials and weights assumed to be analytic and quasi-periodic. Operators of this kind originate from distinct mathematical physics problems, which include the Frenkel-Kontorova model with impurities, the discretization of singular Sturm-Liouville operators, and the Fisher-KPP lattice equation in heterogeneous media.
Paper Structure (11 sections, 13 theorems, 144 equations)

This paper contains 11 sections, 13 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Notations and conventions
  4. Transformation to Jacobi operators
  5. A large deviation theorem for the Green's functions
  6. Initial step
  7. Preparations
  8. Inductive lemma
  9. Exponential decay for the generalized eigenvector
  10. Preparations
  11. Exponential decay

Key Result

Theorem 1.1

Consider the operator given by where the potential $v$ and the non-negative weight $w$ are assumed to be $1$-periodic analytic functions. Moreover, it is assumed that $w \not\equiv 0$, and there exist zeros of $w$ on the real line. Suppose $\omega \in DC_{c_0',A}$ for some $A \ge 3$ and small $c_0'>0$. Then, $\overline{H}$ can be

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 2.1
  • proof
  • Lemma 3.1: Perturbation
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4: Cartan's estimate
  • ...and 11 more