Quantitative Green's Function Estimates for Lattice Quasi-periodic Schr\"odinger Operators

TL;DR

This paper develops precise Green's function estimates for higher-dimensional lattice quasi-periodic Schrödinger operators, leading to proofs of Anderson localization and regularity of the integrated density of states, thus solving a longstanding problem.

Contribution

It introduces new quantitative Green's function estimates for lattice QP Schrödinger operators, enabling proofs of Anderson localization and IDS regularity.

Findings

Proves Anderson localization for certain lattice QP Schrödinger operators.

Establishes Hölder continuity of the integrated density of states.

Provides a solution to Bourgain's problem from 2000.

Abstract

In this paper, we establish quantitative Green's function estimates for some higher dimensional lattice quasi-periodic (QP) Schr\"odinger operators. The resonances in the estimates can be described via a pair of symmetric zeros of certain functions and the estimates apply to the sub-exponential type non-resonant conditions. As the application of quantitative Green's function estimates, we prove both the arithmetic version of Anderson localization and the $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for such QP Schr\"odinger operators. This gives an affirmative answer to Bourgain's problem in\cite{Bou00}.