Quantitative Green's Function Estimates for Lattice Quasi-periodic Schrödinger Operators

In this paper, we establish quantitative Green’s function estimates for some higher dimensional lattice quasi-periodic (QP) Schrödinger 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 $(\frac{1}{2}-)$-Hölder continuity of the integrated density of states (IDS) for such QP Schrödinger operators. This gives an affirmative answer to Bourgain’s problem in\cite{Bou00}.