Green's function estimates for long-range quasi-periodic operators on ${\mathbb{Z}}^d$ and applications

We establish quantitative Green’s function estimates for a class of quasi-periodic (QP) operators on ${\mathbb{Z}}^d$ with certain slowly decaying long-range hopping and analytic cosine type potentials. As applications, we prove the arithmetic spectral localization, and obtain upper bounds on quantum dynamics for all phase parameters. To deal with quantum dynamics estimates, we develop an approach employing separation property (rather than the sublinear bound) of resonant blocks in the regime of Green’s function estimates.