Dynamical localization for finitely differentiable quasi-periodic long-range operators

Abstract

In this paper, we establish the s-power law dynamical localization for a class of finitely differentiable quasi-periodic long-range operators on ${\ell }^2\left(Z^d\right)$ with Diophantine frequencies. This result represents the strongest form of dynamical localization in the setting of finitely differentiable topology which is a generalization of exponential dynamical localization in expectation in the analytic case. Our approach is based on the Aubry duality and quantitative reducibility theorem of the finitely differentiable $\text{SL}(2,R)$ quasi-periodic cocycles in the local regime. The s-power law dynamical localization discussed here also demonstrates strong ballistic transport for finitely differentiable quasi-periodic Schrödinger operators.