Dynamical localization for polynomial long-range hopping random operators on $\mathbb{Z}^d$

Abstract: In this paper, we prove a power-law version dynamical localization for a random operator $\mathrm{H}{\omega}$ on $\mathbb{Z}^d$ with long-range hopping. In breif, for the linear Schr\"odinger equation $$\mathrm{i}\partial{t}u=\mathrm{H}_{\omega}u, \quad u \in \ell^{2(\mathbb{Z}d),} $$ the Sobolev norm of the solution with well localized initial state is bounded for any $t\geq 0$.