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In this paper, we prove a power-law version dynamical localization for arandom operator $\mathrm{H}_{\omega}$ on $\mathbb{Z}^d$ with long-rangehopping. 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 localizedinitial state is bounded for any $t\geq 0$.

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