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Schr\"odinger operators with periodic potential have generally been shown toexhibit ballistic transport. In this work, we investigate if the propagationvelocity, while positive, can be made arbitrarily small by a suitable choice ofthe periodic potential. We consider the discrete one-dimensional Schr\"odingeroperator $\Delta+\mu V$, where $\Delta$ is the discrete Laplacian, $V$ is a$p$-periodic non-degenerate potential, and $\mu>0$. We establish aLieb-Robinson-type bound with a group velocity that scales like$\mathcal{O}(1/\mu)$ as $\mu\rightarrow\infty$. This shows the existence of alinear light cone with a maximum velocity of quantum propagation that isdecaying at a rate proportional to $1/\mu$. Furthermore, we prove that theasymptotic velocity, or the average velocity of the time-evolved state,exhibits a decay proportional to $\mathcal{O}(1/\mu^{p-1})$ as$\mu\rightarrow\infty$.

Slow Propagation Velocities in Schr\"odinger operators with large periodic potential