Slow propagation velocities in Schr\"odinger operators with large periodic potential

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