Slow propagation velocities in Schrödinger operators with large periodic potential

Abstract: 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 $\mu\rightarrow\infty$.