Slow Propagation Velocities in Schrödinger Operators with Large Periodic Potential

Schr odinger operators with periodic potential have generallybeen shown to exhibit ballistic transport. In this work, we investigatewhether the propagation velocity, while positive, can be made arbitrar-ily small by a suitable choice of the periodic potential. We consider thediscrete one-dimensional Schr odinger operator Delta +mu V, where Delta is thediscrete Laplacian,Vis ap-periodic non-degenerate potential and mu>0.We establish a Lieb-Robinson-type bound with a group velocity thatscales likeO(1/mu)as mu ->infinity. This shows the existence of a linear lightcone with a maximum velocity of quantum propagation that is decayingat a rate proportional to 1/mu. Furthermore, we prove that the asymptoticvelocity, or the average velocity of the time-evolved state, exhibits a decayproportional toO(1/mu p-1)as mu ->infinity