Fastest possible dispersion in one-dimensional crystals

Identify the fastest possible dispersive decay rate for Schrödinger evolutions on one-dimensional crystals (ν=1) in the L^1→L^∞ operator norm, by establishing a universal lower bound for ||e^{itH_Γ}||_{L^1→L^∞} and proving its sharpness within this class.

Background

The authors show examples with dispersion as fast as t^{-1/2} and discuss slowing down dispersion using integer powers of the adjacency operator, but a universal optimal rate over all 1D crystals remains unknown.

Determining the optimal dispersive rate would benchmark how long-range periodic connections can enhance or limit dispersion relative to standard lattices.