Existence and construction of sub-ballistic (diffusive or anomalous) transport

Ascertain whether there exists a crystal (Z^d-periodic graph with finite fundamental cell and summable symmetric nonnegative weights) that exhibits sub-ballistic transport for compactly supported initial data; if so, construct one with diffusive scaling ||x e^{-itH_Γ}ψ||_2∼t^{-1/2}, or with anomalous scaling t^{-α} for some α not in {0,1/2,1}.

Background

The paper identifies ballistic and super-ballistic regimes (e.g., for fractional Laplacians in one dimension) and provides criteria for ballistic transport along layers, but does not exhibit sub-ballistic behavior in this periodic class.

This problem asks for true sub-ballistic transport, including canonical diffusive scaling and other anomalous power laws, within the same periodic and summability framework.