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.

References

Problem 9.9. What is the fastest dispersion speed for crystals in dimension d = 1?

— The curious spectra and dynamics of non-locally finite crystals  (2411.14965 - Kerner et al., 2024) in Section 9, Problem 9.9

Whether the intermediate regime of super-polynomial but sub-exponential dispersion can occur remains open.

— Arbitrarily Fast Quantum Dispersion in Long-Range Crystals  (2608.27326 - Leclerc et al., 27 Aug 2026) in Remark in Section 1, subsection “Background and results on dispersion”; Section 6, subsection “Maximal rate of dispersion”

In view of the comparison with fractional Brownian motion discussed above, it is natural to ask whether arbitrarily fast polynomial dispersion holds almost surely when $\lambda$ is sufficiently large compared with $\mu{-1}$.

— Arbitrarily Fast Quantum Dispersion in Long-Range Crystals  (2608.27326 - Leclerc et al., 27 Aug 2026) in Section 3, subsection “Prospects for the theory,” part (b) “Random and multiplicative hopping”