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Dispersive estimates for periodic Schrödinger operators on periodic graphs remain largely open

Develop L^1→L^∞ dispersive estimates for Schrödinger evolutions generated by periodic Schrödinger operators on periodic graphs beyond the currently known special cases, in particular for general periodic potentials on Z^d with d>1.

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Background

The paper proves several sharp transport and non-dispersion phenomena in non-locally finite periodic graphs, yet highlights that, even in the locally finite periodic setting, establishing dispersive estimates is largely unresolved.

Only specific 1D or product-graph cases have been addressed in the literature; extending dispersion results to general higher-dimensional periodic graphs with periodic potentials remains an open direction.

References

Indeed, proving dispersive estimates for periodic Schro¨dinger operators on periodic graphs is already largely open in the locally finite case, even though ballistic transport is understood (see [13] and references therein).

— The curious spectra and dynamics of non-locally finite crystals (2411.14965 - Kerner et al., 22 Nov 2024) in Section 7.2 (Dispersion)