Speaker: Mostafa Sabri (NYU Abu Dhabi)

Title: The curious spectra and dynamics of non-locally finite crystals

Abstact: I will discuss the spectra and dynamics of Z^d-periodic graphs which are not locally finite but carry non-negative, symmetric and summable edge weights. These graphs are shown to exhibit rather intriguing behaviour. We construct a periodic graph whose Laplacian has purely singularly continuous spectrum. Regarding the point spectrum, we construct a graph with a partly flat band whose eigenvectors must have infinite support. Concerning dynamical aspects, under some assumptions we prove that motion remains ballistic along at least one layer. We also construct a graph whose Laplacian has purely absolutely continuous spectrum, exhibits ballistic transport, yet fails to satisfy a dispersive estimate. This provides a negative answer to an open question of Damanik. The fractional Laplacian is viewed as a special  case, and we prove for it a phase transition in its dynamical behaviour. Many questions still remain open, and we believe that this class of graphs can serve as a playground to better understand exotic spectra and dynamics. The proofs rely on a combination of Floquet theory and harmonic analysis.