In this talk, we investigate the spectral theory of periodic graphs which are not locally finite but carry non-negative, symmetric and summable edge weights. These periodic graphs are shown to have rather intriguing behaviour. We construct a periodic graph whose Laplacian has purely singular continuous spectrum. We prove that motion remains ballistic along at least one layer under quite general assumptions. We construct a graph whose Laplacian has purely absolutely continuous spectrum, exhibits ballistic transport, yet fails to satisfy a dispersive estimate. This answers negatively an open question in this regard, in our setting. Concerning the point spectrum, we construct a graph with a partly flat band whose eigenvectors must have infinite support, in contrast to the locally finite case. We believe the present class of graphs can serve as a playground to better understand exotic spectra and dynamics in the future.
Based on a joint work with Joachim Kerner, Olaf Post and Matthias Täufer.
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