The curious spectra and dynamics of non-locally finite crystals

TL;DR

This paper explores the spectral and dynamical properties of non-locally finite periodic graphs with non-negative weights, revealing unusual spectra and transport phenomena, and addressing open questions in the field.

Contribution

It introduces new examples of non-locally finite graphs with exotic spectral types and dynamical behaviors, including singular continuous spectrum and ballistic transport without dispersive estimates.

Findings

Constructed a graph with purely singular continuous spectrum.

Demonstrated a graph with a flat band eigenvector of infinite support.

Provided a negative answer to an open question about dispersive estimates.

Abstract

This paper is devoted to the investigation of the spectral theory and dynamical properties of 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: for example, we construct a periodic graph whose Laplacian has purely singular continuous spectrum. Regarding point spectrum, and different to the locally finite case, 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 in this context. Furthermore, we include a discussion of the fractional Laplacian for which we prove a phase transition in its dynamical behaviour. Generally speaking, many questions still remain open, and we believe that the studied class of graphs can serve as a playground to better understand exotic spectra and dynamics.