The curious spectra and dynamics of non-locally finite crystals

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.

References in corpus (7)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• On the spectra of carbon nano-structures
• Singular Continuous Spectrum for the Laplacian on Certain Sparse Trees
• Stochastic calculus for fractional Brownian motion with Hurst exponent $H>1/4$: A rough path method by analytic extension
• Irreducibility of the Bloch Variety for Finite-Range Schrödinger Operators
• Upper bounds on transport exponents for long range operators
• Robustness of flat bands on the perturbed Kagome and the perturbed Super-Kagome lattice