Zero measure Cantor spectrum for Sturmian metric graphs

17.04.2024, 13:00  –  Haus 9, Raum 0.17 und Zoom
Forschungsseminar Diskrete Spektraltheorie

Gilad Sofer (Technion)

Abstract: A classical result by Belissard et al states that the spectrum of a 1-dimensional Hamiltonian with a Sturmian potential is a singular continuous Cantor set of Lebesgue measure zero. More recently, the same result was proven by Damanik-Fang-Sukhtaiev for a class of aperiodic metric antitrees equipped with the standard Laplacian.

In this talk, we present an analogous result for a large family of metric graphs whose local geometric structure is determined by Sturmian sequences. We prove that almost surely, the spectrum of these metric graphs is a zero measure and purely singular continuous generalized Cantor set. The proof is based on a mixture of Kotani theory with tools from the world of quantum graphs.

Based on joint work with Ram Band.