Mathematical Physics and Harmonic Analysis Seminar

Date: November 11, 2022

Time: 1:50PM - 2:50PM

Location: BLOC 306

Speaker: Gamal Mograby, Tufts University

Title: Topological quantum numbers

Abstract: We present a detailed spectral analysis for a new class of fractal-type diamond graphs and provide a gap-labeling theorem in the sense of Bellissard for the corresponding probabilistic graph Laplacians using the technique of spectral decimation. Labeling the gaps in the Cantor set by the integrated density of states provides a set of topological quantum numbers that reflect the branching parameter of the graph construction and the decimation structure. The spectrum of the natural Laplacian on limit graphs is shown generically to be pure point supported on a Cantor set. However, one particular graph has a mixture of pure point and singularly continuous components.