WG4 Research Seminar - Talk by Mostafa Sabri

On February 2nd, 2022, Mostafa Sabri (University of Cairo) will give a talk at 2:30 pm CET about

"Spectral analysis of large quantum graphs".

The talk will be held online in Zoom as part of the research seminar Analysis of the FernUniversität in Hagen. The zoom URL will be sent to all members of the WG4 and to all further colleagues who send a mail to Delio Mugnolo.

Abstract

Quantum graphs are continuum analogs of discrete (combinatorial) graphs in which the adjacency matrix is replaced by a differential operator on the edges. They arise naturally in chemistry, physics, and engineering when one considers propagation of waves through a quasi-one-dimensional system, and they are also interesting for purely mathematical reasons.

This talk is intended for a wide audience; I will first introduce quantum graphs then discuss four problems: how to define an appropriate notion of convergence for sequences of quantum graphs? Does this notion guarantee the convergence of empirical spectral measures? how about the nature of the spectrum at the limit infinite quantum graph? Can we show that in some cases it exhibits absolutely continuous spectrum? And if yes, what if we consider a sequence of finite quantum graphs converging to such a model, are the eigenvectors delocalized in the sense that they are essentially uniformly distributed over the graph?

This is based on several joint works with Nalini Anantharaman, Maxime Ingremeau, and Brian Winn.