Mina Farag: The Trace Formula for Quantum Graphs

Abstract: In discrete mathematics, a graph consists of a set of vertices and edges. Upon viewing each edge as a separate copy of an interval in \(\mathbb{R}\), one obtains a metric graph. We can then consider functions defined on each edge separately. Working on the appropriate function spaces, we can define operators on the graph. The operator of interest is the Schrödinger operator.

The aim of this talk is two-fold: first we will describe metric graphs; in the second part, we will review the trace formula, which relates the spectrum of the Laplacian on compact (each edge is a finite interval) metric graphs to the closed paths of the graph.