A quantum graph is a metric graph – a collection of intervals of possibly varying lengths, connected at a set of vertices – on which a differential operator such as the Laplacian acts. Such objects have enjoyed considerable and growing popularity in the last 20 years, not only because of the obvious applications to modelling networks of various kinds, but also because they exhibit many features typical of higher-dimensional problems despite their essentially one-dimensional nature, thus serving as useful “toy” problems.
We will start by introducing the Kirchhoff Laplacian, the prototypical differential operator on a metric graph. This operator enjoys all the same basic properties as the Neumann Laplacian (either in one or more dimensions), and in particular has a discrete set of eigenvalues and eigenfunctions, which decompose the operator.
Our goal is to understand how these eigenvalues – in particular the first non-trivial one (equal to the spectral gap, the smallest eigenvalue being zero) – depend on the structure of the graph: its total length, diameter, number of edges/vertices, connectivity, the presence of certain subgraphs, and so on. Despite, or perhaps because of, the seemingly simple nature of quantum graphs, very little is known in this area. We will present a number of results which hopefully demonstrate that such dependencies can be surprisingly complex and subtle, despite being treatable by (relatively) elementary techniques.
This is based on ongoing joint work with Pavel Kurasov (University of Stockholm), Gabriela Malenová (KTH, Stockholm) and Delio Mugnolo (FernUniversität Hagen).