Alice Brolin: Spectral geometry and magnetic fields on quantum graphs

Abstract

We are interested in spectral geometry: the connection between the spectrum of certain operators and the geometry of the domain, as well as inverse problems: determining an operator from its spectrum. In this thesis two papers dedicated to Schrödinger operators on metric graphs are presented. A metric graph \(\Gamma\) (similar to a discrete graph) consists of a set of vertices and a set of edges. But each edge is viewed as a real interval of some length. This gives \(\Gamma\) a metric space structure. The (magnetic) Schrödinger operators on \(\Gamma\) are self-adjoint operators acting on functions that are defined on the graph. Specifically they act as differential operators \[-\left(\frac{d}{dx}-ia(x)\right)^2+q(x)\]where \(q,a\) are real valued functions. We view \(q\) as an (electric) potential and \(a\) as a magnetic potential on \(\Gamma\). Any Schrödinger operator on a metric graph is determined by the potentials \(q,a\) and the vertex conditions describing the domain.

The first paper gives an interpretation of the Colin de Verdière parameter in the context of metric graphs. The original Colin de Verdière parameter for discrete graphs is a number that gives topological information about the graph, such as whether the graph is planar or not. It is defined as the maximal multiplicity of the second eigenvalue for a class of matrices, associated with the graph, called Colin de Verdière matrices. The second paper is about Ambarzumian type theorems for metric graphs. These are theorems that show that a certain exceptional operator, in a family of operators, is uniquely determined by its spectrum. E.g. any Schrödinger operator with non-zero electric potential will have different spectrum than a Laplacian (i.e. an operator with potential equal to 0), provided the vertex conditions are standard. In the article it is shown how some Ambarzumian type theorems can be generalized to not only include the electric potential, but also a magnetic potential. This is the first time magnetic potential has been included in an Ambarzumian type theorem.