Honors Thesis Presentation: "Spectra of Quantum Graphs via a Scattering Matrix Approach"

Abstract: A common approach to study the structure of a graph or a network, G, is through the analysis of the spectrum of linear operators on the vector space of functions on it. In this project, we are interested in quantum graphs, where vertices are connected by length-parametrized edges, together with the self-adjoint Neumann-Kirchhoff Laplacian. Following the ideas of Kottos and Smilansky, we explore the spectrum of this operator using a scattering matrix approach in some specific examples, particularly, the cycles, the line segments and the star graphs. We notice that in star graphs where edges are not rationally related, we have a truly infinite spectrum whereas when edges are rationally related, the spectrum has some sort of periodicity. We also propose the idea of Cayley metric graph in order to understand the infinite multiplicity in the case where the corresponding combinatorial graph of the quantum graph is Cayley.

Host: Blake Thornton

Access Zoom Presentation HERE.