Geometric spectral theory of quantum graphs

TL;DR

This paper provides an accessible overview of the spectral theory of quantum graphs, focusing on eigenvalue dependence and optimization related to graph geometry, suitable for doctoral students.

Contribution

It offers a comprehensive introduction to quantum graph spectral theory, including recent developments on eigenvalue optimization and geometric dependence.

Findings

Analysis of eigenvalue dependence on graph geometry

Methods for optimizing operator eigenvalues

Integration of recent research developments

Abstract

These are lecture notes from a course given at the summer school "Heat kernels and spectral geometry: from manifolds to graphs" in Bregenz, Austria, 2022. They are designed to be accessible to doctoral level students, and include background chapters on Laplacians on domains and quantum graphs before moving on to specialised topics involving the dependence and optimisation of operator eigenvalues on a metric graph in function of the graph geometry, drawn in part from the recent literature.