Anna Maltsev

Topological eigenfunctions of quantum graphs

A quantum graph is a set of vertices and edges, like a usual (combinatorial) graph but the edges are taken to be segments of a real line equipped with a Euclidean metric, which allows us to do analysis on graphs. We study the Schrodinger equation on such graphs, with a Kirchoff condition at vertices, which means that outward derivatives at each vertex add up to 0. Quantum graphs not only arise in a multitude of real-world applications (e.g. electrical networks, roads, pipes, neurons, etc.) but also have very interesting mathematical properties. In this talk, which is based on joint work with Evans Harrell, I will discuss eigenfunctions on quantum graphs that reflect graph topology of quantum graphs and illustrate new connections between vertex and edge scattering matrices via a “quantum” Ihara's theorem.