Spectral and Stochastic Solutions to Boundary Value Problems on Magnetic Graphs

Abstract

A magnetic graph is a graph G equipped with an orientation structure σ on its edges. The discrete magnetic Laplace operator LσG, a second-order difference operator for complex-valued functions on the vertices of G, has been an interesting and useful tool in discrete analysis for over twenty years. Its role in the study of quantum mechanics has been examined closely since its debut in a classic paper by Lieb and Loss in 1993. In this paper, we pose some boundary value problems associated to this operator, and adapt two classic techniques to the setting of magnetic graphs to solve them. The first technique uses the spectral properties of the operator, and the second technique utilizes random walks adjusted to this particular setting. Throughout, we will prove some useful results including a Green’s identity, mean value characterization of harmonic functions, and extensions of the solution techniques to Kronecker product graphs.

Biography: Sawyer Jack Robertson is a Norman native and a sophomore undergraduate student in the Department of Mathematics. He has been participating in undergraduate research for one academic year, and has presented at conferences in four states across the country. He is also a recipient of National Merit, Court, and Rust scholarships and has been recognized nationally for his achievements in academics and research. A passionate mathematics major, he hopes to one day attend graduate school at a top institution and become a research mathematician helping to solve problems arising in research areas across many scientific disciplines.

University Libraries Undergraduate Research Award