Research Interests

• Mathematical Physics
• Spectral Theory
• Quantum Theory
• Graph Theory
• Ordinary and Partial Differential Equations

Most of my research activities involve differential operators acting on both discrete and metric graphs. I am interested in the spectrum of these operators and properties of the corresponding eigenfunctions. My dissertation focuses on the magnetic Schrodinger operator including (1) the stability of eigenvalues on quantum graphs with respect to perturbation of a magnetic flux and (2) touching spectral bands and Dirac conical points in the spectrum of infinite periodic graphs. Recently I have been analyzing spectral zeta functions of the Dirac and Schrodinger operators on graphs. I am also continuing to study Dirac conical points while working with an undergraduate honors student.

Recent Publications

• J.M. Harrison and T. Weyand, Relating Zeta Functions of Discrete and Quantum Graphs, Submitted, arXiv:1612.04273.
• J.M. Harrison, T. Weyand, and K. Kirsten, Zeta Functions of the Dirac Operator on Quantum Graphs, J. Math. Phys., 57 (2016), DOI: 10.1063/1.4964260;  arXiv: 1606.07834.
• R. Band, G. Berkolaiko, and T. Weyand, Anomalous Nodal Count and Singularities in the Dispersion Relation of Honeycomb Graphs, J. Math. Phys., 56 (2015), DOI: 10.1063/1.4937119; arXiv: 1503.07245.
• G. Berkolaiko and T. Weyand, Stability of Eigenvalues of Quantum Graphs with Respect to Magnetic Perturbation and the Nodal Count of the Eigenfunctions, Phil. Trans. R. Soc. A, 372 (2014), DOI: 10.1098/rsta.2012.0522; arXiv: 1212.4475.

My dissertation may be viewed here.

While an undergraduate, I conducted research on fluid dynamics for two years. This experience gave me the opportunity to give two presentations and complete an optional undergraduate honors thesis. I also participated in the Research Experience for Undergraduate (REU) Program on Algorithmic Combinatorics on Words at the University of North Carolina, Greensboro in 2007 which resulted in a publication that can be seen here.