Analysis - Mathematical Physics

Topic: On a universal limit conjecture for the nodal count statistics of quantum graphs

Speaker: Lior Alon

Affiliation: Technion

Date & Time: Friday October 11th, 2019, 3:30pm - 4:30pm

Location: Simonyi Hall 101

Video: https://video.ias.edu/analysis/2019/1011-LiorAlon

We consider Laplace eigenfunctions of a metric graph satisfying Neumann-Kirchhoff conditions on every vertex. The nodal count of a given eigenfunction is the number of points at which it vanishes. The nodal count of the n-th eigenfunction was shown to be bounded between n-1 and n-1+\beta, where \beta if the first Betti number of the graph. The difference between the nodal count and n-1 is called the nodal surplus. Berkolaiko et al. showed that the n-th nodal surplus equals to a magnetic stability index of the n-th eigenvalue. We present recent results on the statistics of the nodal surplus and conjecture a universal behavior for large graphs.  This talk is based on joint works with Ram Band (Technion) and Gregory Berkolaiko (Texas A&M).