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

Mathematical Physics & Probability

Speaker: Lior Alon, Technion
Related Webpage: https://www.lioralon.net
Location: 1147 MSB
Start time: Wed, Oct 16 2019, 3:10PM

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).