Mathematical Physics and Harmonic Analysis Seminar

Date: February 23, 2022

Time: 09:00AM - 10:00AM

Location: Zoom

Speaker: Gilad Sofer, Technion

Title: Differences between Robin and Neumann eigenvalues on metric graphs

Abstract: We consider the Laplacian on a metric graph with Neumann vertex conditions. One may introduce a perturbation by placing δ vertex conditions at a selected subset of vertices. This results in an increase of the Laplacian eigenvalues. The differences between the eigenvalues of the perturbed and original operators are known as the Robin--Neumann gaps. The sequence of Robin--Neumann gaps has been recently studied for planar domains (Riviere--Royen), the hemisphere (Rudnick--Wigman) and star graphs (Rudnick--Wigman--Yesha).

The questions of interest concern the mean value of the Robin--Neumann gap sequence, the limiting values of the sequence and its bounds. We answer those questions for general metric graphs and compare our results to those obtained or conjectured for planar domains. In particular, we make a connection to inverse spectral problems, by showing that the mean value of the aforementioned sequence is determined by some geometric properties of the graph. The talk is based on a joint work in progress with Ram Band, Holger Schanz and Uzy Smilansky.