Patrizio Bifulco: Comparing the spectrum of Schrödinger operators on metric graphs using heat kernels

We study Schrödinger operators on compact finite metric graphs subject to \(\delta\)-coupling and standard boundary conditions often known as Kirchoff-Neumann vertex conditions. We compare the \(n\)-th eigenvalues of those self-adjoint realizations and derive an asymptotic result for the mean value of the eigenvalue deviations which represents a generalization to a recent result by Rudnick, Wigman and Yesha obtained for domains in \(\mathbb{R}^2\) to the setting of metric graphs. We start this talk by introducing the basic notion of a metric graph and discuss some basic properties of heat kernels on those graphs afterwards. In this way, we are able to discuss a so-called local Weyl law which is relevant for the proof of the asymptotic main result. Finally, we will also briefly discuss a first specific class of infinite graphs having finite total length on which the asymptotic result can be generalized.

This talk is based on joint works with Joachim Kerner (Hagen) and Delio Mugnolo (Hagen).