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

**Time: **
Wed 2023-11-01 11.00 - 12.00

**Location: **
Albano, Cramérrummet

**Participating: **
Patrizio Bifulco (FernUniversität in Hagen)

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).