### Speaker

### Description

By now, quantum and metric graphs have become popular models in different areas

of mathematics and other areas of science such as physics. Being a (typically) complex

structure which is locally one-dimensional, they in some sense interpolate between one-

and higher-dimensional aspects known, for example, from the study of manifolds. In this

talk, our main goal is to compare the spectrum of different Schrödinger operators defined

on a given metric graph in a suitable way. Establishing so-called local Weyl laws - which

prove interesting in their own right - we shall derive an explicit expression for the limiting

mean-value of eigenvalue distances. We will first look at finite compact metric graphs

and then move on to a certain class of infinite metric graphs. As we will see, some things

might change in the infinite setting. Furthermore, we shall discuss some application of

the results regarding inverse spectral theory. Namely, we derive some seemingly novel

Ambarzumian-type theorems on graphs. This talk is based on recent work with Patrizio

Bifulco (Hagen).