Jun 24 – 28, 2024
Institut de Mathématiques de Toulouse
Europe/Paris timezone

Spectral comparison results on quantum graphs

Jun 26, 2024, 11:30 AM
Institut de Mathématiques de Toulouse

Institut de Mathématiques de Toulouse

Université Paul Sabatier 118, route de Narbonne - Bat. 1R3 31062 Toulouse Cedex 9


Joachim Kerner


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).

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