# Talk by Pavel Exner

On February 7th, 2024, **Pavel Exner** (Doppler Institute for Mathematical Physics and Applied Mathematics Prag) gave a talk about **"Effects of time-reversal non-invariance in quantum graphs"** as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.

## Abstract

The topic of this talk are quantum graphs with the vertex coupling which does not preserve the time-reversal invariance. As a case study the simplest example with the asymmetry being maximal at a fixed energy will be analyzed. In this situation the high-energy scattering depends crucially on the vertex parity; we will demonstrate implications of this fact for spectral

and transport properties in several classes of graphs, both finite and infinite periodic ones. In particular, we prove the Band-Berkolaiko universality for kagome lattices with this coupling.

Furthermore, we discuss other timeasymmetric graphs and identify a class of such couplings which exhibits a nontrivial PT -symmetry despite being self-adjoint; we also illustrate the role of the Dirichlet component in the vertex coupling. Finally, we show how a square lattice with such a coupling behaves in the presence of a magnetic field when the two time-asymmetry effects compete. The results come from a common work with Marzieh Baradaran, Jiří Lipovský,

and Miloš Tater.