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
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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.
