Mathematical Challenges of Quantum Transport in Nano-Optoelectronic Systems - Abstract

Exner, Pavel

Spectra of periodic quantum graphs and the effect of local perturbations

I am going to review several recent results about spectra of quantum graphs. Speaking first about periodic graphs, I will show how their spectral bands and gaps can behave asymptotically at high energies in case of a general self-adjoint vertex coupling. Furthermore, it will be demonstrated that even for $mathbbZ$-periodic graphs spectral edges may not correspond to the periodic or antiperiodic situation unless the graph is a chain, the components of which are connected by single edges. Using the examples of chain and comb graphs I will also show how local perturbations give rise to eigenvalues and resonances in such systems.