Kochi University of Technology
Unified Quantum Device Laboratory Seminar
⭐Speaker: Dr. Pavel Exner (Czech Technical University).
⭐Date&Time : 11 / 13 / 2019 (Wed) 10:30-12:00
⭐Place: Kami Campus, room B107
⭐Title: Uncommon spectral properties of periodic quantum graphs
The talk is concerned with some unexpected spectra properties of periodic quantum graphs. From the mathematical side, the aim is to illustrate that a nontrivial topology of the configuration space or number-theoretic prop- erties of the parameters involved can lead to a variety of spectral types. We discuss this question with the focus on second-order equations used in physics to describe periodic quantum systems. Such a PDE in a Euclidean space has typically the spectrum which is absolutely continuous, consist- ing of bands and gaps, the number of the latter being determined by the dimensionality. If analogous second-order operators on metric graphs are considered, a number of other situations may arise. Using simple examples, we show that the spectrum may then have a pure point or a fractal charac- ter, and also that it may have only a finite but nonzero number of open gaps. Furthermore, motivated by recent attempts to model the anomalous Hall ef- fect, we investigate a class of vertex couplings that violate the time reversal invariance. We find spectra of lattice graphs with the simplest coupling of this type and demonstrate that it depends substantially on the parity of the lattice vertices, and discuss consequences of this property.