Unconventional features in spectra of periodic quantum graphs

Spectra of periodic quantum systems are usually expected to be absolutely continuous, consisting of bands and gaps, the number of the latter being determined by the dimensionality. Our aim is to show that if the systems in question are quantum graphs, one may encounter a very different spectral behaviour. Using simple examples, we show that the spectrum may then have a pure point or a fractal character, and also that it may have only a nite but nonzero number of open gaps. Furthermore, motivated by recent attempts to model the anomalous Hall effect, we investigate a class of vertex couplings that violate the time reversal invariance. We will nd spectra of lattice graphs with the simplest coupling of this type, the one with ‘maximum’ non-invariance, and demonstrate that it depends substantially on the lattice topology. Finally, we will discuss an interpolation between this ‘maximal’ coupling and the usual $\delta$-type one.