In 2016 a localization dichotomy result for periodic Schrödinger operators has been proved. This result shows that the localization properties of the Wannier functions associated to a periodic Schrödinger operator are deeply connected to the topological properties of the quantum system. In this seminar I will start by discussing the generalization of Wannier functions and Chern number to non-periodic systems. After that, I will present some results regarding the generalization of the localization dichotomy to generic gapped quantum systems.
The talk is based on joint works with H. Cornean, G. Marcelli, D. Monaco, and G. Panati.