Abstract: In 1978 Wannier discovered a Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. I will show how to extend this relation to a gap labelling theorem for any 2D Bloch-Landau Hamiltonian operator and to certain non-covariant systems having slowly varying magnetic fields. The integer slope will be interpreted as the Chern character of the projection onto the space of occupied states. This result will be seen in the perspective of a non periodic generalization of the localization dichotomy for gapped quantum systems, which in the periodic case has been proved in 2016 by Monaco, Panati, Pisante and Teufel.
The talk is based on a joint work with H. Cornean and D. Monaco.