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.
The talk is based on a joint work with H. Cornean and D. Monaco.