# Massimo Moscolari: Beyond Diophantine Wannier diagrams: gap labelling for Bloch-Landau Hamiltonians

**Time: **
Mon 2019-04-15 14.00 - 15.00

**Location: **
Seminar Hall Kuskvillan, Institut Mittag-Leffler

**Lecturer: **
Massimo Moscolari, Sapienza University of Rome

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 is based on a joint work with H. Cornean and D. Monaco. Then, I will briefly discuss the relation between the Chern character and the existence of a localized generalized Wannier basis for the space of occupied states of a generic 2D gapped quantum system, this is a work in progress with G. Marcelli and G. Panati.