A Thouless formula for quasi-periodic long-range Schrödinger operators

This talk is devoted to quasi-periodic Schrödinger operators beyond the Almost Mathieu, with more general potentials and interactions. The  links
between the spectral properties of these operators and the dynamical properties of the associated quasi-periodic linear skew-products rule the
game. In particular, we present a Thouless formula  and some consequences of Aubry duality.