The Floquet theory and the state density of quasi-periodic Schrödinger operators

We study the Floquet solutions of quasi-periodic Schrödinger operators on flows which satisfy a Diophantine condition. Using these solutions, we show that the resolvent of such an operator is smooth with respect to certain derivations in the C^*-algebra associated with the flow, and that every projection corresponding to a spectral gap has a finite localization length. Based on these results, we derive a formula which quantizes the integral components of the integrated density of states for the operator on each spectral gap.