On the Integrated Density of States for Schrödinger Operators on ℤ2 with Quasi Periodic Potential

In this paper we consider discrete Schrödinger operators on the lattice ℤ2 with quasi periodic potential. We establish new regularity results for the integrated density of states, as well as a quantitative version of a “Thouless formula”, as previously considered by Craig and Simon, for real energies and with rates of convergence. The main ingredient is a large deviation theorem for the Green's function that was recently established by Bourgain, Goldstein, and the author. For the integrated density of states an argument of Bourgain is used. Finally, we establish certain fine properties of separately subharmonic functions of two variables that might be of independent interest.