Localization and Regularity of the Integrated Density of States for Schrödinger Operators on Zd with C2-cosine Like Quasi-Periodic Potential

In this paper, we study the multidimensional lattice Schrödinger operators with C 2 -cosine like quasi-periodic (QP) potential. We establish quantitative Green’s function estimates, the arithmetic version of Anderson (and dynamical) localization, and the finite volume version of ( 1 2 - ) -Hölder continuity of the integrated density of states for such QP Schrödinger operators. Our proof is based on an extension of the fundamental multi-scale analysis type method of Fröhlich–Spencer–Wittwer (Commun Math Phys 132(1), 5–25, 1990) to the higher lattice dimensions. We resolve the level crossing issue on eigenvalues parameterizations in the case of both higher lattice dimension and C 2 regular potential.