Localization and Regularity of the Integrated Density of States for Schrödinger Operators on ${\mathbb{Z}}^d$ with $C^2$-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 $(\frac{1}{2}-)$-Hölder continuity of the integrated density of states (IDS) for such QP Schrödinger operators. Our proof is based on an extension of the fundamental multi-scale analysis (MSA) type method of Fröhlich-Spencer-Wittwer [$\mathit{\text{Comm.}}\mathit{\text{Math.}}\mathit{\text{Phys.}}$ 132 (1990): 5–25] 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.