On the Spectrum of Quasi-periodic Schrödinger Operators on \({\mathbb {Z}}^d\) with \(C^2\)-Cosine Type Potentials

In this paper, we establish the Anderson localization, strong dynamical localization and the \((\frac{1}{2}-)\) $(\frac{1}{2}-)$ -Hölder continuity of the integrated density of states (IDS) for some multi-dimensional discrete quasi-periodic (QP) Schrödinger operators with asymmetric \(C^2\) $C^2$ -cosine type potentials. We extend both the iteration scheme of Cao-Shi-Zhang (Commun Math Phys 404(1):495–561, 2023) and the interlacing method of Forman and VandenBoom (Localization and Cantor spectrum for quasiperiodic discrete Schrödinger operators with asymmetric, smooth, cosine-like sampling functions. arXiv:2107.05461, 2021) to handle asymmetric Rellich functions with collapsed gaps.