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 $(\frac{1}{2}-)$-Hölder continuity of the integrated density of states (IDS) for some multi-dimensional discrete quasi-periodic (QP) Schrödinger operators with asymmetric $C^2$-cosine type potentials. We extend both the iteration scheme of \cite{CSZ23a} and the interlacing method of \cite{FV21} to handle asymmetric Rellich functions with collapsed gaps.