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arXiv · 2303.01071

Localization and Regularity of the Integrated Density of States for Schr\"odinger Operators on $\mathbb{Z}^d$ with $C^2$-cosine Like Quasi-periodic Potential

Hongyi Cao·Yunfeng Shi·Zhifei Zhang
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Abstract

In this paper, we study the multidimensional lattice Schr\"odinger 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 $(\frac 12-)$-H\"older continuity of the integrated density of states (IDS) for such QP Schr\"odinger operators. Our proof is based on an extension of the fundamental multi-scale analysis (MSA) type method of Fr\"ohlich-Spencer-Wittwer [\textit{Comm. Math. 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.

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Hongyi Cao, Yunfeng Shi, Zhifei Zhang. 2023-03-02. Localization and Regularity of the Integrated Density of States for Schr\"odinger Operators on $\mathbb{Z}^d$ with $C^2$-cosine Like Quasi-periodic Potential. https://arxiv.org/abs/2303.01071

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