Localization and Regularity of the Integrated Density of States for Schrödinger Operators on ℤ^d with C²-cosine Like Quasi-periodic Potential

2023/03/02 by Hongyi Cao, Yunfeng Shi, Cao, Hongyi +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2303.01071

openalex publication_date 2023/03/02 · openalex created_date 2023/03/05 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the multidimensional lattice Schrödinger operators with C²-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-)-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 [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 C² regular potential.

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