Anderson-Bernoulli Localization on the 3D lattice and discrete unique continuation principle

2019/06/11 by Linjun Li, linjun li, Lingfu Zhang +2 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Probability (math.PR) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math-ph #math.AP #math.MP #math.PR #math.SP

paper · pdf · doi:10.48550/arxiv.1906.04350

Revised according to referee reports. Duke Math. J. to appear

openalex publication_date 2019/06/11 · openalex created_date 2019/06/27 · arxiv created 2021/03/15 · arxiv updated 2021/03/16 · openalex updated_date 2026/07/28

Abstract

We consider the Anderson model with Bernoulli potential on the 3D lattice, and prove localization of eigenfunctions corresponding to eigenvalues near zero, the lower boundary of the spectrum. We follow the framework by Bourgain-Kenig and Ding-Smart, and our main contribution is a 3D discrete unique continuation, which says that any eigenfunction of the harmonic operator with bounded potential cannot be too small on a significant fractional portion of all the points. Its proof relies on geometric arguments about the 3D lattice.

Citations

• On localization in the continuous Anderson-Bernoulli model in higher dimension
• Anderson localization for Bernoulli and other singular potentials
• Random band matrices in the delocalized phase, III: Averaging fluctuations
• Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
• Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice
• Anderson–Bernoulli Localization at Large Disorder on the 2D Lattice
• Random Band Matrices in the Delocalized Phase I: Quantum Unique Ergodicity and Universality

Cited by

• Random Schrödinger operator with singular potentials
• Time evolution of many-body localized systems in two spatial dimensions

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• Anderson–Bernoulli localization on the three-dimensional lattice and discrete unique continuation principle
• Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice
• Localization and unique continuation for non-stationary Schrödinger operators on the 2D lattice
• Anderson-Bernoulli localization at large disorder on the 2D lattice
• An introduction to multiscale techniques in the theory of Anderson localization. Part I