Localization near the edge for the Anderson Bernoulli model on the two dimensional lattice

• 成果类型: Article
• 署名作者: Ding, Jian; Smart, Charles K.
• 署名单位: University of Pennsylvania; University of Chicago
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-019-00910-4
• 发表日期: 2020
• 页码: 467-506
• 关键词: large disorder; Operators; PROOF

摘要

We consider a Hamiltonian given by the Laplacian plus a Bernoulli potential on the two dimensional lattice. We prove that, for energies sufficiently close to the edge of the spectrum, the resolvent on a large square is likely to decay exponentially. This implies almost sure Anderson localization for energies sufficiently close to the edge of the spectrum. Our proof follows the program of Bourgain-Kenig, using a new unique continuation result inspired by a Liouville theorem of Buhovsky-Logunov-Malinnikova-Sodin.