A useful formula for periodic Jacobi matrices on trees

• 成果类型: Article
• 署名作者: Banks, Jess; Breuer, Jonathan; Garza-Vargas, Jorge; Seelig, Eyal; Simon, Barry
• 署名单位: University of California System; University of California Berkeley; Hebrew University of Jerusalem; California Institute of Technology; California Institute of Technology; California Institute of Technology
• 刊物名称: PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
• ISSN/ISSBN: 0027-9539
• DOI: 10.1073/pnas.2315218121
• 发表日期: 2024-05-28
• 关键词: absolutely continuous-spectrum; random schrodinger-operators; anderson model; random-walks

摘要

We introduce a function of the density of states for periodic Jacobi matrices on trees and prove a useful formula for it in terms of entries of the resolvent of the matrix and its half -tree restrictions. This formula is closely related to the one-dimensional Thouless formula and associates a natural phase with points in the bands. This allows streamlined proofs of the gap labeling and Aomoto index theorems. We give a complete proof of gap labeling and sketch the proof of the Aomoto index theorem. We also prove a version of this formula for the Anderson model on trees.