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A useful formula for periodic Jacobi matrices on trees.
- Source :
-
Proceedings of the National Academy of Sciences of the United States of America [Proc Natl Acad Sci U S A] 2024 Jun 04; Vol. 121 (23), pp. e2315218121. Date of Electronic Publication: 2024 May 31. - Publication Year :
- 2024
-
Abstract
- 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.<br />Competing Interests: Competing interests statement:The authors declare no competing interest.
Details
- Language :
- English
- ISSN :
- 1091-6490
- Volume :
- 121
- Issue :
- 23
- Database :
- MEDLINE
- Journal :
- Proceedings of the National Academy of Sciences of the United States of America
- Publication Type :
- Academic Journal
- Accession number :
- 38819999
- Full Text :
- https://doi.org/10.1073/pnas.2315218121