A useful formula for periodic Jacobi matrices on trees

• Published in: Proceedings of the National Academy of Sciences - PNAS Vol. 121; no. 23; p. e2315218121
• Main Authors: Banks, Jess, Breuer, Jonathan, Garza-Vargas, Jorge, Seelig, Eyal, Simon, Barry
• Format: Journal Article
• Language: English
• Published: United States National Academy of Sciences 04.06.2024
• Subjects: Labeling; Mathematics; Physical Sciences; Theorem proving; Theorems; Trees; Jacobi matrices; trees; spectral theory
• ISSN: 0027-8424; 1091-6490; 1091-6490
• DOI: 10.1073/pnas.2315218121

SignificanceThe subject of periodic Jacobi matrices on trees has evoked interest among mathematical physicists, analysts, and number theorists. We introduce a function of use in the study of these objects and prove a useful formula for this function. We illustrate the usefulness of this formula by using it to provide a proof of gap labeling that does not use C*-algebras. We also use it to provide an understanding of the Aomoto index theorem. 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.