A useful formula for periodic Jacobi matrices on trees - Banks, Jess; Breuer, Jonathan; Garza-Vargas, Jorge; Seelig, Eyal; Simon, Barry (Proceedings of the National Academy of Sciences - PNAS, 2024)
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.