A useful formula for periodic Jacobi matrices on trees

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. ; © 2024 the Author(s). Published by PNAS. This article is distributed underCreative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND). ; Research of J. Breuer, J.G.-V., E.S., and B.S. supported in part by Israeli Binational Science Foundation (BSF) Grant No. 2020027. Research of J. Breuer and E.S. supported in part by Israel Science Foundation Grant No. 1378/20. Research of J.G.-V. supported in part by NSF Focused Research Group (FRG) Award 1952777 and Caltech Carver Mead New Adventures Fund under the aegis of Joel Tropp’s award, and Caltech Center for the Mathematics of Information (CMI) Postdoctoral Fellowship. We would like to thank Misha Sodin for pointing out to us that one could view Ψ as generalization of the Marchenko–Ostrovskii theory from cyclic graphs to general finite graphs. ; J. Banks, J. Breuer, J.G.-V., E.S., and B.S. performed research; and wrote the paper. ; There are no data underlying this work. ; The authors declare no competing interest.