A useful formula for periodic Jacobi matrices on trees

Jess Banks, Jonathan Breuer, Jorge Garza-Vargas, Eyal Seelig, Barry Simon

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)e2315218121
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number23
StatePublished - 4 Jun 2024


  • Jacobi matrices
  • spectral theory
  • trees

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