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. 2024 Jun 4;121(23):e2315218121.
doi: 10.1073/pnas.2315218121. Epub 2024 May 31.

A useful formula for periodic Jacobi matrices on trees

Jess Banks  1 , Jonathan Breuer  2 , Jorge Garza-Vargas  3 , Eyal Seelig  2 , Barry Simon  4   5
Affiliations

A useful formula for periodic Jacobi matrices on trees

Jess Banks et al. Proc Natl Acad Sci U S A. .

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.

Keywords: Jacobi matrices; spectral theory; trees.

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Conflict of interest statement

Competing interests statement:The authors declare no competing interest.

References

    1. Angel O., Friedman J., Hoory S., The non-backtracking spectrum of the universal cover of a graph. Trans. Am. Math. Soc. 367, 4287–4318 (2015).
    1. Aomoto K., Point spectrum on a quasi homogeneous tree. Pac. J. Math. 147, 231–242 (1991).
    1. Avni N., Breuer J., Kalai G., Simon B., Periodic boundary conditions for periodic Jacobi matrices on trees. Pure Appl. Funct. Anal. 7, 489–502 (2022).
    1. Avni N., Breuer J., Simon B., Periodic Jacobi matrices on trees. Adv. Math. 379, 107241 (2020).
    1. Banks J., Garza-Vargas J., Mukherjee S., Point spectrum of periodic operators on universal covering trees. Int. Math. Res. Not. 2022, 17713–17744 (2022).

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