A useful formula for periodic Jacobi matrices on trees

What is it about?

Periodic structures lie at the basis of solid state physics as models for crystalline materials. The spectral theory of such structures in the Euclidean case is studied through periodic Schroedinger operators and periodic Jacobi matrices. Both the spectrum and eigenfunctions are well understood for such operators and described through Bloch-Floquet theory, which explains the band-gap structure underlying the transport properties of related materials. Periodic Jacobi matrices on trees are a noncommutative version of such structures, where the symmetry is considerably more complex than in the Euclidean case. In this case, the study of the structure of the spectrum and eigenfunctions is also more challenging and much less is known here. In this paper we present a formula tying the density of states and certain resolvent entries, which generalizes a fundamental one-dimensional formula. We show that this formula easily explains the band-gap structure which exists for trees as well. We also demonstrate this formula's usefulness for understanding the existence of eigenvalues for such operators. We believe this formula is an important step towards a better understanding of the spectral theory of periodic operators on trees.

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