Title: Eigenfunctions of periodic Jacobi operators on trees. Eyal Seelig
Abstract: The theory of periodic Jacobi matrices on Z is rich and well-established, with diagonalization achieved through a set of special eigenfunctions known as Floquet solutions or Bloch waves. These eigenfunctions are governed by a phase parameter, which, as it varies from 0 to pi, reveals the band structure of the spectrum. In this talk we explore the generalization of this theory to periodic Jacobi operators on trees, namely lifts of Jacobi operators on finite graphs to their universal covers. We construct Floquet solutions that are governed by a single phase in certain tree models, despite the non-commutative symmetries of the tree. This contrasts with a general formula we prove for the density of states measure, where a different phase arises and plays a role in a new proof for gap labeling. Based on joint works with J. Banks, J. Breuer, J. Garza-Vargas, and B. Simon.