Point Spectrum of Periodic Operators on Universal Covering Trees

Abstract

A. For any multi-graph with edge weights and vertex potential, and its universal covering tree  , we completely characterize the point spectrum of operators  on  arising as pull-backs of local, self-adjoint operators on . is builds on work of Aomoto, and includes an alternative proof of the necessary condition for point spectrum he derived in [Aom91]. Our result gives a nite time algorithm to compute the point spectrum of  from the graph , and additionally allows us to show that this point spectrum is contained in the spectrum of . Finally, we prove that typical pull-back operators have a spectral delocalization property: the set of edge weight and vertex potential parameters of giving rise to  with absolutely continuous spectrum is open and dense.

Remarks on Periodic Jacobi Matrices on Trees

“…Specifically, we were motivated by the work of Avni, Breuer and Simon [3] (see also [2]) whose notation and ideas we will follow; in particular, we refer the reader to that paper for the definitions from graph theory that we will use. We note also the relevance of some recent preprints by a group at Berkeley [8,4].…”

“…As we were preparing this paper, which represents work mainly done in June 2019, we received an early draft of a very interesting paper of Banks et al [4] that provides a lot of information about point spectrum of periodic Jacobi matrices on trees. It seems to us possible that with the methods of [4] one can extend Theorem 4.1 to a much more general context.…”

Spectra of infinite graphs via freeness with amalgamation

“…In [ABS20] and in previous versions of this paper some questions about the point spectrum of 𝐴 T were asked. These questions were answered in [BGVM20], where a full characterization of the point spectrum of 𝐴 T was provided by analyzing the combinatorial structure of its eigenvectors. Later in [ACSY21], using free probability tools, a general theory about atoms of spectral measures of polynomials in non-commutative random variables was developed, and some of the results in [BGVM20] can be obtained directly from this general theory.…”