Empirical spectral measures of quantum graphs in the Benjamini-Schramm limit

Abstract

We introduce the notion of Benjamini-Schramm convergence for quantum graphs. This notion of convergence, intended to play the role of the already existing notion for discrete graphs, means that the restriction of the quantum graph to a randomly chosen ball has a limiting distribution. We prove that any sequence of quantum graphs with uniformly bounded data has a convergent subsequence in this sense. We then consider the empirical spectral measure of a convergent sequence (with general boundary conditions and edge potentials) and show that it converges to the expected spectral measure of the limiting random rooted quantum graph. These results are similar to the discrete case, but the proofs are significantly different. The density Im

Absolutely Continuous Spectrum for Quantum Trees

“…9 Though it is known that z → ψ, G z ψ is Herglotz for any ψ ∈ L 2 (T) by the spectral theorem, we followed this somehow roundabout argument to deduce the same holds for z → G z (v, v). See the appendix of [8] for a more general result.…”