Absolutely Continuous Spectrum for Quantum Trees

Abstract

We study the spectra of quantum trees of finite cone type. These are quantum graphs whose geometry has a certain homogeneity, and which carry a finite set of edge lengths, coupling constants and potentials on the edges. We show the spectrum consists of bands of purely absolutely continuous spectrum, along with a discrete set of eigenvalues. Afterwards, we study random perturbations of such trees, at the level of edge length and coupling, and prove the stability of pure AC spectrum, along with resolvent estimates.

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

“…The universal cover T endowed with such lifted data is studied in [8], where it is shown that T has bands of absolutely continuous spectra on which the Green's kernel exists and is continuous. It follows from Theorem 3.12 that µ QN (I) −→ 0 Im G λ+i0 (x 0 , x 0 ) dx 0 dλ for I in such band.…”

On asymptotic expansions of the density of states for Poisson distributed random Schrödinger operators

“…For results in this regime we want to recall the expansion results [MPR97; MPR98; Poi99]. Furthermore, for small coupling, expectations of functions of the resolvent were studied for the Anderson model on the Bethe lattice [AK92; Kle98; FHS06; ASW06; FHS07; AW13], for the discrete lattice in the special case of a Cauchy distribution [P69; KK20], and recently for quantum trees [Ana+21]. Moreover, we want to mention deterministic perturbation results about distributions of eigenvalues for periodic Schrödinger operators [FKT90;FKT91].…”