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.

Quantum ergodicity for expanding quantum graphs in the regime of spectral delocalization

“…We showed in [6] that if the perturbation is weak enough, then the bands of AC spectra remain stable, and (Green) holds in such bands. Note that here we assume there is no edge potential…”

“…The next two hypotheses can be seen as a condition of spectral delocalization. Indeed, they imply that P-almost all quantum trees have purely absolutely continuous spectrum in I, see [6,Theorem A.6].…”

“…We showed in [6] that the spectrum of H T consists of bands of pure AC spectra along with a possible set of discrete eigenvalues (outside the bands). We also showed that within the bands, the limits R± λ+i0 (o b ) exist, are finite and satisfy Im R± λ+i0 (o b ) > 0.…”

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) −→…”

Point Spectrum of Periodic Operators on Universal Covering Trees

“…Finally, we prove that the set of edge weights and potentials for which  has point spectrum is a closed set of Lebesgue measure zero. is may be regarded as a spectral delocalization result of the kind long-studied in mathematical physics [Ana18]; see [AISW20] for recent and analogous work in the context of quantum graphs.…”