On the absence of absolutely continuous spectra for Schrödinger operators on radial tree graphs

Abstract

The subject of the paper are Schrödinger operators on tree graphs which are radial having the branching number $b_n$ at all the vertices at the distance $t_n$ from the root. We consider a family of coupling conditions at the vertices characterized by $(b_n-1{)}^2+4$ real parameters. We prove that if the graph is sparse so that there is a subsequence of $\{t_{n+1}-t_n\}$ growing to infinity, in the absence of the potential the absolutely continuous spectrum is empty for a large subset of these vertex couplings, but on the the other hand, there are cases when the spectrum of such a Schrödinger operator can be purely absolutely continuous.

27 pages, 1 figure

References in corpus (6)

• Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder
• Approximation of a general singular vertex coupling in quantum graphs
• The absolutely continuous spectrum of one-dimensional Schr"odinger operators
• Singular spectrum for radial trees
• Eigenvalue estimates for Schroedinger operators on metric trees
• Resonance asymptotics in the generalized Winter model

Cited by in corpus (7)

• Spectral Theory of Infinite Quantum Graphs
• Transparent Quantum Graphs
• Dirac Particles in Transparent Quantum Graphs: Tunable transport of relativistic quasiparticles in branched structures
• Absence of absolutely continuous spectrum for the Kirchhoff Laplacian on radial trees
• Manakov system on metric graphs: Modeling the reflectionless propagation of vector solitons in networks
• Spectra of Regular Quantum Trees: Rogue Eigenvalues and Dependence on Vertex Condition
• Absolutely continuous spectrum for Laplacians on radial metric trees and periodicity