On the absence of absolutely continuous spectra for Schr\"{o}dinger operators on radial tree graphs

TL;DR

This paper investigates Schrödinger operators on radial tree graphs, demonstrating conditions under which the absolutely continuous spectrum is absent or present, depending on the graph's sparsity and vertex coupling parameters.

Contribution

It establishes new criteria for the absence or presence of absolutely continuous spectrum in radial tree Schrödinger operators with variable coupling conditions.

Findings

Absolutely continuous spectrum is empty for sparse graphs without potential.

Certain coupling parameters lead to purely absolutely continuous spectrum.

Sparsity and coupling conditions critically influence spectral types.

Abstract

The subject of the paper are Schr\"odinger 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 Schr\"odinger operator can be purely absolutely continuous.