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

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

2010-04-12

arxiv.org/abs/1004.1980v1

Physics

Mathematical Physics

27 pages, 1 figure

Scientific paper

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.

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