10-59 Pavel Exner, Jiri Lipovsky

On the absence of absolutely continuous spectra for Schr\"{o}dinger operators on radial tree graphs (375K, pdf) Apr 12, 10

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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.

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