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

Pavel Exner Ji Lipovsk

TBD mathscidoc:1910.43188

Journal of Mathematical Physics, 51, (12), 122107, 2010.12
The subject of the paper is Schrdinger operators on tree graphs which are radial, having the branching number bn at all the vertices at the distance tn from the root. We consider a family of coupling conditions at the vertices characterized by (bn1)2+4 real parameters. We prove that if the graph is sparse so that there is a subsequence of {tn+1tn} 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 Schrdinger operator can be purely absolutely continuous.
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@inproceedings{pavel2010on,
  title={On the absence of absolutely continuous spectra for Schrdinger operators on radial tree graphs},
  author={Pavel Exner, and Ji Lipovsk},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020132730921302717},
  booktitle={Journal of Mathematical Physics},
  volume={51},
  number={12},
  pages={122107},
  year={2010},
}
Pavel Exner, and Ji Lipovsk. On the absence of absolutely continuous spectra for Schrdinger operators on radial tree graphs. 2010. Vol. 51. In Journal of Mathematical Physics. pp.122107. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020132730921302717.
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