JournalsjemsVol. 15, No. 4pp. 1167–1222

Resonant delocalization for random Schrödinger operators on tree graphs

  • Simone Warzel

    TU München, Garching, Germany
  • Michael Aizenman

    Princeton University, USA
Resonant delocalization for random Schrödinger operators on tree graphs cover
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Abstract

We analyse the spectral phase diagram of Schr\"odinger operators T+λVT +\lambda V on regular tree graphs, with TT the graph adjacency operator and VV a random potential given by \emph{iid} random variables. The main result is a criterion for the emergence of absolutely continuous (\emph{ac}) spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials \emph{ac} spectrum appears at arbitrarily weak disorder (λ1)(\lambda \ll 1) in an energy regime which extends beyond the spectrum of~TT. Incorporating considerations of the Green function's large deviations we obtain an extension of the criterion which indicates that, under a yet unproven regularity condition of the large deviations' 'free energy function', the regime of pure \emph{ac} spectrum is complementary to that of previously proven localization. For bounded potentials we disprove the existence at weak disorder of a mobility edge beyond which the spectrum is localized.

Cite this article

Simone Warzel, Michael Aizenman, Resonant delocalization for random Schrödinger operators on tree graphs. J. Eur. Math. Soc. 15 (2013), no. 4, pp. 1167–1222

DOI 10.4171/JEMS/389