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Resonant delocalization for random Schrodinger operators on tree graphs

Author(s): Aizenman, Michael; Warzel, Simone

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Abstract: We analyse the spectral phase diagram of Schrodinger operators T + lambda V on regular tree graphs, with T the graph adjacency operator and V a random potential given by iid random variables. The main result is a criterion for the emergence of absolutely continuous (ac) spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials ac spectrum appears at arbitrarily weak disorder (lambda << 1) in an energy regime which extends beyond the spectrum of T. 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 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.
Publication Date: 2013
Citation: Aizenman, Michael, Warzel, Simone. (2013). Resonant delocalization for random Schrodinger operators on tree graphs. JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 15 (1167 - 1222. doi:10.4171/JEMS/389
DOI: doi:10.4171/JEMS/389
ISSN: 1435-9855
Pages: 1167 - 1222
Type of Material: Journal Article
Version: Author's manuscript

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