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Resonances and Partial Delocalization on the Complete Graph

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

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Abstract: Random operators may acquire extended states formed from a multitude of mutually resonating local quasi-modes. This mechanics is explored here in the context of the random Schrodinger operator on the complete graph. The operator exhibits local quasi-modes mixed through a single channel. While most of its spectrum consists of localized eigenfunctions, under appropriate conditions it includes also bands of states which are delocalized in the -though not in -sense, where the eigenvalues have the statistics of eba spectra. The analysis proceeds through some general observations on the scaling limits of random functions in the Herglotz-Pick class. The results are in agreement with a heuristic condition for the emergence of resonant delocalization, which is stated in terms of the tunneling amplitude among quasi-modes.
Publication Date: Sep-2015
Electronic Publication Date: 18-Oct-2014
Citation: Aizenman, Michael, Shamis, Mira, Warzel, Simone. (2015). Resonances and Partial Delocalization on the Complete Graph. ANNALES HENRI POINCARE, 16 (1969 - 2003. doi:10.1007/s00023-014-0366-9
DOI: doi:10.1007/s00023-014-0366-9
ISSN: 1424-0637
EISSN: 1424-0661
Pages: 1969 - 2003
Type of Material: Journal Article
Journal/Proceeding Title: ANNALES HENRI POINCARE
Version: Author's manuscript

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