Resonances and Partial Delocalization on the Complete Graph

Michael Aizenman, Mira Shamis, Simone Warzel

Research output: Contribution to journalArticle

6 Scopus citations

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 Schrödinger 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 ℓ1-though not in ℓ2-sense, where the eigenvalues have the statistics of Šeba 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.

Original languageEnglish (US)
Pages (from-to)1969-2003
Number of pages35
JournalAnnales Henri Poincare
Volume16
Issue number9
DOIs
StatePublished - Sep 18 2015

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics

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