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arXiv · 1405.3951

Resonances and Partial Delocalization on the Complete Graph

Michael Aizenman·Mira Shamis·Simone Warzel
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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 Schr\"odinger operator on the complete graph. The operators 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 $\ell^1$-though not in $\ell^2$-sense, where the eigenvalues have the statistics of \v{S}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.

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Michael Aizenman, Mira Shamis, Simone Warzel. 2014-05-15. Resonances and Partial Delocalization on the Complete Graph. https://doi.org/10.1007/s00023-014-0366-9

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