Resonances and Partial Delocalization on the Complete Graph

TL;DR

This paper investigates how resonances and partial delocalization occur in the spectrum of the random Schrödinger operator on the complete graph, revealing conditions under which states become extended in a specific sense.

Contribution

It introduces a new analysis of resonant delocalization in the complete graph setting, connecting spectral properties with tunneling amplitudes and resonance phenomena.

Findings

Most spectrum is localized eigenfunctions

Existence of delocalized states with -sense delocalization

Spectral statistics follow -Seba spectra

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.