Resonant delocalization for random Schr\"odinger operators on tree graphs

TL;DR

This paper investigates the spectral properties of Schr"odinger operators on regular tree graphs, establishing conditions for the emergence of absolutely continuous spectrum due to resonances, and analyzing the effects of unbounded and bounded potentials on localization.

Contribution

It provides a new criterion for ac spectrum emergence via resonances and extends understanding of spectral phases in tree graph Schr"odinger operators, especially at weak disorder.

Findings

Absolutely continuous spectrum appears at weak disorder for unbounded potentials.

The regime of pure ac spectrum is complementary to localization under certain conditions.

Disproves the existence of a mobility edge beyond which spectrum is localized at weak disorder.

Abstract

We analyse the spectral phase diagram of Schr\"odinger 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 $(\lambda \ll 1)$ 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.