Resonant delocalization for random Schrödinger operators on tree graphs

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 \emph{iid} random variables. The main result is a criterion for the emergence of absolutely continuous (\emph{ac}) spectrum due to fluctuation-enabled resonances between distant sites. Using it we prove that for unbounded random potentials \emph{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 \emph{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.