Boosted Simon-Wolff spectral criterion and resonant delocalization (Q2828467)

The authors consider the spectral effects of disorder associated with self-adjoint operators of the form \(H(\omega)=A+V(\omega)\), where \(A\) is self-adjoint bounded operator and \(V(\omega)\) is a multiplication operator, all acting in the \(l^2\)-space of functions over an infinite graph \(G\). Here, the parameter \(\omega\) represents the disorder and is from the probability space \(\Omega\).NEWLINENEWLINEAmong the existing literature, there exists a dearth of methods for establishing regimes of delocalization in the presence of disorder. On the short list of such are arguments based on the method of resonant delocalization. The main goal of the paper under review is to advance the latter method, combining it with an improved version of the Simon-Wolff criterion for a related sufficiency criterion under which one may conclude the existence of continuous spectrum, and in some situations an absolutely continuous one.NEWLINENEWLINEIn a related application of the Simon-Wolff criterion for the point spectrum, the authors also present an improved result on the simplicity of the point spectrum, providing it for a naturally broad class of random potentials.