Title: Singular Spectrum and Recent Results on Hierarchical Operators,
Abstract: We use trace class scattering theory to exclude the possibility of absolutely
continuous spectrum in a large class of self-adjoint operators with an
underlying hierarchical structure and provide applications to certain random
hierarchical operators and matrices. We proceed to contrast the localizing
effect of the hierarchical structure in the deterministic setting with previous
results and conjectures in the random setting. Furthermore, we survey stronger
localization statements truly exploiting the disorder for the hierarchical
Anderson model and report recent results concerning the spectral statistics of
the ultrametric random matrix ensemble.