Singular Spectrum and Recent Results on Hierarchical Operators

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

Localizationiin the Hierarchical Anderson Model

Presented at the QMath13 Conference: Mathematical Results in Quantum Theory, October 8-11, 2016 at the Clough Undergraduate Learning Commons, Georgia Tech.Quantum Mechanics with Random Features - Sunday, October 9th, 2016, Skiles 005 - Chair: Gunter StolzPer von Soosten is a Scientific Assistant at the Center for Mathematics at the Technische Universität München, Germany.Joint work with Simone Warzel.Based on arXiv: 1608.0160