Monotonicity versus Non-Monotonicity in Random Operators

Abstract

The past two decades have brought widespread interest and a large number of publications in the mathematical theory of random operators. While this has lead to many deep results and considerable mathematical insights, there are still important open problems which are far from being solved. This becomes particularly clear when looking at what is known for models other than the relatively well understood Anderson model and its generalizations. Of course, there are big challenges left in the Anderson model as well (delocalization!), but for other physically significant models (e.g. Poisson model, random displacement model) even the seemingly better understood phenomenon of localization is mathematically far from being settled.