Georgia Institute of Technology College of Sciences

We will discuss work of Michael Aizenman, Francois Germinet, Abel Klein, and Simone Warzel from 2007 on optimal Bernoulli decompositions of random variables and applications thereof. We will briefly discuss the basic properties of such decompositions, and demonstrate the existence of decompositions for which the contribution of the Bernoulli disorder is optimized in various ways.

We will then go through a proof of almost sure spectral localization (at the bottom of the spectrum) for continuous random Schroedinger operators with arbitrary bounded disorder. This proof relies on a Bernoulli decomposition of the disorder combined with a slightly stronger variant of the 2005 result from Jean Bourgain and Carlos Kenig showing such localization when the disorder is Bernoulli.