Abstract,
We review Anderson localization via separation of resonances. We
introduce systems with apriori degenerate bare energies at large
distances. We demonstrate a form of Anderson localization. For a
simple system we discuss dynamical behavior between degenerate bare
energies.

Abstract: The talk is about several tentative results, joint with J. Bourgain and S. Jitomirskaya. We consider a model of two 1D almost Mathieu particles with a finite range interaction. The presence of interaction makes it difficult to separate the variables, and hence the only known approach is to treat it as a 2D model, restricted to a range of parameters (both frequencies and phases of the particles need to be equal). In the usual 2D approach, a positive measure set of frequency vectors is usually removed, and extra care needs to be taken in order to keep the diagonal frequencies (which is a zero measure set). We show that the localization holds at large disorder for energies separated from zero and from certain values associated to the interaction.

These “forbidden” energies do indeed obstruct the localization. One can easily show that, even in the non-interacting regime, zero can sometimes be an eigenvalue of infinite multiplicity. Moreover, in the case of large coupling at interaction and a special relation between the phases of the particles, we show that the interaction can create a “surface" band of ac spectrum, which can be described by an effective 1D quasiperiodic long range operator.

Erdos conjectured that a set of natural numbers of positive lower density contains the sum of two infinite sets. In this talk I will describe progress on the conjecture.  In particular, I will discuss the truth of the conjecture in the “high density” case and how this implies a “1-shift” version of the conjecture in general.  These aforementioned results use nonstandard analysis.  Time permitting,  I will also discuss the conjecture in model-theoretically tame contexts.