Some recent quantitative unique continuation theorems, Semin. Équ. Dériv. Partielles, École Polytech

In this note I will survey some recent work on quantitative unique continuation problems which have had some interesting applications. These results have their origin in recent work with J. Bourgain [B-K] on Anderson localization for the con-tinuous Bernoulli model, a well-known problem in the theory of disordered media. I will start out by describing this work. The problem of localization originates in a seminal 1958 paper by Anderson [A], who argued that, for a simple SchrÄodinger operator in a disordered medium, \at su±ciently low densities, transport does not take place; the exact wave functions are localized in a small region of space. " In this work we have concentrated on continuous models; the corresponding issues for discrete problems remain open. Thus, consider a random SchrÄodinger operator on R n of the form H " = ¡¢+ V"; where the potential V"(x) =