The absolutely continuous spectrum of Jacobi matrices

I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schrödinger operators.These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potentials.In particular, we prove a Denisov-Rakhmanov type theorem for the general finite gap case.The main theme is the following: It is extremely difficult to produce absolutely continuous spectrum in one space dimension and thus its existence has strong implications.