Abstract
I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of onedimensional 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 DenisovRakhmanov 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.

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