Christian Remling
The absolutely continuous spectrum of Jacobi matrices
(124K, LaTeX)

ABSTRACT.  I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schr"odinger 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.