The absolutely continuous spectrum of Jacobi matrices

Pages 125-171 from Volume 174 (2011), Issue 1 by Christian Remling

Abstract

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.

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Keywords
, ,
DOI
http://doi.org/10.4007/annals.2011.174.1.4
MR
2811596
zbMATH
05960698
Milestones
Received: 7 June 2007
Revised: 11 August 2010
Accepted: 6 August 2010
Published online: 1 July 2011

Authors

Christian Remling

Department of Mathematics
University of Oklahoma
Norman, OK 73019-0315