The absolutely continuous spectrum of one-dimensional Schr"odinger operators

Details The absolutely continuous spectrum of one-dimensional Schr"odinger operators The absolutely continuous spectrum of one-dimensional Schr"odinger operators

2007-10-22

arxiv.org/abs/0710.4128v2

Math. Phys. Anal. Geom. 10 (2007), 359-373

Mathematics

Spectral Theory

references added; a few very minor changes

Scientific paper

10.1007/s11040-008-9036-9

This paper deals with general structural properties of one-dimensional Schr"odinger operators with some absolutely continuous spectrum. The basic result says that the omega limit points of the potential under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure. This implies an Oracle Theorem for such potentials and Denisov-Rakhmanov type theorems. In the discrete case, for Jacobi operators, these issues were discussed in my recent paper [19]. The treatment of the continuous case in the present paper depends on the same basic ideas.

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