Mathematical Physics, Analysis and Geometry

, Volume 10, Issue 4, pp 359–373

The Absolutely Continuous Spectrum of One-dimensional Schrödinger Operators

Article

Abstract

This paper deals with general structural properties of one-dimensional Schrödinger operators with some absolutely continuous spectrum. The basic result says that the ω 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 (Remling, The absolutely continuous spectrum of Jacobi matrices, http://arxiv.org/abs/0706.1101, 2007). The treatment of the continuous case in the present paper depends on the same basic ideas.

Keywords

Absolutely continuous spectrum Schrödinger operator Reflectionless potential 

Mathematics Subject Classifications (2000)

Primary 34L40 81Q10 

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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of OklahomaNormanUSA

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