The Absolutely Continuous Spectrum of One-dimensional Schrödinger Operators
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.
KeywordsAbsolutely continuous spectrum Schrödinger operator Reflectionless potential
Mathematics Subject Classifications (2000)Primary 34L40 81Q10
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