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Transactions of the American Mathematical Society

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Wigner-von Neumann type perturbations of periodic Schrödinger operators


Authors: Milivoje Lukic and Darren C. Ong
Journal: Trans. Amer. Math. Soc. 367 (2015), 707-724
MSC (2010): Primary 35J10, 34L40, 47B36
DOI: https://doi.org/10.1090/S0002-9947-2014-06365-4
Published electronically: July 17, 2014
MathSciNet review: 3271274
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Abstract: We study decaying oscillatory perturbations of periodic
Schrödinger operators on the half line. More precisely, the perturbations we consider satisfy a generalized bounded variation condition at infinity and an $ L^p$ decay condition. We show that the absolutely continuous spectrum is preserved, and give bounds on the Hausdorff dimension of the singular part of the resulting perturbed measure. Under additional assumptions, we instead show that the singular part embedded in the essential spectrum is contained in an explicit countable set. Finally, we demonstrate that this explicit countable set is optimal. That is, for every point in this set there is an open and dense class of periodic Schrödinger operators for which an appropriate perturbation will result in the spectrum having an embedded eigenvalue at that point.


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Additional Information

Milivoje Lukic
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005

Darren C. Ong
Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005

DOI: https://doi.org/10.1090/S0002-9947-2014-06365-4
Received by editor(s): May 24, 2013
Published electronically: July 17, 2014
Additional Notes: The first author was supported in part by NSF grant DMS–1301582. The second author was supported in part by NSF grant DMS–1067988.
Article copyright: © Copyright 2014 American Mathematical Society

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