Wigner-von Neumann type perturbations of periodic Schrödinger operators
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- by Milivoje Lukic and Darren C. Ong PDF
- Trans. Amer. Math. Soc. 367 (2015), 707-724 Request permission
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.References
- H. Behncke, Absolute continuity of Hamiltonians with von Neumann Wigner potentials. II, Manuscripta Math. 71 (1991), no. 2, 163–181. MR 1101267, DOI 10.1007/BF02568400
- D. J. Gilbert and D. B. Pearson, On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators, J. Math. Anal. Appl. 128 (1987), no. 1, 30–56. MR 915965, DOI 10.1016/0022-247X(87)90212-5
- Pavel Kurasov and Serguei Naboko, Wigner-von Neumann perturbations of a periodic potential: spectral singularities in bands, Math. Proc. Cambridge Philos. Soc. 142 (2007), no. 1, 161–183. MR 2296400, DOI 10.1017/S0305004106009583
- Alexander Kiselev, Christian Remling, and Barry Simon, Effective perturbation methods for one-dimensional Schrödinger operators, J. Differential Equations 151 (1999), no. 2, 290–312. MR 1669721, DOI 10.1006/jdeq.1998.3514
- Helge Krüger, On the existence of embedded eigenvalues, J. Math. Anal. Appl. 395 (2012), no. 2, 776–787. MR 2948266, DOI 10.1016/j.jmaa.2012.05.075
- Milivoje Lukic. A class of Schrödinger operators with decaying oscillatory potentials. arXiv:1207.5077, 2012.
- Milivoje Lukic, Schrödinger operators with slowly decaying Wigner-von Neumann type potentials, J. Spectr. Theory 3 (2013), no. 2, 147–169. MR 3042763, DOI 10.4171/JST/41
- Günter Stolz, Bounded solutions and absolute continuity of Sturm-Liouville operators, J. Math. Anal. Appl. 169 (1992), no. 1, 210–228. MR 1180682, DOI 10.1016/0022-247X(92)90112-Q
- Joachim Weidmann, Lineare Operatoren in Hilberträumen. Teil II, Mathematische Leitfäden. [Mathematical Textbooks], B. G. Teubner, Stuttgart, 2003 (German). Anwendungen. [Applications]. MR 2382320, DOI 10.1007/978-3-322-80095-4
Additional Information
- Milivoje Lukic
- Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005
- MR Author ID: 947053
- Darren C. Ong
- Affiliation: Department of Mathematics, Rice University, Houston, Texas 77005
- MR Author ID: 845285
- 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.
- © Copyright 2014 American Mathematical Society
- 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
- MathSciNet review: 3271274