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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Riesz bases consisting of root functions of 1D Dirac operators


Authors: Plamen Djakov and Boris Mityagin
Journal: Proc. Amer. Math. Soc. 141 (2013), 1361-1375
MSC (2010): Primary 47E05, 34L40
Published electronically: September 12, 2012
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Abstract: For one-dimensional Dirac operators

$\displaystyle Ly= i \begin {pmatrix}1 & 0 \\ 0 & -1 \end{pmatrix} \frac {dy}{dx... ...0 & P \\ Q & 0 \end{pmatrix}, \;\; y=\begin {pmatrix}y_1 \\ y_2 \end{pmatrix}, $

subject to periodic or antiperiodic boundary conditions, we give necessary and sufficient conditions which guarantee that the system of root functions contains Riesz bases in $ L^2 ([0,\pi ], \mathbb{C}^2).$

In particular, if the potential matrix $ v$ is skew-symmetric (i.e., $ \overline {Q} =-P$), or more generally if $ \overline {Q} =t P$ for some real $ t \neq 0,$ then there exists a Riesz basis that consists of root functions of the operator $ L.$


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Additional Information

Plamen Djakov
Affiliation: Faculty of Engineering and Natural Sciences, Sabanci University, Orhanli, 34956 Tuzla, Istanbul, Turkey
Email: djakov@sabanciuniv.edu

Boris Mityagin
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210
Email: mityagin.1@osu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11611-9
PII: S 0002-9939(2012)11611-9
Received by editor(s): August 20, 2011
Published electronically: September 12, 2012
Additional Notes: The first author acknowledges the hospitality of the Department of Mathematics and the support of the Mathematical Research Institute of The Ohio State University, July - August 2011.
The second author acknowledges the support of the Scientific and Technological Research Council of Turkey and the hospitality of Sabanci University, April - June 2011.
Communicated by: James E. Colliander
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.