Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


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
MathSciNet review: 3008883
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.$

References [Enhancements On Off] (What's this?)

  • 1. N. Dernek and O. A. Veliev, On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operato, Israel J. Math. 145 (2005), 113-123. MR 2154723 (2006b:34188)
  • 2. P. Djakov and B. Mityagin, Instability zones of periodic 1D Schrödinger and Dirac operators, in Russian, Uspekhi Mat. Nauk 61 (2006), 77-182; English translation: Russian Math. Surveys 61 (2006), 663-766. MR 2279044 (2007i:47054)
  • 3. P. Djakov and B. Mityagin, Asymptotics of instability zones of the Hill operator with a two term potential, J. Funct. Anal. 242 (2007), 157-194. MR 2274019 (2007j:34176)
  • 4. P. Djakov and B. Mityagin, Bari-Markus property for Riesz projections of 1D periodic Dirac operators, Math. Nachr. 283 (2010), 443-462. MR 2643021 (2011d:34170)
  • 5. P. Djakov and B. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Doklady Math. 83 (2011), 5-7. MR 2810153
  • 6. P. Djakov and B. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Mathematische Annalen 351 (2011), 509-540. MR 2854104
  • 7. P. Djakov and B. Mityagin, 1D Dirac operators with special periodic potentials, Bull. Pol. Acad. Sci. Math. 60 (2012), 59-75. MR 2901387
  • 8. F. Gesztesy and V. Tkachenko, A Schauder and Riesz basis criterion for non-self-adjoint Schrödinger operators with periodic and anti-periodic boundary conditions, J. Differential Equations 253 (2012), 400-437.
  • 9. I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint operators, Translation of Mathematical Monographs, vol. 18. American Mathematical Society, Providence, Rhode Island, 1969. MR 0246142 (39:7447)
  • 10. A. Makin, On spectral decompositions corresponding to non-self-adjoint Sturm-Liouville
    operators, Dokl. Math. 73 (2006), 15-18.
  • 11. A. S. Makin, Convergence of expansions in the root functions of periodic boundary value problems, Dokl. Math. 73 (2006), 71-76.
  • 12. A. Makin, On the basis property of systems of root functions of regular boundary value problems for the Sturm-Liouville operator, Differ. Eq. 42 (2006), 1717-1728. MR 2347119 (2008g:34216)
  • 13. B. Mityagin, Convergence of expansions in eigenfunctions of the Dirac operator. (Russian) Dokl. Akad. Nauk 393 (2003), 456-459. MR 2088512 (2005i:47073)
  • 14. B. Mityagin, Spectral expansions of one-dimensional periodic Dirac operators, Dyn. Partial Differ. Equ. 1 (2004), 125-191. MR 2126830 (2005m:34204)
  • 15. A. A. Shkalikov and O. A. Veliev, On the Riesz basis property of eigen- and associated functions of periodic and anti-periodic Sturm-Liouville problems, Math. Notes 85 (2009), 647-660. MR 2572858 (2010j:34200)
  • 16. O. A. Veliev, On the nonself-adjoint ordinary differential operators with periodic boundary conditions, Israel J. Math. 176 (2010), 195-207. MR 2653191 (2011i:34154)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47E05, 34L40

Retrieve articles in all journals with MSC (2010): 47E05, 34L40

Additional Information

Plamen Djakov
Affiliation: Faculty of Engineering and Natural Sciences, Sabanci University, Orhanli, 34956 Tuzla, Istanbul, Turkey

Boris Mityagin
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210

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.

American Mathematical Society