Riesz bases consisting of root functions of 1D Dirac operators
HTML articles powered by AMS MathViewer
- by Plamen Djakov and Boris Mityagin PDF
- Proc. Amer. Math. Soc. 141 (2013), 1361-1375 Request permission
Abstract:
For one-dimensional Dirac operators \[ Ly= i \begin {pmatrix} 1 & 0 \\ 0 & -1 \end {pmatrix} \frac {dy}{dx} + v y, \quad v= \begin {pmatrix} 0 & P \\ Q & 0 \end {pmatrix}, \;\; y=\begin {pmatrix} y_1 \\ y_2 \end {pmatrix}, \] 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
- Neşe Dernek and O. A. Veliev, On the Riesz basisness of the root functions of the nonself-adjoint Sturm-Liouville operator, Israel J. Math. 145 (2005), 113–123. MR 2154723, DOI 10.1007/BF02786687
- P. Dzhakov and B. S. Mityagin, Instability zones of one-dimensional periodic Schrödinger and Dirac operators, Uspekhi Mat. Nauk 61 (2006), no. 4(370), 77–182 (Russian, with Russian summary); English transl., Russian Math. Surveys 61 (2006), no. 4, 663–766. MR 2279044, DOI 10.1070/RM2006v061n04ABEH004343
- Plamen Djakov and Boris Mityagin, Asymptotics of instability zones of the Hill operator with a two term potential, J. Funct. Anal. 242 (2007), no. 1, 157–194. MR 2274019, DOI 10.1016/j.jfa.2006.06.013
- P. Djakov and B. Mityagin, Bari-Markus property for Riesz projections of 1D periodic Dirac operators, Math. Nachr. 283 (2010), no. 3, 443–462. MR 2643021, DOI 10.1002/mana.200910003
- P. B. Dzhakov and B. S. Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomials as potentials, Dokl. Akad. Nauk 436 (2011), no. 1, 11–13 (Russian); English transl., Dokl. Math. 83 (2011), no. 1, 5–7. MR 2810153, DOI 10.1134/S1064562411010017
- Plamen Djakov and Boris Mityagin, Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials, Math. Ann. 351 (2011), no. 3, 509–540. MR 2854104, DOI 10.1007/s00208-010-0612-5
- Plamen Djakov and Boris Mityagin, 1D Dirac operators with special periodic potentials, Bull. Pol. Acad. Sci. Math. 60 (2012), no. 1, 59–75. MR 2901387, DOI 10.4064/ba60-1-5
- 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.
- I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142, DOI 10.1090/mmono/018
- A. Makin, On spectral decompositions corresponding to non-self-adjoint Sturm-Liouville operators, Dokl. Math. 73 (2006), 15–18.
- A. S. Makin, Convergence of expansions in the root functions of periodic boundary value problems, Dokl. Math. 73 (2006), 71-76.
- A. S. Makin, On the basis property of systems of root functions of regular boundary value problems for the Sturm-Liouville operator, Differ. Uravn. 42 (2006), no. 12, 1646–1656, 1727 (Russian, with Russian summary); English transl., Differ. Equ. 42 (2006), no. 12, 1717–1728. MR 2347119, DOI 10.1134/S0012266106120068
- B. S. Mityagin, Convergence of expansions in eigenfunctions of the Dirac operator, Dokl. Akad. Nauk 393 (2003), no. 4, 456–459 (Russian). MR 2088512
- Boris Mityagin, Spectral expansions of one-dimensional periodic Dirac operators, Dyn. Partial Differ. Equ. 1 (2004), no. 2, 125–191. MR 2126830, DOI 10.4310/DPDE.2004.v1.n2.a1
- O. A. Veliev and A. A. Shkalikov, On the Riesz basis property of eigen- and associated functions of periodic and anti-periodic Sturm-Liouville problems, Mat. Zametki 85 (2009), no. 5, 671–686 (Russian, with Russian summary); English transl., Math. Notes 85 (2009), no. 5-6, 647–660. MR 2572858, DOI 10.1134/S0001434609050058
- O. A. Veliev, On the nonself-adjoint ordinary differential operators with periodic boundary conditions, Israel J. Math. 176 (2010), 195–207. MR 2653191, DOI 10.1007/s11856-010-0025-x
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
- 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
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 1361-1375
- MSC (2010): Primary 47E05, 34L40
- DOI: https://doi.org/10.1090/S0002-9939-2012-11611-9
- MathSciNet review: 3008883